Leaky modes in parallel-plate EMP simulators

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-20, NO. 3, AUGUST 1978

Leaky Modes in Parallel-Plate EMP SimulatorsALI M. RUSHDI, STUDENT MEMBER, IEEE, RONALD C. MENENDEZ, MEMBER, IEEE, RAJ MITTRA, FELLOW, IEEE, AND

SHUNG-WU LEE, SENIOR MEMBER, IEEE

Abstract-The finite-width parallel-plate waveguide is a useful toolas an EMP simulator, and its characteristics have recently beeninvestigated by a number of workers. In this paper, we report theresults of a study of the modal fields in such a waveguide. Once thesemodal fields and their corresponding wavenumbers are known, theproblem of source excitation in such a waveguide can be solved, at leastapproximately, using the so-called leaky-wave concept.

Key Words: EMP simulators, parallel plate, leaky modes.

I. INTRODUCTION

MANY EMP SIMULATORS make use of a parallel-platefinite-width waveguide (Fig. 1) as a guiding structure for

the electromagnetic field. A complete description of such afield requires a superposition of the TEM mode, higher orderTE and TM leaky modes, and, finally, a continuous spectrum[1]. For low frequencies, the TEM mode is dominant; how-ever, at higher frequencies for which the free-space wavelengthis of the same order as the cross-sectional dimensions of theguide, the TEM mode alone may not be the dominant part ofthe field.

The properties of the TEM mode on a parallel-plate simu-lator have been investigated elsewhere [2] -[4] . Our goal herewill be to obtain the wavenumbers and associated field distri-bution in the region between the plates for the higher orderTE apd- TM leaky modes. In spite of the fact that these leakymodes are not proper solutions of Maxwell's equations in theentire space and do not, in general, form a complete set oforthogonal functions, they can nevertheless be employed toobtain a convergent representation of a dominant portion ofthe source-excited field in the region between the plates [1],[5] . The method of obtaining such a representation is discussedin Appendix A.

Recently, a number of works [6]-[8] dealing with thehigher order leaky modes on a parallel-plate simulator haveappeared. The work described in this paper is an extension ofthe transverse-resonance approach of Mittra and Itoh [6] tothe coupled-mode case. The approach of Marin [7] and [8]has similar aims to those of this paper, but he specializes hissolution to the case of narrow plates [7] or wide plates [8] .

The electromagnetic field inside the parallel-plate simulatoris uniquely decomposed into TEZ and TM, modes derivablefrom two scalar potentials h, and e,, respectively. In addition,

Manuscript received November 4, 1977; revised March 15, 1978.This work was supported by the Air Force Office of Scientific Researchunder grant AFOSR-76-3066.

A. M. Rushdi, R. Mittra, and S. W. Lee are with the Electro-magnetics Laboratory, University of Illinois, Urbana, IL 61801. (217)333-1200.

R. C. Menendez is with Bell Laboratories, Holmdel, NJ.

(a)y Y

/ z z Wx -*

2h

2w(b)

Fig. 1. Two perfectly conducting finite-width parallel plates. (a)Perspective view. (b) Transverse section.

the field may be symmetric (even) or antisymmetric (odd) inthe direction perpendicular to the plates (i.e., with respect tothe plane y = -h). Thus the field is decomposed into fourparts: even-TEe, odd-TE_, even-TM,, and odd-TM, modes.Those four sets of modes are uncoupled [1] and, hence, theycan be investigated independently.

II. FORMULATION OF THE PROBLEM

Consider the auxiliary two-dimensional problem of an elec-tromagnetic wave incident in the transverse plane in the insideof a semi-infinite parallel-plate waveguide (Fig. 2).

Let the scalar potential of the mth mode of this incidentwave be (for exp iwt time convention)

Um + = amf(my) exp (ikxmx)

where

cos (2hyf(my) =

(sin 2hY

TEZfor modes

TMZ

(1)

(2)

0018-9375/78/0800-0443$00.75 i 1978 IEEE

443

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-20, NO. 3, AUGUST 1978

whose lth mode is

VI+ = ( SIn(kt)bn) f(ly) exp (-ik.nX ) (10)

The transverse resonance condition can now be written as

Fig. 2. Open-ended parallel-plate waveguide with a semi-infinite width.

for the even TE, set

vl+(x = -w) =um+(x = W)6lm, for all m

where 61m is the Kronecker delta02,4, ,

= 2, 4, --,

1,3,5, *,

for the even TMz set

for the odd sets

kxm = /k2 - (mnr/2h)2

k=e

(4) Hence

(5)

This incident wave will give rise to a reflected wave whosenth mode is given by

Un=(- Snm(k m) f(ny) exp (-ikxnx) (6)

where Snm(k) is the reflection coefficient from the mth modeto the nth mode (Fig. 2). If we neglect the coupling effectsbetween the two openings at x = 0 and x = -2w of the open

waveguide of Fig. 1, then the reflection coefficient for thisconfiguration also will be given by Snm(k). Expressions ofSnm(k) for the different mode types have been obtained in[9] and are included in Appendix B.

Now, instead of the above two-dimensional problem, con-

sider the three-dimensional problem of an obliquely incidentwave with an axial variation of exp [ikzz]I. Since the structureof Fig. 1 is invariant under translation in the z direction, thereflected wave has the same z variation as the incident one. Infact, it tums out [1] that the above analysis is still valid in thiscase, provided that all field quantities are multiplied byexp [ikzz] and that k is replaced in (4) and (6) by an equiv-alent wavenumber

kt= k2-kk 2. (7)

Now we introduce the change of coordinates x =-(x' +2w) so that (6) becomes

Un-= bnf(my) exp (ik.nx') (8)

where

bn= (Snm(kt)azm) exp (i2kxm w). (9)

Equation (8) represents the nth mode of the wave reflectedat the right opening x - 0. This wave will be incident on theleft opening x = -2w (x' = 0), giving rise to a reflected wave

SIn (kt)Snm (kt) exp (i(k.i + 2k n + kxm )W)n m

=am6Im3 for all m.

If we let

Rnm Snm (kt) exp (i(kxn + kxm )w)

(13)

(14)

then, by interchanging the order of summation in (13), we get

( RinRnm) am -6lmam,m n

for all m. (15)

Rnm can be interpreted as the reflection coefficient from themth mode to the nth mode, evaluated at the center plane x-x =-w.

The equations (1 5) can be combined in a compact matrixform

[R] [R]A = [I]A (16)

where A is the modal amplitude vector (evaluated at the open-

ing x = 0) with elements am at the mth row, [R] is a square

matrix with elements Rnm at the nth row and mth column,and [I] is the identity matrix.

Theoretically, the dimensions ofbothA and [R] are infinite;however, we will truncate them to some finite dimension N.

If we write (1 6) as

{[R] [R] -[I]}A = 0 (17)

then A has a nontrivial solution only if the determinant of thematrix between the curly brackets is zero, i.e.,

Det{[R][R] -[I]} =0. (18)

Making use of the identity

Det {[Q1 I [Q2 ] =}Det [Q1 ] Det [Q2]

: 2h

y

zt x

-w.4) Urn-wv,U n

(3)

(11)

5Im = <0O

l=m

l m.

(12)

A .so3

v .

444

k

(19)

RUSHDI et al.: LEAKY MODES IN SIMULATORS

we can reduce (18) into the following two equations

Det {[R] -[I]} = 0 (20)

or

Det {[R] + [I] } =0 (21)

which correspond, respectively, to the vector eigenvalueproblems

{[R] -[I]}A =0 (22)

and

{[R] +[I]}A=0.

(defined with a branch cut along the negative real axis in thekxm2 plane) belongs to the improper Riemann sheet, i.e., theleaky modes grow exponentially in the x direction.

The roots obtained by the SEARCH and HOMEIN subroutinesof [11] were improved further by the Muller method. Theabsolute value of the complex function was reduced to lessthan 10-6 at the location of a root.

Let kz1 and kzla be the roots of the symmetrical equation(20) and the antisymmetrical equation (21), and let the corre-sponding kx be denoted by kxmIs and kxmia, respectively.Here

i0, 2, 4, ,

I =1, 3, 5, ,

for modes symmetrical in x

for modes antisymmetrical in x(26)

(23)

These, when combined together, are equivalent to the originaleigenvalue equation (17).

Equations (20) and (21) are the modal equations for modeswhose axial field components u = hz are, respectively, symmet-rical and antisymmetrical with respect to the center planex = -w.

and m as given by (5) are the indices associated with the fieldvariations in the transverse directions x and y, respectively.The modal amplitude vectors Al' and Al' can now be obtainedby solving

[R-I] (kz s)Als = 0 (27)

III. SOLUTION OF THE MODAL EQUATION

For specific mode type and physical dimensions, [R] is afunction only of the transverse wavenumber kt. Thus kt is theunknown to be determined from our modal equations (20)and (21). This procedure is similar to the one discussed in [6]and [8]. Once kt is known, k, is readily obtained from therelationship k = Nk2-kt2.

The solution of the modal equations (20) and (21) amountsto finding the complex roots k, of complex transcendentalfunctions. This type of problem has been discussed in [10]and [11]. In [10], Muller discusses an iterative method thathas been programmed under the name ZANLYT for the DECsystem 10 and CDC CYBER 175 Fortran Version 4 compilers.The Muller method requires knowledge of the approximateroot(s) of the complex function and it is unreliable for com-plex functions of exponential behavior. In [11], Cauchy'stheorem is used to find the number and location of zeros of ananalytic function in a given contour. This method was especiallysuitable for our purpose, since it allowed us to search for thezeros of the functions in (20) and (21) within a rectangularregion of the first quadrant in the complex kz(=,B3 + icfz)plane. The rectangle was defined by

0<z <k, 0<X<k (24)

which implies that the leaky modes decay as z -+ +±. Therestriction to cxz < k reflects our interest only in leaky modeshaving the smallest attenuation in the z direction. We digresshere to note that the propagation constant in the x direction

and

(28)

To be specific, let us consider the solution of (27). Thematrix [B] = [R - I] (k_18) is an N X N matrix with a zerodeterminant. If we assume that the order of this zero is Lwhere 1 < L < N, then the rank of the matrix [B] is (N - L).The value L = 1 corresponds to a unique field distributionassociated with k,18. On the other hand, for L > 1, a multi-plicity of L degenerate solutions are associated with an iden-tical kZ s. Although such degenerate solutions are known tooccur in closed waveguides, their presence in open guidingstructures has not been reported, and that topic deservesfurther study. _

The solutions Als of (27) and Al, of (28) can thus beobtained numerically with minor modifications of a subroutinedesigned to calculate the generalized eigenvectors of a complexmatrix.

IV. FIELD DISTRIBUTION OF THE MODES IN THEREGION BETWEEN THE PLATES

The scalar potential u of any of the four basic mode sets isgiven by

eve [ ]s]tAls exp (ikzlsz)I even

U =

[C1a] tjIa exp (ikzIsz)I odd

(29)

kxm =+ [k2 h)]-kz2for sets symmetrical and antisymmetrical in x, respectively.

445

[R + II (kz la)Zla = -0.

(25)

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-20, NO. 3, AUGUST 1978

Cis and Cla are vector matrices whose mth elements are,respectively,

1-f(my4[exp (ikxmi8Xll) + exp (-ikxm,.Is(x ± 2w)]2

(30)

and

1-f(my)[ exp (ikxm lax) - exp (-ikx m la (X + 2w))]. (31)2i

With the use of (30) and (31), the expressions for u can besimplified to

Iz amIsf(my) exp (-ikxmlsw)Im

* cos [kxmls(x + w)] exp (ik lsz) (32)u-

Y amlaf(my) exp (ikxmlaW))Im

* sin [kxmla(x + w)] exp (ik,laz). (33)

TM, modes

Ez = z z sin (2h FmI(x, z)

Et = Wi ;z sin (2 G.Im(x, z)xI m \2h/

+ (-) cos (iY) Fz).kIx/ktz 22hFmi(X, kzla/kt1a2

Ht= iwy [ 2)I m L\2h/

cos (ir) FmI(x,z)X

m7T I_(x l/ktl--2+ sin 2h Y) G,, I (x, z)y I l/kt,a 2 (35)

where

Fmi(X, Z)

amIs exp (-ikX,jsw) cos [kxml8(x + w)] exp (ikzlsz)h

=amla exp (-ikmlaw) sin [kxmla(x + w)] exp (ik_aaz)

(36)It is noteworthy that, once the eigenvector equations (27)

and (28) are solved for A1 a, the parameters {am Isa: vm} arenot all arbitrary, for they can be given in terms of a single arbi-trary constant. With this thought in mind, and apart fromdifferences of notation, we can observe that (32) and (33) areessentially the same as (87) and (84) of Marin [8].Now (32) or (33) can be used to calculate the following

field distribution of the modes in the region between theplates:

TE modes

Hz - ¢cos( -y )FmI(x, z)1m \2h/

and

{-kxm Isam 18 exp (-ikxm IsW)- sin [kxmls(x + w)] exp (ik85sz)

kxm ism, (-ikxmlaSw)Gmi(X, z) = k1maam,a exp (-jkXm,aW)cos [kxmla(x + w)] exp (ikz1az)

symmetricalfor sets i m i in x. (37)

I antisymmetricaltJ

In the special case of wide plates

ht =-4 1.

w

I m [ (2h)

-()sin ( 2 Y) Fm I (x, Z)Y]|kakl

Et = -iLU LI m

[2h i) 2hiF.I(x, z)

(38)

Then, among the higher order leaky modes, only the TEZ modesof order m - 0 have a significant contribution to the totalfield. For these modes, the field components are given by

Hz= Fo(x, z)

kzia/kxoia2H1=i~G01(x,z) kzis/kxO012Hx = iGo(,)Ika kzl xola

cm( r l/ktls2$+ cos 2h Y} GM I(X, Z)W 1k 2ll/kxois2(34) Ey = -iclY. G I(x, z) /kos

I llkxoll2(39)

446

RUSHDI et al.: LEAKY MODES IN SIMULATORS

where the parameters aols in FO(x, z) and GoI(x, z) areobtained by solving the scalar eigenvalue problem that isobtained by truncating the dimensions of [RI andA (SectionIV) of the even TE, set to N = 1. In the following section, wewill consider this problem further and we will develop anapproximate solution for it which is valid for

IkxolwI>I and Ikxolh I<1. (40)

V. APPROXIMATE MODAL SOLUTION FOR SINGLETEZ MODES WITH m = 0

Boersma [131 and Weinsten [14] found that for smallk,Oh they can approximate the self-reflection coefficient

for the TE_o modes by

Under the conditions (40), the root of (45) is obtained as

1 / 47rPoi = 1 +-t In t +(1 -C) +iff

. (l+ 1r + O(Pol2t2)

[1[/ 7r\ irl1-- -~l -(±T [ o±( 23)(

*(I1+ I)7r +O(PO12t2)+ OQ2). (48)

A perturbational solution for Po, is obtained if we substitute( + 1 )7r for it in the right-hand side of (48):

Soo = -exp [i2kxoh n

+ 0(kxo2h2)]

where C is the Euler constant

( 2r \rri\

kxo )+ -C)+2)

(41)

C=Lim 1 + +--- + =-Inn=0.57716.n-*oo 2 n

(42)

PoZ [1 -7 (ln [ ±(1-C)+i-

*(+ 1)ir+ 0(Po02t2) + 0(2). (49)

For I = 0, the first of the conditions (40) is not quite satis-fied, so Poo obtained by (49) is not very accurate.

If only the leading terms of the real and imaginary parts areretained in (49), then it is simplified to

I t < 1.The transverse resonance condition for a single TE0o modecan be written as a special case of (1 1) as

S00 exp (2ikxow) = e 1. (43)

If we let

Pot = 2kXolw = Po, I ei0ol (44)

For the first few lower order modes

(1+ l)7rm= w1.2kw

then the solutions of (43) are given byfunction f(POl) defined by

the zeros for the

1 4rr rif(Po0)=Pol +-Polt In .. + 1-C) +-

Pozt 2

- (I + 1)7r + O(Pol2t2)

Then the axial propagation constant k8 is given by

k81o 2

kzo= (12kw)

(45)

where the logarithm in (45) is interpreted as

In (-) =ln -i0ol1 (46)

A solution P', of (45) is expected to be in the fourth quadrant-r/2 < 0oI < 0. Due to the approximation in (41), our modalequation (45) is now not dependent on both h and w, butrather on their ratio t. Since Limt,O t log t = 0, we get

PO, -1T(I + 1) as t 0

which could have been anticipated.

which has positive and real and imaginary parts.From (39), the transverse field distribution is given by

Hzolsl lcos kxojsxl

- Hzolaf sin kxolaxl

cos 1x3olsx1 cosh c°xolxl )

- i sin Ox3oi8x1 sinh x 0189x

sin 3xolaxl cosh czxosxx l+ i cos fOx018x1 sinh exolsx|

(50)

(51)

(52)

(53)

447

,- kNrl.-X.12 +itX,2

2

Po, -- (I + Olr Ii.

5

2-

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-20, NO. 3, AUGUST 1978

where

X1 =x+w (54)

which, under the conditions (40), can be simplified with theaid of (50) to

Hzo,8 cosgxco 1xl cos (kXzxl))Re s

Hzol" sin I3Ox Glxl sin (ky1xl)(55)

and

(Hzols -ax!OLxl sin xoixiIm ° c

{ t(kXixj) sin (kXzxl)

-f-t(kxj)cos (kxlxl)J(56)

TABLE ITHE WAVENUMBERS FOR A TEzo1 MODE OBTAINED

BY (45), (48), AND (20) OR (21)

0.02 0.0 wk = 10,

R P = 2kxw (43) P = 2kxow (45) k /k (45) k /k (20) or (21)

0 3.027 - i.0314 3.030 - i.0294 .995 + i.907E - 4 .995 + i.874E - 4

1 6.082 - i.0628 6.086 - i.0593 .981 + i.372E - 3 .982 + i.359E - 3

2 9.147 - i.0942 9.153 - i.0894 .956 + i.867E - 3 .957 + i.835E - 3

3 12.220 - i.1257 12.226 - 1.1197 .921 + i.161E - 2 .923 + i.155E - 2

4 15.297 - i.1571 15.303 - i.1500 .873 + i.266E - 2 .877 + i.256E - 2

S 18.378 - i.1885 18.385 - i.1804 .813 + i.413E - 2 .816 + i.397E - 2

I

G1)c-

Table I compares some calculated values of the complexwavenumbers of the lowest order TE,o1 modes, using theapproximate equation (45), the more exact equations (20) and(21), and the perturbational formula (48). Corresponding fieldvariations are plotted in Figs. 3 and 4.

I-

APPENDIX A

Consider a guiding structure which is infinite in the zdirection, but whose transverse dimensions are finite. Let thisstructure be excited by a source confined to a finite region inspace. Using the spectral concept, we can resolve this sourceinto its Fourier (spatial) components, and express the excitedfield in terms of a superposition integral

0

-0.5 -0.4 -0 3 -0.2 -0.1

X/W

0.08,

0

-005 _-0.5

1.2

J f(kz, x) exp (ikzz)dkz (Al)

where kz is the axial wavenumber, and C is the familiarBROMWICH contour. This contour must lie in the upper sheetof the kz plane in order that the field derived from (Al) satisfythe radiation condition.

Let us consider a guiding structure, for which the kz planehas two branch points at k -+Tko (see Fig. 5). If the inte-grand in (Al) does not have poles in the upper half of thelower sheet of the k, plane, it becomes possible to deform theoriginal path of integration C to a new one B (Fig. 5). Thepath B may be modified by letting it cross the branch cuttwice entering into the lower sheet in this process. The portionof the path in the lower sheet, namely, B', may now bedeformed into a new path D shown in Fig. 6, with the purposeof improving the convergence of the integral. In the process ofcarrying out the last step, we may cross some poles of the inte-grand in (Al). These poles, being in the first quadrant of kz,are associated with fields that decay as they propagate in thez direction. However, since these poles belong to the lowersheet of the k, plane, the fields associated with them show agrowth behavior in the transverse or x direction. For this

IG-

0

-0.5

0g-_I

.E

-0.4 -0 3 -02 -0.1

X/w

=0 P= 3.03- 0.029l=2 ----- P = 9.15 -iO0.089=-4- P 15.30-i 0.0150

(a)

'\ s ,,"".7 \K

0. N,

0

-0.4 -0.3 -0.2 -0.1 0

X /w

-0 5 -04 -0.3 -02 -0.1

X /W

= o P = 3.03 -i 0.029=2 P = 9.15 0.089=4 - P 1530-i 0.0150

(b)Fig. 3. (a) The normalized longitudinal magnetic field of the three

lowest symmetric TEzo, modes (t = 0.02). (b) The normalizedtransverse electric and magnetic fields of the three lowest symmetric

TEz(l modes (t = 0.02).

448

0'

0,,

0 -.K

_\ ,,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Ias osE,, .~~~~~~~~~~~~~~~~~is v~~~~~~~~~~~~~~~~s ,, \v .~~~~~~~~~~~s ;~~~~~~~~~~~"I ,' \v

, ~~~~~~~ -Ns~~~~~, I -- I7 -

I~v)

.Z',;)

04,

-oqg

I..cI.:

^-.

-I.

RUSHDI et al.: LEAKY MODES IN SIMULATORS

0

-O.

0.09

0

5 -0.4 -0.3 -0.2 -0.

X /w

0

-0.5 -04 -0.3 -0.2 -0.1 0

X(/w

/ = P=6.09-i 0.0591=3 P=1223-i0120

1 =5 ---------- P= 1838-i 0 180

(a)

upper sheet--- lower sheet

Fig. 5. Illustrating the BROMWICH contours C, B, and B'.

1.2

o~-_I

aL)

-1.2 _-0.5

008

I

Eo

- 0.05 _-0.5

-0.4 -0.3 -0.2

X/w

-0.4 -0.3 -0.2 -0.

X/W

Im (kz)

-0.1 0

0

= P= 6.09- 0.0591 = 3 - -P12.23-i0 120

=5 ---------- P=18.38-i0.180

(b)

Fig. 4. The normalized longitudinal magnetic field of the three lowestantisymmetric TEzoi modes (t = 0.02). (b) The normalized trans-verse electric and magnetic fields of the three lowest antisymmetricTEzo, modes (t = 0.02).

reason, these fields are referred to as the improper or leakymodes. In terms of these leaky modes, the field expression(Al) may be rewritten as

z [residues of {f(kz, x) exp (ikz(z)} at the leaky poles

deforming B' to D]

+ f(kz, x) exp (ikzz)dk . (A2)

upper sheet_} lower sheet

Fig. 6. Deforming the BROMWICH contour to cross the leaky poles.

The residue series in (A2) is often the dominant contributor tothe field in the interior region of the guide [5], [8]. In thisevent, the branch-cut integral which is rather time consumingto evaluate, may be neglected altogether. The above proceduremay be generalized for the case where the branch pointsnumber more than two, as in the situation for the open-wave-guide problem being considered in this paper. However, a dis-cussion of this case is beyond the scope of this paper and theinterested reader is referred to [14] for further details.

T--I

G)

-I1

449

T--I

E

,f' ,/_ ____<""\ ,'K'

NNj- .- .7.

--.I

2

.2

0.07

1..

-r% nr, L

ko

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-20, NO. 3, AUGUST 1978

APPENDIX B

The reflection coefficient Snm(kt) is given by

Snm(kt) = - (kxn + kt)(kxm +kt)-G+(kxG (kx)(kxn +kxm)kxn(1 +6on)

even TEZ (BI)

00

P- (77) = H 1

n=2n even

(1+00

P.,(??)= Hn=ln odd

9'/7

exp (Iner

exp i2/-

n7r/

hk ktl2 km+ktl2Snm ±) m ±kt)'12 L+(kxn)L+(kxm)h7(kxn +kxm)kxn

odd TE, (B2)

Snm = -inn ( \ 2 G+ (kxeven TMn )

t2h )kxn (kxn + kxm )

-imn 2h) L+(kxn)L+(kxm)

hkxn (kxn + kxm )(kxn + kt)l12(kxm + kt)l/2odd TM,

In (Bi 1) and (B12)

In ( =) In (1 ) i(argkt 2r) (B16)

(B3) is specified to have the unique value corresponding to / = 0.(B3) This choice agrees with the definition of the log function in

the limit of arg kt -+ 0.For 77=kxm in (B13)

C(kxn) =exp [-In kY i(arg (kxm +ikym)

-arg kt)] (B17)

(B4)

where the G+ and L+ functions are the "plus" functions [9]obtained by the Wiener-Hopf factorization of

sinh zhG(77) = G+(rU(77) = e-sh

yhL(7) = L+(77)JL(r ) = e-yh coshyh

where

j177.2 k 2)1/2, for I7 <>I ktj

where(B5)

(B6)

arg (kxm + ikym) =

(B7)

For 77 = kxm, we have

-1 /Im (kxm ) + kym\Re (kxm) /

for Im (kxm)+kym >0

-t --Ofm (kx m) + kym)-tan-1_______ B8

Re (kxm)

for Im (kxm)+kym .0.

-y = -i(mrr/2h) = -ikym

sin kth /2=G(7)=G_(-n)= th) =B'(7)C(??}> (q)

L+r)= L4-77) = (cos kth)' I2B0(71)C(77)P000 (72)

(B8)

(B9)

(B10)

In (B14) and (BI 5)

nn=~~~~7$ 2h)

-kx n I

In actual computation, we truncate the infinite products of(B14) and (B1S) toMmultiple factors [11 where

77h2 1

2 1r Emax(B20)

irh7h 277\ irBe(71) = exp-In kh +((-C) + i-

B°(i7)=exp-I[n (k)+(I C)+2]

) [77 kt )(B13) emax < 1 is the maximum permissible relative error intro-

duced due to the truncation.

(B14)

(B15)

where

if Re (kt2) > (-h)

/n7r \if Re (kt2)(j2

(B 19)

(Bi 1)

(B12)

450

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-20, NO. 3, AUGUST 1978

REFERENCES

[1] A. M. Rushdi, R. Menendez, and R. Mittra, "A study of theleaky modes in a parallel-plate waveguide," Univ. of Illinois,Electromagnetics Lab. Rep. 77-16, Urbana, Aug. 1977.

[2] C. E. Baum, "Impedance and field distributions for parallelplate transmission line simulators," Sensor and Simulation Note21, June 1966.

[31 T. L. Brown and K. D. Granzow, "A parameter study of two-parallel-plate transmission line simulators of EMP sensor andsimulation note 21," Sensor and Simnulation Note 52, Apr. 1968.

[4] C. E. Baum, D. V. Giri, and R. D. Gonzalez, "Electromagneticfield distribution of the TEM mode in a symmetrical two-parallel-plate transmission line," Sensor and Simulation Note 219, Apr.1976.

[5] N. Marcuvitz, "On field representation in terms of leaky modesor eigenmodes," IRE Trans. Antennas Propag., vol. AP-4, pp.192-194, 1956.

[6] R. Mittra and T. Itoh, "A study of the modes existing in anopen, finite-width, parallel-plate waveguide used to simulate anEMP." University of Illinois, Urbana, Electromagnetics Lab.

Rep. 73-9, Apr. 1973.[7] L. Marin, "Modes on a finite-width, parallel-plate simulator.

I: Narrow plates," Sensor and Simulation Note 201, Sept. 1974.[8] -, "Modes on a finite-width, parallel-plate simulator. II: Wide

plates," Los Angeles, CA, Dikewood Corp., Westwood ResearchBranch, Mar. 1977.

[9] R. Mittra and S. W. Lee, Analytical Techniques in the Theory ofGuided Waves. New York: Macmillan, 1971.

[101 D. E. Muller, "A method for solving algebraic equations usingan automatic computer," Mathematical Tables and Other Aids toComputation, vol. 56, Oct. 1956.

[11] K. Singaraju, D. Giri, and C. Baum, "Further developmentsin the application of contour integration to the evaluation ofthe zeros of analytic functions and relevant computer programs,"Mathematics Note 42, Mar. 1976.

[12] J. Boersma, "Analysis of Weinstein's diffraction function,"Philips Res. Repts., vol. 30, pp. 161-170, 1975.

[13] L. A. Weinstein, The Theory of Diffraction and the Factoriza-tion Method. Boulder, CO: The Golem Press, 1969.

[14] Ali M. Rushdi and R. Mittra, "Open questions in open wave-guides problems," unpublished.

Short Papers

An Application of Walsh Functions in Radio AstronomyInstrumentation

J. GRANLUND, FELLOW, IEEE, A. R. THOMPSON, MEMBER, IEEE,AND B. G. CLARK

Abstract-Walsh functions can be used to advantage to performphase switching in antenna arrays used for radio mapping of the sky byFourier synthesis. This paper describes the reason for phase switching,the advantage of using Walsh functions, and the implementation ofthese ideas in the Very Large Array (VLA) program of the NationalRadio Astronomy Observatory.1

I. INTRODUCTION

The basic element of a Fourier synthesis array is a pair ofspaced antennas with signal-processing equipment that formsthe time average of the product of the received signals, as

shown in Fig. 1. The outputs of the two multipliers provide thereal and imaginary parts of the complex cross-correlation ofthe received signals. This output quantity is often referred toas fringe visibility. When measured as a function of the two-dimensional spacing of the antenna positions projected onto a

plane normal to the incident radiation, it is related by a Fouriertransform to the two-dimensional distribution of radio bright-

Manuscript received December 22, 1977; revised March 9, 1978.J. Granlund is with the National Radio Astronomy Observatory,

Edgemont Road, Charlottesville, VA 22901.A. R. Thompson and B. G. Clark are with the National Radio

Astronomy Observatory, P. 0. Box 0, Socorro, NM 87801.1 The National Radio Astronomy Observatory is operated by

Associated Universities, Inc., under contract with the NationalScience Foundation.

ness on the sky; see, for example, Swenson and Mathur [ 1] orFomalont and Wright [2]. By using many pairs of antennaswith different spacings, and tracking the sky area under inves-tigation for a number of hours as the earth rotates, an ade-quate density of measurements of the fringe visibility in thenormal plane can be obtained. Fourier transformation thenprovides a radio map of the sky.

The Very Large Array (VLA) now under construction bythe National Radio Astronomy Observatory near Socorro, NM,is an example of an instrument that works on the above prin-ciple. Twenty-seven tracking antennas of 25-m diameterprovide a total of 351 antenna pairs. The antennas are locatedon the three arms of an equiangular Y-shaped array at distancesup to 21 km from the center, to give angular resolution oftenths of an arcsecond at centimeter wavelengths.

The long observing times used in radio astronomicalmapping, typically 6 to 10 hours, result in the detection offeatures in the sky for which the instantaneous signal level isvery small. The lower limit for such signals in the case of theVLA is about 77 dB below the level of instrumental noiseproduced by the RF amplifiers. In a complicated receiving sys-tem, spurious signals of instrumental origin can be introducedinto the signal channels. These signals may be correlated, inwhich case spurious responses are produced. Suppressing theseresponses below those of the wanted signals is clearly animportant consideration. Such unwanted signals can arise inseveral ways: stray signals that occur in the process of local-oscillator synthesis can cause CW components to be injectedduring frequency conversions, and insufficient isolationbetween signal channels from different antennas can causecross coupling. The first of these mechanisms is usually themore serious, and it is often difficult to get the last 10 or 20dB of rejection required. The second involves signals in theform of broad-band noise and is less important because, for

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