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NBSIR 79-1605 A111D4 514065

A THEORETICAL STUDY OF UNBALANCED GROUND EFFECTS

ON RECEIVING DIPOLES

M. T. Ma

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

May 1979

“ QC

100

,U56

79-1605

Mvr$r*

NBSIR 79-1605 I • •

A THEORETICAL STUDY OF UNBALANCED GROUND EFFECTS

ON RECEIVING DIPOLES

M. T. Ma

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

May 1979

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

.1. INTRODUCTION 1

2. CHANGES IN FIELD STRENGTH CAUSED BY PLANE IMPERFECT GROUND 1

3. GROUND EFFECTS DUE TO POSITIONAL IMBALANCE OF THE RECEIVING POLE 4

4. VARIOUS CAPACITANCES BETWEEN THE DIPOLE AND GROUND/TRANSMISSION LINE 9

5. CONCLUDING REMARKS 11

6. ACKNOWLEDGEMENT 11

7. REFERENCES 11

i i i

A THEORETICAL STUDY OF UNBALANCED GROUND EFFECTS

ON RECEIVING DIPOLES

M. T. Ma

Balanced ground effects on the performance of some antenna systems are relatively well known and can be taken into account by the design engineer. Unbalanced ground effects on a measuring system are, however, more complicated and make a thorough understanding difficult.

In this report, specific ground effects on the calibration of a dipole antenna with an arbitrary inclination angle with respect to the ground are analyzed by means of a theoretical

model. Numerical results representing this undesired effect are also included.

Key Words: Dipole antenna; ground effect; unbalanced system.

1. INTRODUCTION

Characteri zation of an electromagnetic environment at a given point in space can best be accomplished by systematic measurements with a broadband antenna. One of the antenna systems may

consist of three orthogonal short dipoles. Another may be one with three orthogonal loop antennas. To measure the incoming field with different polarizations, the mutually orthogonal antennas are usually

arranged in such a manner that they are neither parallel nor perpendicular to the ground plane. Because of this positional inclination of the measuring dipole itself with respect to the ground, the

two halves of the dipole will experience a different ground effect, giving different results at different measurement heights. This, in turn, causes difficulty in calibrating the antenna.

In the actual measurement program, we used a known vertical antenna installed above a ground

system to generate the transmitting field, and employed both the orthogonal dipoles and loops for measurement. The frequency range of interest is about 10 kHz to 100 MHz. Spot measurement results

showed a difference of as large as three decibels in received signal levels for the dipole, when the measurement height was changed from 1 m to 2 m above the ground. No significant changes in measurement

results were observed, however, by the loop antenna for the lower end of frequency range. This

discrepency in actual measurement results by the inclined dipole prompted the study contained in this

report.

Another mechanism that may cause different signal levels at different measurement heights is a change in the transmitted field strength by the plane imperfect ground even though a ground system has been provided for the transmitting antenna.

Theoretical models for each of the above mechanisms and some numerical results are presented

here.

2. CHANGES IN FIELD STRENGTH CAUSED BY PLANE IMPERFECT GROUND

Because the ground plane on which a transmitting antenna is erected may not be perfect, the fielci strength radiation pattern in the vertical plane near the ground surface is quite different from that transmitted by an antenna over a perfect ground. The space-wave radiation pattern of a base- fed vertical linear antenna over a flat earth, as shown in figure 1, may be mathematically expressed

as [1]

E

e

' [A + j B * V*

JB,] - v/"' a)
where

A = cos(kh cose) - cos kh,

B = sin(kh cose) - cose sin kh,

C = a constant,

2 1/2 cose - ( k/k

1 ) [1 - (sine k/k * ) ] R

« "

. 1/2 ' cose + (k/k') [1 - (sine k/k') ]

k' 5 k[e r - j 18 o ( lo3 )/fMHz

]1/2

>

f

= the operating frequency in MHz, MHz

k = 2tt/X,

X = the operating free space wavelength in m,

e r= the dielectric constant of the earth,

a = the earth conductivity in mho/m,

h = the antenna height in m, and

e = the zenith angle measured from the z axis in degrees.

z

Figure 1. A base-fed vertical monopole.

Note that in eq (1) we have assumed a simple sinusoidal distribution for the antenna current,

also that at the ground surface (9 = 90°), we have B = 0, R v = -1, and Eg = 0. A typical space-wave pattern over an imperfect earth is also shown in figure 1.

Note

2
Mow, suppose we have a dipole receiving antenna with the same polarization installed at a distance

from the transmitting antenna as depicted in figure 2. Naturally, when the receiving antenna terminal

Transmitting Antenna Receiving Antenna

Figure 2. A pair of transmsitting and receiving antennas.

is right at the earth surface (H = 0 and 9 = 90°), the measurement reading should be zero in view of

the reason given above, assuming no surface-wave field exists at the measuring site. On the other

hand, when H is finite and small, the angle at which the receiving antenna is used for measurement may

be approximated by:

9 = 90° - tan ^

( H/d ) = 90° (1 - ~) degrees, (2)

TT

where H is the height in meters of the receiving antenna, d is the distance in meters between the transmitting and receiving antennas, and x = H/d< <1

.

Under the above condition, sin9 = 1, cos9 = sin(H/d) = x. In addition, if kh = tt/ 2 (a quarter-wave monopole for the transmitting antenna), we obtain

A = 1 and B = (~ - 1) cose = (~ - l)x.

The field strength at the angle given in eq (2) may be calculated in accordance with eq (1). For

example, if the earth is relatively poor (e.g., e r

= 4 and a = 0.001 mho/m) and f^z

= 30, we

have

1/2

k’/k = (4 - j 0.6) ,

Arg(k'k) = 0 ,

2 1/2

(k/k ' ) [1 - (sin9 k/k ' ) ] = 0.432 + j 0.022 a y,

=

x - y

v x + y’

and

E

q (H) = C[ (1 + R

y )A + j (1 - R

v )B]

= C(2 + jl.l42y)x/(x + y) v/m.

When the height of the receiving antenna is raised to 2H, a similar analysis follows, oro\’.vc

the parameter x in eq (3) is replaced by 2x, where 2x = 2H/d<<l. Then the field strength "eas. vc

the receiving antenna should read:

3
E (4)

q

( 2H) = C(2 + jl.l42y)2x/(2x + y), v/m.

Since x<<y, we obtain

E (2H)/E (H) = 2 n, 6 dB . (5)

W u

On the other hand, when the earth conductivity is very good, such as e r

= 80 and a

= 5 mho/m, we

then have, for the same frequency of 30 MHz,

1/2

k'/k = (80 - j 3000)

'

,

Arg(kVk) = -45° ,

1/2

( k/k 1 ) [1 - sine k/k 1 ) ] = 0.013(1 + j) = (1 + j)y.

R X - y ~ Jy -1 + xy'

1 - jxy1

, v x + y + jy

E (H) = C[y

1 + j (tt - 2 - y] )]x , v/m. (6)

u

When the antenna height is replaced by 2H, we again obtain the same result as that given in eq (5).

This implies that the field strength measured by a receiving antenna at a height of 2H meters above an imperfect earth (a is finite) is approximately 6 decibels more than that measured by the same antenna

at one-half of the previous height, when the distance to the transmitter is very much greater than H. This difference in measurement results is solely due to changes in the transmitted field strength by the imperfect earth.

When a perfect ground is available, such as the case of installing a very good ground screen on the earth surface, we have a->- °°

, R

v ->-+1, and Eq(H) = 2C cos(khx), for kh = tt/2. Under this

ideal condition,

E (2H) = 2C cos(2khx) = E (H), x«l, (7)

0 0

which implies that the measurement results at two different heights should remain essentially the

same.

3. GROUND EFFECTS DUE TO POSITIONAL IMBALANCE OF THE RECEIVING DIPOLE

When a dipole receiving antenna is not oriented parallel to the ground surface, such as that shown

in figure 3, different ground effects on the lower and upper halves of the dipole will occur, yielding

an unbalanced system. We use a capacitance model for the analysis. The capacitances between the lower

part of the dipole and the ground/transmission line are different from the corresponding ones between

the upper part of the dipole and the ground/transmission line. The unequal ground effects will result

in different measurement data at different measurement heights, even though the transmitted field

strengths at these heights have the same level (such as those near a perfect ground), which in turn

will cause some difficulty in calibrating the receiving antenna.

4
h

Figure 3. An unbalanced dipole antenna above a perfect ground with vertical connecting transmission

line.

The severity of this problem of positional imbalance may be analyzed with the preliminary model network as shown in figure 4. Notations in figure 4 are explained as follows:

Figure 4. A model network representing the dipole antenna given in figure 3.

Z = input impedance of the dipole antenna [2]

a

= R

, - jX , ohms, (8)

a a

where

R

a = 80m2(h/X)2,

X

a = 120[Jln(h/a) - 1] cot kh,

h = half length of the dipole in m.

5
and

a = radius of the dipole in m;

V} = voltage induced at the dipole terminal by an incoming field

= Eq h e sin volts.

where

Eq = incoming field strength in v/m, ( 9 )

and

where

h e

= effective length of the dipole

= h for a very short dipole (h < 0.1X);

1\ = an RC divider network of the differential amplifier connected to the dipole

= Ri / (1 + j coR i C i ) = -j (1/ooCi) ohms, (10)

Rl = 3(10®) ohms and = 50 pf, for the frequency range between 10 kHz and 30 MHz;

I'l = impedance per unit length of the transmission line

= r 2/(1 + JC0R2C2 ) ohms/m, (11)

R2 = 8,000 ohms/ft = 26,250 ohms/m and C2 = 5 pf/m;

V2 = voltage induced at the terminal of the transmission line by the same incoming field

= E

0 H/2 , volts, (12)

where H is the length of the transmission line in meters, which is also the measurement height;

Z3 = impedance between the lower half of the dipole and the ground, consisting of the self-capacitance of the half dipole wire, the mutual capacitance between the lower half

dipole and its ground image, and the capacitance between the same half dipole and the

transmission line

where

-j (I/C0C3) , ohms. (13)

with C3 representing the total capacitance in farad;

and

Z4 = impedance between the upper half dipole and the ground, consisting of three similar

capacitances

= - j ( 1 /00C4 ) , ohms. (14)

Note that it is not easy to determine the values for C3 and C4 with a reasonable accuracy, although we know that C3 > C4.

Solving for the three loop currents in figure 4, we can express the voltage drop between points A and B to represent the actual measurement reading:

6
where

4 ' VZ

1 * ?Z1 Z

2 + Z

1 Z

3 + ¥4 + Z2

Z

3 * Z2

Z 4

+ Z

3

Z4>

+ Z

1 (Z1 Z3 + Z1 Z4 + ZZ2

Z

3 + ZZ2

Z4 + 2Z

3

Z4»-

When the measuring antenna is a balanced system such as a horizontal dipole, we have

C 3 = C4 , Z3 = Z4 ; and, according to eq (15), v/\g does not depend on V2

- In other words, no matter what the measurement height H is, v B does not play a role. The

measurement results at two different heights for the same incoming field with the same polarization should be the same.

For an inclined dipole, Z 3

j- Z4

. The measurement result of vAB will depend

on V£ and therefore on H. A numerical example is presented below in order to examine

quantitatively the dependence of vAB on H for a given angle of inclination >1 . In addition to the parameters given in eqs (10) and (11), let us also assume C 3 = 5 pf and

C4 = 4 pf. Then at f = 10 kHz, we have

Z a = -j4. 76(10®), assuming h/a = 25 and h = 0.27 m (10.5 in),

Zi = -jO. 32(106),

Z2 = R2 = 0.026(106) with H = 1 m,

Z 3

= -j3.18(106),

Z4

= -j3.98(106),

V3 = 0.15 volt, assuming Eg = 1 v/m and 'P = 35.2°,

V2 = 0.5 volt with H = 1 m,

and, in accordance with eq (15),

vAB (lm) = 0.016814 + 0.007498 = 0.024312 volt. (16)

If C 3 and C4 decrease respectively to 4.55 pf and 3.64 pf when H = 2m, we have

Z 3 = -j3. 50(106),

Z4 = —j 4.38(1 06 ) 3

and

vAB (2m) = 0.016927 + 0.013833 = 0.030760 volt. (17)

Therefore, we obtain

VAB^

2m ^ /VAB^ lm ^ = 2-04 dB "

Comparing eqs (16) and (17), we note that, when the measurement height is doubled, the — '>

which is the response due to V 3 does not change much while the second term due to positiona

imbalance changes substantially. Obviously, the ratio vAg(2m)/vAg(lm) , depends on

Z 3 (2m)/Z 3 (lm), Z4 (lm)/Z 3 (lm), and Z 4 (2m)/Z 3 (2m) . Specific results as functions of these

parameters are given in figure 5.
Z 4 (2m)/Z 3 (2m)

Figure 5. A functional relationship between v/\g(2m)/v/\[3(lm) and Z4(2m)/Z3(2m) when

Z4 (lm) = 1.25Z3(lm) and f = 10 kHz.

Note that in the above example Z 2 is very small relative to Z 3 and Z4 , making Z3

Z4 the dominating term in eq (15). When the transmission line resistance per unit length is increased to Rg

= 64,000 ohms/ft achievable by a higher resistance line, we then have Z2

s R 2 = 0 . 21 ( 10 ®) ohms for H = lm, which is still insignificant as compared with Z

3 and Z4

. Therefore, the results to be obtained for VAB(2m)/vAB(lm) will essentially be the same as those presented in figure 5. Furthermore, if the operating frequency is increased to 30 MHz, making 0JR 2

C 2

= 24.74»1,

Z 2 = -j ( 1 /

C

0C 2 ) = -j 1061 , and at the same time, 1\ = - j 106, Z3

= - j 1 061 , Z4

= - j 1362.25 , and

Z a = - j 1 574 , we have

v (lm) = 0.016937 + 0.002825 = 0.019762 volt. (18) AB

When H = 2m, Zg(2m) = 1 . 10 Z 3 (lm) = - jl 166 , and Z4 (2m) = 1.25Z3(2m). Under this condition,

v (2m) = 0.017048 + 0.005498 = 0.022546 volt, (19) AB

or

vAB

( 2 ni )/ VAB^

ln1 ^ = 1,14 dB ’

which is smaller than the corresponding value of 2.04 dB obtained previously for f - 10 kHz.
From the above example and many other numerical results which have not been included here, we may

concl ude:

(1) that the transmission line impedance Z

2 itself is not likely to cause any significant change

in vAB for different measurement heights, except that the transmission line with higher

resistance per unit length may reduce the capacitance between itself and the dipole,

(2) that a higher frequency makes a smaller change in v/\g than a lower frequency does for the same structure,

and

(3)

that the basic reason for a substantial change in v/\ B is mainly due to a change in the ratio of Z4/Z3 versus measurement height H.

In view of this conclusion, a closer study of the possible range for the capacitances shown in figure 3 (therefore, for the ratio of Z4/Z3) is given in the following section.

4. VARIOUS CAPACITANCES BETWEEN THE DIPOLE AND GROUND/TRANSMISSION LINE

For the convenience of presentation, let us designate the self-capacitance of the half-dipole wire

by C0 , the mutual capacitance between the lower half dipole and its ground image by Cg , and the capacitance between the same half dipole and the transmission line by C^ (see fig. 3). The total capacitance for the lower half dipole at H = lm may be written as

C Mm) = C + C + C . (20)

3 0 g t

Similarly, the corresponding total capacitance for the upper half dipole is

C, (lm) = C + C' + C‘ (21) 4 0 g t

where

Cg = the mutual capacitance between the upper half dipole and its ground image (which should be less than Cg)

,

C-f- = the capacitance between the upper half dipole and the transmission line (which should be

1 ess than C^-)

.

Because the inclination angle of the dipole with respect to ground is approximately 35.2° and the ground image capacitance is inversely proportional to the separation, we may express:

C'/C = (1 - 0.5 x 10.5 x 2.54 x 0.01 sin 35 .2°) / ( 1 + 0.5 x 10.5 x 2.54 x 0.01 sin 35.2°) = 0.857

or

C' = 0.857 C . (22)

g g

In eq (22), we have used 10.5 inches for the length of half dipole.

An approximate relation between C^ and Cl is, however, not available. Using eqs (20'. CM.

and (22), we obtain

9
( 23 )

Z (2m) C + C + C _3 o g t

Z (lm) ” C + 0.480C + C

3 o 9 t

and

Z (lm) C + C + C 4

V _

o g t

Z 0( lm) C + 0.857C + C'

' 3 o g t

Setting values for Z 3 ( 2 m)/Z 3 (lm) and Z 4 (lm)/Z 3 (lm) , we nay solve for Cg/C 0 and

Ct /C 0 with an assumed range of values for C{/Ct. Numerical results on this are given in table 1. In this table, we have chosen a range of 1.10 to 1.25 for Z3 ( 2m)/Z 3 (lm) and a range of 1.15 to 1.25 for Z 4 (lm)/Z 3 (lm) such that the values obtained for Cg/C 0 and /C 0 have a reasonable range of 0.3 to 0.9. Otherwise, either negative values or values greater than unity for

Cg/C 0 and Ct/C 0 will result, which are certainly not reasonable.

We also include the results of v^B(2m)/v/\B(lm) and VAB(10m)/v/\B(lm) in the table,

where the assumptions that V 2 ( 2 m) = 2 v 2 (lm) and V 2 ( 10m) = 10 v 2 (lm) have been made. It is clear

from this table that in order to keep the change of v/\g due to variation in measurement heights to

a minimum, we should require that Z 3 ( 2 m)/Z 3 (lm) be as large as possible and that Z4 (lm)/Z 3 (lm) be as small as possible. To achieve the latter objective, it may be feasible to insert some dielectric

material between the upper half dipole and the ground for the purpose of increasing C4 (decreasing

Z4 ) to reduce the ratio of Z4 (lm)/Z 3 (lm) . Any other means of reducing V2 such that

V2 (nm) < nv2 (lm) will also help, by reducing the contribution of the second term in eq (15).

Table 1. Realistic ranges for various capacitances.

Z

3 (2m)/Z

3 (lm) Z4 (lm)/Z3 (lm) ,ab

( 2" )/vab ( 1 ">- dB V AB^

10m ^ /VAB^ lm ^ dB *

1.25 2.92 13.51

1.10 1.20 2.58 12.56 1.15 2.12 11.24

1.25 2.64 12.92

1.20 1.20 2.23 11.80

1.15 1.69 10.11

1.25 2.48 12.64

1.25 1.20 2.07 11.41 1.15 1.48 9.52

*In the table above, we have used the following ratios, similar to eqs (23) and (24):

Z_ (10m) C + C + C _3 0 g t Z. (lm) C + 0.093C + C

3 0 g t

Z. (10m) C + C + C 4 0 q t

Z (lm) ” C + 0.092C + C'

3 0 g t

A final remark concerning possible use of a transmission line with a higher resistance per unit

length is in order. As we noted earlier, a higher value of Z 2 should not exert much effect on

v/\b» However, when the resistance per unit length of the transmission line is higher, the capacitance between the line and the half-dipole wire may be substantially reduced. In terms of the

10
notations used, the parameters C t and in eqs (23) and (24) will be smaller, which makes

Z3(2m)/Z3(lm) increase and Z4 (lm)/Z 3 (lm) decrease. These changes are in the right direction to

cause a lesser variation in v/\g. As an example, when Z3(2m)/Z3(lm) = 1.10 and

Z4 (lm)/Z 3 (lm) = 1.25 corresponding to the result of the first line given in table 1, where

Cg/C 0 = 6.304, Ct/C 0

= 0.434, and C^/C ^ = 0.30. If Ct/C 0 is reduced to, say, 0.25

with Cg/C 0 and C-fYC-t unchanged, we obtain Z3(2m)/Z3(lm) = 1.113 and

Z4 (lm)/Z 3 (lm) = 1.164, which are respectively larger and smaller than the values for 1 ow-resistance

transmission line.

Thus, from the consideration of minimizing the unbalanced ground effect, we should use a transmission line with the highest possible resistance per unit length. However, from the practicality

and power supply viewpoints, there is a limitation in this regard. The transmission line with 210,000

ohms/m (64,000 ohms/ft) used in the above illustrative example is probably the limit we can use in this

particular application.

5.

CONCLUDING REMARKS

In this report, we have examined unbalanced ground effects on the performance of a receiving

dipole through a parametric study. Specifically, the voltage drop across the terminal of the receiving

antenna was considered under two different situations. In the first, a variation in the transmitted

field strength due to an imperfect earth was analyzed while the effect due to positional imbalance of

the receiving dipole was neglected. In the second situation, ground effects resulting from the

unbalanced position of the receiving dipole were studied through a theoretical model by assuming no variation in the transmitted field strength. In reality, changes in measurement results with different

antenna heights may be a combined consequence of both. More systematic measurements are required to

help identify the degree of dependence on each of these causes. If the second cause is identified as the real reason for variations in measurement results, possible means for reducing the unbalanced

ground effect on the receiving system being examined were also suggested.

6.

ACKNOHLEDGMENT

The author wishes to express his appreciation to C. Reeve for his help and consultation throughout

this study and to M. Kanda and C. K. S. Miller for reviewing the draft manuscript.

7.

REFERENCES

[1] Ma, M. T., Theory and Application of Antenna Arrays, p. 280 (John Uiley and Sons, New York, N.Y.,

1974).

[2] Jordan, E. C., Electromagnetic Waves and Radiating Systems, p. 312 and p. 512 (Prentice-Ha’ , Englewood Cliffs, N.J., 1950).

11
WBS-114A (REV. 0-78)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 79-1605

3.

4. TITLE AND SUBTITLE

A Theoretical Study of Unbalanced Ground

Effects on Receiving Dipoles

5. Publication Date May 1979

7. AUTHOR(S)

M. T. Ma

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, state, ZIP) 13. Type of Report & Period Covered

15. SUPPLEMENTARY NOTES

| | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or

literature survey, mention it here.) Balanced ground effects on the performance of some antenna systems are relatively well known and can be taken into account by the design engineer.

Unbalanced ground effects on a measuring system are, however, more complicated

and make a thorough understanding difficult. In this report, specific ground

effects on the calibration of a dipole antenna with an arbitrary inclination

angle with respect to the ground are analyzed by means of a theoretical model.

Numerical results representing this undesired effect are also included.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name

separated by semicolons)

Key Words: Dipole antenna; ground effect; unbalanced system.

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