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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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