A hybrid numerical technique for analysis and design of microwave integrated circuits

A Hybrid Numerical Technique for Analysis and Design

of Microwave Integrated Circuits

B.S., Tsinghua University, 1985 M.S., Tsinghua University, 1986

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Electrical and Computer Engineering Department We accept this Dissertation as conforming

to the required standard

0 Dr. R. Vahldieck. Sunervisor

Dr. J.'Bornemadn. Departmental Member

Dr. W. Hoefer, Departmental Member

Dr. C^Efa^ley, Outside Member

Dr. V. K. Tripathi, Exjen^l'Exanrlhen.Oregbn State Univ.)

A ll rights reserved. This Dissertation may not he reproduced in whole or in part by mimeograph or other means,

without the permission o f the author.

in the Department of

© Ming Yu, 1995 UNIVERSITY OF VICTORIA

ABSTRACT

Miniature Hybrid Microwave Integrated Circuits (MHMIC’s) in conjunction with Mono ­ lithic MIC’s (MMIC’s) play an important role in modem telecommunication systems. A c­ curate, fast and reliable analysis tools are crucial to the design o f MMIC’s and MHMIC’s. The space-spectral domain approach (SSDA) is such a numerically efficient method, which combines the advantage o f the one-dimensional method of lines (MoL) with that of the one­ dimensional spectral-domain method (SDM). In this dissertation, the basic idea o f the SSDA is first introduced systematically. Then, a quasi-static deterministic variation o f the SSDA is developed to analyze and design low dispersive 3-D MMIC’s and MHMIC’s. S- parameters and equivalent circuit elements for discontinuities are investigated. This in­ cludes air bridges, smooth transitions, open ends, step in width and gaps in coplanar waveguide (CPW) or microstrip type circuits. Experimental work is done to verify the sim­ ulation.

The full-wave SSDA is a more generalized and he'd theoretically exact numerical tool to model also dispersive circuits. The new concept of self-consistent hybrid boundary con­ ditions to replace the modal source concept in the feed line is used here. In parallel, a de­ terministic approach is developed. Scattering parameters for some multilayered planar dis­ continuities including dispersion effect are calculated to validate this method.

Examiners

y --- -— — — \

f p r . R. Vahldiqck, Supervisor

Dr. J. Sfo^femann, departmental Member

Dr. W. Hoefer, Departmental Member

Dr. C. Bra$Ht?y, Outside Member

Table of Contents

Table o f Contents iii

List of Figures v

List of Tables vii

Acknowledgments viii

1 Introduction I

1.1 Background and G o a l s ... I

1.2 Organization of This Dissertation... 7

2 The Space-Spectral Domain Approach 8 2.1 The Spectral Domain M e th o d ...8

2.2 The Method of Lines... 10

2.3 The Relationship Between the SDM and MoL... 14

2.4 The Space-Spectral Domain Approach... 16

2.4.1 SDM in x-direction... 17

2.4.2 MoL in z-direction ... 19

2.4.3 The Eigenvalue Solution of a Resonator P ro b lem ... 25

3 The Quasi-Static SSDA 28 3.1 Why Quasi-static ? ... 28

3.2 The Quasi-static S S D A ... 28

3.3 On the Nature of the S S D A ... 39

4 The Full-wave SSDA 43 4.1 Eigenvalue A p p ro a c h ... 43

5 Numerical and Experimental Results 54 5.1 Convergence A nalysis o f Quasi-static S S D A ... 54

5.2 Sim ulation Results o f Quasi-static SSDA . . 55

5.3 Convergence Study o f Full-w ave S S D A ... 64 5.4 Sim ulation Results o f Full-w ave S S D A ... 64 5.5 Experim ental R e s u l t s ...67 6 Conclusion 7J 6.1 C o n t r i b u t i o n s ... 71 6.2 Future W o r k ...72 Bibliography 75 Appendix 79

List of Figures

v

Figure 1.1 Exam ple for an M M IC Circuit Figure 1.2 M H M IC discontinuities Figure 1.3 A typical planar discontinuity Figure 2.1 A shielded microstrip line

Figure 2.2 Cross-section view o f a m icrostrip line

Figure 2.3 A m icrostrip discontinuity in a resonator enclosure Figure 3.1 Planar circuit discontinuities

Figure 3.2 Discretization of a C P W discontinuity Figure 3.3 The equivalent circuit

Figure 3.4 A CPW Air Bridge

Figure 3.5 Configuration of general transmission line Figure 4.1 A n eigenvalue approach

Figure 4.2 A deterministic approach

Figure 5.1 Convergence analysis o f the Quasi-static SSDA (w lli= l, cr~9/>) Figure 5.2 Capacitance o f m icrostrip open ends.

Figure 5.3 Equivalent capacitance o f a microstrip gap discontinuity. w /h = I, h=0.508mm , er=8.875

Figure 5.4 S-param eters o f a m icrostrip step. u q -1 m m , u’2=0.2.5//////, u*//;=- /, 8 , - / 0 .

Figure 5.5 Equivalent capacitance o f a CPW open e*id. er=9.6, h -■■0.6.15, ///

d - l , d=w+2.s

Figure 5.6 S-param eter o f a CPW airbridge. w ~ l5\ini,s= IO \im , l=.1\Lrn,

h=200[Lm, b=1\un

Figure 5.7 S-param eters o f a CPW step. w ^-0,4m m , w j= 0.lm m , Vj -•■0.1 mm,

\\'2=0.4mm, er=9.<V, h=0.254mm

d2=2s2+u'2,£r=9.<S’, h=0.635m m , W[lh=0.2, u ’j /d 1=w2/cJ2-0.56,

H'2/ir , =3

Figure 5.9 Equivalent capacitance o f a microstrip and a C P W step/taper. For CPW, W[=0.8mm, s [=0.1 mm, w2-0.2m m , s2= =0.6m m , er=9.6,

li=0.254mm. For m icrostrip, w \= lm m , w2-0 .2 5 m m , z r=9.6, h=0.25rnm

Figure 5.10 S-parameters o f a CPW airbridge versus bridge length /. w=().3mm, s= 0.1 m m , b = 3 ]im , er=9.6, h=().254mm. Figure 5.11 Frequency dependent behavior of m icrostrip step Figure 5.12 Frequency dependent behavior of CPW step Figure 5.13 Convergency behavior o f the full-wave SSDA Figure 5.14 Full-wave S-param eters o f a microstrip step Figure 5.15 Full-wave S-param eters o f a microstrip step

Figure 5.16 S-parameters for a cascaded step discontinuity separated by a transmission line o f length 1. w [=0.4 mm, w o = 0 .2 m m , \\'2=0.8m m , er=3.<S\ h=0.25mm.

Figure 5.17 Measured and com puted S-param eters o f a C PW gap

Figure 5.18 Measured and com puted S-param eters o f end-coupled CPW reso­ nators

Figure 5.19 M easured and com puted S-param eters o f a C PW step discontinu- i tv

Figure 5,20 S-parameters o f a CPW end-coupled filter. w = 0 .2 , s = 0 ,1 5 , g a p width: 25.4\un, resonator length: 2mm.

Figure 6.1 Future application: electro-optic modulator Figure 6.2 An in-line 3-port discontinuity

vii

List of Tables

Acknowledgments

The author wishes to express his acknow ledgm ents to his thesis supervisor, Dr. Ruedi- ger Vahldieck, for his guidance, encouragem ent and invaluable suggestions through­ out the course o f this thesis.

Financial support for this research by Dr. R. Vahldieck (through N SERC ), Science Council of British Colum bia (through GREAT Award) and M PR Teltech Ltd. is also gratefully acknowledged. In particular, I w ould like to thank Dr. J. Fikart and H. Minkus, MPR Teltech, for the fabrication o f the M HM IC prototypes, w hich w ere used to verify the SSDA results.

The author also w ishes to extend his thanks to Dr. K. Wu for his invaluable sugges­ tions. The author is also grateful to his colleagues at the Laboratory fo r Lightw ave Electronics, M icrowave and Com m unications (LLiM iC), U niversity o f Victoria for their support and discussion.

Last, but by no m eans least, the author w ishes to express his thanks to his family, esp e­ cially to his wife, M ei Li, for their support and encouragement.

I

Chapter 1

Introduction

1.1 Background and Goals

M iniaturization o f m icrowave circuits is essential ir Mic evolution of modern com ­ m unications system s. In analogy to the m iniaturization that has taken place in VLSI (Very Large Scale Integrated Circuits), M onolithic M icrowave integrated Circuits (M M IC ’s) but also M iniature H ybrid M icrowave Integrated Circuits (M H M IC ’s) com bine a steadily grow ing num ber o f m icrow ave components on smaller and sm aller chip real estate. M M IC ’s as show n in Figure 1.1 [14] are very expensive in the fabrication and are only justified fo r large volum e applications. M H M IC ’s are a hybrid technology that is in p ar­

ticular suitable fo r sm all to medium volume applications. While M M IC ’s require sem i­ conductor fabrication facilities (circuits are grown on CaAs), which arc capable to integrate active devices like F E T ’s (Field Effect Transistors) in one process on the same wafer, M H M IC ’s are grown on alum ina substrate as shown in Figure 1.2, and active devices are wire bonded into the chip in a final fabrication step. The latter technology is less attractive for large volum e applications because of the additional labor involves, but offers better circuit perform ance and is less expensivu for small to medium applications. Both M M IC ’s and M H M IC ’s play an im portant role in modern com m unication systems.

A serious bottleneck in both technologies is the lack of accurate, fast and reliable design strategies. A lthough com m ercial design software is available, the num erical m eth­ ods used are either com putationally very inefficient or inaccurate at higher frequencies. Several fabrication cycles become necessary to trim the circuit so that it satisfies the design requirem ents. This process is very expensive and tim e-consum ing (scvcra1 m onths). To cut dow n on the processing time and cost, it is necessary to develop accurate design algorithm s for M M IC ’s and M H M IC ’s in order to achieve first-pass success.

In generai, numerical methods can be classified into two categories:

1. M ethods which use an ei genmode approach to describe the electrom agneticJicld <’g-,

• Mode M atching M ethod (M M M ) 111

• Spectral Domain M ethod (SDM ) f 2 o /

2, M ethods which discretize a differential operator

• Finite D ifference M eihod (F DM) (6 j

• M ethod o f Lines (MOL) [7]-[131

T he first category of m ethods use orthogonal modes or basis functions to expand the field directly.

T he second category o f m ethods is applied directly to either M axw ell’s or the H elm holtz equations. The first and second differential operator are approxim ated by finite differences.

B oth category of methods can be subdivided further into

Quasi-static techniques which calculate equivalent network param eters. M icrowave circuits are described by lum ped elements like capacitors and inductor' . which are assumed to be constant over frequency.

Full-wave techniques which describe the electromagnetic f eld directly fro m M axw ell’s equations. Circuits are considered from the fie ld theory

3

the uite faction o f fundam ental and h i»her order modes at discontinuities.

COPLANAR WAVCGUIDE

COPLANAR AIR BRIOCE WAVEGUIDE

SLOTLINE ( a ) C o p l a n a r W a v e g u id e T J u n c t i o n AIR BRIDGE ( d ) S l o t L i n t C o p l a n a r J u n c t i o n a ir b r id g e ( b ) C o p l a n a r W a v e g u id e S l o t l i n e J u n c t i o n SLOTUNE ( e ) S l o t l i n e T J u n c t i o n AIR BRIDGE COPLANAR WAVEGUIDE COPLANAR WAVEGUIDE AIR BRIDGE COUPLED SLOTLINE MHUIC • SLOTLINE (c ) C o p l a n a r W a v e g u i d e / S l o t l i n c T r a n s i t i o n ( f ) MliMIC E m b e d d e d I n to DIODE' AIR BRIDGE I a CPWG S t r u c t u r e AIR BRIDGE 3 BIAS AIR BRIDGE 2 MIM CAPACITOR COPtANAR WAVEGUIDE -SLOTLINE ' ' — OUTPUT

( g ) C i r c u i t C o n f i g u r a t i o n o f U n i p l a n a r MIC B a l a n c e d M u lti p lie r

5

Q uasi-static numerical techniques are traditionally faster than full-wave techniques, in particular, on the serial machines widely used today. These methods do not benefit from the availability of parallel processors. For some applications, their accuracy can rival that o f full wave techniques and, therefore, they arc very useful engineering design tools.

Theoretically speaking, a quasi-static approach only works at zero frequency. H ow ­ ever, as long as the dimensions of circuits are small compared with the w avelength and the dispersion of the transmission line system is weak or non-existing, quasi-static m eth­ ods cun w ork up to the millimeter-wave range. A large num ber o f com m ercially avail­ able softw are is built on quasi-static methods. In most cases the equivalent elem ent values derived from quasi-static methods are assumed to be constant over the frequency. Furtherm ore, it is assumed that the discontinu,ties for which they are derived do not radi­ ate or interfere with each other. This assumption becomes invalid the closer m icrowave elem ents are placed on the chip. In this case the predicted perform ance of the M M IC ’s or M H M IC ’s m ay deviate significantly from the required performance.

For structures, or frequencies, at which quasi-static m ethods do not provide accurate results, either because the circuit density is too high or dispersion effects arc too signifi­ cant, full-w ave modelling of microwave and m illimeter-wave circuits becomes neces­ sary. In this dissertation, a generalized Space-Spectral Domain Approach is first introduced that is suitable for this task. Secondly, a new quasi-static determ inistic tech­ nique w ill be presented. Finally, two SSDA full-wave algorithm s will be discussed.

Typical generalized full-v/a -e approaches are, e.g., the Finite Difference M ethod (FDM ) or Transm ission Line M atrix (TLM) fl6 j method. These methods start directly from M axw ell’s equations with very little approximation and virtually no analytical pre­ processing. They have almost no structural limitations and provide a high degree o f accu­ racy. B eing very flexible, they often require large amounts o f com puter mem ory and long com puter run-time, at least on most of today’s available engineering w orkstations. To speed these methods up, parallel processor machines are required which, for som e time to come, will not be commonly used in engineering laboratories because o f the special program languages necessary to fully take advantage o f the potential of these m achines.

side view top view

▼ x

Figure 1.3 A typical planar discontinuity

W hen dealing with the analysis and design o f M M IC ’s and M H M iC ’s, their quasi- pJanar structure, as shown in Figure 1.3, allows the use o f less generalized but com puta­ tionally more efficient techniques. In the past, the m ost suitable methods for 3-D planar circuit analysis have been the Spectral Dom ain M ethod (SDM ), the M ode M atching M ethod (M M M ), and the M ethod o f Lines (MoL) (full-w ave and quasi-static).

In the SDM , the Fourier transform is taken along a direction parallel to the sub­ strate, and G alerkin’s teetmique is used to yield a hom ogeneous system of equations. To determ ine the eigenvalue problem (propagation constant) in a planar circuit, the 1-D SDM is well know n for its fast com putational algorithm and m inimum m em ory require­ ments. B ut the SD M also requires that the circuit discontinuities fit into an orthogonal coordinate system and, especially, that the basis functions are chosen carefully. For 3-D discontinuity analysis, the 2-D SD M requires usually a large num ber o f tw o-dim ensional basis functions w hich are not easy to chose and handle and w hich increase the com puta­ tion time significantly because o f potential convergence problem s [43].

T he M oL is a space-frequertcy dom ain method sim ilar to the FDM but uses an orthogonal transform . To treat 3-D discontinuities, the 2-D M oL is used, w hich dis­ cretizes the two spatial variables parallel to the substrate plane while an analytical solu­ tion is obtained in the direction perpendicular to the substrate plane. This method requires only a tw o-dim ensional discretization for a general 3-D problem. The advantage o f this m ethod is its easy form ulation, sim ple convergence behavior and the fact that there are no special basis functions necessary. The disadvantages o f the 2-D M oL is that satis­

7

fying all boundary conditions sim ultaneously for arbitrarily shaped circuits may be very difficult or may require a very fine 2-D discretization. In short, when applying the SDM or M oL to an arbitrary 3-D discontinuity problem, each method by itself encounters a num ber o f serious problems which are inherent in the method.

To overcom e the inherent lim itation of each method, a new hybrid numerical method has been developed by Wu and Vahldieck fl7]. This method is called the Space- Spectral Dom ain Approach (SSDA) which combines the 1-D SDM and 1-D MoL. This new m ethod elim inates the shortcom ings o f the conventional 2-D M oL and 2-D SDM and takes advantage o f the attractive features associated with the 1-D SDM and the 1-D MoL. T he SSDA is developed in particular for the analysis of arbitrarily shaped spatial 3- D planar discontinuities.

In this dissertation, we first introduce the generalized SSDA concept which is extended from the work of Wu and Vahldieck [17], Although this analysis can only be applied to calculate resonant frequencies but not discontinuity S-param ctcrs, it is used to explain the basic idea of the SSDA. On that basis, a new determ inistic quasi-static SSDA [18], [20] is presented followed by a full-w ave SSDA [19], w hich is aimed at the calcula­ tion o f S-param eters in structures supporting hybrid modes.

1.2 Organization of This Dissertation

C hapter 2 reviews the SDA and M oL and investigates their relationship, which forms the basis o f the SSDA. The generalized SSDA is introduced in a planar resonator problem which form s the basis o': this dissertation.

C hapter 3 introduces the quasi-static SSDA. C hapter 4 introduces the full-w ave SSDA.

C hapter 5 discusses simulation results and their experimental verification. C hapter 6 concludes the dissertation.

The Space-Spectral Domain Approach

In this chapter, the Spectral Domain M ethod (SDM ), the M ethod o f L ines (M oL) as well as the relationship betw een both m ethods are first reviewed. T h e co n cep t o f the S pace- Spectral D om ain A pproach (SSDA) is introduced as a com bination o f the SD M and the MoL. For a three-dim ensional (3-D) electrom agnetic problem , the SD M is applied to the x-direction, the M oL is applied to the z-direction, and the analytical process is applied to the y-direction (see Figure 2.1).

Figure 2.1 is a 3-D version of Figure 1.3; it shows a typical planar circuit discontinuity in a shielded box. In the Spectral D om ain M ethod the Fourier transform is taken along the x-direction for a 2-D problem and alw ays along the x- and z-direction for a 3-D problem . The analysis in the Fourier transform dom ain was first introduced by Yam ashita and M it- tra [2] fo r c o m p u ta tio n o f the c h a ra c te ris tic im pedance and th e p h a s e v e lo c ity o f a microstrip line b ased on a quasi-static approach. It is one of the m ost popular and w idely used num erical techniques for planar circuits. Numerous publications can be found in the literature, e.g., [ 3 - 5 ] .

For planar transm ission line and discontinuity problem s, the electric and m agnetic fields E and /? are often written in term s o f scalar potentials and ¥ 1 in a C artesian coordinate system, show n in Figure 2 .1 (this is called a TEZ / T M Z form ulation)

2.1 The Spectral Domain Method

VxVxl 4 'c

w here z is the u n it v e c to r in z-direction. h satisfy the wave e q u a tio n - j 2 e , h ^ 2 c, h 0 2 ( -,/! + L - ~ - + - • „ + k 'V ’ = 0 3.v 3 A 3 r , 2 2 k = CO (0.8 (2 .2 )

A

/

Figure 2.1 A shielded microstrip line

The idea o f the SD M is to apply the Fourier transform along the x-diicction in order to elim inate the space variable x and replace it with a spectral term a x

e, h c h ja x .

y = f M' c d x (2.3)

Assum ing the problem is a two-dim ensional one in x- and v-dircction, with the propagation constant p in z-direction, equation (2.2) yields

-,2 e, h

2 e , h a \u _ 2 e,/i , 2 e , h _ a y + — — P ij/ +

k

3y

(2.4)

dif-ferential equation

2 r 2 2) _

a + (3 - A' I q/ = 0 (2.5)

By applying the boundary conditions (m ore details will be given in Section 2.4 ), one finally obtains an algebraic equation in matrix form

e and e . are the Fourier transforms of the electrical field in the a- and r-direction.

/ and /\ are the Fourier transforms of the current in the a- and r-direction. The unknown / ( and / are expanded in term s o f known basis functions with unknown weighting coeffi­ cients. By applying G alerk in ’s technique, the propagation constant and field distribution can be found.

In summary, the SD M has several features:

• Simple form ulation in the form of algebraic equations • Utilization o f a-priori (physical) knowledge of modes • N um erically efficient

The SDM is well know n for its computational efficiency and m inim um memory requirem ent for tw o-dim ensional problems (1-D SDM) because usually only a few basis functions are needed. T he SDM loses som e o f its advantages when applied to spatial three-dimensional discontinuities (2-D SDM ). In particular when these discontinuities are arbitrarily shaped, it becomes generally a problem to find suitable basis functions and to achieve reasonable convergence.

d iffe re n tia l e q u a tio n s. It w as applied to m icrow ave analysis and desig n problem s by Pregla and co-authors [8 - 13].

The concept o f the M oL is as follows: for a given system of partial differential

(2.6)

2.2 The Method of Lines

I t equations, all but one o f the independent variables are discretized to obtain a system o f ordinary differential equations so that the w hole space is represented by a r um ber of lines. This semi-analytical procedure is very useful in the calculation o f planar transm is­ sion line structures.

To dem onstrate the basic steps of the M oL, consider the m icrostrip line cross-sec­ tion in Figure 2.2

x=L Figure 2.2 Cross-section view o f a m icrostrip line

Equation (2.2) is to be solved here. T he discretization is done in the .v-direction as shown above. The figure also shows that two separate line systems are used to represent

£1 /i

lP and T . This shifting schem e has several advantages: the lateral boundary condi­ tions are easily fulfilled, it allow s an optimal edge condition flOj, second order accuracy [11] and sim ple m atrix form ulation.

Let the number o f ¥* and xVn lines in Figure 2.2 be equal to Ari, T he potentials on all the lines are com bined to form a vector ^ and respectively. E quation (2.2) can then be rewritten as

(2.7)

-.2 r A t J l ..2 rA f j l .

+ L ^ _ + t V * = 0

d x dy dz

The first derivative with respect to x is form ed as backward difference quotients for

d*P'

In the difference operator [D] the lateral boundary conditions are included (here a D irichlet-N eum ann boundary condition is used as an example). The second derivatives can also be represented by m eans o f the operator [D]

h232^

OA D

. 202T>/'

a o T T ^ "

o x

E quation (2.7) can then be written as

x x D h_XX (2. 10) 3 2$ ’ ( - 2 _2 ;2W dy (2.11) w here

"

c , h is the eigenvalue m atrix and

7, c , h the eigenvector belonging to h which

can be obtained analytically dependent on the different lateral boundary conditions 1111. For exam ple, for the structure show n in Figure 2.2, the elem ents o f arc

~,e . in n

7 ‘." = s,n/v7 T N (2.14)

Equation (2.12) is called the orthogonal transform because r f-n is a symmetrical matrix and _{f '} is an orthogonal matrix. Equation ( 2 .11) is in the transform dom ain, which is sim ilar to equation (2.5) in the Fourier domain. However, the way to solve equa­ tion (2.5) and equation (2.11) are different in either techniques. By applying lateral boundary conditions, a system equation sim ilar to equation (2.6) can be obtained. A pply­ ing the orthogonal transform a new algebraic system equation can be derived in the (orig­ inal) spatial domain ([11] gives m ore details)

(2.15)

T h e vector notation is used here to represent discretized quantities. Because equa­ tion (2.15) is in the spatial dom ain, it can be simplified by rem oving those lines which do not pass through the m etallization at the interface

0 = Z.

I ,

(2 . 16)

T he propagation constant and field distribution can be calculated by solving the root o f the determ inant o f jzf] , w here subscript r signifies that [Zr \ is a residual matrix.

In summ ary, the 1-D M oL has the following features: • No basis function needed

• Sim ple formulation and efficient calculation • No relative convergence phenomenon

How ever, sim ilar to the SDM, when applied to three-dim ensional problem s, the 2- D M oL becom es numerically less efficient because o f the two-dimensional discretization.

2.3 The Relationship Eetween the SDM and MoL

From th e p rev io u s sections, it is qu ite obvious that both m ethods, the SD M and M oL, have s o m e sim ila ritie s if one c o m p a re s equations (2 .3 ), (2 .5 ) and e q u a tio n s (2 .1 1 ), (2.12). T he follow ing analysis show s that the M oL is indeed related to the SD M and that this relationship helps to com bine the advantages o f both m< ‘hods into one new m ethod, the Space-Spectral Domain Approach.

T his becom es obvious if one rewrites equation (2.2) for the tw o-dim ensional trans­ m ission line problem (the superscripts o f T are removed w ithout loss o f generality)

^ + = 0 (2.17)

d x 2 a y 2 y

w here (3 is the propagation constant.

In both the M oL and SD M , a transformation is perform ed in the .v-direction

M o L ¥ = > $ # = > $ $ = [7] $

(2.18)

SDM 4' => \]f fg = J

.0 0

w hich leads to a onc-dim ensional normal differential equation that corresponds to equation (2.5) and equation (2.11).

From equation (2.18) one may deduce that the orthogonal transform in the M oL represents a discrete Fourier transform in matrix form. A lthough in [11J som e analysis is provided to support this point, there is no clear explanation to prove that the M oL

schem e (discretization and orthogonal transform) and the SDM are truly identical. Fur­ thermore, also the connection betw een the SDM, the MoL and the SSDA has not been investigated in detail. The following analysis is intended to fill this gap.

To dem onstrate the relationship between the SDM and the MoL, the structure in Figure 2.2 is used again. (Dirichlet boundary condition is applied for the electric

potcn-Q

tial T at a - ( ) and at x=L ) The Fourier expansion in region [ 0 , /.] is written as

L V = jT ^ s in axeLx 0 a = in /- (2.19) / = : - D O or

If the potential is discretized into N points in the x-direction, i.e.

n

n N . + 1L , /'= / N:, and N : spectral terms are used,

N .

e . in n

sin-ii = i N + 1

/ = I N_ (2.20)

The subscripts are used to represent discrete quantities and discrete spectral terms. If vectors are used to represent discretization, equation (2.20) can be rewritten as

\\i =

where the elem ent of

J ' J

n e . in n

From here one may compare

M oL SDA .if \U = I rV!r " ^ I.1 J t ] ' / ’ ( 2 .2 1 ) (

2

22

This shows that the MoL is a discrete SDM. The equivalence o f the M oL and the SDM is established under the same finite discretization schem e. From another point o f view, because the SDM (theoretically) uses an infinite num ber o f spectral term s, the SDM gives an infinite number of precise eigenvalues and eigenfunctions w hich can be solved using analytical transforms w hile the M oL yields a finite num ber o f approxim ate eigenvalues and eigenfunctions which can be solved using finite discretization.

In summ ary the following properties are found com paring the SD M and M oL • The SD M and MoL are indeed related to each other.

• Both o f them are numerically very efficient for 2-D problem s and less effi­ cient for 3-D problems.

Com bining the MoL and SDM will take the advantage o f both m ethods to analyze 3-D problems m ore efficiently. This leads to the invention o f a new m ethod called the Space-Spectral Dom ain Approach (SSDA).

2.4 The Space-Spectral Domain Approach

This sectio n d e sc rib e s the basic p rin c ip le s o f the S p a c e -S p e c tra l D om ain A p p ro a c h (SSDA). The SSD A was first introduced by Wu and Vahldieck f 17] and further developed by the author together with Wu and V ahldieck [18 - 20]. First, a generalized introduction is given for a 3-D planar resonator circuit.

The two techniques combined in the SSD A are the S D M to sim ulate the cro ss sec­ tion of transm ission line structures and the M oL to m odel their longitudinal direction. The microstrip line step discontinuity show n in Figure 2.3 is taken as an exam ple to d e m ­ onstrate the basic steps involved, n this chapter only the hom ogenous boundary condition is considered (the discontinuity is enclosed in a shielding box).

First o f all a combination o f electric and m agnetic lines are introduced to discretize the structure in the z-direction. This corresponds to slicing the structure in the x-y plane. Then a set o f conventional basis functions for each slice is introduced w hich satisfy the boundary conditions along the x-coordinate. Every slice is o f regular rectangular shape, so that only well known conventional 1-D basis functions are needed. T he F ourier tran s­ form is performed to replace the x-coordinate in the H elm holtz equation with the spectral

17

term u . Since the M oL procedure is used in z-direction, the resulting wave equations arc coupled. T he orthogonal transform in the spatial domain is utilized to decouple the sys­ tem equations. T he three spatial variables in the Helmholtz equation are now reduced to the rem aining y variables and can therefore be solved analytically. The advantage of this procedure is that fine circuit details such as narrow strips and slots as well as com pli­ cated discontinuity shapes can be easily resolved by discretizing the structure in z-direc­ tion. Furtherm ore, problem s such as complicated basis functions, huge memory space and long C P U tim e know n from the 2-D SD M or MoL (i.e. 3-D problem s) are avoided. The final steps o f the SSD A are: the boundary conditions between layers at the top and bottom of the closed structure are transformed into the circuit plane. Satisfying the boundary con­ ditions at that location leads to a set o f equations which are the G reen’s functions by nature. A fter transform ing these final equations into the spatial dom ain, G alerkin’s tech­ nique is applied so that a characteristic matrix equation is obtained. By introducing hybrid boundary conditions, the S-param eters can be obtained. This will be discussed in Chapter 4.

2.4.1 SDM in x-direction

The c ro s s -se c tio n o f a m icrostrip reso n a to r is shown in Figure 2,3, A lthough a single layer structure is d raw n for the purpose o f simplicity, the following formulations are also valid fo r a m u ltila y er structure (yk and y m are used here for a generalized form ulation). T he e le c tro m a g n e tic field in the p 'h layer can be expressed in term s o f scalar potential functions accoraing to equation (2.1)

VxVx T i

t

I

Figure 2.3 A microstrip discontinuity in a resonator enclosure

V* and H!1' are the solutions o f the partial differential H elm holtz equation pair

L 2 L _ + L 3L _ + L 3 L _ + ^ . y ‘ = 0

O.v dv d : (2.26)

*o = t o V e 0 where

19 applied in .v-direction . . . c , / / c, h (2.27) CO h r n , f ’, h j a x , V = J 1 e d x (2,28) _ 00

where a is the space-spectral variable.

The electric and m agnetic field vectors in equation (2.25) will take the following form in the spectral dom ain

00 CO

r ^ j a x ,

-j-e = _oo

J E c ^ d x J = J E e ^ d x (2.2.9)

The space-spectral dom ain Helmholtz equation can now be written as:

3 2q/e,/l d2y L' h ( 2 / j ,2'') (Uh

~ 7 T - + - T r - 1 a - r oJ = 0 (2-30)

dv oz

2

.

4.2 MoL in z-direction

A fter the Fourier transform has been applied in .v-direction, the M ethod o f Lines (M oL) can be applied in z-direction. The structure is sliced in the x-y plane at each r-coordinatc. T h e electric lines and m agnetic lines are introduced to represent the discretized scalar potentials in the sp atial F ourier transform dom ain. A total nu m b er o f AL lines are used for different types o f transm ission line configuration.

In vector notation the discretized potentials v ^ ' are written as a N .-element vector as described in Section 2.2 .

\ \ h \pc' 11 (2.31)

N on-equidistant discretization [15] can be used here to increase the flexibility. T he non-equidistant discretization can also be considered as a linear transform from (original vector) to cp4’ 1 (non-equidistantdiscretized vector)

_s. <\ h h

X|/ —> cp _(2.32)

^ Q Jj

The new potential cp ’ is defined as

h <P v h (2.33) where re,h = d ia g

(J;M

he lli denotes the discretization interval of electric and magnetic lines, respectively. h() is the limiting case for the discretization interval (equidistant discretization).

The finite difference expression of the first derivative is written in matrix notation

where . 3(p‘ . a?" D. D. D. <P <P (2.35) (2.36)

[D] is defined in equation (2.9). The second order derivatives can be written as

D_ D <P <P (2.37) where D . D. D. D D. D. (2.38)

to transform this m atrix into a diagonal form: h, i r v T c o d: 7 = 6c e 2 ll0 T 7 =

52

Similar to Section 2.2 , a new potential in the (orthogonal) transform dom ain is defined as

s. <\ //

TA h <P.A /l (2.40)

The final 1-D H elm holtz equations in the transform ation dom ain are derived from equation (2.30) a v d v 2 d 2^ f .2 2 2) /, ~ ~ 2 \ Ith + CX - £r k o ) V = 0 dv (2.41)

The analytical solutions o f the above N , decoupled equations can be expressed as transm ission line equations from point y m to y k. T h e /^'com ponent is

V*I dV:

V,.

coshy .d. - sin h y d

el i y ei- 1,1 i

yel.sinhy cid. c o sh y eid.

V, dV, . d y .. (2.42) where a / = coshM ^ /usin h M / dy Y/(,-sinhy/((.d. c o sh y /(/d. V

dV

T he equation (2.42) and (2.43) can also be represented in matrix form as: r p e b p" BP By BPe = Y p. By BP" By By ^ h 1/ _ h V (2.45) w here [Qp] is a 4 N : by 4 N 2 matrix

\P

.

Ce

c,

M [o] H [o]

[o] M [o]

= dia g ( sinh ( y/( .</.) / y /|(.) (2.47)

= d ia g (y,„.sinh (y/l/r//) )

If one uses ex, ez and hx, hz to represent the components of e and Ti defined in equa­ tion (2.29), then from equation (2.1), the transverse electromagnetic field in the p ll‘ layer can be expressed in the spectral domain as

_ a

d\yc d\\rh

a By

23

A pplying the non-equidistant discretization, equation (2.33), and the orthogonal transform , equation (2.40), the fields in the transform dom ain can be written in matrix notation as w here

b =

_dP'

dv

dP_

jjl

_rh_ K

1 =

r

b

f .

k

7J[

r

h

N ote that the vector form ulation is introduced to represent the M oL discretization. U sing block matrix notation

ih z - L R, f

dPh

dy

dP"

dy

R. a b l,e p t o e 0 £ r

-0 [o]

[o]

j<oeQePr

[o] [o]

[o]

0

[o] [o]

6

[o]

[°] -[/]

(2.52)

By com bining equation (2.45) and equation (2.51), the field relationship betw een tw o lay­ ers is J t . - L LR P_ f i p j R. 'h jljz - h x (2.53)

Tli'* expression for m ultiple layers can be obtained by cascading the respective matrices.

W ith equation (2.53) one can always transform the electrom agnetic field from one layer another. By transform ing fields into the layers o f m etallization arid applying the boundary conditions at those interfaces, a matrix equation sim ilar to equation (2.6) and equation (2.15) can be obtained

\ e ~,v n l x = Z b ~z LX! \ Lk (2.54)

[zj ‘s a ^ ’z by 4M, m atrix. Transforming the electric fields and currents back into

the original dom ain by using the same orthogonal transform introduced in equation (2,40), the spectral dom ain algebraic matrix equation becom es

25 \ Jx X = Tzl_{I / d \} Jz w here [Z] is a 2NZ by 2/V, m atrix [z] =

> 1 [o ]

[o]

t

T C

In sum m ary, the following transform s have been utilized in the above analysis —> Fourier transform -> non-equidistant discretization —> orthogonal transform

i.e.

(2.57)

( Ex . y - > ex,y) (Zx,y->2jc.y) ( J X, V j'x, y ) ( a - , .V - * I

T he system equation (2.55) is obtained by following the reverse procedure.

In sum m ary, equation (2.1) to (2.57) represent the generalized procedure o f the Space (from the M oL)-Spectral Domain, (from the SDM) Approach (SSDA). A lthough in this chapter, the form ulation is lim ited to a resonator problem, the foundation o f the SSD A fo r scattering param eter calculation, which will be discussed in Chapter 3 and Chapter 4, is laid.

2.4.3 The Eigenvalue Solution o f a Resonator Problem

T he resonant frequency o f a planar resonator can be calculated by finding the roots o f the determ inant o f the system equation [17 ].

first step is to expand the elem ents o f unknown j x and /_ in term s o f known basis fu n c ­ tions with unknow n coefficients a . and a[.

N , / / N - 1 / / h i = £ a x i X\ x i h i = Z /= 1 / = 0 (2.58)

where / represents the /th basis function, N x is the total num ber o f basis functions, / rep re ­ sents the /th line and

-ah

(2.59)

where >t>;- is the strip width. J0 is the 0 tn o rder B essel’s function. Or, in vector notation

1 /

■n,I

C alculating the inner product betw een basis functions and each elem ent of the sy s­ tem equation (2.55) (further details can be found in Chapter 3) yields

f

the right side o f eq u ation (2.61) is alw ays zero. The left side can be written as

!/■ '(/)]* = 0 (2.62)

*s the result o f the inner product and cl represents the coefficient vector, f denotes the resonance frequency which can be obtained by solving the zeros o f the d e te r­ minant

27

To extend the SSD A to calculate the S-parametcrs o f planar discontinuities the determ inistic quasi-static SSDA and the full-w ave SSDA with hybrid boundary condi­ tions are introduced in Chapter 4 and 5, respectively.

Chapter 3

The Quasi-Static SSDA

3.1 Why Quasi-static ?

Q uasi-static num erical techniques are trad itio n ally faster than fu ll-w a v e tech n iq u es in particular on the serial m achines widely used today. T hese m ethods do n o t benefit from the availability o f parallel processors. For som e applications their accuracy can rival that o f full-w ave techniques and, therefore, they are very useful engineering analysis tools. In the recent literature, the quasi-static analysis has again received m ore atten tio n [21 - 24], because M M IC ’s and M K M IC ’s are usually sm all in dim ension com pared w ith the o p e r­ ating w avelength and, therefore, dispersion is norm ally weak. In [21] a q u asi-static sp ec­ tra l d o m ain approach (S D A ) was used to c a lc u la te m ic ro strip d is c o n tin u itie s . T h is m eth o d is num erically efficient but req uires c o m p lic a ted 2-D b a sis fu n c tio n s , w h ic h so m e tim e s m ay be d ifficult to find. A q u a s i-s ta tic fin ite d iffe re n c e m e th o d (F D M ) is described in [22] to analyze CPW discontinuities. T his m ethod can treat arbitrary d isco n ­ tinuities, but at the expense of large com puter memory. In [23] the q u asi-static m ethod o f lines (M oL ) is em ployed to analyze cross-coupled p lanar m u ltico n d u cto r system s. T his m ethod does not use basis functions and is faster than the FD M but still req uires signifi­ cant am ounts of m em ory and is difficult to apply to arbitrary d isc o n tin u itie s. T he d e te r­ m inistic SSDA eliminates these problems and will be introduced in the n ext section.

3.2 The Quasi-static SSDA

T his approach utilizes the basic idea behind the SSD A but avoids so lv in g an eigenvalue problem by using a new deterministic technique instead. To m inim ize e rro rs in the c alcu ­ lation o f the capacitance param eters, the excess ch a rg e density [24] h as been used and calculated in the space-spectral domain in one step via G alerkin’s m ethod. T h is approach

29

leads to an algebraic equation for the equivalent circuit parameters o f the discontinuities and is com putationally very stable, requires little memory space and is very fast on serial com puters. This m ethod is capable o f treating arbitrarily shaped planar circuit discontinu­ ities. F igure 3.1 illustrates the type of discontinuities this method has been applied to.

M icrostrip Open End

Microstrip Gap

Microstrip Step

CPW Step ■.vfSSKtSTIS*?.;?

CPW Open End

CPW Gap CPW Air Bridge

Figure 3.1 Planar circuit discontinuities

CPW Tai c

I T T

S Z

V

Microsirip Taper

A CPW discontinuity illustrated in Figure 3.2 is used as an example to demonstrate the theory. This discontinuity contains three regions (1, 2, 3) with thicknesses It/, It,, hi and is shielded by a metal housing. The three regions arc defined as:

1. hj+li2<y<h]+h2 +ht

2. hj<,y<ch i+hi 3. 0<y<hj

A s m entioned before, discretization o f the structure in z-direction corresponds to slicing the structure in the x-y plane at each z-coordinatc. Therefore, the potential for each slice must satisfy the 3-D Laplace’s equation

)2 I2 )2

J L y + J L v + J L v = o ( 3 I )

r).v 0 y 2 r):

In this case k=() (o>=0, compared with equation (2.2)). The task here is to simplify L a p lac e ’s equation which depends on the three spatial variables. The electric lines (solid lines) are introduced to represent a discretized electric potential ip, which is independent

of the magnetic potential. T he dashed lines are used to represent the m agnetic potentials as used in the conventional M oL (the magnetic potential is not o f interest here because it is independent o f the quasi-static electric field). The shift in both lines is necessary to reduce the discretization error and can be derived from [8]. Sim ilar to C hapter 2, the first step is to transform the electric potential function 'F into via a F ourier transform along the x-direction. Here the superscript is om itted because only the electric potential is o f interest. The spatial variable x becomes a spectral variable a . The next step is to discretize \(/ by using Nz lines in the z-direction which leads to the vector \p. The tapered region is enlarged in the left part of Figure 3.2, which dem onstrates how a smooth transition is theoretically discretized and approxim ated by a sequence o f abrupt steps.

Top View Cross Section

Enlarged discontinuity

m-line e-line

Figure 3.2 Discretization o f a CPW discontinuity

By utilizing the basic steps o f the SSDA in C hapter 2 with non-equidistant discreti­ zation in the z-direction, L aplace’s equation can be decoupled

^ - j - y 2 P = 0 ( 3 . 2 )

dy where

31

2 _ 2 2

y = o + a

_I']*

D ue to the discretization, L aplace’s equation (3.1.) is now reduced to only one spa­ tial variable, y. 5 is the eigenvalue o f D ,, . ['/’] is the eigenvector matrix. 6 and |y j are defined in Chapter 2 (note: the superscript e is om itted here because only electric potentials are used in the form ulation, y has only two terms instead of three as in C hap­ ter 2 because k=0). Solutions to the above 1-D simplified Laplace’s equation can be expressed in term s of the sum o f hyperbolic functions, and the relationship o f the electric potentials between any two adjacent layers can be expressed in the same way as described in equation (2.42) o f C hapter 2, that -s

X dV, — dy _V* c o sh y .d - sin h y .J 1 i yi ' i y .s in h y ^ , coshy(c/( I-', ()\\ c)v (3.4)

V, is the

ilh

P

equation (3.2) is decoupled, each line is represented by the sam e form o f normal differen­ tial equation. W ithout loss o f generality the subscript / can be removed in the following analysis. Instead, the subscript is now used to represent the potential in the different dielectric region.

F or L aplace’s equation, there is always y = y, = y2 = y.? because k„ 0 in equa­ tion (2.43).

T he boundary condition at the interface is as follows:

at

at

31/,

Sv2

w here

q

V { = 0 a t v = h { + h 2 + /i3

V3 = 0 at v = 0

Substitute equation (3.7) into equation (3.4) yields

dV~ _Y

d y tanh y h2

dV3 _ y

dy tanh y/i3 ' 3

Com bine equation (3.4) - (3.8) provides

8(y) Vi

tanhy/t2 tanhy/i, e 2y e3y tanhy/i2 tanhy/?3

(3.10)

To characterize a discontinuity, one needs to find the solution for the electric charge belonging to the discontinuity part. This is usually achieved by subtracting the total charge o f the discontinuity area and the charge belonging to the connected transm ission line. Since both quantities are often quite small, the errors arising from the subtraction of two electric charges, which are close in magnitude, c a n be significant. To avoid these errors, the excess charge technique [24] is used. This technique can briefly be sum m a­ rized as follow s; the 2-D transm ission line problem is solved first in the spectral domain on either side of the discontinuity, i.e. solving equation (3.2) (homogeneous transm ission line in z-direction) and analytically subtracting the charge distribution o f the fictitious hom ogeneous transmission line from the charge distribution o f the corresponding discret­ ization line o f the transmission line containing the discontinuity. Based on the above for­ mulation, the 2-D problem can be solved by using the solution given in equation (3.9) with 5 = 0 , i.e. y = a

K (o c )l', = ~ £.

33

(3.U)

w nere <■/„ is the charge density o f a fictitious hom ogeneous transm ission line. T he excess charge density p is defined as

( 3 . 1 2 ) Now the excess charge technique is applied, which m eans the two quantities g(y) and g(a ) are subtracted

= U ( Y ) - A'( « ) ) l ~ ( 3 . 1 3 )

and the results are transformed back into the original dom ain

M' = C P

a is the excess charge vector in the original domain and

H = H H

Mag [g

q

(3.14)

( 3 . 1 5 ) where d ia g [g (y )-g (a )l includes the difference [g(y)-g(a)J from all the lines. T h e size o f the m atrix is N z by N z.

G alerk in ’s technique is now applied to obtain the characteristic m atrix •equation. The first step is to expand the elem ents o f the unknown a in term s o f known basis func­ tions with unknow n coefficients c/

a = I I ( 3 . 1 6 )

w here N x is the number of basis functions, N, is the num ber o f discrete lines and

1 r J j a x j

T|/ and <5/ are Fourier transform pairs of basis functions. For CPW circuits, they take on the follow ing form for the center conductor with width ivj (also for m icrostrip c ir­ cuits with w idth i»’j)

cos ( 2 1 - 1) tc-I. I = 1,2, (3.18) nu'j •n, = - r a w . \ ( a w . / = 1 , 2 ,... (3.19) For (C PW ) g ro u n d (sym m etrical) con d u cto rs (iv/( is the ground conductor w idth, is the x-coordinate o f the ground conductor center (one side))

cos In-( x + b w!) w u cos In ( x - b .) v 117' H ’ i / II -Sill 2 ( x + b wt) w i 1 J x < 0 (x + b J /tc -IV 1/ 1 -sin n x + b j w u J .v> 0 (x ~ bJ In-" ’u 11 -2 (X + ! U ) \ w u J 11 -IV ,

x < 0

and w u of each line instead of changing the form of the basis function. Thus the disconti­ nuity shape can be a rb itra ry . This is an advantage of the SSD A and makes it possible to develop contour driven software.

Similar to C hapter 2, the inner product between basis functions and each elem ent of the system equation (3.14) is calculated

/ = 1 ,2 N . (3.22)

In quasi-static analysis, the excitation potential is always a constant across the m et­ allization. This property can be utilized to achieve a sim ple deterministic solution through the use o f Parseval’s theorem

J tj/ • 'q ^ a = 2k J lF • t; clx (3.23) where 1' = \ 1 ! I

V

k

'F is the discretized electric potential (inverse Fourier transform of ij/) and is c o n ­ stant across the m etallization. If this constant is defined as V' , the left side of equation (3.22), which is further processed in equation (3.23), can be written as

J m> • n d a = 2 u V (, | % d x = 2 n V tj \ _{I « = 0} (3.25)

Unlike the full-w ave resonator analysis described in C hapter 2, the left side o f s y s ­ tem Equation (3.22) is known as an • gebraic equation. The determ inistic solution can be obtained by m atrix inversion. Rewiiting equation (3.22) in m atrix form, which contains Nx independent equations, will yield

2 n V ,

A,

- J

a = 0

[c]® J ’1

d a (3.26)

By using equation (3.16) and replacing the continuous integral by a discrete su m ­ mation, equation (3.26) can be written as

(3.27) where Aoc Z| = B lo c k (3.28) a = 0 k m 1 1 2 * i 2 2 2 Nx 1 G k m % n T1/n ^ 2 G k m X S m (3.29) /v. A„ Nx N, ^ k n i ^ n t ^ m ^ k m ^ m ^ m ^ k n t ^ m

^

A a is the step w idth o f the discrete Fourier integration. [Z] is a ,V_/Vr by N:N y. block matrix, which contains N z by Nz subm atrices. Each submatrix f Z ] ^ , is a N x by N x matrix. G \m is the (k, m) elem ent of [G],

The charge density coefficient vector d is defined as

d = _{<i a L} 1 2 _{. .} Nx 1 2 *2 Cl7

A . _{1 2} A.

ClNz V - "A’ (3.30)

Front equation (3.27), it is evident that only a one step m atrix inversion is now required

37

(3.31) In contrast to finding zeros o f a determ inant through an iterative procedure (equa­ tion (2.63)), the quasi-static SSDA is a determ inistic approach and provides the results by a one-tim e m atrix inversion.

The total charge Q can be obtained by integrating the charge density over the dis­ continuity area

The equivalent circuit for different discontinuities is shown in Figure 3.3. The gap discontinuity is characterized by a n network. Open end, step in width, tapered disconti­ nuities and air bridge are approximated by a shunt capacitor. The capacitances Cpl, Cp2, C s or C p are then calculated from C=Q!Vc assum ing different excitation voltages Ve (even mode Fe l= l, Vt2~ 1, odd m ode Fel= l, Vc2= - 1 for a n network, VQ=\ for a shunt capacitor) at both ports o f the strip. Ve=0 is chosen for the ground conductor, The s-paramctcrs can be derived by using netw ork theory.

To calculate the shunt capacitance o f an air bridge, the above form ulation must be slightly modified. A C P W air bridge is shown in Figure 3.4.

The three region form ulation from equation (3.2) to equation (3.31) can still be uti­ lized if the air bridge is approximated as a patch sitting above the CPW as shown in Fig­ ure 3.4. This approxim ation is only valid when h2 is very small (h<w /5), which is true in m ost cases. Also the boundary condition of equation (3.6) is changed to

(3.32)

P

Step, Open End

Airbridge Gap

d V , d V 2 ;2 a 7 d v 2 2 7 7 = d V 3

83

77

W here q { is the charge density (in the transform ed dom ain) o f the bridge and q2 is the charge density o f the CPW area. Following a procedure sim ilar to that described by equations (3.5) - (3.15), a new system equation can be derived

Es ( r) ]

Y \

(3.34)

39 e 2y tanhy/i2 tanhy/i, - e 2y tanhy/z., - ? 2Y tanhy/i„ -s 2y s 3y tanhy/i2 tanliy/^ (3.35)

F inally the system equation in the transform domain has the same form as equation (3.14), but in a block m atrix form ulation. [G\ from equation (3.15) becomes

[c] =

E ach subm atrix is a 2x2 m atrix. The total rank o f fG | is 2NZ by 2N/:

3.3 On the Nature of the SSDA

It has b een show n in C hapter 2 that the M oL and SDM arc indeed equivalent if the same d isc re tiza tio n schem e is used, i.e., the num ber o f lines in the MoL equals the num ber of spectral term s in the SD M . T h is equivalency form s the basis o f the SSDA. A fter having in tro d u ced the q u asi-static S S D A , it is w orthwhile to look back and study the nature of th e S S D A . T h is se c tio n w ill show that by using a T E y/T M y form ulation, the SSDA includes the 2-D SD M and 2-D MoL.

In the analysis o f planar transm ission line problems the most common approach is to express the electrom agnetic field in terms o f Ez and Hz (i.c, TEZ and T M Z wave), w hich g ives a coupled T E Z and TM Z wave form ulation as described in Chapter 2. But w hen the field is expressed in term s o f Ey and H y, a transmission line type of formulation can be obtained by coordinate rotation (in the x-z plane) to a u-v coordinate system, w hich, fo r the SDM , led to the imm ittance approach [4| or a simplified formulation in the M oL [13], [25]. If a T E y and T M y wave formulation is used in the SSDA, som e inter­ esting results can be obtained.

V x£ = -yoopj)

V yj) = j m z t l (3.37)

and rearranging the above eq u ation yields a m ore su ita b le form for the p u rp o se o f this section

d2

?7

+ *

X

d2

d

dxdy J^d=

E

d2

dydz

X

u2_

A variable transformation in x- and z- direction (this can be either the F o u rie r trans­ form o f the SDM or the orthogonal transform of the M oL) is now introduced as follow s

a

dx

dx

d2

U sing the low er case to re p re s e n t the field com ponents after the tra n sfo rm , e q u a tio n (3.38) is written as a decoupled TE/TM to y formulation

0 -a > |i ~j T a y

°

.d

a x and a , can be spectral term s o r eigenvalues of the transform matrix, d e p e n d in g on

which m ethod is used in the x- and z-direction, respectively (note: when an eigenvalue is used, it should be m ultiplied by j ) . T he U matrix corresponds to the coordinate rotation from (x, z, y) to (u, v, y) as show n in Figure 3.5.

B ased on the above form ulation, the SSDA is really a 2-D SDM when a and a , are spectral term s. Similarly, the SSD A becomes a 2-D M oL when a v and a are eigen­ values o f the transform m atrices (discretization). In the SSD A the SDM and Mol can be applied separately to the x- and z-direction, respectively, or one can use any one o f the two m ethods. It is worthwhile to point out that the reason behind the form ulation is that the TEy and T M y modes are independent (not coupled anymore!). On the other hand, this equivalency is only valid from the form ulation point of view. The SSDA has its unique style in solving discontinuity problem because it uses neither 2-D discretization as in the M oL nor 2-D basis function as in the SDM.

Figure 3.5 C onfiguration o f general transm ission line

• T he SSD A combines the M oL and SDM which can be derived from each other

• T h e SSD A can be form ulated from the TEy/TM y wave expansion using a coordinate rotation.

• the 2-D SDM or 2-D M oL are the special case o f a general hybrid m ethod: the SSD A .

4 5

Chapter 4

The Full-wave SSDA

This chapter fo cu ses on the full-w ave SSD A. Two alternative approaches arc presented: an eigenvalue and a deterministic approach.

The foundation of the full-wave SSDA is laid in chapter 2, where only a h o m o g e­ neous boundary condition is used. To calculate the S-param eters o f planar circuits, inho- mogeneous boundary conditions m ust be included. This chapter describes two d ifferent approaches to implement the inhom ogeneous boundary conditions and to extract S- parameters.

4.1 Eigenvalue Approach

The e ig e n v a lu e approach em ploys the con cep t o f se lf-c o n siste n t inhom ogcncou.s (o r hybrid) boundary conditions at the end o f feed lines which are connected to cither side o f the discontinuity. This approach m akes it possible to sim ulate the w hole structure via an eigenvalue equation in which the solution is the reflection coefficient o f the discontinuity. The hybrid boundary conditions have been used before in [ 15] and 116], but in the first case to m odel the forward and reflected w aves individually and in the second case to find the total field at the launching point by using a modal source approach. In the m ethod p re ­ sented here, the reflection coefficient (or S (j) is obtained directly,

If a 2-port discontinuity (Figure 4.1) is under investigation, it is assum ed that at some distance from port I of the discontinuity, there will be a standing wave o f the fu n d a ­ mental m ode only consisting of incident and reflected waves: