Electromagnetic Oscillations from a Bent Antenna-A Dissertation Presente

EXCHANGE

ELECTROMAGNETIC OSCILLATIONS FROM A

BENT ANTENNA

ROBERT CAMERON COLWELL

/

ELECTROMAGNETIC OSCILLATIONS

FEOM A BENT ANTENNA

A DISSERTATION PRESENTED TO THE

FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

BY

ROBERT CAMERON COLWELL

PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER. HA.

1920

Accepted by the Department Physics, February, 1918

! '.

'.;',

f

,'r

' t

;

-',

of

ELECTROMAGNETIC OSCILLATIONS FROM A BENT

ANTENNA The purpose

of this investigation

is

to find the mathematical

equations for the electromagnetic oscillations tenna, which

used

is

is

known

that of Pocklington

Abraham2 and

from a bent an-

to send out directed waves.

particularly

1

The method

which has been developed by M.

by G. W.

Peirce 3

who has

recently

published a remarkable research on the radiation resistance of a flat top antenna. This article is based upon the work of Peirce

and Abraham, but the equations are worked out for the fundamental and not for the forced vibration. The application of the formula

is

new.

The following assumptions are made: 1. That any antenna may be considered to be made up of a large number of Hertzian doublets placed end to end. 2. That the earth is a perfectly conducting plane. 3. That the waves propagated high into the air eventually return to the earth. The reason for this assumption will be shown in the section dealing with the horizontal part of the antenna.

Let a part

d,

top antenna have a vertical part h and a horizontal It will be necessary to discuss the effect of the Fig. 1. flat

EARTH'S SURFACE

FIG. 1

radiation from this antenna in three parts. I. The radiation due to the vertical part h and

its

image

h.

Pocklington H. C., Camb. Phil. Soc. Proc., 1898, p. 325. 2 Abraham, Theorie der Electrizitat, Vol. II. 3 G. W. Peirce, "Radiation Characteristics of an Antenna," Proc. Am. Acad. Arts and Sciences, Vol. 52, No. 4, October, 1916. 1

3

ELECTROMAGNETIC OSCILLATIONS d. The 'radiation from the horizontal part d and its image and horizontal The mutual action between the vertical parts.

II.

III.

I.

RADIATION FROM THE VERTICAL PART

Let the point where the vertical part enters the ground be the origin of co-ordinates, and choose the axes as shown in Fig. 2

FIG. 2

where the x axis

which the free antenna points and the vertical antenna coincides with the direction OZ. The azimuth and co-latitude

end

is

parallel to the direction in

of the horizontal

angles have their usual designations. Let P be any point in space whose polar co-ordinates are r, , 6, where r is very great

compared to the height h of the vertical part of the antenna. P lies at a distance r from the point P which is the position of r

one of the Hertzian doublets postulated in the first assumption. The effect of the doublet at P f on the point P is given by the 1 theory of Hertz

CD where f(t)

V=

=

the

moment

velocity of light.

E

is

PP

f

and and H in and 6', as is

of the doublet edz, r

length

expressed in electrostatic

electromagnetic units; r Q and 6 appear in place of r legitimate because of the great magnitude of r in comparison

with OP. If

the total length of the antenna

a node of current at 1

+

/

and

/.

is

/

=

h

+

d there must be

For the fundamental vibra-

Hertz, Electric waves, Chap. IX, Trans. D. E. Jones; Bateman, Elec-

trical

and Optical Wave Motion,

p. 8.

FROM A BENT ANTENNA tion, the current i at

mum

current

any time expressed

in

5

terms of I the maxi-

must be i

=

I sin pt cos -r

(2)

A

where X

=

the

wave length p

= 2x7. r is A

of the system,

the angular velocity

No w

and the current doublet, that

i is

the rate of change of the charge

e

on any

is

=

l

de

di

dt'

Tt

=

d2 e

df

Therefore d>e

2x71

di

2x7*

^ = ^ = -r~ ^r cos

Substituting (3) in (1)

dE

a

which -r A

in

=

p/sinfl

tyi is

we

get

cos

p (t

/

--r

2x2 cos

(3)

ir

zcos0\ -

y

1

2xz

,

cos -r- dz

given the shorter form p, and

(4)

z cos 6 is

r

written for r as these are approximately equal. z and this doublet The doublet1 at OP f has an image at will

have an

effect at

pi sin dEi = p

The

effect

adding

(4)

P

given by the equation

6

cos

( p [t

--r + z cos 6 -

y

\

Icos

2x2 -y-

dz

(5)

due to the two doublets on the point P is found by and (5), and the action of all the doublets and their

Theory of Images: Jeans, Electricity and Magnetism, Chapter VIII; Maxwell, Electricity and Magnetism, Chapter XI; Webster, Electricity and Magnetism, p. 303. 1

ELECTROMAGNETIC OSCILLATIONS

6

images,

by

integrating from

ind C h

to

Then

h.

2irz[ -

\

r

(

\

pz cos 6

,

-y~ &

The terms

in the square brackets are of the

cose

form [Cos

(x

+ y)

y)] may be simplified by use of a well-known trigonometrical formula: then

cos (x

/

2plsm8

r

Ch

\

2vz cos-r-ds

pzcosd

,

~^^y The

integral in (7)

a standard form and

is

is

found

any table

in

Integrating and putting in the limits

of integrals.

(7)

we

get for

the integral of (7) in

fph cos0

2Trh\

y

I

r

sm

1

( I

2Trh\ r-

,

ph cos

H

I

This reduces to sin

-^- (cos

sin

1)

~ (cos 0-1) Adding these fractions we get X

J

^rx 2 sm 0[ i

_\

cos (2
\X

)

(cos

vA (cos 0+1)

A

2?r

-^

sm ^v-.

\

/27T/J

27T/1

X

/

term

for the integrated

sln

"T~ cos

\X

I

)

/

27T/1

cos "T~ cos

X

Substituting in (7)

27

Vhmt".

X

v

I

Equation

(8)

is

cos

Iy

cos

sin

(

\

~~ cos

)

V X

^^:

.

s

/ i

cos z__-

L

(8)

the basis for the following brief discussion Poynting's Vector Theorem states that

of a vertical antenna.

FROM A BENT ANTENNA

7

the power radiated along the radius at any point of a sphere is V/4:Tr(E H) per unit area. In the case of the vertical E = H.

X

The energy radiated

is

equal to 4&.

is

length cos2 2ir/\(Vt

E The wave The average value of

directly proportional to

27/Fr

is

constant.

.

Therefore the expression for the power radiated at an angle 6 takes the simple form TQ) is |.

P.

(COS (fcosfl) '

= K

(9)

sin 6

The is

angle at which the greatest

amount

of energy

is

radiated

given by cos

7p "

n

fir I

\ cos 6

\

T7

dS

\

/

sin B

sin

^^

(

cos

^ /2 )

2

(

^ cos

8 } cos

\_?

sm

L

cos

sin 6

=

)

.

6

]

ir/2 makes d P ldd negative, and therefore the direction of maximum radiation of energy. represents Therefore a vertical antenna sends out its maximum energy

The

solution B

2

2

K

along the horizontal.

Table 1 is developed from (9), being given an arbitrary value. Diagram A is a graph of Table I, showing the approximate energy distribution. The distribution is such that nearly all the radiation lies within the 30 angle with the horizon. These equations (8) and (9) throw some The radius of action is light on the action of tall antennas.

of energy

increased as the antenna height is increased for two reasons. First, because the lengthening of the antenna automatically increased the wave length and the earth is a better conductor for

long waves than for short ones. Secondly, because the waves from a high antenna are not brought to earth so quickly (see assumption three) as those from a short one. Just as soon, however, as the wave length is reached for which the earth is a perfect conductor, no further effected

by

improvement

increasing the wave.

in

conduction

Similarly there

is

is

a limit to

ELECTROMAGNETIC OSCILLATIONS

8

the second action, so that there is no reason for increasing the height of the sending antenna indefinitely.

TABLE

I

V

Angle

0.00

*$ 0.00

15

.078

30 45 60 75 90

.144

.42

.422

.65

.672

.82

.902 1.00

II.

.22

.95 1.00

THE HORIZONTAL PART

In order to apply Hertz's theory to the horizontal antenna, necessary to measure the co-latitude from the axis X. The

it is

The designated by \[/ and the azimuth by % where the the vertical at antenna enters point origin remains the earth, Fig. 3. It should be noticed here that the action of co-latitude

is

FIG. 3

the horizontal part is dependent upon the length of the vertical for the latter determines the phase of the current at any point

Neither in theory nor in practice can the action of the horizontal be dissociated from the vertical. The positions in the former.

of the doublets

and doublet images are denoted

in Fig. 4.

The

small letters x, y, z (or x, o, h) refer to the doublets, while the position of the point in space P is given by large X, Y, Z.

FKOM A BENT ANTENNA Hertz's theory may now be applied to the effect at doublets at Q and Q'. Then

P

of the

(10)

Now

n= V(X At

(ID 2

z)

+ p+

(Z

+

(12)

great distances (13)

FIG. 4

From

(11), (12), (13)

-

2Zh (14)

2r

-

2Xx (15)

2r

As before

in (3)

2irV -

_

/ cos

27T. rr -^-

Vt

\

2ir

.

(16)

Also,

w(* fit \

r2

^ T7I= V )

pi

COS p I(^t

\

27T

cos

-

M 57

I

V J

cos

ELECTROMAGNETIC OSCILLATIONS

10

2ro

&-j- (i+*)
j

(17)

Substituting (16) and (17) in (10), simplifying by the trigonometrical formula for the sum of two cosines; neglecting a;2 h2

+

in

comparison with 2Xx, we obtain the expression

Jrt

dEj,

47r7 sin

T7 = = dHy ,

A7*o

-

integration of (18)

2irZh

sin ~T

VA

Vt

The

.

\//

T^T TQ

+ x cos ^}

r

is

(h

cosy

+ x)dx 1

very similar to that of

(18)

and gives

(7)

finally

E* =

21 TZ r<>V

.

2wZh [ 2ir sm-r sm-r- (Vt A .

7

sm ^

cos

Xro

X

f

r

s ) i

cos-^-

|_

cos

-T

(

2nd X

f

.

cos

j

\l/

}

H-

^ sm

cos

-r-

2^

)

/2^

sm

-T-

r

(Vt

I

-r

\11 cos

^

(19)

1

equations (8) and (19) are based all the conclusions regarda bent antenna. The total flux of force is obtained by coming pounding EQ of (8) with the E^ of (19) in such a way as to get

Upon

the complete electric intensity E, and the corresponding magnetic intensity H. Although the doublets which give rise to E e are perpendicular to those producing E$,

it

by no means

follows that

E e are at right angles; in fact, they make an angle (a) with one another which varies for every point on the particular sphere under consideration. Professor Peirce, of Harvard UniE# and

versity, has

shown that

this difference in direction gives rise to

a third term in the power radiation of the form (8) and (19)

from

{

the other two

come

of course

}

2 cos

aE & E&,

The average value

where

cos

a.

=

sin 1

1

of this term,

when

\{/

cos 8 cos

r-^-r 2 8 sm

multiplied

<j>

x

cos 2

by

4>

F/47T,

he

FROM A BENT ANTENNA calls the

In the problem under discussion

mutual power.

2K cos sin

4?r

cos

sin

0(1

2

4>

.

sm ""

2irhZ J

L

cos

\r Q

)

sin

where

--- cos

2irh 2wh cos ^: T cos

X

A

A

for convenience

11

K

is

2irh

-r-

A

.

sm

2 put equal to 7 /7rFr

-

ir2h cos

2 .

true only for the average value of (V/2ir) cos the term

-T-

A

(Vt

r

)

=

\ and sin

A

(Vt

r

(9

/rtm (20)

.

r-

f

A

This equation

aE e E^

cos r- (Vt

)

sin

cos

(

is

cos2

cos

[

so that

r

A

)

=

SECOND PART NUMERICAL CALCULATION The power

radiated per unit area in certain zones at distances

and 9,000 meters above the earth's surface, and at a distance 10,000 meters from the origin will now be calculated by means of equations (8), (19) and (20). In these equations = 440. Then let h = 10 meters, d = 100 meters, X = 4(& d) the power radiated from the vertical part may be expressed by (8) in the form 2,000, 6,000

+

:

V E? 47T

= K jcosf

-r-

COS0J 10,000

sin-r-

cos

sin

(

~- cos

0)

cos

~^-

M. jwoo

jr.

"^6000

M.,

ASOOOM.

FIG. 5

The

values in Table II are calculated from this equation:

12

ELECTROMAGNETIC OSCILLATIONS TABLE

VERTICAL

II

Height

X h r

= 440M = 10M = 10000M

9000M 6000M

.0049K .0160

5000 2000

.0164 .0196 .0196

The calculations of E 9 (V/4:ir) for the horizontal part are by no means as simple as this. The value of the angle in formula (19) 2

changes for every point on the zone at the height 9000, 6000 and so on. This change must be calculated. In Fig. 6 ABC

FIG. 6 is

a spherical triangle with sides 8, \l/, irj'2 and angle given. a well-known formula in the trigonometry of triangles: <

Then from

Cos

=

\l/

sin 6 cos

The power radiated from the zone that

<j>

horizontal antenna through the

Z= is,

9000, 6000, etc., will vary for different angles of <j>, the bent part of the antenna has a directive effect on the

oscillations.

In equation (19)

VW = K 47T

5in

,

f

2rZh Xrn

I

sin

}

f

cos 2

r

-T-

j

1

+ cos

2 \l/

2ird 2 sin -r

\l/

\

2ird

2 cos -r

2nd

I

2

cos

cos

1

2 cos

\f/

sin

XX sm

cos

From this equation the values of Table III are calculated for the different zones and different directions. The operations are long and tedious but not

difficult.

The

horizontal part of the

FROM A BENT ANTENNA

13

antenna has a

slight directive effect perpendicular to the direc-

tion in which

it

contrary to what one would show a directive effect in the expect: experiments direction away from the free end of the antenna. However, it has been shown by Pocklington that a circular wire radiates more power perpendicular to its area than in any other direction and the form of equation (19) shows that Pocklington's method of doublets applies to this problem and that the solution is correct to the approximations made.

because

points.

This

is

all

HORIZONTAL PART

TABLE

III

now be shown

that the power arising from the mutual modify the directive effect at high points in the atmosphere in such a way as to give a fore and aft directive It will

effect (eq. 20) tends to

effect.

The

values of Table

IV

are calculated from equation (20).

MUTUAL EFFECT TABLE IV

14

ELECTEOMAGNETIC OSCILLATIONS

Adding up the

vertical, the horizontal

and the mutual

effects

II, III and IV, we obtain the complete power radiated through unit area of the zones. The results are given in Table V, in which K has been set equal to an arbitrary value. Plate B is plotted from Table V. In Plate C, the dis-

contained in Tables

tribution at 9000

M

is

compared

mentally by Fleming. TOTAL EFFECT

The symmetry

to a curve obtained experi-

TABLE

V

developed from the theory can be reconciled with the asymmetrical curve found in the experiof the curves

ments by supposing that the electrical waves brought back to the earth from high in the atmosphere are more intense toward the bend in the antenna than at the free end. Zenneck (Zenneck, Phys. Zeitsch., Vol. 9, p. 553, 1908) has shown that this difference may be due to imperfect conductivity in the earth's surface.

SUMMARY First

:

The equations developed by

Pocklington,

and Peirce have been applied to an antenna with

Abraham

vertical

and

horizontal parts in such a way as to find the energy given out for the fundamental vibration:

X

=

4(fc

+

d)

Second: The intensities so obtained are plotted and are shown to have a fore and aft directive inclination. The intensity fore

and

aft is

symmetrical and not asymmetrical as required by the

experiments.

FROM A BENT ANTENNA

15

Close to the antenna the intensities are symmetrical azimuth agreeing with experiment. Fourth: The conclusion is that the fore and aft asymmetry

Third: in

antenna is caused by the difference in conductivity between the atmosphere and the earth for the electromagnetic of a bent

oscillation.

The forms

Fifth:

of the resulting equations

show that

Peirce's

assumptions regarding the current satisfy Pocklington's criterion for the use of Maxwell's Equations.

ADDENDUM The Integration of Equations

The

sin

I

Expand and

{

n.

Vt

rQ

r

)

A

2irx

sm

r2ir A

A

cos

The

27r /T _ -T-

A

(p

t

(h

+ x)dx

rQ)

fo)

2irh sin 2ir

--

27rxCost~] A

--

2irx

Cos

r

A

-Cos

2irx

\[/

2irx

2irx

\l/

.

COS-T ax A

(21)

,

(22)

Cd 2wx Cos -^ 2wx cos 7 sm-^r- dx A A J A

(23)

,/Q

r

sm -r-

2irh

A

is

r

A

--

27rh

sin^r

integration of (21)

sin

cos -T" dx

A

.

.

sm

)

2

cos^r cos A

cos -r-

TQ)

)

.

r

A

2wh

,.

.

sm^r- (Vt A

-r-

,

A

(Vt

2ir

.

r

2wh C d (Vt

cos

\//}

-- -- + cos 27r.__ (Vt

-sm

2ir -r-

x Cos

27rxCost

cos

cos-^cos^ [2irh

cos

-f-

multiply, thus:

27T. . T sin-r- (Vt

+

(18), p. 10.

integrable part of (18) is

A

.

r sm 2jrxCos\l/- sm 2wx dx .

.

r

t/o

(24)

A

A.

the same as that of (7) on page 4 and

gives

X

2wdcos\l/ 2

2ir sin

The

\//

\_

X

.

2wd X

.

2wdcos\l/^

27rd~\

X

integration of the integral part (22) comes under

X J

Form 360

16

ELECTROMAGNETIC OSCILLATIONS

in Peirce's

Table of Integral and simplifies into

2wd cos

.

+ (23)

comes under form

The complete

To

2irh

.

sin

sin

(360) but in a different

-

cos

(26)

way from

integration follows:

find

r

2irx

Cos

2irx

cos

sin

(Form 360) sin

mx

cos nx ax

cos

cos

(m n)x 2(m n)

=

(m 2(m

+ ri)x

Let 2ir

Then

2ir

cos

integration gives

47T

(1

+

COS

\l/)

\

COS

COS

A

,

2nd

+ sm ^A

_X 47T

2nd (1

cos

\p)

\

A

sm

COS

2wd -r

A

sin'

\I/

r

A

.

sin

2irdcos\l/ -

T

A

2ird cos

cos -r- cos

2w

---

27T(Z

-

^

2ir sin

A

.

sin

2ird cos

\1/

2

(22).

FROM A BENT ANTENNA 2ird cos

2ird

2ir sin

The

2

\p

cos

2ird sin -r sin

\f/

.

(27)

2nd cos ^ r

1

A

A

integrable part of (24)

Cos

r2wx sin

is

,

.

+

17

also a standard

sin

A

form and becomes

>s^cos

2ird cos r

\l/

A

2irx

.

\l/

r

finally

2ird

.

,

dx

A

.

sm

sm^r

A

2ird cos

2nd

\L

COS-T

A

]

)

A

Then

Substitute 25, 26, 27 and (28) in (21), (22), (23) and (24). and obtain:

insert the result in (18)

4wl f-^Ji

,

~~~

2irh

27T

sm -T-

r Q)

(Vt

sin

\//

cos

,

27r /T .

y (Vt

2irh r

)

COS -r

,

2<jr

sin

.__

(Vt

COS

j

+

+

I

)

2irh sin -r

f

sin

T

^~

2nd Cost A

Ivd]

cos -r

f

A

J

2nd cost COS ^

-- cos -- sin 2wh--

2,ird

.

sin

,

r

\l/

a

2ird

.

sin

j

cos

COS

.

7%

2nd cos ^

I

cos

^

X

27rZh

.

Sin

Tr-\

.

\[/

r-

cos

\[/

--r

j

+ cos t sin ~" sin --r 2ird

+ cos

f

2ird cos

2<7rd

cos -r- cos

2ir ,-:-

(Vt

.

TO) sin

2irh

^

f

,

cos

2ird cos

.

cos

2nd cos ~

\l/

~Y~

\l/

j

sm -,

--t COS

2ird cos

^

1

.

2wd

sm "T~ 2<jrd

~X

(28)

J

{~] |

ELECTROMAGNETIC OSCILLATIONS

18

Now X

.

2irh

TT

=

=

2wd

2'T~

4'''T~ so that 2irh

cos -T

=

sin -r

=

2nd sin -T

A

A

When

2ird

cos -r

A

A

these expressions are substituted in (29), the terms in r ) may be TO) and those in cos (2ir/\)(Vt

sin (2irl\)(Vt

added together and give the simple form: 21 r

T7

.

,

..

V sin ^ 1

cos -r

A

which

is

f J

27r 27rZh[ sin Xr L

+ cos

f

.

sin -T

27r. -r-

A

equation (19), page

(

T

L

TO)

cos

.

Vt

(9).

2wd cos ^

.

r

)

j

cos

\f/

sin

i

S

o

3

s I

s

/

S

g

8

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