Numerical Techniques for Microwave and Millimeter-Wave Passive Structures [1 ed.] 0471502839

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Numerical Techniques for Microwave and Millimeter-Wave Passive Structures Edited by

TATSUO ITOH Department of Electrical and Computer Engineering The University of Texas at Austin Austin, Texas

■ WILEY

A Wiley-lnterscience Publication

JOHN WILEY & SONS New York

Chichester

Brisbane

Toronto

Singapore

Copyrig ht

©

1989 by john \Vilcy & & St,n . lnc .

Ubrary1 of Co11gr~ss Ca1aloging in P{1blicatio11 Data: T iitsuo h nh umeric.:al tcchniqu~s for microwave ,ind millim tcr-wavc pl\ · ·ive tructurc I edited by Tat uo ltoh . p. em . .. A Wiley- lnterscicnce puhhcalio11." fnt;ludcs biblic.)gniphic~ and index. 1 8 . 1 0-471-6?563•9

l. Mjcrowave devices-Design a11d eonstruction- D ata proces ing. 2. i umerical cakula.rions. l. lto h . T aL~uo .

TK7876. N86 19~· 62 1.381'3- litb significant adva nces in the capabilities of con1putcr , numerical methods have also been adva nced in the last several car . So1ne methods have bee n n1ade more efficient a nd so1ne that originated in a diffe rent d iscipline uch a engineering n1echanics have found applica tion .. in cJcctron1agnetic ,vave problen1 . Thi · hook i · intended for graduate tuden ts a nd \VOrking e ngineer · ,vho \.Vant to acquire a vlorkiog k.no\vledge of nun1erical n1ethod fo r p·t i c structure . Such a talent i ~ believed t o be one of the n10 t import·1nt a "·ct · for the next gcn~ra ti n e ngi neers. The book is ~1 ritten by a nu111 ber of inte rnationall rccogniz ' d re ·earche r on each ubject. H o,v er. th y recognize the fact thal many novice trying to learn the e difficult ~·u bjcct · have a hard time in catching up with the late ·t dcvclopn1ent . For thi: rea~on . each chapter L ,vritten as con1prehensivcly as the book format allov., . Typical1y. each chapter ·t·,rt with a brief hi torical back~round and a de cription of the n1e thod, fo llowed by detailed formulation .. fr practical exam.pie ~ Cornputer progran, de ·cription. are included where appropriate. The book tart· with an Iniroductio n and O,·crvicw to gi e rhe r a,,(r') dr' = p(r)

( 14)

The second step i to ta ke inner product of (14) with te. ting f unction Xm( r), 111 = 1. 2 . ... ~ N. The re ult a re \

L

11

- 1

K"111 c11 = h,,,

whe re

K,,,,, = ( x,,,(r) b,,,

L

G(r r ' ),,, (r ' ) dr')

= {x

111

(r)~p(r))

(16)

(17)

The symbol ( ) indica te the inner product a nd i · typically an integral with respect to rover the region D. It i clear tha t ( 15) is a set of linear eq uation of size N x N. T here are ever al choice. availa ble for the basis f unctio n ,, (r ) and the testing functi on x,, (r). One of the simples t is the choice in the o-called point-ma tc hing method. In thi method . the folJo,ving e lectio n i made:

11

MOMENT METHODS AND GALERKIN'S METHOD

¢.(r) = U(r,,) =

{

x

11

r = (r,. - 11/ 2 r,, + ~ / 2] otheP..vise

~

(r) =- o(r -

r,J

where U i the unit pulse function, \vhich is zero outside the narrow range of &. around a discretized point r >, in the don1ain ~ and 8 is the Dirac de lta functi on. It is clear now that, if lt1I i sma1l eno ugh , (1 )

( 19) Due to the choice of the function , no in tegral operation are needed . Hence the analytical preprocessing is ex tre me ly simple. The price one ha to pay for this simpJicity is the la rge m a trix ize N for accurate olution . The method is quite structure-independent a nd can be applied to a large class of o dd-shaped geom etries. There are several jmproved version of the point-u1atcbing method. For insta nce . higher-order functions o r pjecewise sinusoidal function can be used for basis functions [19]. Another popular me thod i the GaJerkin·s n1e thod. which e entiaJly results in the same procedures as the Rayleigh- Ritz n1ethod . In Gale rkin· method the ba. is functions and the te ting funcrions are identical a nd are defined over the en tire range .... .

x, (r) = i our mall variation in the adn1i ~ ible function. Subtractino eq. (27) from (28) (and ignoring any tern1s \vith degree 2 in o: dS + J · 2 I I chO dS

(?9)

where now J ·imply de not .I( ) and ol = .I( + S) - J( ). Again we wield Gre n' theorern to expre · the right-hand idc of (29) a

For tationarines .

ol = 0.

and we mu ·t th refort: ha (30)

or. to rca rrange it. (31)

We have not yet di. cussed what function ~ are admi ible. Fina they n,u -· r be sufficiently differentiable ( thi. can be checked by . eeing the con.. equenc~ of an violation) . But in addition:

l. Suppo ·c that for admi ibility l iJn term we can argue that for eq. ( 42) to be true for any S we mu t have

a = O

(4J)

an

These are the natura l bound ary conditi ons re lating to the variational expression , eq . (40). They are rather like ''E ule r equation. at the boundary. ''

E uler eq uation . are atisfied Insistence tha t 51 == 0 =;>

and Natural boundary condition are a l ·o satisfied

On 0ur first dealing with eq . ( 14) \Ve took a hortcut by assurning certain boundary conditions to be sari ·lied and then applying stationarity a rguments. If, however , ,,,_,e do not restrict the trial function , then ,ve fi nd tha t the boundary condition a l r7n == 0 is naturally implied . Note 1. We say implied ~ the boundary conditio n a re not exactly "a ti ficd. any more than (gene rally speaking) the Euler e quation~ arc exactly ati ·fte d by any trial function we choo.. e to u ·c. T h i n,i conceptic n does appear in the literature~ Note 2. lf bo undary conditio ns have to be ·atisfied for a n expre ion to be statjonary, they ure tem1ed principal or essential boundary conditions- in Contrast to the natural bounckuy conditio ns .

The importance of natural boundary conditio ns lie in the fact tha t it i often difficult ~ sometime·s impossible. t ~ arrange for the essentia l con dition to be satisfied . When this applies \Ve ca n find the n atural conditio n . a nd (rather Like the E ule r e quations) if they correspond to . Now inste ad of eq. ( 40) we try a n extra term :

J J('i/ / dS - Ff ( aq,1 de J() = - - - - - - -

JJ ©

2

il11)

(44)

dS

here Fi a con tant factor. to be chosen later. By our standard variationalproving proceduTe we fi nd that the following eq uation must be satisfied:

\ \1

2

ff o(v~q>+

J ) dS

= (2 - F) f ocj> a de - Ff cp an

(i(S) de in terms of q. In genuine 3-D, the field come from. H = curl (A)

E

=-

jwA - grad cf>

whete the vector and scalar potentials are given by

( 135)

(136)

92

THE FINITE ELEMENT METHOD

A= µ,

a nJ

_!

i- dS

( l51)

V · (eVcp) = 0

( 152)

has as E uler equation

Importantly. thi · means that finite element based on (151) can deal. with equal ease, with electro tatic problem involving arbitra,y transver. e ariation of dielectric con tant. Note that this flexibility does not appl to the integral equation version of the problem (Section 6). f n the hollo,v-waveguidc proble1ns. again it is adequate to work in term of one scalar function- the E= or H = "'c;parately for the TM and TE. a · in ( 148) and ( i 49). Unfortunately .. the flexible exten..ion to inhon1ogencou: dielectric con tant doe. nut apply. Thi i ' because (in contrast to the electro tatic situation) the proble1n i fundamentally more invol ed: specifically.. it i no longer po. sible to find the field from any ingle . calar function~ at lea t t\vo calar are needed. The mode. arc not TE or TM. but ome combination (and :o are called hybrid mode. ). One procedure i · to take the total field. to be indeed a straight um of .the field derived fron1 the TM and TE expressions. Specifically. con id r E(x , v. z, t)

= e(x . ..v) exp f j(wt - :B z)l

H(x. v . .... r)

= h(x.

v) exp f i(wt - B z)I

(154

All field component can now be obtained frorn the addition of tho e given in (148) an: field profile along the y-axis symmetry line for the dominant H~, mode. After [19] © 1984 IEEE.

other hand , it i more difficu lt to treat ani ·otropic dielectric: it a pnon approxi111ation can break down (e.g .. when the ,node lack an TE-like or TM-like nature) and at the lea t need the ~exact"' approach to a · ~e it ' accuracy. Much work i -till progressing in thi direction of finit element ·olution of optical and n1icrowa e . tructure . For current and n1ore de tailed in forn1ation, it is be t to refer t r ecent journal (e .g .. IEEE Transacrion 011 Microwa ve Theory and Techniques and IEEE Journal of Liglu Wa, e Technology) .

9.

FINITE DIFFERENCES IN SPACE AND TIME

This section will pre ent the very briefest account of the u c of finite differences, in the coo text of thi work. 1-tinite difference · provide an alternative to finite element for the solution of partial differential equation..

FINITE DIFFERENCES IN SPACE AND TIME

103

. .. ace. But more strongly~ they provide the most common way (Yt'here :· · ~"-' · · elements are not especially appropriate) of dea ling numerically with ~:uarisient time depcnden~e. . . . ··We will be interested 1n using finite differences along (some of) the x, v . .-.i;,d taxes. In all cases, we consider a function to be ·· ampled'' at (usually z, .~ . bl re~ar) intervals along th~ v~1a e x. y . z. or r. . . . . . ··to one dimension. cons1denng, sav. a real function f( ) . a denvat1ve 1 ~µimonly defined by l

. { f (a + h ) - f( a) } f'(a) = hm h

J,- ll

(163)

This ~uggests that any required derivative mav be reasonabl v approximated ,

f'(a) = { f(a

> .

+

11)- f(a)}

( 164)

~qualJy eligible approXlfilation might be

(165) Hquations (1_6 4) and ( 165) are called, re pectively. a / orward difference fo_.rmula and a backward difference formula . A central difference fomzu /a i

f, ( 0 )

= {f (a + h) -

f (a - h) }

?h

( 166)

errors (and justification) of the above formula can be obtained [ Ll qirectly from Taylor s theorem. and are proportional to:

, The

h

h i.

for the forward and backward formulas

for the central formula

. _T he better accuracy of the central formula can be vi ualizcd from Fig. 27 ~ ~ere it is clear which of the three chord i neare t to being parallel to the ~ent . . A central difference formula for the second de rivative follow from ·cc~ssive application of the first derivative formula:

104

THE FINITE ELEMENT METHOD

r

h

a

a- h

Fig. 27 ( 166).

(l

+ I,

Chords B- C, A- B, A- C corresponding to difference formulas ( 164)-

, ) f'(a + 0.5'1) - f'(a - 0.5h) f ' (a = · I,

- [f (a + I1) -

-

f( (I) l //1 -

,,

r/(a) - f (,, - h )111,

f( a + h) - 2f (a) + [( a - h)

: =: : - - - - -1,----

(167)

The error in ( 167) i a 0 ain proportional to lz 1.

9.1.

Finite Differences in Space

Finite difference can be u "'ed a an alternative to fin it elcn1ent in virtua llv all the work described in thi~ chapter. Going back to our uni fied approach u ing weighted re._ idua l ·, it i rather like using regularl spaced Dirac delta function -ju t a in digital filter , work.iJ1g ,vith a sampl d data trearn. Finite difference method can be set up various way. ; o ne elegant way i

to u e a variational expre ion. We will illu trate with an example. Con ider a holk)\v conducting waveguide of arbi trary cro ection. In tcad of the earlier variational fonn, eq. (145). it is . lightly more convenient (allowing use of tho more accurate central difference ) to u ·e another variational form:

105

FINJTE DIFFERENCES IN SPACE AND TIME

-

--

--

--

-

-

~

-

-

-

~

--

•

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

'

.

.

.

.

.

.

.

.

.

.

.

.

.

,,

.

.

.

.

•

.

.

.

.

.

.

.

.

I

I '

.

.

.

.

.

.

I►

•

.

.

.

.

.

I

--

-

-

,_

-

• '

I ,

-

-

--

-

-

-

~

Fig. 28

'

-

' I

I

-

Finite difference mesh of nodes.

' II cbV\f> JI J/

dS

k - = - - --

(168)

dS

To approxima te ( l68) . we uperimpo, e a regu la r q uare me h o er the g u ide cross . cction a nd con ..idc r value , of the u u a l on ly at the node ·.

(See Fig. 28.) To eva luate the int gra l. o f ( l 6t ) ~ the dcno n1i nator wiJJ clearly be ( 169) wit b urn matio n over a ll nod es and with special a daptio n nea r bou ndarie . To approximate t he num ra to r . referring to Fig. 29. ( 167) \-vill l!ive )

a -¢ ...... i: r1x 'I

---

+ \,\

- 2o '

11 -

(170)

-and with a j m ila r \xp re,• ion in they direction ,vc h ave the cla icfive-point

formula ( 17 1) Applying ( 169) and ( 171) to ( 168) n1u t give

(172)

106

THE FINITE ELEMENT METHOD N

0 w---------F-------eE

Fig. 29 Typical five-point star of neighboring nodes.

s

which. after the usua1 Rayleigh- Ritz proced ure [applying e q. (55)] g1ve. (173) So the rn et hod of fini te differences give~ the ·a,ne matrix fo r solutio n as fin ite elen1ents (( 150)] except that ( 173) is lightly simpler in having f Bl =

[I] , the uni t matrix. In ide n tica l style, finite differences could be appli~d to o ur strip]ine / e lectro tatic problem of s~ction 5 . Finite differ ences have been, a nd will continue to be, used a lo ngside finite e le m e nt ·. A comparison of 1he method, i so111ewhat l lurred, b ut son1e of the pros and con can be Ii. tcd.

Finite difference have the fo llowing advantages : 1. T hey can be easier to et up. Unlike the finite e le1nen t approach. th ~ do not need a Oa lerkin a \vcigh ted-rcsi 0 for all x· this typically follows because it con1e fron1 tertns such a 11

A is sometime complex, but numerically the difference is straightforward. so ,ve will consider A to he real.

As an example of ( l92). the vibrating string of Section 4 gave (6?). which, after applying (5:) gives (19 )

\vith igen alue and vectors gi en after eq . (63). 10.3.1.

Types (Sparsity Patterns) of Matrices A and B

The main categorie are: 1. Full 2. Band

(den e)

( par e) 3. Variable band (spar e) 4. Arbitrarily parse (sparse) where alt of che above have their nonzero e lements stored. and. finally ~ .. 5. Any patten1. but where the elements are not stored; that is. e lements are :generatecr~or calculated each time they are needed in the solving algorithm. Method of solution are basically:

MATRIX COMPUTATIONS

113

Direct where the olutio n emerge in a finite number of calculation (if we temporarily ignore roundoff error due to finite word length) . Indirect. ot iterative. where a tep-by-step procedure converge toward the correct solution. Indirect methods can be specially ·uitcd to spar e 1natrices (e. pecially ·when the order i large) becau c they can often be i1nplen1entcd wilhout the need to ·tore the entire 1natrix A (or intermediate fo rn1s or matrice ) in jiigh-speed R AM torage. All the common direct routine are available 111 ofnvare libranes and in books and _journals ~ comm only in Fortra n. Pascal. or Basic.: [3, 22, 23 J. More detailed accou nt · of ma tri x methods have al o been publi hed [1, 3

22-24].

~0.4.

Direct Methods for Solving Ax = y

The classic olution 111ethod of ( 190) i! the Gauss n1etbod. Given the system

( 19➔)

we subtract 2 time the fir t row from the second row. and then we :.. ubtract 3 time the fir t row from the thjrd row to g ive L

[

l 0

4 - 1

0

-6

and then subtracting 2 time the econd ro\ fron1 the third row give..

l - 34 -67] x·~ [

X,

o 0

()

/

.,\ \

[

J]

= - 1

(196)

()

from ( 194) to ( 196) are te rmed rriangu!ation or fo rward elimination. The triangular form of the left-hand matrix of ( 196) i crucial~ it ·allows the next steps . The third row immediately gives The

tep

X-= 0 \

and substitution into row 2 gives

( 197a)

114

THE FINITE ELEMENT METHOD

X ., --

0

.!.~

( l97b)

and then ub t itution into row I give

X , -- _ 31

.0

( 197c)

Th . teps through eq . ( 197) are termed back-substitution. We now ignore a complication of· pivoting'· [l . 22-24] . lrnportant point about this algorithm are Ii. ted belo,v. 1. Computing time i. proportional to ,,,3_ This n1cans that doubling the number ()f finite element node in a n integral equatio n solution ( hich give a fulJ matri.x) will jncrease CPU tiJne by up to 8 time ! 2. The determinant con1es imn1ed1ately as the product ot the diagonal elements of ( l 96).

3. Algorithm that take advantage of the special hand and variable band are very traightforward f3, 73. 24J. ju t changing_ the limit~ of the DO or FOR loop .. , and son1e bookkeeping. For exan1ple. in u rnatrix of Hsemibandwidth .. 4, the first c0Ju1nn has nonzero ele1nents only in the first four row , as in Fig. 30. Then only those tour nun1ber need storing. and onlv the three clcm~nt belo\V the diago nal need to be eliminated in the fir t colun1n. 4. Oddly. it turns out that, in our context, one should ue,,er find the inverse matnx A in orclcr to olve Ax = y for x . Even if it need.. doing for a number of diffcrent right-hand- ide ector y. it i • better to keep a re ord of tbe 'triangular form of ( 196) and back- u b ti tut a · necc ·sar . 5. Other methotJ · vcr similar to Gau s are due to C rout and hol , ki (and. intcre ·tingly. the original ( 'rout paper wa in an lectrkal en gin ering journal). T he Jatte r i · (only) for u. e \Vith . ym ,netric matrice . It advantage i._ that tin1e and storage are half that of the orth dox Gau . For the.

*

*

0

0

*

0

*

0 Fig. 30 Zeros and nonzeros in band matrix of semjbandwidth 4.

0

0

*

*

MATRIX COMPUTATIONS

115

reasons. the Choleski algorithm is chosen for use in t he complete computer program pre ented in Section 11. 6. There i a drastic varia nt of Gauss" method tl1at i ideal for finite element work [24). It accept a n1atrix A of arbitrar_ parsity; it ba.. ically starts with an en1pty n1alrix. inserts just the nonzero elements. and cleverly applies Gauss ,vithout causing too 1n uch ... fi ll•-in''-without preading its arithmetic too much out. ide the original pattern of nonzeros. 7. Another variant of the Gauss a1gorith1n has been especially developed for finite c lement work. In the fro nta l method . elimination takes place in a carefully controlled manner. \Vith intermediate result being kept in backing storage. Again. the method i well studied a nd documented [3. 231.

10.5.

Iterative Methods for Solving Ax = y

·We will outline two method : (1) the conjugate gradient algorithm and (2) the Jacobi (sin1ultaneous di placement) method \.vith the clo cly related Gauss-Seidel ( ucces ive ruspf acement) algorithm.

10..5.1.

The Conjugate Gradient Algorithm

Although known and u ed for decades\ the conj ugate gradient me thod ha come, in the 1980s to be adopted a one of the most popular iterative :~.lgorithms for solving Ax= y. It ~ rationale tart a a matrix version of our weighted-residua l approach where, as in eq. (7). vve introduced and made J ·small an error resid ua l R = Lu v. : The equation to be olved for x.

(198)

Ax = y

can be recast as fin ding x to minimize tbe error residual , a column cctor r defined as a function of x by

r = Ax - y

(l99)

_For an iterative method. thi.. means. from a given value of x,. finding a '~better value x 1"" , .. with a ·· n1alle(' re idual r. The norm llrll of thi residual vector is a convenient measure of r to be minimized,, and the conventional steepest-gradient method of optimization [1, 22] would eern surely the best way of proceeding fron1 one step to t he next toward the bottom of the valley. Unfortunately. in numerical work. the valleys are rarely isotropic in the parameters being varied (all the elements of x) and the teepest-de cent algorithm is too myopic to notice! The conjugate gradient method instead evaluate· the gradient (with 2 respect to x) of~ llr(x)II at a point x: ~

~

116

THE FINITE ELEMENT METHOD

V( ~ Hr(x) {f

2 )

and then minimize II rll along a particular line (x + v bx); that is. it finds the value of v that minimizes II r(x + v Sx) IIIt tran pire [1 22J that

V( 111 r(x)ll

2

)

= A l(Ax -

y)

(200)

and

u=

- oxv( ½11 rll 2)

( 201)

IJA5xll~

An immediate feature of expression (200) and (201) i · that reference to matrix A is only via sin1plc matrix product · for given value of the matrice A y, x;. and ox,. \Ve need o nly fo rm A time a vector (Ar or A ox;) and At times a vector (Ax; - y). The e can be forn1ed from a given sparse A \.Vithouf unnecessary multiplication (of nonzeros) or storage. In Reference 22 and many comn1crcial packages. the user h a to arrange the, e calar product ., while guida nce through the overall str ategy is provided by the package. More robust versions of the algorith m see to a prelin1inary preconditioning of the matrix A to alleviate the problem that t he condition number of (200) is the square of the condition number of A-a seriou problem if A i not afely po itive- 11, + 0 (usually air) and the terminal plane z = -dN. We hall assume that thi plane can be modch;d a an impedance wall forcing a in1 ple relatiotY'hip .between the tangential electric and n1agnetic fie ld existing o n it. nan1ely

( l2)

Electric and magnetic ,vall\; arc incl uded in eq. ( 1) a .. particular ca e given respectively , by z .'i = 0 or z., = oo. The exi tencc of an impedance ,vall implie that the problem can be olvcd without knowing the actual field that can exi t belo\v the tenninal plane - = - d ,. Each layer i , a su1ned to be an i otropic homogcncou and po ibly lo y material ,vith complex permittivity e and con1plcx permeability µ . Similarly \ 'VC a un1e that the e n1bedded conducto r are characterized by a boundary condition on their urfaces:

nxE = Zs nxJ

~

w here ZJ is again a surface impedance (zero for perfect electric conductors)~ n is the oun.,,ards unit normal vector ( Fig. 1), and J, is the electric urface current exi ting in the conductor. · This current i excited by an excitation field E e and: in turn, creates a scattered or diffracted field E ''. The total field in eq. ( 13) is the um of the etcitation and diffracted fie lds.

138

I TEGRAL EQUATION TECH IQUE

z

z=O z=-dl

2

\

J

I

I \

I

I

I

\

I

.

(E-, U-)

1

1

l

I

\

\

I

I

\

'

'

z=-d.1

-+ ' I

e

z

z=-dN

Fig. 1 Multilayered medium tha1 includes several embedded conducting objects. The N layers are sandwiched between a half-space medium (usually the air). 0 < z < -x and an impedance wall at z = - d .

Th di (fr t d ti Id I an l fun tion G ,. hich i. d fin

pre . d th 1in r r Jati n

dE' r

= GE(rlr ·• / r') di

Th t tal iffrl r th . µrf a

In

th

ri o u approach. the

hihitin ari · fr n1 qui al nt c ndu r . Ho'. r. if can b n J t d. Finall t ·

tatn d b ndu t r .

up rp

di

14

iti n int

r tin

nib d ed

ry h(_h ohmic lo . p< rtion 1n n ti · ·urr · n fin d o th u ondu .. ivit i r hi h t ma n th b und r condition 1 b come 1

urr nt.

INTEGRAL EQUATIONS FOR LAYERED MEDIUM PROBLEMS

n x E''(r) = - n x

JGc(r\r') ·J_,(r') ds' + Z_,n x J,

139

( 15)

which is a generalized form of the e lectric field integral equ atio n (EFIE) for the unknown J c An a lterna1jve integral eq uation for J \~u ·ing a boundary condition on the magnetic field can also be derived f2 j. However it has been shown l3J that the magnetic field integral e 4uation (MFlE) i nume ricaHy un table when the en1bedded conductors are thin and faiJs completely for z re-volume conductors. Since integrated circuits often include very thin cond ucting .sheets. this integral eq uation will not be con idered here. Returning to the EFIE there are t\vo.~/u ndamental ways of interpreting the current J~a nd the Green · · function Gr: appearing in eq. (15).

1. In a first approach. we apply equi alence theorems for rcplacj ng the different layers by fictic.:ious electTic anu magnetic currents exi ·ting in the boundaries between layer (the o-called polarization currents). In this manner \Ve solve an equivalent problem ~,here the conductor are e mbedded in an unbounded homogeneous medium. Consequent))'. rhe Green's function in ( 14) and ( 15) take . the sin1plc fonn

(16) where k = w 1JkE a nd i L the unit dyadic. On the other hand. rh e unknown in (15) include fictive as well a true current a nd the integration domain must be extended to the . et of parallel planes _ = - d 1 (Fig. I). Thi approach has been . ucce sfully used in tatic problem [4l 2. The second po sibility deal. directly ,vith the real problem . The jntcgral equation applies only to the urface.. of the conductor. a nd the ·o le uakno,vn i the true urface current. But the boundarv cond ition bet\-vcen .layers must be included in rhe Green ·s functions \\1hose expre ion ace n0\.\1 Qy no means evident. Tn pite of this fact. this econd approach ha .. b en found to be numerically efficient [5 6J~ and ,ve will deal exctu ~i ely \ ith it throughout this chapter. ·.. For . tratified media problems . the spectra l domain technique can be applied in two different way : 1

•

,I

1. The integral equation (I . . ) i written an 0. To o btain an e timatio n of the pole location in a lo:,;:y case . we con ·ider D™ a.., a complex function of two complex variable ~ z and e,., and vvc again perform a Taylor: expansio n ar ound the poi nt z 0 -= A1,n/ k 0 , e, = c~. Keeping only Lhe dominant t nns . we obtain:

Here A and B are the p artial derivatives of DT~,1 with re pect to. respectively, z and Er calculated at the point z 0 == AP1/ k 0 • e, = E~. Replacing no\\'· the variable z by the com plex value of the pole J... P - j"P, \ Ve get the final result

11.

= (E '

-

k 1I, .. ( 1 ) tan o - •-• FI

which bow. that, to the first order. the real part of the pole is inde pe ndent of losses, \VhiJe the imaginary part is proportion al to the lo. s tangent. The integration path for the Som1nerfel (p' ) + q/ p') l = con tant = U

(94)

Instead of sta rti ng ,vith an excitatio n charge , . o lvin g (93) for chc ~- cattered charge,· q... and finally computing the voltage U ,\Tith integral (94 ). it will be frequently easier t(.) ' tare by assun1ing the voltage U kno,vn and con idering (94) a ' an integra l equation for the total charge q ~"' + Q~- T hi last approach tollow · closely the circuit rcpre cntauon of Fig. lUe and

ZERO. LOW. AND HIGH FREQUENCIES

163

correspond to the well-kno,vn static integral equation for the evaluation of capacjtance . T he Green s function to be used in eq. (94) can be found by setting k 0 == 0 in the corresponding dynamic expression . For instance.. in the case of a i□gle layer. eq. (80) become ,vith .A = Re fkl' I:

or, expanding the sum in ide the parcnthe.ses into powers ot exp - 2Ah) and .integrating the resulting infinite series term by cern1:

with 'rJ an 0) and P the residue of Fat the pole . .The functio n P,,;T'_ . - is inteerated anaJvticallv- as

168

INTEGRAL EQUATION TECHNIQUE

2.00 B - - - {a) R~al part

1.50

------ (b) Imaginary part

1.00

.50 (ll}

0.00 - .50 A

-l.00

koP

B

0.00

LOO

2.00

3.00

4.00

5.00

0.2 (Re}

Om) 0.00

1.0

09

1.5

>Jko th)

(Re)

(Im)

- 0.6

Fig. 11 (a) Normalized values of the integrand associated with the scalar potential V of an HED on microstrip. f, = S: k0 h = 0 .2; k0 p = 3. A, Discontinuities in the derivative; B, sharp peaks due to the pole; C, oscillatory and divergent behavior at infinity. Dotted line, real part; dashed line, i·maginary part. (b) Enlarged view of Fig. 11 a in the il)terval A E [0 .9k0 , 1.4k0 ] . After [19] © 1986 IEEE.

l

ko\f"i;

k

F

o

. ' ,ng

d\

"

p

=-

11

+ (k,,, e.'r - .,\ P )~ ·p k,,'\["2,' + J a rctan 12 + (k + ~ )2 1 f> 0 /\ p pp

2

2

ln

+

j P arct an __._ P_ _0

P

A, 1

A - k.

(109)

VP

It i worth mentioning t hat in t he lossless case (vp becom e

= 0)

the above in tegral

NUMERICAL EVALUATION OF SOMMERFELD INTEGRALS

169

( l l 0)

and the refo re the principa l-value formul atio n rcq. (87)1 of the lossle s ca e is include d a a li miting ca e in this numerical te chnique . Figur e 12 depict the real part of the original fun ction F( A) (curve A) and the differe nce F( A) - Fsrng( A) (curve B )~ wher e the ingularity ha. been extracted . T here is . till an infinite dt.:rivati vc in the cu rve B at A = k 0 • With a change of va ria ble A= k n cas h r one finally obta in a very smoo th integrand (the discontinuo u curve in F ig. 12). which is integra tc-.

=

kc,cosh t

- 0.6

Fig. 12 The real part of the integrand of Fig. 11 for the lossy case in the interval [ko, kovs,.]. Curve At before the extraction of the singularity; 8 , after the extraction of the singularity; C, after the change of variables: A = k0 cosh t. After (19] © 1986 fEEE.

170

INTEGRAL EQUATION TECHNIQUE

0.2 A

- 0.2

A: Before the extraction of the static term B: After the extraction of the static term

Fig. 13 The real part ot the lntegrand of Fig. 11 for the lossy case in the interval [k0 \/"s; , -x]. Curve A, before the extraction of the static term : 8 , after the extraction of the static term. After (19)

©

1986 IEEE.

8.2. Integrating Oscillating Functions over Unbounded Intervals Somn1erfeld integrals, a given by eqs. (79) and (80) can be grouped in a more general class of integrals defined by:

l ( p)

=

L

g( >.p )f(A) d>..

( 111)

\.Vhere

l. g( Ap) i a con1plcx fu nction ,vhose real and imaginary part o cilJate with a . trictly pe riodic behavior ( in, co ) or be have a y mploticallv a. the product of a pe riodic function and monoton ic function . A typical exa mple of this clas · of functions which will be te ro1ed from now o n quasi-periodic, are Bessel functi o n. of the fir. t kind . 2. f( A) i. a smooth , nonoscillating function tha l behaves a ·yn1prot ically a. A". exp( - A,8 ) . whe re f3 i the differe nce in heigitt be twee n o urcc and ob. c rvatio n point.. T herefore, when bo th point - are on thl; interface ( (3 = 0) the function I( A) decrease - very slowly o r even dive rge at infinity if a < 0. This situation. co m1nonl v fo und in practical micro trip problcn1s. i the n1osr difficult from a numerical point ot view. For the sake of ·implicity. /( )..) i. as umcd to be real. Complex function. can be ha ndled by \vorking successively with th ir real and in1agi nary part . 3. The lo\ver in tegration hound a has been chosen conveniently to en ·urc that the inte r al la, oc j is far enough from the possible singularitie of f( A). For instance in o ur problem ~ '-tve shall take a = ve;.

NUMERICAL EVALUATION OF SOMMERFELD INTEGRALS

171

It is \.VOrth mentio ning that the general expression (111) int:ludes many integral transforms su ch as Fourier and HankeJ t ra nsforms. Hence. the folJov.,iog techniques can be applied to n1any other problen1s in nurnericaJ aoalysj .

·rhc classical problem invol ing SommcrfeJd integrals i, the problem of radio,\'ave propagation above a lossy ground~ where the comprchen i e monogra ph of L_·tie and Lager [22j is the classic reference. The e author have found an ite rative R o mberg integration sati factory, since h re the integrand displays an ex po nential convergence and its poles have been removed from the real axi . In n1icro 'trip problems R.on1berg integration has also been used . but it effectivenes decreases considerably in the absence of a ,..vell-behaved integrand. In recent vears. there has been a considerable amount of work publi hed on 1.he nun1etlcal evalu ati:on of Fourier tran. form . . which are ,nc\uded in eq. ( 111) a a partjcu]ar case. T he technique involved can be classified into three group, . l. The decomposition [a. ~1= [a, A]+ [A . ~]. Here . filo o algorithm i applied to the finite interval [a. A], whiJe an a yrnptotic exprc sion of the integrand is used to estima te the integral's value over lA, :o] (231. The mo t seTious drawbacks of this approach are the choice of A and the ana lytical work required . features that are difficuJt to incorporate in an automatic computation routine.

2. Another approach a pplies if g( )lp) is a strictly pe riod ic function; then the following dccompo 'ilion i · used:

l (p)

=

r-;•

g(Ap) ; , f( A+ 11p) dA

(112)

where p is the pen od of the t un ction g. T he infi nite sun1 und r the integration sign can be evaluated by tandard devices. such a Euler·· transformation. 1\l. o a more involved techn ique using theoretical Fourjertransform concept ha. been de cri bed in connection with a problem on quantum-1nechanics impact cro ecu on (24). The e method ,vork very well for large valu of p a nd an exp onentially decreasing integrand. Unfortunate ly. t heir exten ion to q ua i-periodic di crging integrand ecm problematic . 3. The third group of teclu1iq ue . introduced by Hurwjtz and Z\veifeJ l25L are defined by the decomposition l{ p) =

±1".

ln - 1/fl 2

11 ~ 11

g( Ap)f(A) d>.

113)

1.T+ n p / "1

The integration over each haJf cycle i performed prior to the series· summation. As before an accelerating device .. such as the nonlinear trans-

172

INTEGRAL EQUATION TECHNIQUE

formation of Shank (Alaylioglu et al. [26] or Sid i [271 can be u e d to um the infinite series. We feel that the decompo~ition ( 113) i· particularl well uited for the ommerfeld integral e nco unte red in micro trip pro blem . and \Ve have use• ~

-

en tl,o (,I

•

90

log k 1p -

1

0

• -t80'---- - - - - - - - - - - - - - - - - - - -- - - - Phase of the normalized scalar potential in a singfe layered (sr = 1O) micros1rip structure as a function of the normatized distance. ( - --) Numerical value; (---) near-field approximation; (-- -) far-field approximation. After [15].

Fig. 20

--- t r = l ----- - · t

- -·-

10- 3

r

= 2.2

t,

= 4 .34

f r

=96

t::::.___....JL_..L-JLL.J..J.J~--L.._J..--1.-.L.1-J...J..J...L..,_,._i__.L.....J~J--'-'.1

10

2

2

4

7 10

I

2

4

7 100

2

4

7 10 ~

kon Fig. 21 Effect of the dielectric constant on the scalar potential (singte,layer substrate), (1) e, = 1: (2} s, = 2 .2: (3) r = 4 .34: (4) e,. = 9 .6 .

180

INTEGRAL EQUATION TECHNIQUE

h i An- ·rhis transition zone appears a a rapid variation of the pha e (Fig. 20) that rise sharply fron1 small positive values a nd aften.vard follO\VS a t 1 pical surface-wave pattern. More in~ight into these phenon1ena can be obtained by keeping /z / An con tant and changing Er . The cases h / A0 = 0.CJ5 a nd e, = L 2.2. 4.34. and 9.6 are pre ented in Fig. 21 . Except fo r a .. caling factor (e,. + 1) / 2. the curve for E, # I follow close ly the ho mogeneou behavior up to a rransitioo zone, after which the urface wave become dominant. A poin t \\•Orth n1e ntioning i · that for e, = 10. th e near-field apptoxirnation cannot he extended beyond p l J\ 0 = 0.03, which is the distance corresponding to the transition between ~ tatic a nd urface wave beha ior . Figure 22 depicts simultaneou ly the n1odulu and pha e of the calar potential in a polar diagran1. For the ·ame substrate of thicknes · h I A11 = 0.05. we have plotted the cases E~ = l and er = 2. 1'he dots and ~ tar · .represent values of the scalar potential ranging from k 0 p = 0.5 to k 0 p = 10 in steps of 0.5. Both curves start at phase zero a nd in the homogeneou ca c the phase decrea es teactily yielding a clockwise spira l rhat con ergctoward the origin. On the other hand, the curve , tart counterclock\vi c in

5

r 4

3

Fig. 22 Polar logarithmic plot of the normalized scalar potential for the cases ( • ) e, = 1 and (*) £ , = -2 .2 of Fig. 21 . After [15).

RESULTS FOR THE POTENTIALS

181

a n inho mogen eous structure. T hi ., is an a no m a lou. . itua tion '-"-'here the phase increase \Vith distance. due to a complex interaction bct\veen pace and surface waves. After a turning point (kt,P = 1.5) the curve resumes the behavior of an o utward propagating wave. pfraJjng cJockwise. \Yhile jrs distance to the origin ( the modulus of the calar potential) d ecreases low Iv as expected fro m a s urface wave. ln conclusio n, the calar po tential is don1inated by the surface wave and no a na lytical approximatio ns a re usefu l in the practical range 0.01 .A 0 < p < A11 . Hence. \ VC n1u t resort to careful numerical evaluatio n of the corresponding Sommerfe ld inte~aL .

9.2. Two-Layer Microstrip For a sub ·trate including two la ·e r . we obtain "ssentiallv . the ~ame behavior for the potential ·. A. in the ·ingl -la er ca.. e . 'Ne can aga in avoid the numerical evaluation of the Son1merfeld integ ral as. ociated to the vector ~

potential. provided the layer.. are electrically thin . Exact exp res ions will then be replaced by homogeneous a pproximatio n .. . For the . ca lar pote ntial. we can ha rdly u c ta tic or homogeneou approxin1atio n. . Figures 23 and 24 give the modulu , and pha e of the Scalar Potential

4 .

2

10° 7 4 ~~

2 10 -

1

7

4 2 10 l

7 4

10- 2 2

4

7 10- 1 2

4

7 100

2

4

7

,,,Ao Fig. 23 Modulus of the normalized scalar potential for a two-layer substrate when the source and the observer are both on the air- upper layer interface. Upper layer: er1 = 2, tan 5 1 = 0, k0 h , = 0 .05. Lower layer: , 2 = 5. tan S, = 0, k0 h2 vanable.

182

INTEGRAL EQUATION TECHNIQUE Sca\a1 potentla\

6.00

2.00

2.00

~ f

- 6.00

~ tO .s::. ~

- 10.00

- 14.00

- 18.00

L.---____J---..L------1---t.___.L.....a......&--L...i.------'-----~----......_,__.

l0 -2

2

4

7

10-t

2

4

7

J) ()-.c:,

Fig. 24 Phase of the normalized scalar potential for a two-layer substrate when the source and the observer are both on the air-upper layer interface. Upper layer: i, 1 =2, tan8 1 = 0~ k0 h, = 0.05 . Lower \ayer: e, 2 = 5. tan 8, = O, k 0 h2 variable.

normalized calar potential 47Te0 ..\,,V as a function of the norn1alized radial distance. Layer·. parameter · are Er 1 = 2. er2 = 5 k 0 h 1 = 0.05 and k 0 h::. = 0.01. 0.025. 0.05. A could be expected. the urface-wave behavior i reached taster for tl11ckcT substrate . In th - ·e t\i'VO figures. ource and observer are both ar the air-dielectric interface . ln fact. there are four ituations of practical intere t. depending on the rclativ po ition of ~ource. and ()bser'ler: l. Source and obser er are both at the air-dielectric interface (z

OL

= z' =

-

2. Source and observer are both at the interface beh,vccn dielectrjc..

(z =z'= - h 1 ) . 3. Source at z' = 0 and oh erver al 2 = - /J 1 • 4. Source a L z' = - h I and observer at z = 0. The la ·t two possib,Hties give identical Tesu\ts for the potential' as can be hown by reciprocity. Figures 25 and 26 gi e the modulus and pha e of the scalar potentiaL calculated in cases 1- 3 for a sub trate of parameter e t = 2. E r?.== 5.

RESULTS FOR THE POTENTIALS

183

Scalar Potential 7 4

2 0

7 4 ~

2

-..

..;..

10 l

7 4

2 10- 2 7

4

10- 2 2

4

7 10 1 2

4

7 10° 2

4

7

fig. 25 Modulus ot the normanzed scalar potential for a two~\ayer substrate. Upper layer: c:r, = 2, tan

1

.I ,I'

,(/

.

- - l"Ul~~Cti!tl

'

....... er,tto c:~ ., ,:c;, ,. i

I 111

1.22

I IS

. ,r...._.,

rro

126

I .JC

(d i

Flg. 40 (a)-(d) Some scattering parameters (moaulus in decibels) ior the configuration presented in Fig. 39 near resonance.

206

INTEGRAL EQUATION TECHNIQUE

close structures. but \vhen the distance bet,iveen patches increa e , the e nd-coupl ing in the x direction ( parallel to the surface currents) decrea e slowly~ due to the surface wave. unu become dominant.

12.3.

A Slot in a Microstrip Line

As the la. t cxamp1e we con ider a microstrip line \Vith a slot in the upper conductor (Fig. 4 1). The lot can be viewed as t\.VO c lose step di continuitie. that cannot be analyzed separately. -rhe rnicrost rip line i printed on a single-layer ub t rate \Vitb Er= 2.33 . h = o.: I mm. tan o = 0.00 l. The line width i w =- l .53 mm. \.Vhicb correspond. to a quasi-TEM characteristic impedance of !l. The s lot i s = 1.223 rnm wide and t = 0. 918 Inn1 deep. ln the previou · exampJe . \Ve obtained a . cattcring n1atrix and an equivalent circuit of the whole tructure, including the coaxial probes. T herefore. the predicted theoretical values can be directl_ compared with mea urement . However. in this example it is intcre ·ting to extract from the overall result an equivalent circuit fo r the geometrical region occupictl b_ the discontinuity a nd lin1ited by wcll-d fine d reference plane.:. For this purpo e. it i. convenient to analyze the tructurc \vithout a di ·continuity. i. e .. to remove the portion in ide the plane PP' in Fig. -+l and to stud the re. ulting uniform n1icro ·trip Line ection of length 2L0 excited by the two coaxial probe . The field ana1y is \Viii then provide a total chain matrix T0 which can be decomposed in two identical part T,,. each one accounting for a half-. ction of length L,1:

:o

lf we now tudy the ·tructure including the di continuity and obtain a

.

.

....

..

-- - - -- - - - - - - - -

, ...

______ ___ _

,,,_

__

.J

__

.,,

____

____

,

p

I

I I I

.

, '--· ----- - -- - - .... ,

. .. ..

•

'

.. .

..

- -

p

,

-

-·

-

-

-

•

_

. ._

-

-

-

-

....

llito,.

\

., I

I

,

~, - - -_ --- - - --_ \, __ _ ___ B_ __ I , -_.,.- -___ _______ ____ _ _ / /

-- .

- - - - - - - --

-------- ---- - - --, - --- -- - - -

. ,.

- . . .

,

,_,.

____

- -. . --------,,,.,.,,~,

- -

__

>

•

________

, , .

-

I

,.

,

•

,.

••

-

-

-

-

.,.

- - - - -

-

.--

- ..

I

'

-

,- ---- ~--- --- - ---------------- - -~ ---------·

......

n

.,.

•

..

'

- -

.

·---.,.. --- - -- ~ - , . •• _, __ _ ,. __ .,.. _____ , __

•

-

Fig. 41 A slot In the upper conductor of a 50 microstrip line (h = 0 .51 mm. w = 1.53 mm. r , = 2 .33) . The line is excited by two coaxial p(obes with currents f and - I. The figure shows the amplitude pattern of the surface current at 1 GHz. (A) Unperturbed contour. ( B) Slot contour.

•

PRACTICAL APPLICATIONS

chain m atrix T. it is reasonable to a ·sun1e that T chain matrix of the discontinuity i

207

= T11 T ,,T,, and therefore the ( J 58)

ln this ,vay. the effects of th e coaxial probes are eliminated a nd t he equivalent circuit will be. a. req uired. independent of the excitation, provided that the line ection arc 1 ng e no ugh to ee bet\veen excitation and discontinuity and typical curre nt distribution of a unifom, line . To check the validity if the model. we first conside r a ·lot with zero deep (contour A in Fig. 41). Then. eq. ( 157) sh o uld give the eq uivalent circuit of an unperturbed 50 fl line section of length 2L 0 . Therefore. we can compare the impedance matrix derived from ( 157) with the theoretical irnpedancc matrix provided by standard transmission Hne theory. In particular t he complex characteristic unped ance j given by (159) and the con1plex propagation factor 'Y by -y

(Zn)

= arcosh z,!

( 160)

PreliTninarv calculation ho\ved that ,vith a decompo ition into 5 x 4-. cells, the real part of Zc and the imaginary part of -y agreed '-''ith the quasi-TEM values \Vi thin 1 ~ > at 1 GHz. Actually. the f requcncy depcnd~ncc ·.@f characteristic in1pedanc ~ effccti e dielectric co1L tant, and attenuation factor can be o btained with this technique. but easier n1eth d · art,; available

for propagation proble1n in uniform stn1ctures. The slot ,vas no" introduced by ren1oving the appropriate celL (contour B in F ig. 41). The curre nt distribution resulting fro1n the field analy is at 1 GHz i a l o depicted in Fig. 41. It i readily appa rent how th current stems radian. fro m the coaxial probe and then fo llows a uniform line patten1 before flo\\1ing acros .. the d i ·cQntinuity. In particular. the transver e depe nde nce of the current in a uniform line is well represented. If tran. ver e currents are neglected. it would be possible to include the tran. ver e depende nce of the \ongitudina\ curre nt in the ba ·1~ function , but thi simpJificatio n fa il for comp lex geome tries. The chain matrices can be easily transformed into scattering matricc. and we can compute the quantity ( 161) which is related to the total dissipated power and eq uals un ity in a los le s two port.

208

INTEGRAL EQUATION TECHNIQUE

Figure 42 gives the value of P co rresponding to the s lot and a n unperturbed line section of the ·a me width. While there is a lmost no di ipatio n in t he unperturbed line ( P = 0 .95 at 10 GHz), a large amo unt of power is r adiated a nd la unc hed in the fo rm of s urface waves by the d iscontinuit (P < Q_75 at 10 GHz). Finally it is a lso possible to tran. fo rm the cha in matrjx of the di continuity into a n in1pedance matrix a nd define a T equivalent circuit \vith a lumped series impedance Z~ = z 11 - z r~ and a lumped paraUel admittance

Y,, = 1/ z!~· The real part of the. c qu a ntitie . account for Josse d ue to radiation . The imaginary parts ,vhich a rc depicted in Fibrs. 43 and 44 ho,v a linear

0

~~======-----------,

~ 1 :-=::: ...~.

...--,

LINE/ SLOT

•• •. '(j

. , . .. . ..... 0

.. ..

lO

A

Ol

.. .. 0

...

Ol

. ·. .. ...

lO

... B .·0.

(I)

Q_

..

...

... ..

0 (I)

.

b.

..

.. .. -..

•.

tn

r"

.

0

" ~-~----r--~------~-...-------------.-------~ 1.00

2.50

4.00

5 . 50

Freq

7.00

8.50

10.00

[GHz]

Fig. 42 Values of P = Is,, 12 +1 s2 1 1, as a function ot frequency for (A), the unperturbed line section between planes PP '. (B ) The slot between the same planes (Fig. 41 ). 2

209

CONCLUDING REMARKS

S TEP

0

·----------------------------

0

.

0 (\J

0

.

lO ~

0

en

N

C\I

~

0

a,

0

.

0 0

. ~------.------,,--.....--...---......--~-........--------------t 1.00

2 . 50

4.00

Freq

5 . 50

7.00

8 . 50

10.00

[ GHz]

Fig. 43 Imaginary part (reactance in ohms) of the series impedance associated to the T-equivalent circuit of a microstrip slot discontinuity (Fig. 41 ). The quasi-linear dependence with frequency shows that this reactance behaves approximately like a lumped inductance.

depe nde nce o n the frequency in the lo~i1 r ange and therefore confirn1 th e quasistatic prediction · Z , - jw Lt !2 a nd YP-- }lu CP n - \ a lid for these di continuitic ~ a t low frequencic~. In additio n, o ur a na ly i. provide, real part for Z s and YI >accounting for surface waves and radiation .

13. CONCLUDING REMARKS The mixed potentiaJ integral equation h a been fo und to be a very powerful and flexible tool for the n umcrical a nalysis of microwave and millimet r wave integrated cjrcujt . including planar conductors em bedded in a la er ed

210

INTEGRAL EQUATION TECHNIQUE

STEP

g

0

,q

fl1

0

('f')

I

co (\J

0 ~

*

0 (\J (\J

(1

>-

0 Ul ,,r-i

0 0

0

~ -+--

1.00

- --

- , - - - - - - - , . - - ~ - ~---.--- - . - -- , - ------.--

2 . 50

4 . 00

Fr eq

5 . 50

7 . 00

r---

8 . 50

~-----i

10 . 00

[GHz]

Fig. 44 Imaginary part (susceptance in siernens) of the parallel admittance associated to the T-equivalent circuit of the microstrip slot discontinuity (Fig. 41 ). The quasi-linear dependence with frequency shows that this susceptance behaves approximately like a lumped capacitance.

medium. Combin~d with a met hod of n1oment using a sub ectio nal basi . this technique ca·n deal vlith conductor._ of quite irregular hape. A lso the MPIE re1nain · valid at any frequency and can be u ed for tudying higherorde r rcsona nc~ a ,vell as fo r characterizing geometrical di ·continuities at re la tively low frequencies. Thu . the technique. described in thi chapter are particularly u eful for prob Jems wh e re th e frequen cy i too high for a urning a qua i-sta tic situatio n hut ·till too l ow for computing the field.. a expan io ns over the unperturbed resona nt modes of the structure. T he MPIE includes surface \vaves a nd rad iarjo n . Multi laverev.·ave Theory Tech ., vol. MTT-_-,,:'. pp. 1036- )( 1-l'2. 19g-. 37. B. D . Po povic. ~1. B. Dragovic. and A. R. Djordjevic . Anal_v. is and yniht)si of \iVfre Anrennas. Wile . New York. 1982. 38. R . C. 1-laJI. J . R . Mo ig, and F. E. Ga rdio l. .. AnaJysis of micro.·tri p antenna array!'i wi th thick ·ub trate ... / 7th Eur. A1icrowai·e Conf.. pp. 951- 9.-6. Rom~ . Italy. Sept. l 987. 3?. J. R . Mo. i.g.- ··Arbitrarily haped microst rip st ructure a nd their analy. i. with a mixed potential integral equ ation:· JEEE Trans. Micro wa, e Theory Tech.. ol. MTT-36, pp. 314-323. 198 '. 1

4 Planar Circuit Analysis K. C. Gupta Department of Electrical and Computer Engineering University of Colorado Boulder, Colorado

M. D. Abouzahra Lincoln Laboratory Massachusetts lnsti1ute oi Technology Lexington. Massachusetts

1.

INTRODUCTION

In the pru t . micro\J a e engineers and re archers ha c gc·ncrally focu ed on three principa l ca tegorie of electric circuit ·. The fir . t and most fa n1iliar category consist. of lumped elements \Vith phy ·ical dimensio n, much n1all r than the 'A'avele noth . T his eta of luinpcd e le ment · ma be referred to a · zero-dilnensional components. E lectric circui ts who c di me n "ion are n1uch smaller in two direction but cornparable to the wa clength in the third direction constitute the eco nd principa l category . This category of dLtributea .

II

27T !IX

•

\Ga

·1n

co

2 ii( I -

111) y

(66)

30

with l = -(rn + 11 ). 4.3.2.

Circles and Circular Sectors

Greens function for a circular egme nt with a shorted boundary alo ng the circumference may be obtained by expan ion into eigenfunctions a. d ' cussed in Section 4.1. The result i • analooo us to the Green \s function 2iven in (52) and may be expres ed as ~

:r

I

t

I

(67) where k mn ( for the case of the shorted boundary) is given by

-

.

1,, (k,,"'a)

=U

Green' function for a circular ·ector v, ith horted boundary 1 al o btained by u ·ing ( 42). The eigenfunction t/1,, may be written a a product of angular and radial factor.. The nornutlizetl angular eigenfunction ( [Qr the geometry shown in Fig. 9) that sati -fie,., the boundary condition at cp = 0 and (f> =

) + jwµd J fJ n

l

1

h fir. t- rd r H nk I fun ti n th n kind. rd r l fun ti n f th c nd kind. J,1 i th urr nt it · int f r =1 . micro tri and r due d-h i ht v id r ii n tripl -t p planar ir uit . Th ariabJ th nt ur · and r i the . tr i ht tin Ji t'lll th · "' and L (gi n b and ·0 ru h n in Fi . l . n 0 i· m rai ht lin j inin int and L with th n rmal t th p riph r at L. Equation l O d crib th r lati n en a th rf v l a nd th r curr nt di tributi n al ng th p ri h r . A pl n· r ir uit d b ( 100 ma ha ·n b und ., . a h rt d b ur dar . boundary. In ith r a int gra1 qu tion giv n b_ l 0 b dividin tl1 ircuit ri h r . into in r m nt l ar itr r width W . W ~ . . . . . W, 1 a ui · d in th · n t o .u ti n . •

5.2.

Ii..;

Open-Boundary Planar Circuit (12]

d cribed rli r. · n imp d·ln m· t.rix ·har riz· tio n i ppr riat f r a planar ircuit wi h an p n unda . Ia o d r t n1 th · imp d· n m· tri .. , h riph r i di id d int n numb r 1 2 ... ha in idth _ r p cf illu ·tr in 1• Fig. I . Th p riph r f th pl 'tnar ircuit i: di i