IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY
The IEEE Press Series on Elcctromagnetic Wave Theory consists of new lilIes as well as reprints and revisions of recognized classics that maintain long-term archival significance in ela:tromagnelic waves and applications. Series Edilor Donald G. Dudley
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BOOKS IN THE IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY
Christopoulos. C. The Transmission-Une MQdeling Me/hads: TLM Clcmmow. P. c.. The Plane WaveSpeclrnm Represemmion o[£leclromagnetic Field~' Collin. R. B.. FieM Theoryo[Guided Waves. Second Edition Collin. R. E.. Foundations [Qr MicrQwave ElJgilJeering Dudley, D. G.. Mathematical FQundatiollS[or Elec/romagne/ic Thl'oJy Elliot, R. S., Electromagnetics: lIis/Qty. TheQry. and Applicll/iQns Felscn. L. 11.. and Marcuvit"z. N.• Radiation and Scallerillg o[Wal-'t's Harrington. R. F.• Field ComplllariQII by MOIllI'Il/ Metho
An IEEE Press Classic Reissue
TIME-HARMONIC ELECTROMAGNETIC FIELDS
Roger F. Harrington Professor ofElectrical Engineering (retiraJ) Syracuse Uni''€rsity
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FOREWORD TO THE REISSUED EDITION
The purpose of the IEEE Press Series on Electromagnetic Wave Theory is to publish books oflong-tcnn archival significance in electromagnetics. Included are new
titles as well as reprints and revisions of recognized classics. Time-Harmonic Elec· lromagrtetic Fields. by Roger F. Harrington, is one of the most significant works in electromagnetic theory and applications. The book has been one oflhe principallcxls in the first graduate course on electromagnetic theory for the past fony years; many would say lhe principal text. This classic volume contains a complete coverage of dynamic fields and is as fresh today as it was when originally published in 1961. Time-HamlOnic Electromagnetic Fields has proved to be popular over the past 40 years with students, professors, researchers and engineers who require a compre~ensive, in-depth treatment of the subject. Indeed, a colleague of mine, Dr. Kendall F. Casey of SRI writes, as follows: "When I begin a new research project, I clear my desk and put away all texts and reference books. Invariably, Harrington's book is the first book to find its way back to my desk. My copy is so worn that it is falling apart." Another colleague, Professor Chalmers M. Butler of Clemson University adds the following: "In the opinion of our faculty, there is no other book available which serves as well as Professor Harrington's does as an introduction to advanced elcctromagnctic theory and 10 classical solution methods in electromagnetics." Professor Harrington has been an internationally well-known contribUlor to eleetromagnetics for many years. He is universally regarded as the "father" of the Method of Moments. His book on the subject, Field Computation by Moment Methods, was added to the series in 1993. Professor Harrington is a Fellow of the IEEE. Prior to his retirement from active teaching, he was a Distinguished Professor at Syracuse University. Among his many awards and honors, he was awarded the IEEE Centennial Mcdal in 1984, the IEEE Antennas and Propagation Society Distinguished Achievement Award in 1989, the URSI Van der Pol Medal in 1996, the Jubilee Tesla Medal in 1998, the IEEE Electromagnctics Field Award in 2000, and an IEEE Third Millenium Medal in 2000. It is with pleasure that I welcome this classic book into the series.
G. DUDLEY University ofArizona Series Edito,' IEEE Press Series on Electromagnetic Wave Theory DONALD
vii
PREFACE This book W8..'l written primarily as a graduate-level text, but it should also be useful lIS a reference book. The organization is somewhat different from that normally found in engineering books. The material is arranged according to similarity of mathematical techniques instead of according to devices (antennas, waveguides, cavities, etc.). This organization reflects the main purpose of the book-to present mathematical techniques for handling electromagnetic engineering problems. In the sense that theorems are proved and formulas derived, the book is theoretical. However, numerous practical examples illustrate the theory, and in this sense the book is practical. The e.'qlerimental lISpect of the subject is not considered explicitly. The term time-harmonw has heen used in the title to indicate that only sinusoidally time-varying fields are considered. To describe such fields, the adjective a-e (alternating-current) haa heen borrowed from the corresponding specialization of circuit theory. Actually, much of the theory can easily be extended to arbitrarily time-varying fields by means of the Fourier or Laplace transformations. The nomenclature and symbolism used is essentially the same lI.8 that of the author's earlier text, "Introduction to Electromagnetic Engineering," except for the following ehange. Boldface script letters denote instantaneous vector quantities and boldface block letters denote complex vectors. This is a departure from the confusing convention of using the same symbol for the two different quantities, instantaneous and complex. Also, the complex quantities are chosen to have rms (roo~mean-l!quare) amplitudes, which corresponds to the usual a-c circuit theory convention. The many examples treated in the text are intended to be simple treatments of practical problems. Most of the complicated formulas are illustrated by numerical calculations or graphs. To augment the examples, there is an extensive set or problema at the end of each chapter. Many of these problems are of theoretical or practical significance, and are therefore listed in the index. Answers are given for most of the problems. Some of the material of the text appears in book form for the first time. References are given to the original sources when they are known. ix
x
PREFACE
However, it has not been possible to trace each concept back to its original inventor; hence many references have probably been omitted. For this the author offers his apologies. Credit has also been given to persons responsible for the origina.l calculations of curves whenever possible. A bibliography of books for supplemental reading is given at the end of the text. The book has been used for a course directly following an introductory course and also for a course following an intermediate one. On the former level, the progress was slower than on the latter, but the organization of the book seemed satisfactory in both eases. There is more than enough material for a year's work, and the teacher will probably want to make his own choice of topics. The author expresses his sincere appreciation to everyone who in any way contributed to the creation of this book. Thanks to W. R. LePage, whose love for learning and teaching inspired the author; to V. H. Rumsey, from whom the author learned many of his viewpoints; to B. Gruenberg, who read the gaUeys; to colleo.gues and students, for their many valuable comments and criticisms; and, finally, to the several secretaries who so expertly typed the manuscript. Roger F. H arringtqn
CONTENTS Foreword to the Revised Edition.
vii
Preface Chapter 1.
1-1. 1-2. 1-3. 1-4. 1-5. 1-6. 1-7. 1-8. 1·9. 1-10. 1-11. 1_12. 1-13. 1-14.
Introduction. Basic Equations. Const.itutive Relationships. The Generalized Current Concept Energy and Power . Circuit Concepts Complox Quantities. Complex Equations. Complex Conatitutive Parametera Complex Power A-C Characteristica of Matter A DilKlU8$ion of Current . A-C Behavior of Circuit Elements Singularities of the Field . Chapter 2.
2-1.
Fundamental Concepts
3-4.
3-5. 3-6.
9 12
13 16
18 19 23
26 29 32
37
The Wave Equation
2-5. Reflection of Waves 2-6. Transmission-line Concepts 2-7. Waveguide Concepts 2-8. Resonator Concepts 2--9. Radiation 2-10. Antenna Concepts . 2-11. On Waves in General
3-1. 3-2. 3-3.
•7
Introduction to Waves
2-2. Waves in Perfect Dielectrica 2-3. lutnusia Wave Constanta . 2-4. Waves in Lossy Matter
Chapter 8.
1 1
41
48 51 54
61 66
74 77 81 85
Some Theorems and Concepts
95
The Source Concept Duality . Uniquenesa . Image Theory The Equivalence Principle.
103 106
Fields in Half-space
110
98 100
xii 3-7. 3-8. 3-9. 3-10. 3-11. 3-12. 3-13.
CONTENTS The rnduct.ion Theorem Reciprocit.y Groon's Functions Tensor Green's Functions. Integral Equation8 . Construction of Solutions The Radiation Field
113 116 120 123 126 129 132
Chapter •.
....1. ....2. 4-3.
,-4.
4-5. 1·6.
....7. 4-8. ....9. 4-10. 4-t l. 4_12.
The Wave Functions Plane Waves . The Rectangular Waveguide Alternative Mode Seu. . The Rect.a.ngular Cavity Partially Filled Waveguide The Dielcctric-slab Guide . Surface-guided Waves Modal Expansions of Fields Currents in Waveguide8 Apertures in Ground Planes Plane Current Sheela Chapter 6.
6-1. 6-2. 6-3. 6-4. 6-6. 6-6. 6-7. 6-8. 6-9. 5-10. 5-11. 5-12. 5-13.
PilUle Wave FunctioDs
143 1'6 "8 162 166 158 163 168 171 177
ISO 186
Cylindrical Wave Functions
The Wave Functions The Circular Waveguide RAdial WaveguidC8. The Circular Cavity Other Guided Waves Sources of Cylindrical Waves. Two-dimensional Radiation Wave Transformations. Scattering by Cylinders Scattering by Wedges . Three-dimensional Radiation Apertures in Cylinders. Apertures in Wedges Chapter 6.
232
238 2'2 245 260
Spherical Wave Functions
6-1. The Wave Functions 6-2. The Spherical Cavity 6-3. Orthogonality Relationships Space as a Waveguide . 6-6. Other Radial WaveguidC8 . 6-6. Other Resonators 6-7. Bources of Spherical Waves 6-8. Wave Transformations. 6-9. Scattering by Spheres . ~IO. DipOle and Conducting Sphere
....
198 20' 208 213 216 223 228 230
26' 269 273 276 279 283 286
289 292 298
CONTENTS
~ll.
6-12. 6-13.
Apertures in Spheres . Ficlds External to Cones Ma.ximum Antenna Gain
xiii 301 303 307
Chapter 7. Perturbational and Variational Techniques 7-1. 7-2. 7-3.
7-4. 7--5. 7+6. 7-7.
7-8. 7-9. .. 7-10. 7-11. 7-12.
Int.roduction. Perturbations of Cavity Walls Cavity-material Perturbations Waveguide Perlurbations . Stationary Formulas for Cavities. The Riu Procedure. The Reaction Concept. Stationary Formulas for Waveguides. Stationary Formulas for Impedance . Stationary Formulaa for Scattering Scatlering by Dielectric Obstacles Transmiaaion through Apertures. Chapter 8.
g..1. ~2.
8-3. g...t.
8-5. 8-6.
8-7. 8-8. 8-9. 8-10. 8-11, 8-12. 8-13. 8-14. 8-15.
A. B. C. D. E.
340 345 348 355 362 365
Microwave Networb
Cylindrical Waveguides Modal Expansions in W&Veguides The Network Concept. One-porl Networn. Two-port Networks. Obstaclea in Waveguides Posts in Waveguides Small Obstacles in Waveguides Diaphragms in Waveguides Waveguide Junctions . Waveguide Feeds . Excitation of Apertures Modal Expansions in Cavitics. Probes in Cavities . Aperture Coupling to Cavities
Appendix Appendix Appendix Appendix Appendix
317 317 321 326 331 338
Veetor Analysis Complex Permittivities Fourier Series and Integrals Bessel Functions . Legendre Functions
381 389
391 393 398 402
406 411
414 420
425 428 431
434 436 447 451
456 460 465
Bibliography .
471
lndu .
473
CHAPTER
1
FUNDAMENTAL CONCEPTS
1-1. Introduction. The topic of this book is the theory and analysis of electromagnetic phenomena that vary sinusoidally in time, henceforth called a-c (alternating-current) phenomena. The fundamental concepts which form the basis of our study are presented in this chapter. It is assumed that the reader already has some acquaintance with electromagnetic field theory and with electric circuit theory. The vector analysis concepts that we shall need are summarized in Appendix A. We shall view electromagnetic phenomena from the "macroscopic" standpoint, that is, linear dimensions are large compared to atomic dimensions and charge magnitudes are large compared to atomic charges. This allows us to neglect the granular structure of matter and charge. We assume all matter to be stationary with respect to the observer. No treatment of the mechanical forces associated with the electromagnetic field is given. The rationalized mksc system of units is used throughout. In this system the unit of length is the meter, the unit of mass is the kilogram, the unit of time is the second, and the unit of charge is the coulomb. We consider these units to be fundamental units. The .units of all other quantities depend upon this choice of fundamental units, and are called secondary units. The mksc system of units is particularly convenient because the electrical units are identica.l to those used in practice. The concepts necessary for our study are but a few of the many electro~ magnetic field concepts. We shall start with the familiar Maxwell equations and specialize them, to our needs. New notation and nomenclature, more convenient for our purposes, will be introduced. For the moSt part, these innovations arc extensions of a.-c circuit concepts. 1-2. Basic Equations. The usual electromagnetic field equations are expressed in terms of six quantities. These are &, caUed the electric intensity (volts per meter)
x, called the magnetw intensity
(amperes per meter) :Il, called the electric flux density (coulombs per square meter) $, called the magnetic flux den8ity (webers per square meter) called the electric current density (amperes per square meter) q., called the electric charge density (coulombs per cubic meter)
a,
1
2
TnIE-B.A1WONlC ELECl'ROMAGNl7I'1C FlELDS
dl
s"---_ _- -
c Flo. 1·1. d1 and ds on an open surface.
FIo. 1-2. ds on a closed eurfaoe.
We shall call a quantity well-behaved wherever it is a continuous function and has continuous derivatives. Wherever the above quantities are weUbehaved, they obey the Maxwell equations il
VX&=--
at
a:n vx:re--+.!l
at
v·m - 0
v .» """
(I-I) q~
These equations include the inIormation contained in the equation of conJ.inuity
aq.
(1-2)
--a/
which expresses the conservation of charge. Note that we have used boldface script letters for the various vector quantities, since we wish to reserve the usual boldface roman letters for complex quantities, introduced in Sec. 1-7. Corresponding to each of Eqs. (1-1) are the integral forms of Maxwell's equations
1!>CIl ds ~ 0 o
1!>:n ds o
=
ffl
(1-3)
q.d,
These are actually more general than Eqs. (1-1) because it is no longer required that the various Quantities be well·beh80ved. In the equations of the first column, we employ the usual convention tha.t ell encircles ds according 1.0 the right-hand rule of Fig. 1-1. In the equations of the last column, we use the convention that ds points outward from a closed surface, as shown in Fig. 1-2. The circle on eo line integral denotes a closed con1.ourj the circle on a 5urface integral denotes a closed 5urface. The integrallorm 01 Eq. (1-2) is (1-4)
3
PUNDA.J£EN'1'AL CONCEPTS
where the samc convention applies. This is the statement of conservation of charge as it applies to a region. We sh&11 use the name field quantity to describe the quantities discussed above. Associated with each field quantity there is a circuit quantity, or integral quantity. These circuit quantities are
" called tbe oolIa{}, (volts) i, called the electric current (amperes) q, called the electric charge (coulombs) !/t, called the magnetic flux (webers) "", called the electric flux (coulombs) u, called the magnetomotive force (amperes) ... The explicit relationships of the field quantities to the circuit quantities can be summarized as follows:
f E·d1 i-II g·ds q - Ilf q.dr
v=
+~ffCB.ds
+' - If D·ds
(J-,S)
u=fx.cn
AU the circuit quantities are algebraic quantities and require reference conditions when designating them. Our convention for a ffline-integral" quantity, such as voltage, is positive reference at the start of the path of integration. This is illustrated by Fig. 1-3. Our convention for a usurfaec-intcgrnl" quantity, such 88 current, is positive reference in the direction of ds. This is shown in Fig. 1-4. Charge is IIr "net-amount" quantity, being the amount of positive charge minus the amount of nega· tive charge. We shall call Eqs. (1-1) to (1-4) jU,ld equations, since all qunntities appearing in them are field quantities. Corresponding equations written in tcrJ1l8 of circuit quantities we shall describe as ciTcuit equation&. Equa-
+
4l
•
A,,,
\
~_---r, I
\
I \
Fla. 1-3. voltap-.
•
Reference convent.ion for
'\
J_
I .-";.---""" ...
I
FIG.
1-4.
en~n~.
,,
Reference
convention
for
4
TIME-HARMONIC ELECTROMAGNETIC FIELDS
tiona (1-3) are commonly written in mixed field and circuit form as ,kt.d1 _ _ d>/-
'Y
f
dt
(1-6)
d>/-' :JC·d1--+i dt
Similarly, the equation of continuity in mixed field and circuit form is
4.1J.ds = _ 'Jr
dq dt
(1-7)
Finally. the various equations can be written entirely in terms of circuit quantities. For this, we shall use the notation that :z denotes summation over a closed contour for a line-integral quantity, and summation over a closed surface for a surface-integral quantity. cuit forms of Eqs. (1-6) are
1>--~~
1>
=
In this notation, the cir-
(1-8)
ddt' +i
and the circuit form of Eq. (1-7) is (1-9)
Note that the first of Eqs. (1~8) is a generalized form of Kirchhoff's voltage law, and Eq. (1-9) is a generalized form of Kirchhoff's current law. It is apparent from the preceding summary that many mathematical forms can be used to present a single physical concept. An understanding of the concepts is an invaluable aid to remembering the equations. While an extensive exposition of these concepts properly belongs in an introductory textbook, let us here summarize them. Consider the sets 01 Eqs. (1-1), (1-3), (1-6), and (1-8). The first equation in each sct i. essentially Faraday's law of induction. It states that a changing magnetic flux induces a voltage in a path surrounding it. The second equation in each set is essentially Ampere's circuital law, extended to the time-varying case. It is a partial definition of magnetic intensity and magnetomotive force. The third equation of each set states that magnetic flux haa no "flux source," that is, lines of
5
FUNDAMENTAL CONCEPTS
can be neither created nor destroyed, merely transported. Lines of current must begin and end at points of increasing or decreasing charge density. 1~3. Constitutive Relationships. In addition to the equations of Sec. 1-2 we need equations specifying the characteristics of the medium in which the field exists. We shall consider the domain of E and ac as the electromagnetic field and express :D, (Bl and n in terms of & and :re. Equations of the general form :D ~ :D(8,3<:) III ~ 1Il(8,3C) 11 - 11(8,3<:)
(1-10)
are called constitutive relationships. Explicit forms for these can be found by experimentation or deduced from atomic considerations. The term free space will be used to denote vacuum or any other medium having essentially the same characteristics as vacuum (such as air). The constitutive relationships assume the particularly simple for.m.s in free space
(1-11)
where fO is the capacitivity or permiUivity of vacuum, and 110 is the inductivity or permeability of vacuum. It is a mathematical conaequence of the field equations that (foJ10)-~ is the velocity of propagation of an electromagnetic disturbance in free space. Light is electromagnetic in nature, and this velocity is called the velocity of light c. Measurements have established that I c = . ,-- = 2.99790 X 10'
v
~
3 X 10' meters per second
(1-12)
foJlo
The choice of either fO or Jlo determines a system of electromagnetic units according to our equatioDs. By international agreement, the value of Jlo has been chosen as lJo = 4r X 10-7 henry per meter for the mksc system of units. fO
ell
8.854 X la-It
(1-13)
It then follows from Eq. (1-12) that
~ 3~ X
lO-g farad per meter
(1-14)
for the mksc system of units. Under certain conditions, the constitutive relationships become simple proportionalities for many materials. We say that such matter is linear
6
TIME-HARMONIC ELJ:crROMAGNJ::TIC PfELDS
in the simple sense, and call it limple m.oUer for short. :1)=,& } (B =- Jl~ lJ = uE
Thus
in simple matter
(1-15;
where, as in the free-space case, f is called the capacitivity of the medium and p is called the inductivity of the medium. The parameter u is called the conducliuity of the medium. We originally made the qualifying statement that Eqs. (1-15) hold "under certain conditions." They may not hold if & or 3C are very large, or if time derivatives of & or:Ie are very large. Matter is often classified according to its values of tT, E, and p. Materials having large values of 11 are called conduaora and those baving small values of u are called insul4toT3 or dUlulria. For analyses, it is often convenient to approximate good conductors by perfect conductors, characterized by u = 00 I and to approximate good. dielectrics by perfed dielectrica, characterized by rT = O. The capacitivity f of any material is never less than that of vacuum to. The ratio t, = tlto is called the dielutric constant or relative capacitivity. The dielectric constant of a good conductor is hard to measure but appears to be unity. For most linear matter, the inductivity p. is approximately that of free space lAo. There is a class of materials, called diamagnetic, for which IA is slightly less than P.o (of t.be order of 0.01 per cent). There is a clagg of materials, called paramagmtic, for which Jl is slightly greater than 110 (again of the order of 0.01 per cent). A third class of materials, caUedferromagmtic, has values of Jl much larger than p.o, but t.hese materials arc often nonlinear. For our purposes, we shan call all materials except the ferromagnetic ones nonmagnetic and tnkc II - Po for them. The ratio p., = p/po is called the rekUive inductivity or rel4tive permeability and is, of course, essentially unity for nonmagnetic matter. Quite often the restriction on the time rate of change of the field, made on the validity of Eqs, (1-15), can be overcome by extending tbe definition of linearity. We say that matter is linear in the general sense, and call it linear matter, when the constitutive relationships are the following lincar differential equations:
in linear matter
(1-16)
Even more complicated formulas for the constitutive relationships may
FUNDAMENTAL CONCEPTS
7
be necessary in some cases, but Eqs. (1-16) are the most general that we shall consider. Note that Eqs. (1-16) reduce to Eqs. (1-15) when the time derivatives of & and 3C become sufficiently small. The physical significance of the extended definition of linearity is 88 follows. The atomic particles of mat.ter have mass as well as charge, SO when the field changes rapidly t.he particles cannot U follow" the field. For example suppose an electron has been accelerated by the field, and then t.he direction of g changes. There will be a time lag before the electron can change direction because of its momentum. Such a picture holds for 9 if the electron is a free electron. It holds for:D if the electron is a bound electron. A similar picture holds for (B except that the magnetic moment of the electron is the contributing quantity. We shall not attempt to give significance to each term of Eqs. (1-16). It will be shown in Sec. 1-9 that all terms of Eqs. (1~16) contribute to an "admittivitylJ and an II impedivity >l of a material in the time-harmonic case. 1-4. The Generalized Current Concept. It was Maxwell who first noted that AmpAre's law for statics, V X X ... :1, was incomplete for time-varying fields. He amended the law to include an elutr* di.3plaament currem O'D/at in addition to the conduction current. He visualized this displacement current in free space as a motion of bound charge in an Uether/' an ideal weightless Buid permeating all space. We have since discarded the concept of an et.her for it bas proved undetectable and even somewhat illogical in view of the theory of relativity. In dielectrics, part of the term aD/at is a motion of the bound particles and is thus a current in the true sense of the word. However. it is convenient to consider the entire O'D/at term as a current. In view of the symmetry of Maxwell's equations, it also is convenient to consider the term (}(B/at as a magnetic di8ploament current. Finally, to represent sources, we amend the field equations to include impre88ed currents, electric and magnetic. These are the currents we view as the cause of t.he field. We shall see in the next section that the impressed currents represent energy sources. The symbols .g and ml will be used to denote electric and ma.gnetic currents in genera.l with superscripts indicating the type of current. As discussed above we define total currents j
j
j
j
j
.!l' - a:D at ol
+ .!l' + .!l; (1-17)
m<' - -+m<;
at
where the superscripts t, c, and i denote total, conduction, and impressed currents. The symbols i and k will be used to denote net electric and magnetic currents and the same superscripts will indicate the type. j
8
TIJlE.-H.A1WONIC ELEcrB.OKAONETlC nELDS
Thus, the circuit (orm corresponding to Eqs. (l-17) is
i,=d¥- +i.+i' dl
(1-18)
k'=#+k' dl The i and k
&re,
of course, related to the
i-II a-ds
k -
a and ml by
II mHs
(1-19)
where these apply to any of the various types of current. In terms of the generalized current concept, the basic equations of electromagnetism become, in the differential (orm,
v
X & = _:nt'
VX:JC=.g'
(1-20)
p:re-dl ~ II a'-ds
(1-21)
and in the integral form,
PS-dl- - II ""'-ds Also, the mixed
fietd~ircuit
form is
PS-dl~-k'
(1-22)
and the circuit form is (1-23)
Note that these look simpler than tbe equations of Sec. 1-2. Actually, we have merely included many concepts in the functions:nt' and a'j so tome of the information contained in the original Maxwell equations bas become hidden. However, our study comprises only a small portion of the general theory of electromagnetism, and the forms of Eqs. (1-20) to (1-23) are well suited to our purposes. Note that we have omitted the H divergence equations" of Maxwell from our above sets of equations. We have done so to emphasize that this information is included in the above sets. For example, taking the divergence of each of Eqs. (1-20), we obtain V-OIl'-O
for V· V X a "'" 0 is an identity. closed surfaces became
1P OIl' - ds -
0
V-lJ'-O
(1-24)
Similarly, Eqs. (1-21) applied to (1-25)
Thus, the totol currents are solenoidal. Lines of total current have no beginning or end but must be continuous.
9
FUNDAliEl\I"TAL CONCEPT8
As an illustration oC the generalized -turrent concept, consider the circuils 01 Figs. l-5 and 1-6. In Fig. 1-5, the Ucurrent source" I)' produces a conduction current I)c through the resistor and a displace-
ment current
a' -
iYD/dl
through
l
~urce l_1P_ _j~
the capa.citor. In Fig. 1-6, the Hvoltage source" mz;C produces an Fro. 1--5. Type.a of electrio current. electric current in the wire which in turn causes the magnetic displacement current 9n" "'" iJ(B/iJt in the ma.gnetic core. In these pictures we have used the convention that a single-headed arrow represents an electric current, a double-headed arrow represents a magnetic current.
-
,.. large
i
Source <;
p.5ll'
FlO.
SlI'
1-.6. Types of magnetic current.
It is not possible at this time to give the reader a complete picture oC the usefulness of impressed currents. Figures 1-5 and 1-.6 anticipate one application, namely, that of representing sources. More generally, the impressed currents are those currents we view as sources. In a sense, the impressed currents are those currents in terms of which the field is expressed. In one problem, a conduction current might be considered as the source, or impressed, current. In another prof>.. lem, a polarization or magnetization current might be considered as the source current. Our understanding of the concept will grow as we learn to use it. 1-6. Energy and Power. Consider a region oC electromagnetic field, as suggested Flo. 1-7. A region contai..oiDg hy Fig. 1-7. The field oheys the Maxwell ..=ea. equations, which in generalized current
10
TIKE-!lARMONIC J:,LECTROKAGNETlC FIELDS
notation are Eqs. (1-20). As an extension of circuit concepts, it e&n be shown that a product t· tJ is a power density. This suggests a scalar multiplication of the second of Eqs. (1-20) by s. Also, in view of the vector identity
v . (E
X :Ie) = :Ie . V X E - E . V X :Ie
a Bcalar multiplication of the first of Eqs. (1-20) by ac is suggested. difference of the l'C8ulting two equations is
v· (E X
:Ie)
+ E·.g' + :Ie. "'"
The (1-26)
- 0
If this equation is integrated throughout a region, and the divergence theorem applied to the first term, there results (1-27)
We shall interpret these as equations for the conun:ation of eneTfl1I, Eq. (1-26) being the differential form and Eq. (1-27) being the integral form. The generally accepted interpretation of Eqs. (1-26) and (1-27) is as follows. The Poynting vector (1-28)
s=&XX
is postulated to be a density-of-powcr flux.
The point relationship
P, = v . S = V . (E X :Ie)
(1-29)
is then a volume density of power leaving the point, and the integral
cP, = 1PSods =
1/>& X :JCods
(1-30)
is the total power leaving the region bounded by the surface of integration, The other terms of Eq. (1-26) can then be interpreted as the rate of increase in energy density at a point. Similarly, the other terms of Eq. (1-27) can be interpreted n.s the fate of increase in energy within the region. Further identification of this energy can be made in particular cases. For media linear in the simple sense, as defined by Eqs. (1-15), the last two terms of Eq. (1-26) become E· .g' 3C • ml' =
where
.'at!. (!2 .E') + .&' + E 'Il'
(1-31)
!at (.!.2 I-lXt ) + :JC • mti
a' and mt' represent possible source currents. w, - Ji't6 1
W.. -
J.i'1J3C1
The terms (1-32)
FUND.ulENTAL CONCEPTS
11
are identified as the electric a.nd magnetic energy densities of static fields, and this interpretation is retained for dynamic fields. The term (1-33)
is identified as the density of power converted to heat energy, called diuipaled power. Fin&1ly, tbe density of power supplied by the source currents is defined as (1-34)
The reference direction for /SOurce power is opposite to that for dissipated power, as evidenced by the minus sign of Eq. (1-34). In terms of the above-defined quantities) we can rewrite Eq. (1-26) as p. - P,
+ p, + ata (w. + w.)
(1-35)
A word statement of this equation is: At any point, the density of power . supplied by the sources must equal that leaving the point plus that dissipated plus the rate of increase in stored electric and magnetic energy densities. A more common statement of the conservation of energy is that which refers to an entire region. Corresponding to the densities of Eqs. (1-32), we define the net electric and magnetic energies within a region as (1-36)
Corresponding to Eq. (1-33), we define the net power converted to heat energy as ll', =
JJJ.&' d,
(1-37)
Finally, corresponding to Eq. (1-34), we define the net power supplied by sources within the region llS (1-38)
In terms of these definitions, Eq. (1-27) can be written as ll'. = ll',
+ ll', + citd (W. + w.)
(1-39)
Thus, the power supplied by the sources within 8 region must equal that leaving the region plus that dissipated within the region plus the rate of incrcuse in electric and magnetic energies stored within the region. If we proceed to the general definition of linearity, Eqs. (l-16), the sepa.ration of power into a reversible energy change (stor!"ge) and ao
12
TIHE-HARMONIC ELECTROMAGNETIC FIELDS
irreversible energy change (dissipation) is no longer easy. Contributions to energy storage and to energy dissipation may originate from both conduction and displacement currents. However, Eqs. (1-35) and (1-39) still apply to media linear in the general sense. We merely cannot identify the various terms. In See. 1·10 we shall see that for a-e fields the division of energy into stored and dissipated components again assumes a simple form. 1-6. Circuit Concepts. The usual equations of circuit theory are specializations of the field equations. Our knowledge of circuit concepts can therefore be of help to us in understanding field concepts. In this section we shall quickly review this relationship of circuits to fields. Kirchhoff's current law for circuits is an application of the equa.tion of conservation of charge to surfaces enclosing wire junctions. To demonstrate, consider the parallel RLC circuit of Fig. 1-8. Let the letter 0 denote the junction, and the letters a, b, c, d denote the upper terminals of the elements. We apply Eq. (1-7) to a surface enclosing the junction, as represented by the dotted line in Fig. 1-8. The result is
. +.~<>lI +'ho +.I."" +,+dq ~J dt =
~OG
0
where the i... are the currents in the wires, i l is the leakage current crossing the surface outside of the wires, and q is the charge on the junction. The term dqjdt can be thought of 88 the current through the stray capaci~ tance between the top and bottom junctions. In most circuit applications both if and dq/dt are negligible, and the above equation reduces to
i ... +i<>ll+ioc+i",,=O This is the usual expression of the Kirchhoff current law for the circuit of Fig. 1-8. Kirchhoff's voltage law for circuits is an application of the first Max. well equation to closed contours following the connecting wires of the circuit and closing across the terminals of the elements. To demonstrate, consider the series RLC circuit of Fig. 1-9. Let the letters a to h. denote
--
------. ,
-R
L
- -" S
--c
,
a) FIo. 1.-8. A parallel RLC circuit.
FUND~TAL
CONCEPTS
13 R
---- c
b a FlO. 1-9. A Bel'ies RLC
circuit.
d
I I I
I
I
I
I
•
h
B
---c
L
I
f
the t.erminnls of the elements as shown. We apply the first of Eqs. (1-6) to the contour abcdefgha, following the dotted lines between terminals. TJ;Us gives
dy,
~+~+v~+~+v~+~,+~+~+~-O
v._
where the are tbe voltage drops along the contour and !/I is the magnetic flux enclosed. The voltages v.., v.... v~, and Vfl are due to tbe resistance of tbe wire. The term dl/I/dl is tbe voltage of the stray inductance of the loop. When the wire resistance and the stray inductance can be neglected. the above equation reduces to
This is the usual form of Kirchhoff's voltage law for the circuit of Fig. 1-9. In addition to I{irchhoff's laws. circuit theory uses a number of l'elcment laws." Ohm's law for resistors. v - Ri, is a specialization of the constitutive relationship .11 -= u6. The law for capacitors, q - Cv. expresses the same concept as ~ - t6. We have from the equation of continuity i .. dq/dt. so the capa.citor lnw ellD also be written as i .,. C dv/dt. The law for inductors, '" - Li. expresses the same concept as m - ",3<:. From the first Maxwell equation we have v - d"'/dt. 80 the inductor law can also be written 8B v - L dijd,t. Finally. the various energy relationships for circuit theory can be considered as specializations of those for field theory. Detailed expositions of the various specializations mentioned above can be found in elementary textbooks. Table 1-1 summarizes the various correspondences between field concepts and circuit concepts. 1·7. Complex Quantities. When the fields are a-e, that is. when the time variation is harmonie, the mathematical analysis can be simplified
14
TIME-HARMONIC ELECTROMAGNETIC FIELDS
TABLil 1-1. CoRRESPONDENCES BETWEEN
CmcUlT CoNCIlPT8 AND FuLO CoNC.E1'1'8
Field concepts
Circuit concepts Voltage
Eleetric intensity 8
II
a
Current i
Electric current. deMity or magnetic intensity Ie
Magnet.ic Bux ""
Magnetic ftux density i13
Cb&rgc q
Charge density q. or electric Awe density n
Kirchhoff's voltage lAw (generalized) Maxwell-Faraday equiltion
L·-Lt·-•
4
ilGl
VX&---
dt
at
Kirchhoff's current law (generalized) Equation of continuity
aq.
v·$Jc.-a,
dq dl
Element laws (linear) 1 Resistors i - - D
R Capacitors q - Gil
.,
..noe)
Conductors.9< - crE Dielectrics :D-
d. i-C-
O<
.s
jJ"-e-
a'
dl
Inductors '" - Li 0<
Coo!5Litutive relationships (linear in the simple
Magnetic properties
di v-Ldt
0<
(B -
ax at
~"-JJ.-
Power flow PI - vi
Power Bow B - & X :JC
Power dissipation in resistors 1 pt \Pol _ til. _ _
Power d.issipo.Lion Pol - & '.g' - erst
R Energy in capacitors 'W. -
liqo - He.,
Electric energy w.-~:D·6
-
~«:I <
Energy in induetore 'W. -
li.. - liLi'
"Magnetic energy w.. -
~~m':fC -
~:JC
~1o'3C1
15
FUNDAMENTAL CONCEPl"8
by using complex quantities.
ei·
The basis for this is Euler's identity %E
cosa+jsina
where j = v'=1. This gives us a relationship between real sinusoid&1 functions and the complex exponential function. Any a-e quantity can be represented by a complex quantity. A scalar quantity is interpreted according to l v =
v'21V1 co, (wt + a)
v'2 Re (Vel~) quantity and V "'"' lYle"" -
(1-40)
where" is called the instantaneoulJ is called the complex quantity. The notation Re ( ) stands for lithe real part of/' that is, the part not associated with j. Other names for V are "phaaor quantity" and flvector quantity," the laat name causing confusion with space vectors. In our notation v represents a voltage, hence V is a complex lJol14ge. Equation (1-40) with IJ replaced by i and V replaced by I would define a complex current, and so on. Note that the complex quantity is not a fUDction of time but it may be a function of position. Note also that the magnitude of the complex quantity is the effective (root-meansquare) value of the instantaneous quantity. We have chosen it so because (1) a.-c quantities are usually specified or measured in effective values in practice, and (2) equations for complex power and energy retain the same proportionality factors as do their instantaneous counterparts. For example, in circuit theory the instantaneous power is p - vi, and complex power is P = V I·. A factor of ~ appears in the equation for complex power if peak. values of v and i are used for IVI and IJI. Complex notation can readily be extended to vectors having sinusoidal time variation. A complex E is defined as related to fln instantaneoui t according to
s
~
v'2 Re (Ee'·')
(1-41)
This means that the spatial components of E are related to the spatial components of t by Eq. (1-40). For example, the z components of E and t arc related by S. =
v'2 Re (E.d·')
-
v'2 IE.I co, (wi + a.)
where E", = IE",lei"'.. Similar equations relate the y and: components of E and t. The phase of each component may be different from the phases of the other two components, that is, a"" a., and a", are not necessarily equal. In our notation & is an electric intensity, hence E is called the com-pia eled.rU; intetuity. Equation (1-41) with E replaced by H and 8 by X I The convent.ion " _ "the imaginary part. or." peak value of II.
~
1m (Vel.') can &lao be used, where 1m ( ) atancla fOT The factar v"2 can be omitted if it is deeircd that IVI be Lb.
16
'I'IJrOioItAJUlONlC ELECTROMAGNETIC nELDS
defines a comple:e magnetic intenrily H, representing the instantaneous magnetic intensity X, and 80 aD. Note tha.t the magnitude of a component of the complex vector is the effective value of the corresponding component of the instantaneous vector. This choice corresponds to that taken for complex scalars and has essentially the same advantages. A real v~tor, sucb as S or :re, can be thought of as a triplet of real scalar (unctions, ne.mely, the %, Y, and t components. At any instant of time, the vector has a definite magnitude and direction at every point in space and can be represented in three dimensions by arrows. A complex vector, such as E or H, is a group of six real scalar functions, namely, the real and imaginary parts of the X, 1/t and z components. It cannot be represented by arrows in three-dimensional space except in special cases. One such s~ia1 case is that for which 0::", = a. - a., 80 that the vector has a real direction in space. In this case the instantaneous vector always points in the same direction (or opposite direction), at a point in spa.ee, changing only in amplitude. We could define a ucomplex magnitude" and a "complex direction" for lJ, complex vector as extensions of the corresponding definitions for real vectors, but these would ha.ve little use. Throughout this book we shall use the following notation. Instantaneous quantities are denoted by script letters or lower-e&se letters. Complex quantities which represent the instantaneous quantities are denoted by the corresponding capital letter. Vectors are denoted by boldface type. 1-8. Complex Equations. The symbol He ~ ) can be considered as a mathematical operator which selects the real part of a complex quantity. A set of rules for manipulating the operator Re ~ ) can be formulated from the properties of complex functions. The following are the rules we shall need. Let a capital letter denote a complex quantity and a lower~ase letter denote a real quantity. Then Re (A)
+ Re (B)
-
Re (oA) =
:x
Re (A) -
f Re (A) dz -
Re (A 0
+ B)
Re (A)
Re(~~) Re (f Adz)
(1-42)
The proof of these is left to the reader. In addit-ion to the above equations we shall need the following lemma. If A and B are complex quanlitiu, and He (AeJ- ' ) - Re (Bei"'l) for aU I, then A = B. We can readily show this by first taking t = 0, obtaining Re (A) ~ Re (B), and then taking wt - T/2, obtaining 1m (A) = 1m (B). Thus, A - B, for the above two equalities are the definition of this. To illustrate the derivation of an equation for complex quantities from
17
FUNDAMENTAL CONCEPTS
one for instantaneous quantities, consider
Expressing v and £ in terms of their complex counterparts, we have
V2 Re (v"·') -
f V2 Re (E';·') . dl
By steps justifiable by Eqs. (1-42), this reduces to
Cancella.tion of the "\I'2's and application of the above lemma then gives
Note tha.t this is of the same form as the original instantaneous equation. We have illustrated the procedure with a scalar equation, but the same steps apply to the components of a vector equation. From our rules for manipulation of the Re ( ) operator, it should be apparent that any equation linearly relating instantaneous quantities and not involving time differentiation takes the same fonn for complex quantities. Thus, the complex circuit quantities V, I, U, and K are related to the complex field quantities E, H, J, and M according to
V-fE.dl I-ffJ·d'
U=fH'dl K=ffM'd'
(1-43)
There is no time differentiation explicit in the field equations written in generalized current notation. The complex forms of these must therefore also be the same as the instantaneous forms. For example, the complex form of Eqs. (1-20) is
v X H ... JI
(1-44)
Even though these complex equations look the Bame as the corresponding instantaneous equations, we should always keep in mind the difference in meaning. As an illustration of the procedure when the instantaneous equation exhibits a time differentiation, consider the equation
alB
VX£=-aT
Again we express the instantaneous quantities in terms of the complex
18
TUlE-1lARJ40NIC ELECTROMAGNETlC FIELDS
quantities, and obtain
The time variation is explicit, Bnd the differentiation can be performed. By steps justifiable by Eqs. (1-42), the above equation becomes
y'2 Re (V X Eel") = - y'2 Re (jwBeI"') By the foregoing lemma, this reduces to V X E - -jwB
It should now be apparent that each time derivative in a linear instantaneous equation is replaced by ajc.J multiplier in the corresponding complex equation. For example, the Maxwell equations in complex (orm corresponding to Eqs. (1-1) are V X E -
v·B - 0 v· D - Q.
-jwB
vXH~iwD+J
(1-45)
The other forms of these can be obtained in a similar fashion. 1-9. Complex Constitutive Parameters. The constitutive relationships for matter linear in the general sense can be specialized to the a-e case by the procedure of the preceding section. To illustrate, consider the first of Eqs. (1-16), which is
:O_(f+fl'!+f,a + ...)& at at' t
The complex (orm of this equation is readily found as D = (f
+;Wft -
The quantity (t + jWEl - Wlft which we shall denote by i(w).
Wlf:
+ ..
+ ...) is just
')E
compler. (unction of w, Thus, the complex equation 8
D - .(w)E
which looks like the form for simple media, is actually valid for media linear in the general sense. The other two of Eqs. (1-16) simplify in a similar manner; so we have the a-c comlitulivt relatiomhip! D - '(w)E B ~ p(w)H J' - '(w)E
(1-46)
for linear media. We call l the complex permiUiuily of the medium, P the compkx permeability of the medium, and 4 the complex tXlnductivity
FUNDAMENTAL CONCEPTS
19
of the medium. Remember that these parameters are not necessarily the d-e parameters, but I:(w), A(w), 8'(w)
_.
--+ E, lA,
CT
The d-e parameters may apply over a wide range of frequencies for some materials but never over all frequencies (vacuum excepted). In terms of tbe generalized current concept, the induced currents (caused by the field) are
+
J - (8 jw.)E - g(w)E M - jwpJI - f(w)H
(1-47)
The parameter ti(w) bas the dimensions of admittance per length and will be called the admiUivity of the medium. The parameter ~(w) has the dimensions of impedance per length and will be called the impedivity of the medium. Note that fj is a combination of the 8' and ~ parameters. A measurement of fj is relatively simple, but it is difficult to separate 4 from i. The distinction is primarily philosophical. If the current is due to free charge, we include its effect in a. If the current is due to bound charge, we include its effect in i. Thus, when talking of conductors, the usual convention is to let fj = 6' + jWf.o. When discussing dielectrics, it is common to let '0 = jWf. To represent sources, impressed currents are added to the induced currents of Eqs. (1-47). Thus, the general form of the a-e field equations IS
- V X E V X H
+ M' ~ g(w)E + J' ~
f(w)H
(1-48)
The z(w) and 'O(w) specify the characteristics of the media. The J' and Mi represent the sources. Equations (1-48) are therefore two equations for determining the complex field E, H. Solutions to these equations are the principal topic of this book. 1-10. Complex Power. In Sec. 1-5 we considered expressions for instantaneous power and energy in terms of the instantaneous field vectors. We shall show now that similar expressions in terms of the complex field vectors represent time-average power and energy in a.-c fields. For this, we shall need the concept of complex conjugate quantities, denoted by·, and defined as follows. If A - a' + jalf = IAlei", the conjugate of A is A· "'" a' - jalf = jAlcJ... It follows from this that AA' ~ IAI', Let us first consider any two a-c quantities a. a.nd CB, which may be scalars or components of vectors. These are in general of the form
a - 01A I cos (wt + a) (II -
01BI cos (wI
+ P)
- 0 Re (A,"') - 0 Re (Be i.,)
20
TlMJ)o-IIABJ,[ONlC ELECTROMAGNETIC nELDS
where A - IAIe'- and B
~
IBI....
The product of two such quantities is
01AI cos (",t + a) 01BI cos (..t + P) - IAllBllcos (a - P) + cos (2",t + a + P»)
Cl
(1-49)
We shall denote the time average of a quantity by a ba.r over that quantity. The time average of the above expression i.e Cl
IAIIBI cos (a - P)
We also note that
AB' - IAIIBI(.os (a - P) 80
+ i sin ~a - p»)
it is evident that Cl
(AB')
(1-50)
This identity forms the basis of definitions of complex power. The instantaneous Poynting vector [Eq. (l-28)} can be expanded in rectangular coordinates as
S = u.(8,:JC. - B.Xy)
+ uvCB.:JC. -
B.:JC.)
+ u.(&.3C, -
BI/:JC.)
This is a sum of terms, each of which is the form of Eq. (1-49). fore foUows that
s=
It there-
& X "" = Re (E X H')
In view of this we define a complex Poynting veet4r S = E X U·
(1-51)
whose real part is the time average of the instantaneous Poynting vector, or S-Re(S) (I-52) We shall interpret the imaginary part of Slater. We can obtain an equation in which S appears by operating on the complex field equations in a manner similar to that used in the instantaneous case. Starting from Eqs. (1-44), we 8calarly multiply the first by H· and the conjugate of the second by E. The difference of the resulting two equations is E . V X H· - H· . V X E "'" E • JI. + H· . M' The left-hand term is - V . (E X H·) by a mathematical identity; so we have (1-53) v . (E X H') + E . + H' . M' - 0
r"
The integral form of this is obtained by integrating throughout
8.
region
21
FUNDAKENTAL CONC&P1'8
and applying the divergence theorem.
This results in
1ft E X H' . ds + fff (E· J" + H' . M') dT -
0
(I-54)
Comp...e these with Eqs. (1-26) and (1-27). We shan can Eqs. (I-53) and (1-54) expressions for the conurvation of complex. pqwer, the former applying at a point and the latter applying to an entire region. The various terms of the above equations are interpreted as foUo\\1J. As suggested by Eqs. (1-29) and (1-52). we define a compla volume cWuity of power kaui1U} a point as
iiI - V . S = V • (E X H')
(1-55)
The real part of this is a time-average volume density of power leaving a point, or Re (fJJ) - PI (1-56) where PI is defined by Eq. (1-29). po'IOeT kauing a region as
PI
=
Similarly. we define the complex
1ft S· ds - 1ft E X H'· ds
(1-57)
It is evident from Eqs. (1-30) and (1-52) that the real part of this is the time-average power flow, or (I-58)
Note that these relntionsrnps are quite different Crom those used to inter. pret most complex: quantities IEqs. (1-40) and (1-41)). This is because 5. P. and 19 are not sinusoidal quantities but are Cormed of products of sinusoidal quantities. To interpret the other terms of Eq. (1-53), let us first specialize to the ease oC a source-Cree field in media linear in the simple sense. We then have J' = OE - (. + j",,)E Mf ... .fH = jWIIH so E . J" = .IEI' - j""IEI' H- . M' - jWIIIHI' where lEi' means E . E- and IHI' means H . H*. In terms of the instantaneous energy and power definit.ions of Eqs. (1-32) and (1-33). we have
'fJ, -
.IEI'
~'IEI' ~"IHI' We can now write Eq. (1-53) as
w, w. -
V •S
} in simple media
+ P' + j2",(w. -
Ill,) - 0
(1-59)
(1-60)
22
TIME-HARMONIC ELECTROMAGNETIC FIELDS
Thus, the imaginary part of 'PI as defined by Eq. (I-55) is 2w times the difference between the time-average electric and magnetic energy densities. The integral relationships corresponding to Eqs. (1-59) arc
in simple media
(1-61)
where C9d, 'W., and '9... arc defined by Eqs. (1-36) and (1-37). The specialization of Eq. (1-54) to source-free simple media is therefore
1ft S· ds + <1', + j2w('W. -
'W.) - 0
(1-62)
corresponding to the point relntionsbip of Eq. (1-60). Note that this interpretation of complex power is precisely that chosen in circuit theory. IC sources arc prescnt, a. complex power density supplied by tM sources ca.n be defined as (1-63) ft. - -(E· J" + H'· M') The real part of this is the time-average power density supplied by the sources, or
He (ft.) - P. where p. is defined by Eq. (1-34). P. - PI
(1-64)
We can write Eq. (1-53) in general as
+ P' + j2w(w. -
w.)
(1-65)
where all terms have been identified for simple media.. Similarly, the total complex power 8upplied by sources within a region can be defined as P. - -
!!! (E . J" + H' . M') dr
(1-66)
where, from Eq. (1-38), it is evident that
He (P.) = <1'.
(1-67)
Then the form of Eq. (1-65) applicable to an entire region is
P. - PI
+ <1', + j2w('W.
- '11'.)
The real part of this represents a time-average power balance.
(1-68)
The
imaginary part is related to time-average energies, and, in conformity with
circuit theory nomenclature, is called reactive power. Note that we have never defined
FUNDAMENTAL CONCEPTS
23
general case of linear media by extending our definitions. This is done The time-average power dissipntion is defined in general as
as follows.
fJ', = Re
[ffl (tilEI' + flHI') dT]
(1-69)
which reduces to the first of Eqs. (1-61) in simple media. The first term of the integrand represents both conduction and dielect·ric losses, and tbe second term represents magnetic losses. The time-avernge electric and magnetic energies are defined in general as
(1-70)
which reduce to the last two of Eqs. (1-61) in simple media. The first of Eqs. (1-70) includes kinetic energy stored by free charges as well as the usual field and polarization energies. More discussion of this concept -is given in the next section. 1-11. A-C Characteristics of Matter. In souree-free regions, the complex field equatiolU5 read - V X
In free space, ! and
E - l(w)H
V X
H - ti(w)E
y assume their simplest forms, being O(w) = jWfo lew) = jWJAO
1
in free space
(1-71)
These hold for all frequencies and all ficld intensities. In metals, the conductivity remains very close to the d-c value for all radio frequencies, that is, up to the infrared frequency spectrum. The permittivity of metals is hard to measure but appeara to be approximately that of vacuum. Thus,
~~:~ : jw~ jWfO }
in nonmagnetic metals
(1-72)
In ferromagnetic metals, JAo would be replaced by fl. We shall consider this case later_ In good dielectrics, it is common practice to neglect" and express 9 entirely in terms of l. Thus, ti(w) - jw. lew) = jWJAo
I
in nonmagnetic dielectrics
Let us now consider l(w) in more detai1. l l
(1-73)
We can express l in both ree-
A. Von Hipple, "Dielectric MaterWa and ApplicatioWl," John Wiley & BoWl,
!I:le_, New York, 1954.
24
TIME-HARMONIC ELECTROMAGNETIC FIELDS
tangular and polar form as ~(w) =
l - jt" =
llie-il
(1-74)
where I, I', and /; are real quantities. We call l the a-c capacitiuity, l' the dielectric loss /Mtor, and 0 the dielectric loss angle. In Sec. 1-13 we
shall see that they are related to the capacitance, resistance, and loss angle, respectively, of an ideal circuit capacitor. In terms of power and energy, we have from Eqs. (1-69) and (1-70) that
Iff <'IEI'd, ~,~ IfI w<"IEI'd,
W.
~~
(1-75)
Thus, I contributes to stored energy (acts like t in simple matter), and contributes to power dissipation (acts like u in simple matter). Measured values of ~(w) are usually expressed in terms of I and tan 0, or in terms of t' and (". We shall use the latter representation. A
3
"/f<J 2
~
0.0012
0.0008
~ /
1
0.0004
o
o
'''/eo
;.0:- -
10
FIG. 1-10. ;(1.,1) "'" f' -
if"
-
1/
102 103 104 10S!Q6 107 lQ8 10' 10 10 Frequency, cycles per sec verllus
frequency for polystyrene at 25"C,
25
FUNDAMENTAL CONCEPTS
0.20
~
4
0.15
3
0.10
f- 2
\
\
K
..
,,/ \
("/'0
,
0.05
1
" o
o
-. --
10 102 103 104 105 106 107 loa 109 10 10 Frequency, cycles per sec
FlO. I-II. :C...) - e' - je" versus frequency for PlcxiglM at 25°0.
class (the latter also being ferromagnetic). Such dielectrics are usually lossy. A qualitative explanation of the behavior of i can be made in terms of atomic concepts, but we shall view i as simply a measured parameter. A table of i for some common dieleckics is given in Appendix B. In ferromagnetic matter, when it can be considered linear, both conduction and dielectric losses may be significant. In addition to these, magnetic losses become important. Thus,
~-u+jw'l f. = jwp.
in ferromagnetic matter
(1-76)
The parameter p.(w) can be treated in a manner analogous to the treatment of lew). Thus, we express jJ. in both rectangular and polar form as
pew)
~
"' - j"" - Iple-;··
(1-77)
where /Jo', ~", and 6.. are real quantities. We call p' the a-c inductivity, the magnetic loS8 factoT, and 6... the magnetic loss angle. In Sec. 1-13 we shall see that they are related to the inductance, resistance, and loss angle, respectively, of an ideal circuit inductor. In terms of power and energy, we have from Eqs. (1-69) and {1-70) that p"
\1>.
~
Hff "'!HI' fff W""IHI'
dr (1-78)
dr
26
TUrn-HARM:ONIC ELECTRO)(AON"ETIC FIELDS
28
l,-,
24 /,-'/1'0
flO
/
;-
• 16
~
c
•
f 12
I
:•
I
8
-"I" I
\
4
o ...{,
lQ2 lQJ
FIG. 1.12. ;'(w) - ",' -
10,
10" la' lQ6 Freqllency, cycles per sec
hi' vel'8Wl frequency
'< k }()8
1()9 10
,
0
for Fcrramte A at ZS-C.
where the above lJ'" is only the time-average magnetic power loss, to which must be added the conduction and dielectric losscs for the total power ~ipation. Thus,~' contributes to stored energy and Jl" to power dissipation. Measured values of p(<4) are usually expressed in terms of p.' and tan ~., or in terms of 1£' and p.". We shall use the latter representation. Ferromagnetic metals are extremely lossy materials (primarily due to u), and also quite nonlinear with respect to fl. They are seldom intentionally used at radio frequencies. However, the ferromagnetic ceramics CaD be profitably used at radio frequencies to obtain high values of p.'. They are lossy in the magnetic sense, in that they also have appreciable 1J". Figure 1-12 shows /J' and /J" versus frequency for Ferramic A, to illustrate the characteristics of ferrite ceramics. These materials become even more usefu1 when magnet.ized by n d-c magnet.ic field, in which case f! assumes t.he form of an asymmetrical tensor. Magnetized ferrites can be used to build II nonreciprocal" devices, such as U isolators" and
C. L. Hogan, The Ferromagnetic Effect. at. Microwave Frequencies, Bell Sf/Item
Tteh. J. t vol. 31, no. 1, January, 1952.
FUNDAMENTAL CONCEPTS
27
even though it is not entirely a. motion of charge. A further genera.lization was made to include magnetic displacement current as a. .. dual" concept of the electric displacement current. Finally, impressed currents, both electric and magnetic, have been introduced to represent sources. Because of the breadth of the concept of current, many different phenomena are included, and the nomenclature used is somewhat lengthy. We shall summarize the notation and concepts in complex form in this section. Consider the complex electric current density. Internal to conductors, the current is, for all practical purposes, due entirely to the motion oC free electrons. Such current is called the conduction current and is expressed mathematically by J = aE. (We shall consider u = a, a. real quantity, Cor this discussion. This is usually true at radio frequencies.) Even in dielectrics there is some conduction current, but it is usually small. In Cree space there is no motion of charges at aU, and we have only a free-space di8placement current, given by J = jWEoE. In ma.tter, in addition to the conduction current and the free-space displacement current, we have a. current due to the motion oC bound charges. This is called the polarization current and is expressed mathematically by J = jw(~ - EO)E. Because the term J = jw~E is of the same mathematical form as the free-space displacement current, it is called the displacement current. For our purposes, still a.nother division of the electric current is convenient. This involves viewing the current in terms of a. component in phase with E, called the di8sipative current, J = (a + w~If)E, and a. component out of phase with E, called the reactive current, J = jwe'E. This is essentia.lly a generalization of the circuit concept of current, where the dissipative current produces the power loss and the reactive current gives rise to the stored energy. All the currents mentioned are classificd as indueed currents, that is, are caused by tbe field. Impre88ed currents arc used to represent sources or known quantities. In this sense, they are independent oC the field and are said to cause the field. The total electric eurrent is the sum of the induced currents plus the impressed currents. The nomenclature used for electric currents is summarized in the first column of Ta.ble 1-2. Both the nomenclature and the concepts of complex magnetic currents Qre similar to those for electric currents. The one essential difference in the two concepts is the nonexistence of magnetic II charges II in nature. Thus, there is no free magnetic charge and no magnetic conduction current. In absence oC matter, we have a. magnetic free-space diaplacement current, M = jWlloH, a.nalogous to the electric case. When matter is present, we have magnetic effects due to the motion of the atomic particles, giving rise to an induced magnetic current in addition to the free-space displacement current. We call this the magnetic polarization
28
TnlE-IlARMONIC ELECTROllAONETIC nILD6 TAJlLIl 1-2. CI.Aaa1J'ICATlOS OJ'
ELunuc
AND MAGNZTlC CtmuN'T11
Complex electric cunent density
Type
Complex magnetic current. densit.y
.E
Conduction
jWttE
;..,p.~
i..,(l - ft)E
j..,(p - pt)H
;..<E
;w;iH
(.. + """)E
",p"B
;wi1J.
;.-.,.'H
Free-epace dillplacement.
Po1Ariul.tion Displacement.
Disaipative Reactive
gE - (.,
Induced
- (0'
+ ;..,a}E + W.!' + j .....)E
JIB - j..,jiH - (...p"
J'
Impressed
J' -
Total
~E
+ ;101,.')8
M'
+ Ji
M' _
fa
+ MI
expressed by M = jw(p. - JoI.)H. The term M = jwpH is called the magnetic dilplacem.enl current. being the sum of the (ree-spaec displacement current. and the polarization current. We find it convenient to divide the magnetic current into a component in phase with H, called the magnetic di38ipative current, M - wp."H, and a component out of phase with HI called the magnetic reactive current, M jWIA'H. The dissipative magnetic current contributes to the power loss, nnd the reactive magnetic current contributes to the stored energy. All the aforementioned magnetic currents arc indtu:ed currents, that is, caused by the field. In nonmagnetic matter, the induced magnetic current is simply the free-space displacement current, M - j"'l-loH, a. reactive current. To represent sources or known quantities, we use impTC88ed currents. The nomenclature for magnetic currents is summarized in the second column of Table 1-2. A convenient classification of matter from the elcctric current standpoint can be made in terInB of a quality factor Q. This is defined as magnitude of reactive current density Q - magnitude of dlSSipative current density
CUrTent.
:IIC
~'
(1-79)
- :.-:+i":~=" In nonmagnetic matter, this involvcs a ratio of stored electric energy to
FUNDAMENTAL CONCEPTS
29
power dissipated. In terms oC the energy and power densities , Eq. (1-79) can be written n.s
_
Q - (a
""lEI'
+ ",")[E]'
_
peak density oC electric energy average density oC power dissipated = 211' peak density of electr\c energy density oC energy dissipated in onc cycle -
W
(1-80)
Thus, the concept of Q in nonmagnetic matter can be considered as an extension of the concept of Q for capacitors in circuit theory. A good dielectric is a high-Q material, while conductors have an extremely low Q. When magnetic matter is considered, there is an additional power dissipation due to magnetic hysteresis loss. The interpretation given to Eq. (1-80) must be modified, since it includes only the power loss due to el~ctric effects. In this case, the Q defined above would be called the electric Q, and an analogous magnetic quality factor Q~ could be defined. Since we deal principally with nonmagnetic materials, we shall not expand this concept further. 1-13. A-C Behavior of Circuit Elements. The complex notation used for a-c fields is the extension of the complex notation used for a-
30
TUlE--H.ARM.ONIC ELECI'ROllAONETlC FIELDS
+ V
~I
+i
t
t1,
I,
1,
G
1
•
I I I
, I
I,'
V
Flo. }·13. A capacitor according to circuit concepts. equivalent, circuit; (c) complex diagram.
(a) Physical capacitor; (b)
element is called a resistor, and when P is primarily imaginary, the element is called a reactor. A reactor is called an inductor or capacitor according as 1m (Z) is positive or negative, respectively. It should be noted that p. and hence Z, is a function of frequency. Thus, the designa~ tion of an element as a resistor, inductor, or capacitor is. to somo degree. dependent upon frequency. We usually classify elements according to their low-frequency behavior. For an explicit discussion, consider the parallel-plate capacitor of Fig. 1~13a. The low-frequency equivalent circuit of tbis element is shown in Fig. 1·13b, where the conductance G accounts for energy dissipation (Lnd the capacitance C accounts for energy storage. The relationship or complex terminal current I to complex terminal voltage V is
I = I.
+ I, =
+ jwC)V
YV = (G
Figure 1-13c shows the complex diagram representing this equation. complex power to the element is l
P =
IVI'(G -
(1-83) The
jwC)
For a H good" capacitor (wC» G) the current leads the voltage by almost 90°. and the power is principally reactive. For a "poor" capacitor (0)> wC) the current and voltage are almost in phase, and the power is principally dissipative. The element in this case could be classified as a resistor. The angle between Ie and I is called the loss angle 8, as shown in Fig. 1-13c. Let US idealize the problem to a capacitor with perfectly conducting plates. Furthermore. we shall approximate the field by
V
E=(f
I
J=A
1 We are using t.he convention P - VI-. Some authors define P .. IV-, in which case the lign of react.ive power is opposite to that. which we get.
31
FUNDAMENTAL CONCEPTS
where A is the area. of the plates and d is their separation. The constitutive relationship for the field between the capacitor plates is
J - fiE
~~
(.
+ wi' + i",,')E
Substituting for E and J from the preceding
where we have taken 4 = u. equations, we have
I =
~
&-0
V = (u
+ we" + j~') ~
V
A eompnrison of this with Eqs. (1-83) shows that
Y -
A
fI d
G - (.
A
A C - l d
+ wi') d
Thus, for our idealized circuit element, the admittance is proportional to the admittivity of the matter between the plates. The equivalency of l
P-
III fI'IEI' d, = fI'IEI'Ad ~ IVI'Y'
We can use this result to define the admittance of a cube and then view admittivity 9 as the admittance of a unit cube. The magnetic properties of matter are similarly related to the circuit behavior of an inductor. To demonstrate this, consider the toroidal inductor of Fig. 1-14.a. The low-frequency equivalent circuit of this element is shown in Fig. l-Ub, where the resistance R accounts for energy dissipation and the inductance L accounts for energy storage. The relationship of c!omplex terminal voltage V to complex terminal current I is (1-84) v = V, + V, = ZI = (R + jwL)I The complex diagram representing this, equation is shown in Fig. 1-14c.
+J V, - - - - -
v
/{
L V,
(a)
v
1
(b)
FIc. 1-14. An inductor aeeording to circuit concepti. equivalent circuit; (c) complex diagram..
(0) Toroidal inductor; (6)
32
TIME-HARMONIC
ELECTROM~GNETIC FIELDS
The complex power to the element is
p
~
JII'(R
+ jwL)
For a good inductor (wL» R) the current lngs the voltage by almost 90°, and the power is principally reactive. For a poor inductor (R» wL) the current and voltage are almost in phase, and the power is princi~ ~pally dissipative. The element in this case could be classified as a resrstor. The angle between VI and V is called the magnetic loss angle (I..., as shown on Fig. 1-14c. We now idealize the problem to an inductor of perfectly conducting wire and approximate the field by
where N is the number of turns, l is the average circumference, and A is the cross-sectional area. The magnetic constitutive relationship for the field in tbe core is M = !H = (wp."
+ jwp.')11
A substitution for 11 and M from the preceding equations gives N2A I I, - •;l -1-
;0
(WIJ"
+.JWIJ ') -1N'A
I
Comparing this with Eq. (l-84), we see that Z _ I N'A 1
R
= WIJ"N2 A
1
L=IJI
N2A
1
Thus, for the idealized inductor, the impedance is proportional to the impedivity of the matter. From Eq. (l-82), the power supplied to the inductor is
p -
III IIHl'dT -IIH!'AI- III'Z
which is consistent with Eq. (1-82). Using this result to define the impedance of a cube, we can think of impedivity as the impedance of a unit cube. This development serves to illustrate the close correspondences between a,..c circuit concepts and a-c field concepts. A summary of the various concepts is given in Table 1-3. 1-14. Singularities of the Field. A field is said to be singular at a point for which the function or its derivatives are discontinuous. Most of our discussion so far has been about wcll·bchaved fields, but we have meant to include by implication certain types of aUowable singularities.
33
FUNDAMENTAL CONCEPT8 TABLE 1-3. COIUtESPONDENCES BETWEEN A-C CIRCUIT CONCEPTS AND A-C Y.'U:I.D CONCEM1I
A~C
circuit conccpts
A-C field concepts
Complcx voltage V
Complex electric intensity E CompleK magnctic currcnt density M
Complex current 1
Complex electric current density Complex magnetic intensity H
Complex power flow VI-
Density of complex powcr now E
Impedance Z(",,)
Impedivity i(",,)
Admittance Y("")
Admittivity D(...) Conductors (IT
Resistors: Admittance, y{",) Current, 1 -
1
Ii
1
-Ii
Powcr dissipation
R1 VV·
1 . Admittancc, y{",) --+,,,,C R
Stored energy
Admittivity, 9{",,) ... tT
J-
tTE
DeJUlity of power dissipation, oE . EDielectrics (",&" »tT):
Capacitors:
Current, I -
x a-
» ",f_):
Current density,
V
J
(~ + j""C) V ~CVV-
Power di!l8ipation
~
VV·
InductOIll: Impedance, Z(",) - R + j",l~ Voltage, V - (R + jwL)l Stored energy, HL1P Power dissipation, Rl1·
Admittivity, 9(...) ... "'f" Current density,
J ...
+ j",l
("'f"
+ j",e')E
Density of stored energy, H.'E· E· Density of power dissipll.tion, "'f"E . E· Magnetic propertics: lmpedivity, !I(",) - "'jJ." + jwjJ.' Magnetic currcnt, M - (wp." + jWjJ.')H Density of stored energy, H,,'H' HDeJUlity of power dissipation, wjJ."H . a-
These can occur at material boundaries (discontinuous ~ and '0) and at singular source distributions, such as sheets and filaments of currents. A13 evidenced by Eqs. (1-44), the total electric and magnetic currents are vortices of Hand -E, respectively. Suppose we have a surfa.ce distribution of currents J_ and M., as represented by Fig. 1-15. By applying (1-1l5)
34
TllIE-lLUUolONIC ELEcrROMAGNETIC FIELDS n
Region (1)
J.
s
M.
FlO. 1_15. Surface currenU.
Region (2)
to rectangular paths enclosing a portion of the surface currents, we obtain 1 (l-86)
where n is the unit vector normal to the surface and pointing into region (1). The superscripts (1) and (2) denote the side of S on which E or H is evaluated. Equations (1-86) arc essentially the field equations at sheets of currents. They express at current sheets the same concept as Eqs. (1-44) express at volume distributions of currents. If J. and M. are impressed currents, Eqs. (1-86) arc tbe Uboundary conditions" to be
satisfied at the source. Equations (1-8(;) apply regardless of whether or not 8. discontinuity in media exists on S. Whenever J. and M. are zero, Eqs. (1-86) state that the tangential components of E and H are continuous across the surface. If z and fJ are finite in both regions 1 and 2, no induced surface current can result. Thus, tangential compomnts of E and Hare e<>nlinuom aero" any material boundary, perfect conductors excepted. If onQ side of S is a perfect electric conductor, say region 2, a surface conduction current J. can exist even though E is zero, since y "'" a is infinite. In this CllSC, Eq•. (1-86) reduce 10 n X H =
J'I
nXE-O
at a perfect conductor
(1-87)
where n points into the region of field. Thus, the "boundary condition" at a perfect electric conductor is vanishing tangential components of E. The perfut magnetic e<>nductor is defined to be So mat.crial for which the tangential components of Hare ?-cro at its surface. This is, however, purely a mathematical concept. The necessary" magnetic conduction current" on its surface has no physical significance. Finally, at a filament of current, the field must be singular such that Eqs. (1-85) yield the current enclosed, no matl-er how small the contour. For example, at a filament of electric current I, the boundary condition for H is
"'H.d!
'fa
,1
radiul of 0-0
(l-88)
A similar limit of the second of Eqs. (1-85) must be satisfied at a filament
of magnetic current. I R. F. Harrington, "Introduction to Electromagnetic EnginC
35
FUNDAMENTAL CONCEPTS
It is often convenient for mathematical and discussional purposes to consider the various singular Quantities aa limits of nonsingular quantities. For example, we can think of an abrupt material boundary aa the limit of a continuous, but rapid, change in '0 and!. Similarly, a sheet of current can be thought of as a volume distribution of current having a large magnitude and confmed to a thin shell. By such expediencies we can avoid much tedium in the exposition of the theory. PROBLEMS
1-1. Using Stokes' theorem and the divergence theorem, show that Eqs. (1-1) are equivalent to Eqs. (1-3). 1-11. The conduction current in conductors is affected by the magnetic field 1\8 well as by the electric field (Hall effect). Using an atomic model, justify that 1I ... tr8
+ tr'kE.
X
(B
where h is the Hall constant. For copper (h _ -5.5 X 10- 11 ), determine the (B {or which the second term of the above equation is 1 per cent of the first term. 1-3. Given g .. u..yl sin wt and :lC .. u"x cos wl, determine n' and ::mI. Determine i' and k' through the disk z - 0, x' + y' ... I. 1-4. For the field of Frob. 1-3, determine the Poynting vector. Show that Eq. (1-26) is satis6ed for this field. 1-6, Starting from Ma.xwell's equations, derive the circuit law for capacitors, i - C dvld4, and the circuit law for inductors, 1/ - L dildt. 1-6. Determine the instantaneous quantities corresponding to (a) 1 - 10 + j5, (b) E - useS + j3) + u,(2 + j3), (c) H - (u. + u,)elt.+.l. 1-7. Prove Eqs. (1-42). 1-8. Given H - u. sin y in a 9Ource-free region of Plexigla.e, determine E And g at a. frequency of (a) 1 megacycle, (b) 100 megacycles. 1-9. Show that Q. - 0 (complex charge density vanishes) in a source-free region of homogeneous matter, linear in the general sense. 1-10. Show that the instantaneous Poynting vector is given by 5 -
Re (8
+E
X Hel"-I)
Why is 5 not related to S by Eq. (1-41)7 1-11. Consider the unit cube shown in Fig. 1-16 which has sl1 sides except the face z - 0 covered by perfect conductors. If B• .. 100 sin (ry) and 11. - e/r'l sin (ry)
z 11"'--<' 1
y
X
FlO. 1-16. Unit cube for Prob. 1-11.
35
TlKE-HARMONIC ELECTROMAGNETIC FIELDS
over the open face aDd no sources exist. within tbe cube, determine (4) tbe timeaverage power d.issipated within the cube, (b) the difference betwccn the time-average electric and magnetic energies witbin the cube. 1-12. Suppose a filament of z-diroctcd electric current Ii - 10 is impressed along the 2: axis from z os 0 to z = 1. U B - u,(l determine the complex power and the time-average power supplied by this source. 1-13. Suppose we have a 10--megacycle field B - u..5, H - u,2, at tome point. in a material having ~ - 10-\ a- (8 - jlO-I).o, and j;J. - (14 - ;)110' at tbe operating frequency. Determine each type of current (except impressed) listed in Table 1-2. I-a. A small capa.eitor has a d-e capacitance of 300 micromicrofaradll when airfilled. When it is oil-lilled, it is found to have an impedance of (500 - 11 X 10' at 0lJ 10'. Determine fl. t', and ~"of tho oil, neglecting conductor IOIllle8. I-Ifi. For a prnctical toroidal induetor of the type shown in Fig. 1.I4a, show that the power IOflll in the wire will usually be much larger than that in a core of 10w·loSli ferromagnetic materiaL 1-16. Assume that ~ - I' - jl" is an 8nalytie function of '" and !Ihow that
+ ,1.
I(w)
-., + - f· 2
101'(10) dw
rOw'",'
__
~
( . 10[1(10) -
r}o
w'
fll dw w'
(Equations of this type are valid for any analytic function regular in the lower b&1f plane.) 1-17. Derive Eqs. (1-86).
CHAPTER
2
INTRODUCfION TO WAVES
2-1. The Wave Equation. A field that is a function of both time and space coordinates can be called a wave. We shall, however, be 11 bit more restrictive in our definition and use the term wave to denote a solution to a particular type of equation, called a wave equation. Electromagnetic fields obey wave equations, so the terms wave and field are synonymous for time-varying electromagnetism. In this chapter we shall consider a number of simple wave solutions to introduce and iIIustra.te various a-c electromagnetic phenomena. For the present, let us consider fields in regions which are source-free Ui = Mi = 0), linear (£ and 1) independent of lEI and 181), homogeneous (z and 1) independent of position), and isotropic (z and y are scalar). The complex field equations are then
v X E - -£H V X H - gE
(2-1)
The curl of the first equation is V X V X E = -zV X H
which, upon substitution for V X H from the second equation, becomes V X V X E - -£gE .
The frequently encountered parameter
'Ii -£0
(2-2)
is called the wave number of the medium. equation becomes
In terms of k, the preceding
k ~
v
X V X E - k'E = 0
(2-3)
which we shall call the complex vector wave equation. If we return to Eqs. (2-1), take the curl of the second equation, and substitute from the first equation, we obtain VXVXH-k'H=O
Thus, H is a solution to the same complex wave equation as is E. 37
(2-4)
38
TIME-HARMONIC ELECTROllLAGNETIC FIELDS
The wave equation is often written in another (orm by defining an operation V'A
~ '1('1 .
A) - V X V X A
In rectangular components, this reduces to
+ u,IVIA, + u.VIA.
vIA _ u"V 2A.,
where u." U II , and u. arc the rectangular-coordinate unit vectors and is the Laplacian operator. It is implicit in the wave equations that
VI
(2-5)
v·H - 0
shown by taking the divergence of Eqs. (2-3) and (2-4). Using Eqs. (2-5) and the operation defined above, we can write Eqs. (2-3) and (2-4) as
V'E+ k'E=0
(2-6)
V~H+k2H=O
These we shall also call vector wave equations.
They arc not, however,
so general as the previous forms, for they do not imply Eqs. (2-5).
In
other words, Eqs. (2-6) and Eqs. (2-5) arc equivalent to Eqs. (2-3) and (2-4). Thus, the rectangular components of E and H satisfy the complex 8calar wave equation or Helmholtz equation I
v'" + k'" -
(2-7) 0 We can construct electromagnetic fields by choosing solutions to Eq. (2-7) for E:r:, Ell, and E. or H:r:, H II , and H., such that Eqs. (2-5) are also satisfied. To illustrate the wave behavior of electromagnetic fields, let us construct a simple solution. Take the medium to be n perfect dielectric, in which case fJ "'" jWf:, .! "'" jwSl, and
k -
",.y;;;
Also, take E to have only an x component independent of x and y. first of Eqs. (2-6) then reduces to
d'E. 2 dz
+ k'E
•
(2-8)
The
~0
which is the onc-dimcnsional Helmholtz equation. Solutions to this are linear combinations of eih and e-i *'. In particular, let us consider a solution (2-9) This satisfies V . E = 0 and is therefore a possible electromagnetic field. I We shall usc the symbol'" to denote "wave functiODs," that i9, solutions to Eq. (2-7). Do not confuse these ",,'s with magnetic flux.
39
INTRODUCTION TO WAVES
The associated magnetic field is found according to iWIAH = - V X E =
u.ikE~
which, using Eq. (2-8), can be written as
E~ = ~Hr
(2-10)
Ratios of components of E to components of H have thc dimensions of impedance and arc cRUed wave impedames. The wave impedance associated with our present solution,
is called tbe intrinsic impedance of the medium. Jlo =
In vacuum,
[;;""" 120.. "'" 377 ohms V;;
(2-12)
We shall see later that the intrinsic impedance of a medium enters into wave transmission and reflection problems in the same manner as the characteristic impedance of transmission lines. To interpret this solution, let Eo be real and determine Sand 3C according to Eq. (1-41). The instantaneous fields are found as E. -
:JC. -
V2 E. cos (wt V2 E. cos (wt -
"
kz) kz)
(2-13)
This is called a plam wave because the phase (kz) of Sand :JC is constant over a set of planes (defined by z = constant) called equiphaae aurfaces. It is called a uniform plane wave because the amplitudes (Eo and EO/JI) of Sand 3C are constant over the equiphase planes. S and X are said to be in phaae because they have the same phase at any point. At some specific timc, B and X are sinusoidal functions of z. The vector picture of Fig. 2-1 illustrates sand 3C along the z axis at t = O. The direction of an arrow represents the direction of a vector, and the length of an arrow represents the magnitude of a vector. If we take a slightly later instant of time, the picture of Fig. 2-1 will be shifted in the +z direction. We say that the wave is traveling in the +z direction and call it a. traveling tDaQe. Thc term polarization is used to specify the beha.vior of & lines. In this wa.ve, the & lines are always parallel to the :r: axis, and the wave is said to be limorly polarized in the :r: direction. The velocity at which an equiphase surface travels is called the phau
40
TUIE-IlARldONIC ELECTROMAGNETIC FiELDS
velocity of the wave.
An cquiphase plane z. = zp is defined by wt -
kzl' = constant
that is, the argument of the cosine functions of Eq. (2-13) is constant. As t increases, the value of z.p must also increase to maintain this constancy, and tbe plane z = z,. will move in the +z direction. This is illustrated by Fig. 2-2, which is a plot of 8 for several instants of time. To obtain t.he phase velocity dz ../dt, differentiate the above equation. This gives
dz
w-k-.!'-O
d'
The phase velocity of this wave is called the intrinsic phase veWdty the dielectric and is, according to the above equation,
.------•
dz,. dt
w
k
1
lip
of
(2-14)
v;;.
In vacuum, this is the velocity of light: 3 X 10' meters per second. The wavelength of a wave is defined as the distance in which the phase increases by 2'11'" at any instant. This distance is shown on Fig. 2-2. The wavelength of the particular wave of Eqs. (2-13) is called the intrinsic wavelength ~ of the medium. It is given by k~ "'" Zr, or X = 2r .,. 2TV. = v. k
w
(2-15)
f
where f is the frequency in cycles per second. The wavelength is often used as a. measure of whether a. distance is long or short. The range of wavelengths encountered in electromagnetic engineering is large. For exa.mple, the free-space wavelength of a 6(k:ycle wave is 5000 kilometers, whereas the free-space wavelength of a lOoo-megacycle wave is only 30 centimeters. Thus, a. distance of 1 kilometer is very short at 60 cycles,
x Direction of travel
~
z y Fro. 2.1. A linearly polarized uniform plane traveling wave.
41
INTRODUCTION TO WAVES Glt =
z,
0 "'t = ",/4 "'t",. w/2
z 1 - - - - - >----1 FIG. 2-2. wave.
&
at several instants of time in alincarly polarized uniform planc traveling
but very long at 1000 megacycles. The usual cireuit theory is based on the assumption that distances arc much shorter than a wavelength. 2.2. Waves in Perfect Dielectrics. In this section we shall consider the properties of uniform plane waves in perfeet dielectrics, of which free space ia the most common example. We have already given a special case of the uniform plane wave in the preceding section. To summarize. E z = Eoe- /h
k =
where
w.y';
~=~
E H w = -;;
e-iJ:.
= 2'11" =
~
"
v,
(2-16)
It is an :v-polarized, +z traveling wave. Because of the symmetry of the rectangular ~coordina.te system, other uniform plane-wave solutions can be obtained by rotations of the coordinate axes, corresponding to cyclic interchanges of coordinate variables. We wish to restrict consideration to +z and -z traveling waves; so we shall consider only the transformations (x,y.z) to (-Y,x,z), to (x, -Yo -z), and to (y,x, -z). This procedure, together with our original solution, gives us the four waves
H w+ = A e-ih
• •e -0 H-= __ • •
-B e- jh
H~+ = - -
E.-
=
Ce ih
lh
H,,- = D
•
e/h
(2-17)
42
TUlE-IL\RlIONlC ELECTROMAGNETIC FIELD8
where the preVlously used Eo has been replaced by A, B J C, or D. The superscript + denotes So +t traveling wave, and the superscript. - denotes a -z traveling wave. The most general uniform plan~ wave is a superposition of Eqs. (2-17). We have already interpreted the first wave of Eqs. (2-17) in Sec. 2-1. This also constitutes an interpretation of the other three waves if the appropriate interchanges of coordinates are made. We have not yet mentioned power and energy considerations, so let us do 80 now. Given the traveling wave
we evaluate the various energy and power quantities as
w• .., ;
e' "'" fE,' cos' (loll -
kz)
w. = ~ X' = fE o' cos' (wt - kz)
g
=
E X H*
(2-18)
2
S = & X ~ =
u. -
•
E,'
cos' (loll
-
kz)
E,' -u.-
•
Thus, the electric and magnetic energy densities are equal, hall of the energy of t.he wave being electric and hall magnetic. We can define eo :JeWcity of propageui<Jn of energy v. as v.
-=
power flow densit.y S energy density ... w. + w..
(2-19)
For the uniform plane traveling wave from Eqs. (2-18) a.nd (2-19) we find j
v.-V;;; 1
which is also the phase velocity [Eq. (2-14)]. These two velocities are not necessarily equal for other types of electromagnetic wa.ves. In gen· eral, the phase velocity may be greater or less than the velocity of light, but the velocity of propagation of energy is never greater than the velocity of light. Another property of waves can be illustrated by the uanding ~ E IIl == Eosin kz
.E. ~ H " =J-COS"'"
•
(2-20)
obtained by combining the first and third waves of Eqs. (2-17) with
43
INTRODUCTION TO WAVES
A
= -
C &",
<E
jE o/2.
The corresponding instantaneous fields are
_M = v2Eosinkzcoswt
3C~
=-
-Eo
V2-coskzsinwt
•
Note that the phase is now independent of 1" there being no tra.veling motion i hence the name standing walle. A picture of E and 3C at some instant of time is sho'm in Fig. 2-3. The field oscillates in amplitude, with & reaching its peak value when 3C is zero, and vice versa. In other words, & and 3C are 900 out of phase. The planes of zero & and 3C are fixed in spa<:e, the zeros of & being displaced a quarter-wavelength from the zeros of X. Successive zeros of 8 or of 3C are separated by a half· wavclength, as shown on Fig. 2-3. The wave is still a plane wave, for equipbase surfaces arc planes. It is still a uniform wave, for its amplitude is constant over equiphase surfaces. It is still linearly polarized, for E always points in the same direction (or opposite direction when 8 is negative). The energy and power quantities associated with this wave are w. =
i
&2 ",.
fE o2 sin 2 kz cos' wt
w... = ~ X' = fE o2 cos 2 kz sin 2 wl
E2
(2-21)
S = & X:JC = -u. 2~ sin2kzsin2wt
'E' 2.
S = E X H$ = -UIU sin2kz
The time-average Poynting vector S = Re (5) is zero, showing no power flow on the average. The electric energy density is a maximum when the magnetic energy density is zero, and vice versa. A picture of energy
x
z y FlO. 2-3. A linearly polarized uniform plane standing wave.
44
TUllE-HARMONIC ELECTROMAGNETIC FIELDS
A+C
A-C
FlO.
Z 2-4. Standing-wave pattern of two oppositely traveling waves of unequal ampli-
tudes.
oscillating between the electric and magnetic forms can be used for this wave. Note that we have planes of zero electric intensity at kz. = nr, n an integer. Thus, perfect electric conductors can be placed over one or more of these planes. If an electric conductor covers the plane z "'" 0, Eqs. (2-20) represent the solution to the problem of reflection of a uniform plane wave normally incident on this conductor. If two electric conductors cover the planes kz = th'F and kz = ntr, Eqs. (2-20) represent the solution of a onc-dimcnsional "resonator." A more general x-polarized field is one consisting of waves traveling in opposite directions with unequal amplitudes. This is a superposition of the first and third of Eqs. (2-17), or E,. = Ae-ih
H II =
+ CeJ'h
! (Ae-ib -
•
(2-22)
Ceil.)
If A = 0 or C 0, we have a pure traveling wave, and if IAI = ICI, we have a pure standing wave. For A ¢ C, let us take A and C reaP and express the field in terms of an amplitude and phase. This gives :Ell
A-C
E.,
=
VAS
-un b + CS + 2AC cos 2kz e_jtan-l ( -A+C
)
(2-23)
The rms amplitude of E is
vAt
+ C2 + 2AC cos 2kz
which is called the 8tanding-wave pattern of the field, This is illustrated by Fig. 2-4. The voltage output of a small probe (receiving antenna) connected to a detector would essentially follow this standing-wave patI This is actually no restriction on the generality of our interpretation, for it corresponde to a judicious choice of 2 and t origins.
45
INTRODUcrtON TO WAVES
tern. For a pure traveling wave, the standing-wave pattern is a const..'l.nt, and for a pure standing wave, it is of the form Icos kzj, that is, a "rectified" sine wave. The ratio of the maximum of the standing-wave pattern to tbe minimum is called the standing-wave ratio (SWR). From Fig. 2-4, it is evident that
SWR~A+C
A
(2-24)
C
because the two traveling-wave components lEqs. (2-22)] add in phase at some points and add 1800 out of pbase at other points. The distance between successive minima is >./2. The standing-wave ratio of a pure traveling wave is unity, that of a pure standing wave is infinite. Plane traveling waves reflected by dielectric or imperfectly conducting boundaries will result in partial standing waves, with SWR's between one and infinity. Let us now consider a. traveling wave in which both E~ and Ell exist. This is a superposition of the first and second of Eqs. (2-17), that is,
+ u,8)....'.. (-u~B + uIIA) !e-
E = (u.A
(2-25)
jh
H =
"
If B = 0, the wave is linearly polarized in the x direction. If A = 0, the wave is linearly polarized in the y direction. If A and B are both
real (or complex with equal phases), we again have a linearly polarized wave, with the axis of polarization inclined at an angle tan- 1 (B/ A) with respect to the x axis. This is illustrated by Fig. 2-5a. Ii A and Bare complex with different phase angles, t will no longer point in a single spatil11 direction. Letting A = IAj&"' and B =- IBleitl , we have the instany
y In this direction
,, I
X
t.lt = 5'11'/4
I
/'
..,t = '11',
"-
2-5. Polaritation of a uniform plane traveling wave.
(b) elliptical polarization.
_
-
~ "- t.lt = /
.,t "" 3"/4
FlO.
e
rotates in this direction
e vibrates
0
X
/' t = TT/4 ..,t"" '11'/2
(a) Linear polarization;
46
TIME-HARMONIC ELECTROMAGNETlC FIELDS
taneous electric intensity given by
v'2IA[ cos (wt
&. -
&" = v'2 [EI cos
- kz (wt - kz
+ a) + b)
A vector picture of S Cor various instants of time changes in both amplitude and direction, going through this variation once each cycle. For example, let IAI = 21BI, a = 0, and b = 1r/2. A plot of & for various values of t in the plane z = 0 is shown in Fig. 2-5b. The tip of the arrow in the vector picture traces out an ellipse, and the field is said to be elliptically polarized. Depending upon .Ii and B, this ellipse caD be of arbitrary orientation in the xy plane and of arbitrary axial ratio. Linear polarization can be considered as the special C!\8e of elliptic polarization for which the axial ratio is infinite. If the axial ratio is unity, the tip of the arrow traces out a circle, and the field is said to be circularly polarized. The polariza.tion is said to be right-handed if 8 rotates in the direction of the fingers of the right hand when the thumb points in the direction of propagation. The polarization is said to be left-handed if 8 rotates in the opposite direction. The specialization of Eq. (2-25) to right-handed circular polarization is obtained by setting A = jB = Eo, giving E - (u. - ju,)E,.-J'·
(2-26)
H = (u", - julI)j Eo e-iA.
"
A vector picture of the type of Fig. 2-1 for this wave would show 8 and:JC in the form of two corkscrews, with 8 perpendicular to :JC at each point. As time increases, this picture would rotate giving a corkscrew type of motion in the z direction. Tho various energy and power quantities associated with this wave are
to..
:or
~ ac' "'" fE o'
2 S "'" t X :JC = u. - Eo'
(2-27)
"2 Eo' " Thus, there is no change in energy and power densities with time or S = E X H· =
UJ
-
space. Circular polarization gives a steady power flow J analogous to circuit-tbeory power transmission in a two-phase system.
47
INTRODUCTION TO WAVES
As a final example, consider the circularly polarized standing-wave field specified by E = (U,. + julI)Eo sin kz (2-28) H = (U,.
+ JUII) Eo, cos kz
This is the superposition of Eqs. (2-17) for which A = -0 = jEo/2, D = -B = E o/2. The corresponding instantaneous fields are t =
(U.,
v'2 Eo sin kz u ll sin wt) V2 Eo cos kz
cos wt - u ll sin wt)
:JC = (u" cos
wt -
,
Note that t and :re are always paraUel to each other. A vector picture of t and :Ie at t = 0 is shown in Fig. 2-6. As time progresses, this picture rotates about the z axis, the amplitudes of t and :re being independent of time. It is only the direction of t and :JC which changes with time. The amplitudes of t and :re are, however, a function of z, giving a standingwave pattern in the z direction. The energy and power densities associated with this wave are
w. = ; &' = f.E o' sin' kz w.. = ; X' = tEo' cos' kz (2-29)
S=tX3C=O
,
5 = -u l l Eo'sin 2kz It is interesting to note that the instantaneous energy and power densities are independent of time. This field can represent resonance between two perfectly conducting planes situated where E is zero. It thus seems that the picture of energy oscillating between the electric and magnetic Corms
x
z y
FIo. 2-6. A circularly polarized uniform plane standing wave.
48
TIME-HARMONICELECTUOMAGNETIC FIELDS
is not generally valid (or resonance. However, the circularly polarized standing wave is the sum of two linearly polarized waves which can exist independently of each other. We actually have two coincident resonances (called a degenerate case), and the picture of energy oscillating between electric and magnetic forms applies to each linearly polarized resonance. 2-3. Intrinsic Wave Constants. When the wave aspects of electromagnetism arc emphasized, the wave Dumber k and the intrinsic impedance 7Jt given by k =
y-zg
(2-30)
play an important role. The second equation is a generalization of Eq. (2-11). obtained in the same manner as Eq. (2-11) when i and g are not specialized to the case of a perfect dielectric. We can solve Eqs. (2-30) for z and y, obtaining (2-31 )
A knowledge of k and" is equivalent to a knowledge of ! and y, and hence specifies the characteristics of the medium. The wave number is, in general, complex, and may be written as k=k'-jk"
(2-32)
where k' is the intrinsic phase constant and kIf is the intrinsic attenuation constant. We have already seen that when k = k', it enters into the phase function of the wave. We shall see in the next section that kIf causes an exponential attenuation of the wave amplitude. The behavior of k can be illustrated by a complex diagram rehting k to ! and y. This is shown in Fig. 2-7. In the 1m expressIons
fJ
= u
+ wE" + jwE' + jwJJ.'
Z = wJJ."
f", and JJ." are always positive in source-free media, for they account for energy dissipation. The parameters E' andJJ.' are usually positive but may be negative for certain types of atomic resonance. Thus, z and f) usually lie in the first quadrant of the complex plane, as shown in Fig. 2-7. The product -zQ then usually lies in the bottom hall of the complex
0',
Re
FIo. 2-7. Complex diagram relating k to I and 1).
49
INTRODUCTION TO WAVES
1m
FIG. 2--8. Ccmp1ex diagram relating"
x
to' and g.
1/9 plane. The principal square root, k "'" V -£0, lies in the fourth quadrant, showing that k' and k" arc usually positive. Even when l or p.' is negative, k" is positivej it is only k' that could conceivably be negative. In losslcss media, g = jWf, I = jr.Jp., and k is real. The intrinsic wave impedance can be considered in an analogous manner. Expressing" in rectangular componenUl, we have (2-33)
where at is the intrinsic wave resistance and X is the intrinsic wave reactance. For a wave in a perfect dielectric, " is purely resistive and is there~ fore the ratio of the amplitude of & to X. We shall see in Sec. 24. that X introduces a phase difference between & and x. The complex diagram relating" to 9 and I in general is shown in Fig. 2-8. In source-free regions, v, I', and p." are always positive, and land p.' are usually positive. Thws t usually lies in the first quadrant and l/g in the fourth quadrant. The ratio I/g therefore usually lies in the right half plane and " in the sector ±45° with respect to the positive real axis. When " or p.' is negative, " may lie anywhere in the right half plane, but at is never negative. In lossless media, the wave impedance is real. There are several special cases of particular interest to us. First, consider the case of no magnetic losses. From the first of Eqs. (2-31), we
bAve I ik· jk·, • - jk - jkk' ~ - I!IlDI
the last equality following from Eqs. (2-30). we have
Now for: = jwp. """ jl~l,
no mBgnetic losses
(2-34)
50
TIlLE-HARMONIC ELEC!'ROAUGNETIC FlELDS TaLE
2.1.
WAVE NUlI8ER
bPEO.t.NCI: (If -
(t - l' - ;1.:") AND tJl j!r. - I.lej~
+
k'
INTRlN81C
k"
'"
til
v::t9
""-R
lm~
General
"" v' -.f
No magnetic IOI!lell
Im~
Perfect dielect.ric
... V;;
Good dielectric
... v;I
--"~ -, 2
Good conducLOr
~
~
if
.R'" .R'"
-1m
n. ....r;;;;;g
k'
-
Ifl
k"
Ifl
J
0
0
"~('
2t'
Separation into real and imaginary parts is shown explicitly in row 2 of Table 2-1. A similar simplification can be made for the ease of DO electric losses. (See Pooh. 2-13.) Three special cases of materials with no magnetic losses arc (1) perfect dielectrics, (2) good dielectrics, and (3) good conductors. The perfect dielectric case is that for which
k=w~
f/=~
This is summarized in row 3 of Table 2-1. A good dielectric is chaneterized by! = jWj.I, fJ = we" + jwl, with f.'» l'. In this case, we have
k-w~""(l-jj)~wv;;;'(l-j;:.) k'
" - IUl
~
r; ( .•" ) 'J7 I + ) 2<'
which is summarized in row 4 of Table 2-1. Finally, a good conductor is characterized by ~ = jWJ,l, Y = f1 + jWf, with ~ »Wf.. In this C8.SC, we have
k =
V
k· "-IUl~
jwp(u
+ jWf.)
"'"
V
jW,uT
WJ.'
•
The last row of Ta.ble 2-1 shows these parameters separated into real and imaginary parts.
51
INTRODUCTION TO WAVES
2-4. Waves in Lossy Matter. The only difference between the wave equation, Eq. (2-7), for lossy media and loss-free media is that k is complex in lossy media and real in loss-free media. Thus, Eq. (2-9) is still a solution in lossy media. In terms of the real and imaginary parts of k, it is Also, H is still given by Eq. (2-10), except that "l is now complex. the H associated with the E of Eq. (2-35) is
H w = Eo ell. = Eo e-He-l".cjl'.
•
where "l = II'JIeir. and (2-36) are
Thus, (2-36)
1.1
The instantaneous fields corresponding to Eqs. (2-35) &.. =
V2 EoC
J<;, =
v2 ~ e-
k
cos (wt - k'z)
".
k
".
cos (wt - k'z -
(2-37)
n
Thus, in lossy matter, a traveling wave is attenuated in the direction of travel according to c k "', and X is no longer in phase with S. A sketch of & and 3C versus z at some instant of time would be similar to Fig. 2-1 except that the amplitudes of & and JC would decrease exponentially with z, and 3C would not be in phase with & (:Je usually lags e). A sketch of &:0 versus z for several instants of time is shown in Fig. 2-9 for a case of fairly large attenuation. A sketch of :Jew. versus z would be similar in form.
c. Direction of travel -
~
--
_ ~ Envelope - e- II".
- -z
---
-
--
(lit = ../2
<>It =
"'t -
"'/4
a
FIG. 2-9. & at sevClal instanUJ of time in a linearly polarized uniform plane traveling wave in dissipative matter.
52
TJME-HAR~[QNtC
ELECTROMAGNETIC FIELDS
The wave of Eq. (2-37) is still uniform, still plane, and still linearly polarized. So that our definitions of phase velocity and wavelength will be unchanged for lossy media, we should replace k and k' in the loss-free formulas, or
A=2'1'=~ k'
(2-38)
f
Then v. is still the velocity of a. plane of constant phase, and X is still the distance in whicb the phase increases by 2,... Two cases of particular interest are (1) good dielectrics (low·1055), and (2) good conductors (high-loss). For the first case, we have (see Table 2-1)
in good dielectrics (til
« i)
(2-39)
Thus, the attenuation is very small, and e and JC arc nearly in phase. The wave is almost the same as in a loss-free dielectric. For example, in polystyrene (see Fig. 1-10), a lo-megacycle wave is attenuated only 0.5 per cent per kilometer, and thc phase difference between Sand 3C is only 0.003 0 • The intrinsic impedance of a dielectric is usually less than that of free space, since usually l > fll and jj = jja. The intrinsic phase velocity and wavelength in a dielectric arc also less than those of free space. In the high·loss case (see Table 2-1), we have
k'=~ k" =
1,1
~
~w;o. fo;j,
in good conductors (<1 » Wf)
(2-40)
"1/. T
!~4
Thus, the attenuation is very large, and 3C lags e by 45°. The intrinsic impedance of a good conductor is extremely small at radio frequencies, having a magnitude of 1.16 X 1O-~ ohm for copper at 10 megacycles. The wavelength is also very small compared to the free-space wavelength. For example, at 10 megacycles the free·spacc wavelength is 30 meters, while in copper the wavelength is only 0.131 millimeter. The attenuation
53
INTRODUCTION TO WAVES
in a good conductor is very rapid. For the above-mentioned lO-mego.cycle wave in copper the attenuation is 99.81 per cent in 0.131 millimeter of travel. Thus, waves do not penetrate metals very deeply. A metal ncts as a shield against electromagnetic waves. A wave starting at the surface of a. good conductor and propagating inward is very quickly damped to insignificant values. The field is localized in a thin surface layer, this phenomenon being known as skin effect. The distance in which a wave is attenuated to lie (36.8 per cent) of its initial value is called the skin depth or depth oj penetration 3. This is defined by k"3 = I, or ! _ ~ 2 WP.d
_,!" _ ~.
(2-41)
2r
k'
where A.. is the wavelength in the metal. The skin depth is very small for good conductors at radio frequencies, for A... is very small. For example, the depth of penetration into copper at 10 megacycles is only 0.021 millimeter. The density of power flow into the conductor, which must also be that dissipated within the conductor, is given by S == E X H* = u,lllo!217..
where H 0 is the amplitude of H at the surface. The time-average power dissipation per unit area of surface cross section is the real part of the above power flow, or watts per square meter
(2-42)
where
" l{~+ -B e-'''~e-f''~ "C H, " H.,- = D e1"'ei1-' " ,
=
= -
e!""eJ1"
(2-43)
54
Tl'-fE-HARMONIC ELECTROl'llAGNETIC FIELDS
The preceding discussion of this section applies to each of these waves if the appropriate interchange of coordinates is made. A superposition of waves traveling in opposite directions, for example
+ CeJ<"'ei1h
E,. ...
Ae-k"le-it'.
H II =
! (Ae-.t"'e-i1"
•
- Cef"'gil")
(2-44)
gives us standing-wave phenomena. However, it is no longer possible to have two "equal" waves traveling in opposite directions. One wave is attenuated in the +z direction, the other in the -z direction; hence they can be equal only at one plane. Suppose that the wave componenta are equal at z = 0, that is, A = C ,in Eq. (2-44). There will then be standing waves in the vicinity of z = 0, which will die out in both the +z and -z directions. This is illustrated by Fig. 2-10 for a material having fairly large losses. Far in the +z direc~ion ~he +z traveling wave has died out, leaving only the -z traveling wave. Similarly, far in the -z direction we have only the +z traveling wave. The standing-wave ratio is now a function of z, being large in the vicinity of z = 0 and approaching unity as Iz] becomes large. For very small amounts of dissipation, say in a good dielectric, the attenuation of the wave is small, and standingwave patterns are almost the same as for the dissipationless case. Other superpositions of Eqs. (2-43) can be formed to give elliptically and circularly polarized waves. In a picture of a circularly polarized wave traveling in dissipative media, the If corkscrews" for E and :JC would be attenuated in the direction of propagation. Also, & would be somewhat out of phase with:re. A circularly polarized standing wave would be a localized phenomenon in dissipative media, just as a linearly polarized sta.nding wave is localized. 2-6. Reflection of Waves. We saw in Sec. 1-14 that the tangential components of E and H must be continuous across a material boundary.
---- ---
-__
e-""a-
---Z
Flo. 2·10. Standing.wave pattern or two oppositely traveling waves in dissipative matter.
55
INTRODUCTION TO WAVES
A ratio of a component of E to a component of H is called the wave impedance in the direction defined by the cross-product rule applied to the two components. Thus, continuity of tangential E and H requires that wave impedance. normal to a material bO'Und-
Region (I)
Region (2)
Incident
Transmitted
•
ary must be continuoua.
Reflected
..
The simplest reflection problem ill that of a uniform plane wave normally incident upon a plane boundary FIG. 2-11. Reflection at a plane dibetween two media. Thill is illustrated electric inte.rfa.ce, norma.l incidenoe. by Fig. 2-11. In region I the field will be the sum of an incident wave plus a reflected wave. The ratio of the reflected electric intensity to the incident electric intensity at the interface ill defined to be the reflection coefficient r. Hence, for region 1 E a (1) c:z Eo{Cik,. + reiA:,.) H (I) = E, _ (e-J-kl.l - reik,l) II
~1
In region 2 there will be a transmitted wave. The ratio of the transmitted electric intensity to the incident electric intensity at the interface is defined to be the transmission coefficient T. Hence, for region 2
E.(t.) = EoTe- jlt•• H
(2)
•
:lO:iI
Eo Te-fl" ~.
For continuity of wave impedance at the interface, we have Z.
I_-0 =
I
E.(I) H (1)
I
.-0 = ~1 I
1/
+ r _ r
= ~.
where '/1 and ~. are the intrinsic wave impedances of media I and 2. Solving for the reflection coefficient, we have
r = From the continuity of E. at z given by
=
T ~ I
~.
-
(2-45)
'/1
'/, + ~1
0, we have the transmission coefficient
+r
~
~.
2'/.
+ '/1
(2-46)
If region 1 is a perfect dielectric, the standing-wave ratio is
SWR
E~ _ 1 =
+ Irl
E~. - 1 - lrl
(2-47)
56
TDm-HARMONIC ELECTROHAGN·ETIC FlELDS
y'
,, ,
z Fro. 2-12. A plane wave propagating at an angle
~
z
with respect to the
%-0%
plane.
because t.he incident and reflected waves add in phase at some points and add 1800 out of phase at other points. The density of power transmitted across the interface is
lio- - Re E X H'·
u.1._0 - s..,(l -
Irl')
(2-48)
where Si.. - E.I/'lJl is the incident power density. The difference between the incident and transmit.ted power must be that reflected, or (2-49)
We have used an x-polarized wave for the analysis, but the results are valid for arbitrary polarization, since the :z: axis may be in any direction tangential to the boundary. Those of us familiar with transmission-line theory should note the complete analogy between tho above plane-wave problem and the transmission-line problem. Another reflection problem of considerable interest is tha.t of 8. plane wa.ve incident at an angle upon a plane dielectric boundary. Before considering this problem, let us express the uniform plane wave in coordinates rotated with respect to the direction of propagation. Let Fig. 2-12 represent a. pla.ne wave propaga.ting at an angle E with respect to the xz plane. An equiphase plane z' in terms of the unprimed coordinates is
z' "'" zcosE+ysin; and the unit vector in the v' direction in terms of the unprimed coordinate unit vectors is Uy' -. u.. cos ~ - u.sin ~
57
U1TRODUCTION TO WAVES
The expression for a uniform plane wave with E parallel to the z "'" 0 plane jg the 6rst of Eqs. (2-17) with all coordinates primed. Substituting from the above two equations, we have E• .... Eoe-itbol..f+'_ fl H = (u.,cos
~
-
u.sin~) ~Orfl(r"f+-_fl
(2-50)
The wave impedance in the z direction for this wave is
Z. "" E. _ _._ HI/
cos
(2-51)
t
In a similar manncr, from the second of Eqs. (2-17), the exprcssion for a uniform plane wave with H parallel to the z = 0 plane is found to be
E - (u"cos t - u.sin E)Eoe-1t (l1oJ.- H .-o H. = _ Eo e-it(l1ol.t1(+._o
(2-.52)
•
The wave impedance in the z direction for this wave is E. Z.=--:a::17C08t
(2-53)
H.
Thus, the z-direet.ed wave impedance for E parallel to the z - 0 plane is always greater than the intrinsic impedance, and for H pnrallel to the z = 0 plane it is always less than the intrinsic impedance of the medium.. Now suppose that a uniform plane wave is incident at an angle t ... 8; upon a dielectric interface at z - 0, as shown in Fig. 2-13. Part of the wave win be reflccted at an angle E...... - 8r , and part transmitted at an angle E - 8,. Each of these partial fields will be of the form of Eqs. (2-50) if E is parallel to the interface or of the form of Eqs. (2-52) if H is parallel to the interface. (Arbitrary polarization is a superposition of these two RegIon (1)
FlO. 2-13. Reflection at a plane dielectdc interface, arbitrary angle of incidence.
RegIon (2)
z
58
T1ME-HARMONIC ELECTROMAGNETIC FIELDS
cases.) For continuity of tangential E and H over the entire interface, the y variation of all three partial fields must be the same. This is so if k l sin 8, = k 1 sin Or = k! sin 6,
From the first equality, we have (2-54)
that is, the angle of reflection is equal to the angle of incidence. second equality, we have
.j§'"'
l sin - -8,= -k= -V,= sin 8; k, VI
-
~2~!
From the
(2-55)
where v is the phase velocity. Equation (2-55) is known as Snell's law of refraction. The direction of propagation of the transmitted wave is thus different from that of the incident wave unless IiWt = fWt. In practically all low-loss dielectrics, 1J.l = Ilt = /le. If medium 2 is free space and medium 1 is a nonmagnetic dielectric, the right-hand side of Eq. (2-55) becomes VEl/EO = V;;, which is called the index of refraction of the dielectric. The magnitudes of the reflected and transmitted fields depend upon the polariza.tion. For E parallel to the interface, we have in region 1 E~(I)
= A(g-il,._.,
H~(I)
=
+ rei
l ,._ ••)
~ cos 8;(e-ll,.-.; -
",
wbereA includes the y dependence. in region 1 at the interface is
reil,.-.,)
Thus, thez-directed wave impedance
Z~(t) = EP) = --!!- 1 + r H~(I) cos 8,1 - r
This must be equal to the z-directed wave impedance in region 2 at the interface, which is Eq. (2-51) with t .... 8,. Thus, r =
711 112
sec 8, sec U, +
111 111
sec U; sec 8;
(2-56)
Note that this is of the same form as the corresponding equation for normal incidence, Eq. (2-45). The intrinsic impedances are merely replaced by the z-direct.ed wave impedances o{ single traveling waves. It should be apparent {rom the form of the equations that, for R parallel to the interface, the reflection coefficient is given by
r ...
112 cos Ul fit cos 0,
-
'11
cos 8;
+ 'Ill cos 0;
(2-57)
59
INTRODUCTION TO WAVES
In both cases we have standing waves in the z direction, the standing-waye ratio being given by Eq. (2-47). Two cases of special interest are (1) that of total transmission and (2) that of total reflection. The flf'S(. case occurs when r = O. For E parallel to the interface, we see from Eq: (2-56) that r = 0 when
Substituting for 9, froro Eq. (2-55) and for the '1'5 from Eq. (2-11) we obtain (2-58)
as the angle at which no reflection occurs. real solution for
(Ji.
This does not always have a
In fact, sin
(Jj _
...,-,..
co
For nonmagnetic dielectrics (PI = PS = po) there is no angle of total transmission when E is parallel to the boundary_For the case of H parallel to the boundary, we find from Eq. (2-57) that r """ 0 when sin 8; =
f.JEI -
Es/Et -
Ill/P.I
EdEs
(2-59)
Again this does not always have a real solution for arbitrary po and But in the nonmagnetic case
E.
(2-60)
There is usually an angle of total transmission when H is parallel to the boundary. The angle specified by Eq. (2-60) is called the polarizing angle or Brewster angle. If an arbitrarily polarized wave is incident upon a nonmagnetic boundary at this angle, the reflected wa.ve will be polarized with E parallel to the boundary. The ease of total reflection occurs when Irl = 1. We are considering lossless media; so the 'l'S are real. It is apparent from Eqs. (2-56) and (2-57) that Irl ~ 1 for real values of 8, and 81• However, whon flJ.l1 > EtJ.lS, Eq. (2-55) says that sin 8, can be greater than unity. What does this mean? Our initial assumption was that the transmitted wave was a uniform plane wave. But Eqs. (2-50) specify a solut.ion to Maxwell's equations relYlrdless of the value of sin E. It can be real or complex. All that is changed is our interpretation of the field. To illustrate, sup-
60 pose sin
Tl1ofE-HARliONIC ELECTROMAGNETIC FIELDS
e> 1 in Eqs.
(2-50) and let
ksin!~~
keos!
=
kv'l
sin'!:= ±ja
(2-61)
If we choose the minus sign for a, Eqs. (2-50) become (2-62)
which is a field exponentially attenuated in the z direction. Note the 90° phase difference between E~ and Hili so the wave impedance in the z direction is imaginary. and there is no power flow in the z direction. A similar interpretation applies to Eqs. (2-52) when sin t > 1. Returning now to our reflection problem, from Eq. (2-55) it is evident that sin 0, is greater than unity when sin 8. > Yf.I}J!!fl}Jl' Thus, the point of transition from real values of 81 (wave impedance real in region 2) to imaginary values of 8, (wa.ve impedance imaginary in region 2) is
.
smB. =
..§;"" EIJ.l.l
(2-63)
The angle specified by Eq. (2-63) is called the critical angle. A wave incident upon the boundary at an angle equal to or greater than the critical angle will be totally reflected. Note that there is a real critical angle only if EJJ.ll > EU'! or, in the nonmagnetic case, if EI > ft. Thus, total reflection occurs only if the wave passes from a "dense" material into a C1less dense" material. The reflection coefficient, Eq. (2~56) or Eq. (2-57), becomes of the form
R-jX
r = R +jX when total reflection occurs. It is evident in this case that IrJ is unity. Remember that the field in region 2 is not zero when total reflection occurs. It is an exponentially decaying field, called a react1've field or an evanescent field. Optical prisms make use of the phenomenon of total reflection. All the theory of this section can be applied to dissipative media if the ,,'s and 8's are allowed to be complex. Of particular interest is the case of So plane wave incident upon a good conductor at an angle 8;. When region 1 is a nonmagnetic dielectric and region 2 is a nonmagnetic conductor, Eq. (2-55) becomes
61
lNTROD'OCI'ION TO WAVES
l+dI
...4-
i}'+V+d~ -~--
l}+v ===j:======ll===",n ~ r-- dz ---i'1
r-- dz
-I
Co)
(b)
FtG. 2-14. A lnmrro..ion line accordi.ng to circuit concepts. equivalent circuit.
(a) Phyaicalline; (6)
This is an extremely small quantity for good conductors. For most practical purposes, the wave can be considered to propagate normally into the conductor regardless of the angle of incidence. 2-6. Transmission-line Concepts. Let us review the circuit concept of a transmission line and then show its relationship to the field concept. Let Fig. 2-14a represent a. two-eonductor transmission line. For each incremental length of line dz there is a series voltage drop dV and a shunt current dI. The circuit theory postulate is that the voltage drop is proportional to the line current I. Thus,
dV--IZdz where Z is a series impeda-n.ce per unit length. It is also postulated t.hat the shunt. current is proportional to the line voltage V. Thus, dl
~
-VYd,
where Y is a shunt admiUance per unit length. thc a-c transm.i8sion-line equation8
dV -d, - -IZ
dl
dz~
Dividing by dz, we have
-VY
(2-64)
Implicit in this development are the assumptions that (1) no mutual impedance exists between incremental sections of line and (2) the shunt current dI flows in planes transverse to z. The transmission line is said to be uniform if Z and Y are independent of z. Taking the derivative of the first of Eqs. (2-64) and substituting from the second, we obtain
d'V dz' - ZYV - 0
d'l - ZYl- 0
dz'
which are one-dimcnsion:ll Helmholtz equations.
(:HIS)
The general solution
62
TIME-HARMONIC ELECTROMAGNETrc FIELDS TABLE 2-2. COMPARISON OF TRANSMISSION-LINE WAVES TO UNIFOIW PLANE WAVES
Transmission line d'V
-d,' - .,,'V -
d'E"
0
+ k'E"
_ 0
d,'
d'/ - - -rtf - 0 d,'
d'H __ I +k'H
a.'
_ II
0
-VZY V.+,,-r· + Vo-e'"
E. _
ik - y'ij 8.+,,-/10 + E.-"iio
+ 1.-e"·
H. _
H.+e-/~'
T
V _
..
Uniform plane wave
1 - I t +,,"""
+ H.-"i
l•
" _ Eo+ __ 8.- _ ~ H.+ 8 0fJ
V,+ Vj)~ Z. - 11+ - - [0- y
S, - E.U:
p - VP
is a sum of a +z traveling wave and a -z traveling wave, with propagation constant (2-66)
Choosing the +z traveling wave
V+ =- Voe--r' we have from Eqs. (2...64) that
/+ = I atr'"
V+ Z 'Y 1+ = -:y = y
Substituting for
'Y
from Eq. (2.-66), we have
z, -
V+ ]+ -
rz
V1'
(2-67)
which is called the characteri8tic impedance of the transmission line. The imaginary parts of Z and Yare usually positive, and it is common practice to write Z=R+iwL Y -G+jwC (2-68) The equivalent circuit of the transmission line is then as shown in Fig. 2-14b. The reader haa probably already noted the complete analogy between the linearly polarized plane wave and the transmission line. This analogy is summarized by Table 2-2. In the circuit theory development, we assumed no mutual coupling
63
INTRODUCTION TO WAVES
between adjacent elements of the transmission line. From the field theory point of view, this is equivalent to assuming that rio E. or H. exists. Such a wave is called transverse electromagnetic, abbreviated TEM. This is not the only wave possible on a transmission line, for Maxwell's equations show that infinitely many wave types can exist. Each possible wave is called a. mode, and a TEM wave is called a trammission-line mode. AU other waves, which must have an E. or an H. or both, are called higher-oroer mode8. The higher-order modes are usually important only in the vicinity of the feed point, or in the vicinity of a discontinuity on the line. In this section we shall restrict consideration to transmission-line, or TEM, modes. For the TEM mode to exist exactly, the conductors must be perfect, or else an E. is required to support the z-directed current. Let us therefore specialize the problem to that of perfect conductors immersed in a homogeneous medium. We assume E. = H. = 0 and z dependence of the form e-r.. Expansion of tbe field equations, Eqs. (2-1), then gives 'YH, -= flE~ 'YH. = -fJEI/
'YE" = -JH. 'YE. = JH1/ aE" _ aE~ = 0
ax
aH" _ all.
ay
ax
ay
=
0
It follows from these equations that 'Y =
jk
(2-69)
The propagation constant of any TEA! wave is the intrimic propagation constant of the medium. The proportionality of components of E to those of Ii: expressed by the above equations can be written concisely as E=.HXu.
1 H--u.XE
(2-70)
• Thus, the z-directed wave impedance of any TEM wave is the intrinsic wave impedance of the medium. Finally, manipulation of the original six equations shows that each component of E and H satisfies the twodimensional Laplace equation. We can summarize this by defining a transverse Laplacian operator V,I =
and writing
at a' ax' + ay1
(2-71)
V,'E = 0
The boundary conditions for the problem are
0)
E, = H. = 0
at the conductors
(2-72)
Thus, the boundary-value problem for E is the same as the electrostatic
64
TrnE-HARMONIC ELECTROMAGNETIC FIELDS
problem having the same conducting boundaries. The boundary-value problem for H is the same as the magnetostatic problem having "a.nticonducting" (no H ..) boundaries. It is for thiS reMon that II static" capaci-
tances and inductances caD beAused for transmission lines even though the field is time-harmonic. To show the relationship of the statio L's and C's to the Za of the transmission line, consider a cross section Flo. 2.-15. Cross sedion of a. transmission Iino.
of the line as represented by Fig. 2-15. In the transmission-line problem, the line voltage and current are related to the fields by V ~ /.c, E·dl
1- /.0, H·d!
where C1 and Ct are as shown on Fig. 2-15. the second of Eqs. (2-70) we have
I -
~ ( '1
le,
u. X E· d!
(2-73)
From the second of these and
_! ( 'I
le.
E. dl
But in the corresponding electrostatic problem the capacitance is
c _ .'L _.!.
( E. dl V}c.
V
Thus, the characteristic impeda.nce of the transmission line is related to the electrostatic capacitance per unit length by Zo
V
=-
T
•
= 7/
C
(2-74)
Similarly, from the first of Eqs. (2-73) and (2-70) we ha.ve
V~'/.c. Hxu"dl~,/. c, H.dl In the corresponding magnetostatic problem we have
L=!._!!. ( H.dl I
1
}e,
Therefore, the characteristic impedance of the line magnetostatic inductance per unit length by Z, -
V L T - , ;;
IS
related to the
(2-75)
Note also that Land Care rela.ted to each other through Eqs. (2-74) and (2-75). The electrostatic and magnetostatic problems h:-ve E and H everywhere orthogonal to each other and are called conjugate problems.
65
INTRODUCTION TO WAVES TAllLE
2-3.
CRARACf£R16TJC bIP£DANCE8 OJ'
Geometry
Line
~'o
wire
Coaxw
Confocal elliptic
Parallel plate
CoUKON TRAl\I6Jo11SSl0N LINES
Characteristic impedanco
~
0
Bolla
t--D-+l
@ ~ I--w-l.L b
Z. __• 101_ 2D
•
Z, --Iog• b 2. •
"
-
I+-D~
->i';;1+-
b + Vb' - (:,
Z. --log 2.. (I+~
b Z. _ ,,w
T
CoUinear plate
1D
• 4D Z, _-log•
Z ... -Iog• 4h
Shielded pair
l@ n ..y
z, .... ;: log d
Wire in trough
T w _h"j/d
~
.!.
2.
»b
D» 1D
w
hi°.!.Td
h
Wire above ground plane
D »d
d
d
h»d
" C' +.',t) D' D'
• (4W .h) Z, .... -log """dtanh2.. .. 1D
D »d • »d
h»d w»d
Once the electrostatic C or the magnctostatic L is known, the Zo of the corresponding transmission line is given by Eq. (2-74) or Eq. (2-75). Table 2-3 lists the characteristic impedances of some common tra.nsmission lines. When the dielectric is lossy but the conductors still assumed perfect, all of our equa.tions still apply. Zo (proportional to ,,) and 'Y ( - jk)
66
TlME-B.ARMONIC ELEC!'ROMAGNETIC FlELDS
become complex. The most important effect of this is that the wave is attenuated in the direction of travel. The attenuation constant in this case is the intrinsic attenuation constant of the dielectric (Table 2-1, column 2, row 4). When the conductors arc imperfect, the field is no longer exactly TEM, and exact solutions are usually impractical. However, the waves will still be characterized by a propagation constant "y = a + ifJ. Hence a +z-traveling wave will be of the form V 1- -
Z.
and the power flow is given by
p,
IV.I' e- lt..
= V I· ,.. - -
Z·
•
,..
Poe-f...
or, in terms of time-average powers,
IJ>, - Re (P,) - Re
(P.).-~·
The fate of decrease in ~I versus z equals the time-average power dissipated per unit length (1)01, or
<J>" ... - d<J>, = 2a(f>,
d,
Thus, the attenuation constant is given by
.--
IJ>, 21J>,
(2-76)
While this equation is exact if d'>" and ~I are determined exactly, its greatest use lies in approximating a by approximating
67
INTRODUCTION TO WAVES
of rectangular cross section. Fields existing within this tube must be characterized by zero tangential components of E at the conducting walls_ Consider two uniform plane waves traveling at the angles ~ and - ~ with respect to the xz plane (see Fig. 2-12). If the waves are x-polarized, we use Eq. (2-50) and write E" = A(cikllth.t _ e,kllalnE)e-ih_t = -2jA sin (ky sin~) e-jh .... E Let Eo denote (- 2j A) and define kc=ksin~
1'=jkcosE
In view of the trigonometric identity sin i l' a.nd k. are related by
~
+ cos
2
E = 1, the parameters
" - k.' - k' The above field can now be written as
(2-77)
E. = E. sin (k,y) r>'
(2-78)
let us see if this field can exist within the rectangular waveguide. There is only an E,,; so no component of E is tangential to the conductors x = 0 and x a. Also, E" """ 0 at y = 0; so there is no tangential component of E at the wall y = O. There remains the condition that E" = 0 at <:::
y = b, which is satisfied if n = 1,2,3, . . .
(2-79)
These permissible values of k. are called eigenvalues. or charaeteri8tic values of the problem. Each choice of n in Eq. (2-79) determines a possible field, or mode. The modes in a waveguide nrc usually classified according to the existence of z components of the field. A mode baving no E. is said to be a tramverse electric (TE) mode. One having no fl. is said to be a transverse magnetic (TM) mode. All the modes in the rectangular waveguide fall into one of these two classes. The modes represented by Eqs. (2-78) and (2-79) have DO E. and arc therefore TE modes. The particular modes that we are considering are TEon modes, the subscript 0 denoting no variation with x, and the subscript n denoting the choice by Eq. (2-79). The complete system of modes will be considered in Sec. 4-3. For k real (loss-free dielectric), the propagation constant 'Y can be expressed as k
n.
>T
(2-80)
k
< n1r b
68
TUllE-HARMONIC ELECTROMAGNETIC FIELDS
where a and fJ are real. This follows from Eqs. (2-77) and (2-79). When = ifJ, we have wave propagation in the 2: direction, and the mode is called a propagating mode. When 'Y = a, the field decays exponentially with z, and there is no wave propagation. In this case, the mode is called a nonpropagating mod€, or an evanescent mode. The transition from one type of behavior to the other occurs at a = 0 or k "'" n1r/b. Letting k = 2rf v'~p, we can solve for the transition frequency, obtaining 'Y
I.
n
= 2b
.y;;;
(2-81)
This is called the cutoff frequency of the TEo.. mode. intrinsic wavelength
The corresponding
x. _ 2b
(2-82)
n
is called the cuwff wavelength of the TEon mode. At frequencies greater than Ie (wavelengths less than >".), the mode propagates. At frequencies less than f. (wavelengths greater than A~), the mode is nonpropagating. A knowledge of /~ or A~ is equivalent to a knowledge of k~i so they also arc eigenvalues. In particula.r, from Eqs. (2-79), (2-81), and (2-82), it is eviden t that (2-83)
Using the last equality and k as
CI
27:/ W in Eq. (2-80), we can express 'Y
I> I. (2-84)
1
X. =
VI
(Uf)'
(2-&5)
showing that the guide wavelength is always greater than the intrinsic wavelength of the dielectric. The guide phase velocity v, is defined as the
INTRODUCTION TO WAVES
69
velocity at which a point of constant phase of & travels. Thus, in a manner analogous to that used to derive Eq. (2-14), we find
...
'. - Ii - Vi
'.
(2-86)
(Un'
where v.. is the intrinsic phase velocity or the dielectric. The guide phase. velocity is therefore greater than the intrinsic phase velocity. Another important property of waveguide modes is the existence of a characteristic wave impedance. To show this, let us find H (rom the E of Eq. (2-78) according to V X E "'" -jwJJI. The result is E. - E. sin (k,y) r<' H" =
.;L Eo sin (keY) tr".
J""
H. =
~ Eo cos (keY)
J"'.
el"
where E. has been repeated for convenience. the z direction is z. _ E. _ j",.
H.
(2-87)
The wa.ve impedance in (2-88)
~
This is ca.lled the characteristic impedance of the mode and plays the same role in reflection problems as does the Zo of transmission lines. If we substitute into the above equation for "y from Eq. (2-84). we find
I> I, (2-89)
v(un'
1
1
Thus, the characteristic impedance of a TEo.. propagating mode is always greater than the intrinsic impedance of the dielectric, approaching '1 as f -. 00. The characteristic impedance of a nonpropagating mode is reactive and approaches zero as f - O. All our discussion SO far has dealt with waves traveling in the +z direction. For each +z traveling wave, a -z traveling wave is possible, obtained by replacing "y by -'Y in Eqs. (2--87). The simultaneous existence of +z and -z traveling waves in the same mode gives rise to standing waves. The concepts of reflection coefficients, standing-wave ratios, etc., used in the case of uniform pla.ne-wave reflection, also apply to waveguide problems. The mode with the lowest cutoff frequency in a particular guide is called the dominant mode. The dominant mode in a rectangular waveguide, assuming b > a, is the TEOl mode. (This we have not shown, for
70
TIME-HARMONIC ELEcrnoMAGNETlC FIELDS
y )lltplt I I 'I
,, , l ,' I
, ,
,
I
,
', ,.., I!.
I,
x e------;... , 9 { - - - - -. .
lines into paper
• ••
Lines out of paper
FIG. 2-17. Mode pattern for the TE ol wAveguido mode.
we have not considered all modes.) From Eq. (2-82) with n = 1, we see that the cutoff wavelength of the TEo! mode is Xc :>:: 2b. Thus, wave propagation can take place in a rectangular waveguide only when its widest side is greater than a half-wavelength.! A sketch oC the instantaneous field lines at some instant is called a mode pattern. The mode pattern of the TEo! mode in the propagating state is shown in Fig. 2-17. This figure is obtained by determining E and JC from the E and H of Eqs. (2-87) and specializing the result to some instant of time. AB time progresses, the mode pattern moves in the z direction. It is admittedly confusing to learn that many modes exist on a given guiding system. It is not, however, so bad as it seems at first. If only one mode propagates in a. waveguide, this will be the only mode of appreciable magnitude except near sources or discontinuities. The rectangular waveguide is usually operated so that only the TE ol mode propagates. This is therefore the only wave of significant amplitude along the guide except near sources and discontinuities. Because of the importance of the TED I mode, let us consider it in a little more detail. Table 2-4 specializes our preceding equations to this mode and includes some additional parameters which we shall now consider. The power transmitted along the waveguide can be found by integrating the axial component of the Poynting vector over a guide cross section. This gives
PI'"
foG
fob
E~H: dx dy
=
IEol! 2i:
which, above cutoff, is real and is therefore the time-average power transmitted. Below cutoff, the power is imaginary, indicating no time-average I We are referring to the intrinaic wlLvelengt,h of the dieleet,rie filling the waveguide, which is usually free space.
71
INTRODUCI'ION TO WAVES
TABi..E 2-4. SUMMARY OF WAVEOUIDE PARAMETERS FOR THE DO),[INANT MODE (TE ol ) IN A RECTANGULAR WAVEOUIDE
E~ ... Eo sin TY e-"f'
Complex field
Cutoff frequency Cutoff wavelength
Propagation constant
b Eo . TY 1 H - -8m -e ' v Zo b Eo I. TJI H ---cos-e"'" • ;" I b
I. -
1
V III
2b
).., - 2b
{j~ "1'-
-
jkVI 2...
a - -
'.
Characteristic impedallce
Guide wavelength
Guide phase velocity
Power transmitted
Attenuation due to lossy dielectric
Attenuation due to imperfect conductor
V1
u.;n'
f >1.
(fll.)!
I
;WII ,IV 1 - (f,ff)' z.-, - ( i"/V (f
A. -
'.
-
VI
,
1>1. I
(f
'. VI
(f
p _ IEoltab
2Z.
a. -
2"VI
WI"
a. - a"
(f
VI m U
[2. + b ~-)'] J 1
power transmitted. (The preceding equation applies only at z = 0 below cutoff unless the factor cta. is added.) It is also interesting to note that the time-average electric and magnetic energies per unit length of guide are equal above cutoff (see Prob. 2-32). In contrast to the transmission·line mode, there is no unique volta.ge and current associated with a waveguide mode. However, the amplitude of a modal traveling wave (Eo in Table 2-4) enters into waveguide reflection problems in the same manner as V in transmission-line problem.s.
72
TIME-HARMONIC ELECTROMAGNETIC FIELDS
To emphasize this correspondence, it is common to define a mode voltage V and a. mode current] such that
Zo =
v
T
P = VI-
,,
(2-90)
From Table 2-4, it is evident that
V
=
EO~e-T.
(2-91)
satisfy this definition. Remember that we are dealing with only 0. +z traveling wave. In the -z traveling wave, I = - V /Zo. When waves in both directions are present, the ratio V/ I is a. function of z. Other definitions of mode voltage, mode current, and characteristic impedance can be found in the litera.ture. These alternative definitions will always be proportional to our definitions (see Prob. 2-34). Our treatment has 80 far been confined to the ideal loss-free guide. When losses are present in the dielectric but not in the conductor, all our equations still apply, except that most parameters become complex, There is no longer a real cutoff frequency, for 'Y never goes to zero. Also, the characteristic impedance is complex at all frequencies, The behavior of'Y = a + jfJ in the low-loss case is sketched in Fig. 2-18. The behavior of 'Y for the loss-free case is shown dashed, The most important effect of dissipation is the existence of an attenuation constant at all frequencies. In the low-loss case, we can continue to use the relationship
provided f is Dot too close to f~,
Letting k = k' - jk" and referring to
Flo. 2--18. Propagation const.a.nt (or & l066y waveguide (loss-free. case shown dashed).
•
o
f
73
lNTRonVC'I'lON TO WAVES
Table 2-1, we find (2-92)
This is t!le attenuation constant due to a lossy dielectric in the guide. Even more important is the attenuation due to imperfectly conducting guide walls. Our solution is no Jonger exact in this case, because the boundary conditions are cbarrged. The tangential component of E is now not quite zero at the conductor. However, for good conductors. the tangential component of E is very small, and the field is only slightly changed, or "perturbed," from the loss-free solution. The loss-free solution is used to approximate H at the conductor, and Eq. (2-42) is used to approximate the power dissipated in the conductor. Such a procedure is called a perturbational mdJwd (see Chap. 7). The power per unit length dissipated in the wall y = 0 is
lP,
L. - (!l f." IH.I' =
~ (!lIE.I' (:i)' J.' d%
(!lIE.I'a (~)'
and an equal amount is dissipated in the wall y "" b. length dissipated in the wall x = 0 is
The power per unit
lP,1_, = (!l jo(. (lH.I' + IH.I') dy
[sin' + (I.), cos' ry] "1/1 - (!lIE,I' [2;., + Gi)' ~] _ (!lIE.I' (. jo
(ry/b)
b
Zit
dy
and an equal amount is dissipated in the wall x "" a. The total power dissipated per unit length is the sum of that for the four walls, or
lP, - (!lIE.I' [i., + (:i)' (20 + b)] Equation (2-76) is valid for any traveling wave; so using the above and ~I "*' P of Table 2-4, we have a, -
~.,
6lZ.[b ab Zo' + (I,)' .1 (2a + b) ]
-a. vI (!l V,If)' [I + b (I.),] 7 2a
This is the attenuation constant due to conductor losses.
When both
74
TIME-HARMONIC ELECTROMAGNETIC FIELDS
dielectric losses and conductor losses need to be considered, the total attenuation constant is
x
z
a""'a<J+a~
(2-94)
for by Eq. (2-76) we merely add the two losses. 2-8. Resonator Concepts. In Sec. 2-2 we noted a similarity between standing waves and circuit theory resonance. In the loss-frceca.sc,clec~ troroagnetic fields can exist within Y a source-free region enclosed by a
a
b
perfect conductor. These fields can exist only at specific frequencies,
FIG. 2-19. The rectangular cavity.
called resona-nt frequencies. When losses are present, a source must exist to sustain oscillations. The input impedance seen by the source behaves, in the vicinity of a resonant frequency, like the impedance of an LC circuit. Resonators can therefore be used for the same purposes at high frequencies as LC resonators nrc used at lower frequencies. To illustrate resonator concepts, consider the "rectangular cavity" of Fig. 2-19. This consists of a conductor enclosing a. dielectric, both of which we will assume to be perfect at present. We desire to find solutions to the field equations having zero tangential components of E over the entire boundary. The TElll waveguide mode already satisfies this condition over four of the walls. We recall that standing waves have planes of zero field, which sugges~ trying the standing-wave TEol field. For E" to be zero at z = 0, we choose E" = E~+ =
For
E~
+ E~- =
Eosin
A sin 'Jr: (e-i'" - ei/l.)
(7) sin pz
to be zero at z = c, we choose pc =
'Jr, which, according to Table
2-4, is
Solving {or the resonant frequency I = In we have
f _
..!. ~b' + c'
r-2bc
tJl
(2-95)
When a is the smallest cavity dimension, this is the resonant frequency of
75
IN'TRODUC'I'ION TO W AYES
the dominant mode, called the TEolI mode. The additional subscript 1 indicates that we have cbosen the first zero of sin f3z. The higher zeros give higher-order modes, that is, modes with higher resonant frequencies. Setting f3 c: rIc in the above expression for E and determining H from the Maxwell equations, we have for the TEo II mode it
·ry·r? BIt = E ,SIDbSlD
C
II II =
jbE,
" Vbt
+ ct
. r1J ..-z Sln-cos-
(2-96)
c
b
jcEo ."y . rz C08-. smt t "V b + c C c Note that E and Jl are 90 out of phase; so & is maximum when 3C is minimum and vice versa. A sketch of the instantaneous field lines at some time when both & and 3C exist is given in Fig. 2-20. Also of interest is the energy stored within the cavity. From the conservation of complex power, Eq. (1-68), we know that 'W. "'" 'W.. Thus, the time-average electric and magnetic energies are ll. = -
'\\l.
~ '\\l. ~ ~
_/
fff lEI' d, ~ i IB.I'abc
(2-97)
cavit.)'
We also know from conservation of energy, Eq. (1-39), that the total energy within the resonator is independent of time. ]( we choose a time for which 3C is zero, '\'1. will be zero, and W. will be maximum and twice its average value. Therefore,
w - 2'W.
i IEol abc t
=
(2-98)
is the total energy stored within the cavity.
, ,• • •• •• •,,•
1 b
J L.-----l ·
• • • • • • • • • • •
e---~.
..-
- .....
~
/// ..... ---~-I I • --...--- • \
I I
,;' •
I I
\.
••
• ""\
'\
I I
JI.I·····I·11 \
•
I
,--~---'
I I
. . ...... / ) ,\..... _-..... -----'
I \
---~--_
I I
.;
II<---.~-, _ _ .I .!J(--~-
FlO. 2-20. Mode pattern for the TE'11 cavity mode.
76
TIME-HARMONIC ELECTROMAGNETIC FIELDS
When the resonator has losses, we define its quality factor as _ w X energy stored w'W Q - average power dissipated = lYd
(2-99)
by analogy to the Q of an LC circuit. If the losses are dielectric losses, we have (2-100)
so the Q of the resonator is that of the dielectric, Eq. (1-79). This is valid for any mode in a cavity of arbitrary shape. Usually more important in determining the Q is the loss due to imperfect conductors. This is determined to the same approximation as we used for waveguide attenuation. We assume H at walls to be tha.t of the loss-free mode and calculate &d by Eq. (2-42). To summarize,
19, ~ m
1f.
IHI' d. ~ 2",~I~~'c') [bc(b'
+ c') + 2a(b' + c')]
c.v.~y
••n.
Sub8tituting this, Eq. (2-98), and Eq. (2-95) into Eq. (2-99), we have 11'"'1
Q, - 2m bc(b'
a(bt + ct)H + c') + 2a(b' + c')
(2-101)
From the symmetry of Qc in band c, it is evident that b = c for mum Q. For a II square-base " cavity (b = c), we have 1.11'1
Q, - m(1
+ b/2a)
maxi~
(2-102)
The Q also increases as a increases, but if a > b we no longer have the dominant mode. As an example of the Q's obtainable, consider a cubic cavity constructed of copper. In this case we have
Q, - 1.07 X 10'/0
(2-103)
which, at microwave frequencies, gives Q's of several thousand. This idealized Q will, however, be lowered in practice by the introduction of a feed system, by imperfections in the construction, and by corrosion of the metal. When both conductor losses and dielectric losses are considered, the Q of the cavity becomes I
I
I
Q~Q,+Q. which is evident from Eq. (2-99).
(2-104)
77
INTRODUcrlON TO WAVES
2-9. Radiation. We shall now show that a source in unbounded space is characterized by a radiation of energy. Consider the field equations
v XE
~
-jw,JI
v XH
-jW<E+J
(2-105)
where] is the source. or impressed, current. These equations apply explicitly to a perfect dielectric, but the extension to lossy media is effected by replacing jw~ by ! and jlNt by g. In homogeneous media. the divergence of the first equation is
v·H = 0 Any divergcnceless vector is the curl of some other vector i so H - V XA
(2-106)
where A is called a magnetic vector 'Potential.l into the first of Eqs. (2-105). we have V X (E
+ jw,.A)
Substituting Eq. (2-106)
- 0
Any curl-free vector is the gradient of some scalar. E
+ jw,.A =
Hence,
- Voj>
(2-107)
where ¢I is an electric acalar potential. To obtain the equation for A. substitute Eqs. (2-106) and (2-107) into the second 01 Eqs. (2-105). This gives v X V X A - klA = ] - jc.>tV4> (2-108) which. by a vector identity. becomes v(v . A) - V'A - k'A =
J - j",voj>
Only V X A was specified by Eq. (2-J06). We are still free to choose If we let v·A = -jWt~ (2-109)
V • A.
the equation for A simplifies to V'A
+ k'A
-
-J
(2-110)
This is the Helmholtz equation, or complex wave equation. Solutions to Eq. (2-110) are ca.lled wave potential~. In terms of the magnetic wave potential, we have E = -jw,.A H~VXA
+ J- V(V . A) J'"
(2-11I)
I In general e.1ectromagnetie theory it is more oommon to let A be the vector paten\ial of B. In homogeneous media i·he two potentials are in the ratio"., a constant.
78
TIJrLE-IL\RMONIC ELEcrROMAGNETIC FIELDS
z
obtained from Eqs. (2-106), (2-107), and (2-109). The principal advantages of using A instead of E or H are (1) rectangular components of A have corresponding rectangular com~ ponents of J as their sources and (2) A need not be divergenccless. Let us first determine A for a. cur~ rent I extending over an incremental II, ;r<'-------;;y length l. forming a current elermnl or x electric dipole of moment Il. Take Fra. 2-21. A z-direct.ed eurrcnt. clement. this current element to be z-dircct.ed and situated at the coordinate origin, at. the coordinate origin. as shown in Fig. 2-21. The current is z-directed; so we take A to have only a z component, satisfying
,
VIA.
+ ktA~
= 0
everywhere except at the origin. The scalar quantity AI has a point source Il and should therefore be spherically symmetric. Thus, let A. = A.(r), and the above equation reduces t.o
.!.'!. r dr 2
(r,dA.) + k'A • _0 dr
This has the two independent solutions
the first of which represents an outward-traveling wave, and the Becond an inward-traveling wave. (In dissipative media, k - k' - ik", and the first solution vanishes as T -+ co, and the second solution becomes infinite.) We therefore choose the first solution, and take
where C is a constant. I As k-+ 0, Eq. (2-lJOY reduces to Poisson's equation, for which the solution is
A._E 4",. 1 To be preciee, C might be a function of k, but the solution must also reduce to the static field u r - O. Bence, C is not Go function of k.
INTRODUcrrON TO WAVES
79
Our constant C must therefore be
and hence
(2-112)
is the desired solution for the current element of Fig. 2-21. The outward-traveling wave represented by Eq. (2-112) is called a. spherical wave, since surfaces of constant phase are spheres. The electromagnetic field of the current element is obtained by substituting Eq. (2-112) into Eqs. (2-111). The result is
E = 2'It11'" r
e-ib
(!L + _._1_) cos 8 r2
JWfCr '
.,(iw"+"+ 1). -. - 8m8 r: JWfCr
Jl E',=_e~r 41r r H. = -Jl 4'11"
e-ib
(ik + -1).sm -
r
r2
(2-113)
'
8
Very close to the current element, the E reduces to that of a static charge dipole, the H reduces to that of a constant current element, and the field is said to be quasi-static. Far from the current clement, Eqs. (2-113) reduce to
E,
= '11
L
2XT 11
e-ib
sin 8
.Il
H. = L e-ib sin 8
l
T» X
(2-114)
2XT
which is called the radiation field. At intermediate values of r the field is called the induction field. The outward-directed complex power over a sphere of radius r is PI =
t:ffi E
= •
X H* . ds =
10'2" dq, 10" d8 r 2 sin 0 E,H:
2; I~ll' [1 - (!,).]
(2-115)
The time-average power radiated is the real part of PI, or
I
~I = .~ ~ll'
(2-116)
This is independent of r and can be most simply obtained from the racliation field, Eq. (2-114). The reactive power, which is negative, indicates that there is an excess of electric energy over magnetic energy in the near field.
80
TIME-HARMONIC ELECTROMAGNETIC FIELDS
z
FlO. 2-22. Radius vector Dotation.
x To obtain the field of an arbitrary distribution of electric currents, we need only superimpose the solutions for each element, for the equations are linear. A superposition of vector potentials is usually the most convenient one. For this purpose, we shall usc the radius vector notation illustrated by Fig. 2-22. The ufield coordinates" arc specified by r=u~+u,y+u.z
and the Hsource coordinates II by
r'
= U~'
+ llllY' + u.z'
In Eq. (2-112), r is the distance from the source to the field point. Jl not at the coordinate origin, r should be replaced by jr -
t'l
-= V(x - x'p
+ (y
- y'p
+ (z -
For
ZI)Z
Note the direction of the vector potential is that. of the current; so Eq. (2-112) can be generalized to a current element of arbitrary orientation by replacing Il by /1 and A. by A. Thus, the vector potential from current element of arbitrary location and orientation is A _ II e-i-lIr-r'1
- 4ilr - r'l To emphasize that A is evaluated at the field point (x,y,z) and II is situated at the source point (x',y',z') , we shall use the notation A(r) and fl(r'). The above equation then becomes Il(r')c.iilr-r'l
A(r) -
4-rlr _ <'I
(2-117)
Finally, for a current distribution J, the current element contained in a volume element dT is J dT, and a superposition over all such elements is (2-118)
81
INTRODUCTION TO WAVES
The prime on dT' emphasizes that the integration is over the source coordinates. Equation (2-118) is called the magnetic vector potential integral. It is intended to include the cases of surface currents and filamentary currents by implication. We therefore have a formal solution for any problem characterized by electric currents in an unbounded homogeneous medium. The medium may be dissipative if k is considered to be complex. 2-10. Antenna Concepts. A device whose primary purpose is to radiate or receive electromagnetic energy is called an antenna. To illustrate antenna concepts, we shall consider the linear antenna of Fig. 2-23. It consists of a straight wire carrying a current I(z). When it is energized at the center, it is called a dipole antenna. The magnetic vector potential, Eq. (2-118), for this particular problem is _ 1
A. - -b
Ir -r'l
where
=
ILI2 I(z')e-Jilr-r'l , -L12 1r r 'I d,
vrt+z't
(2-119)
2rz'c080
(2-120)
The radiation field (r large) is of primary interest, in which case Ir - e'l = r - z' cos 0 and
A.
~
....··ILI2
-41rr
l(z')&"I:o'-' dz'
-L/2
T» Z'
(2-121)
r»L
(2-122)
Note that the second term of Eq. (2-121) must be retained in the «phase term" e-i11r-f'I, but not in the llamplitude term" Ir - e'I-I. To obtain the field components, substitute Eq. (2-122) into Eqs. (2-111) and retain only the l/r terms. This gives E. = J1'W/l sin 0
H. = - E.
•
A.}
z T
large r-r'
(2-123)
This result is equivalent to superimposing Eqs. (2-114) for all elements of current. To evaluate the radiation field, we must know the current on the antenna. An exact determination of the current requires the solution to a boundary-value problem. Fortunately, the radiation field is relatively insensitive to minor changes in current distribution, and mueh use-
•
r
y
X
I(r')
-L/2 FIo. 2-23. The linear antenna.
82
TIME-HARMONIC ELECTROMAGNETIC FIELDS
Cui information can be obtained from an approximate current distribution. We have already seen that on transmission lines the current is a harmonic function of kz. This is also true for the principal mode on a single thin wire. The current on the dipole antenna must be zero at the ends of the wire, symmetrical in <:, and continuous at the source (z = 0). Thus, we choose (2-124)
The vector potential in the radiation zone can now be evaluated as
A. = Iae-ibjLI':J. sin[k(~-lzll)]ejh._tdz' 4.1rT
-L/2
I.e-it.
2 [cos
=~
2
(k~COS 0) - cos (k~)] ksin 2 8
From EQ. (2-123), the radiation field is
rib[COS(k~COS~) - COS(k~)]
E, =jT/I.. 2111'"
(2-125)
sm 8
with H. = E,/'1. Note" that the radiation field is linearly polarized, for there is only an E,. The density of power radiated is the r component of the Poynting vector
s. -
• _ ,11.1' [cos E,H. - (2.-,)'
H
cos
6) - cos H)]'
sin 6
(2-126)
The total power radiated is obtained by integrating S. over a. large sphere, or 2 ~I = ... Sr r 2 sin BdB dtIJ
10 10"
~ ,11.1' 2'lr
r H Jo [cos
cos
r
6) - cos H) sm B
d6
(2-127)
The radiation resistance R r of an antenna is defined as
R. ~
iJ>, llI'
(2-128)
where I is some arbitrary reference current. For the dipole antenna, the reference current is usua.lly picked as J... Hencc,
(2-129)
83
INTRODUCTION TO WAVES
280 240
/
200 Flo. 2-.24. Radiation reIlist.ance or the dipole antenna.
R 160 , 120 80 40
o
1/ ./ 310./2
'/2
210.
L
This integral can be evaluated in terms of tabulated functions (see Prob. 2-44). A graph of Rr versus L is given in Fig. 2-24. The radiation field pattern of an antenna is a plot of lEI at constant r in the radiation zone. For a. dipole antenna, the radiation field pattern is essentially the bracketed term of Eq. (2-125). This is shown in Fig. 2-25 for kL small (short dipole), kL = ,.. (half-wavelength dipole), and kL = 2,.. (full-wavelength dipole). The radiation power pattern, defined as a plot of ISrl at constant r, is an alternative method of showing radiation characteristics. When the radiation field is linearly polarized, as it is for the dipole antenna, the power patt.ern is the square of the field pattern. The gain g of an antenna in a given direction is defined as the ratio of the power required from an omnidirectional ant-enna to the power
FIG. 2-.25. Radiation field paUerm ror the dipole antenna.
84
TIllE-DARMONIC ELEcrROltAGNETIC FIELDS
required from the actual antenna, assuming equal power densities in the given direction. Thus, (2-130) For L :S X, the maximum gain of a dipole antenna occurs at 8 From Eqs. (2-126) and (2-128), we have
g( 2
~).,.
kL)'
,,11.[% ( 1-
= 11
C082"
~19,
EO:
.-/2.
( 1- C08 kL)' ~R,
2
(2-131)
In the limit kL - 0, we have g{7T/2) = 1.5j so the maximum gain of a short dipole is 1.5. For a hall-wave dipole, we can use Fig. 2.24 and calculate a maximum gain of 1.64. Similarly, for a full-wave dipole, the maximum gain is 2.41. The input impedance of an antenna is the impedance seen by the source, that is, the ratio of the complex terminal voltage to the complex terminal current. A knowledge of the reactive power, which cannot be obtained from radiation zone fields, is needed to evaluate the input reactance. The input resistance accounts for the radiated power (and dissipated power if losses are present). We define the input resistance of a lossfree antenna as
R;
~,
... fIJi
(2-132)
where ~I is the power radia.ted and I, is the input current. If losses are present, a "loss resistance" must be added to Eq. (2-132) to obtain the input resistance. For the dipole antenna,
. kD I i"'" , .. 8m
2
and the input resistance is
R. _
R,
- sin'lk(L/2)J
(2-133)
In the limit as kL is mnde small, we find
R. _
.(kL)'
•
24.-
(2-134)
The short dipole therefore has a very SDlali input resistance. For example, if L = X/10, the input resistance is about 2 ohms. For the haUwavelength dipole, we use Fig. 2-24 and Eq. (2-133) and find
R. - R. - 73.1 ohms
(2-135)
85
iNTRODUCTION TO WAVES
For the full-wavelength dipole, Eq. (2-133) shows R o '"'" co. This incorrect result is due to our initial choice of current. which has a null at the source. The input resistance of the full-wavelength dipole is actually large, but not infinite, and depends markedly on the wiro diameter (see Fig. 7-13). 2-11. On Waves in General. A complex function of coordinates representing an instantaneous function according to Eq. (1-40) is called a tooVt function. A wave function y" which may be either n scalar field or the component of a vector field, may be expressed as
'" = A (x,y,z)ei 4l (··r••)
where A and
~
are renl.
V2
(2-136)
The corresponding instantaneous function is
A (x,y,.) cos [wI
+ "(z,Y,')]
(2-137)
The magnitude A of the complex function is the rms amplitude of the instantaneous function. The phase ~ of the complex function is thc init.ial phase of the instantaneous function. Surfaces over which the phase is constant (instantaneous function vibrates in phase) are called equiphase 8'Urfacu. These are defined by (2-138)
Waves a.re called plane, cylindrical, or spherical according as their equiphase surfaces are planes, cylinders, or spheres. Waves are called 'Uniform when the amplitude A is constant over the equiphase surfaces. Perpendiculars to the equiphasc surfaces are called walle normals. These are, of course, in the direction of V~ and are the curves along which the phase changes most rapidly. The rate at which the phase decreases in some direction is called the phase constant in that direction. (The term phase constant is used even though it is not, in general, a constant.) For example, the phase constants in the cartesian coordinate directions arc
a.,
p • - - ax
a
p~-
•
ay
P.
=
(2-139)
These may be considered as components of a vector pha8e constant defined by (2-140)
The maximum phase constant is therefore along the wave normal and is of magnitude IV~I. The instantaneous phase of a wave is the argument of the cosine function of Eq. (2-137). A surface of constant phase is defined as
wt
+ "(x,y,z)
~
constant
(2-141)
86
TIME-HARMONIC ELECTROMAGNETIC FIELDS
that is, the instantaneous phase is constant. At any instant, the sur~ faces of constant phase coincide with the equiphasc surfaces. As time increases, l) must decrease to maintain the constancy of Eq. (2-141), and the surfaces of constant phase move in space. For any incrementds the change in ifl, is V4>· ds
i» d. ax
>::: -
04> + -0<%1 dy + - d, iJy iJz
To keep the instantaneous phase constant for an incremental increase in time, we must have (oj
dt
+ V<}l . ds =
0
That is, the total differential of Eq. (2-141) must vanish. The phase velocity of a wave in a given direction is defined as the velocity of surfaces
of constant phase in that direction. along cartesian coordinates arc
For example, the phase velocities
v" = ~
~
v" = .... fJll!jay = fJM
v.
w
~
= -
a()/az
(2-142)
=
i.
The phase velocity along a wave normal (ds in the direction of -v
which is the smaUeat phase velocity for the wave. Phase velocity is not a vector quantity. We can also express the wave function, Eq. (2-136), as (2-144)
where e is a. complex function whose imaginary part is the phase 1'. A vector propagation constant can be defmed in terms of the rate of change of e as
r - -va - • + j~
(2-1'15)
where (J is the phase constant of Eq. (2-140) and a: is the vector attenuation conatant. The components of IX arc the logaritbmic rates of change of the magnitude of y; in the various directions. In the electromagnetic field, ratios of components of E to components of Hare callcd wave impedances. The dircction of a wave impedance is defined according to the right-band
INTRODUCT10N TO WAVES
rotated into component B.
For example, E. = Z..+ = Z.
(2-146)
H, is
II.
wave impedance in the
87
+z
direction, while
E 1/ , = Z.,,- = Z_.
is a wave impedance in the -z direction. +z direct.ion involving E. and H. is
(2-l47)
The wave impedance in the
-E H.
__" """ Z,.+ = -Z".-
(2-148)
The Poynting vector cnn be expressed in terms of wave impedances. For example, the z component is
s. (2-149)
The conccpt of wave impedance is most useful when the wave impedances arc constant ovcr equiphase surfaces. Let us illustrate the various concepts by specializing them to the uniform plane wave. Consider thc x-polarized z-traveling wave in lossy mntter,
.
E. _
E~·"·c"Jr'·
H
E. c''''cit'.
,
.=
The amplitude of E. is Eae-t'.· and its phase is _k'z. Equiphase surfaces are defined by -k't; .= constant, or, since k' is constant, by z = constant. These arc planes; so the wave is a plane wave. The amplitude of E. is constant over each equiphase surface; so the wave is uniform. The wave normals all point in the z direction. The cartesian components of the phase constant are fl. = fJ,I = 0, fl. = k'; so the vector phase constant is ~ ... u.k'. The phase velocity in the direction of the wave normals is ~JI = wlk'. The cart.esian components of the attenuation constant are a.... a, """ 0, a. = k tl ; 80 the vector attenuation constant is u .= u.k". The vector propagation constant is y ~ •
+ 1~
- u.(k"
+ jk')
- u,jk
Tbe wa.ve impedance in the z direction is Z. = Z*"+ - E.I H w "" 71. Note that the various parameters sJ>C(;ialized to the uniform plane traveling wave are all intrinsic parameters. This is, by definition, the meaning of the word "intrinsic."
88
TIME-HARMONIC ELECTROr..fAGNETIC FIELDS
PROBLEMS
2-1. Show that E. - Er[/b satisfies Eq. (2-6) but not Eq. (Z-5). Show tha.t it does not Hlllisfy Eq. (2-3). This ill nol II. possible c.1ectromagnctic field. 2-2. Derive the "wave equatioDs" for inhomogeneous media VX(f-1VXE)+tiE-O V X (f)-IV X B) !H .,. 0
+
Are these valid for nonisotropie mcdia1 Do Eqs. (2-5) hold for inhomogeneous mediar 2-3. Show that for any lossless nonmagnetic dielectric
,. ,- -v.;
, ,,-v.; where
t. i$ the dielectric constant and k o, '10, ),,0, and.c are the intrinsic parameters of vacuum. 2·4. Show that the quantities of Eqa. (2-J 8) satisfy Eq. (1.-35). &peat {orEqa. (2-21), (2-27), and (2-20). 2-15. For the field of Eqs. (2.20), show tha.t the velocity of propagation of energy 8.9 defined by Eq. (2-19) is
,
_ ~
II. -
v,"
1
sin 2kt sin 2wt < _,_ cos 2h: cos ~ - V;
2-6. For the field of Eqs. (2-22), show that the phase velocity i.e 1
tI" - . . . ; ; ;
2-7. For the field of Eqs.
(A+C A-C.) A _ C cost kz + A + C kz Sln
(2~28),
1
show that the z-direeted wave impedances are
Would you e;\:pect Z~. + - Z.~ + to be true for al1ll....(; fields? 2-8. Given a uniform plane wave traveling in the +z direction, show that the wave is circularly polarized if
E. -
E.
- ±.,
being right-handed if the ratio is +j and left-handcd if thc ratio is -j. 2-9. Show that the uniform plane traveling wave of Eq. (2-25) can be expressed as the sum of a right-hand circularly polarized wave and a left-hand circularly polarized wave. 2-10. Show that the uniform plane traveling wave of Eq. (2-25) can be expressed as
E - (El
+ jE1)C /h
89
n."Tl\ODOCl'JON TO WA YES
where B l and E t are real vecton lying in the %1J plane. ReJate E 1 and E z to A and B. 2-11. Show that the tip of the arrow representing t for 80 arbit.rary complex E i 1m (E) and usc the rcsulta traccs out an elliptl6 in space. [Hint: let E - Re (E) of Prob. 2.10.\ 2-HJ. For the frequencies 10, 100, and 1000 megacycles, determine k - k' - ik" and., _ lJI. + iOC for (a) polyst.yrene, Fig. 1-10, (b) Plexit!u. Fig. I-II, (c::) Fernunic A, Fig. 1-12, f. - 10, and (d) copper, " - 5.8 X 107• 2-1S. Show that. when all1088ell are of the maplctie type (" - e" - 0),
+
2-101. Show th8t for nonm8gnetic dielectrics
Q»I
where Q is defined by Eq, (1-79). i-Hi. Show that (or nonmagnetic conducton
.' ~ ~(I +~) ." ~ ";" (I -~) lJI. ;(1 +~)
Q
«1
h~(I-~) where Q is defined by Eq. (1-79). 2-16. Show that for metals
"'"'" lJI.(1
+11
I '--(1-" •
"'-1
.1
where Gl is the surface resistance, II is the skin depth, and" is the conductivit.y. 2-11. Derive the following formulu
(Jl
Gl (silver) Gl (copper) Gl (gold) (aluminum) Gt (braaa)
-
2.52 X 10-7 VI 2.61 X 10- 7 v1 3.12 X 10- 7 VI
3.26 X 10-' 5.01 X 10-7
where I is the frequency in eycles per lICCond.
Vi VI
90
TlllE-HAlWONlC ELECI'aOJolAONETIC FIELDS
2-18. Find t.he power per square meter dissipated in II. copper aheet if the rma magnetic intensity at it.e lIurfae6 is 1 ampere per meter at (a) 60 cyclCtl, (b) 1 megacycle, (6) 1000 mcgacyeletl. 2,.19. Make a sketch similar to Fig. 2-6 for Il. circularly polarized standing wave in dissipative media. Give a verbal deacription of 8 and :te. 2-20. Given a uniform plane wave normally incident. upon a plMC air-to-dielcctric interface, abow t.hat the standing-wave ratio is
V; - index of refraction
SWR -
wbere ... is the dielectric constant of the dielectric (ILSSUmed nonmaloetic and 100000free). 2-21. Take the index of refraction of water to be 9, and calculate the percentAge of power reflected and tranamitted wben a plane wave is normally incident on a calm lake. 2-22. Calculate the two polarizing angles (internal and external) and the critical angle for a plane interface between air and (a) water, f< - 81, (b) high-density gla8ll, f< - g, and (e) polystyrene, f<- - 2.56. 2-28. Suppose a uniform plane wave in I!Io dielectric just gra.teS a plane dielectric. to-air interface. Calculate the attenuation constant in the air ta .. defined by Eq. (2-61)J for the three caee8 of Prob. 2-22. Calculate the distance from the boundary in which the field ia attenuated to 1/_ (36.8 per ceot) of ita "-lue at the boundary. What i.a the value of a at the critical an&le? 2-f:!. From Eq8. (U)6) and (2-68), 8how that when R «wI. and G «we
R
a-2~
fJ ,.
j,J
+GVLiC 2
....;rc
whcre ")' - or + jfJ. 2-215. Show that '1 &ad C of a transmission line are related by ~" "';'" O--:rC--Z
•
•
when the dielectric II homogcneous. Show that R of a tranamisaion line is appro:!imatelyequal to tbe doC n:cistance pet' unit length of hollow conduclora haviol thickneu , (akin depth) provided H is approximately conatant over each conductor and the radiu8 of curvature of t.he conductora i.a large compared. to ,. 2.26. U8ing results of Prob. 2-25, show that for the lwo-wire line of Table 2-3
R ... 2Gt
.d
d»a »d
D
and that for tbe couialline R . Gl4 + b 2. . .
4»1
and that for the parallel-piate line
.
R.~
U1»b
2-27. Verify Eql. (2-70). 2-28. CoDJIider a parallel-plate waveguide formed by conductor8 covering the planes y - 0 Ilod 11 - b. Show that the field
" - 1,2, 3, .••
IN'I'ItoDUCTION TO WAVES
91
defines a set ot TE" modes and the field
n.,
n
H. - H. COST e-r·
defines a aet
-{I,
1,2, •..
ot TM. modes, where
\l(n;), _k'
y _
in both CMClI. Show that the cutoff trequeneies of the TE,. and TM. modes are n
I. - 2hV;; --Show that Eqs. (2-83) to (2-86) apply to the parallel-plate waveguide modes. 2-29. Show that the power transmitted per unit width (z direction) of the parallel· plate waveguide of Prob. 2-28 ill
P-
bl:;I' >o}1 - (J)'
for the TE.. modes, and
for the TM. modea (n P! 0). 2·30. For the parallel-plate waveguide of Prob. 2.28, Bhow thaL Lbe attenuation due to conductor 1088ca is
for the TE,. modes. and
for the TM. modes (n >L 0). 2-31. Show that the TM. mode of the parallel-plate waveguide 1\8 defined in Prob. 2-28 ia actually a. TEM mode. Show that. for thia mode the atteouation due to con· ductor loaaea is
..
a. - b,
Compare this with a obtained by using the results of Probs. 2·26 and 2-24. 2-32. For thc TEfl reclangWar waveguide mode, ahow that the time-average electric and magnetic energ.ies per unit. length are
w. -
~-
-
i
lEt/lab
Can this equality of W. and '\9.. be predicted from Eq. (1-62)1 2-SS. Show that the time-average velocity of propagation of energy down a rectangular waveguide ill
g. 1 p.----oW V;; for tbe TE.. mode.
I -
G-')'
92
TIME~HARMONIC
ELECTROMAGNETIC FIELDS
2-34. For the TED, rectangular waveguide mode, define 8. voltage Vaa fE' dl acro88 the center of the guide and 8. current. 1 as the total z.directcd current in the guide wall :r _ O. Show that these are
Show that P ¢: V /-, Why? Define a characteristic impedance ZYI - V / I and show that it is proportional to Zo of Table 2-4.
2-36. Let a rectangular waveguide have 8. diseontinuity in dielectric at t _ 0, that is, fl, III Cor z < 0 and fl, III for z > 0, Show that the reflection and transmission coefficients for a TEll wave incident from z < 0 arc
r _
Z .. - ZOI
Zn
where
ZOI
+ ZII
and Zot are the characteristio impedances z < 0 and z
> 0,
respectively.
These results are valid for Bny waveguide mode.
2·36. Show that there ill no rollected wavo for the TE ol mode in Prob. 2-35 when ~
/
/"
-
I PI )
11(/01,1
Joll(s.IU
Jollfl)
where /<1 is the cutoff frequcncy .t < O. Note that we cannot have a reflectionless interface when both dieJectrics are nonmagnetic. This result is valid for any TE mode. 2-37. Take a parallel-plate waveguide with 'I, PI for z < 0 and 'z, p, for z > O. Show that there is no reflected wave for a TM mode incident from z < 0 when
For nonmagnetic dielectrics, thi.8 reduces to
L _ ~'I + 'I
1<1
U
Compare this to Eq. (2-60). Thue rCllu!ta are valid for any TM mode. 2-38. Design a square-base cavity with height one-half the width of the base to reaonate at 1000 meg&cyc!ea (II) when it is air-!illed and (b) whcn it is polystyrenefilled. Calculate the Q in each ClLllC. 2-39. For the rectangular cavity of Fig. 2-19, dcfine a voltage V lUI that between mid-points of the top and bottom walls and a current I &8 the total :H:Iirected cur· tent in tbe side walls. Show that
v-
EtII
Defino a mode conductance G 88 G -
iii~/IVII
+
and show that
+ +
G _ lR(bc(b Z el) 2II{b' 2'l'a' (b' e')
Define a mode resistance
+ e') I
R &8 R - iii./lll' and show tbat R _ r'
+ e + 211(b' + c,)1 32(b' + c')' Z)
93
INTRODUCTION TO WAVES
240. Derive Eqa. (2-123). 2-4.1. Consider the small loop of constant current I that the magnctic vector potential is
A.
-A~
I
__ Ia
._0
where
411"
f"
all
shown in Fig. 2-26.
Show
!c08q/d.p'
0
! _ exp (-ik .yr1+ a l
yr l + a!
2ra sin 8 cos ,,"') 2ra sin 8 cos .p'
Expand I in a Maclaurin scrica about a - 0 and show that
Ira! .• A • _--e-' .......04...
(i-rk + -') rl
.
SID 8
The quantity hal - ISis called the magnetic moment of the loop.
z
, FlO. 2-26. A circula.r loop of current.
I
y
2-4.2. Show that the field of the small current loop of Prob. 2-4.1 is
+-rI) l cos 8 IS e-'l< ( k - -! + ik -r! + -r I) sin 8 411" r
IS (ik H. - -e-'~' 'L.:. 2.. rl li,
e
E•
k ). , 'lIS .• (k! --e-' -r - iSID 411" r!
-
l
Show that the radiation resistance of the small loop referred to I is
R'-'l~e:r 2-48. Consider the currcnt element of Fig. 2~21 and the current loop of Fig. 2-26 to exist simultaneously. Show that the radiation field is everywhere circularly polarized if II - US 244. In terms of the tabulated functioDs Si(;t) ... ( '" ain ;t dx .}o ;t
Oi(x)
__
r-~dx
].
a
94
TIME-HARMONIC ELECTROMAGNETIC FIELDS
show that Eq. (2-129) can be expressed as Rr
-
i [ C + Jog kL -
CikL
+ sin kL(~8i2kL +H
- SikL)
coskL( C
+ log k~ + Ci2kL
- 2CikL)]
where C - 0.5772 . . . is Euler's constant. 2-46. If the linear a.ntenna. of Fig. 2-23 is an integral number of baU-wavelcngthll long, the current will assume the form
regardless of the position of the feed 8.8 long as it is not neM 8. current nulL Such &n antenna is said to be of resonant length. Show that the radiation field of the antenna is
.I
E, _ ,,, .. c 2,",
COB
itr
(7 0) COB
---'-'C.'"-< 8m 8
. (n.
)
I SID 2"c088 E, - ~ e-/tr ,,_.L
----'':c..
21fT
810
n odd
8
'" even
where n - 2L/>. is an integer. 2-46. For an antenna of resonant length (Prob. 2-45), show that the radiation resistance referred to I .. is H. - 4:
Ie
+ log 2n...
- Ci(2nr)J
where 11. - 2L/". C - 0.5772, and Ci is as defined in Prob. 2-44. resistance for a loss-free antenna with feed point at :: - a). is
R(-.
Show that the input
R~
SIn 2r(a
+ n/4}
Specialize t.his result to L - >"/2, a - 0 (the half-wave dipole) and show that R, - 73 ohms.
CHAPTER
3
SOME THEOREMS AND CONCEPTS
3-1. The Source Concept. The complex field equations for linear media arc (3-1) -v X E - £H+M vxH-gE+J where J and M are sources in the most general sense. We have purposely omitted superscripts on J and M because their interpretations vary from problem to problem. In one problem, they might represent actual sources, in which case we would call them impressed currents. In another problem, J might represent a conduction current that we wish to keep separate from the 1]E term. In stln another problem, M might represent a magnetic polarization current that we wish to keep separate from the ~H term, and so all. We can think of J and M as If ma.the· matical sources/' regardless of their physica.l interpretation. For our first illustration, Jet us show how to represent 1/ circuit sources II in terms of the "field sources" J and M. The current source of circuit theory is defined as one whose current is independent of the load. In terms of field concepts it can be pictured as a short filament of impressed electric current in series with a perfectly conducting wire. This is shown in Fig. 3-la. That it has the characteristics of the current source of circuit theory can be demonstrated as follows. We make the usual circuit assumption that the displacement current through the surrounding medium is negligible. It then follows from the conservation of charge that the current in the leads is equal to the impressed current, independent of the load. The field formula. for power, Eq. (1-66), reduces to
---I
FlO. 3-1. Circuit sources in terms nf impressed currents. (a) Current Bouree; Cb) voltage
I
+ V
I'
K'~
D
source. Ca)
9'
(b)
•
+ V
96
TllLE-BAJUlONIC ,£LEcraoMAGNETIC nELDS
the circuit Cormula (artius source. p. - -
We have only electric currents; hence
III E'J"dT -
-I"
I E·d1- VI'
The "internal impedance" of the source is infinite, since 8. removal of the impressed current leaves an open circuit. The voltage &aurce of circuit theory is defined as onc whose voltage is independent of the load. In terms of field concepts it can be pictured as a small loop of impressed magnetic current encircling a perfectly conducting wire. This is illustrated by Fig. 3-lb. To show that it has the characteristics of the voltage source of circuit theory, we neglect displacement current and apply the field equa.tion K "'" -:J'E· d1 to a path
coincident with the wire and closing across the terminals. The E is zero in the wire; 80 the line integral is merely the terminal voltage, that is, K' "" - V. The impressed current, and therefore the terminal voltage, is independent of load. The field formula for power, Eq. (1-66), reduces in this case to p. = -
III H-· MfdT -
-K't H-·dl = VI-
which is tbe usual circuit formula.. The internal impedance of the source is zero, since a removal of the impressed current leaves a short circuit. We can use the circuit sources in field problems when the source and input region are of "circuit dimensions." that is, of dimensions small compared to a wavelength. Given a pair of terminals close together, we can apply the current source of Fig. a-la, that is, 8 short filament of impressed electric currcnt. Given a conductor of 5Inall cross section, we can apply the voltage source of Fig. a-Ib, that is, a small loop of impressed magnetic current. As an example of the use of a circuit source, consider the linear antenna of Fig. 2-23. The geometry of the physical antenna is two sections of wire separated by a small gap at the input. To excite the antenna, we can place a current source (a short filament of electric current) across the gap, which causes a current in the antenna wire. An exact solution to the problem involves a determination of the resulting current in the wire. This is difficult to do. Instead, we approximate the current in the wire, drawing on qualitative and experimental knowledge. We then use this current, plus the current source across the gap, in the potential integral formula to give us an approximation to the field. We shall find much use for the concept of current sheets, considered in Sec. 1-14. As an example, suppose we have a J. over the cross section of a rectangular waveguide, as shown in Fig. 3-2. Furthermore, we postulate that this current should produce only the TEol waveguide mode,
97
SOME THEOREMS AND CONCEI"I"a
x
•
/1
J.
/L __
+ 1,-
/'
/: z
/
I
// /
I ,I ,L __
,-
FlO. 3-2. A sheet of current in a rectangular waveguide.
which propagates outward from the current sheet. Table 2-4, we have the wave
E£+
=
A sin
1r: ci4~
H"+=~sin7rY_i6~ Zo b <> .
H~+
=
Abstracting from
z>O
f
~ cos 1r: e-i6•
where the constant A specifies the mode amplitude. The -z traveling wave is of the same form with {J replaced by -{J and Zo by -Zoo Thus,
E£-
=
B sin
i: eJt'.
B."1/.. H,- = - Zosmb(7~'
z <0'
Bf, ry .•• H •- = .,-COS-(7nf b where B is the mode amplitude of the -z traveling wave. At z = 0, Eqs. (1-86) must be satisfied. Take the (1) side to be z > 0, so that n = U" and obtain -uAH,+ -
H,,-J_o = J.
(E~+
- E£-l_o
=0
Substitution for H, and E£ from above reduces these equations to -u~
Let
A+B."1/ Zo sm b =
J. -
J.
u,J. sin
A - B
7:
c:
0 (3-2)
98
TIME-HARMONIC ELECTROMAGNETIC FIELDS
The preceding equations then have the solution A = B = -JoZo/2. Thus, if the current of Eq. (3-2) exists over the guide cross section z = 0, then JoZ, . •y • - - - sin - e-"" z>o E. ~ 2 b (3-3) • 'TO'!J ~ JoZ o sln z
1--r
It would admittedly be difficult to obtain the current of EQ. (3~2) in practice, but this is not of concern at present. We shall learn how to treat more practical problema la.ter. Note that our approach in this problem was to assume the field and find the current. This we shall find to be a very powerful concept. 3-2. Duality. If the equations describing two different phenomena are of the same mathematical form, solutions to them will take the same mathematical form. The formal recognition of this is called the concept of duality. Two equations of the same mathematical form are called dual cqualion8. Quantities occupying the same position in dual equations are oalled dual quantitie8. Note that the field equationa, Eqs. (3-1), are duals of eaoh other. A systematic interchange of symbols ohanges the first equation into the second, and vice-versa. A duality of importance to us is that between a problem for which all sources are of the electric type and a problem for which all sources are of the magnetic type. The first two rows of Table 3-1 givo the field equations in each case. The last two formulas of column (1) were derived in Se<:. 2-9 for homogeneous space. The corresponding equations for the magnetic source case are evidently the last two formulas of column (2), obtained by systematically interohanging symbols. The particular interchange of symbols is summarized by Table 3-2. The reader should oheok for himseU that a replacement of the symbols of TAllLE
3--1.
DUAL .EQU"-TIO~S ron PaOBLE.'\lS IN Wmell
(I)
ONJ.y ELECTRIC
SOURCES ExiST MW (2) ONLY \I..IAONETIC SoURCES EXIST
(I) Electric sourcell
(2) Magnetic sources
vXH-fJE+J
-v X E - IH
-v XE ... fH
v X H - fJE E - -v X F
H-vXA I A -4r -
+M
If! v""-'" lid., r
r'
F == -I 4r
Ii! Ir
M.-,,"-'"
r'l
d...'
SOME THEOREMS AND CONCEPTS
99
TAIlLE 3-2. DUAL QUANTITIES FOR PR08LE.lfS IN WHICH (I) ONLY ELECTRIC SoURCES ExIST, AND (2) ONLY MAONETtC SoURCES EXIST
(I) Elearic IOUrcu
(2) AIagnetic
=,=
E
H
H
-E
J
M
A
F
•9
9
• k
k
,
1/,
column (1) of Table 3-2 by those of column (2) in the equations of column (1) of Table 3-1 results in the equations of column (2). The quantity F of these tables is called an electric vector potential, in analogy to A, II magnetic vector potential. The concept of duality is important for several reasons. It is an aid to remembering equations, since almost half of them are duals of other equations. It shows us how to take the solution to one type of problem, interchange symbols, and obtain the solution to another type of problem. We can also use a physical or intuitive picture that applies to one type of problem and carry it over to the dual problem. For example, the picture of elect.ric charge in motion giving rise to an electric current can also be used for magnetic case. That is, we can picture magnetic charge in motion as giving rise to magnetic current. Such a picture can serve as a guide to the mathematical development but cannot, of coursc, serve to argue for the existence of magnetic charges in nature. The concept of duality is based wholly on the mathematical symmetry of equations. It is often convenient to divide a. single problem into dual parts, thus cutting the mathematical labor in half. For example, suppose we have both electric and magnetic sources in a homogeneous medium of infinite extent. The field equations, Eqs. (3-1), are linear; so the total field can be considered as the sum of two parts, one produced by J and the other by M. To be explicit, let where and
E = E' + E" H = H' + H" V X H' = yE' + J -V X E' = zH' V X H" = yE" - V X E" = iH" + M
We have the solution for each of these partial problems in Table 3-1. The complete solution is therefore just thc superposition of the two partial solutions, or E ~ - V X F + y-'(v X V X A - J) II - V X A
+ r'(v
X V X F - M)
(3-4)
100
TLME-HAnMONIC ELECTROMAGNETIC FIELDS
Iff If _ ell dr' 1 rrf M(r').-,'1,-'·' F(r) ~ 4rJJ [ r "I d,' J(r').-~,,-,·,
1 A(r) = 411"
where
(3-5)
J
We thus have the formal solution (or any problem consisting of electric and magnetic currents in an unbounded homogeneous region. The above formulas arc meant to include by implication sheets and filaments of currents.
It is instructive to show that an infinitesimal dipole of magnetic current is indistinguishable from an infiniteBimalloop of electric current. We might suspect this from the circuit source representations of Fig. 3-1. However, ratber than rely on this argument, let us consider the fields explicitly. A z-directed magnetic current dipole of moment Kl at the coordinate origin is the dual problem to the electric current dipole (Fig. 2-21). An interchange of symbols, according to Table 3-2, in Eqs. (2-113) will give us the field of the magnetic current element. For example, the electric intensity is E.~
I).
-Kl e-ib (jk __ -+-2
4...
r
r
sm(J
The small loop of electric current is considered in Probs. 2-41 and 2-42 and is pictured in Fig. 2-26. Abstracting from Prob. 2-12, we have the electric intensity given by
E.
IS e- (k' L'k) sin
= -'-
41r
it •
_ r
r2
(J
A comparison of the above two equations shows that they are identical ir Kl = jwp.IS
(3-6)
This equality is illustrated by Fig. 3-3. Thus, effect of an clement of magnctic currcnt can be realized in practice by a loop of electric current. 3-3. Uniqueness. A solution is said to be unique whcn it is the only onc possiblc among a given class of solutions. It is important to have n
C:;::>lS (oj
(bj
FlO. 3-3. These two sourees radiate the same field if Ki - ;/,oJ~IS. (a) Magnetic currcnt element; (b) electric current loop.
FtO. 3-4. S encloses linear matter !lnd sources], M.
101
SOM.E THEOREMS AND CONCEPTS
precise theorems on uniqueness for several rea,sons. First of all, they tell us what information is needed to obtain the solution. Secondly, it is eomforting to know that a solution is the only solution. Finally, uniqueness theorems establish conditions for a one-to-one correspondence of a field to its sources. This allows us to calculate the sourees from a field, as well as the more usual reverse procedure. Suppose we have a set of sources J and M acting in a region of linear matter bounded by the surface S, as suggested by Fig. 3-4. Any field within S must satisfy the complex field equations, Eqs. (3·1). Consider two possible solutions, E", H" and Eb, Hb. (These can be thought of as the fields when the sources Qutside of S are different.) We form the difference field oE, oR according to oE=E"_Eb
Subtracting Eqs. (3-1) for the a field from those for the b field, we obtain
-v X IE - HH v X IH ~ tPE
I
within S
Thus, the difference field satisfies the source·free field equations within S. The conditions (or uniqueness are those for which oE = oH = 0 everywhere within S, for then E" = Eb and H" = Hb. We now apply Eq. (1-54) to the difference field and obtain
effi (m X mO) . ds + III (11!T/1' + UO!IEI'l dT Wheneve' effi (IE X mO) . ds - 0 ovcr S, the volume integrnl must also vanish. then
0 (3-7)
Thus, if Eq. (3-7) is true,
III [Re (1)1!Ii[' + Re (g)laEI'J dT ~ 0 III [1m (zll!HI' - 1m (Ul!IEI'J dT ~ 0
(3-8)
For dissipative media, Re (z) and Re (t/) are always positivc. If we assume somc dissipation everywhere, however slight, then Eqs. (3-8) nrc satisfied only if oE = oH = 0 everywhere within S. Some of the more important cases fOr which Eq. (3-7) is sa.tisfied, and thereCore uniqueness is obtained in lossy regions, are as follows. (1) The field is unique among a class E, H having n X E spccified on S, for then n X. oE- = 0 over S. (2) The field is unique among a class E, H having n X H specified on S, Cor then n X oR = 0 over S. (3) The field is unique among a class E, H having n X E specified over part of Sand n X H specified over the rest of S. These possibilities ean be summarized by the following uniqueness theorem. A field in a lossy region is uniquely
102
TIME-HARMONIC ELECTROMAGNETIC FIELDS
specified by the sources within the region plus the tangential components of E over the boundary, or the tangential components of H over the boundary, or the former over part of the boundary and the laUer over the rest of the boundary. Note that our uniqueness proof breaks down for dissipationlcss media. To obtain uniqueness in this case, we cO'Mider the field in a dissipationleS8 medium to be the limit of the corresponding field in a lossy medium as the dissipation goes w zero. We have explicitly considered only volume distributions of sources and closed surfaces in OUf development, but the results are much more general than this. Singular sources, such as current sheets and current filaments, can be thought of as limiting cases of volume distributions and therefore are included by implication. Surfaces of infinite extent can be thought of as closed at infinity and can be included by appropriate limiting procedures. Of particular importance is the case for which the bounding surface is a sphere of radius r -+ llQ, so that all space is included. If the sources are of finite extent, the vector potential solution of Eqs. (3-4) and (3-5) vanishes exponentially as e-J:"r, l' -+ llQ. We therefore have (3-9)
for this solution (in lossy media). According to our uniqueness proof this must be the only solution for a class E, H satisfying Eq. (3-9). Thus, given sources of finite extent in an unbounded lossy region, any solution 8ati8fying Eq. (3-9) fnWlt be identically equal W the potential integral 80lution. The loss-free case call be treated as the limit of the lossy case as dissipation vanishes. To illustrate the above concepts, consider the current element of Fig. 2-21. Our solution at large r is Eq. (2-114). Let this be the a solution of our uniqueness proof, or j l l " ,rsln . 8 H • • = -e-' 2;\T It can be shown that the inward-traveling wave
-jll",. H .=2;\r&r sm 8 '
E' H.' "'=-'1
is also a solution to the equations at large r. In Sec. 2-9, we threw out this second solution by reasoning that waves must travel outward from the source, not inward. Let us now consider these two solutions in the light of the uniqueness theorem. The difference field in this case is 6H. = H.o - Hl"" j
:~coskrSill 6
II . krSIO . 8 "'E, "'" E ,0 - E', "'" ";\r81n
103
SOME THEOREMS AND CONCEPTS
In dissipationless media (k real), we can pick a sphere r = constant such that either oH. or oE, vanishes. Thus, Eq. (3-7) can be satisfied without obtaining uniqueness of the solution. However, in lossy media, sin kr and cor kr have no zeros T > 0, and Eq. (3-7) cannot be satisfied for any r. In this case, only the a solution vanishes as r --+ 00. It is therefore the dp..sired solution in loss-free media. 3-4. Image Theory. Problems for which the field in a given region of space is determined from a knowledge of the field over the boundary of the region are called boundary~value problema. The rectangular waveguide of Sec. 2-7 is an example of a boundary-value problem. We shall now consider a class of boundary-value problems for which the boundary surface is a perfectly conducting plane. The procedure is known as image theory. The boundary conditions at a perfect electric conductor are vanishing tangential components of E. An element of source plus an "image" element of source, radiating in free space, produce zero tangential components of E over the plane bisecting the line joining the two elements. According to uniqueness concepts, the solution to this problem is also the solution for a current element adjacent to a plane conductor. The necessary orientation and excitation of image elements is summarized by Fig. 3-5. Matter also can be imaged. For example, if a. conducting sphere is adjacent to the plane conductor in the original problem, then two conducting spheres at image points are necessary in the image problem. In other words, we must maintain symmetry in the image problem. The procedure also applies to magnetic conductors in a dual sense. The application of image theory in a-c fields is much more restricted than in d-c fields. It is exact only when the plane conductor is perfect. As an example of image theory, consider a current element normal to the ground (conducting) plane, as shown in Fig. 3-6a. This must produce the same field above the ground plane as do the two elements of Fig. 3-6b. Let us determine the radiation field. The radius vector from each current element is then parallel to that from the origin and given by ro=r-dcosO) r. = r + d cos 0
r»d
_II
where subscripts Q and i refer to original and image elements, respectively. The radiation field of a single clement is given by Eq. (2-114); so the radiation field of the two elements of Fig. 3-6b is the superposition
(e-fl: + -e-il:"). sm 8 r,
r • j II H. = -2A To
"" j Il ~r
rile.
cos(kd cos 8) sin 0
~
FlO. 3-5. A sum-
(3-10)
mary of image theory.
104
TIME-DARMONIC ELECTROMAGNETIC FIELDS
z
Z
r
e
'"
e
n
r;
r
II
n
(a)
FlO. 3..{t A current clement adjacent to a ground plane.
(6) (a) Original problem; (b)
image problem.
and E, = 7JH~. According to image theory, this must also be the solution to Fig. 3-6a above the ground plane. The problem of Fig. 3-6a represents the antenna system of a short dipole antenna adjacent to a ground plane. The total power radiated by the system is
~,
=
JJ B,B: ds
= 2'l1'"7J
fo·
n
11i.lt r t sin () dO
hemi_
.phere
where integration is over the large hemisphere z tuting from Eq. (3-10) and integrating, we have
I
Ill' [1:3 -
is', - 2""i: As kd - t
00.
cos 2kd (2kd)'
> 0,
r --+
2kd] + sin (2kd)'
00
Substi-
(3-l1)
the power radiated is equal to that radiated by an isolated
element [Eq. (2-116)]. As led --+ 0, the power radiated is double that radiated by an isolated clement. The gain of the antenna system over an omnidirectional radiator, according to Eq. (2-130), is g -
4.1rT 211I H40l t
is',
2 J = _'I-_---=C=OS'"2ki
(3-12)
along the ground plane. This is g = 3 at /cd = 0, and g 6 as kd -+ 00. The maximum gain occurs at Jed 0= 2.88, for which 9 = 6.57. Thus, a gain of more than four times that of the isolated element (1.5) can be i.chieved. Figure 3-7 shows the radiation field patterns for the cases =0
SOME THEOREMS AND CONCEPTS
105
FlO. 3·7. Radiation field patterns for the current. clement of Fig. 3-&.
• II
Fro. 3-8. Problema involving multiple images. (0) Current ele.ment. in a conducting tube; (b) current element in a conducting wedge.
106
TIME-HARMONIC ELEC'rnOMAGNETIC FIELDS
d = 0 (element at the gound plane surface) and d = 0.459>-. (maximum gain). Image theory also can be applied in certain problems involving more than one conducting plane. Two such cases are illustrated by Fig. 3-8. In the case of a conducting tube (Fig. 3-Sa), an infinite lattice of images is needed. In the case of a conducting wedge (Fig. 3-8b), a finite set of images results. Image theory can be used for conducting wedges when the wedge angle is 180o/n (n an integer).
3-5. The Equivalence Principle. Many source distributions outside a given region can produce the samo field inside the region. For example, the image current element of Fig. 3·6b produces the same field above the plane z = 0 as do the currents on the conductor of Fig. 3-6a. Two Rources producing the same field within a region of space are said to be equivalent within that region. When we are interested in the field in a given region of space, we do not need to know the actual sources. Equivalent sources will serve as well. A simple application of the equivalence principle is illustrated by Fig. 3-9. Let Fig. 3-9a represent a source (perhaps a transmitter and antenna) internal to S and free space external to S. We can set up a problem equivalent to the original problem external to S as follows. Let the original field exist external to 8, and the null field internal to S, with free space everywhere. This is shown in Fig. 3-9b. To support this field, there must exist surface currents J., M. on S according to Eqs. (1-86). These currents are therefore
J. -
n X H
M.
~
E Xn
(3-13)
where n points outward and E, H are the original fields over S. Since the currents act in unbounded free space, we can determine the field from them by Eqs. (3-4) and (3-5). From the uniqueness theorem, we know that the field so calculated will be the originally postulated field, that is, E, H external to S and zero internal to S. The final result of this procedure is a formula for E and R everywhere external to S in terms of the tangential componentS'" of E and H on S.
(r;)
Flo. 3·9. The equivalent currents original eources.
prod1J~
the 5&ffie field c.xlernal to S
lill
do the
107
SOME TIlEOREJd.S AND CONCEPTS
. ___ I\
E'H'
/--
..... ,
1-... \
s:...
(0)
!n
E',H'
I
--' /
(b)
\,; /~__.Jn (....... " \
\
E"H",
J,J.
-~
S' (0)
M,
Flo. 3-10. A general formulation of t.he cquivalcnce prhu::iple. (a) Original 4 problem; (b) original b problem; (e) equivalent. to a external to S and to b internal to S; (d) equivalent to b exwmal to Saud to a internal to S.
We were overly restrictive in specifying the null field internal to S in the preceding example. Any other field would serve as well, giving us infinitely many equivalent currents as far as the external region is eoncerned. This general formulation of the equivalence prineiple is represented by Fig. 3-10. We have two original problems consisting of currents in linear media, as shown in Fig. 3-1Oa and b. We can set up a problem equivalent to a external to S and equivalent to b internal to S as follows. External to S, we specify that the field, medium, and sources remain the Barne as in the a problem. Internal to S, we specify that the field, medium, and sources remain the same as in the b problem. To suP"' port this field, there must be surface currents J. and M. on S. According to Eqs. (1-86), these are given by
J.
= D X (Ho - H')
M. - (Eo - E') X D
(3-14)
where Ea, Ha is the field of the a problem and E', II' is the field of the b problem. This equivalent problem is shown in Fig. 3-1Oc. We can also set up a problem equivalent to b external to S and to (J internal to S in I\n Analogous manner, as shown in Fig. 3-1Od. In this case the necessary ~urface currents are the negative of Eqs. (3-14). Note that in each case we must keep the original sources and media. in the region for which we keep the field. Note also that we cannot use Eqs. (3-4) and (3-5) to
108
TIME-HARMONIC ELECTROMAONETIC FIEL06
!
E,B /
E,B
....; ; _......
{ I Sources \/
D
Zero field
\
Electric
J
"
S ......- - - (a)
_/
D
s
conductor
E,B
--~
D
.:ftI
M., - EXn (0)
(e)
Flo. 3-11. The field external to S is the Mme in (a), (b), and (e). Ca) Original problem; (b) magnetic current backed by all electric conductor; (e) eledric current backed by a magnetic conductor.
determine the field of the currents unless the equivalent currents radiate into an unbounded homogeneous region. Finally, note that the restricted form of the equivalence principle (Fig. 3-9) is the special case of the general form for which all a sources and matter lie inside S and all b sources are zero. So far, we have used the tangential components of both E and H in setting up our equivalent problems. From uniqueness concepta, we know that the tangential components of only E or H arc needed to de~rmine the field. We shall now show that. equivalent. problems can be found in terms of only magnetie eurrents (tangential E) or only elect-ric currents (tangenlial H). Consider a problem for which all sources lie within S, as shown in Fig. 3-110. We set up the equivalent problem of Fig. 3-11b as foUowa. Over S we place 0. perfect electric conductor, and on top of this we place a sheet of magnetic current MI' External to S we specify the same field and medium as in the original problem. Since the tangential components of E are zero on the conductor (just behind M.), and equal to the original field components just in front of M I , it follows from Eqs. (1-86) that
MI = E X n
(3-15)
We now have the same tangential components of E over S in both Fig. 3-11a and bj so according to our uniqueness theorem the field outside of S
must be the same in both cases. We can derive the alternative equivalent problem of Fig. 3-11c in an analogous manner. For this we need the perfect magnetic conductor, that is, a boundary of zero tangential components of H. We then find that the elcctric current sheet ]. -
D
X H
(3-16)
over a perfect magnetic conductor coveri.ng 8 produces the same field external to S 88 do the original sources. By now, the general philosophy of the equivalence principle should be
109
SOME THEOREMS AND CONCEPTS
appa.rent. It is based upon the one-to-one correspondence bet\"een fields and sources when uniqueness conditions are met. If we specify the field and matter everywhere in space, we can determine all sources. We derived our various equivalences in this manner. Considerable physical interpretation can be given to the equivalence principle. For example, in the problem of Fig. 3-9b, the field interna.l to S is zero. It therefore makes no difference what matter is within S as far as the field external to S is concerned. We have previously assumed that free space existed within S, so that the potential integral solution could be applied. We could just as well introduce a perfect electric conductor to back the current sheets of Fig. 3-9b. It can be shown by reciprocity (Sec. 3-8) that an electric current just in front of an electric current conductor produces no field. (We can think of the conductor as shorting out the current.) Therefore, the field is produced by the magnetic currents alone, in the presence of the electric conductor, which is Fig.3-11b. Alternatively, we could back the equivalent currents of Fig. 3-9b with a. perfect magnetic conductor and obtain the equivalent problem of Fig. 3-He. When matter is placed within S in Fig. 3-9b, the partial fields produced by J. alone and M. alone will change external to 8, but the total field must remain uncha.nged. Perhaps it would help us to understand the equivalence principle if we considered the analogous concept in circuit theory. Consider a source (active network) connected to a passive network, as shown in Fig. 3-12a. We can set up a problem equivalent to this as (ar as the passive network is concerned, as follows. The original source is switched off, leaving the source impeda.nce connected. A current source 1, equal to the terminal current in the original problem, is placed across the terminals. A voltage I
~
Source
tv
Passive network
Source Impedance
(0)
Passive network
Passive network (b)
t"''---_~_"'_tw_O_'_k_.J Passive
I
(d)
FIG. 3-12. A circuit theory analogue to thQ equivalence principle. (0) Original problem; (b) equivlilent sources; (c) source impedance replaced by a short circuit; (d) source impedance replaced by an open circuit.
uo
TIME-HARMON'IC ELECTROMAGNETIC FIELDS
source V, equal to the terminal voltage in the original problem, is placed in series with the interconnection. This is illustrated by Fig. 3-12b. It is evident from the usual circuit concepts that there is no excitation of the source impedance from these equivalent sources, whereas the excitation of the passive network is unchanged. Thus, Fig. 3-12b is the circuit analogue to Fig. 3-9b. Since there is no excitation of the source impedance in Fig. 3-12b, we may replace it by an arbitrary impedance without affecting the excitation of the passive network. This is analogous to the arbitrary placement of matter within S in the field equivalence of Fig. 3-9b. In particular, let the source impedance be replaced by a short circuit. This short-circuits the current source and leaves only the voltage source exciting the network (recall circuit theory superposition). Thus, the voltage source alone, as illustrated by Fig. 3-12c, produces the same excitation of the passive network as does the original source. This is analogous to the field problem of Fig. 3-lIb. Now consider the source impedance of Fig. 3-12b replaced by an open circuit. This leaves only the current source exciting the network, as shown in Fig. 3-12<1. This is analogous to the field problem of Fig. 3-11e. S-8. Fields in Half-space. A combina.tion of the equivalence principle and image theory can be used to obtain solutions to boundary-value problems for which the field in half-space is to be determined from its tangential components over the bounding plane. To illustrate, let the original problem consist of matter and sources z < 0, and free space z > 0, as shown in Fig. 3·13a. An application of the equivalence concepts of Fig. 3-lIb yields the equivalent problem of Fig. 3-13b. This consists of the magnetic currents of Eq. (3-15) adjacent to an infinite
z-o
I I
E,H Sources and
matter
@
E.H
I I I
I I I I
E,H
Zero field
E.H
Ima~e
fie d
~
~
..," C
M. = El
8
M. - 2El
u
'C
11 i;]
~n (a)
z~o
• - 0
n (b)
n (e)
FIo. 3-13. Illustration 01 the otcps uoed to establish Eq. (3-17).
son
111
THEOREMS AND CONCEPTS
z
.
z
,
•
•
~4y
y
x
x
p
FIG. 3-14. A coaxial line opening onto a grcund plane. equivalent problem.
(a) Original problemj (b)
ground plane. We now image the magnetic currents in the ground plane, according to Fig. 3-5. The im3gCS are equ31 in magnitude to, and essentially coincident with, the M. of Fig. 3-13b. Thus, as pictured in Fig. 3-13c, the magnetic currents 2M. radiating into unbounded space produce the same field z > 0 as do the original sources. They produce an image field z < 0, which is of no interest to us. The field of Fig. 3-13c is then calculated according to Eqs. (3-4) and (3-5) with A = O. This cao be summarized mathematically by E(r) - - V X
II 2'-;"'-;1
E(r') X ds'
(3-17)
This is a mathematical identity valid for a.ny field E satisfying Eq. (2-3). The H field satisfies Eq. (2-4), which is identical to Eq. (2-3); so the abovc identity must also be valid for E replaced by H. We can show this by reasoning dual to that used to establish Eq. (3-17). The above result is particul:JJ'ly useful for problems involving apertures in conducting ground planes. AB an example, suppose we have a coaxial transmission line opening into a ground plane (Fig. 3-14a). According to the above discussion, the field must be the same as that produced by Fig. 3-14b. Note that M. exists only over the aperture (coax opening), for tangential E is ECro over the ground plane. Let us asume that the field over the aperture is the transmission-line mode of the coax. that is
E• -
-v
p
log (bfa)
112
TIME-H.AJUlONJC ELECI'aOMAG.:·rETIC FIELDS
where V is the line voltage. rent in Fig. 3-14b is M
To this approximation, the magnetic cur=
•
V p log (bla)
This is a loop of magnetic current which, if b «A, acts as an electric dipole (dual to Fig. 3-3). Visualize this current as a continuous distribution of magnetic current filaments of strength dK = M. dp. The total moment of tbe source is then
(3-18)
The equivalent electric current clement must satisfy the equation dual to Eq. (3-6), or II = -jw.KS (3-19) We have now reduced the problem to that of Fig. 3-& with kd = O. From Eq. (3-10) and the above equalitics, we have the radiation field given by ..,...V(b' - a') . (3-20) H. = 2Ar log (bfa) e-;" 510 8 and E. = flJI._ Thus, the radiation field patted is the d - 0 curve of Fig. 3-7. The gain of the antenna system is g = 3. The power radiated is Eq. (3-11) with kd = 0 and II given by Eqs. (3-18) and (3-19), or _V(b' - a') 2 ~, = 2r. 2X log (bla) :I
I
I
= 4r r'(b' - a')V 3. X'iog (bla)
I'
I'
(3-21)
Note that the power radiated varies inversely as ).4. Note also t.hat our answers are referred to a volt.age, characteristic of aperture antennas. This is in contrast to answers referred to current for wire anten.nas. For aperture antennas we define a radiation conductance according to (3-22)
where V is an arbitrary reference volt-age. In the coaxial radiator of Fig. 3-14 it is logical to pick this V to be the coaxial V at the aperture. Hence, the radiation conductance is 4r' [ b' - a' ]' G. - J,j" X' log (bla)
(3-23)
113
SOME THEOREMS AND CONCEPTS
t
E_E'+gE' n
n
Source
;/
/ J .... Hi Xn
Obstacle
Obstacle
'----c~ M.
-
nxE'
(b)
(a) Flo. 3-15.lilustration of the induction thoorcm. equivalent.
(a) Original problem; (b) induction
For the usual coaxial line, Gr is small, and the coaxial line sees nearly an open circuit. As a and b are made larger, the radiation becomes more pronounced, but our formulas must then be modified.! 3-7. The Induction Theorem. We now consider a theorem closely related in concept to the equivalence principle. Consider a problem in which a set of sources are radiating in the presence of an obstacle (material body). This is illustrated by Fig. 3-15a. Define the incident field E', Hi as the field of the sources with the obstacle absent. Define the scattered field E-, H' as the difference between the field with the obstacle present (E, H) and the incident field, that is, E' - E - E'
H' - H - H'
(3-24)
This scattered field can be thought of as the field produced by the cur· rents (conduction and polarization) on the obstacle. External to the obstacle, both E, Hand E;, Hi have the same sources. The scattered field E', H' is therefore a source-free field external to the obstacle. We now construct a second problem as follows. Retain the obstacle, and postulate that the original field E, H exists internal to it and that the scattered field E', H- exists external to it. Both. these fields are source-free in their respective regions. To support these fields, there must be surface currents on S according to Eqs. (1-86), that is,
J. -
n X (H' - H)
where n points outward from S.
J. -
M.-(E·-E)xn
According to Eqs. (3-24), these reduce to
H' X n
M. - n X E'
(3-25)
It follows from the uniqueness theorem that these currents, radiating in the presence of the obstacle, produce the postulated field (E, H internal to S, and E', H' external to S). This is the illdmlion theorem, illustrated by Fig. 3-l5b. It is instructive to compare the induction theorem with the equival H. Levine snd C. B. Papas, Theory of the Circular Diffraction Antenna, J. Appl. Ph"., voL 22, no. 1, pp. 2H3, January, 1951.
114
TIME-HARMONIC ELECTROMAGNETIC FIELDS
lence theorem. The latter postulates E, H internal to S and zero fieM external to S, which must. be supported by currents
J. -
H X
M. '"'" n X E
0
on S. These currents can be considered as radiating into an unbounded medium baving constitutive parameters equal to those of the obstacle. Thus, we can use Eqs. (3-4) and (3-5) to calculate tbe field of the above currents. However, we do not know J. and M. until we know E, H on S, that is, until we have the solution to the problem of Fig. 3-15a. We can, however, approximate J. and M. and from tbese calculate an approximation to E, H within S. In contrast to the above, the induction theorem yields known currents [Eqs. (3-25». (This assumes that EI, Hl is known.) We cannot, however, use Eqs. (3-4) and (3-5) to calculate the field from J., M" for they radiate in the presence of the obatacle. A determination of this field is & boundary-value problem of the same order of complexity as tho original problem (Fig. 3-15a). We can, however, approximate the field of I., M, and thereby obtain an approximate formula for E, H internal to Sand E', H" external to S. A simplification of the induction theorem occurs when the obstacle is a perfect conductor. This situation is represented by Fig. 3-160. The solution E must satisfy the boundary condition n X E .... 0 on S (zero tangential E). It then follows from the first of Eqs. (3-24) that n
X E'
= -n
X E'
(3-26)
ooS
We now know the tangential components of E' over S; so we can construct t.he induct.ion represent.ation of Fig. 3-1Gb as follows. We keep the perfect.ly conducting obstacle and specify that external t.o S the field E', H' exists. To support this field, there must be magnetic currents on S given by (3-27) MI=E'Xn=nXEi We can visualize this current as causing the tangential components of E The
to jump from zero at the conductor to those of E' just outside M"
t
E_EI+E'
o
n
Source
;/
p,rloct conductor
Perfect
conductor
----~~ M. (n)
nXE'
(b)
FIo. 3-16. The induction theorem a.e applied to a perfeetly conducting obstacle. Original problem; (b) induction eQuivalent
(a)
80ME THEOREMS AND CONCEPTS
115
Ei+ E'
..
Incident wave
M. . . .
Conducting
Conducting
plate Ca) FlO. 3-17. Scattering by a conducting plate. cquiva.lent.
plate Cb) (a) Original problem; (b) induction
tangential components of E in Fig. 3-16b therefore have been forced to be E'. Thus, according to uniqueness concepts, the currents of Eq. (3-27) radiating in the presence of the conducting obstacle must produce E', H' external to S. It is interesting to compare this result with the previous one (Fig. 3-1Sh). We found that, in general, both electric and magnetic currents exist on S in the induction representation. How, then, can both Fig. 3-1M and Fig. 3-lab be correct for a perfectly conducting obstacle? The answer must be that an electric current impressed along a perfect electric conductor produces no field. If the conductor is plane, this is evident from image theory. We can prove it, in general, by using the reciprocity concepts of the next section. To iIlustru.te an application of the induction theorem, consider the problem of determining the back scattering, or radar echo, from a large conducting plate. This problem is suggested by Fig. 3-17a. For normal incidence, let the plate lie in the z = 0 plane and let the incident field be specified by (3-28)
According to the induction theorem, the scattered field is produced by the currents 11{1/ = Eo on the side facing the source and M 1/ = - Eo on the side away from the source. These currents radiate in the presence of the original conducting plate, as represented by Fig. 3-17b. Let the field from each clement of current be approximated by the field from an element adjacent to a ground plane. According to image theory, this means that each element of !If1/ seen by the receiver radiates as 2M1/ = 2E o in free space. Hence, far from the plate, it contributes
116
TIM.E-HARMO:'lo.C ELECTROMAGN1o."T[C FIELDS
in the back«atter direction. Each clement Dot seen by the receiver contributes nothing to the back-scattered field. Summing over the entire plate, we ha.ve the distant back-scattered field given by E,'
-if
dE,' =
-j;~,A e-tk
(3-29)
where A i the area of the plate. The ed&o area or radar cross aeclion of an obstacle is defined a.s the area for which the incident \vave contains sufficient power to produce, by omnidirectional radiation, the same back-scattered power density. In mathemfltical form, the echo area is
A. "'" lim (4Jrr! ~'.) ........ S'
(3-30)
where Sl is the incident power density and S· is the scattered power density. For our problem, = IE,lt/" and, from Eq. (3-29),
gl
S' _ !
•
I I' kE,A
2rr
The echo area of eo conducting piMo is lherefore k'J'P
-hAt
A, = - - -~,r
(3-.11)
valid for large plates and normal incidence. 3-8. Reciprocity. In its simplest sense, a reciprocity theorem stat.es that a response of 8. system to a source is unchanged when source and measurer are interchanged. In a more general sense, reciprocity theorems relate a response at one source due to a second source to the response at the second source due to the first source. We shull establish this type of reciprocity relationship for a-c fields. The reciprocity theorem of circuit theory is a special case of this reciprocity theorem for fields. Consider two sets of a-c sources, J", M- nnd Jb, Mb, of the same frequency, existing in the same Iinenr medium. Denote the field produced by the a sources alone by E-, H·, and the field produced by the b sources alone by Eb, Hb. The field equations are then
v X HG
=
- .. X E'
~
tiE-
+ J-
m' + M"
"XH'-~E'+J'
-v X Eb= iH'+Mb
We mulliply the first equation scalariy by Eb and the last equation by Hand add the resulting equations. This gives
-
~.
(E' X H')
~ ~E'
. E' +
m" . H' + E' . J' + H' . M'
117
SOME THEOREMS AND CONCEPTS
where the left-hand term has been simplified by the identity V . (A X B)
~
B.V X A - A.V X B
An interchange of a and b in this result gives ~
- V . (Eo X H')
OEo . E'
+ SHo . H' + Eo . ]' + H' . Mo
A subtraction of the former equation from the la.tter yields - V •
(Eo
X
H' - E'
Ho) _ Eo . ]'
X
+ H' . Mo
- E' . ]0 - Ho . M' (3-32)
At any point for which the fields are source-free (J reduces to V .
(Eo X H' - E' X Hp)
~
=
M
=
0
0), this (3-33)
which is called the Lorentz reciprocity theorem. If Eq. (3-33) is integrated throughout a. source-free region and the divergence theorem applied, we have
effi (Eo
X H' - E' X Ho) . ds ~ 0
(3-34)
which is the integral form of the Lorentz reciprocity theorem for a sourcefree region. For 0. region containing sources, integra.tion of Eq. (3-32) throughout the region gives
- effi (Eo
X H' - E' X Ho) . ds
- III (Eo.]' -
Ho. M' - E'·]o
+ H'· Mo) dT
(3-35)
Let us now postulate that all sources and matter arc of fInite extent. Distant from thc sources and matter, we have (see Sec. 3-13) E~ =
E, = TjN..
-TIll,
The Icftr-hand term of Eq. (3-35), integrated over a sphere of radius <:0, is then
r -+
-'Il1/> (lJ,GH,b + H.alI. b -
11,bH,a - JJ(>bH.") ds = 0
Equation (3-35) now reduces to
III (Eo.]' -
Ho. M') dT
~
III (E'·]o -
H'. Mo) dT
(3-36)
where the integration extends over all space. This is the most useful form of the reciprocity theorem for our purposes. Equation (3-36) also applies to regions of finite extent whenever Eq. (3-34) is satisfied, For
118
TIME-RAlUIONIC ELEctROMAGNETIC FIELDS
example, fields in a region bounded by a perfect electric conductor satisfy Eq. (3-34) i hence Eq. (3-36) applies in this case. The integrals appearing in Eq. (3-36) do not in general represent power, since no conjugates appear. They have been given the name reaction. I By definition, the reaction of field a on source b is
(a,b) ~
fff (E-· J' -
H' . MO) dT
(3-117)
In this notation, the reeiprocity theorem is
(a,b) - (b,a)
(3-118)
that is, the reaction of field a on source b is equal to tho reaction of field b on source a. Reaction is a useful quantity primarily because of this conservative property. For example, reaction can be used as a measure of equivalency, since a source must have the same reaction with all fields equivalent over its extent. This equality of reaction is a necessary. but not sufficient, test of equivalence as defined in Sec. 3·5. We shall use the term &tll-reaction to denot-e the react.ion of a field on its own sources, that is, (a,a). A valuable tool for expositional purposes can be obtained by using the circuit sources of Fig. 3-1 in the reaction concept. For 0. current source (Fig. 3-1a), we have
(a,b) -
f E'· I'd!- I' f E-·d! ~ -V'I'
where V- is the voltage across the b source due to some (as yet unspecified) a source. For a voltage source (Fig. 3-1b), we have K' = - V', and
where 1- is the current through the b source due to some a source. summarize, the "circuit reactions" are
(a,b) -
J-
V'l'
l + V'I-
b a current source b a voltage source
To
(3-119)
If we use a unit current source (I' = I), then (a,b) is a measure of V(the voltage at b due to another source a). If we use a unit voltage source (V' = I), then (a,b) is a measure of J- (the current at b due to aoother source a). To relate our reciprocity theorem to the usual circuit theory stMement of reciprocity, consider the two-port (Cour-terminal) network oC IV. H. Rum.scy. The Reaction Concept in Electromagnetic Theory, PAy,. Rn., Bel'.
2, vol. 94, no. 6, pp. 1483--1491, June 15, 1954.
119
SOllE THEOREMS AND CONCEPTS
Fig. 3-18. The characteristics of a. linear network can be described by the impedance matrix [z} defined by
_[,,, ,,,] [I,] [V,] V, I, Zu
(3-40)
Zu
Supposc we apply a. current source II at port 1 and a. current source 1, at port 2. Let the partial response Vi} be the voltagc at port i due to sourcc I} at port j. Each current source sees the other port open-eircuitcd (sec Fig. 3-1a)j hence V'J
ti} - -
IJ
In terms of the circuit rcactions [Eq. (3-39)J, (j,tl = - VijI.; honce
z··- -0,') 1)
(3-41)
/;/}
Thus, the clernent<s of the impedance matrix are thc various reactions among two unit current sources. The reciprocity theorem [Eq. (3-38)], applied to Eq. (3-41), shows that (3-42)
which is the usua.l statement of reciprocity in circuit theory. Equations (3-41) and (3-4.2) also apply to an N-port network. The use of voltage sources instead of current sources gives reactions proportional to the clements of the admittance matrix {yI, and reciprocity then states that Yi} = Vit'·
The proofs of many other theorems can be based on the reciprocity theorem. For ex~mple, the preceding paragraph is a proof that any nelwork constru.cted 0/ linear isotropic matter has a symmetrical impedame matrix. This llnetwork" might be the two antennas of Fig. 3-19. Rcciprocity in this case caD be stated as: The voltage at b due to a current source at a is equal to the voltage at a due to the same current source at b. If the b antenna is infinitely remote from the a antenna, its field will be a plane wave in the vicinity of a, and vice versa. The receiving pattern of an a.ntenna is defined as the voltage at the antenna.
v~{~
\ (1)
\
(2) Network
Fla. 3-18. A two-port network.
(b)
Flo. 3-19. Two antennas.
l20
TIME-HARMONIC ELECTROMAGNETIC FIELDS
terminals due to a plane wave incident upon the antenna. The reciprocity theorem Cor antennas can thus be stated as: The receiving patlern of any antenna constructed of linear i80tropic matter i. identical U> it& tranamilling patlem. In Sees. 3-5 and 3-7, we used the fact that an electric current impressed along the surface of a perfect electric conductor radiated no field. The reciprocity theorem proves this, in general, as follows. Visualize a set of terminals a on the conductor and another set of terminals b in space away from the conductor. A current clement at b produces no tangential component of E along the conductor; so V06 (Vat a due to 16) is zero. By reciprocity, Vk (Vat b due to 10 ) is zero. The terminals b ata arbitrary; BO the current element along the conductor (at a) produces no V between any two points in space; hence it produces no E. We can think of I. l\S inducing currents on the conductor such that these currents produce a free-space ficld equal and opposite to the free-space field of I •. 3-9. Green's Functions. Our reciprocity relationships are formulas symmetrical in two field-source pairs. Mathematical statements of reciprocity (symmetrical in two functions) are called Green's theorems. The difference between a Green's theorem and a reciprocity theorem is that no physical interpretation is given to the funcHons in the former. The scalar Greeo's theorem is based on the identity V· (fV~) - fV'~
+. vf· v~
When this is integrated throughout a region and the divergence theorem applied to the left-hand term, we obtain Green's first identity (3-43)
Interchanging y. and ¢ in this identity nnd subtracting the interchanged equation from the original equation, we obtain Green's second identity or Green's theorem (3-44)
This is a statement of reciprocity (or scalar fields y. and ,p. The vector analogue to Grecn's theorem is based on t.he identity V· (A X V X B) = V x A . V x B - A· V X V X B
An integration of this throughout. a region and an application of the divergence theorem yields the vector analogue to Green's first. identity
1ft (A X v
X B) . cis
~
III (V X A· V x B -
A· V X V X B) d. (3-45)
121
SOM.E THEOUEAiS AND CONCEP1'S
d.
FlO. 3·20. Region to which Green's thoorem is applied.
o..c:..-4:---~ r
We can interchange A and B and subtract the resulting equation from the original equation. This gives the vector analogue to Green's second identity, or the vector Green's theorem,
~(A X V X B - B X V X A) ·ds =
fff (B • V X V X A -
A • V X V X B) dT
(3-46)
reciprocity theorem [Eq. (3-3S)}, for a. homogeneous medium, is essentially Eq. (3-46) with A = Ell and B = Ell. For an inhomogeneous medium, still another vector Green's theorem corresponds to our reciprocity theorem (sec Prob. 3-28). Green's theorems have been used extensively in the literature as foUows. Suppose we desire the field E at a point r' in a. region. Instead of solving this problem directly, a point source is placed at r , a.nd its field is called a Green's function G. We then substitute E = A and G = B in Eq. (3-46). This gives a formula for E at r' , as we shall discuss below. What we have done is solve the reciprocal problem (source at the field point of the original problem) and then apply reciprocity. The equivalence principle gives the solution more directly. Let us summarize the various Green's functions used in the literature. Stratton chooses l OUf
(3-47) ~ _
whe'e
CJk1r-r'[
I'
"1
(3-48)
and c is a constant vector. A comparison of Eq. (3-47) with Eq. (2-117) shows that G 1 is the vector potential of a current clement II = 4rc. Hence, G 1 is a solution to Eq. (2-108), or V X V X G1
-
k 2G 1 = V(V . G 1)
r F r'
(3-49)
Now suppose we wish to find E at r' in a source-free region enclosed by S. The source of G 1 is placed at r' and surrounded by an infinitesimal sphere 8, as shown in Fig. 3-20. Equation (3-46) with A = E and B = G 1 is now I J. A. Stratton, "Electromagnetic Theory," p. 464, McGraw-Hill Book Company, Inc., New York, 1941,
122
TI3oIE-llARMQXIC ELEcrROMAGNETIC FIELDS
applied to the region enclosed by Sand -4rc. E -
t
The result. is
$.
(E X V X G. - G. X V X E
+ E v· G.)· ds
(3..,\Q)
which is a. formula. for calculating E at r' in terms of n X E, n X V X E, and n . E on S. Furthermore, it is required that E be continuous and
have continuous first derivatives on S.
This is a severe restriction on
the usefwness of Eq. (3-50), although it can be amended to admit singular E'8 on S. A choice of Grecn's function which overcomes some of the disudvantagcs of Eq. (3-50) is l O 2 = v X ttl>
(3-51)
where tP is given by Eq. (3-48). This is evidently the magnetic field of a current clement 11 = be. Hence, G t is a solution to r ,. r'
(3-52)
We now apply Eq. (3-46) with A = E and B = G t to the region enclosed by Sand, in Fig. 3-20. The result. is' 4.e . V' X E
~
1ft (G. X V X E -
E X V X G.) . ds
(3-53)
s This is a formula for Vi X E (bence for H) at r in terms of n X E and n X V X E on S. Equation (3-53) does not require E to be continuous on S, nor do we need to know n . E on S. Thus, Eq. (3-53) is a substantial improvement over Eq. (3-SO). In fact, Eq. (3-53) can be shown to be identical to the formula obtained from the equivalence principle of Fig. 3-9, applied to a homogeneous medium. Another useful Green's function is
G,=vxvxcq,
(3-54)
where ¢ is given by Eq. (3-48). This is proportional to the electric field of an elect.ric current elementi 80 G, also satisfies Eq. (3-52). An application of Eq. (3-46) would yield a formula for E at r' , similar in form to Eq. (3-53). All of t.he G's considered 80 far are tlfrce-space" Green's functions, that is, they.are fields of sources radiating into unbounded space. We can choose other G's such that t.hey satisfy boundary conditions on S. I J. R. Menber, "Scat.tering and Diffract.ion of Radio Waves," p. 14, P@tgamon Press, Kew York, 1955. I The left-hand aide of this equation is a function only of the primed coordinates. Bence, a prime is placed on v' to indicate operation on r' instead of r.
THEORE~IS
SOME
123
AND CONCEPTS
For example, let Gt = G2
+G
(3-55)
t'
such that G 4 satisfies Eq. (3-52) and n X V X G4
"'"
on S
0
(3-56)
The physical interpretation of Gt is that it is the magnetic field of a current element II = hc rad.iating in thc presence of a perfect electric conductor over S. The G 2 is the incident field, and the G t ' is the scattered field. Application of Eq. (3-46) with A = E and B = G 4 results in Eq. (3-53) with the last term zero, because of Eq. (3-56). Thus, be· v· X E ~
1ft (G. X V X E) . ds
(3-57)
s
which is a formula for V' X E in terms of only n X V X E over S. The same formula can be obtained from the equivalence principle of Fig. 3-11, as it applies to a homogeneous region. Similarly, defining a G~ such that on S
n X G6 = 0
(3-58)
we can obtain a formula 41fc • V' X E =
-1}
(E X V X G 6 )
•
ds
(3-59)
and so on. All these various formulas, and many more, can be directly obtained from the equivalence principle. We have discussed the Green's function approach mel'cly bec:luse it has been used extensively in the literature. 3-10. Tensor Green's Functions. We shall henceforth usc the term /lOreen's function" to meau "field of a point source." Suppose we have a current clement II at r' and we wish to evaluate the field E at r. The most general linear relationship between two vector quantities can be represented by a tensor. Hence, the field E is related to the source 11 by E
~
Ir]!l
(3-60)
where IfJ is called a tensor Green's function. and ma.trix notation, Eq. (3-60) becomes Elf [E.] E.
=
[r.. r.. I'lIz
1'1/11'
r ..]
I',l'
r az rill r..
In rectangular components
[II.] Il"
(3-61)
Il.
Thus, r,j is the ith component of E due to n unit j-directed electric current clement. The E might be the free-space field of 11, ill which case
124
TIME-HARMONIC ELECTROMAGNETIC FIELDS
[rJ would be the "free-space Green's function." Alternatively, E might be the field of 11 radiating in the presence of some matter, and [1') would then be called the uGrean's function subject to boundary conditions." Still other Green's functions are those relating H to 11, those relating E to Kl, and so on. Our principal use of tensor Green's functions will be for concise mathematical expression. For example, the equation (3-62)
where [r] is tbe free-space Green's function defined by Eq. (3-60), represents the solution of Eq. (2-111), which is
-jw,.A
E -
+ Jw, .,!.- v(v . A)
-Iff J.-'"'-'''
=--:;rdT , ':t'lrlf - r I
A -
(3-63)
Equation (3-62) also represents the field of currents in the vicinity of a material body if {rl represents the appropriate Green's function, and so on. In other words, Eq. (3-62) is symbolic of the solution, regardless of whether or not we can find [r]. Even though we shall not use tensor Grecn's functions to find explicit solutions, it should prove instructive to flnd an explicit IrJ. Let us take [rJ to be the free-space GreenJs function defined by Eq. (3-60). If II is z-directed, I Le-Jl:lr-r'l
and
A , -- 4;1, "I . A + 1 a'A, E '" = -Jwp.
'"
-.-
JWt
-a x'
E __1_ alA", jwe ayax
tI
E __1_ alA", I
jWf.
az ax
Comparing this with Eq. (3-61) for 1"" = Il. = OJ we sec that
r"""
=
a')
. 1 , f ( -JwJl+-·-JWf. aX
1 a 1/1 r "'" -jWf. ay'ax
r"" where
=
J- dZiJ"Ytax
JWt
e-Jl:1r-r'1
f -
4rI' _ <'I
(3-64)
SOME THEOREMS AND CONCEPTS
125
The other elements of [r] are found by taking n to be y-dire9-ted' and then z-directed. From symmetry considerations, the other r ./s,will differ only by a cyclic interchange of (x,y,z). The result is theref9re
(-i
r;; _
r01
1CI
w•
+.,!.a')~ JWf a~2
ia" jWf ai aj
with 1/J given by Eq. (3-64). symmetry
.'~J'
(3-ll5)
r-
The reciprocity theorem is reflected in the (3-66)
which can be proved for r's subject to boundary conditions as well. 3-11. Integral Equations. An integral equation is one for which the unknown quantity appears in an integrand. We already have the concepts needed to construct integral equations. For example, the potential integral of Eq. (2-118) is essentially an integral equation when J is unknown. Most problems can be formulated either in terms of integral equations or in terms of differential equations. When exact solutions nre desired, the differential equation approach is usually the simpler one. An important use of integral equations is to obtain approximate solutions. There is good reason for this. Integration is a summation process, and it is not necessary that each element of the summation be correct. Errors in some elements of the summation may be compensated for by errors in otber elements. Also, all elements do not contribute equally to a summation. It is much more important that tbe elements contributing most to the summation be correct than that the elements of minor contribution be correct. This is why we were able to obtain useful results by B.S$uming the current on the linear antenna of Fig. 2-23, by assuming the field of each element of magnet.ic current in Fig. 3-17b, and so on. To illustrate the formulation of an integral equation, consider the induction theorem of Fig. 3-16. Let [r(r,r')J be the tensor relating the E field at r due to an element of M at [' radiating in the presence of t.he conductor over S. In equation form, this is dE(r)
~
[rer,r')) dM(r')
The total scattered field for the problem is then the summation E'(r)
-1} [r(r,r')]M.(r') do'
where M. is given by Eq. (3-27).
When r is on S, Eq. (3-26) must
126
TU.fE-HA.R~lONlC
ELECTROMAGNETIC FIELDS
also be true; hence n X E'(r) - n X
1ft Ir(r,r')]E'(r')
X ds'
ron S
13-67)
s
The incident field E; is assumed to be known; so Eq. (3-67) is an integral equation for determining Irl. As we mentioned earlier, an exact solution to Eq. (3-67) would be difficult even for the simplest specialization. Problems involving a region homogeneous except for small uislands" of matter are commonly encountered. Examples of such problems are the linear antenna of Fig. 2-23 and the obstacle of Fig. 3-15a. To illustrate the general concepts involved, suppOSe we have an inhomogeneous region, possibly containing sources Ji and M', Within this region, the field satisfies
-v
X
E - tH
+ M'
vxH-liE+J'
where ~ and y arc functions of position. We can define normal values of iropedivity and admittivitY,.€1 and '01, which may be any convenient constants (usually the most common .€ and '0 in the region). We can now rewrite the field equations as
-v X E = t,H
+M
v X H = Ii,E
+J
where the effective currents are M = (I - 1,)H J - (Ii - 1i,)E
+ M' + J'
(3-68)
These effective turrents can then be treated as source currents in a homogeneous region. Since J and M are functions of E and H, a solution in terms of them will lead to an integral equation. However, if ~ = .€l and '0 = 01 except in small subregions, we can assume J and M in the su~ regiolls and obtain approximate expressions for E and H elsewhere. (Recall the linear antenna problem, where we assumed I on tbe antenna wire.) Note that, when the normal z and yare taken as the free-space parameters, Eqs. (3-68) reduce to M ~ jw(P - ",)H
J-
jw(' -
+ M'
'olE +.E + J'
(3-69)
The effective currents in excess of the true sources (M' and J') are now just those due to the motion of atomic particles in vacuum. Let us reconsider the problem of scattering by an obstacle in the light of the above concepts. Given the problem of Fig. 3-t5a, we can consider the total field to be the potential integral solution of Eqs. (3-4) and (3-5), with J and M given by Eqs. (3-69). The incident field is that produced
12i
SOME THEOREMS AND CONCEPTS
by Ji and Mi' outside of the obstacle, and the scattered field is that produced by M - jw(ll - ,,)H (3-70) J = jw(' -
To be explicit, outside of the obstacle
E' - -v X F _
where
+ JWfo J- v
iff 1,J iff l
1
X V X A
.-,,,,-el
A - 400
"I
dr
,
oblt-o.c1e
(3-i2)
M,-;'I~el dT , r - r' i
-
l F -k-
(3-il)
obn.ele
with J and M given by Eq. (3-70). If we can guess J and M with reasonable accuracy, then Eqs. (3-71) and (3-72) will give us an approximate solution. For a nonmagnetic obstacle, M, and consequently F, will be zero. For a good conductor, J reduces to uE, and this current resides primarily on the surface of the obstacle. If we assume the obstacle perfectly conducting, then J becomes a true surface current. The solution in this case reduces to E' =
1 11-4
-r"JWfo
effi JI'e-/l:1r-'I 'l ds' t
V X V X
r
r
(3-i3)
8
If wc specialize this equation to S, then Eq. (3-26) must be met, and we have an ibtcgral equation for determining J•. An approximation to J., known as the physical optics approximat1'on, is as follows. Let Fig. 3-210. represent a perfectly conducting obst~cle illuminated by some source. In terms of the t.otal field, t.he surface current on the conductor is given by
J.
= n X H
When the obstacle is large, we assume that thc total field is negligible in n
Incident wave
•
Cb)
Ca)
Fro. 3-21. Thc physical optics approximation. ,-pproximation.
(a)
Original problem; (b) the
t28
TUlE-BARllONlC ELECTROMAGXETIC FJEbDs
the II shadow OJ region. Furthermore, if the obstacle is smooth and gently curved, each element of surface behaves similarly to an element of a ground plane. According to image theory, the tangential components of H at a ground plane are just twice those from the same source in unbounded space. We therefore approximate the current on the obstaclj) by I. "'" 2n x HI over Sf (3-74) where S' is the illuminated portion of S. mation to the scattered field is therefore 1
E' ~ .-:
......Jwto
v
V X
X
If
The physical optics approxi-
(n X H~c"l.-" ---.:--=do' If -
1
I
(3-75)
8'
This approximation is illustrated by Fig. 3-21b. As an explicit application of the physical optics approximation, again consider the large conducting plate of Fig. 3-17a. The incident E is given by Eq. (3-28); hence fl'
,
=-
.
Eo e-ih
The physical optics approximation to the obstacle current [Eq. (3-74») is therefore J. _ 2Eo
•
Each element of this radiates as a current element in Cree spnce, 8B analyzed. in Sec. 2-9. The contribution to the radiation field in the back-scatter direction from each J. ds is • -jkE o ds dE• = 211'T
...-Ilu
e'
The total distant back-scattered field is thereCore
If
dE' , - - jkE,A 2,.,. .-'"
(3-76)
,~w
which is identical to Eq. (3-29), the approximation obtained from the induction theorem. The physical optics a.pproximation to the echo area of the plate is therefore that of Eq. (3-31), This equality of the two approximations to back scattering [Eqs. (3-29) and (3-76)) is no coincidence. It can be shown that the two a.pproaches always give the snme back scattering but do not give the sume scattering in other directions. I I R. F. Harrington, On Scattering by Large Conducting Bodies, IRE Tram., vol. AP-7, no. 2, pp. 150-153, April, 1959.
SOME THEOREMS AND CONCEPTS
129
3-12. Construction of Solutions. So far, we have explicitly considered two types of solutions to the field equations, namely, uniform plane waves and thc potential integrals. In the next three chapters, we shall learn how to construct many other solutions. A general method of obtaining these solutions is considered here. In a homogeneous source-free region, the 6cld aa.tis6ca ~nly
-v
v
X E - ZH X H ~ ~E
(3-77)
In view of the divergcncelcss character of E and H, we can express the field in terms of a magnetic vector potential A or in terms of an electric vector potential F. More important, we can employ superposition and express part of the field in terms of A and part in terms of F. The A must be a solution to Eq. (2-108) with J = 0, and the F a solution to the dual equation. The general equations for vector potentials aro therefore
v
X V X A V X V X F -
k 2A = -9v~ k 2F = -zV'
(3-78)
where cit" and ¥ arc arbitrary scalars. The electromagnetic field in terms of A and F is given by Eqs. (3-4) with J = M = 0, or
E=-VXF+~VXVXA
H~VXA+~VXVXF
(3-79)
Equations (3-78) and (3-79) arc the general form for fields and potentials in homogeneous source-free regions. There is a great deat of arbitrariness in the choice of vector potentials. For instance, we can choose the arbitrary *'s according to
v·F =-w
v . A = -U4>"
(3-80)
This reduces Eqs. (3-78) to V2A V2F
+k A = 0 + k F '- 0 2
2
(3-81)
SolutioDs to these equations arc called wave potentials. Note that the rectangular components of the wave potentials satisfy the scalar wave equa.tion, or Helmholtz cquation, (J.
Also, when Eqs. (3-80) are satisfied, we can alternatively write Eqs.
130
TIYE-DARMONIC ELECTROMAGNETIC FIELDS
(3-79) ..
1 E = -v X F-lA+-v(V·A)
9
1 H - v X A - gF + j v(v, F)
(3-83)
We have yet to decide how to divide the field between A and F. As a word of caution, do not make the mistake of thinking oC A as due to J and F as due to M. This happened to be our choice Cor the potential integral solutioo, where we considered t.he sources everywhere. We afe now concerned with regions of finite extent, and we ctln represent a field in terms of A or F or both, regardless of its actual sourcc. Let us now consider some particulo.r choices of potentials. If we take F = 0 and A = u.y, (3-84)
then
E - -fA +
~ V(V . A)
H - v X A
(3-&)
This can be expanded in rectangular coordinates as
E~ =
1 ii"l-
1I _ ii'" • iiy
gaxaz
E ~ ~ ii"l-
fI=_iJt • iix
) '" E. - Y -1 (ii' -+k' dZ'
H. = 0
,
yayaz
(3-86)
A field with no IJ. is called transverse magnttic W z (T.M). We shall find it possible to choose y, sufficiently general to express an arbitrary TM field in a homogeneous source-free region according to the above formulas. In the dual sense, if we choose A=-O and F ,... u.1/t
then
E - -v X F
(3-87)
1 H - -gF + -v (v . F) f
(3-88)
Expanded in rectangular coordinates, this is E~ = -
a",
oy
H. =
1
ii'~
'i ax oz
E _ iJt
1I _ ~ ii"l-
E.
H. =
•
AX:
=
0
•
fayot
(3-89)
~(::s + kS)Vt
A field with no E. is called tromt'trse electric to z (TE). We shall find it possible to choose Vt sufficiently general to express any TE field in a homogenoous source-free region according to the above formulas.
SOME THEOREMS AND CONCEPTS
131
Now suppose we have a field neither TE nor TM. a. 'It according to
We can dctermine
8"/1' az'
+ k'," ~ "E 'f'1I'
which will generate a field TM to z according to Eqs. (3-86). This TM field will have the same E. as does the original field; so the difference between the two will be no TE field. We cnn therefore determine this difference field according to Eqs. (3-89), where the y, is found from
8'az'" + k'.'.' _ 'f'
!H
•
Thus, an arbitrary field in a homogeneous source-free region can be expressed as the sum of a TM field and a TE field. Explicit expressions for the field would be superposition of Eqs. (3-86) and (3-89), with superscripts a and f added to the y,'s to distinguish between them. Since the z direction is arbitrary, we can express this independent of the coordinate system by defining (3-90) A - c>/l' where c is a constant vector. which become E -
-v
H = V
X
The field is then given by Eqs. (3-79),
X (c>il)
(c>/l')
+~v y
+ 'j1 V X
X
V
X
(c>/l') (3-91)
V
X (C~f)
where the y,'s are solutions to Eq. (3-82). We must therefore study solutions to the scalar Helmholtz equation to Jearn how to pick the y,'s. If the region is not source-Cree but is still homogeneous, our starting equations are -V X E = ZH + M (3-92) vxH=yE+J instead of Eqs. (3-77). General solutions to Eqs. (3-92) call be constructed as the sum oC any possible solution, called a particular solution, plus a solution to the source-free equations, called a complementary solution. We already have a particular solution, namely, the po~ential in~e gral solu~ion of Sec. 3-2. ThereCore, solu~ions in a homogeneous region containing sources are given by
E - E~ + E"
H ~ H•• + H"
(3-93)
where the particular solution (pa) is formed according to Eqs. (3-4) and (3-5), and the complementary solution (cs) is constructed according to Eqs. (3-91). We can think of the particular solution as the field due to
132
TIME-BARlIONIC ELECTROMAGNETIC FIELDS
sources inside the region and the complementary solution as t.he field due to sources outside the region. 3-13. The Radiation Field. It is easier to evaluate tbe radiation (distant) field from sources of finite extent than to evaluate the near field. (See, for example, Sees. 2-9 and 2-10.) In this section, we shall formalize the procedure for specializing solutions to the ra.d.iation zone. Consider a distribution of currents in the vicinity of the coordinate origin, immersed in a homogeneous region of infinite extent. The complete 80lution to the problem is represented by Eqs. (3-4) and (3-5). If we specialize to the radiation zone (r r:.....). as suggested by Fig. 3-22, we have Ir - ['1-+ r - r' cos t (3-94)
»
where t is tbe angle between rand r'. Furthermore, the second term of Eq. (3-94) can be neglected in the Umagnitudc factors/' Ir - r/l-I, of Eqs. (3-5). It cannot, however, be neglected in the flphase factors,1I exp (-jkjr - ell), unless r~ «"-. Thus, Eqs. (3-5) reduce to A
~ : ' JJJ J(r')&"-'dT'
F -
~ JJJ M(r')&"'-'dT'
(3-9.)
in the radiation zone. Not-e tha.t we now ha.ve the T dependence shown explicitly. Many or the opera.tions or Eqs. (3-4) can therefore be performed. Rather thnn blindly expanding Eqs. (3-4), let us draw upon some previous conclusions. In Sec. 2-9 it was shown that the distant field or an electric current clement was esscntinUy outward-traveling plane waves. The same is true of a ma.gnetic current element, by duality. Hence, the
z
To distant
POi"1
field r - ,-
Source r'
r
Fro. 3-22. Geometry for evaluating the radiation field.
y
x
133
SOME THEOREMS AND CONCEPTS
z
Flo.
3~23.
,
Conventional
tlounIiull.W oricutu.tioll.
y
y
x
-------
radiation zone must be characterized by (3-96)
since it is a superposition of the fields from many current elements. We can evaluate the partia.l H field due to J according to H' =- V X A (see Sec. 3-2). Retaining only the dominant terms (r 1 variation), we ha.ve H~ =- (V X A), = jkA. H; ~ (V x A). - -jkA.
with E' given by Eqs. (3-96). Simila.rly, for the partial E field due to M, we have, in the radiation zone, E;' ~ - (V X Flo - -jkF. E~ = -(v X F). =jkF,
with R" given by Eqs. (3-96). The total field is the sum of these partial fields, or E, = -jwj.LA, - jkF. (3-97) E. = -jwj.LA. + jkF, in the radiation zone, with H given by Eqs. (3-96). Thus, no differentiation of the vector potentials is necessary to obtain the radiation field. Also, for future reference, let us determine r' cos ~ as a function of the source coordinates. The three coordinate systems of primary interest are the rectangular, cylindrical, and spherical, as illustrated by Fig. 3-23. For the conventional orientation shown, we have the transformations
x=rsin8cost/J y = rain 8ain t/J z=rcos8 To obtain r' cos
~,
X=PCOSt/J
y=-paint/J z= z
(3-98)
we form
rr' cos ~ = r . r' = xx'
+ yy' + zz'
(3-99)
134
TWE-HARMONlC ELECTROMAGNETIC FiELDS
Substituting for
X,
r' cos
Y. z from the first set of Eqs. (3-98), we obtain ~ =
(x' cos tP
+ v' sin 4»
ein
(J
+ %' C08 8
(3-11lO)
which is the desired form when rectangular coordinates are chosen for the source. Substituting into Eq. (3-100) for x', Vi, z' from the second set of Eqs. (3-98), we obtain
r' cos
~ =
p'ain Seos (41 - q/)
+ z' cos (J
(3-101)
which is the desired form when cylindrical coordin:ltes are chosen for the source. Finally, substituting into Eq. (3-100) for x'. y', z' from the first set of Eqs. (3-98), we have
r
cos E = r'[cos 8 cos 0'
+. sin 8 sin 8' cos (I/>
- ,p')]
(3-102)
wruch is the desired form when spherical coordinates arc chosen for the source. PROBLEMS
S-1. Show that a current sheet
J-
u.J,
over the: - 0 plane produces the out.ward-traveling plane waves
HI< -
- ~"-'-
,> 0
_,
{ -Tel .,.,. 'u
J
<0
in an infinite homogeneous medium. 3-2. Instead of the electric eurrent sheet, suppose that the magnetic current sheet
exi!lt.s over the ctOSll section: - 0 in the waveguide of Fig. 3-2. magnetic current :>roducea a field
,>0 ,<0
s-s.
Suppose now that the two current sheets A . ...
J,-u·Z,SIn"b M, - u r A sin
'7
Show that thiJl;
135
SOME TU.EOREMS AND CONCEPTS ~
exist. simultaneously over the Cf'06l leetion duce a field
0 of Fig. 3-2.
I: -
Show that. thcaa pro-
• >0
•<0 This BOurce is a "directional coupler." 3-4. In Fig. 3-2, suppolle that a "shorting plate" (conductor) is placed over the croaa section I: - -d. Show t.hat the current abeet of Eq. (3-2) now produces a field
J,z. rv _..... -- (1 -e-,•• - oJ' am-e''-
H. _
.> 0
b
2
{ -jJ.z.e-i~ sin T-sin 1tJ(d + %)1
-d
< I: < 0
Note t.hat wben d i. an odd number of guide quarter-wavelengths, E,. for Ii > 0 is twice that for the current sheet alone [aee Eq. (3-3)1. but when d is an integral number of guide haH-wavelengths. no HI erists {or I: > O. 3-6. The TE and TM modes o{ a parallel-plate wavcguido (prob. 2-28) arc almost dual to each other. Show that the field dual to the TE. mode of Prob. 2-28 is the TM. mode for the parallel-plate guide having conductors over the planes 11 _ b/2 and 11 - -b/2. Show that the field dual to the TM. modo of Prob. 2-28 is the TE modo of this new waveguide. 3-6. Obtain the field of an infinitc.llimalloop of magnetic current having z-direeted moment KS. Show that this produces the same field as the electric current clement of Fig. 2-21 if I
3-1. Figure 3-240 aho," the cfOlSllleCtion of a "twin·~o1ot" transmission line. Show t.hat the field distribution is dual to that of the eollinur plate line of Fig. 3-24b. By integrating along the eontours shown in Fig. 3-2-k, determine the line voltages and
~
E
lW
TT
H
c,' {
I"'~ '.
D
H
E
1
c,
cd \
'-
H
(oj
Flo. 3-24. Figures for Prob. 3-7. Iration contoUr!.
(b)
('J
(0) Twin-3lot line; (b) collinear plate line; (c) into-
136
TIME-HARMONIC ELEcrROMAGNETIC FIELDS
currents of both the 810t line and the plate line.
Show t.hat
7i>3~'~' (Z')NolUM - d(Z ) --:;-.. Ipta_U. From Table 2-3. it foUowa that. {Z.)ot.'lI.. - 4. log (WIlD)
D
»
UI
The two tranamission linea are said to be complementary struetUJ'C8 (Bee Babinct', principle, Seo. 7-12), 3-8. Show that the field I~
,> 0
,-,I-
,<0
.... JoZ• sm
E. -
2
{
J-:,
To!
ain
~
ia also Ii mathematical801ution to the problem of Fig. 3-2 with J. given by Eq. (3--2). What do our uniqueness theorems say about tbia l5CCOnd IlOlution? WhaL ean we say about it on physical grounds? Give a couple of other possible solutions to tbe
problem, and interpret them physically. 8-9. Show that the current sheela
II ] ... -til-a-I'. 4.. M.-
II -u.-eU
jh
1). +o· SUl'
(ik (l
(iwp." 1). ~+-+-.- ,m' a
at
J-a l
over the sphere r .. (l produce the field or Eqa. (2-.113) r > a and !loro field r < 4. 3-10. If E is well-behaved ill a homogenoous region bounded by S, and if fH - -v X &, show that the currente
J-
1
-~E-!VXVXE
will support this and only t.hi& field among a claas E, H having identieal lange.nti&! components of E on S. Show that the same H, but different H, can be obtained within this (ll&sfl if magnetic sources K are allowed in addition to J. 8-11. Suppose there exiBte within t.he rectangular cavity of Fig. 2-19 a field ,-rY'h E . - E ,Bln"bBln 'YJ:
where eAD be
V
(-rib)' ,tl and 1 is complelt (lOll8Y dielectric). Show that thia field supported by the lJOurce
'Y -
M, - -u.E,sin
'Z sinh
'YC
at the wall z - c. Show tbat for a low~loS6 dielect.ric, M. almost vanishes at the resonant frequency (Eq. {2-95)J, that iB, a small M. produces a large E. 3-U1. Consider a ~rected current element II a distance It in front of a ground plane covering the l' - 0 pla.ne, aa shown in Fig. 3-25. Show that the radiation field ia given by
E, _ -'Ill .-AI' ain 6 lin (kd sin • ain 6)
"
137
SOME THEOREMS AND CONCEPTS
and 1/110 _ H,. referred to / is
Find the power radiated and show that the radiation resist.a.nce R _ '1""11 [~ • A' 3
For d
_
sin 2kd _ cos 2A:d + sin 2kd] 2kd (2kd)l, (2kd)l
.s: k/4, the ffilH'imurn radiation is in the 1/ direction.
Show that
and that the gain is 7.5 for d small, 4.15 for d - A/4, and approximately 6 for d large.
z
,
• FlO. 3·25. Current element parallel to a ground plaoe.
II y
3-13. In Fig. 3-&, suppose we have a small loop of electric current. with ~irccted moment IS, instead of the current element. Show that the radiation field is given by '1 . (kd cos ' ) .' E • - ;'lj2...IS ~e-"8m Sln
and ?lB, _ -E•. referred to I is
Find tbe power radiated and show that the radiation resistance 2kd ('8)' [1"3 + cos(2kd)l -
R. - 2>01); For small d,
sin 2kd] (2kd)'
i'1""ISkd
. E. ~ - - - e-/" sm 28 u ..... O Air
<'
R. id.:O 15 ('8kd)' -,Thus, ma.ximum radiation is at 6 _ 45" for small d. The gain at small d is 15. For large d, the maximum radiation lies close to tbe ground plane, and the gain i.e 6. 3-14. In Fig. 3-25, suppose we have a small loop of electric current with ~irected moment IS, instead of the current element. Show that the radiation field i~ given by
E. _ I/ktlS e-i~' sin 8 cos (kd sin", sin 8) 2~
138 aod .,H, -
TIME-HARMONIC ELECTROMAGNETIC FIELDS -E~.
Show
R. _
'I'll
th~t
tho radiation resistance referred to / is
(OS)' [~ + sin 2kd + 008 2kd _ sin 2M] h 3 2kd (2kd)' (2kd)'
The maximum radiation is lLlong the ground plane, in t.ne z direction.
For smaJl kd
4,." (OS)'
R'u::QT X
which is twice that for the isolated loop. For d - 0, the gain is 3; for d - ),/4, it ill 7.1; and for d- "", it is 6. 8-1ri. The monopole antenna consists of a straight wire perpendicular to a ground plane, fed at the ground plane, 88 shown in Fig. 3-26. Show that the field is the same 118 that from the dipole antenna (Fig. z...23), fed at. the center. Show that the gain of the monopole is twice that of the corresponding dipole and that the radiation
resist.ance is one-half.
For example, the radiation resistance of the ),/4 monopole is
36.6 ohms.
FIG. 3-2G. The monopole antenna.
f(z.)
3·16. Consider an open-ended e08Jl:ial line (Fig. 3-14a without tho ground plane) of small radii a and b. Treat the problem according to the equivalence principle as applied to a surface just enclosing the coax. Assume n X II is C8S6ntially zero over the entire surface and that tangential E is that of the transmission-line mode over the open end. Show that to this approximation the radiated field is one-half that of Eq. (3-20) and that the radiation conductance is one-half that of Eq. (3-23). 3-17. A slot antenna consist" of a slot in a. conducting grolWd plane, as shown in Fig. 3-27. It is coiled a dipole alot antenna when fed by a voltage impressed across the center of the slot. •The slot and ground plane can be viewed as flo transmission line, and the field in the slot will be essentially a harmonic function of kz. AS8ume
in the slot, and obtain the magnetic current equivalent of the form of Fig. 3-1:k. For w small, show that this equivalent representation is the dual problem to the dipole antenna of See. 2-10. Uaing duality, abow that the radiation field is
jv..e_weoS(k~COS~) ~
-C08(k~) _{
Denne the radiation conductance of this antenna Il8 O. (0) • 010' dl .... l
H,
-H.
mn8
4(R.)'rindlpol• ,,'
(}IdlY.1
y>O
y
and show that
139
SOME THEOREMS AND CONCEPTS
z
,
/r I I I
Flo. 3-27. A slot. antenna..
I
where R. is as plotted in Fig. 2-24. The input voltage Vi is related to V .. by Vi - V.. Bin (kL/2); 80 the input conductance is given by
8-18. For the antenna of Fig. 3-27, lUlaumo
E~
in the slot the same as in Prnb. 3-17,
and show that for arbitrary to
JVMe- i .' 'Iff
where
{I'. oJ
f«(J,~) -
{He
-lit
II> 0 11
<0
.;n (k ~ ". 0'in.) [00' H 00") - 00' (k ~)] w.
k 2 coss1II9
61116
3-19. Figure 3-28 shows an'aperture o.nUmna consistillg of a rectangular waveguide opening onto 8. ground plane. Assume that E,. in the aperture is that of the TE'1
z
, FIG. 3-28. A rectangular waveguide opening onto
a ground plane.
y
140
TUdE-HARMONIC ELECTROMAGNETIC FIELDS
waveguide mode, and show that the radiation field is
H• ... 2jbEte-/ b sin ( k ~ COB'; sin 8) cos ( k ~ C08 8) 'IT C09 4> (,..1 (kb COS 8)11 3-20. Figure 3·29 represents a rectangular condullting plate of width a in the direction and b in the .li direction. Let the incident plane wave be specified by
Use the induction theorem with the aame
appr~ximation as
~
waa used in the problem
r r
FlG. 3-29. Scattering by a rectangular plate.
of Fig. 3-17, and &how that at large E ,'...
Show that the echo
T
the .scattered field in the zy plane is
+ +
kED/lbll-lv sin (k(a/2)(sin .; sin 4Jo)J j21fT k(a/2)(Bio tP sin .po) cos 4>
STelL
is
A• ... 4:.- [ab eos 4>0 si~ (ia sin ,po)]' XkaSln 4>0
3-21.
~pcat
Prob. 3-20 for the ort.hogonal polarization, that is,
and show that. at large T t.ho scattered field in the :cy plane is H
" ...
+ +
jiB .,abe- jlr sin Ik{a/2) (sin 4> sin 4>0) J 2,..,. k(a/2) (sin. sin fla) cos "'a
Show that the eeho area is the same as obtained in Prob. 3·20. 3-22. Usc rcdprocit.y to evaluat.e t.he radiation field of the dipole antenna of Sec. 2-10. To do this, place n 9-direeted current element at. large r, tIond apply Eq. (3-36), obtaining Eq. (2-125). 3-23. By applying voltage lIOurces t.o the network of Fig. 3-18. show that the admittance matril( luI defined by
_["" "u] [v.] [ I,] It 1111 lin VI satisfies the reciprocity relationship Un - ~u when Eq. (3-38) is valid.
141
SOME THEOREMS AND CONCEPTS
Flo. 3-30. lIClatt.ering.
Differential Obstacle
3·24. Let Fjg. 3-30 represent two antennas in the presence of an obst.acle.
Let
VI be the voltage received at antenna 1 when a unit current source is applied at
antenna 2 and V, be the voltage received at antenna 2 when & unit current source is applied at antenna 1. Let VI' and V,I be the corresponding voltages when the obstacle is absent. Define the scattered voltages as
and show that V,' _ V t·. 3·2li. For the problem of Fig. 3-2, define the input impedance of the sheet of current as
z- -(0,0) p where (0,0) is the self-reaction of the currents and I is the total current of the sheet. Evaluate Z when the field is given by Eqs. (3-3). 3-26. Repeat. Prob. 3-25 lor the current sheet and field of Prob. 3--1. 3-27. In the vector Green's t.heorem (Eq. (3-16) I, let A_E· and B - E~ in a homogeneous isot.ropie region, and show that it reduces to Eq. (3-35). 3-28. Use the vector identity v . (A X
1ft
-fff
(B' v X
Let A - E·, B _ E·,
'JY •
-
G t X V X E +EV'GI)'ds
, 4",c·E
If f'I.....O
142
TIME-HARMONIC ELECTROMAGNETIC FIELDS
3-30. Let G, be the magnetic field of a z-directed current element situated 11 > 0 and radiating in the prescnce of a perfect electric conductor covering the 11 "" 0 plane. In other words, let c - u. and S be the y co 0 plane. Show that
where
fl ClO ft -
Vex
vex
+ (11 y')' + ell x')! + (11' + Ill' + (z x'P
il' )1
z')'
3-31. Specialize the G. of Prob. 3-30 to fl - 00, and apply Eq. (3-57) to the problem of Fig. 3-28. Show that this gives the same answer as obtained in Prob. 3-19. 3-32. Apply duality to Eqs. (3-65), and evaluate the magnetic tenoor Green's function 11') defined by H - [rlKI in free space. S-33. Evaluate the I'll for the free-space tensor Green's function defined by
H - [rill 3-34. Repeat Prob. 3-20 using the physical optics approximation, and show that the answer for E,' differs from that of Prob. 3~20 by an interchange of 4i and 4Jo. Show that the echo area is identical to that of Prob. 3~20. 8-35. Repeat Prob. 3~21 using the physical optics approximation, and show that the &nswer for H ," differs from that of Prob. 3-21 by an interchange of 4J and 4Jo. Show that the echo area is identical to that of Prob. 3·2l. 8-36. Let", ... e- /iw in Eqs. (3~86), and evaluate the electromagnetic field. Classify this field in as many ways as you can (wave-type, polarization, etc.). 3-87. Let of ." e- ib in Eqs. (3·89), and evaluate the electromagnetic field. Classify this field in B.8 many waye B.8 you can. 8-38. Let c - U z , "'. - r il.., "" .. je- ii" and evaluate Eqs. (3-91). Classify this field in as many ways as you can. S-39. Derive Eqs. (3~97) by c.'\:panding Eqs. (3-4) with A and F as given by Eqs. (3-95).
CHAPTER
4
PLANE WAVE FUNCfIONS
4-1. The Wave Functions. 'l'he problems that we have considered 80 far are of two types: (1) those reducible to sources in an unbounded homogeneous region, and (2) those solvable by using one or more uniform plnne waves. Equations (3-91) show us how to construct general solutions to the field equations in homogeneous regions once we have general solutions to the scalar Helmholtz equation. By a method called 8sparation of variabw8, general solutions to the Helmholtz equation can be constructed in certain coordinate systems. I In this section, we use the method of scparation of variables to obtain solutions for the rectangular coordinate system. The Helmholtz equation in rectangular coordinates is (4-1)
The method of separation of variables seeks to fInd solutions of the form
f - X(x) Y(y)Z(,)
(4-2)
that is, solutions which are the product of three functions of one coordinate each. Substitution of Eq. (4-2) into Eq. (4-1), and division by,p, yields (4-3)
Each term can depend, at most, on only one coordinate. Since each coordinate can be varied independently, Eq. (4-3) can sum to zero for all coordinate values only if each term is independcnt of x, y, and z. Thus, let
where k", kll , and k. are constants, that is, are independcnt of x, Y, and z. (The choice of minus a constant squared is taken for later convenience.) lIt hM been shown by Eisenhart (Ann. Mmh., vol. 35, p. 284, 1934) that the Hclmholh equation is separable in 11 three-dimensional orthogonal coordinate systems.
143
144 We
TnIE-RARMONIC ELEcrnoMAGNETIC FIELDS DOW
have Eq. (4-1) separated into the trio of equations
d'X dx t d'Y
dy' 2
d Z2 dz
+ k 'X _
0
Z
+ k,'Y _
0
+ k!Z
0
=
(4-4)
'
where, by Eq. (4-3), the separation parameters must satisfy
k. 2
+ k,,2 + k.
t
=
kt
(4-5)
This last equation is called the separation equation. Equations (4-4) arc all of the same form. They will be called harmonic equatiom. Any solution to the harmonic equation we shall call a harmonic junction,! and denote it, in general, by h(kzx). Commonly used harmonic functions arc (4-6)
Any two of these arc linearly independent. A constant times a harmonic function is still a harmonic function. A sum of harmonic functions is still a harmonic function. From Eqs. (4-2) and (4-4) it is evident that
f •.•••• - h(k..)h(k,y)h(k.,)
(4-7)
are solutions to the Helmholtz equation when the Ie; satisfy Eq. (4-5). These solutions llre called elementary wave functions. Linear combinations of the elementary wave functions must also be solutions to the Helmholtz equa.tion. As evidenced by Eq. (4-5), only two of the lei may be chosen independently. We can therefore construct more genera.l wave functions by summing over possible choices for one or two separa.tion parameters. For example,
f ~
LLB••••f ...... .1:. .1:.
- LL
R••••h(k..)h(k,y)h(k.z)
(4-8)
.1:. k.
where the B,j are constants, is a solution to the Helmholtz equation. The values of the k, needed for any particular problem are determined by the boundary conditions of the problem and are called eigenvalues or characteristic values. The elementary wave functions corresponding to specific eigenvalues are called eigenfunctions. I The term harmonic function also is used to denote a rrolut.ion to Laplace's equat.ion. This is not the present meaning of the term.
PLA.NE WAVE FUNcrION8
145
StiU morc general wave functions can be constructed by integrating over one or two of the k a• For example, a solution to the Helmholtz equation is
~~
IIf(k.,k,)~.,•••• dk. dk,
-!l k. 01:.
f(k.,k,)h(k.;<)h(k,y)h(k..) dk. dk,
(4-9)
where f(k",Ie,,) is an analytic function, and the integr3.tion is over any path in the complex k", and Ie" domains. Equation (4-9) exhibits a continuous va.riation of the separation parameters, and we say that there exists a continuous spectrum of eigenvalues. We shall see that solutions for finite regions (waveguides and cavities) a.re characterized by discrete spectra of eigenv3.1ues, while solutions for unbounded regions (antennas) often require continuous spectra. Wave functions of the form of Eq. (4-9) are most commonly used to construct Fourier integrals. We should be familiar with the mathematical properties and with the physical interpretations of the various harmonic functions so that we can properly choose them for particular problems. Keep in mind that wave functions represent instantaneous quantities, according to Eq. (1-4.0). Solutions of the form h(kx) = r ih (k positive real) represent waves traveling unattenuated in the +z direction. If k is complex and Re (k) > 0, we have +z traveling wa.ves which are attenuated or augmented according as 1m (k) is negative or positive. Similarly, solutions of the form h(kx) = fib, (Re (k) > OJ represent -z traveling waves, attenuated or augmented if k is complcx. If k is purely imaginary, the above two harmonic functions represent evanescent fields. Solutions of the form h(kz) = sin kz and h(kx) = cos kx with k real represent pure standing waves. If k is complex, they represent localized standing waves. If k is purely imaginary, say k = - ja with a real, then the" trigonometric functions" sin kx and cos kx can be expressed as Hhyperbolic functions" sinh ax and cosh ax. We should get used to thinking of the various functions as defined over the entire complex kx plane. The trigonometric and hyperbolic functions are then just specializations of the complex harmonic functions. Table 4-1 summarizes the above discussion, (The convention k = fJ - ja with a and fJ real is used.) Note that the degenerate caso k = 0 has the harmonic functions h(Ox) = 1,%. The choice of the proper harmonic functions in any particular case is largely a matter of experience, and facility in this respect will be gained as we use them. 4-2. Plane Waves. Consider an elementary wave function of the form (4-10)
146
TIME-H.ARMONIC ELECTROMAGNETIC FIELDS TABLE 4-1. PaoPERTIES or TIn: HARMONIC FUNCTIONS'
M.b\
Ze~t
Sped.lil... tions of
I!lfi .. ltl~t
Speei_\ l"CJ)r<:eentatlollll
.\:,,6-1<>
i" ..... - j .
h .....
,flo
.h: ..... J .
b ..... -i-
.ink..
COlI ..,,,
,-I/f.
Ie !"(lal
,-I',
f_
A: Im'fri""Y .l: complex .I: rn.1 1 imacinary k eomplu
..., ..... ±i-
k,,'" " ..
b-(n+H)r
k" ..... ±j..
.....-". ,-a• -
_.
+z tT.vclinli: wave Evane.eent field AtUnu .. ted traveling ..II",
,Ie.
-::0:
... ·.ifJ.
Evanucent licld AtteDulUed trlveli", .... n
.l: real k Im..;"....y
lin 11%
.\: complex
.in fb lotIh .. " - i (OfIB.. ainh .."
.l:rnl k imlll'inary
eo. 8% co.h ..% COlI It" cosh ..:II
k\lOmplu
Ph)'.i~a1
interpretatiDa
-j
'inll oJ:
ttaveLin, .......
SUndinl ......e Two lI".IlNCC'" field. Loc&li.ed atandin...,Vel
StIDdi.., ... ave
Two
ev.neacen~
fielde
LocalilO8d 'landi"l WIIVeI
+i,in(jr,inh .."
• For .t ... 0, the harmonic (unetlollll
t
For I"
&l"$
1I(0z) ... t,,,,
_"ti.... Bln.ula.ity. ~his columD &ive. ~h" ..ympwu" behavior.
The Ie. must satisfy Eq. (4-5), which is of the form of the scalar product of a vector (4-11) k = u"k", + uwk.. + u.k. with itself.
Note that in terms of k and the radius vector r=u...x+uwY+u.z
(4-12)
we can express Eq. (4-lO) as
y,
= eft·,
(4-13)
For k real, we apply Eq. (2 140) and determine the vector phase constant ~ ~ -V( -k· r) ~ k M
Hence, the equiphase surfaces are planes perpendicular to k. The amplitude of the wave is constant (unity). Equation (4-13) therefore represents a scalar uniform plane wave propagating in the direction of k. Figure 4-1 illustrates this interpretation. For k complex, we define two real vectors k =
~
- j.
(4-14)
and determine the vector propagation constant a.ccording to Eq. (2-145). This gives y = -v(-jk.r) =jk +j~
=.
We now have equiphase surfaces perpendicular to lJ and equiamplitude
147
PLANE WAVE FUNCTIONS
Z
Equiphase surface
Direction of propagation
, /
Flo. 4-1. A uniform plaDe
/
w ..vc.
y
x surfaces perpendicular to a. Thus, when k is complex, Eq. (4-13) represcnt.s a plane wave propagating in the direction of IJ and attenua.ting in the direction of a. It is a uniform plane wave only if ~ and a: are in the same direction. Note that definitions k """ lJ - ja and k "'"' It - jk" do not imply that {J equals k' or that a equals k" in general. In fact, for loss-free media,
a' -
k' = k . k "'"' {J' -
j2a . \}
must. be positive real. lIenee, either 0: = 0 or a: • I) = O. When a: "'" 0 we have the uniform plane wave discussed above. When a: and 0 are mut.ually orthogonal we have an evanescent field, such as was encountered in tot"l reflection [Eq. (2-62)J. The elementary wave functions of Eq. (4-10) or Eq. (4-13) are quite general, since sinusoidal wave functions are linear combinations of the exponential wave functions. Wave functions of the type of Eqs. (4-8) and (4-9) are linear combinations of the elementary wave functions. We therefore conjecture that all wave functions can be expressed as superpositions of plane waves. Let us now consider the electromagnetic fields that we can construct from the wave functions of Eq. (4~1O). Fields TM to z are obtained if 'It is interpreted according to A = ua,p. This choice results in Eqs. (3-86), which, for the 'It of Eq. (4-10), become H - - uJk'" =
and
v'It X Ua
+ u,ik'" =
Nu,
(4-15)
X k
+ u,ik. + uJk.)~ + n.k'>/+ u.k')~
UE - jk.{uJk. - (- kJ<
(4-16)
For k real, H is perpendicular to k by Eq. (4-15), and E is perpendicular to k, since
Uk· E - (-k.k'
+ kJ<')~ -
0
148
TIME-HARMONIC ELECI'ROMAGNETIC FIELDS
Thus, the wave is TEM to the dirootion of propagation to z). For k complex, define a: and (,\ by Eq. (4-14). that the wave is not necessarily TEM to the direction (that of (,\). It will be TEM to {.\ only if a and (,\ are in tion, that is, if k = l} - ja = (uJ + ulJm u.n)k
(as well as TM It then follows of propagation the same direc-
+
with l, m, n real. In this case, fJ = k', a = kif, and l, m, n are the direction cosines. The dual procedure applies when'" is interpreted according to F = u."'. In this case, Eqs. (3-89) apply, giving E~#kxu.
IH ~ (-k.k
(4-17)
+ u.k')f
which are dual to Eqs. (4-15) and (4-16). For k real, this is n. wave TEM to k and TE to z. Its polarization is orthogonal to the corresponding TM-to-z wave. For k complex, the wave is not necessarily TEM to the direction of propagation. All these fields are plane waves. An arbi. trary electroma.gnetic field in a homogcneous region can be considered as a superposition of these plane waves. 4-3. The Rectangular Waveguide. The problem of determining modes in a rectangular wa.veguide provides a good illustration of the use of elementary wave functions. In Sec. 2-7 we considered only the dominant mode. In this section we shall consider the complete mode spectrum. The geometry of the rectangular wa.veguide is illustrated by Fig. 2-16. It is conventional to classify the modes in a rectangular waveguide as TM to z (no H a) and TE to z (no E.). Modes TM to z are expressible in terms of an A having only a z component ¥t. We wish to consider traveling wavesj hence we consider wave functions of the form
f
~
h(k.x)h(k.y),-i',·
(4-18)
The electromagnetic field is given by Eqs. (3-86). E. -
~ (k'
In particular,
- k.')f
The boundary conditions 011 the problem are that tangential components of E vanish at the conducting walls. Hence, E. must be zero at x = 0, x = a, Y = 0, and y = b. The only ha.rmonic functions having two or more zeros are the sinusoidal functions with k i real. Thus, choose
.
siu k,x
k --~ a
m = 1,2,3, .
h(k,ly) ;;;; sin k,ly
k=~ • b
n
h(k,x)
~
= 1,2,3, .
PLANE WAVE J'UNC'I'lONS
149
50 that the boundary conditions on E. are satisfied. Each integer m and " specifies a. possible field. or mode. The TM... 7rWCU funclicm are therefore .1.
-
.,....'nI:
= sm
mn . nry
a
81D
b e iIt"
(4-19)
with m .. I, 2, 3, ...• a.nd n "'" I, 2, 3•... ,and the separation pal"11m~ eler equation {Eq. (4-5)] becomes
(m;)' + (n;)' + k.' _ k'
(4-20)
The TM.... modo fields are obtained by substituting the l/t.. "TM into Eqs. (3-86).
Modes TE to z are expressible in terms of an F having only a z component J/I. Again. we wish to find traveling waves; so the If must be of the form of Eqs. (4-18). The electromagnetic field this time will be given by Eq•. (3-89). In particular,
E--~ • ay the first of which must vanish at y "'" 0. y '"'" b, and the second at % = 0, a. Harmonic functions SiLtisfying these boundary conditions are
% -
h(k..) - co. k..
.-
k -~ n k _ n...
,
b
m=O.1.2. n=0,I,2,
Each inl.cgcr fn and n. except m = n "'" 0 (in whieh case E vanishes identically), specifies a mode. Hence, the TE.." mode functions are
If.nTE
= cos
m'lr% nry ., a cos be-' ,-
(4-21)
with m "'" 0, 1,2, . . . ; n = 0, 1.2, . . . ; m = n "'" 0 excepted. The separation parameter equation remains the same as in the TM MSC [Eq. (4-20)]. The TE... mode fields are obtained by substituting the If.. "TB into Eq•. (3-89). Interpretation of each mode is similar to that of the dominant TE ol mode, considered in Sec. 2-7. Equation (4-20) determines the mode propa.gntion const.nnt y "" jk.. For k real, the propagation constant vanishes when k is
~("'aT)' + (";)' =
(k.)••
(4-22)
ISO
TlM'E-BAlUlONIC ELECTROMAGNETIC FIELDS
The (k.)•• is called the ctdoff wave number of the mn mode. values of k, we have , ••
~ jk. ~
I
j~ - j yk' a =
(k,)••'
For otber (4-23)
(k.)." - k'
Thus, for k > k. the mode is propagating. and for k < k. the mode is nonpropagating (evanescent). From Eq. (4-22) we determine the cutoff frequencies (j,) •• - 2.
~'" - 2 ~ ~(~)' + (~)'
(4-24)
and the cutoff wavelengths 2. 2 (Xc)... = -k = /T.:7:'W'F'E:iill • y(mla)' (nib)'
+
(4-25)
In terms of the cutoff frequencies, we can ro-oxpress the mode propagation constants as .., "'"' jk. =
l
j~
-jk
a"" kc
(J)' - (J.)'
1 -
I> I• (4-26)
1
where mode indices mn are implied. We can also define mode wavelengths (or each mode by Eq. (2-85) and mode phase velocities by Eq. (2-S6), where mode indices are again implied. It is apparent that "Y = jk. for each mode has the same interpretation as "Y for the TEol mode. It is the physical size "(compared to wavelength) of the waveguide that determines which modes propagate. Table 4-2 gives &. tabulation of some of the smaller eigenvalues for various ratios bla. Whenever two or more modes have the same cutoff frequency, they are said to be degenerate modes. The corresponding TE..... and TM.... modes nrc always degenerate in the rectangular guide (but not in othershaped guides). In the sqU::Lre guide (bla = 1), the TE...., TE..., TM...., and TM.... modes form &. foursome of degeneracy. Waveguides are usually constructed so that only one mode propagates, hence b/a > 1 usually. For b/a - 2, we have a 2:1 frequency range of single-mode operation, and this is the most common practical geometry. It is undesirable to make b/a greater than 2 for high-power operation, since, if the guide is too thin, arcing may occur. (The breakdown power is proportional to for fixed b.) To illustrate the use of Table 4-2, suppose we wish to design an air-filled waveguide to propagate the TEol mode at 10,000 megacycles (>" 3 centimeters). We do not wish to operate too close w I~t since the conductor losses are then large (see Table 2-4). If we take
.yo:
:c1
151
PLANE WAVE FtiNCTJONS TABLE 4-2. (1;,)... _
(k.h.
-•b
(J~) ... _ (>...).1 (/.).. ("-)... TEll
FOR TIlE
RrCUNOULAlt
TE u TM II
TEll TM l1
TEn TM n
2.236 3.162 4.123
2.828
4
2.236 2.500 2.828
6
3.606
6.083
6.325
•
•
•
•
TE••
TEll
TM II
TE H
TEll
I 1 1 1 1
1 l.5
1.414 1.803 2.236 3.162
2 2 2 2 2
2 3
1
1.5 2 3
•
2 3
•
•
,
W.\VEQUlDE, b
3.606 4.472
>c -
TEll 3 3 3 3 3
b = 2 centimelers, then}... - 4 centimeters for the TEol mode, and we are operating welt above cutoff. The next modes to become propagating are the TE IO and TEn: modes, at. a frequency of 15,000 megacycles. The TEll and TM l l modes become propagating at 16,770 megacycles, and so on. The mode pa.tterns (field lines) a.re also of interest. For this, we determine E and H from Eqs. (3-86) and (4-19) or Eqs. (3-89) and (4-21), and then determine S, :JC from Eq. (1--41). The mode pattern is a plot of lines of t and 3C at some instant. (A more direct procedure for obtaining the mode patterns is considered in Sec. 8-1.) Figure 4-2 shows sketches of cross-sectional mode patterns for some of the lowCI'-'Oroer modes. When a line appears to end in space in these patterns, it actually loops down the guide. A more complete picture is shown for the TEo l mode in Fig. 2-17. In addition, each mode is chnrocterized by a constant (with respect to
(a)
TEo1
(c) TMIl
(b) TEn
~~' , '/
(d) TEo,
...
,,'-
..
",
.
'-' ' ... ~"r:f-
(e) TEI2
e
)Io.!J( -
- - - -;.,..
Flo. 4-2. Rectangular waveguide mode patterns.
152
TIME-HARMONIC ELEC"rROMAONETIC FIELDS
y) z-directcd wa.ve impedance. For the TE•• modes in loss-free media, we h.ve from Eq•. (3-89) .nd (4-21)
2:,
·H '"
]WIJ.
"'"
-1·k8~ -. - = -1·k-.0, P
8•
. H .. = -1·kI 8~ fJy == J·kE ••
JWIJ
The TEa. characteristic wave impedances are t.herefore
f > f. (4-27)
f < f. Similarly, for the TM•• modes, we h:lve from Eqa. (3-86) and (4-19)
j~.
-
-jk.: = jk.N,
jwtElI = -jk.
:t
= -jl.,.ll,.
Thus, the TM... characteristic wave impedances are
(Z) ~ 0 ••
E. H"
-E. 11.
k'-l~-.-
-------Wf
a JW'
f > f. (4-28)
f < f.
It is interesting to Dote t.hat the product (Zo}••T&(Zo}...n1 = 'It at all fr~ quencies. By Eq. (4-26), P < k for propagating: modesj so the TE characteristic wave impedances are always greater than 'I, and the TM characteristic wave impedances are always less than '1. For non propagating
modes, the TE characteristic impedances aro inductive, and the TM characteristic impedances are capacitive. Figure 4-3 illustrates this behavior. Attenuation of the higher-order modes due to dielectric losses is given by the same formula as for the dominant mode (see Table 2-4). Attenuation due to conductor losses is given in Prob. 44. 4-4. Alternative Mode Sets. The classification of waveguide modes into sets TE or TM to z is important because it applies also to guides of nonrectanguln.r cross section. However, for many rectangular waveguide problems, more convenient e1a.ssifications can be made. We now consider these alternative sets of modes. If, instead of Eq. (3-84), we choose (4-29)
153
PLANE WAVE FUNCTIONS
\ Zo
1\
t > t, ~ {Ro "jXo t < t,
\RoTE
"
, _Ko TAt
/
X,TE/
1/
/
"-
':;;;,TM
\
1/
/
o
3
2
I
FlO. 4-3. Chnracteristic impedance of wfl.vcguide modes.
we have an electromagnetic field given by a set of equations differing from Eqs. (3-86) by a cyclic interchange of x, y, z. To be specific, the field is given by H. - 0
II ~
,
a.a,
(4-30)
a;, Hr=.-• ay This field is 'I'M to x.
Similarly, if, instead of Eq. (3-87), we chooso (4-31)
we have an electromagnetic field given by
E __
,
E _
•
a.
ay
a.az
(4-32)
154
TUdE-HAR.YONIC ELECTROMAGNETIC FIELDS
This field is TE to x. According to the concepts of Sec. 3-12, aD arbitrary field can be C!'nstructed as a superpositioo of Eqs. (4-30) and (4-32). The choice of ¥'s to satisfy the boundary conditions for the rectangular waveguide (Fig. 2-16) is relatively simple. For modes TM to z (TMx•• modes) we have (4-33)
where m - 0, 1, 2, . . . ; n = 1, 2, 3, ' .. ; and k. is given by Eq. (4-26). The electromagnetic field is found by substituting Eq. (4-33) into Eqs. (4-30). For modes TE to x (TEx..... modes) we have (4-34) where m - 1,2,3, . . . ; n = 0, 1,2, . . . ; and k. is again given by Eq. (4-26). The field is obtsinOO by substituting Eq. (4-34) into Eqs. (4-32). Note that the TAu.. mOOes are the TEo. mOOes of Sec. 4-3, and the TEx. o modes are the TE.o modes. All other modes of Eqs. (4-33) and (4-34) are linear combinations of the degenerate sets of TE and TM modes. Note that our present set of modes have both an E. and H. (except for the O-order modes). Such modes are called hybrid. The mode patterns of these hybrid modes can be determined in the usual manner. (Determine E, H, then S, :JC, and specialize to some instant of time.) The TEa.. o mode patterns are those of the TE.. modes, and the TMxo.. mode patterns are those of the TEo. modes. Figure 4-4 shows the mode patterns for the TF.a: u and TMxll modes, to illustrate the character of the higher-order mode patterns. The characteristic impedances of the hybrid modes are also of interest. For the TM% modes, we have from Eqs. (4-30) and (4-33) H, - -jk,~
Hence, the z-directed wave impedances are
f > f. (4-35)
f
, ..
1:0
,
t
~~
'!I
\
A
I
I
-
P~E
x
\ ,\
~:::
(a)
'~II
TExu
~~~S§i
r-TTTT-' -111
155
WAVE FUNCfIONS
x
bY'/'-l>'1
~
Iif
1 1 \ \,
11
I'
/
--~---~--_
....
h (b) TAbu
PSG. 4-4. Hybrid mode pat.terns.
Eqs. (4-32) and (4-,'34) we find
Z) ~ (
0.. -
- E.
II%.
:;z
kt
~.k. (m,../a)'
=-
!
~.P
k' (m. /a)' T..--~I>:~=.";a,-,, kt (m..';a)'
f> f. (4-36)
f < f.
Note that for a small, the cutoff TEx.." modes all have inductive characteristic impedances. Sets of modes TM a.nd TE to Y can be determined by letting A = U~Vt and F = U1/1/I, respectively. The fields would be given by equations similar to Eqs. (4.30) and (4-32) with x, y, z properly interchanged. The TMy and TEy mode functions would be given by Eqs. (4-33) and (4-34) with .",,/a ond ny/b interchanged, 4-'S. The Rectangular Cavity. We considered the dominant mode of the rectangular cavity in Sec. 2-8. We shall now consider the complete mode spectrum.. The geometry of the rectangular cavity is illustrated by Fig. 2-19.
The problem is symmetrical in x, Y. %j SO we can express the fields as TE or: TM to anyone of these coordina.tes. It is conventiona.l to choose the z coordinate, and then the cavity modes arc standing wa.ves of the usual TE and TM waveguide modes. The wave functions of Eq. (4-19)
156
TIME-HARMONIC ELECI'ROMAGNETIC f'TELD8
satisfy the boundary condition of zero tangential E at four of the walls. It is merely necessary to rcpick h(k.z) to satisfy this condition at the remaining two walls. This is evidently accomplished if ~TK = "'''1'
with m = 1,2,3,
..
j
n
.
m'll'Z
.
n-ry
SID - - SID -
p-rz
(4-37)
COB -
abc
; and Eq.
1,2,3, . . . ; p - 0, 1,2,
=
(4-20) becomes (4-38)
The field of the TM••,. mode is given by substitution of Eq. (4-37) into Eqs. (3-86). Similarly, the TE••JI' mode fUDctioDS Brc given by ,',Till:
"" • •1'
=
mrX nry. prrz a bsm c
(4-39)
C08 - - COS -
with m = 0,1,2, . . . in = 0, I, 2, . . . ; p "'" 1,2,3, im = n - 0 excepted. The separation equation remains Eq. (4-38). The TE....,. mode field is given by substitution of Eq. (4-39) into Eqs. (3-89). AB indicated by Eq. (4-38), each mode can exist at only a single k, given a, b, c. Setting k = 2r/ W. we solve Eq. (4-38) for the resonant frequencies (4-40)
For a < b < c1 the dominant mode is the TE oJ1 mode. Table 4-3 gives the ratio (j.) ...p/(J.)oll for cavities of various side lengths. Note that
TABLE
,
-b -
• • I
1
1
2 2 4 4
1 1 1 1 I 1
8 16
(fr)tlI
TEln TE II1 TM no
1
2 2 4 4 4
(J.)",.. 4-3. -
Fon TilE
TM III TEl. TE,ulT8tll TEl" TE III
1
1.22
1.26
1.34
1.58 1.58 1.84 2.00 2.91 2.91 3.62 3.65 3.88 4.00
2.05 3.00 3.66 4.01
I 1
R ECTANC':t!LAn C ,,"VITT, a
1.73
-1.58 -1.58 1.58 1.26 1.58 1.26 1.58 1.26 1.08
1.84 1.58 1.84 1.58 1.84 1.96
1.84 2.91 3.60 5.71 7.20
1.58
1.26 2.00 2.00 3.16 3.65 7.76 3.91
-
TM I11
-
TM m TM m
TE llt
-1.58 2.00 2.00 2.53 3.16 4.03 4.35
1.58 2.00 2.91 3.68 5.71 7.25
7.83
1. 73 1.55 2.12 2.19 3.24 3.82 4.13
157
PLANE WAVE FUNCTIONS
the TE"'''10 and TM..... 10 modes, mnp all nonzero, are always degenerate. When two or more sides of the cavity are of equal length, still other degeneracies occur. The greatest separation between the dominant mode and the next lowest-order mode is obtained for a. square-base cavity (b = c) with height one-half or less of the base length (b/a ~ 2). In this case, the second resonance is = 1.58 times the first resonance. The mode patterns of the rectangular cavity are similar to those of the TE or TM waveguide modes in a. z = constant plane, and similar to the hybrid mode patterns in the other two cross sections. The most significant difference between the waveguide patterns and the cavity patterns is that E is shifted from JC by 'A Q/4 in the latter case. Also, 8 and :JC are 90° out of phase in a cavity; so E is zero when :JC is maximum, and vice versa. The TEoll mode pattern is shown in Fig. 2-20. To illustrate higher-order mode patterns, Fig. 4-5 shows the TE 123 mode pattern. The quality factor Q of each cavity mode can be determined by the method used in Sec. 2-8 for the dominant mode. The Q due to dielectric losses is the same for all modes, given by Eq. (2-100). The Q's due to conductor losses for the various modes are given in Prob. 4-10. Note that the Q increases as the mode order increases. The Q varies rougWy as the ratio of volume to surface area of the cavity, since the energy is
v%
,
-I 000 000
o
Section A
r
,""
, " 0 0 0
..,•
a
Lgo Section C
"",
Section B
FIG. 4.-5. Rectangular ca.vity mode pattern for the TE iU mode.
158
TIW.E-llAlU:IONIC ELECl'RO!ol.AONETIC FIELDS
x
z
Flo. 4-6. A pa.rtially di· electrio-filled rcclanlU1ar waveguide.
.1"--,...----( d
o
y
b
stored in the dielectric and the losses are dissipated in the conducting walls. 4·6. Partially Filled Waveguide. l Consider a waveguide that is dielectric filled between x = 0 and:z; .. d (or has two dielectrics). This is illustrated by Fig. 4-6. The problem contains two homogeneous regions, o < :r; < d and d < z < a. Such problems are solved by finding solutions in each region such that tangential components of E and H are continuous across the common boundary. An attempt to find modes either
TE to % or TM to % will prove unsuccessful, except for the TE..o case. Most modes arc therefore hybrid, having both E. and B.. An attempt to find modes TE or TM to % will prove successful, as we now show. For fields TM to x. we choose ""s in each region (region 1 is x < d, region 2 is x > d) to represent the x component of A, as in Eq. (4-29). The field in terms of the vis is then given by Eqs. (4-30). To satisfy the boundary conditions at the conducting walls, we take .•. C1 cos k,,1X sIn . -bn'lrtJ ....Ii " 'f'1 = (4-41)
1/It "'" Ct eos [kd(a
- x)] sin n;y e-it ••
with n I, 2, 3, . . .. It has been anticipated that Ie" nff/b and k. must be the same in each region for matching tangential E and H at x = d. The separation parameter equations in the two regions are CII
:IC
k£l t
k ot
1
t
+ (nT)' b + k. t
-
+ (bnT)' + k. t -
kit",", ",tEJJ.ll
(4-42) ktt
=-
",tEU't
L. Pincberle, Electromagnet.ic Waves in Metal Tubes ruled Longitudinally with 00. 5, pp. 118-130, 1944.
Two Dielectrics, Phy•. RG., vol. 66,
159
PLANE WAVE FUNCTION8
From Eqs. (.wIl) snd (4-41) we calculste 1 nT. nq E~l = - -.- C 1k lO1 -b SlD. k..tX cos -b "".
'IWE I
1 nT . n,.-y E~t = -.- Ctkat -5 8m [kat(a - x)] cos -b r
ll ••
JWf.t
E' l
=
1-. C1kdk. Bin k d % sin nb1r'Y trfk··
"'"
Continuity of
E~
..
..!.. C"'..I<. sin lk..(a -
Ed - -
~
.)] sin n"!/ "... b
and E. at x .,. d requires that
J. C,k.. sin k.,d
- -
tl
.!tt Csk.. sin [k•• (a -
a)]
(4-43)
Similarly, from Eqs. (4-30) and (4-41) we calculate
1I~1 = -jk.C 1 cos klll% sin
n;v
ell••
H~t - -jk.Ct cos {k",(a - %)1 sin n~ rfl··
cos k%l% cos n;v r
Hd
-
~T Ct
Hd
,."
~T Ct cos {kdCa - z)J cos n~ rPo.•
ik .,.
Continuity of 11~ and H. at x = d requires that
CI cos kdd = Ct cos (kd(a - d»)
(4-44)
Division of Eq. (4-43) by Eq. (4-44) gives
kill tan kll1d """ _ kill tan [kd(a - d)] Et
tl
(4-45)
Both kd and kill are functions of k. by Eqs. (4-42)i so the above is a transcendental equation for determining possible k.'s (mode-propagation constants). Once the desired. k. is found, kd and kd are given by Eqs. (4-42), snd the ratio C,jC, is given by Eq. (4-43) or Eq. (4-44). For fields TE to z, we choose "",'s in each region to represent the z component of F. To satisfy the boundary conditions at the conducting walls, we take
ill =- C 1 sin kdx "'t -
cos n;t' r
Ct sin [klll(a -
i4 ..
z)] cos n;v
(4-46) e-ik .,.
160
TIME-HARMONIC ELECI'ROMAGNETIC FlELDS
with n "'" 0, I, 2, .. ;. The separation parameter equations are again Eqs. (4-42). The field is calculated from the by Eqa. (4-32). A matching of tangential E and H at % - d yields..the characteristic equation
""s
k.. cot k..d _ _ k.. cot[k..(o 111
III
d)J
(4-471
The kd and kd arc functions of k. by Eqs. (442); so tho above is a tra;scendental equation for determining k.'s for the modes TE to :t. The modes of the partia.lly filled rectangular waveguide arc distorted versions of the TEa and TM% modes of Sec. 4-4. The modo patterns are similar to those of Fig. 4-4, except that tho field tends to concentrate in the material of higher E and Il. In the lossless case, the cutoff frequencies (k. =- 0) of tbe various modes will always lie between those for tho corre·
sponding modes of a. guide filled with a material El, Ill, and those--of a guide filled with a material E" PI. (This can be shown by the pcrturba.~ tional procedure of Sec. 7-4.) In contrast to the filled guide, the cutoff frequencies of the corresponding TE:t and TMx modes will be different. Also, a. knowledge of the cutoff frequencies of the partially filled guide is not sufficient to determine k. at otber frequencies by Eq. (4-26). We have to solve Eqs. (4-45) and (~7) at each frequency. Of special interest is the dominant mode of a partially filled guide. For b > ", this is the mode corresponding to the TJ\uol mode of the empty guide, whicb is also the TE u mode of the empty guide. For a given n, Eq. (4-45) has a. denumerably infinite set of solutions. We shall let m denote the order of these solutions, as follows. The mode with the lowest cutoff frequency is denoted by m = 0, the next modo by m = 1, and 50 on. Tbis numbering system is chosen so that the TMx... partially filled waveguide modes correspond to the TMx... empty-guide modes. The dominant mode of the partially filled guide is then the TMxol mode when b > a. lIenee, the propagation constant of the dominant mode is given by the lowest-ordcr solution to Eq. (4-45) when the k.'s a.re given by Eqs. (4-42) with n = 1. Fign.re 4-7 shows some calculations for the case E ",. 2.451'1). When k l is DOt very different from k l , we should expect k. 1 a.nd k.z to be small (k. is zero in an empty guide). If this is 50, then Eq. (4-45) can be approximated by
k..'d
_-:;k",,-,'(",0_d:=!)
1'1
1'1
--~
(4-48)
With this explicit relationship between kd and kd , we can solve Eqs. (4-42) simultaneously for k.r and k. (given Cal). Note that when kd is real, k.z is imaginary, and vice versa. The cutoff frequency is obtained hy oetting k, - 0 in Eqa. (4-42). Uaing Eq. (4-48), we have for the
161
PLANE WAVE FUNCTIONS
1.6
1.2
•
, ,
T.
1 t-- .b---l :«i
~
~ 0.8
/'
.-
/
/
0.4
o
0.1
0.2
0.3
0.5
0.4
0.6
0/).0
Fla. 4-7. Propagation constant for a rectangular waveguide pll.rtially filled with dielectric, • - 2.45-.. alb - 0.45, dla - O.SO. (Alta Prank.)
dominant mode kd
+ (i)' =
'
WlflJll
fl(~~d) k + (i)' "" zl
l
These we solve for the cutoff frequency r ColC"=-b
fl
valid when Eq. (4-48) applies.
l W ftJll
==
Col
flea d) (a d - ) fUll
When
J11
obtaining
We,
+ ftd
+
==
(4-49)
J
E2UfIJlI
J11
==
J1,
this reduces to (4-50)
Note that thi.s is the equation for resonance of a parallel-plate mission line, shorted at each end, and baving
L ==
J10
~
==
EIES
fl(a - d)
traDS~
+ Etd
per unit width. All cylindrical (cross section independent of z) waveguides at cutoff are tw b. The dominant mode of the empty guide is then the TEz u mode, or TE lo mode. The dominant mode of tbe pa.rtially filled guide will a.lso be a
162
TWE-HARMON1C ELECTROHAONETIC FIELDS
TEz modej SO the eigenvalues are found from Eq. (4-47) with n "'" O. We shall order the modes by m 88 followa. That with the lowest cutoff frequency is denoted by m = 1, that with the next lowest by m = 2, and 50 on. This numbering system corresponds to that (or the empty guide, the dominant mode being the TEx lO mode. When k 1 is not too different. from k" we might expect k Z1 and kd to be close to the empty-guide value k., = ria. An approximate solution to Eq. (4-47) could then be found by perturbing k. 1 and kd about 7/a. For the cutoff frequency of the
t:= d-----j T 1
"I
0
b
Zo -
'II
{j '""' kl
I
(2,1'2
I-d
I
Zo - '12 13 :: k2
I
1
"I'
(oj
I
0-
d----l
(bj
FIo. 4-8. (a) Partially filled waveguide; (b) tran.amission·line resonator. The cutoff frequency of t.he dominant. mode of (0) ia the reIIOnAot frequency of (b).
1.6
12
-
I-d-i
,
~l t--o--j
die
l.-?=' ~ ::..-V V ---
r7k::
fl; / . /V f/ '/1# ~ Iff
... o
~I 0.8
Ie
1
.8 .6 0.5 0.375 0.280 0.167 0
0.4
o
0.2
0.4
0.6 o{>..
0.8
1.0
FIo. 4-9. Propagation constant for a rectangular waveguide partially 6UOO with (After PraM.)
dielectric, • - 2.451..
163
PLANE WAVE FUNCTIONS
x Flo. 4-10. The dielectriclliab waveguide.
dominant mode, Eqs. (4-42) becomo k"'ll k l d
= kl~2 = wc2el~1
= k 2c I
. . We 2e 2/JI
and Eq. (4-47) becomes I
I
- cot k1cd = - - cot (k..(a -
",
",
dlJ
(4-51)
It is interesting to note that this is the equation for resonance of two shortcircuited transmission lines having Z.'s of '11 and 'Ill and P's of k 1c and k'k,
as illustrated by Fig. 4-8. The reason for this is, at eutoll, the TEz lo mode reduces to the parallel-plate transmission-line mode that propagates in the z direction. This viewpoint has been used extensively by Frank. 1 Some calculated propagation constants for the dominant mode are shown in Fig. 4-9 for the case «: - 2.45«:0. Similar results for a centered dielectric slab arc shown in Fig. 7·10, and the characteristic equation for that case is given in Prob. 4-19. 4.-7. The Dielectric-slab Guide. It is not necessary to have conductors for the guidance or localization of waves. Such phenomena also occur in inhomogeneous dielectrics. The simplest illustration of this is the guidance of waves by a dielectric slab. The so-called slab waveguide is illustrated by Fig. 4-10. We shall consider the problem to be two-dimensional, allowing no variation with the y coordinate. It is desired to find z-trllveling waves, that is, rj!~ variation. Modes TE and TM to either x or z can be found, and we shall choose the latter representation. For modes TM to 2, Eqs. (3-86) reduce to E = -k. '"
WE
a"ax
E. _
.,!... (k'
_ k.')"
H, _ - "'"
)we
ax
(4-52)
We shall consider separately the two cases: (1) .p an odd function of x, denoted by ~, a.nd (2) '" an even function of x, denoted by 1/t'. For case IN. H. Frank, Wave Guide Handbook, MIT Rad. L4b. Rept. 9,1942.
164
TIME-HAUMONIC ELECTROMAGNETIC FIELDS
(I), we choose in the dielectric region
[xl
< 2~
(4-53)
and in the air region
a
x>'2 x
<
(4-54)
a --2
We have chosen kuj = u and k&O = jv for simplicity of notation. (It will be seen later that u and v are real for unattenuatcd wave propagation.) The separation parameter equations in each region become
u1 _Vi
+ k.
+
1
k. 2
= kd t = = k ol =
1 W (dJJd
(4-55)
",1£0#010
Evaluating the field components tangential to the air-dielectric interface, we have E. =
~ u ' sin tLX e-p.,. }
[xl < ~
JWEd
2
11" = - Au cos u:c e-jk .-
-8
a
E. - -.- v1e-""e-il ..
x>2
JWEo
E. = .B v1e""e-ik,s ]WED
Continuity of E. and HI/at x =
± a/2 requires that
The ratio of the first equation to the second gives ~tan~ = ~va
2
2
EO
2
(4-56)
This, coupled with Eqs. (4-55), is the characteristic equation for determining k/s and eutoff frequencies of the odd TM modes,
165
PLANE WAVE FUNCTIONS
For TM modes which are even functions of
%,
we ehoose (4-57)
The separation parameter equations are still Eqs. (4-55). The field components are still given by Eqs. (4-52). In this case, matching E. and H. at :t - ± 0{2 yields ua
ua
E.,va
(4-58)
- -cot- = - 2 2 t6 2
This is the characteristic equation for determining the k.'s and cutoff frcquencies of the even TM modes. There is complete duality between the TM and TE modes of the slab waveguidej so the characteristic equations must be dual. For the TE modes with odd 1/t we have ua ua -tan-
2
2
Il4va
( 4-59 )
~--
PO
2
as the characteristic equation, and for the TE modes with even 'It we have _
~cot
2
ua = 2
1A1l~ lAO
(4-60)
2
as the characteristic equation. The u's and v's still s:l.tisfy Eqs. (4-55). The odd wave functions generating the TE modes are those of Eqs. (4-53) and (4-54), and the even wa.ve functions generating the TE modes are those of Eqs.. (4-57). The fields are, of course, obtained from the +'s by equations dual to Eqs.. (4-52), which are, explicitly, H _ - k, a~ •
WI!
ax
fI, _
.,.!.. (k' JWP
_ k,')~
E-~ • ax
(4-61)
These are specializations of Eqs. (3-89). The concept of cutoff frequency for dielectric waveguides is given a somewha.t different interpretation than for metal guides. Above the cutoff frequency, as we define it, the dielectric guide propagates a mode unattenuated (k. is real). Below the cutoff frequency, there is attenu~ atoo propa.gation (k. "'" fJ - ;0.). Since tbe dielectric is loss free, this ll.ttenuation must be accounted for by radiation of energy as the wave progresses. Dielectric guides operated in a radiating mode (below cutoff) are used as antennas. The phase constant of an unattenuatcd mode lies between tbe intrinsic phase constant of tbe dielectric and that of air; tbat is,
166
TUfE-BARMONIC ELECTROMAGNETIC FIELDS
This can be shown as follows. Equations (4-55) require that u and v be either real or imaginary when k. is real. The characteristic equations have solutions only when v is real. Furthermore, II must be positive, else the field will increase with distance from the slab [see Eqs. (4-54) or (4-57)J. When v is real and positive the characteristic equations have solutions only when 'U is also real. Hence, both u and II are real, and it follows from Eqs. (4-55) that ko < k. < ka. This result is So property of cylindrical dielectric waveguides in general. The lowest frequency for which unattenuated propagation exists is called the cutoff frequency. From the above discussion, it is evident that cutoff occurs a.s k. -. ko, in which case v -+ O. The cutoff frequencies are therefore obtained from the characteristic equations by setting u = ylkd' ko' and v = O. The result is
ko')
tan (; ylkd '
=
0
cot (~ylkd'
which apply to both TE and TM modes. when
kO') =
0
These equations are satisfied
n = 0, 1,2,
This we solve for the cutoff wavelengths
>.~
= 2a
n
IfdJld _ 1
'\I fO~O
n = 0, 1, 2,
(4-62)
and the cutoff frequencies
f~
n
= 2a
V fd~d
fO~O
n "'" 0, I, 2, . . .
(4-63)
The modes are ordered as TM.. and TE" according to the choice of n in Eqs. (4-62) and (4-63). Note thatf~ for the TEo and TM omodes is zero. In other words, the lowest-order 7'E and TM modes propagate unattenuated no maUer how thin the slab. This is a general property of cylindrical dielectric waveguides; the cutoff frequency of the dominant mode (or modes) is zero. However, as the slab becomes very thin, k. --+ ko and t1 --+ 0, so the field extends great distances from the slab. This characteristic is considered further in the next section. Finally, observe from Eq. (4-62) that when fdJld» EQJto, the cutoffs occur when the guide width is approximately an integral number of half-wavelengths in the dielectric, zero half~wavelengtb included. Simple graphical solutions of the characteristic equations exist to determine k. at any frequency above cutoff. Let us demonstrate this
167
PLANE WAVE FUNCTIONS
for the TE modes. ou l
Elimination of k. from Eqs. (4-55) gives
+ Vi =
kill -
k ol =
WI(E,iPd -
~llPll)
Using this relationship, we can write tho TE characteristic equations at'!
ua2 \ I(wa)' _.'ua cot ua - V"2 ('m ~uatan
,u." 2
,u." 2
,,,,,) -
(ua)' "2
2
Values of uaj2 for the various modes are the intersections of the plot of the left-hand terms with the circle specified by the right-hand term. Figure 4-11 shows a plot of the left-hand terms for Jld = ,u.ll. A representative plot of the right-hand term is shown dashed. As w or e" is varied, only the radius of the circle changes. (For the case shown, only three TE modes arc above cutoff.) If Pd F Po, the solid curves must be redrawn. The graphical solution for the TM mode eigenvalues is similar. Sketches of the mode patterns afe also of interest. Figure 4-12 shows the patterns of the TEo and TM 1 modes. These can also be interpreted as the mode patterns of the TM o and TEl modes if g and 3C are intercba.ngcd, fOf there is complete duality between the TE and TM cases.
r ••
~
f--
--
~
(UB) -,- ''" (UB)
"
.
[
I
I
I
I
I
/
I
I
/
I I
cot 2/~
,
I
"2
~2
~
1/
\
I I
/
,
I [
I',\\
/i
\
,I
,
. + + (U~.) (U~.)
Fio. 4-11. Grp.phical solution of the characteristic equation for the Blab waveguide.
168
TIME-HARMONIC ELECTRO?,IAGNETIC FIELDS
•
•
0
•
• .1 !t1
~~~
".1.
0 0 0
14
l\~t1 (ltZ; >t Q'\~r2:.i 0)8) jJ~~
~~'4 COCO~.. It)D1.1:\1 Q x
It
0
It
0
•
•
•
•
• I
•
•
•
•
• (b)
FlO. 4-12. Mode pfl.ttern.s for the dieJectric-6lab wll.Vcguidc. dashed); (b) TM 1 mode (t lines IlOlid).
(a) TEo mode (X lines
As the mode number increases, more loops appear within the dielectric,
but not in the air region. 4~8. Surface-guided Waves. We shall show that any "reactive boundary" will tend to produce wave guidance along that boundary. The wa.ve impedances normal to the dielectric-to-air interfaces of the slab guide of Fig. 4-10 can be shown to be reactive. A simple way of obtaining a single reactive surface is to coat a conductor with a dielectric layer. This is shown in Fig. 4-13. The modes of the dielectric-coated conductor arc those of the dielectric 0 plane. These are the TM.., slab having zero tangential E over the x n 0, 2, 4, . . . ,modes (odd if;) and the TE., n = 1,3,5, ...• modea =0
=0
169
PLANE WAVE FUNCTIONS
(even '/I) of the slab. We shall retain the same mode designations for the coated conductor. The characteristic equations for the 'I'M modes of the coated conductor are therefore Eq. (4-56) with 'a/2 replaced by t (coating tl},ickness). The characteristic equation for the TE modes is Eq. (4-60) with a/2 replaced by t. The cutoff frequencies are specified by Eq. (4-63), which, for the coated conductor, becomes
n
f. -
41
V"""
(4-64)
<0,,,
where for TM modes n = 0, 2, 4, . . . , and for TE modes n - 1, 3, 5, . . .. The dominant mode is the 'I'M o mode, which propagates uoattenuatcd at all frequencies. The mode pattern of the 'I'M, mode is sketched in Fig. 4-14. Let us consider in more detail the manner in which the dominant mode decays with distance from the boundary.....hI the air space, the' field attenuates as r·%. For thick coatings, k. -. k4 , and, from Eq. (4-55),
"'_k, 1~-1 I W&fI "EOJ.IO
(4-65)
This attenuation is quite large for most dielectrics. For example, if the coating is polystyrene (to" = 2.56E o, JI.4 - "'0), the field in 0.12~ has decayed to 36.8 per cent of its value at the surface. However, for thin coatings,
x
".1'0 Z Pta. 4-13. A dielcctrie-eoated conductor.
•
o o
0
•
0
•
• • •• • • •
• 0
0
• 0
0
0
Flo. 4-14. The TM, mode pattern for t.he coated conduetor (s lines 1lO1id.)
170
T!KE-JlARI(ONIC ELECTROMAGNETIC FIELDS ]{
z FlO. 4-15. A corrugated conductor.
the field decays slowly.
In this case, k. --+ k~, and
k("' ,,)t
v--:,.2r o - - - flmall Ilo f,,}.,
(4-66)
If the polystyrene coating were 0.0001 wavelength thick, we would have to go 40 wavelengths from the surface before the field decays to 36.8 per
cent of its value at the surfaco. We say that the field is It tightly bound" to a thick dielectric coating and Uloosely bound" to a thin dielectric coating. Another way of obtaining a reactive surface is to flcorrugate" a conducting surface, as suggested by Fig. 4-15. For a simple treatment of the problem, let us assume that the" teeth JJ are infinitely thin, and that there are many slots per wavelength. The teeth will essentially short out any E III permitting only E. and E. at the surface. The TM fields of the dielectric-slab guide are of this type; hence we shall assume that thia field exists in the air region. Extracting from Sec. 4-7, we ha.ve
x>d
where
(4-67)
The wave impedance looking into the corrugated surface is Z
-.
,." E. = ju 1I, W~O
(4-68)
Note that this is inductively reactive; so to support such a field, the interface must be an inductively reactive surface. (The TE fields of Sec. 4-7 require a. capacitively reactive surface.) In the slots of the corrugation, we assume that the parallel-plate transmission-line mode
PLANE WAVE FUNCTIONS
171
exists. These are then shorkircuited transmission lines. of characteristic wave impedance 'l0. Hence, the input wave impedance is
Z_. - iv. tan k,.J For kod < T/2. this is inductively reactive. (4-69), we have • = k. tan k,.J
(4-69)
Equating Eqs. (4-68) and (4-70)
and, from Eq. (4-67), we have k, = k,
Vi + tan' k,.J
(4-71)
It should be pointed out that this solution is approximate, for we have only approximated the wave impedance at x "'" d. In the true solution, the fields must differ from those assumed in the vicinity of x = d. (We should expect E a to terminate on the edges of the ~th.) When the teeth are considered to be of finite width. an approximate solution can be obtained by replacing Eq. (4-69) by the average wave impedance. This is found by assuming Eq. (4-69) to hold over the gaps, and by assuming zero impedance over the region occupied by the teeth. The result is' k,
~ k. ~1 + G~ ,)' tan' k,./
where g = width of gaps and t := width of teeth. While at this time we lack the concepts for estimating the accuracy of the above solution, it has been found to be satisfactory for small kdl. Note that, from Eq. (4-70), the wave is loosely bound for very small kod, becoming more tightly bound as kod becomes larger (but still less than 11/2). Tho mode pattern of the w&ve is similar to that for the TM o coated-eonductor mode (Fig. 4-14), except in tho vicinity of the corrugations. 4-9. Modal Expansions of Fields. The modes existing in a waveguide depend upon the excitation of the guide. The nonpropaga.ting modes are of appreciable magnitude only in the vicinity of sources or discontinuities. Given the tangential components of E (or of H) over a waveguide cross section, we can determine the amplitudes of the various wave-guide modes. This we shall illustrate for the rectangular waveguide. Consider the rectangular waveguide of Fig. 2-16. Let E~ = 0 and E, "" J(x.y) be known over the z = 0 cross section. We wish to determine the field z > 0. assuming that the guide is ma.tched (only outwardtraveling waves exist). The TEx modes of Sec. 4-4 have no Ea; 50 let uS
Ie. C. Culler, Electromagnetic Waves Guided by Corrugated Conducting Surfaces, BeU TdqMru Lab. Rept. MM-44-160-218, October, 1944.
172
TI:M.E-HAR~(ONIC
flLECI'ROKAGNETlC FIELDS
take a superposition of these modes.
. .
'I'
2: 2:
-=
A •• sin
This is
m;z cos n~ rr-'
(4-72)
._1 ...0
where A•• are mode amplitudes and the -r•• are the mode-propagation constants, given by Eq. (4-23). In terms of !fi. the field is given by Eqs. (4-32). In particular, E, at z = 0 is given by
E~
. . '\' '\' ... Lt L..t I 0
=
. m.. a
"r.... A... am
nry cos b
",_I .... 0
Note that this is in the form of a double Fourier series: a sine series in :t and a cosine aeries in y (sec Appendix C). It is thus evident that 'Y.... A ... sre the Fourier coefficients of E" or ll
"roo.A ... - E•• = 2t ab
{"
)0
dx
ior dy E, I._0 sin mn a cos n b
ry
(4-73)
where f .. = I for n = 0 and E. = 2 for n > 0 (Neumann'& number). The .A••• and hence the field, are now evaluated. The solution for E. = /(x,Y) and E, .,. 0 given over the z - 0 cross section can be obtained from the above solut.ion by a rotation of axes. The general case for which both E~ and E, are given over the z .= 0 cross section is a superposition of the two cases E• ... 0 and E, = O. The solution for the ease H ~ and H, givCD over the z = 0 cross section can be obtained in a dual manner. For a large class of waveguides, when many modes exist simultaneously, each mode transmits energy as if it existed alone. We shall show that the rectangular waveguide has this property. Given the wave function of Eq. (4-72), specifying a field according to Eqs. (4-32), the z-direetoo complex power at z ... 0 is
P••
Because of the orthogonality relationships for the sinusoidal {unctions,
173
PLANE WAVE FUNCTIONS
Incident wave
L z ---------FIG. 4-16. A capacitive wa.veguide junction.
this reduces to (4-74)
where (yo) .... are the TEx wave a.dmittances, given by the reciprocal of Eqs. (4-36). The above equation is simply a. summation of the powers for the individual modes. In a lossless guide, the power for a propagating mode is real and that for a non propagating mode is imaginary. To illustrate the above theory, consider the waveguide junction of Fig. 4.~16. The dimensions are such that only the dominant mode (TE IO) propagates in each section. Let there be a wave incident on the junction from the smaller guide, and let the larger guide be matched. For an approximate solution, assume that Ell at the junction is tbat of the incident wave Ell
I { "'"
...0
...
y c
sma 0
(4-75)
From Eq, (4-73), the only nonzero mode amplitudes are E lo
=
E 1.. =
C
'Y10A IO
=b . nll'c
2
'YhAh = -
5ln-
(4-76)
b Thus, only the m = 1 term of the m summation remains in Eq. (4-72). Let us usc this solution to obtain an "aperture admittance" for the junction. From Eqs. (4-74) and (4-76), the complex power at z = 0 is p
~
n~
•
a<:'
2b
(Y)'
0 10
+ 2 '\' (Y)' ~
"-.
where, from Eqs. (4-36), (Yoho =
k' - (~/a)'
(Yoh.. = k
wp.(j
2
-
0 1>1
""
nTc/b
Vi
(Uf)' f/
.(tr/a)' "" J(,JJla
['in (n.c/b)]'j
}..,
j2b(Y ohG (2b/}".) ,
vn'
174
TUIE--HARMONIC ELECl'ROYAQNETIC FIELDS
3
+~o~
•
I'
1\ 0.3
I- 0.2
2 f-
tE
1\
n,
'I
~'
E - sin(1l' % fa}
.\ 1'\\
=r-
rrt X
\. ~
0-
1
"~
""- "~
0-
N
'"
I o
02
0.4
0.6
1.0
0.8
c/b· Flo. 4-17. Susceptance of a capacitive aperture.
The Ie and X. llre those of the TE lo mode. We shall refer the aperture admittance to the voltage across tho center of t.he aperture, whieh is V "'" c. The aperture admittance is then
y
p.
• = lVI' = (
y
[a .J,. 2b
• .20 \ '
sin! (nrc/b)
+ 1 ~. ._1 f.< (nrc/b)' -In'
]
(2b/~.)'
(4-77)
The imaginary part of this is the aperture suscepta.nce B
-
2a \ '
• - ~.Z.
f.<
.-1
sin! (nrc/b) (nrc/b)' n' (2b/~.)'
(4-78)
where A, and Z. are those of the dominant mode. Calculated values for B. are shown in Fig. 4-17. For small c/b, we have'
~2:' B. = -log I0656~ [1 +
1-
(~~),]l
(4-79)
1 This equation ill a quui-static result.. The direct 15peeialization of Eq. (4_78) to IlIlall clb yields a numerical factor of O.37\) instead of 0.656.
PLANE WAVE FUNCfION8
tx
175
tx Incident W<M!-
z ----------Flo. 4-18. An induct.ive waveguide junction.
The aperture susceptance is a quantity that will be useful for the treat,. ment of microwave networks in Chap. 8. Note that the susceptance is capacitivc (positive) i so the original junction is called a capacitive waveguide jurn;tion. Remember that our solution is only approximate,' since we assumed E in the aperture. (We shall see in Sec. 8-9 that the true susceptance cannot be greater than our present solution.) We have assumed that only onc mode propagates in the guide; hence ou.r solution is explicit only for
When a second mode propagates, it contributes to the aperture conductance, and Eq. (4-78) would be summed from 11. = 2 to (lO, and so on. Another problem of practical interest is that of the waveguide junction of Fig. 4-18. Again we assume only the dominant mode propagates in each section. Take a wave incident on the junction from the smaller guide, and let the la.rger guide be matched. For an approximate solution, we assume Ell in the aperture to be that of the incident wave E, 1.·0 ~
. TZ 81D-
{
0
x
c
Z
>c
(4-80)
From Eqs. (4-73), we determine the only nonzero mode amplitudes as
E
2c sin (mrc/a) ., - Tall (me/a)'J
(4-81)
Thus, only the 11. = 0 term of the 11. su.mmation remains in Eq. (4-72). Again we can find an aperture admittance for the junction. From Eqs. (4-74) and (4-81), the complex power at z = 0 is • p _ 2bc' \ ' (Y)'
...'a
..
Lt,
0.0
[sin 1
(mTc/a) ]'
(me/a)!
176
TIME-HARMONIC ELECTROMAGNETIC FIELDS
where, from Eqs. (4-36), Y) _ k' _ ( o 10 - wp.fJ -
(Yo)., == k
l __
VI - U'/f)'
y;:a)t
'IJ
=
~j ~(;t:y
_
m> I
1
The voltage across the center of the a.perture is V ... b. admittance referred to this voltage is therefore
Y. -
The aperture
.~~ ([t~-«;::?,r (Yo)" •
_ 1. "\'
..
"1.<,
[sin (moefa)]' I(mx)' - 11 I - (_fa)' 'V 2a
(4-82)
The imaginary part of this is the aperture susceptance (4-83)
which is plotted in Fig. 4-19. The susceptance is inductive (negative); 90 the original junction is called an inductive waveguide junction. For single-mode propagation, we must. have a < X; so our explicit interpre-
I I
0.2
~
~-"-=i1
0.8
o
0.2
af- 0.5-
F:
'" '" 0.1
I
h
~7
" " "-"- ""
t-,. "-
E
0::
1 II b
sin (yx/c)
X
0.9 ~
0.4
0.6
"' -
...... ::::.; t-.. ~
1::3
0.8
cIa Fro. 4-19. Susceptance of an inductive aperture.
~
1.0
177
PLANE WAVE FUNCTIONS
tation of the solution is restricted to this range. For wave propaga.tion in the smaller guide, we must have c > ),,/2 if it is air-filled. However, if the smaller guide is dielectric-filled, we can have wave propagation in it when c < >./2. Moreover, the aperture susceptance is defined only in terms of E~ in the aperture and has significance independent of the manner in which this E~ is obtained. 4-10. Currents in Waveguides. The problems of the preceding section might be called II aperture excitation >1 of waveguides. We shall now consider" current excitation" of waveguides. This involves the determination of modal expansions in terms of current sheets over a guide cross section. The only difference between aperture excitation and current excitation is that the former assumes a knowledge of the tangential electric field and the l::Ltter assumes a knowledge of the discontinuity in the tangential magnetic field. The equivalence principle plus duality can be used to transform a.n aperture-type problem into a current-type problem, and vice versa. To illustrate the solution, consider a rectangular waveguide with a sheet of z-directed electric currents over the z = 0 cross section. This is illustrated by Fig. 3-2, where J. = uz!(z,y) is now arbitrary. We shall assume that only waves traveling outward from the current are present, thllt is, the guide is matched in both directions. At z = 0 we must have E~, E~, and H ~ continuous. Hz must also be antisymmetric about z = 0; hence it must be identically zero, and it is convenient to use the TMx modes of Sec. 4-4. (Note that J and its images are x-directcdi so it is to be expected that an x-directed A is sufficient for representing the field.) Superpositions of the TMx modes are
. .
L 2:
(4-84)
B.....- cos
7
z <0
sin n;y e"'-'
","'0 .... 1
+
where superscripts and - refer to the regions z > 0 and z < 0, respectively. The field in terms of the ,p's is given by Eqs. (4-30). Continuity of E~ nnd Ell at z = 0 requires that
B....+ = B.....- = B.....
(4-85)
The remaining boundary condition is the discontinuity in H ~ caused by J s, which is 21'....8 ..... cos
a
mr.r . nTy
sm T
178
TIME-HARMONIC ELECTROMAGNETIC FIELDS
This is a Fourier eosine series in x and a Fourier sine series in y. It is evident that 2y.....B..... are the Fourier coefficients of J~, that is, 2'Y.....B..... = Jill.. =
2t", fa [b m1rX . nry fib}o ax io dyJ"cOSaSlllT
(4-86)
This completes the determination of the field. The solution for a y-directed current corresponds to a. rotation of axes in the above solution. When both J" and J II exist, the solution is a. superposition of the two cases J II = 0 and J" = O. The solution for a magnetic current sheet in the waveguide is obtained in a dual manner. A z-directed electric current can be treated as a loop of magnetic current in the cross-sectional plane, according to Fig. 3-3. A z-directed magnetic current is the dual problem. Thus, we have the formal solution for all possible cases of currents in a rectangular waveguide. It is also of interest to find the power supplied by the currents in a waveguide. This is most simply obtained from p -
-
JJE . J: d, - - J: dx J: dy J: E.!._o ._0
We express J. in its Fourier series and evaluate E. by Eqs. (4-30) applied to the above solution. Because of the orthogonality relationships, the power reduces to (4-87)
where (Zo)..." are the TMx wave impedances, given by Eqs. (4-35). This is a summation of the powers that each J '"II alone would produce in the guide. In a lossless guide, the power associated with each propagating mode is real, and that associated with a oonpropagating mode is imaginary. As an example of the above theory, consider the coax to waveguide junction of Fig. 4-20. This is 8. waveguide llprobe feed," the probe being the center conductor of the coax. If the probe is thin, the current on it will have approximately a sinusoidal distribution, as on the linear antenna. With the probe joined to the opposite wa.veguide wall, as shown in Fig. 4-20, the current maximum is at the joint x = a. We therefore assume a current 00 the probe I(x)
R<
cos k(a - z)
(4-88)
The current sheet approximating this probe is
J. - I(.)o(y - c)
(4-89)
PL.~"'E
179
WAVE FUNCTlONS
Matched load
Matched load
I
Coax
Flo. 4-20. A coax to wAveguide junction.
where !(y - c) is the impulse function, or delta function (sec Appendix C). The Fourier coefficients for the current arc then obtained from Eq. (4-86) as
J
=
••
2e",ka sin ka sin nrc/b b[(ka)' (mr)'l
(4-90)
This, coupled with our earlier formulas, determines the field. In terms of this solution, let us consider the input impedance seen by the coaxial line. The power supplied by the stub is given by Eq. (4-87). The impedance seen by the coax is then
Z, -
1:"'- R, + iX,
where, from Eq. (4-88), the input current is
Ii - coska Assume that the waveguide dimensions arc such that only the TEol mode propagates. Then only the m "'" 0, n ... 1 term of Eq. (4-87) is renl, and
R,
(ZO)o1 OOIJ"I' 4 I;
= -
-
= -a (Z) 001
b
ka)' .
(tan -ka
1 5,1I'"C m-
b
(4-91)
All other terms of the summation of Eq. (4-87) contribut.c to X,. However, since we assumed a filamentary current, the series for X. diverges. To obtain a. finite X•• we must consider a conductor of finite radius. For small a, the reactance will be capacitive. In the vicinity of a "'" "A/4, we have a resonance, above which the reactance is inductive. Note that Eq. (4-91) says that the input resistance is infinite at this resonance. This is incorrect for an actual junction, and the error ties in our assumed current. Equation (4-91) gives reliable input resistances only when we are somewhat. removed from resonant points. [This is similar to our linear antenna solution (Sec. 2-10»). Feeds in waveguides with arbitrary terminations are considered in Sec. 8-11.
180
TIME-HARMONIC ELECTROMAGNETIC FIELDS
x
x
=-fl1
T~---;---~
Incldent
z
l
Flo. 4·21. A parallel-plate guide radiating into half-flpace.
4-11. Apertures in Ground Planes. We have already solved the problem of determining the field from apertures in ground planes, in Sec. 3-6. At this time, however, we shall take an alternative approach and obtain a different form of solution. By the uniqueness theorem, the two [orms of solution must be equal. One form may be convenient for some calculations, and the other form for other calculations. Let us demonstrate the theory for an aperture in the ground plane y = 0, illustrated by Fig. 4-21. We further restrict consideration to the case E. = 0, there being only an E., in the aperture. Taking a clue from our waveguide solution (Sec. 4-9), let us consider Fourier transforms (see Appendix C). The transform pair for E% over the y 0 plane is :IE:
(4-92)
where a bar over a symbol denotes transform. The form of the transformation suggests that we choose as a wave function (4-93)
which is a superposition of the form of Eq. (4-9). For our present problem, we take Eq. (4-93) as representing a field TE to z, according to Eqs. (3-89). There is a one-to-one correspondence between a function and its transform; hence it is evident that the transform of '" is
if
~ J(k..k.)....•
(4-94)
PLANE WAVE FUNCl'IONB
181
We also can rewrite Eqs. (3-89) in terms of transforms as
n. _ -kJc. if1""
R _ -k,.k. if,
lW~
(4-95)
R. _ k' :- k.' if-
1"" Specializing the above to the y = 0 plane, we have
D.I,_o =
- jk,!(k.,k.)
A comparison of this with Eqs. (4-92) shows that
-1 !(k.,k.) - ~k D.(k.,k.) 1 ,
(4-96)
where E. is given by the second of Eqs. (4-92). This completes the soluk.' tion. As a word of caution, k, = ± Vk l k. ' is double~valued, and we must choose the correct root. For Eq. (4-9-1) to remain finite as Y"'-'+ 00, we must choose
k'
k.'
k k
+ k.' > v'k.' + k.' < v'k.'
(4-97)
The minus sign on the lower equality is necessary to remain on the sarno branch as designated by the upper equality. The extension of this solution to problems in wh.icb both E. and E. exist over the y=:O plane can be effected by adding the appropriate TE to x field to the above TE to z field. It can also be obtained as the sum of fields TE and TM to z, or to x, or to y. The case of H,. and HII specified over the y = 0 plane is the dual problem and can be obtained by an interchange of symbols. For simplicity, we shall choose our illustrative problems to be twodimensional ones. Let Fig. 4-21 represent a parallel-plate waveguide opening onto a ground plane. If the incident wave is in the transmissionline mode (TEM to V), it is a.ppa.rent from symmetry that H. will be the only component of H. Let us therefore take H. as the scalar wave function and construct
.J.-.
I H. - -2
!(k.)&>.·ei'.· dk.
(4-98)
From this, it. is evident that t.he transform of H. is
n. = !(k.)ei'.'
(4-99)
182
TIME-HARMONIC ELECTROMAGNETIC FIELDS
From the field equations, we relate the transform of E to
E-a Specializing E a to y
E.I
=
.. _0
R. as
-
k'll •
(4-100)
= -
0, we have
~
f-
k, J(k.) -
WE
__
E.(x,O)..-"'· dz
(4-101)
from which f(k.) may be found. For an approximate solution to Fig. 4-21 for y > 0, we assume E. in the aperture to be of the fOfm of the incident mode, that is,
E.
1.-.
1 ~
0
1
Ixl < 2 "
(4-102)
Ixl > "2
Using this in Eq. (4-101), we find
E.I
11-0
~
k, J(k.)
WE
~ k2 sin (k. -2")
(4-103)
:r
To complete the solution, we must also choose the root of kv for proper behavior as y _ 00. From Eq. (4-99), it is evident that this root is
k < k>
Ik.1 Ik.1
(4-104)
The fields are found from the transforms by inversion. A parameter of interest to us in future work is the aperture admittance. To evaluate this, we shall make use of the integral form of ParsevaPs theorem (Appendix C), which is
f-
J(x)O'(x) dz -
i. f-
J(k)U*(k) dk
We can express the power per unit width (z direction) transmitted by the aperture as
From Eqs. (4-100) and (4-102), this becomes
p
~ -!:!. 2T
f-
I, 111.1' dk. = - .! _. kll Al1
f-
_..
sin'Y,,:/2) dk. kllk z
183
PLANE WAVE Jl'UNC!'IONS
4
FJO. 4-22. Aperture admit.Lance of a capacitive dot radiator.
~~
.1
3
T1 E -, I
r-..
2
I, "B.
I
r--..
)
",loG.
r--
I"
o
0.2
0.4
0.6
0.8
1.0
ai' We now define the aperture admittance referred to the aperture voltage V - a as y _ p' = -4 sin' (k.a/ 2) dk • IVII ),11(11 _ _ k.k,.l ,.
f"
Note that, by Eq. (4-104), tbe above integrand is res! rnr Ik.1 < k and We can therefore separate Y. into ita real and
imaginary for lk,.1 > k. imaginary parts as
G. =
....!.a1 >t"1
4 B• -),"1(11 -
f'
-J:
sin' (k.a/2) dk. k,.l v'k1 k,.l
(f-' + J.') k,,'sin'v'k,,' (k.a/2) dk k' .. __
J:
The above integrals can be simplified to give
X,p. - 2 ),"1B.
=
2
f, J.o
w'
tD
dtD
v' (ka/2)'
w'
(4-105)
sin' w dw
b/2
For srna.ll ka, these
ainl
b/2
w'
v'w'
(ka/2)l
0.00 1
x,p. = r [I >..."B.
t>::
(';1'] }
a
I: < 0.1
(4-106)
3.135 - 2 log ka
For intermediate ka, the aperture conductance and susceptance are plotted in Fig. 4-22. For large w, we have I The formula for B. ill a qU&!i-et.at.ic result. The direct specialUation of the seeond of Eq•. (4-lo.s) to small ka gives a numerical facwr of 4.232 instead of 3.135.
184
TIME~BAlWONlC
ELECTROllAGNETIC VJELDS
!
~"o. ~ ~
~.B. ~ (:.)' [1 - ~.faC08(~' + DT] j
~ > 1 (4-107)
The aperture is capacitive, since B. is always positive. Another problem of practical interest is that of Fig. 4-21 when the incident wave is in the dominant TE mode (TE to y). In this case, E. will be the only component of E, and we shall take E. as our scalar wave fundion. Analogous to the preceding problem, we construct 1 E. = 2...
f"_. f(k.)eJk.z,p.· dk.
(4-108)
In terms of Fourier transforms, this is
2. -
(4-109)
f(k.) ....'
From the field equations, we find the trans(orm of H to be
R.
:::s
-k,S.
H, _ k. 2.
WJl
WJl
(4-110)
Thef(k,,) is evaluated by specializing Eq. (4-109) to y "'" 0, which gives
2.1'.0 ~
f(k.} -
f_'. E.(x.O)..-"·· dx
(4-11 I}
For an approximate BOlution, we assume the E. in the aperture of Fig. 4-21 to be that of the incident TE mode, tbat is,
E.
I,-0
!
COS
~
TX
a
(4-112)
0
Substituting this into the preceding equation, we find
E.I,.0
= f(k.} _ 2T' cos (k,af2) w- t - (k,..a)t
(4-113)
The choice of the root for k, is the same as in the preceding example, given by Eq. (4-104). This completes the formal solution. Let us again calcula.te the aperture admittance. The power transmitted by the aperture is P -
f. [E.H:)~.
dx -
~
where we have used Parseval's theorem.
f.
[E.n:),•• dk.
From Eqs. (4-110) and (4-113),
185
PLANE WAVE FUNCTIONS
0.8
tX
I
f-
\La
_E
0.6 f- I-
IT
f- I- \
( .{l<JG.
E """ cos ('lIx/a)
0.4
I I
0.2
V
V
(.;lJB~
V
t-
V
o
1.5
1.0
0.5
ai' FIo. 4-23. Apert.ure admittance of an inductive Blot radiator.
this becomes p
~
...=!.
21fwIJ
f"
_..
k*IE.I' dk. ~ -2..' WIJ
1/
f"
k: cos' (k.a/2) dk.
_ .. 1~2 -
(k.a)2J2
We shall refer the aperture admittance to the voltage per unit length or the aperture, which is V = 1. This gives
y .,., p.
. rvr
= _2~a2
Wa
f"-" [r'cos (k.a)'l' 2
kll
(k~aj2) dk
•
The integrand is real for Ik~! < k and imaginary for Ik.,l > k. Aseparation of Y. into real and imaginary parts is therefore accomplished in the same manner as in the preceding example. The result is '1
1
_
(0/2
2 }o
); G. -
vi (ka/2)2
w 2 cost W
w'J'
[(r/2)'
dw
• vlw% (kaj2)tcos2w -B =-If." -dw }.,.. 2 h/2 [(orr /2)2 w 2J2
For small ka, we have
~G ~ ~(~)' }.,..
1r).,
~ B. ~ -0.194
l
);a < 0.1
(4-114)
(4-115)
For intermediate ka, the aperture conductance and susceptance are plotted in Fig. 4-23. For large ka, • a ~ G.. = 2>.
a
); >
1.5
(4-116)
186
TIME-HARMONIC ELECTROMAGNETIC FIELDS
z
FIo. 4-24. A sheet. of z-di~
J.
rccted currents in the
y
y -
0 plane.
x and B.. is negligible. The aperture is inductive since B" is always negative. 4-12. Plane Current Sheets. The field of plane sheets of current can, of course, be determined by the potentia.l integral method of Sec. 2-9. We DOW reconsider the problem from the alternative approach of constructing transforms. The procedure is similar to that used in the preceding section for apertures. In fact, if the equivalence principle plus image theory is applied to the results of the preceding section, we have complete duality between apertures (magnetic current sheets) and electric current sheets. However, rather than taking this short cut, let us follow the more circuitous path of constructing thc solution from basic concepts. Suppose we have a sheet of z-directcd electric currents over a portion of the y = 0 plane, as suggested by Fig. 4-24. The field can be expressed in terms of a wa.ve function representing the z-component of magnetic vector potential. (This we know from the potential integral solution.) The problem is of the radiation. type, requiring continuous distributions of eigenvalues. We anticipate the wave functions to be of the transform type, such as Eq. (4-93). From Eqs. (3-86), we have the transforms of the field components for the TM to z field, given by
n. - jk,~ D,
~ -jk.~
D. - 0
(4-117)
187
PLANE WAVE FUNCTIONS
We construct the transform of y, as
These are dual to Eqs. (4-95).
if!'" = f+(k..k.) ~
,,-
f-(k.,k.)
y>O y
•• -·
(4-118)
For the proper beha.vior of the fields at large Iyj, we must choose k,/, as in Eq. (4-97), and kll- as the other root. That is,
Our boundary conditions at the current sheet are continuity of E s and Il~, according to Eq. (1-86). The boundary condition 011 Es and Ell leads to 1+ = 1-, and the boundary condition on H ~ then leads to
Ell, and a discontinuity in
(4-120)
where J., the transform of J., is J.(k••k.) =
J-". J-". J.(x,z).......•......··dzdz
(4-121)
This completes tbe determination of the field transforms. The field is given by the inverse transformation. Our two solutions (potential int-egral and transform) plus the uniqueness theorem can be used to establish mathematical identities. For example, consider the current element of Fig. 2-21. The potential in~ gral solution is A := u.!Jt where lle-'h '" - hr r = V x""+'-y""+'-z::;'
(4-122)
For the transform solution, J. - II '(x) '(z)
J. Hencel for y
= -1,
41r
f' f"
> 0 we have A jll '" = 8...,
where k"
= k,+
II, _. J ~-~·~e-ii .• dx dz = 4r
_.
=
u.J/! where
f" f" _.
I _. -k" eJA··~.lIeJA·· dk~ dk.
is given by Eq. (4-119).
(4-123)
In this example,!Jt as well as the
188
TIME-HARMONIC ELECTROMAGNETIC nELDS
field is unique. the identity rib
-
r
Hence. equating Eqs. (4-122) and (4-123), we have
f" f-
1 21f-j _..
= -
e-i,..;l. l.' .1:,'
_. yk 2
k. t
ks t
...••...··dk.dk
(4-124)
•
This bolds for all y, since kll changes sign as y changes sign. We have considered explicitly only sheets of z-dirccted current. The solution for x-directed current can be obtained by a rotation of coordinates. When the current sheet bas both x and z components, the solution is a superposition of the x-directed case and the z-direct.ed case. The solution for magnetic current sheets is dual to that for electric current sheets. Finally, if t-he sheet contains y-directed electric currents, we can convert to the equivalent x- and z-directed magnetic cunent sheet (or a solution. and vice versa for ~irecLcd magnetic currents. A ttl.·o-<Jimensional problem to which we shall have occasion to refer in the next chapter is that of a ribbon of axially directed current. uniformly distributed. This is shown in Fig. 4-25. The parameter of interest to us is the If impedance per unit length." defined by (4-125)
where P is the complex power per unit length and I is the total current. Rather than work through the details. let u.s apply duality to the aperture problem of Fig. 4-22. According to the concepts of Sec. 3-6. the field y > 0 is unchanged if the aperture is replaced by a magnetic current ribbon K = 2V. This ribbon radiates into whole space; so the power per unit length is twice that from the aperture. The admittance of the magnetic current ribbon is thus
z
y ......lb
=
jP:pott 2Vlt =
p. TKft -
lL)" 7:::
."'~
where the aperture admittance
"'"' G. + jB. is given by Eq. (4-105). which we can represent by 1 Y.~" = .X J(k4) Y.pe,~
J.
y
x
By duality, we have the radiation impedance of the electric current ribbon given by FIa. 4-25. A ribbon
or current.
Z"M'" ~
1 11
2 ~ J(ka)
-
11'
"2
Y. M "
(4-126)
ltLANlf: WAVE FUNQTIONS
180
(Compare this with Prob. 3-1. The fo.ctor--
z...... ~ 2~ IT + j(3.135 -
2 log ka)J
(4-127)
This we shall compare to the corresponding Z for a cylinder of current in Sec. 5-6. PROBLEMS
.-1. Show that Eq. (4-9) i.a t\ solution to the !Calar Behnholh equation. • -2. For Ie - fJ - ;a, show that lin b - ain (Jz coah G% COlI b: - COIJ (Jz eoah lEt
-
i C08 (Jz Binh GZ
+ J Bin fJz sinh cr%
4,-3. Derive Eq!!l. (4-17) . ..... Following the method UlIed to establisb Eq. (2-93), sbow that the attenuation con.atant duo to conductor loasca in a rectangular waveguide is giveD by Eq. (2-93) tor aU TE.. modes and by (a)
_
• ....
2m [ "
(a
ab
+ b)U./J)1 + ~ I
vI - (I./J)'
I,)' bm' + an' ] - ( 7 6l m l + ain l
Vi
tor TEo.. modes, m and n nonzero, and by ( ) Qc • • -
IfQb
mfb' + lI'a' vi26l (J.ln'm'b + nla t
t
(or TM•• modes. • -6. An air-filled rectangular waveguide ia needed for operation at 10,000 meg.... oycles. It is desired to have single-mode operation over a 2: 1 froquency range, with center frequency 10,000 megacycles. It. is also desired to havo maximum poWOl'handling capacity under these conditionll. Determino the waveguide dimensions and the attenuat.ion constant ot the propagating mode for copper wa.lIs. 4,·8. For 8. parallel-pllt.te wa.vc~dc formed by conduotors coverins t.he 11 and 11 - b planca, show that.
°
n-I,2,3• . . . are the mode functions generating the two-dimenaional TE.. modes according to Eq•. (3-89), and R -
1, 2, 3, . . .
arc the mode funct.ionB generating t.he two-dimensional TM. modCll aocording to Eq•. (3-86). Show that. the TEM mode ill generated by
190
TU,fE-RARMONIC ELECTROMAGNETIC FIELDS
"·7. Show that an alternative sct of mode functions for the parallel.plate waveguide of Prob. 4-6 are n - 0, 1,2, ..
which generate the
TM~
modes Recording to Eq8. (4-32), and n -= 1,2,3, ..
which generate the TM:t.. mode" according to Eqa. (4-30). Note that n - 0 in the above TEx mode function gives the TEM mode. f.-8. Show that the TEa: and TM:z: modes of Sec. 4-4 are linear combinations of the TE and TM modes of Sec. 4-3, that is, E
TE. _
A (EO'.TE
H
TIoh _
C(H....T£
+ BE....'Bl.) + DH.....T)I)
Determine A, il, C, and D. 4-9. Show that the resonant frequencies of the two-dimensional (no: variation) resonator formed by conducting plates over the x - 0, :e - a, 11 - 0, and y _ b planes arc the cutoff frequencies of the rectangular waveguide. (-10. Following the method used to establish Eq. (2-100, show that the Q due to conductor losses for the various modes in a rectangular cavity arc Q
f/abck?
Til:
(
TK
.) ..0,. -
+ 2ack/ + 2abk.') "aock,' 2lR(ack,' + 2b<:k,' + 2abk,l)
2Gl.(bck.'
Til: (Q.) ..... - 4.tRlbc(k•••
qabck.
Til
(Q').d - 2lR(abk,'
+ 2bck.'' + 2ack.') '1abck..'k.
Til
(Q,)",.,. - 4lR{b(a
where
f/abcku'k,1
+ k"k. l ) + eu:(k,.' + k,'k.') + abk..' k.' ]
+ C)k,1 + a(b + clk,l)
k._ mr
a
k•• = .../k.
'
+ k,l
k
nr '-T
k
pr ·'-7
fr, _ Vk.1
+ k,l + k.'
4-11. Calcull1tc the first tcn higher-order rcsonatlt frequencies for the rectangular cavity of Prob. 2-38. (-12. Consider the two-dimensional parallel-plate waveguide formed by conductors over the z - 0 and x _ a planes, and dielectrics ~I for 0 < x < d and ~I for d < x < a. Show that for modes TM to % the characteristic equation is Eq. (4-45) with
and for modes TE to x the characteristic equation is Eq. (4-47). TEM to z (the direction of propagation) is possible.
Note that no mode
191
PLANE WAVE FUNCTIONS
.-18. Show that the lowest-ordcr TM to % mode of Prob. 4-12 reducC6 to the transmission-line mode either as ' I ..... t l and ~I ..... ~1 or as d ..... O. Show that, if
a«
)..10
for the dominant mode. Show that the static inductance and capacitance per unit widt.h and length of the transmission line are
The usual transmi.ssion-linc formula k, _ Col ..JL{j thereforc applies if {I is small. Also, the field is almost TEM. 4:-14. Consider the dominant mode of the partially filled guide (Fig. 4-6) for b > a. WIlen d is small, Eq. (4-45) can be approximated by Eq. (4-48) for the dominant mode. Denote the empty-guide propagation constant. Cd - 0) by
and show, from the Taylor expansion of Eq. (4-48) about d _ 0 and k. _ tJ., that for Bmall d
4-16. Considcr the dominant mode of t.he partially filled guide (Fig. 4-6) for a Denote the empty-guide propagation constant (d .. 0) hy
>
b.
and show, from the Taylor expansion nf the reciprocal of Eq. (4-47) about d _ 0 o.nd k, - tJ., that for small d k. - tJ.
FI~tfJ. - F1 (.)1 -rIFI (kit + ---a -da + -3ptfJ,
k.')
(d)' a
4:-16. Show that thc resonant frequencies of a partially filled rectangular cavity (Fig. 4-6 with additional conductors covering the z ... 0 and z - c planes) are solutions to Eqs. (4-45) and (4-47) with
whcre n .. 0, 1,2, . . . ; p .. 0, 1,2, . . . ; n _ p _ 0 excopted. 4-17. For the partially filled cavity of Prob. 4-16, show that if c resonant frequency of the dominant mode for smll.11 d is given by
> b > a.
the
192 where
TDtlE-BARIoIONIC ELECTROHAGNETIC FIELDS cq
is the r'e«>Dant frequency of the empty ea.vity,
-. -k
(;J + (;)'
Hint: Usc the results of Prob. 4-14.
4.-18. For the plU'tially filled cavity-of Prob. 4-16, show that if c resonant fr~ucney of the dominant mode for IilUall d i8 given by Col _
"'I
[1 _~ _,_._ ~ - 3,., "., a + c' a l
'1"1'1 (tlJl.I 'Ull
> a>
b, the
1) (~)'] a
wbere All is the reaonant frequency of the empty cavity
Him: Use the roeults of Prob. 4-15. 4.-19. Consider a rectangular waveguide with a cco.tcred dielectric slab, aa shown in the insert of Fig. 7-10. Show that the characteristic equation for determining tho propagation constants of modes TE to :t is
.
k.. tan (,"'.12"d) k-co •• ,('- a-d) - -;; l\.~o-2
and for modes TM to :t it is
where
The dominant mode is the lowest-order TE mode (smallest root for n - 0). 4.-20. Derive Eq. (4-S8). 4.-21. A plane slab of polystyrene (" - 2.56) is ~ centimeter thick. Who.t lllabguide modes will propagate unattenuat.cd at a frequency of 30,000 megacyclea? Calculate the cutoff frequencies of these modes. Using Fig. 4-11. determine the prop&gation constanta of the propagating TE modes at 30,000 mer;4CYcles. Determine t.he propagation constanta of the propagating TM modes by numerical solution of Eq. (4-.56) or (4--68). Bow can the cutoff frequenciee of corresponding TE and TM modee be the same, yet the propagation conlJtants be different. T .-2~. By a Taylor expansion of Eq. (4-56) about II - 0. It - 0, &bow that the dominant TM mode of the lllAb guide (Fig. 4-10) is characleriud by
for small
/I.
..
.
Similarly, show that the dominant TE mode is characterized by 1I - - (k4' - k DI ) " 2
PLANE WAVE FUNCTIOSS
for Iml&l1
/I.
193
.
In each ea.se, the propagation constant is given by
k.-k.+2k. 4.·23. A plane conductor baa been coated witb sbellac (eo- - 3.0) to & thickness of O.OOS inch. It is to be U8ed. in a 3O,()()().megaeyde field. Will any tightly bound surface wave be possible! Calcuhte the attenuation constant in the direction perpendicular to the coated conductor. 4-2'- For the corrugated conductor of Fig. 4-15, it is desired that the field be attenuated to 36.8 per cent. of ita eurfa.ee vaJue at one wlwclength from the eurfaee. Determine the minimum depth of slot needed. ,-~t5. Suppo8e that the elota of the corrugated eonductor of Fig. 4-15 are filled with a dielectric charaeterhed by f4J ~4' Show that. for this case
..
!! k 4 tan k.,d
II -
k. - k.
....
~I + ~ tan
l
k#1
where k 4 - ., "';;;;4. '-~6. Use the TE: mode functioD.ll of Prob. 4-7 for the parallel-plate waveguide 10rmed by conductors covering the ¥ _ 0 and ¥ - b planes. Show that a field baving DO E. is given by Eqs. (4-32) with
•
~-l A.coa!Trt~.
..,
wh~
.. /.' I
A. - -'.,.. .
0
E.
... 0
.>0
eoe-.. orJI d
01-27. Conslder the junction of two parallel-plate tran&niaeion linel of height c for
: < 0 and beight b lor. > 0.
with the bottom plate continuous. (The C10M fJeCtion Using the formulation or Prob. 4-26. show that t.he aperture sWlCCptance per unit width relerred to the aperture voltage is is that of the eecond drawing of Fig, 4-16.)
•
B •
O¥
4 \' sin l (n ..c!b) II>' "" (O"C/b)1 VOl (26/>')1
" .,
where a constant E. has been assumed in the aperture. Compare this with Eq, (4-78). (-28. The centered capacitive waveguide junction is shown i.n }o'ig. 4-26. Show that tbe aperture llU8Ceptanee referred to the maximum aperture voltage fa given by Eq. (4-78) with>" replaced by 2>... It is UlIumed that E. in the aperture is that of the incident mode.
FlO. 4-26. A centered capacitive waveguide junctioD
194
TWE-HABllONlC ELECI'ROllAGNETIC I'lELDS
tx
fx
I
-
Incident
w.W>
FlO. 4·27. A centered inductive waveguide junction. 4.-29. Consider the centered inductive 1I"&veguide junction of Fir,:. 4-27. Aasumin& t.hat B, in t.be aperture is that of the incident mode, &how that the apertl1l'6 IU8CePt.. anee referred to t.he muimum apertuJ't! voltage is giVeD by
B.
_
"
~ (~)' "..'0
./(!!!)' _(~)' 2 ).
[""" (m..r~)]'
k\ '
a
1
"V
(rne/a)!
3,11.7, ••.
•-so. In Eq. (4-83), note that M c/a - 0 the summation becomes similar to an int.egration. Uso the analogy rM./a ,..,. z and cia "'" th to show that
--8._b, 1 0 I _). ,/0_0 ...'/."(";''')' ZI
%d%
Integrate by pArliI, and use the identity'
toahow that
• sin ~ - - d% /.0%1-1 _ b.,
/.2. 0
sin 11 -r- d1l - Si(Z,..)
B.-.... 8i(2
),
c/a_O
) _ 0.226
2...
4.-31. Let there be ••beet of .,..directed current J. over the z - 0 plane of a parallel· plate waveguide formed by conductors over the 11 - 0 and" - b p1aOell. The guide ill matched in both tbe +z and -z direetiotlll. Show that the 6eld produced by the
c\U"rent sheet is
"
• >0 • <0
~ A. cos n...y ,-,..1.1 _ { ll. ~ b -JJ.
.., where
A" -
I"
(.
2b Jo
J.w)
n .....
eos T
dy
'-32. Let. the eurrent. sheet. or Prob. 4-3l be %-directed imteAd cf v-directed. t.hat. 6eld produeed by this Zodirect.ed eurrent. sheet. is
B. -
2:" B" T
.-.
;161' f.
ain
Show
,-T.. lol
. n.....
B" - ..,ob Jo J.(y) 8.ln T
dll
I D. Bierel18 de Baan, "Nouvelles tables d'intkgrales definiCll," p. 225, table 161, no. 3, Halner Publishing Company, New York, lQ39 (reprint).
195
PLANE WAVE FUNCTIONS
'-83. Coosider the cou to waveguide junction of Fig. 4-2&. Only the TE'I mode propagatell in the waveguide, which is matched in both direct.ions. Assume that the aurren\ on the wire varies a8 cos (ll), where I is the diatance from the end of the wire. Show tbat. the input. resistance Ren by the coax ia R _ ~ (Z) ,
Co
, ..
[Bihleoal(c+d) n (re!b) ain id]'
where (Z,)., ia the TE.l ch&raeteristic wave impedance.
Xi--b
-I
T
Ir--::-, L I-- '--i
(aj
Ir---' Coa'
Y
(b)
FlO. 4--28. Coax to waveguide junctions.
'-K Suppc>ee that the coax to waveguide junction of Prob. 4-33 ia changed to that of Fig. 4-286. Show that the input resiatance Ren by the coax is now R _ ~ (z) I
Co
'11
lain (..a/bUrin k(c + d) kacosk(c+d)
sin b:1}'
f.-36. By expanding (sin VJ/w)l in a Taylor aeries about of Eqll. (4-105) becomea
),'1'/1. -
'If
I [ 1 - 6
(ta)' I (to)' 2" + 60"2
tD -
0, sbow that the first
(ta)' + ... ]
I 1008 2"
1m
4.-86. Consider the second of (4-105) as the contour integral , B
" •-
Ile [/.
Eqs.
(I - ,;N)dw ] (14/2)1
c, w' v' w'
where C1 ia shown in Fig. 4-29. Cooaider the closed contour C1 + C, + C. + C" &lid eJ:press M,B. in terma of a contour integral Over C, aDd C.. Show that as ta/2 becomes l&rge, this last conwur integral rOO.ueee to the eeoond of Eqa,
w plane
c, Co
Re
(4-107).
Flo. 4-29. CootoW1l for Prob. 4--36.
196
TIl&E-HAB140NJC ELECTROMAGNETIC FlELDe
4.-11. By expanding coal tD/[(1I'/2)· - will in a Taylor seric. about that the first of Eq•. (4-114) bccomCll
•
..,
tD _
0, abow
(0)"
• 2 \' );0.-; ~ b. >: bl b, 6, b, b, bl
-
_
+1.0 -0.4.67401 +0.189108 -O.05M13 +0.012182 -0.002083
40-88. Specialir;e the second of Eq•. (4-114) to the and U8C the identity (8ee l"rob. 4-30)
f,-
jt
C!\8e
a - 0, integrate by partll,
r
lIin 2z. dz _ ~ rain 1/ d _ ~ SiC..) (11'"/2)' :r:' 1I'}o II 1/ 11'
- !!' 1. SiC...} - ..I ~ - 0.194 ). B. --I> _02..
to show that
4.-St. Show that the 6n;t of Eq•. (4-ll4) reduce. to the contour integral
'G [f
•
(I
+ ''''')",If! dW]
~G. ka_!o"8 Re le, [(.../2)1 where C, ia 8hown in Fig. 4-30.
Conllider the clOlled contour C.
and CXPre&8 G. in terml of a conlour integral over CJ and C,.
+ C. + C. + C.. Evaluate this Ill.8t
contour integral, and ahow that
• '0 -0._4a).
,b-o_
x
1m
ED pltlne
---c+,--E------' Co C,
_/2
-r
h Cl
Fro. 4·30. Conloul1l for Prob. 4-39.
R.
FlO. 4-31. Two parallel-plate trR~ sion lines radiating into half-spllcc.
,-to. Two parallel-plate trans.misaion lines opening onto & conducting plane are excited in oppoeit.e phase and equal magnitude, lUI sbown in Fig. 4-31. Assume E. in
PLANE WAVE FUNCTIONS
197
the aperture is a con.stant lor each line, and show that the aperture susceptance referred to the aperture voltage of one line Us
G • -
B. _
8
J..b
;\"
0
8in' to dto
WI
! (. ;\" J.to to l
V (bip
ttll
sin w dID 4
vw
l
(ka) I
4:·4:1. Construct the vector potential A - UN {or & sheet of t-diteeted currents over tbe 11 - 0 plane (Fig. 4-24) by (0) tbe potential integral method and (b) tbe transform method. Show by use of Grcen'a second identity [Eq. (3-44)] that the twO.p1 are equal. Specialize the potential integral801ution to , _ 10, and show that
f
e-i~
.,.
--+ -4~.
J.( -k cos q, sin
where J.(k.,k.) is given by Eq. (4-121). 4:-4:2. Supposo that tho current in Fig. 4-25 Bnd of magnitude
~
8, -k COB 8)
z.directed rather than z..directed,
..,
J.-C08a
Show tha.t the impedance per unit length, defined by Eq. (4-125), where I is the current per unit length, is given by Eq. (4~126), where Y.p"r' is now the llperture admittance of Fig. 4-23.
C',
CHAPTER
5
CYLINDRICAL WAVE FUNCTIONS
6-1. The Wave FUDctions. Problems having boundaries which coin· cide with cylindrical coordinate surfaces are usually solved in cylindrical coordinates. 1 We shall usually orient the cylindrical coordinate system as shown in Fig. 5-1. We first consider solutions to the scalar Helmholtz equation. Once we have these scalar wave functions, we can construct electromagnetic fields according to Eqs. (3-91). The scalar Helmholtz equation in cylindrical coordinates is
! ~ ( af) + .l.- a.." + a.."2 + k'f p up
p up
p! oq,t
dz
_ 0
(5-1)
which is Eq. (2-7) with the Laplacian expressed in cylindrical coordinates. Following the method of separation of variables, we seck to find solutions of the form
f -
R(p)~(¢)Z(z)
(5-2)
Substitution of Eq. (5-2) into Eq. (5-1) and division by '" yields 1 d (dR) pR dp p dp
1 d'Z
1 d'4>
+ p'4> d¢' + Z dz' + k
,
- 0
The third term is explicitly independent of p and q,. It must a.lso be independent oC z if the equation is to sum to zero Cor all p, q" z. Hence,
.! d'Zt Z dz
_ -k'
(5-3)
•
where k. is a constant. Substitution of this into the preceding equation and multiplication by pi gives
.e.R dp ~ ( dR) + .! d'~ + (k' p dp ~ d~' Now the second term is independent of
p
_ k ') , _ 0 • p
and z, and the other terms are
I The term "cylindrical" is often used in flo more general sense to include cylinders of arbitrary cross section. We are at present using the term to mean "circularly cylindrical."
198
199
CYLINDRICAL WAVE FUNCI'IONS
independent of ¢.
z
Hence,
Id'4> - = '" d~'
-n'
(5-4)
p
where n is a constant. The preceding equation then becomes -p -d ( p -dR) - n l Rdp dp
+ (k l -
• y
k~l)pl
- 0
x
(5-5)
which is an equation in p only. The wave equation is now separated. k.'
+ k.'
Flo. 6-1. Cylindrical coordioat.ell.
To summarize, define k_ as
- k'
(5-6)
and write the separated equa,ions [Eqs. (5-3), (5-4), sud (5-5)1 as d ( p dR) p dp dp
+ [(k.p)' -
n')R - 0
d'''' d~' + n'4> ~~ + k.IZ
- 0 0=
(5-7)
0
The cia and Z equations are harmonic equations, giving rise to harmonic functions. These we denote, in general, by h(n¢) and h(k.,z). The R equation is Buw', equation of order n, solutions of which we shall denote in general by B.(k_p).1 Commonly used solutions to Bessel's equation ar. B.(k.p) ~ J.(k.p) , N.(k.p), H."'(k.p), H."'(k.p)
(5-8)
where J .. (k,p) is the Bessel function of the first kind, N.(k,p) is the Bessel function of the second kind, H.(I)(k,p) is the Hankel function of the first kind, and H.
of••.•.•• -
B.(k.p)h(n~)h(k.z)
(5-9)
lIt iI more usual to denote solutions w Bessel'. equation by Z.(k,llP), but we wiah to avoid conIuaion with our Z(z) function and with impedances.
200
TDU-HAIUlONlC J:LECl'IlO¥AOnrrtC ""LOS
where Ie, and k. Me interrelated by Eq. (lHl). We call these '" ~lenumI<Jry wave Junction,. Linear combinations of the elementary wave functions are also solutions to the Helmholtz equlltioD. We can sum over pO!!llible values (eigenvalues) of 11 and 10" or of 11 and k. (but not over k. and k. for they are interrelatoo). For example,
"/! -
~
f
O.Jo."it.......
- LLO•.•.B.(k.p)h(n.p)h(k••)
(5-10)
•••
where the C•.•• are constants, is a solution to the Helmholtz equation. We can also integrate over the separation constants, although n is usually discrete (this is dieeu!!800 below). We shall, however, have occllsion to integrate over either k. or k.. Thus, possible solutions to Lhe Helmholtz equation are
L1./·(k.)B.(k,p)h(n.p)h(k,z) die. • Y, = LI., g.(k,)B.(k.p)h(nq,)h(k.z) dk, y, -
(5-11) (5-12)
• where the integrations are ovcr any contour in the complcx planc and !.(k,) and g.(k.) are functions to be determined from boundary conditions. We shs.Il use Eq. (5-11) to construct Fourier integrals, as we did in Chap. 4. Equation (5-12) is used to construct Fourier-Bessel integrals. We dillCu8ged the interpretation of the harmonic functions in See. 4-1, a summary being given in Table 4-1. The. coordinate of the cylindriea.l coordinate system is also one of the rectangular coordinates; so the sa.me considcrations that dictated the choice of h(k~) in Chap. 4 apply at present. The.p coordinate is an angle coordinate and, as such, places restrictions on the choice of h(n.p) and n. For example, if we desire the field in a cylindrical region containing all .p from 0 to 2..., it is necessary that y,(.p) - ,,(.p + 2..-) if "it is to be single-valued. This means that h(1I.p) must be periodic in .p, in which case n must be an integer. In most cases, we choose sin (n.p) or cos (n.p) or a linear combination of the two, although in some caBell the exponentisls eI·. and trI·· are more descriptive, or easier to deal with analytically. Thus, the 11 summations of Eqs. (5-10) to (5-12) are usually Fourier serics on .p. Now, consider the various solutions to Bessel's equation. Graphs of the lower-order Bessel functions are given in Appendix D. We note that only the J .(k,p) funotions are nonsingular at p = O. Hence, if a field is to be finite at p - 0, the B.(k,p) must be J.(k.p), and the elementary
CYLINDRICAL WAVE FUNCTIONS
wave functions are of the form y,.,...... = J,,(k,p)el".e fl ••
p = 0 included
201 (5-13)
We have written the harmonic functions in exponential form, which is still general since sines and cosines are linear combinations of them. Note "'from Eq. (5-6) that k, = ± yk' Ie.' is indeterminate with respect to sign. Our convention will be to choose the root whose real part is positive, that is, Re (k,) > 0. 1 Now consider the asymptotic expressions for the various solutions to Bessel's equation [Eqs. (D-ll) and (D-13)]. Note that H .. (2 1 (k,p) are the only solutions which vanish for large p if k, is complex. They represent outward-traveling waves if k, is real. Therefore, if there are no sources at infinity, the B..(k,p) must be H..(tl(k,p) if p ---+ co is to be included. Hence, the elementary wave {unctions become p - . co included (5-14) Other choices ~f cylinder functions are convenient in certain cases, as we shall see when we apply them. Insight into the behavior of solutions to Bessel's equation can be gained by noting their similarities to harmonic functions. It is evident from the asymptotic formulas of Appendix D that, except for an attenuation of l/ykp, the following qualitative analogies can be made: J .. (kp) a.nalogous to cos kp N ..(kp) analogouB to sin kp
H ..W(kp) analogous to ell, H .. (2)(kp) analogous to e- ik ,
(5-15)
For example, J" and N .. exhibit oscilla.tory behavior for real k, as do the sinusoidal functions. Hence, these solutions represent cylindrical standing waves. The H.. Cil and H ..m functions represent traveling waves for k real, as do the exponential functions. They therefore represent cylindrical traveling waves, Il.. w representing inward-traveling waves and H.. C2l representing outward-traveling wavcs. 2 If k is complex, the traveling waves arc attenuated or augmented in the direction of travel (in addition to the l/Ykp factor). When k is imaginary (k = -ja), it is conventional to use the modifu:d Bessel functions J.. and K", defined by I ,Cap) - j'J.( -jap) K,,(ap) =
i (-j) ..+IH,,{2)( -jap)
Ilf k, is imaginary, choose the root according to the limit 1m (k)
(5-16)
-+ O. This direction of wave travel ilJ a consequence of our choice of ,i'" time variation. If we had initially chosen ,-/.,1, then our interpretation of JI"lIl &nd ll"m would be reversed. I
202
Tllol&HARMONIC ELECTROMAGNETIC FIELDS
These are real when ap is real. From their asymptotic behavior, Eqs. (D-19), it is evident that we have the qualitative analogies I.(ap) analogous to eo' K.(ap) analogous to e-'
(5-17)
From these it is apparent that the modified Bessel functions are used to
represent evanescent-type fields.
That the various analogies of EqB.
(5-15) and (5-17) e.'tist is no coincidence. Both Bessel's equation and the harmonic equation arc specializations of the wave equation. In the case of waves on water, a dropped stone would give rise to "Bessel function" waves, while the wind gives rjse to "harmonic Cunction" waves. Table 5-1 summarizes the properties of solutions to Bessel's equation, Our understanding of the physical interpretation, given in the last column, will increase as we apply the various functions to specific problems. When k = 0, we have the degenerate Bessel functions B.(Op) """1, log p 8.(Op) - p., P-
n .. 0
Note that these are essentially the small-argument expressions Cor J. and N •. To express an electromagnetic field in terms of the wave functions 1/1. the method of Sec. 3-12 can be used. The unit z~oordinate vector is a constant vector; 80 we can obtain a field TM to z by letting A = u."", and expanding Eqs. (3-85) in cylindrical coordinates. The result is
E E
,
_!yapa""az 1
.. = fjp
a¢a""az
E. - ~(::,+k')~
(5-18)
H. - 0
which are sufficicntly general to cxprcss any TM (no H.) field existing in a homogeneous source-free region. Similarly, we can obtain a field TE to z by letting F = u.y, and expanding Eqs. (3-88) in cylindrical coordinates. The result is 1
a~
E, - - - P a~
E. _
a~
ap
E. '"'"' 0
(5-19)
TA.8LE 5-1. PnOPERTIEs OF SOLUTfO:<;S TO BESSEL'S EQUATTON
Aher ....livo .... p.......,n~t.lon.
B.(4"...)
H.lIl{.I:,,)
I.(t"l + jN .(t,,)
SmaU-ac&\!II>ent formul ... (.I:........ 0)
z.~
1.781)·t
In1iniliea
PhylllCl&llnterpl'flUlion .I: real--inward-travellng w"ve
l-j~IOg(....:..) r )'.1:" (.1:,,). . 2·{II ---, 2".. r
Lar.. _r&\!II>ent formulas (It,,l .......)
Coy -
"_ 0
1:... ... 0
..)-'i '"
t" ..... j ..
__ j-•• i.p
I) I
" >0
...(.1.:...).
.I.: lm&&;inary_v.. n"",eent field
k" ..... - j... I: complcx_twnuated travelin.. ...·..ve .I.: rul--out ......rd-U'a.vtllnc wave
H.II}{.l.:p)
("pl" . 2'(" - III --+, 2".. 1 ...(t,,)"
, J.(k,,)
....!.... ..)'",
I.(k,,) - jN.(t"l
i:l
2i
..[± ( ...) .1:,,---2
" >0
Infinite number alon..
..)' (1:"-"2-, ...) ,h. r.l:"ain
r(.I:,,)·
kp-o :tj-
I: lm.... inary-two flvall_lIt fi""d. t floropln-loc..Hre
.''''''di". w..ve
.I: real--.b"dinC .... ve
" _ 0
[H.(»(.I:...) _ HoU)(.I:,,)]
2'(" - I)!
4
I"finil.o nuroher ..10... 'hfl ru,1 axi!
axit
" >0
.l:P" 0 .I: ima&illllry-two eva....ee"t 5eldt
,~,
tp-o i j .. t comp\eor-loc&lired .u"di... waves
• "When I: .. -ja. the functiolll'.(jh) .. 1. (a") .. j'J.(-j.,,,) and K.(jtp) .. K.(..,,) .. i{-j)··'H.UI(-j..,,) are "lied. t
fi~d
kp ..... ; ..
...."
-~IO"(~) ., TI:P
-
i~na~van~nt
t complex-attenu.. ted travtllnc wa"e
-flOll
20 .. 1
,
.I.:
t rul_.."di.". wave "_ 0
H[II.(OI{kp) + H.I'I(.I:"lJ
-
.l.:p ..... -j..
j"s-i'p
" >0
(.1:,,).
H.{k,,)
1:..... 0
" - 0
l+j;IOCC:... )
When I: .. O. tbe Beuel f"netion. are I alld lOll", .... O. and
p' .. nd p-' • .....
O.
204 which are sufficiently general to express any TE (no E.) field existing in a homogeneous source-free region. An arbitrary field (one having both an E. and an H.) can be expressed 8S a superposition of Eqs. (5-18) and (5-19). 5-2. The Circular Waveguide. The propagation of waves in a. hollow conducting tube of circular cross flection, called tho circular waveguide, provides a good illustration of the use of cylindrical wave functions. Qualitatively, the phenomenon is Bimilar to wave propagation in the rectangular waveguide, considered in See. 4-3. The coordina.tes to be used are shown in Fig. 5-2. For modes TM to %, we may express the field in terms of an A havin« only & z component 'It. The field is finite at. p - 0; 80 the wave (unctions must be of the form of Eqs. (5-13). It is conventional to express the 4J variation by sinusoidal functions; hence '" _ J.(k ) ,p
ISin cos n~) nq, ,-f'"
(5-20)
is the desired form of the mode functions. Either sin nq, or COB nq, may be chosen; 80 we have 8. mode degeneracy except for the cases n "'" O. The TM field is found from Eqs. (5-18) applied to the above y,. In particularJ 1
E. - - (k' -
9
k.·)~
which must vanish at the conduct-jng walls
p -
/I.
Hence, we must have
J .(k.a) - 0
(5-21)
from which eigenvalues for k, may be determined. The functions J M(Z) afC shown in Fig. D-l. Note that for each n there are a denumerably infinite number of zeros. These are ordered a.nd designated by X"JO. the
x z Flo. 6-2. The circulu waveguide.
y
205
CYLrNDRlCAL WAVE nJNCI'JONS
x I
2 3
•
TABLE &-2. ORDERED ZEROS ~. 0'
0
I
2
3
2.40.5 5.520 8.054 J1 .7fi2
3.832 7.016 10.173 13.324
6.136 8.417 11.620 14.796
6.380 9.761 13.015
J .(:)
•
•
7.688
8.771 12.339
11.005 14.372
first subscript referring to the order of the Bessel function and the second to the order of the zero. The lower order %•• are tabulated in Table 5-2. Equation (5-21) is now satisfied if we choose k
.--
(5-22)
'" a
Substituting this into Eq. (5-20), we have the TM •• mode functions
~. ~ ~ I'
J. ( •••
a
p) Jlcosn¢o sin n¢) .-".,
(5-23)
where n "'" 0, 1, 2, . . . , and p = I, 2, 3, . . .. The electromagnetic field is then determined from Eqs. (5-18) with the above y,. The mode phase constant k. is determined according to Eq. (5-6), that is,
(.~.)' + k.' -
k'
(5-24)
Subscripts np on the k. are sometimes used to indicate explicitly that it depends on the mode number. Modes TE to % are e;'(pressed in terms of an F having only a % component J/I. This wave function must be of the form of Eq. (5-20), with the field determined by Eqs. (5-19). The E. component is 81/I/iJp, which must vanish at p - a; hence the condition J:(k.a) ~ 0
(5-25)
must be satisfied. The J .. are oscillatory fUDctions; hence, the J~ also are oscillatory functions. (For example, J~ "'" -J l .). The J~(%) have a dcnumerably infinite number of zeros, which we order as x~Jl' (The prime is used to avoid confusion with the zeros of the Bessel function itself.) The lowcr-order zeros are tabulated in Table 5-3.
x 1 2
3
•
0
1
2
3
3.832 7.016 10.173 13.324
1.841 5.331 8.536 11.706
3.054 6.706
4.201 8.015 11.340
9.969
13.170
•
•
5.317
6.416 10.620 13.987
9.282
12.682
206
TIME-HAn~rONIC
ELECTROMAGNETIC FIELDS
We now satisfy Eq. (5-25) by choosing
,
k• =~
(5-26)
a
Using this in the wa.ve function of Eq. (5-20), we have the TE..p mode functions
~ TO ~
J
11)1
where n
=
(~)
"a
0, 1, 2, ... ,and p
=
I
sin n¢J
cos nq,
"I',.
1, 2, 3, . . ..
(5-27)
The electromagnetic
field is given by Eqs. (5-19) with the above if. The mode propagation constant is dotermined by Eq. (5-6), which with Eq. (5-26) becomes
(~)' + k.'
- k'
(5-28)
This completes our determination of the mode spectrum for the circular waveguide. The int.erpretation of the mode propagation constants is the same as for those of the rectangular guide and, in fact, is the same for all cylindrical guides of arbitrary cross section if the dielectric is homogeneous. (This we show in Sec. 8-1.) The cutoff wave number of a mode is that for which the mode propagation constant vanishes. Hence, from Eqs. (5-24) and (5-28), we have (k)
,
~"p
If k k~ =
TM
x"p
_
a
-
(k.) .. pTE = ~
(5-29)
a
> k., the mode propagates, and if k < k. the mode is cutoff. 'hI. Y;;, we obtain the cutoff frequencies ~
(I) ."p
Alternatively, setting
=2 k~
=
x.,
_I
:Ira v EIJ
TE_ (I) • "p -
Letting
,
x" P
(5-30)
211'"a VEIJ
'br/X., we obtain the cutoff wavelengths ') TE ( A. "p
= 2'1fa
x'
••
(5-31)
Thus, tho cutoff frequencies are proportional to the X"p for TM modes, and to the x~p for the TE modes. Referring to Tables 5-2 and 5--3, we note that the zeros in ascending order of magnitude are X~l, X01, X~h Xu, and X~I' etc. Hence, the modes in order of ascending cutoff frequencies are TEll, TM o1 , TE u , TM I1 , and TEo I (a. degeneracy), etc. Circular waveguides are used in applications where rotational symmetry is needed. The dominant TEn umode" is actually a pair of degenerate modes (sin 4> and cos 4> variation); hence there is no frequency
207
CYLINDRICAL WAVE FUNC'I'rONS
(a) TEll
(d) TMll
(b)
e-~.~
(c) TBn
TMOl
(e) 2"Eo'l.
9£---
(f)
7M21
FIG. 5-3. Circular wtlveguidc mode patterns.
range for single-mode propagation. (Recall that single-mode operation over a 2: 1 frequency range is possible in the rectangular waveguide.) Note that, except for the degeneracies betwcen TE op and TM 1p modes, TE and TM modes have different cutoff frequencies and hence different propagation constants. The modes of the circular waveguide have HJirected wave impedances of the same form as we found in the rectangular waveguide. For example, in a TE mode, (Z,)" _ E. ~ _ E. _ ~ H. Hp k.
(5-32)
which is the same as Eq. (4-27). The behavior of the Zo's is therefore the samc as in the rectangular waveguide, which is plotted in Fig. 4-3. Attenuation of waves in circular waveguides due to conduction losses in the walls is given in Frob. 5-9. Modal expansions in circular waveguides can be obtained by the general treatment of Sec. 8-2. The mode patterns for some of the lower-order modes are shown in Fig. 5-3. These can be determined in the usual manner (find £ and :JC, and specialize to some instant of time). Field lines ending in the crosssectional plane loop down the guide, in the same manner as they did in the rectangular waveguide. Solutions for cylindrical waveguides of other cross sections also can be expressed in terms of elementary cylindrical wave functions. Representative cross scc;tions arc shown in Fig. 5-4. Note that all or these
208
TIME-HARMONIC ELECTROMAGNETIC FIELDS
b
Ca)
(d)
Cb)
(e)
(.)
(I)
FIo. 5-4. Some waveguide cross sections for which the mode functions arc elementary w&.ve functions. (a) Coaxial; (b) coaxial with baffle; (e) circular with bame; (d) semicircular; (e) wedge; (f) sectoral.
are formed by conductors covering complete p = constant and 4> = constant coordinate surfaces. Wave functions for the guides of Fig. 5-4 are given in Probs. 5-5 to 5-7. 5-3. Radial Waveguides. In the circular waveguide we have plane wa.ves, that is, the cquiphase surfaces arc parallel planes. Wave functions of the form '" - B.(k.p)h(k.z).±i··
with B,.(kpp) and h(k~z) real, have equiphase surfaces which arc int.crsecting planes (the q, = constant surfaces). Such waves travel in the circumferential direction, and we shall call them circulating waves. Examples are given in Prob. 5-10. Finally, we might have wave func· tions of the form
.H.Ol(k,p)j
'" ~ h(k.z)h(n~) (ll."'(k.p)
with h(k.z) and h(nq,) real. These waves have cylindrical cquiphase sur· faces (p = consta.nt), and travel in the radial direction. We shall call them radial waves. l In this section some simple waveguides capable of guiding radial waves will be considered. Radial wa.ves can be supported by parallel conducting plates. DependI These arc true cylindrical waves as defined in Sec. 2-11, but we are using the term "cylindrical wavo function" to mean "a wave function in the cylindrical coordinate system," regardll$S of il.l:l cquiphllJle surfaces.
209
CYLINDRICAL WA VI'J FUNCTIONS
z y
(b)
(0)
FIa. 5-.5. Radial waveguides.
(a) PBrBl1cl plate; (b) wedge; (e) hom.
iog upon the excitation, waves between the plates may be either plane or radial. When the waves are of the radial type, we call the guiding plates a parallel-plate radial waveguide. Figure 5-5a shows the coordina.te sy&tern we shall use. The TM wave functions satisfying the boundary conditions E_ """ E. :z 0 at Z = 0 and z ... a arc l{!••
~
_ (mT) a Z cosn41 IH,O'(k,p») IJ. m (k_p)
- cos
where m = 0, 1,2, . . . ,and n - 0, I, 2, .. k, _
k' _
(5-33)
, and, by Eq. (5-6),
("'.r)'
The electromagnetic field i!5 given by Eqs. (5-18) with the above l{!. TE wave functions satisfying the boundary conditions arc
. . TE -51n _ . (mT JH""(k,p») aZ) cosn411H.. cIJ(k,p)
.,.-..
(5-34)
The
(5-35)
where m '""' 1, 2, 3, . . . , a.nd n -=- 0, 1, 2, . . . , and Eq. (5-34) still applies. The electromagnetic field for the TE modes is found from Eqs. (&-19) with the above!/t. In both the TM and TE cases, the 11..(I)(k,p) represent inward-traveling wa.ves (toward the Z axis), and the IJ... u1 (k,p) represent outward-traveling waves. For a complete set of modes, those with sin nq, variation must also be included. Radial waves are characterized by a phase constant which is a function of radial distance. Following the general definition of Sec. 2-11, we have the phase constants for the above ~'s given by
p ~~ -
[tan-'
N.(k,p)] J .(k,p) 2 1 - TpJ.'(k,p) N.'(k,p) ap
+
(5-36)
210
TW}:-HARJ,lONIC ELECI'RO.l.lAGSETIC FIELDS
Using asymptotic formulas (or the Bessel functions, we find that (or real k, fJI'
II,,,....!
(5-37)
k,
This is to be expected, because a.t large radii the waves should be similar to plane waves on the parallel-plate guide. Note that the phase constant of Eq. (5-36) is that of tbe mode function and not that for the field. Components of E and H transverse to p are not generally in phase. They become in phase at large radii. Each mode of the radial waveguide is also characterized by a single radially directed wave impedance. Using Eqs. (~) and (5--18), we find for outv.·ani-traveling TM modes ~
__ E. _
~
H."'(k,p)
(5-38)
k, H ..Ol(k,p} E. Z -, ~ = 11. = - jWf. H .. (I)I(k,.p}
(5-39)
Z
H.;I.>E H.UI'(k,p)
+"
while for inward-traveling TM modes
Note that for real k, we have Z_,TW = Z+,TI.I*. we find
Z
TIl:
+"
=
E. H• ...
E.
jWIl
Similarly, for TE modes
H,,(l)'(k,p)
T; H.l2>(k,p} -;w~
B ..(II'(kPJ) Z_,TE "" - H • .". ~ H.lll(k,.p)
(5-40)
where the first equation applies to outward-traveling waves and the second equation to inward-traveling waves. Note that the TE wave admittances are dual to the TM wave impedances. It is seen from Eq. (5-34) that k,. is imaginary if mr/a > k. In this case, let k,. - -ja, and
where K. is the modified Bessel function (see Appendix D). The mode functions are now everywhere in phase, and there is no wave propagation. The radial wave impedances become imaginary, indicating no power flow. For example, from Eq. (5-38), if k,. "" -ja,
Z +,.
~ ~ -ja H."'(-jap) ~ ja K.(ap)
j<M. H.(2)'(-jap)
Wf
K~(ap)
(5-41)
which are always capaeitivcly reactive, since K. is positive and K~ is negative. Hence, whenever a < >'/2, the modes m > 0 are nonpropagating (evanescent). For small 0, only the TM o.. modes propagate, for
211
CYLINDRICAL WAVE FUNCTIONS
which Eq. (&-33) reduces to ~,.
)!
~ _ /H.Ol(k p - cos n~ lH.u'(kp)
(5-42)
From Eqs. (5-38) and (5-39) we have the wave impedances for these modes given by Z+~TM
=:;
~
Z_;n.t·
=:;
. H.'''(kp) H..U)'(kp)
-JY]
IH.,,,7(kp)i'
l.ip -
jIJ.(kp)J;(kp)
+ N.(kp)N;(kp)]!
(5-43)
A consideration of the behavior of the Bessel functions (Figs. D-l and D-2) reveals that {or arguments kp < n the N" functions and their derivatives become large in magnitude. Hence, when 2-,;p < n)." the wave impedances become predominantly reactive. Figure 5-6 illustrates this behavior by showing XI R. where Z+~Ttd = R + jX, (or the first five TMo" modes. We shall call kp = n the point of gradual cutoff, the wave impedances being predominantly resistive when kp > n and predominantly reactive when kp < n. Note that these gradual cutoffs occur when the circumference of the radial waveguide is an integral number of wavelengths. From the above discussion it is evident that the TM oo mode is dominant, that is, propagates energy effectively at smaller radii than any other mode. For this mode we have (5-44)
representing inward-traveling waves, and
k'
E.+ = -.-HoW(kp)
JW. H.+ = kH 1(2'(kp)
(5-45)
which represent outward-traveling waves. Note that there are no p components of E or H, the mode being TEM to p. It is called the transmission-line mode of the parallel-plate radial guide, because of its similarity with plane transmissionline modes. For example, at a given radius we can calculate a unique voltage between the plates a.nd a net radially directed current on one of
4 n
3
4
3
1\
\
2
K-
1 0
o
1
2
3
4
5
kp
FIG. 5-6. Ratios of wave reactance to wave resistance for the TM Oto radial modes on the parallel-plate waveguide.
212
TIME-HARMONIC ELEcrnoMAoNETIC FIELDS
the plates. Also, the radial transmission line can be a.nalyzed by the classical transmissioll-line equations with Land C a function of p (Prob. 5-13). Radial waves also can be supported by inclined conducting planes, called a wedge radial waveguide, as shown in Fig. 5-5b. We shall assume no z variation of the field, considering the problem as two-dimensional. TM wave functions satisfying the boundary condition E. "'" 0 at t/> = 0
and t/>
=
cPo are Vtt)'rM "'"
sin (pr ,po
•.(kp»)
~) IHr~/
H,..'4>.(kp)
(5-46)
where p = 1, 2, 3, __ . ,and the electromagnetic field is given by Eqs. (5-18). TE wave functions satisfying the boundary condition E~ = 0 at = tPo are l/!pT";
= cos
(1'cPo< ~) IHf~/··(kp) I Hp.,•• (kp)
(5-47)
where p = 0, 1, 2, . . . , and the elccLromagnetic field is given by Eqs. The interpretation of the modes is essentially the same as that for the TM o.. parallel-plate modes, except that nonintegral orders of Hankel functions appear. This introduces no conceptual difficulties, but if numerical results are desired we would be hampered by a lack of tables for functions of arbitrary fractional order. The radial wave impedances for the wedge-guide modes are of the same form as for the parallel-pla.te guide [Eqs. (5-38) to (5-40)]. We nccd only replace n by 1J7r/q,o and k, by k. These wave impedances exhibit the same characteristic of gradual cutoff for fractional-order Hankel func~ tions as they do for integral-order Hankel functions. Again the transitional point is that for which the argument and order are equal, that is, '[J7f/q,g = kp. The radii so determined correspond to those for which the arc subtending the wedge is an integral number of half-wavelengths long. This is as we should expect from our knowledge of plane waves between p$l.r~lIel plates (the limiting ease rJ'O'- 0). The dominant mode is evidently the TE g mode, in which case, from Eqs. (5-47) and (5-19), we have (5-19).
(5-48)
for inward-traveling waves, and (5-49)
for outward-traveling waves.
This is a transmission-line mode, chamc·
213
CYLINDRICAL Wit. VE Pt1NCTIONB
terised by no E, or H, and possessing a unique voltage and cuceent at aD1 given radii. This mode also can be analyzed by the classical teall&mission-line equatdons for nonunifonn lines (L and C a function of pl. Not< tbat tbe field is dual to tbat of the pamllel-plate line (Eqs. (5-44) Illd (5-45»). Finally, simple radial waves can be supported by the hom-shaped ~e of Fig. 5-5c. called a sectoral horn w<MJ6fluiM. The TM modes are specified by the wave functions 1/I...TM
_
where m - 0, 1, 2,
cos
(~z) sin ("", ~) (H~2,··(k,p)1 a q,o H •." •• (k,p)
, and p ". 1, 2, 3, . ..
(5-50)
The field is given by
Eq•. (5-18), and
k, =
~k' _
(m:)'
(5-51)
The TE modes are specified by the mode functions
fo,T< _
..
sin(~%)C
(5-,';2)
where m - 1,2,3, . . . J and p - 0, 1,2. . . .. The field is given by (~19), and k, by Eq. (5-51). These modes are qualitatively similar to the hybrid modes of the rectangular waveguide (Sec. 4-4). There is,
F.qs.
of course, no transmission-line mode, because of the single conducting boundary. Only the TMOJo modes propagate if a < >./2; these plus the TM 1, and TEl. modes propagate if >./2 < a < X; and so on. Each propagating mode has a radius of gradual cutoff, tws being the radius z at which the guide cross section is about the same size as a rectangular waveguide at cutoff. The TMo l mode is usually considered as the dominant mode. (If a > >./2 one might argue that the TE lo mode is dominant at small radii.) 64. the Circular Cavity. U a 1d eeetion of circular waveguide is closed by conductors over two cross sec..... tions, we have a resonator known &8 y \he circular cavity. This is shown p in Fig. ~7. It is a simple matter X to modify the circular waveguide mode functdona to satisfy the addiFlo. 5-7. The. circulAr uvity.
T
•
--
214
TIME-HARMONIC ELECTROMAGNETIC FIELDS
tional boundary conditions of zero tangential E at The result is a set of modes TM to z, speci.:'ted by
Vt~:q =
J ft
?
= 0 and
(X;p) {:~::} cos (~ z)
Z
E:
d.
(5-53)
where n - 0, 1, 2, . . . ; p =: 1, 2, 3, . . . ; and q = 0, 1, 2, . . . . The field is given by Eqs. (5-18). The set of modes TE to z is specified by
~,. ~ J (x~,p) ·)If
wheren
0=
a
II
0, I, 2, ...
j'P =
field is given by Eqs. (5-19). (5-6)] becomes
sin n~) sin ('1" ,) 1Icos nib d
j and the 1,2,3, . . . ;q = 1,2,3, The separation constant equation IEq.
for the TM and TE modes, respectively. solve Cor the resonant frequencies Tlil {fl r IIPO
.y;; ~X"p
1
_
-
(&-54)
2rG
(j,)':. ~ 2••
Setting k = 2701 V;;, we can
2+
' () d qra
(&-55)
'v,. ~x~" + (~.)'
Each n except n = 0 denotes a. pair of degenerate modes (cos n¢ or sin nIP variation). The X Il , and x~p are given in Tables 5-2 and 5-3. The resonant frequencies for various ratios of d/a are tabulated in Table 5-4. )""""-... FOR THE TABLJ; 5-4. --=-,Ct", -;U.)do_;u~1
-d
•
TM Q1t
TE IlI
0 0.' 1.0 2.0 3.0 4.0
1.0 1.0 1.0 1.0
2.72 1.50
~
1.13 1.20 1.31
~
1.0 1.0 1.0 1.0
e e L rnCULAJ'l. ,\VITT OF RADIUS a AND ENOTJl d
TM ut TM oII 1.59 1.59 1.59 1.59
1.80 1.91 2.08
TEsli
TM ul TE t "
TE I l I
~
~
~
~
2.80 1.63
2.00 1.80
3.00 2.05
5.27 2.72
1.19 1.24 1.27 1.31
1.42 1.52 1.57 1.66
1. 72 1.87 1.96 2.08
1.50 1.32 1.20
1.0
TM lIO
TMoio
2.13 2.13 2.13 2.13 2.41 2.56 2.78
2.29 2.29 2.211 2.29
2.60 3.00 3.00
CYLINDRICAL WAVE FUNCTIONS
/;:;---::::"
0"\'
I / . /~" ./ 1 . 0\
•• •• •• •• •• ••• •• \
.t. To ':0) "fl \ \ ,.. r .\ ...... ~ ;/ /
,""
........ -"./ -// ~-
215
.!J(---.
e--....~
• ° • • • • • • • • • • • • •
Fla. 6-8. Mode pattern (or the TM1 • 1 mode (dominant when dlo :$ 2).
Note that for dla < 2 the TM tII mode is dominant, while for dlo, ~ 2 the TE IIl mode is dominant. If dlo, < 1, the sceond resonance is 1.59 times the first resona.nt frequency. Note that this is very similar to the square-base rectangular cavity of small height (the mode separation is 1.58 in that ease). The TM OIO mode corresponds to the first resonance of a short-circuited radial transmission line. The field pattern of this mode, which is dominant for small d, is shown in Fig. 5-8. The TE IIl mode corresponds to the first resonance of a short-circuited circular waveguide operating in the TEn mode. Its mode pattern is thus that of a standing wave in a circular wnveguidc , similar to Fig. 5-30,. The case dlo, --t 0;0 corresponds to that of a two-dimensional circular resonator, for which the resonant frequencies are the cutoff frequencies of the circular waveguide. The last row of Table 5-4 therefore is also the cutoff frequency spectrum of the circular waveguide. The Q'S of the circulnr cavity are also of interest, especially the Q of the TM OIO mode (dominant for small d). From Eqs. (5-53) and (5-18) we determine the field components of the mode as
Following the procedure of Sec. 2-8, we calculate thc stored energy in the cavity as
w
-~. -. _ !5: 2.d
ffl IEI'dT
(. pJ.' (,,,p) dp a
""f}O
216
TIME-HARMONIC ELECTROMAGNETIC FIELDS
This is a. known integral,' the result being rk 4 cIa' W = 2 J 1'(XOI)
(5-56)
""
The power dissipated in the conducting walls is approximately
1f> IHI' d.
rf>, -
(aX'')' 211" [adJ( l
t
%01)
+ 2 J.'0 pJl'(-(X"p) a dP]
where /R is the intrinsic wave resistance of the metal walls. integral is again known,l and we obtain
iJ>d
=
at
(X~l Y2Ta(d + a)J 1'(XOI)
The above
(5-57)
The Q of the cavity is therefore wOW Q- -
cJl..
~
k 4 da' n--=-::"";';--.---,-;c
2wt(]b;o,'(d
+ a)
Recalling that the condition for resonance is ka = simplify this to 1.202. Q -
X01
=
2.405, we can (5-58)
where f/ is tbe intrinsic impedance of the dielectric. This can be compared to the Q of a square-base rectangular cavity [Eq. (2-102»). It is seen that, for the same height-to-diameter ratio, the circular cavity has an 8.3 per cent higher Q than the rectangular cavity. This i.s to be expected, since the volumc-to-area ratio is higher for a circular cylinder than for a square cylinder. The Q's for the other modes of the circular cavity are given in Prob. 5-16. 5-5. Other Guided Waves. The geometries of some other cylindrical systems capable of supporting guided waves are shown in Figs. 5-9 and 5-10. We treated the analogous plane-wave systems in Chap. 4. The methods of solution for the systems of Figs. 5-9 and 5-10, as well as their qualitative behavior, are similar to those of Chap. 4. For the partially filled radial waveguide of Fig. 5-9a, we can obtain fields TM to z which satisfy the conditions E p E. = 0 at z = 0 and z = a by choosing ::II
Vtl
,p!
= C. cos k.,z cos n~ H ..{!l(kpp) = c! cos Ik.,(a - z)J cos nlf> H ,.(!l(kpp)
(5-59)
1 E. Jahnke and F. Erode, "Tables of FunctioD.ll," p. 146. Dover Publications, New York, 1945 (reprint).
217
CYLINDRICAL WAVE FUNcrlONS
z
z
Conductor
z
(c)
FlO. 6-9. Some radial waveguides. (a) Partially filled; (6) dielectric slab; (c) coated conductor; (d) corrugated conductor.
where n = 0, 1, 2, . . .. The subscripts 1 and 2 refer to the regions z < d and z > d, respectively. We have anticipated that the p and 4J variations must be the same in both regions to satisfy boundary conditions at z ... d. Equations (5-59) represent outward-traveling waves. Inward-traveling waves would be of the same form but with H .(!) replaced by 8.(1). The k's in each region must, of course, satisfy the separation relationships k,! k,!
+ +
kIll kl!!
DO
kl! = k!! =
W!~1PI
(5-00)
W 1t!}l1
The field vectors themselves arc obtained from Eqs. (5-18), using the y,'s or Eqs. (5-59). To evaluate the G's and k" we must satisfy the conditions that E" E., H" and H. be continuous at z = d. For E, we ha.ve
1[ a' (1- tJll - -1)] tJI!
[B,I - E"J..., = -:-
JW
~
up uZ
«=l
fl
.-1
=
0
which reduces to
(5-61)
, kit CI sin kold = -k., C sin k. , (a - Ii) ft
For E. we have
fl
1[ (1
1)]
(E. l - E. , ]-.:I = -.- - a' -tJlt - -1/11 Jwi/. iJ4J 81. fl ~1
.-1
=
0
218
TIME-HARMONIC ELECTROMAGNETIC FIELDS
which also reduces to Eq. (5-61).
IH,. -
For H, we have
H,,]..... - ;
[aa~ ("'. - "',)]... . ~ 0
which reduces to (5-62)
0 1 cos kdd = C S COB kd(o - d)
Finally, for H. we have
- [!-ap ("'. -",,)] which a.gain reduces to Eq. (5-62). yields k.. tan k"d tl
~
.-4
~0
Division of Eq. (5-61) by Eq. (5-62)
_ k.. tan Ik.,(a - d») ts
(5-63)
The kd and kd are fun~tions of k, according to Eq. (5-60); so Eq. (5-63) is a. transcendental equation for determining possible k,'s. Once k, is evaluated, the ratio CI/Ct may be obtained from either Eq. (5-61) or Eq. (5-62). For fields TE to z we can satisfy the condition E, = E. = 0 at z = a by choosing 1/11 = C1 sink. 1zcosnq,H..U)(k,p) (5-
where n
= 0, 1,2, . . . j and Eqs. (5-60) must again be satisfied. The
field components are found from these ""s by Eqs. (5-19). tangential components of E and H at z = d yields k d cot k.1d __ kd cot [k 12(a - d)] J,ll
Matching
(5-65)
J,l2
as the equation for determining k, for TE modes. It is interesting to note that the characteristic equations for the partially filled radial waveguide lEqs. (5-63) and (5-65)] are of the same form as those for the partially filled rectangular waveguide {Eqs. (4-45) and (4-47)}. This we could have anticipated, since at large p the Ha.nkel functions reduce to plane waves, as shown by Eqs. (D-13). The modes of the partially filled radial guide can be ordered in the Mme manner as were the modes of the partially filled rectangular waveguide. The dominant mode is the lowest-order TM mode (logically designated the TM oo mode). It reduces to the radial transmission-line mode in the empty guide and has no cutoff frequency. For a « 11. it can be analyzed by conventional transmission-line concepts. It should be apparent from our treatment of the waveguide of Fig. 5-9a that the characteristic equations for the radial waveguides of Fig. 5-9b, c,
CYLINDRICAL WAVE FUNCTIONS
219
and d will be of the same form as those for the plane waveguides of Figs. 4-10,4-13, and 4-15. We need only to repla.ce the k.'s by k,'s. Hence, for the dielectric-slab radial waveguide of Fig. 5-9b, the characteristic equations are
~
for modes TM to z, and
"2 "" taD "2 ""
"" ""
(~)
-2"cot2"
~taD~ 2 2
l
(5-67)
- un 2" cot "" 2"
for modes TE to
z. The u and v are related ut _Ill
+ k,t _
+ k,t =
to k, by
k~t
..
wtf~
kot
=
wtE~O
(5-68)
Possible solutions to these equations can be obtained graphically by the method of Fig. 4-11. Just as in the plane-wave case, the lowest TE and
TM modes have no cutoff frequencies.
The cutoff frequCDcies of the
mades in general are given by Eq. (4-63). The modes of the coated~nductor radial waveguide of Fig. 5-9c are those of the slab waveguide having E, = E. = 0 over the mid-plane of the slab. The dominant mode is the lowest TM mode, which has no cutoff frequency. The cutoff frequencies of the modes in general are given by Eq. (4-&1). Finally, for the corrugated-eonductor radial line of Fig. 5-9d, the characteristic equation for the dominant mode is
k, = k o VI
+ tan' kod
(5-69)
This is analogous to Eq. (4-71) in the plane-wave case. The circular waveguide systems of Fig. 5-10 are interesting, because, except for rota.tionally symmetric fields, the modes arc neither TE nor TM to any cylindrical coordinate. The systems of Fig. 5-100, b, and c have the common property that they are If two-dielectric" problems. We can consider them all at oncc, as follows. Let region 1 be the inner dielectric cylinder in each case and region 2 the outer one. We then choose electric and magnetic ""s ~
in region 1, and
",.. I :z A B... I(k,lP) cos nq, cj1l.. y,-l ... BB•• I(k,lP) sin -n.; cit..
(5-70)
CB."'(k,tP) cos -nq, e-fk.. DB.·t(k,tP) sin nq, r i " ••
(5-71)
",.., = ~' =
220
TLME-1lAIWONIC ELECl'ROHAGNETIC FIELDS
X
X
Z
Z f~"2
/
y
..,., cU'"
y
Conductor (a)
(b)
X
Z
Z
y
y
(,)
(d)
Pto. ~10_ Some ci«:ular waveguides. (4) Partially filled; (6) dielectric alab; (e) coated conductor; (d) corrugated conductor.
in region 2. The "'- determine partial fields according to Eqs. (5-18) and the 1ft' determine partial fields according to Eqs. (5-19). The total field is the sum of the two partial fields in each region. The B ..(k,p) denote appropriate solutions to Bessel'l:! equation of order n, chosen 80 as to satisfy all bounda.ry conditions except those at the interface p = a. In each region the ,p'S must satisfy the separation relationships
+ k. k k,t' + k.' "'" k k,I'
t
-
t
1'
"'"
W El/ll
1'
=
",'ElIlt
(5-72)
The requirements that H" E., H., and E. be continuous at p E,k,l'AB.-l(k'la) = E1k"tCB.-t(k,sa> IAtk,I'BB..'I(k,ta)
=>
a lead to
pJk"IDB.·t(k,za)
Ak,tB.._l'(k,t a)
+ Bk.n B.d(k.. 1a) = wp.la
Ck..1B...1/(k..sa)
AKin B."'(k.. 1a)
+ Bk.. 1B.oI'(k.. 1a) _
Ck.n B.-1(k..sa)
WEta
=
+ Dkln B.d(k.. sa) ClIp.~
WEJ
+ Dk..1B"d'(k,sa)
These equations have a nontrivial solution only if the determinant of the
CYLINDRICAL WAVE FUNcrlON8
eoefficien ts of A, B, C, and D vanishes. F I "" B Fa = B
I(k.la) S(k.ta)
221
Hence, defining,
F s "'" B ..-I(k.ta) F. = B..ot(k.sa)
(5-73)
The characteristic equation in determinantal form is f.sk.t'F I 0 k,IF:
k.n F , --
"",a
When n = 0, the field characteristic equation
f.lk.,sF s 0 0 0 Illk,s2F IlJc,I'Fs 4 k.n F k.n F (5-74) k,sF; =0 W""Ia W",,' k.n F , k,sP~ k'IF~ W'" separates into modes TE and TM to z, and the is much simpler. It is
-
,
--
.
-
k,tFIF~
(5-75)
- k,IFiF. = 0
lor TM modes (n ." 0), and
k,sF tF~ - kplF;F'.
=
0
(5-76)
for TE modes (n = 0). We must now pick the proper F functions for the various cases, For the partially filled circular waveguide (Fig. 5-10a), the field must be finite at p = 0; hence
FI To satisfy E. = 0 at
p =
= P, = JII(k,la)
(5-77)
b, we choose
F, - J.(k,..)N.(k,ob) - N.(k,..)J.(k"b)
Furthermore, to satisfy E. "'" 0 at
p "'"
(5-78)
b, we choose
F. ~ J.(k,..)N;(k,ob) - N.(k"a)J:(k,ob)
(5-79)
The dominant mode is the lowest-
For the dielectric-rod waveguide (Fig. 5-lOb), the field must again be finite at p = 0; so Eqs. (5~77) still apply. However, external to the rod, the field must decay exponentially above the cutoff frequency and represent outward-traveling waves below the cutoff frequency. Hence, we choose (5-80)
Once again, the dominant mode is the lowest n
= 1 mode, and its cutoff
TIME-HARMONIC ELECI'nOMAGNETIC FIELOS
222
3
l--/
@J
/
1
" "
.
°
0.4
0.2
1.0
0.8
0.6
alb
Flo. 5-11. PhlLSC constant (or the part.ially filled circular waveguide, b _ 0.4).1. (After H. Seidel.)
II -
10£1,
frequency is zero. 1 Some solutions for the k. of the dominant mode are shown in Fig. 5-12 for the case E1 = EO and ~l = Jl2 = po. Note that ko < k. < k l , which is the same relationship that applies to the dielectric-slab guide of Sec. 4-7. For the coated conductor of Fig. 5-lOc we must again have exponential decay of the field as p -+ CIO; so Eqs. (5-80) still apply. However, to 1 S. A. SchelkunofT, "Electromagnetic Waves," pp. 425-428,.0. Van Nostrand Company, Inc., Princeton, N.J., 1943.
10 '0
f -
3
V
I
00:4(0
V
• I
°
02
0.4
0.6
2.5 4:1
~G 0.8
1.0
al>'
FIo. 5-12. Phase constant for the circular dielectric rod.
(After M. C. GrG1I.)
223
CYLINDRICAL WAVE FUNc-rIONS
satisfy the condition E. = 0 at
p =
b, we should choose
F, = J.(k.,a)N.(k.,b) - N.(k.,a)J.(k..b)
and, to satisfy E• .", 0 at
F1
-
P .",
(5-81)
b,
J.(k,la)N~(k,lb)
- N.(k,la)J~(k ..lb)
(5-82)
For this guide the dominant mode is the lowest n r: 0 TM mode, which bas no cutoff frequency. (Compare it with the dominant mode of the plane coated conductor of Seo. 4-8.) Copper wire with an enamel coating can be used &8 an efficient waveguide for some applications. I Finally, the corrugated wire of Fig. 5-1Od can be analyzed in a mAnner similar to t.hat used for the corrugated plane (Fig. 4-15). The field extel'nal to the corrugated wire will be essentially the dominant TM (n ... 0) mode of t.he coated wire. The field in the corrugations will be essentially that of tho shorted parallel-plate radial transmission line. The characteristic equation is obtained by matching wave impedances at the corrugated surface. As the radius of tbe corrugated cylinder becomes large, the solution approaches tha.t for the corrugated plane. fi·6. Sources of Cylindrical Waves. In this section we shall consider two-dimensional sources of cylindrical waves, that is, sources independent of the z coordinate. The extension to three dimensions can be effected by a Fourier transformation with respect to z (see Sec. 5-11). Suppose we h.&ve an infinitely long filament of constant &.--Q current along the z axis, 0.8 shown in Fig. 5-13a. From the theory of Sec. 2--9, we should expect the field to be TM to z, expressible in terms of an A having only a z component "'. From symmetry, '" should be independent 1 G. Goubau, Surface-wave Tra.nlfmission Linee, Proc. IRE, vol. 39, no. 6, pp. 619624, June, 1951.
z
y
I
'-I" Y
,
P'
X
p
X Ca)
(b)
FIo. 5--13. An infinite filament of collltaDt a-e current (0) along the I" axis aDd (b) placed parallel to the I" axis.
m..
224
of ¢ and z.
TIM.E-BARMONIC ELECTROMAGNETIC FIELDS
To represent outward-traveling waves, we choose A. - ~ - CH,"'(kp)
where C is a constant to be determined according to lim '+'H.pd¢ = I ~o'f
Evaluating H
:=
V X A, we find
H. _ - Of _ -C 2. [H,"'(k,)] ~ j2C 8p
dp
t_O -rp
The preceding equation then yields I
C=4j I A. - ~ - 4j H,"'(k,)
Hence,
(5-83)
is the desired solution. The line current is the elemental two-d.imensional source, just as the current element (Sec. 2-9) is the elemental threedimensional source. The electromagnetic field is obtained from Eqs. (5-18), using the 1ft of Eq. (5-83). The result is (5-84)
Thus, lines of electric intensity run parallel to the current, and lines of magnetic intensity encircle it. Equiphase surfaces are cylinders, but E and H are not in general in phase. However, at large distances we have E. -
-.kl ~s;rp rT
H. = kl
'-;"1
(5-85)
~8.,;kp j g-li,
which is essentially an outward-traveling plane wave. The amplitude of the wave decreases as p-~t, in contrast to the r 1 variation in the threedimensional case. The outward-directcd complex power crossing a cylinder of unit length and radius p is
P,
1PE H··ds 102'r E.H:pd4J _";! Ikl!'H,"'(kp)[H,""(kp)]'
=
X
= -
The real part of this is the time-average power flow
(SJ"
(5-86)
which, by virtue
225
OYLINDRICA.L WAVE FUNcrIONB
of the Wronskian [Eq. (0-17)1, reducee to /PI -
Re (PI)
_ .k Ill'
(5-87)
4
Hence, the time-avemge power is independent of the distance from the source, as we should expect. It could be more simply obtained from Eqs. (&-85). If the current filament is not along the z axis but parallel to it, we can extend Eq. (5-83) by replacing p by the distance (rom the current to the field point. In radius vector notation, we specify the field point by p-uzX+UI/Y
and the source point (current filament) by '1' - Uz:z;'
as shown in Fig. 5-13b. point is then
+ u,y'
The distance (rom the source point to the field
I. - .'1 - V(x = .,;pI
x')'
+ p'l
+ (y
if)'
2pp'
cos (I/>
1/>')
We emphasize that A. is evalua.ted at '1 by writing A.(p) and that J is located at p' by writing I(ri). We can now generalize Eq. (5-83) to read
A.(.) ~ l~./ H."'(kl. - .'1)
(5-88)
This is our (re&space Green's (unction (or two-dimensional fields. The solution for two or more filaments o( z...directed current can be represented by a summation of the A/a from each current clement. Suppose we have two filaments of equal magnitude but opposite phase, as represented by Fig. 5-14a. As the separation 8--+ 0 and the magnitude 1_ 00 such that 18 remains constant, we have a two-dimensional dipole y
y
-I
j.-."'-lO""'+:-I;-'---X"" (aj
Flo.
~I4.
Sources of higher-order wave-.
(0) Dipole aource; (6) quadrupole aoUl'Ce.
226
TIME-HARMONIC ELECTROMAGNETIC FIELDS
source. Note that A. at a point (xtY) due to a current filament at (x',O) is the same as A. at (x - X',Y) due to a current filament at (0,0). Hence, for Fig. 5-14a, the vector potential is
A. - A+ -~.y) - A.'(Z +~.y) where A,l is that due to a single current filament at the origin [Eq.
(5-83)].
In the limit 8 --+ 0 the above equation becomes A,_ _0
(JA.l
-8-
ax
18
a
= - -. - (H o(2)(kp)] 4J ax
The differentiation yields kl. A.... 4j H 1 (2 J(kp) cos ¢
(5-89)
Thus, the vector potential of a dipole line source is a cylindrical wave function of order n """ 1. For the quadrupole BOUTee of Fig. 5-14b we have, by reasoning similar
to that above,
where (5-89).
is the vector potential of the dipole 8Ouroe, given by Eq. Hence, \ -kIsts! a A. 4; oy [H,"'(kp) cos ~J
A.(2)
which reduces to
A. =-
k2~?8t
1:l 2(2l(kp) sin 2q,
(5-90)
Thus, the vector potential of a quadrupole line SOurce is a. wave function of order 11. = 2. This procedure can be continued to obtain sources for the higher-order wa.ve functions. It can be shown (Prob. 5-29) that, when A. is a wave function of order 11, a possible souree consists of 211. current filaments equispa.coo on an infinitesim.al cylinder. We shall call such a SOurce a multipole source of order n. The dual analysis applies to the case of magnetic current filaments. It is merely necessary to replace I by K and A by F in the various vector-potential formulas of this section. For example, from Eq. (5-88), the electric vector potential at p due to a magnetic current filament at p' is F.(p} =
K~r) H ,'''(kip
- p'[)
(5-91)
Using both electric and magnetic multipoles, we can generate an arbitrary source-free field in homogeneous space (P > 0).
CYLINDRICAL WAVE PUNCI'lON8
z
Fro. 5-15. A cylinder of uniform cmrcnt.
y
The field due to a cylinder of Currents can be obtained quite simply by treating the problem as a boundary-value problem. We shall consider here only a cylinder of uniform z-directed surface current. (The general case is considered in Prob. 5-30.) The geometry of the problem is illuSotrated by Fig. 5-15. Because of the rotational symmetry, we choose
of
I A.- l A.+
~ C,J.(kp)
~ CI H.(t'(kp)
p
p>a
The boundary conditions to be satisfied are
where J. is the density of the z-dirccted current sheet. Using Eqs. (5--18) with the above ¥to and satisfying the boundary conditions, we obtain
- ; .kaJ.H,'''(ka)J.(kp) E. -" 1- -2 ,kaJ,J.(ka)fl,"'(kp) T
p
p>a
as the only component of E. Let us calculate an impedance per unit length for this source, as we did for the ribbon of current in Sec. 4-12. By definition, p Z - jl[i where P is the complex power per unit length r
p = - Jo'J E.J:ad4l =
-2raJ:E.I..._
228
TIME-HARMONIC ELECTROMAGNETIC FIELDS
a.nd I is the total z-directed current
f
J =}o
2.
J.a
d~ ~ 2~aJ.
Hence, the impedance per unit length is
z ~ ,k4 J.(ka)H,"'(ka)
(5-93)
Using small-argument formulas for J a and H 0(2) I we obtain
. r ka ) z __O2"-, ( "--J21og2
(5-94)
la.....
where"Y = 1.781. Compare this with the Z of a ribbon of current [Eq. (4-127)]. Tho resistances (real parts) are identical. The reactance of a cylinder of current of small diameter d is approximately equal to the reactance of a ribbon of current of width w = 2d. More generally, it CaD be shown l by a quasi-static approximation that the impedance per unit length of a small elliptic cylinder of minor axis a and major axis b is the same as that of a circula.r cylinder of diameter d ~ ).f(a
+ b)
)
A ribbon is the special case a = 0 and b = w. 6-7. Two-dimensional Radiation. We can construct the solution {or an arbitrary two-dimensional distribution of currcnts by dividing the source into elemental filaments of current and summing the fields {rom all elements. For example, if' we have a J~, independent of z, each element J ds' produces &. vector potential 0
dA.
~J418' Ho"'(kle
-e'l)
where ds' is an element of area perpendicular to z. entire sourcc, we have
A.
~
;j II
Summing over the
J.(e')H."'(kle - e'l) dB'
where the integration extends over a. cross section of thc source. Since the equations for A z due to J~ and for Al' due to Jl' a.re of the same form as those for A~ due to J o, the above equation also applies for z replaced by x or y. Combining components, we have the vector equation
A(e) -
;j II
J(e')H,"'(kl. - e'l) d8'
(5-95)
I R. W. P. King, "The Theory of Linear Antennas," pp. 16-20, Harvard University PreM, Ca.mbridge, M88S" 1956,
CYLINDRICAL WAVE PUNcrlONB
229
representing the solution for an arbitrary two-dimensional distribution of electric currents. The cases of surfa.ce currents and current filaments are included by implication. The electromagnetic field is obtained, as usual, from H - V X A. The electric vector potential due to two-dimensional magnetic currents M is given by the formula dual to Eq. (5-95), or F(p) -
i ff j
M(p')H."'(klp - p'l) M
(5-96)
'the electromagnetic field in this case is given by E = - V X F. When the field point is distant from the source, our formulas simplify to a. form similar to those for three-dimensional radiation (Sec. 3-13). For klo - fl'l large, the Hankel function can be represented by the asymptotic formula
Furthennore, when p» p', as shown in Fig. 5-16, we have
If' - f"I--+
P -
p'
cos
(~
-
~')
(5-97)
The second term must be retained in the phase factor, exp (-jkle - £1'1), but not in the magnitude factor, 10 - (l'l-~. Hence, the vector potential. of Eq•. (5-95) and (5-96) reduce to
(5-98)
provided p »p~w
These arc the radiation-zone formulas corresponding
to Eqs. (3-95) in the three-dimensional ca.sc.
y Fio. ,&-16. Geometry (or determining the radia.-
So",,,,
tion field.
x
230
TIME-HARMONIC ELECTROMAGNETIC FIELDS
We now have the p variation explicitly shown in Eqs. (5-98), and simplified formulas for the radiation field can be obtained. A13 evidenced by Eq. (5-85), the distant field of a single current filament is essentially an outward-traveling plane wave; so the superposition of fields from all current elements should also be of this type. Hence, in the radiation zone, E, - .H.
E. - -.H,
(5-99)
which can be verified by direct expansion of Eqs. (3-4), using Eqs. (5-98). To obtain the field components, let us again divide the field into that due to I. given by H' = V X A, and that due to M, given by E" = - V X F. Retaining only the dominant terms (p-~ variation), we obtain
H.
= jkA.
H~ =
-jkA.
E': = -jkF. E~' =
jkF.
in the radiation zone. The corresponding E~, E~, H~', and H:' can be determined from Eqs. (5-99). The total field is simply the sum of the primed and doublo--primcd components, or )
E. = -jwp.A. - jkF. E. - -jwp.A. + jkF.
(5-100)
in the radiation zone, with H given by Eqs. (5-99). These formulas correspond to Eqs. (3-97) in the three-dimensional case. Note that, except for the contrasting p-J1 and r- 1 dependences, the radiation fields are of similar mathematical forms in two and three dimensions. 6-8. Wave Transformations. It is often convenient to express the elementary wave functions of one coordinate system in terms of those of another coordinate system.' We refer to expressions of this type as wave transformations. Some representative wave transformations are derived in this section. Others will be derived as they arc needed. Suppose we have the plane wave e-iz , which we wish to express in terms of cylindrical waves. (The conventional coordinate orientation of Fig. 5-1 is assumed.) This wave is finite at the origin and periodic in 2'11' on 1/>. Hence, it muet be expreesible 80S rf:r< = e-i,.-.:o::
• L ,,-- ..
a..J .. (p)e"'·
where the a.. are constants. To evaluate the a.., multiply each side by r~ and integrate from 0 to 2'11" on
10
2 ..
rip -.e-"". d
= 2'11"a""/...(p)
I Two coordina.te systems are considered to be distinct if their origins or orienta· tions are different, even though they may be geometrically t.he same.
231
CYLINDRICAL WAVE FUNcrlONB
The left-hand side is actually a well-known integral, but we need not recognize this, The mth derivative of the left-hand side with respect to p evaluated at p - 0 is j--
..
2r_j-cos·
J.
Tbe mth derivative of the right-hand side evaluated at p ... 0 is 2";0../2-, Hence,
and we have shown that
•
e-P
= e-1- - - =
l j-..J .. .. --.
(p)e iro•
(5-101)
and also that (frI02)
J.(P)
Equation (5--101) is the wave transformation expressing the plane wave ria in terms of cylindrical wave functions. It is closely related to the so-ealled "generating function" of Bessel functions.! Another wa.ve transformation of interest is that which corresponds to a tmnslation of cylindrical coordinate origin. Consider the wave function '" ~ H,(I)(I(l -
(I'D
= Ha(ll(Vpl
+ p"
2pp' cos (41
41')]
where p and p' are as defined in Fig. 5-13b. We can think of '" as the field of a line source at p' in terms of a cylindrical wave function having its origin at the source. We shall reexpress '" in terms of wave functions referred \:.0 p ".. O. In the region p < p', permissible wave functions are J.(p)ei"., n an integer, for'" is finite at p "'" 0 and periodic in 2... on 41. In the region p > pi, permissible wave functions are ll.. (I)(p)e;"·, n an integcr, for'" must represent outward-traveling waves. Also, '" must be symmetric in primed and unprimcd coordina.tes (reciprocity). Rcnce, f is of the form • b.H."'(p')J.(p)ei",....., p < p'
L L• b.J.(p')H•.,'(p)ei",.....,
It _ _ •
p> p'
It _ _ •
wbere the b.. are constants. To cvalua.te thc b.. , let p' -+ 00 and 1/>' - 0, and use the asymptotic formulas for the Hankel functions. Our original I R. V. Churcbill, "Fourier Series and &undary Value Problems," p. 141, MeGrawHill Book Company, Inc., New York, 1941.
232
TIM.&RAR.MONIC ELECTROMAGNETIC FIELDS
and our constructed expression for !f becomes
These are now representations of a plane wave, and, from Eq. (5-101),
it follows that b.. ... 1. Thus, •
2:
H ."'(p')J.(p)""f-'"
F
<
p'
(5-103)
•
2:
J.(p')H."'(p) ...,.....,
p> p'
This equation is known as the addition theorem for Hankel functioDs. It is also valid for superscripts (2) replaced by superscripts (1), since H.Ul = H.(!)·, Adding the addition theorem for BoU) to that for Bo(l), we obtain J .(1.
-
.'1) -
2:• J .(P')J.(p)...,....., .--.
(5-104)
which is the addition theorem for Bessel fUDctions of the first kind. An addition theorem for Bessel functions of the second kind is obtained by subtracting that for H O(I) from that for HoOl. 5-9. Scattering by Cylinders. A source radiating in the presence of a conducting cylinder is onc of the simplest "wave-scatter" problems (or which an exact solution can be obtained. We shall at present consider only two-dimensional cases. Extension to three-dimensional cases can be efleeted by the method 01 Sec. 5-12. Let us first consider a plane wave incident upon a conducting cylinder, as represented by Fig. 5-17. Take the incident wave to be z-polarized, that is, (5-105)
Using the wave transformation or Eq. (5-101), we can express the incident field as • E.' = E.
.--.2:
i-·J .(kp)....
233
CYLINDRICAL WAVE FUNCTIONS
y
p
Flo. 6-17. A plane wave incident upon a conduct... ing cylinder.
-
Incident
wave
x
The total field with the conducting cylinder present is the sum of the incident and scattered fields, tha.t is,
E.
=
E.'
+ Eo'
To represent outward-traveling wa.ves, the scattered field must be of the form Eo' .,.
Eo
2:•
j-"a..H .. m(kp)ejr,,~
(5-106)
hence the total field is E.
~ E,
•
l
..... - .
j-[J.(kp)
+ a.H."'(kp»)e"'·
(5-107)
At the cylinder the boundary condition E• ... 0 at p - a must be met. It is evident from the above equation that this condition is met if -J.(k.)
'. ~ H.'''(ka)
(5-108)
which completes the solution. The surface current on the cylinder may be obtained frOID
Using Eqs. (5-107) and (5-108), and simplifying the result by Eq. (0-17), we obtain (5-109)
In a thin wire the n = 0 term becomes dominant, and we have essentially a filament of current. Using the sma.ll-argument formula for H o(2), we
TDdE-RA.R.IolONIC ELEcraOKAGNETIC FIELDS
find the total current as h 2rE, 1- o J,ad~~. JWp. Iog ""
f,
(1;-110)
Hence, the current in a thin wire is 900 out of phase with the incident field. The pattern of the scattered field is also of interest. At large distances from the cylinder we can use the asymptotic formulas for H ..(I), and Eq. (1;-10ti) becomes • i E I ' ---+ Eo "\j;kp [2Jk eft' L, ~ a.tI-...~,..... a __ •
where the a. arc given by Eq. (5-108). The magnitude of the ratio of the scattered field to the incident field is therefore
_(21 ,,--'\''-< •
~ IE.'1 - V:;kp
This is the scattered-field pattern. dominant and
J .("") H.'" (ka)
""'1
(1;-111)
For small ka, the n = 0 term becomes (1;-112)
The scattered-field pattern for a thin wire is therefore a circle, whicb is to be expected, since the wire is essentially a filament of current. When the incident field is polarized transversely to %, it can be expressed as • H.' = Hrril:s "'" H o
L j-J.(kp)e""·
(1;-113)
Again, the total field is considered as the sum of the incident and reflected fields, that is, H, = H.' + H.' fo represent outward-traveling waves, the scattered field is of the form
no'
•
= H.
l
a __ •
j-b.H."'(kp)""·
6Dd the total field is given by •
H. - H.
l
. ---
j-·[J.(kp)
+ b.H.'''(kp»)''''·
(1;-114)
235
CYLINDRICAL WAVE FUNCTIONS
= 0 at p = a.
This time our boundary condition is E. equations 1 E, - -. (V X a,H,),
From the field
J~
=
~ H,
L:.-0 •
j-'[J;(kp)
+ b.H.""(kp)),;·'
and the boundary condi tion is met if -J;(ka)
(5-115)
b. = H"C2l'(ka)
An incident wave of arbitrary polarization can be treated as a superposition of Eqs. (5-105) and (5-113), When the incident wave is polarized transversely to z, the surface current on the cylinder is (5-116)
For small ka, the n = 0 term becomes dominant. However, the n = ± 1 terms radiate more efficiently and cannot be neglected, as we shall now show. At large distances from the cylinder, the scattered field becomes • H,' - + H 0
/2j e-I",
..,......
with b" given by Eq.. (5-115). to incident field is thus
V1fkp
b"eJlof
The magnitude of the ratio of the scattered
~ _ I2
IH,'I -
\'
....'-'- .
•
"JTkp
I 1..<'\'
"00-"
J;(ka) M H.""(ka) '
I
(5-117)
For small ka we find
jT(ka)'
n=O
4
J;(ka) H,,(ll'(ka) =
jT(ka)'
Inl
4
jT(ka(2),,"1
InJl(ln -
1) 1
= 1
Inl >
1
Hence, for thin wires the scattered-field pattern is
]H,'l"I- T(ka)' ~2 IH.. "a-oO - 4 - ."-kp 11 -
2 eos ~I
(5-118)
The n "" 0 term of Eq. (5-116) is equivalent to a ~-directcd magnetic
236
TlME-HARMON1C ELECTROMAGNETIC FIELDS
y p
Current p' filament
Fto. 5l"18. A current filament. parallel to a conducting cylinder.
x
Conductor
current filament, while the n = ± 1 terms are equivalent to a y..<J.irected electric dipole. A more general problem is that of a current filament parallel to So can..
dueting cylinder, as shown in Fig. 5-18. (Plane-wave incidence is the special case p' -. co,) When the filament is an electric current I, the incident field is
-ktl E.' = - - H,'''(kl. - .'1) 4w. For p
< p' we have,
(5-119)
by the addition theorem (Eq. (5-103»),
E.i =
~:I
l: •
H,.(2)(kp')J ..(kp)ei"<......·)
To this we must add a scattered field of the same form, but with the J. replaced by H II (2), namely• • E..
=
4:12:
c..H"Ul(kp')H..{2)(kp)ejro(·-")
(5-120)
From the preceding two equations it is evident that J.(ka)
(5-121)
c. - - H.U'(ka)
satisfies the boundary condition E. '"" E.l solution is •
L: ~:I 2: ,,-- .
-kif 4w.
E.=
+ E.' =
O.
Thus, our final
Il.'''(kp')[J.(kp)
+ c.H.u'(kp)]e!"''-'"
p
< pi
H,,(t)(kp)[J ..(kp')
+ c.H.(2)(kp')]e"'(......·)
p
>
•
p'
(5-122)
237
CYLINDRICAL WAVE FUNCTIONS
Note that our answer is symmetrical in P, tP and pi, ~' (reciprocity). Note also that the "reflection coefficients" of Eq. (5-121) are equal to those of Eq. (5-108) and are, in general, in· dependent of the incident field. Specializing the second of Eqs. (~122) to the far zone, we have •
E, 0;:;: I(p) _
2: . ---
j. [ J .(kp')
J.(ka) H '''(k ')] ,,"''-''J B ..Ul(ka)· p
The magnitude of this is the radiation field pattern. Figure 5-19 shows the radiation pattern of a current filament O.25~ away from a conducting cylinder of radius 3.75A. The radiation pattern of a current filament 0.25" in front of a plane reflector is Fla. 5-19. Radiation pattern for a. cur· rent filament 0.25). away from a cylinshown for comparison. The patterns drical reflector of radius 3.75), (plane of Fig. 5-19 are also valid for current reflector case shown dllllbed). elements of finite length as long as the reflector is of infinite extent. If the line source of Fig. 5-18 is a magnetic current filament K, we have
instead of Eq. (5-119). The problem is dual to the electric current case, except that the reflection coefficients at the conducting cylinder must be those of Eq, (5-115) instead of those of Eq, (5-121), Therefore, the final solution will be dual to Eq. (5-122), or
2: . --. ~~:K 2: ,,--~~:K
H.=
•
H.U'(kp')(J.(kp)
+ b.H."'(kp»)e'·''-'"
p
H.'''(kp)[J.(kp')
+ b.H.U'(kp')[,,"''-''
p> p'
< pi
•
where the b. are given by Eq. (5-115).
(5-123) According to the equivalence
238
TIME-HARKONIC ELECTROMAGNETIC PlELDS
principle, the field of a nATrOW slot in conducting cylinder is the same as the field of a magnetic current on the surface of a conducting cylinder. Specializing the second of Eqs. (&-123) to the case p' "'" a, ~' - 0, p --+ co, we have • \' j"e i'" II. - !(P) n __ ~ • 11.""(1<0) 8.
The magni tude of this is the radiation pattern of a "stitted cylinder." Figure 5-20 shows a. slitted-cylinder pat.tern for the ease a = 2>.. The pattern for a slit in an infinite ground plane is shown for comparison. The patterns of Fig. 5-20 are also valid for slits of finite length as long as the conductor is of infinite extent. 5-10. S
E. =
I aJl."'(kP')J.(kp) sin ,(~. - 0) sin ,(~ I a.J.(kp')H.'''(kp) sin ,w - sin ,(~ -
l
p
0)
p> p'
•
•
I
< p'
0)
0)
(&-124)
Problema involving conductol"l over tol'l1phk coordinate surfaces are usually easy t....o .. - constant. coordinate surfaces.
to solve. In this cue the wedge covel'lJ
239
CYLINDRICAL WAVE FUNCTIONS
which satisfies reciprocity and insures continuity of E. at p = p'. To satisfy the boundary conditions E, = 0 over ¢ =- a and ¢ = 211' - a, we choose mr (5-125) m = 1,2,3, .. v = 2('lf a) The a. Ilre determined by the nature of the source. To evaluate the a., we view the current element as a.n impulse of current on the surface p = p'. The boundary condition to be satisfied at a current sheet is J. - H.(P'+) - Il.(p'-) Using the field equations and Eq. (5-124), we find
--!:- "Lt a.H.U)(kp')J~(kp) sin v(¢' -
JWIJ
H. =
a) sin v(¢ - a)
p
< p'
p
>
•
~ Lt " a.J.(kp')1l.(1l'(kp) sin v(¢'
JWIJ
- 0:) sin v(¢ - a)
p'
•
Thus, using the Wronskian [Eq. (D-17)J, we have the surface current given by J. =
-2, \ ' a. sin v(¢' - a) sin v(¢ - 0:)
W/,l1rp
1...1 •
This is simply a Fourier series for the current on p = p'. The Fourier sine series for an impulsive current of strength I at ~ = r/l' on p = p' is
J~ =
('II" I a)p"
2:
sin v(¢' - a) sin v(r/l - 0:)
•
By comparison of the preceding two equations it is evident that y
a. =
-WIJ,7rI
2(r - a)
(5-126)
This completes the solution. To obtain the radiation pattern of a current I near a wedge, use the asymptotic formula for H.(2)(kp) in the second of Eqs. (5-124). This, with Eq. (5-126), gives
E.~ I(p) 2>"J.(kp') •
sin v(r/l' - a) sin v(¢ - a)
pi
Current filament
~'
Condudor
•
x
Flo. 5-21. A current filament adjacent to a conducting wedge.
240
TIME-HARMONIC ELEcrRQMAGNETIC FIELDS
FIG. 5-22. Radiation patteroe for an electric current filament adjll.Cent to a. conducting hal! plane, p' _ a, .' ... ~/4. (Ajur J. R. Wail.)
where v is given by Eq. (5-125). Figure 5-22 shows some ra.diation patterns for the special case a = 0 (the conducting half plane). Another special case of interest is that of plane-wave illumination. This is obtained by letting the source recede to infinity. In this case, the incident field becomes
This is recognized as the pla.ne-wave field
E.' E
where
,
=
Eoei t ,
~
-4-
- fJJJJI
-
c,-,')
f2T e-ItII V;:kpi
(5-127)
The total field in the vicinity of the wedge is obtained by specializing the first of Eqs. (5-124) to Ia.rge p'. This gives E.
kp,__ l
~T"%' e-i"'"
L: •
a"j-J.(kp) sin v(rI,' - a) sin v(.p - a)
241
CYLINDRICAL WAVE FUNCTIONS
Finally, substituting fnr a" from Eq. (5-126) and for I from Eq. (5-127), we obtain E.
~ r2rE. " j'J,(kp) sin vW - 0.4
- a) sin v(. - a)
(5-128)
,
where 1/ is given by Eq. (So.125). This is the solution Cor a plane z-polarized wave incident at the angle 41' on a wedge of angle 2a. For 0 we have •
0. ...
E.
=
2Eo ~ jf'/'Jf'/!(kp) sin
.. ,
n:' sin n24J
(5-129)
which is the solution for a plane wave incident on a conducting half plane. The Halmost dual" problem (dual except for boundary conditions) is that of a magnetie-current filament K at p', 4J' in Fig. 5--21. We construct a solution
L
HI""
!
b.H,"'(kp')J,(kp) cos vW - a) co. v(. - a)
p
< p'
•
2: b.J.(kp')H.(t)(kp) COBV(41' ,
a) cosv(4J - a)
p> p'
(5-130)
which is similar to Eq. (5-124) except for the sines replaced by cosines. The boundary conditions E~ - 0 at q, "" a and 4J = 2r - a can now be satisfied by choosing 1/ -
mr 2(... a)
111. -
0, 1,2, . . .
(5-131)
The coefficients b. are determined by the nature of the source, in a manner analogouB to that used to obtain Eq. (5-126). The result is v
~
0 (5-132)
v>O whicb completes the solution. The radiation pattern of a magnetic current K near a wedge is obtained from the second of Eqs. (5-130) by using the asymptotic expression for H.I!)(kp). The result is
H. ...... ---+ 1(P) L, " ..j'J,(k,') cos vW - a) co. v(. - a) ,
242
TIME-HARMONIC ELEC'l'ROYAGN"ETIC FlELDS
where Neumann's number t. i.s 1 (or", - 0 and 2 for 1/ > O. Figure 5-23 shows some radiation patterns (or tbe special case a = O. When q,' - a we have the solution (or a radiating slit in a conducting wedge. Finally, for plane-wave incidence we can specialize the first oC Eqs. (5-130) to the case p' --+ co. The procedure is analogous to that used to establish Eq. (5-128), and the result is H. = TH, '\' ••j'J.(kp) cosvW - a) 7f -
a
1::1
cosv(~ -
a)
(5-133)
•
This is the field due to a plane wave polarized orthogonally to z incident at an angle tIJ' on a wedge of angle 2~ The case a = 0 gives (5-134)
which is the solution for a plane wave incident on a conducting half plane. 6-11. Three-dimensional Radiation. A thrro-dimensionnl problem having cylindrical boundaries can be reduced to a two-dimensional problem by applying a Fourier transformation with respect to z (the cylinder
Flo. 5-23. Radiation patterM for a magnetic current filament adjacent. to a conducting half plane, p' - II, . ' _ ../4. (A.fUr J. R. Wait)
243
CYLINDRICAL WAVE FUNCTIONS
z
axis). 1 For example, if t/;(x,y,z) is a solution to the three-dimensional wave equation
a' a' a' ) ( ax' + ay' + az' + k' '" then J,(x,y,w) =
f
-- -
r'
l(z)
- 0
_*".. t/;(x,y,z)e-i¥· dz
y p
will be a solution to the two-dimensional wave equation
(:;, + :;, + «,) J, =
x
0
where «' = k' - w'. Once the twodimensional problem for J, is solved, the three-dimensional solution is obtained from the inversion 1 ~(x,y,,) - 2.
f"_"
FlO. 5-24. A filamCDt of curreDt aloDg
tho z a.xis.
~(x,y,w)"" dw
This is usually a diffioult operation. Fortunately, in the radiation zone the inversion becomes quite simple. We shall now obtain this far-zone inversion formula. Consider the problem of a filament of z-directed current along the z axis, as illustrated by Fig. 5-24. The only restriction placed on the current l(z) is that it be Fourier-transformable. In the usual way, we construct a solution H-VXA (5-135) A = u.t/; where t/; is a wave function independent of ¢ and representing outwardtraveling waves at large p. Anticipating the need for Fourier transforms, we construct
~ - i. J-"" f(w)H,"'(p yk'
w')...• dw
which is of thegenersJ form of Eq. 105·)1). Tne Fourier transform 0/ jP is evidently ~ - f(w)H ,'''(p yk' w') The I(w) is determined by the nature of the source, according to
10'1.. R. p d¢ -;::t lew) I
Thill applies to cylinders of arbitrary cross sectioD as well as to circular cylinders.
TIME-HARMONIC ELECTROWAGNETIC FIELDS
where D. and I are tDe transforms of HI and I. ment formula for a,en, we have
!If
From the small-argu-
2·
B. - - -iJp --+ ..1.f(w) ,......0 rp and the preceding equation yields few) -
It')
Hence, the "transform solution" to the problem of Fig. 5-24 is
f"
-1.., 8rJ
f
_ ..
l(w)H,"'I...v'k'
lew) -
where
w')...• dw
f-"" I(z')clw'dz'
(5-136) (5-137)
The field is obtained from !/I according to Eqs. (5-135). Compare the equations of t.his paragraph to those of the second paragraph of Sec. 5-6. The transformed equations in t.he three-dimensional problem are of the same form as the equations in the twCHiimensional problem. Another solution to the problem of Fig. 5-24 is the "potential integral solution II oC Sec. ~9. This is J.
•
~
f"_.
I(z')
.-J"/~+<. 4.< V p' + (,
o'l'
")'
dz' (5-138)
with the field given by Eqs. (5-135). It can be shown that the.p is unique in this problem. Hence, Eqs. (5--136) and (5-138) are equal, giving us a mathematical identity. For example, if 1(:) is & short current. element of moment Il, then 1(10) "., Il and Eq. (5-136) becomes
!/I
gO
.!i.. f" IJoUl(p Vk 8"1 -. f -
and Eq. (5-138) becomes
1
wt)ei"'-dw
Ile-/!r
~
Equating these two !/I's we have the identity
f"
elk' 1 _ = -; II ,(t)(p v'k t r 2J-"
w 1 )ei"'- dw
(5-139)
Many other identities can be established in a similar fashion. It is convenient to have two forma for 'f because some operations are easier to perform on one form than on the other. For example, it is simple to specialize Eq. (5-138) to the radiation zone, and we did so in Sec. 2-10. In particular, the specialization is given by Eq. (~l22), which
CYLINDRICAL WAVE PUNcrION8
can be written as
ri"
"'~ 4n l(-k COO B) where lew) is given hy Eq. (5-137).
E, __ ~.
By Eq. (3-97) we have
-jw~A, =jw~sin8'"
e-p,·
E, -;:::;: jw~ 4Tr sin 8 1( - k cos 8)
or
(5-140)
(5-141)
Hence, the radiation field is simply related to the transform of the source evaluated at 10 - -k cos 8. More important, the specialization of Eq. (5-140) must also be the corresponding specialization or Eq. (5-136). We therefore have the identity
f
-
_.l(w)H''''(py'k·
rik, w·) ...·dw~2j-r-l(-kcooB) (5-142)
which holds for any function 1(10). Equation (5-142) can also be estal>-lished by contour integration, using the method of steepest descent. I Finally, we shall need a formula similar to Eq. (5-142) valid for Hankel functions of arbitrary order. The desired generalization can be effected by considering the asymptotic expression
f2i j"~ --'Vn
H ..U}(:x:) ----+ from which it is evident that
--
8 ..(2)(%) ---+ j"B ,(I) (:x:) AAlong as" ¢ 0 or 7', we have p -+ 00 as r --+ co I since P = r sin 8. Also, if k is complex (some dissipation assumed), then .y k' - 10' is never zero on the path of integration. We are then justified in using the asymptotic formula for Hankel functions and can replace the HoC') of Eq. (5-142) by ;....8 ..(1). The result is
f-·.
rI'O'
l(w)H.U'(P y'k'
w·)...·dw--+ 2-j'+'1(-k cos B) ~.
r
(5-143)
We shall have use for this formula in the radiation problems that follow. 6-12. Apertures in Cylinders.' Consider a conducting cylinder of infinite length in which one or more apertures exist. The geometry is I
A. Erde1yi, "Aaymprotic Expanaiona," pp. 26-27, Dover Publieationa, New York,
1956. t Silver &nd Saunders, The External Field Produced by a Blot in an Infinite Cir· cular Cylinder, J. Appl. Ph"., vol. 21, 00. 5, pp. 153-158, February, 1950.
246
TDUi-HAlWONIC ELECTROMAGNETIC FIELDS
z r
FlO. &-25. An aperture in a condu.ctiog cylinder.
y
p
x
------ -
shown in Fig. 5-25. We seck a solution for the field external to the
cylinder in terms of the tangential components of E over the apertures. Anticipating that we shall use transforms of the fields, let us define the "cylindrical transCorms l l of the tangential components of E on the cylinder as
J." d~ f" E.. (n-,tD) = -2 1 J." dljl f" B.(n,1D) = -2 1 r
0
T
-.
0
_.
dz E.(a.~.z)c~e-/W· (5-144)
dz E.(a,~,z)r~rjw·
The inverse transformation is
2: ".. f.", 2~ 2: ,;"' !-", .. --. 1
"
= 2T
E.(n,.,),·" dw (5-145)
,
E.(.,¢,z) -
E.(n,")&-· d.,
Note that these nre Fourier series on 41 and Fourier integrals on %. The field external to the cylinder can be expressed as the sum of a TE component and TM component. According to the concepts of Sec. 3-12, the field is given by E - -'0 X F -
;w"A +.,!.. '0'0' A
H ."" V X A - jwiF where
A "" u.A.
JW'
+ JW. J- vv· F F - uJ'.
(5-146)
(5-147)
247
CYLINDRICAL WAVE FUNCTIONS
We now construct the wave functions A. and F. as
(5-148)
"
F.
=1. \'
~ ,, __ 4w
eM
f"-. g.(w)H."'(p Vk' - w')e'" dw )
which are of the form of Eq. (5-11). We choose the Bessel functions as to represent outward-traveling waves. We choose the rp and z functions such that the field will be of the same form as Eqs. (5-145). To determine the f,.(w) and g.. (w) in Eqs. (5-148), let us calculate E. and E. according to Eqs. (5-146). The result is H,.(I)
" 1..(
E.(p,~,,)
= -2 1 \'
E.(P,tP,z)
"".!. \' 2r L.
rJWf
tJ·. f"-_ (k' -
"
.... --
ei ••
f"__ [- ~w f .. JWf
(w)H .. (ll(p Vlc 1
+ g.. (w) v'lcl -
Since these equations specialized to have
f.(w) _
w')!.(w)H."'(p Vk'
p =
-
w 1 1I,,(I)'(p Vk 1
w')t!-· dw
Wi)
-
Wi)] eiw' dw
a must equal Eqs. (5-145), we
jw.E.(n,w) (k 1
_
w 1)H"U)(a Vk2
g.. (t.r) = vk! _
Wi
fI..
Wi)
(~l'(a Vlcl _
Wi) [
2.(n,w)
(5-149)
+ a(k,nw w') E.(n,W)] This completes the solution. The inversions of Eqs. (5-148) are difficult except for the tar zone, in which case we ean use Eq. (5-143). Hence, we have
(5-150)
248
TWE-HARMONIC ELECI'ROKAGNETIC FIELDS
z
Z
-a3
f-a':::
I
I
-J.~I--'
I
II 'f. ---
Flo. 5-26. A conducting cylinder and (a) an a,;ia.I slot, (b) a circumferential slot.
11--·a--1
v--
I
I
-~,
-~,
(b)
(a)
Finally. in the radiation zone Eqs. (3 97) apply j hence R
E, --+ jwp ,.......
rib ain
rr
8
-
\ ' eJrl.j"+!j.. ( -k cos 6)
Lt
n--"
(&-151)
1t _ _ OO
Thus, the radiation pattern of apertures in cylinders is relatively easy to calculate. The only difficulty is that the number of significant terms in the summation becomes veri)' large for cylinders of large diameter. To illustrate the theory, let us consider the thin rectangular slot in the two orientations shown in Fig. 5-26. For the axial Blot we shall assume in the aperture
E.
=
a. V
rz
-cosL
!
-~
a
2
2
(5-152)
--<~<
and E. = O. (This approximates the case of excitation by a rectangular waveguide.) For a very narrow slot (a -+ 0) the transforms of Eq. (5-144) become
'" ( ) _ VL cos (wL/2) n,w - a ... 2 (LtD)!
J!J.
249
CYLINDRICAL WAVE FUNC"I'lON&
.nd B.(n,w) = O. g.(w) -
Frnm Eqs. (5-149) we then have /.(w) - 0 and
(~.
VL cos (wL(2) (LtD)']a yk' _ w' H.Ul'(a yk'
w')
Finally, by Eqs. (5-151) we have the radiation field given by E, -= 0 and
~ .....sin 9) E. - Vr.......[COS(¥COS8)] (kL)t '-' H.(tl~(1uJ -cos9 11 _ rJar
1-
(5-153)
~
which can be further simplified to a cosine series in';. The radiation pattern in the plane B = 90° is identical to that of the stitted cylinder; 80 for a 2>. the pattern is given in Fig. 5-20. The "vertical" pattern in the .; 0 plane is almost indistinguishable from the radiation pattern of the same slot in an infinite ground plane. l For the circumferential slot of Fig. 5-26b, we assume in the aperture
E....
1
v
~. -C08-
W
a
-~<'<~ a
a
(5-154)
-"2<.<"2
and E• ... O. (Again this approximates excitation by a rectangular waveguide.) For a narrow slot (W --+ 0) the transforms of Eq. (5-144) become - ( ) _ Va cos (na(2) E• n,1D r1 (na)' and B,(n,w) - O. Then Cram Eqs. (5-149) .nd (5-151) we can calculate the radiation field as kVae-iir " E, =.JrT'sm . e
•
Lt [rt 11---
j" cos (na/2) t/". (nc:r)']H.. (t)(1uJsin 8)
E __ Va"- cot'8 ~ nj' cos (na(2) .... • ...,.ka sin 8 Lt [rt - (na)')HII
(5-155)
11---
In the principal planes 8 - r /2 and ~ - 0, the field is entirely B-polarized. However, in other directions, the cross-polarized component E. may be appreciable. The radiation patterns for circumferential slots in reason~ ably large cylinders are very close to the radiation patterns for the same I L. L. Bailin, The Radiation Field Produced by a Slot. in a Large Cireular Cylinder, IRE Trani., vol. AP-3, no. 3, pp. 128-137,.July, 1955.
TIME-ElAlWOllo'lC ELEcraOllAGNETIC FIELDS
Fla. &-27. Radiat.i.on pat.tern for a circumferential slot of length 0.65>" in a conducting eylinder of diameter 3>" (same slot in a ground plane shown dashed).
slot in an infinite ground plane. To illustrate t.his, Fig. 5-27 shows the radiation pattern in the plane 0 = 11/2 for a circumCerential slot O.65X long in a cylinder 3X in diameter. The radiation pattern for the same slot in an infinite ground plane is shown dashed. 5-13. Apertures in Wedges. The problem of diffraction by a conductor is reciprocal to the problem of radiation by apertures in the conductor. By this, we mean that a solution to one of these problems is readily converted to a solution to the other by using the reciprocity theorem. We shall illustrate the procedure Cor the case of conducting wedges. Figure 5-28 shows the reciprocal problems of (a) a current element and a conducting wedge and (b) an aperture in a conducting wedge. To keep the theory simple, we shall consider only the case of a distant current element and the radiation field of the aperture. For the z-directed elcctriccurrent clement of Fig. 5-28a the field will be TM to z, expressible in terms of an A = u.~. The incident field is
e-1l1r-r'1 "" - Il
4;1,
<'I
which, when r» r', reduces to
-
"-'''' "" "'" Il--ei"..
_·tJi,·...,-(......,
(5-156)
This is simply a plane wave incident upon the wedge. The'" in this three-dimensional problem is subject to the same boundary eondition (w = 0) on the wedge as is E. in the two-dimensional problem of Sec.
251
CYLINDRICAL WAVE FUNCTIONS
Hence the solution must be of the same (orm as Eq. (5-128), that
5--10. is,
f - 2<>1-, "\' j'J,Ckp' sin 0) sin vW - 0) ,in v(~ - 0) ?< a 4 , "4'0
where
e-jh hr
= I l - &"1:1'000'
(5-158)
m.
V
(5-157)
=
"2(".=""a")
In terms of y" the field is given by Eqs. (5-18). This completes the solution to Fig. 5-28a. To obtain the solution to Fig. 5-28b, we apply reciprocity (Eq. (3-35» to the region bounded by the conducting wedge. Because of the bound· ary conditions on E at the conductor, Eq. (3-35) reduces to
- JJ E,bH: ds .,."
(5-159)
!lE,·
where the superscripts a and b refer to the fields of Figs. 5-28a and b, respectively. From Eqs. (5-18) and (5-157) we calculate H: = p,(;'7f'/!o a)
z
L•
vjoJo(kp' sin 9) cos v(4)' - a) sin v(4) - a)
z
n
y
y
x
x
(a)
(6)
FlO. 5-28. The reciprocal problems of (a) a current clement and a conducting wedge and (b) an nperture in a conducting wedge.
252
TlME-HA..BMONlC ELECTROMAGNETIC FlELOS
Specializing this to the surface ,p' = a, we can reduce Eq. (5-159) to
f" dz' f." dp'
IlE. - -
Finally,
1/10
__
0
~T/ooE.,) a
p r
\ ' vj'J,(kp' ain B) sin '-'
,
is given by Eq. (5-158), and in the radiation
.(~ -
a)
ZODB
E, __ ,E. SlD
9
Hence, the 8 component of E in the radiation zone is given by Jl
E, - 2 (
r",
e-
a
;
1.(10,11,)
where
.
8m sa
fJ \ ' vj" sin v(,p - a) J,(k cos 8, k sin 0) '-'
(5-160)
,
f"
&v'dz
-.
10" J .(up) dp -1 E.(p,o,z)
(5-161)
P
0
Note that f.(tlJ,u) is of the form of a Fourier transform on z and a FourierBessel (or Hankel) transform on p.l In a similar manner, the E. component of the radiation field caD be obtained by applying reciprocity to Fig. 5-284 with Il replaced by Kl. This z-directed magnetic-eurrent element gives rise to a field TE to z, expressible according to F = u.". The incident field is then specified by Eq. (5-156) with 1 replaced by K. Again the three-dimensional problem is essentially the lS&IDe as t.he two-dimensional problem of Sec. 5-10. The solution is then of the form of Eq. (&-133), t.hat is, >/-
~ T ~. \ ' •• j'J,(kp' Bin B) coo ,(~' ex Lt ,
where
- a) COB ,(~ - a)
(5-162)
....'"
';'0 = K l - eft.. _· hr
v = 2(...
ex)
m =
(5-163)
.9,
I, 2,
The electromagnet.ic field is found from'" according to Eqs. (5-19). To relate this solution to the field from an aperture in a conducting wedge, we again apply reciprocity [Eq. (3-35)}. This reduces to
ff
.,..,
(E,'H: - E,'lI.·) d.
~ KIH,'
(5-164)
where superscripts a and b refer to the fields of Fig. 5-2& with Il replaced by Kl, and of Fig. 5-28b, respectively. From Eqs. (5-19) and (5-162) we 11. N. Sneddon, "Fourier Tranaforma," p. 6, McGraw-Hili Book Company, Inc., New York, 1951.
253
CYLINDBJCAL WAVE FUNctIONS
calculate
H: ...
1fkl sin /1 cos /1 ( ) wp. 1f a
l H.' = .1fk ( sin I 8) 1f1J JWIJ. 1f a
L:. L: ¥t,
,
.
f.J·J~(kP' Stu 8)
cos v(t/>' - a) cos vet/> - a)
f.j·J.(kP' sin 8) cos v(t/>' - a) cos vet/> - a)
•
Finally, we evaluate Eq. (5-164) and use the radiation-zone relationship
E. _ -"fB,
= ~H. Sin
8
The result is
"'-~. E. - 4r(1f a)
where
'\' e.j· cos vet/> L, ,
Q'.(w,u) =
- a)(cos 8 g.(k cos 8, k sin 8)
(&-165)
+ j sin 8 h.(k cos 8, k sin 8)]
J--.. e
it..
dz fo - J~(up) dp E.(p,a,z)
h.(w,u) ~ /_"" .... dz / ; J.(up) dp E.(p,a,z)
(5-166)
We now have a complete solution for the radiation field from apertures in conducting wedges. As an example, let us calculate the radiation from a narrow axial slot of length L, as shown in Fig. 5-29. We shall assume that in the slot
E.
~
..
VI(p - 0) cas L
(5-167)
is the only tangential component of E. The I, Q', and h functions [EQs. (5-161) and (5-166)] are then found to be I.~O
8,-0 cas (wL/2) J ( ) h• ~ 2.VL 7'1 (Lw)! • ua From Eq. (5-160) we see that E. - 0, and from Eq. (5-165) we have E•
""
I() T
· 8 caslk(L/2) cas 8) ... 1 (kL cos 8)1
8m
L: .,j·cos,(~
- a) J .(kosin 8)
(5-168)
•
where v =~, 1,
%, . . ..
Plots of
FlO. 6-29. A narrow axial slot in a conductine halr plane.
254
TIME-HARMONIC ELEctROMAGNETIC Fl.ELDS
Flo. &-.30. Radiation patterns for axial slot.ll in a conducting half plaoe (the slot in fLn infinite ground plane is shown dashed).
the radiation pattern in the plane 8 - 90 0 arc shown in Fig. 5-30 for the case ex - 0 (half plane). The cases a ... 0.16A and a = 0.96>. are sbown, with the infinite ground-plane pattern shown dashed for comparison. PROBLEMS
6-1. Show that Eq. (5-12) is a IlOlution to the ecalar Bclmholtl equation. 6-2. Show that", - Oog p)e-'k ia a 1I01ution to the sealar llelmholu equation. Determine the TM field generated by this" according to Eqa. (5-18). Sketch the t and :Je Jines in a ~ - constant plane. What pbysicalllystem IIUpporla this wave? Repcat for the TE case. 6-S. For two-dimensional fields (no :e variation) ahow that an arbitrary field in a llOurce-frce homogeneous region can be eJl:prCMC'd in ten.Ila of two lICalar wave fuoctioDa, ,p, and lJ-1, according to Eqs. (3-79) whcro A - u"py.,
Note that. ihis corresponds to choosing
*
j;P(~~)
~
_ _j~(F:)
illlltead of Eqs. (3-80). 6-4. A circular waveguide has & dominant mode cutoff frequency of 9000 megacycles. What ia ita inside diameter if it. is air-filled! Determine the cutoll frequencies for the next ten loweJ~rder modeJ. Repeat for the case f. _ 4. 6-6. All the waveguides whose CtOSll eeetiona are shown in Fig. f).4 are characl.erized by wave functions of the form 'I- - B.(.c,p)h(n.)eslA'••
where TM modes are detenninoo by Eqs. (&-18) and TE IDOdea by Eqs. (&-19). phue constant ia given by
The
CTLTh"DRICAL WAVE FUNCTIONS
255
Let. a denote the inner ndius and b the outer radius of the eoaxiaJ. waveguide of Fig. 5-411. Show that for TM modes B..(i",) _ N ..(i,o)J.(i"p) - J ..(i,a)N..(i"p)
h(n.) _ &in n.
or
COlI
n•
..here n - 0, 1.2, . . . , and i, is a root of
Show that for TE modes B.(k",) - N~(kptJ)J..(k,p) - J:(k,a)N..(k,p) h(n.) - sin n. or cos n •
..here n _ 0, 1.2, . . . ,and k, is a root of
G·8. Show that the modea of the coaxial waveguide with a bame (Fig. 6-4b) are ehal'$Cterizcd by the lAme B.(k",) fWlction. u the coaxial guide (prob. &-5), but for Tl\f modes n - ~. 1.
H.2, .
&lid for TE modes n -
..here the baffle iI at • - O.
0, }i, 1, M•.•.
The dominant mode is !.he lowen TE mode with
• -)i.
5·7. Show that the ,.,.edge waveguide of Fig. 6-k supportA TlI,l modes specified by ~na
....... &lid k,IJ is a
_ J .(l-".) ain "'. cua,T 2... :w
n--.-,-.·· . • a .... KJ'O
of J.(lllS).
Show that it aupporta TE modes specified by
....
I/-TZ _ J.(i".) COIn. e*ia..
• 2• n-O-.-,··· '
• h""
and kpa is a tero of J~(k,a). The guides of Fip. 5-4c and d are the special cases ... - 2.. and T, respectively. G-8. Show that the cutoff wavelengtb for tbe dominant mode of the circular waveguide witb bame (Fig. 5-4c) iI
1-'. Using the perturbational method of Sec. 2-7, ahow tbat tbe attenuation eonIl&nta due to conductor loaaes in a circular waveguide arc given by
.. '" "'1
~ - :;;-:;7i~:7:ff,iii for all TM modes, and by
.. - '" VI"
V.tn'
V.tn'
[(>:,>:'- .' + GY]
256
TU.I];-RARMONlC
ELEC'I'ROMAGN'ETlC Fl,ELDS
(or all TE modes. Note that for the "circular electric" modes (n - 0) the attenua,.tion decreases without limit as f -. ... 6-10. Consider the two-dimensional "cireulatinc waveguide" formed of concentric conduct.iDg cylinders, - CJ a.nd p - b. ShOw that the wave function
specifies circulating modes TM to
I:
according to Eqa. (5-18) if n is a root. of
Show that tbe above wave function specifics modes TE to z according to Eqa. (6-19) if n ill a root of J~(ko)
B .A
J:(kb) N .(kb)
------= N~(ko)
6-11. For the TM radial wave specified by Eq. (5-33), show that the radial phaae constant of E. is given by Eq. (b-36), while tbe radial phase constant of H. i.
IJ'-.![l-(~)']. 1, k r, IJ.(k,,)!' + IN.(},,»)' Tp
Show that Eq. (6-37) is also valid for this phase constant.
6-12. Consider the TM radial wave impedances of Eqs. (5-38) and (5-39). that for luge radii Z+,.nI _ Z_..TN _ 11
Show
.......
and that for small radii
Z+.~ - Z_.~·~ .......
{
~k"(d;I"+) k [(2• )'(k..»~ + J'J ",p ft.! "'2 n
.-0 •
>0
where y _ 1.781. 6-18. Conaider the radial parallel-plate w&veguide of Fig. 5-50. For the transmission-lino mode IEq8. (5-45)J. one can define a voltage and current. lUI V(P) _ -oB.
Show that V &nd I sati8fy the transmiMion-line equatiool!l dV dp
-
, LJ
-J(jI
dl -jwCV dp -
where Land C are the ".static" parametera L
_.e 2.,
c _ 2rtp
•
Why.should we expect circuit conecpta to apply for this mode? 6-1'. Cooaidenhe wedge guide of Fig. 5-Sb. For t.he dominant mode (Eq. (5-49»). one can define a voltage and eurre.nt. as 1(P) - H~
257
CYLINDRICAL WAVE FUNCTIONS
Show that V and 1 aatiafy tbe tranamission-liDe equation (prob. 6-13) with
c-~
•••
6-16. Show that the re8Qnant frequenciel of the two-dimensional cylindrical cavity (no I: variation, conductor over p - 0) arc equal to the cutoff frequencies of tbe circular waveguide. 6-18. Following the perturbational metbod used to derive Eq. (5-58), sbow tbat the Q due to conductor lOIlllel for tbe various modes in the circular cavity of Fig. 6-1 are
1-17. The circular cavity of Fig. 6-7 baa dimensions II - d - 3 centimetera. Detennine the first ten resonant. frequeDciei and tbe Q of the dominant mode if the ...alls are copper. 6-18. Consider tbe dominant mode of the partially filled radial waveguide of Fig. &-90. Show tbat for small a and large ,. the phase constant is
Compare tbill to the uniform transmiasion-line formula IEq. approximations
L _ lAid
+ "1(0
(~)I,
using the .tatic
- d)
2r. 6-19. Conlider tbe dieleetric-elab radial guide of Fig. S-9b. Let II - 41. and ,.. and 0 - )... Which model can propagate unattcnuated in the a1ab? Rcpeat the problem for the coatcd~onduetorguide of Fig. 5-9c witb t - 0/2. 6-20. For the partially IDled circular waveguide (Fig. 5-10a), show tbat tbe characteristic equation (Eq. (5-74)] for the n - 1 modell reduces to
PI -
IANI(A:_.l!) where
+ BJI(k_.l!)IlAN;(k,ib) + BJI(k,tb)l
- 0
A - k,IJ;(k,10)J1(k_lO) - k,J'I(k_IO)J1(k_10) B - k,aN';(k,tO)J 1 (k,la) - k_IJ;(k,lo)N.(k,,a)
6-21. Consider the dominant. (n - 1) mode of the dielectric-rod waveguide of Fig.5-1Ob. Show tbat for small a tbe cbaracteriatic equation beeomes (,.1
+ ,.f)(11 + If) 2,... ,K.(PXl)
Note tha.t t.bere ia no cutoff frequency.
258
TIME-HARMONIC ELECTROMAGNETIC FIELDS
6-22. The field external to a dielectric-rod waveguide varies as K1{vp). Using the results of Prob. 5-21, show that for a small (4 « A,), nonmagnetic &01 - #1) rod I 2 1 ~l + t l og:;va ... (k1o)l h - h
where,. - L 781. Take I I - 9'1 and a - 0.1).1, and calculate the distance from the lUis for which the field is 10 per cent of its value at the surface of the rod. 6-23. Consider the circular cavity with concentric dielectric rod, as shown in Fig. &-310. Show that the dominant resonant frequency is the smallest root of
~ J;(kc) "" ~ [No(ktIJ)J;(kl$:) - Jo(kotl)N;(ktcJ] "Jo(ke)
1'/0
N.(koa)Jo(koe)
Jo{koa)N.(koc)
For small c/o., show that resonant frequency"" is related to theempty-eavity resonll.nce %"
2.405
:1:01 -
according to
where
I,. -
./to.
1
~a
I
I
--a
l ~ ------ 'J -----....
r
b L
d
• (b)
(a) FlO.
6-31.
Plirti~ly
filled cavities.
6-24. Consider the circular cavity with a dielectric slab, as shown in Fig. 5-31b. Show that the characteristic equation for tbe resonant frequency of the dominant mode is
-~tank,b
-
•
k.t _ kt _
where
(X;IY
Show that when both d and b are small Iolr ...
lola
•
lr.l;:::;J(~l::::JIZ/.~, )~bId l)bjd
-V~l + ~
where lola is the cmpty-cavity reaooant frequency, given in Prob. 5-23, and and ~ - /JI/J •.
f. -
.1"
259
CYLINDRICAL WAVE FUNClIONS
FlO. 5-32. Wedge jn A circular cavity.
1-26. Consider the circular cavity "dth a conductiDg wedge, as showD in Fig. 5-32 Show that, for daman, the resoDant frequency of the dominant mode is given by
whcre 1D is the firet root of J.(w) - 0 and values of ware
•
11 -
.. /(2..
- ..,. Somc representative
0.5
0.6
0.7
0.8
0.9
1.0
3.14
3.28
3.4.2
3.56
3.70
3.83
1-26. Figure 5-330. shows a linear den.e.ity of z-directcd current elementa alODg the r axil. Show that the field is given by H - V X A where
Show that t.be field is idcntical to that. produced by t.he magnetic dipole formed of a-direct.ed magnet.ic currents +K at 11 - -'/2 and -K at 11 - ,/2 in tbo limit, _ O. &-27. Show that the field of the magnetic-dipole source of Fig. s-33b in the limit ._ 0 is given by E _ -v X u.'" where
6-28. Consider the quadrupole 80uree of Fig. s-33c in the limit '1 - 0 and '1 _ O. Show that the field is given by H - V X where
u."
a-2ft Figure &-33d represents a 50urce of 2n current filaments, equal in amplitude but alternating in sigo, on a cylinder of radius p _ a. Show t.hat, in the limit a --. 0,
260
TIME-HARMONIC ELECl'nOMAGNETIC FIELDS
y
y
•
J.l
, +
T
X
••_K ...1.
CG)
(b)
Y
+1
Y
A
,A ,
X
- I -I
f.--,,-.:.j
+I
X
- I, + I, - I' + I' - I' , +1
,+1 ,- I G
'+1 '-I •-'+1 I
(e)
X
(d)
FIo. S.33. Some two.
the field is given by H - V X
u..'" where
,. .I 1)1 r-2j(n
(kG)" 2" II •'''(k)
.
p sinn';
6-80. Let the cylinder of current in Fig. &-15 be an arbitrary function of 41. but still independent of z. Show that the field is given by H - V X u,y, with
•
••2j +-
I 11--. • •• I 2j
AJ.(ka)H.(t)(kp)e i "
,>.
A.H.(ll(ka)J.(kp),''''
p
<.
I I " - ..
wh""
1 /." A. - 2.. 0 J. ,-/'" d41
A cylinder of z-dircct.ed magnetic currents is dual to this problem. 6-31. Show that the radiation field from a ribbon of uniform z..directed current (Fig. 4-25) i.e given by
E.-
CYLINDRICAL WA.VE FUNCTIONS
261
6-32. Consider the a10t antenna of Fig. 4-21, and make the Mn1l'Opiion that tangential E in the a10t is ~B .. a cooaia.nt. Show that the radiation field is
(""2
. -1_
sin cos .) H ... -1""Iae ' E • 2rjkp 0 (ka/2) COR • And E. - TiH •. li-83. Derive the following wave tra08formations:
•
COR
l
cP.!!in 41) -
I.J IoocP) cos 2n•
• -0
•
L
lin cPain 41) - 2
JIoo'tlcP)ain (tzn
+ 1) •
••0
li-34. Let the cylinder of Fig. 5-17 be dieloctric witb parameters '4, J.I4. For a TM incident plane wave fEq. (5-105»), show thllt the scattered ficld is given by Eq. (5-106) "itb -J.(ka) [
a.. - JI.CII(ka)
~J:(k.ra)/Jc4aJ.(k4a) - J'.(ka)/laJ.(ka) ] 14J~(kllG)/lk4aJ.(k,J(J) H.Ul'(ka)/kaH.ltJ(ka)
Ind that the field internal to the cylinder ill given hy
•
l
B. - B.
j~..J.(k4P)ei'"
.--.
witb
1
c. - J.(k4C) fJ.(ka)
+ a.1J.U'l(k4))
Note that tbis solution reduces to the solution for the conducting cylinder when foI .... 00.
li-Sli. Repeat Prob. 5-34 for the opposite polarization, that ill, when the incident field is given by Eq. (5-113). Note that thiB problem ia completely dual to Prob. 5-34; 80 the solution is obtllinable by using the interchange of symbols of Table 3-2. Noto that the solution reduces to the solution for a conducting cylinder as Jl4 - O. 15-36. Show that the solution of Prob. 5-34 in the nonmagnetic csse reducea to
E.·_ --..
-jrE. (ka)l(fo. _1)H,CIl(k,) 4
- ~/I" Repeat for the opposite polarization, using the result of Prob. 5-35. Con&idcr a conducting half plAne eovering the. - 0 surface and a ..polariud plane wave of m~nitude B, incident at. an angle Q'. The IOlution is given by Eq. (5-129). Show that. the current on the half plane is
where Il6~7.
:8, ..'L\' , •
J. _
]W~P
nj./IJ./1(kp) sin n:'
262
TIME-llMUIONIC ELECTROMAGNETIC FIELDS
.
Show that near the edge of the half plane
.'
=aio• .,.....0 ., ~ 1-
1. _ _ 8. &0 d
B • ;;:0' 2£
p
jkp --J2---;-
. IUD
.' Il1D . "2 1/1 2"
Bence, E. vanishes lUI VfP, and J. bceomCll infinite as 1/v'kP. This is a geoc.ral characteristic of knife odgea. 6-38. ColUlider the half plane of Prob. 5-37 with the incident plano wave polarized tral1llverse to %. The solution is given by Eq. (&-134) Show that the currcDt on the
half plano is
•
'\'
..,
J, - 2H. ~ ..;..IIJ.II (I:,,)
Show that.
DeAl"
the knife edge
CO!!
..
T'
J,_2H, .~,
E,k;:O -"llo
2 1/1'. 1/1 jrkpC08"2Sln"2
where .' i$ the angle of incidence and 4> the angle to the field point. Nol-e that J, is finite at p - 0, while E, bceoroCll infinite as l/...;t;. This is abo a gencc!l1 charlot-
teristic of knife edges. 6-39. Figuro 5-344 show. a conducting cylinder with an aria11y pointing magnetic dipole Kl on ita surface at. • - 0, Z - O. Show that the radiation field is given by
where r. ill Neumann'. number.
z
z
Z
--
~
~
KI
f!'. n
>II
X
r- b-l --- .
(a)
(b)
X
X
---.. (e)
Flo. 5-34. Conducting cylinder with to) axial magnetic dipole on its surface, (b) uial eleetric dipole a distance b from t.he AXis, and (e) radial electric dipolc on its surface. li-4.0. Consider the uill.lIy pointing electric dipole a dilltance b from the axill of II. conducting cylinder of radius G, 8.8 shown in Fig. 5-34b. Show that t.he radiation field is given by
E, _/(,) sin'
•
\'
.--~
where a - ka sin 8 and fJ - 1b ain 8.
J .(a)N.~) - N .(a)1.~) ,'~/" H.!II(a)
263
CYLINDRICAL WAVE FUNCTIONS
6-41. Consider the radially pointing electric dipole on a conducting cylinder of radius a, M shown in Fig. 5-34<:. Show that in the z _ 0 plane (in which Illics) the radiation field is given by
•
~
nj" sin nq,
E. - f(P) ~ H ~(I)'(ka) • -I
The field in other directions hM both 9 and q, components. 6-42. Figure 5-35a shows a conducting half plane with a magnetic dipole parallel to the edge, a distAnce a from it, and on the side 4> - O. Show that the radiation field is
Eo -
j~~1 e-;lr sin
•
9
l
'..itO/IJ.11(ka sin 8) eos n:
•• 0 where~.
is Neumann's number. 6-43. Suppose that the magnetic dipole of Fig. 5-35a points in tho x direction instead of the % direction. Show that the radiation field is then given by K/,e-/lr E, - 4r . 9
ar
E. -
j:t
81n
I•
.-,
e-il' sin 8
n, nj"ItJ"u(ka sin 8) sin2
2:.-0 ~,.j •
z
..I'.t"II(k4 sin 9) cos n2
x p
II
x
FlO. 5-35. A eonducting half plane with a magnetie dipole on the side 4> - 0 a distance a from the edge.
Fro. 5-36. Electric current element on the edge of a conducting wedge.
6-U. Consider the z-directcd electric dipole on the edge of a. eonducting wedge, M shown in Fig. 5-36. Show that in the plane of the element the radiation field is given by
For a half plane, the pattern is a cardioid with a. null in the 4> - 0 direction.
CHAPTER
6
SPHERICAL WAVE FUNCTIONS
6-1. The Wave Functions.
The spherical coordinate system is the simplest one for which a coordinate surface (r = constant) is of finite extent. The usual definition of spherical coordinates is shown in Fig. 6-1. Once again we must determine solutions to the scalar Helmholtz equation, from which we ma.y construct electromagnetic fields. In spherical coordinates the Helmholtz equation is
f, :r (r ~) + r* ~n 9 :9 (sin 9 ~) + r* s;n* 9 ;~ + kty, l
0
(6-1)
Again .....e use the method of separation of variables and let (6-2)
Substituting this into Eq. (6-1), dividing by 'It. a.nd multiplying by r* sin' 9, we obtain
,dR) + sinII 9!!..d9 (. 8 dH) + ell! d¢* d~ + k'r' sm . '8 = 0 d9
~ 2. ( R dr r dr
SID
Tbe ¢ dependence is now separa.ted out, and we let (6-3)
where m is a constant. Substitution of this into the preceding equation and division by sin l 9 yields
,dR) + Hsin9d9 1 d ( . dH) m' + k'r' "'" 0 sm9 d9 -sin'9
1 d ( Rdr T dr
This scparatal the rand 0 dependence. An apparently strange choice of separation constant n is made according to
dH) -
1 d (. H sin Od9 SID 0 d8
...
m' sin' 8 = -n(n
+ 1)
(64)
because the properties of the H functions depend upon whether or not n
265
SPHERICAL WAVE FUNCTIONS
z r I
Flo.
~L
I
•
The spherical
I I
coordinate system.
y
x is an integer.
With this choice the preceding equation becomes
a( aR)
1 _ 1'2_ -n(n+l)+k 2r 2 ..,O R dr dr
(6-5)
which completes the separation procedure. Collecting the above results, we have the trio of separated equations
a( aR) + (kr)' - n(n + I)IR - 0 a(.am edi aH) + [n(n + 1) - f:in2m' e] H"., 0 sin1 8 d8 - r'd1' dr
a"l> - + m"l> a~'
~
(6-6)
0
Note that there is now no interrelationship between separation constants. The ~ equation is the familiar harmonic equation, giving rise to solutions h(m41). The R equation is closely related to Bessel's equation. Its solutions are called spherical Bessel functions, denoted b.(k1'), which are related to ordinary Bessel functions by (&-7) (see AppendiX" D). The 8 equation is related to Legendre's equation, and ita solutions are called a3sociaied Legerulre functiona. We shall denote solutioDS in general by L."(C08 8). Commonly used solutions are
L..... (C08 0) "..." P .... (C08 8), Q.... (cos 8)
(6-8)
where P ."(cos 8) are the associated Legendre functions of the first kind and Q.-(C09 8) are the associated Legendre functions oC the second kind. These are considered in some detail in Appendix E. We caD DOW form
266
TIME-JiARMONIC ELECTROMAGNETIC FIELDS
product solutions to the Helmholtz equation as
f ... - b.(kr)L.-(cos
9)h(m~)
(&-9)
These are the elementary wave functions (or the spherical coordinate system.
Again we can ;construct more general solutions to the Helmholtz equation by forming linear combinations of the elementary wave functions. The most general form that we shall have occasion to use is a summation over possible values of m and n
~
• •
II C_.•b.(kr)L.-(cos 9)h(m~)
-.
(6-10)
where the C.... are constants. Integro.tions over m and n are also solu· tions to the Helmholtz equation, but such forms are not needed (or OUf purposes. The ho.rmonic functions h(mlj) have already been considered in Sec. 4-1. If a singlc-valued y, in the range 0 to 2.. Oil 4J is desired, we must choose h(m;.) to be a linear combination of sin (mQ) and cos (m.p), or of ~ and r~, with m an integer. A study of solutions to the associated Legendre equation shows that aU solutions have singula.rities a.t 6 - 0 or 6 = 'I' except the P.-(cos 6) with n an integer. Thus, if oJ- is to be finite in the range 0 to 'I' on 6, then n must also be an integer and L.-(C08 6) must be P.-(cos 9). The spherical Bessel functions behave qualitatively in the same manner 88 do the corresponding cylindrical Bessel functions. Thus, for k real, i.(1cr) and n .. (kr) represent standing waves, h.(II(kr) represents au inward-traveling wave, and h..(l)(kr) represents an outwnrdtraveling wave. IncidentaUy, it turns out that the spherical Bessel functions are simpler in form than thc cylindrical Bessel functions. For examplc, the zero-order functions are . (k r ) .... sin kr Jo kr
(&-11)
no(kr) .... _ cos kr kr
The higher-order functions are polynomials in l/kr times sin (kr) and C06 (kr), which can be readily obtained from the recurrence formula. The only spherical Bessel functions finite at r = 0 are the i.(Jrr). Thus, to represent a finite field inside a sphere, the elementary wave functions are (&-12) r 0 included "'_.• - i.(kr)P.-(cos 9).... :IZ
SPHERICAL WAVE FUNCTIONS
267
with m and n integers. To represent a finite field outside of a sphere, we must choose outward-traveling waves (proper behavior at infinity). Hence, '!'",.... = h"U)(kr)P,,"'(cos 8)ei"'. r --+ !Xl included (6-13) with m and n integers, arc the desired elementary wave functions. To represent electromagnetic fields in terms of the wave functions ,!" we can use the method of Sec. 3-12. This involve.s letting y, be a rectangular component of A or F. The z component is most simply related to spherical components; hence the logical choice is A = u.y, = Ury, cos fJ - uey,sin fJ
(6~14)
which generates a field TM to z. Explicit expressions for the field components in terms of,!, are given in Prob. 6-1. The dual choice is F = u.'" = ur'!' cos 8 -
U6'"
sin 8
(6-15)
which generates a field TE to z. Explicit expressions for the field components are given in Prob. 6-1. An arbitrary electromagnetic field in terms of spherical wave functions can be constructed as a superposition of its 'I'M and TE parts. An alternative, and somewhat simpler, representation of an arbitrary electromagnetic field is also possible in spherical coordinates. Suppose we attempt to construct the field as a superposition of two parts, one TM to r and the other TE to r. For this we choose A = urA, and F = urF'" with the field being given by Eq. (3-79). The A, a.nd F, are not solutions to the scalar Helmholtz equation, because 'V 1 A. ¢ ('V 2A),. To determine the equations that A. and F r must satisfy, we return to the general equations for vector potentials [Eqs. (3-78)]. For the magnetic vector potential we let A = u,A, and expand the first of Eqs. (3-78). The 0 and q, components of the resulting equation are, respectively,
where
Substituting this into the r-componcntequation obtained from the expansion of Eq. (3-78), we have
c'A,
or'
+ r_1_ ~ ( . CA') 1 c'A, k'A2 sin 0 00 SID (J 00 + rl sin' 8 a¢' + ,- 0
(6-17)
268
TruE-HARMONIC ELECTROMAGNETIC FIELDS
It readily caD be shown that this equation is (V'
+ k')
A, _ 0
(6-18)
r
so Ar/r is a solution to the scalar Helmholtz equation.
ment applies to the electric vector potential.
A dual develop-
To be explicit, if we take
F = urFn substitute into the second of Eqs. (3-78), and choose
-zif>'> ~ aF,
(6-19)
ar
(v. +k')F,=0
we find that
(6-20)
r
is the equation for Fr. by choosing
Thus, electromagnetic fields can be constructed (6-21)
where r = Urr is the radius vector from the origin and the y,'s Brc solutions to the Helmholtz equation. The field is found from the above vector potentials by Eq. (3-79), which is explicitly E ~ -V X ""
+ -g1 V
X V X
1 H = V X r.p +!V X V X
fOP (6-22)
'of'
These we shall find sufficiently general to express any a-e field in a sourcefree homogeneous region of space.
The 1//a of Eqs. (6-22) arc always multiplied by T, and, because of this, it is convenient to introduce another type of spherical Bessel function, defined as ' r;;;;. "';2 B.+M(kr)
A
".(kr) - krb.(kr) -
(6-23)
These arc the spherical Bessel functions used by Schelkunoff. l Their qualitative behavior is the same as the corresponding cylindrical Bessel function. The differential equation that they satisfy is
.'!: + k' [ dr2
_ n(n
+ 1)] h.
r2
_
0
(6-24)
which can be obtained by substituting for b.. in terms of lJ.. in the first of Eqs. (6-6). General forms for the AT and FT in terms of the spherical lB. A. Bchelkunoff, "Electromagnetic Waves," pp. 51-52, D. Van Nostrand Company, Inc., Princeton, N.J., 1943.
269
SPHERICAL WAVE FUNCTIONS
Bessel functions of Eq. (6-23) are
I C_.•ll.(kr)L.-(cos 9)h(m~)
(6-25)
m••
where the e...... are constants. The considerations involved in choosing specific forms for 13..(kr), L,,"'(cos 0), and h(mq,) are the same as those used in Eqs. (6-12) and (6-13). For future reference, let us tabulate explicit formulas for finding the field components in terms of A, and F,. Letting A = u~A, and F = u,F'" and expanding Eqa. (3-79), we obtain E,
=
E, =
(a.
1i1 ar2 + kt
--=-!...- aF~ + l
r sin
0 a¢
_! of, +
E •
r ao
f)r ar ao
a' A,
1
(a. + k )F.
= _1_
(6-26)
t
aA. + ..!. a2P.
rsin 8 aq,
H __ •
a2 A,
f)r sin 0 ar aep
H. = '1£ ar 2
H,
) A,
! aA, + r ao
~rara8
1
a'F,
ir sin 8 aT aq,
When F, ::u 0, that is, when only A, exists, we have a field TM to r. Similarly, when A. = 0, the .$.bove equations represent a field TE to r. 6-2. The Spherical Cavity. Figure 6-2 shows the spherical cavity, formed of a conducting sphere of radius a enclosing a homogeneous dielectric t, p. We shall find it possible to satisfy the boundary conditions (tangential components of E vanish at r = a) using single wave functions. For modes TE to T we choose
F, _ J.(kr)P.-(cos 9)
m~) 1J cos smm,+,
z
(6-27) wbere m and n are integers. The J.. is chosen because the field must be finite at r "'" 0; the P,,'" is chosen because the field must be finite at 8 = 0 and 'lI'. The field components are tben found from Eq. (6-26) with A, = 0 and F. as given above. Note X that E, = E. = 0 at r "'" a if
J .(kG)
- 0
(6-28)
y
FlO. 6-2. The spherical cavity.
270
T(ME-HARMONlC ELECTROMAGNETJC FIELDS TABLE
~n 1 2
3 4
• 6
6-1.
ORDERED ZEnas 1.1•• 0'
1
2
3
4.493 7.725 10.904 14.006 17.221
5.763 9.095 12.323 15.515 18.689
6.988
8.183
10.417 13.698 16.924 ZO.122
11.705 15.040 18.301 21.525
20.371
21.&54
J.. (U)
•
6
7
8
9.356 12.967 16.355 19.653
10.513 14.207 17.648 20.983
11.657 15.431 18.923
12.791 16.641 20.182
4
22.295
22.905
Hence 1m must be a zero of the spherical Bessel fUDction. The denumecably infinite set of zeros of J.. (u) are ordered as U.. p • Table 6-1 givC8 the lower-order zeros. We now satisfy the boundary conditions by choosing k = u""
(6-29)
a
which is the condition for resonance.
are (Fr ) ...." =
Hence, the TE to r mode functions
J( a') P.. II
?Lot,.
"'(cos 9)
Jcosm¢) l sin m¢
(6-30)
where m = 0, 1, 2, . . . ; n = 1, 2, 3, . . . ; and p = 1, 2, 3, . . . . The field is given by Eqs. (6-26) with A. = O. If an A. is chosen of the form of Eq. (6-27), we generate a field TM to T. The boundary conditions E, = E. = 0 at T = a arc then satisfied if J~(ka) ~ 0
(6-31)
so ka must be a zero of the derivative of the apherical Bessel function for TM modes. The denumerably infinite set of zeros of j~(1L') are ordered as U~PI and the lower-order ones are given in Table 6-2. TAllLE 6-2. ORDERED ZEROS
>-; 1 2
3 4 5
• 7
u~" Of' J~(u')
1
2
3
4
5
•
2.744 6.117 9.317 12.486 15.64.4 18.796 21. 946
3.870 7.443 10.713 13.921 17.103 20.272
4.973 8.722 12.064
6.062 9.968 13.380 16.674 19.915 23.128
7.140 11.189 14.670 18.009 21.281
8.211 12.391 15.939 19.321 22.626
15.314
18.524 21.714
7
8
9.275 13.579 17.190 20.615
10.335 14.753 18.425 21.894
8PHERICAL WAVE FUNCTIONS
271
Our boundary conditions are now satisfied by choosing
, k = u .."
(6-32)
a
which is the condition for resonance. therefore (A r ) ..... , =
The TM to r mode functions are
r) P .."'(cos8) 1I sinmq, m~)
, J .. ( u.... a
cos
(6-33)
where m = 0, 1, 2, . . . in"" 1, 2, 3, . . . ; and p = 1, 2, 3, . . . . field is giv.e:n by Eqs. (6-26) with F r = O. The resonant frequencies of the TE and TM modes are found from Eqs. (6-29) and (6-32), respectively. Letting k = Ztrfr V;, we have
~ The
TE u..,. (f,) r ...... =2 .r-
'll'a
')TM = (J. "'....
,v
fJJ
(6-34)
'U .."
21fa
v_ ~ fjJ.
Note that there are numerous degeneracies (same resonant frequencies) among the modes, since fr is independent of m. For example, the three lowest-order TE modes are defined by (Fr )
0,1,1
= J
t (
4.493 ~) cos 8
(F.)iTl =
JI(4.493~)sin8cosq,
(FrH~t.t
Jl (4.493~) sin 8 sin ¢
=
where superscripts "even" and "odd" have been a.dded to denote the choice cos mq, and sin m¢, respectively. These three modes have the same mode patterns except that they are rotated 900 in space from each other. The next higher TE resonance has a fivefold degeneracy, the modes being ordered (0,2,1), (1,2,1) even, (1,2,1) odd, (2,2,1) even, and (2,2,1) odd. In this case there are two characteristic mode patterns. For each integer increase in n, the degeneracy increases by two, since P..-(cos 8) exists only for m S; n. The situation for TM modes is analogous. We see by Eqs. (6-34) that the resonant frequencies are proportional to tbe u .." and u~p. Hence, from Tables 6-1 and 6-2 it is evident that the modes in order of ascending resonant frequencies are TM""I.l, TM.. ,2,l, TE,..,I,l, TM... 1.1, TE",.1,11 and so on, The lowest-order mode~ ar~ then;-
272
ELECTRm,UONETIC FIELDS
TIME-llAR~tONlC
e
•
. 9 ( - - ...... -
Flo. 6-3. Mode pattern for the dominant. modes of the spherical cavity.
(ore the three TM",.I.I modes. Except for a rotation in space, these three modes havo the same mode pattern, which is sketched in Fig. 6-3. The Q of the lowest-order modes is also of interest. For this calcula.tion, consider the TM u . 1 mode. The magneLic field is given by
~ 1 (2.744~) sin IJ
H. -
1
Following the procedure of Sec. 2-8, we calculate the stored energy as
w
=
2W. -"
=JJ
III IHl'd.
Jo2·d~ for dB !o"drH.2rtsinIJ
The 8 and q, integrations are easily performed, giving
w -
S;" 1.'J,' (2.744~)dr
This last integral is evaluated as l (. a J. J,'(kT) dr - 2 [J,'(ka) - 1.(ka)J,(ka)] which, for ka energy is
II:
2.744, is numerically equal to 1.14/k.
Thus, the stored
w _ 8;: (1.14)
(1)-3.>)
The power dissipated in the conducting walls is approximately
tl', -
Illlff> 11l1'd. = ill 8; 1,'(2.744)
(1)-36)
Hence, the Q of the cavity is _ Q _ ~W _ ~
~"(1.14)
kIllJ ,'(2.744)
= 1.01
l! ill
(1)-37)
I E. Jahnke and F. Emde, "Tablea of Functiona." p. 146, Dover Publications, New York, 1945 (reprint).
273
SPHERICAL WAVE FUNCTIONS
Comparing this with Eqs. (5-58) and (2-102), we see that the spherical cavity has a. Q th3.t is 25 per cent higher than the Q of a circular cavity of height equal to its diameter and 35 per cent higher than the Q of a cubic cavity. The Q's of higher-order modes are given in Prob. 6-4. 6-3. Orthogonality Relationships. In many ways the Legendre polynomials are qualitatively similar to sinusoidal functions. For example, the P.. (cos 8), sometimes called zonal harmonics, form a complete orthogonal set in the interval 0 to 11" on 8. An arbitrary function can therefore be expanded in a series of Legendre polynomials in this interval, similar to the Fourier series in sinusoidal functions. The functions P.... (cos 8) cos mq, and P.."'(cos 0) sin 1»4', sometimes called te88eral harmonics, form a complete orthogonal set on the surface of a sphere. Hence, an arbitrary function defined over the surface of a sphere can be expanded in a series of tessel'al harmonics. We shall, in this section, derive the necessary orthogonality relationships. For our proof it is convenient to use Green's theorem [Eq. (3-44)], which is
1ft (~, ~~ - ~, ~:,) ds -
Iff (~,V"", - ~,V"",)
dT
(6-38)
The right-hand side vanishes if 1ft and 1f, are well behaved solutions to the same Helmholtz equation. Assuming this to be the case and applying Eq. (6-38) to a sphere of radius r, we have rl
10
2 .,
d¢
fa" dO sin 0 ( 1fl 0;1 -
1f,
at)
=
0
(6-39)
In particular, choose '" ~ j.(kT)P .(cos e)
~, ~
j.(kr)P.(cos e)
which are solutions to the Helmholtz equation. becomes
27rkT'(j,.j~ - jqj~) This must be valid for all Hence,
Tj 80,
10" p,.pq sin 0 dO
Equation (6-39) then = 0
if n ,e q, the integral itself must vanish.
fo" P ..(cos O)Pq(cos 0) sin 0 dO =
0
(6-40)
When n "'" '1, we have
for [P.. (cos O)p sin 0 dO = 2n ~ 1
(6-41)
which can be obtained by using Eq. (E-lO) and integrating by parts.
274
TIM:E-BAltMONIC
ELECTROMAGNETIC FIELDS
To obtain a Legendre polynomial representation of a function /(8) in
o to 'It: on 6, we assume
fee) -
l•
.-.
(6-42)
a.P.(COB e)
Multiply each side by P ,(cos 8) sin 8 and integrate from 0 to
'If
on B.
•
J: f(e)p,(c,," e) sin ede - l a. J: P.(c,," e)p,(c,," e) sin ede
.-.
Each integral on the right vanishes by Eq. (6-40), except the one n - P, which is given by Eq. (6-41). The result is a. -
2.+11.' 2
(6-43)
• f(B)P .(c,," B) sin B de
Equation (6-42) with the coefficients determined by Eq. (6-43) is called a
Fourier-Legendre &erie.. It converges in the same sense as the usual Fourier series. For a more general result, define tbe tesseml harmonics as
T••·(e,~) - P.-(cos e) COB m~ T...~(81t/J) = P..... (cos 8) sin mt/J
(6-44)
and assume two solutions to the HelmholLz equation as
These are well behaved within a sphere of radius r; hence Eq. (&-39) applies and reduces to
kr2(j,J~ - ivj~)
102• d¢ 10" dB T....·1',/ sin Bde "" 0
The term outside the integral vanishes for arbitrary hence
T
only when n - q; n"q
For the t/I integration, we have the known orthogonality relationships
102" sin m
m"p m-p¢O (6-45)
275
SPHERICAL WAVE FUNOl'IONS
Hence, the final orthogonality can he expressed
I:' d~ I; d9 T,.'(9,~)T ..'(9,~) sin 9 -
0
10 ... dq, /0" d8 T .....'(8,q,) T
0
2
where i ... e or o.
pq l(8,q,)
sin 8
=
88
(6-46)
m,n
d~ f,
p,g
\Vhen m, n "" p, <1, we have
4T
10
¢
d9 [T•• '(9,~)J 8m 9 -
2. . . . .
t'
i
2n+1 2r !n 2n
m=O,i=e
+ m)l
+ 1 (n -
m)!
m '" 0 (6-47)
which can be obtained by using Eq. (E-16) for P ..... and integrating on 8 by parts. A two-dimensional Fourier-Legcndrc series can now be obtained for a. function !(O,q,) on n spherical surface. For this we assume
f(9,~)
-
. . l l
(a.S••'
+ b.S,.')
... 0 ... _0
•
=
•
l L (a .... cos mq, + b..... sin m41)P.....(cos 0)
(6-48)
.. ·0 ... -0
multiply each side by T 1/) sin 0, and integrate over 0 to 2r on rP and 0 to on 8, All terms except those ha.ving tn, n = p, q vanish by Eqs. (6-46), and by Eqs. (6-47)
'lr
a,. a,. = b. o
2n
(.
d~}o d9 f(9,~) P.(coa 9)
2n + 1 (n - m) I (', (' . 2. (n m)!}o d~}o d9f(9,~)T,.'(9,~)8m 9
+
2n -
+ 1 }o("
4...
(6-49)
+ 1 (n - m)! (2'" r'" . (n + m)l}o d~}o d9f(9,~)T,.'(9,~)8m 9
2.
The series Eq. (6-48) with coefficients Eqs. (6-49) converges in the same sense as the usual Fourier series. Still another orthogonality relationship is of interest when dealing with vector fields. To establish the desired relationship, we start from the Lorcntz reciprocity theorem (Eq. (3-34)], which is
1P (Eo X H' -
E' X Ho) . ds - 0
valid when no sources are within the surface of integration.! I
We could just as well use the vector Green's thoorem, Eq. (3.46).
(6-50)
l<'(lr the
276
TIME-HARMONIC ELECfROMAGNETIC FIELDS
a and b fields, choose those obtained from Eqs. (6-26) with F r
At respectively.
~
J.(kr)T•• ;(B,~)
A,' -
"""
0 and
J.(kr)T,,;(B,~)
Applying Eq. (6-50) to a sphere of radius r, we obtain
!(J'J -J'J) ("d~ ('dB(' BaT.jaT,.' +_1_aT••'aT",)_o o . q '''}o )0 sm ao a8 sin 8 84> oq, For arbitrary T and n '" q this equation CRn be satisfied only if the integral vanishes. Also, by the orthogonality relationships of Eqs. (6-45) the integra.l vanishes if m ¢ p and i ¢ j. Thus,
('1'" dq, f" dB (Sin 8 aT"'../ aTpl }o}o ao ao
+ _._1_ aT....' iJT p,/) ~ 0 smO iJq,
o,p
m, n, i
When m, n, i
r
;::>I
:;I!
P, q, i
(6-51)
p, q, j, we have
d~ j,' dB [Sin B(a~;.;)' + do e~;';)'] "",,(n + 1) 2n + 1 ~ 2",,(n + I) (n + m)! 2n + 1 (n m)!
l
m = 0, i"'" e
(6-52)
which can be obtained by integrating once by parts and using Eq. (6-47). 6-4. Space as a Waveguide. We have seen that in a complete spherical-shell region (0 ::; 8 ::; 71,0 ::; q, ::; 211") only spherical wave functions of integral m and n give a finite field. The fields specified by these wa.ve functions can be thought of as the Hmodes of free space." When viewed in this manner, the space is oft-en called a 8pherical waveguide, even though there is no material guiding the waves. The spherical coordinate system is defined in Fig. 6-1. There exists a set of modes TM to T, generated by
n.'O)(h)!
, - 7' '(B) J ( A) r "''' .... ,q, \ 11,,(21(kr)
(6-53)
where n = I, 2, 3, . . . ; m = 0, 1, 2, ' , . , ni and i = e or o. T functions are defined by Eqs. (6-44), and the field is given by
The
(6-54)
a..
Inward-traveling waves are represented by the (!) and outwardtraveling waves by the 11.. (2), In the dual sense there exists a set of
277
SPHERICAL WAVE FUNCTIONS
modes TE to r, genera.ted by
,- , In.W(Ier»)
(F,)•• - T.. (e,~)
(6-55)
n.m(kr)
where n = 1, 2, 3, . . . ; m = 0, 1, 2, . . . , n; and i = e or o. field is given by HT£;
••
1 = --.-VXE~~i
!w.
The
(6-56)
Thc set of TM plus TE modes is complete, that is, a summation of them cRn be used to represent an arbitrary field in a source-free region. Mode patterns for the TM ol and TE ot modes are sketched in Fig. 6-4. The 'fM and TE modes are dual to each other; so an interchange of E by H and H by -E in Fig. 6-4 gives the TEo I and TM ot mode patterns. The spherical modes are qualitatively similar to the radial modes of Sec. 5-3. There is no well-defined cutoff wavelength but rather a Ir cutoff radius." To illustrate, consider the radially directed wave impeda.nces for the TM modes E,+
E.+. B .. C21 / (kr)
Z+,'" = H.+ = - H,+ "'" 1'1 B,,(2)(kr)
E,E.. B,,{l)/(kr) = - H.- = H,- = -JY] O,,(I)(kr)
(6-57)
where the superscripts + and - denote outward- and inward-traveling waves, respectively. Note that, for real Y] a.nd k, Z_/'M = (Z+r™)*. For
9(---_
Fro. 6-4. Mode patterns lor the
(a) TM ol
and
(b) TE ol
(b)
modes ollrce space.
278
TDlE-HARMONIC ELEC'rROYAGNETlC FIELDS
the TE modes the radially directed wa.ve impedances are
Z
n:
0::
+r ZTE
-
E,+ __ E.+ '"'" _j." B.(t)CJ.;r) lJ.+ H.+ O.Ull(kr) o. E.- E.- J":;11~("ik::rl H.H."' 11 ,,(l)'(kr)
(6-58)
The behavior of these wave impedances is qualitatively similar to the behavior of the twtrdimensional wave impedances, illustrated by }~ig. 5-6. In other words, the wave impedances of Eqs. (6-57) and (6-58) are predominantly reactive when kr < n, aod predominantly resistive when Icr > n. The value kr = n is the point of gradual cutoff. Nate that this cutoff is independent of the mode number m. The frequency derivative of the various wave impedances is of interest for determining the bandwidth of various devices (see Sec. 6-13). A Dovel way of representing this frequency derivative, which also illustrates the above cutoff phenomenon l was devised by ProCessor Chu. 1 He took the wave impedances and, using the recurrence formulus for spherical Bessel functions, obtained a partial fraction expansion. For example, for the TM impedance of outward-traveling waves Z+rTV. "'" 1'1
{j~ + 2n _ jkr
1 +l
1
2.3+ jkr (6-.\9)
1 +-31 -+~ jkr j~ + 1
This can be interpreted as a ladder network of series capacitances and shunt inductances, as shown in Fig. ~5a. The equivalent circuit (or the TE.. modes is shown in Fig. 6-5b. Those of us familiar with filter theory will recognize thc equivalent circuits as high-pass filters. The dissipation in the resistive element at the end of the network represents the transmitted power in the field problem. It is therefore apparent that, for fixed r, the higher the mode number n the less easily power is transmitted by a spherical waveguide mode. I L. J. Chu, Physical Limitations of Omnidirectional Antennas, J. Appl. Phl/., vol. 19, pp. 1163-1175, December, 194.8.
279
8PHERICAL WAVE F'ONcrION8
"
2n-3
- - - - f--...,.--i"f---,-- - - --ZTII _ _
••
"'
L 2n-I '--
la,
"
"
C· 2n-1
2n-5
~------,,-""'H"--..,...--i'f-- -
-.-
z'£ __
••
'"
FIG. 6--5. Equivalent circuits for the (0) TM.. And (b) TE... model of free apaee.
A quality factor Q. for modes of order
n
can now be defined
88
(6-60)
'W.. > 'W. where W. and W.. are the average electric and magnetic energies stored in the C's and L's, and (J' is the power dissipated in the resistance. In TM waves 'W. > OW.., while in TE wa.ves 'W. > OW.. However, the two eases are dual to each other; so the Q's of TM waves are equal to the Q's of the corresponding TE waves. An approximate calculation of the Q's for Q > 1 is shown in Fig. 6-6. Note that for kr > n the wave impedances are low Q and for kr < n they arc high Q_ This again illustrates the cutoff phenomenon that occurs at kr - n. 6-6. Other Radial Waveguides. A number of structures capable of supporting radially traveling waves can be obtained by covering 8 = con.. stant and ~ - constant surfacee with conductors. Such II radial waveguides" are e.ho·.rn in Fig. &-7. We can have waves outside or inside eo single conducting cone, 88 shown in Fig. &-70 and b. These two cases are actually a single problem with two different values of 81• The fields must be periodic in 2.. on 1/1 and
280
-
TWE-RARMONIC ELECTROWAGNE'rIC I'l.ELDS
kr 10'10.6-6. Quality factor'll Q. for the TM•• and TE... modes of free llP:t.ee.
finite at 6 = O.
Hence, we choose the TM to
r
mode functions
I I '"
(A,)" - P,'(eos 9) cos . mq, • n,"'(kr)
am m",
(6-61)
where m = 0, 1,2, . . .. To satisfy tho boundary condition E r = E. - 0 at () = 01, the parnmet.er v must be a solution to P ,,(cos 9,) - 0
Also, we choose the TE to
T
(6-62)
mode functions
'" cosm,pl B,'~(lT) (F,),. - P,'(eos 8 ) (.
smmep
(6-63)
where m = 0, 1, 2, .. _ _ To satisfy the boundary condition E. = 0 at 8 = 81, the parameter" must be a solution to (6-64)
281
SPHERICAL WAVE FUNCTIONS
zl
~
"
(0)
(0)
(e)
(d)
(.)
(f)
Flo. 6-7. Borne spherically radial waveguides. (a) Conical (wavC8 external); (b) conical (waves internal); (c) biconical; (d) couial; (e) wedge; (f) born.
Because of a scarcity of tables for the eigenvalues v, it is difficult to obtain numerical values. The field components are, of course, obtained from the A, and F, by Eqa. (1)-26). The biconical and coaxial guides of Fig. 6-7c and d are again a single mathematical problem. Now both 8 = 0 and 8 = 'If are excluded kom the region of 6eldj so two Legendre solutions, P.-(cos 8) and Q.-(cos 8), or P.-(cos 8) and P.-( - coe 8), are needed. Choosing the latter two sohfiions, we find modes TM to T defined by (A,),. = [P.'(eoa 8) p."( -eoa 8,) - p."( -eoa 8) P."(eoa 8,)]
lc?S m4» 18lD m4J where m = 0, 1,2, . . .
I
fJ.l::(kf')
and the v are determined by the rootB of
P.-(eos e,)p,,(- cos e,) - P.-(- cos e,)P.-(eos e,) = 0 I
(6-65)
(6-66)
282
TW&-IIARMONIC ELECTROMAGNETIC :rtELDS
Similarly, for the modes TE to r we have
(F,)_, _ [p,-(COS 8) dP,-( ~8'cOS 8,)
P,_( _ cos 8)
dP,-~~~s
lc?Sm4>\ sm mq,
where m
=
0, 1,2, . . . ,and the
1:1
8,)]
n.al(kr)
are determined by the
TOOts
dP,-(cos 8,) dP,-( - cos 8,) _ dP,-( - cos 8,) dP,-(cos 8,) _ 0
de!
d(h
dB,
(~7) of (6-65)
dB I
Again the field components are found from 'the A r and F r of Eqs. (6-65)
and (6-67) according to Eqa. (6-26). The dominant mode of the biconical and coaxial guides is a. TEM, or
transmission-line, mode. The eigenvalues m = 0, v - 0 satisfy both Eqs. (6-66) and (6-65), but the A, and F, of Eqs. (6-65) and (6-67) vanish. We could redefine Eq. (6-65) such that the limit v - 0 exists, but instead let us separately define the TEM mode 88 a TM oo mode defined by
B
(II
(A,) .. - Q,(cos 8)B,"'(kr) - log cot 2 ('l'J)~tt
(6-69)
The field components of this mode, determined from Eqs. (6-26), are (
Eif' _
'k J. e±iAr uxr 810 8
H
+ 1'Sl08 ---i- e±i
•
'f
=
(6-10)
kr
where the upper signs refer to inward-traveling waves and the lower signs to outward-traveling waves. The wave impedance in the direction of travel is
(6-11)
which is the same as for TEM waves on ordinary transmission lines. The characteristic impedance defined in terms of voltage and current. is of great-er interest.. At a given r, t.he volLage is defined as
V
1:3
J.
'-h
I,
E d ' 1 cot (8./2) .", ' r 8 -= 311 og cot (8J2) e
(6-72)
and the current as
1
=
f02~ H.,. sin 8 d,p - +21r;je±/i'r
(6-13)
SPHJ!:RICAL WAVE FUNCTIONS
283
At small r these are the usual circuit quantities. The characteristic impedance is v+ V-, cot (8,/2) (6-74) Z, - [+ = - [_ = 2r log cot (8.12) Note that the various equations are the same as for the usual uniform transmission lines. For this reason the biconical and coaxial radial lines are called uniform radial transmission lines. Spherical waves on the wedge waveguide of Fig. 6-7e exist for all fJ but only for restricted fjI. Hence, the wave functions will contain only the PIl"(cos 0) with n an integer and to determined by the boundary conditions. We then find TM modes defined by
CA,)., = P.'CC08 8) 'in w~ n.lUCkr)
(6-75)
where n = 1, 2, 3, . . . ,and
pr
w=-
C6-76)
~,
with p = 1, 2, 3,
The TE modes are defined by
'"
(F,)_ = P .'(C08 8) coo w~ l1.'''(kr)
(6-77)
where n - I, 2, 3, . . . , and to is given by Eq. (6-76) with p "'" 0, 1, 2, . . .. There is no TEM spherical mode, the TEM mode being a cylindrical wave defined by Eq,. (5-48) and (5-49). Finally, the spherical-horn waveguide of Fig. 6-7/ will require Legendre functionsL."(cos fJ) of nonint.egral v and w. The TM modcs can be defined by Eqs. (6-65) and (6-66) with m changed to wand only the sin wq, functions allowed. The values of to are those of Eq. (6-76). Similarly, the TE mode, can be defined by Eq,. (6-67) and (6-68) witb m changed to w and only the cos wq, functions allowed. Again, to is given by Eq. (6-76). There will, of course, be no TEM mode. 6·6. Other Resonators. Resonators having modes expressible in terms of single spherical wave functions can be obtained by closing each of the radial waveguides of Fig. 6-7 by one or two conducting spheres. Some examples are shown in Fig. 6-8. The fields in each case can be expressed in terms of mode functioll8 which are the same as for the radial wave-guides of the preceding section, except that the traveling-wave functions 11.(l)(kr) and 11..(I)(kr) are replaced by standing-wave functions J.(kT) and IV.. (kr). Numerical calculations are hampered by a scarcity of tables of eigenvalues. Let us calculate the Q's for the dominant modes of the first three cavitics of Fig. 6-8. For the hemispherical cavity of Fig. 6-80, the dominant mode is the dominant TM to T mode of the complete spherical cavity,
284
TWE-BARMONIC ELECTROMAGNETIC FIELDS
z
"
zrt
zl ~a---ol
NJ f---a---l
(a)
(b)
'r
~!J
(0)
Z!
....~ ;, (.)
(d)
f (f)
FlO. &-8. Some cavities having modes expressiblo in terms of singlc spherical wft.ve functions. (a) Hemispherical; (6) hemisphere with cone; (e) biconical; Cd} conical; (e) wedge; (f) !legmen".
considered in Sec. 6-2. The magnetic field is ll. -
~ JI (2.744~) sin 8
and the stored energy is one-half that for the complete spherical cavity [Eq. (6-35»); hence
w
4<.
= 3k (1.14)
The power dissipated in the hemispherical part of the walls is one-half that dissipated in the walls of the complete spherical cavity; hence 4r (el'.) •••,...... =
The power dissipated in the plane wall is
I."
(el'.)_ -
285
SPHERICAL WAVE FUNCTIONS
Thus, the Q of the resonator is ,
~'W
(6-78)
Q - i$>, - 0.573 (ii
If we compare this with the Q of a rectangular cavity (Eq. (2-102») and with the Q of a circular cavity IEq. (5-58)1 we sec that, for the same height-to-diameter ratios, the hemispherical cavity Q is only 3.2 per cent higher than the rectangular cavity Q, and 4.5 per cent lower than the circular cavity Q. The hemispherical cavity Q is 54 per cent less than the spherical cavity Q, but we have removed the mode degeneracy. From Tables 6-1 and 6-2 we find that the second resonant frequency is 1.41 times the lowest resonant frequency for the hemispherical cavity, compared to approximately 1.58 for the rectangular and circular cavities. The cavities of Fig. 6-8b and c arc theoretically important because they have circuit terminals available. In other words, a voltage and current calculated at the cone tips have the usual circuit theory interpretation. The dominant mode H
_ . A sin ken - r) E:,'-3"1 rsin8
_ A cos k(n - r) r sin 8
.. -
will be excited if the cavity is fed across the cone tips. secn by the source is VI. = lim [" E. r d8 = 2'WjAZ o sin lea
The voltage
r-o}"
where Zo is the characteristic impedance [Eq. (6-74)}. the source is II. = lim ~O
j.2r H. r dq, = 0
The current at
2.. A cos ka
Hence, the input impedance seen by the source is
..
la
ZI. = V. = jZo tan ka
I
(6-79)
which is the usual formula for the input impedance of a short-circuited uniform transmission line. (We saw in the preceding section that the TEM mode of the bieonieal guide is a uniform transmission-line mode.) The resona.nces occur when ka = n7:/2, or
nr
w. =2 "n-"'-y;;;r.="
(6-80)
In the loss-free casc, the input impedance is infinite for n odd (antiresonance) and zero for n even. When small losses are present, tbe input impedance is large for n odd and Bmall for n even. where n = 1, 2, 3, . . ..
TIllE-HARMONIC ELEcrnOYAONETIC nELDS
Let US consider the lowest resonance (n =- 1) in more detail. The input conductance at resonance can be determined from the power losses as G
wOW
"
I. -
rv:fi - Ql vl.I'
The energy stored 'W is simply calculated as
Thus
VI
=.
Gla
"""
Iff
IHI'd. = ~ IAI'Z,
T·Z.
T
(6-31)
-Q(2rZ,)' - 4Z,Q
where Z. is given by Eq. (6-74) and Q can be calculated in the usual manner as 1 T'
I+
Q - 4
csc 8 1 +
CSC
8,
0.824 iOi!{cot (0,72) tan (0,/2)]
This Q is maximum when 81
.,. T
-
8,
:=
\-.
(6-32)
33.5°, in which case
Q - 0.350;
Note that this is smaller than the Q's of other cavities that ,we have considered because of the introduction of the biconical feed system. In the special case 8, =- 90°, we have the cone-fed hemispherical cavity of Fig. 6-&, for which T' [
Q - 4
1
+ CSC 8 1
1 + 0.824 log cot (0';2)
]-'
(6-33)
This Q is maximum when 8• .,. 24.1°. in which case Q - 0.276;
This is a lower Q than that for the hemispherical cavity without the cone [Eq. (6-78»). because of the feed system. The input conductance [Eq. (6-81» is not minimum when Q is maximum, because Zo is also a function of 8. and 8,. For the biconical resonator (Fig. 6-&), the input conduct.ance is minimum when the cone angles are 81 "'" T - 8, =- 9.2°. For the cone-fed hemispherical cavity (Fig. 6-8b), the minimum conduct-ance is obtained when 81 _ 7.5°. 6-7. Sources of Spherical Waves. The sources of the lowest.-order spherical waves are current elements, treated in Sec. 2-9. For exam· • B. A. Schclkunorr, "Electromagnet.ic Waves," pp. 288-290, D. Van rJ08trand Company, Inc., Princeton, N.J., 1943.
287
SPHERJCAL WAVE FUNcrJON6
Z
z
,
Z
r
y
•
y X
X
In
Z
(0)
z
Z
n II
Y
n 01
Y
X
X
(d)
y
X
(b)
(0)
llt),
n
r
Xl
.1.
X (.)
•t
.t ..
y
\0" .j (f)
FlO. &-9. Some 8OU~ of spherical wavcs.
pte, the electric-current element of Fig. 6-9a radiates a field given by H ... V X A with (6-84)
where 1i.U> is the spherical Hankel function of Eq. (6-11). Alternatively, the field can be represented by a radially directed A given by
(6-85)
The field of the current element is discussed in detail in Sec. 2-9. The dual source is the magnetie-eurrent element of Fig. 6-9b. The field of this source is given by E ,. - V X F where F. or F~ is the same as AI or A. with I replaced by K. Tbe fields of the dipole and higher-multi pole sources, rcpresented by Fig. 6-9c to I, can be obtained by tbe same metbod as used in Sec. 5-6. For example, for the dipole source of Fig. 6-9c,
288
TIME-HARMONIC ELECrROMAGNETIC FIELDS
where A,I is the potential from a single current element [Eq. (6-84»). AI! the separation 8 is made small,
aA'
A • --+ _0 where r =
-8 - - ' -
IJz
-v!x t + yl + Z',
jklls a
-- -
4..- iJz
ho(t)(kr)
Also,
Hence for the dipole of Fig. 6-9c
A. -
k'Ils
(6-86)
4.j h,U'(kT)P,(cos 8)
and H = V X A. Thus, the vector potential is a first-order spherical wave function. For the dipole source of Fig. 6-9d, we have
which caD be written as
This is a first-order wave function of n dipole source of Fig. 6-ge, we find A. -
kllla
co;
1, m - 1.
Similarly, for the
.
4.j h,U'(kr)p,'(cos 8) sm ~
(&-88)
Thus, all wave functions of order one can be interpreted as the A, of dipole sources. This procedure can be extended to higher-mullipole sources in & straightforward manner. For example, for the quadrupole source of Fig. 6-9/, we have
A• =
alA,! all dz
81"-- -
clA.(t) -8,-iJy
where A.m is for the dipole of Fig. 6-9c, given by Eq. (6-86).
We also
289
SPHERICAL WAVE l'UNm'IONS
MV.
!.. [h,'"{kr)P,{cos 9») ~
=
~!.. [h,<"{kr) ~] rar T ky,
- - -
-
r'
ht(t'(kr) ... -kh,(I)(k1') sin B cos B sin
~
~ h,"'(kr)P,'(cos 9) sin ~
Hence tbe vector potential of the quadrupole of Fig. 6-9/ is
A.
=
jk l IlJh8t l( . 12:11" ht(t kr)Ptl(cos 6) sm ifJ
(6-89)
In this manner we can identify each wavc function of order n with the A. of a multi pole source of 2n z-directed current elements. 6-8. Wave Transformations. Now that we have wave functions in three basic coordinate geometries available, the number of possible wave transformations becomes very large. We shall here establish only a few representative transformations involving spherical wave functions. A convenient method of obtaining the desired results is that of Sec. 5-8. Let w first consider the plane wave eJ- and express it in terms of spherical wave functions. This wave is finite at the origin and independent of +i hence an expansion of the form • e> = e>-' a,.j.(r)P.(cos 9)
I
•••
must be possible (see Fig. 6-1 for tbe coordina.te orientation). To evaluate the a.., multiply each side by P,(cos 8) sin 6 and integrate from 0 to T on 6. Because of orthogonality (Eq. (6-40)], all terms except q "" n vanish, and by Eq. (6-41) we have
1," c" _. P .(cos 9) sin 9 d9 -
2:;~
1 j.(r)
The nth derivative of the left-hand side with respect to r evaluated at ,=Ois . ('r . j" 2,,+t(n!)l j. cos' 9 p.(cos 9) sm 9d9 = (2n + 1)1
r
The nth derivative of the right-hand side evaluated at r - 0 is 2"+I(n!)I
(2n
+ 1)(2n + 1)1"
Hence, equo.ting the preceding two expressions, we obtain
a. - j'(2n
+ 1)
290
TIME-HARMONIC ELECTROMAGNETIC FIELDS
which, substituted back into our starting equation, gives eP = .. - . -
I•
j·(2n
+ l)j.(,)P.(cos8)
(6-90)
• •0
Note that we have also established the identity ; ..(r) =
i;" h"
eir ...o ' P .. (cos 8) sin 6 dB
(6-91)
Equation (6-90) is the desired transformation expressing a plane wave in terms of spherical wave functions. Transformations from cylindrical waves to spherical waves can be obtained in a similar fashion. For example, consider the cylindrical wave Jo(p), which is finite at T = 0, independent of ,p, and symmetrical about 8 = 7f/2. Hencc, there exists an expansion J,(p) - J,(, sin 8) ~
I•
b.j,.(,)P,.(cos 8)
•• 0
As before, we multiply each side by P a(cas 8) sin 8 and integrate from to 11' on 8. The result 18
o
for Jll(r sin 8)P",(cos 8) sin 6 dB = 4;~ 1 ;I..(r)
/
To determine the b", we differentiate each side 2n times with respect to r and set r = O. This gives b _ (-I)·(4n
+ 1)(2n -
2 1 ..- l n!(n
..
1)1
1) I
Hence the desired wave transformation is
•
+
. "\' (-I)·(4n 1)(2n - I) I . J,(p) ~ J,('8>08) ~ 2'" 'nl(n 1)1 ],.(,)P,.(cos8)
L..
•• 0
(6-92)
Note also that the two equations preceding Eq. (6-92) establish an integral formula for iz.. (r). Now let us consider wave transformations corresponding to changes from one spherical coordinate system to another. To illustrate, consider the field of a point source at r' e-Jlr-r'l
h,'''(I' - r'1) - -
11' ,'I
291
SPHERICAL WAVE FUNCTIONS
z source
FlO. 6-10. Spherical coor· dinates or r and r'.
where rand r' are defined in Fig. 6-10. We desire to express this field in terms of wave functions referred to r = O. The field bas rotational symmetry about the r' axisj so let us express the wave functions in terms of the angle t where cos t = cos (J cos 8'
+ sin 8 sin 8' cos (
(6-93)
Allowable wave functions in the region r < r' are j,.(r)P,.(cos t), and allowa.ble wa.ve functions r > r' are h,.c2l(r)P,.(cos Furthermore, the field is symmetric in rand r' j hence we construct
n.
•
hOU1(jr -
r'D =
L c.h."'(")j.(,)P.(cos I) ••• • L c"j.(")h.",(,jP.(cos I)
•••
,<" , >"
where the c,. are constants. If we let the source recede to infinity, the field in the vicinity of the origin is a plane wave. Using the asymptotic formula
we have for the leCto-hand side of the preceding equation holU (1r - rl')
and for the right-hand side - je~ir' r'-o.. r .'_0
je-ir'
>-.
-I' -,-
r'_oo
L-
.-.
r
eir-'
c,.j"(r)P ,,(cos 9)
292
TutE-HAlUlONlC ELECTROMAGNETIC FIELDS
z r
y
I
Flo. &.11. A plane wave incident on a conducting sphere.
1 ,I
x
-J
t Incident plane wave
A comparison of these two expressions with Eq. (6-90) shows that c" = 211. + 1j hence
-
h,"'(lr - r'J) =
L: (2. + l)h."'(r')j.(r)P.(cos!l ._0 - (2. + l)j.(r')h."'(r)P.(cos n L: ._0
r
< r' (6-94)
r> r' I
This is the addition theorem for spherical Hankel functions. Since ha(l) = ham., Eq. (6-94) is also valid for superscripts (2) replaced by (1). The real part of Eq. (6-94) is nn addition theorem for io(lr - rD, and the imaginary part is an addition t.heorem for nD(lr - t'l). Finally, one can express the zonal harmonics P .(cos t) in terms of the tesseraJ harmonics P.-(cos 8)h(m4». In other words, a wave function referred to the t = 0 axis of Fig. 6-10 cnn be expressed in terms of wave functions referred to the 8 = 0 axis. The identity is
.
\'
.
(n - m)' P .(co. !) ~ __ '-'I '_ (n m)! P .-(co. 8)p.-(co. 8') co. m(~ - ~')
+
(6-95)
where too is Neumann's number (1 for m = 0 and 2 for m > 0). The proof of Eq. (6-95), plus some ot.her wave transformations that we have not treated explicitly, can be found in Stratton's book. 1 Equation (6-95) is an addition theorem for Legendre polynomials. 6-9. Scattering by Spheres. Figure 6-11 represents a conducting sphere illuminated by an incident plane wave. Take the incident wave I J. A. Slrat.ton, "Elec:l.romagnetic Theory," pp. 406-414, MeGra.w-HiIl Book Company. Inc., New York, 1941.
293
SPHERICAL WAVE FUNCTIONS
to be x-polarized and z-traveling, t.hat. is, E~;
=
Ecl.... _
H;
=
Eo eft. = Eo c11ir_'
EI1~~-flir_'
'. .
(6-96)
For convenience in applying boundary conditions, we express t.his incident field as the sum of components TM and TE to r, that is, in terms of an Fr and an Ar _ From Eqs. (6-26) we see that AT can be obtnined from Er, nnd Fr from Hr. The r component of E' is
Er; - cos q, sin 9 E~' _ Eo c~:/ :9 (e-i l'r_') Using Eq. (6-90), we can write this as
•
Ei = E,
ej~r~ 2>-'(2n + l)j.(kr) :0 P.(eDs 8)
._0
Finally, using Eq. (6-23) and the relationship
E,' - - jEt:;:
~
2>-'(20
.-,
aP./ao
=
P.I, we obtain l
+ I)J.(kr)P.'(eDs 8)
Noting the form of E r \ we construct. the magnetic vector potential as
•
w.
Ai = E, CDS
~ L." a.J.(kr)P.'(cos 8) ,
..
and evaluate E,' by Eqs. (6-26). we obtain •
E,'
~
-
jEtk:~: ~
(6-97)
Simplifying the result by Eq. (6-24),
L .-,
a.n(n
+ I)J.(kr)P.'(cDs 8)
Compa.ring this expression with the preceding formula for Er', we see that
a. =
j-'(20 + I) n(n + I)
(6-98)
A similar procedure using H,' and F r ' gives
Pi = where the a. I
Me
Note that. t.he
~. sin ~
•
L .-,
(6-99)
a.J.(kr)P.'(CDS 8)
again given by Eq. (6-98).
II. _
0 term of the lIummation drops out because
ptl -
O.
294
TIME-HARUmnC ELECTROUAGNETIC FIELDS
Now that the incident field is expressed in terms of radially TE and TM modcs. the rest of the solution parallels the cylinder problem (Sec. 5-9). The scattered field will be generated by an A. and P. of the same form as the incident field with J. replaced by 0 ..<11. Hence, we construct scattered potentials as
• A r" == Eo cos
W"
F r" =
q, '\' b.. B"u)(kr)P.. I(cos 6)
~o sin q,
'-< .-1 •
(6-100)
.-2:,
c"fl.. (S)(kr)P.. I(cos 8)
The total field is, of course, the sum of the incident a.nd scattered fields. Therefore E and H arc given by Eqs. (6-26) where
•
A, = ~ E cos ~
W"
F, -
~. s;n ~
...2:,
[a.J.(kr)
+ b.B."'(kT)jP.'(cos 6) (6-101)
2: [a.J.(kT) + c.B.m(kr)]p.'(cos 6) I
.. _1
The boundary conditions are E, "'" E. "'" 0 at r = a, which require that b -
.. -
-a
c --a
.. -
J:(ka)
.. B.. IW(ka)
(6-102)
J.(ka)
.. B..Ul(ka)
This completes the solution. Note that the problem call be viewed as a. short-cirouited radial transmission line (Sec. 6-4) with many modes superimposed. The surface current on the sphere can be found according to J. = U r X H at r = a. The result is J, =
iE 11
• cos Q \ ' a [Sin 6 P.I'(COS 6) 0 ka O.(t"(ka)
..•
4· ,
8iD412: a [P.. I(C088)
J .. -- i- E , - • ka
..,
+
jP.I(C08 8) ] sin 8 O.(t)(ka)
(6-103)
- sin 8P.I'(COS 8)] • s;n 6 B.(~'(ka) iB.m(ka)
where the a.. a.re given by Eq. (6·98).
The distant scattered field can be
295
SPHERICAL WAVE FUNCTIONS
found from the general expressions by using the asymptotic formula O..U>{kt) ~ j ..+te-1u and retaining only the terms varying 88 1/r. • j~o e-ib cos.; j" sin P.. I'{C08
e.. -
E•., •
2:.-,
[b.. B
..,
[b
The result is
B) - c.. P.:~~o; B)]
• o jE e-ikr sin.; \ ' j" P"l{COS 8) _ c sin 8 !'(cos kr 4 " sin B " "
P
where the b" and c" are given by Eqs. (6-102). the back-scattered field
(6-104)
B)]
Of particular interest is
E.' ~ E"I ,_.-- E.' I'_r ._r
• __ r/2
From this we can calculate the echo area according to Eq. (3-30), which is
(._,IE.'j') ·IE l
,. A • = ~~
,..-,
ol
Making use of the relationships P.'(cos 0) ~ (-I)' n(n 8m B I-r 2
....
+ 1)
sin 0 P ."(cos 0) --+ (-21)' n(n and the Wronskian of the spherical Bessel functions, we find A _ ~, ,
4..-
• \'
..'-<,
(-I)'(2n
+ 1)
10
,
B."'(Im)B.''''(ka)
/
1
(6-105)
'"
A plot of A.I>.l is shown in Fig. 6-12. For small ka, the n = 1 term of Eq. (6-105) becomes dominant and
~
A • --+ 9X' (1m)' .to--oo 4....
+ 1)
f1
--.
V V Lr
0.1
(6-106)
which is a good approximation when a/~ < 0.1. Equation (6-106) is known as the Rayleigh scattering law. It states that the echo area. of small spheres varies as >.-4 and was
0.0 1
o
02 0.4
0.6 0.8
1.0 1.2
a/' FlO. 6-12. Echo Ar'e& of a ooDdue;ting apbere of radiua a (optical approrima. tien shown dashed).
296
TIME-HARMONIC ELECTROPoL\ONETIC FlELDS
first used to expla.in the blueness of the sky. A.
kG......
J
For large spheres
lI'a 2
(6-107)
which is the physical optics solution. The region between the Rayleigh and optical approximations is called the resonance region and is charac· terized by oscillations of the echo area. Let us now look at the field scattered by the small conducting sphere. Using small-argument formulas for the spherical Bessel fUllctions, we find from Eq. (6-102) aod (6-98) that b.. _ _ la_O
n
+1
---c.~ n la-O
(2n)! 1)']' (ko)"+' . ),,+1
[2'(n -
(6-108)
so the n = I terms of Eqs. (6-104) become dominant for small /ro. Hence, at large distances from small spheres,
e- jkr B,' - + Eo - k (ka)l cos 4> (cos 8 k..-O
r
(6-109)
rrlJ..
E.'
-10
la-O
~)
Eo - k (ka)! sin 4> r
(~
cos 0 - 1)
A comparison of this result with the radiation field of dipoles shows that the scattered field is the field of an x-directcd electric dipole Il = Eo
~~{ (ka)'
(6-110)
plus the field of a y-directed magnetic dipole
2.
Kl - E. jk' (ka)'
(6-111)
The ratio of the magnetic to electric dipole moments is IKif Ill. = '1'//2. Figure 6-13 illustrates the origin of these two dipole moments. A surface
z
z J.
x (aJ
x
(bJ
FlO. 6-13. Components of surface current giving rise to the dipole moments of a conducting sphere. (a) Electric moment; (b) magnetic moroent.
297
SPHERICAL WAVE FUNCTIOr.."S
current in the same direction on each side of the sphere gives rise to the electric moment, while a circulating current gives rise to the magnetic moment. In general, the scattered field of any small body can be expressed in terms of an electric dipole and a magnetic dipole. For a conducting body, the magnetic moment may vanish, but the electric moment must always exist. Now consider the case of a dielectric sphere, that is, let the region r < a of Fig. 6-11 be characterized by t.I, lSd, and the region r > a by to, 1010. In addition to the field externnl to the sphere, specified by potentials of the form of Eqs. (6-101), there will be a field internal to the sphere, specified by • A,- ~ E, cos ~ \ ' d.J.(k.,.)P.'(cos 0)
..L.t,
WJlO
F,- -
~: sin ~
(6-112)
•
2: ..
.-,
J.(k.,.lP.'(cos 0)
The superscripts - denote the region r < 0, and superscripts + denote the region r > a. Boundary conditions to be met at r :.: a are H,+ = lJ,lJ.+ = H.-
E,+ = E,E.+ = E.-
that is, tangential components of E and H must be continuous. Determining the field components by Eqs. (6-26), using Eqs. (6-101) for r > a and Eqs. (&-112) for T < 0, and imposing the above boundary conditions, we find b.. "'"
- v;;,;; J~(koa)J.(k,a) + ...r.;;;; J.(koa)J~(k,al v;;,;; fl.''''(koa)J.(k,a) - vi"., fl."'(koa)J~(k,a)
a
•
- vi;;;;; J.(koa)J~(k,a) + ...r.;;;; J~(koa)J.(k,a)
c. d.. =
e _
vi,,,,, fl."'(koa)J~(k,a)
- j
_
vi,,,,, fl.,n'(koa)J.(k,a) a.
v';;;;;
vi"", fl.''''(koa)J.(k,a) - vi,,,,, n."'(koalJ~(k,al
(6-113)
0"
i~ 0 ''''' n.''''(koalJ .(k,al •
• v;;,;; fl."'(koalJ~(k,a)
where a" is given by Eq. (6-98). The conducting sphere can be obtained as the specialization J.ld _ 0, Ei --. co, such that k" remains finite. ote that, in contrast to static-field problems, t.I- co is not sufficient to specialize to a conductor. In the special case of a small dielectric sphere, the n = 1 coefficients
298
TIM"L-HARYOSIC ELECrROaLAGNETIC FIELDS
are dominant nod reduce to
(IH14)
tlt
~ 2jl,(2
+ p,)
where f r cc fd!fG and /J. "'" IldllJo. A calculation of the scattered field reveals that it is the field of the two dipoles _
47rj
1(,-1 2
II - u.Eo .k' (ka) <,
+
(IHI5)
4rj (k }'"' - 1 KI = u~ E tF a 11-,+2
Note tbat the magnetic dipole vanishes if the dielectric is nonmagnetic, that is, if IJr "'" 1. Similarly, a magnetic material with IT "'" 1 would scatter no electric dipole field. The field internal to the sphere is uniform in both E and H for tbe small sphere. In fact, the specialization represented by Eqs. (6-114) is the lfquasi·static" solution'! It can be obtained by taking the d-e electric and magnetic polarizations and assuming that they vibrate in phase Quadrature with the incident field. 6-10. Dipole and Conducting Sphere. Figure 6--14a shows a radially directed electric dipole near a conducting sphere. Figure 6-14b shows a problem reciprocal to that of Fig. 6-14a in the following sense. The component of E- in the direction of Ilb equals the component of Eb in
z
z II
x
x
(a) FIG. &-14. The conducting sphere and a radially directed dipole.
lem; (b) reciprocal problem.
(0) Oripnal prob-
299
SPHERICAL WAVE FUNCl'IONS
the direction of /la. (Superscripts reter to Fig. 6-14a lLnd b.) If the II of Fig. 6-14b recedes to infinity, we have the plane-wave scatter problem treat.ed in the preceding section. Hence, the radiation field of Fig. 6-14a can be simply obtained from the results of Sec. 6-9. In particular, in the vicinity of the conducting sphere we have (E%,i)t - t o -jwjJIl e-/trcitr'_r 4rr
r-o_
which is a plane wave.
Letting -jwp./l cJkr 4n'
(6-116)
we have the wave of Eq. (6-96). Hence, the field of Fig. 6-14b is specified by Eqs. (6-101) with coordinates primed. To relate this solution to that of Fig. 6-14a, we need the r' component of E, which is
(0'
1 •" Er' - ]Wt -. ur
+ k' )Ar' ~: cos ¢'
-
•
..L:,
n(n
+ 1)[..J.(kb) + b.fl.'''(kb)]P.'(cos U')
Finally, by reciprocity, E.,) evaluated at r' = b, 8' = equals -Efa at r, 8, q,. Hence,
'If -
8, q,'
'='
0
•
Eo' -
j:', ..L:,
n(n
+ 1)[a.J.(kb) + b.fl."'(kb»)(-I)·P.'(cos U)
(6-117)
where all, b and Eo are given by Eqs. (6-98), (6-102), a.nd (6-116), respectively. In the special case b ". 4, that is, when the current clement is on the surface of the sphere, Eq. (6-117) reduces to ll ,
• E _ ,
,II ....", \ ' j'(2n thjkr
.., Lt
+ 1) P
0,,(1)'(100)
,(cos U) "
(6-118)
This is the radiation field of a radially directed electric dipole on the surface of lL conducting sphere. Figure 6-15 shows the radiation patterns for spheres of radii 4 = >"/4. and a = 2>... The pattern for the very small sphere is the usual dipole pattern. For a very large sphere it approaches the pattern of a dipole on a ground plane but always with some diffraction around the sphere. The radiation field for dipoles of other orientations, and also for magnetic dipoles, can be obtained in a similar manner. The ficld in t-he entire region r > b can be determined from the radiation
300
TIME-HARMONIC ELECTROMAGNETIC FIELDS
I FlO. 6-Hi. Radiation patterna for the radially directed dipole on a conducting Bphere of radius a.
field as follows. From symmetry considerations (Fig. 6-14a) we conclude that H = u.H+, and therefore the field can be expressed in terms of an A = urA r. Also, A, must be independent of q, and represent outward traveling waves; hence A. ~
• L... .., fl.'''(kr)P. (cos 0)
r>b
(6-119)
From this we can calculate E, by Eqs. (6-26), obtaining
(6-120)
The a.. are then evaluated by equating this expression to the radiation
301
SPHERICAL WAVE FUNCTIONS
field previously determined. For example, in the special case b = a we equate Eq. (6-120) to Eq. (6-118) and obtain a" _ 1l(2n 1) (6-121) 41fkB ,,(2)'(ka)
z
+
•
Tbe field everywhere can now be obtained from Eq,. (6-26), (6-119), nud (6-121). x 6-11. Apertures in Spheres. In Sec. 4-9 we saw how to express the field in a matched rectangular waveguide in terms of the field over a FlO. 6-16. Slotted conducting sphere. cross section of the guide. In Sec. 6-4 we saw that space could be viewed as a spherical waveguide. A given sphere r = a is a cross section of the spherical guide. If r > a contains only free space, then the guide is matched, that is, there are no incomir.g waves. By writing the general expansion for outward-traveling waves and specializing to r - G, we obtain the field r > G. When apertures exists in a conducting sphere of radius r = a, tbe tangential components of E are zero except in the apertures. Our f9rmulas for the field r > a then reduce to ones involvinl?, only the tangential components of E over the apertures. A general treatment of the problem is messy; so let us restrict consideration to the rotationally symmetric TM case, that is, one having only an H.. The slotted. conducting sphere of Fig. 6-16 gives rise to such a field if there exists only an E, independent of ~ in the slot. The field is expressible in terms of nn A r of the form
•
•
A, -
..l ,a.n.'''(h)P.(co, 0)
(6-122)
From Eqs. (6-26) we calculate
E.
=
• k \' a jWf.T' L, a"J1.(t)I(lcr) a8 P.(cos 8)
.. ,
(6-123)
Noting iJP./a8 = p.l, we multiply each side of the above equation by P.,I(COS 8) sin 8 and integrate from 0 to 'II' on 8. By the orthogonality relntionship [Eqs. (6-46) nnd (6-47»), we obtain
+ 1) 4,.n,,('I'(kr) 2T1l(n ",::;2:'';:'-.=-' fao• E.P"l(COS 8) sm. 8 d8 -- )r..,...."" 2n+l
302
TIME-HARMONIC ELECTROMAGNETIC FIELDS
I
FIG, 6-17. Radiation patterns for the slotted sphere, 8D .. 11:/2.
Specializing this to r = a, we have the coefficients a.. determined as
+
r'
ja(2n 1) aft = 172'1m(n + l)fl,,(z)'(ka»)o E,
I ' r_a
.
P .. (cos 8) sm 8 de
The field simplifies to some extent in the radiation zone. asymptotic forms for fl.. m in Eq. (6-123), we obtain
(IH24)
Using the
• E, - . ! l e-II:< '\' a,.j"P"I(cos 0) kr-o .. r
i..J
(6-125)
"-,
This result could also be obtained from the plane-wave scatter result of Sec. 6-9, using reciprocity. For tho slotted sphere of Fig. 6-16, let us assume a small slot width, so that E, is essentially an impulse fundiOIl at r = a. Hence. we assume (6-126)
303
SPHERICAL WAVE PUNcrlONS
where V is the voltage across the slot.
+ l)P.. J(C08 8 sin 8 .2rn(n + I)O.""(ka)
jV(2n G. -
Then Eq. (6-124) reduces to 0)
0
and the radiation field [Eq. (6-125)] becomes • jV....." . 9 \ ' ;'(20 + 1)P.'(cos 9.) P '( 9) E, 2S1" SID 0 Lt n(n + l)fl..uJI (ka) .. cos
(6-127)
.-,
Figure 6-17 shows radiation patterns for the case 80 = -r/2, that is, when the conductor is divid~d into hemispheres. Patterns for sphcres of radii >../4 and 2" are shown. Very small spheres produce a dipole pattern, while very large spheres produce an almost omnidirectional pattern with severe interference phenomena in the 8 "'" 0 and 8 = -r directions. In the limit 80 """, 0 we obtain tho patterns of Fig. 6-15, which is to be expected in view of the equivalence of a small magnetic current loop and an electric current element. The general problem of finding the field in terms of arbitrary tangential components of E over a sphere is treated in the literature. l 6-12. Fields External to Cones. The general treatment of the probz lem of sources external to a. conduct-. ing cone is also messy but can be found in the literature. l We shall here l"C3trict consideration to the e, rotationally symmetric case of "ringsource" excitation of a conducting y cone. The geometry of the problem Current filament is shown in Fig. 6-18. The special case of a magnetic current ring on e, the conical surface gives the field of X a slotted cone. The limit as the magnetic current ring approaches the cone tip gives the field of an axially directed electric current element on FlO. 6018. Ring excitation of a conductiog oooe. the tip. Consider first the case of an electric current ring. From symmetry considerations, it is evident that E will have only a 4> componentj so the field is TE to T. The modes of the "conical waveguide" are considered in Sec. 6-5, Eqs. (6-61) to (6-64). In the region r < a we have standing waves, while in the region r > a we have outward-traveling waves. I L. Bailin and B. Silver, Exterior Electromagnetic Boundary Value Problems for Spheres and Cones, IRE TroBl., vol. AP-4, no. I, ~p. 5-15, January, 1956.
304
TIME-HARMONIC ELECrROllAGNETIC FIELDS
Hence, we construct
F~
1a.P.(eas o)B.ln(kT) "'" • I 1 b,P.(eo, O)J.(kT)
r>a (6-128)
r
•
where the" are ordered solutions to
Continuity of E. at
(6-l21l)
a.B.ln(ka) = b.J.(ka)
(6-130)
a requires thaf
=
T
t.,
= 0
[fop.(eas 0)
Finally, Il, at r = a must be discontinuous by an amount equal to the eurface-eurrcnt density (in our case it is an impulse function). Thus, k J. - -,-
JW1-JB
r •
a P.(eo, O)la.B.""(ka) - b.J;(ka)!' '0 v
which, using Eq. (6-130) and the Wronskian of the spherical Bessel functions, becomes
Ira
J. - -
""
'0 P.(e<)rl)
•
a
..
-J-
(6-131)
.(ka)
By the methods of Sec. 6-3 the following orthogonality relationship can be derived:
f,"(:op·)(:op·),;nOdO = where
N. "'" -
1~.
(6-132) w= "
.(.+1)[, a'p,] 21' + 1 am 8 P. iJ8 iJu ,_"
(6-133)
Hence, multiplying each side of Eq. (6-131) by P.(cos 8) sin 8 and integrating from 0 to 6, on 8, we obtain
,a J
.. - N. .(ka)
iJ j.f" J. ao IP.(eas 0)) s;n 0 dO
(6-134)
This completes the solution (or an arbitrary 4Kiirected current sheet at r = Q. For the current filament, I
J. = - 6(0 - 0.) a
(6-135)
and Eq. (6-134) reduces to
.. -
~~ J.(ka) sin 0, a~, P.(cas 0,)
(6-136)
305
SPHERICAL WAVE FUNCTIONS
Numerical calculations are difficult because of the problem of obtaining the eigenvalues '" and the eigenfunctions P•. When the ring source of Fig. 6-18 is a magnetic current, the problem is dual to the eleetrie-current case, except for boundary conditions. Henee, we construct
\l c.P.(cos O)J1.u'(kr)
I
A, - /
r>a (6-137)
d.P.(cn. O)J.(kr)
r
•
wbere the u a.re ordered solutions to P .(cos 0,) - 0
(6-138)
in contrast to the v which were solutions to Eq. (6-129). ll. at T = a requires that
Continuity of
c.J1.'''(ka) - d.].(ka)
(6-139)
At T = a we have E, discontinuous by an amount equal to the surfacecurrent density. Thus, analogous 1.0 Eq. (6-131), we have M. -
- 2 \' :. P.(cos 0)
a ,-,..
•
-.(ka) c.J
(6-140)
The orthogonality relationship for the eigenvalues defined by Eq. (6-138) is
j,"(:Op·)(:Op·)·inOdO where
M. _
u(u 2u
-l~f.
w-u
+ 1) [.in 0aP. ap.] +1
ao au '_',
(6-141) (6-142)
Multiplying each side of Eq. (6-140) by P ..(cos 0) sin 0 and integrating from 0 to 0 1 on 0, we obtain (6-143)
This completes the solution for an arbitrary ~irected magnetic current. sheet at r - a. For the magnetic current filament, K
M. - -0(0 - 0,)
a
(6-144)
and Eq. (6-143) rednces to
-K
a
J (ka) sin Ot aO P..(cos Ot) c. = '11 M ... t
(6-145)
306
TWE-HAn~ONIC
ELECTROMAGNETIC FiELDB
•
FlO. &19. Radiation patterns for the slotled conducting cone.
(ilfler Bailin and
SUVff.)
Again a. calculation of the eigenvalues u and the eigenfunctions P.. is difficult. If we now let 82 = 91 and set K = V in the magnetic current solution, we have the case of a cone slotted at r = a with a voltage V across the slot. For T > a Eq. (6-137) becomes A r = : sin! Ih
LA}u p~
(cos 81)P.. (cos 8)J..(ka)Il,,(l)(kr)
• Using the asymptotic form for 8 m and evaluating E, by Eq. (6-26), we find for the radiation field 10
+ l)(.P.(cos O)/.oJ J.(ka) + 1)(.P.(eo, O,)/QuJ
E. _ V ,_;'" \ ' j·(2u Jr u(u
L< •
(6-146)
Some radiation patterns for slotted cones with cone angle 30° are shown in Fig. 6-19. A discussion of the problem of plane-wave scattering by a cone is given by Mentzer. I I J. R. Mentzer, "Scattering and Diffraction of Radio Waves," pp. 81-93, Pergamon Press, Inc., New York, 1955.
307
SPHERICAL WAVE PUNCTIONS
6-13. Maximum Antenna Gain. The general form of the field in a spherical spnce external to aU sources is Eqs. (6-26) with A, -
2: a••fl.,o(kr)P.·(cos 8) cas (m~ + a••) 2: b••fl.'O(kr)P.'(cos 8) cos (m~ + P••)
~.
F, -
(&-147)
-.'
Given an arbitrary field at T = Ti, the field can be projected backward toward tbe origin as far as desired. At some sphere T - a we can determine sources by the equivalence principle (Sec. 3-5), which will support this field. Hence, it appea.rs that sources on an arbitrarily small sphere can support any desired radiation field. The gain of an antenna. is defined by Eq. (2-130) in general. We shall here consider the largest gain g -
4n"(S,)._ ~I
(&-148)
where (8.)... is the maximum power density in the radiation zone and ~I is the power radiated. By the discussion of the preceding paragraph, it appears that arbitrarily high gain can be obtained, regardless of antenna size. In practice, however, the gain of a directive antenna is found to be related to its size. A uniformly illuminated aperture l type of antenna is found to give the highest practical gain. This apparent discrepancy betwoon theory and practice can be resolved if the concepts of cutoff and Q of spherical waves are properly applied. Let us orient our spherical coorclinatc system so that maximum radiation is in the tJ = 0 direction. The radially directed power flux in this direction is then (&-149) (8.)... = E.H: - E.H: From Eqs. (&-147) and (&-26) we find
e-ik'
E~ = 2jr
\' L..t n(n + l)j"(l1 alii cos
"'Ill -
h. sin (hI!)
• (&-150)
I The term "uniformly illuminated aperture" ill u.sed to describe antcnnss for which the &Duree (primary or aeeondary) is COl1Iltant in 3mplitudc and phase over a given area on .. plane, and zero e1aewhere.
308
TU.£E-.HARMONIC ELECTROBLAONETIC FIELDS
in tbe 8 = 0 direction of the radiation zone. The total radiated power is found by inwgrating tbe Poynting vector over a. large sphere. The result is
~ _ ,- '\' n(n+ I)(n+m)! (I ~/- ~ '-' ..(2n + 1)(n m)!
.a...1'+!lb .'.I')
~
(&-151)
where f . - 1 for m - 0 and t. - 2 for m > O. We used the ortbogonality relationships of Eqs. (6-51) in the derivation of Eq. (6-151). Equations (6-148) to (6-151) give a. general formula for gain in terms of spherical wa.ves. We shall now consider under what conditions g is a. maximum. Note that the Ilumerator of Eq. (6-148) involves only the aIR and bhl coefficients. Hence, the denominator caD be decreased without changing the numerator, by setting
a.... = b... =0 Also, both numerator and denomina.tor of 9 arc independent of
(&-152) 0'1"
and
PI.. ; so they may be chosen for convenience without loss of generality. In particular1 let al. = T and Pl. = T/2 1 and the gain formula reduces to
12: (A. +E.) [' g - ---.=.----'-;'- - - - 22: 2n ~ 1 (IA.I' + IB.I')
(6-153)
•
(&-154)
where
The denominator of Eq. (6-153) is independent of the phases of A. and B.; so we ean maximize the numerator by choosing A. and B. real. Furthermore1 g is symmetrie in A. and B.. ; hence the maximum exists when A .. """ B .. = real
(&-155)
The maximum gain thorefore will be found among those specified by
(&-156)
• where A .. is real. As long as n is unrestricted, this g is unbounded, as we anticipated earlier. If the field, specified by Eqs. (6-147), contains only wave functions of order n ~ N, then an upper limit to g exists. Setting iJgjaA i = 0 for
SPHERICAL WAVE FUNCTIONS
309
all A i, we find N
U_ -
and also
.l-.
(2n
A .. =
+ I) - N' + 2N 2n + 1 A 3 I
(6-157) (6-158)
Equation (6-151) represents the highest possible gain using spherical waveguide modes of order n 5" N. A similar limitation to the nearzone gain also exists. l To relate gain to antenna si7.e, we define the radius a of an antenna M the radius of the smallest sphere that can contain the antenna. We saw in Sec. 6-4 that spherical modes of order n were rapidly cut 01T when ka < n. Hence, it is reasonable to assume that modes of order n > ka are not normally ·present to any significant extent in the field oC an antenna of radius a. We define the normal gain of an antenna oC radius aas (6-159) u_••• = (ka)' + 2ka which is obtained by sett-ing N "'"' ka in Eq. (6-151). Hence, the normal gain is maximum gain obtainable when only uncutoff modes are present. It is interesting to note that, for large 1M, a circular, uniformly illuminated aperture of radius a h3S the same gain as the above-defincd normal gain." It is thereCore not surprising that the uniformly illuminated aperture gives the highest antenna gn.in in practice. The normal gain is not an absolute upper limit to the gain of an antenna. Antennas having higher gain are a distinct possibility and will be called supergain antennQ.$. We shall use the Q concept of Sec. 64 to show that (1) supcrgain a.ntennas must necessarily be narrow-band devices, and (2) supergain techniques yield only a smaU increase in gain over normal gain for large antennas. Other characteristics which we shall not demonstrate here are (3) supcrgain antennll8 have high field intensities at the antenna. structure and (4) they tend to have excessive power loss in the antenna structure. The Q of a loss-free antenna is defined as
'11.>'11.
w. > W.
(6-160)
I R. F. Harrington, Effect of A.ntenna Siu on Gain, Bandwidth, and Efficiency, J. ReuGrcA NBS, vol. 640, no. I, pp. 1-12, January, 1960. t S. Ramo and J. R. Whinnery, "Fields and Waves in M.odern Radio," 2d ed., p. 533, iohn Wiley &: Sons, Inc., New York, 1953.
310
TlllE-HARMOXIC ELECTROI.LAGNETIC FIELDS
Ill'
e-
10'
0
Ill'
30
0'
25 \ Ill'
20
10
10
I
i~5
5
\
\ \ 1\
\.
\';'-\ o
\
FlO. 6-20. Quality factofft for ideal loss-free anten· nas adjusted for mlUimum gain using modes of order n ::; N.
10
15
20
25
ka
where 'W. and OW. are the time-average electric and magnetic energies Bnd {j>, is the power radiated. 'Vc shall define an ideal loss-free antenna of radius a as ODC having no energy storage r < a. The Q of this ideal antenna. must be less than or equal to the Q of any other loss-free antenna of radius a having the same field r > a, since fields r < a can only add to energy storage. If the Q of an antenna is large, it can be interPreted as the reciprocal of the fractional bandwidth of the input impedance. If the Q is small, the antenna has broadband potentialities. Antennas adjusted for maximum gain according to Eq. (5-158) have equal excitation of 'I'M and TE modes. The Q.. of spherical modes, defined by Eq. (6-60) and plotted in Fig. 6-6, involve OW. for 'I'M modes slid "XI", for 1'E modes. We need Q's defined in terms of the same energy for aU modes, and it is convenient to deal with Q's for equal TM and TE modes. The Q for equal TM,. and TE" modes is
ka < N
(6-161)
because the 'W, is essentially that of the TM.. mode alone a.nd the rJJ is twice that of the TM II mode alone. When QII < I, we take it as unity. Because of the orthogonality of energy and power in the spherical modes, the tolal encrgy and power in any field is the sum of the modal energies and powers. Hence, the Q of our ideal loss-free antenna is
Q~
2: p.Q.",m - ='=o---'='i:--=--:::t.....,,.2: A.' (2n ~ I) Q. 2: p. 22:A"(2n~l)
311
SPHERICAL WAVE FUNCTIONS
where P. is the transmitted power in the TM. and TE. modes. Eq. (6-158), thie becomes
Using
N
l
(2n
+ 1) Q.(ka)
.-1 ::-:---:2"'N""·;-+-;--;4"N,.------
Q-
(6-162)
where the Q. are given in Fig. 6-6. Curves of antenna Q for several N nre shown in Fig. 6-20. Note that the Q rises sharply for ka < N, showing that supergain antennas must necessarily be high Q, or frequency sensitive.·, The Q of Fig. 6-20 is a lower bound to the Q of any loss-free antenna. of radius a. By picking a Q, we can calculate an upper bound to the gain of an antenna of radius a. Figure 6-21 shows the ratio of this upper bound to the normal gain. Note that for large ka the increase in gain over normal gain possihle by supergain techniques is small. For small ka supergain can give considerable improvement over normal gain. In fact, ns ka --+ 0 the supergain condition is unavoidable. All very small antennas are supergain antennas by our definition. The problems of narrow bandwidth and high losses associated with small antennas are well-known in practical antenna. work.
10
9 ~
II .~
"
8 7
Ii
6
~
5
3
..
4
.!!
3
0
2
g ~
~
.'\
\"-
'"
--
~l~ '- ........ lO'" '-..
1
0
Q = 10·
10
....
20
"-
30
40
50
60
70
80
Fro. 6-21. Ma.ximum poasible iocrea.ee in gain over normal gain tor a given Q.
90
312
TIME-HARMONIC ELECTROMAGNETIC FIELDS
PROBLEMS
6-1. Use Eqs. (3·85) p.nd t.he wave potential of Eq. (6-14) to show that a general expression for fields TM to t: is . E. - -1fsJJJy, cos
8 ali} . + 1<.l~ -:-1 -iJr• [COB -- (rti) - . - - (V- sm' 6) ] r t Ur r aln (I iJ9
(I
+ Jw€1' iJ (r-", ") -1. - a - [COS - -8 aD ,1 iJr
.• . E , -JfsJj.I.,ylUn
1 a [COS // B. - jw6' sin 8 iJ(/I --;:t
R,
,a.
_! a",
..
ara (r:l,J.) -
1 -a (>/-Sin .')] --. 8 r Sin 8 iJ(J
a
1
.
r sin 8 a8 (y, san' 8)
1
H, _ cot 8 ay,
,
H. -
~ 1 [ sin
(J : ,
(rof)
+ aag (", COB 9) ]
where'" is a solution to the scalar Helmholtz equation, 6-2. Verify that Eqa. (6-17) and (6-18) are identical. 6-3. Consider an air-filled .spherical resonator of radius 5 centimeters bounded by copper walls. Determine the first ten resonant. frequencies and the Q of the dominant mode. 6-4. For the spherical cavity of Fig. 6-2, show that the Q due to conductor losses is, for TM modes, TO
,[,
(Q').A" -
' In
UA" -
n(n
_Ut
+
,
u."
1)]
where the u~" are given in Table 6-2, and, for TE modes, T&
"U A"
(Q.)...." "" 2m
where the u.." are given in Table 6--1. 6-6. Consider the cavity lying between concentric conducting spheres r "" a and r - b, with b > a. Show that the characteristic equation for modes TM to r is
And for modes TE to
r
it is flo(kb) Jo(kb) Jo(,",) - fI.(,",)
6-6. In the concentric-6phcre cavity of Prob. 6-5 let a «b, and show that the resonant frequency w is related to the empty cavity resonant frequency w. by w -
"'I
- - ... %(2.744)'
n;(2.744)
J"
_ 1(2.744)
(a)' b
where
f(k,a)
SPHERICAL WAVE FUNCI'IONS
313
8-'1'. Consider the partially Iilled spherical cavity formed by 8. conductor covering r _ b and containing: a dielectric II, 1'1 for r < a and a dielectric III 1'1 for a < r < b. Show that the characteristic equation for the dominant mode is h:(k,b)J~(k,a) - J~(ktb)n;(k,a) n:(k,b)J I (k,a) J;(k,b)n I(ktll)
where k l - <01 ~ and k l - <01 V;;;;. 6-8. In the partially filled spherical cavity of Prob. 6-7, let 11« b and II - la and III - 1'1. By expanding the characteristic equation in 8. Taylor series about the empj.,Y-cavity resonant frequency ""0, show that the resonant frequency", is given by
~
ItS
3i(2.744)'
WI
where
I, '"'
.d•• and
<010 -
~~(2.744) .. - 1 (~)' 11 (2.744) + 2 b f.
2.744/b V;;;.
Compare this with the answer to Prob.
6-
6-9. Consider the function [(',.)
:
-{
and determine the coefficients 4.. and b•• for the two-dimensional Fouricr-Legendre scriea of the form of Eg. (6-48). 6-10. Let A and B be two vectors and B be the angle between the1ll. Define C _ A - B and show that, for B > A,
•
I
C
2AB
COB
B
6-11. Consider the characteristic impedances of the spherical modC$ of f:Ipa.ce IEqs. (6-57)1. Bhow that
and zrE ... '11/ZTM. Show also that the change from primarily resistive to primarily reactive wave impedances occurs at kr ... n. 6-12. Show that the field of an electric current elemcnt II is the dominnot TM spherical mode of space, and the field of a magnctic-currcnt clement Kl is the dominant TE mode. 6-1S. Using the usual perturbational method, show that the attenuation COtl8tant due to conductor losses for the TEM mode of the biconical or coaxial radial guide (Fig. 6-7c and d) ill given by
.-lR
2"
cae
01
+ esc 8,
I0 0 cot- Bal2 -cot 01 /2
6-101. Show that the dominant spherical TE mode of the wedge guide (Fig. 6-7e) is the free-rspa.ce field of a z-directed magnctie..current element.
314
TUn;-HAIU,(Ol'o'lC ELECTROMAGNETIC J"IELDS
6-1fi. Uae the qualitative behavior of the tpbtrieal Hankel funetioJ\l, to iustify the statement. t.hat the apherieal.horn guide of Fig. 6-7} has a "cutoff radiw" approxi. mately equal to Uat. radiWi for which the Cf'OlllI aection is t.he \lame aa II. rect.anlUlar guide at. cutoff. 6-16. Considtr a hemispherical eavity (Fig. &-Sa) constructed of copper with a - 10 eentimet.el"8, and air-filled. Determine the fint ten resonant. frequeocie. and the Q of the dominant. mode. 6-1'1. Conaidcr the lleCOnd rCllOoance In - 2 in Eq. (6-80)! of the biconieal cavity of Fig. 6-&. Caleulate the Q of the mode and the input resistance seen at t.he CODe tipa. 6-18. Coll8idcr the conical cavity of Fig. ()..8d. Show that modes TM to r arc given by H _ v X u,.A. where (A.) ..... - P .-:(C04 I) COl m
J.
(10:, ~)
where 10:, is the pth I:cro of J:(Ul) and II is a solution to Eq. (~2). that modes TE to r are given by E - -v X u,.p. where (F.}..~ - p."'(coe 8)
coe "'«IJ.
Similarly, show
(Ul•• /i)
to.,
where ia the pth zero of J.(tlJ) aod 1.1 is a 101utiofl to Eq. (6-64). For a complete lIet of modes tho sin m'; variation mUllt allO be included. 8-19. Let. tho eumlDt. elemcntll of Fig. G-9c be replaced by magnetic-current. ele-ments Kl. Show that., in the limit.' ..... 0, the field is given by B - -v X uJ. where P. -
k'Kla 4.j 1a.(I)(b')P1(eoa,}
z
II
8-20. Consider Fir;. 6-22 where tric current. ll. '\"'" 0 and " .....
J.
T ..
,,'
H ... v X u.A. where
y
t~
the quadrupole tIOW'Ce of each element is an elecShow that., in the limit 0, the ficld it! given by
+i'
x
FIG. 6-22. A quadrupole source. 6·21. Derive tho following wave transformat.ion: e-llr-rJ
l•
1
~ - -.-, __ 0 (2" Ir-r, 1"
"here f is t.be angle between rand
+ 1).!.(r')B.lfJ(r)P.(cos U r.
6-21. Derive the following wave transformation:
L•
J.(P) - __ 0 A.,jr..... (r}P,. . .·(eoa,)
wh""
A
_ (-1)·..... (4"1 •
+ 2ft + 1}(2m}1 + ,,)1"11
2r..-(m
315
SPHERICAL WAVE FUNCfIONS
6-23. Derive the following formula:
r'
>r
where ~ is the aogle between rand r'. 6·24. CoIlBider the scattering of a plane-polariled wave by a smrill conducting sphere (Fig. 6-11). Show that the distant. llCatt.ered field is plane pol/lri1ed in t.he direction 8 _ 60'. 6-26. Considcr an z.polariled, z traveling plane wave incident on a conducting sphere encased in llo concentric dielectric coating, IUl shown in Fig. 6-23. Show that the 6eld ill given by Eqll. (6-26), o where for r > b the A. and F. are given by Eqs. (G-lOl), and for (1 < .,. < b
,
z
• A. _ E,
COli
i.J .-,
"'I"
d.IG~(k(1)J.(u)
- J~(ka)R_(kr)JP.'(COlI B)
~·!in
p. -
I
•
•
...l,
e.[G.(ka)J.(u)
x
'.
,
- J.I..)G.lb-)[P.'looo')
t
y
I
~
Incident
W
•
where a. is given by Eq. (6-98) and c.. by Eq. (6-102). 6-27. Con8ider a radially dircc:tcd eleetric dipole adjacent to a dielectric sphere (Fig. 6-14 with the sphere DOW dielectric). Show that the radiation field i8 then given by Eq. (6-117) if b. is given by Eq. (6-113) instead of Eq. (6-102). 6-28. Consider a loop of uniform current I of radius 0, as shown in Fig, 2-26. Show that the radiation field ~ given by
.
E. - ; e-
wbu. and
"H, - -8•.
jlrt
A. - i.Cm)
..2:,
2:<:' ~ l ;-A.P_ ' (O)P.ICCOlI B) U
316
TWE-HARMONIC ELECI'nOMAGNETIC FIELDS
6-29. Figure 6-24 shows & conducting sphere of radius R concentric with a loop of uniform current 1 of radius (I. Show that the radiation field is of the SlUllO form as given in Prob. 6-28 except that
..1.-' _
8.''''(ka) _ J.(kR)R;(ka) - R,(kR)J;(ka) 8.")(ka) J.(kR)R.(ka) - R,(kR)J,(ka)
Show that. this reduces to the answer for Prob. 6-28 aa R - O.
z
z
e
r
y
x FIa. 6-24. A conducting sphere with a concentric ring of electric current.
FlO. 6-25. Current element at the tip of a conducting cone.
6-30. Figure 6-25 shows a current element II at tbe tip of a conducting cone. Show that. the radiation field is given by E, - I(r) sin 8 ~(C08 8) where u is the first root of P .(C08 81) - O. r
-',
u
Some approximate eigcnvalue8 are
I"I I I I I I I I I O.l
'0' 0.2
24' 0.3
37' 0.4
49' 0.5
60' 0.6
69' 0.7
77' 0.8
84' 0.9
90' 1.0
6-31. By considering t.he equivalent circuit of Fig. &-5 and the definition of Eq. (6-60) for Q. show that the Q of the n - 1 epherica.l mode is
If equal TE and TM waves are present, the total Q is approximately one-half this value. A small antenna (say ia < 1) will have minimum Q if only the n - 1 modes are present in ita field. Hence, the minimum possible Q for a smalll088-free antc.nnll is
where a is the radius of the smallest. ephere that can contain the antenna.
CltAPTER
7
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
7-1. Introduction. The differential equation approach of the preceding three chapters leads to an exact solution of the mathematical problem. However, many problems cannot be treated by this method. We saw in Sec. 3-11 that electromagnetic field problems caD be expressed in integral equation form. This form is particularly useful for (1) obtaining approximate solutions and (2) for general expositions of theory. In this chapter, we shall consider two techniques uscful for integral equations arising in electromagnetic theory. Perturbational Methods. The word "perturb" means to disturb or to change slightly. The perturbational methods arc useful {or calcula.ting changes in some quantity due to small changes in tbe problem. Usually two problems are involved: the "unperturbed" problem, for which the solution is known, and the It perturbed" problem, which is slightly different from the unperturbed one. We have already used perturbational methods for calculating resonator quality factors and waveguide attenuation constants. Further uses are given in Sees. 7-2 to 7-4. Variational Methods. The variational methods are useful for determining characteristic quantities, such 85 resonant frequencies, impedances, and SO on. In contrast to the perturbational procedure, the variational procedure gives an approximation to the desired quantity itself, rather than to changcs in the quantity. The variational procedure differs from other approximation methods in that the formuJa is Ustationary" about thc correct solution. This means that the formula is relatively insensitive to variations in an assumed field about the correct field. If the dcsired quantity is real, the variational Cormula may be an upper or lower bound to the quantity. Furthermore, iC an assumed field is expressed as a series oC functions with undetermined coefficients, then the coefficients can be adjusted by the Ritz procedure (Sec. 7--6). In Cact, if a complete set of Cunctions is used for the assumed field, the exact solution can sometimes be obtained, at least in principle. 7-2. Perturbations of Cavity Walls. Figure 7-1a represents a resonant cavity formed by a conductor covering S and enclosing the loss-free region T. Figure 7-1b represents a deformation or the original cavity 317
318
TIME-HAR~{oNIC ELECTRO~[AGNETJC
FIELDS
D
s
D
S'
Eo. H o
E, H
(b)
(a)
FIG. 7-1. Pert.urbation of cavity walls.
(a) Original cavity; (b) perturbed cavity.
such that the conductor covers 8' = S - lJ.S and encloses -r' = 'T - AT. We wish to determine the change in the resonant frequency due to the change of the cavity wall. Let Eo. H o, WI) represent the field and resonant frequency of the original cavity, and let E, H, w represent the corresponding quantities of the
perturbed cavity. that is,
In both cases the field equations must be satisfied,
-v X E =jwlJH
- V X Eo = jWoJlH o V X HI) = jWflEE o
vXH=jwfE
(7-1)
We sealarly multiply the last equation by E~ and the conjugate of the first equation by H. The resulting two equations aTC
Et·V X H =jwEE·Et -H· v X Et = -jwo,l.lHci· H Adding these and applying the identity V • (A X B)
~
B •V X A - A. V X B
we have
v·
(H X Et) = jWfOE·
ES -
jWaJ.lHS· H
By analogous operations on the second and third of Eqs. (7-1), we obtain
V . (H: X E) = jwp.H . H: - jwotE: . E These last two equations arc now added, and the sum integrated throughout the volume of the perturbed cavity. The divergence theorem is applied to the left-hand terms, one of which vanishes, because n X E = 0 on 8'. The resulting equation is
t
H X Et· ds = j(w - wo)
JJJ (fOE· Et + p.H . HS) dT
Finally, since n X Eo = 0 on S, we have
c/fHXEt.dS=O
(7-2)
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
319
and the left-hand side of Eq. (7-2) cau be written as
1ft
H X
E:·ds =
r
1ft H X E;·ds = -1ftH X E;.ds
s-s
d
The last term is taken as negative, to conform to the convention that ds points outward. We can now rcwrite Eq. (7-2) as
ieffiH X Et·ds .s
(7-3)
This is an exact formula for the cha.nge in resonant frequency due to an inward perturbation of the cavity walls. Note that our development assumes that E and}J are real, that is, wc have assumed no losses. Problem 7-1 gives the gcneral formulation in the lossy case. The crudest approximation to be made in Eq. (7-3) is that of replacing E, H by the unperturbed field Eo, H o. For small perturbations this is certainly reasonable in the denominator and should be valid in the numerator if the deformation is shallow and smooth. With this approximation the integral in the numerator of Eq. (7-3) becomes
...effi
H X
E~ . ds "'" =
1f> ..iw.111
(H o X E~) . ds
••
(.IE.I' - _lll,I') dT
The last equality follows from the conservation of complex power [Eq. (1-62)]. Substituting this into Eq. (7-3), and also substituting Eo. H o for E, H in the denominator, we have
III WI,I' - .IE.I') dT III (_lll,I' + .IE.I') dT
W-WDR<~"
w,
(7-4)
•
Note that the terms in the numerator are proportional to the electric and magnetic energies II removed" by the perturbation, while the denominator is proportional to the total energy stored. Hence, Eq. (7-4) can be written as (7-5)
where tlv>. and a'W. are time-average electric and magnetic energies is the total energy stored in the original originally contained in AT and
,v
320 cavity.
TIMErfiARMONIC ELECTROl\lAQNETIC FIELDS
Finally, if AT is of small extent, we caD approximate the AW'S
by aT times the energy densities at the position of liT. Furthermore, W caD be written as T times a. spoce-average energy density Ul. Thus, Eq. (7-5) can be written as W -
000 ...,
(tV...
--= 10,)
AT =
C
Wr
000
AT T
(H)
where C depends only on the cavity geometry and the position of the perturbation.
It is evident Crom the preceding equations that an inward perturbation will raise the resonant frequency if it is made at a point of large H (high U>..), and will lower the resonant frequency if it is made at a point of large E (high We). The opposite behavior results from an outward pertur-
batiOn. It is also evident that the greatest changes in resonant frequency will occur when the perturbation is at a. position of maximum E and zero H J or vice versa. Numerical calculations using Eqs. (7-4) to (7-6) are easy for the cavities treated previously, because we calculated W when we determined the Q'5. For the dominant mode of the rectangular ca.vity of Fig. 2-19, W is given by Eq. (2-98), or 'II> ~
-
-IE,I'T 4
For tJ.T located at the mid-point of the base (maximum E) we usc Eqs. (2-96) to find aW... = 0, and
6oW.
~
-
:lIE,I' 60T
Hence, from Eq. (7-5) we find w - Wo ::::: -2 tl:r
w,
T
(7-7)
If the perturba.tion occurs at the mid-point of the longer side wall (maximum H), we have 11"1». = 0 and
6oW. ~
2(1
-IE,I'
+ c'/b') 60T
Hence, from Eq. (7-5) we fInd
2117
w-wo Wo
:::::
1
+ (c/bp7
(7-8)
Note that for a square-base cavity (b = c) the cbange in resonant frequency due to 117 at maximum H is only one-half as great (and in the opposite direction) as that due to a7 at maximum E.
PERTURDATIONA.L AND VARlATIONAL TECHNIQUES
'fABLE 7-1. TUE
E
PAB,A)lETER
AND
(b)
C
OP
EQ. (7-6) FOR DEFOnMATIONS (a) AT MAXJllU1ll H 011' THE DOMiNANT MODE
AT MAXlloIU){
Cavity
Geometry
Rectangular (a S b ;5 c)
I (O!>---L-- -L
C
/' 1A1 /" (a)
t--b--!'"
Short cyliDder (d
< Za)
Long cylinder (d
~ Za)
~(;'~jI d~
~(b)
"-
(a>
Spherical
Hemispherical
321
tE"(b)
'a...,' (a> . /
':U
(a) -2
b) ( 1
2
+ (c/b)l
(a) -1.85
(b) 0.'
(a) -0.843
(b)
2.80
1
+ (1.714/d)'
(4) -0.3Gl (b> 0.680
(a) -2.02
(b) 0.680
Table 7-1 gives the value of C in Eqs. (7-6) for cavities of several geometries for a., located at (a) maximum E and (b) maximum H. These values have been obtained using the crude approximations of replacing E, H by Ee, H o in Eq. (7-3). They are therefore valid only for smooth, shallow deformations. In general, the frequency shift depends on the shape of the deformation as well as on the shape of the cavity. The formulas for deformations of the form of small spheres or small cylinders caD be obtained from the results of the next section by letting E _ 00 and 11- O. 7-3. Cavity-material Perturbations. Let us now investigate the change in the resonant frequency of a cavity due to a perturbation of the material within the cavity. Figure 7-2a represents the original cavity containing matter E, IJ, Figure 7-2b represents the same cavity but with the matter changed to E + AE , IJ + All.
322
TDrE-RAIUIONIC ELECTROMAGNETIC FIELDS
n
'.
,
n
Flo. 7-2. Perturbatwn of mat.ter in a cavit.y. (4) Original cavity; (b) per· turbed cavit.y.
s
(a)
(6)
Let E Ot HOI Wo represent the field and resonant frequency of the original cavity. and let E, H, (oj represent the corresponding quantities of the perturbed cavity. Within S the field equations apply, that is, - V X E = jw(p V X H = jW(E
- V X Eo = jwopHo V X H o "'" jwotE o
+ .1p)H
+ ..6.E)E
(7-9)
AB in the preceding section, we 8calarly multiply tbe last equation by E: and the conjugate of the first equation by H, and add the resulting two equations. This gives
v . (H
+ AE)E • E:
X E:) = jW(E
- j(J},pB: . H
Analogous operation on the second and third of Eqs. (7-9) gives V.
CD:
X E) = jwCp
+ dp)R· H:
- jW(ltE: . E
The sum of the preceding two equations is integrated throughout the cavity, and the divergence theorem is applied to the left-hand terms. The left-hand terms then vanish, because both n X E = 0 on Sand n X Eo = 0 on S. The result is 0=
III IIw(. + 6.) -
w"jE· E:
+ [wu. + 6,)
- w.,IH· H:I dT
Finally, this can be rearranged as w - Wo w
- IIIIII
(llEE· Et (tE. E:
+ ~pH· Hn dT
+ pH . Ht) d,.
(7-10)
This is an exact formula for the change in resonant frequency, due to a change in t and/or p within a cavity. Once again our development bas assumed the loss-free case, that is, E and p are real. The general formulation when losses are present is given in Prob. 7-5. In the limit, as ~t -+ 0 and IIp -+ 0, we can approximal-e E, H, w by Eo, H o, Wo and obtain
III (6·IE,I' + 6,IH.,') dT III (·IE,,' + ,IH.l') dT
(7-11)
PERTURBATIONAL AND VARL-\TIONAL TECHNIQUES
323
This slates that any small increau in E and/or ~ can only decrease the resonant frequency. Any large change in E ,and/or ~ can be considered as n succession of mnny small changes. Henco, any imrease in f and/or ~ within a cavity can only decroou the reMnant frequ.emy. We can recognize t.he various terms of Eq. (7-11) as energy expressions and rewrite it as w -
w,
w, ::::s -
-
1
'"
fl! (d' - + d" - )d -10. E
-"'_
~
T
(7-12)
where W is the total energy contained in the original cavity. Now if the change in E and ~ occupies only a small region AT, we can further approximato Eq. (7-12) by
(7-13) where tb is the space average of W. The parameters C1 and C! depend only on the cavity geometry and the position of aT. Note that 0. small change in E at a point of zero E or a small change in p. at a point of zero IJ does not. change the resonant frequency. If we compare Eq. (7-13) wit.h Eq. (7-6), it is evident that. C - C! - CJ. For the cases considered.in Table 7-1, aT' is either at a point of zero H, in which case C! = 0, or at a point of .tero E, in which case CI = O. To be explicit, for a material perturbation at (a) of Table 7-1 we have C1 "'" -C and C! = 0, while for a material perturbation at (b) of Table 7-1 we have C 1 = 0 and C! = C. The preceding approximations require that AE, Ap., and AT all be small. We shall now cODsider a procedure for removing these restrictions on AE and Ap.. This introduces the further complication that the change in frequency depends on the shape of tl.r, as well as on its location. The modification is accomplished by using a. quasi-static approximation to the field internal to tl.T. This assumes that the ficld internal to AT is related to the field external to AT in the same manner as for static fields. The procedure is justifiable, because, in a region small compa.red to wavelengt.h, the Helmholtz equation can be approximated by Laplace's equation. There are Cour types oC samples for which this quasi~static modification to the pert.urbat.ional solution is very simply accomplished. These are shown in Fig. 7-3 for the dielectric case. For the magnetic case, it is merely necessary to replace E by Hand E by p.. For the thin slab wit.h E normal to it (Fig. 7-3a), we must have continuity of t.he normal com-
324
TIME-HARr.l0NIC ELECTROMAGNETIC FIELDS
O'----_.:.JD (cj
(d)
FlO. 7-3. Some small dielectric objects {or which the quasi-.static solutions are simple.
ponent of D. so that (7-14)
This approximation is valid regardless of the cross-sectional shape of the cylinder. For the long thin cylinder with E tangential to it (Fig. 7-311), we must have continuity of the tangential component of E, so that (7-15)
Again this approximation is independent of the cross-sectional shape of the cylinder. For E normal to a long thin circular cyliuder (Fig. 7-3c), we can use the static solution, I which is (7-16)
Finally. for E normal to a small sphere (Fig. 7-3d), we can use the static solution,' which is (7-17)
The static solution for a dielectric ellipsoid In a uniform field is also known but is not very simple in form.' To use the above quasi-static approximations, we approximate E (and H in the magnetic case) in the numerator of Eq. (7-10) by E 1D , of the preceding equations. In the denominator we can stilJ use the approximations E = Eo and H = H o, because the contribution from AT is small compared to that from the rest of T. Hence, our quasi-static correction to the perturbational formula is (oj
-
WD
~
JII AtE
IDI •
Eci dT
2jjj.IE.I'dT
(7-18)
I W. R. Smyth, "Static and Dynamic Electricity," pp. 67-68, McGraw-Hill Book Company, Inc., New York, 1950. I J. A. Stratton, "Electromagnetic Theory," pp. 205-213, McGraw-Hili Book Com.
pany, Inc., New York, 1941.
325
PERTURBATIOYAL AND VARIATIOXAL TECHNIQUES
,
A , - -71
1 d--W=-a-----..j
(a)
(e)
(b)
FIo. 7-4. CAvities used t.o illustrate t.be pcrturbatKlnal fonnulas.
for the case t1p. "'" O. (The denominator bas been simplified by equating W.. to 'W•. ) The corresponding formula for the frequency shift due to a magnetic material would be of same form, but with E replaced by H and E by p. throughout. Equation (7-18) is, of course. most valuable for problems for which the exact solution is not known. However, so that we may gain confidence in the results as well as pr:lCuce in the procedure, let us apply Eq. (7-18) to problems for which we have the exact solution. These are illustrated in Fig. 7-4. For a dielectric slab on the base of a rectangular cavity (Fig. 7-44), we have E l ., given by Eq. (7-14). The field and energy expressions for the unperturbed cavity are given in Sec. 2-8. Application of Eq. (7-18) then yields lE,-ld
(7-19)
- 2-,-,-.
where d is the slab thickness and a is the cavity height. A comparison of this with the result of Prob. 4-17 for P.l = P.I "'" p.o and EI = EO shows that our answer is identical to the fU'St t-erm of the expansion for cal in powers of dla. ~n fact, if tip. is also nonzero a.nd we treat it to the same degree of approxima.tion (match tangential H), we again get the correct first term of the expansion. To illustrate the improvcment obtained by using the quasi-static field, we can compare Eq. (7-19) to the result obtained from Eq. (7-11), which is I)
~ o
It is apparent that the above formula is accurate only {or when tu is small.
Er F:S
I, that is,
A nonmagnetic dielectric slab at a. side wall of the rectangular cavity (Fig. 74b) has but little effect on the resonant frequency, because E is zero at the wall. In this case E is tangential to the air-dielectric interrace; so Eq. (7-15) should apply. Note that Eqs. (7-18) and (7-Jl) give
326
TIME-HARMONIC ELECTROMAGXE'I'lC FIELDS
identical approximations in this case. ",-wo
In particular, we obtain
".. - (t~-l)f,J.t'll'"Xd sm :t:
a
Wo
0
~ _ T' (. _ 3'
1)
a
(~)'
(7-20)
•
A comparison of this with the answer to Prob. 4-18 shows that we again ha.ve the correct first term of the expansion when tip. "'" O. As a final example, consider the spherical cavity with a concentric dielectric sphere (Fig. 7-40). The field of the unperturbed cavity is defined by H.
=~JI(2.744&)Sin9
and tbe stored energy is given by Eq. (6-35). using the quasi-static Eq. (7-17), we obtain w "'0
Wo ===
-0.291
Applying Eq. (7-18),
fOr 1 (2.744 ~)' t.+2 b
where a is the radius oC the small dielectric sphere and b is the radius of the conductor. This we can compare to the exact solution (Peob. 6·8), which is the same. The perturbational method used in conjunction with the quasi-static approximation gives excellent a.ecurncy when properly used. This sbift in resonant frequency caused by the introduction of & dielectric sample into a resonant cavity can be used to measure the constitutive parameters of matter. 7-4. Waveguide Perturbations. We shall now consider waveguides cylindrical in the general sense, that is, all z: = constant cross sections are identical. Figure 7-5a represents a cross section of the unperturbed wavegUide, Fig. 7-5b represents a wall perturbation, and Fig. 7-5c repre.sents a material perturbation. All perturbations must, of course, be independent of z. The guide boundary is taken as perfectly conducting in aU cases.
'.
,
S
.c
S-
C (0)
E. H
E.H
Eo.Bo
•
n
n
C' (b)
Flo. 7-5. Perturbat.ions of eylindrieaJ waveguides. (b) wall per~urbalion; (c) material perturbation.
(,j
c
PERTURBATIONAL AND VARIATtoNA.L TECHNIQUES
327
At the cutoff frequency a cylindrical waveguide is a two-dimensional resonator. We should therefore expect formulas similar to those for perturbations of cavities to apply to waveguides at cutoff. In fact, we can apply the cavity derivations directly to the region formed by the cyundrical waveguide bounded by two z "'"' constant planes, changing only some of the explanations. For example, in deriving Eq. (7-2), the left-hand side results from the integral
1P(H X E: +H: X E)·ds taken over the perturbed surface. For lL length of a cylindrical waveguide at cutoff, the ficlcle are independent of Zj 80 the surface integrals over the two z = constant cross sections cancel each other. This leaves only the surface integral on the left-hand side of Eq. (7-2) taken over the wall of the waveguide. Following the derivation further, we find that Eq. (7-3) applies directly for calculating the change in waveguide cutoff frequency. But both numerator and denominator involve an integration with respect to z. which reduces to the length of the segment of thecylindrical waveguide. Hence, from Eq. (7-3) we obtain the change in cutoff frequency 6wc due to an inward perturbation of the waveguide wall as
6",c "'"
j,{.. H X E:·ndl _"-l.'Y-",,c'-__'_
fJ
(oE • E:
+ ,H· H:> ,u
(7-21)
B
where 6C is the contour about the volume of the perturbation and S' is the cross section of the perturbed waveguide (see Fig. 7-5b). The crude approximation of replacing the perturbed fields E, H by the unperturbed fields Eo, H o in Eq. (7-21) gives good results for smooth, shallow perturbations. This leads to
6",c
-
w,
~
If (,IH,[' - ,[E,[') _d. II (,IH,['+,IE,!'),u 11~~-,--,-_-,7
(7-22)
8
which is analogous to Eq. (74). Hence, an inward perturbation of the waveguide walls at a position of high E will lower the cutoff frequency, while one at a position of high H will raise the cutoff frequency. For perturbations not shallow and smooth, we can obtain a better approximation to .6.wc by using a quasi-static approximation for H in the nwner· ator of Eq. (7-21). An example of the perturbation of waveguide wallE is the "ridge waveguide." formed from the rectangular waveguide by
328
TIM~HARMONIC
ELECTROAlAGNETIC FIELD8
adding ridges along the center of the top and bottom walls. 1 Such ridges will lower the cutoff frequency of the dominant mode and will raise the cutoff frequency of the next higher mode (sce Prob. 7-12). Hence, a greater fange of single-mode operation caD be obtained. The ridges also decrease the characteristic impedance of the guide; hence, they arc used for impedance matching. The formulas for material perturbations in cavities can also be specialized to the case of material perturbations in waveguides at cutoff. The reasoning is essentially the same as that used for the wall-perturbation case. Hence, from Eq. (7-10) we can obtain tbe exact formula for the change in cutoff frequency due to a change of matter with the waveguide. It is
if (alE· E: + d~' H:> dB ff (oE· E: + ,.H. H:> d.
A",.
"'.
(7-23)
where the integrals are taken over the guide cross section. Note that an increase in either f or ~ can only decrease the cutoff frequency of a waveguide. If.o.E and !:J.,.,. are small, we can replace E, H by Eo, H o and obtain A",.
-~
"'.
ff (A,IE,I' + A"IH,I') d' ff (,IE.I' + "IH.I') d.
(7-24)
This is analogous to Eq. (7-11). If at' and OjJ are large, but of small spatial extent, we caD improve our approximation by using the quasistatic method of Sec. 7-3. For example, analogous to Eq. (7-18) we bave in the nonmagnetic case A""
Jf
""
2
-~
!:J.a;la,·
E: dB
(7-25)
ff ,IE.I' d.
where Elu is given by the appropriate one of Eqs. (7-14) to (7-16). AB long as the perturbed guide is homogeneous in f and P, we can determine the propagat-ion constant at any frequency from the cutoff frequency according to
~-
l
j~ -jk~la - k,
(:')'
1 - (;.)'
(7-26) w
< CAl,
IS. B. Cohn, Propertiea of Ridge Waveguide, PNX.. IRE, vol. 35, no. S. pp. 783-788, August, 1947.
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
329
(This is proved in Sec. 8-1.) If the perturbed guide is inhomogeneous, no such simple relationship exists. In such cases we can obtain perturbational formulas for the change in 'Y. In the loss-free case we can express the unperturbed fields as
Eo "'" to(z,y)e-i~" K o = a:o(x,y)e-i~••
(7-27)
E = t(x,y)e-i6a H = :a(x,y)e-i~a
(7-28)
and the perturbed fields as
The perturbational formulas are then (j);: X
j,
~ -
,.,,0
=
1l:) • n dl
-) ---;-:;'-'Y=6C"--::--;;_--::r
QQ'
f!s
(j);: X
1l:
+ j);
X
_
1l::> . u. d8
(7-29)
in the case of a wall perturbation, and
~ -
ff (6,j);. j);: + 6.ll: . 1l::> d, ~. ~ w T,s~:----:;--:;---;:--- ff (j);: X 1l: + j); X ll::) . u. d8
(7-30)
•
in the case of a material perturbation. The perturbational formulas in the lossy case are given in Probs. 7-15 and 7-16. To illustrate the derivation of the above formulas, consider a material perturbation. The unperturbed and perturbed fields satisfy Eqs. (7-9) with w = wo, for the frequency is kept unchanged. The two equations following Eqs. (7-9) are still valid, and, with Wo = w, their sum becomes V . (H X E:
+ H:
X E)
~
jw(AEE • E:
+ ApR· H:)
Integrating this equation throughout So region and applying the divergence theorem to the left-hand term, we obta.in
1ft (H X E: + H: X E) ·ds =
jw
IfI (AEE·E: +ApH·H:)dT
(7-31)
This is an identity for any two fields of the same frequency in a region for which E and .... Il are changed to E + At and p + Ap. For material perturbations in a cylindrical wav&- dzI guide, we express the fields according to Eqs. (7-27) and (7-28) and apply Eq. (7-31) to FlO. 7·6. DilTe.rential slice or a the differentia.l slice of Fig. 7-6. On the cylinde.r.
----- -- •
330
TIME-HARMONIC ELECTROMAGNETIC FIELDS
waveguide walls both n X E and n X Eo vanish; so this part of the surface integral vanishes. Also, since the thickness of the slice is a differential distance,
The right-band side of Eq. (7-31) caD be expressed as the integral over the cross section t.imes dz; hence Eq. (7-31) reduces to -j(ji -
~.)
II (a x ~: + a: x E). u. d• • - jw II (6&:' li: + 6 a .a:) d• p
•
Rearrangement of this equation gives Eq. (7-30). In the derivation of Eq. (7-29), the right-hand side of Eq. (7-31) is zero, and the left-hand side equated to zero leads to the desired result. Equations (7-29) and (7-30) as they stand are exact. To usc them, we must make various approximations for :2 and H, just as we did in the cavity problems of Sees. 7-3 and 7-4. For example, in the case of shallow, smooth deformations of waveguide walls, we can approximate t, ft by to, o in Eq. (7-29). Using the conservation of complex power (Eq. (1-62)], we arrive at the result
a
[f (P[O.I' - .[2.1') ds fJ - fJo """
Cal
".-'''---;:---;0----;:-,--II (li: ao+ li. 1l::) . u. d•
•
X
(7-32)
X
(The denominator is twice the time-average power flow in the unperturbed guide.) If the perturbation is not shallow and smooth, better results can be obtained using a quasi-static modification. Similarly, for small .o.E and .o.p. we have the approximation for material perturbations
fJ - fJ. ==
II (6·12.1' + 6pl0.[') d. II (li: ao+ li. an· ds
Cal " , " ' - : - - - - ; - - - : - - - : - - -
•
X
(7-33)
X
For large .o.e and .o.p. we can obtain better result-s by using the quasi-static approximation for the fields within .o.E and .o.p.. As an example of the perturbational approo.ch applied to a waveguide problem, consider a circular waveguide of radius b containing a concentric dielectric rod of radius a. The exact solution to this problem was
331
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
@ ...
1.0
0.9 -
FJO. 7-7. Comparison of the perturbational 801u· tion ""iUt the eu.ct lJOoo lution for the partially filled circular waveguide, I _ lOc., b - 0.44
~~/ /
• b
~ 0.8
/
./' L?'Perturbation solution-
0.7 0.6
o
0.1
0.2
03
o/b
considered in Sec. 5-5, a.nd a numerical example is shown in Fig. 5-11. For the perturbational SoluLion we shall use Eq. (7-30) with tip. = O. In thc numerator we make the quasi-static approximation of Eq. (7-16), and in the denominator we approximate t, by ~lll 11 0 • The unperturbed field of the dominant TEll mode for the circular waveguide is
:a
E, - ; J I
(
E. = l.:U
1.841
£) sin
J~ (1.841~) cOS ¢
H. =
~:
where Zo is the characteristic impedance [Eq. (5-32)]. of Eq. (7-30) tben becomes 2 Zo
where
We
1," 1,' d41
0
dp p(E,l
0
+ E.l)
is the cutoff frequency.
P-P. ko
"'"
vi
The denominator
= 0.7892 ~
1J
The numerator is easily evaluated as
;:: ~~ EO( 1.841 and Eq. (7-30) reduces to
~:
H, = -
rP
iY
2.146(w./w)1 •. +-1(.)' b E.
1
(7-34)
Figure 7-7 compares this solution to the exact solution of Fig. 5-11. OUf approximations give good results for small a/b. At frequencies near the unperturbed cutoff frequency, the We in Eq. (7-34) may be taken as that of the perturbed guide. 7-5. Stationary Formulas for Cavities. Suppose we have a resonant cavity formed by a perfect conductor enclosing a dielectric. possibly inhomogeneous. The {<wave equations" arc v X p.-'v X E -
V X
r v 1
X H -
~IEE
= 0
",~l~
= 0
(7-35)
332
TIME-BARMONlC ELECTROMAGNETIC FIELDS
where w. is the resonant frequency. These reduce to the usual Helmholb equations when f and ~ are constants. If the first of Eqs. (7-35) is scalarly multiplied by E and the resulting equation integrated throughout the cavity, we obtain
III E· V X .-·v xEd,
6.1..'
=
(7-36)
'-'-'--/'/'I-'-,-E-'-d-'---
Similarly, multiplying tbe second of Eqs. (7-35) scalarly by H and integrating throughout the cavity, we obtain
III H·v X ,-'V X Hd, w,' ~ !..L!--/"'-'/I-'-.-H-' d,--
(7-37)
Equations (7-36) and (7-37) arc identities, but, even more important, they are useful for approximating w,. by assuming field distributions in a
cavity. They are particularly well-suitcd for this latter application because of their Ustationary" character. which we shall now discuss. We take Eq. (7-36) and substitute for the true fitl4 E a trial field (7-38)
where 'P is an arbitrary parameter. w'(p) -
where we show sion of ",t is
wt
This procedure gives
III (E+pe)·V X .-·V X (E+pe)d,
'-'--''--77,-----------
1II,(E+pe).(E+pe)d,
as a. function of 7J for fixed e.
ow'l p2 uw +p+--t up .... 0 21 up 2
",2(P)=Wr t
t
(7-39)
The Maclaurin expan-
I
.... 0
+ ...
(7-40)
Note that the first term is the true resonant frequency, because "'rt. In the variational notation l the above expansion is written
",t(O)
::::I
as ",t(p) = wrt
+ p&."t + p' 21lPwt + ...
(7-41)
By definition, each term of Eq. (7-41) equals the corresponding term of Eq. (7-40). The term 8",t is called the first variation of ""t, tbe term 8t ",t is called the second l!ariaiwn, sDd so on. A formula for w t is said to be IF. B. Hildebrand, .. Metbodaof Applied Mathemat.iea," p. 130, Prentice-Hall,Inc., Englewood Cliff" N.J., 1952.
333
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
stationary if the first variation of
",,2
vanishes.
This is equivalent to (7-42)
The extension to more tha.n one p parameter is straightforward. We now wish to show that Eq. (7-39) is stationary. The derivative of the numerator N(p) evaluated at p - 0 is
N'(O) =
JJJ (E·V X p-IV X e +e·v X p-IV X E)d'1
It is a vector identity that
111
E· V X p-IV X e dr -
JJI p-IV X e· V X E d'1 + 1ft [(.-·v X e) X E)· d.
The last term vanishes, because n X E "'" 0 on S. states
JJf p-tv X !; V
X EdT"'"
IfJ
A similar identity
e· V X ",.-IV X EdT
-1ft [(.-·V X E) X eJ· d.
Using these two identities and the fi.rst of Eqs. (7-35), we obtain N'(O) -
2w; ff f .e· E d. -1ft [(.-'v X E) X e)· d.
The derivative of the denomina.tor D(P) of Eq. (7·39) is, for p - 0, D'(O) - 2
fff .e·Ed.
We then obtain
aW'1 ap
- N(O)D'(O) ._0 _ D(O)N'(O)D'(O) 1ft [(.-IV X E) X eJ· d.
--
fffoE'd.
(7-43)
which has been simplified, using Eq. (7-36). The above equation vanishes if n X e = 0 on 8, which requires n X EbloJ. "'" 0 on 8. Hence, Eq. (7-36) is a stationary formula. for the resonant frequency if the tangential components of the trial E vanish on the cavity walle.
334
TIME-IlARMONlC ELECTROMAGNETIC F1ELD8
Equation (7-36) can be put into a more symmetrical form by applying
the identity
III E· V
X .-'V X
EdT - III .-·V X E· V X EdT
+ # l<.-·V
The last term vanishes, because n X E
=
0 on S.
X E) X E] . ds
Substituting this
identity into Eq. (7-36), we obtain
w,' -
III .-·(V X E)'dT l.LL-f"'fI".E=--'d:-T-
(7-44)
This formula proves to be stationary, provided n X E lrial =: 0 on S. If we look carefully at the first variation of Eq. (7-44), it is evident that the requirement n X E kW = 0 on S caD be relaxed if the term 2# [(.-·V X E) X E]· ds
is added to the numerator. ..' -
This gives
III .-·(v X E)' dT + 2# [I;c'v X E) X E!· ds lll.E'dT
(7-40)
which is stationary, even if n X £"1&1 yf 0 on S. This is an important modification, because it is not always easy to find a trial field with vanish· ing tangential components on the cavity walls, especially if the geometry is complicated. Still further modifications in OUf formulas are required if n X E or n X (/r1v X E) are discontinuous over some surface within the cavity. All such modifica.tions can be quite simply effected by the reaction concept of Sec. 7-7. A similar procedure shows that EQ. (7-37) is a st8.tiooary formula. in terms of H, provided n X (ciV X H) = 0 on S. The H-field formula corresponding to Eq. (744) is
..' _ f",f",I<"7-'7(V_X_H_)_'_dT JJJ .fl'dT
(7-46)
which turns out to be stationary subject to 00 boundary conditions on S. Further modifications to account for discontinuities in n X H or n X (f- 1V X H) over surfaces within the cavity can be made. These modifications aga.in follow directly from the methods of Sec. 7-7.
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
335
-,'
o
p,
p
o
(a)
p,
p
(b)
Flo. 7--8. Illustration of Wi Ver1!US 'P for (0) a stationary formula and (b) a nOlUltationary formula.
Let us now briefly consider the advantages of a stationary formula over a nonsLationary one. Figure 7-8 shows pictoraUy the primary ndvao,tage. Given a class of trial fields of the form of Eq. (7-38), the parameter w' (p) determined from a stationary formula such as Eq. (7-39) will have a minimum or maximum at p = 0. 1 This is shown in Fig. 7-8a. The parameter 61 ' determined from a nonstationary formula must have some definite slope at P =* 0, llS shown in Fig. 7-8b. For a givcn error in the assumed ficld, say .dE - pie, the corresponding error in the resonant frequency is WI' - w,.'. It is evident that for small PI the stationary formula gives a. smaller error in 61' than does the nonstationary formula. This property is sometimes summarized as follows: fl A paramctcr determined by a stationary formula is insensitive to small variations of the field about the true field." An crror of the order of 10 per cent in the assumed field gives an error of the order of only 1 per cent in the parameter. In some cases the true field can be shown to yield an absolute minimum or maximum for the parameter. The stationary formula then gives upper or lower bounds to the parameter. Our formulas for 61 ' give upper bounds, as we shall show later, We might also inquire about the general procedure of establishing stationary formulas. One characteristic of all such formulas is that the numerator and denominator contain squares of the trial field. This insures that amplitude of the trial field will have no effect on the calculation. Classically, the method of establishing stationary formulas is to construct formulas of the proper form and then separate the stationary ones from the nonstationary ones by testing the first variation, In Sec. 7-7 we shall give a general procedure which leads directly to the various stationary formulas. I
A complex parameter would have
0.
saddle point at 'P - O.
336
TIME-HARMONIC ELECTROMAGNETIC FIELDS
Now let us apply some of
OUf
stationary formulas to a problem Cor
which we have an exact answEl):, so that we may get an idea of the accu-
racy obtainable. Consider the dominant mode of the circular cavity (Fig. 5-7), for the case d < 2a, The TMo,o mode is dominant and the exact resonant frequency is 2.4048
(7-47)
a VEP.
w, =
The field is sketched in Fig. 5-8 and is given mathematically by E. =
J o (2.405~)
j
-H~
...
2.~05 J (2.405~) 1
Substitution of this true field into any of our stationary formulas must, of course, give us Eq. (7-47). Suppose we first try a formula that requires no boundary conditions [Eq. ('1-46)1. Assume as a trial field VXH=u.2 Equation (7-46) then becomes wi
=
foG 4pdp
10
4
EIJ.
8 =_
pVJP
Epa'
and our approximation is w
~ 2.818
(7-48)
a~
r
This is 16 per cent too high, which is a relatively poor result. This suggests that our trial field was too crude an approximation. We can improve our trial field by assuming H
=
%)
14 (p -
V X H = u.2 (1 -
~)
which is chosen to satisfy the condition n X E = 0 on S. (7-46) then yield. 1012
=
l"
fll
4(1 -
~)' pdp
j,o' ( 23ap')' pdp p--
Equation
180 ElI
__
,.31a'
4.nd oW' approximation is now Wr
"'=
2.410 _ /a v EjJ.
(7-49)
PERTURBATIONAL AND VARIATIONAL TECUN1QUES
337
This is only 0.2 per cent in error. Even though a formula is stationary, we must use care in choosing trial fields. It is advisable to meet the physical boundary conditions 88 closely as possible, for this will hclp to obtain a trial field close to the true field. If the same trial field is used in Eq. (7-37), we again get Eq. (7-49), since n X E "'" 0 on S. Now consider a stationary E-field formula, say Eq. (7-44). This formula requires n X E = 0 on S; henee we choose
1
VXE-u.-a Substituting this into Eq. (7-44.), we obtain
{o!... dp
Jo a l
6
w' ~ -r"'."i('--=-~)"'- - ,.a' Ell Jo l-~ pdp Our approximation is therefore Wrt::S
2.449
_l_
(7-50)
OVE",
which is 1.8 per cent too high. If we had chosen a trial E field not satisfying n X E .,. 0 on S, we would have had to use Eq. (745). Note that all our approximations are too high. This suggests that the true resonant frequency is an absolute minimum, which we shall now sbow. For example, take Eq. (7-39), and, by means of various identities, put it into the form
Ilf
pe' (v X Il-IV X pe - W/Epe) dT
w' - w,' - '-L'----j'j'j-,--,-(E-+-pe-)-'-:a,----
(7-51)
It is known thR.t, the eigenfunctions, that is, the fields of the various modes, form a complete set of orthogonal fUDctions in the cavity space.! Hence, the error field pe can be expanded in a series
where the Ali are constants and the Eli are the various mode fields. Substituting the above equation into Eq. (i-51), making use of the wave I Philip M. Morse and Herman Fcshbach, "Methods of Theoretical Physics," part I, Chap. 6, McGraw~HiI1 Book Compa.ny, Inc., New York, 1953.
338
TIME-HARMONIC ELECTROMAGNETIC FIELDS
equation and the orthogonality relationshipl5, we obtain
l wI -
w
r
'
=
(~;'
- w.')A;' JJ! .E;'dT
~;--'f"!"!'-.(:-E-"-'"-)-'-d-T---
(7-52)
w.
where the arc the resonant frequencies of the ith modes. Since we ha.ve chosen W r as the lowest eigenvalue, Eq. (7-52) is always positive. Hence, any w calculated from Eq. (7-36) will be an upper bound to the true resonant frequency. Also, if we choose a trial field orthogonal to the field of the lowest mode, we have an upper bound to the next higher resonant frequency, and so on. This, of course, requires that tbe dominant modo be known exactly, which is seldom the case for complicated geometries. . Look now at Eq. (746). The trial field H = constant vector is a permissible trial field, since DO boundary conditions are required. The result is W r "'" 0, which is less than the true resonant frequency IEq. (7-47»). Why do we not ha.ve an upper bound in this case? The answer lies in the fact that we have overlooked the "static mode." A static magnetic field (wr = 0) can exist in a cavity bounded by a perfect electric conductor. Fortunately, it is easy to insure that our trial field is orthogonal to all static fields, thereby obtaining an upper bound to the dominant a-e mode. Any trial field satisfying
(7-53)
,Jf.-OonS is orthogonal to aU static fields, as we shall now prove. orthogonality is
where, in general, H ,t• lI • =
-va.
The desired
By virtue of the identity
". (U.,H) - .,H. "U
+ U,,· pH
the preceding equation becomes
This requirement is met for all U by the conditions of Eq. (7-53). Our choices for H in the foregoing examples satisfied Eq. (7-53); so we obtained upper bounds to the dominant TM oio mode, as desired. 7-6. The Ritz Procedure. A further advantage of the variational formulation is that one can choose the best approximation to a stationary quantity obtainable from a given class of trial fields. This is done by
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
339
including adjustable constants, or variational parameters, in the definition of the trial field and then choosing those parameters which will give a minimum or maximum of the stationary quantity. For example, if we choose (7-54)
where the Ai are variational parameters, and substitute into the stationary formula Eq. (7-36), we obtain (7-55)
The best approximation to w. 2 will be the minimum value of can be chosen by requiring
a",'
-=0 aA,
i
::::0
1,2, ... ,n
w\ which (7-56)
This general method is known as the Ritz procedure. I The most common way to include variational parameters is to express the trial field as a linear combination of functions (7-57)
Since the labor of the calculations increases approximately as the square of the number of terms in EQ. (7-57), it is desirable to keep n small. However, it is also necessary that some choice of the Ai will give a reasonably close approximation to the true field. When a complete set of functions E, is used, the method may, in principle, lead to an exact solution. It is also sometimes convenient to choose the E; as an orthogonal set. For an example of the Ritz method, let us again consider the circular cavity of Fig. 5-7 and trial fields of tbe form H = .. (p
+ Ap')
v X H
~
u.(2
+ 3Ap)
(7-58)
where A is a variational parameter_ Note that H satisfies no boundary conditions on S; so we choose Eq. (7-46) as the stationary formula.. Substituting the trial field into Eq. (7-46), we obtain
f:
(2 + 3Ap)'p dp ",' _ J.:""",--_
+ A p2)!p dp 15 8 + 1Ma + 9(Aa)' a"~ 15 + 24Aa + lO(Aa)' E~ foG (p
= I
The method is also referred to as the "Rayleigh-Ritz procedure."
(7-59)
340
TU!E-HAIW:ON1C ELECTROllAGNETIC FIELDS
Note that the approximation of Eq. (7-49) is the special case Aa ",. To determine A by the Ritz method, we set
3i'.
a",'
- -0 aA 24 + 55Aa + 28(Aa)' - 0
and obtain
This can be solved (or Aa as A a
_ -55 ± v'ffi ~ 56
( -1.3100 -0.6543
(7-60)
A substitution of the second of these values into Eq. (7-59) gives
'"
2.4087
(7-61)
aw
~--
which is smaller than what the first of Eq. (7-60) gives. Hence, Eq. (7-61) is the desired "best" approximation to t.he true resonant frequency {Eq. (7-47)J. The solution Aa - -1.31 gives ka = 7.191, which is an approximation to the next higher eigenvalue 5.520. If the trial field bas two variational parameters, we obtain approximations to the lowest three eigenvalues, and 80 00. The Ritz procedure also gives us an approximation to the true field, but it is difficult to esta.blish the nature of tbe approximation. 7-7. The Reaction Concept. 1 A general procedure for establishing stationary formulas can be obtained, using the concept of reaction 88 defined in Sec. 3-8. To reiterate, the reaction of field a on source b is
(a,b) -
f (Eo. dJ' -
Ho . dM')
(7-62)
H all sources can be contained in a finite volume, the reciprocity theorem (Eq. (3-36)1 ;,
(a,b) - (b,a)
(7-63)
The linearity of the field equations is reflected in the identities
(a,b + c) - (a,b) + (a,c) (Aa,b) - A{a,b) - (a,Ab)
(7-64)
where the notation Aa means the a field and source are multiplied by the number A. Many of the parameters of interest in electromagnetio engineering are proportional to reactions. For example, the impedance parameters of a IV. H. RumllCy, The Reaction Concept in Electromngootic Theory, PhlJ•. Reo., lief. 2, vol. 94, no. 6, pp. 1483-1491, June 15, 1954.
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
341
multiport Unetwork" are proportional to reactions, as shown by Eqs. (341). Approximations to the desired reactions can be obtained by a.ssuming trial fields (or sources) to approximate the true fields (or sources). It is then argued that tbe best approximation to a desired reaction is that obtained by equating reactions between trial fields to the corresponding reactions between trial and true fields. To be specific, suppose we want an approximation to the reaction (c.,c.). (The symbol c stands for II correct.") The approximation (a,b) is then best if we 15ubject it to (7-1i5)
(a,b) - (c.,b) - (a,c,)
because we have imposed all possible constraihts. Equation (7-65) caD be thought of as the statement that all trial sources look the same to themselves as to the correct sources. The reaction (a,b) obtained from Eq. (7-65) is also stationary for small variations of a and b about c.. and c.. This we can prove by letting and showing that a(a.b)
ap..
I P.-JlIlo-O -
a(a,b) aP/l
I ""'-Pl-O
0 =
(7.00)
Substituting for a and b ioto Eqs. (7-65), we have the three relationships
+ P.«.,,,) + p,(c.,<.) + P.p,« ....) + PIJ,(c.,e.) (c.,,,) + p.«.,,,)
(a.b) - (c.,,,) = (c.. ,c.)
-
Using the last two equations in the first equation, we obtain (a,b) = (c..,c.) - p..p.(e..,e.)
It is now evident that Eqs. (7--66) are satisfied, proving the stationary character of (a,b).
We have a slightly different case when the reaction concept is used to determine resonant frequencies of cavities. The true field at resonance is a source-free field; so the reaction of any field with the true source is zero. Hence, if we let a = b represent a trial field and associated source. Eq. (7-65) reduces to (7-f>7) (a,a) - 0 We can think of this as stating that the resonant frequencies are zeros of the input impedance. To apply Eq. (7--67), we assume a trial field and determine ita sources irom the field equations. For example, an assumed E field can be sup-
342
TIME-HARMONIC ELECTROMAGNETIC FIELDS
---'8
(b)
(a)
(c)
FIG. 7·9. Sources needed to support (a) a trial E field, (b) a trial H field, and (el both a trial E field and a. trial H field.
ported by the electric currents
]
~
~V
-jw,E -
}w
X (.-'V X E) •
(7-68)
However, if the trial field docs not satisfy n X E "" 0 on S, we need the additional magnetic surface currents
M.
= n
on S
X E
(7-69)
This is illustrated by Fig. 7-9a. We now substitute from Eqs. (7-68) and (7-69) into Eq. (7-67) and obtain
to support the discontinuity in E at S.
o-
(a,a)
= -jw
III J. 1P Iff HII ~
E dd
M, . C~. V X E) ds
,E·Edd
E· V X (.-'V X E) dT
- ~ 1P (n X E) . (.-'V X E) d' If n X E ~ 0 on S, the above equation roouces to Eq. (7-45). If 8. stationary formula. in terms of the H field is desired, we consider the trial field to be supported by the sources
If n X E = 0 on S, this reduces directly to Eq. (7-36).
M - -jw.II -
~ V X (,-'V X II)
}w
M.""nxC~VXH)
(7-70)
onS
as represented by Fig. 7-9b. Application of Eq. (7-67) now lea.ds to Eq. (7-46), or to Eq. (7-37) if M. - O. Statiomiry formulas in terms of both E and H are also possible. This time we consider both electric a.nd magnetic currents, as shown in Fig. 7-9c. They are found from the trial fields according to
J
= -jW<E
+VXH
M = -jwllH - V X E M.... nxE ODS
(7-71)
PERTURBATIONAL AND VABlATIONAlo TECHNIQUES
343
Equation (7-67) then gives
0- fII (E. J -
- Iff
(-jW<E'
H . M)
d, -<jp H . M. da
+ E· v
XH
+ H· v
X E
+ j".H') d,
-<jpE X H·cls which can be rearranged to
.fII (E· V X H + H· V X E) d, -1P E X H· ds fff (.H' - ,E') d,
" - 1
(7-72)
This is sometimes called a "mixed-field 1/ stationary formula. The minus sign in the denominator might seem strange, but it. is easily shown that E and lJ are 90° out of phase in the loss-free case (see Sec. 8-4). Henco, the denominator is twice the stored energy in the cavity. Finally, if the trial fields have discontinuities in n X E or n X Hover surfaces within the cavity, we must add the appropriate surface currents to support the discontinuities. This procedure leads to additional surface integrals in the stationary formulas, as shown in Probs. 7-Zl and 7-28. Earlier we showed that reactions constrained according to Eq. (7-65) were stationary. But in the above cavity formulas we calculated w by forcing the reaction to vanish. We shall now prove that the w so determined is stationary about the true resonant frequency. In the usual manner, we let the trial field be the true field plus 8. parameter times an error field, represented by
a=c+pe For fixed e the reaction (a,a) is a function of both wand p. Equation (7-67) constrains (a,a) to vanishj hence, as wand p are varied, we have a(a,a) ~ iJw..... p-o
;w + a(a,a) ~ iJp
p _ 0
..... p.. o
The second term of this equation vanishes because (a,a) is stationary about p - O. The coefficient of the first term is not in general zerOj SO ow = 0 Thus, the first variation of w vanishes, and all formulas for w derived from Eq. (7-67) are stationary. The reaction concept also provides US with an alternative way of viewing the Ritz procedure for improving the trial field or source. We
344
TIME-HARMONIC ELECTROMAGNETIC FIELDS
assume the trial field or source to be a linear combination of functions, represented by a"'" Uu
+ Vv + ...
(7-73)
where U, V, ... arc numbers to be determined. According to the reaction concept, all trial fields should look the same to themsel ves as to the true source; hence we should enforce the conditions (a,u) - (c,u) (a,v) - (c,v)
(7-74)
Substituting from Eq. (7-73), we obtain the set of equations U(u,u) U(u,v)
+ V(v,u) + + V(v,v) +
~
(c,u) (c,v)
-
(7-75)
which can be solved for the parameters U, V, . The solution 80 obtained is identical to that obtained by the Ritz procedure. To illustrate, let us reconsider the example of Sec. 7-6, which was the Ritz procedure applied to the circular cavity (Fig. 5-7). Our trial field was Eq. (7-58); so for the same approximation by the reaction concept we choose (7-76)
The sources of these fields, according to Eq. (7-70), are 2· M." =..1
w,
M .. = 3ia
(7-77)
w,
Calculating the various reactions according to Eq. (7-62), we obtain (u.u)
:=
a' 211'da 2 ( jwp."4
2 ) + jWf:
a' (u,v) = (v,u) = 211'da 3 ( jwlJ, 5" (v,v) = 211'da t ( jwp. a' '6
2) + jWf
(7-78)
9)
+ jwri
All reactions with the correct source Ilre zero, because the true field is source-free. Bence, (c.lt) = (c,v) = 0 and Eqs. (7-75) reduce to U(u,u) U(u,v)
+ V(v,u) + V(v,v) =
0 0
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
345
These equations can have a .J,lOntrivial solution only if the determinant of the coefficients of U and V vanishes. Hence, (U,UXll,ll) - (U,ll)1 = 0
(7-79)
ia the equation from which w is to be found. The solution of Eq. (7-79), with the reactions 01 Eq,. (7-78), yields Eq. (7-61). 7-8. Stationary Formulas for Waveguides. At cutoff, a waveguide is a two-dimensional resonator; so we should expect stationary formulas for the cutoff frequencies to be of the same form as those for the resonant frequencies of cavities. We must, of course, be careful in applying the reciprocity theorem, because the sources of our trial fields are not of finite extent. However, if we take a slice of the waveguide, as was done in Sec, 7-4, surface integrals over the top and bottom just cancel at resona.nce. The height of the slice is common to all terms and therefore cancels. Starting from Eq. (7-67), we arrive at stationary formulas differing from our cavity formulas only in that volume integrals are replaced by surface integrals and surface integrals by line integrals. Renee, the E-field formula corresponding to Eq. (7-45) is
w,' -
II .-'(V X E)'
ds
+ 2 f lc.-'v
II tE'da
X E) X E] . n dl (7-80)
where n is the outward-pointing unit vector normal to the waveguide walls. The H-field formula corresponding to Eq. (7-46) is
.
w ' -
II ,-'(V X H)' d.
II·H'd.
(7-411)
and the mixed-field formula corresponding to Eq. (7-72) is
.fI (E· V X H +IH· V X E) d. - f E X H· n dl I (.II' - ,E') d,
w, - J
(7-412)
None of the above formulas require boundary conditions on the trial fields. Corrections for discontinuous trial fields can be made as outlined in the preceding section. As an example, consider the partially filled rectangular waveguide of Fig. 4-84. In See. 4-6 we obtained a transcendental equation for the cutoff frequency [Eq. (4-51»). For a variational solution, let. us use Eq. (7-80) and a trial field E
-
. "" U.Sln-
a
346
TIME-HA1U£ONIC ELECTROMAGNETIC FIELDS
which is the empty-guide field.
w.=
The result is l
r [1 +" - "(da 2r1.
2r~]-" ---SIn -
o Vt!'ut
(1
a
(7-83)
Noto that this is an explicit fonnula for We, in contrast to the exact equation, which is transcendental. Table 7-2 compares the above result with the exact solution for the case fl . . 2.45(. and It ... Eo. We should expect the approximation to become \Vorse as tdEt becomes larger, since the field then tends to concentrate more in the dielectric. TABLE 7-2. RATIO OF W ...VEoumE WIOTlt TO CU'J'()I'P WAVELENGTH FOR TIll: Rr:cTANOOLA"lt WAVEGUIDE WITH DII~LECTaIC SLAB
("ElI:aet" valuce read from curvell by Frank) dfa
a~
(cnct)
0 0.167 0.286 0.500 0.600
0.500
1.000
al>..
(approximate)
0.'1.85
0.500 0.486
0.4.50 0.375 0.350
O.4sa 0.383 0.352
0.319
0.319
A knowledge of the cutolT frequency of a waveguide homogeneous in and ~ is sufficient to determine the propagation constant at any other frcqucncy according to Eq. (7-26). If the guide is inhomogeneously filled, as for example the above-treated rectangular waveguide with dielectric slab, there is no simple relationship betwecn the cutoff frequency and the propagation constant. We therefore have need of stationary fonnulas for propagation constants. In all of the previous examples, the field equations were given by an operator which was self-adjoint with respect to the desired integration. I For inhomogencously filled waveguides, the field equations lead to an operator which is not self-adjoint Hence, an appropriate adjoint operator must be fowid and the derivation of the stationary formulas suitably modified. It turns out that the operator for waves traveling in the -z direction is the adjoint of the operator (or waves traveling in the +z direction, and the derivation proceeds as follows. Define +2: traveling waves 88 t
E+ :t+(::,y)e-J6' == (~I + u.B.)e-ii• H+ "'" ft+(x,y)e-J~' == Ca, u.O.)e-i '. :::I
+
(7-84)
I A. D. J3erk, Variational Principles for Electromagnetic RCl'Ionatotli and WllV~ guides, IRE Trana., vol. AP-4, no. 2, pp. 104-110, April, 1956. t B. Friedman, "Principles and Techniques of Applied Mathematics," John Wiley "nd Son.} [nc., New York, 1956, p. 44.
347
PERTURBATIONAL Al'Ii"D VAlUATIONA.L TECHNIQUES
Substituting these into the field equations, we obtain V X ~+ V X it+
+ jW.f!+ -
jPU.
X ~
- jW&:+ "'" ilJu.. x ft+
Using analogous definitions for -z traveling waves, we find V X ~V X
+ jw.fl- =
a- - jw.P:- =
-jpu. X ~-jpu. X iI-
By direct substitution, it can be shown that for any +z traveling wave solution there exists a -2 traveling wave solution given by E- .. ~-(x,y)e~" .,.. (tl - u..2.)e»· (7-85) H- = f!-(x,y)e'" = (-fl, + u.B.)e"· where the t/, tt, and 9 .. of Eqs. (7-84) and (7-85) are the same functions. Now multiply the first of the +z wave equations scalarly by :A-, and t.he second of the -z wave equations by ~+, and add the two resultant equations. This gives Ji-. V X ~ + ~+ . V X iI- + jw.fl- . fl+ - jw'£+ . ~= -2iP~, X l • u. which, when integrated over the guide cross section and rearranged, yields
e.,
a
II
(w.t+ . ~- - w.tt+ . fl- + Jll-. V X ~ + J"t+ . V X a-) dB P = <..<....---------,7"""C--,--------2 II~, X
a,. u.dB
(7-86)
This is a mixed4field formula, stationary if n X E = 0 on C. For the E-field formulation, eliminate it from the +2 and -2 wave equations, and proceed as in the derivation of Eq. (7-86). The resultant formula is
pi
II lA-IE/Ids - j21J JJ dB + II I.-'(v X t+) . (V X ~-) p-IEI' vEl
w',£+ . t-] d8 = 0
(7-87)
stationary if n X E = 0 on C. Tho H-field formula is given by Eq. (7-87) with E, lA, E replaced by P, E, H, and it is stationary with no boundary conditions on H. Equations (7-86) and (7-87) remain stationary in the lossy case, for which ilJ should be replaced by 'Y = (I + iP· For an example of the calculation of propagation constants, consider the centered dielectric slab in eo rectangular waveguide, as shown in the insert uf Fig. 7-10. AJ< a trial field, take
..
. n a
r.,-Uwsm-
348
TIME-HARMOllo'lC ELECTROMAGNETIC FIELDS
1.6
y+ --Idl+-
BEI
1.4
1-- --1 • 0
•
12
2.45
~
/.
d/o- 1.0_0.5 0.3
V
~:-
~~
1//' /
0.8
0.4
(
a
02
0.4
--
l=: 0.1 ~O
Exact-Approximate ----
0.6
0.8
1.0
1.2
of>., Fla. 7-10. Compariaon of approximate and enet. propagat.ion col1$lanta for the rectangular waveguide with centered dielectric slab, t - 2.454. (After BeT.I:.)
and use Eq. (7-87).
p. ~ kG
[1
The result is l
+~ (~ +! Sin!'!) - (..!...)']~ E,a'll" a koa
(7-88)
The exact solution is given in Prob. 4-19 and requires the solution of a transcendental equation. A comparison of a values obtained (rom Eq. (7-88) with the exact values for pjk a is shown in Fig. 7-10 for the case E = 2.451:0. 7-9. Stationary Formulas for Impedance. A formula for impedance in terms of reaction is given by Eq. (3-41). Such a formula. when constrained according to Eq. (7-65), is a stationary formula for impedance. Figure 7-11 represents a perfectly conducting antenna excited by a current source. The resultant current on the antenna will distribute itself so that ta.ngential components of the total electric field vanish on the conductor. The antenna. terminals are close togetherj so the reaction of any field with the current source is of the form - VI. If a trial-eurrent distribution J.- is assumed on the antenna, the formula for input imped· ance [Eq. (3-41)J is (7-8lJ) I
Berk. lJp. cU.
349
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
where I is the input current. The impedance as calculated by Eq. (7-89) is stationary about the true current, as we shall now show. On the antenna surface, the tangential components of tbe true field E~ arc zero except at the input; hence (c,a)
lC:
-
I
Vel = - PZI • = (a,a)
Also, (c,a) = (a,c) by reciprocity; 50 the constraints of Eq. (7-65) have been met, and Eq. (7-89) is a stationary formula. Equation (7-89) was used to calculate impedance before its stationary character was noticed.' This method should not be confused with the induced emf method (7-90)
FIG. 7·11. An antenna excited by a current &(lurcc.
which is based on the conservation of complex power. Equation (7-90) is not stationary unless both the true current and the trial current are real. When the trial current is MSumed real, we get tbe same answer from Eqs. (7-89) and (7-90). Hence, the input impedances for waveguide feeds calculated in Sec. 4-10 a.re also variationa.l solutions to the same problems. If we have two sets of input terminals, as, for example, in the case of the two linear antennas shown in Fig. 7-12, the variational formula for mutual impedance is (7-91)
where I .. and 16 arc the input currents at terminals a and b, respectively. The demonstration that the constraints of Eq. (7-65) are met is similar to that for self-impedance. Note that Eq. (7-91) involves the assumption of currents due to both sources, since E" is the field of J.... The extension to N sets of terminals is straightforward. The calculation of mutual impedance is usually simpler than the calcu~ lation of self-impedance because the source and field points are separated. Let us therefore take a mutual-impedance problem as our first example. Consider the parallel linear antennas of length >./2 as shown in the insert of Fig. 7-12. No appreciable error will be incurred by assuming the currents as filamentary, as long as the antenna diameters are small compared to wavelength and compared to antenna separation. Let the z axis lie I P. B. Carter, Circuit Relations iD Radiating Systems and Applications to Antenna. Problems. PrO(;. IRE, vol. 20, DO. 6, pp. 1004-1041, June, 1932,
350
TIME-HARMONIC ELECTROMAGNETIC FIELDS
BO
60
i\
'/l~di
\ R.,
40
1\
o
c§
20
o -20
\
\ x..
\
/ \~/
1"-./
-40
K /1.0
""
~.5
V'----'
FlO. 7-12. Mutual impedance Z.. _ R.. free space.
+ ;X.. between parallel 1+./2 linear anlennu in
along antenna a, and assume /- .. I. cos
2",
T
2", P - I. cos-
(7-92)
X
Our formula for mutual impedance [Eq. (7-91») become8, in this case,
Zn = - -1// ••
fV'
-),j4
E."I'd,
By the usunl vector-potential method we have
E,' _ }WE ~(:'. + k') A,' v3 where, at antenna. b, I jW4
A .. - • 4-..
["(t')
-'A/4.
e-i1v"'+(0-/)'
V d' + (z
Substituting for E.- and I' in our expression for Z.... where
'" f
I
Z.,
dz'
we obtain
2rz
2rz'
1\
1\
_),j" dz' cos "'" cos ~ G(z,z') 1 (a. ) e- i1,;'d'+<0-,'jI - - - + k' kjwE az' V d' + (z z')'
-Vol
G(.,') -
f)"/4
%')2
dz
(7-93) (7-94)
PERTURBATIONAL A.ND VARIATIONAL TECRNlQUES
351
The integrations of Eq. (7-93) can be expressed in terms of sine integrals and cosine integrals. The details of the integration can be found in the literature. I Letting Z.. -
R.. +iX..
we obtain for the result
+ .' + .') Silv'(kd)' +.' + .'J -
R.. = 4: (2 Ci(kd) - Cilv'(kd)' X..
= ;"'!2Si(kd)
-
+ .' - .'11 Silv'(kd)' +.' - .'11
Cilv'(kd)'
(7-95)
where Ci(x) nnd Si(x) are as defined in Prob. 2-44. Figure 7-12 shows 0. plot of Eqs. (7-95). The mutual impedance between linear antennas of otber lengths and orientations can be found in the literature.l,s The evaluation of the self-impedance of a linear antenna is more difficult because oC the singular integrands encountered. Let us use this problem to illustrate the use of adjustable parameters in the trial current. The geometry of the center-fed linear antenna is shown in the insert of Fig. 7-13. Let the current on the antenna be represented by two functions, according to Eq. (7-73). Our trial current is then a surface current of the form
J. - UJ." + VJ."
(7-96)
where U and V are adjustable parameters. According to the reaction concept, the trial functions should look the same to the assumed current as to the true current; hence we enforce the conditions (a,u) ~ U(u,u) (a,v) - U(u,v)
+ V(v,u) - (c,u) + V(v,v) ~ (c,v)
where (e,u) and (e,v) can be calculated, as we shailialer show. for U and V, we have in matrix notation
Solving
Substituting for U and V into Eq. (7-96) and calculating the self-reaction, we obtain
[(U,U)
(a a) = {(c u) (cu») , "(11.,1')
(v,u)]-. (v,v)
[(C,U)] (e,v)
(7-98)
I P. S. Camt, Circuit Relations in Radiating Systems and Applications to Antenna Problems, Proc. IRE, vol. 20, no. 6, pp. 1004-104.1, June, 1932. I G. Brown and R. King, High Frequency Models in Antenna Inveatigations, Proe. TRE, vol. 22, no. 4, pp. 457-480, April, 1934.
352
TIME-HARMONIC ELECTROMAGNETIC FIELDS
5000
z
T
f--
Lla
L . %=0
f--
4000
1+i
a::
22,000
~2a
3OO0
J
2000
Lla = 1800
1000
Ah..
"-~
V
~
o
/
Lla"" 150
/"
(oj
24oo
I
rl II
Lla = 22,000
1600
/
800
V
E
~
o
'"\
I#" £V
L/'
o -BOO
~Lla = 1800
~ Ljo ~ 150\
I/j,
'// --/ 7
1//
/
IV II
-1600
I V
- 2400
o
2
4
6
B
10
12
kL (b)
FlO. 7-13. Variational solution for the input impeda.nce of the symmetrical cylindrical antenna. (Afttr Y. Y. Hu.) (a) Input resistance; (b) input reactance.
PERTURBATIONAL AND VARIATIONAL TECHNlQUES
353
Equations (7+97) and (7-98) also apply to the case of N adjustable constants if the various matrices are extended to N rows and/or columns. Expanding Eq. (7-98), using the reciprocity condition (u,v) = (v,u), we obtain ( ) (c,u)'(v,v) - 2(c,uXc,vXu,v) + (c,v)'(u,u) a,a .,.
(u,uXv,v)
Now note that n x E·
(u,v)t
= 0 on the antenna surface except at the feed; 80 (c,x) "'" - VI.!"
(or any z, where VI. is the input voltage and [" is the z current at the input. Using the above two relationships in Eq. (7-89), we obtain Z
Z I. """
t I.
1.1 (11.,'1') - 21..1.(u,v) + ! ..S(v,v} (u,v}S (u,u}(v,v)
which can be rearranged to read (7-99)
where I. and I. are the values of the u and v trial currents at the input. Let us now look at the form of the reactions. The currents will be rotationally symmetric %-dircctcd surface currents on the cylinder p = a, where a is the antenna radius. These currents can be expressed as
r." -
1
-2 .a l"(,) u,
(7-100)
where I" is the total current and x .... 11., V. By the potential integral method we can calculate the field of the current J.s as ( kS E.'" "" - S18 ' 1t
Jwt
dz' 10'" dfj/J.rG + a2a') ILI2 -L/2 0
(7-101)
%
where
(7-102)
The various ren.ctioos of Eq. (7-99) are then given by (x,y) =
L/2
I-£/2
dz
10'" adq,E,"'"J.' 0
(7-103)
where Eo'" is given by Eq. (7-101) with p =* a. Note the singular nature of the Grecn's function [Eq. (7-102») a.t p - a. A precise evalua.tion of Eq. (7-103) would be difficultj SO the following approximation is usually used. The field of the current is approximated by the field of a filamentary current of the same magnitude. This is
354
TIME-HARMONIC ELECTROMAGNETIC FIELDS
equivalent to replacing Eq. (7-102) for
p =
a by (7-104)
For thin antennas, the error introduced by Eq. (7-104) is negligible, as caD be shown by the following argument. The field of the filament of current is a source-free field in tbe region external to the linear antenna. We can therefore assume that this field exists and calculate the equivalent currents on the surface of the antenna. As long as the equivalent magnetic currents are negligible, as they will be for thin antennas, we can take the equivalent electric currents for our trial currents. The resultant current is essentially that of Eq. (7-100). Using the above approximation for G, we obtain from Eq. (7-103) (x.y)
~
,.!; fLI'
"X~JWE
-L/2
d'
fLI'
-L/2
liz' 1,(,')1-(,)
(k' + :',) G
(7-105)
uZ
where G is given by Eq. (7-104). Note that, to this approximation, the self-reaction is equal to the mutual reaction between two identical antennas fed in phase and separated by a distance B. Hence, Eqs. (7-95) with d replaced by a give the first-order (one trial function) variational solution for the input impedance of a >../2 linear antenna. In particular, note that for very small a = d, Eqs. (7-95) reduce to
R,.
= 73.1
X,.
= 42.5
(7-106)
as is evident from Fig. 7-12. Resonance (X = 0) occurs for L slightly less than >../2. For trial functions in the second~order solution,
I" -
Sill
k(~ -1'1)
I' - 1 - co, k (~ -
(7-107)
1'1)
have been used in the literature. The evaluation of Eq. (7~105) for (x,y} = (u,u}, (u,v}, and (v,v) is long and involved, and formulas in terms of sine integrals and cosine integrals have been given by Storer' and Hu. t Numerical values of the input impedance are given in Fig. 7-13. The antenna is said to be resonant when X is zero and kL n'lf, n odd. It is said to be antiresonant when X is zero and kL Rl n'll'", n even. Note that. 0:;
I J. E. Storer, Variational Solution to the Problem or the Symmetrical Cylindrical Antenna, Cruft LaO. Rep. TR 101, Cambridge, Mass., 1952. I Y. Y. Hu, Back-llCa.ttering Cross Sections of a. Center-loaded Cylindrical Antenna, IRE TraM., vol AP~6, no. I, pp. 140-148, January, 1958.
355
PERTURBATIONAL AND VARIATIONAL TECHNIQUE8
in the vicinity of resonance, R is inE _ El + E~ sensitive to antenna thickness. It / Obstacle is in these regions that the analysis tSource of Sec. 2-10 gives good results. Both trial currents of Eqs. (7-107) are zero at the input for kL = 41r. Hence, FIG. 7-14. Wave seat.t.ering by an the input impedance calculated there- st.acle. from cannot be valid in the vicinity of kL = 47. Perhaps a better choice for the v current would be
I" -
~
-
D
00.-
Izi
which is finite at z = 0 for all L > O. However, calculations have not been made for this choice. 7·10. Stationary Formulas for Scattering. Let us first treat the ba.ckscattering, or radar echo, type of problem by the variational method. The problem is represented by Fig. 7-14. It consists of a source and one or more obstacles, and we wish to determine the field scattered back to the source. For simplicity, the obstacle will be considered a perfect conductor and the source a current element n. The more general case of dielectric obstacles is considered in Sec. 7-11. Let the incident field, that is, the free-space field of the source alone, be denoted by Ei. The total field E with the obstacle present is then the sum of the incident field Eo plus the scattered field E·. The reaction of the sca.ttered field on the current element is (8,i)
~
liE,'
~
-IV'
(7-108)
where V· is the scattered voltage appearing across l. Let the echo be defined as the ratio of E,- to n. Then, using reciprocity, we ha.ve
EI' (S,1) (i,s) Echo - If - (II)' ~ (II)' -
(I~)' 1P E; . J. d,
(7-109)
where J. is the current induced on the perfectly conducting obstacle. The boundary condition at the obstacle is n X E = 0, or on S
n X E; - -n X E'
(7-110)
Hence, Eq. (7-109) ca.n be written as (c,c) -1 Af.. Echo - (II)' 'Jr E' . J. d, - - (ll)'
where (c,e) stands for the self-reaction of the on the obsta.cle hy the source.
U
(7-111)
correct" currents induced
356
TI~RARMONIC
ELEC1'ROMAONETIC FIELDS
For a stationary formula, we assume a current J- on S and approximate (c,c) by (a,a), subject to the constraint (7-112)
(a,a) - (e,a) - - (i,a)
The last equality results from Eq. (7-110). To express tbis constraint in a form for which (ala) is insensitive to the amplitude of J., we take
(i,a)' (a,a) - - ( a,a ) and, replacing (c,c) by {a,a} in Eq. (7-111), we have -(i a)' Ecbo = (Tl) '(a,a) =
(1fE"
d.)'
Jo
(TI)' 1f Eo.
JO d.
(7-113)
where E- is the field produced by the assumed currents Ja, This is the variational formulation of the problem. Note the close similarity of the echo problem to the impedance problem of the preceding section. The impedance problem is essentially an echo problem for which the source is at the obstacle. A more general formulation of the echo problem can be made by replacing I l with an arbitrary source. The tensor Green's functions alSee. 3-10 can be used to put Eq. (7-113) into a more descriptive form. Define [r(r,r')] as the tensor of proper· tionality between a current element dJe at r' and the field dEe that it produces at r. that is, dEo(r) _ (r(r,r») dJo(r) Then Eq. (7-113) can be written as
- [i11f E'(r) . Jo(r) ds
Echo """
r
-,,---'7;:-:=-------""--
1f d.1f d.' J"(r) . (r(r,r')] Jo(r') This equation is in a form characteristic of variational solutions in general. A commonly calculated parameter is the echo area, defined by Eq. (3-30). For linearly polarized fields, the echo area is given by (7-114) If, in Fig. 7-14, we let TZ be z4rected and located on the.:z: axis, and then
let r = ::c: -+ co, we have, in the vicinity of the obstacle,
E' = u. i"ll tJh 2~T
=
u.BoeJb
357
PERTURBATIONAL AND VAlUATIONAL TECHNIQUES
2.0
••
1:, m
1.5
-1
..
( \ Llo -
150
"
1'"'20
1.0
r'\
~\Lla - 1600
0.5
o
~ .L: \\: 2
........
\
L~o =
4
8
6
2f.OOO
10
12
1L Flo. 7-]5. Broad,ide echo
.rea A. of .. wire.
(Aflu Y. Y. Bu.)
Also, by definition, we have echo = E,'fll; hence from Eq. (7-113)
E J•
~E.
(1f> u, . J'e d8)' -'1'21\r 1f> E' . J'd.
= _--"-il-,,
i"
Therefore, by Eq. (7-114), our stationary formula for echo area is
." (1f> J.·oi" d.)' ,
A. = .. - X
1P E'· J'd.
(7-115)
when the incident plane wave is .-polarized and -x traveling. As an example, consider the scattering of a plane wave by a thin conducting wire, as represented by the insert of Fig. 7-15. The integral in the denominator of Eq. (7-115) is just the self-reaction of the assumed current on the wire. This is the same type of reaction that we encountered in the linear-antenna problem, approximated by Eq. (7-105). Defining A as the self-reaction, we have A
= A E- . J- dB 'it'
=
-~- fL/2
- LI2
do
fL12L/2 dz' [-(z)1'(zI) (k' + ~) G a.' -
(7-116\
358
TIME-HARMONIC ELECTROMAGNETIC FIELDS
where G is given by Eq. (7-104). For the current on the wire we should expect a constant current "forced II by the incident field plus a" naturalmode" sinusoidal current. At the ends of the wire, the current should be practically zerOj hence we assume for our trial current j-
= cos kz
L - cos k '2
(7-117)
Equation (7-116) can then be evaluated as Re (A) - ~: (kL
+ kL cos kL - 2 ,in kL) Si (kL) + log 2ykL - Ci (2kL) -
~: 1(kL + kL co, kL -2'in kL) [ Ci (kL) + log ,~n]
1m (A) -
+ Si (2kL) where y
'in' (kL))
ZE
1.781.
- (I
+ cos kL) ,in kL
(7-118)
I
The integro.l in the numerator of Eq. (7-115) evalu-
ates to
I
Ll' -Ll2
which defines B.
B [-(z) dz = -I (2k sinL - - kL cos -kL) - -
k
2
2
k
(7-119)
Hencc. the echo area is
A. =
I~
I:;.1'
(7-120)
with A and B given by Eqs. (7-118) and (7-119). This solution gives good accuracy out to about kL = 8. Figurc.7-15 shows a plot of A./).! for the second-order solution (two trial functions), as calculated by Y. Y. Hu. t The results for plane wa.ves incident at an arbitrary angle are given by Tai.' He also shows the effect of choosing different trial functions. In two-dimensional problems, the quantity echo width L. corresponds to the echo area of the three-dimensional problems. The echo width is defined as the width of incident wave which carries sufficient power to produce, by cylindrically omnidirectional ra.diation, the same backscattered power density. In equation form, the ceho width is
L. - ,-_ lim (2ro gl ~')
(7-121)
1 Y. Y. Bu, Back..catlering CroM Section of a Center-loaded Cylindrical Antenna, IRE TraM., vol. AP-6, no. I, pp. 14.0-14.8, January, 1958. I C. T. Tai, Electromagnetic Back-scatlering [rom Cylindrical Wires, J. Appl. Ph"., vol. 23, no. 8, pp. 009-916, August, H152.
359
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
orj for linear polariza.tionj
. ( IE'I')
L c = ~~
2rp E'
8
(7-122)
6
where superscripts 8 and i stand for llscattered U and H incident,lI respecti vely. Going through a development similar to that used for Eq. (7-115), except that a line source is used j we obtain
( J. J
L, _.!. • 'f 2X
"h
•
dl)' ,
f EtJ,'dl
>•
tHi ITI-t'"
E'
7 /
o
---
0.2
../
0.4
0.6
0.8
1.0
aI'
FIG. 7-16. Echo width L, of a conduct,.. ing ribbon of width 4.
(7-123)
if the incident field is z-polarized and
-:J:
traveling.
(f J.,"h dl)'
SimilarlYj
, (7-124)
fE'. J' dl
•
if the incident field is y-polarized and -x traveling. From symmetry, J" in Eq. (7-124) should have no z component. In both Eqs. (7-123) and (7-124), it is assumed that thc scatterers are cylinders generated by elements parallel to the z axis and the line integrals are in a transverse (z = constant) plane. For an example of a two-dimensional problemj consider a z-polarized plane wave normally incident on a conducting ribbon of width a. This is illustrated by the insert of Fig. 7-16. Assume that the current induced on the ribbon is uniform j that is,
J.' = 1
(7-125)
Because the current is real, the integral in the denominator of Eq. (7-123) is
J'/2
- ../2
E.oJ.fJ dy =
J'/2
- ../2
E."J.fJ$ dy = -P
where P is the complex power per unjt length supplied by J .-. But we have already analyzed the ribbon oC uniform current in Sec. 4-12, the result being
P =- 1[21Z
=
where Y.~rt is plotted in Fig. 4-22.
at
I• Y_~rt
The echo width j according to Eq.
360
TIME-HARMONIC ELECTROMAGNETIC FIELDS
Receiver
./ FIG. 7-17. scattering.
Differential
Transmitter
(7-126)
A plot of this is shown in Fig. 7-16. and obtain
For large a we can use Eq. (4-107) (7-127)
which is also the physical optics approximation (see Fig. 3-21). The more general case of differential scattering, or transmission,' is represented by Fig. 7-17. The problem consists of a transmitter, which illuminates the obstacle, and a receiver at which we wish to evaluate the scattered signal. For simplicity, let us consider both the source and receiver to be unit electric currents. Then, according to Eq. (3-39), the voltage across the receiving current due to the transmitting current is (7-128)
where t and r refer to the source or field of the transmitter and receiver, respectively. The total signal received is the superposition of the inci· dent field, due to the transmitter alonc. plus the scattered field. due to the currents c on the obstaclc. Hencc, (7-129)
where (t,r) is calculated with the obstacle absent and (c,r) involves the free-space field of the currents on the obstacle. The transmitter and receiver currents are assumed to be known (they are current clements in our simplified case); so V r' can, in principle, be ealculatcd exactly. Our problem is to obtain the variational formula for V r -. We shaH here consider only the simple case of a pcrfectly conducting obstacle, the general case being considered in Sec. 7-11. Applying reciI A traDsmission problem involves the evaluation of the total field at the receiver, while a. scattering problem involves the evaluation of only the scattered field.
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
361
procity. we have, for the scattered voltage at the receiver, - V.' ~ (c,r) - (r,c)
-1ft (E')-· a,')' tU
(7-130)
where 0/)' is the surface current induced on the obstacle by the transmitter and (Ei)r is the field of the receiver current calculated with the obstacle absent (the incident field). Our boundary conditions on the various true fields are n X E = 0 at the obstacle boundaryj hence n X (E~- - -n X (E')' n X (E')' - -n X (E') ,
(7-131)
where superscripts i and 8 refer to incident and scattered components, and t and r refer to transmitter and receiver sources. Hence, by Eqs. (7-130) and (7-131), we have
V.'
-1ft (E')-· (J,')' d8 -
(c.,c,)
(7-132)
where (er,c,) stands for the reaction between the field of the II correct " currents induced on the obstacle by the receiver and the" correct" currents induced by the transmitter. For our stationary formula, we approximate (c"c,) by (a,.,4,), where the a's denote assumed currents on the obstacle, and constrain the latter according to Eq. (7-65), which is
(a.,..) = (c.,a,) - (..,c,)
(7-133)
In the language of t·he reaction concept, Eq. (7-133) says that the assumed currents look the same to each other as to their respective true currents. By Eqs. (7-131) and reciprocity, Eqs. (7-133) become (a"a,) = (Cr,a,) = - (r ,4,) (a"a,) = (a"c,) = (CI,a,) = -(t,a,.)
(7-134)
Substituting from Eqs. (7-134) into Eq. (7-132), we have for our variational formula a) -= (r,4,}(t,4,) V ,, = (a r,' Gr,a,
(>
=
[1ft (E~' . (J,o), d8] [1ft (E')' . a:>- dB] 1ft (EO)' . a:)' d8
(~lM)
where (Ea)r is the field due to the assumed currents (J~.)", which approximate the currents induced by the receiver. Note that Eq. (7-135) involves the assumption of currents on tho obstacle due to sources at both the transmitter and receiver. Note also that Eq. (7-135) reduces to the formula for back-scattering [Eq. (7-113)J when the transmitter and receiver coincide.
362
TIME-HARMONIC ELECl'RQAlAGNETIC FIELDS
7-11. Scattering by Dielectric Obstacles. I The problem of differential scattering by a. dielectric obstacle is represented by Fig. 7-17 if the obstacle is now considered a.s a dielectric body. We shall assume it to be nonmagnetic (}ol = 1£0), but it may be lossy if E is complex. The extension to magnetic obstacles is given in Prob. 7-42. When the obstacle is excited by a source, there will be induced in it polarization currents given by
J' - jw(. -
(7-136)
SuperscripUi tor r will be added to the various quantities to indicate that the exciting source is at the transmitter or receiver, respectively. The treatment of differential scattering of the preceding section made DO assumptions about the nature of the obstacle in the derivation of Eq. (7·130) j hence for unit currents at t and r - V,, - (r,c) -
JJJ (E~'· 0')' dr
(7-137)
where the notation is the same as in the preceding section. Using the relationship E' = E - E' and Eq. (7-136), we can rewrite Eq. (7-137) as
- V,, -
JJJ .-'0')"
0')' dT
-
JJJ
(E')', O'}' dT
= F(c"c,) - (c"c,)
(7-138)
which defines the functional F. Note that F is symmetrical in Cr and and is actually the reaction between Er and aC)1 with the obstacle prescnt. To obtain a stationary formula for the scattered voltage at the receiver, we approximate the true currents c by trial currents a and set
Cl
- Vr '
~
F(a"a,) - (ar,a,) = G(a,.,al)
subject to constraints of the form of Eq. (7-65) applied to G. straints are G(""a,) = G(c"a,) = G(a"c,)
(7-139)
Such con(7-140)
and we find G(c"a,)
~ (r,a,) -
G(.. ,c,) ~ (t,..) -
JJJ (E~'. 0')' dr JJJ (E')'· 0')' dr
(7-141)
Combining the preceding equations to render Vr ' insensitive to the ampliI
M. R. Cohen, Application of the Reaction Concept to Scattering ProbJelll8, IRE
TraM., vol. AP-3, no. 4, pp. 193-199, October, 19M.
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
363
tudes of the trial functions, we have the variational formula
_ V' _ (r,aj)(t,a.) • - F(a"a,) (a..,a,)
[fJI (E')'· (J')' dr] [fJI
- III, 'G')"
(Ery" (J')' dr]
III (E')'· G')' dr
G')' dr -
(7-142)
For the lossy case, Ie = ,"WE + fT. For a perfectly conducting obstacle, t ! - 00 i hence ,e' --+ 0 and Eq. (7-142) reduces to Eq. (7-135). When the transmitter and receiver are represented by the same source, we have the back-scattering problem. Using the definition of Eq. (7-109) for echo, when the source is a unit current, we have
Echo _ -
v/ _ I'
=
JJJ
«i.a)/l)' F(a.a)
GJJJ
(a.a)
E'· J'dr)'
JJJ
'-'(J')'dr -
(7-143)
E'·J'dr
The echo area, defined by Eq. (7-114), can be obtained from Eq. (7-143) by letting the source recede to infinity. The steps parallel those used to obtain Eq. (7-115). For a z-polarized, -2: traveling incident wave, we obtain
(7-144) In two-dimensional problems, the echo width, defined by Eq, (7-122), is found to be
,
(7-145)
II ,-'(J,')' d. - II ENN. if the incident wave is -z traveling and z-polarized, and 7
L--
, - 2X
"(JJ J,'e/ d.)' h
II ,-'(J')' d. -
, (7-146)
liE" J'd.
if the incident wave is -:t traveling and v-polarized. 'Inc surface integrals in Eqs. (7~145) and (7-146) are over the cross section of the oootacle in a z = constant plane.
364
TIllE~BAJU.IONIC
EL:ECTROlLAGNETIC FIELDS
To illustrate the accuracy that we might expect from the variational formulas, let us consider a problem for which the exact solution is available, the circular dielectric cylinder. The incident wave is z.polarized, and the cylinder is defined by p "'" a = >"0/2, as shown in the insert of Fig. 7-1.8. For our first approximation, let us take (7-147)
where k - (oJ V;;; is the wave number of the dielectric. This very crude assumption yields curve (b) of Fig. 7-18. For a better approximation, which yields curve (c) of Fig. 7-18, take (7-148)
where A is a variational parameter to be determined either by the Ritz procedure or by the reaction concept. While Eq. (7-148) is a better approximation than Eq. (7-147), it is still crude. The integrations occurring in the various reactions were accomplished by expressing the exponentials and Hankel functions as Bessel function series, according to Sec. 5-8. The resulting series converged fairly rapidly. An alternative procedure for treating dielectric obstacles can be given
~
0.00012
I
I
,
hi
/
,, i
--+l",1+-
~ 0.00008
---~
/.
-, r-, ,
~--
V
'"
/
0.00004
/
o1.00
1/ 1.04
1.08
1.12
..
(a)
r--.. .... f-j
1',
1.16
Ie,) ,(b)
~
1.20
"-
1.24
1.28
./
Flo. 7-18. Scat.tering by a dielectric cylinder (0) exact. aolut.ion, (b) Jirsw,rder varia· ~ionalllOlution, and (c) aec:ond..<Jrder variat.iollAl solution. (AfUr' Cokm.)
365
PERTURBATIONAL AND VARIATIONAL TECRNlQUES
in terms of cQuivalent currents over the surface of the obstacle. I This method leads t.o more t.han one formula for the desired parameter, and Rumsey discusses how to choose the best approximation according to the react.ion concept. 7-12. Transmission through Apertures. The problem of transmission through apertures in an infinitely thin, perfectly conducting plane is closely related to the problem of scattering by plane obstacles. The precise interrelationship is shown by the following extension of Babinet's principle for optics. Consider the three cases of a given source (a) radiating in free space, (b) radiating in the presence of an electrically conducting screen, and (e) radiating in the presence of a magnetically conducting screen, as shown in Fig. 7-19. The electric and magnetic screens are said to be eomplemen/.aTY if the two screens superimposed cover the entire V = 0 plane with no overlapping. (The apert.ure of one is identical to the obstacle of the other.) Let the fields V > 0 be designated (EI,H'), (Eo,Ho), and (E"',H"') for the cases (a), (b), and (e), respectively. Then Babinet'a principle for complementary screens states that H'
+ H" =
H'
(7-149)
proved as follows. Let S. be the screen surface of Fig. 7-19b, and S. be the aperture surface of Fig. 7-19b. The total field in each case is the incident field E' plus the scattered field E' produced by the currents on the screen. An element of electric current produces no components of H tangential to any plane containing the element (see Sec. 2-9). The currents induced on the screen thus produce no tangential H over the V - 0 plane; hence n X H- ... n X HI over S.
On the screen itself we havc the boundary condition n X E- "" 0
over S.
For the complementary magnetic screen, following similar reasoning, we find nXE"'=nXEi over S, n X H- = 0 over S.
By the above four equations, the sum E' + E-, H- + H- satisfies nx(E'+E-)=nxE' n X (H' + H-) "" n X Hi
overS. over S.
IV. B. Rumsey, The Reaction Concept in Electromagnetic Theory, PAil" Rtl1., vol. 94, no. 6, pp. 148&-1491, June 15, 1954.
2 aer.,
366
Tl!ld.E-HARldO~'lC
ELECTROMAGNE'l'IC FIELDS
+
Hence, the e m. field has the same n X E as the incident field over part of the y - 0 plane and the same n X H over the rest of the y .... 0 plane. These conditions are sufficient to determine E, H in the region y > 0 according to the uniqueness theorem (Sec. 3-3); so Babinet's principle [Eq. (7-149») follows. An alternative statement of Babinet's principle can be given in terms of the dual problem to Fig. 7-19c, shown in Fig. 7-19d. If the original repla.ced by K), the magnetic screen source is replaced by its dual replaced by an electric screen, and the medium replaced by its "reciprocal" (11 by 1/,,), then E will be numerically equal to -H- and H numerically equal to E'" (see Table 3-2). If the field of this dual problem is
a
I I I
Electric conductor S.
I I
EJ. HI 1JO
E-, He,
I
I
t
Is. I
I I I
Source
I
~n
I I
r-+- n
1
,-0
,~O
(a)
(6)
Is.
Is.
II
I
I
t
Source
'10
I
EM, Hili. "10
I S.
Magnetic conductor
I
*
S. Electric conductor
Dual source
I
I
IS.
~n
Is. ~n I
I
,-0
,-0
(0)
(d)
Flo. 7-19. Illustration of Babinet'a principle.
367
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
Electric conductor
E' M,g"t;c cood"to,
"I.
,,-1,
",I,. Transmitter
I
E'+&
1,/ Receiver
Transmitter
Receiver
(6)
(0)
Flo. 7-20. The trnll8mitted field E' of (a) ill equal to the scattered field E' of (b).
denoted by E", R", Babinet'e principle [Eq. (7-149)} becomes H'" - E" = Hi
(7·150)
The problem of Fig. 7-19d is more easily approximated physically than is the problem of Fig. 7-19c. The direct application of Babinct's principle to the problems of Fig. 7-20a and b shows that the field transmitted by an aperture in a planeconducting screen is equal to thc negative of the field scattered by the complementary obstacle. Hence, stationary formulas for..the signal at a receiver on the shadow side of a screen are of the same form as the stationary formulas for the scattered signal at a receiver in the complementary problem. In Fig. 7-20b, let the sources at the transmitter and receiver be magnetic currents across the "terminals" I, and 1.. Then, dual to Eq. (7-135), we have at the receiver
H- ·1.
= -
[fJ
(H')'· (Moo)' dS]
[fJ (H')'· (MtV dS]
ff (Ho),. (Mt)' ds
(7-151)
where Moll denotes the assumed magnetic current on the obstacle. It approximates the true magnetic current M. = (E+ - E-) X n = 2E' X n
(7-152)
where E+ and E- denote E in the regions y > 0 and y < 0, respectively, and n -= U lt• The interrelationships between Fig. 7-20a and b can be expressed as
368
TDlE-H.A.IU10NIC ELEcraOMAGr.'"ETIC FlELDS
Hence, from Eq. (7-151), we obtain for the aperture problem
[JJ
(H~'. (n X Eo), (H~" (n X Eo)- d. ] H'· t. - (Ho),. (n X Eo)' d.
II
(7-153)
where E- is an assumed field in the aperture and H- is the magnetic field calculated from the E-. The sources of Hi are magnet.ic current elements across I, and l~. and, to apply Eq. (7-153), we must assume an n X E in the aperture due to (H')' alone and due to (H')' alone. If ~ and 1. are
images of each other, as they appear in Fig. 7-20, then the aperture problem becomes the same as an echo problem, because of the symmetry of the plane screens about y = o. Sometimes it is the total power transmitted through the aperture that is of interest. We define the tranfmission coefficient T of an aperture as the ratio of the power transmitted through an aperture to tho power incident on the aperture, that is,
T _
If E' X H'··ds Re If E' X ds
Re
~,
apeR
0-
Hi- •
=
(7-154)
~;
Note that T depends on both the nature of the source and the geometry of the a.perture. Another quantity sometimes defined is the trammisrion area, which is the transmission coefficient times the area of the aperture. We shall explicitly consider uniform plane waves normally incident on an aperture in a plane screen, as shown in Fig. 7-21a. Let the incident
Electric conductor
Complete electric conductor
Incident
plane wave
M.
~
D
,-0
,-0
'a)
(b)
FIG. 7-21. {a) Transmission through an aperture, and (b) equivalent problem for the region 'V
> o.
PERTURBATIONAL AND VARiATIONAL TECHNIQUES
369
wave be specified by
H' =
Ue-ih
E' = '7H' X
UI/
(7-155)
"1/'
In the proof of Babinet's
n X H' - n X H'
(7-156)
where u is any unit vector orthogonal to principle, we noted that in the aperture
because the currents on the conducting screen produce no tangential components of H in the y = 0 plane. Equation (7-155) chooses Hi to be real in the y "'" 0 plane; so by Eq. (7-156) n X Hi is real in the aperture. Hence,
ff E' X H'· . ds
, = Re
= Re
.. per~
ff EI X H' • ds
(7-157)
..pen
Now consider the problem of Fig. 7-21b, which for
M.=E'xn is equivalent to Fig. 7-21a in the region y > 0, problem,
~ - Re
(7-158) lIenee, in the equivalent
II M•. H' . ds ~ Re (c,c)
(7-159)
where (c,c) is the self-reaction of the correct magnetic currents radiating in the presence of an electric conducwr covering the entire y=-O plane. For a variational formulation, we approximate (c,c) by (a,a) and constrain (a,a) according to Eqs. (7-65), that is, (c,c) ~ (a,a) - (c,a) - (a,c)
where nIl sources radiate in the presence of the conducting plane. We have (a,c) = (c,a) by reciprocity, and (c,a) can be calculated because we know n X He o:z n X Hi. Hence, our stationary formula for (c,c) is (c,a)' _
(eel - ,
- (a,a)
--
(fI H'· M.' dS)'
II H' . M.' ds
(7-160)
where H" is the field of the assumed current M.... For the incident field of Eq. (7-155), we have the power incident on the aperture given by
where A is the area of the aperture. (7-161L we have T =
"'!-Re[W ff
17A
(7-161)
Hence, combining Eqs. (7-154) to
XN8)']
u·n H4. n X Eo dB
(7-162)
370
TI1lE-HARMONIC ELECTROMAGNETIC FIELDS
2.0 1.5 h 1.0
\.
1
E,t 1
I
"'act Variational
0.5
o
0.2
0.4
0.6
0.8
1.0
where E· is the assumed tangential electric field in the aperture and H· is the magnetic field calculated from E· by the methods of Sec. 3~. As an example, let us consider tbe two-dimensional problem of transmission through a slot, u.s shown in the insert of Fig. 7-22. If we assume E· in the slot to be real, then E" X H"* = (E- X H·)*
01" FIG. 7·22. TransmissiOll coefficient. for a slot.ted conductor, incident. wave polarized traosvcnro to slot. ≪U!. (Alter
and the denominator of Eq. (7-162) is
II HO • n X Eo dB =
llfiJu.)
(ff Eo X Ho' • ds)' 10 Sec. 4-11 we defined the admittance of an aperture as
y.~.. -I~I'
ff
EX H' ·ds
and calculated it for a slot for particular assumed E's. Hence, applying Eq. (7-162) to a unit length of our two-dimensional slot, we have
[U
, I u·Eo X 7 - - Re ---r;;r.;;V 'y' I'l-~_ ..
"a
dJ)']
(7-163)
where a is the width of the slot. When the incident wa.ve is polarized transverse to the slot, we have the case of Fig. 4-22; hence we take
Eo _ I in the slot.
(7-164)
Now Eq. (7-163) reduces to T =
where Y.""•• = G. have for smaH a
~ He (+) "a y.""••
+ jB" is shown in
Fig. 4-22.
r'
To;:::, kolog ko
(7-165)
From Eqs. (4-106) we (7-166)
and from Eqs. (4-107) we have for large a
.....
T_I
(7-167)
371
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
This last result is the geometrical optics approximation. The variational solution is compared to the exact solution, which can be obtained by solving the wave equation in elliptic coordinates l (Fig. 7-22). The case of a. plane wave at an arbitrary angle of incidence is considered by Miles. t If the incident wave is polarized parallel to the axis of the slot, we have the case of Fig. 4-23; so to make use of the analysis of Sec. 4-11 we would assume
..
Ed = cos-
(7-168)
a
in the slot.
Equa.tion (7-163) then reduces to T = 4a 1r;"
where Y.... ,t = G have for small a
d
Re(_1) Y:...
+ jB.. is shown in Fig. T
(7-169)
fI
4-23.
From Eqs. (4-115), we
~ 6.85 (~y
(7-170)
For large a we should expect the field in the aperture to be uniform. Hence, we should not expect the trial field of Eq. (7-168) to give good results for large a, say a > X. Equation (7-169) actually approaches 0.81 for large a, instead of the expected value 1. PROBLEMS
7-1. Suppose the cavities of Fig. 7-1 cOlltaiu lossy material characterized by /l. Show thlLt thc pertllJ'batiolllll formula. corresponding to &t. (7-3) is
IT, t,
and
jdbH X Eo·ds
~CT~lf~.,-
"'-"'0·-;
_
///I.E' E, -.H· Hold.
Note that both wand <.to must be complex. can be interprcted according to
A complex resonaoce ill the low-loss case
whcre .... is the rClll resonant frequency and Q is the qunJity factor (see &e. 8-14). 7-2. Consider the perturbation of a cavity (say Fig. 7~14) from onc having perrectly conducting walls to one having a wall impedaoce Z, defined by nXE-ZHj I Morsc and Rubenstcin, The Diffraction of Wavcs by Ribbons and Slits, PhYI. J?etJ., vol. 54, no. 11, pp. 895-898, December, 1938. tJ. W. Miles, On the Diffraction of an Elcetromagnetic Wave through a Plane Screen, J. Appl. Phy., vol. 20, 00. 8, pp. 760-770, August, 1949.
372 at the wan..
TlUli-HAIDJONIC ELEC1'ROlolAGNETIC FIELDS
Show that the exact. perturbational formula is
1ft
"
-
WI -
-j 'LB. • H. dt 7i'T-.!L------
fff (oE-E.-,H-H.ld.
where the aubecript 0 denotes unperturbed qtw1tities. Note that "'- is real but (If iJ complex if Z haa a real part.. 7-3. Use the resuIt.s of Prob. 7-2 and the approximatioIlll
to show that
Use the relatiollShipa Z -lSI.
+;!J.'.
and show that the perturbational formula gives
Note that the formula for Q is identical to the one that we have been uain& if Re ('f), where ., it the intrinsic impedance of the conducting walls. 1-" Use the results of Prob. 7-3, and show Lhat the fractional change in reftOn.nee duo to metal walls is (ft -
"'r ---•• WI
1 2Q
where "'I is the resonant frequency of the cavity with perfcctly conducting walla. 7-6. Suppose the cavities of Fig. 7-2 &ro chart\Cteri1.ed by IT and
JJf -jtM!Col)E·E. - .o.JiB·B.Jdr ColO --. • JJf [Co - j,,/w)E· E. - JiB· H.I dr
W
-
[(.0.0
Bot.b we and Col are complex if" and IT + 4lr are not identieaUy zero. '1-6. Use the result of Prob. 7-5 for the ease IT - 0, and let E ... E. - lB.l. H - Ho - jlH.l. Col - Col, + ;..../2Q. to show that
•• ./K'!f,-'I_B.I_'d.
fff "lBoi'd.
Q -"
and that Eq. (7-ll) still applies witb Col changed to """
PERTUR.BATIONAL AND VARIATIONAL TECHNIQUES
373
7~7. Suppose that a small sample of 10ll8Y dielectric is introduced into a. clwit.y whose unperturbed resonant frequency is 101.. Show that.
",.-6Or ---2(/--f" '"
.'-..
where l - .. -;." is the complex permiLlivity of the sample and w,. is the perturbed telIOnant frequency. [f the 101l8ell of the unperturbed cavity are significant, then 1
1
I
Q-Q,-Q, where Q. and Q. are the Q's of the cavity with and without thc samplo, respectively. 7-8. Considcr a rectangular cavity with a Imall centered dielectric cylinder, &8 ,hown in Pir;. 7-230. Show tbat tbe change in the resonant frequency of the domi· nant mode due to the introduction of the dielect.rie is Col -
"'.
2A (1
----;;- - be
-
)
~
wbere A is the crol5lJ-fJC(ltional al'(l& of the cylinder.
Use a quasi-atatie approximation.
Area A
I I
I
fo
1
I· (b)
(0)
Flo. 7·23. Rectangular cavity with (0) dielectric cylinder and (b) dielectric sphere. 7-9. Conaider t.he rect:t.ngular cavity with a Imall centered dicleet.ric sphere, as Show that tbe cb.a.nge in the re800AIlt frequency of the domi· nant mode due to "the intro
shown in Fig. 7-23b.
d',.-l ---- ---obc,.+2
W-WG
...
101
where d is the diameter of the sphere. Use a quasi-etatic approximation. 1-10. CollJlider the circular waveguide of Fig. &-2. Suppose the wall is s1ir;btly fl&tt.ened at the point. _ 90-. Show that the chanr;e in cutoff ffflquency for the z.-polariJ:ed (E in the center pointe in the % direction) dominant mode is Aw. _ -O.4IS W.
A1 ...0
wbere A ill the erofl8-600tional aroa of the deformation and w. - 1.841/a";;; is the unperturbed cutoff (requeney. For tbe y-polariloo dominant mode. &..
-
1If.
A - 1.42-
..-0'
Bence, the mode degeneracy has been removed.
374
TIME-HARMONIC ELECTROMAGNETIC FIELDS
7-11. Figure 7-240 shows 8. small centered dielectric cylinder in II. rectangular waveguide. Show that the change in cutoff frequenc)' of the dominant mode from that for the empty guide is
.....
-~
••
where We
_
rlb.y;;. Usc a perturbational method and a quasi-static approximation.
'I
b
I'
T 1
T
0
0
b
I'
'I
~
0
1
--jd}-
~
-Idl-(b)
(0)
FlO. 7-24. Rectangular waveguide with (4) dielectric cylinder nnd (b) conducting
ridges. 7-12. Consider tho rectangular waveguide with smaJ.1 semicircular ridges, as shown in Fig. 7-24b. Use a. perturbational method I!l.od a qutuli-s1,atic approximation to show that the dominant-mode cutoff frequency differs from the TEo! rcetangu1:l.r guide cutoff, according to
••• ••
-~
,d'
-2ob
where l.
"d'
"'e
4D.b
-~-
w.
where "'. - 2r/b Hence, the mode separat.ion is increased. 7-13. Consider t.he rectangular waveguide with t.he bottom covered by fI, t.hin dieleetric slab (Fig. 4-6 wit.h d Sa). U8C 8 pert.urba.t.ional method and quasi-stat.ic approximatiou to show that the phase constant is
v'l
where (10 - k, the same 88 the first given in Prob. 4-14. 1-14. Consider the shown in Fig. 7-240. show that
(J.In' is the empty-guide phase constant. Note that this is term of an expansion of the exact characteristio equation, sa rectangulnr waveguide with 8. centered dielectric cylinder, all Use a perturbational method and quasi-6tatic approximation to
(1-fJ.
rdle.-l
1
-,-,- ~ 2-00- ,-,-+-, :v'~"l=~(.=.'il.=)'"
where
til.
can be taken
88
the cutoff frequency of the perturbed guide, given in Frob
375
PERTURBATIONAL AND VAIUATIONAL TECHNIQUES
7-11, if u il cloac to u 4 •
Sbow thA\ at the unpert.urbed TE,u cutoff frequency
/J .. k. 7-16. Suppose t.hat a waveguide ill filled wit.h 108lIY material, and comrider .. pert.\U'batKJD of illl pcrfect.ly conducting walls. ReprC8eot the unperturbed 6eldl (lUbecript OJ and tbe perturbed fiddll (no lubscripto) by toe....,.~
E. -
E - £e,,,.
H _
Alii....,...
H. -
Note t.he oppoeit.e directions of propagation. to Eq. (7-29) ia
'Y -.,.. -
II
Ae'"
Show t.hat the formula corTCllponding
,,(. 1:o Xft'ndl 'f .c ct. X 11- 2 xli.) 'u.do
s'
Show that thia reduces to Eq. (7-29) in the lose-free case. 7-16. Consider t.he pert.urbation of material in a lossy waveguide from I, 1/-, , to I + 111, I/- + 6p, , + two Represent t.he fielda lUI in Prob. 7-15, and show t.hat. t.he formula corresponding to Eq. (7-30) ia
.fJ(••• -;..
)2,
.,. -.,.. -
-1
IJ (t.
X
to - ....11 ,11.1 do
11- t xli.) 'u.do
Show that. t.his reduees to Eq. (7..:30) in the lOllS-free ease. 7-11. Use the reeults of Prob. 7-16, and let. t.he unperturbed guide be loa-free. Denote t.he propagation constant. of the pert.urbed guide by .,. - a + j/J, and let E ... E: and H .. -~. Show that. t.he resultant. approximation for /J is Eq. (7-33) aod a "
_~IJT"_III...,..I'_d'_
2Refft.x:a:.u.d4
Note that this is an Approximate form of Eq. (2-76). 7-18. Consider the perturbation of the walla of a waveguide from ductor to an impedance sheeL Z. 8uch t.hat.
D.
perlcet con-
nXE-Z.H
Represent the unperturbed and perturbed fields a.s in Prob. 7-15, and show that
7-19. Uee lobe reeulta of Prob. 7-18 and let the unperturbed guide be lOllS-free, 110 t.hAt. ,.. - ;/J.. In the pert.urbed guide, let Z - (J( + j~, .,. "'" a + jfJ, E - &:,
376
H - -
TIME-HARMONIC ELEGraOMAGNETIC PIELDS
a:, and abow that
f "'19.1' JJ t, f 1ll19.1· •• ---A,------
fJ, ...
(J -
dl
--17---,---
2Re
Xa:'u.da dl
2Re / /
t.
X
a: ·u,'"
If Z - 11, the intrinsic impedance of metal walls, the above formula for a is the approximation that we have been using to calculate attenuation in metal wAveguides.
7-110. Show that
/ / / .-'Iv
.... -
X
EI' d,
~//"/-.IB-I·d-,-
is a. stationary formula for the resonant frequency of a lOIl-lru cavity, provided n X E _ 0 on S, but is not stationary if 108!le8 a~ pre&ent. 7-21. Show tluLt. Eq. (7-46) is a stationary formula for ..... 1, with no boundary conditions required on H. 7-22. CoOllider the rectangular cavity (Fig. 2-19) and the stationAry fonnula IEq. (7-44)). Use a trial field E - u.. ~z(~ - b)(z - c)
and abow that.
F~.
(7-4.01) gives w,. ..
Vfij ~Ol + cl be ..
In the exact 1lO1ution (Eq. (2-95)1, the numerical factor is 1T inatead of V'iO, 7-23. Consider a arnaU ddormation of the walls of a cavity, such all represented by Fig. 7-1. Tako tho variational formula IEq. (7-45)), which requires no boundary conditions on E, and take the unperturbed cavit.y field E, as a trial field. Show that. Eq. (7-45) reduces to
.
Wi - "': --
•:
/// (,.lll.l· - .IB.I') d, -'~''-''77----/ / / .IB.I'd,
Show that thia formula is essentially the same as Eq. (7-4). '1-111, Figure &-31b shows a partially lilled circular cavity. trial field
H- u.,J
1
(2.405~)
to show that the dominant. mode re8On&nce ia ........ 2.405 a
V;;;
Compare with the results of Prob. 6-24.
1 _ ~(, __ .')
UIMl Eq. (7-46) and ,.
PERTURBATIONAL AND VARIATIONAL TECHNIQUES
377
7·2l5. Consider a waveguide whose cross section is an equilateral triangle of aide length 11. Use variational formulas to approximate the lowest. cutoff frequency. The exact solution ia
.. ---."
3o.y';;
7-26. Considcr the rectangular cavity (Fig. 2-19) and the mhed-field variat.ional Connula lEq. (7-72)1. Choose a trial ficld E
. ... y ....11 -u~smbsmc
H
- U r A ISln1)COS
·..-y
".r,+A
c
ry.".Z u'lcosb'Bln
c
where AI and AI are variational parameters. Determine Al and AI by the Ritz method, and show that the resultant formula for "', is the exact formula IEq. (2-95)1. Why do we get an exact solution in thia case? 7-27. In Fig. 7-25, the surface S represents a perfect electric conductor enclosing a cavity. A variational solution is desired in terms of a trial field aatisfying n X E _ 0 n
I
I n "'-I
t.
(1)
(2)
s
FlO. 7-25. Trial fields are disconLiJJuous over a. on Sand n X (p-lV X E) continuous at a, but with n X E discontinuous at a. that the stationary E-ficld formula is
Show
where subscripts 1 and 2 refer to regions] and 2 (Fig. 7-25). Show also that a varia-tional solution in terms of trial fields satisfying n X E - 0 on Sand n X E continuous at a, but with n X (,l.I-IV X E) discontinuous at 8, is given by Eq. (7-44). 7-28. Show that the variational H-field formula. for Prob. 7-27 is of the same form 9.8 the above E-Beld formula, given by replacing E by H, • by ,l.I, and ,l.I by t. Show tbat no boundary conditions e.t S are required in the H-ficld formula. 7-29. Consider a perturbation of material in a cavity, auch as represented by Fig. 7-2. Take the mixed-Bcld variational formula fEq. (7-72)], a.nd take t.he unperturbed cavity field Eo, H o as a trial field. Sbow that Eq. (7-72) then reduces to Eq. (7-11). 7-30. Repeat Prob. 7-26, using the reaction concept of Sec. 7-7. 7-31. Consider the partially filled rectangular waveguide of Fig. 4-&. Use the E-ficld variational formula [Eq, (7-8)1, and the trial field E
. '" -u.SID'B
378
TWE-BARMONlC ELECTROMAGNETIC FIELDS
and show that Col. _
!
a
[1 + ~ (!! - ..!.. Bin 2lt'd)J-~ flo2ra
Compare BOrne calculated points with the exact eolution (Fig. 4-9). 7-82. Use the reaction concept to derive the mixed-field variational formula ror waveguide phase constants
which corresponds to Eq. (7-85) if n X E _ 0 on C. No boundary conditions are required in the above formula. 7-53. Consider the variational formula of Prob. 7-3'2 and II perturbation of waveguide walls, &8 illustrated by Fig. 7-5« and b. Use the unperturbed field E.. H. as a trial field, and ahow that the formula of Prob. 7-32 reducetl to Eq. (7--32). 7-34. Consider the variational formula of Eq. (7-85) and a perturbation of matter in a waveguide. represented by Fig. 7-00 and c. Use the unperturbed field E" H. lUI a trial field, and show that Eq. (1-85) reduces to Eq. (7-33). 7-3li. Figure 7-26 shows a coaxial stub to parnllcl·plat.e waveguide feed 8yst.em. Aasume II « "- eo that fL rensonablo trial current is a uniform current. Show by the variational method that. the impedance !een by t.he coax is
• Z _ -ka 4
where
l' -
(1
.2 -,-log -7,",) ... 4
1.781.
____---=j=.I o-d matched
~!
~Ioad---'1111
matched
;
load
p
FlO. 7-26. Coax to parallel-plate feed.
7-36. In Prob. 7-35. remove the restriction on .tub
-
II
and assume a trial current on the
I - COli 1:(11 - z) Obtain the input impedance secn by the coax by the variational method. 7-37. Repeat Prob. 7-36 for the sccond-order variationalllOlution, assuming trial currents
/- - cos 1(4 - :I)
/- - 1
Note that only one new reaction is needed in addition to those obtained in Probl. 7-35 and 7-36. Speeialhie the result to II _ "-/4. T-38. Consider tho two-dimensional problem of planc.wave scattering by a conducting ribbon, shown in the insert of Fig. 7-16, but with the opposite polariZl\tion.
PERTURBATIONAL AND VARIATIONAL TEClL.'ilQ1JE8
'In other words, Hi ia pa.rallel to the aria of tho ribbon.
] ... u~ cos
379
Ulle the trial current
'Z
and .how that the variational solution ia L
32"1 "Y...", I' _
•
1
r~
where" Y.pu~ ia given in Fig. 4-23. Show that lLS la - ... this answer reduces to 0.66 times t.he physical optics solution. Why should we expect the above formula to be inaccurate for large ka1 1·59. Consider plane-wave scattering by a wire, represented by Fig. 7-15. At the first resonnnce (L ., )./2), the current ilJ
1- ..
cO.!
b
and we know that. (lJOO Fig. 2-24) (0,0) ., 73
The imaginary put of (0,0) is BelO becaWle the length is adjUllt.cd for naonanee. Using Eq. (1-115), show that. at resonance the echo area is A• .. 0.86).1
This is relatively inaensitive to the diameter of the win. 7-tO, Figure 7-27 represents a re8Onaot length of wire illuminated by a uniform plnne wave at the angle 8, polarized in the r-z plane. Using the approximations of Prob. 7-39, show that the back-ecattcring area is
[ ~,,)']' ooa !co. ~
A • ., 0.86).'
Apin this is relatively inaeollitive to the diameter of the wire.
zl ~'" L.,
T L 1
r (to
receiver) r"(to transmitter)
I
Flo. 7-27. Scattering by a resonant wire (L .. )./2). 1-4.1, Repeat Prob. 7-40 for the ease of differential ecattering, showing that the differential echo area ia
A• .. 0.86).1
[
'o, (~ao,,) ao, (~ao, ")]' "
SID
whl!re .d. ill defined by Eq, (7-114) with
'
lIln
"
E' evaluated in the I' direction.
380
TIME-HARMONIC ELECTROMAGNETIC FIELDS
7-i2. Consider differential scattering by a magnetic obsf.aele (Fig. 7-17) and define
Show tbat, instead of Eq. (7.143), we have
Echo _ where
(',a) FCa,a) (a,a) ....
«i,a}jl)1 F(a,a) - (a,a)
JJJ (E;· J' - H' . M') d. JJJ (,.-'(J')' - '. -'(M')') d.
JfJ
(E·'
1'> - KG.
M") dr
In the above formulas, EI, H' is the incident. field, Jft and M· are the assumed electric and magnetic polarization currents on the obstacle, and E", H" is the field from J", MG, 7-43. Figure 7-2& represents a metlLl. antenna cut from a plane conductor and fed across the slot abo Figure 7-28b represents tho aperture formed by the remainder of the metal plane left lLftcr the metal antenna was cut. The aperture antenna, fed
, , (b)
(a)
FlO. 7-28. (a) A sheet-metal antenna. and (b) its complementary aperture antenna.
across cd, is said to be complementary to the metal a.ntenna. Let Z", be the input impedance or the mctalllntenna and Y. be the input admittance to the slot antenna, and show that
Hint: Consider line integmls of E BOd H from a to band c to a, and use duality. 7-'U" Consider a narrow resonant slot of approximate length "'/2 in a conducting
8creen.
Show that the transmission coefficicnt ill
,
T "" 0.52w where to is the width of the slot. similar to those or Prob. 7-39.
Hinl.: Use the result of Prob. 7-43 and IUIsumptioD8
CHAPTER
8
MICROWAVE NETWORKS
y 8 -1. Cylindrical Waveguides. Several special cases of the cylindrical waveguide, n such as the rectangular and circular guides, already have been considered. We now wish to give a general treatment of cylindrical x s (cross section independent of z) waveguides '---------'c consisting of a homogeneous isotropic dielectric bounded by a perfect electric conductor. FIG. 8-1. Cross section of a cylindrical waveguide. Figure 8-1 represents the cross section of one such waveguide. Our formula.tion of the problem will be similar to that given by Marcuvitz. 1 A1J shown in Sec. 3-12, general solutions for the field in a homogeneous region can be constructed from solutions to the Helmholtz equation
(8-1)
In cylindrical coordinates, this equation can be partially separated by taking '" - i'(x,y)Z(z)
(8-2)
The resultant pair of equations are
+ k. ' '!' ... 0 dtZ + k tZ = 0 dz t •
'v'.t'!'
(8-3) (8-4)
where the separation constants k. and k. arc related by ke t
+ k.'
= kt
(8-5)
and 'Vr is the two-dimensional (transverse to z) del operator
a ,az
l",-l"-u-
(8-6)
IN. Marcuvitz, "Waveguide Handbook," MIT Radiation Laboratory Series, vol. 10, sec. 1-2, McGraw-Hill Book Company, Ino., New York, 1951. 381
382
TJi\lE-HARMONIC ELECTROMAGNETIC FIELDS
Solutions to Eq. (8-4) are of the general form Z(z) - Ae-f',-
+ BeI',-
(8-7)
which, for k. real, is a superposition of +z and -z traveling waves. The k. are determined from Eq. (8-5) aIter the k e (cutoff wave numbers) are found by solving the boundary-value problem. For TE modes, we take F = u•.p (superscript 6 denotes TE) and determine (8-8)
The component of E tangential to the waveguide boundary C is E 1" = 1· (u. X VI'!") = (n . V,'1")Z'
where 1is the unit tangent to C and n is the unit normal to C (sec Fig. 8-1). The boundary is perfectly conducting; hence E l = 0 on C and a",' _ 0
an
on C
(8-9)
The associated magnetic field is given by
H" = - -.-1 v x E' = -.-1
JW~
JWp.
(a'fa'"," u. iJxiJz -- + u - + u.k.~1f·) ayaz W
For morc concise notation, we define a tran8verse field vector as H, = H - u.II. and rewrite the above as I dZH,' = -.- (V,i") -d JWIJ
(8-10) (8-11)
Z
It is evident from Eqs. (8-8) and (8-11) that lines of 8 and :Je, are everywhere perpendicular to each other. For TM modes, we take A = u,'Jt'" (superscript m denotes TM) and, dual to Eq. (8-8), we determine (8-12)
Defining the transverse electric field vector E, by Eq. (8-10) with H replaced by E, we have, dual to Eq. (8-11), 1 dZ' E,'" = -.- (V1'I'''') -d JWE
(8-13)
Z
From the second of these equations, it is evident that for E. to vanish on C we must meet the boundary condition on
C
(8-14)
383
MJCROWAVE NETWORKS
provided Icc ':F O. Note that EQ. (8-14) also satisfies the condition 1 . E. = 0 on C. When the waveguide cross section is multiply connected, such as in coaxial lines, it is possible to have k. = O. In this case, the necessary boundary condition is '1'- - constant on each conductor. The corresponding ficld is TEM to z and is a transmission-line mode. It should be kept in mind that Eq. (8-3) subject to boundary conditions is an eigenvalue problem, giving rise to a discrete set of mades. These modes can be suitably ordered, and the various equations of this section then apply to each mode. It is convenient to introduce mode !urn;tions e(x,y) and h(x,y), mode voltages V(z), and mode currents I(z) according to E' = eoVo H," =- hOI"
(8-15)
Comparing Eqs. (8-15) with Eqs. (8-8) and (8-11), we see that we may choose Vo ... z. eo "" u. X V.i" = h' X u. ]. I_dZ· (8-16) JWIt dz
foc TE modes. and, comparing Eqs. (8-15) with Eqs. (8-12) and (8-13), e- -
1 dZv-= -jwt dz
-V,i'- = b- X u.
h-- -u.XV,i'--u.Xe-
for TM modes. to
z-
1-. =
(8-17)
:Furthermore, we normalize the mode veetors according
II (o')' d. - II (h')' d. II (,-)'d, - II (h-)'d. -
I
(8-18) I
where the integration extends over the guide cross section. Hence, all amplitude factors are included in the V's and l's. We shall now show that all eigenvalues aTe real. Consider the tW(r dimensional divergence theorem
II V,·Ad. and let A - i'·v,\{I.
~A.ndl
Then,
v,· A "'"' V,'I'·· V.V
+ i'·Vli'
"'"
Iv,'!'11 - k/j"irl l
and the divergence theorem becomes
ff (IV,'I'I' - k.'I'I'I')
d. -
~ '1"
: : dl
384
TIllE-HARMONIC ELECTROi\(AGNETIC FIELDS
But the boundary conditions on the eigenfunction "if are either Ilf =- 0 or (Jllfjan = 0 on C. Hence, the right-hand term vanishes a.nd
!! IM'I'd. !!I<-I'ds
(8-19)
k' - '-',-,---
,
The eigenvalue kef is therefore positive real. There is also no loss of generality if we take all eigenfunctions ..y to be real. To justify this statement, suppose ir is not real, and let '1t = u + jv. Then the Helmholtz
equation is V, 2'1'
+ kc 2'l'
"'" Vb'
+ kctu + j(V,tv + k/v)
= 0
which, since kc l is real, represents two Helmholtz equations for the real functions 'U and v. The boundary conditions, either
onC
'I'=u+ju-O or
a'1' = oU+j{Jv_ O
an
an
an
on C
are satisfied independently by 'U and v; so u and v arc solutions to the same boundary-value problem. Hence, u and v for a particular k. caD differ only by a cODstant, and '1' is in phase over a. guide cross section. We caD take it to be real and include any phase in the V and I functions. Let us now look at the propagation constant 'Y = jk.. For f and J.l real, we have a cutoff wavelength
~,
-t
(8-20)
k,
(8-21)
and a cutoff frequency
i -
1-21l"Vf~
Then, from Eq. (8-5), we have the propagation constant given by j
> j, (8-22)
j
< j,
These are, of course, just the relationships that we previously established for the rectangular and circular waveguides. Figure 2-18 illustrates the behavior of a and {:J versus f. When the mode is propagating (f > fo), the concepts of guide wavelength, 2.
~ - -
~
• - P -- ---r,=iTjj'
(8-23)
385
JoLICROWAVE NETWORKS
where X is the intrinsic wavelength in the dielectric, and guide phau velocity, W
v, "'" - -
P
_,
Vi
(8-24)
(f./fl'
where tI, is the intrinsic phase velocity, are useful. These parameters are discussed in Sec. 2-7. Turning now to the mode voltagea and currents, we see from their definitions IEqs. (8-16) nnd (8-17)] that V and I satisfy Eq. (8-4). Hence, in general they are of the form of Eq. (8-7), or
V(z) "'" V+e- Y ' + V-e Y ' 1(z) .,. J+e-r· + I-eY '
(8-25)
where superscripts + and - denote positively Bod negatively traveling (or attenuating) wave components. Also, from Eqs. (8-4), (8-16), and (8-17) it i.s apparent that V+ [+
=0
V-
Zo
(8-26)
1-'" -Z.
where the cMraderidic impedance Zo is, for TE modes,
Z.' _ jw. y
-l
w; - Vi JW~
•U./fl' JW~
Q" - k.
VI
(fIf.),
f > f. (8-27)
f < f.
and, for TM modes,
Zo- -
:L JW'
-1 ~. -·~I .;2.. _
JWE
(J)' k. /1 _ L)' V Ie
f > f.
-
JWE
(8-28)
f < f.
Note that these are just the characteristic wave impedances that we previously defined for rectangular and circular waveguides. Figure 4-3 illustrates the behavior of the Za'S versus frequency. Finally, from Eqs. (8-4), (8-16), and (8-17), we can show that V and 1 also satisfy the tram-
missWlIrlim eqootiom (8-29)
where Yo=- 1/Zo is the charaderi'lic admittance.
Hence, the analogy
386
TIKF;-UAJUlONIC E.LECTROYAGNETIC FIELDS
I joI'
I I
VI;'
i-
I I
I-
-I
dz (0)
1----
~/jOH
---~
I
jOJI£
I
i-I
I
I I I I
I
I I I
I
I I
I
I'
·1
dz (b)
FlO. 8-2. Equivalent. transmission lines Cor waveguide modes (IICries element. labeled in ohms, 'bunt elements in mhos). (0) TE modes, (6) TM modes.
with transmission lines is complete, and all of the techniques for analyzing transmission lines caD be applied to each waveguide mode. l We may define an equivaknt trammiuion lim for each waveguide mode as one (or which.., and Zo are t.he same as those of the waveguide mode. Such an equivalent circuit may help us to visualize waveguide behavior by presenting it in terms of the more familiar transmission-line behavior. For a dissipationless transmission line, we have
Zo ~
=
/Z IX VY = VB
- v'ZY
-jyXB Equating the above Zo and 'Y to those of a TE waveguide
(see Sec. 2-6). mode, we obtain
jX "" jWIJ
) 'B
.
=)~
+ -.k.'
JW~
(8-30)
Thus, the transmission line equivalent to a TE mode is as shown in Fig. Similarly, for a TM mode we obtain
&-2a.
.
.
k~'
]X=]Wp+-.-
1""
jB = jWt
(8-31)
I For u&mple, see Wilbur LePage and Samuel Seely, "General Network Analysis," Chape.9 and 10, McGraw_Hill Book Company, Inc., New York, 1952.
387
YICROWAVE NETWORKS
The transmission line equivalent to a. TM mode is therefore 88 shown in Fig. 8-2b. H the dielectric is lossy, the equivalent transmission will also have resistances, obtained by replo.cing jWE by (1 + jWf. in Eqs. (8-30) and (8-31). In the light of filter theory, we can recognize the equivalent t.ransmission lines as high-pass filters. The power transmitted along tlie wavcguide is, of course, obtained by integrating the Poynting vector over the guide cross section. Hence, for the +z direction, p, =
JJ E X H··uld, -
= Vl·
IJ e'da
=
VI·
JJ e X h··ulda (8-32)
VI-
and the time-average power transmitted is
/P. - Re (V[')
(8-33)
Hence, in terDl5 of the mode voltage and current, power is calculated by the usual circuit-theory formulas. It is also worthwhile to note that the modc patterns, that is, pictures of lines of Sand :JC at some instant, can be obtained directly from the -v's. For TE modes, H, is proportional to v.'I", and E is perpendicular to H,. Hence. lines 0/ constant '1" are auo linea of instantaneous S. Lines of instantaneous :re, are everywhere perpendicular to lines of instantaneous 8. Similarly, for TM modcs. lines of constant 'It. aTe aUo linea of instanlaneoua :re, and lines of instantaneous 8. are everywhere perpendicular to lines of instantaneous:JC. It is therefore quite easy to sketch the mode patterns directly [rom the eigenfunctions 'It. Recognizing that the gcneral exposition of cylindrical waveguides has been quite lengthy, let us summarize the results. Table 8-1 lists the more important relationships that we have derived. Those equations common to both TE and TM modes are written centered in the table. Keep in mind that all of the equations apply to each mode and tha.t many modes may exist simultaneously in any given waveguide. Finally, for future reference, let us tabulate the normalized eigenfunctions for the special cases already treated. For the rectangular waveguide of Fig. 2-16. we can pick the w's from Eqs. (4-19) and (4-21) and normalize them according to Eq. (8-18). The result is 'l'••C = _1 T
ab,_,. + (na)' cos
(mb)l
(rnT) a x cos (nT) bY
_,.2/ ab _(rnT)_(nT) = ; V(mb)' + (nap SID a:Z: SID bY
(8-34)
't" .......
where m, n = 0, 1, 2, . . . , (m = n = 0 excepted).
SimilarLy. for the
388
TIME-HARMONIC ELECTROMAGNETIC FIELDS TABLE
8-1. Smoo.RY OF EQUATION15 "OR TUE CTLINDBlCAL WAVEGtllDZ (TEM MODES NOT INCLUDED)
I
TE modes Transverse Helmholtz equation Boundary relations
+ k c i1t -0
Vt 1 '1'
a,,'
-an -0
TM modes 1
on C
qi''' -
0
00
C
." -
-'Vt'l''' -u. X'V,'I:'h" -
e' - u. X'Vt'lt· h' - -'Vj'l" Mode vectors
e-hXu. h - u. X e
JJ
Normalization
Propagation constant
Characteristic Z and Y
,.-jk.-
,'do -
I
"do-l
j~ -jkVl - U,If)' a - k, Vi UII,)' 1
jfoli'
Z," - -,. - Y,' dV
Z."
I> I, I
,
1
j-
Y."
-- --
+ ,.Z,l -0
dz
Transmission-line equations
-dl + . . Y,V-O
liz
-
V _ V+e-'Y' + V-e'Y' 1 _ - (V+e-l" - V-e7')
Mode voltage and current
I
Z,
E, - eV
Transverse field
Longitudinal field z~ect.ed
JJ
power
1!,-h1
H..
k,'
- -.- 't'V' )W_
P. -
E,"
vr-
k,' - -.+-r~
389
MICROWAVE NETWORKS
circula-r waveguide of Fig. 5-2, we can pick the if's from Eqs. (5-23) and (5-27) and normalize them. The result is ':lr' ..p
I
t..
= "'Ir[(x~p)2
'!' • _ "p
g
V-;'
n']
J .. (x~pp/a)
J.(•• ,p/a) X ... J .. +l(x..,)
J .. (x~,)
(sin n~l cos
J
sin n~ I
1cos n¢
(8-35)
n¢
where n .,. 0, I, 2, . . . , and p 1,2, 3, . . .. The X. p are given by Table 5-2, and the x~J> arc given by Table 5-3. Normalized eigenfunctions for the parallel-plate guide are given in Prob. 8-1. Normalized eigenfunctions for the coaxial and elliptic waveguides arc given by Marcuvitz. 1 8-2. Modal Expansions in Waveguides. An arbitrary field inside a section of waveguide can be expanded as a sum 'over all possible modes. This concept was used in Sec. 4-4 for the special case of the rectangular wa.veguide. We now wish to consider such expansions for cylindrical waveguides in general. The equations in Sec. 8-1 apply to each mode. Henceforth, to identify a particular mode, we shall use the subscript i to denote the mode number. Let us first show that each mode vector e; is orthogonal to all other mode vectors. For this, we shall use the divergence theorem in two dimensions, co;
Green's first identity in two dimensions,
ff (v,~
- v,~ +
~VN) ds
-
¢ ~ :: dl
and Green's second identity in two dimensions,
If (~VN
-
~VN) ds ~ ¢ (~:: - ~ :~) dl
First, consider two TE modes and form the product Letting If =
et· et = hi" hi" = V1W," V1wt '1',' and ¢ = iI!,' in Green's first identity, we obtain
II
_t- _;,d, - -(k,;,)'
IJ
'!'t'!';'ds
Using the same substitution in Green's second identity, we have [(k.:)' - (k.')'1
IJ'!'t'!';' d, ~ 0
IN. Marcuvitz, "Waveguide Handbook," MIT Radiation Laboratory Series, vol. 10, chap. 2, McGraw-Hill Book Company, Illc., New York, 1951.
390
Rence, if kd ' ¢ becomes'
'l'IME-HARi\IONIC ELECTRO&IACNETlC FIELDS k~t.
the integral must vanish, and the preceding equation
If et· ej'ds
(8-36)
= 0
A dual analysis applies to the TM modes, and we have
If c;",·e/"ds=O
i¢j
(8-37)
Finally, we must consider the TE-TM cross products
et· er = h:· hI"
= - (u. X Vl'lt,") . VI 'It/"
If we let A = 'It/'u. X VI'!': in the divergence theorem, the contour integral vanishes because of the boundary conditions, and we obtain
If
V,'lt/,,' u. X V/I'tds =0
Comparing the preceding two equations, we see that
II e,··erd, - 0
for all i, j
(8-38)
The orthogonality relationships (Eqs. (8-36) to (8-38)} also arc valid for the c's replaced by the h's. At any cross section along a cylindrical waveguide, the field can be expressed as a summation over all possi ble modes:
E,
=
HI =
L, ei'Yt + e..-V,L, btl..' + h.-I....
(8-39)
Because of the orthogonality of the mode vectors, we can determine the mode voltages and/or mode currents at any cross section by multiplying each side of Eqs. (8-39) by an arbitrary mode vector and integrating over the guide cross section, Noting that the mode vectors are normalized, we obtain
II eiPds ff Hjoh;pds= liP El
,
=
ViP
(8-40)
where p = e or m. Since there are two independent constants in V and I for each mode, as shown by Eqs. (8-25) and (8-26), we need two u crOS8I A discrete spectrum of eigenvalues i8 a88umOO. functions for degenerate case8 can also be found.
However, orthogonal sets of mode
JdlCROWAVE NETWORKS
391
sectional" boundary conditions. These may be (1) matched waveguide and E , over one cross section, (2) matched waveguide and H, over one cross section, (3) E , over two cross sections, (4) H, over two cross sections, and (5) E , over one cross section and H, over another eross section. The solutions of Sec. 4-9 are examples of case (1). Furthermore, when we have currents in a waveguide, we can obtain additional cases involving discontinuities in E, and/or H, over waveguide cross sections. The solutions of Sec. 4-10 are examples of this situation. It is also of interest to note that, when many modes exist simultaneously in a cylindrical waveguide, each nwde propagates energy as if it exi813 alone. Hence, the equivalent circuit of a section of waveguide in which N modes exist is N separate transmission lines of the form of Fig. 8-2. To show this power orthogonality, we calculate the z-directed complex power P. -
II
E X H'-.,ds -
. . 2: V;1j II ei·ei ds ij
IIO:e,V,) L V;1f
(8-41)
i
which is a summation of the powers carried by each mode. (We have used the indices i and j to order both TE and 1'1\1 modes in the above proof.) The energy stored per unit length in a waveguide is also the 8um of the energies stored in each mode (see Prob. 8-3). 8-3. The Network Concept. In Sec. 3-8, we saw that, given N sets of "circuit" terminals, the voltages at the terminals werc related to the currents by an impedance matrix. This impeda.nce matrix was shown to be symmetrical, that is, t-he usual circuit-theory reciprocity applied if the medium was isotropic. We shall now show that the same network formulation applies if, instead of circuit voltages and current.s, the modal voltages and currents of waveguide "ports" a.re used. Let Fig. 8-3 represent a genero.l U microwave network," that is, a system for which a closed surface separating the network from the rest of space can be found such that n X E = 0 on the surface except over one or more waveguide cross sections. Suppose that only one mode propagates
PIO. 8-3. A microwave network. (1)
392
TIM&-HARMONIC ELECTROMAGNETIC FIELDS
in each wa.veguide,1 Then, assuming we are far enough along each waveguide for higher-order modes to die out, only the dominant mode exists in each guide. A knowledge of the mode V or I in the guide is equivalent to
a. knowledge of E t or H" respectively, since the mode vectors depend only on the geometry. Hence, according to the uniqueness concepts of Sec. 3-3, a knowledge of V (or I) in all guides is sufficient to determine I (or V) in all guides. Furthermore, the relationship must be linear if the medium is linear, and an impedance matrix [z] is defined by
V,] [zu Zn zn] [V,Va Zu Zu Zu =
Z21
Zn
Zu
[I,] [2
(8-42)
I.
where V .. and In are the mode voltage and current in the nth waveguide. The inverse relationship to EQ. (8-42) defines an admittance matrix (y] according to
I,] [YU Yn yu][V,] I, = Y21 Yn Yn V, [I a Un Un Yu Va
(8-43)
Equations (8-42) and (8-43) have been written explicitly for. the threeport network of Fig. 8-3 but, of course, can be similarly written for any N-port network. Now that we have established these linear sets of equations, we can use all the usual techniques for solving linear equations. The electrical engineer knows these techniques by the name of U network theory."2 It is also of interest to show that, for isotropic media, %;J
=
Yij = Yj;
%j;
(8-44)
that is, microwave networks arc reciprocal in the same sense as are the usuallumped-element networks. To prove this, let us apply the Lorentz reciprocity theorem [Eq. (3-34)]. It states that
"'"
for two fields E", a" and E6, H· in linear, isotropic media. We visualize a surface surrounding an N-port microwave network such that E I = 0 on S except ovor the waveguide cross sections, where (E.). - •• V.
(H,). - h.[.
II{ N modes propagate in a single waveguide, then that guide will be represented by N porn on the equivalent network. I For example, see C. D. Montgomery, R. H. Dicke, and E. M. Purcell (eds.l, "Principles of Microwave Circuits," Chap. 4, MIT Radiation Laboratory Serics, vol. 8, McGraw-Hill Book C<Jmpany, Inc., New York, 1948.
393
MICROWAVE NETWORKS
(The n here refers to the nth waveguide, not the nth mode.) desired surface integrals become
cj>E-X H'·ds =
N
Hence, the
N
._1~ V.-I.'cj>e"Xh"'ds= .-1I V,,"I.'
and the Lorentz reciprocity theorem reduces to N
N
L: V.·I.' - L: V.'I.·
Il_l
(8-45)
._1
To show that Eq. (8-45) is equivalent to Eqs. (8-44), it is merely necessary to consider the special CMes (1) all f." = 0 except It and (2) all f,,' = 0 except 11. Then Vt = zJ;Jt and Vf ... %tifl, and Eq. (8-45) reduces to 'Z(j = zJi. Similarly, taking all V,," = 0 except VI", and all V,,, - 0 except V tin Eq. (8-45) establisbes y" ~ Yii. 8-4. One-port Networks. A one-port network is characterized by a single impedance or admittance element. Visualize a surface enclosing the network such that the field is zero on the surface exccpt where it crosses the input guide, as shown in Fig. 8-4. We then have
P l"
=
-effiE X H··ds = -VI*1Pe X h·ds - Vf*
where Vand f are the mode voltage and current !lot the II reference plane," that is, at the cross section cut by the surface enclosing the network. Because of the conservation of complex power [Eq. (1-62)], we have
V 1° - p •• - i!',
+ j2",(W. -
(8-46)
'N.)
where ~~ is the power dissipated, OW. is the magnetic encrgy stored, and w. is the electric energy stored in the network. The input impedance to the network is therefore
z-
Gf. = dr [i!', + j2w(W. -
(8-47)
W.)J
which is well known for lumped-element network theory.
Similarly, the
FIG. 8-4. A one-port net.-
work and a. surface en· closing it.
~s
394
TtM.&HARMOl\'1C ELECTROMAGNETIC FIELDS
input admittance is
Y -
!Vi' - rvr1 (
1'2",('». - '»,)]
(8-48)
As usual. we define the real and imaginary parts of Z to be resistance and reactance, and the real and imaginary parts of Y to be conductance and susceptance, respectively.
z~
R +1'X
From Eqs. (8-47) to (8-49) we (1) A dissipationless network has be negative in the lossy case. (3) and magnetic energies arc equal.
z·(-"') -
Y - G +1'B
(8-49)
can draw the following conclusions. R - G - O. (2) The Rand G cannot At resonance (X = B = 0) the electric (4) The Z and Y satisfy Y'( -"')
Z("')
~
Y(",)
and hence nand G Brc even functions of wand X and B are odd functions
of w. In the losslcss case, V 1* is imaginary, and hence V must be 900 out of phase with I. We shall now show that everywhere within the network E is in phase with V and B is in phase with I. Hence, E is 900 out of phase with H. Suppose we choose our reference plane such that V is real. Then n X E is rcal over the reference cross section of the input guide and zero over the rest of the enclosing surface (sec Fig. 8-4). These boundary conditions, as well as the field equations
v X E
=
-jwpoH
VXH=jWfE
(8-50)
can be satisfied by assuming E rcal and H imaginary. This is therefore a possible solution, a.nd, assuming uniqueness, I it must be the only solution. Let us now considcr the effect of a change in frequency. The frequency derivatives of Eqs. (8-50) are V X
oE
ow
"H"
= -JJl
- JWJl
oH "E+"JWE oE VXaw=J(
oH
aw
(8-.11)
aw
If we scalarly multiply the first of these by H* and the conjugate of the 1 It. may be recalled that the uniqueDel!Ill theorem of See. 3-3 required some di!sipa.. tion for ita proof. Bence, our eonclusxms apply only if we visualize some slight loa However, eve.n in the loss-free ease, any field baving n X E - 0 over the entire boundary would be uncoupled to the input ports, and would have no influence on the external behavior of tbe network.
395
M1CROWAVE NETWORKS
second of Eqs. (8-50) by dE/ow, and subtract, we obtain
v . (aE
~
X
H.)
-jpIHI" _ jwp aH. H. + jw.J,•. aE
=
~
~
Similarly, if we scalarly multiply the second of Eqs. (8-51) by E· and the conjugate of the first of Eqs. (8-50) by aH/aw, and BuLtract, we obtain
v. (oR X E.) aw
=
jelEI" + jWf: aE. E. aw
_ jw,JI••
iJH aw
We now subtract the above equation from the preceding one and obtain
V. (aE X H' _ ow
aH iJw
X E') ~ -j"IHI' - j,IEI'
(8-52)
Finally, this equation is integrated throughout a region of space, and the divergence theorem applied to the left-hand term.
effi(~~ X H'
-
~~
fff
X E')'dS - - j
("IHI'+'IEI')dT
(8-53)
Note that the right-hand side is proportional to the total electromagnetic contained within tbe region. Equation (8-53) is now applied to the one-port network (Fig. 8-4). The field vanishes over the enclosing surface except where it crosses the input port, and the lefL-hand side of Eq. (8-53) becomes ~nergy
11 (avow T' + alaw v.)
~ _
e X h. ds
(I' av + v. al) aw ow
",here V and I are the mode voltage and current at the input reference ?lane. Hence. we can write Eq. (8-53) as
l ' : : + V·
:~ -
j
fff (MI' + ,lEI')
- 2j(~.
+ ~.)
dT
(8-54)
rhe input reactance X and susceptance B are given by jX-
f
= _.1 B
fheir frequency derivatives are therefore dX_ ck" - -
javi
7 aw
dB j aI dw ... - V ow
l-e.a:a\
I
VCOIlltu>\
(8-55)
396
TIME-HARMONIC ELECTnOMAGNETIC FIELDS
Hence, from Eq. (8-54), it follows that dX
2
dB
2
dw - [If' dw -
(w. + w.)
(8-56)
fiil' (w. + w.)
Equations (8-56) state that the slope oj lM reactam:e or su.sceptance for a l08s-fru one-port network is always pontive. This is known as Foster's
reactance thtwrem.
From Eqa. (8-47) and (8-48) we also have for loss-
free networks 2w
X - [If' (w. B -
- w.) (8-.\7)
2w
fiil' (''''. - <».)
Solving Eqs. (8-56) and (8-57) for the energies, we obtain
(8-56)
Because the energies are positive, it follows that dX
X
dw> w
dB B ->dw w
(8-.\9)
that is, the slope of the reactance or susceptance is always greater than the slope of a straight line from the origin to the point of consideration. IWlationships (8-.\6) to (8-59) were first establisbed in lumped-elemenl network theory. I An important consequence of Eqs. (8-56) and (8-57) is that all polu and zeros of the reactance or 8U8Ceptance function for a lon-free one-porl network are simple. To prove this, suppose X vanishes at a resonant frequency WOo The Ta.ylor series about is then
w,
X(w) -= al(w - w,)
+ a2(w -
w,)'
+ ...
and X'(wo) = ai, which must be positive hy Foster's reactance theorem, Hence, X has a simple zero at Wo and B "'" l/X bas a simple pole at '4 Similar reasoning shows that the zeros of B are simple; hence the poles of X are simple. Furthermore, the poles and zeros for the rea.ctance or susceptance function of a loss-free one-port network must alternate along I R. M. Foster, A Reactance Theorem, BdZ Svdem Tw., J" vol. 3, pp. 259-267, April, 1924.
397
MICROWAVE NETWORKS
x 0'
B
•
(0)
---~ o
(b)
(0)
Flo. 8-.5. (a) Typical reactance or susceptance function, (b) a Foster equivalent work of the first type, and (e) a Foeler equivale.nlo network of the eecond type.
ne~
the w axis; else X'(w) will not. always be greater than zero. Figure g·5a illustrates the general behavior of a reactance or susceptance function. Equivalent circuits for reactance functions of the Foster typel arc illus~ tratcd by Fig. g..5b and c. Other equivalent circuits of the Cauer type, I or of mixed Foster-Cauer type, can be found. An important difference between microwave networks (distributed elements) and lumped--element networks is that the former have infinitely many resonances, while the latter have a finite number of resonances. The loss-free network is, of course, only an approximation to physical networks. It is therefore desirable to know how the behavior of networks with small losses differs from the behavior of loss-free networks. It is known from the usual network theory that a slight amount of dissipation shifts the poles and zeros of the impedance function from the Co) axis to points above it. Hence, the reactance (imaginary part of Z) of a slightly dissipative network would not become infinite for any real Co) but would be somewhat like that shown in Fig. 8-6. Also, since Z(w) is an analytic function of WJ the resistance (real part of Z) is not independent of X. A study of the resistance corresponding to the reactance of Fig. 8-6 reveals that it would behave somewhat like the dashed curve of Fig. 8-6. An exa.mple of a lossy one-port network is the linear antenna. of Fig. 7-13, for which the power ulossJl is actually radiated power. The effect of small losses cnn be shown in the equiva.lent circuits by adding I For 8luunple, see M. Van Valkenburg, "Network Anal)'si5," Chap. 12, Hall, Inc., Englewood Clift's, N.J., 1956.
~Dtico
398
Tum-HARMONIC ELECTROMAGSETlC J'IELDS
",
R
••,
,,
,,
, ,, ,
, •
large resistances in parallel with the LC resonators of Fig. 8-.5b aDd by adding small resistances in series with the LC resonators of Fig. 8-5c.
8-5. Two-port
Networks. The
primary uses of two-port networks in microwave theory are (1) transmission of energy from one place to FlO. 8-6. The effect. of small losses on the another and (2) filtering of signals impedance or tl. microwave network. from one another. While much of the theory can be presented in terms of the impedance matrix [%1, defined by (8-60)
or in terms of tbe admittance matrix (8-(H)
it is often more convenient to use other matrices which emphasize the waveguide character of the ports. The port voltages and currents can be considered to be tbe superposition of incident and reflected components. Hence, {or port 1,
'+ VI' II' + I{:E _1_ (V.i Z..
VI = V1 II =
V{)
(8-62)
and similar equations apply to port 2. Figure 8-7 suggests this travclingwave concept. Mathematica.lly, Eqs. (8-62) are merely a linear traMformation from the two quantities VI, [1 to VI', V {, and it is apparent that ZOl can be arbitrarily chosen. However, it is usually convenient to make the natural choice that Zlll is the characteristic impedance of the waveguide connected to port 1. Another choice, convenient from a mathematical viewpoint, is to normalize the characteristic impedance by choosing all Zo's equal to unity. We shall make the former choice. From the traveling-wave viewpoint, a possible matrix for describing
_1,
+It-:,.. 7.0,
FlO.
~7.
()l
Network
(2)
z.,
Tra.veling waves for a two-port network.
MlCROWAVE NETWORKS
399
Flo. &.8. N two-port networks cascaded.
two-port microwave networks is the transmission matrix (T], defined by
[~::] - [~:: ~::][ t::]
(8-63)
This matrix is particularly convenient when microwave networks are cascaded, as illustrated by Fig. 8-8. The incident and reflected waves at the input of network 7t + 1 are the reflected and incident waves, respectively, at the output of network n. Hence, the T matrix of the over-all network is the product of the T matrices of the individual networks, that is, (8-64)
Another matrix commonly used to describe microwave networks is the scattering matrix [8] defined by
_[S" S,,] [Vi] [V,-] V,r 8 21 Sn V,l
(8-65)
This matrix is convenient for considerations of jmpedance matching. It can also be easily extended to the case of multiport networks. Note that SII is the reflection coefficient seen at port 1 when port 2 is matched and Sn is the reflection coefficient seen at port 2 when port 1 is matched. The various matrices defined for a two-port network are, of course, related to one another. For example, [yJ is the inverse of (z], as stated by Eq. (8-61). The relationship of fSJ to IzJ is more compUcated. Defining the matrix
["~I - [go> ~.J we have
[S] ~ [z -
'.1I' + ',J-'
(8-66)
Similarly, the transmission matrix is related to the scattering matrix by [T) =
su _ SuE" su] Sa Sa Sl1 [ - Sa S121
(8-67)
The derivation of Eqs. (8-66) and (8-67), along with other relationships among the various matrices, can be found in vol. 8 of the Radiation
400
TDlE-IlA.R),[ONIC ELEcrRO»AGNETIC FIELDS
Laboratory Series. I For networks constructed of linear isotropic matter, the reciprocity relationships (Eqs. (8-44») apply. From Eq. (8-00), it is evident that reciprocity requires (lHl8)
in the scattering matrix.
From Eq. (8-67), it follows that reciprocity
fp.quires
TuTu - TuTu =
zZ"
(S-ll9)
"
in the transmission matrix. Equations (8-66) and (8-68) also apply to multipart networks. There are realizability conditions imposed on the matrices by the con~ servation of energy theorem. These conditions can be obtained from the corresponding one-port conditions by terminating the two-port network in various ways to form a one-port. For example, if port 2 is opencircuited (II - 0), then til is the input impedance. Similarly, whon port 1 is opco-i:ircuit-ed, Zu is the input impedance looking from port 2. Hence, by Eq. (8-47) we know
Re (zu)
~
0
Re (z,,)
~
0
(8-70)
Similarly, using the y matrix and short circuits on the ports, we obtain from Eq. (8-48) that
Re (Yu)
~ 0
Re (y,,)
~ 0
(8-71)
More generally, since Eqs. (8-47) and (8-48) must be valid for any passive termination, we can show that
Re (Zl1) Re (zu) - Re (za) Re (Ztl) Re (VII) Re (Yn) - Re (Ya) Re (Y21)
~
0
~ 0
(8-72)
Finally, when the network is loss-frec, the clements of the impedance and admittance matrices become imaginary, and restrictions on them can be obtained from the corresponding restrictions in the one-port case. Such considerations are particularly useful in the theory of filters. l Our principal concern for thc remainder of this chapter win be to obtain equivalent circuits for microwave networks. For any particular network, an infinite number of equivalent circuits will exist. One of oW' tasks will be to choose a .. natural" equivalent circuit, that is, one which suggests the physical nature of the network. For example, a section of I C. D. Montgomery, R. H. Dieke, and E. M. PuN:cll (eda.), "Principles or Microwave Circuita," Chap." MIT Radiat.ion Laboratory Series, vol. 8, McGraw-BiD Book Company, IDe., New York, 1945. • M. Van Valkenburg, "Network ADalYliJI," Chap. ]3, Prentice-Ball, lne., Englewood CIiBI, N.J., 19M.
401
WCROWAVE NETWORKS
FIo_ 8-9. A typical equivalent cireuit tOt a 1000·tree t.wo-port mierowave network.
waveguide would not be represented by an equivalent tee or pi circuit, since t.his would hide the transmission-line character of the guide. For loss-free networks, we shall use the symbolism of Table 8-2 in equivalent circuits. It should be emphasized that it is only the 8ign of a reactance or susceptance that dictates whet.her an inductor or capaoitor is ohosen. The reactance or susceptance does not, in general, have the simple frequency dependence of a lumped-element inductor or capacitor. Figure 8-9 illustrates a t.ypical equivalent circuit for a loss-free two-port network. TABU; 8-2. SUlllOLl&W: Element.
C1RCutTS
USED IN EQutVAL£ST
Lo&&-FREE
NETWOU8
Represents
Symbol
.n
or
jX
P08iLivc reactance
Inductor ---,
.. , jB
Negative S1.l8Ccptance
---4~
Negative reactance
---4ri!!-
Positive lJusceptal)ce
Capacitor
n:l
Ideal transformer
Transmission line
~C Z,
,
~l-----l
Change in impedance level
Waveguidc section
402
TIME-HARMONIC ELECTROMAGNETIC FIELDS
T
T
T
Z.
Z.
~
Zo
Z.
(0)
(b)
FIG. 8-10. (a) A 8ymmetrical obstacle in a cylindrical waveguide, and (b) an equivalent circuit.
In the case of dissipative networks, resistors in series with X or in parallel with B can be used to represent the losses. Similarly, the characteristic impedances and propagation constants of the equivnlent trans· mission lines can be assumed complex to account for losses. Most of the networks used in microwave practice are DIlly slightly lossy, and the small losses introduce only second-order corrections to the reactances calculated on a loss-free basis. 8-6. Obstacles in Waveguides. An object in a cylindrical waveguide can be represented as a two-port network. Figure 8-lOa shows an obstacle, symmetric about the cross section T, in a waveguide. Figure 8-lOb shows a possible equivalent circuit. In the more general case of an unsymmetrical object, the two Z6'S would probably be different from each other, and it might even be desirable to choose two reference planes T. In the loss-free case, the Z's will all be jX's. Before considering the obstacle problem, let us consider "dominantmode sources" in cylindrical waveguides. Figure 8-11 shows the electric source ]. in a waveguide terminated at z = 0 by a magnetic conductor and matched as z -+ - 00. The method of treating this problem is that used in Sec. 3-1 for rectangular guides, as, for example, Fig. 3-2. Let superscripts (I) denote the region -I < z < 0, and superscripts (2) denote the region z < -l. Then in region 1 there will be an incident wave plus a reflected wave such that HI = 0 at z. = O. Hencc, Ej(l)
-= A (e- il"
A
+ 61")e
Hll) ... - (e- I,. Z,
-
= 2A cos (fJz)
e
2A ei,s')b =- -,- sin (ftz) b
l ')
(8-73)
JZ,
where e and h are the mode vectors, fJ is the phase constant, and Zo is the characteristic impedance, all of the dominant mode (see Table 8-1). In region (2) there will be only a wave in the -z. direction; hence Elm = Be;'~e HI(I) _
-8 el" h
--
Z.
Continuity of E l at z ".. -I requires that 2A cos fJl =- Be- uJi
403
I.flCROWAVE NETWORKS
which determines B in terms of A. The boundary condition on H at -l is uJ X tRw - H(ll] = J.
t: "'"
J. - - ~~ ej~le
which leads to
A quantity of interest to us is the self-reaction of the current sheet
If
(8,.) ~
E . J. d. - -
2A' Z, (1
+ &~')
(8-74)
We shall use dominant-mode current sheets as mathematical II waveguide probes" to determine the equivalent circuit im"pedances. Now return to the original problem, Fig. 8-10a. We define even excitation of the waveguide as the case of equal incident waves from both z < 0 and z > 0, phased so that E , is maximum and H, is zero at z = O. By symmetry arguments, the H, scattered by the obstacle will also be zero in the z = 0 cross section i so a magnetic conductor can be placed over the z = 0 plane without changing the field. This divides the problem into two isolated parts, one of which is shown in Fig. 8-12a. The excitation is provided by the dominant-mode source J., which we have just analyzed. The equivalent circuit of Fig. 8-12a is shown in Fig. 8-12b. (The magnetic conductor is equivalent to an open circuit, and the J. is equivalent to a shunt current source I.) We now further restrict the problem to the loss-free case. Then the dominant mode will be a pure standing wave in the region - l < z < 0 of Fig.8-12a. If J. is located where E , = 0, then by the usual tro.nsmissionline formulas
Z Zo ~'or
=
Z.
+Zo2Z. =
the source of arbitrary l, the total reaction on
Reaction -
(8-75)
-jtanpl
J.
is
ff E . J. do = ff (E' + E') . J. d.
- (".) + (c,.) where E' is the field of
~
J.
alone, and E' is the field of the current on the
Matched guide
J·r
Magnetic conductor
-----IIi:.==-;:,==l.17 Fto. S.11. A domiuant-mode source in a waveguide terminated by a magnetio conductor.
404
TIME-IIARMONIC ELECTROMAGNETIC FIELDS
T
Matched ~
guide
T
T
t1-1.-,--.1 J
Mag. condo
T
-----.;-------'..z
1----,-_.1 (b)
(a)
T Matched ~
guide
Za
I
•
V T
T
1I.
Elect. condo
M•
Za
- - - - - . ; - - - - - - - ' +-
,-_.1
T
1+1,--1--'<
z
(d)
(e)
Flo. 8-12. Even excitation of Fig. 8-IOa is represented by (a), which bas an CqUivlllCll~ network (6). Odd excitation of Fig. 8-100 is rcpre8entcd by (e), which bu an equivaIcnt network (d).
obstacle alone, both radiating in the waveguide terminated by the magnetic conductor at z = O. H l is adjusted to a cross section for which E, .= 0, then the reaction vanishes and the above equation becomes
(C,8) - -(8,8) -
2A'
Z, (1
where the last equality is Eq. (8-74). Re (C,8) 1m (c,.) -
+ -"")
Taking A as real, we have
2A'
Z, (1
+ cos 2~1)
2A'
Z. (.in 2~1)
and, using the identity (I
tan 2 = 1
Eq. (8-75) becomes
X.
+ 2X. Z,
sin a
+ COS a
-
(8-76)
We have replnced the Z.. and Z. by jX. andjX. because only the loss-free case is being considered. By reciprocity, (C,.) - (s,c) -
...J,
E'· dJ'
(8-77)
where E· is the incident field, given by Eq. (8-73), and J'" is the current OD
MlCnOWAVE NETWORKS
405
the obstacle.' Note that the problem is now identical to the echo problems of Sees. 7-10 and 7-11, except that all currents radiate in the environment of the waveguide plus the magnetic conductor. For the case of a perfectly conducting object, the obstacle current is eo surface current J.\ and n X E = 0 on its boundary. Hence, n X E' =
and
-0
(s,c) - -(c,c) - -
X E"
ffE" r.' ds
(8-78)
where (c,e) represents the self-reaction of the currents induced on the obstacle. By Eqs, (8-76) to (8-78), we therefore have
Xb
+ 2X" Z.
1m (c,c) - - Re (c,')
(8-79)
Our problem is now one of finding the self-reaction of the currents induced by the incident field of Eq. (8-73) with A real. For a stationary formula, we assume currents J." on the obstacle and calculate (a,a) subject to the constraints
(a,a) - (c,a) - (a,c) (see Sec. 7-7). The last equality is met by reciprocity, and, since n X E' = - 0 X E~ on the obstacle surface, (c,a) - -(s,a) Hence, our stationary formula for (c,c) is (s a)' (cc)"" ', (a,a)
(8-80)
This, coupled with Eq. (8-79), represents the variational solution to the problem. If the trial current is taken as real, then (s,a) is real because E' is real. Equation (8-80) can then be written as
(c,c) and Eq. (8-79) becomes
X.
I I'
~ ~~,~
+ 2X. Zo
~
(a,a)'
1m (a,a) Re (a,a)
(8-81)
This formula applies only when J." is real, which is usually the case. The change of sign in going from Eq. (8-79) to Eq. (8-81) can be explained by noting that J.~ is not real for the given E', but is usually at some constant phase. I The obstacle may be a conductor, a nonmagnetic dielectric, or a magnetic dielectric (,. ~ "0). In the hUer case the term - fH"' dM< must be added to the righthand 8ide of Eq. (~77).
406
TIME--HARMONIC ELECTROMAGNETIC FIELDS
We define odd excitation of the waveguide (Fig. 8-lOa) as the case of equal incident waves from both z < 0 and z > 0, phased so that E l = 0 and H j is maximum at z = O. By symmetry, the E l scattered by the obstacle must also be zero in the z = 0 cross section, and so an electric conductor caD be placed over the z = 0 plane without changing the field. This divides the problem into two isolated parts, one of which is shown in Fig. 8-12c. The excitation is provided by a dominant-mode magnetic
source M., which, together with the electric conductor covering the z = 0 plane, is dual to Fig. 8-IL The equivalent circuit of Fig. 8-12c is shown in Fig. 8-12d. (The electric conductor at z = 0 is equivalent to a short circuit, and the M. is equivalent to a series voltage source V.) The analysis of Fig. 8-12c is dual to that used for Fig. 8-12a. Hence, dual to Eqs. (8-73), in the region - l < z < 0 we have a source field Hi - 2C cos
~C
El' = -
jY,
(~z)
h
(8-82)
sin «(3z) e
where Yo = I/Z o is the characteristic admittance of the dominant mode. Dual to Eq. (8-79) we have j
1
YoZ, - YoX, -
1m (c,c) Re (c,c)
(8-83)
where {c,c} is the self-reaction of the obstacle currents radiating in the presence of an electric conductor over thc z = 0 cross section (sec Fig. 8-12c). Finally, for a variational solution, curt:ents J," are assumed on the obstacle, and their self-reaction {a,a} is calculated. If the J." is real, then dual to Eq. (8-81) we have
1
--~
YoX,
1m (0,0) Rc (0,0)
(8-84)
where (a,a) is calculated with an electric conductor over the z = 0 plane. 8-7. Posts in Waveguides. Some variational solutions for circular posts in rectangular waveguides can be carried out relatively simply. Figure 8-13 illustrates three classes of obstacles: (1) those cylindrical to y,
--
LlIL..-........I X
(a) FIG. 8-13. Posts in and (c) otherwise.
II.
-X
(~
X
00
rectangular waveguide, (a) cylindrical to y, (b) cylindrical to %,
407
),(lCROWAVE NETWORKS
-J,
o
J,
o
-J,
- J,
o
o
J,
o
-J,
O-X
FlO. g..14. Image system for the circular post in a rectangular waveguide.
(2) those cylindrical to x, and (3) all other eases. [The c)·tinders are not necessarily eircular, and case (1) is different from case (2) only because of the excitation.] It is assumed that the incident wave in each C3.SC is the dominant mode with E parallel to y and HI parallel to x. Then the field of case (1) will be TM to y, expressible in terms of a single wave function Ail ",. '" (see Sec. 4-4). The field of case (2) will be TE to x, expressible in terms of a single wave function F" ",. ljI. Type (3) problems require two scalar wave functions to express the field (see Sec. 3-12). We shall consider only the centered circular post, as shown in the insert of Fig. 8-15. For even excitation (Fig. 8-124), assume a constant current on the post ] ,
e _
U
r
_I 1fd
(8-85)
The field produced by J.- in the waveguide closed by the magnetic conductor will be the same as the free-space field from the image system of Fig. 8-14. Hence, we can write
where the first term is the free-space field of J.- and the second term is the frcc-space field from all its images. The self-reaction of J,- in the waveguide with magnetic conductor is one-half that for the complete post in a waveguide; hence 1 (.
("d
(.,,) - 2}0 dy}o 2 d~ (J,E,) . f ("
= 41r}o
(Ev"""
+ E.l_) drp
(1l-ll6)
408
TUlfr.ElARMONIC ELECTROMAGNETIC FIELDS
Now the upost" term is independent of 4> since the I." is independent of ¢.
The "image " term is a source-iree field in the vicinity of the post and
can therefore be expressed as ElIlm....
=
. l
A ..J.. (kp)e f ...
"--00
(see Sec. 5.-8).
r
Thus,
E.'·..·d¢ - 2< A
oJ'(k;) - 2rJ. (k;) E,'·..··I,.o
and Eq. (8-86) reduces to
(a,a) - at [
Er L~ + J. (k;) E,'..... Lo]
(8-87)
The field of a. single cylinder of constant current was calculated in Sec. 5-6. Abstra.cting Crom Eq. (5-92), we have E II I1OH"",
-
~kIJo(k~)Ho(2)(kp)
p
~~
The field from each image is also of the above form, with p replaced by the distance to the image. Hence, Eq. (8-87) becomes
•
(a,a) - K [ H 0'" (k ;) where
K=
-
+ J. (k;) 2
§ka[!Jo(k~)
L:..
,
(-I)"H,"'(nkb)]
(8-88)
is an unimportant constant. Equation (8-88) is an exact evaluation of (a,a) for the assumed current of Eq. (8--85). Unfortunately, the Hankel function summation in Eq. (8--88) converges slowly and is not. convenient for computation. However, we shall now show that it can be transformed to
..
\ ' (-I)'H,'''(nkb) ~
,-.
~~[
1
~ ~(2b/X)'
1
4
+iG10g2r
b
-I
+8)]
(8-89)
where"Y "'" 1.781 and S is the rapidly convergent summation
1(2b/X)' I] n
(8-90)
409
MICROWAVE NETWORKS
The free-spacc field of a fllament of current is given by Eq. (5-84). Hence, the left-hand side of Eq. (8-89) is the E" from all images of the filament
-2
1--
,k
across the center of the original waveguide. ~This problem is Fig. 8-14 with J." replaced by the above f.) Then, by the method of Sec. 4-10, we can find the total field in the z = 0 cross section due to the above f. It is • E ow ~ ~ [ (n/b) (n./2) (nn/b)] (8-91) , • V(2b/X)' IlL. Vn' (2b/X)'
sin
+. \' sin
sin
•• 2
where only the first term is real bccause it is assumed that 1 < (2b/X) < 2. For large 11., the above summation has terms equal to those of
•
•
2:~sin(';)sinH;+ I)] - 2:
... )
C05no
n
.. _1.3.5....
_ Re
• . \'
4 ..... 1.3.5, ...
(e")" _ Re
(!2 log 1I + eei,') _ Re (!2 log I j sincosoI ) 1 j) _ I I - Re ( 2 10g tan (1/2) - - 2 log tan 2 1
11.
Hence, letting x = (b/2) + p in Eq. (8-91) and 8 = 7rp/b in the above identity, we can add and subtract the latter from the former and obtain
E
~[
00' _
"
_0 r
1 V(2b/X)'
1
+j
(!2 log 2b _ 1 + S)] rp
The free-space Ell from the same filament I is
When this is subtracted from the total Ell, and p set equal to zero, we have the right-hand side of Eq. (8-89). Returning now to the self-reaction, wc substitute Eq. (8-89) into Eq. (8-88) and obtain
Re(aa}-C
,
2
V (2b/X)'
X
-C-' 1 b
.N,(kd/2) 1m (a,a) = C [ - 2J ,(kd/2)
2,b
+ log T - 2 + 2S
]
(8-92)
410
TIME-HARMONIC ELECI'ROMAGNETIC FiELDS
where C is the unimportant constant,
C__ .ka l'J"(k~) 4r
2
Equation (8-92) is still exact for the current assumed in Eq. (8-85). However, because of the crudeness of OUf initial trial current, we caD expect our result to be valid onll:. for small d/>.. Hence, we use smallargument formulas for the Bessel functions and obtain
1m (a,a)
~ C (lOg ;~ -
2
+ 28)
(8-93)
Now, substituting from Eqs. (8-92) and (8-93) into Eq. (8-81), we ha.ve X.
+Zo2X. ~ ~).,....d [lOg 46 _ 2 + 28 (~)] ).
(8-94)
where S is given by Eq. (8-90). For odd excitation (Fig. 8-12c), we llSSume a current
J.- "'" induced on the post. 12
u"sin ~
The appropriate variational formula is Eq. (8-84),
T
-
I
1.0
X';',/Zob 0.8
7>
~
0.6
0.4
~
"lb=2.0
IB.ld ,
.l
End view
Top view
jXII
Zo
l'-..
'f-l
jX. 'X Zo J •
R::: ~I.O ~ ~ I'-....
Equivalent circuit
......
"'" '" "R
1.2-
- J4>,/Zoh
I 0.05
-lot-
L.j
1.4 -
O. 2
o
(8-95)
0.10
--
L--
g;; ~ 0.15
.-
~ :::::
11
0.20
dth FlO. 8-15. The centered circular inductive poet. in a rectAngular waveguide. Mamaih.)
(Afkr
411
!lucnoWAVE NETWORKS
the exact evaluation of which follows steps similar to those used to derive Eq. (8-94). The result is (8-96)
---f--6
I
FlO. S-16. A small obstacle
Figure 8-15 shows X .. and X& as calculated in a waveguide. from a second-order variational solution. 1 Our solution [Eqs. (8-94) and (8-96)] is accurate for small d/b, the error being of the order of 10 per cent for d/b = 0.15. Formulas and calculations for off-centered posts are also available. I A solution for the circular ca.pacitive post (Fig. 8-13b) is given in Prob. 8-12. 8-8. Small Obstacles in Waveguides. Figure 8-16 represents a small obstacle in a waveguide of arbitrary cross section. If the obstacle is symmetrical about a transverse plane, the equivalent circuit is as shown in Fig. 8-lOb. If the obstacle is loss-free, the Z's arejX's. The formulation of the problem for a conducting obstacle is that of Sec. 8-6. An approximate evaluation of tho reactions, made possible because the obstacles are small and not too near the guide walls, will now be discussed. Consider even excitation of the guide (Fig. 8-12a). The effect of a small obstacle is small; hence Z& is small and Z.. is large. Equation (8-81) is then X.. 1 1m (a,a) Zo = 2 Re (a,a)
(8-97)
where (a,a) is the self-reaction of the assumed currents in the waveguide. Let us first make some qualitative observations. In a rectangular waveguide, the reaction (a,a) is the free-space self-reaction of the obstacle plus the mutual reaction with all its images. For real current, the imaginary part of the free-space self-reaction becomes extremely large as the obstacle becomes smaU. Hence, for sufficiently small obstacles, we can let 1m (a,a) "'" 1m (a,a)/reo ..._
(8-98)
In contrast to this, the real part oC the Cree-space reaction approaches a constant, independent of the size of the obstacle, as the obstacle becomes small. The mutual reaction between the obstacle and its images therefore cannot be neglected. However, because the real part of the reaction is independent of the size and shape of the obstacle, we can calculate the dipole moment Il of the free-space obstacle and let
Re (a,a) = Re (Il,Il)
(8-99)
IN. Marcuvitz, "Waveguide Handbook," MIT Radiation Laboratory Series, vol. 10, pp. 257-263, McGraw-Bill Book Company. Inc., New York, 1951.
412
TIME-HARMONIC ELECTROMAGNETIC FIELDS
xt J... to
4-- b----j
T~
-
L~~ End view
z
----'----Side view
FIG. 8-17. A small conducting sphere centered in!L rectangular waveguide.
The right-hand term represents the self-reaction of a current element Il in the waveguide. As a.n example, consider the small sphere of radius c in the ccnter of a rectangular waveguide, as shown in Fig. 8-17. As our trial current, assume J." is that which produces the dipole field external to the sphere. This current, even though we shall not need it explicitly, is approximately
J."
II
.
(8-100)
= -U'-228m 8 ~c
where (J is measured from the x direction. Because the above current produces the same field as an x-directed element of moment Il, the ima.ginary part of the free-space self-reaction is the imaginary part of Eq. (2-115) evaluated at r = c. Hence, 1m (a,a)f'" ...-
= -." 23~
(~l)' (L)' 1\
/Ii(;
Equation (8-98) is therefore .~(Il)'
1m (a,a) "'" 12rJ c
(8-101)
'
For the real part of (a,a), we can use the analysis of Sec. 4-10 for a current sheet
J.-Il+-~),(y-n Because the current is real, we can set Re (!l,ll) ... - Re (P) of Eq, (4-87) and obtain
Re
(Il,If) -
where, from Eq. (4-86), 2
Jo 1 =-Il ab Hence, Eq. (8-99) becomes
Re (a a) ~ - Z. ,
ab
(If)' = -
.~. (II)'
ab).
(8-102)
MICROWAVE NETWOR.KS
413
Sub,tituting from Eq,. (8-101) aod (8-102) into Eq. (8-97), we bave
x.
Atab
Z. ~ - 24~'l.,c'
(8-103)
8ma1l~bst.acle approximation Cor a centered sphere in 8. rectangular waveguide. Our free-space reaction is the Rayleigh approxi· mation (Eq. (6-106)], which is valid for c/). < 0.1. Hence, we should expact Eq. (8-103) to be accurate wheo c/~ < 0.1 and c« a/2.
This is the
Now consider odd excitation of the guide (Fig. 8-12c). The evaluation of X. can then be made according to Eq. (8-84). Taking the current as real, we evaluate the imaginary part of (a,a) according to the free-space approximation [Eq. (8-98)]. However, because of the symmetry of the obstacle and of the excitation, there can be no net electric dipole moment, and Eq. (8-99) does Dot apply. There will be a magnetic moment KI (unless the obstacle has zero axial thickness), which can be calculated from the MSumed current. Then, analogous to Eq. (8-99), we use the approximation Re (a,a) ~ Re (KI,Kf) (8-104) where the right-hand term represents the self reaction of a magnetic current element Kl in the waveguide. Return now to the specific problem of a conducting sphere in a rectangular guide (Fig. 8-17). It is evident from symmetry that, for odd exeitation, the resultant magnetic dipole will be y--directcd. For the trial current. assume that which produces the magnetic dipole field external to the sphere. The free-spaee sclf~rcaction of this current is then just the dual of that for the electric dipole. given by Eq. (8-101). Hence. 1m {a,a)I ... 01'_ =
~(Kl)'
12 2 I (8-105) "re For the real part of (a,a), we evaluate the right-hand side of Eq. (8-104) by methods dual to those used to establish Eq. (8-102). For the centered y-directed magnetic current element iu tbe rectangular guide, we obtain 1m (a,a)
1'::
Y, Re (a,a) ~ Re (KI,KI) - ab (Kf)' -
~
ab.'. (Kl)'
Substituting from this and from Eq. (8-105) into Eq. (8-84), we have
Zo ahA, X, ",. - 12..-2e'
(S.106)
The accuracy of this formula is at least as good as that of Eq. (8-103). The evaluation of ot-her small-obstacle equivalent circuits can be found in the literature. I I A. A. Oliner, Equivalent Circuits for Small Symmetrical Longitudinal ApertW1!l aDd Obstacles, IRE TraM., vol. MTT-8, no. 1, January, 1960.
414
TlME-HAlUIONIC ELECTROMAGNETIC FIELDS
----['------
T
---------..!.. (a)
T
~ (b)
FIG. 8-18. (0) A diaphragm in a waveguide, and (6) an equivalent circuit..
8-9. Diaphragms in Waveguides. Figure 8-184 represents a cylindrical waveguide of arbitrary cross section with an infinitely thin electric conductor covering part of the z - 0 plane. This conductor is csJled a diaphragm, and the opening in it is called a. window. The diaphragm pill! the window cover the entire: = 0 cross section. The exact equivalent circuit is just a shunt element, as shown in Fig. 8-1&. Depending upon the shape of the diaphragm or window, the susceptance may be p
2Y,
-~-
B
1m (a,a) Re (a,a)
(&-107)
where (a,a) is the self-reaction of the assumed current
J.- on the dia--
Diaphragm
B/2¢
~Matched
guide
Mag. condo
IJ· I.
+-
.1 z
I
$ PI I.
(a)
Matched
~guide
Elect. condo
I. (0)
.1
(b)
~+.
Mag. condo
M·t
I
I
B/2¢
-+-
.1 z
I.
I
.1
(d)
FlO. 8-19. Symmetrical excitation of Fig. 8-IOa is represented by (a), which h.. all equivalent circuit (6). Symmetrical excitation of Fig. ~lOa ill also represented by (e), which has an equivalent circuit (d).
415
MICROWAVE NETWORKS
phragm. We can think of Fig. 8-19a as being constructed by placing pieces of electric conductor on top of a magnetic conductor. Because the diaphragm problem is self-dual, we have the alternative representat.ion of Fig. 8-19c. This can be viewed as a const.ruct.ion of the window by placing pieces of magnetic conductor on top of an electric conductor. The source has been changed to a magnet.ic current sheet, instead of the electric current sheet of Fig. 8-19a, so that complete duality is preserved. Then, dual to Eq. (8-107), we have
B 1m (a,a). 2Y, """ Re {a,a)..
(8-108)
where the subscripts m are added to emphasize that (a,a)", is the selfreaction of assumed magnetic currents M,o on the window, that is, (a,a). -
-
ff H'· M,' d.
(8-109)
Because the M,o is related to the tangential E in the window of the original problem according to M.'
~
u. X E
(8-110)
Eq. (8-108) is known as an aperture-field formulation of the problem. This is in contrast to Eq. (8-107) which is an obslade-currentformulalion. Note that Eq. (8-108) can also he viewed as a specialization of Eq. (8-84). To illustrate the theory, consider a capacitive diaphragm in a rectangular waveguide (Fig. 8-20). (Note that it must be capacitive, because it is a special case of Fig. 8-l3b.) Take the E-field formula {Eq. (8-108)] and, noto that
ff H' . M,' ds - - ff E X H . u, d. - (- ff E X H'· d8)' - p'
(a,a). - -
U,
because E is real. Hence, the problem is the same as tbrnsc treated in
--,-- T T
b
Side view
1 (0)
yt
a
·1
I~x End view
~
T a
y, 0
T a
Is y,
r' (b)
FlO. 8-20. (a) Capacitive diaphragm, and (b) an equivalent circuit.
0
416 Sec. 4-9.
TIME-HARMONIC ELECTROMAGNETIC FIELDS
In particular, if we assume Ell'
I.-0 {
.
~x
Y
sma 0
=
(8-111)
y>c
in t.he window, we then have precisely t.he problem of Fig. 4-17.
Hence,
from Eq. (4-77), we have
(a,a). -
P' - IVI'Y. ~
lVI' (Y.
where Y. is the aperture admittance. above into Eq. (8-108), we obtain
~ 4b
.!!.
Yo
;b + jB.)
Finally, substituting from the
B. = 8bX, (~.B.) 2aY o
(8-112)
a Yo
where the quantity in parentheses is plotted in Fig. 4-17. A more general treatment of the problem proceeds as follows. We know from the discussion of F'ig. 8-13b that the ficld must be TE to x, and so the most general form for the tangential field in the window is
E:l _
r {f(Y)'in : 0
, .. 0
Y
(8-113)
y> c
Then, by the methods of Sec. 4-9, we calculate • ab \ ' 1 (a,a). - P' ~ "2 1.; ;: (Y,h.IE,.I'
.-0
where, by Eq. (4-73), the Fourier coefficients E I .. are E,. =
n1rY t )0[e fey) cos Ii"" dy
f
and the characteristic admittances of the TEXt" modes are
(Y) o
1..
=
X,
j2bY, (2bjX,p
V nt
(8-114)
The Yo and X, pertain to the dominant mode, which is the only mode having real characteristic impedance, because of our assumption that only the dominant mode propagates. Hence, Eq. (8-108) becomes
•
B 2-Y-,
lIY.h.IE,.I' ,. ... 1 ~ ="'2"Y'."IE",',I"'-
417
WCROWAVE NETWORKS
which, upon substitution from the preceding equations, becomes
B
8b
-Yo
~ >.,
• \'
1 [ « f1. =-V7n:C'C=;'("'2b"'/"'~.T.), J. /(y) cos nTl/]' b dy
=---7[l'.><J-(Y-)-dY~]"'----
(8-115)
Equation (8-112) represents the special case fey) = 1. Better approximations to B/Yo can be obtained by using a better choice for fey), or by applying the Ritz procedure. The stationary formula. in terms of obstacle current [Eq. (8-107)] is specialized to the capacitive diaphragm as follows. The field is TE to x, given by Eqa. (4-32) witb
• !J' = sin ~ \ ' A cos nry e''''~ aLi" b
.-0
where The current on a diaphragm backed by a magnctic conductor (Fig. 8-194) is then
•
nry J.""'HlIl.-0 _abjrtwp cos~\' nA"sin b a '-'
J" - -H.
I
_.0
=
(T/a)' . JWP
•• 0
2: •
k' sin "" a
.-0
A" cos n-' _"_V b
Hence, the current has both x and y componenta, but the A" can be determined from the y component alone. The x component then adjusts itself to make the field TE to x. If we assume a current
J,. ""
g(y) sin ~ a
(8-116)
and define Fourier coefficienta (8-117)
then
418
TIME-BAnMONIC ELECTROMAGNETIC FIELDS
5
,.
(b~/
,,
4
-- -
~~
---
../
-- ---
~ CO) ' Cd)-
. rx \ '
Ell = - sm
0.1
0.2
0.3
0.4
b/'A, Flo. 8-21. The capacitive dia.phragm with c - 2b. (a) Exact solution, (b) crude aperture-field variational solution, (e) crude obstAcle-currcnt variational solution, and (d) crude qU8llistatic solution.
a ..~, ""A" cos nry -b-
Hcnce, in the same manner as Eq. (4-74) was derived, we find the sclfreaction of J." as • (a,a) ~
1
o
Also, at z = O. the tangential electric intensity is given by E" = 0, and •
ab \ ' 1
..,
"2 L.; '.
(Z,).J.'
where the characteristic impedances (Zo)" are the reciprocals of Eqs. (8-114). Because only the dominant mode propagates, only the n = 0 term of the summation is real, and Eq. (8-107) reduces to • (Z,).J.' 2Y, ,.~.~1=~~_ B"" 2ZJo"
L:
Substituting for In from Eq. (8-117) and for (Zo) .. = l/(Ya)l.. from Eq. (8-114), we finally have
(8-118)
This is the stationary formula in terms of obstacle current for the capacitive diaphragm of Fig. 8-20. Figure 8-21 compares various solutions to the capacitive diaphragm problem for the case of a diaphragm covering half the guide cross section. Curve (0) is called the exact solution because the estimated error is less than the accuracy of the graph. This solution is obtained by finding" quasi-static field and then using it in the variational formula, Eq. (8-115).1 Curve (b) is the crude aperture-field variational solution, Eq. (8-112), which is also Eq. (8-115) with f(y) ~ 1. Cu'v. (e) is • crud. I N. Marcuvit.r:, "Waveguide Handbook," MlT Radiation Laboratory Scries, vol. 10, secs. 3-5 and 5-1, McGraw-Hill Book Company, [nc., New York, 1951.
MICROWAVE NETWORKS
419
obstacle-current variational solution, Eq. (8-118), with . 'K(y - c) g(y) - ,m 2(b _ c)
(8-119)
(If the case g = 1 is tried, the solution diverges, because the boundary condition that the current vanishes at 11 = c is violated.) Curve (d) is
a. first-order quasi-static solution to the problem 1 B 8b 'KC - """ -logcscYo X 2b
(8-120)
Q
In practice, waveguides are usually operated with b/XQ < 0.25; so this last solution is a good approximation for most purposes. Note that the aperture-field variational solution, curve (b), is above the true solution, and the obstacle-current variational solution, curve (c), is below the true solution. That this is so for any trial functions /(y) and g(y) follows from the fact that Eqs. (8-115) and (8-118) are positive definite and hence are an absolute minimum for the true fields. Since Eq. (8-115) gives B/Yo and Eq. (8-118) gives Yo/B, the former yields upper bounds and the latter yields lower bounds to the true B/Yo. The existence of variational formulas for both upper and lower bounds is not very common and is a consequence of the self-duality of the problem plus the positive-definite nature of the resulting variational formulaa. Our crude variational solutions give an error of the order of 20 per cent, but it is remarkable that they are as close n.s that. A quasi-sLatic solution to the problem is f( ) _ co, (Ty(2b) y ""in' (Tc(2b) 'in' ("Y(2b)
(8-121)
which actually has a singularity at y = c. Hence, our approximation [(V) = 1 was an exceedingly crude choice, yet it led to usable results. Our approximation to g(y) [Eq. (8-119)] is equally crude. If we were to use Eq. (8-121) in Eq. (8-115), the result would be very close to the true solution. It is interesting to note that the three diaphragms shown in Fig. 8-22 all have the same equivalent circuits. This is evident, because the image systems for all three cases are identical. The treatment of the inductive diaphragm (Fig. 8-23) is similar to that of the capacitive diaphragm. The general variational formulas for upper and lower bounds are given in Probs. 8-14 and 8-15. For a crude aperture-field solution, we ~an assume Eq. (4-75) for E,t in tbe aperture. W. R. Smythe, "Static and Dynamic Electricity," 2d ed., Sec. 15-10, McGraw· Hill Book Company, lne., New York, 1950.
I
420
TDlE-HARAlONJC ELECTROMAGNETIC FIELDS
Ir--iI
,
L.-_ _--l..!. (6)
(aJ
(c)
Flo. 8-22. These three diaphragms give rise to the same shunt. capacitance.
xt-- 6 --f
_ _ _T'---_ _
I
T
1T
Jl
Stde view (a)
T
y,
-
jB
Y,
Y
End view
(6)
FIo. &-23. (0) Induct.ive diaphragm, a.nd Cb) an equivalent circuit..
This procedure gives
~ ~ Yo
_X.a [~ 1(c/a)']' (,6 B) 5m (7fe/a) X C
(8-122)
1I
where B. is the aperture susceptance plotted in Fig. 4-19. The values of -BIY. calculated from Eq. (8-122) will be higher tban the true values (of the order of 20 per cent higher). The problem can also be treated by quasi-static methods) a first-order solution being! B Yo
t::>< _
Xlai' (1 + esc' 2a ~) cot' 'Ire 2a
(8-123)
A combination of the quasi-static and variational methods can be used to obtain solutions of high accurney.' 8-10. Waveguide Junctions. We shall now consider waveguide junctions formed by butting two cylindrical guides together, possibly with a diaphragm covering part of the:;:: = 0 cross section. Figure 8-24 represents the general problem. No longer is there symmetry about the:;:: = 0 cross section; 60 the methods of Sec. &-6 do not apply directly. We there1 W. R. Smyt.he, "Statie and Dynamic Electricity," 2d cd., p. 555, McGraw-HiU Book Company, Inc., New York, 1950. IN. Mareuvitz, "Waveguide Handbook," M1T Radiation Laboratory Series, vol. 10. !leC. &-2, McGraw-Bill Book Company, Inc., New York, 1951.
421
MIcaOWAVE NE'TWORXB
(ore take the more fundamental a.pproach of const.ructing complete solutions in each region and enforcing
.If
E+ X H+· ds
-.£1
(8-124)
E- X H- . ds
where superscripts + and - refer to regions z > 0 and z < 0, respectively. In terms of the reaction concept, we caD think of Eq. (8-124) as stating that the reaction is conserved at the junction. An equivalent network for the junction is shown in Fig. 8-24b. It is evident that only a shunt element is required to represent the junction, because an electric conductor placed across the entire z = 0 cross section presents a short circuit to both waveguides. The characteristic admittances of the equivalent transmission lines are taken to be the characteristic wave admittances of the guides, and the ideal transformer represents the change in admittance level. If the characteristic admittance of the right-band transmission line were chosen as n t times the characteristic wave admittance of the guide, then the transformer would not be needed. We shall usc Eq. (8-124) to obtain stationary formulas for Band n t • It is assumed that the excitation is at z = - gQ ; hence in the region , <0 E,- - (...~" H ,- -
+ ref'") 1 :. r
y.-(e-i~~ -
.,
+
LV...,"., L:, YIV.-e""'~h;:
rei·''') 1 ~. rho -
(8-125)
where ei, h; are the mode vectors, a. are the cutoff mode-attenuation constants, Y; are the characteristic admittances, and r is the reflection coefficient for the dominant mode. The subscripts 0 denote dominantmode parameters. Matched conditions are assumed at z = gQ i hence in T
-I
T
__1-Side view (a)
End view
l:n
T
Y~C: (b)
FIG. &-24. (0) A waveguide junction, and (b) an equivalcnt. circuit.
422
the region z
TIME-HARMONIC ELECTROYAGNETIC }O'l.ELDS
>
0
~oe-i~zeo +
E,+ =
l: ~;~r-"I'ei + 2: ;
H,+ "'" Y o+9"oe-i8 'h,
(8-126)
f;9".e-';'·b.;
;
where the carets distinguish the various parameters from their z < 0 counterparts. The applicat.ion of Eq. (8-124) to the above field expresSiODS yields Yo+9'o'
+
I 1",9",1 .. ~ ~ ~
Yo-V,t -
•
L:
(8-127)
Y,V;%
i
Now the relative admittance seen from the left-hand guide is
Y
l-r 1 r
+ -
Remembering that the Yo real Vi and 1'"; we have
G
Y,- ~ Y,-
aTC
. B
+ J Y,-
(8-128)
~
OJ are imaginary, for
real and the Yi, i
l: Y;V,' + l: 1',1'"
jB _ -'-;_~~~_
Y.-
Y o+9',t
G
Y.-
(8-129)
Yo ¥o'
=
Yo-Yo'
From OUT equivalent circuit, with matched conditions at evident that
Z
::I
<Xl,
it is
(8-130)
hence
Finally, to obtain the Vi and 9'" we need only specialize Eqs. (8-125) and (8-126) to z .. 0 and, using the methods of Sec. 8-2, obtain
.....JJ 1'; - !! ..... Vi
=
E, · c, dB (8-131) E, .•; d•
Note that the integration extends only over the aperture, because Ee - 0 on the conductor. EquatioDs (8-129) and (8-130), with Vi and 1". given by Eq. (8-131), arc formulas stationary with respect to small variations
HIcnOWAVE NETWORKS
423
in the aperture E j about the correct field.
Alternative stationary formulas in terms of current on the conducting wall at % =- 0 can also be obtained (see Prob. 8-18). Note that Eq. (8-129) specialized to the case of two identical guides is the diaphragm solution of the preceding section. To illustrate the theory, consider the rectangular waveguide junctions of Sec. 4-9. For the capacitive junction (Fig. 4-16), the dominant-mode vectors are
e. -
Uif
. ""a ac ..ff<-sln-
Hence, regardless of our assumed tangential E in the aperture
E,d _ ulff(y) sin ~ a
(8-132)
we bave by Eqs. (8-130) and (8-131) (8-133) This is therefore the exact transformation ratio of the ideal transformer. In Sec. 4-9, we calculated the aperture susceptance corresponding to the crude choice 1(Y) - 1. The first summation in the numerator of Eq. (8-129) then vanishes, and the second summation is related to the aperture susceptance of Eq. (4-78) by
-
ll',~.' ilVl'B. - ie'B. , But, for I(y)
c:
I, we have V.t "" ac/2j hence, by Eq. (8-129),
.!! = Yo
t
2c B. =
acYo
4cX, ("'o2aZO B")
(8-134)
where the quantity in parentheses is plotted in Fig. 4-17. The general expression IEq. (8-129)] is positive definite in our particular casej so Eq. (8-134) gives values of B/Y 0 higher than the true values. However, because the field in the aperture is less singular at the edge of a step than at a knife edge, we should expect the 868umption /(V) = 1 to give better results in the junction problem than in the corresponding diaphragm problem. Our approximate answer (Eq. (8-134)] gives an accuracy of the order of 10 per cent, as illustrated by Table 8-3. This can be compared to the 20 per cent accuracy in the corresponding diaphragm prolr lern, illustrated by Fig. 8-21. The inductive junction of Fig. 4-18 is treated in a similar manner. In general, the field in the aperture is of the form E, "'" /(x), and for the
424
TIME-HARMONIC ELECI'ROMAGNETIC FIELDS
TABLE
t
8-3.
CoMPARISON OP
EQ. (8-134) 'to THE ExACT c/b - 0.5
SoLUTION I FOR THE CAeE
Exact
Approximate
o
1.57
0.' 0.3 0.4
1.69
1.63 1.84
1.93
2.10 2.67
2.44
N. Marcuvib:, "Waveguide Handbook," MIT RncliationLaboratory Series, vol. to,
sec. 5-24, McGraw-Bill Book Company, Inc., New York, 1951.
solution of Sec. 4-9 we assumed E,G = u~f(%) = Uwsin ~
(8-135)
c
By Eq. (8-130), we then find the transformation ratio of the ideal transformer as n,
-ic '/r'a
~-
[Bin (7fe/a) ]' 1 - (cia)'
(8-136)
a.nd, by Eq. (8-129), the normalized shunt susceptance as
_2X,(_ s)
~ .b (8-137) YoC A" where the quantity in parentheses is plotted in Fig. 4-19. Note that, in contrast to Eq. (8-133), the transformation ratio [Eq. (8-136») depends on the assumed aperture field and is therefore approximate. Note also that the characteristic wave impedances of thc two guides, z < 0 and z > 0, are now differentj so the superscript - has been retained on Y o- in Eq. (8-137). Finally, the value of -B/Y o- obtained from Eq. (8~137) will be larger than the true solution, because of the positive definiteness of the variational formula. The alternative equivalent circuit of Fig. 8-25 illustrates a very useful way of viewing the waveguide junction T 1 :n T of Fig. 8-24a. We bave separated the shunt susceptanee into two parts, which, by Eq. (8-129), can be identified as
J!...-
:IDCl: FIG. 8-25. Alternative equivalent circuit lor Fig. 8-24a.
LY.V.'
jn-
Yo-
=
-,,~~~
Yo-Vo t
jB+ Y o+ =
425
MICROWAVE NETWORKS
T
T
n : 1
Yo,
r
II Side view
T
Yo,
r
End \liN (0)
(b)
Fla. 8-26. (a) A thin coax.to-waveguide feed, and (b) an equivalent circuit.
where the Vi and V"i are given by Eq. (8-131). Note that B- depends only on guide z < 0, and in particular is one-half tbe shunt susceptance of a diaphragm, assuming E t in the aperture is unchanged. This assumption is, of course, incorrect, but. our formulas are stationary; so B- in the junction problem is approximately B/2 in the corresponding diaphragm problem. Similarly, B+ is approximately B/2 for the diaphragm problem corresponding to the guide z > O. Hence, by defining a.perture susceptances according to Eqs. (8-138), we effectively divide the problem into two parts, each part relatively insensitive to the other. An aperture susceptance calculated for the aperture and ODe guide, such as Figs. 4-17 and 4-19, thereby becomes useful for a wide variety of problems. 8-11. Waveguide Feeds. We shall DOW consider thin coax-to-waveguide feeds, as illustrated by Fig. 8-260. By thin, we meaD that the dimension in the axjal (z) direction is small. The analysis will be exact only for zero-thickness junctions. An equivalent circuit when only one mode propagates is shown in Fig. 8-26b. When morc than one mode propagates, Bay N modes, there will be N ideal transformers in series, each coupling to one mode. The justification for this equivalent circuit will be found in the analysis. Let the feed be viewed as l\ sheet of current J. in the z = 0 cross section. (This neglects the effeet of the gap. which is usually small.) Then, in the region z > 0, we have
(8-139)
where
r,+ is the +z reflection coefficient of the ith mode referred to
426 Z -
TIME-HARMONIC ELECTROMAGNETIC FIELDS
O.
Similarly, for z
< 0,
(8-140)
where rr is the -z reflection coefficient of the ith mode referred to z = O. We have ensured continuity of E, at z = 0 by choosing coeflia cients Vi the same in both Eqs. (8-139) and (8-140). The boundary con~ dition on H at z = 0 is
J.
= u. X (H,+ - H ,-)
-2:, v..
Y..
1.... 0
(~ ~ ~:= + ~ ~ ~::) u. X h ..
(8-141)
Multiplying each side bye, and integrating over the guide cross section, we have
v,y,
G~ ~:= + : ~ ~::) - - II J. -
e, d8
(8-142)
The field is then completely determined if the r's and J. are known. We now use the stationary formula of Eq. (7..s9) to determine the impedance seen by the coax. This formula is
ZOo
~
I~.'
-
II
E-J.d.
where the integration extends over the z = 0 guide cross section and II.. is the current at the reference plane T'. Using the first of Eqs. (8-139) for E, and Eq. (8-141) for ]., we obtain 1 '\'
•
(1 -
ZI.. = II"s ~ V, Y i 1
•
r,-
1 -
r.+)
+ r,- + 1 + r..+
Finally, substituting for Vi from Eq. (8-142), we have
z. (ff J•. e, d.)'
1 '\'
Z,. - I,.'
L, (1
-
r,
)(1
+ r,-)-' + (1
r,+)(l + r,+)-'
(8-143)
where Z, is the characteristic impedance of the ith mode. This is a sta.tionary formula for the input impedance of a zero-thickness COQx-to-waveguide feed. We can put it into a slightly different form by noting that
427
?tfiCROWAVE NETWORKS
the wave impedance of an ith mode referred to z = 0 is ~. ~ Zl
b,
•
+ r,r,
(8-144)
1_
Hence, Eq. (8-143) can also be written as (8-145)
This shows that the guides z > 0 and z < 0 appear in parallel for each mode. Nonpropagating modes decay exponentially from the junction and their r. may be taken as zero unless some obstacle is close to the feed. If we assume that only one mode propagates, then all Z. are imaginary except i = 0, and all r. = except i = 0, provided the terminations are not too close to the feed. Equation (8-143) or (8-145) can then be written as
°
(8-146)
where
(8-147) (8-148)
Equation (8-146) is, of course, just that for the equivalent circuit of Fig.8-26b. As an example, consider a probe in a rectangular guide (Fig. 8-27). Assume
J. -
1~. sin ked -
x) '(y - c)
where k = 2r/X is the wave number of free space. vector is eo
=
xd
(8-149)
The dominant-mode
/2 . "11 b
u., Vab sm
Equation (8-147) is therefore
n "'"
~ foci dx 10
6
dy sin ked - x) a(y - c) sin
(d)
?
TC (8-150) n' - - 2 .SlOt - tan t kk'ab b 2 The summation for X [Eq. (8-148)] divergcs, because the current was
giving
TIME-HARMONIC ELECTROMAGNETIC FIELDS
taken as filamentary. If the probe is taken as circular in cross section, the reactance can be evaluated by methods similar to those used in Sec. 8-7. How~ ever, if the probe is very thick, we shall have to modify the equivalent circuit of Fig. 8-2Gb. The reactance of a short probe can be estimated by the smallFlO. 6-27. Probe in fl, rectangular obstacle approximation of Sec. 8-8. It wAveguide. is evident from the sn;u:t.ll-obstacleanalysis that X is capacitive (negative) for a short probe and is of the order of magnitude of X for a probe over a conducting ground plane. Note that our present solution [Eqs. (8-146) to (8-148)], specialized to a rectangular waveguide matched in both directions, is the same problem treated in Sec. 4-10. From our equivalent circuit (Fig. 8-26), it. is evident that the coax sees
R ,. = under matched conditions.
n! ~o 2
Hence, !
2R I •
(8-151)
n -Z.
where R l • is the quantity calculated in Sec. 4-10. For example, when the probe is connected to the opposite wall of the waveguide, as in Fig. 4-20, we have from Eq. (4-91) ,
2. (tan
lea)! . ~
n=b~
'll"C
SInb"
(8-152)
Other possible feeds are shown in Fig. 4-28. 8-12. Excitation of Apertures. We now wish to consider conducting bodies containing apertures excited by waveguides. The general problem is represented by Fig. 8-28a. As far as the waveguide is concerned, the aperture appcars simply as a load across the reference plane T. A variational solution to the problem can be obtained by assuming tangential E in the aperture, calculating the resultant fields on each side of the aperture, and then conserving the flux of reaction by
II (E X H . ds)... ~ II (E X H . ds),•• ..pen
(8-153)
.. pe~t
This is the same approach that we took in Sec. 8-10 for the waveguide junction. Indeed, we can think of our present problem as a junction between the waveguide and external space.
)(ICROWAVE NETWORKS
Once the tangential E in the aperture is assumed, the problem separates into two parts, external and internal. We have anticipated this separation by taking the equivalent circuit as shown in Fig. 8-28b, where jB represents the internal susceptance of the diaphragm and Yap••, the external admittance of the aperture. The ideal transformer accounts for possible differences of impedance reference in the internal and external problems. The internal problem is identical to one-half of the waveguide-junction problem. Let us therefore abstract from Eq. (8-138)
LY,V,' jB
,
(8-154)
Y ..... Y.V.I
where
-
JJ Eo' . e, d&
V, -
(8-155)
These formulas give the internal shunt susceptance B in terms of an 8.S8umed E,& in the aperture. For tbe external problem, we dofine an aperture admittance as
Y.~.. ~ ~,
ff .,."
E,' X H'· d.
(8-156)
where V is some reference voltage and H- is the external magnetic field calculated from the assumed E 1-. Examples of some aperture-admittance calculatioll8 are given in Sec. 4-11. (These calculations were made on a conservation of power basis, but, beeause E- was assumed real, they are the same as variational solutions.) To determine we note that the dominant-mode voltage coupled to the aperture is V.. but we have referred the aperture admittance to V j hence
"I
n'- -VV',,
(8-157)
where V, is given by Eq. (8-155) applied to the dominant mode.
T Conductor
,-
, ,--.,
----
Side view
,:
,
l' n
I
I
jS
y.. Aperture
I
I
End view Ca)
(b)
FlO. S-28. (a) An aperture excited by a waveguide, and (6) an equivalent circuit..
430
TIME-HARMONIC ELEC'l'ROMAGNETIC FIELD8
0.004
I alb
0.002
1-.,.
/ 'V
o
,-
0.2
/i'--
C
/
-0.002
~G
---- -
-;:;'Ib - k.25
1/ 0;6
alb - I'
-];:JL 0;8
.J 1.0 a/~~
, ,
I I
CEJI
,~I
-0.004
I-a-ol
"
, ,
"
, ,
-0.006 FIG. g..29. Aperture admittance for rectangular apertures in ground planes, referred to the dominant-mode voltage of a rectangula.r waveguide of the same dimcDsiona. (Ailer Cohen, Crowley, and Levi,.)
An aperture of practical importance is the recta.ngula.r aperture in a conducting ground plane, as shown in the insert of Fig. 8-29. The aperture admittance has been calculated for the assumed field E •• =
7'
. U 1I 810-
a
(8-158)
in the aperture, referred to the voltage
v=jJ
(8-159)
which is the dominant-mode voltage for a. waveguide of the same dimensions as the aperture. Hence, when the aperture is simply the flanged open end of a rectangular waveguide, then n - 1. The field due to E,· in the aperture can be found by the methods of Sec. 3-6, and the aperture admittance calculated by Eq. (8-156). The mathematical details are tedious but can be found in the literature.! Figure 8-29 shows the aperture admittances for a square aperture and for a rectangular aperture with Bides in the ratio 1 to 1 and 2.25 to 1. 1 1 Cohen, Crowley, and Levis, The Aperture Admittance of a Rectangular Waveguide Radiating into Half-6paee, Qhw Stal~ UnilJ. Anten1UJ Lab. Rept. ac 21114 SR no. 22, H153. , Additional calculations have been made hy R. J. Tector, The Cavity-backed Slot Antenna, Univ. IUiMi4 Antenna Lab. &pt. 26, 1957.
431
!,(JCROWAVE NETWORKS
As an example, suppose we have a square waveguide of height and width 0 feeding 8. rectangular aperture with sides in the ratio alb - 2.25, as shown in Fig. 8-30. The waveguide is excited in the dominant y-polarized mode, for which
V2 . "" eo = u, sm -
•
•
Hence, by Eqs. (8-155) and (8-158), we have
V o ...
01" a
dz
0
1~ dy sin' ~ 0
=
0
_b_
V2
and eo, hy Eqa. (8-157) and (8-159), n'
IC1
•
b - 2.25
The shunt susceptance B is one-half that for the diaphragm of Fig. 8-22b. An approximation to B is therefore given by Eq. (8-120) with B replaced by B/2, b by 012, and c by b/2, giving B Sa rb a R:: -Iogcsc- = 3.54Y,).. 2a X.
-
Hence, the terminating admittance seen by the waveguide is y
~ j3.54 :. + 2.25Y...,.
where Y.,.•• is given hy the alb""" 2.25 curves of Fig. 8-29. 8-13. Modal Expansions in Cavities. Consider a cavity formed by a perfect conductor enclosing a dielectric medium. Each mode must y
Side view
End view
Fla. 8-30. A square waveguide £eediDg a reeuDgula.r aperture in
&
ground plane.
432
TIME-HARMONIC ELEC'1'ROMAGNETIC FIELDS
satisfy the field equations (8-160)
V X Eo' "'" - jWiJlH..
where i is a mode index. Either Eo or Hi may be eliminated {rom the above pair of equations, giving the wave equations V X (1l- 1V X E,:> - ",-'"E. = 0
(8-161)
V X (elv X Hi) - w,IJlH; - 0
valid even if E and p are functions of position. tions, coupled with the boundary condition n X E, - n X (c'v X H,)
~
Each of these wave equa-on S
0
,<8-162)
where n is the unit normal directed outward from the cavity boundary S, is an eigenvalue problem in the classical sense. l Hencc, for f and Jl real (no dissipation), the eigenvalues Wi (resonant frequencies) are real, and the eigenfunctions Eo, Hi form a complete orthogonal set in the Hermitian sense. Furthermore, we wish to normalize the mode vectors, so that the orthogonality relationships are
ir&j i
=
j
which can be derived from Eqs. (8-160) in the usual ma.nner. ization of the E, also normalizes the Hi, because
(8-163) Normal-
tha.t is, the time-a.verage electric and magnetic energies are equal. Hence, the orthogonality relationships for the H, corresponding to the orthonormal Ei are i¢j i=j
(8-164)
We have alrea.dy shown in Sec. 8-4 that if E; is chosen real, then the cor· responding Hi is imaginary, and vice versa.. Now suppose that electric sources exist within the cavity, as suggested by Fig. 8-31a. The field equa.tions are then
v X E = -jwpH
v XH
~jw.E+J
and the wave equation is V X (p-lV X E) - w'EE "'" -jwJ
(8-165)
I Philip M. Morse and Herman Feshbach, "Methods of Theoretical Physica," chap. 6, part I, McGraw-Hili Book Company, New York, 1953.
433
HICROWAVE NETWORKS
D
Flo. 8-31. A cavity containing (a) electric sources, and (b) magnetic sources. (0)
(h)
Because the E; are a complete set, we can let (8-166)
SUbstituting this into Eq. (8-165), we h&ve
~ A;[V X (.-'V X E;) - w',E;] - -jwJ ;
which, by Eq. (8-161), caD be written as
L
A;(w;2 -
W
2
)fE; = -jwJ
;
If each side is now multiplied Bcalarly by Ej and integrated over the volume of the cavity, &11 terms except i = j va.nish beca.use of orthogonality [Eq. (8-163)], and we have
(w,' - w')A; - -jw
which determines the A"
fff J. E~ dT
(8-167)
Hence, Eq. (8-166) becomes (8-168)
and the corresponding H, obtained from the field equations, is (8-169)
Note that the field becomes extremely large as W a.pproaches Bome resonant frequency. In fact, the field becomes infinite at a resonant frequency in the loss-free case, which is to be expected. Actually, in any physical problem there will always be some dissipation; so the Wi are complex. Hence, the field is large, but finite, at all real resonant frequencies. The dual problem is that oC magnetic sources in a cavity, represented by Fig. 8-3Ib. In this case, the wave equation in H is (8-170)
434
TIME-B.ARMONIC ELECTROMAGNETIC FlELOS
\Ve then expand H in terms of the orthonormal mode vectors
a
as (8-171)
where, dual to Eq. (8-167), the B; are given by (w;' - ( 1 )B; = -jw
Iff M· Ht d.,.
(8-172)
Hence, the expansion of H due to magnetic currents M is l H _ '\'
jwH,
rrr M.H~dT
(8-173)
rrr M.H~dT
(8-174)
Lt, w' ",,' Jll
and the corresponding E field is E- '\'
jw,E,
i..J , W'
w;t
Jll
If both electric and magnetic sources exist within the cavity, we can superimpose Eqs. (8-168) and (8-174) for a solution. 8-14. Probes in Cavities. Mathema.tically, we cnn represent a probe in a cavity in terms of electric currents in the cavity, as shown in Fig. 8-31a. The impedance seen at the input terminals to the probe can then be calculated by the varia.tional formula (8-175)
where J. is the assumed current distribution on the probe, and 1 is the corresponding input current. All mode vectors E, will be chosen realj 80 the field produced hy JO is given hy Eq. (8-168) with the' dropped. Substituting this equation into Eq. (8-175), we obtain (8-176)
where
(8-177)
The analysis neglects the effect of the aperture through which the probe is fed. This effect is usually negligible and can be taken into account by the methods of the next section. AB long as there is no dissipation, the input impedance will be purely reactive. However, if t.he cavity is lossy but high Q, the effect of dissipaI The eigenvalue Wf - 0 must be included in both EqIJ. (~168) and (8-173). The modee a.ssociated with Coli - 0 account for the irrotlltional parte of E and H. See, for example, Teichmann and Wigner, J. Appl. Phll., vol. 24, March, lOSt
435
raaCROWAVB NETWORKS
tion can be taken into account by letting the resonant frequencies be complex, according to l
w,' ~
W.'(l +~)
(8-178)
FIG. 8-32. An equivalent circuit ror a where Q is the quality factor. In the probe-red cavity in the vicinity or vicinity of a resonant frequency, say resonance. Wo (not necessarily the dominant resonant frequency), we can approximate Eq. (8-176) by
ZIII "'" jX -
w2
jw(ao/ f)' wo 2(1 j/Q)
+
(8-179)
where X is the reactance due to all modes except the i = 0 mode (8-180)
The effect of dissipation in modes not near resonance is negligiblej hence, it is not included in Eq. (8-180). An equivalent circuit which represents Eq. (8-179) is shown in Fig. 8-32. To determine the values of R, L, and C, we need only compare the formula for the impedance of the parallel RLC circuit
wo'(1
I
LC
wo! =
jW/C
+ j/Q)
R
wL
It is then evident that
with the last term of Eq. (8-179).
R _ Q
wo
(afo)'
R woL
Q--~-
L _
(.!1.!.)' fwo
C=
U.)'
(8-181)
where ao is obtained from Eq. (8-177). To illustrate the theory, consider a probe in a rectangular cavity (Fig. 8-33). The normalized mode vector of the dominant mode is Eo =
2.1I'y.1fz
U",
--= Sin
-
Sill -
VEabc b c where the normalization factor was obtained from Eq. (2-97). current on the probe, we assume
J",o=
{
I sin k(d - x)!( _ b')!( _ c') smkd Y y
x
o
x>d
(8-182)
For the
(8-183)
I M. E. Van Valkenberg, "Network Analysis," p. 364, Prentice-Hall, lne., Englewood Cliffs, N.J., 1955.
436
TIME-HARMONIC ELECTROMAGNETIC FIELDS
ty
r----I+---,,-i~T
1
TO b' I
Z
-11=,~,;==:=l.1 tx
x
-'----'
.-----iT ,.
.!-L-__--, Ir
II
J"
oL "---'
FlO. &-33. Probe in a rectangular cavity.
Then, by Eq. (8-177), we have a,=
I
2
kYtabc
tan
(kV'(b)'("') 2 b
(8-184)
-slnw--smc
The other parameters needed to evaluate R, L, and C are the resonant frequency fr = wo/2r, given by Eq, (2-95), and the Quality factor Q, given by Eq. (2-101). The evaluation of the series reactance X is a much morc difficult problem. We cannot, of course, use the filamentary current of Eq. (8-183) to evaluate X, since the resulting reactance would be infinite. The actual diameter of the stub must be considered. To a very rough approximation, X will be of the same order of magnitude as for a stub over a ground plane. Hence, for short stubs, the reactance is capacitive. When the stub is bent into a small loop and joined to the cavity walll the system is often called a loop feed. The treatment or loops in cavities is essentially the same as the treatment of stubs, once a current is assumed on the loop. The series reactance X for small loops is inductive l in can· trast to the small-stub case, for which it is capacitive. Some explicit loop feeds are considered in Probs. 8-24 and 8-26. 8-15. Aperture Coupling to Cavities. The general problem of coupling a cavity to a waveguide through an aperture is represented by Fig. 8-34a. For a variational treatm~nt or the probleml we assume an aperture field E," and conserve reaction according to
.rf (E," X H"· ds)..". - ff (E,· X H"· ds)... apert
apert
u.
(8-185)
437
IOCROWAVE NETWORKS
For a given E,-, each side of t.his equation can be considered separately, which amounts to dividing the original problem int.o t.wo parts, as shown in Fig. 8-34b and c. The equivalent current
M.-=nxE,-
(8-186)
in the cavity part is the negative of the terminating current in the waveguide part. The waveguide part of the problem is identical to the problems treated in Secs. 8-10 and 8-11, and is therefore of the form
JJ (E,' X H··ds) ••,•• =
-YVo'
+
_pert.
l
Y.V.'
.... 0
where the V II are the various mode voltagcs, the Y II are the mode-characteristic admittances, and Y is the admittance seen by the dominant mode. Hence, we can rewrite :Eq. (8-185) as
~o = jB, - Y.~ot
.Jf
(8-187)
(E l- X H- . dS).. ri'7
where Yo is the characteristic admittance of the dominant mode and
.L:
B, = -1
(v.)' V
Y. -
•••
Y.
(8-188)
o
..._---...;1. T
(a)
(6)
T
R
1 :n
~ r----~""N'C.........,L (el
(d)
FIo_ 8-34. (a) Aperture coupling from a wavelUidt! to a cavity caD be divided into two p&rts, (b) the cavity, And (e) the waveguide. An equivalent. circuit in tbo vicinity of re8Clnanee is Abown in (d).
438
TIME-HARMONIC
ELECTRO~tAGNETIC
FIELDS
is the shunt susceptance introduced by the waveguide pa.rt of the problem. The.calculation of B, was treated in Sec. 8-10. For the cavity part of the problem, we caD determine the field by Eq, (8-173) with the current given by Eq. (8-186). Taking the mode vectors Hi as real, we obtain
The right-hand side of Eq. (8-185) is then given by
\'
1f
(E ," X HG • ds)..... ,t.. = '-'
.pet~
jwb.' w2
W;2
(8-189)
.:
where
b. -
If Eo" X H.ods
(8-l90)
In the vicinity of a. resonant frequency, we again take losses into account by Eq. (8-178), and Eq. (8-187) can be written as
The first term in the brackets represents the susceptance due to all nonresona.nt modes in the cavity, and the second term gives the resonant;.. mode effect. The above equation caD therefore be written as
y
oS + n' [oS
Yo ~J
II
Yo J
w1
0 -
jw(bo/V)'] wo'(1 +ilQ)
(8-191)
where the .susceptance due to nonresonant cavity modes is
S--~~ c V' w' b;' Wi!
(8-192)
••
and, to account for an arbitrary reference-voltage V, we have introduced the ideal transformer
n' -
(:,y
(8-193)
Finally, we can represont the last term of Eq. (8-191) as a series RLC circuit, as shown in Fig. 8-34d. The formula for admittance of a series RLC circuit is 1
y =
-,,-_--;Jr.;;°w"'/L:;-,= w'
wo'(1
+ j/Q)
Wa' =
LC
1 1 Q = -wC-R- = w-.-C-R
439
MICROWAVE NETWORKS
Comparing thi:5 with the last term of Eq. (8-191), we see that
.!. ~ R
Q "'0
(bVo)'
c - (~)' Vwo
L
(:.y
~
(8-194)
where bo is obtained from Eq, (8-190). Let us illustra.te the above theory by a. treatment of the rectangular waveguide to rectangular cavity junction, shown in Fig. 8-35. The waveguide part of the problem is identical to problems previously considered. In particular, B, will be approximately one-half of Eq. (8-120) with the appropriate interchange of symbols, or B,
4a'
1l'd
Yo
A,
2a'
-~-Iogesc
(8-195)
For the cavity part of the problem, let us make our often-used assumption Et =
in the aperture.
. ry
(8-196)
U.,SlD1J
Also, let us refer the eavity admittances to
v-.f;
(8-197)
which is the waveguide dominant-mode voltage that would be excited by Eq. (8-196) if the waveguide were the same height as the aperture (n 1 would be 1 in that case). In OUf pa.rticular problem, the waveguide dominant-mode voltage is V o = v'ba'/2; hence
n1
d a'
=-
(8-198)
Rather than calculating Eq. (8-192) directly, let us view the aperture as the junction between two waveguides of height a' and a. The susccptT Waveguide FIG. 8-35. Aperture coupiing from a rectangular waveguide to & rectangular cavity.
T
Cavity
~
Top view
~ •
-±-I d
Side view
,
b
·1
~
II
440
TIME-HARMONIC ELECTROMAGNETIC FIELDS
ance Be referred to the mode voltage of a waveguide of height a could then be approximated by Eq. (8-195) with af replaced by B. But we wish to refer it to the V of Eq. (8-197); so we should multiply by d/a and obtain B~
R:
4d .d X. log esc 2a
(1)-199)
Finally, to determine the R, L, and C, we need the normalized dominantmode vector, which is
Ho =
2
vi J1abc(b 2 + 0 1)
. •y . . .y. ") ( u~sln-bcos--u.ccoo-b6InC
(8-200)
C
Hence, from EqB. (8-190), (8-196), and (8-197), we obtain
2d (b.), V ~ .ac(1 + c/b)'
(1)-201)
The resonant frequency f~ = wo/21r is given by Eq. (2-95) and the quality factor Q by Eq. (2-101). Hence, all parameters of the equivalent circuit (Fig. 8-34d) have been evaluated. PROBLEMS
8-1. Consider the parallel.platc waveguide formed by eonductors covering the Show that the eigenfunctions, normalited on a per unit width basis, arc 1J - 0 and y - b planes.
+,. _ ..L
Vb v'2b . nw-y +... - --,mn< b +.' = where
V2b nrll --'0'b n<
1, 2, 3, . . . . Consider an 2:·directed current element II at the point z', y', z' in II. rectangular waveguide (Fig. 2-16). Show that the field is given by formulas of Table 8-1 where 'It's are given by Eqa. (8--34) and, for 11., m 'po! 0, 11. ..
8-2.
v....... - ~f••
v....• - -
..Jif.. .
where
v(mbP + (nap cos m..x' sin flry' -7_10-.'1 f ... _ [1(Z) 0 ... fllb'+na' abe and, for m - 0,
MICROWAVE NETWORKS
441
8-S. For the general cylindrical waveguide (Fig. 8-1), show that the titne-average electric energy per unit length of guide is
and the time-average magnetic energy per unit length of guide is
Note that these are just the sum of the energies in each mode alone. 8-4. Let the T equivalent circuit of Fig. 8-1Ob represent a section of waveguide of length I, propagation constantjJ3, and characteristic impedance Zoo Show that
Z. - -jZo esc fJl z. - jZ. tan 81/2 8-15. Using the usual perturbational method, show tbat, for general cylindrical waveguidCil, the attenuation constant due to eonduct
1'" k ¢('''-)' 'an --
t
dl
2"fl" for TM modes, and
1 III ~ [,{. a.'" 2';i; j ('''')' iff dt
,{. (+"Pdt ] + "'. flt "y
for TE modes. <JI. denotes inVin.sic resistance of tbe metal walls, "intrinsic impedance of the dieleetric, and the other symbols correspond to their usage in Table 8-1. 8-6. Use the above formulas to determine the attenuation in rectangular waveguides (Prob. 4-4) and-in circular waveguides (prob. 5-9). 8-7. Consider a one-port network, and dc6.ne the reflection coefficient r - V~/Vf. Show that, for Z. real,
80.. and
~ ..
(1 1
lrtt)IVflt/Z.
- 'W. - _lV l 1m (r)/z.
•
f t
Bence, in a BOurce-free network, Wit S 1, and, at resonance, r is real. 8-8. Derive Eqs. (8-72). 8-9. Let the characteristic impedances of ports (1) and (2) of Fig. 8-7 bo normalized to unity. Show that the transmission matrix IT] is related to the impedance matrix [zl by
2T Il
-
2T 1t
-
2Tu -
2T.. -
1111
+ -I
(1 - ,III)(:U - 1)
'" I -Zu + - (1 + ZII)(ZU - 1) 1 '" 1Iu + - (1 - Z")(:II + 1) '" + -I (1 + + 1) '" -Zll
ZU)(:II
Show that in the loss-free case Til - T:~ and Tit - 7';'1'
442
TIJdE-liARMONlC ELECI'ROHAQNETlC FIELDS
8-10. Add a magnetic current. .beet M. coineident. with the electric current. sheet.]. of Fig. 8-11. Det.ennioe M. and ]. such t.hat. they are & uni~t.ional dominan~ mode lIOurce, sending waves in the +~ direction only. Determine the Idf-reaetioa of this source in the presenu. of the magnetic conductor terminatioa t.he euide. 8-11. Derive Eq. (8-96). 8-12. Consider the centered capacitive poet in a rectangular w&velUide. llhown in Fig. 8-36. Show that t.he equivalent. nctwork parameters &nl
B.
Y,
Y, .. The approximations are good for die
14 -
..fdl 2011.,
< 0.3 And at>.. < 0.2.
,
T
f ,
Side view
End view
Equivalent circuit
Flo. 8-36. Centered capacitive post in a rectangular waveguide.
8-13. Conaidu the inductive diaphragm of Fi&. 8-23. aperture by
Approrimating B, in the
ehow that Eq. (8-122) is a crude variationallOlution (or the shunt IU8Oeptance. 8-If. The inductive diaphragm (Fig. 8-23) baa boundaries cylindrical to r. The incident. mode ill TM to 1/; hence, the entire field must be TM to Jj'. Expreee t.he 6eId aa H - V X uri- where
• tj;-
'\' L
.-1
a '''··
A .IID . .""
In the aperture, tangential E must be of the form
Show that
is a variational (onnu1& (Of the ahunt IU8Ceptance. Note that it gives upper bouads to -B/Y.. Problem 8-J3 is the special cu.ej(%) - sin (77./e). 8-16. Consider the indlJt:t.ive diaphragm (Fig. 8-23) and the variational fonnulaill terms of obstacle current {Eq. (8-107»). On the diaphragm, the t:urrent is of tbeform
].
-...
(.)
443
ltIICROW AVE NETWORKS
Show that
• Y.
a
[!.c [!.,. a d:l:
/:12 v(n/2)'1 'I;'
- 11 - 2>.
0
(4/X)'
c g(x) Sin
TZ
•
,,(x) sm
n.z
4 en
J'
]'
is the variationallormulo.lor lower bounds to -B/Y•. 8·16. Show that the shunt susceptance of the capacitive diaphragm of Fig. 8-37 is given by the same formula 88 applies to Fig. 8-228.
FIG. 8-37. A capacitive diaphragm (metaJ shown dashed).
8-17. Consider the capacitive junction of Fig. 8-38. Show that the parameters of the equivalent circuit are B+ 4b+ ...c - - -log CIlCYo ~ 2b+ B4b...c --I,.,~ Yo ""' 2b-
-
n
b-
l
- b' -
Use the approximation of Eq. (8-120).
-1-- {~;;I---- --j Side view
End view
Equivalent circuit
FlO. 8-38. A capacitive junction. 8·18. Considc{ the waveguide junction of Fig. 8-24a and the equivalcnt circuit of Fig. 8-25. Show that, Analogous t-o Eqs. (8-138),
Y,jBand n l
-
101/1,t. The mode current.e arc given by
1, -
ff H," !i,do
where H,+ and H,- denote tangential H on the +z and -z sides of the junction, retlpcct.ively. Variational formul!l8 are obtained by M5uming H.+ IUld H,- subject to the restriction H, + - H,- in the aperture.
444
TIME-HARMONIC ELECTROMAGNETIC FIELDS
8-19. Let 0#(Z,1I) - !(P,.) be a 8OIution to the two-dimensional source-free Helmholtz equation p < a. Prove that
where
,i..e is
an operator defined by . D
Sin
,.
1 •
COllD-..,....--
-..,.....-
}k
ax
}k
and ,i.D.J-(O) means e'·D.p(z,y) evaluated at value theorem.
:t -
ax
0, II - O.
This is a. kind of mean-
8-20. Consider the eoa'll: to waveguide feed of Fig. 4-ZO. Let d denote the diameter of the coa.xialstub, and let a «>.. Show that, for the equivalent cireuit of Fig. 8-26b, 2a
1 "r<: n""l),nn"b a "l'kd X AI -,,-)og• 2
where 'Y ... 1.781. 8-21. Let the rcctangular aperture of Fig. 8-29 be thin (b« A) and of resonant lengt.h (0 ... ),,/2). Show that
Him: Usc the duality concept of Prob. 7-43 and the approximat.ioD8 of Prob. 7-39.
Note that the aperture radiates only into half-space. 8-22. Figure 8-39 represents a parallel-plate transmi$8ion line radiating through & slot into balf.space. Let Fig. 8-28b represent the equivalent circuit, and evaluate the parameters, using the aperture admit.tance of Fig. 4-22.
•
-.
b ----'----,~
FlO. 8-39. A parallel. plate t.ransmi88ion line radiating into half~pace.
445
MICROWAVE NETWORKS
8-23. Figure 8-40 represents a rectangular waveguide having aides a, b radiating into half-apace through a narrow resonant slot. Uaing the results of Frob. 8-21, show that reflectionless tranamisaioD through the slot occurs when 4
0.54 coa' (..-A/4b)
;; ~ HI - (~)']jl When blA
< 0.7, the above formul& can
be approximated by
~~~..)l-(~)· )" 3 2b The waveguide is excited in the dominant mode.
b r-rr------..,'I
FIG. 8-40. A rectangular
waveguide radiating into half·space through a resonant slot.
t1
I I
U ..
! I
J
8-24. Considcr the loop-fed rectangular cavity of Fig. 8-41. AMlumc that the loop is arnall, 80 that the current on it may be flS8umed constant. Show that the eJements of the equivalent circuit (Fig. 8-32) are given hy Eqs. (8-181) where
a, - .v;a:bC 2• SiD . T
(b').(") b sm ..- c ..-
Whcn c' «c, this reduces to ao
-/ ...
2...A SID . ( b') _r=c "-b
cv~c
where A - c'd is the area of the loop.
FlO. 8-41. A loop-fed rectangular cavity.
446
TUtE-HARMONIC ELECTROJl,lAGNETIC FJELDS
8-26. Show that the normalized mode vector for the dominant mode of the circular
cavity (Fig. s..42) ill
where
ZOI -
2.405.
o
(bl
FlO. 8-42. A circular cavity with (4) probe feed, and (6) loop feed. 8-26. Figure 8-42a represents a probe-fed circular cavity. AMume sinusoidal de. tribution of current on the probe, and show tha.t the elements of the equivalent circuit (Fig. 8--32) are given by Eqa. (8-181) where
~_ I
1
ka y;;:b 1 1 (%01)
tan(k~)J,(zo,~) 2
a
and Xu - 2.405. 8-27. Figure 8-42b represents a loop-fed circular cavity. Assume unifonn CUm!llt on the loop. and show that the e1ementll of the equivalent circuit (Fig. 8-32) are given by Eq8. (8-181) where
7- - a v;;bd1,(ZOI) J o (ZIIE) a Show that, when c "" a, this reduces to
-a. 1
~
A:tol al~
where A - (4 - c)d is the area of the loop. 8-28. Recollllider Fig. 8-41 for the case of a Bmallioop. Represent the loop by • magnctic-currcnt element Ki, according to Fig. 3-3, aDd evaluate
R- _Re{a,a} .. Kl·B [I
[I
The result i.e. the asme as the small·loop answer in Prob. 8-24. 8-29. Reconsider Fig. 8-42b by the method outlined in Frob. 8-28. Show that the result is the 8I1me 1UJ the small-loop answer of Prob. 8-26. 8~SO. Show that the normalized H mode vector for the dominant mode of the spherical cavity (Fig. 6-2) is
H - U+
O:fJ~ J,
'-va,.
(2.744!) Bin 8 a
APPENDIX A.
VEcrOR ANALYSIS
We shall normally orient rectangular (:r;,'Y,z), cylindrical (p,
+
p "'"
tP
z:
tan- t
(A-I)
1!
•
+ tit + z' t t 8 """ tan-I Vx + 1/ _ r _ y:r;'
•
+ z! tan-1 e
vpt
•
Transformations of the coordinate components or a vector among the three coordinate systems are given by A. - A" cos tP - A. sin tP - Arsln8 cos tP A,l =0:0 All SiD tP + A. cos q, "'" Arsin 8 SiD tP
+ A,eos8c08tP-
A.sintP
+ A,cos8sin tP + A. cos q,
A, "'" A r cos 8 - A, sin 8 A" "'" A.C08¢+ A,sin¢
=0:0 A,sin8+A,eos8 (A-2) A. =- - A. sin tP + A, cos cfJ A. = A.sin 8eGS ¢ A,lain 8ain tP + A,cosS - All sin 8 + A. cos B A, - A. cos 8 cos tP + A. cos 8 sin tP - A, sin 6 "'" A" cos 8 - A, sin 8
+
The coordina~unitvectors in the three systems are denoted by (u.,u.,u.), (u",u.,u.), aDd (ur,u"u.). Differential elements or volume are d, - dzdytb - pdpd~dz - "siD qd,dqd~ 447
(A-3)
TIJLE.-ElARMONIC
ELECTRO~QNET1C
FIELDS
z z
........................... r
X
z -------
Flo. A-I. Nomul coordinate orientation.
I /' Pl..../
y
differential elements of vector area. are
+ u.dzdz + u.dzdy u.P d~ dz + u. dp dz + u.p dp d~ uyr sin e de dt/J + u,r sin 6 dr d4> + U ..1' dr de
ds - u.dydz
-
(A-4)
t
and differential elements of vector length are d1 = u.dz = u.dp = u.,.. dr
+ u,dy + u.dz + ..pd~ + u.dz + u,r de + u,.r sin e d,p
(A-5)
The elementary algebraic operations are the same in all right-handed orthogonal coordinate systems. Letting (Ut,Ut,u,) denote the unit vectors and (AI,At,At) the corresponding vector components, we have addition defined by A
+B
= u.{A.
+ B.) + u,{A, + B,) + u,(A. + B.)
(A-6)
scalar multiplication defined by A· B = A IB I
+ AtB t + AlB,
(A-7)
and vector multiplication defined by (A-8)
The above formula is a determinant, to be expanded in the usual manner. The differential operators that we have occasion to use are the gradient (vw), divergence (V· A), curl (V X A), and Laplacian (Vito). In rectangular coordinates we can think of del (v) as the vector operator (A-g)
449
¥ECI'OR ANALYSIS
and the various operations are
U.
Ur
u.
VXA=!..
d
d
(A-lO)
ay iiz A. A. A.
()x
V'w ~ d'w ax l
+ d'w + d'w alii
az l
In cylindrical coordinates we have
In spherical coordinates we have law I aw + U.-+ ur.SID - .0-a¢ r ao la (r S A) 1 a (A ,smO . ) +-.-1 dA. • +-.-v, A =--
vw
=
iiw
u.iir T10T
T SID 0 00
1 [d ( . ) dA.]
r 510 0 o¢
VXA=u.-.- A.sm8 - r 8m e ae of/>
I. dA. + U,-T1 [ - - -a (rA.) ] 8m8 o¢ ar
Vlw ==
.!:. ~ (TS ()w)
,1
ar
(A-l2)
+u.-r1 [a-iJr( r A . )aA.] -a8
1 ~ (Sin 0 iJw) + l 1 l alvJ ar + ,1 sin 0 as ao r sin 0 a¢1
A number of useful vector identities, which are independent of the choice of coordinate system, are as follows. For addition and multiplica-
450
TIM&-HA1UtONlC ELECTROMAGNETIC FlELDS
tiOD we have
A' - A· A
IAI'
= A·A'
A+B=B+A A·B - B·A A
x
B
~
-B
x
A
(A-13)
(A + B) . C = A· C + B . C (A + B) X C ~ A x C + B X C A·BxC=B·CXA-C·AxB A X (B X C) = (A· C)B - (A· B)C For differentiation we have
V(. + w) ~ V. + Vw V . (A + B) = V • A + V . B V X (A + B) = V X A + V X B v(vw) - • Vw + w v. V· (wA) = wV . A + A· Vw V X (wA) - wV X A - A X Vw V . (A X B) - B . V X A - A . V X B V'A ~ V(V· A) - V X V X A v X (v Vto) "'" Vv X Vw v X Vw = 0 V·VxA-O
(A-14)
For integration we have
JJJ V·Ad< - effi A . ds JJ V X A·ds - ¢A.dl JJJ V X A dr - - effi A X ds JJJ Vwdr = effiwds JJ n X vwd. = ¢wdl
(A-IS)
Finally, we have the Helmholtz identity 4rA -
-V
'fJ.(y Irv'·Arl
dr'
+V
X
f'(yv' X A
J. 1r
r[dr'
(A-16)
valid if A is well-behaved in all space and vanishes at least as rapidly as ,.-1 at infinity.
APPENDIX B
COMPLEX PERMITTlVITIES
The following is a. table of relative a-c capacitivities dielectric loss factors E~' where ~, "
"~
= -
EO
= -EOE' -
. E"
J-
Eo
= E,,
E~
and relative
. 11
- JE,
is the relative complex permittivity. The measurements, along with many others, were reported in II Tables of Dielectric Materials" (vol. IV, Mass. Inst. Technol., Research Lab. Insulation, Tech. Rept.). They also appear in Part V of U Dielectric Materials and Applications," Technology Press, M.LT., Cambridge, Mass., 1954.
Frequency, cycles per second
Material
Amber (fOSflil resin) ............
Bakelite (no filler) .............
Beeswax (white) ......... _.....
TOO
,
25
" 10";' ,
24
" lO'e:.' ,
23
Clay soil (dry) ................ Ethyl alcohol (absolute) ...... ..
Fibergla.a BK 174 (laminated) . .
lead~barium .............
25
Glass,
10'
a x 10'
3 X 10'
2.7 34
2.7 49
2.7
2.7 116
2.65
84
148
2.65 180
.... ....
2.0 223
2.6 234
8.2 1100
7.15
6.5 410
5.9 330
5.4 320
4.9 360
4.4 340
.... ....
3.64 190
3.52
585
2.65
....
130
2.43
2.41
2.39
2.35
165
145
....
2.35
205
120
113
,
2.17
2.17
2.17
2.17
2.17
2.1?'
0.9
1
2.17 6
2.17
17
2.17 1
2.17
130
0
3
8
35
4.73 570
3.94
3.27
2.79
2.57
280
170
.... ....
2.38 48
2.16
390
2.44 98
2.27
470
34
28
. ... ....
... . ....
.... . ...
24.5
24.1
23.7
22.3
220
80
150
600
6.5 165
1.7 10
9.8 255
7.2 115
5.9 52
5.3 24
5.0 17
4.8
4.M
4.40
4.37
365
12.5
10
13
16
,,
14.2
lOtf~'
-,,,
tOtO
470
, ... . " lO'f~' ... .
24 25
10'
2.48
" lO'e:.'
25
Glass, pho8phate .. ............ (2 per cent iron oxide)
10'
2.56 680
,
25
10'
2.63 310
10";'
::;
10'
360
,
~
10'
10'f;'
"
Carbon tetrachloride . .......... 25
10'
5.25
5.25
5.25
5.25
5.25
115
95
85
80
75
5.25 85
5.24 105
6.23 130
6.17
10j~'
240
5.00 210
,,
6.78
6.77
6.76
6.75
6.73
6.72
6.70
6.69
160
120
100
65
85
95
115
130
.... ....
6.64 470
lO'f~'
,
Gut.ta-.percha . ................
Loamy soil (dry) ... ...........
Lucite RM-1l9 ................
Myca.lex 400 (micA, gl!U1S) . .....
~
25 25
- 23 25
Neoprene compound .......... (38 per cent ON)
24
Nylon 66 .....................
25
Paper {Royalgrcy) .... ........
Paraffin 132 0 ASTM ...........
25 25
81 Plexiglas . ....................
Polyethylene (pure) ........ ....
27 24
,, 10·':" ,, ,,
2.61
2.60
2.47
2.45
2.40
2.38
s<
2.53 105
2.50
10
2.58 23
2.55
13
200
300
270
145
120
3.06 2100
2.83 1400
2.69 950
2.60 760
2.53 460
2.48
.... ....
2.47
160
2.44 27
2.44
104t~'
2.84 1250
2.75
2.68
865
.so
2.63 360
2.60 260
2.58
10"-:'
3.20 2000
175
.... ....
2.57 126
2.57 82
7.47
7.45
7.39
7.38
105
95
95
, ... ... .
. ... ....
....
140
7.42 120
7.40
220
. .. .
7.12 235
6.70 1070
6.60 730
6.54 750
6.47
6.26 2400
5.54 6600
4.5 4050
4.24 2700
4.00 1350
4.00 1050
3.88 560
3.75
3.60
3.33
840
860
3.24 700
3.16 660
.... ....
3.03
725
3.45 880
3.30
3.29
3.10
2.86 1600
2.77
2.75
2.70
2.62
620
2.99 1150
1600
1600
1500
1050 2.24 5
,
" 10'':'' ,
" 10'':' ,
" 10":' ,
970
360
34
300
100
250
3.22 360
2.25
2.25
2.25
2.25
2.25
2.~5
2.25
5
5
5
5
5
6
6
.... ....
2.25 4.5
2.02 10
2.02 2.4
2.02 1
2.02
2.02
2.02
4
6
.... ... .
. ... ....
2.00
4
" 10·~'
3.40 2050
3.12 1450
2.95
2.84 570
2.76
2.71 270
....
885
.. , .
2.66 165
2.60 150
2.59 175
2.25 11
2.25
2.25 7
2.25 11
2.25
.. , .
10'~'
.... ....
2.25 7
2.25 9
" 10";' , , lOt,;' ,
" lOt,;'
,
,,
7
385
•
2.25 7
... .
lOA
Frequency, cyclee per lCCond
Material
Polystyrene Cahoot 8tock) ....... Porcelain, "'et process ..........
Porcelain, dry process ..........
Pyranol 1478..........•.......
T'O 25 25 25 26
~
10'
10'
10'
" 10<.:'
2.56 1.3
2.56 1.3
" 10'':'
•
6.47
e.24
1800
1100
•
5.50 1200
5.36
•
" 10'';'
....
,• ,• ,• 10'; ,• 10'';' ,, 10'';' ,• 10 1,:'
QuartJ, ruled .........•....... Rcaio No. 90S.................
Rubber, pale crepe (Revea) ....
Sandy soil (dry) ...............
Sealing w~ (Red Empress).. ..
-
25 25 25 25 25
750
lO'~'
101~:'
10 1t
9
2." 8.5
2.fi4. 11
5.80 780
6.75 805
.... ....
5.51
1i-04
5,04.
5.02
350
390
'90
.. , . ....
4.74 740
ID-
10'
10'
10'
2.56 1.3
2.56
2.56 1.8
2.56 5
2.55 3
2."
1.3
6.08 800
6.98
5.87
630
530
6.82 670
5.23 550
5.14
440
5.08 380
--
850
'.53 64
4.53
'.53
'.53
'.53
9
23
9
55
.... ....
'.50
G80
1700
3.80 8800
3.78
3.78
3.78
.... ....
3.78 2.3
3.78
23
8.78 7.5
3.78
28
S.78 15
3.78
32 3.25 3500
2.94 1450
2.80 770
2.72
2.64 300
2.61
.50
2.58 215
.... ....
2.54 160
2.53 145
2.4 67
2.4 43
2.' 34
2.' 34
2.' 43
2.4
2.' 120
...
2.15
77
3.42 6700
2.91 2300
2.75 940
2.65 530
2.59
2." 410
.... ....
2." 250
2. .55 160
3.68
3.62 530
3.40
3.32
3.2 380
.... ....
3.0<1
260
3.29 260
3.27
340
920 •
a x 10'
ax
10'
,
,
440
•
•
•
240
330 •
, 0
•••
•
65
380
2.53 92
Shellac, natural XL" .......... (3.6 per cent wax)
28
10'';' 70
Styrofoam 103.7 ............... Sulfur, lublimed ............ , ..
,,
25
25
,,
IO'.~'
,, 10',:' ,, ,, ,,
lO'f~
Tellon ........... , ... , ........
22
lO'f~'
I:
Vaaelioe .....................
25 80
Wflter ...•.......•....•.......
1.5
3,66 82.
3.47
3.26
3.10 '
1I00
1I50
930
....
2.86 730
6.50 6800
6,65 4850
tLIO
4.60 2100
4.33 1700
4.00 2200
3.80 2700
.... ....
3.45
3300
1.03 2
1.03 1
1.03 1
1.03 1
1.03
1.03
2
. ... .. , .
1.03 1
3.69
3.69
3.69 8
3.69 8
8
. ... ....
3.62
8
3.69 8
3.69
1I
.... · - .. .... ....
1.03
2
1.5
3,58 6,5
2.1 11
2,1 7
2.1 7
2.1 7
2.1 4
2,1 4
2.1 4
2,1 3
2,1 3
2,08 8
.... ....
2.16 14
2.18 22
. ....
2.10
2.10
19
46
80.6 2500
38 3000
2500
...
2.16
2.16
2.16
2.16
2.16
2.16
6,'
4,3
4
2
2
7
9
,,
2.10
2.10
2.10
7.6
2
2.10 2
2.10
34
.... ....
....
... . ... .
....
....
....
87.0
87.0
16.
87 17
87 61
86.6
1650
. .. ....
.... ....
.... ....
78.2
78.2 310
78.2
78 39
77.r;
3100
.25
76.1 1200
55 3000
.. , . ... .
....
.... ....
.., . ....
68.2 490
·... ....
68 63
67.6 600
2200
.... .
.... I.... I I I
58 18
57
66.5 310
1400
10'.:'
,, ,
"
,, 10 t:' ,, 10 t:' f
85
3.76 480
2.16
101.:' 55
3.81 280
10'.:'
10 1.:'
25
....
3.86 200
f
,
'"
....
....
•.••
....
58
7200
2
36 . ...
....
58
58
720
73
·-
..
'"
280
42
60
04
APPENDIX C
FOURIER SERIES AND INTEGRALS
A periodic function f(x) with period a and aatisfying the Dirichlet conditions can be expanded in a Fourier series • f(x) ... ~o + cos + b" sin (0-1)
..2:, [a. e:1l" x)
where
a" = b.. =
~ ~G f(x) cos J, ~ foG f(x) sin
e:1l" x)]
(2:'" x) d. (2:'" x) dx
(0-2)
Such a. series converges to f(x) at ench point of continuity and to the mid-point of each discontinuity. Also, a finite Fourier series (n ~ N) is a least-mean-square error approximation to l(x). Alternatively, the Fourier series can be writ.ten as •
L:
fCZ) =
" where
.. -.
C..ei(Z....
fo" f(x)e-i(2....
e.. ",. -1 a ,
/G)~
'.)~
dx
(C-3)
(0-4)
A comparison of Eq. (C-l) with Eq. (e-3) reveals that 2c. = a.. - jb..
(Q.,I)
Equation (C-1) is called the trigonometric form, and Eq. (0-3) the exponential farm of the Fourier series. Now consider a nonperiodic function, as represented by Fig. 0-1a. In a given interval, say 0 < x < a, the fUDction caD be represented by Eq. (e-l). However, outside the given interval, the series does not equal f(x) , but instead the series gives a periodic extension of l(x) , as represented by Fig. e-lb. Moreover, we can represent f(x) in the interval o < x < a in terms of a Fourier series of arbitrary period b ~ a, but the series will not be unique until we specify the manner of extending the function beyond x = a. In particularJ if we choose a period 2a and take 456
457
FOURIER SERIES AND INTEGRALS
-. -
-.....
•
o
2•
(Ol
--.
2• •
~-_.
(d)
Flo_ C-1. (0) A function can be repreeented in the interval 0 < :I: < 0 hy (b) a "com· plete Fourier &eries, (e) a Fourier cosine series, a.nd (d) &. Fourier sine aeries. It
the even extension of f(x) from a to 2a, as shown in Fig. C-lc, we have the Fourier cosine series f(.) -
where
A.
=
~. + ~ foil
•
L:
.-,
(':>) (n; x)
A. cos
f(x) cos
lb;
(C-6) (C-7)
Similarly, if we choose n. peciod 2a and take the odd extension of f(x) from a to 2a, as shown in Fig. C-ld, we have the Fourier sine series
f(.) where
B.
~
..L:,
=
1 [. f(x) a}o
•
B. sin
(n; .) (nra )
sin
Z
(C-6)
dz
(0-9)
458
TIME-HARMONIC ELECTROMAGNETIC FIELDS
The representation of Eq. (C-6) converges to f(x) on the closed interval ::5 a, while Eq. (C-8) converges to f(x) on the open interval o < x < a. At x = 0 and x = a, Eq. (C-8) converges to zero, which is the mid-point of the discontinuity in the extended function (see Fig.
o ::5 x
O-Id).
A function f(x) can also be represented as a superposition of sinusoidal functions in an infinite interval, say - OQ < X < 00. In this case, the summation must be replnccd by an integration, and we have
where
ff-
f(x) - 21r
_.
lew) -
f(x),-;" dx
l(w),;" dw
(0-10) (0-11)
The J(w) is called the Fourier tram/arm of f(x). Equation (C-ll) is called the direct transformation, and Eq. (C-1O) is called the inverse trans/ormation. Sufficient conditions on I(x) for lew) to exist arc f_-_lf(x)ldx
<~
(C-12)
and f(x) satisfies the Dirichlet conditions. The inversion {Eq. (C-IO}J then converges to I(x) at all points of continuity and to the mid-point of points of discontinuity. Fourier integrals corresponding to the trigo-Dometrio series of Eq. (C-l) also exist, but we shall not consider them here. A useful relationship between the Fourier coefficients all, bll , 0" and the integral of If(xW over its period, known as Parseval's theorem, is 1
(0
aJ,
•
If(x)I'dx -
1 '\'
la.I' + 2 L..; (la.I' + Ib.I')
.-,
(0-13)
This is readily proved by substituting for f(x) in tbe left-hand term from Eq. (C-l) or (C-3) and interchanging summation and integration. All cross-product terms drop out because of orthogonality. Similarly, for the Fourier integral, we have a Parseval theorem
f__- If(x)I'dx - 2r f-__ II(w)I'dw
(0-14)
1 fdx - 2r _.!(wW(w) dw f_.!(x)g'(x)
(0-15)
1
or, more generally.
459
FOURIER SERIES AND INTEGRALS
The proof of Eq. (0-15) is summarized as follows
f.
f(x)u*(x) dx
= =
f. [2~ f. .!. f· [f·
J(w)e;·· dX] U*(x) dx
_. J(w)
2...
_. u*(x)e;·· dX] dw
A similar generalization of Eq. (C-13) can also be given. Finally, the impulse function (delta function) is useful in Fourier analysis. By definition, the impulse function ~(x) satisfies the integral equation
J:
f(x) o(x - x') dx
I
= ~(X')
a < x' < b x'b
for all f(x). It is evident that ~(x) is not a function in the usual sense, but its use can be justified by rigorous means.' It is helpful to visualize the impulse function as c
c
--<x<2 2 c
Ixl > 2 where c is an appropriately small number. Such a picture gives an intuitive justification of Eq. (0-16). From Eqs. (0-11) and (0-16), it follows that a(w)
=
1--. o(x)e-;·· dx = 1
that is, the transform of the 6 function contains all frequencies in equal amounts. The inverse of Eq. (0-18) is -1 211"
f·
_.
(C-19)
elv'dw = o(x)
which is a particularly simple and useful result. primarily as shorthand notation for Eq. (0-17).
Our use of
~(x)
will be
1 L. Schwart.z, j'TMorie deB dist.ribut.ions,'· Actualitie' ,cienti~uelJ et indu,tritlk,. nos. 1091 and 1122, Hermano' & Cie., Paris, 1950-1951.
APPENDIX D
BESSEL FUNcrroNs
Bessel's equation of order v is
y x .!!- (x a )
ax
ax + (x
2 -
v')y
=
0
(D-I)
Solutions may be obtained by the method of Frobenius, the result being • '\'
J.(x) "'"
(-I)-(x)'-"
Lt m!(m + v)! 2 ",+. _.0 2
(D-2)
where ml = rem + 1) in general. As long as v ia not nn integer, these are two independent solutions to Bessel's equation. However, when v = n is an integer, we have (D-3)
and Eqs. (D-2) a.re no longer two independent solutions. In this case a second solution may be obtained by a limiting procedure. It is conventional to define another solution to Bessel's equation as N •() x ""'
J.(x) co, "" - J_.(x)
.
am V1r
(D-4)
where, for integral v "" n, N.(x) - lim N.(x)
(D-5)
~.
This limit exists and esta.blishes a second solution to Bessel's equation of integra.lorder. The J.(x) are called Bessel functions of the first kind of order v, and the N.(x) are called Bessel functions of the second kind of order v. 460
461
BESSEL FUNCTIONS
Of particular interest are integral orders of Bessel functions. Eq. (D-2) and (D-5), one can determine
•
From
(z)'-
~ (-1)J,(z) - ~ (ml)' 2
.-. 'z
•
+;;2 ~ ~
2
.-.
N .(z) - ;; log "2 J .(z)
(-I)-H (ml)'
(0-6)
(z)' 2 " ~(m)
for the zero-order functions, and
.
J.(z) =
~ (-1)~ m!(m + 10)1
(z)'-" 2
21 '"2
12:
•••
_-l
J .() N .() x=-ogx--
...
'II"
•
. (X)'+'-
.-
ml
1 ~ (-1)- ;; ~ m!(m + 10) I
for n > 0, where
.-.
log
(2)--'-
(10 - m - I)! -
2
(0-7)
x
[~(m)
+
~(m
+ 10)]
(Euler's constant)
'Y = 0.5772
, - 1.781
~(m)
- 1
(0-8)
+ M + M + ... +.!. m
The Bessel functions have been tabulated over an extensive range of orders and arguments, and tables are available. Figure D-l shows 1.0
,\j.
0.8
J,
0.6
V\ i')/r-..-
0.4
/
02
o
1/
/ ./
/ 1\
\
-0. 4
o
1\ 1'\ 1/ h
K \
-0. 2
-0.6
J,
2
I
lX.. )<
I\. ) 4
1/ ~ 1/
~ l'6
8
"\ 1/ l\-
I><
/
K
I/<
17 !'J f' 17 rs:; V 1"!-10
Flo. 0-1. Bes8eJ fUDctioDl of the firtt kind.
12
14
16
462
TIlLE-HARMONIC ELECTROMAGNETIC FI.ELDS
0.6
No
0.4
/
02
o -0.2
~,
N,
J\ /'
\
1/ X
/ / I
-0.4
/ 1\
X. \
N,
h ., ---1\ )\ V IV ~
/ '\.
t----' "-
./
1/
-0.6
'"
/ )<
'"
/ '\. ;X ''j'\, ')< ~
V
~
/
-0.8
/ 1/
-1.0 -1.2
~
o
2
6
4
10
8
12
14
16
Fto. 0..2. Bes8el functiona of the llCCond kind.
curves for the lowesk>rder functions of the first kind, and Fig. D-2 shows those for the second kind. For small arguments, we have from the series Jo(x) __ 1 _0
(D-9)
2 'Y" No{:z:) - -log-2 _0 r and, (or v
> 0,
J.(x) -;::t
~ (~y
N.(x) --+ _ (v - 1)1 (~)' _0
'Ir
(D-lO)
:z:
provided He (tI) > O. For large arguments, asymptotic series exist, the leading terms of which are
cos (x _!4 _ or) 2 N.(x) _ . [2 ,in (x _! _ or) _.. v;Z 4 2 /2x _. '\j...
J.(x) --+
(D-ll)
provided Iphase (x)1 < r. For the expression of wave phenomena, it is convenient to define linear combinations of the Bessel functions
+
H.Ol(X) ~ J.(x) jN.(x) H.Ol(X) - J.(x) - jN.(x)
called Hankel functions of the first and second kinds.
(D-12)
Small-argument
BESSEL FUNCI10NS
463
and large-argument fonnulas are obtained from those for J. and N.. particular, the large-argument formulas become
In
(D-Ia)
whicb place into evidence the wave character of the Hankel functions. Derivative formulas and recurrence formulas can be obtained by differentiation of Eqs. (0-2). Letting B.(x) denote an arbitrary solution to Bessel's equation, we have B;(x) = B_ 1
! B•
-
•
8;(x) .... -8-+ 1
(D-14)
+!
•
B•
which, in the special case v .., 0, become (D-IS)
B;(.) - -B,(.)
The difference of Eqs. (0-14) yields the recurrence formula B.(x) "'"
2(, - I)
•
8._ 1
-
B._ 2
(D-16)
which is useful for calculating 8 ..(x), n > I, from a knowledge of Bo(x) and 8 1(x). The Wronskian of Bessel's equation is often encountered in problem solving. This is (D-17)
from which Wronskians for other pairs of solutions cnn be easily obtained. When x = ju is imaginary, modi.fied Bessel functions of the first and second kind can be defined M I.(u)
~
j'J.(-ju)
K.(u) - ; (-J)o+'H."'( -ju)
(D-IS)
These are real functions for real u. General formulas for I. and K. can be obtained from the' corresponding formulas for J. and H.(2). Figure 0-3 shows curves of the zero- and 6rst-order modified Bessel functions. The large-argument formulas, obtained from Eqa. (D-11) and (D-12),
e"
I.(u)~ _,_ ........ V 2'11"'U
K.(u)
~ /72 . --V2U T
(D-19)
464
TUlE-BARMONIC ELECTROMAGNETIC PlELDS
, • • 2
/
'I
,;/ "
.\-K.
\''''
V
/
<:-
I~
o
•
Fro. 0-3. functions.
•
2
Modified
illustrate the evanescent character of the modified Bessel functions. Derivative fonnulas and recurrence formulas ca.n be readily obtained {rom Eq., (D-i4) to (D-t6), Bessel functions of order n + J.i' are used in the solution of the Helmholtz equation in spherical coordinates. In scalar-wave problems, it is conventional to define spherical Bessel functions as
Bessel
(D-20)
The b. are given the Damc and letter as the corresponding B..+ H . (For example, j .. is the spherical Bessel function oC the first kind, h,.ll) is the spherical Hankel function of the second kind, etc.) In a-c electromagnetic field problems, it is convenient to define the alternative spherical Bessel functions
(O-2i) where 8. is given the same name and symbol as the corresponding B-+ K . The various fonnulas (or b. and B. can be obtained from the corresponding formulas for B..+~. or particular interest is the fact that asymptotic expansions for B..Hi become exact, giving j .(x) _ C.(x) sin (x -
$ .(x) - D.(x) sin (x -
n;) + n;) _
D.(x) cos (x _ C.(x) co. (x _
n;) n;)
(D-22)
n.",(x) - j-·ID.(x) - jC.(x)]"· n.,n(x) - j·ID.(x) + jC.(x)jr'·
:-s:.. where
C ( ) • x -
'\'
'-<
(-i)o(n
(2m) I(n
+ 2m)! 2m) !(2x)"
0-0
D.(x) -
'02:<'-' 0-0
(2m
(D-23)
+
+
(-i)o(n 2m i)! i) I(n 2m 1) 1(2:<)·.... •
+
Note tbat which is of interest in radiation problems.
(D-24)
APPENDIX E
LEGENDRE FUNCTIONS
The associated Legendre equation is
Si~ O:O(siO O~~) + [V(V + I) -
Si::
0] y -
0
(E-I)
This can be put into another common (orm by using the substitution
=- cosS
u
in Eq. (E-l).
(E-2)
The result is
(I - u.) d'y _ 2u dy du' du
+ [.(V + 1) _
1
m'
u'
] y _ 0
(E-3)
When m =- 0, the associated Legendre equation reduces to the ordinary Legendre equation
(I - u.) d'y _ 2u dy
du'
du
+ 0(0 + I)y ~ 0
(E-4)
We shall first consider solutions to this special case and later generalize to the associated Legendre equation. s ~ 1r; so we shall be interested In the spherical coordinate system, 0 in solutions over the range -1 ~ u. S 1. Inparticu]ar,forI1- ul < 2, the Legendre function of the first kind caD be expressed as
s:
N
P ( ) _ " (-1)'(v •u L. (m!)'(v ,-0 _
+ m)1 (I m)!
•
si~.. ~ •
.... +1
-
2
u)'
(m - 1-(m!)' v)!(". + v)! (I -2 u)'
(E-5)
where N is the nearest integer N ~ v. As long as v is not an integer, P.(u) and P.( -1£) are two independent solutions to Legendre's equation [Eq. (E-4)]. If. - n is ao inuge" Eq. (E-5) becomes a liniu series called the Legtndre polynomial of degree n. In t-his case,
...
P.(-u) - (-I)·P.(u)
~)
466
TIME-HARMONIC ELECTROMAONE'l'IC FIELDS
and we no longer have two independent solutions. Another solution, called the Legendre function of the second kind, is defined as = !p,(U) cos", - P.(-U) 2 . sm '"
Q, (u ) When v
=
(E-7)
n is an integer, the limit
Q.(u)
~
(E-S)
Hm Q.(u) ~.
exists and defines a second solution to Legendre's equation. The Legendre polynomials arc of particular interest, because these are the only solutions finite over the entire range 0 ~ 0 ::s:; 'If. In this case, only the first summa.tion in Eq. (E-5) remains, which caD be rearranged to M
l:
p u) .( -
(-I)-(2n - 2m)! 2'm!(n m)!(n 2m)!
(E-9)
U ..- 2M
moO
where M = n/2 or (11. - 1)/2, whichever is an integer, An alternative, and sometimes more convenient, expression for the Legendre polynomials is given by Rodrigues' formula I d' (u' - I)' 2"n! du"
P (u) - "
(E-IO)
Some of the lower-degree polynomials are P,(u) - I P,(u) - u P,(u) - }
+
(E-II)
or, in terms of 6, Po(cos 8) - 1 Pl(COS 6) = cos 6 P,(cos 8) ~ ~(3 cos 28 I) P,(cos 8) - )i(5 cos 38 + 3 cos 8) P ,(cos 8) - )i4(35 cos 48 20 cos 28
+
+
(E-12)
+ 9)
Figure E-1 shows curves of the Legendre polynomials plotted against 8. The Legendre functions of the second kind for integral II "" n are infinite at 8 - 0 and 8 = r, or at u - ± 1. They can be expressed as Q.(u) - P.(u) [ }
~~-
¢(n)]
•
+ \' '-< m_'
(-I)-(n (m!)'(n
+ m)! ¢(m) (I m)!
- u)2
(E-13)
467
LEGENDRE FUNCTIONS
1.0
0.8
~~
\\
0.6
.\
0.4
~ 2
~\3\
0.2
/ 1/
1\
'r-.
V
j
'\ 1\ _/2
4
o
1/7
n-O
\/ / /\ /
\ \I 1 \ 1\/ ' i\ \ 1\V 11\ 1 \ "J '-... V \ \
- 0.2 - 0.4 - 0.6
1""-
- 0.8 - 1.0
-
~~
Fla. E-l. Legendre functions of the first kind, P.(cos 6).
where are
~(m)
is defined by Eq. (D-8).
Q.(u) Qt(u)
=
~
Some of the lower-order functions
l+u
1.£
l+u
log - Q.(u) - -log - - - 1 1-1.£ 2 1-1.£ t 3u - I log 1 + u _ 3u 4 1- u 2
(E-14)
or, in terms of 8, Qo(C08
9
8) "'" log cot '2
Q,(cos 9) -
~(3
Q1(C08
9
8) "" cos 8 log cot '2
9 co,, 9 - I) log cot 2 - % cos 9
-
1 (E-15)
Figure B-2 shows curves of these functions plotted against 8. Now consider the associated Legendre equation, Eq. (E-3). For simplicity, we first take 111. to be an integer. If Eq. (E-4) is differentiated 111. times, there results - 2u(m + 1).'£ + (n - m)(n + m + I)] d"'Y - 0 [ (I - u')..'!.. du' du du" t Letting w = (1 - U) ..l J d-y/du" in the above equation, we obtain Eq. (E-3) with y replaced by w. Hence, solutions to the associated Legendre equation are 1 I
Smythe and others omit the factor (-1)- on the right-hand side of these definitioD.l.
468
TIME-HARMONIC ELECTROMAGNETIC FIELDS
P.-(u) _ (-1)-(1 _ u')-" d.%:~u)
~
Q.-(u)
(-1)-(1 - u')-"
Note that aU p ..... (u) ... 0 for m Legendre polynomials afC
P,'(u) - -(1 - u')~ P,'(u) - -3(1 - u')~ P,'(u) - 3(1 - u')
(E-16)
d.J:~U)
> n. Some of the lower-ordcr associated P,'(u) - %(1 - u')~(1 - 5u') P,'(u) - 15(1 - u')u P,'(u) - -15(1 - u')~
(E-17)
while the P"O(u) = p .. (u) are given by Eq. (E-ll). Some of the lowcrorder associated Legendre functions of the second kind aTC Q11=
-(1-U2)~(HI0g~+:+1 u
Q,' _ -(1 _ Q2' _ (1 _
'U
u')~ (%U log 1 + u
[u
1- u
')~ n Iog 11 + u _ 'U
u')
+ 3u' - 2) 1 - u'
+ 5u (l _
(E-18)
3U'] u'p
while the Q.'(u) - Q.(u) are given by Eq. (E-14). When m is not an integer, the situation becomes even more complicated. A standard formula. for Legendre functions of the first kind, 5
4 3 2
1
\\ \\
~n_O
'\ !'-1
......
i' I'---.
r--.
/ '/y
V ...... t--.. V
-2
-3
N
1\' • \
f"."
1\
-4 -5 FIG. E-2. Legendre functioIl9 of the second kind,
Q~(C08
f).
469
LEGENDRE FUNCTIONS
valid for
11 - ul <
2, is then
(u I)"" (
sin"", p"(u)---(w-I)I -+• 'II" 'u-l
I-
u) (E-19) -vv+ll-w-"'·2
F
where F is the hypergeometric function
F(a,~,~,,) - 1+ (a For real as
11.,
(~- 1)1
I)!(P
"
'\' (a
1)1
L.
_-0
+ m)!(~ + m)! m!(~ + m)!
_H
'
(E-20)
the associa.ted Legendre function of the second kind is defined
Q "(u) _ ! p,"(u) cos (0 + w)~ - p,"( -u) , 2 sin (v + w)~
(E-21)
The solutions P ."'(11.) and P."'( -u) arc linearly independent, except when u + w is an integer. In this latter case, the limit of Eq. (E-21) provides a second solution. Perhaps the simplest way to calculate the Legendre functions is through the recurrcnce formulas. Letting L..... (u) denote an arbitrary solution to the associated Legendre equation, we have
(m - n - I)L:., + (2n + l)uL,- - (m + n)L:-,
~
0
(E-22)
"'" 0
(E-23)
A recurrence formula. in m also exists and is L.. ",+1
+
(1
2m:tp~ L",· +
for the range 111.1 < 1. which arc
(m
+ n)(n -
+ t)J..,,,,"'~1
Ma.ny formulas for derivatives also exist, some of I
L.....'(u) = '1---''-u''2 I-nuL..... - I
m
I
u' (n
+ (n + m)L:...
+ l)uL,- -
(n - m
1]
+ I)L:..]
= ~ L _ + (n + m)(n - m + I) L _. 1 - 11. 2 .. (1 - utpi .. 1
mu
ut
L"''"
(l
(E-24)
1 L"'+t u2»)4i"
If m = 0 in the last formula, we have L,'(u) - -(I - u')>>L;(u)
(E-25)
which is a useful special case. Finally, some specializations of the argument will be of interest to us.
470
TIME-HARMONIC ELECTROMAGNETIC FIELDS
At 8 = 0, that is, at u = 1, the Qnm functions are infinite and m = 0
(E-26)
m>O At 8 = 1r/2, that is, at u = 0, P nm(O)
Qnm(O) =
=
{
(_I)(n+m>l2 1 . 3 . 5 0 2 .4 .6
{~_1)(n+m+ll/£2'14·6 .3.5
(n
+m
1)
-
(n - m)
(n
+m
-
(n - m)
1)
+ meven n + m odd n + m even n + m odd n
(E-27)
Some specializations involving derivatives are
(E-28)
BWLlOGRAPHY A. Clauical Books 1. Abraha.m, A'I and R. Becker: "The Classical Theory of Electricity," Blackie &; Son, Ltd., Glasgow, 1932. 2. Heaviside, 0.: "Electromagnetic Theory," Dover Publications, New York, 1950 (reprint).
3. Jeans, J.: "Electric and Magnetic Fields," Cambridge University Press, London, 1933. 4. Maxwell, J. C.:" A Treatise on Electricity and Magnetism," Dover Publications, New York, 1954 (reprint). B. lnlrodudory Book8 1. Attwood, S.: "Electric and Magnetic Fields," 3d ed., John Wiley & Sons,
Inc., New York, 1949. 2. Booker, H. G.: uAn Approach to Electrical Science," McGraw-Hill Book Company, Inc., New York, 1959. 3. Harrington, R. F.: "Introduction to Electromagnetic Engineering," MeGraw.Hill Book Company, Inc., New York, 1958. 4. Hayt, W. H.: "Engineering Electromagnetics," McGraw-Hill Book Company, Inc., New York, 1958. 5. Kraus, J. D.: "Electromagnetics," McGraw-Hili Book Company, Inc., New York, 1953. 6. Neal, J. P.: "Electrical Engineering Fundamentals," McGraw-Hill Dook Company, Inc., New York, 1960. 7. Page, L., and N. Adams: "Principles of Electricity," D. Van Nostrand Company, Inc., Princeton, N.J., 1931. 8. Peck, E. R.: "Electricity and Magnetism," MeGraw~HilI Dook Company, Inc., New York, 1953. 9. Rogers, W. E.: "Introduction to Electric Fields," McGraw-Hill Book Company, Inc., New York, 1954. 10. Sears, F. W.: "Electricity and Magnetism," Addison-Wesley Publishing Company, Reading, Mass., 1946. 11. Seely, S.: "Introduction to Electromagnetic Fields," McGraw-Hill Book Company, Inc., New York, 1958. 12. Shedd, P. C.: "Fundamentals of Electromagnetic Waves," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1955. 13. Skilling, H. H.: "Fundamentals of Electric Waves," 2d ed., John Wiley & Sons, Inc., New York, 1948. 14. Spence, D., and R. Galbraith: "Fundamentals of Electrical Engineering," The Ronald Press Company, New York, 1955. 15. Ware, L. A.: "Elements of Electromagnetic Waves," Pitman Publishing Corporation, New York, 1949. 16. Weber, E.: "Electromagnetic Fields," John Wiley & Sons, Inc., New York, 1950.
4"
472
TIME-HARMONIC ELECTROMAGNETIC FIELDS
C. lnUr'mtdiate and Advanced Boo~
1. Jordan, E.: "Electromagnetic Waves and Radia.ting Systems," PrenticeHall, Inc., Englewood CliiTs, N.J., 1950. 2. King, R. W. P.: "Electromagnetic Engineering," McGraw-Hill Book Company, Inc., New York, 1953.
3. Mason, M., and W. Weaver: "The Electromagnetic Field," Univenity of Chicago Press, Chicago, 1929. 4. Ramo, S., and J. R. Whinnery: "Fields and Waves in Modern Radio," 2d ed., John Wiley & Sons, Inc., New York, 1953. 5. Scbelkunoff, S. A.: "Electromagnetic Waves," D. Van Nostrand Company, Inc., Princeton, N.J., 1943.
6. Smythe, W. R.: "Static and Dynamic Electricity," 2d ed., McGraw-Hill Book Company, Inc., New York, 1950. 7. Strat.ton, J. A.: "Electromagnetic Theory," McGraw-Hill Book Company, Inc., New York, 1941.
D. Books on Special Topics 1. Aharoni, J.: HAntennae," Clarendon Press, Oxford, 1946. 2. Bronwell, A., and R. E. Beam: /fTheory and Application of Microwaves," McGraw-Hill Book Company, Inc., New York, 1947. 3. Kraus, J. D.: "Antennas," McGraw-HiU Book Company, Inc., New York,
1950. 4. Lewin, L.: IIAdvanced Theory of Waveguides," Illiffc and Sons, London, 1951. 5. Marcuvitz, N.: "Waveguide Handbook'" MIT Radiation Laboratory Series, vol. 10, McGraw-Hill Book Company, Inc., New York, 1951. 6. Mentzer, J. ft.: "Scattering and Diffraction of Radio Waves," Pergamon Press, New York, 1955. 7. Montgomery, C. G., R. H. Dicke, and E. M. Purcell (eds.): "Principles of Microwave Circuits," MIT Radiation Laboratory Series, vol. 8, McGrawHill Book Company, Inc., 1948. 8. Moreno, T.: HMicrowave Transmission Design Data," Dover Publications, New York, 1958 (reprint). 9. Reich, H. J. (ed.): "Very High Frequency Techniques," Radio Research Laboratory, McGraw-Hill Book Company, Inc., New York, 1947. 10. Schclkunoff and Friis: l
INDEX Boldface numbers in parenthcsC8 refer to problems A-e phenomena, 1 Addition theorems, 232, 292 Admittance matrix, 119, 392 Admittivity, 10,23-26 Ampl!ro's law, 4 Antenna concepte, 81-85 Antenna gain, 83 maximum, 307-311 Apertures, 11 admittance of, 173, 428-431 in cavities, 436--440 in conell, 306 in plane conductors, 11, 138(17), 139(18, 19), I&H86, 366-371, 42S-431,444(21-23) in spheres, 301-303 transmission through, 366-371 in waveguides, 174, 176 in wedges, 250-254 Associated Legendre functions, 265, 468--470 Attenuation constant, 48, 66, 73, 86 of biconical guides, 313(13) of ciroular-gttides, 255(9) of guides in general, 376(19), 441(6) intrinsic, 48 of parallel~plate guides, 91(80) of rectangular guides, 86, 189(') of tranamission lines, 90(24)
Babinet's principle, 365-367 Bailin, L. L' J 249, 303, 306 Belk, A. D., 346, 348 Bessel functions, 199-203, 4~64 modified, 201, 463
Bessel functions, spherical, 265, 268,
464 zeros of, 205, 270 Bibliography, 471-472 BiconicnJ cavity, 284-28.6 BiconicnJ wa.veguide, 281-283, 313(13) Bierens de Haan, D., 194 Boundary conditions, 34, 55 Boundary-value problems, 103 Bounds, upper and lower, 335 Brewster angle, 59 Browo, G" 351
Capacitivity, 5 a-e, 24 relative, 6 Capacitor, 13, 30 Carter, P. S., 349, 351 Cavities (set Resonators) Characteristic impedance, 62, 65 of waveguides, 69, 152, 154, 385 Characteristic values, 67, 144 Chu, L, J" 278 Churchill, R. V., 231 Circuit elements, 13, 29 Circuit quantities, 3 Circular cavity, 213-216 plUtially filled, 258(23, Z') with wedge, 259(26) Circular polarization. 46, 88(8) Circular waveguides. 204-208, 389 with bafBe, 208, 255(6, 8) plUtially filled, 220, 257(20), 321, 331 Circulating waves, 208. 256(10) Circulator. 26 473
474
TIUE-BARMONIC ELECTROMAGNETIC FIELDS
Closed contour, 2 Closed surface, 2 Coated conductor, 168,219 Coaxial line, 65 junction, with cavity, 434-436, 445(24-27) with waveguide, 179, 195(~, 34), 425-428, 444(20) opening onto plane conductor, III spherical, 281 waveguide modes of, 254(5) Cohen, M. H" 362, 364, 430 Cohn, S. B., 328 Complementary antennas, 380(43) Complementary solutions, 131 Complementary structures, 136(7), 365Complex quantities, 13 constitutive relationships, 18 pcrmittivities, 18, 451-455 power, 19-23 Poynting vector, 20 Concentric spheres, 321(6, 6) Conducting cone, 279 spera-tures in, 306 current element on, 316(30) 88 waveguide, 279-281 Conducting cylinder, 232-238 and current elements, 262(39-43) Conducting sphere, 292-297 apertures in, 301-303 and current element, 298-301, 315(26) and current loop, 316(29) with dielectric coating, 315(26) Conduction current, 6, 27 Conductivity, 6 complex, 18 Conductors, 6 perfect, 34 Conical cavity, 284, 314(18) Conical waveguide, 279-281 Conjugate problems, 64 Conservation, of charge, 2, 4 of complex power, 21 of energy, 10, 11 Constitutive relationships,S complex, 18
Corrugated conductor, 170, 193(26) circular, 223 radial,219 Critical angle, 60 Crowley, T., 430 Current, 1, 7,.27, 34 in cavities, 431-434 conduction, 6, 27 near cones, 303, 316(30) near cylinders, 262(39-41) displacement, 7, 27 elements, 78-81, 287 filament (set Filament of current) near half-plane, 263(42, 43) impressed, 7, 27 loops, 93(41,42), 100, 315(28) near planes, 103, 136(12-14) reactive, 27, 28 ribbon of, 188, 260(31) sheets, 34 source, 95, 118 near spheres, 298, 315(26, 27), 316(29) surface, 33 in waveguides, 97, 106,134(1-4), 177-179,194(31,32),425-428, 440(2) near wedges, 263(") Cutler, C. C., 171 Cutoff frequency, 68, 150, 166, 169, 206,384 Cutoff wavelength, 68, 150, 206, 384 Cylinder of currents, 227, 260(30) Cylindrical coordinates, 198, 447 Cylindrical waveguides, 381-391 Cylindrical waves, 85
Degenerate modes, 48, 150, 390 Delta function, 179, 459 Depth of penetration, 53 Diamagnetism, 6 Diaphragms, 414-420, 442(13-16) Dicke, R. H., 392, 400 Dielectric, 6, 24, 451-455 Dielectric cylinders, 220, 261 (34, 36) DieJeetric loss angle, 24
INDEX
Dielectric obstacles, 362-365 Dielectric rod guide, 221, 257(21) Dielectric slab guide, 163-168, 192(22), 219 Dielectric spheres, 297 Differential scattering, 360 Dipole, 78 antenna, 81-85 in conducting wedge, 105 ncar ground plane, 104 magnetic, 259(26, 27) two-dimensional, 225 Directional coupler, 135(3) Displacement current, 7, 27 Dissipated power, 11 Dissipative current, 27 Dominant mode, 69, 75 Dominant-mode source, 402 Duality, 98--100
Echo, 355, 363 Echo area, 116, 128, 357 Echo width, 358, 359, 364 Effective value, 15 Eigeniunctions, 144,384 Eigenvalues, 67, 144, 383 Electric quantities, charge, 1, 3 current, 3, 7, 15,27 dipole, 78
475
Equivalent circuit, of obstacles in waveguides, 402 of resonant cavities, 435, 437 of spherical w&ves, 279 of transmission lines, 62 of waveguides, 386 Erdclyi, A., 245 Ether, 26 Euler's identity, 15 Evanescent field, 50, 147 Evanescent mode, 68
Faraday's law, 4 Ferromagnctism, 6, 25 Feshbach, H., 337, 432 Field coordinates, 80 Field quantities, 3 Filament of current, 34, 81, 223, 243 near cylinder, 236-238 near haU·plane, 241-242 near wedges, 238--242 Foster's reactance theorem, 396 Fourier series, 456-458 Fourier transforms, 145, 180, 458-459 Fourier·Legendre series, 275 Frank, N. H., 163 Free space, 5 Fundamental units, 1
flux, 1, 3 intensity, I, 15 scalar potential, 77 vector potential, 99, 129 Elementary wave functions, 144, 200,
266 Elliptical polarization, 46 Emde, F., 272 Energy, 9, 10, 21, 23 conservation of, 10, 11 velocity of, 42 Equation of continuity, 2 Equiphase surfaces, 39, 85 Equivalence principle, 106-110 Equivalent circuit, 62 of coax-to-waveguide feeds, 425 of microwave networks, 401
Gain, 83 antenna, maximum, 307-311 normal,309 of dipole near ground plane, 104, 137(12-14) of dipoles, 84 supergain, 309 Gauss' law, 4 Good dielectric, 24 Goubau, G., 223 Gray, M. C., 222 Grecn's functions, 120-123 tensor, 123-125 Greeo's identities, 120 modified vector analogue, 141(28) in two dimensions, 389
476
TIME-HARM,ONIC ELECTROMAGNETIC FIELDS
Green's identities, vector analogue, 121 Guide phase velocity, 68, 385 Guide wavelength, 68, 384
Hall effect, 35(2) Hankel functions, 199-203, 462 spherical, 266, 464 Harmonic functions, 144, 199
Harrington, R. F., 34, 128,309 Helmholtz equation, 38 in cylindrical coordinates, 198 in rectangular coordinates, 77 in spherical coordinates, 264 Helmholtz identity, 450 Hemispherical cavity, 284 Hildebrand, F. B., 332
Instantaneous quantity, 15 Inaulatof'li, 6 Integral equations, 125-128, 317 Intrinsic parameters, 39, 40, 48, 87 Isolator, 26 Isotropic matter, 37
Jahnke, E., 272 Junctions, coax-to-eavity, 434-436, 445(24-27) coax·to-waveguide, 179, 195(33,34), 425-428, 444(20) waveguide-.to-eavity, 436-440 waveguidc-to-waveguide, 172-177, 193(27-29), 42lH25, 443(17,18)
HOgflD, C. L,t 26 Homogeneous matter, 37 Hu, Y. Y., 352, 354, 358 Hybrid modes, 154, 158
1m operator, 15 Impedance, of apertures, 428--431 characteristic, 62, 65, 69, 152, 154, 385 of circuit elements, 29 of current loop, 93(42) input, 84 intrinsic, 39, 48, 87 of linear antenna., 82, 94(44-46), 352 matrix, 119, 392, 398 surface, 53, 371(2), 375(18) wave, 39, 55, 69, 86, 152 Impedivity, 19, 23-25 Impressed current, 7, 27 Impulse function, 179, 459 Incident field, 113 Index of refraction, 58 Induced emf metbod, 349 Induction field, 79 Induction theorem, 113-115 Inductivity, 5, 6, 25 Inductor, 13, 31 Input impedance, 84 Instantaneous phase, 85
King, R., 351 Kirchhoff's laws, 4, 12
Legendre functions, 265, 465--470 associated, 265, 468-470 LePage, W. R., 386 Levine, H., 113 Levis, C. A., 430 Linear antenna, 81-85, 94(0U--46), 349355 Linear matter, 6, 18 Linear polarization, 39, 45 Loop of current, 93(41, 42),100 in cavity, 445(24), 446(27-29) near conducting cone, 303-306 near conducting sphere, 315(28, 29) Loosely bound wave, 170 Lorentz reciprocity tbeorem, 111 Loss angle, of capacitor, 30 of dielectric, 24 of inductor, 32 magnetic, 25 Lossy dielectric, 24
Maeroscopic standpoint, 1 Magnetic quantities, conductor, 34 current, 7, 27
INDEX
Magnetic quantities, dipole, 100
flux, I, 3 intensity, I, 15 1068 angle, 25 vector potential, 77, 129 Magnetomotive foree, 3 Magnitude, 85 Marcuvits, N., 381, 389, 410, 411, 418, 420,424 Matrix impedance, 119 Maxwell's equations, 2, 18 Mentzer, J. R., 122,306 Microwave networks, 39l-402 Mksc unit8, 1 Modal expansions, 171-177,389-391 in cavitics, 431-434 Mode, 63, 68, 69, 75 Mode current, 72, 383 ~{odefunction,383
Mode patterns, 70 for circular cavity, 215 for circular guide, 2fYl for coated conductor, 169 for free space, 277 in general, 387 for rectangular cavity, 75, 157 for rectangular guide, 70, 151, 155 for slab waveguide, 168 for spherical, cavity, 272 Mode voltage, 72, 383 Modified Bessel functioJ1!, 201, 463 Monopole antenna, 138(15) Montgomery, C. G., 392, 400 Morse, P. M., 337, 371, 432 Multipoles, cylindrical, 226, 259(29) spherical, 28&-289, 314(19)
Neumann's number, 172 Nonpropagating mode, 68 Normal gain, 309 Normalization, 383, 432 Notation, 16
Obstacles in waveguides, 402-418 Ohm's law, 13
477
Oliner, A. A., 413 Orthogonality, 273, 390, 432
Papas, C. H., 113 Parallel·plate waveguide, 90(28), 189(6, 7), 440(1) and coaxial feed, 378(36-37) opening onto ground plane, 181-186 p.,ti.lIy filled, 190(12, 13), 257(18) md;ally, 209, 256(13) Paramagnetism, 6 Parseval's theorem, 182, 458 Partfally filled cavities, circular, 258(23, 2') perturbational formulas for, 321-326, 371-373 rectangular, 191(16-18), 325 spherical, 313(7, 8), 326 variational formulas (or, 331--345, 376-377 Partially 6lIed waveguides, circular, 220,257(20),321,331 perturbational formulas for, 326-331, 374-375 rectangular, 158-163, 191(1~16), 345,348 variational formulas for, 345-348, 377-378 Particular solution, 131 Pattern, mode, 70 radiation field, 83 receiving, 119 stl1nding.wave, 44 Perfect conductor, 34 Perfect dielectric, 24 Permeability, 5, 6, 18 Permittivity, 5, 18 Perturbational methoda, 73, 76, 317331, 371-377 Phase, 85 Phase constant, 48, 85 Phase velocity, 39, 40, 68, 86, 385 Phasor, 15 Physical optics method, 127 Pincherle, L., 158 Plano waves, 39, 85, 143, 145-148
~78
TIME-HARMONIC ELEcrROMAONETIC FIELDS
Polarization. of matter. 27 of waves, 39, 45, 88(8) Polarizing angle, 59 Porta, 391 Posts in waveguides, 406-411, 442(1~) Potentials, 77, 99, 129 Power, 9, 19,22 Poynting vector, 10, 20 Probes, in cavities, 434-436, 446(26) in waveguides, 178, 425-428 Propagating mode, 68 Propagation constant, 62, 68, 86, 384 stationary formulas for, 346-348, 378(32) Purcell, E. M., 202, 400
Q (au Quality factor) Quadrupole, cylindrical, 226, 259(28) 8pherical, 288, 314(20) Quality factor, of biconical cavity, 285 of cavities in general, 372(3, " 6, 7) of circular cavit.y, 216, 257(16) dielectric, 28 of hemispherical cavity, 285 of loss-free antenna, 309 magnetic, 29 minimum antenna Q, 310, 316(31) of rectangular cavity, 76, 190(10) of spherical cavity, 272, 312(4) of spherical waves, 279 Quasi-stntic, definitions, 79, 298, 419, 420
Radar cross section, 116 Radar echo, 115, 355 Radial waveguides, 208-213, 216--219, 279-283 ~ation, 77-81, 242-245 two-dimensional, 228-230 Radiation condoctance, 112 Radiation field, 79, 81, 132-134 R&diation resistance, 82, 93(42, 44), 94(46) Ramo, S., 309 Rayleigh scattering, 295
Rayleigh-Ritz procedure, 339 Re operator, IS, 16 Reaction, 118, 340 Rcacti ve currcn4 27, 28 Reactive power, 22 Realiz.ability conditions, 4()() Receiving pattern, 119 Reciprocity, 116-120 (or antennas, 120 for circuits, 119 for microwave networks, 392 Rectangular cavity, 74-76, 155-157 partially filled, 191(16-18),325,373 Rectangular waveguide, 66--74, 148155,387 partially filled, 158-163, 191(14-16), 192(19), 348, 374 Reference conventions, 3 Re8ection of waves, 54-61 Reflection coefficient, 55, 421 Resonance, 74 Resonant antennas, 94(4.6, 4.6) Resonant slots, 444(21), 445(23) Resonaklrs, circular cavity, 213-216 concentric sphercs, 284 one-dimcnsional, 44 rectangular cavity, 74, 155 sources in, 431434 spherical cavity, 269-272 spherical secklr, 284 Ribbon of current, 188, 260(31) Ridge waveguide, 327, 374(12) Rightrhand rule, 2 Ritz procedure, 338, 344 Rodrigucs' formula, 466 Rubenstein, P. J., 371 Rumsey, V. H., 118, 340, 365
Saddle point, 335 Saunders, W. K., 245 Scattered field, 113 reciprocity for, 141(24.) Scattering, by conducting plate, 115, 128, 140(20,21) by conducklrs, 355-361 by cylinders, 232-236, 261 (34., 36), 364
INDEX
Scattering, by dielectrics, 362-365 differential, 360 by haIr-planes, 241-242, 261(37, 88) by magnetic obstacles, 380(22) by ribbons, 350, 378(88) by spheres, 292-298 stationary formulas for, 355-365 by wedges, 238-242 by wires, 357, 379(39-41) Scattering matrix, 399 SehelkunofI, S. A., 222, 268, 286 Schwa.rtz, L., 459 Secondary units, 1 Secklral horn, 213
8«>ly, S., 386 Segmental cavity, 284 Seidel, H., 222 SeIr·reaction, liS Separation of variables, 143, 198, 264,
381 Silver, S., 245, 303, 306 Simple matter, 6, 18 Singular 6eld, 32 Skin depth, 53 Slot in ground plane, 138(17, 18), 181186,261(32),370,430,444(21,22), 445(23) Slotted cone, 306 Slotted cylinder, 238 Slotted sphere, 302 Smythe, W. R., 324, 419, 420, 467 Sneddon, 1. N., 252 Snell's l3W, 58 Source coordinatcs, 80 Source-free regions, 37 Sources, 7, 12,19,95,96 Spherical Bessel functions, 265, 268, 464 Spherical cavity, 269-273 partiaUy-611ed, 313(7, 8), 326 Spherical coordinates, 265, 447 Spherical waves, 79, 85, 276, 286-289 Standing wave, 42-47, 69 Standing-wave pattcrn, 44 Standing·wave ratio, 45, 55 Static mode, 338
479
Stationary formulaa, 317, 34.1 for aperture admitlance, 428-431 for cavities, 331-345 for cavity feeds, 434-440 for impedance, 348--355 for obstacles in waveguides, 402-406 for scattering, 355-365 for transmission, 365--371 for waveguide feeds, 425-428 for waveguide junctions, 420-425 for waveguides, 345-348 Storer, J. E., 354 Stratton, J. A., 121, 324 Supcrgnin antennas, 309 Surface of constant phase, 85 Surface currents, 33 Surface guided waves, 168-171,219 Surface impedance, 53,371(2),375(18)
Tai, C. T., 358 TE, TM, TEM, 63, 67, 130, 202, 267, 382 Tector, R. J., 430 Teichmann, T., 434 Tensor Green's functions, 123-125,356 Tesseral harmonics, 273 Tightly bound wave, 170 Total reflection, 59 Transmission, 360 Transmission area, 368 Transmission coefficient, 55, 368 Transmission lincs, 61-66 biconical, 284-286, 313(13) equivalent, 386 modC8,63 parallel·plate, 90(28), 01 (31), 189(6, 7), 440(1) radial, 211 twin-slot, 135(7) wedge, 212 Transmission matrix, 399 Transverse 6eld vector, 382 Transvcrse fields, 63, 67. 130, 202 Traveling waves, 39 Trial field, 332 Twin-51ot line, 135(,>
480
TIME-HARMONIC ELECTROMAGNETIC FIELDS
Uniform plane wave, 39, 147 Uniform waves, 85
Waveguide junctions, 172-177, 193 (27-29), 420-425, 443(17, 18)
Uniqueness, 100-103 Units, 1
Waveguides, 66 biconical, 284-286, 313(13)
circular (see Circular waveguides) Van Valkenburg, M. E., 397, 400, 435
Variation, 332 Variational methods, 317, 331-380 Vector analysis, 447-450 Vector Green's theorems, 121, 141(28) Veclor potential, 77, 99
Velocity, of energy, 42 of light, 5 of phase, 39, 40, 88, 86, 385 Voltage, 3, 15
Voltage source, 96, 118 Von Hipple, A" 23
Wait, J. R., 240, 242 Wall impedance, 371(2, 3), 375(18) Wave equation, 37 for inhomogeneous matter, 88(2)
Wave functions, 85 cylindrical, 199-204 plane, 143-145 spherical, 264-269
Wave impeda.nce, 39, 55, 86 characteristic, 69, 152 Wave number, 37 Wave potentials, 77, 129 Wave transformations, 230-232, 289292 Waveguide feeds, 179, 195(33,34),425428, 444(20)
corrugsted conduclor, 170, 193(25)
corrugated wire, 223 dielectric slab, 163, 192(22) in general, 381-391 psrallel-plate (see Parallel-plate waveguide) posts in, 406-411, 442(12) probes in, 178, 425-428, 446(26) radial, 208, 279 partially filled, 216 rectangular (see Rectangular waveguide) Wavelength, 40 cutoff, 88, 150,206,384
guide, 68, 384 intrinsic, 40 Waves, in dielectrics, 41-48 in general, 85-87 in lossy matter, 51-54
Wedge cavity, 284 waveguide, 208, 255(7), 256(14) Whinnery, J. R, 309
Wigner, E, P" 434 Windows, 414
Zeros, or Bessel functions, 205 or spherical Bessel functions, 270 Zonal harmonics, 273
