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Title Application of the Boundary-Element Method to Waveguide Discontinuities (Short Paper)

Author(s) Koshiba, M.; Suzuki, M.

Citation IEEE Transactions on Microwave Theory and Techniques, 34(2), 301-307

Issue Date 1986-02

Doc URL http://hdl.handle.net/2115/6040

Rights©1986 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material foradvertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists,or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.IEEE, IEEE Transactions on Microwave Theory and Techniques, 34(2), 1986, p301-307

Type article

File Information ITMTT34-2.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

IEE£ TR .... "'SACTlO"'S ON MI CROWAVE THEORY AND TECHN IQUES, VOL. MTI-34, "'0. 2, fF.BRU ... RY 1986 301

Application of the Boundary-Element Method to Waveguide Discontinuities

MASANORI KOSH IBA, SENIOR MEMBER. IEEE, AND

MICH IO SUZUKI , SENIOR MEMBER, IEEIl

Ab$l~ud -A numerical method for lbe solution of scattering of the H ­and E -plane waveguide junctions is Otscribed. 'The approach is II combina­tion of th~ boundary-dellM'nt method .nd tbe anal)1i.cal method, A genets! computer program los been de\·eloped. using the quadratic elements (higher order boundary elemenlS). To show the val idity and usefuln,ess of this foonulatioll, eomputw results . rf given for a right-angle rom er bend, II T·junction, Iln Inducti"e strip-planar circuit mOOntw in II wlII'eguide, II ,,'a"egui(fe_type dielectric fil ter , and .n inhomogeneous waveguide junction, and a linear taper, Comparl§Ol1 of lhe present results .. ith lhe results of the finite-element method shows good agreement.

I. bmtODUCT!ON

Waveguide discontinui ties play an important role In designing microwave circuits [I], [2], and theoretical and experimental studies of waveguide discontinuity scattering problems have oc­CUpied the atten tion of numerous researchers for several decades. Recently, a numerical approach based on the rini te-element lIlethod (FEM) has hem developed for the analysis of planar circuits [3], [4], and H- and E-plane waveguide junctions [5]- (7j. The FEM is very useful for the arbitrarily shaped discontinuities. Rowevcr, it requires a large computer memory and long compu-

Manu..cripi reuived May 14, 1985; September 27, 1985. U ~ aUthors an. with the Dc-pamnc=nt or Electronic En&i~rin&. Hokkaido

1I'~t~i ty. SappOro. 060, Japan. IEEE l og Number 8406478.

talion time to solve the fi nal matrix equation. More recently, the boundary-element method (BEM) [8), (9J has been applied to the H-plane junctions [10]-[12] and the planar circuits [13]. The BEM is one of the ' boundary'-type methods based on the integral equation method which has already been successfully applied to open-boundary planar circuits in 1972 [14J and to short-boundary planar circuits in 1975115], [16]. It is therefore possible to reduce the matrix d imension and to use computer memory more ec0-

nomically compared with the 'domain'-t~ method, such as the FEM, However, in [IOJ, (111, [1 4J, and (151, it is assumed tbat the waveguide propagates a single mode only and the evanescent modes are neglected. Therefore, it seems to be difficult to obtain accurate results over a wide range of frequencies. Furthermore, in (10]- [16J, the constant elements [8], [91 or the linear elements [8], (9] are used to divide the boundary o f the two-dimensional region. Generally, it is difficult to reduce tbe energy error with these boundary elements. In (12], the linear elements are used and the condi tion of power conservation is satisfied to an accuracy of about ± 4 percent. In order to obtain more accurate results, fairly many elements are necessary, and, thus, the merits of the BEM are lost. In the FEM analysis using the quadratic triangular elements (higher order finite elements), on the other hand, the energy error is less than 0.1 percent [5]- [7j.

In this paper, the combined method of the BEM wi Lh the q uadratic line elements (higher order boundary elements) and the analytical method is described for the analysis of scattering by the 11- or E-plane waveguide junctions. To show the validi ty and usefulness of this formulation, compllted results are given for various fI- and E-plane waveguide discontinuities, Comparison of the results of the BEM with those of the FEM [5]- [7J shows good agreement. In the present BEM analysis, the power condi­tion is satisfied to an accuracy of ± 10-· to 10- 3.

n. BAS~C EQUATIONS

In order to minimize the detail, we consider the waveguide junction as shown in Fig. 1, where the boundary r, connects the discontinuities to the rectangular waveguide j (i - I , 2), d, is the width a, or the heigh t bl of the waveguide i for the H - or E-plane junction, respectively; the region n surrounded by f l ' f 2 • and the short-circuit boundary fo completely encloses the waveguide discontinuities. and the waveguide i is assumed to be filled with dielectric of relative permittivity (" .

Considering the excitation by the dominant TEUl mode, we have the following basic equation:

k ' , 0 -1.01 (olLo

fa t H-plane junction

for E-planc junction

(I)

(2)

(3)

( 4)

('" 1 - , T (,-(n/ koa ) ,

for If-plane j unction

for £-plane junction

(5)

where 1.01 is the angular frequency, E; and H: arc the electric and

0018-9480/86/0200-0301$01.00 (')1986 IEEE

'.

• FiJ, I . a-..."JoI",otl"~.

magnetic fields, respectively, and ' 0 and 1'0 arc til< permittivity and permeability of IItt space, =~vely.

Ill. MAnIDUncAL FOIUofU1ATlON

A. /Wrmdary·£lt,"rttl APPrQaCh'

Con~d~ring tile ~&ion surrounded by the boundary r as shown in Fi" 2, and using the fundamental solution 9> ' [8]. (9] and Green's formula. from (1) we obtain the following equation [loj- [i3]:

(6)

I . ' - 4/!Jl1

(kr) (7)

ojI ' _ ~ klf/n ( l:r) eo. a. (0)

Here +, is ti>c value of .,. I' the nodal point p, 1}. and 1/> ' are the outward nonnal derivatives of .,. and . ' . mp«ti~ly, ttf' and Hf) are the urofu. and fil'$l.'()fdcr Hankel functions of the sroond kind, respectively, and <I is the angk between the vector , and the outward \lml normal vector _.

Noli", that the nodal point, is plac:a1 on the boundary r and cons.idmng thc inteJ.fation path rA &oina around the nodal point p as shown in Rio 2, we: obtain (or (6)

...!.-.,+!. or·</> dr- !..·.;,dr (9) 2.. r r

where f denotes the Cauchy'. principal value of integration, !lamely I r '" lim, _ofT_r." Dividin& the boundary r inlo quadratic line clements as shown in Fi" 3, .,. and t within ~ach element are defined in t~rm$ of 4>. and +. at tbe nodal points q (q - 1. 2, 3). =~tiveJy, as fOI1OWl!

where

4>_ {N }T{4» . (IO)

+_ {N)T{+} , (11)

( 4) }.-IM'lof>J]T

{+l, - [+t'ob'hr {N} _ [N)N)NJ]T,

(12)

(13)

(14)

no dal poin t p

n

1 3 2 • • • .. (

( =-1 ( =0 ( =1 Fi&- 1. Qo.oatI,.rit: u ... _nL

H~rt T, (. ) , and (_ JT deno t~ a transpose, a oo",m" ."K". ,,", row vector, respectively, and \he shape function N. is pllCn

N. _ A.€l + D.f + C.

A,-I / l ,

C, -0,

with !be normaliud coordinate € d~fined on the e !b .1""",,, Substituting (10) and (11) into (9), we obtain

",here

• -, 4>~ +E(II ); {.) .- [ {I');(oIo). '. .

{Ir )._[ 1I, 1I1Ir , ]T

(I'L - [1',1'11') IT. H~n: r. extends over all dilfermt elemen1$. When the point p does oot b¢long 10 the tilt dement, h. and I , calculated willt Gaussian intqrlltion ..

Ll' I I ~ . h,--, N,- II ,-I(kr)cou. df ., . Ll' I g __ N - utl'(iu )d(

• 1 - 1 ' 4J " O

",here L is the length or the element. When th~ nodal belongs to the e th el~ment. calculations of II , and I . i",~'"'' limitation of to ..... O. In lhis ~a.se, COS a'" 0, so that

h," O.

For the case where !be nodal point p coincide, with the point q - I, 1, or 3 of the eth element in Fig. 2, gt is p''''

1'. - (L/ 2J[ A. J, (2) - (2A , - D, ) ( ',(2) - 2/( ... kILl )}

+(A , - S, + C.) /0(2)]

1', - (Lf2) I A, l , (2) - (2A, + 8,H l,(2) -2/( .PL l»)

+(A, + S,+C.) lo(2)j

'so_ • """,al 'onn~"'hooo or tho i EM wi"'~""" doe ......... rOlf ""alyzi", or ,......dime"""""l cIec,..,........rit: r .. 141 is p..~ i . (! I ~ OI>Iy 'M ""u;"" 01 tho BEM willi quod."!i< ._ .. win be _ toe...

rnpectively. Hel"tC 10, I,. and 11 are calculated as follows:

("')

(24b)

< -t · ,

• r ~ , (2«) ; f whoere l' is the Euler's oumoo. r 10 the mauiK notatiOll. (\7) is rewritten as follows (8}-1n]:

t II/H ' ) -IGH'). (25)

[ From (25). the foHowine equation is obtained for the waveguide t junction in Fig. 1: ,

fUH]o [Hit , -• · [ (. ),] [Hhl {t}t

{toh

• , , [(n] -[!Glo (G], [Ghl ("'l, (26)

(th , , ~~here tM subscripts O. 1. and 1 denote the quantities correspond­

't"'og to the boundaries ro, r" and r1 in Fig. 1, respectively.

. AIUI/ylku( Approac~

: Assuming that the dominant TE'G mode of uni t amplitude is ~iocidenl from the wavquide j (j - I , 2) in Fig. I, ., on rl t, -I. 2) may be « pressed analytically as

i of>(Xl'I_ O,y('»_ 2"i)!/I{y(i l)

- [ ,: 1"1,.(~"){,.{ .. ") '" /1' , ... 0

, '. , •

"'" t,,,,( y<'» "' .;Va: sin m .. /')jll"

P, .. '" I*l,c .. - (m.j ",)l,

m-l,2,3.·· ·

m- I ,2,3,···

(17)

(28)

(29)

for the H·plane junction

Ii ... (l') .. ..jrJ"jb, cos n "/" 1 hi ' ~-O,I,2,···

(lO)

n-O,I,l,'"

, _ (I, • 2,

,-0 ,.0

for the E-plane junction, and 8,) is the Kroneeker B. Using (10) and (11), (17) can be di§cll:titcd as follows:

(31)

(32)

(+L-B,j {/}j +!Zjdolt), (33)

who"

{1J , -2Ud j (34)

[ ZI," - E ( l/jP,,,,) { t .. } ,E !/,.,( >Ili) - . . .( N(x('I _ O, >11'1 )] li)\jil. (l5)

Here tbe components of the {/ .. }, nctor an: tbe values of [, .. <,),1;') al the nodal points on r, and E .. «tends over (he elements related \0 r,. C. CombiiUllicm of Boundary-Ekrrwnl aNi Anolylica( RtlatlON

Using (33), from (26) we obtain the following final matro: equation:

--------- -------------------------101 III 101 101 -IZJ, 10) 101 101 II I 101 101 - I Zh

{ ., lo {OJ {". )1 {OJ {., h (0)

{ "')0 - {OJ (36) ------

("'), "Ii{!}) {+h "2i{/ L

wbert 111 is a unit matrix. 10] is I null maim, and {OJ is a null vector. In (36), {.}o'" {OJ and {t)o" {OJ should be considered for the H- and E-plane junctions, respectively.

The values of + .1 nodal points on r" namely (+),. are computed from (36). and then .,(x('! - 0, /') on r, can be calculated from (10). The solutions on r, aUow the determination of the se&ttering parameters S" of the TEIO mode as foHows:

S;, - !o#'.(x!J) - O. yfjJ)J;, (yli' ) dyfJ '_ l (37)

5'1 - {P" i 'i IPJ t("

·fo#'+(xm"O./")/,, (ylll ) dy(i), ''''j. (38)

In (38), for the H·Plane junction, both i:'i and 'i'l should be replaoo:! by 1.

LEU T""NSACTlONS 0" MICWWAVE TIl~OU ANI) TECHNIQUE!, VOl" WlT,) 4, NO, 2. """'"

•• " . "

m

..

" po," I po<I 1

TE,, " : TE,.O -Sj-S] I~PO':t·~· "P~ o

~"" 2 pori 2 , " " " _,a If

Fl, . • . p"""" ,,..,,.,,.i ... on «><:fl;c;<n' of rifh"O%I$l< CO""" b<-nd.

(>J

IV. COMPUTED REsULTS

In lhis section, we present th<: computed results for various H­and £.plane waveguide disoonlinuities. Convergena: of the solu­lion is checked by increasing m in (35) and the number of the dements. Although Ihe convergence is obtained by using the first Ihree Or four evanescent higher modes, in this analysis, Ihe first si~ evane~nl roWer modes are used in (35). The results 01 Ihe REM agree well wilh those of the FEM (51- I?] and agree well

~ ..

~ ..

• • ' " .. "

- ". po" I po" 1 , , • ,

L.J po,' I

' .. ,. '" ' .. ' .. , .. , . " _,a /I

"J

.. " . .. .. " . 0, 30

_~'2 o ' : 0

L.J p"..' I .. ' ..

'" '"

' .. , "

,. " ' , " /I

'>J

' .. ' ..

\'\';--~--~""" .--~--~~" ".0 11

"J p"""" .-.IItOlion aod ' .... n'mi ... "" «>enici",,, 01 T.juno'i<>n,

, , • • .; • ,

wilh Ihe olher Iheon:tical results [151-(20) and th<: e~perimenUl~ resuhs (18], (19J, (21J. For Ihe H-plane waveguide discontinuitieS;-l the experimenlai rcsulls (18], [19] and the results of the inlegral i equation method [IS], (16), the normal·mode method [17]- [19].1 and Ihe moment melbod (20) are nOI shown in lhis paper (lhesl::

4 resuhs are citw in [51 and 16D· 1

, ~lH~ TU"OACTION$ ON M,C1r.OWAVE TIlEORY ..,;0 TECHNIQUE$, VOl.. MTT-34, NO. 2, HJ!.RU ...... ~ 1986

,

, ,

~ ,

I ~ ~ " '.:

t i ~ " : !t\

• •

, ,

(.)

'" Fi5. 1. Ekmenl divUion fo< T.junc.ion.

~r---~----------,

"" , ...

"

"r"_''-__ J'~O'' I _ I 1[ " a ,

1 1

',L,c--_~ __ -,'", __ --''-L~~" Il!' EF" K Power ".n.mi"i"" «<[mien' 01 indue'; ... ><rip-pl"".r ei reuit ~ moun l«l in . w .... i"~¢

1(" H·PIQneJ~nClion

~~ FIg 4 ~hows the power transmission coefficient (lSld1) of a f,.nght.angle comer bend. Fig. S(a) and (b) shows the element ~divi5ions for the type a and tile type b in Fig. 4, respectively. i!':. The present approach can be applied easily 10 the analysis of ~!Jluhj-port junctions. Fig. 6 shows the power reflection coefli­~en!s (lS,,11 and IS:uf) and the power transmission coefficients ilISld1 and ISd') of a T'jIlDC!lOn, Fig. 7(a) and (b) shows the ~ement divisions for the T-junclion in Fig. 6(a) and the T-junc­~n with wedge in Fig. 6(b), respectively. From Fig. 6(a)--(c), it is

Fii- 9. El<m<n1 divi. ion f", indoc';"" ''';p-pl"" .. cir=it mounted in a wove· &~id<:,

• • ,

,

if. , o , ,

,

,

,

,

,

" , "

,

- "" ". po" , p." I

E, . :.61 O·l ~g

- O,la

T[ " ~ 0 _3.

0..10

I I 0 -15 . , , 0·1. 0 _350 O-]a

'-

" " a "

found thai over a wide range of frequencies, lhe renection al port I is reduced with a linear wedge (FiS- 6(b») and thaI lhis «,nee­lion al pori 1 may be further reduced with a sinusoidal wedge (Fig. 6(c».

FiS· 8 shows Ihe power transmission coefficient of an inductive strip-planar circuit mounted in a waveguide. Fig. 9 shows Ibe element division for this circuit In this case, the boundary condition 4> - 0 should b. considered on slrip conductors.

The present approach can also be applied \0 the analysis of multi-media problem~. A procedure of programming for handling multi-media problems is given in Ill]. Fig. 10 shows lhe power transmission coefficient of a wa~guide'lype dielectric filter. Fis. II shows the element division for this filter. In this case, tbe boundary conditions 4>.;, " 4'd;,,""'ri< and ¥> .. , .. - ¥>dielt<<ri< sbould ~ considered on !he intcrf~ between air and dielectric.

The present approacb is applicable to the frequen cy range in which waveguide propagates multi·modes. Fig. 12(a) and (b) shows the magnitudes of reflection and transmission coefficients of an inhomogeneous waveguide junclion, respeclively. Fig. 13 sbows lhe element division [or this junction. For bOlb reflection and transmission coefficients, tbe results of the BEM asrce well witb those of the FEM [6]. The results of the moment method [201 for the transmission coefficient are different from tbo~ of

J06 lEa TMN$A.CT\Om ON WICaoWAV£ mtOU M'D nCKNIOIIES. VOL. t.01"-J4, NO. 2, "''''m,;

V'" 11.

D ~

.,.. L!, Ek",.." divisioft lor w.vt",~dNYP< did«tri<: l;ltet.

, .1 • ! j

' .• ~---------, • FE ",

".

, "G£' M "I - . I E .. . to ~ .

.,~--------~.------------c.,.,---J " (.,

" ~

IE"

to., , T€ " ,

I I • ". " . • .. .. , ,

"" '.". I.", <

"" Mopll..dot 01 ",1It<;tioB.,.d or .............. eoeflitio»u ot ..... .".,...,...

....,... W'0V<f,uid< i"""'ion.

Ihe REM and the FEM. In Ihe moment method, the transmission coefficients of the hisher order modes are not zero a1 the cutoff values of (,.

Table I shows W number of I~ nodal points USJed in !he BEM and FEM analyses of H-pJane juoctions. Hue, in both BEM .nd FEM analyses, tbe ~Iry of .. eimti t 10 mtuce tbe dimeo­lions of \be matrices is nol UKd. The accuracies of Ihe presen t boundary-clement calculations are almost identical 10 those of the earlier finite-e lement calwlationl [51. (6), and yet the BEM

TABLE! NUlolij~' OF NODAL f'oI..-rs U R n IN THE BEM .. ''0 FE.,\{

A"AUSE,S

,,"pl ... J_tlon ~ '" f lt . 4 (1)'1'1 I ) • '" fI,.' ( typo b) " '" fig . ' (.) .. ., 119.S (b) • '" Fl9.a ,. •• f',. 10 ,ro '" flt·1I " ••

• fo~.'_' . "" ........ ~ • H~

• -- n . ~ CC.... s-.......

"::! " M ,,,,,,,@~ ' ::-'-'-'-+}I-' """ • L- < ---I

'·'''011. "

< I",,,, )

1'",.. 1. . VSWR ...... "",«i>."'" of ij""u ji".pl ...... aper .

allows 1M malru: dimeDSion to tIC' reduccd by a (E'~ ol.t~' · 10 3.

B. £·l'fQn~ Juncfi(Ht

A comparison of Ihe rt$ults obtained applying lhe BEM 10 lin~ar £ ·plane lapen of various lengths with Ihe ,~p"im'''. results (21) and the resulu 01 !be FEM (7) i, given in vel)' good ~menl is obwned.

v. CONCLUSION

A method of analysis, based on Ihe~~~~i~~:~~.1 proach and tM analytical approach, was lion of the H. and £.plane junctio!l$. "The validity

~Hf TIV<NSACTlON$ ON M'C"OWAVF. THEOn AND TEC~NIO\JIS, VOl. ),fTI • .l4. NO, 2. FEUIlUHY 1986 , , ~was confirmed by comparing numerical r~sults for various H· )and £ ·plane waveguide discontinuities with the rtsults of the rlinilc.element melhod. f This approach can be applied easily to the planar circuits (3), d4]. [13]. Tho: problem of how to deal with waveguide Junctions "Wi.h lossy media or anisotropic media hereafler still «:mains. f ~ ~; ACKNOWLEDGMENT

J' The authors wish to thank T. Mild for his assistance in numeri· :-'eal computations. ~.

f ~ REf ER£NCES <

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~. • "

N. MU<."I>vilZ, Waw,..-d< Hu"dbcok, New You : ).lcGrow-Hill, 1951. R. E Col~n. FI.IJ Th~ of GoJJM W"",.. N .... You: McGraw-Hill. ,_. 1'. SOl",,,,,, " Fin;' •• I.mm, on.alysl, or ptanlll' miaowa", <><'work ... " IfEf T"",u. Mi",_~ 17>-,. T",~ .. vol MIT·2l. pp. 104_ los, fob, '91). T. M;y.><tU, "TIl< <>p"""" 01 .1«""",,,&,,<,;0 roo\(!;n plaut 0;'00 ;'." r","", "",. EI«,,,,,,. C...."."~. EnJ. Jopa~. "'I. J5S-S, pp. g,o_9t. Fob . • 97S (in Japane><j. M. Ko.iUb •• M, Sato, and M . S",-"4 ~API'I"'tion or f;n;,<-,i<m<n, ",.:<1\(>(1 ", H·pl_ "'_""o;de d{,""",inui';"''' E~""" . Lm .• vol, tS. pp. 36<1- 365. Apr. 19U M. Koo:tUb.a, M . Sal<> •• nd M. S",-olci, "F;ru .... l,m<Bl .n&!y.i. of vbi· ,,,,rily .hOp«! H. pl1no ",,'<&u;d, d;w; .. " ;nu;';,,,- T",",. I,,,,. EI",,,,,,, C"",,,,,.,.. Enf' Japan. vol. EM, PI', 82- 87. Feb, 1983. M. K<><hib.a, M . Sat", and M. Suwlci, ""'pplicotion of linit<-<i<mcnt m<'Md to £.pl""" w .... oi<It diorontinuitie>,~ Tf~n'. I .. ,. Ekm<>H (""""""~. En,. Juptm. vol. E66, pp. ~"- ~5S, Jul~ '983. C. A. Bn:bb;., 171< B""",*,ry EI ..... m M"W, ... E"gi"«". Londmt: P.n'eclt P", .. , 1978. C. A. B",bbia..,d S. Walk.,. ~ £1_, Tffl,nl'l"L' i~ Engl,,_. 1"$ . London: But1c""o,\h. 1980, S. Wuhisu and I, Fuk&i. "AD .n.I~'" 0/ .I .. "om",,,,,,ic ""bounded roold problo",. by boundary ' kmo .. "",'hOO." T",,,, . In$( . £1"""",. C""'~tU~, F."8' Jo",,~ , vol, J6<I· B, I'P' 1l~9-136~, Dec. 19$1 (i~ lopan<",), S. K oso"'; .nd 1. FUk';, "Appl;",h<m of boundary·do"",nt m<thod '" ckc"omavt<!ic f;'ld prookm>,- IEEE fun" Mi,'.--'I"" Th-,. T«h" vol. MTf·12. pp. 45~_461. Apr. 199,o. K, K OOK> , nd S. Ku"","no. ··W,v<&o id<·,yp<: did .. ,,;., liI«f," T,"", . I~". EI"",."., . C""""u • . En,. JIl(JW<. vol, 161·B, pp. 1177_1178, 00,. 191<4 (,n 1"1> ....... j. E. Tool'" and H, B_fOnd, "Mol.imo<it S'por.,.,.".", of planor mulli· _. june.,,,,,,, by boundary .i<m<n( method," EIUlt-... Lm" vol, 20, pp. 199-802. Sep~ 191<4. T. Oko.ki and T Miyoohi, "n.: pt...., ci,ruit _ "'n approach to m;em· "ave ;",,&,"(<<1 o;tro,,,)' ,~ In;E Tm,.. . 1.1,,,,,,,,,_ 1M"')' T.ch .. ..-oJ MTf·20. PI'. 24S_2S2. ApI. 1972. T OkO<l>i Ind S. Ki' ........ . -Cornpu"" analym of shor'·boundary planar dr¢l.oi,." IEEE T",,,-, , M i"""""", n.-,. Tn-/I ~ vol. MIT.2l. PI' 299- 306. Mar. 1975. T. Ok.,.tU ond S. K, .... wo, "Cornpult' analysi, of >hor'.I>o"ooa,), planar ';.-oul,." T«h . Rff. R.I" / .. ,. l'I.",,,,, . C ... ,"'"". E"~. Jap''". MW1S·H. Oct. .91S (in Ja~) T. A"w. ond J. P. H.u. "A.-..J~si. of 1'1."., cire",' w;(h ;hot, ore";t boundary by """mal .....,. mc:lhod '"'oust> ,mpedance """ii.," TT~ ... "lSI, £1"""",. C_"". h,. J"I"'''' vol. l61 · B, PI'. 646_653. luly 1978 (in hp"""..,j. T. Anoda, M. H>tayoma. and l. P. H .... "Wide·b.nd rT«l ... n<~ cit .. • ""ltri",,,,, or "'O""lul., ",. ",,"ido "'plan< T·ju",,"OtI •. " T«h . R<J R~p. "Uf. Elo",,,,., . C"""".~, t:.,. J"p"", MW79·68 . SeP'- .919 (in I'p_n_j. R, Mo",on, N, 0,""", T , "'nod&,.nd J. P. !b ... "[)c,iva'ion 0/ nor"'" modo lor di<l«trio plan.,- d=i. _ Wa""uide.,yp< di.lectrio JiI, .. ," T ... ~, /I~s. /1.1" I", •. Eke""", C"""""". E",. Jol"'", !>IW79-69. Sep'-1979 (in 1ap.>""$<). y, L. Chow and S. C. Wu. -'" """""'" """Iood with ",i<od boas;. ru""'i",,. ro, >eau«in, by w''''St>id< JUnc,;on,:' IEEt: T,a",. M,cro­"""" TIt"")' T.<lo .. v<>l. MIT·,!. PI' 333_340. M.~ t 973. 1( , M .. ",,,...,,,, "R.Ilt<"o<> """fr.ci •• , 01 E·pl ... 'apered ... o""uo1«:' IRf: T",,,-,. Mkn,.""", Th"")' T«h .. ",1. MlT.{i. I'P' 141_149, Ap', 19S8.

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