Transcript
.
-..
,. .,.
-.
,.%E Sensor,and Simulation Notes Note 259i December 1977 . .
Transient Response of a Loaded Horizontal Antenna Over Lossy Ground with Application to EMP Simulators Y. Rahmat-Samii, P. Parhami l?.Mfttra FORPUBLiCMl@
University of Illinois Urbana, Illinois 61801 .
..
~L/
?04
~=’ fle:”y~
- ..
Abstract A novel technique is developed for evaluating the Sommerfeldtype integrals encountered when analyzing a horizontal current element over 10SSY ground.
The integration results are compared
with their asymptotic,values, making it possible to efficiently compute the currents-of a horizontal antenna over lossy ground. The frequency domain results are compared with those available in the literature, and the Fast Fourier Transform (FFT) technique is used for obtaining the current transient response due to an input pulse excitation.
The effect of loading is also investigated for
reducing the undesirable ringing behavior present in the late time antenna current.
Extensive numerical results are computed and
.
repeated by using an efficient user oriented computer program developed for this work.
Acknowledgement This work was supported by Harry Diamond Laboratories, Adelphi, MD. ._. .,
,
,., .... ..-.
+
..-
—.
,-.
.
.:
,’”:.,
>..
,------
Sensor anclSimulation Notes Note 259
Transient Response of a Loaded Horizontal..Antenna Over Lossy Ground with Application to EMP Simulators Y. Rahmat-Samii P. Parhami R. Mittra e
University of Illinois Urbana, Illinois 61801
.,
Abstract A novel technique is developed for evaluating the Sommerfeldtype integrals encountered when analyzing a horizontal current element over lossy ground. The integration results are compared
9)
with their asymptotic values, making it possible to efficiently compute ‘the currents of a horizontal antenna over 10SSY ground. The frequency domain results are compared with those available in the literature, and the Fast Fourier Transform (FFT) technique is used E-o-r–obtaining the current transient response due to an input pulse excitation. The effect of loading i{ also investigated for reducing the undesirable ringing behavior present in the late time
i
. .
antema
f i
current-. Extensive”numerical resul-tsare computed and
repeated by using an efficient user oriented computer program a
developed for this work.
81 49//, & q \ Th:s work was supported by Harry Diamond Laboratories, Ade phi, MD. {“ ... —.-— —— . . .. ‘m’980 ~\ 25 —L<. .— ~, A5LB1. t l.ib~tw +’ -p
Acknowledgement
9)
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A[ ; ?.a,p;fi~ ~b,ora{orv / p /(’/[l_,\\’”
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TABLE OF CONTENTS Page 1.
INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . .
6
2.
DERIVATION OF i FOR A HORIZONTAL CURRENT ELEMENT . . . . . . .
9
Solution in the Transform Domain. . . . . . . . . . . . . Space-Domain Integral Representations . . . . . . . . . .
13 15
EVALUATION OF THE INTEGRALS . . . . . . . . . . . . . . . . .
23
3.1 Asymptotic Evaluation . . . . . . . . . . . . . . . . . . 3.2 Numerical Evaluation. . . . . .’. . . . . . . . . . . . .
23 24
LOADED HORIZONTAL ANTENNA OVER AN IMPERFECT’GROUND . . . . . .
46
4.1 Integral Equation Formulation . . . . . . . . . . . . . . . . . . 4.2 Matrix Equation Formulation . . . . . . . . . . . . . . . . . . . 4.3 Numerical Results and Discussion. . .
46 48 50
2.1 2.2 3.
4.
●
✎
5.
TRANSIENT BEHAVIOR OF THE ANTENNA CURRENT. . . . . 5.1 5.2
6.
Current Transfer Function H(x,f). . . . . . . . Computation of i(x,t) . . . . . . . . . . . .
CONCLUSIONS . . . . . . . . . . . . . . . . . . . .
✎
✎
✎
✎
.
.
.
.
56
. .
. .
. .
. .
59 61
.
.
.
.
84
● <“ ._
APPENDIX 1. ASYMPTOTIC EVALUATION ................................ 85 APPENDIX 11.
........................ 90
‘AT@2=0 ‘vALuATION ‘F Onlx
REFERENCES ........................................................ 92 APPENDIX 111. LOADED HORIZONTAL ANTENNA OVER AN IMPERFECT GROUND . 94
2
t
o
LIST OF FIGURES
Page
Figure G~ornetryand the coordinate systems for the current element P radiating over imperfect ground. . . . . . . . . 1
18
2
Integration path I’in the complex C-plane . . . . . . . . .
21
3
Branch point locations as a function of–frequency for three sets of ground parameters. . . . . . . . . . . . . . .
26
P.o.le l~cations as a function of frequency, for three sets of--groundparameters . . . . . . . . . . . . . . . . .
28
1
\
4
5
The steepest descent path (SDP) as a function of e2“’””
29
6
Low frequency”examples of-the Olllxintegrand. . . . . . . .
32
7
Low frequency examples of the azil~zintegrand . . . . . . .
33
8
Low frequency examples of the oII~xintegrand. . . . . . . .
34
9
Low frequency examples of the azll~zintegrand . . . . . . .
35
10
Mid frequency examples of the ~ll~xintegrand. . ..*.
.
36
11
Mid frequency examples of the azfl~zintegrand . . . . . . .
37
12
Mid frequency examples of the o~~-xintegrand. . . . . . . .
38
Mid frequency examples of the azil~zintegrand . . . . . . .
39
.
..
High frequency examples of the oll~xintegrand . . . . . . .
40
’15
High frequency examples of the azII~zintegrand. . . . . . .
41
16
‘Highfrequency examples of the OlT~xintegrand . . . . . . .
42
17
High frequency examples of the azlT~zintegrand. . . . . . .
43
18
Exact and asymptotic evaluation of oll~x.. . . . . . . . . .
44
19
Exact–and asymptotic evaluation of fzil~z.. . . . . . . . .
45
20
Loaded horizontal antenna over.imperfect ground . . . . . .
47
21
Input resistance of an unloaded dipole antenna (2L = 10m) radiating in free space. . . . . . . . . . . . .
51
Input reactance of–an unloaded dipole antenna (2L = 10m) radiating in free space. . . . . . . . . . . . .
52
1.4 ?
22
3
Page
Figure ’23
Comparison of the presented techniques with tha~ of Milleretal. . . . . . . . . . . . . . . . . . . . . . .
53
Comparison of the presented technique with that of Milleretal. . . . . . . . . . . . . . . . . . . . . . .
55
25
The Gaussian pulse used for excitation. . . . . . . . . .
57
26
A complete diagram for the pole and branch cut locations as a function of frequency and the steepest descent path for minimum and maximum 82 values . . . . . . . . . . . .
24
● ✿
4 60 &
27
The unloaded transfer function H(x,f) . . . . . . . . . .
62
28
The unloaded transfer function H(x,f) . . . . . . . . . .
63
29
The unloaded transfer function H(x,f) . . . . . . . . . .
64_
30
Loaded (A. = 40 ohms/m) transfer function H(x,f). . . . .
65
31
Loaded (AO = 40 ohms/m) transfer function H-(x,f).. . . .
~~-
32
Loaded (AO = 40 ohms/m) transfer function H(x,f). . . . .
67
33
Unloaded frequency domain current I(x,f). . . . . . . . .
68
34
Unloaded frequency domain current I(x,f). . . . . . . . .
69
35
Unloaded frequency domain current I(x,f). . . . . . . . .
70
36
Loaded (A = 40 ohms/m) frequency domain current I(x,f).. o
71
37
Loaded (A. = 40 ohms/m) frequency domain current I(x,f). . . . . . . . . . . . . . . . . . . . . . . . . .
72 *
Loaded (AO = 40 ohms/m) frequency domain current I(x,f), . . . . . . . . . . . . . . . . . . . . . . . . .
73
39
Unlo-adedcurrent transient response i(x,t). . . . . . . .
74
40
Unloaded current transient response i(x,t). . . . . . . .
75
41
Unloaded current transient response i(x,t). . . . . . . .
76
42
Loaded (A. = 20 ohms/m) current transient response i(x,t). . . . . . . . . . , . . . . . . . . . . . . . . .
77
Loaded (A. = 20 ohms/m) current transient response i(x,t). . . . . . . . . . . . . . . . . . . . . . ! . . .
78
Loaded (A. = 20 ohms/m) current transient response u i(x,t). . . . . . . . . . . . . . . . . . . . . . . . . .
79
38
,
43
44
4
Figure 45
Loaded (AO = 40 ohms/m) current transient response i(x,t). . . . . . . . . . . . . . . . . .$ . $ Q . $
80
Loaded (A. = 40 ohms/m) currenc transient response i(x,t). . . . . . . . . . . . . . . . . . . . 0
81
●
46
●
●
.
47
●
s
●
●
Loaded (A. = 40 ohms/m) current transient response i(x,t). . . . . . . . . . . . . . . . . . . . . . . . 0 0
.
5
82
1.
o.
INTRODUCTION
An electromagnetic pulse (EMF) generated by a nuclear burst typically has a fast rise time (%10 nsecs), a slow decay time (%100 nsecs) and a high peak amplitude (on the order of several kilovolts per meter) [1]. Such a pulse has deleterious effects on most systems, such as transmission lines, ,
transmitting and receiving installations, and missiles, causing temporary or permanent damage. In recent years, several designs have been proposed
●
and built for EMT testing of vulnerable systems. Simulators may be classified as open or closed types depending on the nature of propagation mechanism they support. For example, a parallel-plate structure is considered as a closed simulator while a horizontal antenna is referred to
as
an open simu~ator. A need for an accurate analysis of these simulators is apparent
as
their construction is very expensive. In this report, we investigate the performance of an open simulator which is composed of a horizontal antenna over lossy ground. The radiated field of this structure behaves differently from the driving pulse mainly because of the multiple reflections occurring at the two ends of the antenna, which, in turn, cause the undesirable ringing phenomenon in the time domain (transient) response. This ringing effect may be suppressed by employing an extremely long antenna in which the source current pulse becomes negligibly small as it travels to the end points. Such a long simulator has in fact been built and has a length of approximately 300 meters [1]. However, lengthening the size of the wire is not the only way to reduce the ringing effect. In this report we investigate an alternate approach based on appropriately loading the antenna to suppress the ringing effect. _This approach has the merit of keepi~g the antenna size to reasonable length. 6
●
The radiation characteristics of loaded and unloaded linear antennas in free space have been studied in great detail in both the frequency and time domains. A recent book edited by Felsen [2] covers these topics in depth, and contains pertinent references. Frequency domain responses of linear antennas over..lo.ssy ground have originally been investigated by .
Miller, et al. [3,4] and more recently by others [5,6]. The basic difficulty observed in incorporating the ground effects is the accurate and effi-
.
cient evaluation of Sommerfeld-type integrals. These integrals appear in the exact solution of the radiateclfield of a current element over ground, and their original forms were first obtained by Sommerfeld [7] almo.5t70 years ago.
Since that time, these integrals have been studied extensively
and approximations for their evaluation have been developed for them by many authors [8-13]. Since the integral equation for the antenna current requires the rep-eatedcomputation of these Sommerfeld integrals, an efficient and accurate numerical method for their evaluation would be extremely useful. Moreover, one must repeat:the frequency domain calculation for many frequencies in the process of=constructing the time domain (transient) response making the computation even more time-consuming. To the best of ,
our kndwledge a thorough study of the time domain responses of antennas over lossy (imperfect) ground has not yet been reported in the literature. The present work is an attempt to address the aforementioned problem, i.e., to investigate the transient response of a loaded horizontal antenna over lossy ground. Initially, in Section 2, we derive the vector potentials due to a horizontal current element in terms of Sommerfeld-type integrals. A novel and efficient procedure is then outlined in Section 3 for numerically evaluating these integrals. Also included in this section is the asymptotic
o
evaluation . of the integrals in terms of Fresnel reflection coefficients. 7
Extensive results are presented to justify the accuracy of the technique developed along with a comparison between the asymptotic solution and the numerical evaluation of the integrals. Section 4 develops an integral equation formula~ion for the antenna current in the frequency domain and discusses the numerical procedure used to solve this integral equation, based on the application of the method of moments and the finite difference
.
technique. The results obtained are compared with the available data to .
verify the accuracy of the method.
In the final section, the Fast Fourier
Transform (FFT) algorithm is used to convert the frequency domain results into the time domain responses. Extensive numerical results are presented for the transient current induced on both the loaded and unloaded antennas placed over a Iossy ground. Also included is Appendix 111, a self-contained report which investigates the loading characteristics of linear antennas over lossy ground.
,
.——.
●
2.
DERIVATION OF t FOR A HORIZONTAL CURRENT ELEMENT
@
The geometry of–a current element-Pi over an imperfect ground is depicted in Figure 1. (E1 =
&lr&o,Pl=
Regions 1 and 2 are characterized by
Po) and (E2 = E2r&o,P2 = Uo), respectively, where co and
P. are free-space parameters. The current element is in the x-direction (horizontal) and its coordinates are (O,O,h). The geometrical image of PI with respect to the interface is designated by P2, and the distances of Pl_and P2 to the observation point O, with coordinates (x,Y,z), are labelled as PIO = rl and P20 = E2, respectively. Our objective is to determine ‘thefield radiated by Pl”at the ob-servationpoint O in the presence of the imperfect ground. Starting with Maxwell’s equation and the suppressed time convention exp(jut), viz.,
one may define the vector potential ~ as (2.2)
Introduction of-a scalar potential O from
V’$=t
-
LAoco+i
(2.3)
and application of–the Lorentz gauge v“
o
7i-
(2.4)
@=o,
allows one to finally express Maxwell-!sequation as 9
—
● ✎
P, ‘w
‘o*PO
o
h
●
I
x o
%
2 co~Po
I /~ Iez, /2 h i,/
p2 --L
Figure 1. Geometry and the coordinate systems for the cu-rren~element Pl radiating over imperfect gr~und,
10
(V2 +k2)
(2.5)
fi= -(jw&Oer)-l ~
(2.6) --
o 0c~. where k2 = ti2ps *
regions 1 and 2.
~“v.”+k2)fi
~=
(2.7)
The preceding ;esults are general and valid for both
To solve the vector differential equation (2.5), one
determines the proper boundary conditions by simply enforcing the conti.
nuity of the-tangential ~ and A fields at the interface resulting in
&
i-$x+%
az
IZ
=%-I +% ax 2X az
22
(2.8b)
‘lx = ‘n2x n
lZ
(2.8a)
(2.8c)
= ‘n2.z
(2.8d)
‘here K = ‘2r1zlr”
The unique physical solut-ionis then obtained by
imposing the radiation condition. For the horizontal current element PI with moment I dx’, the source current ~ may be set as b
3,=;1
dx’ 6(X)
6(Y)
6(Z - h)
(2.9)
where h is the height of the currenrelement from the interface. As originally observed’by Sommerfeld [ 7 ], two components of the Hertz p-otentialfiare needed for a complete description of the horizontal current element problem. This, however, is not the case for the vertical current element, in which only one component is needed. The two components
o
are chosen to be in the & and ; directions, i.e., along the current element .—— 11
and along the normal to the interface, and are designated as follows
ii
2,1
=
I-Q + Q
(2.10)
Solution in the Transform Domain To determine lIxand Hz from (2.5) and (2.8), respectively, the Fourier .
transform technique is used. This technique has been employed extensively in the literature for solving infinite-interface-typeproblems. Here, the
.
technique is only briefly discussed and the final results are presented. The two-dimensionalFourier transform pair is defined as co fi=
fiexp[-j(ax + 13y)]dx dy H -m
{2.lla)
m
if=+
41T
ifexp[j(ax + fly)]da dd H -~
(2.llb)
The transform of (2.5), in terms of its components, takes the following forms in regions 1 and 2, viz., . -10
d(z-h)
.
(2.12) 4 {
0
and
(2.13)
where yi (i = 1,2) and 10 have been defined as 12
=
‘fi
_a2
_
#11/2
,
~m(yi)
i
10
where
[~2
=
-1
(jucoslr)
.
—
()
(2,14)
Idx’
(2.15)
k2 = U2UosirE0. i
.
The general solution of (2.12) and (2.13) which satisfies the radiation condition may be expressed as .
-.*
n
10
{) { lx
exp(-jyl]z-h/)/(2jyl)-
N}
.
;17
A“ lx
+
o
exp(-jylz),
‘lZ
Z>o —
(2.16)
Z<() —
(2.17)
—
{) n2x
exp(jy2z)
’22
where’A’s are constant coefficients in terms of u and ~. {
transform otthe
boundary condition (2.8) is a -%
o
-K
o
ii
lx
’12 .
1
The Fourier
0
-K
n
o
i
2x
2Z
13
0
(2.18)
Substituting (2.16) and (2.17) into (2.18) and solving the resulting equation
o_
for A’s, one finally arrives at
A=IO lx
‘1-Y2 exp(-jylh) 2jy1 (Y1+Y2)
A=IO lZ
j(Y1+Y2) (KYl+Y2)
A
=10
1 jK(Y1+Y2)
=1
(K-l) exp(-jylh) jK(yl-@2) (KY1+Y2)
A
2x
22
a(K-1)
(2.19b)
exp(-jylh)
.
(2.19c)
exp(-jylh)
Ci
0
(2.19a)
(2.19d)
.
Having obtained the A’s, one can then determine’the ~’s from (2.16) and
o
(2.17) and find
‘xp(-jylI’2-hl)+ 10
.
10 o
J
j(y~+y2) (2.20)
. exp[-jyl(z+h)] and
I
1
(’‘2X J = 10
‘\i.
) ;2Z
“
1
1 exp(-jylh) exp(jY2z).
j~(yl+yz)
(2.21)
a(K-1) {) KY,+Yq AL
It is worth mentioning that the counterparts of (2.20) and (2.21) can easily be obtained for a vertical current element. Using (2.20) and (2.21) in the transform versions of (2.6) and (2.7), one finds
14
i’ x
Ex”
i Y
j$
o
22. (k2+ — ) as
E. z
‘
-&
.fl
z’ .
and
Hx
H Y
c1
=
a -z
a .az
jwocr
Hz
-j6
o
ja
-j
jl
‘iix
(
o
o
(2.23:}
ilz
2.2 Space-Domain Integral Representations Since in this work the primary interest lies in the evaluation of the fields in region 1, attention is therefore focused on the determination 1
of space-domain integral representations for.II and R These integral lx lZ“ representations take many different forms and a comprehensive discussion is given in [ 9].
In this paper an attempt is made to use the form which has
the contributions from the incident and reflected parts in an explicit manner. To obtain this form, one first splits IIlxfrom (2.20)”into -.
II = rr;x+ lx
Ii:x
‘r n = ir + ir lx m lx o lx
(2.24:1) (2.24b)
where ~.
1 i’?;x=I — O 2jy1
(2.25a)
~xp(-jyllz-ht)
.
(2.25b)
and then redefines II 12 as .
’12 =
‘r n 12
(2.26)
“
-.
In the preceding equations, II~xmay be interpreted as the source contribution
0“
when no ground is present, ~~~x is the itiagecontribution when the ground is a perfect conductor, and finally
“r ‘r and E are the correction terms Onlx 12
for the imperfect ground. The objective is to determine the inverse Fourier transforms of ~’s by using (2.llb). By substituting ~;~x from (2.25b) into (2.llb) and
c
introducing the following spherical-type change of variables:
-X=r
sine Cos$ = p 2 2 2 c0s~2
2
[
. Y =
r2
(2.27)
sine2 sin$2 = P2 .si@2
{
(
z+h = r
2
Cose =Z 22
c1= -h Cos<
--
(2.28)
(
16
●
one arrives at
(2.2.9) 1
with the requirement–tha&Im
In deriving the preceding
F’””
equation the fo-llowingidentity was used, viz.,
~n [-jzcos(~’-~)cos(n~”)d~’
cos(n~)Jn(z) = ~n
(2.30)
–IT
o
where J is the n n
th -order Bessel function. Expression (2.29) can be
integrated in a closed form to yield -jk1r2
lTL =-I e W-lx 0 4Tr2
.-
(2.31)
“
Similarly”, “the inverse Fourier transform of IIL defined in (2.25a), can lx‘ be constructed to give
(2.32)
or
. —
-jklrl ~i e “lx ‘-10 47fr ‘-”-”” 1
(2.33)
In (2.33) and (2;31), (rl,til,$l) “and (r2,02~@2) are the spherical coordinates
17
erected “atthe source and its image point , respectively. The geometry of these coordinates is shown in Figure ~.. Although, as was expected, il~xand
lTr could be expressed in a ~ lx r closed form, this 5s not che case for #~x and IIlz. Substituting ~;~x from (2.25b) into (2.llb) and incorporating the transformationsgiven in (2.27)
.
and (2.28) along with (2.30), one finally arrives at
-jz2 ~ JO(p21.)e
210 ~ r =—J’ Onlx 4Ttj ~
d~
(2.34)
?k~-A2 + _
Similarly, after some manipulations, it is found that
(’2.35)
n-~ 0and‘%’W- 0 ‘“’d” ‘_
where in (2.34) and (2.35) relations Im k
,
,Integral,representations(2.34) and (2.35) are the well-known Sommerfeld’s integrals for a lossy ground [7]. Attempts have been made to employ these equations in numerical evaluations.,and some degree of success has been achievecj[3-6], [10]. IrIthis work, however, other forms of these integrals that appear to be numerically more tractable are used. Incorporating the well-known identities between Bessel and Hankel
o
fun-ctions,viz.,
18
Ji(x) = ;
J1)
(x) =
0..
[H(l) i
-F&
(x) + H ‘2) (x)] i
i = 0,1
;
(2.36a)
(-x)
(2.36b)
(2.36c)
(-x)
— .
into (2.34) and (2.35),one arrives at
r
Onlx
-jz2
“m
10 4mj
=—[’-
Jq-7+vq7
‘o
‘,L
‘2)(p2A)e
da
(:!.37)
and
Jz) 1
.e
.LL
L
d~
.
(2.38)
The behavior of (2.37) and (2.38) at P2 = O is discussed in Section 3, and it is shown that these integrals are indeed bounded at Pv = O, which can be observed directly from (2.34) and (2,35). To recast (2.37) and (2.38) to yet another form which is of considerable interest in this work, th-e—followingchange of variable 1s —
—
.—
introduced.:
A=klsinc
(2,39)
19
Substituting (2.39) into (2.37) and (2..38)and simplifying the result, one finally arrives at
.
-jk1z2Cosg me
dc
(2.40)
and
●
-jk1z2cos& H~2)(k1p2 sin~)8 d~
,
(2.41)
o
where the integration path ~ is depicted in Fig. 2,and on this path the following conditions are enforced
Im(cos&) —< 0 ; Ire(-)
~ O
.
(2.42) r
Some discussions on the pro_perinterpretation of the location of poles and branch cuts of the integrands of (2.41) and (2.42) in the proper Riemann sheet are given in Section 3.
Since in constructing the integral
equation for the horizontal antenna problem the knowledge of A IIr is 2Z 12 important, one evaluates this term using (2.41) along with the fact that z’) = z + h to obtain
20
Mt.
-7T/2
.
J
r
Fig. 2.
Integration path 1’in the complex C=plane.
21
c H(2~k1p2 sin&) e 1
–jk1z2 cos.g d<
.
(2.43)
#
In the following sectioti,asymptotic and numerical evaluations of (2.40-43) are presented.
.-
●
.
“
——
22
3.
o
EVALUATION OF THE INTEGRALS
In this section, an analysis is given for asymptotic and numerical evaluation of–the integrals obtained in the preceding sections. In ‘particular,a novel numerical procedure is presented for the evalution of–(2.40) - (2.43). A comparison is then made between asymptotic and numerical results to evaluate the domain of their validities. 3.1 Asymptotic Evaluation The asymptotic expansion of (2.40) is obtained by employing the results summarized in Appendix 1. Using (2.27) in (2.40),one may express it as
-jklr2cos(g-62) r _ lokl P(g) e d( Onlx 4rj I ~
(3.1)
where sin ~ cos ~
P(g) =
Cos g +
●
L-
H(2)(klr2 sin e2 sin g) o
sin2.$
jk1r2sin62sinc e
.
(3.2)
Let 62 = Oc designate a critical angle forwhich the steepest descent path (SDP) of (3.1) isnear the branch points of 4
- sin2 g (see Section 3.2).
For the situation in which k1r2 is large and the domain of 0z belongs to o < 02 < e one may use the results given in Appendix I to determine the c’ dominant asymptotic expression of (3.1). In this Appendix the result of the higher-order asymptotic terms is included along with a discussion regarding the behavior of the asymptotic solution at—62 = O.
The final
result for the dominant term, which is commonly referred to as the Fresnel reflection result, takes the following form
23
-jk1r2
2 cos e2
r Onlx % 10
(3.3) Cos ‘2 ‘ m
“*’2
“
Similarly, one can determine the asymptotic expansion of H:z and &
H;z
from (2.41) and (2.43), respectively, to arrive at
. r
Cos % 2
II;z% 21~ cos $2 sin @ COS 8 2 2 K
COS
6
2
+
--2---
-jk1r2 02 e 4m2 9 K – sin’ 0 / 2 K
–
sin
.
(3.4)
and (3.5)
The preceding results are valid when k1r2 is large and 82 < ec < Tr/2. For cases in which klrz is sufficiently large, one nay extend the domain of (3.2) - (3.4) to 62 < T/2
by
noting
that
the
contribution to the
integral from the portion of the path near the branch points is of the second order [14]. 3.2 Numerical Evaluation A survey in the literature reveals that numerous attempts have been concentrated on the numerical evaluation of
r r and Ii Most of the Onlx lZ”
available techniques are based on the application of Sommerfeld’s integral$, given in (2.34) and (2.35). It is noted that the integrand of these integrals is highly oscillatory for large p2 or Z2, and, therefore, special consideration must be given for an accurate evaluation of the integrals. In order to avoid the inaccuracy observed in the results when the pole of the integrand of (2.35) approaches the real axis (integration path), some authors [3,4,6] have deformed this pa~h to one which first travels along the imaginary axis and then runs parallel to the real axis. Though on this new path the effect of the pole singularity will be decreased, nevertheless, the decay raLe and the oscillatory nature of the integrand would not be 24
0
changed appreciably. It is worth mentioning that for solving the antenna problem, one has to evaluate #~x
and R~z repeatedly. Therefore, application
of–an accurate and efficient numerical technique is indeed of great importances ‘In this section a novel numerical technique is presented for .
determining ~H~x and H:z in a“highly accurate and efficient manner. This technique is also advantageous to other available techniques as it can be easily related to the asymptotic results given in the previous section. In contrast to other techniques,
r a r are evaluated using and — Onlx’ ‘;Z az ‘lZ
their integral representations as given in (2.40), (2.41) and (2.43), respectively. Here, computational details are given only for
r OHIX”
At first, ~ll~xfrom (2.40) is expressed in the form given in.(3.1), .=
then the integration path 1’“isdeformed to the steepest descent path (SDP) passing through t = 8L.
deformation one.has to guard Since in.thi.s.
against intercepting any of the branch points of /K - sin2 t, the location OE these branch points
is derived as
(3.6)
where, as before, K = c Taking region 1 as air and r~gion 2 as a 2r’&lr” lossy ground with a relative dielectric constant-c and conductivity o, $3 one finds
.-
(3.7)
where o and f are in rnho/rn_ayd MHz, respectivel~~ It is clearly evident tha~: and 17 demonstratethat–theintegrandshave decayed to ne~ligibleV.,IL!L:, LhI_l>r.lncl~ around where.the branck~cut is intercepted. It is (:onciudedt-hatpoint contributionis of second order and can be neglected. This appr(,ximation is also verfif-ied and shown by [14]. 59 —
Steepest
o
Descent Path
–—–——
Pole Location
—.—.
Branch Point Location For O < f < @
—
For 0< f < @
Im t
)
-2
t
-77
Figure 26. A..complete diagram for the pole and branch cut locationsas a funct?onof frequencyand the steepestdescent path for minimum and maximum 6 values. 2 60 ..= ____
o
The antenna transfer function H(x,f) is computed at the total.of 127 frequencies in the range of 1 ~ f ~ 400 MHz with more samples concentrated at the resonance frequencies. Figures 27-29 show the transfer functions of the unloaded antenna at three observation points. Using the loading function given in (5.1), and choosing a proper loading of A
o
= 40 ohms/m, based on the results of Appendix III, the loaded
transfer response is computed and shown in Figs. 30-32. 5.2
Computation of i(x,t) The input pulse of Fig. 25 is shifted by T = 30 nsec so that the
applied voltage would start at t = O and reach its peak at t = 30 nsec. The transform of the”input pulse is multiplied by the antenna transfer functions shown in Figures 27-32 to obtain the corresponding I(x,f)’s which are showm in Figs. 33-38. Note that at f = 400 MHz, the resultant
o
frequency domain currents have decayed sufficiently so that zeros can be added for f >-400 Wz.”
Also using (5.6 - 5.8), the negative frequency
domain currents can be directly constructed as the conjugate of the positive frequency values. Now that the entire frequency domain is constructed numerically, a Fast Fourier Transform routine is e-mployedto obtain the time response i(x,tj. Since many points are needed to assure prop-ersampling, a linear interpolation is made through the available I(x,f) data points, and from that, 2048 equi-distance samples are fed into the FFT routine. Figures 39-41 contain the unloaded time response at.three observation points, while Figs. 42-47 demonstrate the corresponding current transient responses when A = 20 ohm/m and A. = 40 ohm/m loading parameters are employed. o It is interesting to point out that the causality is satisfied very convincingly in Figs. 39-47. Also note that the sharp peaks in Figs. 39-41
61
o R E A L
.01
.0075
.005
.0025
0.
-.
1
002s
t
I
I
250.
1’50.
50.
FREQUENCY
I M : I N A R Y
I
I
I 400.
300.
200.
100.
Cl.
1
350.
●
[MHZ]
.0075
.00!5
.0025
0.
-.0025
~
~
-.005
50.
4oa
.
loa .
o.
250.
150.
FREQUENCY
350. LMHZI
Observationpoint x is ll(:<,F). Figure 27. The unloaded transferfunct.i{jrl on the feed patch. 62
.
O
R
●
01
.005
0.
-.005
0
-~i .
●
50.
400.
●
150.
250. FREQUENCY
“
350.
[MHZ]
.006 ii
.004
I .002
%7
0.
–.002
-.004
o
~~i
100.
. 50-
. 150.
. 250.
FREQUENCY
4CI0 . 350.
[MHZ]
Figure 28. The unloaded transfer function H(x,f). Observation point x is midway between the feed and the end patches, 63
0 R E A L
.
.(304
. 002
.
0.
—. 002
-.004’ o.
100.
SCl.
2cla . 150.
300.
FREQUENCY
+00 . 350.
250.
[ MHZ 1
e
.cio3
. 002
. 001
cr.
-.001 \ —. 002
-.
i
003 [
1
100. 50.
I
1
200. 15Q.
300. 250.
FRECiLJENCY
29.
I
I
—
+ 400.
350. [ MHZ 1
The unloaded transfer functionH(x,f). Observationpoint x is on t’hecenter of the end patch.
64
.
“o
~
.
—-—
005
.004
.003
,002
.002
0. 0.
200.
100. !50 .
250. FREQUENCY
1 M A G 1 N
400.
300.
150.
350. [MHZ]
.002
.001
0.
—. 001
0.
100. 50.
4130.
300.
200.
350.
250.
150.
FRECIUENCY
[MHZ]
Fi&ure”30. Loaded _(AO= 40 ohms/m) transfer function H(x,f). Observation ‘point-x is on the feed p-atch, 65
o .002 E A L . 002
-.001
t
-.002 0.
I
1
2.00. 5a.
1
I
200. Isa.
1
I
300. 250.
FREQUENCY
i 400.
350. [MHZ]
.002
.001
0.
I -.002
1
0.
I
1
2oa. so.
I
t
200. 150.
250. FREQUENCY
I
t
i 300.
i 400.
350. [ MHZ.—1
Figure 31. Loaded (AO = 40 ohms/m) transfer f&ction H(x,f). Observation Point x is midway between the feed and the end patches,
66
o
.0004 : A L .0002
Q.
—. 0002
–.0004 o.
--
--’
t~+ 200.
50.
. 1 !S0 .
. 350.
FREQUENCY
●
400.
250. [MHZ]
0004
. 0002
0.
–.0002
I
–.0004 o.
I
I
100. 50.
I
I
1 200.
I
I
300.
1s0.
250. FREQUENCY
350. [MHZ]
Figure 32. Loaded (AO = 40 ohms/m) transfer function H(x,f). point x is on the center of the end patch, 67
--i 400.
Observation
o
.3OOOE-10
. 2oooE-lcl
. loaoE.la
cl.
-.3.000E-10
u.
ma. 50.
200. 150.
3ao. 250. FREQUENCY
Ii tI N A R Y
. 350.
400.
[MHZ]
.2OOOE-10
@
.IUOCJE-10
0.
-.1OOOE-1O
2aa .
100.
u. 50.
250.
15U.
FREQUENCY
350. [ MHZ 1
Figure 33.‘-Unloadedfrequencydomain current I(x,f). Observationpoint x is on the feed patch.
68
0
o ,.1OOOE-10
a.
I
-.2-OOOE-10 0.
---J-~-l 400.
50.
-
150.
-
250.
●
FREQUENCY
350. [MHZ]
o . 15OOE–10
Ii A G
I N
.IOOOE–2.O
. 5000E–11
‘lr---
o.
–.5000E-11
-.1OOOE-2O Ct.
100. 50.
Figure 34.
200. 150.
300.
400
“250.
350.
FREQUENCY
[MHZ]
Unloaded frequency domain current I(x,f). Observation point x is midway between the feed and the end patches.
69
.
.1OCKIE-10 E “t .5CJCIOE-11
.
cl.
-.5000E-11
-.
$
1OOOE-10
Q.
I
Loo 50.
1
.
I
I
200. 15C).
I
I
300. 250.
I 4ao
.
350.
FREQUENCY
[MHZ]
o
/’
cl.
-.2000E-11
-.400CJE-11 a.
J ,,,,
100. 50.
200.
150.
300.
250. FREQUENCY
400.
35C3. (MHZ)
Figure 35. Unloaded frequencydomain ctirrent .I(x,f]. Observationpoint x is on the center of the end patch.
70
.
QacJoE-21
: f.
_ .eoooE-11
. 4000E-21
.20cloE-ld.
u
0.
1
1
o.
I
I
1
100.
200.
50.
150.
1
300. 250.
-i ’400.
350.
FREQUENCY
. 5000E-11
I
[MHZ?
T
. 40CIOE-11
.3000E-11
.20claE-l-l-
. 2000E-11
0.
I
–.lCIOOE-11 0.
I
I
I
100. 50.
(
200. 150.
I 300.
250. FREQUENCY
Figure 36.
I 350. [MHZ)
Loaded (AO = 40 ohms/m) frequency domain current I-(x,f). Observation_point x is on the feed patrh. 71 IL
-1 Ztcjo -
. 4000E-11 : i! .zaac)E-11
o.
-.200QE-11
I
-.4uaaE-2z
t
a.
i
I
zao. 50.
150.
1
I
zaa.
I
300. 250.
I 400.
350.
FREQUENCY
[MHZ]
.
a.
-.200aE–11
1
-.4aaaE-12
I
u.
I
I
100. m
.
I
200. 150.
t
!
300. ..
250. . . .. . FREQUENCY
1 4ocl .
35cl. [MH”2]
b’
.
Figure 37. Loaded (AO = 40 ohms/m) frequencydomain current I(x,f). Observationpoint x is midway between the feed and the end patches.
72
e
R
.1OOOE-11
c
.
5000E-12
o.
.
—. 5000E-12
-.lOOOE-ll0.
200.
100. 50.
4(30 .
300. 250.
150.
350.
FREQUENCY
( MHZ 1
@ I M A G I N A R ‘i
. 1OOOE-11
.
5000E-12
o.
-.
5000E-12
~~+
—. 1OOOE-1I o.
300.
200.
100. 50.
250.
2.50.
FREQUENCY
Figure 38.
400. 350. [ MHZ 1
Loaded (AO = 40 ohms/m) frequency domain current I(x,f), Observation point x is on the center of the end patch. 73
.004 : R R E N T
A M P s
. 00s
●
002
.001
0.
-.002 I
-.
i
0CJ2! -100.
TIME
Figure 39.
I
1 4oa
i . 45(I.
350.
[NSECI
Unloaded current transient response i(x,t). Observation point x is cm the feed patch.
o
o i!
I 300.
250.
150.
so .
-sCl.
1
200.
100.
c).
1
I
I
I
‘,
o II
II
. one : R R E N T
A M P s
.0015
.001
. Oaas
I
o. i
rQ----
—. 0005
1
–.0(31
I
1
-100.
I 50.
1
1
E!oo . 150.
TIME
Figure 40.
I
100.
0. -50.
1
I
1
300. 250.
1
I
400. 350.
450.
[NSECI
Unloaded current transient response i(x,t). Observation point x is midway between the feed and the end patches.
.tlol :
E Y
2
.0005
-.0CI05
-.001
-.
1
OCIIS -1
1.
1
o. -50.
I J
1
I
I
10(2 . xl.
.
1 Soo
200. l.so
TIME
I 2s0.
I .
350.
I
{
4oi3 . ‘ 45n
.
[NSECI
Figure 41. Unloaded current transient response i(xYt). Observation point x is on the center of the end patch.
o
2’
:
c
:
. CJ03
I
I
I
I
k! R E N T .,
,1I
I
A M
II
:
‘1 ,,
II
‘,
o.
i
– .001
I
1
-50.
Figure 42.
1
1
200. 150.
50.. TIME
,,
[
100.
0.
-100.
I 1
1
I
4no
300250.
I
3scl.
4 . 450.
[NSEJ21
Loaded (An = 20 ohms/m) current transient response i(x,t). Observation point x is on the feed patch.
1, !!
c
.0015
u R R E N T .001 A M P s .0005
-4
m
o.
I
I
–.0005 –100
1
-s0.
1
I
I
I
400. 350.
250.
I i
300.
200. 150.
50. TIME
Ftgure 43.
I
100.
0.
.
I
450.
[NSECI
Loaded (A = 20 ohms/m) current transient response i(x,t). Observation point x is midway between tRe feed and the end patches.
,,
:
1,
.0004 “: R, R E~ NI T
I
II
.0002 A M P, s
I 1! o.
-1,
w
-.0002
1
I
–.0004
-100.
I
1
I
I
250.
1
I +00
300.
200. 250-
50. TIME
??igure44.
I
100.
0. -50.
1
350.
I . 4s0.
[NSECI
Loaded (AO = 20 ohms/m) current transient response i(x,t). Observation point x is on the center of the end patch.
.003
. Ocle
.001
o. II
—-
1
001 –100
1
I
0.
. -50.
1
I
100. 50.
I
200. 150.
TIME
1
1
I
:,
I
400.
300. 250.
I
350.
450.
[NSECI
Figure 45. Loaded (A = 40 ohms/m) current transient response i(x,t). observation point x is on the feed patel?. .:
I
,, h,
‘!
“,
I
E
.001
N T
, I
!, II
A
I
II
. Ooo+
II
: s . ooos~l I !
1,
.0002 II
,, 0.
P’ -.
-———
UUC12
1
-100.
1
I
0. –50.
I
50.
1
1
200. 15a.
TIME
‘Figure46.
I I
100.
I
I
300. 250.
I
i
400. 35J3.
450-
[NSECI
Loaded (A = 40 ohms/m) current transient response i(x,t). Observation point x is midway between t!e feed and the end patches.
.aoo3
. CIO02
.0001
Jt-.)
——— ~..— ..— .. —,-----..—. —,
Ii
““-————ml
cl.
-------4----
-.aalal
“J
r
---l-—–—--k-––
-100.
cl.
.—6..
.—
,,1
-t-t-”’ 100.
–50.
.—.
50.
“1–--– --}-
,
‘---l---–---- +--
2!00 . Isa.
TIME
--- 1--
300.
!
--
I
-{
4aCl .
250.
350.
4!5Q.
[NSECI
I
,,, !,,,
O ,,,,
,0
I
,,
,,,
,11,
:
1,,,
,,
1 II
‘1
I
can be justified if we assume that the feed excitation travels along the antenna as a TEM mode while being multiply reflected from the antenna endpoints. For example, at the center of the antenna (Fig. 39),
the
reflections from both ends return simultaneously and only one peak is detect-edper reflection. However, in Fig.
40,
the observation point is
midway between the feed and the end so that–the reflect-ionsfrom the two ends are clearly distinct. The ringing effect present in the unloaded current transient responses of
Figs. 39-41
is
partially overcome by the AO = 20 ohm/m loading
(Figs. 42-44) and has p.ract-ically disappeared for the A. = 40 ohm/m case (Figs. 45-47).
This result agrees well with those reported in Appendix
III iriwhich the latter loading seemed to be a good-approximation to the optimal loading parameters computed for a wide range of -frequencies. @
83
6.
●
CONCLUSIONS
A novel procedure has been developed for accurately evaluating the Sommerfeld-type integrals and the integration results are compared with .
that of the asymptotic approximations. linefficient and fast useroriented computer program is then developed f&
computing”the current -
transient response of an arbitrarily loaded, horizontal, ~inear antenna over
a
lossy
ground. Selective results have compared convincingly with
those reported in the literature. This program is currently accurate as long as no poles are intercepted during the path . deformations, and the possible branch cut contribution is negligible. Note that for larger 82 (see
Fig. 5) and as u+
O
(see
Fig.
4),
chances of capturing a pole and/
or a branch point increases, therefore, additional analytical and numerical analysis is needed to take their contributions into account. This work can then be extended to compute the near field of the antenna qnd will be reported in the near future.
● 84
APPENDIX I ASYMPTOTIC EVALUATION
In this appendix a general formulation is developed for higher-order asymptotic evaluation of an integral with the following format
-—
-“k~cos(~-e)
1
‘-4nj
dg
,
P(g) e J ~
(1.1)
~
where it is assumed that kr is a large parameter”,-Tr/2< 0 < T/2 and P(g) is a slowly varying function, For large values of kr one is usually interested in determining the asymptotic expression of (1.1). This is done by employing the method of the steepest descent path integration. At.the saddle point < = 6, one can deform the integration path r
o
steepest descent path–(SDP) defined by Re[cos (& - e)] = 1.
to
the
Assuming that
in this deformation no poles or branch points are encountered, one may express (1.1) as
1 4nj
-:u=—
-jkrcos(L-0) dg P(c) e ~
s~p
.
(1.2)
,Since on SDP the relation Re[cos(& - 8)] = 1 holds, one introduces the change of variable
in which t is a real variable taking the domain [-m,~]. Substituting (1.3) into (1;2), one arrives at
u=
-jkr-jn/4 @ 2 e Q(t) e-krt dt I-m ,2&17
85
(r4)
—
o
where
Q(t)= P(E)
(1’.5a)
sec~
in which c is replaced “with
~=+
+jLn(t2+j
1
+, ltl~)
[
+e
,
t~o
(1.5b)
and Ln is interpreted as being its principal value. The corrtplete asymptotic =–.
expansion procedure [23] is now used for the asymptotic evaluation of (1.4). In this procedure, one first expands Q(t) in a Taylor series
(1.6)
where
Q
(n)(o).,:
and ~ is the Gamma function. Then (1.6) is
Q(t)
●
t=o substituted into (1.4) to finally result in
u =
e-jkr-j~/4 m ~–2n -—-= (kr)-n-1’2 Q(2n)(0) I ~ n=o “ 2G
.
(1.7)
In constructing the preceding equation, the following identity was used, ...
viz.,
a! J
–.. ~kr)-(wlm
2 [ n -krt te dt=~
-m
: 17[(1 +
n)72]
.—.-
for n even .
o
..=
--(1.8)
for n odd
( The task is now to determine Q
(2n), s in terms of P.
differentiating (1.5a) and arriving at
86
This is achieved by
.,
.=-
.-
( = Q(0)
P(0)
The higher-order terms can also be determined in the same fashion. It is worth emphasizing here ,that--in deriving the preceding equations, the following assumption has been made:
neither the poles nor the branch
points of P(E) are near the path SDP. To present an example, the higher-order asymptotic expansion of-the fo-llowingHankel function of the second kind and order v is derived.
o
(1.10) v
“
Jr
Comparing (1.10) with (1.1), one obtains
-jV&+jvn/2
P(C) =4je
where it is assumed that $2>> v.
(1.11)
Y
Substituting (1.11) into (1.9) and
simplifying the result, one finally arrives at .
f
Q(o) =
4j
~jv~/2
1 (2)(0) = -8 -V2+~
[1
(, Q
Q(4)
[
(0)
ejvT12
V4 -~V2
= -16j
++
[
1
ejvm’2
.
(2) is then determined using (1.7) to be The -asymptot-ic expansion of HV
87
(1.12)
+o(Q-7/2) .
(1.13)
The purpose of this appendix has been to formulate the necessary steps for determining the asymptotic expansion of (3.i).‘ Comparing (3.1) and (3.2)
with (1.1) and using (1.13) in (3.2), one can then find the asymptotic
expansion of (3.1) from (1.7). Since the final results take
a
complicated
form, the following notations are introduced for-the ease of representation, namely,
e2
(1.14a)
sin 0 2
(1.14b)
c = Cos
s =
q=d==5
.“-
(1.14c)
●
=
Using (1.9) and performing a rather tedious differentiation, one finally arrives at
(1.15)
and Q(2)(0) = Iokl-& r
+(1-K)
ejn/4
(Zj)
{,
C3
-
12
( 4s2(c+q) 2) 2 4qs2+3c’q -1-2CS .
2q3(c+q)2
+ 2c2-3c~ 4q(c+q)
(1.16)
) ——
88
Clearly, the terms for Q4(0) get very involved and therefore are notincluded here.
It is interesting to note that both the second term in (1,15)
and the first term in (1,16) are singular at s = O, i.e., 8 = O. 2 Substitut-ing(1.15) and (1.16) back-into (“1.7)and simplifying the result, one ’clearly observes that these singularities together produce a bounded .. result at 6 = O, 2
In other words, though the asympt-oticconstruction was
not originally valid for 02 = O, the final.solution can be used for this angle. This solution takes the following form -jk,ro , . J-L e + ~ k-2r-2 4~r2 12’
-o~:x =10c2:q
()
(1.17)
and
r OHlx=L.
2C — 0 c+q
-jk1r2 e 4nro L
—
-jk1r2 e + o . 4nklr-; -(
.
‘;10 [ : ‘-+
k-3r-3 1. 2 )
L-A+ C+q)
2
2 + 2CS 2 C13(C+CI) 1
(l_K) 4qs2+3cq
.
(1.18)
It must–be r“e-al-ized that the preceding asymptotic results are valid only for observation angles .02< B where 8 is the critical angle for which c’ c the saddle path of (3.1) intercepts the branch point of the integrand. For the cases where the interception happens, one has to determine the asymptot-iccontribution of the branch point near the saddle point. -.
The
—
procedure is discussed in [14] and it–can be shown that the final result-2 would be of_the order–k-2 ~ r2 , i.e., the same order as the second term in (1.18).- ‘Therefore,as far as klr2 is large and the branch points and .
pole-sare sufficiently away from the saddle point, one can use (1.17) with confidence for almost all angles of observation.
APPENDIX 11. EVALUATION OF ~lfilx ‘AT02=0
In this appendix the behavior of (2.37) and (2.40) at P2 = O is studied. In their present forms, these integrals are not defined at p2 = 0, although it is clear that their equivalent form in (2.34) is
‘-
bounded. Equation (2.37) may be expressed here for convenience as ..
where p2 = r2 sin 0 2“ < x/2,
0502-
From Fig. 1 it is observed that r2 ~ h and
and therefare the difficulty arises at f12= O. (~)
with its expansion
To circumvent this difficulty, one replaces Ho
where y is the Euler’s constant and ~(m) represents Lhe harmonic series> i.e.,
$(m)=l+l/2+1/3+...+
l/m
.
(11.3)
Using the fact that P2 = r2 sin 67, L one may express the “En” term as
90
—-. ..
.=.
-—.—-
.———
It is noted that both the JO and the summation terms in (11.2) are even functions of A, hence, their contributions to the integral (11.1) will zero. Using (11.4) in (11.2) and substituting the result into (II-1), one finally arrives at
A)
7 .0
(11.5)
Introducing the change of=~=riable
which is obviously bounded at 8 = O. 2
A = kl sin ~ into (11.5) and setting 82 = O, one f-inds
-jk1r2cose2cosc “e dc .. .—.
.—.——.
The above . result wa.s.ysed.i_n(3.12)
,
a, =0
.
(11.6)
L —
for defining P(L).
In a similar
f-ashion,one can show that the following re-suitsalso hold at@=O 2
(11.7)
91
>——
REFERENCES
—.— —
-. .*__
1.
Spe(j 1 Joint Issue on the Nuclear Electroma-gnet~cPulse, IEEE Trans. {U Antennas and P~opagation, January 1978 and a’lsoin IEEE Trans. on Electr@II!%l&ticcomP~t~EilitY> ~ebruav 1978”
2.
L.B. Felsen, Ed., Transient Electromagnetic Fields, Topics in Applied Physics, Vol. 10, New York;” SprinIodellingAntennas Near a Half-Space,” Electronics Letters, Vol. 13, pp. 690-691, Nov. 1977
,,
.-
—
,,.,
_@_ ~
92
.
.—
14.
J.A. Kong, Theory of Electromagnetic Waves, New York:
15.
L. Grun, Y. Rahmat-Samii, “Polynomial Approximations of Bessel, Neumann and Hankel Functions for Complex Arguments,” University of Illinois at Urbana-Champaign, Electromagnetic Laboratory, Technical Report No. 77-15, July 1975.
16.
R.F. Barrington, Field Computation by Moment Methods, New York: McMillan, 1968.
17.
P. Parhami, Y-:Rahmat-Samii, R. Mittra, “A Technique for Calculating the Radiation and Scattering Characteristics of Antennas Mounted on a Finite Ground Plate,” Proc. IEEE, Vol. 124, pp. 1009-1016, Nov. 1977.
18.
R.F. Barrington, “Matrix Methods for Field Problems,” Proc. IEEE, pp. 136-149, February 1967.
19.
E..C.Jordan and K.G. B-almain,E1.ectromqgneticWaves and Radiating Systems, pp. 566-567, New Jersey: Pren~ice Hall, 1968.
20.
C.M. BuP1.erand D.R. Wilton, “Analysis of Various Numerical Techniques Applied to Thin-Wire Scatterers,” IEEE Trans. Antennas Propagat-.Vol. AP-23, pp. 534-540, July 1975.
21.
Y.P. Liu, D.L. Sengupta, and C.T. Tai, “On the Transient Wave Forms Radiated by a Resistively Loaded Linear Antenna,” Sensor and simulation Notes, Note 178, February 1973.
22.
F.M. Tesche, “Application of the Singularity Expansion Method to the Analysis of Impedance Loaded _Line_ar Antennas,’l.. Sensor and Simulation Notes, Note 177, May 1973,
Wiley, 1975.
23. L.B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, New Jersey: PrenCice Hall, 1973. 24.
R.F. Barrington, Time Harmonic Fields, New York: McGraw-Hill, 1961.
93
●
APPENDIX III
Y. hallrnat-S~~mij , P. Pall] arni,and R. MiLtra
.—
●
.Ai3STF!ACT
loading
radistion patterns versus differfintantenna dimensions and Sround perrrrit~i-
.
vi~ies and cond(lctivities.
-
-.
.-
0. .=
— —
94
—
o
LIST OF FIGURES
,, Figure
Page
1
Horizontal current element PI over an imperfect ground . . . 94
2
Integration path r in the complex g-plane . . . . . . . . . . 97
3
Nonuniform resistively loaded horizontal’antenna over an imperfect ground . . . . . . . . . . . . . . . . . . . . .101
4
Determination of least2square error for current phases, i.e., ER(AO) =Min26.. . . . . . . . . . . . . . . . . . .104 33
5
Magnitude and phase of–the current for’three diff-erent loading parameters. . . . . . . . . . . . . . . . . . . . . .107
6
Effects of imperfect ground on ER(AO). . . . . . . . . . . . 111
7
Effect of AO on input resistance, no ground and imperfect ground cases. . . . . . . . . . . . . . . . . . . . . . . . . -.
i
112
8
Input resistance versus frequency for loaded and unloaded antenna infreespace . . . . . . . . . . . . . . . . . . . .113
9
Magnitude and phase of current for no loading and critical loading. . . . . . . . . . . . . . ... . . . . . . . . . . .114
10,
Radiation pattern in plane X-Z for no ground, imperfect ground, no loading and critical loading. . . . . . . . . . .115
11
Radiation pattern in plane Y-Z for no ground, imperfect ground, no loading and critical loading. . . . . . . . . . , 116
12
Radiation pattern in X-Y plane for no ground, i-rnperfect ground, no loading and critical loading. . . . . . . . . . .117
LIST OF TABLES P-age 1.
2.,
CRITICAL LOADING PARAMETER A: AND THE MINIMIZED ERROR” “ ER(Aj) VERSUS FREQUENCY FOR tiO ANTENNA RADII . . . . . . . . . 108 CRITICAL LOADING PARAMETER Ac AND THE MINIMIZED ERROR ER(A:) VERSUS FREQUENCY FOR ~IFFERENT GROUND PARAMETERS
_ _A95-
. . . . 109
I.
● ✎ ✎
INTRODUCTION
Loaded wire antennas, which have attracte~ attention in,designing EMP simulators, often are regarded as open simulators. The basic design requirement is to generate a temporal signal which matches the waveform
.
of an exoatmospheric burst [ 1 ] which is typically characterized by rise and fall times of approximately tens and hundreds ofnsecs, respectively. The basic obstacle in using any pulse radiating finite antenna is the undesirable effect due to the reflection from the antenna ends.
The effec~ of these
reflections is predominantly seen in the early ~ilne’behaviofof the radiat<-d --
—.
pulse, Usually the antenna dimension (length) is s“everalA/2(wavelength) for the upper part of the frequency spectrum of-the pulse.
One possible way
to overcome these difficulties is to load the antenna”with a non-uniform
● ✍
resistive loading (frequency independent) which eliminates the end reflections to a great extent. Many different loading functions have been proposed and rested for an~ennas in free space.
Liu, Sangupta and Tai [ 2] have analyzed and compared
the effect of different loadings and have concluded that the following continuous loading is the most suitable one: .
(1) ‘(x) = * where x is measured along the antenna from ,, its center and 2L is the antenna length. They used the method of moments with quadratic basis function and determined the critical value of AO by trial and error algorithm, i.e., by looking at the current distribution and searching for a traveling wavetype behavior. Tesche [ 3 ] has arrived at almost the same conclusion by applying the singularity expansion method.
96
● ✎
In this paper, we analyze the effect of loading with charact-e-r-istic function (1) for antennas over an imperfect graund, and specif-icallydevise a novel approach for the determination of the critical loading. We use the Fresnel reflection coefficient approach and derive an E-integral equation for the ant-ennacurrent. The effect of loading then appears as a diagonal term in the matrix equation f-ormulation. Since for the critical loadings, the current phase distribution is almost linear along the antenna [ .2], we use this fact–to develop our procedure.
For a given value of AO, the current
phase distribution is determined. On one side of the feed, the phases are interpolated numerically by a best fit–straight line and the resultant least square error is then associated _with the J!Oparameter.
The procedure
then is simply a search for an optimized value of AO (critical loading) by minimizing the least square error.
The phase obtained in this manner is
almost linear except at the feed and end points where radiation occurs, which guarantees an almost reflectionless current distribution on the antenna. Results are given for the critical loading p-arameters,antenna currents, input impedances and radiation patterns, versus’various differentant-ennadimensions and ground paramters. --
97
11.
BASIC FORMULATION
In deriving an integral equation for the antenna current, knowledge of the radiated field from a current element is needed.
The major steps
in constructing this field will be briefly discussed in this section. The geometry of a current element Pl over an imperfect ground Ls depicted in Figure 1.
al=
Regions 1 and 2 are characterized by (El = CO, PI = UO,
O) and (C2 = c=
●CO,
P2 = PO, a2 = a), respec~ively, where co and NO
are free–space parameters. The current.element is in the x–direction (horizontal) and its coordinates are (x’, y’, z’).
Furthermore, the .
geometrical image of P~ is designated by P2 and their distances from the observation point O (x, y, z) are labelled as PIO”= Rl”amd P2CI= R2, respectively. Our objective is to find the radiated field of PI at the observation point O in the presence of the imperfect ground. As originally observed by Sommerfeld [ 4], two components of the Hertz potential are needed for current element problem.
a
0.
complete description of the horizontal
However, for the vertical case only one component
would be sufficient. The two components are chosen to be in the x and z directions, i.e., along the current element and along the normal to the interface, and are designated as follows A
ii=i-qi+nzz
.
Later in this paper, quantities like TIlx,112x,etc., will be used which specifically define these quantities in regions 1 and 2, respectively. Employing Max-welltsequation and using the Lorencz ,guage,one arrives at . —-—= (3a)
“i!=” JOCv’X~
(3b)
:=vv”fi+~%
(3C)
98
9=
“z
o_, I
‘0’PO
rlll 1111 Illlllllr
Y
pl
1, I I Itltlfillllll
Fig. 1.
1111+111111
1
“x
7’
‘02 I
o2
Ill
I
R2
‘2
Horizontal current element PI over an imperfect ground.
+
where 1, H and ~ are the electric field, magnetic field and current source, respectively. Furthermore, throughout this work, the time convention
O
exp(jut) is used and suppressed in the formulations. For the problem at hand, the only source term is tl=
;~
dx- 6(x- x“) 6(y - y’) 6(Z - Z’)
,
(4)
where I dx’ is the current element moment. The continuity of the tangential ~and
. —— .—
fifields at the interface .— .-—>
results in the following boundary conditions
(5a)
(5b)
‘lx = K ‘2X
—
.. Jr 12 = K ’22
+?lx=
(5C)
a
K -z‘2X
“-
:
(5d)
-1 - jo(ucO) . Our goal at this point is to determine =——The boundary value problem (3) - (5) has an exact solution -.
where K = c /c 2 l=cr 11 and rl lx 12”
g
in terms of the well–known Sommerfeld integr-als. These integrals talizmany differen~ forms [5,6,7], and the following versions will be used for simplicity
ni+n~x lx = ‘lx where
(6]
-jkR1 —— ‘;x = 10 ‘4nR,
(7)
.
Z’nd
● 5) exp(-jkR2
cos
e2
cos
~) dE
(8)
where k = u%,
I
o
(2) 02 is shown in Figure 1, and HO
= (jwcO)‘1 I dX’,
is the Hankel function of zero-order and second kind. path I’is shown in Figure
2.
The integration ,
We have split ITlxinto ll~xand JTr in,order to lx
emphasize the contributions obtained from the source at point 1, i.e., incident, and from its image at point 2, i.e., reflect-ed,respectively. Furthermore, II takes the f-ollowingform lZ
;H(2) (kR2 sin 02 sin g) exp(-jkR2 cos 62 ,COS ~) dg 1 where
kl[z)is
the Hankel function of first order and second kind, and ~
shown in Figure 1. In (8)
0
(9)
Expressions (8)
and
and (9), (9)
2
one must retain Im[~K - sinz & 1 < 0.
cannot–be evaluated in a closed f-orm,and
recently there have been s.pmeatt=empt-s to evaluate an equivalent version of them in the numerical sense [5,6,7]. The present authors have also developed an efficient method for numerical evaluation of (8) and (9), The results of—this inve-stigat-ion will be reported in their future work . In this paper, only the asymptotic det-erminationof
and (9) would be of
(8)
interest, because”it is assumed”that the current element height z = h > A/2 and the observation point—are away f-remthe interface.
It is’known that
under these conditions asymptotic values of (8) and (9) would be an accurate approximation [5,6,8].To thisend, one-replaces the Hankel functions in (8) and (9) by their asymptot-icexpressions and employs the standard saddlepoint integration technique to finally arrive at
..”[
~r ““lx‘“ 10
-jkR2
2 Cos ez
-1. e
—
Cos e z +,/lK
-
4TR2
2
sin
‘2
101
1
(lOa]
_-
0
lrrx
-77/5
d
r’ -. .
Fig. 2.
Integration path r in the complex ~-plane.
—
0. 102
..
and )
cos
0.
.zV
m 21 cos 42 sin 82 cos 6 II ,2 lZ .0
r Kc,ose+K-sin’e
$
..
lllz
Q
/’
2
-jk cos 62 lllz
-jkR2 e 41TR0
sin2 0. L .
/K -
-
(10b)
L
2
.
(1OC.)
Since for angles 02 w O the argument of the Hankel functions in (8) and (9) is small, it might appear that these Hagkel functions cannot be replaced by .
their asymptotic expressions. “However, if one proceeds to use,the asymptotic expressions and then finds the limit as e2 m O, the results of (10)
be recovered. Therefore, (10) is valid f-orall 62 as long as the
will
aforementioned conditions for asymptotic approximations are satisfied. Substituting (lOa) through (IOC) into (3c) and employing the far-f-ield approximation, one arrives at the following expression for-the ‘r E (reflected) 4)
f-ield:
K
“E:62=
COS
6
-k2 10 COS 8 COS $2
2
-
/
K
Sin2 %2
-
-jkR2 e
62
4nR2
-.—
-~ ——
..=[
K Cos ‘2-+’J’K
-
(11)
sin2 82 ---1
and
~; ~2 = -k2 10 sin 42
[.’ Cos 8
2-~K-sin2’2_
e-jm2
—
4nR2
~
2
“
(12)
Cos e .1 2+#-sin292
In (11) and (12j-~2 and ~2 are the unit vectors of the spherical coordinates centered at the image point 2, shown in Figure 1, and the expressions in the brackets are the well-known Fresnel reflection coefficients [ 9].
It is, of
course, an easy exercise to find the Cartesian components of ~r from (11) and (12). Furthermore, substituting (7) into (3c), one can readily arril~e at–the f~llowing f-ar-fieldapproximation for the current element 103
0 +i
E
= k2 10 -L
where (R , 6 , $ ) are the spherical coordinates centered at the source 111. point 1, shown in Figure 1.
0
.
,..
@
III.
INTEGRAL EQUATION FORMULATION
The geometry of a thin linear antenna of length 2L and radius a mounted horizontally with height h over an imperfect ground is shown in Figure 3. It is assumed that the antenna is loaded with a resistive loading function A(x), given in (l), and is fed from a finite source gap located at the midpoint of the structure. .
In this section, our ob-jectiveis to establish an
E-integral equation for the antenna current. Let us denote the incident tangential electric field produced by the source gap as E‘nc and the tangential scattered field by Esca.
The total
tangential E-field may then be expressed as
o
Application of-Ohm’s law allows one to relate t~e EtOt”and the induced current
I(x) as
E
tot
= A(x) I(x)
where A(x) is defined in (l). To find Esca, we first define some new functional,
Since both the
observation and source points are on the antenna, from (6), (7) and (9),
Gh(x,
X“)
=
1~1 ~lx = g(x, x’) - gi(x,
x’)
+ gh(x,
x’)
(16)
,
A e-j kR g:(x, x’) = —;~= .1. 4iTR
F
(x-x
105
x+
+
az
)2+4h2
(17a)
(17b)
—
.—
.
-—. .
(x)=
A. l-lxl
/l_
20
x
..
-....=
~.
_-
.=.. -— .
–*
Fig. 3.
Nonuniform resistively loaded horizontal antenna over an imperfect ground.
—
2 Cos 02 +%
x’) = Cos e, +=
and 0
2
= Arctan (lx - X ‘1 /2h).
“(X’
‘“)
In (16) subscripts h and i are used to denote
“horizontal” and “image” terms, respectively. Furthermore, it should be mentioned that in defining (17a), we have incorporated the usual thin-wire approximation: Finally, from (lOb)-,the-following functional (vertical) is
defined
Gv(x, X’) = ljl ~
IIlz =
-2jkw
sin 0
2
L
COSL e
2 K
COS
‘d
2
+K
L 2 “--sin 02
“ ‘i(x, x’)(18)
where (x - x“)/jx - x“l accounts for the change of the sign 05 cos +2 in (lOb). It--isapparent
for the perfect ground, i.e., K + ~, both gll
and G’ vanish. v The tangential scattered field Esca can be determined by substituting (16)
and (18) into (3c). Finally the desired integral equation for the
antenna current is obtained from (14) as foll”ows L E‘nC(x)
d2 + ~2
= -(ju&O)-l
—
“(
dx2
-1 d -(jucO) x
~ . -L ) I
Gh(x, X’)
1(x”) dx”
L ( j
Gv(x, x’) 1(x’) dx’ + A(x) I(x),
-L
-L ER(A02). This rendition staLes ~hat A; is within the interval [AOl, A03]. Note-that we areinterested in tilefirst local minimum as the smallest value of fi.c, which is most desirable. Sll(ccs>ive o parabolic interpolation ~llrou:hthe cl]reepaints and the finding of a new minimum will finally converge LO the optimal A ;’
It has been found tha~
bett=erconvergence is obtained by ignoring the fe~d and Lllernd patches in coxp~ltinsER(fi.O).This observation is n_ottoo surprisi~igas the radiation OCCUI-Smainly at these poi~.~saildwe expect the current bella\~ior to deviate from its characteristic ~rsveling wave,
. .
. .
-. -.
. .==
-_
● 110
‘-
—
.— .—
. ___
v.
NUMERICAL R_ESULTSAND DISCUSSION
The first example will consider a two-meter dipole antenna radiating in free space at frequency f = 50 MHz (A = 2/3 m).
Figure 5 shows the
magnitu~e and the phase of the induced current for three different loading > parameters A
o“
The critical loading A; = 247.2 $2was det-erminedin the
manner described earlier. As the loading parameters take the no-loading value A
o
= O, the intermediate.value A. =
120,
and finally the critical
loading A. = 247.2, the phase behavior becomes more linear. On the other .—— hand, the amplitude curve assumes a non-oscillatory behavior and varies similar to the current of a nonref.lettingstructure, We have chosen the example in order to compare our results with those given in [ 2 ].
In this
reference the value of the critical loading was reported as Ac = 318 0, 0 though no specific procedure for the”determination of Ac was described. o We believe that the main difference between our result and the one in reference [ 2 ] lies in the application of—diff-erentnumerical schemes f-or reducing the integral equation into a matrix equation. In order to fully investigate the effect of ground and loading on the .
antenna performances, we consider a center-fed dipole antenna of length 2L = 10 m located at a height of h = 5 (Er
=
10, u = .01 mho~m).
m
above an imperfect ground
Table l_contains the values of critical loading
A; as a function of 5requency for two different radii of the antenna in frcze space.
In this’table, the minimum error, i.e., ER(A~), is also tabulated. .—.
It is interesting to note that for the entire frequency range, a smaller A: is needed as the radius changes from .025 m to .05 m. var.iatibnof—At a> a function of o
E
r
Table 2 shows the
and o for a fixed antenna radius a = .05 m.
To illustrate th”efunctional dependence of–ER(Ao) on A 111
o’
two plots are
III (ma) 5
——.--4
4
—.—.
--
—.
A<
o
AO=120
3 2 I o
d
I A
18($ 135° g O“
45° .
0
-45° -90” - I35” -180° }-ig.5.
Magnitudeand phase of the currentfor three.differentloading paramet-ers.2L = 2m.and~ = 2/3m. .—. . 112
‘
‘“
.
TABLE 1 CRITICALLOADINGPARAMETERA; AND THE MINIMIZEDERRORER(A;) VERSUSFREQUENCYFOR TWO ANTENNARADII..2L = 10 m.
,,1
,,
II ~~
Radius= .025 (m) Freq. (MHz)
P P w
c
Radius= .05 (m) c
ER(A:)
(~ym)
ER(A:)
(lpm)
Radius= .025 (m) Freq. (NHz)
c
ER(A;)
(lYm)
Radius= .05 (m) f c ER(A;) (~7m)
28.4
4.
54
89.0
2.94
72.3
2.40
6.4
39.2
7.9
57
45.0
11.6
60
1.94 1.81
69.7
9.6
88.2 88+2
70.6
1.75 1.19
61.7
12.3
52.7
14.5
63
85.2
1.50
70.7
1.38
30
67.0
14.3
59.8
16.2
66
82.8
1.63
67.4
1.43
33
75.1
16.7
63.4
16.5
69
82.5
1.63
64.6
2.06
36
82.3
16.4
- 70.0
15.3
72
82.8
1.94
64.0
39
81.6
11.9
71.9
11.0
75
80.6
2.2
63.1
‘2.31 2.88
42
86.3
9.98
73.1
9.75
78
80.3
2.38
63,2
3.06
45
88.9
7.81
74.4
1.67
81
80.1
2.63
62.0
3.56
48
89.6
5.56
74.4
5.5
51
89.8
4.5
73.1
3.9
84 87
78.2 79.5
3.31 3.38
60.6 59.6
4.56 5.06
90
76.3
4.18
59.6
5.56
95
74.8
4.94
60.1
7.0
100
73.6
6.56
40.3
10.1
18
35.3
3.2
21 24
47.6 55.E
27
~
TABLE 2 CRITICAL LOAl)l_NC PARAllll’L’LR A; AND TIIE}[INI}IIZED LRROR lIR(A:)VERSUS FREQUENCY FOR DIIWIEiJ’1’ GROUN])PARIWIETERS . 2L = 10 m, AIWIh = 5 m.
.—— I E
r
(J=O
=1,
E
mllos/m c
Freq. (MUz)
LR(I’$)
(19m)
-1 30
59.8
~
16.2
59.4
I .I.6.9I
60.6
\
c
IVZ(A;)
(J7m) — ------53.5 I 15.9
c
ER(A~)
(:?nl)
+a--++-
I
63.4
33
16.5 I [
36
u 5.9
39
71.9
52
73.1.
i
11.0
,5 I i54
16.4
76.2
:1.8.0
72.6
18.6
1.3.2
79.2
! 14.7
L
9.75 —
45 ——
74.4
48
74.4
51
54 57
7,67
82.1
11
+-----J 77.2
1.2
66.6
83 “O—l:fl85.9 ‘;
8.2
——c
—
.
8.5
——
2.6
_ 1.6 I
2.4
‘
. II
67.8
19.9
80.9
17.9
89.,7
14.4
.
97.4
10.1
2.9
151.1
16.2
~~-+ . ,, —— 160.2 —— :139.6 i ——. .— L 87.L ! . . —,+ 69.2 1
++-*1 ~::: ———.——
. ,,
~
5.5 — 73.1 3.9 . —.. .-— 72.3 2.4 Pll,.1 .. .— —.. .—-— — — —-—— 70.1 69.7 1.75 70.0
/
a
60”’ 20.0 68.1
11.6 L
—
60
I
79.6
I :18.9
79.b .—,
1.Y . —..”
69.9
2.3
61.3
3.4
60.9
I
[
3.4 --——
1.1
4.0
x ~o(} .’(JI ~— \ 7~.8 ~
8.8
I
1.6
59.7
. 1,
=I,u=.,
-r
I
1“’ 1.9 4.1 9.0
-
constr-uct-ed in Figure 6. g,r~~ndon ER(AO).
These p-lotsshow the effect of the imperfect
Since the antenna height is h = A and the ground has
low conductivity, the two results do not differ markedly. The effect of AO on the input resistance of the aforementioned antenna is depicted in Figure 7, where the two cases of no ground and imperfect ground are considered. This figure suggests that as AO becomes larger, in this case for instance A > 30 Qlm, the input resistance levels off and o does not–vary significantly. For yet--anotherexample, Figure 8 displays the variation of the inp-utr.eslstanceversus frequency for loading parameters !, = O, i.e., no loading, and A o
o
= 40
Q/m, though this figure is for the no-
gro~n-dcas_e.(cr_ = 1, CI= O mho), it is clearly seen that when the antenna is not loaded, the input resistance varies with the resonant frequencies of the antenna, whereas when it is sufficiently loaded, the input resistance does not vary markedly as the antenna ends are not seen from the feed point. To invest-igatethe radiat-ionp.atterncharacter.isticof a loaded :~nte~,na over an imperfect ground, a center-fe~’tli~oleantenna of length \ 2L = 10 m and radius a = .05 m located at the heigh;’h = 5 m over ,an imperfect groiiiid (Ew = 10, u = ‘;O1.-mhos/m) is cc)nsidered. Furthermore, it–-is L
.
assumed that the antenna is radiating at frequency f = 60 MHz (A = 5 m). From Tables 1 and 2, it is found that the critical loading is A; = 70.6 (fl/m) in the ab-se-nc-eof the ground and A: = 61.3 (0/m) in its presence, resp.eccively. Figure 9 shows amplitude ._. _ the ——.—... __ .-: . and phase distributions of the ..... ., . . ., z?;~.:>nz current I fur the c,lsesof–both no loading and critical loading. Agsin the Iiilearphase behavior is obtained f-orthe critical loading case. The ladiation pat:erns of the aforementioned antenna are plotted in
a
-’-.p~[!r,?s 10, ]1, :~lt.. <.,; ~J—(j–t2L~
.?~l(j
‘s}Iohm
~-.
All p:itternsare no-rmalizedto the maximum value
in F-igure‘lo. In each figure radiation patLerns of– 115
—
ER NO
20C
GROUND
IMPERFECT
-—.—.
‘Y =
.
GROUND
10,a= 0.1(mhos/m)
I 5C
I 00
NG
50
A:
0
20
0
40
60
80
.
I 00
Ao(~/rm)
6.
Effects of imperfect ground on ER(AO). 2L=10m, h=5mand X= 5m.
-
0 116
.
. ., A
0
!-m —
m w
270
——
260
—.—-
‘Y=10,
lx
--
z
220
~=
GROUND 0,01 (mhos/m)
\
240
230
IMPERFECT
—-
\
250
t3
NO GROUND
—..-
---
CAL
Q
LOAD NG
210
o
20
40
60
80
I00
f10 ( Q/m)
Fig
c-t 7. Effect of A. on inpu-tresistance, no ground and imp-e-tie 10 m, h = 5 m and A = 5 m. cases. 2L =
A;
1
1000
-
cj
800
-
u
l\
(J
l\
z
K
II I
I
/
I
I / /
400
\ \ \
/’
t-
‘\
,
3
“n
>Z –
200
100
10
I
I
20
30
Fig. 8. ~npu~ resistance
●
f
40
versus
●
I
I
50
60
FREQUENCY
(MHz)
frequency
I
70
for loaded and unloaded antenna
●
I
I
80
90
in free
sp’ace.
.
b
“
2L = 10 m.
I
0
,, ,,,,
I
III (mA),
A; = 70.6(~/m) Ao= o (0/m)
.——.
3
~/
/
-\
/
2
I
\
,
\
\
-11
●
\ \ \ \ \ d/’
I
1
o 0
5
x (meter)
A,
\ \
18d \
135°
\
\
\
90° 45°
●
0°
!
I
1
I
I
I
#
t t
,
I
I
‘5
x (meter)
-45° -90°
-135° -180° Fig. 9. Magnitudeand phase of currentfor no loadingand crit-ical loading. 2L=10m, and A=5m. —
119
o
Az E
CRITICAL ——— —
LOADING
NO LOADING .
/
\\\\ \-(
NO GROUND
IMPERFECT
%
=
GROUND / O,c=O.Ol(rnhos/m) /
&.
1.0
q= 90°
-J \ /
\
/
\ \
(+’ /
+=180
Fig. 10.
”’4/=[80”
in plane X-2 for no ~round, imperfecL ground, Radiation pattern loading. 2L = 10 m, h= 5 m, and A = 5 m. no loading and critical
● 1.20
CRITICAL --—-
NO
LOADING
LOADING
*
NO ,
GROUND
IMPERFEC 6r=10, c=0.01(mhos/m) * /
w
II
-1--+ \l
‘--‘y-@
●
= 180’ __.4..____
Fig.
11.
Radiation pattern in plane Y-Z for no ground, imperf-ectground, no loading and critical loading, 2L= 10 m, h = 5 m and A = 5 m.
121
CRIT!CAL -—-.
/
GROUND
er=lO, a’O.Ol(mhos/m) f
/
[
NO
o
GROUND
‘~,
1,
\
\
‘\
\ /
NO LOADING
Yf~
E IMPERFECT
.
LOADING
1
/ /
—’.
\
/
It / / /
1 0.73
f
‘+
1
\“
\
=
\ \
i \
+---
\,
\ \
\
/
/
Y+
= 180°
Fig. 12.
Radiation pattern in X–Y plane for no ground, imperfect gmmd, no loading and critical loading. 2L= 10 m, h = 5 mand A = 5 m.
122
90°
the antenna in both the presence and absence of the ground for no loading and critical loading are plotted. results
This allows the reader to compare the
simultaneously. Except for the antenna pattern in the plane
perpendicular to the antenna (Figure 11),.the shape of the pattern is changed due to the loading effect.
Except for the antenna pattern in the
plane parallel to the ground interf-ace(Figure 12), the pattern is effectively influenced by the presence of the imperfect ground. results can intuitively be verified for most cases.
●
These
REFERENCES [1]
Special Joint Issue on the Nuclear Electromagnetic Pulse, IEEE Trans. on Antennas and Propagation, January 1978 and also in IEEE Trans. on Elec&ramagnetic Compatibility, February 1978.
[2] Y. P. Liu, D. L. Sengupta, and C. T. Tai, “On Eh-eTransient Wave Forms Radiated by a Resistively Loaded Linear Antenna,” —Sensor .—— and Simulation Notes, Note 178, February 19.73. [3] F..M. Tesche, “Application of the Singularity Expansion Method to ~he Analysis of Impedance Loaded Linear Antennas,” Sensor and Simulat-ion Notes, Note 177, May 1’373. — [4] A.Sommerfeld, Partial Differential Equations in Physics, New York: Academic Press, 1964.
i
.
[5] E, K. Miller, A. J, Poggio, G. J. Burke and E. S, Selden, “Analysis of Wire Antennas in the Presence of Conducting Half-S.p.ace:Part II The Horizontal Antenna in Free Space,” Canadian J. or Physics, Vol. 50, pp. 2614-2627, 1972. [6] T. K. Sarkar and B. J. Strait, “Analysis of Arbitrarily Oriented ThinWire Antenna Arrays Over Imperfect Ground Planes,” Scientific Report No..9 on contract F19628-73-C-0047, AFCRL-TR-75-0641, Syracuse University, Syracuse, New York, December 1975, [7] A. Banes, -Dipole Radiation in the Presence New York: Pergamon Press, 1966.
of Conducting
Half-Space,
[8] J, A. Kong, Theory of Electromagnetic Waves, New York:
Wiley, 19754
[9] E. C. Jordan and K. G. Balmain, Electroma~tic Waves and Radiatin& Systems, New Jersey: Prentice ~~~68. ‘———” [10] R. F. Barrington, Field Computation by Moment ?Iethods,ItewYork: llcMillan,1968. [11] R. F. Barrington, “Matrix ?!ethodsfQr Field Problems,” Proc. PP. 136-149, February 1967.
IE~,
[12] J. D. McCannon, “Numerical Analysis of a T1~in-WireAnt-~nnaliea~Lossy Ground,” M.S. Thesis, University of Illinois, Urbana, Illinois, 1972. [13]
}fit~ra
and
w,
“A Finire Difference Approach to the Wire Junction Problem,” IEEE Trans. Antennas Propagat., Vol. AP–23, pp. 435-438, May 197~— R+
L,
Ko,
(14] P. Parhami, Y, Rahmat-Samii, and R, ?fittra,llATechnique for Calculating the Radia~ion and Scattering Characteristics of Antennas Mounted on a Finite Ground Plane,” to appear in . Proc. —.—— IEE .
124
i? U.S.
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●
