Lumped Element Quadrature Hybrids
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Lumped Element Quadrature Hybrids David Andrews
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10 9 8 7 6 5 4 3 2 1
Contents ix
Preface 1
Overview of Quadrature Hybrids
1
1.1
From Distributed to Lumped Element Design
1
1.2
Applications
8
1.3
Analysis of Quadrature Error
13
1.4
Conclusion
15
References
16
Basic Theory
19
2.1
Lossless Circuits
19
2.2
Hybrid Circuits
20
2.3
Lumped Element Quadrature Hybrids
24
2.4
Approximate-Phase Quadrature Hybrids
30
2.5
Conclusion
33
2
v
vi
Lumped Element Quadrature Hybrids
References
34
Approximations
35
3.1
Optimization Goal
36
3.2
Polynomial Form of F ( )
36
3.3
Rational Form of F ( )
38
3.4
Second-Order Optimum Rational Function
45
3.5
Higher-Order Optimum Rational Functions
48
3.6
Transfer Functions in Terms of the Complex Variable
64
3.7
Numerator Part of the Optimum Transfer Functions
67
3.8
Transfer Functions of Optimum Approximate-Phase Hybrids
71
Conclusion
72
References
73
Passive Synthesis
75
4.1
Even- and Odd-Mode Analysis
76
4.2
The First-Order Quadrature Hybrid
79
4.3
Higher-Order Lowpass Prototypes
81
4.4
Use of Transmission Lines
93
4.5
Optimum Rational Function Second-Order Prototypes
96
4.6
Higher-Order Optimum Rational Function Hybrids
104
4.7
Hybrid Synthesis Using Cascaded Sections
107
4.8
Cascades of First-Order Sections
109
4.9
Cascades of Second-Order Sections
112
4.10
Further Cascade Arrangements
113
4.11
Approximate-Phase Hybrids
117
3
3.9
4
Contents
4.12
vii
Approximate-Amplitude Hybrids Based on Phase Delay Networks
124
Conclusion
126
References
127
Practical Design
129
5.1
The First-Order Circuit Coupled Inductor Design
130
5.2
A 435-MHz Ground Inductor Hybrid Design
135
5.3
A 1.27-GHz First-Order Microstrip Hybrid
137
5.4
A 100–200-MHz Third-Order Polynomial-Based Hybrid
138
5.5
A 1–3-GHz Mixed Element Hybrid
140
5.6
A 2.5–6-GHz Hybrid with Coupled Transmission Lines 144
5.7
Optimum Second-Order Hybrids
146
5.8
A 50–150-MHz Second-Order Hybrid
150
5.9
Higher-Order Symmetrical Optimum Hybrids
151
5.10
A 2–32-MHz First-Order Cascade Hybrid
152
5.11
A 10–100-MHz Second-Order Cascade Hybrid
154
5.12
A 10–100-MHz Approximate-Phase Hybrid
157
5.13
A 1–300-MHz Approximate-Amplitude Hybrid Based on Phase Delay Networks
164
Conclusion
167
Reference
168
6
Special Topics
169
6.1
Active Circuits
169
6.2
Unequal Division and Nonquadrature Hybrids
181
6.3
Power Handling
190
4.13
5
5.14
viii
Lumped Element Quadrature Hybrids
6.4
Graphical Techniques for Cascaded Couplers
192
6.5
A 50–550-MHz Hybrid with Sections of Different Topology
196
Conclusion
201
References
202
Glossary
203
Bibliography
207
About the Author
211
Index
213
6.6
Preface Quadrature hybrids have wide application in radio frequency (RF) and microwave circuits and systems. In answer to this need, considerable attention has been paid to distributed circuits with the quadrature properties, particularly for microwave applications. RF engineers too find quadrature hybrids useful, although they prefer lumped element circuits for reasons of size. However, they have been disappointed with the availability of material on the subject, which has received far less attention than the distributed counterparts. This is a pity as there is much to say on the subject. In addition, with the trend towards miniaturization of microwave circuits, lumped element circuits are of increasing interest. Microwave engineers will be surprised by the breadth of applications for lumped element quadrature hybrids, which offer the prospects of reduction in circuit size, ease of fabrication, and remarkable performance. RF engineers will also find the material presented in this book useful as, at the time of writing, there has been no other substantial text written on the subject. The inspiration for this work grew originally out of my experience with RF and microwave amplifier design. Quadrature hybrids are a useful component for this purpose, where two amplifier stages placed between two quadrature hybrids form what is known as a balanced stage, after an article by K. Kurokawa, “Design Theory of Balanced Transistor Amplifiers,” in the Bell System Technical Journal, Vol. 44, No. 8, October 1965, pages 1675–1698. The structure exhibits enhanced input and output match characteristics and double the power output of a single stage. In a system, the performance of a ix
x
Lumped Element Quadrature Hybrids
cascade of balanced stages is greatly enhanced, as the reflection between stages is much reduced. However, the desire for wider operating frequency ranges places severe limitations on the use of distributed circuits, so other solutions were sought. The quest for improved bandwidth led to my personal discovery that lumped element circuits of modest complexity could perform the quadrature hybrid function over much greater relative frequency ranges. It was discovered too that there was a lack of information on the subject, and this became the grounds for a research project at the University of Surrey, England. During the course of the research, the lack of information was confirmed, although it became evident that there was a little information on the subject that had not been publicized well. This book is based largely on my Ph.D. thesis of a similar title, although more material has been added since, particularly on the subject of phase delay circuits. Some material found in the thesis has been omitted from this book, as the focus has been on equipping the designer with the most useful aspects of the subject. There was no value in adding “man of straw”arguments to the book. The book is structured in a similar manner to the treatment of filter theory, because the subjects have much in common. Chapter 1 gives an overview of the various forms of quadrature hybrids and their applications, then shows a method for assessing the relative performance of a particular design. Chapter 2 examines the constraints that theory places on quadrature hybrid circuits and, more particularly, lumped element forms. Chapter 3 is a treatment of the subject of approximation, a concept familiar to the filter designer. Quadrature hybrids are also filter circuits, and their performance is one of optimization rather than perfection. Chapter 4 deals with the subject of circuit synthesis and shows how the various approximation functions can be given their expression in electrical networks. Chapter 5, titled “Practical Design,” might also be titled “Realizations” and shows how the theoretical circuits can be made in practice. A number of concept circuits are described, illustrating most of the aspects described in the theoretical chapters. The final chapter, “Special Topics,” shows how the theory and application of quadrature hybrids can be extended to related matters, which are of themselves also useful. Like any other text on filter theory, this book makes considerable use of mathematics. The reader is encouraged to grasp the theory as it will improve his or her understanding of the subject. However, in recognition that many engineers prefer a quick route to design, graphs and tables have been provided to help with the simpler applications. To illustrate the mathematics, a number of worked examples are provided. The reader might notice a lack of reference to computer-aided design (CAD) tools in the book. This is because
Preface
xi
these vary widely in scope and capability. CAD may be used at any stage to check element values, optimize numerical solutions, and compensate for practical limitations. Such tools are no substitute for an understanding of the underlying principles. This is not to say that CAD tools are not useful in the design of quadrature hybrids. Indeed, mathematical and circuit design software were both used in the solution of numerical problems and optimization of element values for circuits described in this book. The reader is assumed to have a graduate-level knowledge of RF and microwave circuits and theory. I never intended to take the material back to first principles. For further treatment of the underlying principles, the references and bibliography should be consulted. Where references have been used, they usually refer to the original work. However, this is not always the case, as the original work is sometimes contained in an obscure publication, or it contains numerous errors and, as such, is likely to lead to confusion. In such cases, the original reference is replaced with a more readily available or accurate one. In the preparation of circuit diagrams, use has been made of an international style of symbols such as might be found in electronic magazines. The main reason for this is cosmetic; in addition, because engineers have used them for some time now, the eye more readily appreciates them. Thus, the rectangular box has been discarded in favor of the zigzag pattern for the resistor symbol, and the former has been reserved to denote generic impedances. The curly form of the inductor symbol is used rather than a series of semicircles, which was only introduced for the convenience of the drawing offices of a bygone era. The book makes frequent use of mutual inductance, indicated by a double-headed arrow between the coupled inductors. The coupling is indicated either as an absolute value or a coupling factor. To avoid confusion, coupled inductors are always drawn in such a way as to make the dot notation unnecessary. I would like to acknowledge the help given during the preparation of this book. Mr. Allen Podell, who himself has authored several patents on hybrid circuits, has helped me particularly with the practical aspects of phase delay circuits. I would like to thank Mr. Mike Davis, a fellow director at Vectawave Technology Limited, who fabricated several of the circuit boards used in the original research, the results of which are featured in the practical design section of this book. I would like to acknowledge the contribution of my former supervisor, Professor Colin Aitchison, who guided the development of the original research upon which this book is based. Finally, I would like to acknowledge the help of those many engineers and teachers who have inspired me along the way.
1 Overview of Quadrature Hybrids 1.1 From Distributed to Lumped Element Design The reader is likely familiar with distributed versions of quadrature hybrids, as these have been covered extensively in the literature, both in books and technical papers. A simple example of these is the branch line hybrid, as shown in Figure 1.1 [1]. At center frequency, power applied to the input port is transferred equally to the two output ports, with the fourth port in isolation. Input match is perfect at this frequency. The upper output as shown in the diagram leads the lower output by 90°. Acceptable performance Z0 /√2 −90°
Input
Z0
Z0
−180°
Isolated Z0 /√2
Figure 1.1 Quarter-wave branch line hybrid circuit.
1
2
Lumped Element Quadrature Hybrids
is achieved at frequencies on either side of the center frequency, but performance degrades rapidly thereafter. Figure 1.2 shows the insertion loss to the two outputs, Figure 1.3 shows the phase balance between outputs, and Figure 1.4 shows the input match and isolation. Another form of distributed quadrature hybrid makes use of an inphase divider with a quarter-wave line extension to one output. The in-phase divider may be of the Wilkinson type, as shown in Figure 1.5. In this circuit, a quarter-wave line has been added to the lower output to give a 90° offset. The circuit gives good input match over a reasonable bandwidth, but phase balance deviates rapidly from quadrature on either side of the center
Insertion loss (dB)
1.0 2.0 −180°
3.0 4.0
−90° 5.0
0.7
0.8
0.9 1.0 1.1 Normalized frequency
1.3
1.2
Figure 1.2 Branch line hybrid insertion loss.
Relative phase (°)
100
90
80 0.7
0.8
0.9
1.0 Normalized frequency
Figure 1.3 Branch line hybrid phase balance.
1.1
1.2
1.3
Overview of Quadrature Hybrids
3
Figure 1.4 Branch line hybrid port match and isolation. √2Z0, λ/4 −90°
Input
2Z0 −180° Z0, λ/4
Figure 1.5 Wilkinson divider–based quadrature hybrid.
frequency. Phase performance is the chief limiting factor for this design. This circuit suffers the disadvantage of not having an externally available isolation port. For many applications, this is not important. Like the branch line coupler, this circuit is suitable only for narrowband applications. Where wider bandwidths are needed, a proximity coupler can be used, as shown in Figure 1.6 [2]. Coupled power is transferred to the coupled line port found at the same end as the input, with remaining power arriving at the through port. With suitable element values, this circuit exhibits perfect match and isolation at all ports. Imperfections only arise at the realization stage, through discontinuities at the interface between the coupling section and interconnecting lines, and through the use of inhomogeneous media. The principal difficulty with this circuit is in achieving the strong coupling necessary for equal power division. In order to preserve the match and
4
Lumped Element Quadrature Hybrids Through −90°
Input Z0eZ0o = Z02, λ/4
Isolated
Coupled 0°
Figure 1.6 Distributed transmission line proximity coupler.
isolation properties, it is necessary to use conductors of small cross-sectional area closely spaced together. In addition to exhibiting perfect match and isolation, this circuit also exhibits exact quadrature characteristics, with the coupled port leading the through port by 90° up to the half-wavelength frequency. The only quantity that changes with frequency is the coupling. A circuit designed to give exact power division at the center frequency will still give acceptable amplitude performance on either side of this. For improved bandwidth at the expense of amplitude balance, the circuit can be designed to give stronger coupling at the center frequency. Figure 1.7 shows the
Insertion loss (dB)
1.0
2.0 Coupled 3.0 Through
4.0
5.0
0.4
0.6
0.8
1.0 Normalized frequency
Figure 1.7 Proximity coupler insertion loss and coupling.
1.2
1.4
1.6
Overview of Quadrature Hybrids
5
frequency response of such an overcoupled coupler, where a center-frequency coupling of 2.7 dB yields an octave bandwidth. All the circuits above can be embellished to improve their performance in various ways. The branch line coupler of Figure 1.1 can be designed with more than two branch lines for increased bandwidth; Figure 1.8 shows an example with three branch lines. The penalty for this improvement is much higher impedance in the outer branch lines. The proximity coupler of Figure 1.6 can be developed into a multisection design, as shown in Figure 1.9. It is found necessary to increase the coupling factor considerably in the center section as compared with the single-section design. The outer sections have much more modest coupling requirements. There is considerable literature on these topics, which are outside the scope of this book [2, 3]. The previous distributed circuits, relying as they do on quarterwavelength structures, become unacceptably large at lower frequencies. For frequencies from around VHF down to even audio frequencies, it is desirable to change to lumped element design. Conventional wisdom in circuit design has attributed distributed circuits to microwave frequencies in the
Input
−180°
Isolated
−270°
Figure 1.8 Multisection branch line hybrid. −540°
Input λ/4
−360°
Figure 1.9 Multisection proximity coupler.
Isolated
6
Lumped Element Quadrature Hybrids
region of 1 GHz and above. This leaves a dilemma for designers in the UHF range, where designs vary between the two. This policy has been modified since the advent of integrated circuit design, so that what was once a distributed design of acceptable size is now unacceptably large. Lumped element design is now demanded at higher and higher frequencies. A lumped element version of the branch line quadrature hybrid has been devised, and this is shown in Figure 1.10. Like its distributed counterpart, this circuit exhibits a restricted bandwidth. It is exact only at a spot frequency, and the quantities of match, isolation, insertion loss, and phase balance all degrade on either side of this. However, it is composed of simple elements and is acceptable for this reason and in narrowband applications. Referring to the circuit diagram of Figure 1.10, exact quadrature properties will be seen at the normalized frequency of = 1 and termination impedance of 1Ω for shunt capacitor values of Cs = 1/(√2+1), coupling capacitor values of Cp = 1, and series inductor values of L = 1/√2. These values should be scaled for frequency and impedance in any practical application. As this circuit does not exhibit ideal match and isolation characteristics at all frequencies, it will not receive further attention in this book. Figure 1.11 shows a superior lumped element circuit, where the phase information is given for operation at the center frequency. With suitable element values, this circuit exhibits perfect match, isolation, and phase balance at all frequencies. In this regard, it is similar to the proximity coupler; however, its amplitude response varies more rapidly with frequency. It requires the use of a perfectly coupled inductor, so fabrication is difficult beyond UHF. This circuit will be described in detail in subsequent chapters, together with a technique for overcoming the limitations of the coupled inductor. Where wider bandwidths are required, the lumped element circuit of Figure 1.11 can be used with the addition of transmission lines to form the L Input
0°
Cs
Cs Cp
Cp −90°
Isolated
Cs
L
Figure 1.10 Lumped element version of branch line hybrid.
Cs
Overview of Quadrature Hybrids
7
L −45°
Input
k=1
C
+45°
Isolated
L
Figure 1.11 First-order lumped element hybrid.
circuit of Figure 1.12 [4]. It is found that the electrical length required for the transmission lines is much shorter than a quarter-wavelength at the center frequency, so this circuit is useful in the lower UHF and VHF bands. With suitable choice of element values, it can be made to operate over an octave bandwidth. Such circuits, composed of a mixture of lumped elements and transmission lines, will be considered in detail. Although this strays a little from the topic of this book, consideration of them here is appropriate as a full treatment might not be found elsewhere. At much lower frequencies, distributed elements are entirely unacceptable, so only purely lumped element designs can be used. Figure 1.13 shows a circuit topology for use at low frequencies. In this design, an in-phase divider distributes an input signal to its two outputs. Each output comprises a phase delay network, the difference between the two approximating quadrature over an appreciable bandwidth. It is possible to specify element values for this circuit to give multidecade bandwidth. Applications of this circuit have largely been superseded by first active circuits and then digital techniques, particularly at audio and lower RF bands. This book considers circuits of this kind and others with potential for very wide bandwidth operation. Such circuits are useful up to UHF bands. The distributed couplers and many of the lumped element hybrids exhibit the properties of good match and exact quadrature at all frequencies, L
L
Z0 ,l
Through
Input
k=1
k=1
C
Coupled
Isolated
Z0 ,l
Figure 1.12 Mixed element hybrid.
C
8
Lumped Element Quadrature Hybrids
Input
Phase delay network φ + 90°
Q output
Phase delay network φ
I output
In-phase divider
Figure 1.13 Differential phase hybrid.
but with ripple amplitude balance characteristics. These circuits, which derive their hybrid properties from coupling action, will be referred to as approximate-amplitude hybrids. In addition, there exists a range of hybrids that also have good match at all frequencies; however, they possess exact division properties and have ripple phase characteristics. These will be described as approximate-phase hybrids. Most of the circuits described in this book are of the approximate-amplitude type, though approximate-phase hybrids are also considered as they are a useful solution in wideband applications. The choice of quadrature hybrid depends on the application and specification.
1.2 Applications Quadrature hybrids find their application in radio frequency systems for analog signal processing and transmission. To illustrate the function of a quadrature hybrid in transmission applications, consider the diagram in Figure 1.14. The ideal quadrature hybrid has the properties of perfect match at all ports and perfect isolation between ports 1 and 3 and between ports 2 and 4. Port 1 is designated the input, ports 2 and 4 are terminated with arbitrary, though identical, loads, and port 3 is the output. Consider a signal at the input port 1 of the hybrid. An equal division occurs between ports 2 and 4, though the signal at port 4 lags behind that at port 2 by 90°. The loads on ports 2 and 4 reflect identically. The reflections are divided by the quadrature hybrid in such a manner as to give a cancellation at the input port 1 and a
Overview of Quadrature Hybrids Input
9 Output
1
Quadrature hybrid
3
2
ZL
4
ZL
Figure 1.14 Balanced circuit.
summation at the output port 3. The outcome then is a circuit that remains matched at the input but whose transmission coefficient is the same as the reflection coefficient of the identical loads, except for a phase delay. The properties of the circuit depend on the nature of the loads at ports 2 and 4 of the quadrature hybrid. For example, if the loads are varactor diodes, a bias voltage can vary their capacitance, hence, the phase of their reflection coefficient. Thus, the transmission phase from input to output can be adjusted, and a voltage-variable phase shifter is realized. If, in the circuit of Figure 1.14, the varactors are replaced with PIN diodes, then bias current variation varies their dynamic resistance. The circuit then becomes a current-variable attenuator. Maximum attenuation occurs when the PIN diode resistance is equal to the characteristic impedance. On applying bias, attenuation begins at a low level and increases to a maximum before decreasing again as the PIN diode resistance reduces below the characteristic impedance. In order to achieve a monotonic function, it is sometimes convenient to place a matched load in parallel with the PIN diode. That way, attenuation starts at a high level and reduces with bias. The principle of the balanced circuit of Figure 1.14 may be extended to enhance the properties of two-port networks, as shown in Figure 1.15. In this arrangement, the two-port reflections from an input signal are combined at the input hybrid isolation termination. The transmission components behave in a similar fashion to the reflection components and combine at the output port of the output hybrid. A similar description of the circuit holds when signals are presented at the output port. The circuit exhibits low reflection at both input and output ports, provided the two-port networks are nearly identical. The operation of the circuit depends on the nature of the two-port. Another form of voltage-variable attenuator is realized if shunt
10
Lumped Element Quadrature Hybrids Output termination
Input
Two-port network #1
Two-port network #2
Output
Input termination
Figure 1.15 Balanced transmission circuit.
PIN diodes are used. This has the advantage over the single hybrid form in that imperfections in the hybrids do not compromise high attenuation. A very useful circuit is realized when the two-port networks are amplifier stages, giving the so-called balanced amplifier configuration [5]. The circuit maintains the gain characteristic of the individual stages, but power reflected from the two-ports combines only at the isolation ports. The combined stage gives twice the power capability of the individual stage, subject to hybrid loss. Their good input and output match improves the properties of multistage amplifiers. Quadrature hybrids are used in various mixer circuits to remove unwanted sidebands or image signals and to determine or provide phase information [6]. Figure 1.16 shows a simple single balanced mixer circuit [7]. Here, the quadrature hybrid provides a degree of isolation between the local oscillator (LO) and RF signals. The mixing products generated by the diodes contain both the sum and difference frequencies, but the orientation of the diodes gives a short circuit to the sum frequency, with only the difference or intermediate frequency (IF) transmitted to the output. Another mixer circuit using a quadrature hybrid is the in-phase/quadrature (IQ) demodulator, as shown in Figure 1.17. This circuit is used, for example, in a vector analyzer to determine phase information from a signal. In this case, the two input frequencies are the same, though their relative phase may vary. One of the signals provides a reference, and the other is the
Overview of Quadrature Hybrids
11
RF in
Σ
IF out
LO in
Figure 1.16 Single balanced mixer.
test signal. Individual spectral components of the test signal can be resolved into in-phase and quadrature components when compared with the reference signal. The mixer outputs separate these components and give vector information. The circuit principle may be reversed to become a vector modulator. In this orientation, the I and Q ports become inputs, and the output is the test port. The mixers may need to be reconfigured if they are not reciprocal components. In the IQ demodulator circuit of Figure 1.17, where the reference and test signals are two different frequencies, outputs are present on both the I I output
Mixer #1 In-phase divider Mixer #2 Test
Q output
Figure 1.17 IQ demodulator.
Ref
12
Lumped Element Quadrature Hybrids
and Q channels. The individual components are in quadrature, but the relative polarity depends on whether the received signal is above or below the test signal. This property is exploited in a modification of the circuit, called the image reject mixer [7]. Here an additional quadrature hybrid combines the I and Q signals, as seen in Figure 1.18. Two input RF signals can potentially mix with the reference or LO signal to give the desired IF signal. In typical applications, one is the desired RF signal, while the other, if present, constitutes an unwanted interference. The image reject mixer separates the two in the final quadrature hybrid so that the wanted signal is transmitted and the unwanted signal is suppressed. In Figure 1.18, signals below the LO appear at the lower sideband (LSB) output of QH2, while signals above the LO appear at the upper sideband (USB) output. An alternative use of the circuit of Figure 1.18 is a single sideband modulator, in which case the outputs marked LSB and USB become the inputs, with RF IN becoming an output consisting of either the lower or upper sideband. In this application, the quadrature hybrid QH2 is likely to operate at baseband and may require a solution with considerable bandwidth ratio. All of the applications described above are potential candidates for lumped element quadrature hybrids. Mixer #1
RF in
LSB
LO in
QH 1
QH 2
Mixer #2
Figure 1.18 Image reject mixer.
In-phase divider
USB
Overview of Quadrature Hybrids
13
1.3 Analysis of Quadrature Error The applications discussed in Section 1.2 assume perfect balance in the amplitude and phase of the quadrature hybrid. It will be shown subsequently that it is theoretically impossible to achieve these two quantities simultaneously over an arbitrary bandwidth. In addition, practical circuits contribute additional errors through component tolerance, stray effects, and circuit losses. As a consequence, the response of a practical quadrature hybrid departs from quadrature and amplitude balance, and operation within a system will be affected. A method of predicting the expected impact of phase and amplitude errors is required. In addition, this method can be used to provide a means of comparison between those quadrature hybrids designed to ripple in phase and those designed to ripple in amplitude. Figure 1.19 shows the operation of an imperfect quadrature hybrid, where the vectors V1 and V2 represent the two outputs. With reference to the diagram, V1 has a magnitude of v1 and makes an angle α1 with the horizontal axis, while V2 has a magnitude of v2 and makes an angle α2 with the vertical axis. It can be seen that the two vectors deviate from quadrature and are not of the same amplitude. It is possible to resolve each vector into two components, as shown in the diagram. One component of each vector constitutes a vector pair of equal amplitude and exact quadrature. One of these vector pairs, the most significant, has been aligned with the axes for convenience and also normalized to unit magnitude. This component pair represents the desired hybrid operation. The second vector pair is much smaller, and its phase, relative to the major vector pair, can take any angle. It differs too from the major vector pair in the direction of rotation. Whereas the major component of V2 leads that in V1 by 90°, the minor component lags by 90°. In the diagram, the minor vector pair has been given an amplitude r and relative angle to the axes of θ. The smaller vector pair denotes the error operation of the hybrid, and its contribution to system error is usually determined solely by its amplitude and not by its relative angle. It is usual for theoretical hybrids to exhibit error in either magnitude only or phase only, whereas practical circuits exhibit errors in both. In Figure 1.19, if the relative angle is zero, the resultant vectors are in exact quadrature but are imbalanced in amplitude. This is typical for most quadrature hybrids of coupler form. Where the relative angle is 90°, the outputs are equal in amplitude, but there is a phase error from quadrature. This characteristic is typical for circuits based on differential phase networks. The error quantity of interest is the value of r. However, this quantity is not directly measured, so it is necessary to relate it to the quantities that are,
14
Lumped Element Quadrature Hybrids
Figure 1.19 Vector analysis of quadrature hybrid error.
that is to say, the relative amplitude and phase of the main vectors. The relative amplitude is given by v1/v2, and the error from quadrature is given by 1 + 2, which is defined as . It is possible to construct relationships between the lengths and angles of the figures in Figure 1.19 using trigonometry. There is redundancy in the quantities, so it is convenient to eliminate from the resulting equations, as this quantity is not of interest. After some manipulation, the following relationship results: 1 v1 v 2 1+ r 2 + sec α = 2 v 2 v1 1− r 2
(1.1)
From (1.1), it is possible to determine r, given the amplitude balance and quadrature error. A graphical representation of this equation is useful. So put 1 v v x = log 1 + 2 2 v 2 v1 y = log ( sec α) 1+ r 2 R = log 1 − r 2
Overview of Quadrature Hybrids
15
then x2 + y2 =R2 Equation (1.1) has been transformed into a relationship describing a system of circles in the x, y plane, where the x-coordinate is a function of amplitude imbalance only, the y-coordinate is a function of quadrature phase error only, and the distance from the origin is a function of the magnitude of the error vector pair. In the graphical representation shown in Figure 1.20, the amplitude imbalance scale is graduated in decibels, as is the error vector magnitude, which becomes a definition of rejection ratio. The quadrature phase error scale is graduated in degrees. While the scales appear linear, they are not, although the departure from linearity is only slight over the range shown. The graphical representation in Figure 1.20 is applicable to a range of quadrature hybrid circuits. For example, in the image reject mixer of Figure 1.18, the magnitude of r represents the degree of suppression of the image signal, given an error in one quadrature hybrid only. In the balanced circuit of Figure 1.15, it represents the proportion of signal arriving at the output isolation termination, given an error in one of the quadrature hybrids. Although (1.1) can be used to determine the equivalence between amplitude and phase errors, it would be useful to express this more clearly. Putting β = 90°− α, where is the phase difference between outputs, sec α =
1 β β tan + 1 tan 2 2 2
(1.2)
Substituting this expression for sec in (1.1) shows that the amplitude imbalance measurement v1/v2 is equivalent to the phase difference measurement tan /2. This relationship is useful in the comparison of quadrature hybrids that ripple in amplitude with those that ripple in phase.
1.4 Conclusion Lumped element quadrature hybrids offer the prospect of a practical design solution for RF and microwave applications. At lower RF frequencies, they may be the only practical solution for reasons of size, and at microwave frequencies they may give an advantage in circuit size. They can be used in any circuit application requiring quadrature hybrid operation.
16
Lumped Element Quadrature Hybrids
14 13 12 Rejection (dB)
11
18 10
Phase error (°)
9 20 8 7
22
6 5
25
4
27
3
30
2 1
35 40 45 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Amplitude imbalance (dB)
Figure 1.20 Hybrid error nomogram.
A method has been described whereby the vector error of a quadrature hybrid can be related to its amplitude imbalance and phase error. This method can be used to compare theoretical and practical circuits.
References [1]
Collin, R. E., Foundations for Microwave Engineering, 2nd ed., New York: McGraw Hill, 1992, pp. 432–434.
[2]
Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Dedham, MA: Artech House, 1980.
Overview of Quadrature Hybrids
17
[3]
Cristal, E. G., and L. Young, “Theory and Tables of Optimum Symmetrical TEM-Mode Coupled Transmission Line Directional Couplers,” IEEE Trans. Microwave Theory and Techniques, Vol. MTT-13, No. 5, September 1965, pp. 544–558.
[4]
Cappucci, J. D., and H. Seidel, “Four Port Directive Coupler Having Electrical Sym metry with Respect to Both Axes,” U.S. Patent No. 3,452,300, June 24, 1969.
[5]
Kurokawa, K., “Design Theory of Balanced Transistor Amplifiers,” Bell System Technical Journal, Vol. 44, No. 8, October 1965, pp. 1675–1698.
[6]
Maas, S., Microwave Mixers, Dedham, MA: Artech House, 1986.
[7]
Pozar, D. M., Microwave Engineering, 3rd ed., New York: John Wiley & Sons, 2005.
2 Basic Theory Practical quadrature hybrids are constructed of physical components whose properties are governed by circuit theory. This chapter considers the fundamental constraints circuit theory imposes on practical circuits. Much of this theory is applicable to both distributed and lumped circuits, though attention is focused on lumped element circuits. Passive circuits are the principal focus of this book, more particularly, those circuits that operate at RF and microwave frequencies. At these frequencies, it is convenient to analyze circuits using scattering, or “s,” parameters. Should practical circuits require testing, the test equipment used will likely measure these parameters directly. In addition, it is found that these parameters are particularly convenient for the analysis of quadrature hybrids and lead to concise forms of equations to describe their operation.
2.1 Lossless Circuits In many practical applications, it is desirable that the quadrature hybrid should introduce a minimum of loss. For example, the loss of a quadrature hybrid at the beginning of a receiver system degrades noise figure. In the output of a balanced amplifier stage, hybrid loss reduces output power. For high-power applications, the loss of power may appear as excessive dissipation in the quadrature hybrid and even cause its failure. Lossless circuits are desirable in theory as well as they are easier to analyze than those with loss. 19
20
Lumped Element Quadrature Hybrids
Should it be necessary to introduce a lossy element, it can be added as the termination to an extra port of an otherwise lossless circuit. Consider the properties of a lossless, but otherwise arbitrary, n-port circuit. In the steady state, it must emit power exactly equal to that entering it; it can be neither a source nor a sink of power. If the power wave entering port i is designated ai and that exiting port i is designated bi, then b ∗b = a ∗ a
(2.1)
whereb is the column vector of emanating power waves, and a is the column vector of incident power waves. The star superscript denotes the transpose vector of conjugate elements. Now, the column vector b is also given by the s-parameter matrix S of the circuit in terms of the column vector a by b = Sa
(2.2)
Next, substitute forb andb ∗ in (2.1), giving a ∗ S ∗ Sa = a ∗ a
(2.3)
Equation (2.3) must be true for arbitrary choice of excitation, so it follows that S ∗S = I
(2.4)
This property (an S-matrix having an inverse that is the transpose of the matrix formed from the conjugates of its elements) is known as the unitary property [1]. It will be useful later.
2.2 Hybrid Circuits A desirable characteristic of the circuits in view is that they should exhibit perfect match at all ports to a reference impedance; in other words, sii = 0. (It is not necessary for the reference impedance to be the same at every port, though it often is.) In addition, the passive circuits in view are assumed to be constructed using only reciprocal elements, such as capacitors, inductors (self and mutual), perfect transformers, and even transmission lines. A circuit comprising only reciprocal elements is also reciprocal, so that sij = sji.
Basic Theory
21
Consider now a three-port circuit, designating port 1 as the input and the other two as outputs, the transfer functions of which are, for the present, undefined. It is immediately assumed that the circuit is both matched and reciprocal so that 0 S 3 = s 21 s 31
s 21 0 s 32
s 31 s 32
(2.5)
0
If it is also required that the circuit should be lossless, then the unitary property will apply so that ∗ ∗ ∗ ∗ ∗ ∗ s 21 s 21 + s 31 s 31 = s 21 s 21 + s 32 s 32 = s 31 s 31 + s 32 s 32 = 1 ∗ ∗ ∗ s 32 = s 21 s 32 = s 21 s 31 = 0 s 31
(2.6)
It is evident that the equations in (2.6) are inconsistent. The first line requires that all elements be nonzero, and the second line requires that at least two of them be zero. It is not possible, therefore, to form a three-port matched reciprocal and lossless circuit. If it is matched, then either it is lossy (for example, a resistive divider) or it is nonreciprocal (as in the case of a circulator). Having seen that a three-port circuit is incapable of satisfying the matched, reciprocal, and lossless requirements simultaneously, consider now a four-port circuit. Figure 2.1 shows such a circuit schematically. In the schematic, the ports are shown as two-terminal connections. At this time, it is not certain whether one terminal of any particular port is connected to a common ground. For the purpose of circuit analysis, this constraint is not necessary, although, in practical circuits, a common ground is often desirable. The matched and reciprocal conditions immediately give rise to the s-parameter matrix as follows: 0 s 21 S4 = s 31 s 41
s 21 0 s 32 s 42
s 31 s 32 0 s 43
s 41 s 42 s 43 0
(2.7)
If the four-port is lossless, the unitary principle applies, and multiplication of the ith row of the transpose conjugate of the above matrix with the ith column of the above matrix gives four equations as follows:
22
Lumped Element Quadrature Hybrids Port 4
Port 3
Port 1
Port 2
Figure 2.1 Generic four-port hybrid. ∗ ∗ s 21 s 21 + s 31 s 31 + s 41∗ s 41 = 1 ∗ ∗ s 21 s 21 + s 32 s 32 + s 42∗ s 42 = 1 ∗ ∗ s 31 s 31 + s 32 s 32 + s 43∗ s 43 = 1
(2.8)
s 41∗ s 41 + s 42∗ s 42 + s 43∗ s 43 = 1 A process of elimination and back-substitution on these equations gives the following relationships between the magnitudes of the elements: s 32 = s 41 s 43 = s 21
(2.9)
s 42 = s 31 It can be seen from the above equations that the magnitude of the transfer function between two arbitrarily chosen ports is equal to that between the remaining two ports. Further constraints upon the four-port are discovered when the elements equating to zero in the product of the unitary S-matrix with its transpose conjugate are considered. Multiplying the first row vector of the transpose conjugate of the S4 matrix with the column vectors, other than the first, of the original S4 matrix gives
Basic Theory
23
∗ s 31 s 32 + s 41∗ s 42 = 0 ∗ s 21 s 32 + s 41∗ s 43 = 0
(2.10)
∗ ∗ s 21 s 42 + s 31 s 43 = 0
Now, the set of equations in (2.9) allows the s-parameters to be described in polar form as s 21 = a∠α s 31 = b∠β s 41 = c∠γ
s 43 = a∠( α + θ )
(2.11)
s 42 = b∠( β + φ)
s 32 = c∠( γ + ϕ) In the set of equations in (2.11), a, b, and c are all purely real quantities. Now, substitute these values into the first equation in (2.10), giving b∠( − β )c∠( γ + ϕ) + c∠( − γ )b∠( β + φ) = 0 In the above, the two terms on the left-hand side must be equal in magnitude and opposite in phase so as to sum to zero. The difference in the angle expressions must therefore be ±180°. The resulting relationship between the angles then simplifies to ϕ − φ β − γ = ±90°+ 2
(2.12a)
Substitutions may be made in a similar manner for the remaining two equations in (2.10), giving θ − ϕ γ − α = ±90°+ 2
(2.12b)
φ − θ α − β = ±90°+ 2
(2.12c)
24
Lumped Element Quadrature Hybrids
Now, taking the sum of the three equations of (2.12), it can be seen that the left-hand side sums to zero. On the right-hand side, the terms in brackets also sum to zero, leaving only the ±90° terms, which cannot sum to zero. It appears then that the equations are contradictory. The only way to resolve the contradiction is to set one pair of parameters in (2.9) to zero. The choice at this stage is arbitrary, so we choose to let s31 and s42 both equal zero. Having done this, it is only necessary to satisfy (2.12b). The result of the preceding paragraph needs emphasis. We have discovered that a matched, reciprocal, and lossless four-port must be directional [2]. Applying a signal at one port gives rise to outputs at two other ports, with isolation at the remaining port. Figure 2.1 illustrates this; arrow lines show the active signal paths. Equation (2.12b) reveals interesting phase characteristics. It shows that once the phase components of two parameters with a common port have been specified, the phase characteristics of the remaining two parameters are constrained (except for an arbitrary offset). The subject of interest is quadrature hybrids so that − equals ±90°. This means that the term in brackets must equal zero. (It may also equal 180°, but this implies a congruent condition in either or .) A particularly interesting case is when both and equal zero. This gives rise to a symmetrical circuit; many quadrature hybrids exhibit this characteristic. It is also possible that both and equal 180°, and this is sometimes seen in quadrature hybrid designs. Where equals , the hybrid becomes an in-phase type. In this case, there must be an inversion between the signals appearing at the two outputs, given an excitation at the isolated port. Equation (2.12b) also proves that it is impossible to achieve a hybrid with in-phase characteristics between all ports.
2.3 Lumped Element Quadrature Hybrids Chapter 1 noted that there are two kinds of quadrature hybrid: those where the amplitude of the two outputs varies with frequency but the phase remains in exact quadrature, and those whose phase varies about quadrature but with exact amplitude balance. We now consider the kind with exact quadrature characteristics with the further restriction that they be composed of only lumped elements. The circuits under consideration will comprise lossless, reciprocal, and passive elements. In addition, it is assumed that the circuits will be finite. This entails, in practice, a combination of capacitors, self- and mutual inductors, and perfect transformers. Such a circuit exhibits s-parameters that
Basic Theory
25
can be described as rational polynomials of the Laplace transform variable s. For the purpose of this analysis, it will be supposed that the input signal is applied to port 1, with outputs present at ports 2 and 4, with port 3 in isolation, as described in Section 2.2. The two transfer functions may be defined as s 21 (s ) =
N 2 (s ) D 2 (s )
(2.13)
s 41 (s ) =
N 4 (s ) D 4 (s )
(2.14)
The two outputs are in quadrature at all frequencies only if the ratio of these two s-parameters is a purely imaginary function when s = j . This is equivalent to saying that the ratio must be an odd function of s, which in turns means that the numerator and denominator of the ratio, once common factors have cancelled, must be odd and even (or even and odd). It has been assumed that the denominators of each transfer function are different. The denominators represent the characteristic frequencies of the entire circuit and, so, are likely to be equal. The only possibility of inequality that exists for s-parameter formulation is where one or another path includes all-pass functions. Suppose then that a first-order all-pass function is added to the path from port 1 to port 2 but not from port 1 to port 4. The new ratio will be N (s ) D 4 (s ) ( σ − s ) s 21 (s ) = 2 D 2 (s ) N 4 (s ) ( σ + s ) s 41 In the above equation, in order for the denominator to remain either even or odd, the ( + s) factor must combine with another factor in the denominator of the form ( − s). Such a right-hand zero must be assigned to the N4(s) part. Similarly, the ( − s) factor in the numerator must combine with another factor of the form ( + s). This factor will not appear in N2(s); otherwise, it would already have cancelled in forming s21(s) before the ratio was taken. It must be assigned to D4(s). Thus, placing an all-pass function in the signal path from port 1 to port 2 has forced the same all-pass function to appear in the signal path from port 1 to port 4. So, in order to remain in quadrature, both signal paths must contain the same all-pass functions. (The
26
Lumped Element Quadrature Hybrids
analysis is identical for a second-order all-pass function with conjugate zeros.) They may be found at the common input or supplied equally to the two outputs. The result is that whether or not all-pass functions appear, the denominators of the two transfer functions must be the same in a quadrature divider. All-pass functions serve only to complicate the circuit with no performance benefit. It would be better to remove them from the synthesis. The simplest and most desirable solution is to specify a common denominator and odd and even numerator components in (2.13) and (2.14). As the transfer functions share a common denominator, it is convenient to drop the denominator’s subscript. The numerator function subscripts “2”and “4”may be replaced with “o”and “e,”respectively, in order to bring clarity to the analysis. This modification attributes the port 1–to–port 2 transfer function with the odd numerator, which until now was an arbitrary choice. Now consider the lossless characteristics of the hybrid. As isolation exists between ports 1 and 3, and all ports are matched, any signal applied at port 1 will be transmitted to ports 2 and 4 only so that s 41 ( j ω) + s 21 ( j ω) = 1 2
2
(2.15)
It is possible now to substitute the assumed transfer functions for the s-parameter expressions, giving Ne D
( j ω)
N Ne ( − j ω) + o D D
( j ω)
No ( − j ω) = 1 D
(2.16)
Equation (2.16) merits further inspection. The expression on the left-hand side is analytic and equal to a constant when s = j , so it must be equal to the same constant for all values of s. Making this substitution with a slight rearrangement leads to N e (s )N e ( −s ) + N o (s )N o ( −s ) = D (s )D ( −s )
(2.17)
Now, Ne is even, so N e ( −s ) = N e (s ). Also, No is odd, so N o ( −s ) = −N o (s ). Therefore, N e2 (s ) − N o2 (s ) = D (s )D ( −s )
(2.18)
Basic Theory
27
As the even numerator polynomial has been assigned to the transfer function from ports 1 to 4, it will be finite at zero frequency. In fact, at dc, all the power is transmitted to this port. In common parlance, it becomes the through port. Correspondingly, the odd numerator polynomial yields a zero transmission from port 1 to 2 at zero frequency. This becomes the coupled port. Port 3 receives no power at any frequency and becomes the isolated port. Consider now the magnitude of the transfer function from port 1 to 4. This is given by s 41 ( j ω) = 2
Ne D
( j ω)
Ne ( − j ω) D
(2.19)
Let s = j in (2.18), and substitute for the denominator functions in (2.19), giving, after rearrangement, s 41 ( j ω) =
1
2
=
( j ω)
N 1− o N e
2
(2.20)
1 1 + F 2 ( ω)
where jF ( ω) =
No Ne
( j ω)
In (2.21), F ( ) is an odd function of for the coupled port, it will be found that s 21 ( j ω) =
and is purely real. Similarly,
1
2
1+ 1
(2.21)
(2.22)
F ( ω) 2
The specification of F( ) is a convenient starting point for the design of quadrature hybrids. For equal power division, its ideal value is ± unity at all frequencies. However, this condition is contradictory to that of a finite odd function, so an approximation is necessary over the passband. The nature of this approximation determines the kind of physical circuit required
28
Lumped Element Quadrature Hybrids
for realization. In recognition of its similarity to filter theory, the function will be referred to as the filtering function. Once specified, the numerator and denominator functions can be determined by means of (2.21) and then (2.18). The preceding analysis can be demonstrated by means of an example.
)
Example 2.1 Given F ( ω) = 2 ω (1 + ω 2 , determine the corresponding
transfer functions for the through and coupled ports of a quadrature hybrid.
Solution Figure 2.2 plots the given filtering function. As can be seen from
the plot, it reaches a maximum value of 1 when = 1, so it achieves a good approximation to equal power division for frequencies in this region. Substituting = s/j in (2.21) gives N o (s ) 2s j = j N e (s ) 1− s 2 Thus, by inspection, N o (s ) = 2s and
1
F (ω)
0
1
ω
2
Figure 2.2 Plot of the filtering function used in Example 2.1.
3
Basic Theory
29
N e (s ) = 1 − s 2 Application of (2.18) then gives D (s )D ( −s ) = s 4 − 6s 2 + 1 For stability, the left-hand zeros must be taken to form the denominator function, which then becomes D (s ) = s 2 + 2 2s + 1 The two transfer functions follow immediately. ❂❂❂
It should be pointed out that the sign of F( ) is arbitrary, so it is appropriate, given F( ), to perform the synthesis of the transfer functions using –F( ). Furthermore, there is an ambiguity of sign in determining No(s) and Ne(s) in (2.21), so for a given No(s) and Ne(s), it would also be appropriate to use –No(s) and –Ne(s). Both of these considerations mean that the sign of both No(s) and Ne(s) can be chosen arbitrarily. The choice depends on how the quadrature hybrid is eventually synthesized, as one particular selection might be more easily realized than another. The ambiguity comes as no surprise, as a change in sign can be realized easily simply by changing the polarity definition of the terminals at any given port. The filtering function has a further significance that can be seen on taking the ratio of the magnitudes of the transfer functions in (2.22) and (2.20). The ratio of coupled to through amplitude is F( ). This ratio was also considered in Section 1.3, where it was expressed as v1/v2. In terms of decibels, the amplitude imbalance is given by ∆(dB) = 20 log 10 F ( ω)
(2.23)
Having determined the transfer functions from an input port to the two outputs, the properties of signals applied at the isolated port need consideration. Section 2.2 showed that the magnitudes of the transfer functions from the isolated port to the two outputs are the same as those of the transfer functions from the input to the two outputs. In order to be the same, they must share the same zero and pole locations, with the possibility of an all-pass function common to the isolated port. Once again, such an all-pass
30
Lumped Element Quadrature Hybrids
function serves only to complicate the design and is best omitted so that the remaining transfer functions are the same as those determined for the input port. The only ambiguity remaining is in the polarity of the termination to the isolated port. (The schematic of Figure 2.1 underlines this point. The polarity of the terminations is arbitrary where no common ground is assumed.) Finally then, the s-parameter matrix for an exact quadrature hybrid is as follows: N o (s ) 0 N e (s ) 0 N (s ) ±N e (s ) 0 0 1 o S= ±N e (s ) 0 ±N o (s ) D (s ) 0 N (s ) 0 ±N o (s ) 0 e
(2.24)
Note that in (2.24), the ± option must be taken consistently throughout the matrix. The majority of quadrature hybrids take the positive option, and the result is a doubly symmetric circuit. The designation of input port is arbitrary; any of the ports may be given this designation with the remaining ports taking their appropriate designation. Hybrids where the negative option is taken do exist, and there are occasions when this choice is preferable.
2.4 Approximate-Phase Quadrature Hybrids The starting point for quadrature hybrids exhibiting approximate-phase characteristics is a pair of in-phase signals of equal amplitude. These signals are frequently provided using a 0°/180° hybrid. Such hybrids have no theo retical frequency limit. The hybrid provides the required isolation between outputs. Depending on the hybrid design, the isolated port may have to be terminated internally. The approximate-amplitude exact quadrature hybrid design has no need of all-pass circuits, but the approximate-phase hybrid requires them inherently. Each output of the 0°/180° hybrid is connected to a different all-pass circuit, the design and complexity of which depends on the specification. The purpose of these all-pass circuits is to give a differential phase characteristic as close to quadrature as possible over the passband. The principle was introduced in Section 1.1, and Figure 1.13 shows a schematic for the circuit. The in-phase divider is shown as a three-port, and this corresponds with the use of a 0°/180° hybrid with the 180° port terminated internally. Figure 2.3 shows a similar arrangement where the input and output ports have been numbered for the purpose of this analysis.
Basic Theory
Port 1
31
Phase delay network a
Port 2
Phase delay network b
Port 3
In-phase divider
Figure 2.3 Differential phase circuit.
A first-order all-pass section of lumped element form gives rise to a transfer function contribution given by σ −s σ+s where is a purely real and positive quantity. Substituting s = j into the above expression, it can be seen that the numerator and denominator are conjugates of each other and that the magnitude is always unity. In the first-order case, the zero and pole are purely real. It is possible to implement complex zeros and poles, provided that they occur as conjugates. As a result, a second-order section is required to have a complex conjugate pair. The transfer function contribution is given by
(s 1 − s )(s 1∗ − s ) (s 1 + s )(s 1∗ + s ) where s1 takes a complex value with positive real part. Once again, substituting s = j gives unity magnitude. Implementing higher-order all-pass functions requires only a cascade of first- and second-order sections. Inspection of these two transfer function contributions indicates that the phase of an all-pass cascade has a monotonic characteristic. It is not possible to achieve a quadrature response over a significant bandwidth using a single cascade, hence, the requirement for two cascades, a and b, one at each
32
Lumped Element Quadrature Hybrids
output of a two-way divider, as shown in Figure 2.3. An approximate differential quadrature response is then achieved by adjusting the phase characteristics of each cascade. To analyze the differential phase characteristics, consider the ratio of the transfer functions to each output. This will give a function of the form
(a − s )(a 2 − s ) K (b 1 + s )(b 2 + s ) K s 21 (s ) = 1 (a1 + s )(a 2 + s ) K (b 1 − s )(b 2 − s ) K s 31
(2.25)
It can be seen from (2.25), upon substitution of j for s, that the numerator and denominator contributions to phase are equal. Considering the numerator only, half the phase difference between outputs is given by its argument. Now, substitute s = j and let [3] β = arg Q (ω 2 ) + j ωP (ω 2 ) 2
[
]
where is the angle between outputs, P and Q are polynomials with real coefficients, and the term in square brackets is equal to the numerator of (2.25) upon the substitution s = j . Evaluating the argument gives 2 β ωP ( ω ) tan = 2 Q (ω 2 )
(2.26)
In Section 1.3, it was found that the quantity tan /2 in a hybrid varying only in phase was equivalent to v1/v2 in a hybrid varying only in amplitude. As v1/v2 is identified with the filtering function F( ), as introduced for the purpose of specifying approximate-amplitude hybrids, this same function can be used as a starting point for the design of approximate-phase hybrids. To further the approximate-phase quadrature hybrid design, it is necessary to evaluate the characteristic frequencies in (2.25). The numerator is also equal to Q ( −s 2 ) + sP ( −s 2 ). This function should be factorized, with the factors giving left-hand plane roots assigned to one all-pass cascade (where they become its denominator polynomial) and those giving right-hand plane roots to the other all-pass cascade (where they become its numerator polynomial). Further analysis will confirm that the pole locations of the all-pass sections are the same as those described by D(s), as defined in (2.18) for the approximate-amplitude case.
Basic Theory
33
Example 2.2 Given the same filtering function as in Example 2.1, determine
the transfer functions of the two all-pass sections of a differential phase quadrature hybrid. Solution From (2.26), substituting the filtering function for the left-hand
side gives 2ω
(1 + ω 2 )
=
ωP ( ω 2 ) Q (ω 2 )
The functions P and Q are easily determined by inspection. The numerator of (2.25) becomes Q (−s 2 ) + sP (−s 2 ) = 1 − s 2 + 2s
= ( 2 + 1+ s
)(
2 −1− s
)
The solution requires a first-order all-pass section in each phase delay network. The transfer functions are given by Τa = Τb =
2 +1− s 2 + 1+ s 2 −1− s 2 − 1+ s
It can be verified that multiplying the denominators of the above transfer functions gives the D(s) function found in Example 2.1. ❂❂❂
2.5 Conclusion In order to be perfectly matched on all ports, a lossless hybrid requires four ports. Such a hybrid is directional so that any given port couples to two other ports and is isolated from the remaining port. Quadrature hybrids with exact quadrature response at all frequencies exhibit coupler characteristics, with zero coupling at zero frequency and approximate-amplitude balance across the passband. With suitable choice of termination polarity, they exhibit two-directional symmetry.
34
Lumped Element Quadrature Hybrids
Quadrature hybrids with approximate quadrature response and exact amplitude balance require two all-pass cascades to optimize phase response across the passband. Both types of hybrid, when implemented using lumped elements, can be specified using a filtering function F( ) that is an odd function of frequency and must be optimized to approximate unity across the passband. In the case of approximate-amplitude hybrids, the function is equal to the ratio of the amplitudes of each output. In the case of approximate-phase hybrids, the function is equal to the tangent of half the phase difference between the two outputs.
References [1]
Chen, W. K., Theory and Design of Broadband Matching Networks, Oxford, England: Pergamon Press, 1976, pp. 88–91.
[2]
Montgomery, C. G., R. H. Dicke, and E. M. Purcell, (eds.), Principles of Microwave Circuits, New York: McGraw-Hill, 1947, p. 437.
[3]
Darlington, S., “Realisation of a Constant Phase Difference,” Bell System Technical Journal, Vol. 29, No. 1, January 1950, pp. 94–104.
3 Approximations Chapter 2 demonstrated how a perfect quadrature hybrid, with equal power division and exact quadrature performance over a finite passband, is impossible in theory. Some approximation over a specified passband is required. We found that the properties of a quadrature hybrid can be determined from a function F( ), which must be an odd function of frequency. This function forms the starting point, whether an approximate-amplitude or approximatephase hybrid is the target circuit. There is little value in examining every conceivable form of the filtering function F( ). We need only consider the function forms that lead to practical realizations. In its general form as applicable to circuits containing distributed as well as lumped components, F( ) will comprise trigonometric functions as well as powers of the radian frequency. However, as only lumped elements are in view, the analysis will be restricted to functions no more complicated than the ratio of two polynomials. This chapter begins with the polynomial form and gives numerical solutions to the filtering function. Next, it describes the optimum rational form, with the solution provided by analysis. The filtering function is then used to derive solutions to the transfer function in terms of the Laplace operator s. The polynomial form gives solutions applicable to approximate-amplitude quadrature hybrids, whereas the rational form applies to both approximate-amplitude and approximate-phase hybrids. 35
36
Lumped Element Quadrature Hybrids
3.1 Optimization Goal The value of F( ) cannot equal its ideal value of unity at all frequencies, so it must be allowed to ripple somewhat about this value across the passband. The limits to F( ) must be specified in such a way as to minimize the imbalance in power or error in phase, according to whether an approximate-amplitude or approximate-phase quadrature hybrid is in view. For the approximateamplitude case, this is equivalent to saying that the ratio of maximum and minimum amplitude of the outputs [an alternative definition of F( )] must be minimized over the passband. This can be achieved by determining a value m > 1 such that F( ) ripples between 1/m and m over the passband. The value of m is optimized to be as close as possible to unity, subject to the constraints imposed by the order and nature of F( ). There are two forms of F( ) to consider. The first is where F( ) is simply a polynomial of odd powers of . The second form is where F( ) is a rational pair of polynomials of , with an odd numerator and even denominator. By this means, it will always be true that F(0) = 0. It will be appreciated that for every function F( ) associated with the transfer function to the through port (where hybrids of coupler form are in view), there is a complementary function 1/F( ) with the same performance associated with the coupled port.
3.2 Polynomial Form of F ( ) Where F( ) is a polynomial of odd powers of , the transfer function becomes that of a lowpass filter. With suitable specification, this function can be made to express the insertion loss of a Bessel, Butterworth, or Chebychev filter of odd order. However, these filter functions are configured to give a low insertion loss up to a particular frequency, with attenuation increasing rapidly thereafter. In contrast, the present purpose requires a function giving as close as possible to 3-dB attenuation over the passband; insertion loss outside of this range is of little consequence. The notation for this form of F( ) is as follows: F ( ω) = a 1 ω + a 3 ω 3 + K + a r ωr + K + a n − 2 ω n − 2 + a n ω n
(3.1)
Figure 3.1 shows a typical response for F( ) for the particular case of n = 5 and an optimum passband of 1/3 < < 1.
Approximations
37
m F (ω)
1.0 1/m
0.5
0.0
ωa
0.5
1.0 ω
Figure 3.1 Fifth-order polynomial example of F ( ).
The evaluation of this form of F( ) is not a new concept as it occurs, for example, during the synthesis of symmetric transmission line couplers [1]. Although tables of solutions exist for this problem, the results cannot be taken directly as optimization limits are specified to give a minimum deviation of insertion loss on a decibel scale, while geometric limits are preferred here. In addition, the tables of solutions are specified in terms of ripple factor rather than the more usual requirement of bandwidth. Irrespective of the optimization goal, the optimization requires numerical analysis. The optimization technique can be inferred by inspection of Figure 3.1. At the lower passband limit a, F( ) takes a value of 1/m. It then makes a tangent to the ordinate values of m and 1/m in an alternate fashion before taking the value of m at the upper passband limit of 1. It takes a value of m at the upper passband limit in cases when n equals 1, 5, 9 … . Where n equals 3, 7, 11 … , F( ) = 1/m at the upper passband limit. The optimization algorithm must be configured to make F( ) equal to the appropriate values at the
38
Lumped Element Quadrature Hybrids
band limits and to either m or 1/m at the points in between where the derivative equals zero. The process is usually helped by a good guess at starting values of ar. Tables 3.1 to 3.4 give the polynomial multipliers for the cases of n = 1 to 7. Bandwidth has been specified as a percentage (difference between upper and lower frequency divided by the average) in addition to the ratio of upper to lower frequency, where n = 1. The tables also give the corresponding amplitude imbalance expressed in decibel form. Solutions up to a 2-dB amplitude imbalance are sufficient to cover most applications. The polynomial multiplier values correspond to (3.1), and these will be required for synthesis. It can be seen by comparison of these tables that bandwidth improves approximately arithmetically with order.
3.3 Rational Form of F ( ) The polynomial form of F( ) leads to transfer functions where right-hand plane zeros of transmission are certainly excluded. The lowpass prototype circuit derived from it can therefore be synthesized as a ladder network, which has the advantage of simplicity. Extending the scope of F( ) to include Table 3.1 First-Order Polynomial Function Bandwidth Ratio
Bandwidth (%)
Imbalance (dB)
a1
1.05
5
0.21
1.0247
1.1
10
0.41
1.0488
1.15
14
0.61
1.0724
1.2
18
0.79
1.0954
1.25
22
0.97
1.118
1.3
26
1.14
1.1402
1.35
30
1.3
1.1619
1.4
33
1.46
1.1832
1.45
37
1.61
1.2042
1.5
40
1.76
1.2247
1.55
43
1.9
1.245
Approximations
39
Table 3.2 Third-Order Polynomial Function Bandwidth Ratio
Imbalance (dB)
a1
a3
1.2
0.054
1.6443
–0.6505
1.4
0.182
1.7789
–0.7997
1.6
0.352
1.9057
–0.9455
1.8
0.545
2.026
–1.0868
2.0
0.748
2.1407
–1.2233
2.2
0.957
2.2505
–1.3548
2.4
1.165
2.3559
–1.4815
2.6
1.371
2.4575
–1.6036
2.8
1.573
2.5557
–1.7213
3.0
1.771
2.6506
–1.835
Table 3.3 Fifth-Order Polynomial Function Bandwidth Ratio
Imbalance (dB)
a1
a3
a5
1.5
0.044
2.3023
–2.2423
0.9451
1.8
0.13
2.5292
–2.8669
1.3528
2.0
0.209
2.6716
–3.2863
1.639
2.5
0.456
3.0033
–4.3256
2.3762
3.0
0.745
3.3076
–5.3326
3.1145
3.5
1.048
3.5905
–6.2965
3.8342
4.0
1.354
3.856
–7.2144
4.527
4.5
1.653
4.1069
–8.0875
5.1903
4.8
1.828
4.2513
–8.591
5.574
rational functions introduces the prospect of transmission zeros, with the possibility that these should occur as right-hand plane zeros. As a consequence, the prototype circuit can no longer have ladder form and must
40
Lumped Element Quadrature Hybrids
Table 3.4 Seventh-Order Polynomial Function Bandwidth Ratio
Imbalance a1 (dB)
2.0
0.061
3.1225
2.5
0.172
3.4959
3.0
0.328
3.8476
–10.003
3.5
0.515
4.175
–12.0398 16.6167
–7.8094
4.0
0.719
4.4817
–14.0335 20.233
–9.7617
4.5
0.932
4.7742
–15.9717 23.8049
–11.7092
a3
a5
a7
–5.9111
6.2301
–2.4385
–7.9467
9.4999
–4.0687
13.0093
–5.891
5.0
1.149
5.0516
–17.8486 27.3017
–13.6287
5.5
1.365
5.3169
–19.6626 30.7073
–15.5071
6.0
1.579
5.5715
–21.4146 34.0142
–17.3374
6.5
1.789
5.8166
–23.1067 37.2204
–19.1164
involve coupling, bridging, or lattice networks. It is anticipated that the increase in complexity required for the lowpass filter prototype will yield a commensurate increase in the complexity of the final quadrature hybrid. It has already been noted that quadrature hybrids of the approximate-phase variety must make use of nonladder forms, but the question to be answered is: Does the improved performance justify the increased complexity? As an introduction to the consideration of rational forms of F( ), consider its form as rendered here:
F ( ω) =
ω2 ω2 Aω1 + 2 1 + 2 K ωn 1 ωn 2 ω2 ω2 1 + 1 + K ω 2d 1 ω 2d 2
(3.2)
For the purposes of this investigation, we will suppose that the orders of the numerator and denominator differ by only one. Furthermore, the natural frequencies specified in the equation will be magnitude-ordered d1, n1, d2, n2, and so forth. These, and the constant multiplier A, must
Approximations
41
be chosen so that the function evaluated between specified frequency limits should ripple between geometric limits m and 1/m about unity. For this analysis, the order of the function will be defined as the higher of the orders of the numerator or denominator functions. Consider the function in a qualitative sense. As increases from zero, F( ) initially increases approximately linearly from zero, being dominated by the first-order nature of the numerator. As increases further, the first quadratic factor in the denominator then begins to take significant effect, causing the function first to level off, then eventually begin to decrease again. If F( ) were just a second-order function, it would diminish asymptotically towards zero. In order to increase bandwidth, we would like the first quadratic factor in the numerator to take effect before F( ) has significantly decreased in value. This causes the function once again to level off and then increase. Further alternate factors in the denominator and numerator increase bandwidth even more. Eventually, the function tends towards zero for a higher-order denominator or infinity for a higher-order numerator. F (ω)
m 1 1/m
0.5
0
ωL
1
ω
Figure 3.2 Fifth-order rational function example of F ( ).
ωU
42
Lumped Element Quadrature Hybrids
To illustrate the effectiveness of such a function, consider a fifth-order function with natural frequencies and constant multiplier determined for optimum performance over a 25:1 bandwidth. Figure 3.2 shows the function; the frequency scale has been normalized for a geometric center of unity. The ripple corresponds with a 0.164-dB amplitude imbalance or 1.1° phase error. This is significantly better than the polynomial forms described in Section 3.2. It can be shown that the function generated by this procedure is optimum, in the sense that the ripple is minimized within the passband and for the order n given. Suppose there is another function G( ), an odd rational function of frequency equal to zero when = 0, of undefined order. Suppose, too, that this new function is an improvement on the original function in that its ripple is less. Figure 3.3 shows such an improved function, where the original function is third order. Let F ( ω) =
p ( ω) q ( ω)
F (ω) m
G(ω)
1/m
0
ωL
ω
Figure 3.3 F ( ) with G( ) “improvement.”
ωU
Approximations
43
and G ( ω) =
r ( ω) s ( ω)
In order to improve upon the first, the new function must cross it within the passband n times. As both functions are odd, there will be a further n crossings of opposite polarity. In addition, the two functions must cross when = 0. The crossings represent points at which the two functions are equal (see Figure 3.3). Therefore, there will be at least 2n + 1 solutions to the equation: p ( ω)s ( ω) − q ( ω)r ( ω) = 0 As the function F( ) is only of order n, either the numerator or denominator of G( ) must be at least of order n + 1 to give the required number of solutions to the above equation. Hence, a better solution must be of higher order than the original solution, which therefore must be optimum. This analysis can be extended to consider only functions that are equal in performance to the original function. Once again, the number of crossings requires higher order for the new function, so the original must be unique as well as optimum. Further properties can be deduced regarding the optimum function. To see these, assume the passband is normalized such that the lower and upper frequency limits L and U are geometrically placed about unity, so that L U = 1. Figure 3.4(a, b) shows the options of odd and even order, respectively. Consider now the lowpass-to-highpass transformation of F( ), given by
1 F = ω
A
1 1 1 1 + 2 2 1 + 2 2 K ω ω ωn 1 ω ωn 2 1 1 1 + 2 2 1 + 2 2 K ω ωd 1 ω ωd 2
(3.3)
The new function can be restored to the ratio of two polynomials by multiplying the numerator and denominator by n. The new function shares certain properties with the original function in that it ripples between the same limits and the same number of times within the passband.
44
Lumped Element Quadrature Hybrids ∞
∞
m
m
1/m
1/m
0 0
ωL
∞
1 (a)
ωU
∞
0
m
1/m
1/m
0
ωL
1 (c)
ωU
∞
0
ωL
1 (e)
ωU
∞
∞
ωL
1 (b)
0
ωL
1 (d)
∞
m
0
0
0
ωU
∞
ωU
∞
m 1/m
0
Figure 3.4 Transformations of the optimum rational function: (a) F (ω), n odd; (b) F (ω), n even; (c) F (1/ω), n odd; (d) F (1/ω), n even; and (e) 1/F (1/ω), n odd.
Approximations
45
Suppose now that n is odd. F (1/ ) decreases from infinity as goes from zero to L, whereupon it takes a value of m. Thereafter, it oscillates between 1/m and m until = U, where it takes a value of 1/m, as shown in Figure 3.4(c). Consider now the reciprocal of F (1/ ), which increases from zero to 1/m as goes from zero to L. It also oscillates between 1/m and m until = U, where it takes a value of m, as shown in Figure 3.4(e). Its oscillation pattern is precisely the same as the original function seen in Figure 3.4(a). Now, it has been established that such a function is unique; therefore,
( )
1 F 1ω = , n odd F ( ω)
(3.4a)
Suppose now that n is even. This time the pattern of F (1/ ) is exactly the same as the original unique function, as shown in Figure 3.4(d). It must therefore be identical, so
( )
F 1 ω = F ( ω), n even
(3.4b)
It was noted during the analysis of the polynomial form of the filtering function that the solution depended on numerical analysis. The same is not true of the optimum rational function. However, before the solution for general n is described, the particular case of n = 2 will be investigated as it may be solved algebraically.
3.4 Second-Order Optimum Rational Function When n = 2, (3.2) simplifies to F ( ω) =
Aω ω2 1+ 2 ωd 1
From (3.4b), Aω ω2 +
1 ω d2 1
=
Aω ω2 1+ 2 ωd 1
46
Lumped Element Quadrature Hybrids
It is necessary therefore that d1 = 1. The only remaining quantity to determine is the numerator multiplier A. This needs to be chosen in a manner to optimize the filtering function over the passband. The optimized function gives F (1) = m and F ( U) = 1/m. Substituting these conditions into the equation for F( ) gives rise to a pair of simultaneous equations in A and m, the solutions to which give 1 A = 2 ωU + ωU and ωU + m=
1 ωU
2
Now, U2 is the bandwidth ratio, and m is a measure of the maximum amplitude imbalance according to (2.23), so it is possible to tabulate imbalance against bandwidth ratio as was done for polynomial forms of F( ). Table 3.5 shows the result. Comparing the result with the polynomial forms Table 3.5 Second-Order Rational Function Bandwidth Ratio
Imbalance (dB)
A
1.5
0.177
2.0205
2.0
0.256
2.0598
2.5
0.441
2.1041
3.0
0.625
2.1491
3.5
0.801
2.1933
4.0
0.969
2.2361
5.0
1.276
2.3166
6.0
1.55
2.3907
7.0
1.795
2.4592
8.0
2.017
2.5227
Approximations
47
given in Tables 3.1 to 3.4, we see that the new function is an improvement over the third-order case for any bandwidth ratio, better than the fifth-order case with bandwidth ratios greater than 2.5:1 and better even than the seventh-order case with bandwidth ratios greater than 6:1. These results indicate the potential bandwidth advantage of rational forms of F( ). It is interesting to compare the second-order rational function with the corresponding function in the formula for the transfer function of a singlesection transmission line coupler. The magnitude of the through transfer function for a transmission line coupler, with electrical length and centerfrequency coupling factor c, is given as [2] s 41 =
=
2
=
1− c 2
(1 − c 2 )cos 2 θ + sin 2 θ
1 c sin 2 θ 1+ 1− c 2 2
1 tan θ 2 1+ 2 (1 − c ) 1 + tan 2 θ 2 4c
2
2
Now, compare this equation to (2.20). In the above result, a filtering function corresponding to F( ) with similar characteristics to the secondorder rational function can be inferred. The transmission line version can be derived from the lumped element version by the following substitutions: A=
2
1− c 2 ω = tan θ 2 The frequency transformation is of the Richard’s type. It can be seen that the lumped element version gives a bandwidth improvement over the distributed case. For example, a distributed coupler designed to operate over an octave bandwidth will be designed for an electrical length ranging from 60° to 120° (assuming the media is not dispersive). Substitution of these val ues into the Richard’s transformation gives values for of 1/√3 and √3, or a 3:1 bandwidth ratio.
48
Lumped Element Quadrature Hybrids
3.5 Higher-Order Optimum Rational Functions Where the optimum rational function is of the third order or higher, algebraic solutions become impractical. However, it is possible to address the solution analytically. Let F ( ω) =
p ( ω) q ( ω)
as before, and differentiate with respect to , giving F ′( ω) =
q ( ω) p ′( ω) − p ( ω)q ′( ω) q 2 ( ω)
(3.5)
In the original F( ), either p( ) or q( ) is of order n, with the other of order n –1, so the numerator of (3.5) must be of order 2( n –1). There is a maximum of 2(n –1) zeros of F ′( ω). The zeros of F ′( ω) can be identified as the turning points of F ′( ω). Now, consider for example the plot shown in Figure 3.2. The method used to construct the function gives rise to n –1 turning points in the portion of graph shown. As the function is odd, there must be a further n –1 turning points in the negative half of the function. There are therefore 2(n –1) real zeros of F ′( ω), and they are all simple. The next stage in the analysis examines the functions that arise when certain constants are added or subtracted from F( ), so that chosen maxima and minima points of the new function make a tangent with the x-axis. Consider first of all the function obtained by subtracting m from F( ), giving F ( ω) − m =
p ( ω) − mq ( ω) q ( ω)
(3.6)
Inspection of the numerator of the right-hand side of (3.6) reveals it to be of order n. Inspection, too, of the graph in Figure 3.2 shows there to be zeros of this function (formed by shifting the x-axis to the line y = m) interpreted as places where the function either makes a tangent to or crosses the line y = m. The points where a tangent occurs are double zeros. If n is of odd order, there will be an additional single zero corresponding to the point where = U. Regardless of whether the order is odd or even, there will be n real zeros (with double zeros counted twice). Therefore, all the zeros of (3.6) are real.
Approximations
49
A similar analysis can be made by subtracting 1/m from F( ). The resulting equation is F ( ω) − 1 m =
mp ( ω) − q ( ω) mq ( ω)
(3.7)
Once again, the numerator is of order n, and the zeros correspond to points where F( ) meets or crosses the line y = 1/m. These will occur as double zeros, except for a single zero where = L and a further possible single zero where = U, should n be even. With double zeros counted twice, there are also n real zeros of (3.7). Two more functions are of interest when the negative portion of the graph is considered. The two equations become F ( ω) + m =
p ( ω) + mq ( ω) q ( ω)
(3.8)
F ( ω) + 1 m =
mp ( ω) + q ( ω) mq ( ω)
(3.9)
and
Both of these equations also have n real zeros by the same arguments given for (3.6) and (3.7). The next stage in the analysis considers what happens when the functions in (3.6) to (3.9) are multiplied together. The multiplied function has double zeros at all the turning points, single zeros at the band edges, and quadruple poles corresponding to the poles of F( ). Compare this with the differential of F( ) as given in (3.5). This also possesses single zeros at the turning points and double poles corresponding to the poles of F( ), but no zeros at the band edges. As all of the poles and zeros are accounted for, it follows that the square of F ′( ) is equal to the product of (3.6) to (3.9), when divided by factors corresponding to the band edges and multiplied by some constant. The result is a differential equation given by
[F ′(ω)]
2
=
1 C
2
⋅
( F (ω) −1 m )( F (ω) + 1 m )( F (ω) − m )( F (ω) + m ) (ω − ω L )(ω + ω L )(ω − ωU )(ω + ωU )
(3.10)
50
Lumped Element Quadrature Hybrids
It is convenient to normalize the function such that making this substitution and simplifying gives dω ω2 2 2 1 − 2 (1 − ωU ω ) ωU
=
L
CdF F 2 2 2 1 − 2 (1 − m F ) m
= 1/
. So
U
(3.11)
The solution of (3.11) requires the integration of both sides and elliptic integrals. It is convenient for the purpose of analysis to define a parameter equal to the integral of each side of (3.11), which we shall call z in anticipation of examining its properties in the complex plane. The desired solutions to (3.11) express F in terms of a rational pair of polynomials in . Such solutions only exist for specific values of C and m. The starting point for the solution of (3.11) is to consider the incomplete elliptic integral of the first kind given by sn −1 ( y , k ) =
∫
y
0
dt
(1 − t 2 )(1 − k 2t 2 )
(3.12)
The integral is an inverse definition of sn(x,k), the Jacobian elliptic sine function, the properties of which depend on the value of k, known as the modulus. (When omitted, it is assumed to be some predefined value k.) The function simplifies to sin x when k = 0 and to tanh x when k = 1. For all values of k in between (and only values between 0 and 1 are relevant), the function is periodic for real x, oscillating between –1 and 1. A quarter-period of sn(x,k) is equal to the value of (3.12) when y = 1. This value is denoted by the letter K and is known as the complete elliptic integral of the first kind. When k = 0, K is equal to /2 and increases to infinity as k increases to 1. The function sn(x,k) is plotted for several modulus values in Figure 3.5. We can see that the function appears sinusoidal until the modulus approaches a value of 1. The complete elliptic integral can be defined as a function of the modulus k and is plotted in Figure 3.6. This function remains close to a value of /2 when k is small and only increases rapidly as k approaches 1. The reader should not be intimidated by the use of these functions. Elliptic functions give rise to more complicated expressions than, say, trigonometric functions, but for the purpose of this analysis, only a modest use is required. Only the aspects required for the present analysis will be described. For a more in-depth treatment, consult [3].
Approximations
51
1
0
1
3
2
4
5
x sn(x, 0.3) sn(x, 0.75) sn(x, 0.9) −1
Figure 3.5 The elliptic sine function.
Complete elliptic integral K (k)
3
2 π/2
1
0
0.5
k
1.0
Figure 3.6 The complete elliptic integral of the first kind.
The Jacobian elliptic sine function may be used to evaluate each side of (3.11). Consider first the left-hand side. This may be solved using (3.12) by the substitutions t = U and k = k1 = 1/ U2. Integration then gives
52
Lumped Element Quadrature Hybrids
z =
1 sn −1 ( ωU ω) + c 1 ωU
The constant of integration in this equation can be chosen arbitrarily at this stage as the variable z has been chosen arbitrarily as well. As sn–1 (0) = 0, it is convenient to assign a value of zero to c1. The formula for z in terms of then becomes z =
1 sn −1 ( ωU ω) ωU
(3.13)
As sn(x,k) oscillates between –1 and 1, real solutions to (3.13) only exist for values of between –1/ U and +1/ U. Such a restricted range is insufficient for analysis, as the properties within and above the passband are omitted. In addition, inspection of (3.2) indicates purely imaginary zero and pole locations. In order to encompass all values of of interest, it is necessary to consider the properties of z in the complex plane. It is well known that the trigonometric function sin x is periodic. The periodicity is evident for real argument. When the argument is complex, the periodicity can be seen as vertical strips in the complex plane, repeating with a period of 2 . The hyperbolic function tanh x, on the other hand, is not normally considered periodic because a real argument is assumed. However, it exhibits periodic characteristics when the argument is imaginary as it can be proved readily that tanh jy = j.tan y. The Jacobian elliptic function sn(z,k) becomes one of these two functions when k = 0 (sin z) and the other when k = 1 (tanh z). For all other modulus values, sn(z,k) has the curious property of being periodic in both the real and imaginary directions. It is said to be doubly periodic. In the real direction, the function sn(z,k) has a period of 4K. The period in the imaginary direction depends upon a related quantity, K ′. This √(1 – k2), the complementary can be determined by replacing k with k modulus, in (3.12) and integrating between the limits 0 and 1. The period of sn(z,k) in the imaginary direction is equal to 2jK . A rectangle in the complex plane with a length of 4K in the real direction and a height of 2K in the imaginary direction constitutes a period-parallelogram for sn(z,k). The term parallelogram is used because, for elliptic functions in general, the sides may not be aligned with the complex plane axes. In the case of the Jacobian elliptic sine function, the parallelogram simplifies to a rectangle. It is possible to divide the whole z-plane into a grid of adjacent period-parallelograms
Approximations
53
so that, for every point within any given parallelogram, there exists a corresponding point in every other parallelogram such that its position relative to its own parallelogram sides is exactly the same as the original point, whose elliptic sine is exactly the same. Such points are said to be congruent. The alignment of the grid is arbitrary, although it is usually convenient to place the sides coincident with integer multiples of the real and imaginary quarter-periods. In every period-parallelogram, there are two points whose elliptic sines take the same value. Thus, there are two zeros and two poles. The zeros occur at z = 0 and z = 2K and all congruent points, and the poles occur at z = jK and z = 2K + jK ′ and all congruent points. It is sometimes useful to divide the period-parallelogram further into regions that map onto the entire complex plane only once with the operation sn z. Such a region is known as a fundamental region, and two are encompassed within a period-parallelogram. In general, complex z will give complex sn z. However, there are lines in the complex plane along which sn z is either purely real or purely imaginary. When z is real, sn z is real. In addition, when y in z = x + jy is an integer multiple of K ′, sn z is also real. When x is an even multiple of K, sn z is imaginary, and when x is an odd multiple of K, it is real. The evaluation of sn z in these circumstances makes use of two further elliptic functions related to the elliptic sine function. One of these, known as the elliptic cosine function, is given as cn z = √(1 – sn2 z). Its properties for real arguments are thus rather cosinelike, just as the elliptic sine function is sinelike. The other function, known as the elliptic difference function is given as dn z = √(1 – k2sn2 z). For real arguments, this function oscillates between 1 and k′. Having introduced the properties of sn z in the complex plane, it is possible to trace a locus in the z-plane that maps onto a line from zero to infinity under the operation. The first segment is where z = 0 to K, and this maps onto the line segment 0 to 1. At this point, sn z reaches a maximum for real z, so in order to increase the value, it is necessary to branch into complex values. The next segment is where z goes from K to K + jK . Along this line, sn (K + jy) = 1/dn(y,k ). Thus, the segment maps onto the line segment 1 to 1/k. If the locus were to continue along this line in the z-plane, the mapping of sn z would repeat along the line segment 1 to 1/k. This property will be of interest later. For now, the value of sn z as z goes from K + jK to jK is of interest. Along this segment, sn(x + jK ) = 1/(k sn x). Thus, the segment maps onto the line 1/k to ∞. The locus of points as described in the z-plane maps onto a line from zero to infinity under the operation, with the value of sn z increasing monotonically as the locus is traced. Table 3.6 shows this behavior.
54
Lumped Element Quadrature Hybrids
Table 3.6 Behavior of sn z Along Straight Lines in the Complex Plane
z
sn z
0→ K
0→ 1
K → K + jK
1 → 1/k
K + jK → jK
1/k → ∞
The properties of the elliptic sine function in the complex plane can now be applied to ω as a function of z, as given by (3.13). This may be inverted to give ω=
1 sn ( ωU z , k1 ) ωU
(3.14)
The first segment of interest is along the line 0 to K1/ U, which maps onto the segment 0 to 1/ U in the -plane; this is shown in Figure 3.7(a) as OA. This segment corresponds to frequencies up to the lower passband edge. The second segment AB continues with z increasing in the imaginary direction up to the point (K1 + jK1′)/ωU, and this maps onto the segment 1/ U to -plane. This segment corresponds to the passband frequencies. U in the The final segment, BC, with z decreasing in the real direction to the point jK1′/ωU, maps onto the segment U to ∞ in the -plane. This segment corresponds to frequencies above the upper passband edge. Figure 3.7(a) shows the locus for a value of k1 equal to 0.25 ( U = 2). Figure 3.7(b) shows the mapping to the -plane. The locus has been unfolded into a straight line running along the abscissa line of a Cartesian graph; thus, the transfer function between the two is depicted as a graph. Having examined the solution to the left-hand side of (3.11), we now turn our attention to the right-hand side, which, similarly, may be solved by means of (3.12), this time using the substitutions t = mF and k = kn = 1/m2. Integration gives z =
C −1 sn (mF ) + c n m
(3.15)
Approximations
55
C
jK1′/ωU
B
z-plane
A O
K1 /ωU (a)
8 ω-plane
C′
ωU
1/ωU O
B′
A′
A
B (b)
Figure 3.7 (a) z-plane locus. (b) z to ω transfer function.
C
56
Lumped Element Quadrature Hybrids
The constant of integration cn cannot be assigned arbitrarily this time, but must be consistent with the evaluation of z in terms of . However, as F = 0 when = 0 and z = 0, cn must be either 0, 2Kn, or any congruent value. Setting cn = 0 means that F is positive going when z and are positive going close to zero; this is the preferred option. Thus, (3.15) immediately simplifies to z =
C −1 sn (mF ) m
(3.16)
The function F can also be expressed in terms of z as F =
1 zm sn m C
(3.17)
Each of the segments in the z-plane corresponding to specific segments in the -plane also have significance in the F-plane. Consider first the segment along the real axis. Along this segment, the mapping in the -plane was from zero to the lower band edge. In the F-plane, the desired mapping is from zero to 1/m. It is possible to arrange this by scaling the graph such that z = CKn/m at the right-hand end of this segment. Continuing the locus in the imaginary direction, as z goes from CKn/m to CKn/m + jCKn′/m, the mapping into the F-plane increases from 1/m to m. At this point, it would be desirable for the locus to continue in the imaginary direction so that F can ripple between 1/m and m the desired number of times. We might suppose this to be possible with a suitable choice of C and m. Once the locus in the z-plane has reached its limit in the imaginary direction, it is necessary that it correspond with an integer multiple of CKn′/m so that the final locus segment back to the imaginary axis will correspond with a mapping to real values in the F-plane. In fact, the integer multiple is equal to the order of F. If the integer is even, the locus terminates at a place on the imaginary axis that maps onto zero in the F-plane. If the integer is odd, the locus terminates at a place on the imaginary axis that maps onto infinity in the F-plane. To illustrate the nature of this function, suppose F has been devised, by suitable choice of C and m, such that its locus when mapped back onto the z-plane is the same as that shown in Figure 3.7(a). Figure 3.8 shows the new mapping. Figure 3.8(b) shows the mapping to the F-plane, where the ordinate axis is used once again to represent real values in the F-plane. The line segment OA maps onto the line segment 0 to 1/m. The line segment AB
Approximations
57
C
jnCKn′/m
B
z-plane
A O
CKn /m (a)
8
F-plane
C″
m 1/m
B″ A″
O
A
B (b)
Figure 3.8 (a) z-plane locus. (b) z to F transfer function.
C
58
Lumped Element Quadrature Hybrids
maps n times onto the line segment 1/m to m. The line segment BC maps onto the line segment m to infinity. Once again, the locus in the z-plane has been unfolded for convenience to illustrate the transfer function between the two as a graph. The new function has been configured to ripple three times. The imaginary axis in Figure 3.8(a) has been marked with small circles and crosses at integer multiples of jCKn′/m. These correspond with the zeros and poles of F and are of interest later. It is interesting at this point to compare the analysis so far with the lowpass elliptic filter problem [4]. The differential equation given by (3.11) is very similar to the one required in the course of the lowpass elliptic filter problem’s solution. The difference is in the mappings used. For the quadrature hybrid solution, the conversion from to F is achieved by the compression of cycles in the imaginary direction in the z-plane. If solutions to the elliptic filter problem are required, then the compression takes place in the real direction. The problem then is to determine the values of C and m that give the desired result. We arrive at a partial solution by considering the ratio of the lengths of the sides of the rectangle described by the locus in the z-plane. This is either K1′/K1 or nKn′/Kn, depending on whether the solution of (3.11) is for the left-hand or right-hand side (see Figures 3.7 and 3.8). These two ratios must be the same so that nK n′ K 1′ = Kn K1
(3.18)
Equation (3.18) is worthy of closer examination. The left-hand side is a function of both n, the order of the proposed solution, and kn, hence, of m, the ripple factor. The right-hand side is a function of k1, hence, ωU, the specified bandwidth. If the order n and the bandwidth are specified, then it is possible to use (3.18) to determine the ripple. The solution requires consideration of the ratio of complete elliptic integrals. Fortunately for this analysis, these ratios have received a certain amount of attention in the world of mathematics as they appear in other mathematical problems. They are required, for example, in the solution of elliptic filter problems and in conformal transformation problems. In principle, it is possible to solve (3.18) in terms of the modulus by translating it into algebraic equations. Unfortunately, these solutions become of high order for even modest n. The solution is relatively simple when n = 2, where it becomes
Approximations
k2 = Now, substituting k1 = 1/ ing gives m=
2 U
59
2 k1
(3.19)
1 + k1
and k2 = 1/m2 into (3.19) and simplify-
1 1 + ωU 2 ωU
(3.20)
Equation (3.20) is the same as determined in Section 3.4 for the second-order case. Approximate formulae exist for determining K ′ K ; the following are accurate to within 3 ppm: K′ 1 1+ k ′ (k ) ≈ ln 2 , k <1 K π 1− k ′
(3.21a)
2
−1
1+ k K′ (k ) ≈ π ln 2 , k >1 K 1− k
2
(3.21b)
These equations may be inverted to give π KK ′ − 2 e k′ ≈ K ′ πK e + 2
2
π KK ′ − 2 e k≈ K πK ′ e + 2
2
, K ′ K >1
(3.22a)
, K ′ K <1
(3.22b)
A simpler approximation is available; though less accurate, it is useful for illustrating a point and is sufficiently accurate for many applications. The approximation is K′ 2 4 ≈ ln K π k
(3.23)
60
Lumped Element Quadrature Hybrids
This approximation is accurate to within 1 part in 300 for a value of k of 0.2, and it improves as k decreases to zero. Such an approximation is appropriate for the right-hand side of (3.18) for a bandwidth ratio greater than 5:1. As k1 = 1/ U2, then the bandwidth ratio BW = 1/k1. Making this substitution into (3.23) and then substituting into (3.18) for the right-hand side leads to 1 BW ≈ e 4
n π K ′n 2 K n
(3.24)
Equation (3.24) shows that bandwidth for the optimum rational function increases exponentially with order. The improvement in bandwidth for a unit increase in order will vary according to the demanded ripple. This property makes these functions of special interest for broadband quadrature hybrids and contrasts with many other families of design, such as distributed circuits, where the improvement in bandwidth with order is relatively slight. Another solution to (3.19) is available that is theoretically exact. The solution depends upon another function called the elliptic nome function q, defined as a function of the modulus by the equation q (k ) = e
−π
K ′ K
(3.25)
This function is sometimes available in mathematical software, or it may be available in tables of elliptic functions, albeit expressed in the form log10q versus sin–1 k. Having defined this function, it is then possible to express the elliptic modulus k itself as a function of q, as k(q). Now, take (3.19), multiply by , take the exponent, and make the substitution using (3.25) to give
(q n )
n
= q1
(3.26)
From (3.26), taking the nth root gives qn, which can be inverted using the elliptic modulus function to give kn, hence, the ripple factor m. This is the maximum value of F( ) in the passband, so by using (2.23), the amplitude imbalance can be determined. In summary,
{
∆(dB) = −10 log 10 k ( n q 1
)}
(3.27)
Approximations
61
and 1 q 1 = q (k1 ) = q BW Most designers will have little interest in the mathematical details. For them, a simple graph showing the amplitude imbalance or phase error against relative bandwidth is of more value. Figure 3.9 shows such a graph, as well as the exceptional bandwidth achieved with a modest order of circuit. It is useful to illustrate the analysis so far by means of an example. Example 3.1 Determine the minimum order of function necessary to achieve
a 0.5-dB ripple specification, where the target quadrature hybrid is required to operate over a decade bandwidth. Determine the theoretical ripple. Solution The maximum value of F( ) is m, and so, by (2.23), m is required
to be less than 1.059. Therefore, kn = 0.891 [see the text preceding (3.15)]. The bandwidth ratio of 10:1 corresponds with k1 = 0.1. Determination of the minimum order of function requires the solution of (3.18). With the required values of k1 and kn available, complete elliptic 1.8
n=2
1.6
4
1.2
8
5 6
1.0
6
7 0.8 8 0.6
4
0.4 2 0.2 0.0
1
10
100 Bandwidth ratio
1,000
10,000
Figure 3.9 Graphs of quadrature error with bandwidth for rational functions.
Quadrature error (°)
Amplitude imbalance (dB)
10
3
1.4
62
Lumped Element Quadrature Hybrids
integrals may be computed, or the approximations of (3.21) may be used. The approximations will be sufficiently accurate, giving K1′/K1 = 2.347 and Kn′/Kn = 0.7407. This gives a value for n of 3.17. As fractional order is not permitted, the minimum integer value is 4. To determine the actual ripple over a 10:1 bandwidth, (3.27) may be used. Evaluation of the elliptic nome function gives q(0.1) = 0.000628, for which the fourth root is 0.158. Now, take the elliptic modulus function for this value, giving kn = 0.9629, hence, ∆ = 0.164 dB. ❂❂❂
The previous example illustrates the improvement that can be made by an increase in order (0.164-dB ripple against the specification value of 0.5 dB). In practice, given the specification, the designer might well have considered relaxing the ripple specification in order to simplify to a third-order solution. In this case, the ripple is 0.63 dB. Rather than going through the complication of computing all of the functions, it would have been much quicker to look up the required information using Figure 3.9. The value of C is not of interest at present (in fact, in practical circuit design, it is not needed at all). Comparing (3.2) with (3.11) when and, hence, F tend towards zero, it can be seen that C = 1/A. This constant multiplier will be determined later. The mappings between the -, F-, and z-planes can be used to locate the zero and pole locations of F( ). The first step is to identify a fundamental region of the z-plane for the mapping implied in (3.14) in order to be certain to include all possible values of without repetition. A convenient region for this purpose is the rectangle bounded by the lines x = ±K1/ U and y = ±K1′/ U. As the mapping of F onto the z-plane involves n cycles in the imaginary direction for every cycle of the mapping, there will be repetitions in the z-plane region just defined. Of particular interest are points that map onto F = 0 and F = ∞, the zero and pole locations. These occur at regular intervals alternately along the imaginary axis. The interval is given by CKn′/m, or K1′/n U [see Figure 3.8(a)]. For the zeros, the required solutions to z are given by z zr =
2rK 1′ nωU
1 1 r = − (n − 1) K (n − 1), n odd 2 2 n n r = − − 1 K − 1 , 2 2
n even
(3.28)
Approximations
63
Having determined the locations of the zeros in the z-plane, (3.14) can be used to determine the corresponding values of . There is a complication in that it is necessary to determine the elliptic sine of an imaginary quantity. Such a function can be determined in terms of functions of a real variable, just like the trigonometric sine function. The elliptic sine of an imaginary quantity is given by sn ( jy , k ) = j
sn ( y , k ′ )
(3.29)
cn ( y , k ′ )
The ratio of the elliptic sine and cosine is tangentlike in appearance. It is sometimes abbreviated to the shorthand notation sc x. As we will use this function frequently, we will adopt this convention here. The solutions to are purely imaginary, and inspection of (3.2) indicates that this is the desired result. The zeros occur in pairs of opposite polarity, so it is sufficient to consider positive solutions. Thus, making the substitution for z in (3.14) gives the zero locations as ω nr =
1 2rK 1′ sc , k1′ ωU n
1 r = 0K (n − 1), n odd 2
(3.30)
n r = 0K − 1 , n even 2 The pole positions in the z-plane occur at points exactly halfway between the zero locations, so they are given by z pr =
( 2r − 1)K 1′ nωU
1 1 r = − (n − 3 )K (n − 1), n odd, and ≥ 3 2 2 (3.31) r=−
(n − 2 ) 2
n K , 2
n even
The corresponding values in the -plane can be determined in the same manner as for the zero locations. They are given by ω dr =
1 ( 2r − 1)K 1′ sc , k1′ n ωU
1 r = 1K (n − 1), n odd 2 (3.32) n r = 1K , n even 2
64
Lumped Element Quadrature Hybrids
It has been discovered therefore that the zeros and poles of F ( ) are all imaginary. This confirms the assumption made in the form of (3.2). Having found the zero and pole locations, the value of the constant multiplier may now be determined. The procedure is simpler when n is odd. Owing to the reciprocal frequency properties of the optimum function, as given by (3.4a), for every pole location, there is a corresponding reciprocal zero location; therefore, zr = 1/ dr. In addition, F(1) = 1, which allows us to write F ( ω) = ω ⋅
1 ( n −1 ) 2
Π r =1
ω 2 ω 2dr + 1 , n odd ω 2 + ω 2dr
(3.33)
For the case when n is odd then, there is no need to evaluate the constant multiplier or the zero positions separately. When n is even, the constant multiplier requires evaluation. Evaluating z in terms of and F separately at the end of the first segment in the locus of z, and recalling C = 1/A, it follows that A=
ωU K n mK 1
(3.34)
The solution of F( ) when n is even is thus complete and is given by (3.2), where A is given by (3.34), nr by (3.30), and dr by (3.32).
3.6 Transfer Functions in Terms of the Complex Variable The expressions of approximation used in this chapter up to now have been in terms of the real frequency variable . Such expressions are useful in analysis to determine the amplitude against frequency of the approximation under consideration. However, when the problem of synthesis is considered, a more useful variable is the complex variable s. Techniques for solving the transfer functions in terms of s were demonstrated in Section 2.3, where approximate-amplitude hybrids are concerned, and in Section 2.4 where approximate-phase hybrids are in view. The transfer functions have an additional quality useful in analysis in that the insertion phase can be determined upon substitution of s = j . For approximate-amplitude hybrids based on polynomial forms as described in Section 3.2, (2.21) may be used to determine the numerator functions, followed by (2.18) to determine the common
Approximations
65
denominator. Chapter 4 will show that synthesis of element values for this kind of hybrid can be made more directly using the filtering function, so the transfer functions are only useful for analysis. Like the polynomial expressions of the filtering function, the transfer functions require numerical analysis. On the other hand, the optimum rational function forms, having already been examined using analytic functions, can also have their transfer functions determined by analysis. For problems that use the optimum rational function, the denominator part of the transfer functions is determined first. For the transfer functions of approximate-amplitude hybrids, it will be shown that the numerator parts can be derived very simply from the denominator. Where approximate-phase hybrids are concerned, the numerator follows directly, using (2.25). Consider first the approximate-amplitude case. The denominator function determines the poles of the hybrid, or the contrived frequencies where infinite response occurs. In terms of the real frequency expression, this is given by [see (2.20)]: 1 + F 2 ( ω) = 0
(3.35)
Thus, it is necessary to solve the equation F ( ω) = ± j Just as the parameter z was found useful as an intermediate variable to map between F( ) and to find the zeros and poles of the former, so it is also useful in determining the zeros of (3.35). Making use of (3.17) and noting that m = 1/√kn, we have j mz sn = ± C kn The expression in brackets, fortunately, has a simple solution, given by jK n′ jK n′ mz , 2K n ± =± 2 2 C and all congruent points. Only the solutions within the fundamental region with respect to the mapping to the -plane are of interest, that is to say, only a rectangle
66
Lumped Element Quadrature Hybrids
bounded by the lines x = ±K1/ U and y = ±K1′/ U or, equivalently, x = ±CKn/m and y = ±nCKn′/m. The only solutions within this rectangle are along the imaginary axis, and when expressed in terms of the mapping to the -plane, they are given by z = ± j (r −
1 2
)
K 1′ nωU
r = 1K n
(3.36)
Equation (3.14) may be used to map onto the -plane, and then, upon substitution of s = j , the solutions in the s-plane become σ ±r = ±
1 sc (r − ωU
1 2
)
K 1′ , k1′ n
r = 1K n
(3.37)
In (3.37), the positive subscript on the left-hand side is assumed to correspond to the positive sign on the right-hand side. Equation (3.37) gives positive and negative values, and as the sc (x) function is odd and within the range given, the positive sign indicates a net positive value for . We see then that the poles of the transfer functions are all real. Equation (3.37) gives 2n solutions, corresponding to both D(s) and D(–s). The solutions required are those appearing in the left-hand s-plane so that the denominator function is given by n
D (s ) = Π( σ r + s ) r =1
(3.38)
A simplification of the procedure to determine the roots is available. It can be confirmed by algebraic manipulations of elliptic functions that σr =
1 σ n +1 −r
(3.39)
The number of elliptic function calculations is thus divided by exactly 2, where n is even, and by slightly more than 2, where n is odd, as the central value is always 1. For exact solutions, the formula of (3.37) must be used to determine the frequency parameters r. In practical applications, approximate solutions are sufficient as errors are usually dominated by component tolerance. It is
Approximations
67
more useful to express (3.37) in graphical form, as shown in Figure 3.10. Use of (3.39) means that only the first few values need to be plotted.
3.7 Numerator Part of the Optimum Transfer Functions One method of determining the numerator polynomials, as in (2.13) and (2.14), is to make use of the zeros and poles of F( ). The required expressions in terms of s can be determined using (2.21), with the substitution s = j . If the values haven’t already been determined, then this will involve additional calculations of elliptic functions. Having found the zeros of the numerator functions, it will then be necessary to determine the constant multiplier for the coupled port transfer function. The constant multiplier for the through port transfer function is easily determined at dc. Similarly, when n is odd, the constant multiplier for the coupled port transfer function is 0.07
0.7 σ2, n = 4
Frequency parameter
0.6
0.06
0.5
0.05
0.4
0.04 σ1, n = 2
σ2, n = 5
0.3
σ 1, n = 5
0.03
σ1, n = 3
0.02
0.2 σ1, n = 4
0.1
0.01
σ1, n = 5
1
10
100 Bandwidth ratio
Figure 3.10 Graph of frequency parameters.
1,000
10,000
68
Lumped Element Quadrature Hybrids
easily determined as, when frequency tends towards infinity, the magnitude tends towards one. When n is even, the constant multiplier for the coupled port is more difficult to evaluate. Fortunately, there is another way to determine both numerator polynomials, using just the information already obtained for the denominator. To see how this is done, consider first the qualitative properties of the numerator polynomial function. When mapped onto the z-plane, the zero positions corresponding to the zeros and poles of F( ) alternate along the imaginary axis. Now, mapping these points onto the -plane, the points also alternate along the imaginary axis as the mapping function is monotonic over the locus of interest. With the substitution s = j , the zeros alternate along the real axis. This means that the zeros of both transfer functions are all real. Consider now the relationship between the numerator and denominator functions arising from the lossless criterion, as given by (2.18). Factorizing the left-hand side gives
(N e (s ) + N o (s ))(N e (s ) − N o (s )) = D (s )D ( −s )
(3.40)
The zeros of the first factor on the left-hand side of (3.40) correspond with instances of s where the two numerator functions are equal in magnitude but opposite in sign. The zeros of the second factor correspond with instances where they are equal in magnitude and sign. Now, consider the s-plane real axis divided into consecutive segments defined by the zeros of the numerator functions, as shown in Figure 3.11. In any particular segment, there must be a point where the numerator functions are equal in magnitude and either equal or opposite in sign. In the adjacent segment, the opposite is true of the sign of this equality. For example, in Figure 3.11, the line segment OA has a point s = 1, where the two numerator functions are equal, whereas the line segment AB has a point s = 2, where the two numerator functions are equal and opposite. The solutions to the first factor in (3.40), therefore, occur in every other segment. Their values can be determined by selecting the appropriate zeros on the right-hand side of (3.40), and these solutions are given by (3.37). Whether the equality of magnitude in any particular segment is then either equal or opposite in sign depends on the polarity chosen for each numerator function, as discussed in Section 2.3. A possible set of solutions for a fourth-order function is – 4, + 3, – 2, + 1. Having chosen a set of solutions, the individual numerator functions can be determined by first multiplying the chosen factors and then separating into the even and odd parts.
Approximations
69
200 100 50 20 10 5
No(s)
2 −σ2
−σ3
1
A
σ3
B
O −5
−2
−1
−σ1
σ1
σ2 1
2
−1 −2
s
5
Ne(s)
−5 −10 −20 −50 −100 −200
Figure 3.11 Graph of numerator functions.
Although theory allows the numerator functions to be of arbitrary sign, one or the other is preferred in practice. We might imagine a practical quadrature hybrid composed of capacitors, both self- and mutual inductors, and ideal transformers. The ambiguity in sign of the transfer functions corresponds with the ambiguity in polarity of the terminations. In practice, it is preferable to arrange the circuit such that all terminations share a common ground plane. Considering, then, the through transfer function, it would be convenient if the magnitude were plus unity at very low frequencies, thereby removing the initial need for a perfect transformer. Consider, too, the coupled transfer function. Coupling at low frequencies will be the first-order effects of capacitance from the input to the coupled output and mutual inductance. Although mutual inductance can be configured for either polarity, the capacitance part makes it necessary for the first-order term in the numerator of the transfer function to have a positive coefficient if the initial necessity of a perfect transformer is again to be avoided. These two
70
Lumped Element Quadrature Hybrids
requirements can be satisfied by defining a function equal to the sum of the numerator functions given by n
(
N (s ) = N e (s ) + N o (s ) = Π σ r + ( −1) r =1
r −1
s
)
(3.41)
The function N(s) will be referred to subsequently as the numerator function as it describes the individual numerators of the two transfer functions by separating into even and odd parts. Having determined the numerator function by this method, we need devote no further effort to determining any constant multipliers as the form in (3.41) gives the required magnitude directly. The complete solution to the transfer functions can be determined using (3.38), with the assistance of (3.39) to make the work easier. The analysis can be illustrated by means of an example. Example 3.2 Determine the through and coupled transfer functions for an
optimum fourth-order quadrature hybrid operating over a decade bandwidth, normalized to a geometric center radian frequency of unity. Solution With the geometric center radian frequency normalized to unity,
ωU = 10, and k1 = 1 / ωU2 = 0 .1. Therefore, k1′ = 1 − k12 = 0.995. Application of (3.37) to the first two frequency parameter values gives . σ 1 = 015139 σ 2 = 0.59702 Equation (3.37) may be used for the remaining two parameters, or (3.39) may be used with the values already calculated, giving . σ 3 = 167498 σ 4 = 6.60526 From (3.38), the denominator function is given by D (s ) = ( σ 1 + s )( σ 2 + s )( σ 3 + s )( σ 4 + s ) Using (3.41), the numerator function becomes
Approximations
71
N (s ) = ( σ 1 + s )( σ 2 − s )( σ 3 + s )( σ 4 − s ) Separating this into its even and odd parts (using N e (s ) = 1 −
{( σ
4
1 4
=
2 3
= 1) gives
}
− σ 1 )( σ 3 − σ 2 ) + 2 s 2 + s 4
N o (s ) = ( σ 4 − σ 3 + σ 2 − σ 1 )s (1 − s 2 )
Having determined the expressions required for the transfer functions, the specific values of the frequency parameter may be substituted, and assuming the port designations of Figure 2.1, the transfer functions may be evaluated as s 41 (s ) = s 21 (s ) =
1 − 8.95701s 2 + s 4 . + s )(0.59702 + s )(167498 . + s )(6.60526 + s ) (015139 . s (1 − s 2 ) 537591
. . + s )(0.59702 + s )(167498 + s )(6.60526 + s ) (015139 ❂❂❂
In practical applications, it is likely that operation over a much higher frequency range is required, so the transfer functions will normally be scaled for frequency. Although the frequency parameters were computed individually in this example, in practical applications, the graph of values in Figure 3.10 will usually give sufficient accuracy. We will see subsequently that circuit element values can be determined directly from the frequency parameters, so the evaluation of transfer functions is only useful for analysis.
3.8 Transfer Functions of Optimum Approximate-Phase Hybrids Continuing the analysis from Section 2.4, we must find solutions to the zeros of the numerator part of the ratio between the two hybrid outputs. This is given by Q (−s 2 ) + sP (−s 2 ) = 0 Now, making the substitution s = j
(3.42)
and rearranging (3.42) give
72
Lumped Element Quadrature Hybrids
ωP ( ω 2 ) Q (ω 2 )
= j
(3.43)
Now the expression on the left-hand side of (3.43) is the same as that on the right-hand side of (2.26). The left-hand side of (2.26) was identified in the text following it as equal to F( ). It is necessary then to find solutions in which the filtering function is equal to j. The problem is very similar to that required to determine the denominator part of the optimum approximate-amplitude transfer functions, except, in this case, the points where F( ) = –j are excluded. As the required points alternate with those that are not required, we need take into account only every other solution given in (3.37). In summary then, the required solution to (2.25) is given by substituting a1 = 1, a2 = 3, b1 = 2, b2 = 4, and so forth. The hybrid is constructed by adding all-pass functions of increasing center frequency alternately to each cascade. This solution is consistent with intuition. Note that only first-order all-pass functions are required in each cascade. Using the circuit references of Figure 2.3, the transfer functions become s 21 (s ) =
( σ 1 − s )(σ 3 − s ) K ( σ 1 + s )(σ 3 + s ) K
(3.44)
s 31 (s ) =
( σ 2 − s )( σ 4 − s ) K ( σ 2 + s )( σ 4 + s ) K
(3.45)
3.9 Conclusion Both polynomial and rational function expressions can be used as the starting point in specifying the performance of quadrature hybrids. Rational functions offer improved bandwidth potential at the expense of more complex circuits in the anticipated realization. The solution of the polynomial expressions requires numerical analysis both for the real frequency and complex frequency transfer functions. On the other hand, the optimum rational function expressions can be solved by analytic methods. The performance of an optimum quadrature hybrid and the solution of its transfer function can be described in terms suitable for graphical solution; thus, it is not necessary for the designer to solve difficult mathematical equations.
Approximations
73
References [1]
Cristal, E. G., and L. Young, “Theory and Tables of Optimum Symmetrical TEM-Mode Coupled Transmission Line Directional Couplers,” IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-13, No. 5, September 1965, pp. 544–558.
[2]
Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Dedham, MA: Artech House, 1980, p. 778.
[3]
Bowman, F., Introduction to Elliptic Functions, London, England: English Universities Press, 1953.
[4]
Zverev, A. I., Handbook of Filter Synthesis, New York: John Wiley & Sons, 1967, pp. 112–114
4 Passive Synthesis The term synthesis denotes the stage in circuit design at which topology and element values are determined. This chapter will describe only the synthesis of passive circuits because, for many applications, quadrature hybrids are required for power transmission and may well operate at frequencies excessive for conventional active design. The synthesis stage takes the transfer functions derived from the approximation functions devised in the previous chapter and determines circuits and element values that express them. The synthesis technique used depends upon the form of the transfer functions. We will see that circuits of coupler form can be synthesized to exhibit transfer functions derived from the polynomial form. Circuits of coupler form can also exhibit approximateamplitude transfer functions derived from the rational form. Alternatively, circuits based on phase delay networks can be used to synthesize quadrature hybrids with approximate-phase characteristics, once again using approximation functions of rational form as the starting point. Whether the approximation function is of polynomial or rational form, and whether the final circuit is of approximate-amplitude or approximatephase form, determines the synthesis technique required. Various synthesis techniques will be described in this chapter. Chapters 2 and 3 illustrated the equivalence between approximate-amplitude and approximate-phase hybrids. This chapter shows how this equivalence can be expressed at the synthesis stage as well. 75
76
Lumped Element Quadrature Hybrids
4.1 Even- and Odd-Mode Analysis This analysis technique, particularly associated with distributed coupler design, is appropriate also for coupler-type lumped element quadrature hybrids [1]. This analysis uses the circuit in Figure 4.1, in which the horizontal dashed line represents a plane of symmetry. The circuit follows the port numbering convention used in Chapter 2; that is, port 1 is the input port, port 2 the coupled port, port 3 the isolated port, and port 4 the through port. This convention will be followed whenever a coupler form of quadrature hybrid is used. This circuit assumes that all ports share a common ground terminal, so only a single terminal is needed to represent each port. The reference impedance Z0 is, in general, an arbitrary value and may not be equal at all ports. In all practical applications, it will be equal and purely resistive, and the analysis uses this restriction. The analysis begins with the s-parameter matrix given by (2.24), using the positive sign option, giving the doubly symmetric form. Consider first an even-mode excitation, where equal excitation is applied at both port 1 and port 2. At port 1, the match property means that the generator at this port does not excite a reflection. However, a signal emanates from port 1, owing to the excitation at port 2 and the coupling to port 1. From the point of view of port 1, it appears that there is a reflection equal to the port 1–to–port 2 coupling, given by No(s)/D(s). At port 4, a signal emanates owing to the excitation at port 1. The excitation at port 2 does not contribute to the signal emanating from port 4 as the ports are isolated from each other. The transmission coefficient in the even mode is therefore Ne(s)/D(s). Similar expressions can be found when even-mode excitation is placed at ports 3 and 4. The circuit can be reduced to an equivalent even-mode circuit, involving only ports 1 and 4, given by the s-parameter matrix: SE =
1 N o (s ) N e (s ) D (s ) N e (s ) N o (s )
(4.1)
In the even-mode case, the plane of symmetry in Figure 4.1 constitutes a virtual open circuit between the two halves. Consider now an odd-mode excitation, where the excitation at port 2 is of equal magnitude but opposite phase to that at port 1. Once again, the excitation at port 1 does not give rise to any reflection from this port. However, the excitation at port 2 generates a signal emanating from port 1. As the generator is opposite in phase, the emanating signal appears equal and opposite in phase to the coupling between ports 1 and 2, given by –No(s)/D(s).
Passive Synthesis
77
Figure 4.1 Hybrid circuit showing even- and odd-mode analysis.
The transmission coefficient in the odd mode from port 1 to port 4 remains unchanged as Ne(s)/D(s). Similar expressions arise once again for odd-mode excitation at ports 3 and 4. The reduction to an equivalent odd-mode circuit gives rise to an s-parameter matrix involving only ports 1 and 4, given by SO =
1 −N o (s ) N e (s ) D (s ) N e (s ) −N o (s )
(4.2)
In the odd-mode case, the plane of symmetry in Figure 4.1 constitutes a virtual short circuit between the two halves. The odd-mode equivalent circuit has a similar set of parameters to the even-mode equivalent circuit. The transmission parameters are identical, but the reflection parameters are of opposite sign. This is equivalent to saying that the odd-mode circuit is the dual of the even-mode circuit with the same reference impedance (dual circuits do not necessarily have the same reference impedance). The principle of duality holds true for distributed couplers, although this terminology is not generally used in the literature. The dual of a length of transmission line between terminations Z 0 of characteristic impedance Ze is the same length of transmission line, except with characteristic impedance Z02/Ze. This is the condition for a matched distributed coupler. The reflection coefficients of these two lines are equal and opposite. Note that the s-parameter matrices given by (4.1) and (4.2) both describe lossless circuits. Therefore, the matrices are unitary, as discussed in Section 2.1. Provided that the circuits are symmetric, the reflection and transmission coefficients are in quadrature. It is only necessary to synthesize the reflection coefficient, and the correct transmission coefficient is achieved
78
Lumped Element Quadrature Hybrids
automatically. The coefficients will not necessarily be in quadrature if the circuit contains all-pass networks. In practice, it is safer to employ a synthesis technique that constrains the circuits to be symmetrical. The reduction into even- and odd-mode circuits indicates a technique for synthesizing quadrature hybrids as follows: 1. Synthesize a circuit whose reflection coefficient is the required coupling. (The correct transmission coefficient is achieved automatically.) 2. Form the dual circuit by inspection. 3. Combine the two circuits into a four-port hybrid. Of these three stages, the last is the most demanding and may require a fair degree of ingenuity. Whether the original circuit or its dual is the evenor odd-mode equivalent circuit is arbitrary, although one combination is usually easier to synthesize than the other. A conceptual quadrature hybrid can be formed using the system shown in Figure 4.2. In this circuit, one 0°/180° hybrid is used to separate into even and odd modes, the prototype circuit and its dual generate the required reflection and transmission coefficients, and the second 0°/180° hybrid recombines the two transmission components. This circuit may not be practical as it is necessary to fabricate additional hybrids as well as the two prototype circuits. However, it serves to illustrate that Step 3 above can always be achieved. Throughout this chapter, we will describe better synthesis techniques tailored to suit the underlying approximation function that initiates the design.
1
Σ
1
Prototype circuit
0°/180° Hybrid
2
2
∆
Σ
1
4
2
3
0°/180 ° Hybrid
Dual circuit
Figure 4.2 Quadrature hybrid synthesis using 0°/180°hybrids.
∆
Passive Synthesis
79
4.2 The First-Order Quadrature Hybrid The simplest coupler form of quadrature hybrid is that of the first order. The approximation function constitutes the simplest case of both the polynomial and rational forms, as given by (3.1) and (3.2), respectively. If we use the normalized rational function form of Section 3.6, then the normalizing frequency is referenced to the geometric mean of the lower and upper passband frequencies, the prototype filtering function is given by F( ) = , 1 = 1, and D(s) = 1 + s. Section 3.7 discussed the selection of suitable numerator functions, and in this case, (3.41) gives N(s) = 1 + s. The required transmission and coupling coefficients for the hybrid follow as s 41 (s ) =
σ1 σ1 + s
s 21 (s ) =
s σ1 + s
(4.3)
(4.4)
The coupling coefficient of (4.4) is also the apparent reflection coefficient of the even mode. An equivalent even-mode circuit can be determined by considering the input impedance corresponding to this reflection coefficient using the following equation: Z = Z0
1+ Γ 1− Γ
(4.5)
It is useful at this stage to normalize the characteristic impedance to unity. So taking (4.5) and substituting the even-mode coupling coefficient for Γ give Ze = 2
s +1 σ1
(4.6)
This impedance can be synthesized using an inductance of value 2/ 1 in series with the 1-Ω terminating resistance. The input reflection coefficient of the odd-mode circuit requires the coupling coefficient of (4.4) with a reversal of sign. Substituting this into (4.5) (remembering that Z0 = 1) gives
80
Lumped Element Quadrature Hybrids
Zo =
1 2s σ 1 + 1
(4.7)
This impedance can be synthesized by a capacitor of value 2/ 1 in parallel with the 1-Ω terminating resistance. The final stage in the synthesis is to combine the even- and odd-mode circuits. As the even-mode circuit contains no capacitance, the odd-mode capacitor must terminate on the virtual earth between the coupled lines. The two capacitors in each half appear in series and, so, can be replaced by a single capacitor of value 1/ 1. The odd-mode circuit contains no inductance, so it is necessary to configure the even-mode inductance such that its effect is cancelled in the odd mode. There are two ways to do this. The first is to place a series inductor of equal value in each line and then perfectly couple the two. In the odd mode, each inductor passes a current of opposite polarity, leading to flux cancellation. Thus, the inductance element contributes zero reactance. The self-inductance of the coupled inductor has a value of 1/ 1 so that it appears as a value of 2/ 1 to each half of the even-mode circuit. Figure 4.3 shows the final circuit [2]. The capacitor is connected to the midpoint of each inductor to illustrate the symmetric nature of the hybrid. In fact, it may be placed at any point along the circuit and is sometimes shown divided into two halves, one on each end of the inductor. For simplicity, the ground terminals required on all ports are not shown explicitly. This convention will be followed subsequently for circuits where all ports share a common ground terminal. The circuit of Figure 4.3, with its use of a coupled inductor, is suitable for frequencies up to around UHF. Above this, the distributed nature of practical circuits becomes significant, and the performance deviates significantly from first-order theory. The second way to implement the inductor is to place it in the ground line between ports 1 and 2 and ports 3 and 4, as shown in Figure 4.4 [3]. The ports of such a hybrid do not share a common ground terminal, so special L 1
4
k=1
C
2
Figure 4.3 First-order hybrid with coupled inductor.
3
Passive Synthesis
1
L
81
C
2
4
3
Figure 4.4 First-order hybrid with ground line inductor.
techniques are required to prevent the screen terminal of connecting transmission lines from short-circuiting the inductor. In the odd mode, there is no net ground current from left to right, so the inductor has no effect. Once again, the normalized inductor value is 1/σ1 so as to appear as a value of 2/σ1 to each half of the even-mode circuit. Placing the inductor in the ground line removes the upper frequency restriction imposed by the coupled inductor of Figure 4.3, so this circuit offers the prospect of microwave operation. This is the simplest form of quadrature hybrid, requiring only two components. Such simplicity offers the prospect of ready alignment.
4.3 Higher-Order Lowpass Prototypes Section 3.2 described the approximation functions that initiate the design of quadrature hybrids based on lowpass filters. One synthesis method is to derive the numerator functions using (2.21) and then the denominator function using (2.18). Having determined the transfer functions, a circuit can be synthesized with a reflection coefficient equal to the required coupling. Darlington synthesis may be used for this purpose [4]. The dual can then be formed by inspection, and the final hybrid devised by some appropriate means. Although this procedure is valid, it is more useful to make use of a technique that exploits the vertical plane of symmetry that exists in the equivalent even- and odd-mode circuits [5]. It will be found that this approach reduces the burden of calculation. Consider the circuit of Figure 4.5, which represents either the even- or odd-mode equivalent circuit (at this stage it doesn’t matter which) and includes the terminations. The circuit is divided in the middle at its plane of symmetry such that the same impedance is seen looking into either half at the
82
Lumped Element Quadrature Hybrids
midpoint. As the circuit is lossless between terminations, the magnitude of the reflection coefficient is constant throughout the length of the circuit. Two positions are of interest: one at one end and the other in the middle. The magnitude of the reflection coefficient at both positions is the same as the coupling in the envisaged quadrature hybrid. This may be related to the filtering function using (2.22). At the midpoint, the source impedance is complex, so the reflection coefficient is given by Γ=
Z L − Z O∗ Z L + ZO
(4.8)
In this case, the source impedance ZO and load impedance ZL are equal. Define this impedance as R + jX, where R is the real part and I the imaginary part of the impedance, as shown in Figure 4.5. Both parts are functions of frequency. Substituting the real and imaginary parts of the impedance in (4.8) gives Γ=
jX R + jX
(4.9)
Using (4.9), the magnitude of the reflection coefficient is given by Γ2 =
1 R 1+ X
R + jX
Z0
Left-hand 2-port
(4.10)
2
R + jX
Γ
Right-hand 2-port
Figure 4.5 Prototype circuit with division at plane of symmetry.
Z0
Passive Synthesis
83
Comparing (4.10) with (2.22), it is possible to identify the ratio of the imaginary to real part of the impedance, as seen at the midpoint, with the filtering function of the target quadrature hybrid. Thus, F ( ω) = ±
X R
(4.11)
The presence of the ± sign indicates two possible solutions, one the dual of the other. The problem of synthesis now becomes that of realizing an impedance with a ratio of the imaginary to real part given by (4.11). Suppose the impedance looking into one-half of the circuit is given by Z (s ) =
n (s ) n e (s ) + n o (s ) = d (s ) d e (s ) + d o (s )
(4.12)
In (4.12), the e subscript refers to the even part and the o subscript to the odd part of the appropriate numerator and denominator polynomials. From (4.12), it follows that Z ( −s ) =
n e (s ) − n o (s ) d e (s ) − d o (s )
Dividing this expression into (4.12) gives Z (s ) n + no d e − d o = ⋅ = e Z ( −s ) d e + d o n e − n o
no d e ne d e nd 1− o e ne d e 1+
− ne d o − no d o − ne d o − no d o
(4.13)
Now, when s = j ω, we have Z ( j ω) = R + jX . As the real part is an even function of , and the imaginary part an odd function of , it follows that Z ( j ω)
Z ( − j ω) Now, substitute for
=
R + jX 1 + jX R = R − jX 1 − jX R
= s/j, giving Z (s ) 1 + [ jX R ]ω =s = Z ( −s ) 1 − [ jX R ]ω =s
j j
(4.14)
84
Lumped Element Quadrature Hybrids
The quantity in brackets in (4.14) may be identified with the ratio of polynomials in (4.13). Define this quantity as / , where ℵ and ℜ are both polynomials in s. Now, take the sum of these polynomials to give ℜ + ℵ = n o d e − n e d o + n e d e − n o d o = n (s )d ( −s )
(4.15)
The left-hand side of (4.15), derived from the ratio of imaginary to real parts of the impedance, may be factored to give the right-hand side. As the numerator and denominator of a realizable impedance must each consist of the product of factors with left-hand zeros, it is possible to determine the numerator and denominator directly by inspection. Using this technique, the circuit may be synthesized without the intermediary requirement of solving for the transfer functions. This technique can be used directly in the synthesis of lowpass circuits. A problem is encountered when the technique is used to synthesize bandpass circuits, and this can be seen from (4.13). It is possible for the required impedance to have an equal and opposite pair of zeros. Such a zero pair is an even factor of both the even and odd parts of the numerator and cancels in the ratio (node – nedo)/(nede – nodo). There is a similar cancellation if the impedance has an equal and opposite pair of poles on the imaginary axis. An odd factor cancellation occurs if there is a zero or pole at the origin. Continuing the procedure to the synthesis stage will produce a circuit of lower complexity than actually necessary. Of these possibilities, only the presence of a zero pair on the real axis is to be found in quadrature hybrid prototype circuits, and, even then, only where the rational approximation function is used. We will see later in this chapter that other synthesis techniques are more appropriate for this class of function. A circuit with impedance Z(s) may be synthesized by extraction of elements. The circuit so formed is reflected, and the two halves are joined back to back to form either the even or odd equivalent circuit. The dual is then formed by inspection. The combination of the even- and odd-mode equivalent circuits into a quadrature hybrid requires some ingenuity. Usually, it is desirable to synthesize a circuit with a common ground terminal at all ports. It is also desirable to synthesize the circuit without using perfect transformers. Figures 4.6 and 4.7, respectively, show practical circuits for third- and fifth-order quadrature hybrids, both of which satisfy these two criteria. It would be useful to illustrate the synthesis procedure for polynomialbased quadrature hybrids by means of a couple of examples.
Passive Synthesis
Ce
Ce
Le
85
Le 3
1
Co
k=1
Lo
Lo
Co
k=1
2
4
k=1
Le
Ce
Le
Ce
Figure 4.6 Third-order polynomial-based hybrid.
L1
C2
L2
C2
L1
1
4
C1
M1
C3
k=1
C1
M1
3
2
L1
C2
L2
C2
L1
Figure 4.7 Fifth-order polynomial-based hybrid.
Example 4.1 Design a quadrature hybrid based on a third-order polynomial
filtering function operating over a 100–200-MHz bandwidth and matched to a 50Ω system. Solution The first step in the solution is to normalize to a 0.5–1-rad/s band width so that the data in Table 3.2 can be used. The normalized filtering function is therefore
86
Lumped Element Quadrature Hybrids
F ( ω) = 21407 . ω − 12233 . ω3 This filtering function is identified with the ratio of imaginary to real parts of the impedance at the center of the prototype circuit, as given by (4.11). Next, determine the quantity in brackets in (4.14), defined as / , by setting = s/j. Thus, 21407 s + 12233 s3 ℵ . . (s ) = 1 ℜ Using (4.15), the numerator and denominator functions can be introduced, giving ℜ(s ) + ℵ(s ) = n (s )d ( −s ) = 1 + 21407 . s + 12233 . s3
= (1 + 2.3603s )(1 − 0.2196s + 0.5183s 2 )
The first-order factor on the right-hand side of the above equation represents a left-hand zero and, so, is identified with the numerator function. The quadratic factor represents a pair of right-hand zeros and is identified with the denominator function. The required impedance is therefore Z (s ) =
1 + 2.3603s 1 + 0.2196s + 0.5183s 2
The above impedance tends toward zero as frequency tends toward infinity, so its end element is a capacitor. In order to extract this, it is easier to consider the admittance, given by the reciprocal of impedance as Y (s ) =
1 + 0.2196s + 0.5183s 2 1 + 2.3603s
The value of the capacitor is the limiting value of Y(s)/s as s tends towards infinity, or 0.2196. The remainder admittance becomes Y 1 (s ) = Y (s ) − 0.2196s =
1 1 + 2.3603s
Passive Synthesis
87
By inspection, the remainder admittance is the series connection of an inductor of value 2.3603 and a resistor of value 1, the terminating impedance. See Figure 4.8. The circuit of Figure 4.8 is one-half of one of the prototype circuits. In order to complete one prototype circuit, it is necessary to reflect the half-circuit and join the two. The end capacitors combine to become twice the value calculated for Figure 4.8, and the prototype circuit becomes like that shown in Figure 4.9(a). The dual circuit is formed by inspection and is shown in Figure 4.9(b). The next stage in the synthesis is to assign the prototype circuit and its dual to either the even or the odd mode. There is more flexibility with the inductor elements in this process as the polarity of mutual inductance can be chosen to suit. However, without the use of perfect transformers, any capacitor appearing in the even mode also appears in the odd mode. It is therefore necessary that the circuit of Figure 4.9(a), with its lower capacitor value, be assigned to the even mode, leaving the circuit of Figure 4.9(b) to the odd mode. The circuit of Figure 4.6 has been devised to implement a third-order polynomial-based quadrature hybrid. Analyzing this circuit in the even mode first, the inductors represented by Le, with their perfect coupling, appear as a value of 2Le from the point of view of port 1. By symmetry, the inductor represented by Lo, again having perfect coupling in the windings, has equal and opposite currents in each winding, so flux cancellation occurs. In the even mode, this inductor appears as a short circuit. Consider now the operation of the capacitors in the even mode. The capacitors with a value of Co experience zero potential difference across their terminals and, so, do not contribute. The capacitors represented by Ce in total contribute a value of 2Ce to the even mode. 2.3603
Y1(s) Y (s)
0.2196
Figure 4.8 Half-prototype circuit of third-order polynomial derivation.
1
88
Lumped Element Quadrature Hybrids 2.3603
2.3603
(a)
0.4392
0.4392
(b)
2.3603
2.3603
Figure 4.9 (a) Prototype and (b) dual circuits for third-order polynomial example.
In the odd mode, flux cancellation occurs in the inductors represented by Le. The inductor represented by Lo appears as a value of 2Lo in the odd mode. The capacitors represented by Co contribute a value of 2Co to the odd mode as they straddle a virtual earth. The value is augmented by a single capacitor of value Ce. Thus, the total value of the odd-mode capacitor is 2Co + Ce. Having assigned the various component values of Figure 4.6 to the component values of Figure 4.9, the normalized component values become Ce = Lo = 0.2196 Le = 1.1801 Co = 1.0703 The specification requires operation over a 100–200-MHz bandwidth with a system impedance of 50Ω. The normalized values1 must be scaled for a reference frequency of 2π × 200 × 106 rad/s and an impedance of 50Ω, giving practical components values as follows:
1. This text makes considerable use of normalized element values, referenced to a characteristic impedance of 1Ω and frequency of 1 rad/s. These normalized values will be dimensionless. Practical values can be distinguished by the use of dimensions, such as pF, nH, Ω, and so forth. If the normalized component values of inductance and capacitance are L′ and C , respectively, and the reference frequency and impedance are 0 and Z0, respectively, then the final circuit components are given by L = Z0L / 0 and C = C ′/(Z0 0).
Passive Synthesis
89
Ce = 3.495 pF Lo = 8.737 nH Le = 46.96 nH Co = 17.04 pF ❂❂❂
The previous example shows that determining the normalized element values of the third-order polynomial-based quadrature hybrid of Figure 4.6 requires considerable mathematical effort. Having determined the normalized values, scaling for frequency and impedance is an easy task. It would be useful for the designer to have these element values available in a range of bandwidth ratios, and some are given in Table 4.1. Example 4.2 Design a quadrature hybrid based on a fifth-order polynomial
filtering function operating over a 30–90-MHz bandwidth and matched to a 50Ω system. Solution The procedure follows a similar pattern to Example 4.1. Normaliz-
ing to a 1/3–1-rad/s bandwidth, the information given in Table 3.3 may be used to determine the filtering function, which is F ( ω) = 3.3076 ω − 53326 . ω 3 + 31145 . ω5 Table 4.1 Normalized Element Values for the Circuit of Figure 4.6 Bandwidth Ratio
Ce
1.2
Lo
Le
Co
0.19275
0.91853
0.82215
1.4
0.20349
0.9912
0.88946
1.6
0.21102
1.05837
0.95187
1.8
0.21618
1.12109
1.01301
2.0
0.21958
1.18014
1.07035
2.2
0.22167
1.23608
1.12524
2.4
0.22278
1.28937
1.17797
2.6
0.22315
1.34035
1.22877
2.8
0.22295
1.3893
1.27783
3.0
0.22233
1.43646
1.32529
90
Lumped Element Quadrature Hybrids
Determine /
by setting
= s/j ; thus,
3.3076s + 53326 s 3 + 31145 s5 ℵ . . (s ) = 1 ℜ Identify this result with the numerator and denominator parts of the impedance to be synthesized: n (s )d ( −s ) = 1 + 3.3076s + 53326 . s 3 + 31145 . s5
= (1 + 3.7112s )(1 + 0.4227s + 0.8062s 2 )(1 − 0.8263s + 1041 . s2)
The last quadratic factor of the above equation is assigned to the denominator of the required impedance, with the remaining factors assigned to the numerator, which becomes Z (s ) =
1 + 4 .134 s + 2.3751s 2 + 2.9919s 3 1 + 0.8263s + 1041 . s2
Element extraction gives a series inductor of value 2.8742, followed by a shunt capacitor of value 0.8263, and another series inductor of value 1.2597, followed by the terminating resistor of value 1. Figure 4.10 shows the half–prototype circuit. Figure 4.11(a) shows the complete prototype circuit formed by reflecting the half-circuit and cascading the two. Figure 4.11(b) shows the dual formed by inspection. The circuit shown in Figure 4.11(a), with its smaller capacitor values, must be assigned to the even mode. With reference to the target circuit of Figure 4.7 and the prototype circuits of Figure 4.11, we observe that the outer inductor in the even mode is larger than the outer inductor in the odd mode. Thus, the mutual inductance given by M1 should be of a polarity that will add in the even mode and subtract in the odd mode. In the even mode, therefore, the circuit of Figure 4.7 appears as a series inductor of value L1 + M1, a shunt capacitor of value C2, a series inductor of value 2L2, with the remaining elements determined by symmetry. In the odd mode, the circuit of Figure 4.7 appears as a shunt capacitor of value 2C1, a series inductor of value L1 – M1, a shunt capacitor of value 2(C2 + C3), and the remaining elements determined by symmetry. The central coupled inductor of Figure 4.7 is a short circuit in the odd mode.
Passive Synthesis
91
2.8742
1.2597
Z (s) 0.8263
1
Figure 4.10 Half–prototype circuit of fifth-order polynomial derivation. 1.2597
5.7484
(a)
0.8263
0.8263
(b)
1.2597
1.2597
0.8263
0.8263
5.7484
1.2597
Figure 4.11 (a) Prototype and (b) dual for fifth-order polynomial example.
This analysis is sufficient to determine the required normalized elements values in Figure 4.7. Thus, L1 + M1 = 1.2597 C2 = 0.8263 2L2 = 5.7484 2C1 = 1.2597 L1 – M1 = 0.8263 2(C2 + C3) = 5.7484 Solving for the element values not given explicitly gives
92
Lumped Element Quadrature Hybrids
L1 = 1.043 M1 = 0.217 L2 = 2.8742 C1 = 0.6299 C3 = 2.0479 The element values must be scaled for the required high-frequency reference value of 2 π × 90 × 10 6 rad/s and a reference impedance of 50Ω, giving C1 = 22.28 pF L1 = 92.24 nH M1 = 19.16 nH C2 = 29.23 pF L2 = 254.1 nH C3 = 72.43 pF ❂❂❂
If it is desirable to tabulate normalized element values for the thirdorder case, the desire is even greater for the fifth-order case. Table 4.2 gives element values for the circuit of Figure 4.7.
Table 4.2 Normalized Element Values for the Circuit of Figure 4.7 Bandwidth Ratio
C1
L1
M1
C2
L2
C3
1.5
0.44326
0.81618
0.07034
0.74584
2.16162
1.41578
1.8
0.48629
0.87729
0.09528
0.78201
2.33864
1.55663
2.0
0.51306
0.91213
0.11399
0.79814
2.44364
1.64549
2.5
0.57457
0.98487
0.16428
0.82026
2.67443
1.85417
63.0
0.62986
1.04303
0.2167
0.82634
2.87423
2.04789
3.5
0.68024
1.09206
0.26843
0.82363
3.05369
2.23006
4.0
0.72661
1.13464
0.31858
0.81606
3.21887
2.40281
4.5
0.76964
1.17255
0.35648
0.80582
3.37344
2.56762
4.8
0.79408
1.19355
0.3946
0.79895
3.46211
2.66316
Passive Synthesis
93
4.4 Use of Transmission Lines The third-order lowpass-derived quadrature hybrid described in Section 4.3 requires three perfectly coupled inductor pairs. Such a circuit is suitable for operation up to UHF, but at microwave frequencies, coupled inductors perform poorly. We may overcome this difficulty partially by using a ground line inductor instead of the perfectly coupled inductors in the outer two positions. It is not possible to use a ground line inductor in the central position owing to the connection arrangement. However, as this component has a relatively small value, extended frequency range is possible. As frequencies increase further into the microwave region, even the small value of the central inductor element will be difficult to realize. However, a slight change in the topology of the circuit overcomes this limitation. Consider the even- and odd-mode equivalent circuits of the third-order polynomial-based hybrid, as shown in Figure 4.9. If a short length of transmission line replaces the central component in each case, then, in the even mode, its capacitive properties dominate, whereas, in the odd mode, its inductive properties dominate. In order to maintain the dual nature of the two circuits, the transmission line impedance in the even mode and the transmission line
L
Z0e, θ
L
(a)
Z0o, θ
(b)
C
C
Figure 4.12 Even- and odd-mode equivalent circuit for third-order mixed hybrid: (a) prototype circuit, and (b) dual circuit.
94
Lumped Element Quadrature Hybrids
impedance in the odd mode must have a geometrical mean equal to the characteristic impedance of the terminations. Purely distributed couplers share this particular requirement. As capacitance in the even mode and inductance in the odd mode are required, it is desirable to minimize even-mode impedance and maximize odd-mode impedance. The limits of these two requirements are achieved when both impedances equal the characteristic impedance, and the transmission lines are uncoupled. When the central element is replaced with a length of transmission line, the prototype circuits, as shown in Figure 4.9, are transformed into those shown in Figure 4.12. Now the two circuits of Figure 4.12 may be combined to form a quadrature hybrid using a first-order section on each side of two transmission lines [6]. Either form of first-order section may be used, subject to the application. Higher frequencies favor the ground inductor form. However, the circuit of Figure 4.13 shows the use of a coupled inductor. The determination of element values for the third-order mixed hybrid requires the solution of transcendental equations as the filtering function becomes a mixture of powers and trigonometric functions of frequency. Some writers have used numerical analysis to solve for element values [7, 8], but their information is of limited value. Table 4.3 gives normalized solutions for the circuit of Figure 4.13, together with the amplitude balance the theoretical circuit achieves. The amplitude balance may be compared with
Table 4.3 Element Values for Third-Order Mixed Element Hybrid Bandwidth Ratio
Amplitude Imbalance
L
1.2
0.052
0.828
22.84
1.4
0.178
0.896
24.11
1.6
0.343
0.959
25.0
1.8
0.531
1.02
25.6
2.0
0.732
1.077
26.0
2.2
0.937
1.132
26.24
2.4
1.142
1.185
26.36
2.6
1.346
1.236
26.39
2.8
1.546
1.285
26.35
3.0
1.742
1.333
26.26
C
(
1)
Passive Synthesis
95
Z0, θ
L
L
1
4
k=1
C
k=1
Z0, θ
C 3
2
L
L
Figure 4.13 Third-order mixed element quadrature hybrid.
the corresponding column in Table 3.2, where it can be seen that the mixed hybrid actually achieves a slight improvement over the purely lumped element version. The normalized values use an upper radian frequency of 1. The transmission line sections are electrically short for any choice of bandwidth ratio, so a compact design can be expected in practice. The principle may be extended to higher-order circuits, and the final arrangement is an alternating pattern of first-order sections and transmission lines. Figure 4.14 shows the circuit for the fifth-order case. Table 4.4 gives normalized element values for this circuit. Larger element values, both for the center lumped element section and the length of the transmission lines, are the penalty for wider bandwidth. The large value of the central first-order section of the fifth-order mixed element hybrid can cause a difficulty in practice, in that it may not be possible to confine its size within the limits of the adjoining transmission lines. This limitation becomes a consideration when the center section uses a ground line inductor for higher-frequency performance. We can overcome this problem by replacing the transmission lines and outer first-order section on each side of the center section with a length of coupled transmission line. In order for the odd-mode circuit to remain the dual of the even-mode circuit, the even- and odd-mode impedances of the coupled transmission line Z0, θ
L1
Z0, θ
L2
L1
1
4
C1
k=1
Z0, θ
C2
k=1
Z0, θ
C1
k=1 3
2
L1
L2
Figure 4.14 Fifth-order mixed element quadrature hybrid.
L1
96
Lumped Element Quadrature Hybrids
Table 4.4 Element Values for Fifth-Order Mixed Element Hybrid Bandwidth Ratio
Amplitude Imbalance
L1
1.5
0.038
0.253
1.859
48.41
1.8
0.112
0.286
2.027
50.73
2.0
0.181
0.308
2.128
51.71
2.5
0.403
0.364
2.356
52.95
3.0
0.668
0.418
2.559
53.11
3.5
0.955
0.469
2.743
52.69
4.0
1.246
0.518
2.914
51.97
4.5
1.536
0.564
3.075
51.09
4.8
1.707
0.59
3.167
50.53
C1
L2
C2
(
1)
must have a geometric mean equal to the characteristic impedance of the system. Figure 4.15 shows the circuit. The frequency response resembles that of a fifth-order, purely lumped, polynomial-based hybrid, except the bandwidth is slightly improved. Table 4.5 gives normalized element values for this circuit. The coupling factor for the coupled lines is modest when compared with the requirements for purely distributed 3-dB couplers.
4.5 Optimum Rational Function Second-Order Prototypes Section 4.2 dealt with the trivial first-order case of the optimum rational function. The next case to consider is the second-order function. This case will be dealt with in detail as not only is the synthesized circuit of practical Z0o,Z0e, θ
Z0o,Z0e, θ
L
1
4
C
k=1
3
2
L
Figure 4.15 Coupled line with lumped element section quadrature hybrid.
Passive Synthesis
97
Table 4.5 Element Values for Coupled Line, Lumped Section Hybrid Bandwidth Ratio
Amplitude Imbalance
L
1.5
0.031
2.0
0.151
2.5
Z0 e
(
1.649
1.284
89.91
1.876
1.305
99.12
0.337
2.067
1.352
105.77
3.0
0.561
2.233
1.391
110.63
3.5
0.803
2.383
1.43
114.31
4.0
1.052
2.52
1.47
117.18
4.5
1.302
2.649
1.508
119.47
5.0
1.548
2.77
1.546
121.33
5.5
1.79
2.886
1.583
122.88
C
1)
value in its own right, but later in this chapter we will see how it can be used as the building block for higher-order hybrids. The synthesis begins with the specification of the frequency parameters as determined by (3.37). Using (3.38), the common denominator of the transfer functions is D (s ) = ( σ 1 + s )( σ 2 + s ),
n=2
(4.16)
The numerator parts of the transfer functions are determined by means of the numerator function given by (3.41). For the second-order case, this becomes N (s ) = ( σ 1 + s )( σ 2 − s ) = σ 1 σ 2 + ( σ 2 − σ 1 )s − s 2 , n = 2 (4.17) The numerator part of the through transfer function requires the even part of N(s), while the coupled transfer function requires the odd part. The required transfer functions may then be determined by inspection and are given by s 41 (s ) =
N e (s ) σ1σ 2 − s 2 = D (s ) ( σ 1 + s )( σ 2 + s )
(4.18)
98
Lumped Element Quadrature Hybrids
s 21 (s ) =
( σ 2 − σ 1 )s N o (s ) = D (s ) ( σ 1 + s )( σ 2 + s )
(4.19)
Even- and odd-mode analysis may again be used to determine the equivalent even- and odd-mode circuits, but the synthesis can be further simplified by making use of the vertical plane of symmetry in the target quadrature hybrid. Thus, the form of s-parameter matrix given by (2.24), with the positive sign option, is assumed. Even- and odd-mode analysis makes use of two modes of operation, with excitation applied at ports 1 and 2, as in Figure 4.1. It is possible to apply excitation at all four ports to exploit the vertical plane of symmetry, and this leads to four modes of analysis. The first mode of analysis, which we shall call Mode 1, describes the condition where an equal excitation is applied at all four ports, as shown in Figure 4.16(a). Now consider the apparent reflection coefficient at port 1. This is the sum of components excited by each of the generators divided by S/C
O/C +
1
4
+
+
1
4
2
3
O/C
O/C +
+
3
2
+
+
(a)
(b)
S/C
O/C
1
+
4
1
4
2
3
+
S/C
S/C
3
2 (c)
+
(d)
Figure 4.16 Hybrid circuit analysis using four modes: (a) Mode 1, (b) Mode 2, (c) Mode 3, and (d) Mode 4.
Passive Synthesis
99
the incident wave at this port. As the hybrid is perfectly matched, the generator at port 1 does not contribute to the emanating wave. As the hybrid possesses isolation properties between ports 1 and 3, the generator at port 3 does not contribute either. The only contributions come from the generators at ports 2 and 4. Using the doubly symmetric four-port matrix of transfer functions given by (2.24), the apparent reflection coefficient becomes ΓM 1 =
N o (s ) N e (s ) N (s ) + = D (s ) D (s ) D (s )
(4.20)
The second mode of analysis, Mode 2, occurs where the generators at ports 1 and 2 are equal, as are the generators at ports 3 and 4, but the two pairs are of opposite polarity, as shown in Figure 4.16(b). The apparent reflection coefficient at port 1, using (2.24), becomes ΓM 2 =
N o (s ) N e (s ) −N ( −s ) − = D (s ) D (s ) D (s )
(4.21)
The third mode of analysis, Mode 3, occurs where the generators at ports 1 and 3 are equal, while those at ports 2 and 4 are of opposite polarity, as shown in Figure 4.16(c). The apparent reflection coefficient becomes ΓM 3 = −
N o (s ) N e (s ) −N (s ) − = D (s ) D (s ) D (s )
(4.22)
Finally, the fourth mode of analysis, Mode 4, occurs where the generators at ports 1 and 4 are equal, while those at ports 2 and 3 are of opposite polarity, as shown in Figure 4.16(d). The apparent reflection coefficient becomes ΓM 4 =
−N o (s ) N e (s ) N ( −s ) + = D (s ) D (s ) D (s )
(4.23)
These four modes are sufficient to define any excitation to the circuit; for example, an excitation at port 1 only can be simulated by a quarter of the sum of Modes 1 to 4. Even-mode excitation is half the sum of Modes 1 and 2, and odd-mode excitation is half the sum of Modes 3 and 4. Figure 4.16
100
Lumped Element Quadrature Hybrids
shows the virtual open and short circuits that appear in each of the four modes. This enables the circuit to be defined using only one-quarter of it. Having determined the required apparent reflection coefficients, the apparent impedances can be determined using the relationship between them, given by (4.5), once again assuming a normalized reference impedance of 1Ω. These impedances are as follows, where the subscript refers to the respective mode of excitation: D (s ) + N (s ) D (s ) − N (s )
(4.24)
D (s ) − N ( −s ) D (s ) + N ( −s )
(4.25)
D (s ) − N (s ) D (s ) + N (s )
(4.26)
D (s ) + N ( −s ) D (s ) − N ( −s )
(4.27)
Z M1 =
ZM2 =
ZM3 =
Z M4 =
When applied to the general order n cases, the denominator and various numerator functions possess common factors. These readily cancel to give simplified expressions. These impedances may now be applied to the second-order case. For Mode 1, substituting for D(s) and N(s) in (4.24) from (4.16) and (4.17) gives Z M1 =
σ2 s
(4.28)
This impedance is equivalent to a capacitor of value 1/ 2. For Mode 2, substituting again for D(s) and N(s) in (4.25) gives ZM2 =
s σ1
(4.29)
This impedance is equivalent to an inductor of value 1/ 1. These two modes give sufficient information to derive the equivalent circuit for the even
Passive Synthesis
101
2/σ1
2/σ2
(a)
1/σ1
1/σ1
−1/2σ1
2/σ2
(b)
Figure 4.17 Even-mode equivalent circuit of second-order hybrid: (a) physical circuit, and (b) analytic form.
mode. When the input and output ports of the even-mode equivalent circuit have the same polarity, the circuit between them must appear as a shunt capacitor, whereas when they are of opposite polarity, it must appear as a series inductor. The solution to these two conditions is a center-tapped inductor between the two ports, with a capacitor from the center tap to ground. The value of the capacitor must be 2/ 2 in order to appear as 1/ 2 from the point of view of each port. The value of the inductor must be 2/ 1 in order to appear as 1/ 1 from the point of view of each port. Figure 4.17 shows the even-mode equivalent circuit. The center-tapped inductor of Figure 4.17(a) can be expanded into a “T” network of inductors, where the vertical element has a value of –1/2 1, as shown in Figure 4.17(b). The negative inductor in series with the capacitor gives two zeros of transmission where s = ±√( 1 2), as required by (4.18). Having determined the even-mode equivalent circuit, the odd mode may be determined by first considering the remaining two modes of the four-mode analysis. For Mode 3, using (4.26), the equivalent impedance is
102
Lumped Element Quadrature Hybrids
ZM3 =
s σ2
(4.30)
This impedance is equivalent to an inductor of value 1/ 2. Finally, for Mode 4, the equivalent impedance is Z M4 =
σ1 s
(4.31)
This impedance is equivalent to a capacitor of value 1/ 1. Mode 3 represents the odd-mode circuit where the input and output are of opposite polarity, so (4.30) indicates there is an inductor of value 2/ 2 between the two ports. Mode 4 represents the odd-mode circuit where the input and output are of the same polarity, so (4.31) indicates a capacitor to ground of value 2/ 1. The topology of the circuit is therefore exactly the same as was found for the even-mode circuit using Modes 1 and 2. The component values are different and are shown in Figure 4.18 for the two elements. If the centertapped inductor is once again expanded into a “T”network, the combination of the vertical negative value with the capacitor gives the same zeros of transmission as seen in the even-mode equivalent circuit.
2/σ2
2/σ1
(a) 1/σ2
1/σ2
−1/2σ2
2/σ1 (b)
Figure 4.18 Odd-mode equivalent circuit of second-order hybrid: (a) physical circuit, and (b) analytic form.
Passive Synthesis
103
The circuits of Figures 4.17 and 4.18 are particular examples of what are known as Brune sections, or Darlington Type C sections, for the special case of a center tap. Note that although the even- and odd-mode circuits are electrical duals of each other, the physical form is the same. These sections belong to a class of circuit elements that are self-dual. The final part of the synthesis is to combine the even- and odd-mode circuits into one coupled circuit. This is done by considering the capacitive and inductive elements separately. The capacitor value in the odd mode is larger than it is in the even mode. The solution, then, for the final circuit is to add a capacitor between the two tap points of the equivalent circuits for each half of the final circuit. The additional capacitor must be of value 2(1/ 1–1/ 2) to the virtual short circuit of the odd mode, and two such in series yield a value of 1/ 1–1/ 2. The inductor value in the even mode is larger than it is in the odd mode. To achieve this, magnetic coupling is required between the two inductors, the polarity of which is such that the mutual inductance adds in the even mode and subtracts in the odd mode. The two conditions become L + M = 2 σ1 L − M = 2 σ2 From these two conditions, the self- and mutual inductances can be determined; thus, L=
1 1 + σ1 σ 2
(4.32)
M =
1 1 − σ1 σ 2
(4.33)
Furthermore, k=
M σ 2 − σ1 = σ 2 + σ1 L
(4.34)
It is interesting to observe that the coupling factor k between the inductors is equal to the center-frequency coupling of the hybrid. All the
104
Lumped Element Quadrature Hybrids
required information is now available to specify the second-order quadrature hybrid, and this is shown in Figure 4.19. In practice, calculation of the second-order hybrid element values is simplified using 2 = 1/ 1. However, it is useful at this stage to retain the frequency parameters separately as the circuit will be used later as a building block for higher-order hybrids. The self- and mutual inductances are as given in (4.32) and (4.33), and the capacitor values, as determined previously, are C1 =
1 1 − σ1 σ 2
(4.35)
2 σ2
(4.36)
C2 =
The circuit of Figure 4.19 is much simpler than the alternative synthesis using the even and odd equivalent circuits of Figures 4.17 and 4.18 combined with in-phase hybrids, as shown in Figure 4.2. This illustrates that the synthesis of quadrature hybrids is not unique. A degree of ingenuity is usually required to compose the best topology.
4.6 Higher-Order Optimum Rational Function Hybrids The same principles outlined in Section 4.5 can be used to determine the circuits for hybrids based on the third-order and higher optimum rational L 1
4
C2
M
C1
C2
3
2
L
Figure 4.19 Second-order rational function–derived quadrature hybrid.
Passive Synthesis
105
functions. As order increases, the composition of the hybrid becomes a more difficult task. So far, no canonical technique for determining the topology has been determined. Solutions for the third- and fourth-order prototypes are shown in Figures 4.20 and 4.21, respectively. Element values have been added to the figures, and these are normalized to a center frequency of 1 rad/s and a system impedance of 1Ω, so use of (3.39) gives simplified expressions for the element values. Observe, particularly with reference to Figure 4.21, that the circuits become quite sophisticated for higher-order hybrids. One might suppose, therefore, that the prospects for wideband operation, offered by filter functions of higher order, come at a high price. This is not the case, and Section 4.7 describes techniques that simplify the synthesis of higher-order hybrids.
C3
L1
k=1
1
L2
C2
L2
M1
C1
4
L2
C2
2
L2
3
k=1
C3
1/σ1 +1/σ1+1+σ1+σ1 2
L1 =
L2 =
C1 =
L1
2
1/σ1+σ1 1
1
2
1/σ1+σ1
1/σ12−1/σ1+1−σ1+σ12 1/σ1+σ1
1/σ1 −1/σ1+3−σ1+σ1 2
M1 =
C2 = 2
C3 =
1/σ1 + σ1
1
1
2
1/σ1 + σ1
Figure 4.20 Third-order rational function–derived quadrature hybrid.
2
106
Lumped Element Quadrature Hybrids k=1 4
1
L2
L2
C3
C4 L1 C2
M2
M1
C1
C3
C2
L1 C4
L2
L2
2
3
k=1
L1 =
1 1 + +σ2+σ1 σ1 σ2
M1 =
1 1 − +σ2−σ1 σ1 σ2
L2 =
σ2+σ1 4(1+σ1σ2)
M2 =
σ2−σ1 4(1+σ1σ2)
C1 = (σ2−σ1) 1+
C3 =
1 2(1/σ1+σ2)
1 σ1σ2(1+σ1σ2)
C2 = 2
C4 =
1 +σ1 σ2 σ2−σ1 2(1+σ1σ2)
Figure 4.21 Fourth-order rational function–derived quadrature hybrid.
Passive Synthesis
107
4.7 Hybrid Synthesis Using Cascaded Sections The use of cascaded sections in quadrature hybrid design is a familiar technique to the distributed circuit designer. The principle can be applied to both coupled line [9] and branch line [10] designs. It is useful in coupled line designs in that it allows looser couplers to be cascaded to give a tighter coupling. For example, two 8.34-dB quarter wave couplers can be cascaded to give a 3-dB coupler. The cascaded design is more complicated than a single coupled section and is in fact slightly inferior in bandwidth. However, the use of the technique may be desirable in that it allows the construction of a coupler in a medium where the tight coupling of 3 dB in a single section is not practical. In this context, the technique is sometimes referred to as a tandem connection. The principle of cascaded sections can be applied to lumped element circuits as well. Consider the cascade connection of two quadrature sections, as in Figure 4.22. The manner of connection between the two is the same as in the more familiar distributed versions; that is, the through and coupled ports of the first hybrid are connected to the input and isolated ports of the second. As the second hybrid is perfectly matched, there is no reflection to the first hybrid, and the isolated port of the first hybrid remains isolated in the cascade. The through port of the second hybrid becomes the through port of the cascade, and the coupled port of the second hybrid becomes the coupled port of the cascade. The response of the cascade to an excitation at the input port should now be considered in detail. It may be supposed that the cascade consists of two symmetrical hybrid sections, a and b, as shown in Figure 4.22. Each section is defined by its own numerator function, which can be divided into its even and odd constituents, and its own denominator function. Consider first
Input
I/P
Thro
I/P
Hybrid a
Isolated
Isol
Thro
Through
Coup
Coupled
Hybrid b
Coup
Isol
Figure 4.22 Cascade connection of quadrature hybrids.
108
Lumped Element Quadrature Hybrids
the response from the input to the through port. Two signal paths contribute to this response, one via the through transmission path of each hybrid and the other via the coupled transmission paths of each hybrid. The total response is formed by superposition and is given by N N + N ao N bo Ne (s ) = ae be (s ) D a Db D
(4.37)
The use of subscripts to indicate whether the whole circuit or the hybrid a or b is referred to, and then whether the even or odd part of its numerator function is referred to, is tacitly understood. Note from the numerator on the right-hand side of (4.37) that the combination yields an even-order result, indicated by the subscript on the left-hand side. The response from the input to the coupled port is similarly the combination of two signal paths. Each is the through path of one hybrid and the coupled path of the other. The complete response is given by N N + N ae N bo No (s ) = ao be (s ) D a Db D
(4.38)
In this case, the combination yields an odd-order numerator. The cascade gives quadrature outputs, as can be verified by the substitution s = j . Now examination of the subscripts in (4.37) and (4.38) shows that the transfer functions would have been exactly the same if the order of the hybrids had been reversed. The cascade is therefore electrically symmetric, even though it is not physically symmetric. Furthermore, the individual hybrids in Figure 4.22 may themselves also be formed of the cascade of more elementary hybrids. The principle may than be used recursively so that the cascade of an arbitrary number of (electrically) symmetric quadrature hybrids is also a (electrically) symmetric quadrature hybrid. Consider now the numerator function of the cascade hybrid of Figure 4.22, expressed as the sum of the numerators in (4.37) and (4.38). This is given by N =Ne +No = N ae N be + N ao N bo + N ao N be + N ae N bo = (N ae + N ao )(N be + N bo ) = N aN b
Passive Synthesis
109
The numerator function of the cascade is thus the product of the numerator functions of the constituent hybrids. Clearly, the same is true of the denominator function. This principle may be used recursively for a cascade of arbitrary length so that the numerator and denominator functions are the respective products of the constituent hybrids. The respective numerator and denominator functions for an arbitrary-length cascade become N = N aN b N c K
(4.39)
D = D a Db Dc K
(4.40)
4.8 Cascades of First-Order Sections The investigation of cascades was made with the object of simplifying the design of high-order rational function–based quadrature hybrids. The response of a first-order hybrid is given by (4.3) and (4.4) and is repeated here in terms of the numerator and denominator functions: Ne σ (s ) = σ+s D
(4.41)
No s (s ) = D σ+s
(4.42)
In the case of a first-order hybrid, the numerator function is given by N (s ) = σ + s = D (s )
(4.43)
In a cascade of first-order hybrids, the numerator function equals the denominator function. However, the equations governing the response of an optimum quadrature hybrid require that they be different, as described by (3.38) and (3.41). On their own, then, a cascade of first-order hybrids cannot yield an optimum hybrid. It is possible to modify a first-order cascade such that an optimum quadrature hybrid can be formed. The technique, invented by J. D. Cappucci [11], is to insert an inverter at a crucial point in the cascade. To show how the technique operates, we will consider the inclusion of an inverter as a modification to the circuit of Figure 4.22. Figure 4.23 shows the new cascade. The effect of the inverter is to change the sign of Nao(s). The outputs of the
110
Lumped Element Quadrature Hybrids
Figure 4.23 Hybrid cascade with inverter in coupled line.
combined circuit remain, respectively, even and odd, but the responses have been otherwise modified. Evaluation of the combined numerator function gives N = N ae N be − N ao N bo − N ao N be + N ae N bo = (N ae − N ao )(N be + N bo )
(4.44)
= N a ( −s )N b (s )
No change occurs in the common denominator. Now compare (4.44) with (3.41). If hybrid a is composed of a cascade of first-order sections, its individual numerator function can be identified with factors in (3.41) where the negative sign appears, that is to say, where r is even. Hybrid b should comprise sections expressing the factors where the positive sign appears, that is, to say where r is odd. The modified cascade can then form an optimum quadrature hybrid. Note that the hybrid in this case is not symmetric. Reflection in the horizontal plane changes the sign of both outputs, while reflection in the vertical plane changes the sign of the coupled output but leaves the through output unchanged. The hybrid is an example of (2.24) where the negative sign option is taken. The hybrid can be made electrically symmetric by reversing the polarity at the isolated port. Example 4.3 Design a fourth-order optimum quadrature hybrid based on a
cascade of first-order sections, operating over the high-frequency band of 2 to 32 MHz with a system impedance of 50Ω.
Passive Synthesis
111
Solution The required bandwidth ratio is 16:1, with a geometric center fre-
quency of 8 MHz; thus, ωU = 4 when normalized to a center frequency of unity. Using (3.27), the maximum amplitude imbalance is 0.301 dB (the graph of Figure 3.9 would have been sufficiently accurate as well). The frequency parameters, as given by (3.37), are 1 = 0.13604, 2 = 0.57154, 3 = 1.74965, and 4 = 7.35099. These values are required to determine the element values of the various first-order sections. Figure 4.24 shows the circuit giving a solution to the problem. According to the previous argument, the sections corresponding to 2 and 4 are on the left-hand side of the inverter, while those corresponding to 1 and 3 are on the right-hand side. In this circuit, coupled inductor versions of the first-order sections are chosen, and this may well be appropriate for the frequency range. The inverter is an inverting transformer. In practice, an inverting balun might well be used instead. Scaling the frequency parameters for center frequency and impedance, the element values become L1 = 7.312 µH, C1 = 2.925 nF L2 = 1.74 µH, C2 = 696.2 pF L3 = 565.5 nH, C3 = 227.4 pF L4 = 135.3 nH, C4 = 54.13 pF The frequency parameters can also be used to predict the center frequencies of the individual sections, which in this case are 1.088, 4.572, 13.997, and 58.808 MHz. These values are useful at the development stage as each section can be aligned individually before the cascade is formed. ❂❂❂
Although it is important that the sections occupy their designated positions with respect to the inverter, the order in which the first-order sections are placed within their own subsections is not important. L2
L4
L1
L3 4
1
k = 1 C2
3
k=1
L2
C4
k=1
k=1
C1
L4
Figure 4.24 Fourth-order cascade of first-order sections.
L1
C3
L3
2
112
Lumped Element Quadrature Hybrids
4.9 Cascades of Second-Order Sections If a cascade of second-order sections is formed, its numerator function can be determined using (4.39) and (4.17), except the frequency parameters of (4.17) will vary according to the design of each section. The numerator function will have an equal number of factors with negative and positive signs. Examination of (3.41) shows this to be the desired result, provided that an even-order hybrid is required. Thus, no inverting transformer is required in the design. The synthesis of this type of hybrid requires allocating the frequency parameters to the various second-order sections in pairs as 1, 2; 3, 4; and so on. If the target hybrid is of odd order, then the frequency parameter n can be allocated to a first-order section. As the cascade is of symmetric hybrid sections only, they can be assembled in any order. This design principle is best illustrated by means of an example. Example 4.4 Rework the problem of Example 4.3 using a cascade of second-
order sections. Solution As the problem is of the fourth order, a cascade of two second-
order sections is required, as shown in Figure 4.25. Taking the first two frequency parameters, substituting them into (4.32), (4.33), (4.35), and (4.36), and scaling for frequency and impedance gives element values of
L1
L2 4
1
C4
C2
M1
C1
C2
M2
C3
L1 3
Figure 4.25 Fourth-order cascade of second-order sections.
C4
L2 2
Passive Synthesis
113
L1 = 9.053 µH M1 = 5.572 µH C1 = 2.229 nF C2 = 1.392 nF Similarly, element values for the second section take the last two frequency parameters, giving L2 = 703.8 nH M2 = 433.2 nH C3 = 173.3 pF C4 = 108.3 pF It may be confirmed that both second-order sections have the same coupling between the inductors, which, in this case, is k = 0.615, or 4.22 dB, and is also the center-frequency coupling of the individual sections. The individual sections have a center frequency determined as the geometric mean of the frequency parameters. When scaled for frequency, these become 2.231 and 28.69 MHz. Once again, these figures are useful at the development stage to align sections individually. ❂❂❂
4.10
Further Cascade Arrangements
The cascade arrangements of first- and second-order sections, as described in Sections 4.8 and 4.9, are each sufficient for the synthesis of optimum quadrature hybrids of any order. Other arrangements are possible, giving practical benefits in certain circumstances. Consider once again the use of first-order sections, as described in Section 4.8. One may wonder whether there is a more efficient way of synthesizing the transfer functions of a cascade of first-order sections only. To do this, return to the techniques of Section 4.3 to use a quadrature hybrid section divided at its midpoint such that the impedance looking into one-half of the even-mode equivalent circuit is as given by (4.11). Section 4.3 assumed a lowpass ladder circuit, but for the purposes of the present analysis, suppose that the half-circuit of the even-mode equivalent circuit is formed by the terminating impedance in series with a pure reactance. For convenience, the terminating impedance may be normalized to 1Ω. Therefore, by (4.11), F( ) = X. Now, use (2.21) to relate the filtering function to the transfer function numerator parts so that substituting for F( ) gives
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Lumped Element Quadrature Hybrids
jX ( ω) =
No Ne
( jω)
(4.45)
The problem then becomes one of establishing whether the function No(s)/Ne(s) describes a pure reactance. Now, in a quadrature hybrid based solely on first-order sections, as in Figure 4.23, where two cascades are placed on either side of an inverter, the numerator function of the cascade of first-order sections is given by either N (s ) = ( σ 2 + s )( σ 4 + s ) K
(4.46a)
N (s ) = ( σ 1 + s )( σ 3 + s ) K
(4.46b)
or
In both cases, the numerator function is a Hurwitz polynomial, and, as such, its ratio of odd to even parts can be synthesized as a pure reactance [12]. Thus, where (4.45) defines the numerator functions of a cascade of first-order sections, it also describes a pure reactance. Of particular interest is the case in which two first-order sections are cascaded. To synthesize an equivalent circuit using just one section, the inductor of a first-order section becomes a second-order reactance by the addition of a capacitor in parallel. Similarly, the first-order capacitor is augmented with an inductor in series. The reactance of the parallel inductor/ capacitor for the even-mode equivalent circuit is as given by (4.45). The required reactance in the second-order hybrid section is doubled for the complete even-mode circuit, but then halved again when the even- and odd-mode circuits are combined into the final hybrid circuit. Thus, the reactance given by (4.45) is also the reactance between terminations in the final section. As the odd-mode circuit is the dual of the even-mode circuit, its shunt susceptance is equal to the series reactance of the even mode. So (4.45) may be used for both quantities, provided normalization to a 1Ω reference impedance is assumed. Figure 4.26 shows a circuit making use of second-order reactances We can appreciate this circuit’s practicality by considering either the even-mode reactance Le, Ce or the odd-mode susceptance Co, Lo. Consider the series inductor of the circuit of Figure 4.4. In practice, the inductive element will be complicated by stray capacitance, so it will appear more like the parallel combination of the inductor and capacitor shown in the circuit of Figure 4.26 at
Passive Synthesis
Le
1
115
Co
4
Ce Lo
3
2
Figure 4.26 Single-section equivalent of first-order cascade.
higher frequencies. As the practical first-order circuit exhibits second-order properties to some degree, it may be better to exploit these to make them part of an intentional second-order section instead of taking the trouble to create a more complex hybrid composed of two individual first-order sections. The same is true of the capacitor in Figure 4.4, which in practice is complicated by stray inductance. The element values of the second-order circuit can be determined using the two factors of either of the equations in (4.46). At this point it may be supposed that the section is intended to replace a cascade of two first-order sections only, such as appear in Figure 4.24 on either side of the inverter. Consider the first two factors of (4.46b), which, by (4.45), gives an impedance function of
( σ 2 + σ 4 )s No (s ) = Ne σ 2 σ4 + s 2 =
1
(4.47)
1 1 s+ σ 2 + σ4 (1 σ 2 + 1 σ 4 )s
This reactance function requires the parallel combination of an inductor and capacitor, whose values are given by Le =
1 1 + σ 2 σ4
(4.48)
116
Lumped Element Quadrature Hybrids
Ce =
1 σ 2 + σ4
(4.49)
The series combination of capacitor and inductor is the dual of the parallel combination, and as a 1-Ω impedance is used, their values are directly given by the inductor and capacitor values already determined above. Thus, C o = Le
(4.50)
Lo = C e
(4.51)
The equations above for the left-hand replacement section in the circuit of Figure 4.24 can be adapted for the right-hand section by replacing the sigma values appropriately. In addition, the design can be simplified to the third order by allowing 4 to increase to infinity. The left-hand section then becomes a first-order section. The cascade connection of two second-order sections to form a fourth-order hybrid is an efficient implementation. It is possible to redistribute the frequency parameters so that one second-order sections uses 1 and 4, and the other uses 2 and 3. If this is done, then an inverter is required, just as in a first-order cascade. The solution then appears inferior to the version that does not require an additional inverter. However, there may be circumstances in which this new implementation is preferred. Consider the coupling in each section. For the version without an additional inverter, the coupling is the same in each. For the implementation with the frequency parameter distribution just described, the sections require different coupling. For the frequency range of Example 4.4, the coupling of each section would have been 0.964 (0.32 dB) and 0.507 (5.89 dB). The tight coupling of one section may not necessarily be difficult to achieve, but the specific coupling of the sections in the original example may for some practical reason be difficult to achieve. This technique is also used for hybrids when the section on either side of an inverter is something other than a rational function–based section but is of comparable center-frequency response. For example, two mixed element sections, as in Figure 4.13 except with ground line inductors, may be combined to achieve a much wider bandwidth at frequencies where suitable magnetic materials are not available. The technique has been applied to the use of coupled transmission lines, both having a quarter-wavelength at center frequency [13].
Passive Synthesis
4.11
117
Approximate-Phase Hybrids
Figure 2.3 shows the form of an approximate-phase hybrid. The hybrid requires two constituent parts in its construction: an in-phase hybrid and phase delay networks. There are several options for both of these constituents. It is desirable for the in-phase divider to have the isolation properties of a hybrid so as to reduce the interference from one output to the other caused, perhaps, by a poor output match. In addition, the hybrid should be based on a lossless design if the complete circuit is required to exhibit low loss as well. Figure 4.27 shows the basic in-phase hybrid. Matched operation of this hybrid requires different input and output reference impedances. Usually, systems require that they all be the same, so a step-down transformer is necessary at the input. The 2R resistor provides isolation across the outputs. In most applications, the imbalance component caused by output mismatch only needs to be dissipated. If the reflected signal is important, then this resistor must be replaced with an isolation transformer to make the signal available externally. O/P 1 Z0=R I/P Z0=R/2
2R O/P 2 Z0=R
Figure 4.27 Basic in-phase hybrid.
O/P 1 Z0=R
R/2 O/P 2 Z0=R
I/P Z0=R/2
Figure 4.28 Variation on the basic in-phase hybrid.
118
Lumped Element Quadrature Hybrids
Figure 4.28 shows a variation on the basic hybrid circuit. This circuit performs the same function but has an isolation resistor of value R/2. Where high operating frequencies are needed, the hybrid circuits of both Figures 4.27 and 4.28 suffer from leakage inductance. In practice, this characteristic can be reduced by the use of twisted pairs, but it can never be completely overcome. At high frequencies, a transmission form of circuit may be preferred, such as that shown in Figure 4.29 [14]. Magnetic loading is required on the transmission lines (shown here as coaxial cable), and this is indicated by the dashed lines on either side of each transmission line. This circuit is based on a resistive bridge with two isolated ports joined together to form the input. This hybrid too requires an input transformer if the input port impedance needs to be the same as the outputs. If it does, the same reasons that made it necessary to use a transmission form of hybrid will make it necessary to use a transmission form of transformer, such as those described in [15, 16], as well. There is no theoretical upper frequency limit to the transmission properties of this hybrid, provided the transmission lines at the input have the same electrical length. In some applications, the isolated port needs to be made available. Using the basic in-phase hybrid, this requires an additional transformer. In these circumstances, a cross-coupled pair of transformers, as shown in Figure 4.30, can provide hybrid operation. This circuit is usually associated with directional coupler applications [17] with a high turns ratio for weak coupling. In these applications, there is little variation in the voltage along the main line from port 1 to port 4, so the position of the transformer tap is not important. Where the turns ratio is low, considerable voltage variation occurs along this line, so it is important to place the tap at the center of the
Z0=R I/P Z0=R/2
R
Z0=R
Figure 4.29 Transmission line hybrid.
Z0=R
O/P 1 Z0=R
Z0=R
O/P 2 Z0=R
R
Passive Synthesis
119
N1 4
1
N2 N2
3
2
N1
Figure 4.30 Transformer directional coupler.
winding to which it is attached in order to preserve match and isolation. The transfer functions for this coupler are given by s 41 =
4N 2 − 1 4N 2 + 1
(4.52)
s 21 =
4N 4N 2 + 1
(4.53)
where N =
N2 N1
The transfer functions are not a function of frequency, so bandwidth is limited only by construction. An approximate equal power division occurs where N = 6/5. The second constituent in the approximate-phase hybrid is the two phase delay networks. These must give transfer functions as described by (3.44) and (3.45). The conventional passive topology implementing a phase delay is the lattice network, as shown in Figure 4.31. The network has an input impedance equal to the terminating impedance, provided that Y = Z R 2 . If this condition is met and the terminating impedance is normalized to 1Ω, then Y = Z, and the transfer function is given by s 21 (s ) =
1 − Z (s ) 1 + Z (s )
(4.54)
120
Lumped Element Quadrature Hybrids Z
Y Port 1
Port 2
Y
Z
Figure 4.31 Balanced lattice network.
The problem is now to determine whether the form of (4.54) can give transfer functions of the form of (3.44) and (3.45). This is very similar to the problem described in Section 4.10 regarding a single-section equivalent to a cascade of first-order sections. Dividing either (3.44) or (3.45) by the common even part of the numerator and denominator, Z(s) becomes equal to the ratio of the odd-to-even parts of the denominator. As the denominator is a Hurwitz polynomial, the ratio must be realizable as a pure reactance. In fact, the impedance Z(s) is the same as the series-connected impedance of the single-section equivalent to the cascade of first-order hybrids. Similarly, Y(s) is the same as the parallel admittance. Thus, if the denominator function of one all-pass section is given by D (s ) = D e (s ) + D o (s ) = ( σ 1 + s )( σ 3 + s ) K
(4.55)
then both the impedance Z(s) and admittance Y(s) are given by Z (s ) = Y (s ) =
D o (s ) D e (s )
(4.56)
In the use of the lattice network, measures must be taken to isolate the ground terminal of the input to that of the output. Usually, this is done using a balun. However, the resulting lack of a common ground may cause difficulty in some circumstances. It may be preferable to seek a phase delay
Passive Synthesis
121
network with a common ground. One possibility is the circuit of Figure 4.17. Its transfer function is given by (4.18). If 2 is made to equal 1, the transfer function simplifies to s 41 (s ) =
σ1 − s σ1 + s
(4.57)
This is the same as (4.54) for a first-order phase delay. Inspection of (4.19) reveals that the circuit is perfectly matched as well. The circuit can be made to provide higher-order phase delays by increasing the order of the reactive elements. Clearly, a higher-order reactive network can replace the capacitor. The same is true of the inductor if it is recalled that a centertapped inductor is equivalent to a center-tapped perfect transformer with an added parallel inductor. A higher-order reactive network can then replace this parallel inductor as well. If the reactive network replacing the inductor has an impedance given by Z(s), and the terminating impedance is 1Ω, then the admittance of the capacitor replacement network is given by Y(s) = Z(s), and the transfer function is given by s 21 (s ) =
1 − Z (s ) 2 1 + Z (s ) 2
(4.58)
The required reactive networks can always be synthesized as they are just scaled versions of those required for the lattice implementation [compare (4.54)]. Example 4.5 Determine element values for the phase delay networks of a sixth-
order approximate-phase quadrature hybrid operating over a 1–300-MHz bandwidth with a reference impedance of 50Ω. Solution Using (3.37), the frequency parameter values are
= 0.03613 = 0.165 = 0.5525 3 4 = 1.80996 5 = 6.06046 6 = 27.6763 1
2
122
Lumped Element Quadrature Hybrids
The geometric center frequency for the design is 17.32 MHz. The odd subscript parameters are assigned to one phase delay network, and the even subscript parameters are assigned to the other. Assume that each phase delay network consists of a second-order all-pass lattice in cascade with a first-order lattice. For the phase delay network assigned to the odd subscript parameters, use 1 and 5 for the second-order lattice. With reference to Figure 4.31, the impedances denoted by Z(s) consist of a parallel combination of an inductor and capacitor, with normalized element values given by (4.48) and (4.49) and with the new parameter values substituted. The normalized inductor value is given by LZ =
1 1 + = 27.8413 σ1 σ 5
The normalized capacitor value is given by CZ =
1 . = 0164 σ1 + σ 5
Scaling for frequency and impedance (assumed to be 50Ω) yields element values of LZ 1 = 12.79 µH CZ 1 = 30.14 pF The admittance denoted by Y(s) consists of a series combination of an inductor and capacitor with element values given by (4.50) and (4.51). Thus, the normalized element values are given by CY = LZ and LY = CZ. Scaling for frequency and impedance yields element values of CY 1 = 5.117 nF LY 1 = 75.36 nH The first-order lattice requires an inductor for Z(s) and a capacitor for Y(s), both of which have a normalized value of 1/ 3 = 1.80996. Scaling for frequency and impedance yields element values of LZ 3 = 831.6 nH CY 3 = 332.6 pF
Passive Synthesis
123
The procedure for determining element values for the second phase delay network is the same, except parameter values of 2 and 6 are used for the second-order lattice, and 4 is used for the first-order lattice. In the second-order lattice, the element values for the parallel inductor and capacitor combination of Z(s) are given by LZ =
1 1 + = 6.0966 σ2 σ6
CZ =
1 = 0.03592 σ2 + σ6
and
Once again, the normalized values of the series capacitor and inductor combination of Y(s) are given by CY = LZ and LY = CZ. Scaling these values for frequency and impedance gives LZ 2 = 2.801 µH CZ 2 = 6.601 pF CY 2 = 1.12 nF LY 2 = 16.5 nH The first-order lattice requires an inductor for Z(s) and a capacitor for Y(s), both of which have a normalized value of 1/ 4 = 0.5525. Scaling for frequency and impedance yields element values of LZ 4 = 253.8 nH CY 4 = 101.5 pF This example may be difficult to realize in practice, owing to the large range of element values and the frequency range over which the components are required to operate. It may be better to cascade a second-order lattice with a first-order lattice, as was done in the example, rather than to use a cascade of first-order lattices only. The large inductor values required in the series arms of the second-order lattices are likely to be accompanied by a substantial interwinding capacitance, so the practical value of the capacitor in parallel will be less than the calculated value. ❂❂❂
124
4.12
Lumped Element Quadrature Hybrids
Approximate-Amplitude Hybrids Based on Phase Delay Networks
The basic in-phase hybrid and more elaborate designs all require an extra transformer if input and output impedances must be the same. This component may impose a bandwidth limitation. The need for extra transformers can be removed by the use of a second in-phase hybrid at the output of the phase delay networks, as shown in Figure 4.32. The first in-phase hybrid typically divides either the input voltage or the input current between the two outputs. Thus, its output impedance may be either Z0/2 or 2Z0, respectively. Suppose it divides the current. This means the phase delay networks must be designed for a reference impedance of 2Z0 in order to remain matched. Now, when their outputs are transmitted to the second hybrid, they meet a reference impedance of 2Z0 as well, as the second hybrid is placed the opposite way around. However, the second hybrid maintains its output impedance at Z0, so the whole circuit is matched to Z0 at all ports. The action of the second hybrid not only corrects for output impedance but modifies the output signals as well. Consider the voltage vectors of signals as they propagate through the circuit. Suppose, first of all, that the input hybrid contributes an arbitrarily small phase delay to the input signal. As it divides current, the voltage applied to each phase delay network is the same as the source voltage. The phase delay networks impart a phase rotation to the source voltage such that their outputs are nearly, but not quite, in quadrature. Any common phase delay is not important, so we might suppose the output from the first phase delay network to have a relative phase output
1
Σ
1
Phase delay network a
1
Input in-phase divider
3
∆
Σ
4
∆
2
Output in-phase divider
2
Phase delay network b
Figure 4.32 Dual in-phase hybrid circuit.
2
Passive Synthesis
125
of , where is presumed to be small. Furthermore, the output from the second delay network has a relative phase of 90° – . These two outputs may be denoted by the vectors VP1 and VP2, respectively, and expressed in complex form as V P 1 = V S cos α + jV S sin α
(4.59)
V P 2 = jV S cos α + V S sin α
(4.60)
The two equations have been arranged such that the first term in each is the wanted component, and the second is the error component. The relative error is tan . As regards the second hybrid, the sum port transmits a signal that is the average of the two inputs, and the difference port is half the difference of the input ports. Thus, the two outputs become V O 1 = 12 V S (1 + j )(cos α + sin α) =
1 2
V S ∠45° cos α +
1 2
V S ∠45° sin α
V O 2 = 12 V S (1 − j )(cos α − sin α) =
1 2
V S ∠ − 45° cos α −
1 2
V S ∠ − 45° sin α
(4.61)
(4.62)
Once again the two equations have been arranged such that the first term is the major wanted constituent, and the second is the error component. The relative error remains unchanged at tan . The phase of the error component this time is in phase with the wanted component. Thus, the outputs are that of an approximate-amplitude hybrid. It is thus seen by practical means how approximate-amplitude and approximate-phase quadrature hybrids are equivalent. Any 0°/180° hybrid design may be used, subject to practical con straints. A useful form of the hybrid is that shown in Figure 4.33. The advantage of this circuit is that it is composed entirely of baluns and, so, avoids the high-frequency constraints imposed by a conventional transformer, such as that used in the basic hybrid of Figure 4.27. Practical baluns behave like transmission lines. In the circuit of Figure 4.33, the optimum characteristic impedance of the balun transmission lines is 2Z 0. In addition, all ports are available externally. If the output ports have an impedance of 2Z0, then the inputs have an impedance of Z0.
126
Lumped Element Quadrature Hybrids
I1/2+I2 /2 I1
1 Z0
V1+V2
Σ 2 Z0
V1
I1/2−I2 /2
I2 2 Z0
V1−V2
∆ 2 Z0
V2
Figure 4.33 Transmission line transformer hybrid.
The operation of the circuit is as shown in the diagram. Provided the circuit is terminated with loads of equal impedance, a current at each input divides equally between the two outputs. The output current components arising from a signal at port 1 are of the same polarity, whereas those arising from a signal at port 2 are of opposite polarity. The choice of showing sum and difference outputs on the right is arbitrary. It is equally valid to exchange the port designations 1 and 2 with Σ and ∆, respectively, and to operate the circuit in the opposite direction. The orientation shown is useful in practice as the inputs can share a common ground terminal. However, in other circumstances, a step down in impedance is preferred. Care must be taken in handling the outputs as they are floating. It is not possible to ground more than one terminal without introducing a short circuit to a balun output.
4.13
Conclusion
A narrowband first-order quadrature hybrid can be synthesized in two forms, each with its own advantages, with the potential of operating up to microwave frequencies.
Passive Synthesis
127
Quadrature hybrids based on a polynomial function can be synthesized using coupler-type circuits and without recourse to perfect transformers. Transmission lines may be substituted for lumped elements where high-frequency operation is anticipated, with only slight departure from the purely lumped element response. Quadrature hybrids based on an optimum rational function can be synthesized both as coupler and phase delay circuits. The c oupler form is more easily synthesized for high-order specification using a cascade circuit. Circuits based on phase delay networks are more elaborate than the coupler form but can be made without the need for coupled inductors.
References [1] Reed, J., and G. J. Wheeler, “A Method of Analysis of Symmetrical Four-Port Networks,” IRE Trans. Microwave Theory and Techniques, Vol. PGMTT-4, No. 4, October 1956, pp. 246–252. [2] Kurokawa, K., “Design Theory of Balanced Transistor Amplifiers,” Bell System Technical Journal, Vol. 44, No. 8, October 1965, pp. 1675–1698. [3] Monteath, G. D., “Coupled Transmission Lines as Symmetrical Directional Couplers,” IEE Proceedings, Vol. 102, Part B, No. 3, May 1955, pp. 383–392. [4] Darlington, S., “Synthesis of Reactance 4-Poles Which Produce Prescribed Insertion Loss Characteristics,” Journal of Mathematics and Physics, Vol. 18, No. 4, September 1939, pp. 257–353. [5] Guillemin, E. A., Synthesis of Passive Networks, New York: John Wiley & Sons, 1957, pp. 467–470. [6] Cappucci, J. D., and H. Seidel, “Four Port Directive Coupler Having Electrical Sym metry with Respect to Both Axes,” U.S. Patent No. 3,452,300, June 24, 1969. [7] Fisher, R. E., “Broad-Band Twisted-Wire Quadrature Hybrids,” IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-21, No. 5, May 1973, pp. 355–357. [8] Ho, C. Y., “Design of Quadrature Hybrids— Part II,” 1979, pp. 32–39.
RF Design, September/October
[9] Shelton, R., J. Wolf, and R. van Wagoner, “Tandem Couplers and Phase Shifters for Multi-Octave Bandwidth,” Microwaves, April 1965, pp. 14–19. [10] Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Dedham, MA: Artech House, 1980, pp. 812–814. [11] Cappucci, J. D., “Networks Using Cascaded Quadrature Couplers, Each Coupler Having a Different Center Operating Frequency,” U.S. Patent No. 3,514,722, May 26, 1970.
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[12] Weinberg, L., Network Analysis and Synthesis, Huntington, NY: Robert E. Kreiger Publishing Company, 1975, p. 238, Theorem 6-1. [13] Ho, C. Y., and J. H. Weidman, “A Broadband Quadrature Coupler,” Microwave Journal, Vol. 26, No. 5, May 1983, pp. 216–219. [14] Edwards, R. L., and B. L. Martin, “Wideband Transmission Line Signal Combiner/ Divider,” U.S. Patent No. 4,774,481, September 28, 1988. [15] Sevick, J., Transmission Line Transformers, Atlanta, GA: Noble Publishing, 1996. [16] Abrie, P. L. D., Design of RF and Microwave Amplifiers and Oscillators, Norwood, MA: Artech House, 2000, pp. 181–188. [17] Sontheimer, C. G., and R. E. Fredrick, “Broadband Directional Coupler,”U.S. Patent No. 3,426,298, February 4, 1969.
5 Practical Design The previous chapters have dealt with the theoretical aspects of quadrature hybrid design. Synthesis has assumed the use of perfect components. In reality, a practical quadrature hybrid requires construction from imperfect components, exhibiting characteristics such as stray capacitance, leakage inductance, and loss mechanisms of various kinds. In addition, the theoretical aspects do not allow for the practical consideration of construction. The assembly of components for a particular design may be so crowded as to be very difficult, if not virtually impossible, to construct, or the component values may be quite unsuitable for the frequency of operation. This chapter looks at practical issues regarding the construction of quadrature hybrids. The techniques described have all been tested in practice, and measurement results of actual circuits are presented. At every stage, only commercially available components are used, reflecting the environment of the prospective designer. Although under certain circumstances custom materials might be of advantage (particularly for magnetic materials), these will not be readily available to the designer. The bandwidths of the examples given extend from HF to lower microwave frequencies. Quadrature circuits below this frequency range will usually be implemented using active circuits, and Chapter 6 will describe such implementations. Lumped element quadrature hybrids operating at moderate to higher microwave frequencies require the use of microtechnology and entail greatly increased equipment costs. The examples given here can be constructed using a modest equipment level and cover a broad range of 129
130
Lumped Element Quadrature Hybrids
potential applications. Should designers require operation at higher microwave frequencies, it should be possible to adapt the techniques described here for those purposes. The description of practical design techniques will follow the same pattern of development used in Chapter 4.
5.1 The First-Order Circuit Coupled Inductor Design The circuit to be realized is that shown in Figure 4.3. Of the two elements required in its construction, the capacitor is the more easily procured. For the circuit’s anticipated bandwidth, the capacitor may be implemented using a leaded or surface mount component. Typically, two capacitors selected from a preferred range of values must be connected in parallel to give a reasonable approximation to the required value. Otherwise, an adjustable component may be used. In practice, the use of a single capacitor joined to the centers of the coupled inductor windings may be inconvenient. For this reason, a practical circuit may use two individual capacitors of half the total capacitance placed each end of the coupled inductor, as shown in Figure 5.1. In this position, they may be attached to the hybrid terminals, which are typically more substantial than the inductor windings. The operation of the circuit remains unchanged, provided that the coupled inductor is a perfect component. This can be appreciated by considering the even- and odd-mode equivalent circuits, both of which remain unchanged by the arbitrary distribution of the capacitor along the length of the coupled inductor. Capacitors are available in a range of dielectric materials. We will consider the use of only ceramic capacitors as they offer the best electrical properties over the range of frequencies considered. Ceramic materials are available in a range of permittivity, broadly described as low-k, med-k, and high-k. The lowest loss materials come in the low-k category and include materials described as “COG,” “NPO,” or “P90.” Although materials in the L 1
4
C/2
k=1
C/2 3
2
L
Figure 5.1 First-order circuit with divided capacitor.
Practical Design
131
med-k category give tolerable electrical properties in most applications, the range of capacitor values available using low permittivity materials is sufficient for designs covering HF up to microwave. High-permittivity materials exhibit higher loss and poor temperature stability and, so, are unsuitable for quadrature hybrids. At moderate microwave frequencies, high-Q porcelain capacitors offer high performance. Their exceptionally low loss characteristics make them suitable for high-power operation. Higher microwave frequencies may require single-layer capacitors. These are available in a range of materials, sizes, and capacitor values and must be selected according to the particular application. The coupled inductor element will have to be constructed for the specific application. The ideal circuit requires perfect coupling between the inductors, and only an approximation to this requirement will be achieved in practice. A common technique to improve coupling is to construct the inductor using two insulated wires twisted tightly together, the “twisted pair” or “bifilar winding.”This construction exhibits distributed capacitance as well as leakage inductance and, so, behaves as a transmission line. The impedance depends on the degree of twisting, thickness of the insulation, and diameter of the wire, but it is typically in the region of several tens of ohms. This distributed behavior is usually undesirable, so it is preferable to make the wire length small. In order to achieve the desired inductance, the wire is typically wound around a magnetic core. Magnetic materials for this purpose have a range of relative permeability depending on the type of material used, with increasing permeability generally associated with greater magnetic loss. Various mixes of carbonyl iron material range in permeability from around 4 to 60, though only the lower values exhibit the most desirable properties for quadrature hybrids. Nickel zinc ferrite material is available with a relative permeability in the range of around 10 to a couple of thousand. Within this range, the lower permeability material, though exhibiting desirable magnetic properties, is available in only a few core shapes, while the higher values exhibit too high a loss. The most useful range of permeability for nickel zinc material is in the region of 100, where a reasonable choice of core shapes is available. However, the loss characteristics of nickel zinc material in this range make it suitable only up to frequencies of a few megahertz. Manganese zinc ferrite material exhibits loss characteristics that make it unsuitable for use at frequencies where quadrature hybrids are typically employed. A particularly useful core shape for the magnetic part of the quadrature hybrid is the toroid. The enclosed magnetic path gives high inductance for a particular core material. In addition, the core shape is available in a wide range of materials. To achieve the required inductance with low loss, it is better to
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Lumped Element Quadrature Hybrids
choose a lower permeability material and apply more turns to the core. This plan is particularly suitable for the design of first-order quadrature hybrids that operate as a single element in a narrowband application close to their equal power division frequency. Although the wire length is greater, the increase in conductor loss is more than made up for by the reduction in core loss. However, where the first-order quadrature hybrid is the low-frequency element in a more complex hybrid, such as a cascade, then it will have to exhibit acceptable properties throughout the passband of the total circuit. The high number of turns on a low-loss core may lead to excessive interwinding capacitance. If this should lead to a resonance within the passband, then the performance of the hybrid will be degraded. We observed previously that it is difficult to attach a single capacitor to the windings of a coupled inductor and that splitting the value into two halves and placing one on each end of the coupled inductor is more convenient for construction, as shown in Figure 5.1. This technique is suitable for narrowband operation but introduces a difficulty for wideband applications. To illustrate this point, consider the actual equivalent odd-mode circuit. In reality, the coupled inductor behaves as a transmission line in the odd mode, giving an equivalent circuit more like that shown in Figure 5.2, where Zt denotes the characteristic impedance of the twisted pair. Although the preferred odd-mode circuit is of the first order, the actual circuit is of the third order, and at some frequency above the center frequency, it will exhibit zero insertion loss. The even-mode circuit is not affected and will continue to exhibit first-order characteristics. The even- and odd-mode circuits are therefore not duals of each other, and isolation and match properties will degrade as frequency increases, with an acute degradation at the transmission frequency of the odd-mode circuit. To illustrate this point, a first-order quadrature hybrid was constructed with a center frequency of operation of 20 MHz. The inductor element was Zt /2,θ
C/2
Figure 5.2 Odd-mode circuit with divided capacitor.
C/2
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constructed using a bifilar winding of 0.2-mm-diameter enameled copper wire wound as an eight-turn winding on a Micrometals T22-2 toroidal core. The carbonyl iron core material has a relative permeability of 10 and exhibits low loss characteristics at the center frequency of operation. At first, two capacitors, each of value 82 pF, were placed at either end of the inductor element. The hybrid was measured at frequencies up to 200 MHz, and Figure 5.3 shows a plot of coupling and input return loss as a sample of its characteristics. At low frequencies, particularly around 20 MHz, the hybrid exhibits excellent match properties. However, as frequency increases, the return loss decreases steadily and reaches a minimum of only 10.4 dB at a frequency of 151 MHz. The coupling, which should steadily approach 0 dB, exhibits a dip to 3 dB at this frequency. This hybrid, though it gives good performance at its center frequency, is only of limited use in more complex designs. If it were the low-frequency section of a cascade design, as described in Section
Figure 5.3 Coupling and return loss of hybrid with separate capacitors.
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4.8, then it might only have suitable performance to allow an upper passband limit of 120 MHz. In typical designs, the low-frequency section has a center frequency at around half the lower band edge, so only designs operating from around 40 to 120 MHz could be designed using this section. The capacitors were then attached at the center of the coupled inductor, as shown in Figure 4.3. The two parameters of coupling and return loss were again measured, but this time to the much higher frequency of 500 MHz. Figure 5.4 shows the responses. Operation at the center frequency of 20 MHz remains virtually unchanged. The lump in return loss and dip in insertion loss, as seen in the previous example at 150 MHz, have been removed. The return loss is better than 20 dB up to 483 MHz, and insertion loss has only reduced to 0.6 dB. The properties of this section make it suitable for cascade designs operating at more than a decade in bandwidth. The insertion loss will add to the total loss of the design and is likely to be the greatest contributor to
Figure 5.4 Coupling and return loss of 20-MHz hybrid with single capacitor.
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loss as the remaining sections of higher frequency design will exhibit superior loss characteristics at the upper passband limit. It should be noted that some capacitance had to be added to the ports of the hybrid to compensate for leakage inductance up to 483 MHz. However, no such compensation would have been possible for the design with divided capacitors. The construction of the hybrid with a single capacitor is more difficult and less robust as attachment has to be made at the midpoint of the winding. As a result, this technique should only be used when necessary.
5.2 A 435-MHz Ground Inductor Hybrid Design The distributed nature of the coupled inductor considered in the previous section continues to influence performance, even when using a single capacitor. In the odd mode, a delay is present between the terminals and the capacitor. At lower RF frequencies when a core with appreciable permeability is used, this delay is of little significance. However, as design frequency increases, lower-permeability material becomes necessary, and the relative electrical length of the bifilar winding increases. At frequencies in the UHF region, any magnetic core material may become the source of unacceptable loss, and an air-spaced inductor becomes necessary. Inductor properties may be improved by coiling the bifilar wire into a solenoid, but as frequency increases still further, this element becomes more difficult to construct as dimensions decrease. An additional limitation can be found in the discontinuity between the transmission line at the ports of the hybrid (typically microstrip) and the wire of the inductor. Once again, at modest RF frequencies, this discontinuity is not significant, but its effect becomes more marked as frequencies increase. As quadrature hybrids using a coupled inductor are difficult to construct at microwave frequencies, designers have typically resorted to distributed circuits. For lumped element quadrature hybrids to continue to be useful at microwave frequencies, the ground inductor design, as shown in Figure 4.4, is preferred. It is possible in theory to design such a hybrid to operate at lower RF frequencies as well by winding transmission lines around a magnetic core. The transmission lines act as waveguides and introduce an additional electrical delay to the circuit. Provided that all four ports are equipped with an equal delay, the quadrature properties are not compromised. Usually, however, this form of quadrature hybrid is used at higher RF and microwave frequencies, where the inductor element is air spaced and has no enhancement of mutual inductance from multiturn winding.
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Lumped Element Quadrature Hybrids
The circuit of Figure 4.4 requires additional components to prevent the ground terminals of opposite ports from giving a short circuit to the ground line inductor. The most convenient method of doing this is to add transmission lines to each port and use the screens of the transmission lines to form the inductor. Figure 5.5 shows a circuit diagram of the arrangement where the transmission lines have been drawn as coaxial cables. In other circumstances, the transmission lines may be realized in microstrip. Figure 5.6 shows an example of a ground line inductor hybrid using coaxial cable. This hybrid was designed for UHF radar applications at a center frequency of 435 MHz. The coaxial cable has a diameter of 2.2 mm and is
Z0 ,θ
Z0 ,θ
1
4
C Z0 ,θ
Z0 ,θ 3
2
L
Figure 5.5 Practical circuit of ground line inductor hybrid.
Figure 5.6 435-MHz ground line inductor hybrid.
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supported at the center of the structure by a substrate bridge, which also supports the lumped capacitor. The inductor element comprises the screens of the coaxial cable, together with a cavity underneath. The lumped capacitor is a high-Q, porcelain-chip type, and this, together with the substantial dimensions of the inductor element, leads to very low loss. Equal power division was measured at a frequency of 445 MHz rather than the intended value of 435 MHz. Part of the reason for this was the decision to use a single capacitor. With the choice of value limited to preferred values, center frequency is adjustable only in discrete steps. The actual value was 6.8 pF, which, in theory, should give a center frequency of 468 MHz as the characteristic impedance of the terminations was 50Ω. It is found in practice that the capacitor element requires a slightly smaller-than-calculated value as stray capacitance and series inductance conspire to make the value appear larger.
5.3 A 1.27-GHz First-Order Microstrip Hybrid For operation at microwave frequencies, the use of microstrip transmission lines is often preferred. In this medium, the continuous ground plane must be interrupted to form the inductor. Figure 5.7 shows one implementation in which the various elements of this 1.27-GHz concept design have been separated. At the left is the aluminum carrier with a cavity machined out. In the middle is the upper side of the substrate, showing the microstrip lines with a lumped capacitor across them. At the right is a second substrate, showing the underside ground plane structure. This structure can be modeled as two short circuit stubs in series that are electrically short and, therefore, inductive at the design frequency. Each stub occupies the volume from the center of the cavity to one end and is defined by the width of the ground line strip and cavity depth. Provided that the ground line strip is wide compared with the microstrip lines on the upper side of the substrate, it provides an effective screen. The characteristic impedance of the ground line stub can be determined by treating it as another microstrip line with air dielectric. The length of the ground plane structure has to be determined to give the correct inductive reactance. Part of the solution to this problem is determining the characteristic impedance of the ground line strip. The reactance of the two stubs in series is given by X = 2 Z S tan θ
(5.1)
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Lumped Element Quadrature Hybrids
Figure 5.7 1.27-GHz microstrip hybrid.
In (5.1), X is the reactance of the series stubs, ZS is the characteristic impedance of the ground line strips, and is the electrical length. At the center frequency, X is equal to the system characteristic impedance, which is usually 50Ω. This equation is complicated in practice as the cavity walls add inductance. In addition, there is a small gap between the two strips of the ground line structure. This is necessary to provide minimum stray inductance in the odd mode; however, in the even mode, this small gap introduces a parallel capacitance. Both of these factors tend to increase the ground line inductive reactance so that determination of the correct dimensions requires experimentation. As there is no inductance enhancement through multiturn windings or magnetic materials, the ground line inductor techniques described above give a restricted bandwidth. The distributed nature of the ground line inductor structure, together with the stray inductance across its terminals, gives rise to a relatively low parallel resonance frequency. As a consequence, the multioctave operation of quadrature hybrids based on the optimum rational function using cascaded circuits is not practical using these elements.
5.4 A 100–200-MHz Third-Order Polynomial-Based Hybrid To illustrate some of the techniques required in the construction of hybrids based on higher-order polynomial approximations, consider Example 4.1.
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Figure 4.6 shows the target circuit, and element values were determined to give an optimum response in the 100–200-MHz bandwidth. The capacitor elements can be made up using either a single component or two in parallel. Often a single component in the E12 series is sufficiently accurate. As a first estimate of the capacitor value, the next size down on the calculated value should be tried as stray capacitance in the circuit tends to add to the fitted components. For this application, an initial value of 15 pF was used for the instances of Co, and 3.3 pF was used for the instances of Ce . The type of capacitors used was commercial EIA 1206 size package, with NPO dielectric. For the design of the inductor elements, a vector analyzer configured to measure impedance was used to measure some test components. All of the coupled inductors were wound with a bifilar pair of 0.2-mm-diameter enameled copper wire. This size has been found useful in many applications. Although finer wire reduces stray capacitance, it is more difficult to handle. For the coupled inductor in the center position, an air-spaced, two-turn solenoid of 1.6-mm diameter was found to give the best performance. The coupled inductors in the outer positions, being larger, were wound on a Micrometals T12-6 core. This core is made of carbonyl iron material, having a relative permeability of 8.5. Its diameter is about 3 mm. Although its specific inductance is only 1.7 nH, four turns were sufficient to give an inductance in the region of the required 46.96 nH. The specific inductance quoted by the manufacturer relates to a full winding. Where the winding is partial, magnetic flux extending outside the core adds to the inductance. Figure 5.8 shows the completed hybrid. The circuit has been constructed on 1.6-mm-thick FR4 substrate, with a continuous ground plane on the underside. Although the dielectric constant is not well controlled for this material, the width of the tracks is approximately correct for a 50Ω line, and the lines are not long enough to suffer from a slight impedance error. Two of the original 3.3-pF capacitors were replaced with 2.7-pF ones. Although this slightly imbalances the circuit, it is still an improvement on having them all of one value or the other. A production version might use a value of 3 pF throughout. The inductors were adjusted in value by altering the separation of the turns. Optimum performance was seen using capacitor values slightly lower than that calculated. This is to be expected, considering the layout of the circuit. For example, the pads to which the 15-pF capacitors are attached add a capacitance of around 0.5 pF to each instance of Co. The wire-to-wire capacitance of the outer coupled inductors adds to each instance of Ce. The final dimensions of the hybrid were about 15 mm × 15 mm. The tracks extending away from the hybrid components are only a convenience for measurement and might be reduced in size for an integrated component.
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Lumped Element Quadrature Hybrids
Figure 5.8 100–200-MHz third-order polynomial hybrid.
Figure 5.9 shows the insertion loss to coupled and through ports. From Table 3.2 we learn that a 2:1 bandwidth should give 0.748-dB amplitude imbalance. The markers at 150 MHz are placed approximately at the point of maximum imbalance and show an imbalance of 0.78 dB. This close agreement with theory is consistent with the modest frequency range of the design. Some insertion loss due to dissipation can be inferred from this plot. For a perfect design, the two insertion-loss measurements should cross at a value of 3.01 dB. The two crossing points indicate an excess loss ranging from 0.1 to 0.2 dB, although a small part of this is due to reflection at the input port and leakage to the isolated port. The phase error from quadrature was less than 1°. Port match and isolation were both better than 20 dB.
5.5 A 1–3-GHz Mixed Element Hybrid Section 4.4 described the synthesis of mixed element quadrature hybrids. To illustrate their practical design, we will consider the specific case of a fifth-order hybrid operating over a 1–3-GHz bandwidth. Normalized
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Figure 5.9 100–200-MHz hybrid insertion loss.
element values can be determined from Table 4.4 using information for a 3:1 bandwidth ratio. Once again, a characteristic impedance of 50Ω is assumed. Scaling the normalized values for frequency and impedance, the outer sections require a ground inductor of 1.11 nH and a capacitor of 0.444 pF. The center section requires a ground inductor of 6.79 nH and a capacitor of 2.72 pF. A microstrip version of this coupler is likely to be most useful in practice. As the center inductor section is expected to be large in size, a relatively low dielectric constant for the substrate is desirable so that the outer sections are further removed from the center. For this design, a substrate with a relative dielectric constant of 3.28 and thickness of 0.3 mm was chosen. Although materials with a lower dielectric constant are available, these are typically PTFE based and exhibit poor structural qualities. Such materials were thought too weak for use over the unsupported area above inductor cavities. With this substrate, 50Ω transmission lines require a line width of 0.67 mm. As microstrip lines have an open structure, some coupling will occur between lines in close proximity. In order to be uncoupled, as the design requires, the lines should be far apart. However, as the ground line inductor
142
Lumped Element Quadrature Hybrids
requires the use of the screen as part of its structure, and as this performs better if it is narrow, the microstrip lines are required to be close together. These two conditions necessitate a compromise. In practice, as the coupling due to the ground line inductor and coupling capacitors is much greater than the proximity coupling of the microstrip lines, they can be put relatively close together. For this design, a 0.5-mm line spacing was chosen. For the inductor of the center section, the technique used in the 1.27-GHz first-order design was repeated here. The ground line strips had a width of 3 mm. Although some of the RF fields due to the microstrip mode will extend into the cavity, most will be constrained by the substrate. The depth of the cavity was 5 mm, over which the 3-mm strips have a predicted impedance of 154Ω. The total length of the cavity was 12.5 mm, and given the characteristic impedance of the ground line strips, this indicates an inductance of 6.4 nH. However, owing to the distributed nature of this element, its reactance will increase more rapidly than an inductor’s linear relationship with frequency. At 3 GHz, each strip has an electrical length of about 22.5°, for which the reactance of each 154 Ω shorted line is 63.8Ω and is 127.6Ω in total. Such a reactance at this frequency is equivalent to an inductor of value 6.8 nH. In addition, the actual reactance will be complicated by the cavity walls and the capacitance of the gap between the two strips. In fact, analysis can only give a guide to the initial length of the strips. The final length is arrived at by experiment. One method of determining the correct dimensions is to construct the center section as a separate unit and measure the coupling. The component values correspond to a first-order section with a center frequency of 1.17 GHz. In practice, it may be better to optimize the design for performance at 3 GHz as this will tend to degrade with increasing frequency. At 3 GHz, the theoretical coupling is 0.62 dB. As dissipation will be comparable to this value, allowance must be made for it by, for example, considering the loss to the through port as well. The outer sections require a much lower inductance value, so the technique of forming it by ground line strips is not appropriate. In this case, a pattern on the ground plane is preferred. The solution in this case was to introduce a “bow tie” pattern very much like two slot-line short circuit ele ments with a short section of slot-line between them. As slot-line elements, they are intended to give low impedance over a wide bandwidth; at lower frequencies, however, they behave as inductors. Although a cavity is required to prevent a short circuit to the inductor element, the inductance value is not sensitive to its dimensions. Information on slot-line discontinuities is not readily available, so this element had to be determined by experiment. Once again, an outer section can be constructed and measured in isolation. The
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component values correspond with a first-order section with a center frequency of 7.18 GHz; at 3 GHz, its coupling is 8.3 dB. Figure 5.10 shows the complete hybrid, although it has been dismantled to show all the constituent elements. At the left is the aluminum carrier showing the three cavities. We can seen that there is very little space between them, with only two thin webs as separation. At the center we can see the top substrate metallization comprising 50Ω microstrip lines with lumped capacitors placed at the center of each lumped element section. The practical values were 2.4 pF for the center section and 0.4 pF for the outer sections. At the right, we can see the underside of the substrate showing the ground line strips of the center section and the bow tie arrangement of the outer sections. A restriction with this design is that there is very little scope for tuning. In fact, only the capacitors can be adjusted, both in value and position. Figure 5.11 shows the insertion loss to the through and coupled ports. Table 4.4 gives a theoretical maximum imbalance of 0.668 dB. The measured performance does not achieve this value below about 1.2 GHz. Part of the reason for this is that the inductor elements were optimized for 3-GHz operation, and their variation of inductance with frequency leads to a comparatively low inductance at the lower frequencies.
Figure 5.10 1–3-GHz microstrip hybrid.
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Figure 5.11 1–3-GHz hybrid insertion loss.
5.6 A 2.5–6-GHz Hybrid with Coupled Transmission Lines Section 4.4 described the synthesis of quadrature hybrids employing coupled transmission lines in conjunction with lumped elements. They are useful in applications where the close proximity of lumped element sections in mixed element designs become difficult to realize. Although it is possible in theory to increase the frequency of operation simply by reducing the size of the hybrid, there may be practical difficulties in doing this. Consider the 1–3-GHz example of Section 5.5. This already used a thin substrate, so for reasons of mechanical strength, it is undesirable to consider a thinner material. Even with the substrate as thin as it was, positioning of the cavities of the lumped sections was only just possible. A design based on a single lumped element section in conjunction with coupled transmission lines removes this problem and extends the application of lumped element sections to higher microwave frequencies. To illustrate the technique, consider the requirement of a quadrature hybrid operating over a 2.5–6-GHz bandwidth. Suppose too that this design
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is intended for integration into a microstrip circuit. Such an application will often be served by the use of a Lange coupler, but these are difficult to implement without recourse to thin-film technology. The design procedure should anticipate degradation in performance at the upper passband limit, so it may be better to skew the center frequency slightly high and accept a poorer amplitude balance at 2.5 GHz. The theoretical design may therefore be optimized for amplitude balance over a 3–6-GHz bandwidth. Table 4.5 gives prototype element values for quadrature hybrids employing coupled transmission lines. Using the information for a 2:1 bandwidth, the lumped element section has a center frequency that scales to 3.2 GHz. The even-mode impedance of the coupled line sections corresponds to an 11.7-dB coupling. As circuit discontinuities and stray elements are likely to make a significant contribution at these frequencies, it is necessary to test the lumped element section independently. The particular application for this circuit placed a constraint on the depth of the lumped element section cavity, which was limited to a maximum of 2 mm. A test circuit was constructed using 0.8-mm substrate having a dielectric constant of 3.38. Two 50Ω microstrip lines were placed over a cavity having a length of 8 mm. The ground line structure was similar to the 1.27-GHz example described in Section 5.3, and a strip width of 5 mm was used. A parallel capacitor was added, and this was adjusted in value for optimum match and isolation. The best performance was achieved using a value of 0.68 pF, and this gave a center frequency of 4.21 GHz. This value indicates that the cavity dimensions are too small. An additional property of the first-order test circuit was a resonant dip in through loss at 8.8 GHz. This property is likely to be the result of stray capacitance between the ends of the ground line strips and will require compensation for operation to 6 GHz. The information derived from the first-order experiment was used to model a complete quadrature hybrid, which could then be optimized. The optimization procedure predicted that a stronger coupling in the coupled line sections was required. The new coupling value was too strong for manufacture using conventional etching techniques, necessitating a four-finger interdigitated design. This relaxed the manufacture to a line width of 0.363 mm and spacing of 0.322 mm. The coupled line length was 7 mm on each side of the lumped element section. Figure 5.12 shows an assembly with the elements of the 2.5–6-GHz quadrature hybrid. At the left, we can see the aluminum carrier with its cavity. In the center is a substrate showing the ground line structure. At the right is the top circuit, showing the interdigitated pattern. The isolated port is locally terminated for practical applications. Bonding between the lines is
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Figure 5.12 2.5–6-GHz microstrip hybrid.
similar to a four-finger Lange coupler, although an additional link is required in the center as the lines are half a wavelength long within the bandwidth of the design and exhibit a parasitic resonance if not bonded. Figure 5.13 shows the insertion loss to the through and coupled ports. The response does not follow the normal fifth-order response of the prototype circuit, but resembles more a third-order circuit. Owing to the presence of parasitic elements, hybrid performance degrades rapidly above 6 GHz. This design exhibited port match and isolation typically around 20 dB. Quadrature response varies with construction and can be in error by as much as 10°.
5.7 Optimum Second-Order Hybrids Section 4.5 described the synthesis of these hybrids. These hybrids are of interest not only for their own sake but also as building blocks for higherorder hybrids. It is therefore necessary to consider their design for operation at much higher frequencies than their mean coupling. Figure 4.19 shows the circuit to be constructed. The capacitors are readily implemented as fixed values, but the coupled inductor element requires consideration. Although the coupled inductor is drawn as two centertapped inductors, in practical circuits it is necessary to minimize leakage
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Figure 5.13 2.5–6-GHz hybrid insertion loss.
inductance between each inductor. Note that the inductor elements on each side of the center tap must be perfectly coupled. The element is not equivalent to two inductors in series. The technique for doing this is the same as that used in the first-order circuit with coupled inductor; that is to say, a bifilar twisted pair should be used. Figure 5.14 shows the practical connection. The link shown between each half of the inductor should represent only a small leakage inductance. This is usually made possible by placing the ends of the twisted pair close together. The indicated inductance refers to the selfinductance from one end of the inductor to the center tap. It is sometimes easier to design a center-tapped inductor based on this value in the knowledge that the inductance from one end to the other is four times this value. Having determined the configuration for a center-tapped inductor, it is then necessary to couple two together partially. To this end, enhancement of coupling using magnetic materials is essential for two reasons: first, the coupling factor required in typical designs is difficult to achieve using an
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CT
Figure 5.14 Practical center-tapped inductor.
air-spaced design, and, second, it is necessary to minimize the contribution of leakage inductance in each inductor. The most readily available core shape in magnetic materials of suitable permeability is the toroid. Various techniques can be used to achieve the required coupling between the two inductors. The preferred solution is to wind the inductors on the same core, but this may not always be possible. At low frequencies, relatively high-permeability materials are usually preferred in order to restrict size. A single core may give excessive coupling between the two inductors no matter how its wound, so additional cores are necessary. Figure 5.15 shows a practical arrangement where a common core is used to provide the mutual inductance M. Secondary cores are used to provide the remaining self-inductance L – M. As frequency increases, lower-permeability materials are required. In the arrangement of Figure 5.15, the mutual core exhibits progressively more leakage inductance in each winding, and the value of the additional inductors Common core
Separate cores
Figure 5.15 Practical inductor coupling at low frequencies.
Bifilar windings
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becomes relatively lower. Eventually, there will be no need at all for the additional inductors, and a single core will be able to achieve the required selfand mutual inductances. Indeed, it may be better to choose a lower permeability core deliberately in order to make the design simpler. It is possible to arrange the windings of each inductor to adjust for coupling. Where the permeability is relatively high, for example, at lower frequencies, weaker coupling can be achieved by maximum separation of the two inductors, as shown in Figure 5.16(a). Where the permeability is lower, stronger coupling can be achieved by distributing the turns around the core and bringing the end turns of each winding close together, as shown in Figure 5.16(b). In practice, the strongest coupling occurs between turns that are close together, so a large core with many turns will exhibit less coupling than a small core of few turns, even though they are composed of the same material. Very strong coupling can be achieved by winding the coupled inductors in parallel. The required self-inductance can be measured easily on a single winding. The mutual inductance is best measured indirectly. If both windings are connected in parallel, the measured inductance equals 0.5(L + M ). With the knowledge of the self-inductance, mutual inductance can be derived.
(a)
Decrease inductance, increase coupling
(b)
Figure 5.16 Variable coupling on a single core: (a) minimum coupling, and (b) moderate coupling.
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The twisted pair of each inductor exhibits distributed qualities, and these become significant even at lower UHF frequencies as the use of lower permeability materials becomes necessary. If the characteristic impedance of the twisted pair is equal to the termination impedance, the eventual effect as frequency increases and the capacitors tend toward a short circuit is that the two sides of the hybrid become uncoupled, and the transfer function to the through port is equivalent to a length of transmission line with an inversion. At moderate frequencies, hybrid operation is compromised.
5.8 A 50–150-MHz Second-Order Hybrid To illustrate the design of a second-order optimum hybrid, we will consider the specific requirement of a mid-VHF-band hybrid operating over a 50–150-MHz bandwidth, using the circuit of Figure 4.19. For this 3:1 bandwidth ratio, the normalized frequency parameters may be determined from Figure 3.10, or, for greater accuracy, (3.37) may be used to give 1 = 0.39332. Either (3.37) may be used again, or (3.39) may be used to give 2 = 1/ 1 = 2.5425. Equations (4.32), (4.33), (4.35), and (4.36) give the normalized element values. Scaling for a geometric center frequency of 86.6 MHz and an assumed characteristic impedance of 50Ω leads to element values of L = 270 nH M = 197 nH C1 = 79 pF C2 = 28.9 pF The capacitors were implemented as fixed values. For C1, the preferred value of 82 pF was used, and the parallel combination of 27 pF and 2.7 pF was used for C2. All capacitors were multilayer chip components. The inductor element was wound with 0.2-mm bifilar enameled copper wire. The required self-inductance was achieved using five turns on a Micrometals T16-2 carbonyl iron core. In order to achieve the required mutual inductance, it was necessary to overlap the end turns of each winding. A design of this type requires a relatively strong coupling in order to achieve the overcoupling at center frequency. The center-frequency coupling factor is equal to the ratio of mutual to self-inductance, which in this case is 0.732 or 2.7 dB. Having assembled the hybrid, further adjustment was necessary for optimum response, and Figure 5.17 shows the insertion loss measurements.
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Consulting Figure 3.9, the theoretical imbalance is 0.625 dB. The measured result of an 0.8-dB maximum imbalance is therefore close to theory. The loss of this circuit can be deduced where the two traces cross over, and this increases from around 0.15 to 0.3 dB with increasing frequency. Isolation and match for this hybrid were both better than 20 dB, and departure from quadrature was less than 1°.
5.9 Higher-Order Symmetrical Optimum Hybrids Although the third-order circuit as shown in Figure 4.20 is of manageable complexity for practical construction, the fourth-order circuit of Figure 4.21 is of little practical value. For higher-order optimum hybrids, the symmetric form should be abandoned in favor of cascade circuits.
Figure 5.17 50–150-MHz hybrid coupled and through insertion loss.
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5.10
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A 2–32-MHz First-Order Cascade Hybrid
Section 4.8 described the synthesis of hybrids based on a cascade of firstorder sections. In order to illustrate their practical construction, we will use the data derived in Example 4.3 for a quadrature hybrid operating over a 2–32-MHz bandwidth. The example given consisted of four stages, and the circuit diagram of Figure 4.24 is applicable to the problem. The coupled inductors may all be wound on carbonyl-iron toroids using cores of suitable dimensions and permeability. In this case, all inductors and transformers were wound using a bifilar twisted pair of 0.2-mm-diameter enameled copper wire. The inductors were wound on Micrometals parts, as described in Table 5.1. In every case, the capacitors were placed at the center of the winding for reasons given in Section 5.1. For the larger inductance values, where a relatively high-permeability core is required, the manufacturer’s data on specific inductance may be used to give a close approximation to the correct value. However, where a lower inductance is required, using a relatively lowpermeability core, the measured value is much larger than that calculated, so more experimentation is required. The capacitors fitted were either single values or combinations in parallel to give approximately the correct value. The inverting transformer of Figure 4.24 was replaced with an inverting transmission line transformer. This is just as easy to construct as a conventional transformer but gives superior high-frequency performance. For this component, an Epcos B62152-A4-X1 dual hole bead was wound with 10 turns, using the bifilar twisted pair. With a specific inductance of 190 nH, this component appears as a shunt inductor of value 19 µH. At the lower passband limit, the inductance has a susceptance of 4.19 mS and will not load the circuit significantly. In addition to its shunt inductance, the transmission line transformer also contributes a phase delay. In order to equalize the
Table 5.1 Inductor Details for 2–32-MHz Fourth-Order Hybrid Inductance
Core Type
Permeability
Number of Turns
7.312 µH
T44-1
20
26
1.74 µH
T25-1
20
16
565.5 nH
T22-6
8.5
10
135.3 nH
T16-2
10
7
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transmission paths, a second identical winding was made except that no inversion was used, and this was placed on the other side of the circuit. Figure 5.18 shows the insertion loss measurements of the circuit. The measurements are shown with a logarithmic frequency scale, showing the geometric symmetry of the underlying theory. The maximum imbalance of 0.4 dB is only slightly greater than the theoretical value of 0.3 dB. The excess loss is more significant, increasing from 0.2 dB at 2 MHz to 0.7 dB at 32 MHz. Figure 5.19 shows the measured insertion phase to the coupled port relative to the through port. The response is only close to quadrature from the mean frequency and upward. The departure from quadrature in the lower half of the passband is caused by the admittance of the inverting transformer’s magnetizing inductance. The 4° error at 2 MHz is equivalent to an amplitude imbalance of 0.6 dB (see Figure 1.20) and represents a more severe error than the amplitude imbalance. Whether this is of concern depends on the application. The circuit may be refined in several ways, for example, using a compensating inductor in the noninverting channel, a compensating capacitor in series with the inverting transformer (exact value 7.6 nF), or a much increased inductance in the inverting transformer design.
Figure 5.18 2–32-MHz hybrid coupled and through insertion loss.
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Lumped Element Quadrature Hybrids
Figure 5.19 2–32-MHz hybrid relative insertion phase.
Port match and isolation were typically better than 20 dB in this construction, though this varied according to the measurement port combination, and only degraded from this benchmark value at frequencies in excess of 20 MHz.
5.11
A 10–100-MHz Second-Order Cascade Hybrid
Section 4.9 described the synthesis of hybrids comprising a cascade of second-order sections. As a comparison to the example of Section 5.10, consider the design of a fourth-order hybrid operating over a 10–100-MHz bandwidth. Figure 4.25 shows the topology of the circuit, which consists of the cascade of two second-order sections of the form of Figure 4.19. The design commences with the determination of the frequency parameters, which, by (3.37), for a 10:1 frequency range and fourth-order design are 1 = 0.15139, 2 = 0.59702, 3 = 1.67498, and 4 = 6.60526. The first two parameters are associated with the low-frequency section, and the second two with the high-frequency section. Using (4.32), (4.33), (4.35), and (4.36), the prototype element values may be determined. Scaling
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these for frequency and a characteristic impedance of 50Ω, they become as shown in Table 5.2, where the element values refer to the circuit of Figure 4.19. The section center frequency and coupling information is useful at the development stage in that it is possible to align each section separately before forming the cascade. Both inductor elements were wound using a bifilar twisted pair of 0.2-mm enameled copper wire, as was used previously. With such a winding, it is easier to determine the correct configuration using a value of inductance a quarter of that above and with a single wire. As the wires of a bifilar winding are connected in series, the inductance will be four times that of a single wire. For the low-frequency section, the winding consisted of six turns on each side of a Micrometals T25-3 toroidal core. For the high-frequency section, four turns on each side of a Micrometals T16-2 core were used. The small number of turns on both cores meant there was significant scope for inductance adjustment. This was useful for tuning. The capacitors fitted were fixed values. Figure 5.20 show a photograph of a test circuit. The construction has been made on 1.6-mm-thick FR4 substrate as the dimensions of the circuit do not require a close tolerance on the dielectric constant. At the left, we can see the low-frequency section inductor element. For convenience, the through connection is from top left to bottom right. The high-frequency section at the right is wired similarly. Midway through the circuit, we can see a capacitor from each connecting line to ground. Although these components do not appear in the theoretical circuit, they were found to be beneficial in the practical circuit. Their function is to compensate for stray inductance in the wound components. The value used was
Table 5.2 Element Values for the 10–100-MHz Cascaded Section Hybrid
Parameter
Low-Frequency Section
High-Frequency Section
L
2.084 µH
188.3 nH
M
1.241 µH
112.1 nH
C1
496.3 pF
44.9 pF
C2
337.2 pF
30.5 pF
Section center frequency
9.51 MHz
105 MHz
Section center coupling
4.5 dB
4.5 dB
156
Lumped Element Quadrature Hybrids
Figure 5.20 10–100-MHz fourth-order hybrid.
6.8 pF, though the optimum value may vary according to construction. The hybrid capacitors are fitted on the underside of the circuit and are not visible in this view. Figure 5.21 shows the insertion loss to the through and coupled ports. The theory of Chapter 3 predicts an amplitude imbalance of 0.164 dB over this bandwidth (see Example 3.1), so the measured performance does not significantly deviate from this, except at the lower band edge. Figure 5.22 shows the measured insertion phase to the coupled port relative to the through port. The deviation from quadrature of 2.3° at 100 MHz is more significant than the amplitude imbalance at this frequency, which is equivalent to 0.35 dB. Both port match and isolation were better than 20 dB at any frequency and port combination. It is interesting to compare the performance of this circuit with that described in Section 5.10, in which a 2–32-MHz hybrid was realized using a cascade of first-order sections. Even though the 10–100-MHz hybrid oper ates at a higher frequency, it exhibits a lower excess loss of only 0.5 dB at 100 MHz, partially due to its more concise circuit topology. Only two wound components occur in each transmission path, whereas there are five in the first-order cascade. It appears, then, that the second-order cascade realization offers a lower-loss solution for comparable construction techniques.
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Figure 5.21 10–100-MHz hybrid coupled and through insertion loss.
5.12
A 10–100-MHz Approximate-Phase Hybrid
Section 4.11 stated that approximate-phase hybrids require two constituent parts, an in-phase divider and phase delay networks. To show one method of construction, we will consider the problem of a fourth-order hybrid operating over a 10–100-MHz bandwidth with a reference impedance of 50 Ω at all ports. For this application, an in-phase hybrid of the form shown in Figure 4.27 may be used, together with an input transformer to correct for port impedance. The ideal turns ratio is √2:1, but a 7:5 practical value is close enough. As magnetic isolation is not required, this component may be wound as an autotransformer. The performance of the complete quadrature hybrid depends in part on the performance of the in-phase hybrid. This component may be constructed and aligned first. To achieve the required frequency range, the autotransformer was wound using 0.71-mm-diameter enameled copper wire on a Ferroxcube MHB2-13/8/6-4B1 dual hole bead. The material exhibits high magnetizing impedance over the specified frequency range and, so, will contribute low loss. The hybrid transformer part was formed using a twisted
158
Lumped Element Quadrature Hybrids
Figure 5.22 10–100-MHz hybrid relative insertion phase.
pair of 0.71-mm enameled copper wire threaded through four Ferroxcube BD5.1/2/4-3S1 beads. This material exhibits predominantly resistive impedance across the passband, but the hybrid transformer contributes little to quadrature hybrid loss as no flux is generated in normal operation. The presence of the resistive part reduced the required conductance in the isolation component, so a 150Ω resistor achieved a better isolation than the theoretical 100Ω one. Figure 5.23 shows the in-phase hybrid circuit. The circuit of Figure 5.23 shows a 15-pF capacitor to ground at the 25Ω output of the autotransformer. This was found to be desirable to compensate for leakage inductance in the wound components. RF performance of the in-phase hybrid showed a gradual degradation with frequency. However, at the upper passband limit of 100 MHz, excess insertion loss was only 0.11 dB, and input port match was better than 30 dB. The second constituent of the quadrature hybrid is the phase delay network. As the in-phase hybrid outputs are referenced to ground, a groundreferenced phase delay network, rather than a lattice, is more convenient. As
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T1 7:5 I/P T2 1:1
O/P 1
C1 15 pF
O/P 2
Figure 5.23 In-phase hybrid with input transformer.
a fourth-order solution is required, each phase delay cascade needs to be second order. One solution, therefore, is to cascade two sections of the form of Figure 4.17 with component values chosen for all-pass operation ( 2 = 1). However, in Section 4.11, we suggested that the circuit of Figure 4.17 might become higher order by replacing the inductor and capacitor components with higher-order reactive networks. So, to make the circuit second order, an inductor may be placed in series with the capacitor, and a capacitor may be placed in parallel with the inductor. The extra inductor may be combined with the existing inductive element, the advantage of this being that the two sides of the center-tapped inductor no longer need to be perfectly coupled. Figure 5.24(a) shows the equivalent circuit. The required transfer function of the circuit of Figure 5.24(a) is the second-order version of (3.44); that is, s 21 (s ) =
( σ 1 − s )(σ 3 − s ) ( σ 1 + s )(σ 3 + s )
(5.2)
Determination of the element values of Figure 5.24(a) is aided by exploiting its symmetric properties. Applying an in-phase excitation to the circuit, the apparent reflection coefficient is equal to s21. The apparent impedance may be determined using the relationship between the reflection coefficient and impedance given by (4.5). So, normalizing the reference impedance to 1Ω,
160
Lumped Element Quadrature Hybrids C1
L1
L1
L2
C2
(a)
C1
M
LS
LS
C2
(b)
Figure 5.24 Second-order ground-referenced phase delay network: (a) equivalent circuit, and (b) practical circuit.
( σ 1 − s )(σ 3 − s ) ( σ 1 + s )(σ 3 + s ) Z e (s ) = ( σ 1 − s )(σ 3 − s ) 1− ( σ 1 + s )(σ 3 + s ) 1+
=
σ1 σ 3 + s 2
(σ 1 + σ 3 )s
=
1 1 1 + s σ1 σ 3
(5.3) +
s σ1 + σ 3
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161
This impedance is equivalent to the series combination of an inductor of value 1/( 1 + 3) and a capacitor of value 1/ 1 + 1/ 3. Inspection of the circuit of Figure 5.24(a) shows that the in-phase impedance is equal to an inductor of value L1 + 2L2 in series with a capacitor of value C2/2. Therefore, 1 1 C 2 = 2 + σ1 σ 3
(5.4)
1 σ1 + σ 3
(5.5)
and L 1 + 2L 2 =
Determination of the remaining element values requires an analysis with antiphase excitation. This time, the apparent reflection coefficient is given by –s21. As the reflection coefficient is opposite in polarity to the in-phase excitation, the apparent impedance is the dual, given by the parallel combination of a capacitor of value 1/( 1 + 3) and an inductor of value 1/ 1 + 1/ 3. Inspection of the circuit of Figure 5.24(a) shows that the antiphase impedance is equal to a capacitor of value 2C1 in parallel with an inductor of value L1. Therefore, C1 =
1
2( σ 1 + σ 3 )
(5.6)
and L1 =
1 1 + σ1 σ 3
(5.7)
The value of L2 may be determined by substituting for L1 in (5.5), giving σ 12 + σ 1 σ 3 + σ 23 = −M L 2 = − 2 σ 1 σ 3 ( σ 1 + σ 3 )
(5.8)
This negative inductor value may be identified with the mutual inductance of two coupled inductors, as indicated in the above equation. The self-inductance is given by L1 – L2, so from (5.7) and (5.8),
162
Lumped Element Quadrature Hybrids
Ls =
σ 12 + 3 σ 1 σ 3 + σ 23
2 σ 1 σ 3 (σ 1 + σ 3 )
(5.9)
The coupling coefficient between the coupled inductors is given by k=
σ 12 + σ 1 σ 3 + σ 23 M = 2 Ls σ 1 + 3σ 1 σ 3 + σ 32
(5.10)
It can be seen from (5.10) that the coupling is always less than unity, so the circuit is realizable. Figure 5.24(b) shows the practical circuit. Testing of candidate coupled inductors in any design requires a similar strategy to the case of coupled inductors for the second-order coupler hybrid, that is, testing for self-inductance Ls by measurement of one winding and testing for mutual inductance indirectly by measuring the inductance of both windings in parallel, which equals 0.5(Ls + M ). Although the required coupling is less than unity, it is nevertheless very high in practice. Thus, the solution will stop short of requiring twisted pairs but will still require overlapping of the turns. The element values for the particular case of a 10–100-MHz hybrid may now be calculated. The normalized frequency parameters were determined in Section 5.11. Use of (5.4), (5.6), (5.8), and (5.9) gives element values for each second-order all-pass network, where the circuit is that of Figure 5.24(b), given in Table 5.3. The capacitors were implemented as combinations of fixed values. For the inductor element belonging to phase delay network a, two windings of
Table 5.3 Element Values for 10–100-MHz All-Pass Networks
Parameter
Phase Delay Network a
Phase Delay Network b
C1
27.56 pF
6.99 pF
C2
1.45 nF
367.7 pF
Ls
975.1 nH
247.3 nH
M
837.3 nH
212.3 nH
k
0.859
0.859
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163
eight turns each on a Micrometals T20-3 core were found to give the correct inductance. The windings were formed using 0.315-mm enameled copper wire, with the first five turns of one winding separately wound and the remaining three alternating with the second winding. It was also found necessary to distribute the winding widely round the core as turns collected together gave excessive inductance. The inductor element belonging to phase delay network b was wound on a Micrometals T25-2 core. This time two windings of seven turns each were used, with only two turns wound separately and the remaining five alternating with the second winding. Close spacing of the turns was required for the correct inductance and coupling. Figure 5.25 shows the phase response of the hybrid. This shows the insertion phase at the output of phase delay network b with respect to the output of phase delay network a. The response shows the double hump characteristic of a fourth-order hybrid. The theoretical deviation from quadrature can be read from Figure 3.9 or calculated as 1.1°. The measured response shows some departure from this at the upper passband limit.
Figure 5.25 Phase delay hybrid response.
164
5.13
Lumped Element Quadrature Hybrids
A 1–300-MHz Approximate-Amplitude Hybrid Based on Phase Delay Networks
The form of hybrid, as described in Section 4.12, where two phase delay networks are placed between 0°/180° hybrids, is an elaborate solution. It does have its benefits though, and which the next example will illustrate. With this technique, the phase delay networks have a reference impedance of either half or double the terminal impedance. For exceptionally wide bandwidths, greater demands are placed on the inductor elements, so it is desirable to choose smaller values. This means that the preferred reference impedance for the phase delay networks should be half the terminal impedance. The 0°/180° hybrids must be configured accordingly. A suitable form of hybrid is that based on the resistive bridge [1], shown in Figure 5.26. The hybrid is similar to the one shown in Figure 4.33, except no common ground terminal is used, with isolation applied at all ports. The resistors of the bridge have been removed from the bridge network itself by means of loaded transmission lines. All the transmission lines share the same characteristic impedance Z0. The parallel combination of two otherwise isolated arms of the bridge form each output.
O/P 1 Z0 /2
I/P 1 Z0
I/P 2 Z0
Figure 5.26 Bridge form 0°/180°hybrid.
O/P 2 Z0 /2
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165
The more elaborate circuit can only be justified if it offers advantages in other ways. The advantage for this circuit is that it can be used in the design of exceptionally wideband circuits. Consider, then, the problem of designing a circuit operating over a 1–300-MHz bandwidth, as described in Example 4.5. The first part of the design is to determine suitable ferrite loading for the various arms of the in-phase hybrid. The loading must offer sufficient impedance at 1 MHz, while also exhibiting acceptable properties at 300 MHz. All arms of the hybrid used a winding of eight turns of 0.047-in., 50Ω semirigid cable on a Ferroxcube TN13/7.5/5-4A11 toroid core. The element values determined in Example 4.5 were all referenced to 50Ω. They must be rescaled to 25Ω, which is done by simply halving the inductor values and doubling the capacitor values. The required all-pass networks are shown in Figure 5.27. These were assembled onto two separate circuit boards. Transmission lines were required between the cable connection and all-pass sections, as well as between the individual sections. These were constructed using a parallel plate transmission line formed of 5-mm-wide tracks on each side of a 0.8-mm FR4 substrate. For the capacitors, fixed values were used, either singly or in combination. Surface mount COG capacitors were used for the most part, although, for the larger 10.24-nF capacitor, an X7R 10-nF part was used, and for the 2n24 capacitor, a 2n2 leaded part was used. The 6.4-µH inductors were wound using 17 turns on a FairRite 596100101 core. The 1.4-µH inductors used 13 turns on a Micrometals T25-1 core, and the 416-nH inductors used 13 turns on a Micrometals T16-2 core. All inductors wound on a core used 0.2-mm enameled copper wire. The 127-nH inductors were wound as an air-cored solenoid using six turns on a 4-mm diameter. The 37.7-nH inductors were wound with three turns on 3-mm-diameter air-cored solenoids. These solenoids were wound using 0.315-mm enameled copper wire. Finally, the 8.25-nH inductors were implemented as links of 0.6-mm diameter tinned copper wire. Referring to the circuit of Figure 4.32, the high-frequency all-pass network was used in the position of phase delay network a, and the lowfrequency network was used in the position of phase delay network b. The positions are arbitrary in that the circuit will still perform as a quadrature hybrid with the positions reversed. The particular configuration chosen makes the sum output of the output in-phase divider the through port and makes the difference output the coupled port, with the coupled port exhibiting a quadrature phase lead as it would in a symmetric coupled circuit. All the inductors were capable of fine adjustment by varying the spacing of the turns. This was the only adjustment necessary to tune the complete hybrid. Figure 5.28 shows the amplitude response. The maximum imbalance
166
Lumped Element Quadrature Hybrids 1.4 µH 203 pF
13.2 pF
127 nH
8.25 nH 2.24 nF Port 1
Port 2 2.24 nF 8.25 nH 127 nH 13.2 pF
203 pF 1.4 µH (a) 6.4µH 665 pF
60.29 pF
416 nH
37.68 nH 10.23 nF
Port 1
Port 2
10.23 nF
37.68 nH
416 nH 60.29 pF 665 pF 6.4 µH (b)
Figure 5.27 1–300-MHz phase delay networks: (a) high- and (b) low-frequency sections.
deviates only slightly from the theoretical value of 0.53 dB. The maximum deviation from quadrature was measured as 2°. Port match was generally better than 20 dB across the passband, and isolation was better than 20 dB at all frequencies.
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Figure 5.28 1–300-MHz coupled and through insertion loss.
This example demonstrates why the use of phase delay networks between in-phase hybrids is a valid technique. Such a bandwidth is extremely difficult to achieve at these frequencies using the simpler coupler circuits because of leakage inductance. The lattice form of phase delay network preserves its all-pass characteristics very effectively over a wide frequency range. Provided that suitable 0°/180° hybrids can be designed, use of this form of hybrid may be the only method to realize exceptionally wide frequency ranges.
5.14
Conclusion
First-order couplers may be designed using either a ground line or coupled inductor. The ground line inductor is useful for microwave frequencies, while the coupled inductor is more effective for size reduction in RF circuits. Higher-order polynomial-based hybrids can be designed to operate at moderately high RF frequencies and offer moderate-bandwidth solutions.
168
Lumped Element Quadrature Hybrids
The principle can be extended to microwave frequencies by the inclusion of transmission lines. Both first- and second-order coupler sections can be used on their own or in a cascade in hybrids operating over a significant bandwidth ratio. The response corresponds to the optimum rational function. The solutions are particularly concise where second-order coupler sections are used, but they are limited in frequency range through the need for a magnetic core. Hybrids based on all-pass networks can be designed to give optimum rational function response. The circuits suffer from greater complexity but offer the prospect of exceptionally wide bandwidths.
Reference [1]
Sontheimer, C. G., “Electrical Connector for Transmission Lines and the Like,” U.S. Patent No. 3,325,587, June 13, 1967.
6 Special Topics The focus of the development of the theory to practical circuit design, as described in Chapter 5, was with a view to realizing passive quadrature hybrids operating at RF and microwave frequencies. The theory is not, however, restricted to these applications, so this chapter looks at other applications. In addition, it will investigate other aspects of quadrature hybrid design. This chapter considers the topics of active circuits, weak couplers, nonquadrature dividers, power considerations, and graphical techniques.
6.1 Active Circuits It will always be necessary to use passive (rather than active) circuits in transmission applications, regardless of operating frequency. Passive circuits are convenient at RF and microwave frequencies as component dimensions are small and active elements only serve to increase the scope for imperfections. As frequencies decrease, the component dimensions increase in size. This is particularly true for inductors. For signal-processing applications, it is desirable to consider the use of active elements as their contribution to imperfections diminishes, and their use allows the elimination of inductors. In addition, the low-frequency requirement for increased capacitor values can be moderated by the use of higher circuit impedances. The active circuit may use discrete elements, but, more usually, an integrated design is preferred. The active elements may then be amplifiers, switched capacitors, or even logic gates in the case of digital signal processing. 169
170
Lumped Element Quadrature Hybrids
This chapter considers only analog design using active elements in the form of operational amplifiers. Other active design techniques, such as digital signal processing, are beyond the scope of this book. It is inappropriate to refer to the final circuit as a hybrid, so, for the purposes of this section, it is better to refer to it as a quadrature divider. In general, the circuit is unilateral. There are many forms of the operational amplifier, only one of which will be considered here— the ideal voltage amplifier. This consists of an amplifier with two inputs, noninverted (+) and inverted (–). The amplifier has only one output, with an assumed zero output impedance. The output is positive going when the noninverted input is more positive than the inverted input. However, the gain of the ideal voltage amplifier tends towards infinity, so in order for the output to be finite, the potential difference between the two inputs must tend toward zero as well. This can only be achieved by the use of feedback. Practical operational amplifiers exhibit properties that deviate from the ideal characteristics listed above. In reality, they exhibit finite input impedance, an output impedance and finite gain with a frequency roll-off. In addition, they exhibit nonlinear characteristics, so their outputs include harmonic and intermodulation products in addition to the wanted signals. These imperfections will ultimately limit the frequency and amplitude capabilities of the active circuit. Managing these imperfections is a matter of cost. A wide range of operational amplifiers is available from the manufacturers, with higher performance in one aspect or another gained at a premium. For the purposes of this analysis, we will consider only perfect amplifier operation. The next stage in the development of active circuit theory is to consider the form of approximation function to use. Chapter 3 considered two forms, polynomial and rational. The polynomial form is inferior but was considered because, in certain circumstances, it was easier to realize in passive circuits. However, where active circuits are in view, there is great flexibility in the specification of the transfer functions, so the superior rational form is the only one that needs to be considered. This class of approximation function leads to either an approximate-amplitude or an approximate-phase solution. Either solution may be realized using active circuits of comparable complexity. It is customary in active circuit design to limit the choice of components to amplifiers, resistors, and capacitors. Components requiring wound elements, such as inductors and transformers, are to be avoided as they are costly to manufacture and large in size at the frequencies where active elements perform well.
Special Topics
171
Consider first of all approximate-amplitude dividers. Chapter 4 demonstrated that circuits distributed in a cascade were preferable to high-order, directly synthesized circuits. This principle can be extended to active circuits. What is required then is a section with two inputs and two outputs, exhibiting either the first-order transmission properties given by (4.3) and (4.4) or the second order as given by (4.18) and (4.19). These equations were expressions for s-parameters, but for active circuits, the concept of voltage transfer functions is more appropriate. The transfer s-parameter and voltage transfer function amount to the same thing in matched passive circuits, so these equations can be used for active circuits as well. Having determined the required section topology, the principle of cascading sections, as described in Section 4.7, can be used. The design of active circuits with many inputs and outputs and arbitrary response has received little attention in the literature, so the design of the required section requires special consideration. If the section is to be cascaded, then either the inputs must have infinite input impedance or the outputs must have zero impedance. Either choice is valid, with the final design having buffers on the output or input, respectively, for integration into a system. In this analysis, the output impedance of zero is the chosen option, so the output of an ideal operational amplifier becomes the output of a section. Figure 6.1 shows a concept section. The two-port will be analyzed by its s-parameters and is assumed to have a common ground terminal at both R5 = r R1 = r
Vi1
U1
Vo1
U2
Vo2
R2 = R 1 Two-port 2 R3 = R
Vi2 R4 = r
R6 = r
Figure 6.1 Active quadrature divider section.
172
Lumped Element Quadrature Hybrids
ports. The analysis proceeds by considering the response to the individual excitation of Vi1 and Vi2, then determines the total response according to the principle of superposition. Consider first the response to Vi1 only. The inverting input to U1 has infinite impedance, so given that R1 = R5, its voltage must be the average of Vi1 and Vo1, given by V 1− =
1 2
(V i 1 + V o 1 )
(6.1)
Now, the impedance at the noninverting input to U1 is also infinite; so its voltage is determined by Vi1 and the potential division created by R2 and the input impedance of the two-port. If this input impedance is defined as Zin, then the voltage at the noninverting input to U1 is V 1+ =
Z in V i 1 Z in + R
(6.2)
Now, the gain of the amplifier is infinite, so its two inputs must be at the same potential. Otherwise, the response will not be finite. Equating (6.1) and (6.2) yields a response at the output of U1 of V o 1 Z in − R = V i 1 Z in + R
(6.3)
Now, compare this equation with (4.8), which describes the reflection coefficient in terms of the input impedance. The right-hand sides are identical, where R is the same as the characteristic impedance. As R terminates the two-port at its second port (for the present, Vi 2 is zero), then this reflection coefficient is equal to the s11 of the two-port, provided that the s-parameters of the two-port are referenced to R at both ports. It follows, then, that Vo1 = s 11 Vi 1
(6.4)
The response to the output of U2 may also be determined using the s-parameters of the two-port. Now the voltage at the noninverting input of
Special Topics
173
U2 with the two-port in place, divided by the voltage at this point when the two-port is removed and a direction connection is made, is s21. The voltage at the noninverting input of U2 when the two-port is absent is half the input voltage Vi1. The voltage at the noninverting input of U2 is therefore given by V 2+ = 12 s 21V i 1 However, with Vi2 equal to zero, the output of U2 is twice the voltage of its noninverting input, leading to the following result: Vo 2 = s 21 Vi 1
(6.5)
Equations (6.4) and (6.5) reveal an interesting characteristic of the circuit of Figure 6.1, even without a signal applied to the second input. It operates as a network analyzer without the cumbersome requirement of a reflection bridge. The circuit has the attractive characteristic that it can easily be adjusted for common arbitrary reference impedance simply by changing the values of R2 and R3. The total response of the circuit requires an excitation at the second input. As the circuit is symmetrical (except for the two-port for now), the response to Vi2 can be determined from (6.4) and (6.5) by transposing the “1”and “2”subscripts. The response can be put into matrix form as follows: V o 1 V i 1 V = S V o2 i2
(6.6)
The form of (6.6) is analogous to the two-port case of (2.2), with incident and emanating power waves replaced by input and output voltages. The response of the circuit is governed by the s-parameters of the two-port. The problem then reduces to that of determining a two-port network that simultaneously exhibits the through and coupled response of a quadrature hybrid in its reflection and transmission coefficients. This problem has already been addressed in Section 4.1, where it was found that the even or odd equivalent circuits of a quadrature hybrid exhibited the desired quantities, as given by (4.1) and (4.2). For the circuit of Figure 6.1, there is more flexibility in that the even and odd numerator functions may, in some circumstances, be interchanged.
174
Lumped Element Quadrature Hybrids
For the two-port in Figure 6.1 to be the same as the even- or odd-mode equivalent circuit of a quadrature hybrid, it must be lossless and symmetrical. In addition, it must not contain inductors for reasons of size and cost. This leads to three possibilities of interest. It must be either a shunt or series capacitor in the first-order case; in the second-order case, it must be a symmetrical pi-network (or equivalent) of capacitors. It will be necessary to search for solutions where the reflection and transmission parameters satisfy (4.3) and (4.4) in the first-order case and (4.18) and (4.19) in the second-order case. If the two-port is a series capacitor, as in Figure 6.2(a), then its s-parameters are given by s 11 = s 22 =
σ σ+s
(6.7)
s 21 = s 12 =
s σ+s
(6.8)
and
where σ=
1 2C s R
(6.9)
Cs
2
1
C1
C1
2
(b)
C2
(a)
1
Cp
1
2
(c)
Figure 6.2 Two-port options using capacitors: (a) series capacitor, (b) shunt capacitor, and (c) pi-capacitor network.
Special Topics
175
Substituting (6.7) and (6.8) into (6.6) gives the response of a first-order divider with the output equivalent to a through response on the same side as the input. The next case is a shunt capacitor, as in Figure 6.2(b). Its s-parameters are given by s 11 = s 22 =
−s σ+s
(6.10)
s 21 = s 12 =
σ σ+s
(6.11)
and
where σ=
2 C pR
(6.12)
In this case, the through response is on the opposite side as the input. The form of the numerator function, defined in Section 3.7 as the sum of the individual numerator functions of both transfer functions, is of interest. For the series capacitor version, it is + s, whereas for the shunt capacitor it is – s. Both the series and shunt capacitor options give a quadrature response, and both are useful in a cascade. Recall that the numerator function of a cascade is the product of the individual numerator functions of each section, as given by (4.39), but for an optimum hybrid, it must be of the form given by (3.41). One possible solution, therefore, is a cascade using both series and shunt capacitors options in the individual two-port networks, with the series capacitor element values calculated to give the odd-order frequency parameters and the shunt capacitor elements to give the even-order frequency parameters. The cascade can be constructed with sections in any order. Another solution is to use series capacitor two-ports for the implementation of the even-order frequency parameters and shunt capacitor two-ports for the implementation of the odd-order frequency parameters. The cascade gives an inversion of the coupled transfer function. This option is permissible as, unlike with passive circuits, it does not require the use of a perfect transformer.
176
Lumped Element Quadrature Hybrids
Example 6.1 Determine circuit elements for a quadrature divider operating
over the speech band of 300 Hz–3 kHz, assuming a fourth-order solution comprising a cascade of first-order active sections. Solution A 300-Hz–3-kHz quadrature divider is appropriate for single-
sideband modulation systems, where the circuit of Figure 1.18 is used, and QH1 operates at speech frequencies. The speech band hybrid is better implemented with a less cumbersome active solution. A fourth-order solution achieves a theoretical amplitude imbalance of 0.164 dB [see (3.27)], corresponding to an unwanted sideband rejection of 40.5 dB. The normalized frequency parameters for a fourth-order 10:1 solution have already been determined in Section 5.11 as 0.15139, 0.59702, 1.67498, and 6.60526. The odd-order frequency parameters are associated with sections having a series capacitor, for which (6.9) may be used. Assuming a reference impedance of 10 kΩ and a center radian frequency of 5960.8, the series capacitor values required are 55.41 nF and 5.008 nF. The even-order frequency parameters are associated with sections having a shunt capacitor, for which (6.12) may be used. With a reference impedance of 10 kΩ again used, the shunt capacitor values required are 56.2 nF and 5.08 nF. Figure 6.3 shows the solution, where the sections are arbitrarily cascaded in order of increasing frequency of operation. With a signal applied at the input shown, Vo2 corresponds with the coupled output and Vo1 with the through output of the equivalent passive hybrid. ❂❂❂
r
r
r
r
r
r
r
r
Vi
Vo1 10k
10k
10k
10k
56n2
5n08
55n4
5n01
10k
10k
10k
10k
Vo2 r
r r
r r
Figure 6.3 300-Hz to 3-kHz quadrature divider.
r r
r
Special Topics
177
The final case to consider is the pi-network of capacitors (or equivalent) required for a second-order solution. This is shown in Figure 6.2(c), and the s-parameters are given by s 11 = s 22 =
σ1σ 2 − s 2 ( σ 1 + s )( σ 2 + s )
(6.13)
s 21 = s 12 =
( σ 2 − σ 1 )s ( σ 1 + s )( σ 2 + s )
(6.14)
and
The transfer functions given by (6.13) and (6.14) will be realized for capacitor values given by C1 =
1 σ 2R
(6.15)
and C2 =
1 2R
1 1 − σ1 σ 2
(6.16)
The transfer functions correspond with those required for the secondorder case, where the through response is on the same side as the input. These sections may be cascaded in a similar fashion as the passive sections described in Section 4.9. The frequency parameters must be distributed through the cascade in the same way; thus, one section uses 1 and 2, another 3 and 4, and so on. The cascade may be constructed with the sections taken in any order. Example 6.2 Design an eighth-order quadrature divider based on a cascade
of second-order active sections, operating over a 20-Hz–15-kHz audio band. Solution The solution requires a cascade of four sections. For this 750:1
bandwidth, the amplitude imbalance will be a maximum of 0.25 dB. Application of (3.37) gives the following normalized frequency parameters:
178
Lumped Element Quadrature Hybrids
= 0.019044 = 0.077853 3 = 0.22137 4 = 0.60581 5 = 1.6507 6 = 4.5173 7 = 12.845 8 = 52.509
1 2
Figure 6.4 shows the circuit, with element values added, as determined using adjacent pairs of frequency parameters, (6.15) and (6.16), and scaling for frequency and impedance. The resistor terminations for the two-ports are adjusted through the circuit so as to give capacitor values within the compass of high-quality ceramic types. ❂❂❂
Approximate-phase quadrature dividers can also be implemented with active circuits [1]. The all-pass network of Figure 4.31, for example, is 33k
100k 33k
100k
3k3
10k 3k3
10k
Vo1
Vi 33k 14n53
100k 37n32
57n63
12n62
37n32
100k
3k3 1n677
10k 6n433
5n585
2n589
14n53
6n433
1n677
33k
10k
3k3
Vo2 33k
100k 100k
3k3
10k 33k
Figure 6.4 20-Hz to 15-kHz quadrature divider.
10k
3k3
Special Topics
179
replaced with an equivalent active section, as shown in Figure 6.5. The response of this circuit is only first order. A higher order could be achieved using a higher-order reactive network in place of the capacitor, but this would violate the criterion for a circuit without inductors. In order to construct a useful wideband quadrature divider, it is necessary to use two cascades with suitable element values. The complication of providing isolation in the passive circuit is removed in the active circuit as the output of a buffer amplifier provides adequate signal for both cascades, with isolation provided by the unilateral properties of the amplifiers in the cascade. The frequency response of the all-pass section is given by Vo σ −s (s ) = σ+s Vi
(6.17)
where σ=
1 CR
(6.18)
Comparing these equations with the required transfer functions of phase delay networks, as given by (3.44) and (3.45), we see that the required phase delay sections can be determined using the odd frequency parameters in one cascade and the even frequency parameters in the other. Example 6.3 Design a sixth-order quadrature divider using the differential
phase of two cascades of all-pass sections operating over a 100-Hz to 4-kHz bandwidth.
r r Vo
Vi R C
Figure 6.5 All-pass active section.
180
Lumped Element Quadrature Hybrids
Solution The solution requires two cascades of all-pass sections, each of
which comprises three first-order sections. Solving for m in (3.18), using the techniques of Section 3.5, gives a value of 1.012. Section 2.4 showed that the value of m is equal to the maximum value of tan /2, where is the phase difference between the outputs. Therefore, max = 90.67°, so the maximum deviation from quadrature is 0.67°. For a sixth-order optimum rational function with a 40:1 bandwidth ratio, the normalized frequency parameters are = 0.068893 2 = 0.25908 3 = 0.64733 4 = 1.5448 5 = 3.8598 6 = 14.515
1
The odd subscript parameters are associated with one cascade, and the even subscript parameters are associated with the other. Figure 6.6 shows the 100k
33k 33k
100k
15k 15k
Vo1 33k
100k 36n53
15k 4n346
11n78
Vi 47k
6k8
22k
47k
22k
6k8
Vo2 47k 20n67
22k 7n404
Figure 6.6 100-Hz to 4-kHz quadrature divider.
6k8 2n55
Special Topics
181
suitable circuit, where a buffer is placed at the input to divide the source signal. The resistor and capacitor values are determined using (6.18), where the resistor values are adjusted for the convenience of capacitor selection, when scaled for frequency. ❂❂❂
6.2 Unequal Division and Nonquadrature Hybrids Section 2.3 described the coupling of an approximate-amplitude hybrid in terms of a filtering function F( ). For equal power division, its value should be ± unity. In practice, it is devised in such a way as to ripple about the ideal value. If the value is lower than unity, then the power to the coupled port is diminished, as shown by (2.22). It is possible that the coupling structures described in Chapter 4 might be configured with the intention of realizing a coupling less than 3 dB. This could be the case where the hybrid properties are less important than the directional coupler properties, such as in a power measurement circuit. There may be other reasons why a weaker coupling is required, one of which is described in Section 6.5. In the case of approximate-phase hybrids, a filtering function less than unity corresponds with a phase difference between outputs of less than 90°. Although the quadrature case is by far the most important application of approximate-phase hybrids, there may be instances where another phase difference is required. The treatment of these cases is similar to the quadrature hybrid cases in that the polynomial or rational function form of the filtering function is appropriate to circuits of coupler form, whereas only the rational function form need be considered in connection with differential phase delay circuits. In all cases, it is convenient to modify the filtering function by multiplying it by the factor . With reference to (2.22), the nominal coupling defined by setting F( ) = 1 then becomes ε2 C 0 (dB) = 10 log 10 1+ ε2
(6.19)
Where coupling is weak, the nominal coupling is given approximately by C 0 (dB) ≈ 20 log 10 ε
(6.20)
182
Lumped Element Quadrature Hybrids
Equation (6.20) is exact where the coupling is referenced to the through port rather than the input port. In Section 2.4, we determined that the filtering function was also equal to the tangent of half the phase difference between the two outputs of a differential phase delay circuit. With the filtering function modified to its new value, the nominal phase difference is then given by β 0 = 2 tan −1 ε
(6.21)
The variation in coupling is a function of both the modifier and the bandwidth, so it has to be determined on a case-by-case basis. The same is true of the variation in phase of a differential phase delay circuit. Weak couplers based on the optimum rational filtering function are of limited practical use. The circuits require magnetic coupling, which restricts their upper frequency limit. At the frequencies where optimum lumped element quadrature circuits are practical, in-phase coupling circuits such as that shown in Figure 4.30 may be preferred, particularly as there is no complication of coupling variation with frequency. At higher RF frequencies, increasing to microwave frequencies, quadrature circuits employing transmission lines may be configured to give weak coupling. Circuits employing lumped element quadrature hybrid techniques may be appropriate in circumstances where a moderate coupling is required, but fabrication is still difficult using purely distributed circuits. One application is for microstrip circuits giving coupling in the region of 3 to 10 dB. Consider, first, couplers based on the polynomial filtering function. It is possible to analyze the third-order case and to determine element values for a given center-frequency coupling. The solution does not involve bandwidth in the calculations. Consider, then, the half-prototype circuit of the form shown in Figure 4.8, for which the element values have yet to be determined. Analysis of this circuit yields an input impedance of Z ( jω) =
R + jωL 1 − ω 2 LC + jωCR
(6.22)
The coupling of the final circuit depends on the ratio of the imaginary to real parts of this impedance, using (4.11) to relate the filtering function to the ratio and (2.22) to relate the coupling to the filtering function. The ratio of the imaginary to real parts of the impedance can be determined from (6.22) and is given by
Special Topics
ω(L − CR 2 − 3 ω 2 L 2C ) Im ( ω) = R Re
183
(6.23)
The center frequency is given by the maximum of (6.23), which may be determined by taking its differential and setting it to zero. This gives ω0 =
L − CR 2 3L 2C
(6.24)
It is convenient to normalize the center frequency and termination resistance to unity so that the capacitor and inductor values become related by the following equation: C=
L 3L 2 + 1
(6.25)
Now, substituting for capacitance, normalized frequency, and termination resistance in (6.23) gives Im 2L 3 (1) = 2 Re 3L + 1
(6.26)
In (6.26), the left-hand side is calculated from the required coupling. The right-hand side is monotonic for positive real values of L and, so, leads to a unique solution. The form of the prototype circuit is as shown in Figure 4.9(a), where the capacitor value determined from (6.25) must be multiplied by 2. Table 6.1 gives element values for various coupling factors. The third-order mixed element coupler, as shown in Figure 4.13, also has a single maximum coupling factor, so, once again, bandwidth is not required in the analysis. However, the presence of trigonometric terms makes analysis more difficult, although the procedure is the same in principle as for the third-order polynomial case. Table 6.2 gives element values normalized to unity center frequency and termination resistance over coupling ranges where the circuit may be useful in a microstrip environment. The circuit is potentially useful not only in the moderately strong coupling region but even for loose coupling applications, owing to its reduced electrical length when compared with a purely distributed circuit.
184
Lumped Element Quadrature Hybrids
Table 6.1 Prototype Element Values for Third-Order Polynomial Coupler Maximum Coupling (dB)
L
C
3.01
1.67765
0.355301
4.77
1.27735
0.433377
6.02
1.10321
0.474374
10.0
0.77645
0.552904
15.0
0.5596
0.577069
20.0
0.42672
0.551936
If the filtering function modified by the factor is the optimum rational function, then the mathematics may be solved analytically. Chapter 3 showed that the transfer functions could be determined from a number, equal to the order of the filtering function, of frequency parameters. These frequency parameters were also the characteristic frequencies of the transfer Table 6.2 Prototype Element Values for Third-Order Mixed Element Coupler Maximum Coupling (dB)
L (H)
3.01
1.51043
21.05
4.0
1.2308
24.15
4.77
1.07161
26.25
5.0
1.03102
26.84
6.02
0.87675
29.24
7.0
0.75923
31.28
8.0
0.66079
33.14
9.0
0.57848
34.8
10.0
0.50859
36.3
12.0
0.39655
38.85
15.0
0.27655
41.79
20.0
0.154
45.0
0 (°)
Special Topics
185
functions, given by the solution of (3.35), upon the substitution s = j . The same technique can be used in the weak coupling or nonquadrature cases in the approximate-amplitude and approximate-phase realizations, respectively. The problem is modified and becomes the solution to the following equation: 1 + ε 2 F 2 ( ω) = 0 or F ( ω) = ±
j ε
(6.27)
Once again, it is convenient to make use of the mapping to the z-plane as a stepping-stone to the solution of (6.27). Using the mapping of the F-plane to the z-plane, as given by (3.17), and the boundary defined in Section 3.6, the only solutions lie along the imaginary axis of the z-plane in the range z = − jnCK n′ m to + jnCK n′ m . The procedure then is to determine the solutions to z, then to apply the mapping from the z-plane to the -plane, as given by (3.14), to determine the corresponding solution to , hence, s. It is helpful in this respect to consider a typical graph, as shown in Figure 6.7, which has been plotted for the case of a 10:1 bandwidth and order of three. Solutions occur at points where the imaginary part of F intersects with the lines of imaginary part ±1/ . In general, there are 2n solutions, so for the case of n = 3 in Figure 6.7, there are six solutions. As F is an odd function of z, the solutions occur in equal but opposite pairs. Figure 6.7 also shows the imaginary part of , so it is possible to read off its value corresponding to the value of z. Of the 2n solutions to z available, the only ones of interest are those that give a positive imaginary part of , so only the portion of the graph to the right of the vertical axis needs to be considered. The first solution, which might be considered the principal value, occurs for an imaginary value of z between 0 and CK n′ m . Define the proportion of its value to the given interval as the multiplier . Now, this interval occupies a half-period of the function F, which is also an odd function, so the next solution is obtained using the new multiplier 2 – , the next by 2 + , and so on. Thus, the solutions to z are given by
{
mz r r = jK n′ r − 12 + ( 12 − λ )( −1) C
}
r = 1K n
(6.28)
186
Lumped Element Quadrature Hybrids
σ3 7 6 5
Im(ω)
Im(F ) 4 3 1/ε
2 σ2 σ1
1
K ′1/ωU
z2 z1
CK ′n /m
2CK ′n /m
z3
3CK ′n /m
−1/ε
Figure 6.7 Plot of imaginary part of F and
with respect to z, BW = 10, n = 3.
The mapping procedure identifies the value n CK n′ m as equal to K 1′ ωU , so (6.28) may also be written as zr = j
K 1′ nωU
{r −
1 2
+ ( 12 − λ )( −1)
r
}
r = 1K n
(6.29)
Now, the z-plane to -plane mapping of (3.14), followed by the substitution s = j , may be used to give the solution set as σr =
1 r sc r − 12 + ( 12 − λ )( −1) ωU
{
} Kn ′ , k ′ 1
1
r = 1K n (6.30)
Equation (6.30) reduces to the solution of frequency parameters for optimum hybrids, as given by (3.37), when = 0.5. This value of occurs
Special Topics
187
when = 1. For other values of the modifier, it is necessary to calculate its value. To this end, suppose the solution at the principal value of z1 is given by jv = mz1/C. Suppose, too, that y = m/ . From (3.17) and (6.27), we have sn ( jν, k n
) = jy
Hence, y=
sn ( ν , k n′ )
cn ( ν , k n′ )
Now, from the theory of elliptic functions (see Section 3.5): cn ( x ) = 1 − sn 2 ( x ) So, on rearrangement of the above, sn ( ν, k n′ ) =
y 1+ y 2
or y ν = sn −1 , k n′ 1+ y 2 The unknown v is related to the known y by the incomplete elliptic integral, as defined by (3.12). It is otherwise denoted as F(x,k) and is evaluated in tables and mathematical software. Now, at the required solution, v = λK n′ , so m , k n′ F 2 2 m +ε λ= K n′
(6.31)
Example 6.4 Determine the normalized frequency parameters of a weak cou-
pler operating over a 10:1 bandwidth ratio using a fourth-order solution and = 0.1.
188
Lumped Element Quadrature Hybrids
Solution The chosen modifier for the filtering function gives an approximate coupling of –20 dB, by (6.20). The solution is given by (6.30), but it is first necessary to determine the value of , as given by (6.31). This in turn requires the determination of a number of elliptic functions and integrals. First, the 10:1 bandwidth gives
ωU = BW = 31623 . hence, k1 =
1 = 01 . w U2
Now, from tables or mathematical software, q 1 = q (k1 ) = 0.00062815 and from (3.26), q n = 4 q 1 = 015831 . Then, k n = k (q n
) = 0.9629
and k n′ = 1 − k n2 = 0.26987 Then, K n′ = K (k1′ ) = 16006 . and m=
1 kn
= 10909 .
Special Topics
Now, all the variables are available to determine becomes
189
by (6.31), which
λ = 0.93654 In order to determine the frequency parameters of (6.30), it is also necessary to determine k1′ = 1 − k12 = 0.99499 and K 1′ = K (k1′ ) = 3.6956 The variables may be substituted into (6.30), giving σ 1 = 0.30963 σ 2 = 0.36409 The remaining two parameters may be determined using (3.39) or evaluated directly as σ 3 = 2.74657 σ 4 = 3.2297 The parameters required in the problem have thus been determined. More information that may be useful can be determined from the calculations made. The maximum value of the filtering function when modified is given by m , whereas the minimum value is given by /m. Using (2.22), the coupling limits for a coupler form of hybrid are 0.10138 and 0.09766, or –19.88 dB and –20.21 dB. For a phase delay form of circuit, the maximum and minimum values of the modified filtering function correspond with a phase difference between the outputs varying from 11.21° to 11.64°. If the coupler form used a cascade of two second-order sections, as in Figure 4.17, the coupling factor of the inductor in each section is 0.0808 [see (4.34)]. Such a weak coupling will require a different approach to inductor design, as the techniques discussed in Section 5.7 were devised assuming the strong coupling required for quadrature hybrids. ❂❂❂
190
Lumped Element Quadrature Hybrids
6.3 Power Handling In many applications of quadrature hybrids, only their function as signalprocessing components is of concern. In these applications, the signal strength under consideration is a small proportion of their dynamic range capability. Low loss is still of concern, but so is small size and ease of fabrication. Where quadrature hybrids are required for transmission purposes, they may be required to handle considerable power levels. In these applications, careful design may be necessary to ensure that the hybrid does not suffer damage during operation. The power handling of a quadrature hybrid depends on various quantities, which we will examine. The first example for examination is the 435-MHz first-order hybrid of Figure 5.6, which was described in Section 5.2. This hybrid was actually designed for high-power operation and demonstrates the two sides of the power-handling problem; that is to say, it was designed to operate under large signal pulse and continuous conditions. Consider first the limitation under pulse conditions. This is a question of voltage breakdown. Provided that the hybrid is well constructed, the component with the lowest breakdown voltage is the porcelain capacitor. Even this component is rated at 500V. For this calculation, it will be assumed that the hybrid is supplied at the input port only. In practice, power may be supplied at two ports for combination at a third, but this situation is comparable. The voltage across the capacitor is equal to the voltage present at the through port minus that at the isolated port as these ports share a common ground, and the capacitor is connected across them. This voltage varies with frequency for a given supplied power and tends toward the supply voltage as frequency tends towards zero. At the frequency of equal power division, the voltage across the capacitor is 1/√2 times the supply voltage. With a reference impedance of 50Ω, this equates to a power level of 5 kW before the peak voltage of the RF cycle exceeds 500V across the capacitor. Under continuous operating conditions, average power-handling is likely to limit the hybrid. The coaxial cable was RG405 and is limited to around 200W at 435 MHz. However, this value is based on a single cable with free air convection cooling. If the outer jacket were cooled by conduction, for example, the ground point at one end of each length, then power rating could be considerably higher. The porcelain capacitor is capable of conducting in the region of 5A root mean square (rms). This equates to a power level of 1.25 kW. Once again, this value assumes effective conduction cooling of the component. In its operating environment, the hybrid was
Special Topics
191
required to handle a continuous power of 100W and a pulsed power of 200W. Neither of these values posed a challenge to its construction. The environment may determine the power rating of a hybrid. For example, the 1.27-GHz hybrid described in Section 5.3 uses microstrip connecting lines. On the chosen substrate, these can support a power of several hundred watts. However, at these power levels, the copper tracks will be significantly elevated in temperature and may be up to 100°C above ambient temperatures. Such a temperature difference will limit the upper operating temperature of the hybrid. The limit due to the capacitor is in the region of 400W. The limit is similar from both power and peak-voltage considerations for the porcelain capacitor used. Where the hybrid uses magnetic materials in the inductive elements, the cores will be the limiting factor to power handling. It is usual for the operating frequency to be above the high-Q range of the core material. This is particularly so for cascaded designs, where the low-frequency section has to transmit RF power up to the upper passband limit. Conventional analysis for the loss of magnetic materials focuses on hysteresis loss. However, this loss mechanism is of little concern at the frequencies where quadrature hybrids are required. At radio frequencies, magnetic material causes loss because of not only the changing magnetic fields but also the electric fields. Magnetic materials exhibit a relative permittivity in the region of 10, accompanied with a poor loss tangent. Consequently, the loss of magnetic materials is best considered as electromagnetic in the general sense. The core material used in the low-frequency section of a cascaded design is likely to behave in the manner of a radio absorptive material at the upper passband limit. As an illustration of the effect of core loss, consider the 50–150-MHz second-order hybrid described in Section 5.8. The excess insertion loss of this design was measured at 0.3 dB. We may suppose that this loss is almost entirely due to the magnetic core, with other loss factors such as capacitor loss and copper loss being small in comparison. Under large signal conditions, the average power rating of the hybrid will be determined by the permissible temperature rise of the core. The manufacturers provide some information on the prediction of temperature rise with the following formula: P (mW ) T (°C ) ≈ 2 A(cm )
0. 833
(6.32)
In (6.32), P (mW) is the power dissipated in the core and A(cm2) is the surface area of the core, which for the T16-2 core used is 0.8 cm2. Suppose
192
Lumped Element Quadrature Hybrids
that the temperature rise of the core is limited to 50°C above ambient. This value will allow operation up to an ambient temperature of at least 75°C, assuming an upper temperature limit of the core of 125°C. Under these conditions, (6.32) predicts a dissipation of 87.6 mW. Now, the excess insertion loss of 0.3 dB may also be represented as a transmitted power of 93.3% of the incident power, leaving 6.7% dissipated in the core. The maximum input power for the given dissipation is therefore 1.31W. This relatively low power handling is the price paid for the advantage of small size.
6.4 Graphical Techniques for Cascaded Couplers The analysis of Chapter 2 concluded that there was equivalence between approximate-phase and approximate-amplitude quadrature hybrids and that their respective responses were determined by a common filtering function. This equivalence can be used to determine the response of a cascade of coupler sections in a simple manner [2]. Consider the cascade of two sections, as shown in Figure 4.22. It may be supposed that the operation of the first section is described by the filtering function Fa( ), while that of the second is determined by Fb( ). Now, the response at the through port of the first section to an excitation at the input port is given by (2.20), with the filtering function replaced by Fa( ). If this function is now replaced by the equivalent approximate-phase representation of tan a /2, then the response to the through port is given by s 41 a ( ω) =
1 1 + tan 2 ( β a 2 )
= cos( β a 2 )
(6.33)
Similarly, the response to the coupled port is given by s 21 a ( ω) =
tan ( β a 2 ) 1 + tan 2 ( β a 2 )
= sin ( β a 2 )
(6.34)
In (6.33) and (6.34), only the magnitude functions have been given. However, as the section is defined to be in exact quadrature, the response to the coupled port will always either lead or lag the through port by 90°. It is convenient at this point to keep the through response as a purely real quantity and to indicate a 90° phase lead at the coupled port by multiplying the
Special Topics
193
magnitude response by j. Any common phase delays are not important for the relative transmission response. Having presented the first section with an excitation, the second section will receive an excitation at both the input and isolated ports. Its response to an excitation at the input port has a form similar to the expressions given by (6.33) and (6.34), except that the subscript is changed from a to b. The response at the through port of the total circuit to an input excitation can be deduced by inspection, with phase leads taken into consideration, and becomes s 41T = cos( β a 2 ) cos( βb 2 ) + j sin ( β a 2 ) j sin ( βb 2 ) β + βb = cos a 2
(6.35)
Similarly, for the response to the coupled port, the expression becomes s 21T = cos( β a 2 ) j sin ( βb 2 ) + j sin ( β a 2 ) cos( βb 2 ) β + βb = j sin a 2
(6.36)
Equations (6.35) and (6.36) show how it is possible to describe the complete response of a cascade by adding together the equivalent-angle expression of the constituent sections. The sum becomes the equivalentangle expression of the cascade. This technique may be used recursively so that the response of an arbitrary-length cascade may be calculated. It is possible to determine the coupling of a cascade of two sections by means of a nomogram, and Figure 6.8 shows one. All the scales have been marked in terms of decibel coupling. The outer scales represent the constituent sections, and the inner scale represents the cascade. Figure 6.8 shows the example where one section has a coupling of 10 dB and the other has one of 7 dB. The coupling of the cascade is approximately 3 dB. The technique of converting a coupler section’s response to an equivalent phase angle can be extended to the design of wideband hybrids. In cascaded designs, the greatest demands are placed on the section with the lowest operating frequency. Although the section may operate well at the lower passband limit, its characteristics may be far from ideal at the upper limit. It is likely to have been constructed using magnetic materials whose properties vary considerably through the band. In addition, its performance is likely to
194
Lumped Element Quadrature Hybrids 3
0
3.5
3 3.5
0.1 4 4.5 5 5.5
4 0.3 0.5 1
6
4.5 5 5.5 6
1.5 7
2
7
8
2.5 3
8
9 10
9 4
10
5 12
6
12
8 15 17
10
15 17
20
15
20
25
20
25
Inf
Inf
Inf
−20log10s21a
−20log10s21T
−20log10s21b
Figure 6.8 Nomogram for quadrature hybrid cascades.
be altered considerably by the presence of its leakage and stray properties. However, provided that the section can be tuned for good match in a symmetric manner, the quadrature, isolation, and match properties of the cascade design can often be maintained throughout the passband. The leakage and stray properties, in concert with the tuning elements used for compensation, will nevertheless cause the through and coupling responses to depart from the values of the ideal circuit. As a result, the through and coupling responses of the cascade will also vary from the ideal. Consider the specific case of a fourth-order cascade, comprising two second-order sections, intended to operate over an ambitious frequency range. It may be supposed that the low-frequency section will be constructed using a magnetic core for the inductive component. The ideal circuit, as shown in Figure 4.19, has a coupling characteristic that starts from zero at zero frequency and reaches a maximum at the center frequency of the
Special Topics
195
section. Thereafter, it smoothly decreases towards zero as frequency increases towards infinity. However, in the practical circuit, leakage and stray components will be present, so the inductors exhibit interwinding capacitance, and the capacitors exhibit lead inductance. These parasitic elements increase the order of the reactive components and introduce a resonance at a finite frequency. The effect on the hybrid is that its frequency response will suffer distortion. This will not become evident until well above the section’s center frequency; thereafter, however, the coupling will reduce more rapidly than predicted by an analysis of the ideal circuit. Having considered the response of the low-frequency section, consider now the contribution of the high-frequency section. Its purpose is to complement the coupling of the low-frequency section, contributing little at low frequencies and the majority of the coupling at high frequencies. However, if the low-frequency section makes a feeble contribution at high frequencies, then it will be necessary to reconsider the specification of the high-frequency section to make up the shortfall. In practical circuit design, the low-frequency section should be constructed and measured first. It is usual in the course of development to make trial circuits, evaluate their performance, and make improvements. Having developed the circuit subject to the constraints of the project budget, an acceptable design is produced. It is at this point that graphical techniques are useful. If the coupling is converted to the equivalent phase angle, using (6.34), its response may be plotted against frequency. A logarithmic frequency scale should be used, the reason for which will be shown next. The next stage is to specify the high-frequency section. Unlike the low-frequency section, its behavior within the passband will be close to theory. As the ideal section for the specified bandwidth is likely to give insufficient coupling when cascaded with the low-frequency section, it will be necessary to adjust its specification. One technique of doing this is for the designer to prepare a selection of graphs, where the theoretical response is converted to the equivalent phase angle. If the vertical scale is inverted, and the frequency response is plotted on the same logarithmic scale when compared with the low-frequency section plot, the test response can be compared with the low-frequency section response. The aim is to line the two up, such that the 0° line of one is coincident with the 90° line of the other. Where the two plots cross, the sum of the equivalent angles is 90°, corresponding to the optimum 3-dB coupling [see (6.36)]. The reason for using a logarithmic frequency scale is now apparent. The designer will only have to make test plots for individual coupling values, with the center frequency of the test plot
196
Lumped Element Quadrature Hybrids
marked on the graph. Once a good fit has been achieved, the optimum center frequency can be read off the scale of the low-frequency section plot. With this technique, it is possible to mix the types of coupler section. For example, it is most likely that the low-frequency section will make use of a magnetic core and will be designed to push the first resonant frequency as high as possible. In typical designs, the center frequency of the high-frequency section will be centered in the region of the upper passband limit. This frequency may well be so high as to make the use of a magnetic core inappropriate. Rather than use a second-order section, such as shown in Figure 4.19, it may be better to use a third-order polynomial-based section, or one that uses transmission lines. All these types of coupler section exhibit a similar frequency response up to their center frequency. Having compared trial plots with the low-frequency section, the designer may decide on a particular solution or reconsider the design of the low-frequency section.
6.5 A 50–550-MHz Hybrid with Sections of Different Topology To illustrate the use of graphical techniques, consider the particular example of a quadrature hybrid designed to operate over a 50–550-MHz bandwidth. It may be supposed that the design will make use of coupled sections. This frequency range is significantly higher than the concept demonstrators described in Chapter 5. Consider first of all the prospect of a fourth-order optimum rational function solution. Consulting the graph of Figure 3.9, the predicted imbalance is less than 0.2 dB for this 11:1 bandwidth ratio. The frequency parameters may be solved using (3.37), giving 1
=0.148,
2
=0.595,
3
=1.681, and
4
=6.757
The design of the low-frequency section should be considered first. The prototype element values can be determined from (4.32), (4.33), (4.35), and (4.36) using the first two frequency parameters, then scaled for frequency and a characteristic impedance of 50Ω to give L = 410 nH M = 244 nH C1 = 97.4 pF C2 = 64.5 pF
Special Topics
197
It may be verified using (4.34) that the coupling factor for the inductor is k = 0.602. The center-frequency coupling for the section is therefore 4.4 dB. A trial section was constructed with the inductor implemented as two four-turn windings of bifilar 0.2-mm enameled copper wire on a Micrometals T30-2 carbonyl iron core. As regards the capacitors, fixed values of 100 pF and 68 pF were used for the coupling and shunt elements, respectively. Although the trial section exhibited close-to-predicted performance in the region of its center frequency of 49.2 MHz, coupling reduced more rapidly than simple theory predicts. At 550 MHz, the coupling was measured as 28 dB, whereas it should have been 17.6 dB. The ideal circuit would not have exhibited such a small coupling until a frequency in the region of 1.8 GHz. With such a performance, a hybrid using this section can be expected to achieve a response well short of the optimum circuit. Although the response of the accompanying high-frequency section might be adjusted to correct for balance at the upper passband limit, the coupling mid-band is still likely to be incorrect. It will be necessary to adjust the center frequency of the low-frequency section as well in order to achieve a compromise solution. Having seen the degradation of the trial second-order section, a new section was designed with a view to improve performance at higher frequencies. It is unlikely that the predicted amplitude balance of less than 0.2 dB will be achieved in a practical circuit at these frequencies. A 0.4-dB amplitude balance is more realistic, corresponding with a bandwidth ratio of around 20:1. A distortion of the center frequency of the low-frequency section according to this bandwidth may give a more practical circuit. The center frequency of the low-frequency section increases to around 60 MHz. A new set of elements was calculated, with a coupling capacitor value of 82 pF and a shunt capacitor value of 56 pF. The inductor element was reduced in size to a Micrometals T25-2 core with four turns of 0.2-mm bifilar enameled wire in each winding. Further improvements for high-frequency performance were made, including the use of thinner substrate material, 0603 size capacitors instead of 1206, and 1.5-pF tuning capacitors at each port to compensate for leakage inductance. The new low-frequency section was measured separately, and this showed a minimum of 28-dB port return loss and 29-dB isolation. The center frequency was measured at 60.7 MHz, where a coupling of 4.4 dB was measured. In addition, the coupling at 550 MHz was considerably improved to 20.8 dB, which is much closer to the figure expected of the ideal circuit. The next stage in the design is to relate the coupling of the lowfrequency section to the equivalent angle, using (6.34). Rather than using the
198
Lumped Element Quadrature Hybrids
directly measured coupling alone, both the through and coupling losses were measured so that the total loss of the circuit could be calculated. The total loss could then be subtracted from the coupled loss in order to give a better representation of its level relative to the through port. With the modified value transformed to its equivalent angle, the response was plotted and is shown in Figure 6.9. The center frequency of the high-frequency section of an optimum fourth-order hybrid is 559 MHz. At this frequency, magnetic materials exhibit poor characteristics. A better solution might be a low-pass third-order solution, as shown in Figure 4.6. However, the design solutions presented in Section 4.3 were all given for a center-frequency coupling corresponding with 3 dB or stronger. In the present application, the low-frequency section coupling reinforces that of the high-frequency section, so the required coupling is likely to be weaker than 3 dB. In addition, the response relative to the center frequency is required so that a normalized set of responses can be 90
0
80
10 Low-frequency section
70
20 High-frequency section
60
30
50
40
40
50
30
60
20
70
10
80
βb(°)
βa(°)
0
50
100
200 Frequency (MHz)
300
400
Figure 6.9 50–550-MHz hybrid sections equivalent-angle response.
500 600
90
Special Topics
199
plotted. This requirement calls for a weak coupler solution, as described in Section 6.2. The coupling with frequency may be determined by use of (3.1), limited to the third-order case. The maximum coupling can be determined by considering the turning points of (2.22). Differentiating with respect to and setting to zero gives the following two possibilities: F ( ω) = 0 or F ′( ω) = 0 Of these two possibilities, the case where the differential of the filtering function equals zero is the one required. Where the filtering function equals zero is clearly a minimum point as there is no coupling at all. Applying this criteria to the third-order case of (3.1) gives a 1 + 3a 3 ω 2 = 0 It is required that the turning point occur at a normalized radian frequency of unity, so a 3 = − 13 a 1 Substituting for a3 in (3.1) gives F ( ω) = a 1 ω(1 − 13 ω 2 ) The maximum coupling, occurring at c0 =
(6.37)
= 1, is given by
2a 1 9 + 4 a 12
or, inversely, as a1 =
3c 0 2 1 − c 02
(6.38)
200
Lumped Element Quadrature Hybrids
Using (6.38), it is possible to determine the constant multiplier in (6.37). The filtering function may then be substituted into (2.22) to give the coupling as a function of frequency. The equivalent phase angle, as defined by (6.34), may then be plotted against frequency. For the particular case of a candidate function for the high-frequency section of the 50–550-MHz hybrid, various trial functions were compared with the low-frequency section plot, and the best solution was found to be given by a center frequency of 560 MHz and 4-dB coupling. This is shown as the second trace in Figure 6.9. The crossing points of the two traces correspond with frequencies where amplitude imbalance is zero. The points of maximum deviation are approximately 3°, corresponding with an amplitude imbalance of about 0.5 dB. These occur at the lower band edge and at a frequency in the region of 80 MHz. Now, 4 dB expressed in magnitude is 0.631, so from (6.38), a1 = 1.22. With this information, either the synthesis technique of Section 4.3 may be followed, or (6.37) may be used at unity radian frequency, in turn, to solve (6.26) and follow the procedure outlined in Section 6.2. In either case, the equivalent circuits of Figure 4.9 are the result, with a normalized outer element value of 1.421 and an inner element of 0.4026. These should be combined into the circuit of Figure 4.6, following the procedure of Example 4.1. Once element values have been scaled for frequency and impedance, they become as follows: Le = 10.1 nH Lo = 2.86 nH Co = 3.47 pF Ce = 1.14 pF These element values were used to construct a section on the same substrate as the low-frequency section, which was 0.8-mm-thick FR4. With such low element values, the practical values had to be adjusted to account for stray effects. In fact, the best performance was found using Co = 1.2 pF and Ce = 0.82 pF. The much smaller value required for Co is attributed, in part, to significant interwinding capacitance in the even-mode coupled inductor Le. For this element, a solenoid consisting of two turns with an internal diameter of 2 mm was formed using bifilar 0.2-mm enameled copper wire. The odd-mode coupled inductor Lo was formed using a short loop consisting of 3.5 twists of the same bifilar wire, which was twisted at the rate of 72 full twists per 100 mm. No magnetic core material was used for either coupled inductor. This section was adjusted separately, before being cascaded with the low-frequency section.
Special Topics
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Figure 6.10 shows the insertion loss of the cascaded hybrid, where the amplitude imbalance maxima predicted at the lower band edge and at 80 MHz can be seen. Both port match and isolation were better than 20 dB at all frequencies in band, but the phase reached a deviation from quadrature of 2° by 550 MHz. This in itself is equivalent to a 0.3-dB amplitude imbalance and is a significant contributor to quadrature error. The example is an illustration of how performance suffers at more demanding frequencies.
6.6 Conclusion The transfer functions describing quadrature hybrids can be realized using active circuits, and these are particularly useful at low frequencies in signalprocessing applications. It is possible to configure circuits without the use of inductors. With a change in the definition of the filtering function, it is possible to synthesize circuits with weak coupling or nonquadrature phase response.
Figure 6.10 50–550-MHz hybrid insertion loss.
202
Lumped Element Quadrature Hybrids
These two properties are equivalent in approximate-amplitude and approximate-phase circuits. The power handling of quadrature hybrids varies widely, according to the construction technique used. Circuits with air-spaced inductor components exhibit the highest power handling, with circuits using magnetic material suffering considerably in power handling. The equivalence of approximate-amplitude and approximate-phase circuits can be used to determine the response of cascaded circuits, using graphical techniques. The techniques can be used to determine response quickly at a spot frequency, or more elaborate graphs can be drawn to help design hybrids operating at high relative and absolute frequencies.
References [1]
Keely, T. A., “Design of Constant Phase Difference Networks,” RF Design, April 1989, pp. 32–42.
[2]
Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Dedham, MA: Artech House, 1980, pp. 812–815.
Glossary The following is a list of terms and abbreviations as found in the text. The meaning given is that used in the text, which may not necessarily be the entire meaning of the word in general usage. Approximate amplitude Two signals whose amplitudes are identical in the ideal case but actually differ slightly. Approximate phase Two signals whose phase difference is 90° in the ideal case, but the actual phase difference varies slightly from this value. Bandwidth ratio (BW) The upper and lower frequency limits of a specification described as a dimensionless ratio. Coupler A circuit that transfers energy between two systems of conductors using the linking of electric or magnetic fields or both. Decade A 10:1 ratio, usually used to describe the ratio of two frequencies but that may also be used to describe a range of element values. HF (high frequency) The electromagnetic spectrum between the frequencies of 3 MHz and 30 MHz; a frequency band that is substantially within these limits, such as “2 to 32 MHz.” Hybrid A four-port passive circuit wherein a signal applied at one port is transmitted to two others and not the fourth. 203
204
Lumped Element Quadrature Hybrids
Isolation A condition where two ports of a network exhibit a small or nonexistent transfer coefficient. Lumped element A circuit element that does not require the dimension of length (or the equivalent) in its description. Match A load with an impedance equal to the complex conjugate of the impedance of a generator to which it is attached. Microwave The electromagnetic spectrum between the frequencies of 1 and 300 GHz. Narrowband A comparative term used in the text to denote bandwidths where the difference between the upper and lower frequencies is less than one-fifth of the center frequency. Numerator function A polynomial in s that is the sum of the numerator parts of both through (even) and coupled (odd) transfer functions or is simply one or the other. Octave A musical term denoting an interval of eight notes applied to general spectrum analysis to denote instances where the frequency of one signal is twice that of another. Passband The specified frequency range of operation and not necessarily frequencies where transfer functions represent low loss. Perfect coupling Magnetic coupling between two inductors where the entire magnetic flux of either inductor links the other. Quadrature Two signals whose spectral components exhibit a 90° phase difference. Return loss The ratio of power not delivered to a load divided by the maximum available power from a generator. UHF (ultra high frequency) The electromagnetic spectrum between the frequencies of 300 MHz and 1 GHz; also applies to circuits whose response is substantially, though not entirely, within this range. VHF (very high frequency) The electromagnetic spectrum between the frequencies of 30 and 300 MHz; also applies to circuits whose response is substantially, though not entirely, within this range.
Glossary
205
Wideband A comparative term, synonymous with “broadband,” that denotes bandwidths where the difference between the upper and lower frequencies is more than half the center frequency.
Bibliography The following is a list of publications used in the preparation of this book and other publications containing relevant material. Abrie, P. L. D., Design of RF and Microwave Amplifiers and Oscillators, Norwood, MA: Artech House, 2000. Andrews, D. P., and C. S. Aitchison, “Design of Cascaded Lumped Element Quadrature Hybrids,” IEE Proceedings, Microwaves, Antennas and Propagation, Vol. 148, No. 5, October 2001, pp. 275–279. Andrews, D. P., and C. S. Aitchison, “Microstrip Lumped Element Quadrature Couplers for Use at Microwave Frequencies,” IEE Proceedings, Microwaves, Antennas and Propagation, Vol. 147, No. 4, August 2000, pp. 267–271. Andrews, D. P., and C. S. Aitchison, “Optimum Approximation Functions for Lumped Element Quadrature Hybrids,” IEE Proceedings, Circuits, Devices and Systems, Vol. 148, No. 1, February 2001, pp. 5–9. Andrews, D. P., and C. S. Aitchison, “Synthesis of Symmetric Lumped Element Quadrature Hybrids,” IEE Proceedings, Microwaves, Antennas and Propagation, Vol. 148, No. 5, October 2001, pp. 269–274. 207
208
Lumped Element Quadrature Hybrids
Andrews, D. P., and C. S. Aitchison, “Transfer Functions of Opti mum Lumped Element Quadrature Hybrids,” IEE Proceedings, Circuits, Devices and Systems, Vol. 148, No. 3, June 2001, pp. 135–139. Andrews, D. P., and C. S. Aitchison, “Wide-Band Lumped Quadra ture 3dB Couplers in Microstrip,” IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-48, No. 12, December 2000, pp. 2424–2431. Bode, H. W., Network Analysis and Feedback Amplifier Design, New York: Van Nostrand, 1945. Bowman, F., Introduction to Elliptic Functions, London, England: English Universities Press, 1953. Brune, O., “Synthesis of a Finite Two Terminal Network Whose Driving-Point Impedance Is a Prescribed Function of Frequency,” J. Maths and Phys., Vol. 10, No. 3, October 1931, pp. 191–236. Cappucci, J. D., “Don’t Overspecify with Quad Hybrids,” Microwaves, Vol. 12, No. 1, January 1973, pp. 50–54. Cappucci, J. D., “Don’t Overspecify with Quad Hybrids,” Microwaves, Vol. 12, No. 2, February 1973, pp. 62–64. Cappucci, J. D., and H. Seidel, “Four Port Directive Coupler Having Electrical Symmetry with Respect to Both Axes,”U.S. Patent No. 3,452,300, June 24, 1969. Cappucci, J. D., “Lumped Parameter Directional Coupler,”U.S. Pat ent No. 3,452,301, June 24, 1969. Cappucci, J. D., “Networks Using Cascaded Quadrature Couplers, Each Coupler Having a Different Center Operating Frequency,”U.S. Patent No. 3,514,722, May 26, 1970. Chen, W. K., Theory and Design of Broadband Matching Networks, Oxford, England: Pergamon Press, 1976. Collin, R. E., Foundations for Microwave Engineering, 2nd ed., New York: McGraw Hill, 1992. Cristal, E. G., and L. Young, “Theory and Tables of Optimum Sym metrical TEM-Mode Coupled Transmission Line Directional Couplers,” IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-13, No. 5, September 1965, pp. 544–558.
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Darlington, S., “Synthesis of Reactance 4-Poles Which Produce Pre scribed Insertion Loss Characteristics,” Journal of Mathematics and Physics, Vol. 18, No. 4, September 1939, pp. 257–353. Darlington, S., “Realisation of a Constant Phase Difference,” Bell System Technical Journal, Vol. 29, No. 1, January 1950, pp. 94–104. Edwards, R. L., and B. L. Martin, “Wideband Transmission Line Sig nal Combiner/Divider,”U.S. Patent No. 4,774,481, September 28, 1988. Fisher, R. E., “Broad-Band Twisted-Wire Quadrature Hybrids,” IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-21, No. 5, May 1973, pp. 355–357. Firestone, W. L., “Coupling System,”U.S. Patent No. 2,972,121, February 14, 1961. Guillemin, E. A., Synthesis of Passive Networks, New York: John Wiley & Sons, 1957. Hilbert, W., “From Approximations to Exact Relations for Characteristic Impedances,” IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-17, No. 5, May 1969, pp. 259–265. Ho, C. Y., “Design of Quadrature Hybrids— Part I,” July/August 1979, pp. 49–54. Ho, C. Y., “Design of Quadrature Hybrids— Part II,” September/October 1979, pp. 32–39.
RF Design, RF Design,
Ho, C. Y., “Design of Wideband Quadrature Couplers for UHF/VHF: Part I,” RF Design, November 1989, pp. 58–61. Ho, C. Y., “Design of Wideband Quadrature Couplers for UHF/VHF: Part II,” RF Design, December 1989, pp. 51–55. Ho, C. Y., and J. H. Weidman, “A Broadband Quadrature Coupler,” Microwave Journal, Vol. 26, No. 5, May 1983, pp. 216–219. Keely, T. A., “Design of Constant Phase Difference Networks,” RF Design, April 1989, pp. 32–42. Kurokawa, K., “Design Theory of Balanced Transistor Amplifiers,” Bell System Technical Journal, Vol. 44, No. 8, October 1965, pp. 1675–1698. Maas, S., Microwave Mixers, Dedham, MA: Artech House, 1986.
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Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Dedham, MA: Artech House, 1980. Monteath, G. D., “Coupled Transmission Lines as Symmetrical Di rectional Couplers,” IEE Proceedings, Vol. 102, Part B, No. 3, May 1955, pp. 383–392. Montgomery, C. G., R. H. Dicke, and E. M. Purcell, (eds.), Principles of Microwave Circuits, New York: McGraw-Hill, 1947. Orchard, H. J., “Computation of Elliptic Functions of Rational Frac tions of a Quarterperiod,” IRE Trans. Circuit Theory, Vol. CT-5, No. 4, December 1958, pp. 352–355. Podell, A. F., “Transmission Line Quadrature Coupler,”U.S. Patent No. 3,484,724, December 16, 1969. Pozar, D. M., Microwave Engineering, 3rd ed., New York: John Wiley & Sons, 2005. Reed, J., and G. J. Wheeler, “A Method of Analysis of Symmetrical Four-Port Networks,” IRE Trans. Microwave Theory and Techniques, Vol. PGMTT-4, No. 4, October 1956, pp. 246–252. Sevick, J., Transmission Line Transformers, Atlanta, GA: Noble Publishing, 1996. Shelton, R., J. Wolf, and R. van Wagoner, “Tandem Couplers and Phase Shifters for Multi-Octave Bandwidth,” Microwaves, April 1965, pp. 14–19. Sontheimer, C. G., “Electrical Connector for Transmission Lines and the Like,”U.S. Patent No. 3,325,587, June 13, 1967. Sontheimer, C. G., and R. E. Fredrick, “Broadband Directional Cou pler,”U.S. Patent No. 3,426,298, February 4, 1969. Toker, C., “Design Criteria for a Lumped-Element Directional Cou pler,” Int. Journal of Electronics, Vol. 42, No. 3, March 1977, pp. 209–227. Weinberg, L., Network Analysis and Synthesis, Huntington, NY: Robert E. Kreiger Publishing Company, 1975. Zverev, A. I., Handbook of Filter Synthesis, New York: John Wiley & Sons, 1967.
About the Author David Andrews studied electrical and electronic engineering at Brunel University, England, in conjunction with a student apprenticeship at Marconi Space and Defense Systems in his hometown of Portsmouth, England. He graduated with a B.Sc. honours degree in 1984. After graduation, Dr. Andrews worked at Marconi Space Systems in Portsmouth on microwave active circuits. In 1986, he moved to Plessey Microwave in Towcester, England, and worked on microwave amplifier design. In 1989, he moved to Milmega on the Isle of Wight, England, where he eventually became the chief design engineer, responsible for the design of high-power RF and microwave amplifiers. In 1997, Dr. Andrews became a cofounder of Vectawave Technology, a company specializing in the design and manufacture of wideband, high-power RF and microwave amplifiers. He currently holds the position of technical director. From 1999 to 2002, he undertook a period of part-time study at the University of Surrey, England, researching lumped element quadrature hybrids, and was awarded a Ph.D. in 2003. He has written several papers on the subject and has been a speaker at International Microwave Symposia organized by the IEEE Microwave Theory and Techniques chapter. Dr. Andrews is a member of the Institution of Electrical Engineers.
211
Index Active circuits, 169 All-pass function, 25, 30, 31 Amplitude imbalance, 29 Approximate amplitude hybrid, 8, 24 Approximate phase hybrid, 8, 30 Approximations, 35 polynomial, 36 rational, 38 second-order, 45
double periodicity, 52 fundamental region, 53 modulus, 50 nome, 60 period parallelogram, 52 principal value, 185 sn(z,k), 52 Equivalence, 15, 75, 125 Even and odd mode analysis, 76
Balanced amplifier, 10, 19 Bandwidth ratio, 60 Branch line hybrid, 1 Brune section, 103
Filtering function, 28, 32 polynomial, 36 rational, 38 uniqueness, 43 Four mode analysis, 98 Frequency parameters, 66, 67
Capacitors, 130 Cappucci, 109
Graphical techniques, 192
Darlington synthesis, 81 Directional coupler, 24, 33, 118 Duality, 77
Image reject mixer, 12 In-phase hybrid, 24, 78, 124, 157, 164 IQ demodulator, 10
Elliptic integrals, 50, 187 Elliptic filter, 58 Elliptic functions, 50 cn z, 53 complementary modulus, 52 congruency, 53 dn z, 53
Ladder network, 38 Lange coupler, 146 Lattice network, 40, 119, 166 Loss-less circuits, 19 Microstrip, 141, 145 Mixers, 10 221
222
Lumped Element Quadrature Hybrids
Numerator function, 70 Operational amplifier, 170 Optimization, 36
mixed element, 93, 140 polynomial-based, 81, 138 rational function–based, 104, 152 second-order rational, 96, 150
Phase delay network, 119, 165 active, 178 Power handling, 190 Proximity coupler, 3
Richard’s transformation, 47
Quadrature error, 13 Quadrature hybrid approximate phase, 117, 157 cascaded section, 107 coupled transmission line, 47, 95 first-order, 79, 130
Transfer functions, 64 Transmission line coupler, 47
Scattering parameters, 19 Synthesis, 75
Unitary matrix, 20 Weak couplers, 181 Wilkinson divider, 2
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