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

код для вставкиСкачатьA 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 in the Department of 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.) © Ming Yu, 1995 UNIVERSITY OF VICTORIA A ll rights reserved. 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B o h av io ra l............................. Clinical . . . Developmental Experimental Industrial Personality Physiological . P sychobiology .......................... Psychometrics Social 0621 0384 0622 0620 ,062.) 0624 0625 0989 0349 0632 0451 Supervisor: Dr. R. Vahldieck 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. Ac curate, fast and reliable analysis tools are crucial to the design of MMIC’s and MHMIC’s. The space-spectral domain approach (SSDA) is such a numerically efficient method, which combines the advantage of the one-dimensional method of lines (MoL) with that of the one dimensional spectral-domain method (SDM). In this dissertation, the basic idea of 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. Sparameters 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 --------------------------------------------------- -— — —\ fp r . R. Vahldiqck, Supervisor Dr. J. Sfo^femann, departmental Member Dr. W. Hoefer, Departmental Member Dr. C. Bra$Ht?y, Outside Member Dr. V. K. Tripathi,j2$eiHftH3Xarniner (Oregon State Univ.) Table of Contents Table of Contents List of Figures List of Tables Acknowledgments 1 Introduction 2 iii v vii viii I 1.1 Background and G o a ls ....................................................................................... I 1.2 Organization of This Dissertation..................................................................... 7 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 rob lem ............................ 25 3 The Quasi-Static SSDA 28 3.1 Why Quasi-static ? ..................... 3.2 The Quasi-static SSD A .................................................................................... 28 3.3 On the Nature of the S S D A ............................................................................ 39 4The Full-wave SSDA 28 43 4.1 Eigenvalue A p p ro a ch .................... 43 4.2 Deterministic A p p ro a ch ................................................................................ 48 IV 5 Numerical and Experimental Results 54 5.1 Convergence Analysis o f Quasi-static S S D A ............................................... 54 5.2 Sim ulation Results o f Quasi-static SSDA . . 5.3 Convergence Study o f Full-wave S S D A ........................................................ 64 5.4 Simulation Results o f Full-wave S S D A ........................................................ 64 5.5 Experim ental R e s u l t s ........................................................................................67 6 Conclusion 55 7J 6.1 C o n tr ib u tio n s ..................................................................................................... 71 6.2 Future W o r k ..........................................................................................................72 Bibliography 75 Appendix 79 v List of Figures Figure 1.1 Exam ple for an M M IC Circuit Figure 1.2 M HM 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 microstrip line Figure 2.3 A microstrip discontinuity in a resonator enclosure Figure 3.1 Planar circuit discontinuities Figure 3.2 Discretization of a C PW 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 microstrip 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-parameters o f a m icrostrip step. u q -1m 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-parameter of a CPW airbridge. w ~ l5\ini,s= IO \im , l=.1\Lrn, h=200[Lm, b=1\un Figure 5.7 S-parameters o f a CPW step. w^-0,4m m , w j= 0.lm m , Vj -•■0.1mm, \\'2=0.4mm, er=9.<V, h=0.254mm Figure 5.8 Equivalent capacitance o f a CPW gap, d {—2,Vj + it'| , vi d2=2s2+u'2,£r=9.<S’, h=0.635mm, 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 PW step/taper. For CPW, W[=0.8mm, s [=0.1 mm, w2-0.2mm , s2==0.6mm, er=9.6, li=0.254mm. For microstrip, 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=0.1m m , b=3]im , er=9.6, h=().254mm. Figure 5.11 Frequency dependent behavior of microstrip 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-parameters o f a microstrip step Figure 5.15 Full-wave S-parameters 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 computed S-param eters o f a CPW gap Figure 5.18 Measured and computed S-param eters o f end-coupled CPW reso nators Figure 5.19 Measured and computed S-param eters o f a CPW step discontinui 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 Figure 6.3 Arbitrary multi-port discontinuity vii List of Tables Table 4.1. B oundary conditions46 viii Acknowledgments The author wishes to express his acknowledgm ents to his thesis supervisor, Dr. Ruediger 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 NSERC), Science Council of British Columbia (through GREAT Award) and M PR Teltech Ltd. is also gratefully acknowledged. In particular, I would like to thank Dr. J. Fikart and H. Minkus, MPR Teltech, for the fabrication o f the MHM IC prototypes, w hich were used to verify the SSDA results. The author also wishes to extend his thanks to Dr. K. Wu for his invaluable sugges tions. The author is also grateful to his colleagues at the Laboratory for Lightw ave Electronics, M icrowave and Com m unications (LLiMiC), University o f Victoria for their support and discussion. Last, but by no means least, the author wishes to express his thanks to his family, espe 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 microwave circuits is essential ir Mic evolution of modern com m unications system s. In analogy to the miniaturization that has taken place in VLSI (Very Large Scale Integrated Circuits), M onolithic Microwave integrated Circuits (M M IC ’s) but also M iniature Hybrid M icrowave Integrated Circuits (M H M IC’s) combine a steadily grow ing num ber o f microwave components on smaller and sm aller chip real estate. M M IC ’s as shown in Figure 1.1 [14] are very expensive in the fabrication and are only justified for large volum e applications. M H M IC ’s are a hybrid technology that is in par ticular suitable for small 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 alumina 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 volume 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 important role in modern com m unication systems. A serious bottleneck in both technologies is the lack of accurate, fast and reliable design strategies. Although commercial design software is available, the num erical m eth ods used are either computationally 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-consuming (scvcra1 months). 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. 2 In generai, numerical methods can be classified into two categories: 1. M ethods which use an eigenmode 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, Methods which discretize a differential operator • Finite Difference M eihod (FDM) (6 j • M ethod o f Lines (MOL) [7]-[131 The first category of methods use orthogonal modes or basis functions to expand the field directly. T he second category of methods 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. Both category of methods can be subdivided further into Quasi-static techniques which calculate equivalent network parameters. Microwave circuits are described by lumped 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 point o f view and may be described by S-parameters, which include 3 the uite faction o f fundam ental and hi»her order modes at discontinuities. Figure 1.1 Example for an MM1C Circuit 4 COPLANAR WAVCGUIDE AIR BRIOCE COPLANAR WAVEGUIDE AIR BRIDGE SLOTLINE (a ) C o p la n a r W a v e g u id e T J u n c t i o n a ir ( d ) S l o t L i n t C o p la n a r J u n c t i o n br id g e SLOTUNE (b ) C o p la 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 (e ) S lo tlin e T J u n c tio n COPLANAR WAVEGUIDE COPLANAR WAVEGUIDE AIR BRIDGE AIR BRIDGE COUPLED SLOTLINE • SLOTLINE (c ) C o p la n a r W a v e g u id e / S l o tli n c T r a n s i t i o n MHUIC ( f ) MliMIC E m b e d d e d I n to DIODE' a CPWG S t r u c t u r e AIR BRIDGE 3 AIR BRIDGE I BIAS AIR BRIDGE 2 MIM CAPACITOR COPtANAR WAVEGUIDE -SLOTLINE ' ' — OUTPUT (g ) C ir c u it 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 Figure 1.2 MHMIC discontinuities 5 Quasi-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 wavelength and the dispersion of the transmission line system is weak or non-existing, quasi-static m eth ods cun work up to the millimeter-wave range. A large number o f comm ercially avail able software 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. Furthermore, 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 microwave elem ents are placed on the chip. In this case the predicted performance of the M M IC ’s or M H M IC ’s may deviate significantly from the required performance. For structures, or frequencies, at which quasi-static methods 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 millimeter-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 deterministic tech nique w ill be presented. Finally, two SSDA full-wave algorithms will be discussed. Typical generalized full-v/a -e approaches are, e.g., the Finite Difference M ethod (FDM) or Transmission Line Matrix (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 computer memory and long com puter run-time, at least on most of today’s available engineering workstations. 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 of the special program languages necessary to fully take advantage of the potential of these m achines. 6 side view top view ▼x Figure 1.3 A typical planar discontinuity W hen dealing with the analysis and design of M M IC ’s and M H M iC ’s, their quasipJanar 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 most suitable methods for 3-D planar circuit analysis have been the Spectral Domain M ethod (SDM), the M ode M atching M ethod (MM M ), and the M ethod of Lines (MoL) (full-wave 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 determine the eigenvalue problem (propagation constant) in a planar circuit, the 1-D SDM is well know n for its fast com putational algorithm and minimum m emory require ments. But 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 SDM requires usually a large number of two-dim ensional basis functions w hich are not easy to chose and handle and which increase the com puta tion time significantly because o f potential convergence problem s [43]. The MoL 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, which 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 two-dim ensional discretization for a general 3-D problem. The advantage o f this method 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 simultaneously 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 number 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 SpaceSpectral Dom ain Approach (SSDA) which combines the 1-D SDM and 1-D MoL. This new m ethod elim inates the shortcomings of the conventional 2-D MoL 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 3D 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-paramctcrs, it is used to explain the basic idea of the SSDA. On that basis, a new deterministic quasi-static SSDA [18], [20] is presented followed by a full-w ave SSDA [19], which 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. Chapter 3 introduces the quasi-static SSDA. C hapter 4 introduces the full-wave SSDA. C hapter 5 discusses simulation results and their experimental verification. Chapter 6 concludes the dissertation. Chapter 2 The Space-Spectral Domain Approach In this chapter, the Spectral Domain M ethod (SDM), the M ethod o f Lines (M oL) as well as the relationship betw een both m ethods are first reviewed. T he concept o f the SpaceSpectral Dom ain A pproach (SSDA) is introduced as a com bination o f the SDM 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). 2.1 The Spectral Domain Method 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 Dom 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 Yamashita and M ittra [2] fo r c o m p u ta tio n o f the c h a ra c te ristic im pedance and th e p h a se v e lo c ity o f a microstrip line based on a quasi-static approach. It is one of the m ost popular and w idely used numerical techniques for planar circuits. Numerous publications can be found in the literature, e.g., [ 3 - 5 ] . For planar transmission line and discontinuity problems, the electric and m agnetic fields E and /? are often written in term s o f scalar potentials and ¥ 1 in a Cartesian coordinate system, shown in Figure 2 .1 (this is called a TEZ/ T M Z form ulation) VxVxl 4 'c (2 . 1) w here z is the unit v ecto r in z-direction. e,h - j2 ^2 h satisfy the wave equation c, h 02 (-,/! + L- ~ - + - • „ 3A 3r 3.v ,2 k = + k 'V ’ = 0 (2 .2 ) 2 CO (0.8 A y / Figure 2.1 A shielded microstrip line The idea o f the SDM is to apply the Fourier transform along the x-diicction in order to eliminate the space variable x and replace it with a spectral term a x y e, h = c h jax . f M' c d x (2.3) Assuming the problem is a two-dimensional one in x- and v-dircction, with the propagation constant p in z-direction, equation (2.2) yields 2 a y e ,h -,2 e, h a \u _2 + — — P ij/ 3y e,/i ,2 + k i|/ e ,h _ =0 (2.4) The above equation (2.4) can be further simplified as a one-dim ensional normal dif- 10 ferential equation 2 r2 2) a + (3 - A' I q/ _ =0 (2.5) By applying the boundary conditions (more details will be given in Section 2.4 ), one finally obtains an algebraic equation in matrix form ( 2 .6 ) 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 terms o f known basis functions with unknown weighting coeffi cients. By applying G alerkin’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 • Num erically efficient The SDM is well known for its computational efficiency and m inimum memory requirement for two-dim ensional problems (1-D SDM) because usually only a few basis functions are needed. The SDM loses som e of 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. 2.2 The Method of Lines T he Method o f Lines (M oL) was first developed by mathematicians (7 J in order to solve differential eq u ations. It w as applied to m icrow ave analysis and design 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 It equations, all but one of the independent variables are discretized to obtain a system o f ordinary differential equations so that the whole space is represented by a r umber of lines. This semi-analytical procedure is very useful in the calculation o f planar transmis sion line structures. To demonstrate 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 microstrip 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 lP /i and T . This shifting scheme has several advantages: the lateral boundary condi tions are easily fulfilled, it allows an optimal edge condition flOj, second order accuracy [11] and simple matrix formulation. 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. Equation (2.2) can then be rewritten as -.2 r A t J l dx ..2 rA f j l dy +L ^ _ +tV * dz . = 0 (2.7) The first derivative with respect to x is form ed as backward difference quotients for 'V1 and forward difference quotients for XV" 12 d*P' r — ' • o a r ^ N 11' (2.8) >/J with 1 ... 0 0 -1 ... 0 ... £> = (2.9) 1 0 ... 0 -1 In the difference operator [D] the lateral boundary conditions are included (here a Dirichlet-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] h 232^ Dxx OA (2. 10) 202T>/' hX ao T T ^ " DX ox . Equation (2.7) can then be written as 32$ ’ (-2 _2 "= h ;2W ( 2 . 11) dy w here (2. 12) c ,h is the eigenvalue matrix 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 shown in Figure 2.2, the elem ents o f ~,e . in n arc N 7 ‘." = s,n/v7 T (2.14) Equation (2.12) is called the orthogonal transform because rf-n is a symmetrical matrix and is an orthogonal matrix. Equation ( 2 .11) is in the transform domain, f' 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 similar to equation (2.6) can be obtained. Apply 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 he vector notation is used here to represent discretized quantities. Because equa tion (2.15) is in the spatial domain, it can be simplified by rem oving those lines which do not pass through the m etallization at the interface 0 = Z. (2 . 16) I, T he propagation constant and field distribution can be calculated by solving the root o f the determinant o f jzf] , where subscript r signifies that [Zr \ is a residual matrix. For three-dimensional problem s also the z-variblc is discretized (2-D fvloL). 14 In summary, the 1-D MoL has the following features: • No basis function needed • Sim ple formulation and efficient calculation • No relative convergence phenomenon However, similar to the SDM, when applied to three-dimensional problem s, the 2D M oL becomes numerically less efficient because o f the two-dimensional discretization. 2.3 The Relationship Eetween the SDM and MoL From the previous sections, it is quite obvious that both m ethods, the SD M and M oL, have so m e sim ila ritie s if one co m p ares equations (2 .3 ), (2.5) and e q u a tio n s (2.11), (2.12). The follow ing analysis shows that the MoL 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 becomes obvious if one rewrites equation (2.2) for the two-dim ensional trans m ission line problem (the superscripts o f T are removed w ithout loss o f generality) ^ + dx2 ay2 = 0 (2.17) y w here (3 is the propagation constant. In both the M oL and SDM , a transformation is perform ed in the .v-direction MoL ¥=>$ #=>$ $ = [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 some 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 between 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 potcntial T Q at a - ( ) and at x=L ) The Fourier expansion in region L a V = jT ^ s in axeLx n the potential n L , /'= / N. + 1 is in /- / 0 If = /.] is written as [0 , discretized into =: -D O N (2.19) or points in the x-direction, i.e. N:, and N: spectral terms are used, N. ii = i e . in n sinN +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 = ( 2 .2 1 ) J'J ne . in n ( 2 .22) From here one may compare M oL SDA From equation (2.14) .if r " \U = I rV! t ] ' / ’ ^ I.1 J (2.23) 16 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 which 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 summary the following properties are found com paring the SDM 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. Combining the MoL and SDM will take the advantage o f both m ethods to analyze 3-D problems more efficiently. This leads to the invention o f a new method called the Space-Spectral Domain Approach (SSDA). 2.4 The Space-Spectral Domain Approach This section describ es 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 Vahldieck [18 - 20]. First, a generalized introduction is given for a 3-D planar resonator circuit. The two techniques combined in the SSDA are the SD M to sim ulate the cross sec tion of transmission 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 dem 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 magnetic 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 which 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. The Fourier trans form is performed to replace the x-coordinate in the Helm holtz equation with the spectral 17 term u . Since the M oL procedure is used in z-direction, the resulting wave equations arc coupled. The 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 PU tim e know n from the 2-D SDM or MoL (i.e. 3-D problems) 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 domain, Galerkin’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 ss-se c tio n o f a m icrostrip resonator is shown in Figure 2,3, Although a single layer structure is draw n for the purpose o f simplicity, the following formulations are also valid fo r a m ultilayer structure (yk and ym are used here for a generalized formulation). The ele c tro m ag n e tic field in the p 'h layer can be expressed in terms o f scalar potential functions accoraing to equation (2.1) (2.25) VxVx T i 18 t y=y« 1 y=yn 2 e r r x=a M/ ' >[/' ‘hi I its Figure 2.3 A microstrip discontinuity in a resonator enclosure V* and H!1' are the solutions of the partial differential H elm holtz equation pair L 2 L _ + L 3L _ + L 3 L _ + ^ . y O.v dv d: ‘ = 0 (2.26) *o = to V e 0 where ppr is the relative dielectric constant in the plh layer. The Fourier transform is 19 applied in .v-direction c, ...c ,// h (2.27) CO V h r n , f ’, h j a x e = J 1 , dx (2,28) _ 00 where a is the space-spectral variable. The electric and magnetic field vectors in equation (2.25) will take the following form in the spectral dom ain 00 r ^ CO ja x , e = J E c^dx -j- J= J E e^dx (2.2.9) _oo The space-spectral dom ain Helmholtz equation can now be written as: 32q/e,/l d2y L' h ( 2 /j,2'') (Uh ~ 7 T - + - T r - 1 a - r oJ = 0 dv oz (2-30) 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 he electric lines and m agnetic lines are introduced to represent the discretized scalar potentials in the spatial F ourier transform dom ain. A total num ber o f AL lines are used for different types o f transmission 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) Non-equidistant discretization [15] can be used here to increase the flexibility. The non-equidistant discretization can also be considered as a linear transform from (original vector) to cp4’ 1 (non-equidistantdiscretized vector) 20 _s. <\ h X|/ —> cp h (2.32) ^ Q Jj The new potential cp ’ is defined as h h <P (2.33) v where re,h = d iag (J;M (2.34) I A/ >le<hi) 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 . 3(p‘ D. <P (2.35) . a?" D. <P where D. (2.36) [D] is defined in equation (2.9). The second order derivatives can be written as D_ <P (2.37) D <P where D. Because [/T D. D. D D. D. (2.38) is a symmetrical matrix, the following transforms can be applied 21 to transform this matrix into a diagonal form: i r v Tc h,o d : = 7 6ce 2 T = 7 52 (2.39) ll0 Similar to Section 2.2 , a new potential in the (orthogonal) transform domain is defined as s.<\ // T Ah .A /l <P (2.40) The final 1-D Helm holtz equations in the transform ation dom ain are derived from equation (2.30) a v dv2 (2.41) d 2^ f .2 2 2) /, ~ ~ 2 \ Ith + CX - £r k o ) V = 0 dv The analytical solutions o f the above N , decoupled equations can be expressed as transmission line equations from point y m to y k. T he /^'com ponent is V*I V, coshy .d. - sin h y d el i y ei1,1 i dV: yel.sinhy cid. V,. (2.42) dV, coshy eid. . dy. . V coshM = a // dy ^ /usinhM Y/(,-sinhy/((.d. co sh y /(/d. dV (2.43) d y. where 2 £ 2 2 p .2 y ei, = 6 „ + a - B r k Q 2 2 2 „ 2 y>ih = 6/,/,+ a - e/ o dl = (2.44) 22 T he equation (2.42) and (2.43) can also be represented in matrix form as: pe r bp" By = BP By BPe By ^ h 1/ _ (2.45) Y p. BP" By V h w here [Qp] is a 4N : by 4N 2 matrix Ce L°J ^ c, \P p. M 3h2 [o] H [o] M = d ia g (c o sh y .d.) S el = d ia g ( sinh (y(,.t/(.) / y j 3cl = d ia g (y ^ sin h (y e|.d .)) (2.46) [o] [o] C, = diag (coshyh.d.) ’/ii = diag ( sinh ( y/( .</.) / y /|(.) (2.47) h2 = diag (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 (oeQeP r" d : e, = ^ dy ' Sa 2\v < i>.2 e — T + e, k oV j(£>£0£P r _. _ lx ~ a By By + m p 0 dz ( -a h jth cop V s V dz N / (2.48) 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 b = ab toe0e r P 1,2 $ = _dP' dv y- ££pe P j a>£0£r . dP_ - I = ' 3v a 5, y copn J ',. 2 * r 0 ~ °/j /i f / 1 J l = 00(X (2.49) w here t = jjl -1 _rh_ K 1 = I b~z = - 11 k f. = 7J[ r, r -I (2.50) 'h N ote that the vector form ulation is introduced to represent the M oL discretization. U sing block matrix notation f R, ih z -L dPh dy dP" dy a /' w here [R/;] is a 4NZ by 4NZ matrix (2.51) 24 a b l,e -0 [o] [o] [o] [o] [o] 0 [o] [o] cop. [o] [°] -[/] p to e 0 £ r P .2 j, r 0 " ee j<oeQeP r R. - (2.52) 6 a 5 ,/l cop0 By com bining equation (2.45) and equation (2.51), the field relationship betw een tw o lay ers is J t. LR P_ fip j R. (2.53) jljz - L 'h -hx Tli'* expression for m ultiple layers can be obtained by cascading the respective matrices. With equation (2.53) one can always transform the electrom agnetic field from one layer another. By transform ing fields into the layers o f metallization 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 lx = LZ X! \ b ~z Lk [zj ‘s a ^ ’z (2.54) by 4M, m atrix. Transforming the electric fields and currents back into the original domain by using the same orthogonal transform introduced in equation (2,40), the spectral domain algebraic matrix equation becomes 25 \ X = ITzl /d Jx \ Jz (2.55) w here [Z] is a 2NZ by 2/V, matrix -l > 1 [o ] [z] = [o] 1r T 1-1 (2.56) tTC In summ ary, the following transforms have been utilized in the above analysis —> Fourier transform -> non-equidistant discretization —> orthogonal transform i.e. (2.57) ( Ex . y - > ex,y) ( J X, V j'x, y) (Zx,y->2jc.y) ( a - , .V - * I The system equation (2.55) is obtained by following the reverse procedure. In summ ary, equation (2.1) to (2.57) represent the generalized procedure of the Space (from the M oL)-Spectral Domain, (from the SDM) Approach (SSDA). Although in this chapter, the formulation is lim ited to a resonator problem, the foundation o f the SSDA for scattering param eter calculation, which will be discussed in Chapter 3 and Chapter 4, is laid. 2.4.3 The Eigenvalue Solution of a Resonator Problem T he resonant frequency of a planar resonator can be calculated by finding the roots o f the determ inant o f the system equation [17 ]. G alerk in ’s technique is applied to obtain the chaiactcristic matrix equations. The 26 first step is to expand the elements o f unknown j x and /_ in term s o f known basis func tions with unknow n coefficients a . and a[. N, N - 1 / h i = £ / / a x i X\ x i = h i /= 1 / Z (2.58) /=0 where / represents the /th basis function, N x is the total num ber o f basis functions, / repre sents the /th line and (2.59) -ah order B essel’s function. Or, in vector notation where >t>;- is the strip width. J0 is the 0tn 1 / (2.60) n "ipr ■n,I / 'H.wv, / V Calculating 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 K ' < da = f da (2.61) the right side o f equation (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 deter minant d e ( (/•'(/> ]] = 0 (2.63) 27 To extend the SSDA to calculate the S-parametcrs o f planar discontinuities the deterministic quasi-static SSDA and the full-wave SSDA with hybrid boundary condi tions are introduced in Chapter 4 and 5 , respectively. 28 Chapter 3 The Quasi-Static SSDA 3.1 Why Quasi-static ? Q uasi-static num erical techniques are traditionally faster than fu ll-w a v e techniques in particular on the serial m achines widely used today. These 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 attention [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 er ating wavelength and, therefore, dispersion is norm ally weak. In [21] a quasi-static spec tral do m ain approach (SD 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 ethod is num erically efficient but requires co m p licated 2-D basis fu n c tio n s, w h ich som etim es may be difficult to find. A q u a si-sta tic finite d ifferen ce m eth o d (F D M ) is described in [22] to analyze CPW discontinuities. This method can treat arbitrary discon tinuities, but at the expense of large com puter memory. In [23] the quasi-static m ethod o f lines (M oL) is employed to analyze cross-coupled planar m ulticonductor system s. T his m ethod does not use basis functions and is faster than the FD M but still requires signifi cant am ounts of mem ory and is difficult to apply to arbitrary d iscontinuities. T he d e te r m inistic SSDA eliminates these problems and will be introduced in the next section. 3.2 The Quasi-static SSDA T his approach utilizes the basic idea behind the SSD A but avoids solv in g an eigenvalue problem by using a new deterministic technique instead. To m inim ize errors in the calcu lation o f the capacitance param eters, the excess charge density [24] has been used and calculated in the space-spectral domain in one step via G alerkin’s m ethod. T his 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. Figure 3.1 illustrates the type of discontinuities this method has been applied to. M icrostrip Open End CPW Step Microstrip Gap CPW Open End CPW Tai c ■.vfSSKtSTIS*?.;? IT T SZ V 3 Microstrip Step CPW Gap CPW Air Bridge Microsirip Taper Figure 3.1 Planar circuit discontinuities 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 mentioned before, discretization of 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 JL y + JLv + JLv =o r).v 0y2 (3I) r): In this case k=() (o>=0, compared with equation (2.2)). The task here is to simplify L aplace’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 30 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 ]. Similar to C hapter 2, the first step is to transform the electric potential function 'F into via a Fourier transform along the x-direction. Here the superscript is om itted because only the electric potential is of 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 approximated by a sequence of 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, Laplace’s equation can be decoupled ^ - j - y 2P = 0 dy where (3.2) 31 y 2 _2 = o +a 2 (3.3) I']* D ue to the discretization, Laplace’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 omitted here because only electric potentials are used in the form ulation, y has only two terms instead of three as in Chap ter 2 because k=0). Solutions to the above 1-D simplified Laplace’s equation can be expressed in terms of the sum o f hyperbolic functions, and the relationship of the electric potentials between any two adjacent layers can be expressed in the same way as described in equation (2.42) o f Chapter 2, that -s X — dV, dy V, is the co sh y .d - sinhy.J 1 i yi ' i y .sin h y ^ , coshy(c/( V* ilhelem ent o f P and I-', ()\\ (3.4) c)v corresponds to the ilh line o f discretization. Because equation (3.2) is decoupled, each line is represented by the same form of 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. For Laplace’s equation, there is always y = y, = y 2 = y.? because k„ 0 in equa tion (2.43). T he boundary condition at the interface is as follows: at y = l>2 + ^h at 31/, £ ,-^ v = h .j (3.5) Sv2 E 0 -=— -'2 dv - — L at y = h 2 -f //3 (3.6) dV2 "2 3v w here tion q is 5 dy at the charge density in the transform dom ain. At the top and bottom m etalliza 32 at V{ = 0 v = h { + h2 + /i3 (3.7) V3 = 0 v = 0 at Substitute equation (3.7) into equation (3.4) yields dV~ Y tanh y h2 dy dV3 _ dy v = h2 + /i3 (3.8) y tanh y/i3 ' 3 at v = h-. Com bine equation (3.4) - (3.8) provides (3.9) 8(y) Vi where £2y c 2y S(Y) = tanhy/t 2 + e iY tanhy/i, tanhy/i 2 e 2y e3y tanhy/i 2 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 transmission line. Since both quantities are often quite small, the errors arising from the subtraction of two electric charges, which are close in magnitude, can 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 transmission line problem is solved first in the spectral domain on either side of the discontinuity, i.e. solving equation (3.2) (homogeneous transmission line in z-direction) and analytically subtracting the charge distribution o f the fictitious hom ogeneous transmission line from the charge distribution of 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 33 K (o c )l', = ~ £. (3.U) wnere <■/„ is the charge density o f a fictitious hom ogeneous transmission line. The excess charge density p is defined as (3.12) Now the excess charge technique is applied, which m eans the two quantities g(y) and g(a) are subtracted = U (Y ) - A'( « ) ) l ~ (3.13) and the results are transformed back into the original dom ain M' = (3.14) CP a is the excess charge vector in the original domain and H = H H Mag [g (y) - q (a )] ‘ ’‘ [?J' (3.15) where d iag[g(y)-g(a)l includes the difference [g(y)-g(a)J from all the lines. T he size o f the matrix is Nz by N z. G alerkin’s technique is now applied to obtain the characteristic matrix •equation. The first step is to expand the elem ents o f the unknown a in term s of known basis func tions with unknown coefficients c/ I I a = (3.16) w here N x is the number of basis functions, N, is the num ber o f discrete lines and 1 r J 4, = | V ja x j dx (3.17) 34 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 microstrip cir cuits with w idth i»’j) cos ( 2 1 - 1) tcI. aw. nu'j •n, = - r \ I = 1, 2 , (3.18) ( aw. /= 1 ,2 ,... For (CPW ) g round (sym m etrical) conductors (iv/( is the ground conductor width, (3.19) is the x-coordinate o f the ground conductor center (one side)) cos ( x + b w!) In- (v x - b 117.)' cos In wu H’i/ II I = 0,2,... n x+ bj 2 ( x + bwt) 1- - w i1 J wu x <0 Sill .v> 0 (x + b J sin In- 1/ " ’u 2 (X + ! U ) 11\ wu x<0 (3.20) (x ~ bJ / tc IV J J 11 / = 1,3, - IV , n / .v>0 J11/ = —n w x -"c o s a b, wi I = 0, 2, ... (3.21) a iv 1(. - I n 11/ = s i n a 6 w/ / = 1,3, ... F or different microstrip or CPW discontinuities, one only needs to adjust ivp b wi 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 Chapter 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 = 2 k J lF • t; clx (3.23) where 1! \ 1' = (3.24) I V1 k 'F is the discretized electric potential (inverse Fourier transform of ij/) and is c o n stant across the metallization. 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-wave resonator analysis described in C hapter 2, the left side o f sy s tem Equation (3.22) is known as an • gebraic equation. The determ inistic solution can be obtained by matrix inversion. Rewiiting equation (3.22) in m atrix form, which contains Nx independent equations, will yield 36 [c]® J 2 nV, ’1 -J A, da (3.26) a =0 By using equation (3.16) and replacing the continuous integral by a discrete sum mation, equation (3.26) can be written as (3.27) where (3.28) Z| = B lo c k Aoc a =0 1 1 2 * Nx G km% n i 2 2 2 ^ 2 G km XSm km /v. 1 T1/n (3.29) Nx N, A„ ^kni^nt^m ^ km ^m ^m ^knt^m ^ A a is the step w idth of the discrete Fourier integration. [Z] is a ,V_/Vr by N:N y. block matrix, which contains Nz by Nz submatrices. Each submatrix fZ ]^ , 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 = 1 2 <i a L .. Nx 1 2 *2 Cl7 A. 1 2 ClNz V A. - "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 determinant through an iterative procedure (equa tion (2.63)), the quasi-static SSDA is a determ inistic approach and provides the results by a one-time m atrix inversion. The total charge Q can be obtained by integrating the charge density over the dis continuity area (3.32) 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 of the strip. Ve=0 is chosen for the ground conductor, The s-paramctcrs can be derived by using netw ork theory. P Step, Open End Airbridge Gap Figure 3.3 The equivalent circuit To calculate the shunt capacitance o f an air bridge, the above formulation must be slightly modified. A C PW 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 h 2 is very small (h<w/5), which is true in most cases. Also the boundary condition of equation (3.6) is changed to 38 dV , dv2 at 27 7 = dV2 ;2 a 7 dV3 8377 v = h2 + h3 (3.33) , ch at V = !u Cross Section ill ill h3 approximate Figure 3.4 A CPW A ir Bridge Where 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 (3.34) Es (r)] Y\ where 39 e 2y tanhy/i 2 - ? 2Y tanhy/i„ tanhy/i, (3.35) - e 2y tanhy/z., -s 2y s 3y tanhy/i 2 tanliy/^ Finally 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 -if./ [c] = [g (y)] " [g (ex) 0 (3,36) 0 E ach submatrix 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 been show n in C hapter 2 that the MoL and SDM arc indeed equivalent if the same d iscretization schem e is used, i.e., the number of lines in the MoL equals the num ber of spectral term s in the SDM . T his equivalency forms the basis of the SSDA. A fter having introduced the quasi-static SSD A , it is worthwhile to look back and study the nature of the S S D A . T his se ctio 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 transmission line problems the most common approach is to express the electrom agnetic field in terms o f Ez and Hz (i.c, TEZ and TM Z wave), w hich gives a coupled TEZ and TM Z wave formulation as described in Chapter 2. But w hen the field is expressed in term s o f Ey and Hy, a transmission line type of formulation can be obtained by coordinate rotation (in the x-z plane) to a u-v coordinate system, which, for the SDM, led to the immittance 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. Starting from M axw ell’s equations 40 Vx£ = -yoopj) (3.37) V yj) = jm z tl and rearranging the above equation yields a more su itable form for the purp o se o f this section d2 .2 d2 . d dxdy J^d= d2 . a dydz -/ 0)fia.v X 57+ * E . (3.38) ,2 d X ?7 +* u2_ . a a2 _/ Ea= a ja ? . a a2 a ^ //. A variable transformation in x- and z- direction (this can be either the F ourier trans form o f the SDM or the orthogonal transform of the M oL) is now introduced as follow s a dx T z ^ - ja * d2 dx 2 d* (3.39) 2 — =- —> -e x . d z2 U sing the low er case to rep re sen t the field com ponents after the transform , eq u ation (3.38) is written as a decoupled TE/TM to y formulation where 0 -a> |i coe ~j T ay ° 0 0 -j .d (3.40) a x and a , can be spectral term s o r eigenvalues of the transform matrix, d ep ending 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 shown in Figure 3.5. Based on the above form ulation, the SSDA is really a 2-D SDM when a are spectral terms. Similarly, the SSDA becomes a 2-D M oL when a v and a and a , are eigen values o f the transform matrices (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 TM y modes are independent (not coupled anymore!). On the other hand, this equivalency is only valid from the formulation point of view. The SSDA has its unique style in solving discontinuity problem because it uses neither 2-D discretization as in the MoL nor 2-D basis function as in the SDM. Figure 3.5 Configuration o f general transmission line In sum m ary 42 • The SSD A combines the M oL and SDM which can be derived from each other • T he 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 method: the SSDA. 45 Chapter 4 The Full-wave SSDA This chapter focuses on the full-wave SSDA. 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 hom oge neous boundary condition is used. To calculate the S-param eters o f planar circuits, inhomogeneous boundary conditions must be included. This chapter describes two different approaches to implement the inhom ogeneous boundary conditions and to extract Sparameters. 4.1 Eigenvalue Approach The eig e n v a lu e approach em ploys the concept 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 whole structure via an eigenvalue equation in which the solution is the reflection coefficient of the discontinuity. The hybrid boundary conditions have been used before in [ 15] and 116], but in the first case to model the forward and reflected waves 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 assumed that at some distance from port I of the discontinuity, there will be a standing wave o f the funda mental mode only consisting of incident and reflected waves: 44 <« <■( -/Pr' V = %,{<* yP|*) - r(1 J h h( - f t i 2 jp A V = v 0{ c +rc J where |3| is the propagation constant at the boundary o f port 1 calculated separately by using the SD M , r is the voltage reflection coefficient and are the incident T E / TM potentials at z = 0, which are solutions o f the homogeneous connecting transm ission line. Connecting Transm ission Line Discontinuity Region Connecting Transmission Line 1 z=0 Discretization Figure 4.1 N, An eigenvalue approach The inhom ogeneous boundary (z-direction) conditions can be derived indepen dently without considering the spectral dom ain factors (x-direction). With reference to the matched, open- and short-circuited conditions, at port 2 three different cases for the boundaries exist, these are the Dirichlet, Neumann and hybrid boundary conditions. F or the matched condition at port 2 , there are two choices for the discretization schem e depending on w hether to assign an e or h line as the first line. In the following, the d is cretization schem e begins with an m (m agnctic)-line (open-circuit). When using an mline as first line, only the boundary condition for \^'0 is specially treated, while the boundary condition for is implicitly included. For the same reason, only the bound 45 ary condition for vjc(J is specially treated w hen using an c-line as the first line. In case of the matched and opcn-circuit conditions, the hybrid boundary condition at port 1 can be expressed as: .„ ( d '/ ;lv /( V ' l h (4 .2 ) jx V(> -/*<• and at z = 0.5 h combining equation (4.2) and equation (4.3) /0.5/ip, d \ |/ Tz •o e J V\ - 0 .5 h e ./os/ip. -re " + 70.5/i|t, h 70 5/at.Vi . A (4.4) equation (4.4) can be simplified as By dz h s (4.5) z = 0.5// where " = l-y-ctan(0.5/iP1) 1+/ ~r~—j'ta.i (0.5/,f5,) T= T 7 The voltage reflection coefficient r is thus explicitly involved in the hybrid bound ary conditions. At port 2 the matched condition corresponds to: J ' = -,/T V lV (4-7) where (32 is the propagation constant at port 2 if a two-port circuit is considered. The propagation constants and p 2 can be derived from the l-D SDM or MOL. Note that the matched condition corresponds to the discretization schem e o f the opcn-circuit condition. In a sim ilar way, the hybrid boundary conditions obtained for the short-circuit situation is as follows: 46 3\}/ dz e = -v y , (4.8) ’ = (Nz + 0,5) h w here x -./ta n (0 .5 /j P ,) v = / P i 71-yxtan (0 .5gi,B~~ /;p [) 1 + /. T = 7— T 1 - r: (4 -9) Obviously, the potential functions and their first derivatives constitute the character istic solutions o f the w hole circuit. It is interesting to see that the com plex functions of the inhomogeneous boundary conditions at the input described in the above equations are not only expressed in term s of the propagation constant Pi but also in terms of the dis cretization interval h and the unknown voltage reflection coefficient r (or ,vn ). In other words, the inhom ogeneous boundary conditions are no longer “static” and strongly depend on the unknown scattering parameter, which in turn depends on the geom etry o f the structure o f interest as well as the operating frequency. This is why the inhom oge neous boundary conditions are said to be self-consistent. In sumrr ry, w hen the load o f port 2 is m atched, open or short, the corresponding boundary conditions are listed in the following table (homo=homogeneous boundary con dition, inhomo=inhom ogeneous boundary condition) port 2 M'*1 v hi H'8 2 A m atched hom o inhomo inhomo homo open hom o inhomo homo homo sh ort in h om c homo homo hom o Table 4.1. Boundary conditions It is noted that only the case of inhom ogeneous boundary conditic-P need to be spe cially treated here, because the case of hom ogeneous boundary condition is discussed in Chapter 2. Sim ilar to C hapter 2, the determ inant equation will be derived from the SSDA procedure. The solution o f this determinant equation is the unknown reflection coeffi cient r. The matched condition is taken as an exam ple in the following analysis. The inhomogcncous boundary conditions are 47 V dz = "V i 2 = 0 . 5 ,'i (4.10) chjr dz : = (A /i+ 0.5) h 'Va -2 in which u and v are the coefficients defined in equation (4.6) and (4.9). in order to m aintain the essential transformation properties (known from the M OL procedure), sym m etric second-order finite-difference operators are required to deal with the Helmholtz equation and, in particular, the field equations tangential to the interfaces. Using the con cept and algorithm described in Chapter 2, the electric and magnetic potential vectors in the original discrete dom ain are normalized by diagonal matrices: <P = (4.11) with 1 J ii h 1 f //] = l/’J r !_ (4.12) 1 Jvh Therefore, the first and second derivatives o f the potential functions arc approxi m ated by formulae sim ilar to the ones in Chapter 2. Note that the unknown voltage reflec tion coefficient r is directly involved in the first elem ent of [/X',/,J and its related matrices. Applying the continuity condition at each dielectric interface leads to a matrix rela tionship between the tangential field components o f two adjacent subregions in the inter face plane. Next, by successively utilizing the continuity condition and multiplying the resulting matrices by the transmission line m atrices associated with the m ultilayer subre gions, the boundary conditions from the top and bottom walls can be transformed into the interface plane o f the discontinuity. This leads to a kind of space-spcctral Green’s function in the transform dom ain which must be transformed back into the original dom ain. This step can be performed by the conventional MoL and SDA procedures inde pendently. From the m athematical viewpoint there is no difference which procedure is applied first. However, applying the MoL first leads to a better physical understanding and easier mathematical treatment. As a result, the matrix elements of the resulting 48 G reen ’s function in the space-spectral domain are once again coupled to each other through the reverse transformation back into the original domain \ it . ex b \ A \ A - .. G alerkin’s technique is again used together with an appropriate choice o f basis functions which will be defined on the conductor surface for each slicing line in the zdirection. This leads to a characteristic matrix equation system which m ust be solved for the zeros o f its determinant, whereby the determinant is a function o f the reflection coeffi cient r. (4.14) F or irregularly shaped discontinuities, the geom etric param eters becom e a function of the z-coordinate and, therefore, are different for each line. In general, this does not com plicate the analysis of planar structures at all, as long as the circuit contour can be described mathematically or by a set of coordinates. In addition, singularities o f the cir cuit in the x-direction are automatically considered in the form ulation o f the basis func tions. O nce the voltage reflection coefficient r is known, an arbitrary constant for the first elem ent o f the x-oriented current coefficients can be assumed. Applying a singular value decom position technique yields all the current coefficients for the chosen basis functions assigned to each discrete line. Therefore, the S-param eters can be extracted from incident and reflected currents on the strip. 4.2 Deterministic Approach A lthough the eigenvalue approach for the full-wave SSD A has the advantages that no 2D field distribution calculation is required, the root o f the determ inant, w hich is derived from a large matrix, must be found. For practical applications, a “o n e -s te p ” solution is most desirable. In this section, a deterministic full-wave SSDA is presented, which avoids solving an eigenvalue equation by iterative computation. A deterministic approach was used earlier in the 2-D M oL [12] [26]. In [12] a “three-step” approach was presented rather than a “one-step” approach because open and 49 short conditions were utilized. In 126] inhomogeneous boundary conditions were intro duced. B ut based on the a u th o r’s experience, the resulting algorithm does not provide a stable solution because a good m atching condition for S-parametcr calculation can not be realized. T he key steps in the follow ing approach is to express the field distribution on the connecting transmission line (far from the discontinuity) as a superposition of incident and reflected waves, then derive the inhomogeneous boundary conditions for incident and reflected waves (sim ilar to the previous section, the eigenvalue approach), and finally com bine the incident and reflected waves to satisfy the tangential field condition at the m etallization plane. By solving the 2-D transm ission line problem first, the inci dent w ave distribution is know n. The reflected wave distribution is derived by using the know ledge o f the incident wave. F igure 4.2 illustrates the deterministic approach. Region B t and C together arc called region B. Port 2 is alw ays matched. Region B Connecting Transmission L ine Region A Discontinuity Region B j Connecting Transmission Line Region C 2 z =0 Nz Figure 4.2 A deterministic approach It is assum ed that only one propagation mode exists on the transmission line connected to port 1, w hich is so far from the discontinuity region B j, that the higher order modes excited by the discontinuity have vanished at port I . The same assumption also applies to 50 region C. Using i and r to represent the incident and reflected waves, the inhom ogeneous boundary conditions are expressed as <V' incident IPi°-5/' h -yPjf dz portl -yp, 0 .5/1 h = .iP f Vj dy reflected vL z = 0.5/1 (4.15) z = 0 .5 h -yp2o.5/. in c id e n t = -JP2C z reflected port 2 -yp20 5/« 3 i|/ dz V,\.\- = (Nz + 0.5) // = -./(V ( 4 .1 6 ) v,Vr : (yvz + o .5 )/i Similar to equation (4.10), equation (4.15) and equation (4.16) can be simplified as f i, r chy dz = “ h Vi z = 0 .5 h (4.17) r Tz z = (A 'z + 0 .5 ) h where ,. u = y P j-c -yp,o.5/i ,• u = yp 0 .5/1 -jfif - / p . 0 .5/1 v = - / (32 c ( 4 .1 8 ) As shown in Chapter 2, two system equations can be obtained for incident and reflected waves respectively Z (4.19) Rearranging equation (4.19) and using subscripts A and B to represent fields and currents in different regions, yields 51 h 11 \i Ja i _ A H_ vl hi •/ A11 Ja Jn Once again G alerkin’s technique is applied (described in chapter 2 and 3) to expand the unknow n incident and reflected currents in terms o f known basis functions q and unknown coefficients C ',r \ ‘' r ./ = 1] (4.21) Calculating the inner product of basis functions and using equation (4,20) yields r -| w' _ 11_ w‘ L 21 -i . w‘12_ ii <*■'#'w w‘ _ 22] \yr L C'a = . M/r 12 K w r21 wr Yy27 _ < 4 V k C \4 is the know n (from 2-D SDM ) coefficient for the current distribution at the connect ing transm ission lin e at port 1 (region A), C rA is the unknown coefficient of the reflected current distrib u tio n at port 1. A lthough i and r are used here, both C'n and C n are the unknown coefficients o f the outgoing current distribution at port 2 . In region A o r B, on the m etallization, the total tangential electric field must be zero e+ er = 0 (4.24) Com bining equation (4.22), (4.23) and (4.24) yields . w *I <1 + . . w‘21. 4 + wr . w1 . 12 - *L \\h21. W*22 - , w r12 ) c » + w 22 r y„+ wr12 wr12 52 where c‘r = c' +cr (4.26) C " ^ is more useful fo r determ ining the S-param eters. Since C 'A is know n, equation (4.25) can be solved by eliminating C B, W\ IV w",12 W<21 IV,. i v ] !w ' 1V22 ^ - d t n 2i IV11 C (4.27) where IV. iv 12 IV12 iv:22 iv: 22 (4.28) In equation (4.27), C A and C'rB are unknown. Because C B is valid in region B t and C, equation (4.27) can be further simplified by replacing C ‘rB by the known coeffi cients of region C (solution o f a homogeneous transmission line) multiplied by an unknown factor. Thus the number of unknowns is reduced to I in region C no m atter how many lines are used there. Applying the same principle to region A will also reduce the number of unknow ns in region A from NA to I. Finally, C A and C 'rB can be obtained by simple matrix algebra which is described in the appendix. Since the coefficients of the reflected current at port 1 and outgoing current at port 2 are known, the currents can be calculated accordingly. They are defined as I' j and I '2. Also the incident current obtained from a 2-D SDM analysis is defined as 1+ The S- parameter can then be calculated by S.. = (4.29) ^12 = (4.30) 53 w here Z; is the characteristic impedance o f the connecting hom ogeneous transmis sion line at port / ( /= / , 2). Chapter 5 Numerical and Experimental Results 5.1 Convergence Analysis of Quasi-static SSDA A c o n v e rg e n c e analysis is perform ed for a m ic ro strip open-ended d isc o n tin u ity , as show n in Figure 5.1 (er is the dielectric constant and h is the thickness o f the substrate). Nx is the num ber o f basis functions in the x-direction. Nz is the num ber o f lines in the zdirection. The convergence behavior depends on the num ber o f spectral term s, Nz, and Nx. W hen the num ber of basis functions is increased, the number o f spectral term s must be increased accordingly. A good convergence behavior is obtained w hen N z >40, N x>2, and the spectral term is greater than 80. dashdot: tiz= 40 N x= l Nz=40 Nx=2 Nz=40 Nx=3 solid: Nz=60 Nx=l Nz=60 Nx=2 Nz=60 Nx=3 LLf N z=J0 Nz=10 Nz=10 N x=2 Nx=3 dashed Nx=l M icrostrip Open End ! 100 30 # ol spectral terms Figure 5.1 Convergence analysis of the Quasi-static SSDA (ir//;= /, e r=9.<5) 55 5.2 Simulation Results of Quasi-static SSDA First o f all the microstrip (M S) and CPW open end, gap and step in width as w ell as the CPW air bridge are analyzed by the quasi-static SSDA. Results are shown on Figure 5.2 to Figure 5.8. Those results are used to validate the SSDA. Figure 5.2 shows the equivalent capacitance o f a microstrip open end. The solid line is calculated by the SSDA. A ll markers are results from literature. 10 x x x Full-W ave[33| o o o Quasi-static[ 22 ] + + + SD M | 19 J er= l 6 10 ar=9,6 + o - 4 *'X io' * * * SDM [39] SSDA 10“ 10 w 10 w/h 10’ Figure 5.2 Capacitance of microstrip open ends. Figure 5.3 shows the equivalent capacitances o f a microstrip gap, which is repre sented by a n network as shown in Figure 3.3. The circles represent the SD M sim ulation and stars represent measurements [21]. The solid line is the results of the quasi-static SSDA sim ulation. 56 45 SSDA * * * M easurem ent [19] o o o SDM [19] 40 35 30 GT25 0.20 15 10 I 10 20 30 g a p width (x0.01 mm) 40 50 60 Figure 5.3 Equivalent capacitance o f a microstrip gap discontinuity. w/h=l h =0 508m m er=8.875 09 0 8 S12 V> &0 7 SSDA W, 0 6 W2 o o o Full-w ave[28| 0 ,5 S11 0 4 0 3 10 . . 1- 15 20 25 30 35 F requency (GHz) , 4. , , 40 .... t .. 45 50 Figure 5.4 S-param eters o f a m icrostrip step. w ,= l/n m , \\'2=0,25mm, w /h= l, 57 Figure 5.4 illustrates the S-parameters of a microstrip step discontinuity, which was derived from network theory, and its shunt capacitance value. The circles represent the full-w ave SD M simulation [30]. The solid line is calculated by the quasi-static SSDA. Figure 5.5 presents the equivalent capacitance o f an open-ended eoplanar waveguide com pared with published results. The solid line is calculated by the quasi static SSDA. The circles and stars represent finite difference simulation and m easure ment [22], respectively. Figure 5.6 shows the S-parameters for a eoplanar air-bridge. The circles represent the results from a full-wave analysis using the Frequency Domain Finite Difference (FDFD) m ethod [31]. Figure 5.7 illustrates the S-parameters of a CPW step discontinuity analyzed in [32], The solid line is calculated by the SSDA. The circles arc calculated by the fullwave SD M [32]. Figure 5.8 presents the S-parameters of a CPW gap analyzed in |2 2 |. The solid line is calculated by the SSDA. The circles represent the results from the quasi-static SDM [22], In summ ary, Figure 5.2 to Figure 5.8 present a variety o f planar discontinuities. The SSD A sim ulation agrees well with either published results or measurements. 58 30 25 SSDA o o o Finite Difference [22] * * * M easurem ent [22] tr 20 n3 o(tJ Q. 03 % c 15 w 10 02 03 04 0.5 0,6 0.7 0,8 0.9 w/d Figure 5.5 0 06 Equivalent capacitance o f a CPW open end. er=y.6, h=0.635, /;/</=/, cJ=\y+2s I 0 ,0 5 t SSDA o o o FDFD [29] 0 04 £2-o 03 w 0 02 0 01 °0 10 20 30 40 so ' 60 F re q u e n c y (GHz) 70 8 0 9 0 100 Figure 5,6 S-param ctcr of a CPW airbridge. w = l3\im , x= l()\lm , I=3 [On, h~200[W i, b~3\xni 59 S21 SSDA o o o SDM [30| S11 28 30 32 38 36 34 40 F re q u e n c y (GHz) Figure 5.7 S-parameters of a C PW step. wj-OAimn, u w2=0.4 mm, e r~0.S, h=0.254mm Sj-O .lm m , 40 J SSDA o o o [221 35 tSo* W, 30 w2 sl 25 c<d 20 O § Cp2 15 O 10 5 Cs 0 0 05 Figure 5.8 i. 01 i Cp1, , 0 15 02 0 25 GapWidth/SubstratoThicKnoss 0*3 Equivalent capacitance o f a CPW gap, (/|»2s j t-H-j, h~0,635mm, w^lh-O 2, \ v \ l d l -w ^ d l'-O A d , u'2/ " 7 2+tt’2.‘V'* y "4 60 The main reason why the quasi-static SSDA works extrem ely well for M H M IC ’s is because the circuit dimensions are small compared with the wavelength, which m eans the dispersive effect is normally very small (the dispersive effect m ay become obvious at very high frequencies). When this condition is not valid (Figure 5.12), visible discrepan cies to full-wave techniques may occur at lower frequencies. The com putation time of the quasi-static SSDA is typically a few m inutes on a SUN SPARC-2 station (this will give the results over the whole frequency range). It is noteworthy to p oint out that results shown from Figure 5.2 to Figure 5.8 do not only d e m onstrate the good agreement with published results, but also show that this method is flex ible. The accuracy o f this deterministic quasi-static SSDA com pares well with full-w ave methods and measurements. To show the ability of modeling arbitrary discontinuities, microstrip and C PW M HM IC tapered discontinuities are calculated. The transitions are represented by their equivalent capacitances as illustrated in Figure 5.9. When the length o f the slope, W s, increases, the equivalent capacitance decreases. This makes perfect physical sense. When the length o f the slope equals zero, it represents an abrupt step in width (this was proved by calculating the capacitance o f a step). The whole capacitance curve is quite smooth which show s the good numerical stability of this method. The S-param eter o f a CPW air bridge is also calculated in terms of air bridge length, which is show n in Figure 5.10. One can observe that when the air bridge length L decreases, S j ( decreases too, because the capacitance to the ground decreases. Figure 5.11 and Figure 5.12 show the limitation of the quasi-static SSDA. F igure 5.11 shows the S-param eters of a microstrip step discontinuity up to 100GHz. A lthough at frequencies below 50GHz the comparison with full-wave techniques is excellent, higher order m odes start to propagate on this structure at 60G H z which is not accounted for in the quasi-static method. This leads to an increase of S n and a decrease in S2i at about 50GHz. 20 wi Ws substrate. Q. Microstrip — w CPW Slope Length Ws (m m ) Figure 5.9 Equivalent capacitance o f a m icrostrip and a CPW step/taper. For CPW tv1=0.8mm, s ,=0.1 mm, w2=0.2mm, s2==0.6mm, er=9.6, h=0.254m m . For microstrip, w 1=1 mm, w2=0.25mm, s r=9.6, h=0.25mm 62 <N 20 40 60 80 100 Frequency (GHz) Figure 5.10 S-param eters of a CPW airbridge versus bridge length /. \\'=03m m , s-O .lm m , b=3\lm, s r- 9 ,6 , h=0.254mm. 1.0 —■ -------r " -t ’ i ... -r ..t ... 1 ...... o .y - S ' 0.8 - 0.6 „ ............. Full Wave TLM [33] 0.5 • 0.4 0.3 - 1 Quasi-Staiic SSDA 0.7 - *** ,, s ,, - - - | 0.2 - Wi ‘■ ”1 ........... Wi 0.1 nn • t , ,1 — I---------- 1-------- w 1=0.25 mm w2=lmm h=0.25 Er=9.6 i Frequency (GHz) Figure 5.11 Frequency dependent bcba\ lor of microstrip step 63 1.0-1 f ‘ 1 ■ 1 I <1 1 ■ I » ‘ I >"l » I < '!- [ i i I I I I I i I j | - | - | T - |- | I i I I I I I i 1 i i i i r~pr~rT-r-w- 0.8- - SSDA S -P a ra m e te rs - MMM 1121 0 .6 •FDTLMI3.1] - S11 r. — 0 .4 £r=9.6, h=0.254 0.2 - 0.L 0.0 - 0.8 10.2 I I I I I 1II l l l l l l l l l l l l l l . l l l l . i l 10 20 30 ■ l-i-i i 40 I i i i,, i,_ 50 Frequency (GHz) Figure 5.12 Frequency dependent behavior of CPW step 64 5 .3 C o n v e r g e n c e S tu d y o f F u ll-w a v e S S D A A c o n v e r g e n c e an alysis is perform ed for a microstrip step discontinuity as sh o w n in F ig ure 5 .1 3 . 1 ........................ t ------- •*-T--................. t ■- - ~"T.......... . *---- -t- - -* 09 0,8 - ■ ...... 0.7 t2 06 w — 1- - - *■— ■* ----- r .......... -1 - - .......r -----------T -. ;— . ■ * : ..................... r i - ~ - ■ .... . ■ -t n z= 3 0 nz=40 n z= 5 0 o o o + + + [29) [2 9 ] s 21 n z= 6 0 1 05 rS 09 0 .4 ------— - --------------- — ----------- ----- _ 03 S 11 ..,* 1 1 1 0.2 W i ' n W 2 0.1 - .................— -t-— °c 2 - - J------- 4 ... F igu re 5 .1 3 x.......................X ------- --------- i~ ~ --------- 6 8 - X ............ 10 12 Freq (GHz) i ............. 14 J 16 , - _ 18 -----------x. ... 20 , C on vergen cy behavior o f the fu ll-w ave S S D A w 1—1mm, w2=025m m , h=0,25mni, er=l() W h en the num ber o f lin es (in the propagation direction z) is greater than 5 0 , co n verg en ce is a ch ieved . A s pointed o u t in C hapter 4, using the approach presented in [26] (u sin g hybrid boundary co n d itio n s) w ill lead to an unstable solution, because the m atching co n d itio n is applied to o n ly o n e line. U sin g the fu ll-w ave S S D /i, a stable solution can a lw a y s be obtained. 5.4 Simulation Results of Full-wave SSDA A co m p a riso n o f S-param eters obtained by the fu ll-w a v e S S D A and by others (i.e |3 0 |) fo r a m icrostrip step d iscontinuity are show n in Figure 5 .I 4 and F igu re 5 . 15. A g ood agreem en t can be ob served o v er the frequency range up to 20G H z. 65 1 09 SSD A [29] 08 ooo 07 + + + [29] S 12 J j j 1 | cD o0 6 C CD § 05 C_D Q ^04 W, . , L.... w2 j } ii i 03 0 2- +L,_„__~ ....... 01 Su 4 i 2 4 6 8 10 12 Fieq (GH/) 14 1r> 18 ,*i) Figure 5.14 Full-w ave S-parameters of a microstrip step W j ^ h n m , u’2=0 .5mm, 11=0 .25nun, e,.= I0 i .™~— ----- r... . . T, .... . . T’ r •* " *-*----- ‘r s I2 os 08 t A -*» wi *r• m k: ■ z n . ;. AA*-. 1 - i*1* ..... . ... ... r r + ♦ SSDA [29] fOOl ooo 111 0.7 t2 a» 0.6 OJ § 0.5 ro a. «• 0.4 t r w2 .. S i, 0.3 0.2 0.1 0D.. --- A2 • .... A4 .... X 6 i 8 i * 10 12 Freq (GHz) . 14 j 16 i 18 Figure 5.15 Full-w ave S-paramcters o f a microstrip step w2=0.25mm, h=0.25mm, er =l() i 20 66 Transmission characteristics o f two closely spaced m icrostrip step discontinuities arc shown Figure 5.16 (from 2 to 40 GHz). It is evident from the Figure 5.16 (a) that there is a strong interaction between both steps since their separation is less than half a guided wavelength. This type o f structure is widely used in filter design. In general, it behaves like a low-pass filter as in the case of l=1.20mm. The interconnecting stub also contributes to the dip around 5GHz for the case o f 1=1.20m m. T he shorter the length 1, the further the dip moves towards higher frequencies (as shown in the case o f 1=0.15mm). Hence the stop band effect occurs only at higher frequencies. A lso the Q factor decreases because the coupling between two hom ogeneous transm ission lines increases. The phase characteristics o f the S-parameters are show n in Figure 5 .16(b). 0 .6 -, 0.5- - - I = 1.20 rnm ... I = 0.60 mm — I = 0.30 mm I = 0.15 mm 0.3 - 0.2 f (G Hz) (a) 100-, 50- X, o’ 100-1---10 20 fto ii/, •b) 40 Figure 5.16 S-parameters for a cascaded step discontinuity separated by a transm ission line of length 1. W]=0.4mm, \\’o=0.2mm, \v2=0.Smm, er=3.8, h -0 ,2 5 tn m . (a) M agnitude of S ,,, (b) Phase of S , j 67 5.5 Experimental Results By using a netw ork analyzer at microwave and millimeter wave frequencies, it is often im possible to directly measure the scattering parameters of a device under test (DUT). D e-em bedding techniques must be used to obtain correct S-parameters (36 - 42]. Som e M H M IC discontinuities have been fabricated for experimental investigation. Figure 5.17 to Figure 5.20 show the calculated (solid line) and measuremcnt(dashed line) results perform ed at the University of Victoria. Five M HM IC’s with via holes are tested. The de-em bedded S-param eters are compared with quasi-static SSDA simulations. All M H M IC ’s are built on 0.254 mm conductor-backed substrate with dielectric constant 9.6. Ground plane via holes are used to suppress the microstrip mode. Figure 5.17 shows the S-param eters o f a single CPW gap. Figure 5.18 investigates CPW double gaps with two different resonator lengths. The SSDA calculation shows how the resonant peaks move w hen the resonator length changes. Figure 5.18 illustrates the S-param ctcrs o f a CPW step discontinuity. 68 [ i i-n-| T i"i r — j"i— i-i i | in i| in i| i i | i i i i | i i i-i-q i i (dB) Experim ents SSDA S-parameters 0.0254 C i-9.6, h= 0.254m m 0 10 20 Frequency (GHz) 30 40 Figure 5.1.7 M easured and com puted S-parameters o f a CPW gap SSDA L=3mm SSDA L=1,5mm Experiment U 3mm Experiment L= 1.5mm S12 (dB) 0.0508, L | er=9.6, h=0.254nim Figure 5.18 M easured and com puted S-param eters of end-coupled CPW resonators 69 S11 (dB) SSDA Experiment er=9.6, h=0.254mm 1 0 0 -<L * i i i l <i i i I i i u I i i i i i ■ ■ ■ ■i i .............. Frequency (GHz) Figure 5.19 M easured and computed S-parameters o f a C PW step discontinuity 70 /S -10 -20 *o --3 0 V \ -40 SSDA -60 3 □ El □ O l - - M easurem ent -50 26 27 28 29 \ l 31 32 Freq (GHz) Figure 5.20 S-param eters of a CPW end-coupled filter. it'=(J.2, s= 0.I5, gap width: 25.4\im, resonate, length: 2mm. Figure 5.20 show s the response of a CPW end-coupled filter. Via holes are also used to short-circuit the ground-plane and the back-metallization. The SSDA simulation exactly predicts the position and magnitude of all four resonant peaks which are gener ated by this 4-pole filter. In summary, all experim ental results agree with our quasi-static SSDA simulation very well. 71 Chapter 6 Conclusion 6.1 Contributions A fter introducing the generalized form o f the S p ace-Spectral D om ain Approach (S S D A ), a new d eterm inistic quasi-static S S D A has been d ev elo p ed to an alyze planar cir cuit d iscon tin uities.T his n ew approach extends the S S D A to calcu late q u asi-static capaci tances and S-param eters o f arbitrarily shaped planar d iscon tin u ities, T h e d iscontinuity param eters are derived from an algebraic matrix equation instead o f an e ig e n v a lu e matrix. T w o new approaches based on the fu ll-w a v e Space-S p ectral D o m ain Approach (S S D A ) have been proposed to calculate scattering param eters for three-dim ensional d is continuity problem s in M M IC ’s and M H M IC ’s. T h e theory presented in this thesis d em onstrates how to im p lem ent self-con sisten t hybrid boundary c o n d itio n s and how to derive the determ inistic approach. A com parison o f the results with other m ethods and m easu rem en ts sh o w s e x c e l lent agreem ent up to 40G H z. R esults o f so m e com p licated structures su ch as a M HM IO m icrostrip and C P W taper, C P W air bridge and C P W en d -co u p led filter arc g iv e n . Exper im ental validation o f the theoretical results are presented. A lth ou gh the lim ited num ber o f exa m p les g iv en is not representative for all type o f d isco n tin u ities, this technique by nature can treat arbitrary planar two-port circuit contours, T h is is an advantage o f this new m ethod w hich m akes it an attractive C A D tool for en g in eerin g ap p lication s. 72 In sum m ary, the major contributions o f this dissertation arc, • A g en eralized S S D A w as developed, and a com plete field form ulation w as g iv en , • A d eterm in istic quasi-static S S D A w as introduced. • T h e nature o f the S S D A w as investigated. • T he s e lf c o n siste n t hybrid boundary con d ition s have been d ev elop ed and im plem ented into the full-w ave S S D A . • A determ inistic fu ll-w ave SS D A w as d evelop ed . • E xperim ental w ork w as done to verify the S S D A and to investigate other planar structures. • A user frien dly com puter-aided design (C A D ) software package, M H M IC 2 .0 , w as d e v e lo p e d based on the work o f this thesis. 6.2 Future Work A lth ou gh e x te n siv e sim ulation and validation w ere performed in this th esis and a variety o f d iscon tin u ity m o d u les were included in our C A D software, a uscr-oricnted so ftw a r e can be d e v e lo p e d for m ore com plicated discontinuities. Another approach is to use the S S D A algorithm to generate libraries for certain popular CA D p ack ages such as T O U C H S T O N E ™ or S U P E R C O M PA C T ™ . T h e quasi-static S S D A is a very efficient tool to analyze most planar circu its. It can be ex ten d ed to o p to -e le c tr o n ic applications, e.g., for the analysis o f field distribution o f an electro -o p tica l m odulator as sh ow n in Figure 6.1. Electrode Waveguide F igu re 6 . 1 Future application: electro-op tic modulator 73 S in c e the S S D A in its present form d ocs not include the effect o f finite m etallization thick ness and conductor lo sse s, tw o different approaches arc proposed to incorporate finite m etallization: • u tilize the concept o f surface im pedance as it is d on e in the SD M : add sur fa c e im pedance into the G reen ’s function to incorporate finite m etallization and conductor loss; the form ulation w ill stay m o stly the sam e as in the SSD A . • u sin g the m ode m atch ing m ethod (M M M ) in conjunction w ith the M oL: u se M M M in the transverse sectio n to incorporate finite m etallization and conductor loss, in the propagation direction (u su ally called /.-direction) the M oL is used as sam e as in the S S D A . T h e S S D A presumed in this th esis w as applied to a 3-port application in the c a se o f the qu asi-static S S D A only (C hapter 5 ). 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Pollard, “Improved calibration and measurement o f the scattering param eters o f microwave integrated circui ts,” IEEE Trans, on M icrowave Theory Tech., Vol. MTT-37, pp. 1675-1680, Jan. 1989. (40] R. Lane, “De-embedding device scattering param eter,” M icrow ave J., Vol.27, pp. 149-156, Aug. 1984. ]41] H. J. Eul and Shiek, “Thru-M atch-Reflect: An improved technique for calibrating the dual six-port automatic netw ork analyzer,” Technical Report, RuhrUniversity Bochum, Institut fu r H och and flockstfrecpienztechnik, Bochum, Germany. [42] M. Yu, “ M icrowave Device D e-em bedding Technique,” Project Report o f ELEC454, M icrowave and O ptical Communication System s, Department of Electrical and Com puter Engineering, University o f Victoria BC, Canada, 1992. [43] A. Hill and V. K. Tripathi, “An efficient algorithm for the three-dimensional analysis o f passive microstrip com ponents and dicontinuities for microwave and millimeter-wave integrated circuits,” IEEE Trans, on M icrow ave Theory Tech., Vol. M TT-39, pp. 83-89, Jan. 1991. 71 Appendix The total num ber o f lines in the z-dircction: N = N ,+ N /() Nfl=Nfl,+Nf (F.4) In region A, one unknown is enough to represent C / because o f the traveling wave assumption. In region C, the same principle applies. The total number o f unknowns: C /: I C/jir: Nfl/-f 1 C / and C flir can be expressed as /[}// e '« r: - a2' (K5) r aN, +\'(lN„i +2 1. (' '|W'- (Kt>) Com bining all left terms of equation (4.27) into one matrix | W ,) yields (K 7) M* - K K where M'',.'i MV' A = \a,t a2, a i,...,aN +2 j is the unknown coefficient vector. When N/t>N ^/+2, equation (4.27) can be solved. (m VITA Surname: Yu Place of Birth:B eijing, C h in a Given Names;Ming ______ _____________ Date o f Birth:June 1962 _____ 1iducational Institutions Attended; University of Victoria, Victoria, BC, Canada Tsinghua University, B eijing, P.R. China Degrees Awarded: • M S. in Electrical Engineering, July 1986, Tsinghua University, Beijing, P.R. China Thesis: 11GHz D ual-m ode Linear Phase Filter • B.S. with Honor in Electrical Engineering, July 1985, Tsinghua University, Beijing, P.R. China 1 l o nou rs and Awards: - G.R.E.A.T. Aw ard, B.C . Science Council ( i 992-1993) • Graduate Studies Fellow ship, University of Victoria (1990) • Graduate Teaching Award, University of Victoria (1991) • Graduate Teaching Fellowship, University of Victoria (1991,1992) Publications: ReFereed Jo u rn a l P a p e rs Published 1, M . Yu, R. Vahldieck and K, Wu, “Theoretical and Experimental Characterization of Coplanar Waveguide Discontinuities”, IEEE Trims. on Microwave Theory and Tech., MTT-41, No.9, Sept. 1993. 2, K. Wu, M . Yu and R. Vahldieck, “Rigorous analysis of 3-D planar circuits disconti nuities using the space-spectral domain approach (SSDA)”, IEEE Trans, on Micro wave Theory and Tech., MTT-40, No,7, pp. 1475-1483, July 1992. 3, M . Yu, K. Wu and R. Vahldieck, “A Deterministic Quasi-static Approach to Micros* trip Discontinuity Problem in the Space-Spectral Domain”, IEEE Microwave and (lid d ed Wave Led., Vol.2, No.3, pp. 114*116, Mar. 1992. Partial Copyright License I hereby grant the right to lend m y thesis (or dissertation) to users o f the U n iversity o f Victoria Library, and to m ake sin gle co p ie s o n ly for such users or in resp on se to a request from the Library o f any other university, or sim ilar institution, on its b eh alf or for o n e o f its users. I further agree that p erm ission for ex ten siv e c o p y in g o f this thesis for scholarly p u ip o ses m ay be granted by m e or a m em ber o f the U n iv ersity designated by m e. It is understood that co p y in g or publication o f this thesis for financial gain shall not be allow ed w ithout m y written perm ission. T itle o f T hesis/D issertation: ^ Hybrid Nuniei ical rechni(|iio lor the Analy sis and Design of Microwave Integrated Circuits Author (Signature) M IN G Y U (N am e in B1o_ k Letters) (D ate)

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