Library, N,W,^Bldg ' ' JAN 5 1 1964 ^

NBS MONOGRAPH 72

National Burssii Stardarti? JUL 8 8 IS'2 ^/.1201

Curves of Input Impedance Change

Due to Ground for Dipole Antennas

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UNITED STATES DEPARTAIENT OF COMA-IERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Curves of Input Impedance Change

Due to Ground for Dipole Antennas

L. E. Vogler and J. L. Noble

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

Most of the work in this Monograph was sponsored by the Jet Propulsion Laboratory, Pasadena, Calif., as part of the Lunar Point-to-Point Communication Study

National Bureau of Standards Monograph 72

Issued January 31, 1964

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price 35 cents
Library of Congress Catalog Card Number: 63-62354
Contents

Page

1. Introduction 1 2. Input impedance 1 3. Limiting expressions . 3

4. References 5

5. Appendix 5

6. Graphs of normalized input impedance change AZ/Rf 11 m

CURVES OF INPUT IMPEDANCE CHANGE DUE TO GROUND FOR DIPOLE ANTENNAS

L. E. Vogler and J. L. Noble

Graphs of the change in input impedance of electrically short dipole antennas

in the presence of an isotropic and homogeneous ground are presented, considering

four types of antennas: horizontal and vertical, electric and magnetic dipoles.

Curves of the change in both the input resistance and reactance are shown for a wide range of values of the frequency, antenna height above the ground, and electromagnetic ground constants,

1. INTRODUCTION

A vast amount of literature exists on the subject of dipole antennas, a considerable portion of which is devoted to impedance properties. Tabulations and curves of input impedance are available

for restricted ranges of the parameters involved and for particular types of dipoles; however, no comprehensive numerical work on the subject is known to the authors. It is hoped that the present

paper, although not comprehensive in the final sense, may serve as a step towards providing complete numerical information on dipole impedance in the presence of conducting ground.

Impedance evaluations are important not only in considering propagation and antenna performance [Norton, 1959; Vogler and Noble, 1963 ], but are also coming into use in some of the newer

methods of geophysical exploration [Wait, 19 51; 19 59 ]. Data from impedance measurements

obtained by sending out lunar or planetary probes that relay back the appropriate information can be

used to help answer questions affecting communication requirements of manned exploration projects

[Vogler, 1963 ]. The curves in the present paper give the change in both the resistance and reactance of four

types of antennas: vertical and horizontal, electric and magnetic dipoles. Emphasis has been placed

on the more poorly conducting materials that are expected in lunar communication systems; however,

estimates of input impedance for highly conducting grounds may also be obtained from the curves.

2. INPUT IMPEDANCE

It is assumed that an elementary dipole of length dl is located at the origin of a cylindrical

coordinate system (p , 6, z) at a height h above an isotropic and homogeneous conducting halfspace characterized by a dielectric constant and conductivity cr^. The interface between the

tv/o media is the plane z = -h, and for z > - h the dielectric constant is assumed to be equal to the

free space value t . The magnetic permeability of the whole space is taken as the free space value

o

|jL^. (MKS units are used throughout this paper. ) In the case of a vertical dipole, the antenna is

oriented in the z direction, whereas, for a horizontal dipole, the orientation is in the 9=0 direction.

The elementary magnetic dipole may be considered equivalent to a small circular loop with its axis

in the direction of the dipole and with an area dA = di/(3 , where p is the free space wave number

(= Zir/X). For practical short dipoles having a linear distribution of current, dJl should be replaced

by an effective length - dl/Z.

For a dipole antenna with the current varying as I^ e^'^*, I^ a constant, expressions for the

field components have been derived by various authors [ Sommerfeld and Renner , 1942; Wait, 1953,

1961; King, 1956 ]. A brief discussion of the subject is also contained in the appendix to this paper.

The components of interest in the present impedance considerations are given by equations (A- 11) of

1
the appendix, where E denotes the electric vector (in volts per meter) associated with electric '

dipoles, and B denotes the magnetic vector (in webers per square meter) associated with magnetic

dipoles. < Using the "emf method" [King, 1956 (p. Z58); Wait, 1962 ], one may find the input imped- I

ance Z in terms of the field components at the antenna by taking the limits bf (A- 11) as both z and |

p approach zero. Thus, for the electric dipole,

j

Lim z—

o

p-o

Edi

= AZ , (2. la)

and for the magnetic dipole,

Z = Lim

z—o

p—

o

ojBdA

~l

s Z^ + AZ (2. lb)

Here, Z is defined as Z^ = Lim Z, and AZ is then that portion of the impedance that is affected

f f h—

M

by the presence of the ground (z < - h).

If one now substitutes the variable x = Zhu in (A-12) of the appendix and normalizes the o impedance change AZ by dividing by the free space resistance of the antenna, = R e Z^, the

following expressions for the normalized input impedance change due to the presence of the ground is obtained (VED, HED, VMD, HMD refer to the initial letters of the four types of dipoles considered);

VED: AZ/R^ = i^^S/Za-^^ |'l^(N^) + l2(N^) R^ = 20 (3^^(di)^ , (2.2a)

HED: AZ/R^ = i( 3/4a 1^(1) + I^IN ) R. = 20 p^di)^, (2.2b)

VMD: AZ/R^ = i[3/2a j j 1^ (1) + 12(1)

3A r. R = 20 p^(dA)^, (2.2c)

HMD: AZ/R^ = i( 3/40-"') I^(N^) + I^d)"! , R^ = 20 (3^^(dA)^ , (2.2d)

where

c

[j(6) = J

, 2 2 2 6x - J X - (N -1)

6x +1 x^ - a^(N^ -1)

e dx

,

(2.3a) I

1,(6) = r

2 X •< la

12 2 2 6x - X - Q- (N -1)

I 2 "Z 2 6x +. ) X -a (N -1)

>- e dx

,

(2.3b)
a - 2h a = 2h(2-n-/M. N^-{£,/e )-i(o-,/aje ). o i o i o

The variable x in (2. 3) assumes purely imaginary values from the lower limit to zero and from

there ranges over real values to the upper limit. The sign of the square root is chosen such that

the real part is greater than zero. The expressions in (2. 2) were programmed for an electronic computer, Gaussian quadrature

methods being used for the integrals of (2. 3). Graphs of the normalized input impedance change

AZ/R^ for the four different antennas are shown in the figures with the parameters | N| , 4)/2, and

a defined by:

r 2 2 1 1 N[=!e^+s

j

, <i)/2 = |tan" (s/e^), a = (2h) (2Tr/X) , (2.4)

where

€,/e , s = cr,/co€ = 60 \ cr (mhos /m)

,

1 o 1 o

(2. 5)

and (r{mhos/m) is the conductivity of the ground; X denotes the free space wavelength, and h

denotes the antenna height, both measured in meters.

3. LIMITING EXPRESSIONS

From a study of the curves, it is apparent that there can be a large variation in impedance as

the antenna height is varied. In fact as the antenna approaches the ground, the impedance, and

especially the reactance, become very large. This latter effect, in the limiting case of h e^; 0, is a mathematical result rather than a physical one, due to the infinitesimal dimensions assumed for the

elementary dipole.

An attempt was made in the curves to show impedajice detail for both large heights and small

heights (large a and small a). However, for values not obtainable from the graphs, the following

expressions derived from an integration by parts of the integrals involved may be used at the two

extremes.

For a N > > 1

:

(3. la)

HED: ie"^"*, (3.1b)

VMD: AZ/R,

6 / 2N+1

N^

(3. Ic)

HMD: AZ/R 3 (Nf ^ 3 1 N+

2 a

2 , , > . r 6 A, 2N+1

6/N) + x^«+— (l^ N

ie"^". (3. Id)

3
For a N < < 1

f 4 ^.^ 3(1+A)D

; ^-A

(3. 2a)

(3. 2b)

VMD: AZ/R-£^ 3(N + 1) f ~ 4 ^.|3(N_i2)D|^-A

(3. 2c)

HMD: AZ/R^^^ 3(N + 1)

where

2. .2 Fj(X)=X+{l-X(l+X)}d + i(l+X)(l-X )d , (3. 3)

F^{X) = - ^ + {1+X(l-X)}d + {1-X(l+X)(2-X)}d^+ i{l + X{l-X){2-X^)}d^ - 1 (1 - X) (1 - X^)^ d^ ,

(3.4)

(3.5)

N - 1 _ N - 1 . , N + 1

For the case of dipoles above a perfectly conducting plane (N—«), equations (3. 1) reduce to the well-known expressions:

r 3 VED: AZ/R^ \

~3 )

^ ^^^^ a - a cos a} + i {cos a + a sin or} (3. 6a) i

HED: AZ/R — ,^ 2a

VMD

f{(l-Qi )sinQ'-Q;cosa}+i{{l-Q;^)coSQ;+asinQ:}J , (3. 6b)

: AZ/R^ — - "T y

^ {sino- - a cos a} + i {cos a+asina}"! , (3.6c)

HMD: AZ/R — - ,^ Za

{(l-ff )sina!-acosQ;}+i{{l-a^)cosa+asinff}J . (3, 6d)

4
4. REFERENCES

fpCing, R. W. P. (1956). The theory of linear antennas, {Harvard University Press, Cambridge,

Mass. ).

'Norton, K. A. (J\ily-Aug. 1959). System loss in radio wave propagation, J. Res. NBS 63D (Radio

Prop.), No. 1, 53-73.

Bommerfeld, A., and F. Renner (1942). Strahlvmgsenergie xind Erdabsorption bei Dipolantennen,

Ann. Phys. 41_, No. 1, 1-36.

fVogler, L. E. (Jan. -Feb. 1963). Point-to-point communication on the moon, J. Res. NBS 67D

(Radio Prop. ), No. 1. 5-21.

Vogler, L, E. , and J. L. Noble (1963). Curves of ground proximity loss for dipole antennas^

NBS Tech. Note 175.

Wait, J. R. (Nov. 1951). The magnetic dipole over the horizontally stratified earth. Can. J. Phys

29, 577-592.

Wait, J. R. (May, 1953). Radiation resistance of a small circular loop in the presence of a conducting ground, J. Appl. Phys. 24, No. 5, 646-649.

Wait, J. R, (1959). Overvoltage research a-nd geophysical applications, (Pergamon Press, New

York, N. Y. ).

Wait, J. R. (July, 1961). The electromagnetic fields of a horizontal dipole in the presence of a conducting half-space, Can. J. Phys. 39, 1017-1028.

Wait, J. R. (Sept. -Oct. 1962). Possible influence of the ionosphere on the impedance "of a groundbased antenna, J. Res. NBS 66D (Radio Prop. ), No. 5, 563-569.

5. APPENDIX

The electromagnetic field of an elementary dipole above a conducting plane (see section 2 for

orientation and location) may be obtained from the electric and magnetic Hertzian potentials, FI e and n , the components of which, if the vectors are referred to a cartesian coordinate system (x, 2 2

z), satisfy the scalar wave equation (V + p ) FI = 0. Using the cfylindrical coordinate form of the wave equation, particular solutions are found by the method of separation of variables, resulting in

, expressions for the n components involving combinations of the eigenfunction

cos nO J (\p) C(\, P) exp J ± J \ -p zl, (A-1)

Iwhere \ denotes the eigenvalue (following Sommerfeld's notation), is a Bessel function of the

first kind and order n, and the C's are functions determined from the boundary conditions. These

conditions, in the present instance^ require that the tangential components of the electric vector E

and the magnetic vector B be continuous at the interface between the two media (air and ground).

In the case of vertical dipoles, E and B may be expressed in terms of Hertzian potentials

having a z -component only. Thus, in the upper mediurn.

If " =n T=(n^ + n^)T, {a-2)

vert z \ z z /

where the superscripts d and r refer to the direct or primary stimulation and the reflected or secondary stimulation, respectively. After solving for the coefficients C through the boundary
conditions, one finds that for

where

VED: n iK

IK

VMD: n

ez p

OO

o o

.N^u - u -(z + Zh)u

u + u J (^/u )d\.

o

o 1 IK

mz p

\ J (\p) e (\/u )dX., J o o

mz p

IK ^ ,u - u

^o o ^ o 1

u - -{z + 2h) u (\/u^)d\.

I 2 F I 2 2 2

(A-3)

(A-4)

{A-5)

p and Pj^ are propagation constants of the upper and lower media, respectively, and the K's are

amplitude factors given by

P I (di) p |j. I (dA)

K = . K = . ° °

°

e 4truj€ m 4-n- {A-6)

For horizontal dipoles, the Hertzian potentials require both an x- and a z-component. Thus,

corresponding to (A-2),

TT, =n "x + n "z = (n^ + n^)x'+n "z". hor X z X X z

After solving for the C's through the boundary conditions, one obtains:

HED: n = -—^ \ J (\p) e °(\/u )d\, ex p J o o o o

(A-7)

iK

i K cos

e

1 V-'le^}

-(z + 2h)u

{\/u )dX,

o

y (A.8)

2{u - u.) , -{z + 2h)u

°{\^/p^^)dK,

o 1

6
HMD: n iK ^« ml , , o.

„ , J (Xp) e (\/u )d\, mx p J o o

iK ^" mx p

^ ^ N u -u -(z + 2h)u

(X/u )d\,

o

iK cos 9 ^« m r j(xp){^^|

^ In U +n J

2{u -u,K -(z + 2h)u

o 1

(A.9)

where u , u,, N , K , and K are defined in (A-5) and (A-6).

o 1 e m

With the Hertz vector IT determined, the electromagnetic field in the upper medium is derived through the following relationships:

E = grad div + p^^ 11^ , (electric dipoles)

B = grad div + p^^ ^rn*

(magnetic dipoles)

(A-lOa)

(A-lOb)

The components (in cylindrical coordinates) of interest in this paper are found to be

VED: E (p, z) = ( a^/9z^ + p

^

) n = ( 9^/az^ + p

^

) fn*^ ) z \ o / ez V o / V ez ez/

IK

P J o o o

-z u , N u - u, - -(z + 2h) u , oil o N u + u J o 1

{\ /u )d\,

o (A-1 la)

HED: E (p, z) = (a/9p) div n +p cos 9 n

p 6 O

iK cos 9 e

+ ^P^^JJ\p)-\^^ JJ\p)

J^(\p). N ,u^-u^. -(z + 2h)u^

xp \u^+u^

/ J ().p)N 2\^u^u -u^) -(z + 2h)u

^ >-B<Nu+u. P^^N^^+u^)W J (A-1 lb)

/^2, 2 Z\ / 2, 2 2\/d „r \ VMD: B (p,z) = ( a /az +p )n = ( a /az +p jin + n ) z V o / mz \ o / \ mz mz /

iK -zu u -u -(z + 2h)u

)d\, (A-llc)

7
HMD: B (p, z) = (9/ap) div n +B cos 9 n

p m o mx

iK cos e

' C

2 zr Ji(^p).\

2 2r "^i^^P' ~i \ r ^ ^r^"^i ^ -(z + 2h)u

N u +u,

o 1

2 2.

J, Xp)\ . 2\ u u -u K - z + 2h u ^

)J L e (\/u )d\, (A-lld)

To calculate the dipole input impedance using the emf method [King, 1956 (p. 258); Wait,

1962 ] requires a knowledge of the axial field at the antenna. This may be obtained by taking the limits of (A- 11) as p and z approach zero (and with 0 = 0 in the case of horizontal dipoles). The

impedance Z is then given by

VED: Z = Lim Je (p, z) di/I

p^o 1 z o

iK di

e N u - u. -2h u

o "^o L l-N u +u

o 1

° ' (X^/u^)d\, {A-12a)

HED: Z = LimiE (p,z) di/I L p-o I P o j

iK di - 1 i^i-^'i^y*

O O O L

u - u, ^ -2h u oil o

, e u +u,

o 1

2 2 X u (u - u ) ^ -2hu

+ ^ °/ ^ \ e °1 (X/u )dX.

o or

(A-lZb)

VMD: Z = Lim j coB (p, z)dA/I

p->o

icjK d A m

u -u ^ -2hu

o qL ^ O \ J J (\^/u )dX.

o (A-12c)
HMD: Z= Lim^ wB (p,z)dA/I

p->o z—©

P P

2

oo oL ''Nu+uJ o 1

2 2

, \ u (u - u ) , -2hu

+ ^ ^ °^

°

I e ° (\/u ) dX. (A-12d)

3 (N u +uj J J °

o o 1

Notice that the effect of the ground is contained in terms involving the factor exp(- 2hu ); thus, the

o change in impedance AZ due to the presence of the grovmd may be defined as

AZ s Z - Lim Z = Z - {A-13)

h—00

and the free space antenna resistance is obtained from the relationship:

R = Re ^ Lim Z I . (A-14)

With the use of (A-13) and {A-14), one can now derive the expressions for the normalized

input impedance change AZ/R^ given by equations (2. 2) of section 2; here, the change of variable

, 2 2 X = 2hu - 2h\ X - (3 o ^ "^o

has been made and the following relationships used:

{\d\/u ) = {dx/2h), a ^ Zh^ , u = \|x^ - q;^(N^ - l)/2h, \^ = (a^ +x^)/(2h)^ . (A-15)

O O 1 '

9

6. GRAPHS OF NORMALIZED INPUT IMPEDANCE CHANGE, AZ/R

for

Vertical Electric Dipole (VED)

Horizontal Electric Dipole (HED)

Vertical Magnetic Dipole (VMD)

Horizontal Magnetic Dipole (HMD)

r 2

€ r + s J

s = 60 \ (r(mhos/m), a = {2h) (2-n-/\),

e : relative dielectric constant of ground

r "

o-(mhos/m): conductivity of ground

h: height in meters of antenna above ground

k: wavelength in meters

11
12


Figure 4,

15
16
Figure 6.

17
Figure 7»

18
19
20

Figure 11.

22
Figure 12.

23
Figure 13.

24
25

Figure 16.

27

29
30

32
33



Figure 26o

37
Figure 27.

I; 38
HED. I N j = 10

01 0 2 0.5 0.7 I 2 5 7 10 20 50 70 100

Figure 28.

39
40
41
42
Figure 32,

43

B- 401081
44
1.6

1.4

-(f)

Figure 34„

45
46


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