THE VALIDITY OF APPROXIMATING CURRENTS ON A RESISTIVE STRIP
USING PHYSICAL OPTICS
Randy Hanpt
Ah. Force Insritute of Technology
The U n i w s i of Michigan
Ann Arbor, Echigan 48109
Val& V. Liepa
RadiatiDn LaborDept of Electrical Ehg and Computer Science
The UnivMiky of Michigan
A m Arbor, Michigan 48109
Introduction
Calculating the current8 induced on an object when excited by an electromagnetic
wave,usually inmlvea approximate methodn. R m l y do exact solutions exist. Physical
0 p & 1(PO) offers a simple approach to the problem at the expense of accnracy. S i c e
PO ia a local phenomenon, it does not take into account interactions between surfaces,
d i h c t i i n from edges, tips, corners, ek. On the other hand, more accurate techniques
require complicated computer solutions.
This paper investigates the validity of applying PO to 6nd the currents induced on
a nsistive strip by an incident plane wavc Prim in~e~tigations
did not coneider PO aa
a viable method. %or and Liepa obtained the currents induced in resistive strip by
&ng the method of momenta (MOM)[l]. More recently, Ray and Mittra used the spectral
iterakive technique to 6nd the -nrS
in such ntrips[2]. In these two studies the atrip
either had a uniform or a tapered resistivity. In thio study currents on strips having vations
reSisrMtim were computed ming MOM and PO expresaom. The results indicate that PO
provides simple,
upressiono for the current when the resistiviw on the striphaa a
smooth taper with large valuea of reeistance at the edges.
=curate
Problem Formulation
Figure 1 is a model of the problem. The resistive strip is 4
. wide and lies in the xy
plane centered on the B axis. Although both Epolarired (E. = ezp(-ik(zcosdo
ysin&)))
and H-polari.ed (Hz= e z p ( i k ( z c o s + o lain&,))) incident waves were inwtigated, only
the Epolarised d
b
sare detailed here.
+
+
The bound-
condition at the r
e
s
s
it
v
iesurface for an Epolarised incident plane wave
is givan by111
E, = J=R
and
e -HF = - J ,
Plas and mimas mgna on E, indicate upper and lower faces of the strip respectively. R ie
the resistivity of the atrip in ohms. When R=O the str;P is perfectly conducting.
The integral equation for 6nding the cnrrant on a resistive strip resulting from an
Epolarbed incident field is
CH2325-9/86/0000-0137$01.00 @ 1986 IEEE
137
where EP’ is the zero orderHankel h c t i o n of the k t kind and Z is the inttimic
impedance of free. space. This equation w a programmed and mlved using MOM.
A much simpler method is to use the PO cxprrssions for the cment[3]given by
Thi. equation provides an easp aolution for the problem, but does not take into account
edge -ion
nor interactions between anxfacea. Thus, PO is appropriate in casea where
edge d i h c t i o n s and snrface intersctiona can be ignored.
Comparieon of MOM and PO Solutions
The MOM and PO solutions for surface fields are compared for various strip resistivities
and angles of incidence. The snrface field for Epolariration is given by ZJ.(z) and for Hpolarisation J,(z)/Z. Fignres Z through 5 ihatrate the comparison. Figure Z shows the
surface 6eld induced on a perfectly conducting strip due to an E-polarired plane wave at
normal incidence. Agreement between MOM and PO is good at the center of the strip and
poor at the edges. Changing the angle of incidence from normal creates greater disparity
between the ~ 0 h t i 0 ~ .
Figare 9 shows the d a c e fields induced on a nsistive d r i p that has a uniform nom a l i d resistance of K , = R/Z = 2, where R is the strip reaistivitJr and Z the intrinsic
impedance of free space.PO and MOM have bettar agreement on a resistive atrip, but PO
still faibat the edges.
Figure 4b shows the d
a
c
e field9 on the strip w
i
&resistive edge loading. The strip is
4A wide. The center of the strip is perfectly conducting and the outer ends .5X from either
edge have a normabed resistivity, R,, = 2 (Figure 4a). Although the edgea are loaded,
there ia an abrnpt discontinuity in the resistivity of the strip. As a reanlt, PO and MOM
do not agree well.
Figures 5a,b, and c demonstrate the effececta of a smooth resistive taper. Figure 5a shows
the normabed resistive taper vn. distance along the strip. The correspondingd a c e fields
induced by an Epolarired plane wave are shown in Figure 5b. Now, there ia a remarkable
agreement betweenPO and MOM solutions. Figure 5c shown the solution for an H-polarized
plane wave. In both cases PO providea a reasonable alternative for finding the currents on
a resistive strip.
Condnaion
Under two conditiona PO offers an alternative to MOM when calculating the snrface
fields (oraments) on a resistive strip. However, there are restrictions: &at, the reaistivitp.
at the edgcr must be very high and second, the W v e taper acroea the atrip mnst be a
smooth u possible.
RCfVenCcS
1.
2.
3.
T. B. A. Senior and V.V. Liepa, nBadEscattering from Tapered Reaistive Strips,’
IEEX lhna A P , vol32, no 7, July 1984, pp 751-754.
S. Ray and R. M
i- ‘Scattering from Rwiative Strips and from Metallic Strips
with Resistive Edge Loading,’ IEEE 1983 A€’*
Intl Sym Digest, pp 805606.
T.B. A. Senior, ‘Scattering by Resistive Strips,” Radio Science,vol14, no 5,
SepOct 1979, ‘pp 911-924.
138
SURFRCE F I E L D ON STRTP
I
I
INCIDENT
FJELD
SOLD
LINE
PO
ORSHEO LINE MOM
H
U.
\
I
.\
\
\
-1.96
POSITION ON S T 8 : I N
I.%
URVELENCTHS
2. Surface Fields on aPerfectly
Conducting Strip E!-PoIarixation
Figure 1. Geometry of theProblemFigure
SURFACE F I E L D ON STRIP
SOLID LINE PO
ORSHEO LINE non
1
x
N
0
J
w
H
LL
3 -1.96
POSITION ON S?i' ?!
1
I N UAVELEhg?tlS
Figure 3. Surface
Fielda on a ResistiveFigure
Strip ( R , = Z), Epolarisation
139
4a. Strip with Edge
Loading
SURFACE FIELD ON STRIP
SOLIDLINE PO
DRSHEO L I N E ROB
2
0
-I
W
H
LL
W
s
a
U-
a
VI
0
3
Figure 4b. Surface F'ield
Figare 5a. Resistive Taper
SURFRCE FIELD ON STRIP
SURFRCE FIELD ON STRIP
S O L I 0C I N E
SOLID L I N E PO
ORSHEO U N E BOB
PO
ORSHEO LINE ROB
-1.96
O
NITP
O
S
I
Figure 5b. Surface F'ield, Epolarisation
ON ST%
N
I
UAVELE~~~HS
Figure 5c. Surface Field, H-polarisation
140