I

A Simple Technique for Obtaining the Near Fieldsof Electric Dipole Antenm+s frorn Their Far Fields

Bharadwaja K. Singaraju.. -.;- Carl E. Baum

Air Force Weapons Laboratory

... ..... . .... ..: ,,:,. .:i.’:,, ;..!’ .

Pi/i?~ +;5/97

Abstract

In this report a technique of obtaining the near fields of axially andlengthwise symmetric electric dipole antennas from their far fields isdiscussed. .In particular it is shown that if the 6 component of the farelectric field is known either in time or frequency domains, all otherelectric and magnetic field components including their near fields can beobtained very easily. An example involting the fields of a resistivelyloaded dipole antenna driven by a step function generator is discussed indetail.

L Introduction

In the calculations of fields due to simple current distributions such

as involving electric or magnetic dipoles, the far electric or magnetic field

have a very simple behavior. In general these fields behave like the radiated

fields and are proportional to 1/r, r being the distance from some reference

point on the antenna to the observation point. Because of the ease with which

the far fields can be obtained and also because of their applications in most

linear antenna problems, far field representation in general is sufficient.

However, in cases such as large vertically polarized dipole simulators used

in the testing of aircraft, etc. , the observation point in general cannot be

considered to be in the far field of the antenna (simulator). At most fre -

quencies of interest, and within useful distances from the simulator, near

field components contribute to varying degrees with their contribution at the

Iow frequencies being predominant. As such, a simple way of calculating

the near fields of such structures was desirable. This note deals with one

such simple technique. An example involving an impedance loaded conical o

dipole over a ground plane is discussed in detail.

2

@II. Electric and Magnetic Dipole Moments and Their Far Fields

Consider a volume Vt bounded by a closed surface S!. The coordinate

system is as described in figure 2.1. Let us assume that the volume Vf

contains electric” current density 7(J’ ) and charge density ;(~’ )0 Here a

tilde (-) over the quantity represents the bilateral Laplace transformed

quantity. Assuming that the charge and current densities are zero before

time t=to and requiring no current density to pass through S!, the electric

dipole moment can be defined as1,2

;(t) z 1 ;I p(;}, t)dV’

v!

or in terms of Laplace transformed quantities as

(2. 1.)

e Notice that ~(t) is a time-dependent dipole moment while ~

dependent. Equivalently, in terms of the current density

is frequency

(2. 3)

s being complex radian frequency. Similarly, the magnetic dipole moment+ 2m can be written as

in the time domain and

-+ 1m=-—

4

in the Laplace

o can be written

domain. In the low frequency limit these dipole moments

as

(2.4)

(2.$}

3

a Position vector

.+Is

outward normal

Position vector tothe observation point

~1: surface boundingthe sources

volume containingthe sources

●

✎

v

.

Figure 2.1. Coordinate system describing the source and radiation region

4

*

J 4-+;= rlp(r!)dV[ = $

J-~(:’)dVl (2.6)

VI v!-

where ~(;’ ) and ~(;’ ) are charge and current densities respectively. The

magnetic dipole moment can be written as

. +

Low frequency

then be written as

far fields due to electric and magnetic dipoles 1 can

@+

where lr is a unit vector in the radial direction and

No = free space permeability

e = free space permittivityo

c= 1/~ = speed of light in free space

ruZ. =

sYc n—

u— = 377Q = characteristic impedance of free spaceeo

= complex propagation constant

(2. 7)

(2.9)

(2. 10)

It is clear that if the electric or magnetic fields due to an electric or

magnetic dipole moment are known, other components can be simply cal -

culated using the complementary and far field relations of the electric and

magnetic quantities.

If we consider an electric dipole source term alone, electric and

magnetic fields due to an electric dipole~ with 1/r2 and 1/r3 terms included

9 can be written as

From the above equations, a general relationship exists

1/r term to the 1/r2 and 1/r3 terms. This relationship

(2.11)

(2. 12).

which relates the .

can be thought of

as a frequency scaling along with a simple proportionality constant. It is

clear that in the case of the electric field, the 1/r3 type field is more.

important at low frequencies. If ?r is perpendicular to ~+certain simpli-

fications result in (2.1 1).

If we consider a magnetic dipole instead of an electric dipole, its

fields can be obtained by setting *

++

P~~ (2. 13)

ip(~) - zo~m(x) (2. 14)

++Hp(r) _ - + Em(;) (2.15)

o

in (2. 11) and (2. 12).

A general source distribution may contain electric and magnetic *

dipoles along with their higher order counterparts. Relationships similar

to (2. 11) and (2. 12) may be found for multipole terms. To include these

multipole terms, one may expand the Green Ts function in series involving

spherical Bessel and Hankel functions and spherical harmonics. The order

of these functions is related to the order of the multipole under considers- :!

tion. This procedure is quite involved and is considered outside the scope

.

of this report.

o III. Expansion of the Fields in Spherical and Cylindrical Coordinates

—In practical applications it is convenient to express the electric and

magnetic fields in terms of their vector components in spherical or cylin-

drical components. Let us consider the special case of the electric current

density symmetrically situated as shown in figure 3.1. such that only a z

directed dipole moment is present. In terms f spherical coordinates theY

electric dipole moment can be written as

(3.1)

Using this representation, the electric field given by (2. 11) can be written

as

Note that the radial electric field does not have a 1/r component. Suppose

that

this

the e component of the far electric field is known, let us represent

by ~fo as

s2pEf ( ()

- -yr= -& sinepe )

e

then the 1/r2 component ~20 can be w:ritten as

ZosE2 ( ()

--yr =&& .~~= - sin flpe )

e 4m2sr f

eyr f~

3while 1/r component can be written as

(3. 3)

(3.4)

7

h

X,X1

r

Figure 3.1. Axially and lengthwise symmetric dipole antenna

8

.

1: =——

--yr_l*30

4~r3_c=_= .= -___ .= -=_=( () )sint)pe - --; E2 = ~E

22fo -=- !Q 7? 8

Representing the total 0 component of the electric field by ~. , it can be

written as

Hence in the dipole approximation of an antenna, if the 6 component of

the far electric field is known, the total 0 component of the electric field

near the antenna can be calculated with the help of (3. 6).

In a similar fashion, the 1/r2 part of the radial electric field can

be written as

ZosE2 = --y (cos(@)~e

. _ 2cot(e)~“l’r) n 2cot(6)E2 - -yr

2?rrf.

r e

(3.5)

(3.6)

(3. 7)

awhile the 1/r 3 component is given by

E3 = —A(cos(0)pe- 2f20t(e)~- ‘~r) . 2cot(e)E3 =

213 ‘fe

(3.8)r o e y2r2

and the 1/r component is zero. Hence the total radial electric component.

of the electric field Er can be written as

E ( 1= 2c0t(e) *+— )Ef

ry2r2 e

The only non-zero component of the magnetic field fi$ is given by

(3. 9)

(3. lo)

and.

.+ (1+‘4 o

# Efe

(3* 12)

Note that the radial electric component falls off at a faster rate than

the 6 component. Not e also that the @component of the far electric field is

sufficient to reconstruct the total radial and @components of the electric

field and the @ component of the magnetic field.

Let us now consider the case when the 0 component of the far electric

field is known in the time domain. Using basic principles of the Laplace

transform we can write

,

.

(3.13)

(3.15)

A.s should be expected, if the @component of the far electric field is known

in time domain, a simple process of integration yields the total near fields.

If Er and E ~ are known, z, fI and ~ components of the electric and

magnetic fields in cylindrical coordinates can be written as+

Ez(~) = Er(;)cos (e) - E6(F)Sin(O)

Ep(;) = Er(~sin(6) + Ee(I)cos (0)

(3. 16)

(3. 17)

10

(3.18)

In the case of dipole type antennas it is generally expedient to calculate

the far fields. Using the procedure described above, the near fields can be

very easily obtained. If the far fields are measured, it suffices to make one

good measurement namely Efa(;) either in time or frequency domain. Using

this measurement,_ the n.ea.r.f~eld calculation becorn.gs a trivial exercise.

In this connection it should be pointed out that this procedure is accurate

only if r >> h, i. e., the observation point is far compared to the antenna

height .’ For distances of the order of h, one may have to include higher

order multipole terms in the expansions for the field. In the high frequency

regime, the reconstruction of the field is not accurate, however, the errors

are small.

11

N. Calculation of the Near Fields of an Impedance Loaded DipoleAntenna from its Far Field

Consider a typical axially and lengthwise symmetric dipole antenna

as shown in figure 3.1. The coordinate system is as shown. The impedance

loading on the antenna is assumed to be A(z’ ) per unit length.3-7

In our present analysis of the dipole antenna we also use the trans -

mission line model for calculating the current distribution on the antenna.

We assume that the generator feeding the antenna has a generator capacitance

Cg and a voltage source VoU(t) in the time domain or Vo/s in the frequency

domain. Here U(t) is the unit step while s is the complex radian frequency.

In the transmission line model, the antenna in figure 3.1 can be approxi-

mated as shown in figures 4.1 and 4.2. Elements of the incremental section

of the equivalent transmission line are related to the characteristic impedance

of the bicone. If the bicone has cones at # = 01 and 0 = ~ - @l , the charac-

teristic impedance Z ~ is given by

Z. o

z03

= —ln[cot(el /2)]7

(4. 1)

The geometric factor fg is defined as

zf

m~_g ZO

= *In lcot (@1/2)] (4.2)

for a biconical antenna. If the biconical antenna is of half height h and 01

radius a << h, then the antenna can be considered to be thin and the geo -

metric factor f can beg

approximated

where the mean radius of the antenna

as

(4. 3)

can be used. Assuming that the

biconical antenna is in free space, parameters of the incremental section

of the equivalent transmission line are given by

12

.—

e

- + I

OpenCircuit

Generator I TransmissionI

ILine

Figure 4.1. Transmission line with generator

‘-r~---—.

J?igure 4.2. Incremental section of the transmission line

13

~1 . /JfOg

c! ‘c/fOg

and

Zf(f) = a~(~)

We define normalized retarded time Th as

_ct-r7 _—‘h

where

h=

r=

t =

‘h =

h

half height of the antenna

distance to the observer from the center of the antenna ,

time measured in seconds

h/c

We also define

‘h

c!

Cac

~

sh=— = normalized radian frequencyc

c=1+$

g

= antenna capacitance

= generator capacitance

(4.4)

(4. 5)

(4. 6)

(4.7)

Using transmission line equations the current on the transmission

line can be calculated, this current distribution can be used as the current

on the antenna. For a special impedance loading of the form

A(F) = w-w(4.8)

Baum5 has calculated the 6 component of the far electric field in frequency

domain to be

●

14

(4. 9)

d

where

rqe) . sine;-2_

(Sh + CY) sin (0) shsin4 (e)

[

-Sh(l-cos(e)) -Sh(l+cos(e))

cos2Fj e ()+ sin2 ~ e

+2sh sin4 (6)

1]

(40 10)in the frequency domain.

It has been found very cliffic ult to cbtain the near fields of this antenna

analytically. However, if we use the technique discussed in section 111,the

anear fields can be obtained very easily. Considering frequency domain

first, we can write the 0 component of the electric field as

while

[

VothE3 = —

e2?i-f

g

-yorh2 e——

r2r 1+F\(e)

‘h.

[1Vt-yor

oh h2e=2?i-f 5Y [1

Y; (e)gr

(4. 11)

(4. 12)

-. —

15

. .

We note that the normalization factors ~! ( 6), ?;(6) and ~~(e) are all functions

of normalized frequency, @and a only. These are plotted for several values

of a by varying (3and setting sh

= jwh. Notice that at low frequencies ~~( 0)

is more predominant while at high frequencies F!(6) is more important.

~’ {e) is insignificant at most frequencies compared to the contributions of-2~\(6) and ~~(d). We can write the total O component of the electric field as

.1“Voth e-~or——2Tf

gr

From figures 4.3 through 4

-1

(4.13)

11, if a, h, r and 8 are known the total 0 com-

ponent of the electric field can be easily visualized. If r >> ~ , the contri-

butions of ~;(e) and ~~( 6J)can be neglected to varying degrees.

In a similar fashion the total radial component of the electric field can

be written as

These can also be visualized from figures 4.3 through 4.11 given a, h, r

(4. 15)

and O. ln using I~1 1, \~~ I and I?’~ I graphs

exercised because they should be added with

In time domain the 6 component of the5

as

shown earlier care should be

the proper phase factor. .

far e~ectric field can be written

(4. 16)

16

1

oI 1 I I I I I I I I I I I I I I I I I I 1 I I I 1

g=7r/2

01 I 1 i I I\

I I I I I I I. 1 I I I I I i I I I 1 1 I I

1

. 1

.001

k 1 1 1 I I I I I I I 1 I I I I I 1 I I I I I 1 I I 1 4f3=7r/2%

I I

-!

.0001 I I I I 1 I I [ I I I I I I 1 1 I I I I I I t I 1 I 1 u

.1 1

‘h

Figure 4.4. Effect of @● &’;(e)j CY

10

= 1

100

0.

—“

10

1

.1

.01CD22”

w

.001

.0001L

.00001

.000001II I 1 I I 1 I I I I I I 1 I I I I I I I I I I I I I 1

● 1 1 10 ,n -J.Wu‘h

Figure 4.5. Effect of e on ~~(e), a = 1“

10

1

.1

.01

001

~-”’ ‘‘ ‘I I I I I I

I 1 1 I I I I I I I I I I ! I [ I I I10 100.1 1

Figure 4.6. Comparison of =1

i,,!Ii,, q

o

,,I

K!

10

1

.1

. 001

.0001

I I I i I I I 1 I I I 1 I I I I I I I I I I I I I 1-

1 I I I I I I I I I 1 I I I I I I I 1100

=1

10

1

.1

I .01 -–

.001

● 0001 I I I I I I I I I I ! I 1 1 I I I I i10 100

‘h. 1 1

Figure 4, 8. Comparison of

‘NGJ

1 1’ I I I I I I I I I 1 I I I I I I I I I I I I I I

---

1 I I I t I I I 1I I I I I I I I I I I I I I. 01 I I I.1 1 10

Figure 4.9. Effect of generator capacitance on 1~}1, (3 = ~/2

100

1

.

.00

.000 —

.1

CY=l

@=l.50=2

\

1

‘h

10

Figure 4.10. Effect of generator capacitance on lr~l, e=~/2

o

100

10

1

.1

.01

b’IL*—rJ-

.001

● 0001

.00001

.000001

12 I I I I I i 1 1 I I I I 1 I 1 I I 1 1 I I I I f I 1

ck’=2.57\\S\

b \ 4

II I I I I 1 1 1 1 I I I 1 I I I 1 I I I I I I I I 1 1. 1 1 10 100

‘h

Figure 4.11. Effect of generator capacitance on lgJ f)=7f/2..J

with

[

<Th

-m

?’;(0) = sin($) e2

{1l-e

hIJ(rh) - u(Th)

c1sin2 (0) sin4 (0)

-a{7h-(1-cos(e))}

( )/

26 l-e \+ 2 Cos j(Urh- (1 - Cos (0)))

T

{+ { ~:-(l-cos(e))}~

()(2~ l-e

+ 2 sin2 a j(

u Tll-

1’.(~+ COSMJ))) “’

(4. 17)

and

()et-r‘h= h

Using (3. 13) we can write the l/r2 part of the 6 component of the electric

field as

o(4. 18)

where

26

r1

= sin (e)sin2 (0)

L.

2

sin4 (6)

-(IT

{11h

-e u(7h)c1

+ 2 sin(

20z

a while the 1/r 3 part of

1‘h - (1 + Cos(e))--

a

+{~h-(l-cos(e))}l-e

~2

)

u (Th- (1-CclS(e)))

-a{7h-(l+cos(e))}1 -e

~2

the @component of the electric field as

f

t 2t ‘TE3 (;$t) = $

IfE2 (;, t)dt = ~ dT Ef (;, t)dt

e -w e r -m -m e

u (Th

1

-(l+ccls(e)))”

(4. 19) ‘

(4. 20)

where

2’7

+

+

Using (4. 16

field can be

si~(,j){, { II2

‘h ‘h: l-e4 ‘h

E-CY 2u(Th) ‘

a

26 1 2

1

(Th -(l - Cos(e))

2 Cos –2~Th - (1 - CoS(e))z - (1 - CoS(@)) a I

- (1 - Cos(e))‘a{7h-(1-cos(@)}

‘h +l-e2 ~2

\

u (Th - (1 - cow))

CZ

01{ 1

‘h- (1 + Cos(e))

2132 sin ~ & 7~- (1 + COS(6))2 - (1 + CoS(o)) c? I ●

- (1 + Cos(e))

‘1 1

-Czph-(l+cos(e))}

‘h~2

+’-e ~z Uph - (1 + Cos(e)))

(4.21)

through 4.2 I ) the total time domain 6 component of the electric

written as

(4.22)

In figures 4.12 through 4.20 ~;(6)$ 5~(0), and ~~(6) are plotted for various

angles and a. As can be seen ~fi(e) and ~;(6-) become more important at

late tfies. u a, r, h and 6 are known, one can reconstruct the total ~

component at the given point.

The total radial component of the electric field can be obtained to be

28

.-.

1.9

1.5

.a-1-1

1.1

.7

.3

-.1

-.5

I I I 1 I I I I I

#I=7T/6

\ !9=7r/3

F(3=7i/2

I 1 I I I I I I I

,,

I

‘1,, ‘1

.4

-03-mlw

.2

(1 2 3

‘h

4 5

●

1’‘,

\

.4

.

,2

(1-0

1 [ ) I \ 1- 1 I I

1 3 4 5

‘h

Figure 4.14. Effect of @ on $;(e) using Q = 2

1.1

.9

.7

.1

-.1

5-. 3 1 1 I I I I I 1 I I

o 1 2 3 4‘h

4.15. Comparison of $\(@), ~#0) and ~~(0) for e = 7r/2$a = 1

0

.

o *

coGJ

1.3

1.1

.9

.7

.5

.3

.1

‘. 1

-. 3

e)

‘ye)

I I I I I Io

1 I I4 51 2

‘h

4.16. Comparison of~~(0), ~~(0),—

1,

1.

-ccl 1.(.03w

.*W

.2

-. 2

I I I 1 I I I I I

.

E;(f)) ‘E;(e)

1 I Ii 1 I2

1 1 13

I4 5

L

Figure 4.17. Comparison

‘h

=1

I

coCJl

1.1 I I I I I 1 I I I

.9

.7

.5

m.4w

.3

.1

-* 1

-. 3 0 1. .

Figure 4.18.

th

4

Effect of generator capacitance on~~(o), 0 = ~/2

w01

.4

-a ,2:&lw

o 1 2 3‘h

Figure 4.19, Effect of generator capacitance cm&~(6),

4 5

0 = 7r/2

,

.

.4

o

.4

co-.l

.2

00 1 2 + 3 4

I

n

5‘h

Figure 4.20. Effect of generator capacitance”on ~;(e), o = w/2

As should be expected, the radial component becomes more important

late times. In a similar fashion the total @ component of the magnetic

is given by

[][

vH(;,t)=~~ –

# 1~ ‘i($)+ $ ~~(e)o ~

Both Er(;, i ) and Hd(;, t ) at a given point can easily be obtained if h, r,

o(4. 23)

at

field

(4. 24)

e

and a are known.

As an example of the procedure discussed above, let us consider an

impedance loaded dipole antenna having the characteristics of ATHAMAS 118,9

(large proposed Vm%i.calelectric dipole). The antenna capacitance is

s 3.8 nF, the generator capacitance s 4 nF, the charge equivalent height

of the antenna s 50 m while the half angle of the bicone = 40. 4~. Since the9

antenna is assumed to be over a perfectly conducting ground plane, the

generator voltage is taken as 10MV to include the effect of perfectly con-

ducting ground. Using these characteristics, the @and r components of

the electric field and the #Jcomponent of the magnetic field are plotted in

figures 4.21 through 4.26 in both time and frequency domains. Since the

pulser is assumed to have a step function input, the high frequency fields

behave as 1/f, f being the frequency, while the late time domain fields

reach a steady state value. A similar procedure can be used to obtain the

field components in cylindrical coordinates. Ii should be noted that if

e<olor(n - 6) <6 ~ the formulae for the fields are not valid because of

the shadowing, the diffraction from the edges, etc. If r >> h and o satisfies

the above condition the fields as calculated here should compare well with

the measured quantities. For low frequencies or late times if r >> h the

results are valid for all @.

38

~ccl

2X104

1● 5X104

1X104

● 5X104

o

-. 5X104

i I I 1 I I I

1

f3=50”/

I I I I I I 1

0

Figure

.fj10X1O

4.21. Ee at r=300m using

–c2OX1O “

Time (seconds)

an ideal. puls er and

3OX1O-6

the antenna on a ground plane

40X10-6

0 -6 -6 -6 -6loxlo 2OX1O 3OX1O 4OX1O

Time (seconds)

Figure 4.22. Er at r=300m using an ideal pulser and the antenna on a ground plane

o 0 ●. .

2 ● 5X104

1. 5X104

. 5X104

o

-. 5X104

I 1 I I I 1 I

I I IP *o loxlo-” 2-X10-D 3oxlo-” 4OX1O

-6

i,,,,,,,,,

.N

-z2

U)2!3.

0

10-2

10-3

10-4

10-5

I I I I I I I I I I 1 I I I I I 1I I I I 1 I I I I I I I I I I I

L

1 1 I 1 I I 11I 1 I 1 1 I I I I I I 1 i 1 I I I I I 1 I I I 1 1 J I

104 105 106 107 108Frequency (hertz)

Figure 4.24. E. at r=300m using an iw

pulser and the antenna on a ground plane

10-2

10“3

10-7

0 aE I I I I I I 1 I I I I I I I I I I

II I I I I I I I I I I I I ! I 1 I

10 -8 I 1 I I I i 1 I I I I 1 I I I 11I I I I I I I 1 I I I I I 1 I I I I ,104 105 106 107 108

Frequency (hertz)

Figure 4.25. Er at r=300m using an ideal pulser and the antenna on a ground plane

10-3

10-4

I I I 1 I I I I I I 1 1 I 1 1i I 1 1 I I I 1 1 I I I 1 I I 1 I I I

.’-”” - ‘ “

‘+9=900

,6=70”

r\x 6=50°

10-5 [ 1 1 1 I I 1 I I104

I I 1 I I I I 1 I 1I I 1 I I I I I 1 I I I 1 I I I105 106 107 1

1

Frequency (hertz)

o Figure 4.26. aZofi@ at r=300m using . deal pulser and the antenna on a ground plane ●

!)~ .1

*v. Conclusions

A procedure has been developed which yields the near field electric

and magnetic components of an axially and lengthwise symmetric electric

dipole if the far field components of the same are known. In particular, it

has been shown that if Ef6, i. e., the @ component of the far field is kno~n>

all other components of the electric and magnetic fields including their near

fields can be determined. This procedure is shown to have been applicable

both in time and frequency domains. In frequency domain the near field

components can simply be obtained by frequency scaling the far field com-

ponents while in time domain they can be obtained by integrating the far

fields with respect to time.

Although we have only discussed the case of analytical calculation,

this procedure is equally applicable to measured data. If the far field com-

ponents of an electric dipole type source are known, the near fields can be

*obtained easily by using the procedures discussed here.

If the antenna is a magnetic dipole instead of an electric dipole, the

procedures developed in this report can still be used by noting that a

magnetic dipole is a complement of an electric dipole. Immediately it is

clear that if Hf6 , i. e., the 6 component of the far magnetic field is know n,,

all other components of the electric and magnetic fields including the near

field terms can be obtained by using the principle of duality.

45

References

1.

2.

3.

4.

5.

6.

8.

9.

C. E. Baum, “Some Characteristics of Electric and Magnetic DipoleAntennas for Radiating Transient Pulses, 1‘Sensor and SimulationNotes, Note 125, January 1969.

J. Van Bladel, Electromagnetic Fields, McGraw Hill, 1964.

T. T. Wu, R.W. l?. King, “The Cylindrical Antenna with Nonreflect-ing Resistive Loading, “ IEEE Trans. G-AP, AP-13, May 1965,pp. 369-373.

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