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Jatou A thesis submitted in conformity with the requirements for the Degiee of Master of Applied Science in the Department of Electrical and Computer Engineering in the University of Toronto © Copyright by Firas F. Jatou, 1995 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1*1 National Library of C an ad a Biolioth6que nationale du C an ad a Acquisitions a n d Bibliographic Services Branch Direction d e s acquisitions et d e s services bibliographiques 395 Wellington Street Ottawa. Ontario 395. rue Wellington Ottawa (Ontario) K1A 0N 4 K1A0N4 Our M<* Nor*<* The author h a s granted an irrevocable non-exclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sell copies of his/her thesis by any m eans and in any form or format, making this thesis available to interested persons. 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N om e F ~ lR N > -> Q \ 1 ----------------------------------- .— ------------------------- — , Dissertation Abstracts Intemational is arran g ed by broad, general subject categories. Please select the one subject which most nearly describes tfie content of your dissertation. Enter tbe corresponding four-digit code in tbe spaces provided. t. ETrv j iric o / X 4 P /-~ V SUBJECT C O O t SUBJECT TERM UMI Subject Categories THE HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS AND THE ARTS Archriecture................................. 0729 Art History................................... 0377 Cinema ........................................0900 D an ce ...................... 0378 0357 fine A r ts .......................... 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Urban a n d Regional Pfenning — 0999 W omen's Studies........................0453 THE SCIENCES AND ENGINEERING BKKOaCAl SCIENCES Agriculture G eneral................................. 0473 Agronomy.............................0285 Animal Culture and Nutrition ....... 0475 Animal Pathology................. 0476 Food Science a id Technology........................ 0359 Forestry ona Wildlife............0478 Plant Culture . 0479 Plont Pathology.....................0480 Plant Physiology................... 08117 Range M anagement ....... 0777 W ood Technology................ 0746 BW*SS)«rol................................. 0306 Anatomy .............................. 0287 Biostotistics ....................0308 Botany... . 0309 C ell. 0379 Ecology 0329 Entomology ............. 0353 Genetics .. 0369 limnology ....................0793 Microbiology ...................0410 Molecular ....................0307 Neuroscience ..................... 0 3 17 Oceanography ................. 0 4 16 Physiology .............. 0433 Radiation 0821 Veterinary Science................ 0778 Zoology 047? 8«ophys*ci General 0786 Medicol ... 0760 EARTH SCIENCES Biogeochemntry ......................0425 Geochemistry 0996 G eo d esy ................ - .................. 0370 G eology...................................... 0372 G e o p n ja s a ................................. 0373 Hydrology................................... 0388 Mineralogy.................................. 0411 Paleobotany :::::::::::::::::.:o345 Poleoeoology............................... 0426 P oleontalo^................................0418 Fofeoxoology .......................0985 Pahmology ........................ 0427 Physical Geography................... 0368 Physical O ceanography.............0415 HEALTH AND ENVIRONMENTAL SCIENCES Environmental Sciences..............0768 Heolth Sciences G eneral................................. 0566 A udidogy................. 0300 Chemotherapy 0992 Dentistry ............................. 0567 Education................... 0350 Hospital M anagem ent 0769 Human Development 0758 Immunology...................... 0982 Medicine and Surgery 0564 Mental H ealth.......................0347 N ursing................................. 0569 Nutrition................................0570 Obstetrics and Gynecology .0380 Occupational Health and Therapy..............................0354 Ophthalmology 0381 PaihologyZT. ................. 0571 Phormocology.......................0419 Pharm acy..............................0572 Physical th e ra p y .................. 0382 Pubfic Health......................... 0573 Rodiology..............................0574 Recreation ........................... .0575 Speech Pathology................. 0460 Toxicology............................ C383 Home Economics........................ 0386 PHYSICAL SCIENCES P u re Sciences Chemistry G eneral.................................0485 Agricultural .......................... 0749 Anolytico!....................... 0486 Biochemistry ........................ 0487 Inorganic...............................0488 N uclear.................................0738 O rganic................................. 0490 Phormoceutkal......................0491 Physical.................................0494 Polymer.................................0495 Radiation...............................0754 Mathematics................................0405 Physics G en eral.................................0605 Acoustics...............................0986 Astronomy ond Astroohysics....................... 0606 Atmospheric Science.............0608 A tom ic.................................. 0748 Electronics and Electricity 0607 Elementary Particles ond High Energy.......................0798 Fluid and Plasma.................. 0759 Molecular ............................0609 N uclear.................................0610 O p tic s...................................0752 Rodiafion...............................0756 Solid S tate.............................0611 Statistics................................. 0463 A p p lie d Sciences Applied Mechanics.....................0346 0984 Computer Science................. 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Further reproduction prohibited without permission. THE UNIVERSITY OF TORONTO LIBRARY MANUSCRIPT THESIS - M ASTER’S AUTHORITY TO DISTRIBUTE SO TE: The A U TH O R will sign in one o f the two places indicated. It is the intention o f the University that there be NO RESTRICTION on the distribution o f the publication o f theses save in exceptional cases. a) Immediate publication in microform by the National Library is authorized. Author’s signature-. D a t e *{l. -OR- THIS FORM SHOULD BE INCLUDED WITH UNBOUND COPY b) Publication by the National Library is to be postponed until. (normal maximum delay is two years) Date__________________ 4 Author's signature________________________________________ D a te __________________ This restriction is authorized for reasons which seem to me, as Chair of the Graduate Department o f , to be sufficient. Signature of Graduate Department C h air___________________________________________ D ate____________________________ B O R R O W E R S undertake to give proper credit for any use made of the thesis, and to obtain the consent of the author if it is proposed to make extensive quotations, or to reproduce the thesis in whole or in part. Signature o f Borrower Address Date i i i | C-I (1993) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table Of Contents AKNOWLEDGMENTS L IST OF FIGURES..................................................................................................... iv LIST OF TABLES .................................................................................................................................vii 1. IN TR O D U CTIO N .....................................................................................................I 1.1 Motivation 1.2 Objectives 1.3 Methodology 1.4 Overview 2. A P P R O A C H ....................................................................................................................... 7 2.1 Formulation o f the G reen’s Function 2.2 M ethod of Moments 2.3 Determination o f Network Parameters 2 .3 .1 Characteristic Impedance (Z0) 2.3.2 Propagation Constant 2.3.3 Impedance and Scattering Parameters 3. NUMERICAL IM PLEM ENTATION................................................................ 17 3 .1 Computer Program 3.1.1 Input File 3.1.2 Impedance M atrix Formulation 3.1.3 Network Parameters 3.2 Parallel Implementation 3.2.1 Domain Partitioning 3.2.2 Load Balancing 3.2.3 Performance Evaluation 3.3 Conclusions ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. VALIDATION 4.1 Results 4.1.1 4.1.2 4.1.3 4.1.4 31 o f Uni-Planar Discontinuities Series Gap Right Angie Bend with Non-Uniform Discretization Four-port Cross Coupled Line Filter 4.2 Results o f Multilayered Discontinuities 4.2.1 Stripline Gap Junction 4.2.2 Tw o-iayer Resonator Band-Pass Filter 4.3 Sum m ary o f Numerical Results 5. TIM E-DOM AIN ANALYSIS 42 5.1 The M ethod 5.2 The C om puter Program 5.3 Simulation of a Linear System 5.3.1 The Cross Junction 5.3.2 L-Type Coupling Structure 5.4 Conclusion 6. CONCLUSIONS 6. 1 Concluding Remarks 6.2 Improvements and Recommendations Appendix A. INPUT GENERATION PROGRAM Appendix B. TIM E-DOMAIN ANALYSIS PROGRAM in Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 60 LIST OF FIGURES Figure 1-1 (a) Edge-coupled filter, (b) M eandering edge-coupled filter, (c) Two-layer coupled filter using less area. 2 Figure 1-2 Three types of multilayered m icro strp layouts (a) Open (b) C overed (c) Shielded. 4 Figure 2-1 A homogeneously filled cavity il.ustra.ing the effect o f an electric current J s and its contribution to the electric field at point P. 8 Figure 2-2 Side view illustrating how multilayers t f dielectric slabs are reduced to one source layer sandwiched by equivalent impedance walls. 9 Figure 2-3. (a) Typical interconnect discretizition into Method of M om ent elem ents, (b) Non-uniform piecewise sinusoidal basis function, (c) Basis function shape used for ports. 10 Figure 2-4 Voltage excitation at the input port of a discretized interconnect geometry. 13 Figure 3 -1 Flow chart illustrating the main com ponents of the computer code. 18 Figure 3-2 (a) Typical interconnect discretization, (b) Non-uniform piecew ise sinusoidal basis function described in Table 3-2. 20 Figure 3-3 Description of the output matrix for a three-layer exam ple (a) G eneral impedance matrix, (b) Impedance submatrix (c) Physical interpretation of one of the im pedance terms. 21 Figure 3-4 The conducting strips extending from the ports towards the discontinuity are extracted for the network parameter measurement. In turn the nature o f the discontinuity or the location does not affect the routine (a) Complete interconnects structure (b) Extracted current distributions. 24 Figure 3-5 Com parison of three com putational techniques: (a) serial implementation, (b) integral partitioning, and (c) frequency partitioning. 25 Figure 3-6 Illustration of data movement between processors with and w ithout affinity. 29 Figure 3-7. Performance results o f the different techniques using a sim ple geom etry with 1(H) cells. 30 iv Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Figure 4-1 (a) Top-view o f the series gap structure analyzed (£r = 9.7, w = h = 0.025". b = c = 0.25". The gap is 0.009" [ 10]). (b) M agnitude of S for the series gap discontinuity. 33 Figure 4-2. (a) Layout and discretization of the right angle bend analyzed with £,^9.8, h = 0 .6 3 5 mm. a = 3.18 mm, vv-2.545 mm, (b) Magnitude of the S j2 parameter. 34 Figure 4-3 (a) Top-view o f the 4-port cross circuit analyzed with £r = 9.8. h = 0.635 mm, a = 3.18 mm. b = I = 5.0 mm, u j = 1 mm, u'2 = 2 mm (b) Magnitude of the S ln parameters. 36 Figure 4-4 (a) Sketch o f the two resonator coupled line filter analyzed [9|. (b) Magnitude o f the S [2 parameter o f the initial filter with resonator separations o f 0.025”, (c) Magnitude o f the S 12 parameter of the optimized filter (er = 9.7, w = h = 0.025", / = 1.002", b = 0.25” , a = 0.4” , seperations are 0.005", 0.025", 0.005") 37 Figure 4-5 (a) Sketch of the two-layer stripline gap junction analyzed (b) The magnitude o f the S ^ parameter (£r = 2.2, t = 0.254 mm, vv = 1.75 mm, a = b = 5.0 m m , / = 10.1 GHz). 39 Figure 4-6 (a) The suspended stripline bandpass filter analyzed [12], (h) Magnitude o f the S [2 parameter compared with FDTD results [40] and measurements. 40 Figure 4-7 The magnitude of the current for a microstrip transmission line as a function of the distance from the sidewalls. 41 Figure 5-1 Obtaining an equivalent circuit from the transfer function. The first LC pair can be evaluated based on one o f the transfer function’s zeros, such as z l in this case. 44 Figure 5-2 Ttme-Domain analysis program flow chart. 45 Figure 5-3 (a) The m icrostrip cross structure used for the time-domain analysis (b) The schematic layout of the cross using LIBRA. 47 Figure 5-4 Comparison o f the computed S-parameter data and the polynomial transfer function obtained using this method, (a) S j2 and S 14, (b) 48 s 13Figure 5-5 Time-domain response of the cross structure for an input excitation (a) Gaussian pulse (b) Step with 5xlOy Volts/sec risetime. 49 Figure 5-6 The transmitted and the reflected waves o f the cross junction based on transmission line theory ( T = -0.5 since the lines have the same characteristic impedance). 50 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5-7 Time-domain response to a 1 kilovolt human-metal electrostatic discharge, fa) Response showing complete discharge, fb) The high-frequency com ponent of the discharge is show n to occur only at the initial rise. 51 Figure 5-8 The L-Type coupling structure. 52 Figure 5-9 The output response, S ^ . showing a gradual decline in transmission as frequency is increased. The approximated transfer function is shown to deviate at higher frequencies. 53 Figure 5-10 Time-domain voltage on main line showing the effect o f coupler loading. 54 Figure A -l The GUI drawing program used to create the input file. 59 vi Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Table 3-1 Input tile description Id Table 3-2 M om ent method element description Id Table 3-3 Program descriptions 23 Table 5-1 M atlab routines used in the com puter code 44 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT MODELLING OF MULTILAYERED INTERCONNECTS IN MICROWAVE AND HIGH SPEED CIRCUITS Firas F. Jatou Master of Applied Science 1995 Department of Electrical and Com puter Engineering University of Toronto The increasing com plexity of the functions performed by microwave and VLSI circuits has made it necessary to resort to three dimensional m ultilayer interconnects. In this thesis, a fullwave integral equation technique is applied in the frequency domain to study multilayered structures. A computational code is written describing general multilayered circuits in terms o f their electrical characteristics, such as scattering and im pedance parameters. Computer sim ulations are conducted to validate the code. Various parallel processing techniques are successfully implemented and com pared in terms of computational efficiency. A method is also presented that allows for the determ ination of an equivalent analytical system representation o f multilayered interconnects from frequency-domain data. The equivalent system can then be used for time-domain analysis of the interconnects. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AKNOWLEDGMENTS I w ish to extend my sincere gratitude to a kind supervisor. Prof. T.E. van Deventer for her invaluable guidance and support throughout the course of this work. Thanks are due to my lawyer, Susan Nessan, for her constructive criticism of this document. I would like to thank my parents and all my colleagues in the Electrom agnetics Group for their help and encouragement. This work was also supported by the Bell Canada/NSERC Chair in Electrom agnetics at the University o f Toronto. i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 INTRODUCTION 1.1 Motivation The birth o f electrical circuitry in the early tw entieth century has proved to be a significant source of technological advancement. Prior to the late 1960’s most known electrical components were categorized into three principal classes: lum ped, distributed, and w aveguide. The mid 1970’s marked the beginning o f a new modus operandi w ith the introduction o f the planar circuit. Thin conductors printed on top of dielectric slabs becam e significantly popular. Today, such planar circuit technology is widely used in all types o f applications ranging from the com m on logic circuit to the high frequency microwave and millimeter-wave integrated circuit. Recent advances in material technology and fabrication have allowed designers to include the integration o f digital functions with analog m icrow ave circuits. The benefits from this increase in integration include system size and weight reduction, enhanced performance, and reduced system cost [I]. However, the interconnects of integrated circuits limit the density of active devices and cause propagation delay, power consumption, and noise problems [2]. T h e increasing complexity o f the functions perform ed by microwave and V L SI circuits has made it necessary to resort to 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. three dimensional m ultilayer interconnects. The coupling among various layers in multilayered circuits occurs either by direct connection using via holes, o r by vertical electrom agnetic coupling. Whereas vertical coupling is viewed as a parasitic effect in fast digital networks, it has found a wide range of applications in high frequency circuits such as passive filters and directional couplers. In recent years, good results have been obtained by fabricating passive filters directly onto substrates, such as G aAs, along with other devices [31. In microwave filter design however, high coupling is difficult to obtain from edge coupled planar microstrip elem ents (Fig. 1 (a)) due to the need for im practical spacing between conductor edges. In order to obtain good performance, edge-coupled planar filters may end up being long and narrow, and the amount of circuit real estate taken up is large compared to the rest of the circuit. Depending on the (b) Figure 1-1 (a) Edge-coupled fiUcr. (b) Meandering edge-coupled fil ter. (c) Two-layer coupled filter using less area. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fabrication technique employed, an additional problem o f such circuits arises when considerable shrinkage o f the interconnect structure occurs during production, which could raise the filter’s unit cost appreciably [3]. Another technique employs the m eandering edge-coupled filter geometry in a limited package size is, as shown in Fig. 1-lb. Although this modification does reduce the length, higher modes m ight appear since the filter is housed in a more rectangular cavity. W ith the use of multilayered architecture the aspect ratio (ratio o f length to width o f substrate) can be kept small, as w ell as keeping housing narrow so undesired modes are elim inated (Fig. 1-lc). There are three basic classes o f m icrostrip circuits: open, covered, and shielded, as shown in Figure 1-2. In order to prevent radiation losses and electrom agnetic interference with the outside environm ent, most microwave integrated circuits (MIC’s) are placed w ithin a metallic packaging structure. Such housing introduces an additional conducting body which can significantly affect the behavior o f the circuit. Most o f the Computer Aided D esign (CAD) models used in the available CAD tool, LIBRA, do not adequately account for such effects [4], For this reason, accurate analysis and modelling o f multilayered circuits are im portant for the design o f any com ponent or system. 1.2 Objectives The general goal of this research is to develop and apply a computational code using a full-wave analysis technique to describe general multilayered circuits in terms of their electrom agnetic characteristics, such as scattering and im pedance parameters. These parameters can then be used to obtain accurate m odels fo r basic multilayered building blocks used to describe more com plex systems. The specific objectives of this research are • A pply a fullwave method in an efficient computer program for analyzing general shielded m ultilayered circuits. " D evelop a method to describe multilayered geometries as generalized N-port networks. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1-2. Three types of multilayered microstrip layouts (a) Open (b) Covered (c) Shielded. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Establish a technique to obtain time-domain responses o f such multilayered circuits. • Validate the method by obtaining numerical results for various well know n planar and multilayered structures. 1.3 Methodology During the past tw o decades, several numerical techniques have been developed to investigate planar interconnects for MIC applications. The selection of one technique over another is primarily driven by the desire to obtain accuracy and'or computational efficiency. Some of the techniques applied can be categorized as 1 me-<-oirain methods such as: finite difference time-domain method [!}, the time-domai i nite-e ement method [3], and the transm ission line method [5], while others fall under the frequency domain category such as: conformal-mapping technique [I], hybrid quasistatic analysis [6], and fullwave analysis using integral equation techniques [7,8]. O f the methods developed to date, fullwave techniques provide the most accurate results, at the expense of high computing cost [1,5]. Nevertheless, the accuracy demanded from the latest MIC technology [9] has made it necessary tc resort to computationally intensive methods in order to obtain more accurate CAD tools. In this thesis, a fullwave space-domain integral equation (SDIE) technique is applied to study multilayered structures. This involves a rigorous electrom agnetic description o f planar interconnect geom etries with die following assumptions: • the thickness o f the interconnects are assumed to be negligible • the ohmic loss o f the conductors as well as the dielectric losses are not taken into account The fullwave approach has been previously applied successfully to the study o f uniplanar microstrip [10] and coplanar waveguide [11] structures. For the sake of clarity, the theoretical analysis presented in this thesis is applied to the particular problem o f planar m etallic intercon nects, as shown in Figure 1-1, although the generality o f the method allows fo r the analysis o f microstrips, patches, as well as slots. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.4 Overview In Chapter 2, the theoretical approach is described, where the dyadic Green's function for the electric field in the structure is used in an integral equation formulation. The choice o f basis and testing functions resulting from the application of the method o f moments is discussed along with the formulation o f the resulting im pedance matrix. In addition, the determination o f the network parameters from such impedance matrices is also provided. C hapter 3 outlines the numerical implementation o f the theory discussed in Chapter 2. Validation of the developed code is given in Chapter 4 for various uni-planar as well as multilayered geometries. In Chapter 5, an approach is described which uses previously obtained frequency-domain results to compute time-domain responses as well as an equivalent continuous transfer function. The conclusions o f this thesis and recommendations for future work are presented in Chapter 6. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 APPROACH In this chapter, the theoretica approach is described, where the G reen’s function for the electric field in the structure is used in an integral equation formulation [12]. The choice of basis and testing functions resulting from the application o f the m om ents method is discussed along with the formulation of the resulting impedance matrix. A fter performing the matrix inversion the electric current distribution on the interconnects is obtained and is combined with conventional transmission line theory to solve for the network parameters of the structure [13]. 2.1 Formulation of the Green’s Function For a general cavity, the governing integral equation for the electric field at any point P is given by: E (*, y, z) = JJG * (x, y , z| x \ / , z ') • 7, (x , y ', z') d y 'd z ' (2- 1) S where G*is the electric field dyadic Green s function and 7*denotes the unknown electric current 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. flowing in th e m icrostrip conductors. Both y and z directions o f current are considered allowing the analysis o f a wide range o f planar structures. The electric dyadic G reen 's function o f the problem represents the electric field at point P caused by an infinitesimal electric current source at point P '. T he total field E {r) is the sum o f all the fields caused by arbitrary planar currents present in the volum e V (Fig. 2-1). The G reen’s function for tl problem is obtained by satisfying the boundary conditions o f the structure. Since the walls of the cavity are assum ed to be perfectly conducting, the tangential electric field m ust vanish at y = 0, b, z = 0. I and at .r = 0. a. The X Figure 2-1. A homogeneously filled cavity illustrating the effect of an electric current Ts and its contribution to the electric field at point P. boundary problem pertinent to multilayered interconnect problems is solved here by deriving a generalized G reen’s function that accounts for any number o f dielectric layers within a shielded structure. D ue to the existence of various dielectric layers, boundary conditions o f continuity of electric and m agnetic field tangential to the interfaces between them must be met [ 141. The remaining boundary condition to be applied is the Dirichlet condition of vanishing tangential 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. electric field on the surface of the metallic interconnects. T o obtain a general form ulation o f the problem, the G reen’s function is expressed with reference to a point source w ithin one o f the homogeneous layers, i.e., the source layer (Fig. 2-2). The layers above and below this source layer are replaced by impedance boundaries obtained through the application o f transm ission line theory [ I 6J. Impedance z Walls /’ is®* [ / Source Layer Figure 2-2. Side view illustrating how multilayers of dielectric slabs arc reduced to one source layer sandwiched by equivalent impedance walls. 2.2 Method of Moments In order to obtain the surface current distribution current isexpanded into J s o f an arbitrary interconnect layout, the a set o f orthogonal functions [15]. Each o f these expansion functions is defined over a subsection of the layout, as shown in Fig. 2-3(a). One suitable expansion series is: Nr V r) = 5 / y P f ^ 8P ^ (2' 2 a) Ni Vr>=2/* v ° q V y) ( 2 - 2 b ) where Iy and l z are unknow n amplitude coefficients, and Np, N q are the total num ber o f segments in the y and z directions respectively. The expansion fu n c tio n /is used to describe the current 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Z-dlrected bails function Y-dlractad basis function z (a) y (c) Figure 2-3. (a) Typical interconnect discretization into Method of Moment elements, (b) Non-uniform piecewise sinusoidal basis function, (c) Basis function shape used for ports. distribution in the longitudinal direction of current while g describes the current distribution in the transverse direction (Fig 2-3 b). T he subsectional sinusoidal basis function, /, and the pulse distribution function, g are as follows: sin M y - y , - t ) . y ; . / < y < yi sin M y[ /( * ) = sin ks (yi + l - y ) • y i < y < y i+ i sinAr.L 5 /i*l 0 , elsewhere 10 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (2-3 a) where ly = y t - and ks a scaling param eter which is a function o f the frequency of operation and the dielectric permitivity o f the source layer and is defined based on [16]. To satisfy the edge conditions at the ports, a modified version o f / i s used where current exists only in the region Z j\ - as show n in Fig. 2-3 (c). Figure 2-3(a) shows that the orthogonal surface currents Jy and J z do not share a common centre. The centres o f the two sets o f subsections need to be offset w ith respect to each other because a z-directed current cannot induce current in a y-directed subsection located in the same position. This modal layout has been show n to provide good results for its ability to handle the current com ponents at conductor edges and com ers [17]. The integral equation (2-1) can now be w ritten in the following form: p ' L I ^ y ) - L{f g ) = (:M ) P w here L is the integral operator, and h is the vector function o f the electric field associated with the current J s given in equation 2-2. A suitable inner product (a, b) is chosen for the problem in term s o f an integral over the conducting surface for which the distribution o f equation 2-2 is defined: (a,b) = j a • b ds (2-5) S are defined in the range o f the operator L, and T hen a set o f weighting, or testing, functions the inner product of equation 2-5 is used with each Wj to get p l a p if- g) ,wt) ■I (xy) = ig (£ ) .w,) (2-6) A G alerkin’s method is applied to this problem, the weighting functions are taken to be identical to the basis functions. In this way, the problem has been reduced to a set o f linear algebraic equations which can be w ritten in matrix notation as [ 10]: 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. M oM where Umom\ *s (2-7) t M oM vector ° f unknown y and z current amplitudes, [Vexc] is the excitation vector which is identically zero everywhere except at the position of the source, and [ZW(),vfl is the moment method im pedance matrix, which has the form: ( 2 - 8) The subm atrices represent the coupling between the y and ^-directed current elements. Since only planar currents are considered in this thesis, there are no .t-components in the moment method im pedance matrix in equation (2-8). 2.3 Determination of Network Parameters The current distribution is obtained from the adm ittance matrix by exciting the structure at the port locations. T he voltage excitation used, often called the delta gap model, is equivalent to creating an infinitesimal gap at the port cell and placing a voltage generator at this location (Fig. 2-4). A ll other elem ents o f the voltage vector are set to zero to maintain a short-circuit at the interfaces between the elements. In other words, the excitation vector may be written as: (2-9) where 8 . is the discrete delta distribution vector with the /th element being equal to I and all others are 0. Although such a m odel for the excitation mechanism is an idealization, it does provide accurate results [18]. It w as shown that the use o f a voltage gap generator instead o f a coaxial line feed excitation does not affect the phase nor the m agnitude of the reflection coefficient [ 10). Once the current distributions are obtained, the circuit can be described through several different netw ork param eters such as: 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2-4. Voltage excitation at the input port of a discretized interconnect geometry. • Input impedance • Reflection coefficient • Scattering matrix • Z, ABCD, Y, T, H matrices These network param eters depend greatly on tw o characteristics that describe the current distribution along a uniform line; namely the characteristic impedance and the propagation constant. 2.3.1 Characteristic Impedance (Zo) There exist various definitions of ZQ for m icrostrip line problems [10] and sim ilarly for multiconductor problem s [20], The method used in this research does not depend on a certain Zq definition. As a result, the characteristic impedance o f the various ports is not considered; all values presented are normalized parameters. T he normalized impedance at a certain port corresponds to the characteristics of the line at the port, and not some external ZQ. It has been shown that the norm alized 5-parameters, for exam ple, can be used to obtain 5-param eters relative to any arbitrary characteristic impedances using a linear transformation [21]. T he method allows for such a transform ation provided that the true characteristic impedances of the lines are known. This is achieved in this work by a post-processing Matlab program which converts scattering parameters to generalized scattering parameters. G eneralized scattering param eters provide the electrical scattering param eters o f a multiport system having different characteristics impedances 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. at different ports [21]. 2.3.2 Propagation Constant T h e expression for the phasor current at any coordinate z along the line is: (2-10) The values o f the phasor coefficients // and /? are determ ined by four factors: the voltage excitation at z= 0, the attenuation o f the line, the line length, and the load impedance. The com plex propagation constant y may be w ritten as ( 2 - 11) y = a + ;P where a is the attenuation factor w hich, for shielded geom etries, represents the ohmic losses of the interconnects and the dielectric losses o f the layers, causing the wave to diminish steadily. The phase constant P is a function o f the whole geometry. Using an optim ization technique based on the Levenberg-M arquardt method [22], the expression in equation (2-10) is curve-fitted to the m om ent method solution. From this expression, the reflection coefficient is easily computed, providing the normalized input im pedance [10]. For a given current distribution on a microstrip line i, the normalized input im pedance would be: 2.3.3 Impedance and Scattering Parameters A s previously outlined, follow ing the method o f m om ents solution, a matrix equation is obtained as: (2-13) [ V l " [ZAfoAf] [7AfoAf) w here Z m 0m is the moment m ethod matrix, [Vejtc] and are l^ e excitation vectors and the 14 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. current distribution respectively. The size of the matrix Z m 0m depends on the num ber of cells used in the discretization o f the interconnect layout. F or example, the layout in Fig. 2-3, would require 16 z-directed basis functions and lO y-directed basis functions to adequately describe the layout with the grid discretization chosen. T he discretization procedure is automated for uniformly discretized layouts by using the com puter program ‘M AKEINPUT’ described in Appendix A. From equation (2-13), the network im pedance parameters are obtained as follows: (2-14) where Z imp is the im pedance matrix o f size NxN, with N representing the num ber o f ports of the network. In order to extract Zjmp from equation (2-13), the network has to be excited with N independent voltage excitations, as described in [23], where the nth excitation is given by: T (2-15) where V, is the ith elem ent o f the excitation vector v exc, and n For each excitation, N associated input im pedances are generated. As a result, N 2 input impedance terms arc created. The transformation from the input impedance vector to generate the impedance matrix elem ents is as follows: (2-16) where [A] is the transform ation matrix outlined in [23]. O nce the normalized im pedance matrix is found the scattering matrix [S] can be evaluated by: (2-17) 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the particular case o f two port networks, the choice of excitations given in equation 2-15 will not provide independent sets as shown: vexc(1) = f I -U (2-18 a) Vexc(2)= [-l 1} (2 -18 b) In this particular situation, even and odd mode excitations are used represented by [ I 1] and [1 -1), for the even and odd modes respectively. The input im pedance, Z in, at each port is found for both the even and the odd excitations. Together they are used to find the two-port impedance matrix: , yodd ye v e n "11 /n ~ yeven " z 12 = in (2-19 a) 2 _ yodd " in ^2 ( 2 ' , 9 b ) T he scattering parameters are found from equation (2-17). Other network parameters are also obtained through the use of linear transformations o f the im pedance matrix [24]. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 NUMERICAL IMPLEMENTATION In this chapter the com putational implementation o f the method discussed in chapter 2 is presented. A n explanation is provided o f how the geometry is described by means of the input file to the program and how the algorithm efficiently produces the im pedance matrix. The electrical com putation of the characteristics is explained in detail. Also, various parallel processing im plem entations [251 developed and compared in terms o f computation and storage efficiency. Convergence of the method is also addressed briefly. 3.1 Computer Program The program is a user-oriented code based upon the m ethod of m om ents, for treating perfectly conducting thin dielectric planar interconnects. The general flow o f the com puter program written in FORTRAN can be subdivided into the steps shown in Fig. 3-1. The program employs a non-uniform discretization o f the interconnects in order to accurately represent the la y o u t The m atrix is used to compute the electrical quantities of interest such as current distribution, input im pedance, or scattering parameters. 3.1.1 Input File The param eters o f the input file that describe the multilayered geom etry to be analyzed are n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. READ CAVITY MODEL &DEFINE STRIP GEOMETRY | / COMPUTE IMPEDANCE^ SUBMATRICES FOR I EACH LAYER / _______________ I _______________ PLACE SUBMATRICES IN GENERAL IMPEDANCE MATRIX r_____ INVERT MATRIX I 1----------- I 1_____________ ^ ^ ( COMPUTE CURRENT DISTRIBUTION V FOR GIVEN EXCITATIONS ( CALCULATE NETWORK V. PARAMETERS Figure 3-1. Flow chart illustrating the main components of the computer code listed in Table 3-1. Additional parameters in the input file include the frequency o f operation as well as the level o f accuracy of the impedance matrix elements which is governed in part by the summation limits m stop and nstop, which are further explained in section 3.1.2. The metallic interconnects are subdivided in rectangular cells (see Fig. 3-2) which are described by the parameters in Table 3-2. To input the layout definition in a user friendly manner, an interface program w as written to convert bitmap images o f interconnect layouts into the format o f Table 3-2 (see also A ppendix A). Since bitmap images do not provide length o r width o f the mapped biLs, this program can only be used to create uniformly discretized elemenLs. One may use this tool to create a ‘skeleton’ layout and manually add a few non-uniform elements at edges for optimizing the response o f the interconnects. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3-1: Input file description Physical variable Description a Height b Width I Length Dielectric nl N um ber of layers layers si Source layer number C avity dimensions Relative permittivity of layer n Metallic Interconnects Operational Frequency Accuracy It Loss tangent o f layer n nU Num ber of interconnect layers ne N um ber of elements ###### Elem ent description (Table 3-2) start Starting frequency nf N um ber o f frequency points to analyze increment Frequency increment mstop, nstop Summation limits Table 3-2: Moment method elem ent description Parameter Description (y,z) Coordinates of the test function front The length o f the cell directly in front o f the above point back T he length o f the cell directly behind the above point width The width o f both cells port ‘ 1’ if the cell is a port. ‘O’ if it is not. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) Figure 3-2. (a) Typical interconnect discretization, (b) Non-uniform piecewise sinusoidal basis function described in Table 3-2 3.1.2 Impedance Matrix Formulation T he impedance matrix is com puted based on the geom etry o f the structure. This entails the com putation o f the impedance subm atrices corresponding to each layer at the operating frequency (Fig. 3-3). A s illustrated in Fig. 3-3, the impedance submatrix Z*/y (/, j = y, /.; k, I = 1,2 w here N is the number o f cells) represents blocks o f the impedance matrix given by: m sto p n s to p tfil - ^ m ^ (m , n ) f - k ( m ) T y k ( n ) ^i(n) n 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N, LAYER (1.1) LAYER <U> LAYER (1.3) LAYER (2.1) LAYER (2.2) LAYER (2.3) z “12 LAYER (3.1) LAYER (3.2) LAYER (3.3) S, _ _ 12 VV 7? 12 7P 12 (a) Figure 3-3. Description of the output matrix for a three-layer example (a) General impedance matrix, (b) Impedance submatrix (c) Physical interpretation of one of the impedance terms. where T and B represent the testing and basis function integrals over the discretized cell elem ents k and /, respectively. Each matrix elem ent requires the com putation o f the two basis and testing function integrals to account for the coupling between any two elem ents. A Galerkin procedure is used in this algorithm where the testing and the basis functions are the same with: y^i -v/ y y f) = | COSky y ' f ( y ' ) d y + J COSky y ' f ( y ' ) d y ' (3-2 a) ■v r - 1 T fy ) = J s in kz z ’g ( z ) d z ' (3-2 b) <7-1 w here the basis fu n ctio n s,/an d g, shown in equation (3-2) are defined in chapter 2 and kv k y are the w avenum bers in the z and y directions respectively, which are dependent on the cavity size [16] as follows: 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Similarly, the basis functions in the case for a --directed current source: (3-4 a) (3-4 b) Upon exam ining the above equations, it can be noticed that the integrals are only dependent on one variable, either the z or the y coordinate. This means that interconnect elements having a common coordinate (aligned in the z or y direction) as well as having the sam e length (integration limits) w ould produce the same integral. In order to reduce computation time, an initial search is conducted in the program to find the cells containing similar integrals to eliminate any redundancy in computation. In many cases, this initial step reduces the computation time of the integrals by approxim ately 50-75%. The output file from the program contains the impedance matrix as w ell as the z and y coordinates o f each cell which are necessary to compute the network parameters. This is further explained in the next section. 3.1.3 Network Parameters In chapter 2, it was shown how the im pedance matrix can be used to obtain the current distribution on the conducting strips within the shielded geometry. From the knowledge of the current distribution, the various network param eters are obtained using the steps shown in section 2.3. The m ethod is implemented using a function, Networkp, written for the MATLAB 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. environment, which requires the input parameters shown in Table 3-3 . The portion o f the current Table 3-3: Program Descriptions Program Description 1 A utom atic (using KAP) Automatic parallelization software 2 Sum m ation partitioning Parallelizing the loops over m and n 3 Individual elements Integral evaluation and m atrix tiling 4 Individual elements with locality Same as above with layout consideration 5 Frequency partitioning One frequency per processor at a tim e 6 Frequency partitioning with inte gral sharing The M oM integrals are evaluated once and then shared by all others Program No. distribution extending from the ports, whic is represented by the variable strip, is extracted from the admittance m atrix, y. This current constitutes only a portion of the total currents obtained. This allows the method to be versatile in obtaining the network parameters for arbitrary interconnect structures without relying on the shape, location of the discontinuity, or the location o f the ports (Fig. 3-4). In other words, the network parameters can be computed from the current distribution away from the discontinuity. As a result, only the current along the strips extending from the ports is needed. 3.2 Parallel Implementation Recently, parallel processing techniques have been successfully applied to a w ide range of electromagnetic problem s, ranging from the study o f transients in pov/er system s [19] to their implementation in w idely available codes such as the Numerical Electrom agnetics C ode (NEC) [26]. This section discusses the effectiveness o f several parallelization techniques as applied to multilayered structure problems, and compares their performance from a hardw are point o f view. The formulation o f the problem is presented along w ith its numerical im plem entation. Various techniques are successfully implemented and com pared in terms o f efficiency. The m ost efficient 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. current distribution Interconnect Discontinuities port mici Dstrip extraction port n (a) (b) Figure 3-4 The conducting strips extending from the ports towards the discontinuity are extracted for the network parameter measurement. In turn the nature of the discontinuity or the location docs not affect the routine: (a) Generic interconnects structure (b) Extracted current distributions at the ports. techniques for shared-merr.ory m ultiprocessors are discussed in detail. T h e parallel processing techniques applied in this research were developed to run on the Kendall Square Research (KSR1) shared memory machine available at the University of Toronto. The KSR1 is a highly parallel com puter that can be configured with up to 1,088 processors [271. These processors are superscalar RISC devices with separate integer and floating point processing units. The processors have the ability to perform separate add and multiply instructions in each clock cycle, giving the processors a maximum rating o f 40 M FLOPS. The available m achine is equipped with 32 processors, providing a maximum rating o f 1.28 GFLOPS. The total m em ory of the system is 1 Gbyte. F o r com plex structures containing a large number of cells, the parallel implementation overhead becomes very small relative to the computation time o f the large impedance matrix [ 19, 26]. To allow for a true com parison o f the overhead tim e caused by the various parallel im plem entations, all perform ance tests over domain partitioning and load balancing were run on a 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. simple structure containing 100 cells in total. 3.2.1 Domain Partitioning As one can deduce from the serial implementation o f the im pedance formulation outlined in Figure 3-5(a), partitioning of the w orkload can be done effectively over three sets of variables: the sum m ation parameters m and n, the individual discretized cell elem ents that make up the interconnects, and the operating frequency. To take advantage o f these different partitioning schem es, several FORTRAN program s were written and are listed in Table 3-3. Input Data Frequency loop [ Input Data Input Data Frequency loop MoM integrals Summation loop 1 MoM integrals Summation loop 1 MoM integrals -J l Summation loop 2 i i • 45 i i Frequency loop I I I I I I Summation loop 1 1 H Summation loop 2 i l T I 1 Il Summation loop 2 MoM integrals MoM integrals \ \ : .T 7 (a) \ ' (_L \ \ I * \ V f 1 M I \ v. A s. s / (c) (b) Figure 3*5. Comparison of three computational techniques: (a) serial implementation, (b) integral partitioning, and (c) frequency partitioning. First, a program was created by the KSR automatic parallelization pre-processor, called K AP [28], fo r comparison purposes. T he second program is parallelized with respect to the m ode sum m ations m and n. This version attem pts to provide good overall efficiency by achieving better 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. load balancing, and w ill be discussed in further detail in the next section. In the third program, the parallelization is done with respect to the discretized geometry (i.e. over the individual cell elements) and is illustrated in Figure 3-5(b). The m ain drawback of this method is that processes are created through every summation iteration which effectively reduces the efficiency by increasing the overhead cost of process creation. Program 4 is an enhanced version of program 3. Here memory locality is taken into consideration by minimizing data movement. Alternatively, when parallelization is performed over frequency, no data dependence is encountered in the matrix formulation. This ensures that invalidation latency [29] does not occur. Such frequency partitioning is im plem ented in programs 5 and 6. T his latter program first com putes the moment integrals before initiating the frequency dependent calculations. Taking advantage of the shared memory architecture, the integrals are placed in global memory so that all processes can access them independently, as illustrated in Figure 3-5(c). 3.2.2 Load Balancing The overall efficiency o f all the techniques described above is highly governed by the balance of the workload allocated to each processor. In program ., the summation limits m stop and nstop, not only represent the extent of the partitioning dom ain but are also large relative to the number of available processors. In this end, the w ork can be distributed evenly across the different processors, w hich ensures an adequate load balance. To achieve a favorable load balance in program s 3 and 4, the memory organization must be taken into consideration. From the viewpoint of a single processor on a KSR/Serics computer, the memory is organized in a hierarchical fashion w here data that resides in the low est level of the hierarchy, although limited, can be accessed faster by the processor. For m ost o f the structures analyzed, the data representing the number o f cells can be maintained in the local cache, giving the program a good locality o f reference. If the complexity of the geom etry requires a larger number of discretization cells, some data will reside in the next level o f m em ory which may cause higher subcache misses. The latency generated would then occur evenly across all processors 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. while still maintaining an optimum load balance. Furtherm ore, because the com putation required for each impedance elem ent is independent of the geographic location of the cells, load balancing is ensured as long as the elements are partitioned evenly across all processors. In most microwave applications, characterization o f the M IC ’s is only required at a num ber of frequencies com parable to the number of available processors. For such problems, load balancing in the frequency partitioning case (programs 5 and 6 ) m ight not be optimum. Nevertheless, this imbalance is reduced as larger number o f frequency points are analyzed, for exam ple, in the case o f transient analysis. 3.2.3 Performance Evaluation In all cases considered, parallel implementation provides better speedup for m ore complex structures. This is m ainly due to the fact that having m ore cells means that a larger portion of program time is spent on true computation. O verhead tim e owing to process creation becomes negligible relative to the impedance matrix computation. The parallelization techniques of programs 1 and 2 are not effective in providing any speedup to the overall com putation. In the case of program 1, performance monitoring program s [27] available on the K SR m ultiprocessor show that m ost o f the processors remain idle 87% of the time. The reason fo r the poor performance o f this program is that the autom atic parallelism preprocessor (KAP) only transforms loops with com pletely independent data, w hich occur solely at the input and output stages o f the program. On the other hand, the poor perform ance o f the second program, is caused by invalidation latency as w ell as memory contention, as various processors attempt to update the same impedance elem ents for different values o f m and n. Loop parallelization o f integral computation is achieved by programs 3 and 4 through the use o f tiling, the process w hereby the execution o f a single do loop is transformed into parallel execution o f multiple tiles, o r segments, of that oop [28J. M oderate speedup is realized with this method because the integral computation is only a portion o f the total com putation. Therefore, 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. much o f the original program is maintained in its serial form (Fig. 3-5). Another reason for the m odest performance of program 3 was observed through the use o f available performance m onitors. Computation o f some o f the impedance elements w as performed on a different processor during each summation iteration. This problem was solved in program 4 by the creation of affinity regions [31], These affinity regions force the same part o f the impedance matrix to be executed on the same processor for every loop iteration, thus im proving cache performance by m inim izing data movement. For the purpose o f illustration. Figure 3-6 shows the data movement between processors throughout the mode summation process. A s an example, the data being worked on by processor 1 during the first iteration will be subsequently updated by processor 9, unless data affinity is used. In addition, a KSR compiler directive using a wavefront strategy [311 was used in program 4 in order to avoid deadlock when all processors try to access data belonging to one particular cell at the sam e time. Results in Figure 3-7 show that program 4 outperforms program 3 in term s of speed and scalability. O f the techniques employed, frequency partitioning implemented in programs 5 and 6 provides the best overall speedup. This is obviously due to the fact that the analysis is performed in the frequency-dom ain, allowing the computation to be geared tow ards computing one frequency at a tim e; thus providing a natural decoupling in terms of frequency. This method provided near-linear response even for small structures as shown in Fig. 3-7. The figure also shows that the rate o f speedup seems to decrease as the num ber o f processors is increased. This tendency is due to the poor load balance caused by the increasing number of processors relative to the num ber of frequency points used (which w as 20 points in this case). In program 6, the same frequency partitioning technique is applied w ith the modification described in section 3.2.1. This provides better than linear performance as shown in Fig. 3-7. Som e performance degradation is encountered w hen larger numbers o f processors arc used due to the same reason ascribed above for program 5. In term s o f portability of the code, all parallel versions o f the method were implemented through the u se of KSR-specific com piler directives. Therefore the programs cannot be directly 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i S um m ation lo o p c o u n te r = 1 Impedance Matrix | j T With y data affinity / \ Without \ \ a' \ S um m ation loop c o u n te r ^ 2 Processors 1 2 3 4 5 6 7 8 S ... Figure 3-6. Illustration of data irovement between processors with and without affinity. compiled on other multiprocessor machines. T he techniques used in these program s however, are quite portable and can be adapted to other shared-m em ory architectures. Timing resu io may difler slightly depending on the network topology and the memory organization used. 3.3 Conclusions The feasibility and the efficiency o f parallel processing techniques for the fullwave frequency-domain problem have been demonstrated. Comparison of the various techniques shows that static frequency partitioning o f the work provided the most noticeable speed im provem ent in all tested cases. Tests also illustrated that the larger the interconnect structure being analyzed, the greater the need for parallel computing. Speedup was found to approach linearity for more complex structures. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — ■- Linear 14 q Integral partitioning X With affinity X Frequency partitioning + With integral sharing Speedup 2 4 6 10 8 Number o f processors 12 14 16 Figure 3-7. Performance results ol the different techniques using a simple geometry w,'hl(X) cells. Discretization of the interconnect layout was made simple w ith the use of the interface program (bm2in). However, care m ust be taken in choosing the level o f discretization. Convergence studies o f [18, 30] were used as a guideline in discretizing the structures, which range from 12 to 100 cells per wavelength. In general, the length o f the individual cells did not exceed X J2 0 , where Xq is the propagation wavelength at the centre frequency of the operating range. F or example, the wavelength associated with 15 C H z is used for frequency analysis in the range o f 1 - 30 GHz. The summation limits were kept constant at 800 for all structures analyzed. T hese figures of m erit produced adequate results. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 VALIDATION In this chapter, numerical results obtained using the present code are compared with various published data. The structures analyzed are chosen to highlight the m ethod’s abilities in handling various configurations o f interconnects that may be encountered. The comparisons are m ade to available CAD models where published measured data is not available. The CAD models used are based mostly on closed form solutions and look-up tables obtained from experimental data [4], The geom etries analyzed fall under two sub-categories: uni-planar and multilayered discontinuities. 4.1 Results of Uni-Planar Discontinuities U ni-planar discontuinuities are considered here because they constitute a large portion of modern-day M IC’s. In addition, the uni-planar geometries w hich form a sub-class of shielded multilayered structures are studied to allow further com parison with other full-wave techniques [ 10]. 4.1.1 Series Gap 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4-1 show s a top view o f the series gap discontinuity analyzed in this work. This represents one o f the simplest planar discontinuities. Results of the transmission coefficient S 12 compare more closely with measurements [ 10] than with the commercial CAD program LIBRA [4], T his discrepancy is m ost likely due to the fact that the available CAD model only takes the effect o f the side walls into consideration aw ay from the discontinuity. The gap analyzed was discretized using 200 uniform overlapping elem ents o f length 1.8 mil each. This allows for the analysis o f several gap widths by removing elements from the impedance matrix which corresponds to increasing the gap width by 0.9 mil increments. This first step in the validation of the method dem onstrates the ability to analyze two-port planar structures with uniformly discretized interconnects. 4.1.2 Right Angle Bend with Non-Uniform Discretization T he structure depicted in Fig. 4-2 is analyzed to extend the verification to non-uniformly discretized structures. In addition, the right angle bend provides a further numerical test o f the developed code: the transverse direction o f current is tested by having the second port perpendicular w ith the first port. The structure is shown in Fig. 4-2 along with the comparison of S j2 to that obtained from LIBRA, which agree reasonably well with the accepted model. The ripples shown in the response obtained from LIBRA are believed to be due to a mismatch between the port reference impedance of 50 Q and the true impedance o f the line, which is a function of the line width, the substrate height, and the proximity o f the shielded structure to the microstrip line. W ith the availability of non-uniform discretization, the interconnect subsections can be further refined to monitor the current density as it flows through the bend. The cell width used in this case was 2.54 mm and the length was 0.254 mm. With the application o f fine discretization at the bend location, an approxim ation of a mitered bend, which is common in today’s circuits, is achievable. 4.1.3 Four-port Cross A s discussed in Section 3.1, structures w ith more than two ports are handled differently using 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. port port 2 (a) - 10- -1 5 - IsJ' 20 (dB) -2 5 - - LIBRA x Measurements [I2 | o Present method 20 Frequency (GHz) (b) Figure 4-1. (a) Top-vicw of the series gap structure analyzed. (er =9.7, W=h = 0.025”, b = 0.25", g= 0.009” [10)]. (b) Magnitude of S12 for the series gap discontinuity. I■ 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 | y-etemen! r~ *~'* z-etem > ent f.-:i 1 port 2 (a) -2 -12 -14 S 12 (LIBRA) S 12 (present method) S „ (LIBRA) Sn (present method) -16 -18 -2 0 20 35 40 30 Frequency (GHz) 45 50 (b) Figure 4*2. (a) Layout and discretization of the right angle bend analyzed with e^=9.8, h=0.635 mm, a=3.18 mm, vv=2.545 mm, (b) Magnitude of the S12 parameter. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 the method outlined in 3.1.3. To test the N-port parameters extraction method, a four-port cross junction was analyzed (Fig 4-3). A s shown, the results obtained from this work agree w ith those o f the available CAD package LIBRA. The small deviation is most likely due to the inability of the available CAD tool to truly m odel the interaction or coupling among the four branches o f the cross. In LIBRA, the complete circuit is modelled by cascading the electrical representation o f each sub-component, i.e. the microstrip lines, and the cross junction. 4.1.4 Coupled Line Filter T he final planar circuit presented here is a two resonator coupled line filter described by Fig. 4-4. The response of this edge-coupled filter is compared with LIBRA and with measurements for validation [10]. First a simple approximation of the filter w as simulated where the resonator inter-spacing was kept constant. A comparison of the non-optimal filter is shown is Fig. 4-4(b) to the analysis made by LIBRA. Fig. 4-4(c) shows the comparison o f the optimized filter prediction to measured results o f the two resonator filter. Close inspection o f the computed response shows that there exists a sudden jum p in the S parameters which occurs at approximately 18 GHz. This may be due to the fact that the discretization used was not fine enough relative to the guided wavelength at the high frequency region. The average subsection length used was one tenth that o f the wavelength at the centre frequency o f 10 GHz. A possible improvement of the results may be achieved by varying the discretization according to the w avelength o f operation. Nevertheless, the approximation achieved here already provides good results. To verify that the code is correctly handling the existence o f additional dielectric substrates, the substrate for this structure was divided into two equally thick substrates having the same relative permittivity and the sam e total substrate height. The code was executed for several frequency points. The S-paramcters produced w ere exactly the same as those o f the single substrate case. 4.2 Results of Multilayered Discontinuities 4.2.1 Stripline Gap Junction A n extension of the planar series gap discussed in section 4.1.1 is the two-layer overlapping 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. port 2 port 4 (a) 0.7 — 0.65 UBRA x • o Present method 0.6 |S| 0.55 0.45 0.4 0.35 0.3 Frequency (GHz) (b) Figure 4-3. (a) Top-view of the 4-port cross circuit analyzed with e^=9.8, /t=30 mil, «=75 mil, b=t=200 mil, wt=25 mil, W2=50 mil (b) Magnitude of the S |Bparameters. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) •10 •20 -60 •70 LIBRA Present method -60 •00 6 16 10 Frequency (GHz) (b) -to •20 x Measurement - Present method -«0 -7 0 10 12 Frequency (GHz) (C ) Figure 4-4. (a) Sketch of the two resonator coupled line filter analyzed [9] (b) Magnitude of the S 1 2 parameter of the initial filter with j j = s2 = s$ = 0.025” (c) Magnitude of the SI2 parameter of the optimized filter.(er =9.7, w = h = 0.025” , / = 1.002”, b = 0.25”, a = 0.4”, J, = j 3 = 0.005”, J2 = 0.025”) 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transm ission line ends depicted in Fig. 4-6. This simple structure is very important in the design o f m ultilayered microwave circuit because it offers increased coupling relative to end o r side coupled planar lines, and may be used as a basic building block for more complicated circuits [33]. Figure 4-6(b) shows the transm ission scattering param eter S 12 as a function of overlapping distances d at an operating frequency o f 10 GHz. The results are compared with a method based on a combination o f spectral dom ain immittance matrix approach and standard CAD m ethods [33]. Although some slight deviation is shown in the transmission region the overall response resem bles very much that obtained by the above mentioned method. 4.2.2 Two-layer Resonator Band-Pass Filter A s an application o f this m ethod for the analysis o f m ultilayered circuits, the suspended stripline filter shown in Fig. 4-7 is simulated and compared to m easured data obtained from [351. W hile, in the past, such circuits could only be modelled by cascading uni-planar microstrip param eters with overlap param eters described in section 4.2.1, the present method allow s for consideration of the com plete circuit as a single unit w ithin a multilayered geometry, i.e. according fo r the total electrom agnetic interact’on occurring within the filter. The results shown in F ig 4-7(b) agree well, except for the slight frequency shift at roll-off beyond 10.5 GHz. T he shift could be due to the fact that the fabricated circuit was subjected to slight adjustments for tuning purposes w hich are not taken into account In this situation. In addition, it has been shown that the grid size chosen to discretize the filter could slightly shift the frequency response [ 18]. A nother set of tests was conducted with this type o f geom etry to study the effects o f the packaging, and in particular the side walls, on the performance o f shielded circuits. This is especially important for suspended substrate structures where the electromagnetic interaction with side walls can affect the circuit’s performance as much as the cover shields above and below. F igure 4-8 shows the am plitude o f the current on a uniform stripline inside the shielded geom etry used for the filter. The current is created by a voltage excitation at one end of the line. Packaging analysis results of this type are useful for the designer in determ ining the placement o f side walls 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 X Menzcl Present Method 0 0 0 0 0 0 0 0 Overlap d (mm) (b) Figure 4-5. (a) Sketch of the two-layer stripline gap junction analyzed (b) The magni tude of the Sp parameter (er = 2.2, t = 0.254 mm, w = 1.75 mm, a = b = 5.0 m m ,/= 10.1 GHz). 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — Measured FDTD x Present Method -5 -10 -1 5 IX -3 5 -4 0 -4 5 -5 0 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 Frequency (GHz) (b) Figure 4-6. (a) The suspended stripline bandpass filter analyzed [12]. (b) Magnitude of the Si2 parameter compared with FDTD results [42] and mea surements. 40 R e p ro du ced with permission o f the copyright owner. Further reproduction prohibited without permission. without affecting circuit performance. Several CA D packages which handle suspended substrates do not take the side walls into consideration. They are simply added later for shielding and packaging purposes. D ata in Fig. 4-8 shows that placing the side walls a minimum o f 1.5 cm away from the stripline would maintain the circuit p*'rform ance estim ated by available CAD packages. 0.02 0.018 - m 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0 .0 02 0.5 1.0 i.5 2.0 2.5 3.0 Distance to the walls (cm) Figure 4-7. The magnitude of the current for a microstrip transmission line as a function of the distance from the sidewalls. 4.3 Summary of Numerical Results In this chapter, several structures were analyzed to provide such parameters as effective dielectric constant, S-param eters, guided wavelengths, and current distributions. Com parison was made to other fullwave analysis results and available C A D packages. The deviations encountered suggest that further study is required to establish a discretization scheme that w ould ensure optimum results consistently. Comparison to geom etries with more than two layers o f interconnects is required to fully validate the com puter program. The lack o f published measurements o f these circuits hinders such analysis. However, the method is show n to be an effective tool in the analysis o f multilayered circuits. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 TIME-DOMAIN ANALYSIS Present day h:gn speed digital circuit design relies heavily on time-domain analysis o f the signals along the circuit interconnects to study such effects as propagation delay and cross talk. D iscontinuities along the interconnect paths can introduce signal delay as well as reflections that not only affect the response along the path but can also affect other areas of the system. The latter consequence is one form of electrom agnetic interference (EM I). In the previous chapters, the application and the validation o f a fullwave method was presented which allows the user to obtain the desired data at discrete frequency points. In the present chapter, a method is presented that allow s for the determination o f an analytic representation o f the interconnects in the form of a continuous frequency domain transfer function. The state-space [36] representation of the transfer function is then used to perform the required time-domain analysis. 5.1 The Method Tim e-dom ain analysis based on frequency-domain characteristics is usually performed [37] by transform ing the input signal V(t) into the discrete frequency-domain signal V/n(CO) via the Fourier transform : 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vin (co) = j V ( t ) e jm d t (5-1) —oa Vin((H) along with the discrete transfer function G (CO), obtained from the frequency-domain analysis, are then used to produce the output Vouf(CO) as follows: Vou i« °) = ^ ( ( 0 ) 0 ( 0 ) ) (5-2) The output voltage is then usually transformed back to the time-domain by using the inverse Fourier transform. In this thesis, a different method is used where the system frequency response data, C (CO), is fitted to a continuous line: r transfer function,G(s), using a least squares optimization method [22]. Once the transfer function is obtained, a built-in MATLAB routine, LSIM, is used to provide the time response of the polynomial function to any arbitrary input. LSIM converts the num erator and denom inator o f the transfer function to a state-space representation [22] o f the system A B U( t ) C D Ly (0. G(s) (5-3) where x represents the states o f the system, u(t) the input, y(t) the output. The state-space system is then simulated for a discrete input u to provide the desired output y. This method provides several advantages. First, the transfer function obtained from this procedure can be easily represented by lumped elements [32]. Hence an equivalent circuit representation is produced in addition to the time-domain output (Fig. 5-1). Second, since the analog transfer funrtion obtained is defined through the entire region o f interest, problem s o f matching the Fourier transformed input signal points with the original discrete transfer points are avoided. Third, once the analog transfer function is obtained, any arbitrary input signal can be subjected to the system without resorting to Fourier transformation for every input. Finally, t h : transfer 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. function can be correlated to the physical dimensions o f the analyzed structure to find an equivalent model representation. This can be done by altering the physical dimensions and m onitoring the change in transfer function parameters. G(c)_ A(s?-#Z1XS2- z 2 ) ( s 2- z 3 )( s 2- z 4) / s 2- p1)(s2- p2)(s?-p3) Output Input Figure 5-1. Obtaining an equivalent circuit from the transfer function. 'ITie lirst LC pair can be evaluated based on one of the transfer function’s zeros, such as zl in this case. 5.2 The Computer Program T he computer code, written for the MATLAB environm ent, acts mainly as a front- ‘nd to various built-in functions within the commercial MATLAB p ickage. The functions used am show n in Table 5-1. Table 5-1: Matlab routines used in the com puter code. -............ Function Description INVFREQS Analog filter leastsquare fit to frequency response data LSIM Simulation of continuous-time linear systems to arbitrary inputs ROOTS Computes the roots o f the polynomial A s shown in the flowchart of Fig. 5-2, the program uses an iterative technique to find the orders for the num erator and the denom inator for the transfer function to best fit the computed data. The 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. variance in the transfer function elem ents is used as a measure o f how w ell the fitted functions approximates the actual data. Since the function is created based on discrete data provided, it is only valid for the frequency range o f the input. In addition, a larger num ber of frequency points used in the input would produce a m ore accurate interpolation o f the true transfer function. The input requirem ents for the program are further explained in Appendix B. Input frequency data & initial guess for num. den INVFREQS (MATLAB) Change num. den NO Good ^ approximation? YES RCOTS (MATLAB) signal LSIM (MATLAB) Poles &zcros output Time-domain output Figure 5-2. Time-Domain analysis program flow chart 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 3 Simulation of a Linear System 5.3.1 The Cross Junction In order to validate the method, a simple geometry was chosen for the analysis. The planar microstrip cross w as designed to be symmetric in order to test the method for reciprocity (Fig. 5-3). For exam ple, die scattering parameter describing the transmission from port 1 to port 2 must be identical to that o f port 1 to port 4. Similarly, the following conditions m ust hold: S 11= S 22= S 33= S 44 Sl2=Sl4=S23=S34 (5-4) S 13=S 24 The linear system m ust satisfy reciprocity. Therefore the reciprocal o f the above conditions must also be equal to each other. The cross was analyzed and validated for 10 frequency points from 100 M Hz to 12 GHz with a commercially available package which handles planar microstrip cross junctions. Additional frequency points were obtained using this package in order to save time as well as to show that the tim e-dom ain analysis method can be used with any frequency-domain results (Fig. 5-4). The program was executed for three scattering param eters, namely S 12, S 13, and S 14. Third order polynom ials fit the frequency-domain data very well as shown in Fig.5-4. F or example, the polynomial fitting the S 14 data is given by: „ , , S 14(s) 14 - 0.3985s3 + 75.97s2 - 8796.3s + 420230 = ------- =---------------5---------------------------------s 3 + 158.07s2 + 19792s + 845120 (5' 5) The associated roots o f the numerator and the denom inator of the polynomial are then obtained by using the ROOTS function to produce: , S t j (s) 14 0 2 - 132320) 0 - 7 9 . 7 ) ^ (s 2 - 13873) ( s + 60.92) (5 6) The transfer function above can then be used to find an equivalent LC representation. The LC network representation lies beyond the scope of this thesis but is a relatively straightforward 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) USCB-MSUBI U C G V C < t« U C 0 /T port u lin pr pflf* •! tL i p o t «»3 W -50 L-100 ur;uB-M5uBi - msubi UCOVCR-UCOVt wsub U CO vCP-M CO V MWALl-*#Al. « DATA U SU B vsuei C H -tO I USUB-USUB1 | U C G V C R ttU C O vt I UWAll-WWAl^ -2 0 0 BMQ-1 » C H-< ? SOr t •4 (b) Figure 5-3. (a) The microstrip cross structure used for the time-domain analysis (b) The schematic layout of the cross using LIBRA. extension o f the above transfer functions [32]. Note that there are no resistors in the representation since the analysis w as performed assuming perfect conductors, and lossless dielectrics. Fig. 5-5 shows the response o f a gaussian pulse injected at port 1 using the above m ethod. This type of pulse is used often in time-domain analysis tools, such as finite difference time-domain (FDTD) methods because it provides simple transform ation to the frequency-dom ain. As discussed in section 5.1, this method offers the advantage o f allowing the sim ulation o f any arbitrary signal with very little additional work. Fig 5-5(b) shows the response o f the sam e cross structure w hen subjected to a 200 psec rise-time step. The response at the three ports approached 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 .1 - -5 0 -1 0 0 - 6 .3 - 6 .4 - 6 .5 - -2 5 0 6.6 - 6 .7 -3 0 0 ,s 10 ' 10 o S-param eter data - Transfer function 10 10,o Frequency (Hz) 10 Frequency (Hz) 10 (a) - 3 .5 -5 0 -1 0 0 at CL -5 -2 5 0 -3 0 0 - 5 .5 -3 5 0 — 6 .9 10,10 10.9 Frequency (Hz) 1010 10 ' Frequency (Hz) (b) Figure 5-4. Comparison of the computed S-paramcter data and the polynomial transfer function obtained using ibis method, (a) S12and Si4. (b) S|3. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P o rt 1 (input) P o rt2 (output) 0.8 P o rt3 (output) 0.6 0.2 0.2 - 0.05 0.1 0.1 5 0.2 0.25 0.3 Tim e (nsec) 0.35 0.4 0.45 0.5 (a) P o rt 1 (input) Port2 (output) 0.8 Port3 (output) 0.6 o> 0.4 0.2 0 0.1 0.2 0 .3 0.4 0.5 0.6 0 .7 0.8 0 .9 1 Time (nsec) (b) Figure 5-5. Time-domain response of the cross structure for an input excitation (a) Gaussian pulse (b) Step with 5xl09 Voits/sec risetime. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. half o f the incident signal which agrees w ith transmission line theory approxim ations. Incident _n_ i _ r Reflected Transmitted Figure 5-6. The transmitted and the reflected waves of the cross junction based on transmission line theory, r = -0.5 since the lines have the same characteristics impedance. Fig 5-7 show s the response to a typical electrostatic discharge o f a human body charged to I kilovolt [34]. This figure demonstrates the versatility o f the m ethod in handling not only operational signals but also interference signals on integrated circuits. Fig. 5-7(a) shows the difference betw een the response at pert 2 and port 3 to be substantial, regardless of the fact that the two ports should be at the same potential since they are connected w ith perfect conductors that are equidistant from the source. Figure 5 -7(b) shows the high frequency com ponents of the same electrostatic discharge of Fig. 5-7(a). All frequency components below 90() MHz have been removed to explain the existence o f a potential difference only at the beginning o f the discharge. The S-param eters, as shown in Fig. 5-4, start to deviate from the quasistatic value of 0.5 at approxim ately 900 MHz. Therefore, only the high frequency com ponent o f Fig 5-7(b) creates the difference show n in Fig. 5-7(a). The difference is shown to occur slightly after the 1 nsec occurrence o f the input discharge which is due to the delay in travelling along the microstrip lines. This small difference can be quite critical in high speed digital circuits if, for exam ple, the input to a high speed active device is based on the potential between the two ports. Such analysis can serve 50 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. - -Port 1 (input) - Port 2 (output) — -Port 3 (output) (Port 2 - port 3) 3.5 2.5 < It 3 0.5 I - 0 .5 10 Time (nsec) (a) 0.25 0.2 0.15 0.1 0.05 1.5 2 .5 3 .5 4.5 Time (nsec) (b) Figure 5*7. Time-domain response to a 1 kilovolt human-metal electrostatic discharge, (a) Response showing complete discharge, (b) The high-frequency component of the discharge is shown to occur only at the initial rise. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as a powerful tool in the design of low noise high speed digital circuits. 5.3.2 L-Type Coupling Structure Further validation of this method w as achieved by analyzing the circuit shown in Fig. 5-8. The layout is shown to consist o f a m icrostrip transmission line on a 0.006” thick dielectric substrate w ith relative permittivity o f 9.8, loaded by an inverted ‘L’ shaped coupler. The separation, w, used in the analysis was set equal to the w idth of the microstrip lines o f 0.006” , and the length, /, o f the coupled section is 0.79”. The analysis may be done in practice to study the effects o f loading, caused by adding couplers on the data transmission line extending from port 1 to port 2. Such analysis can be useful in determ ining the maximum num ber o f allow able couplers that can be placed on the line without the generation o f unacceptable deterioration to the signal at port2. (b) Figure 5-8. The L-Typc coupling structure (a) Top view (b) 3-dimentional view /r1=0.01”, A2 = 0.006”. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A s with the cross junction discussed in section 5.3.1, this structure w as also analyzed at a few frequency points with the present full-wave method of chapter 2, and additional points were obtained using LIBRA (Fig. 5-9). As shown, the response does not exhibit the smoothness that the cross junction has, as shown in Fig. 5-3. This is due to the m ism atch in the reference port im pedance used of 50 ohm and the characteristic impedance o f the 0.006” microstrip line. This width is chosen to demonstrate that for more complex responses, a higher order filter approxim ation is required to model the response adequately. Fig. 5-9 shows the 16th order transfer function used to model the response which required 40 iterations to obtain. . S-param eter data - Transfer function 0.9 0.8 0.7 0.6 0.5 v 0.4 0.3 0.2 Frequency (GHz) Figure 5-9. The output response, S12, showing a gradual decline in transmission as fre quency is increased. The approximated transfer function is shown to deviate at higher fre quencies. Figure 5-10 shows the response o f a ‘square’ pulse having sm ooth rise and fall times o f 350 psec injected at port 1. This pulse input is chosen to demonstrate the effects loading may have on a digital signal as it propagates on a bus line for example. The advantage o f having an analytic transfer function o f the single coupler geom etry is that the output (the ‘O ne coupler’ curve o f Fig. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5-9) can be re-injected into the mathematical system to obtain the output o f a tw o-coupler circuit. This recursive process can be carried out as m any times as required to study the effects o f having many couplers as show n in the 20 coupler exam ple in Fig. 5-9. Although the recursion technique ignores the electrom agnetic interaction among the various couplers, it nevertheless serves as a good low frequency model if the couplers are kept far apart. One coupler — Five couplers Twenty Couplers 0.8 Voltage (v) 0.4 0.2 20 40 60 80 100 120 Time (nsec) 140 160 180 200 Figure 5-10. Time-domain voltage on main line showing the effect of coupler loading. 5.4 Conclusion In this chapter, a method for obtaining tim e-dom ain signatures from frequency response data was presented. T h e approach was demonstrated for a generic microstrip cross junction, where the transmitted signals were acquired for various tim e-dom ain input signals. The linear simulation of a 500 point tim e-dom ain input requires less than 4 seconds to compute on a SPARC 2 workstation using MATLAB, show ing its numerical efficiency. 54 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. The only disadvantage o f this method is its inability to handle certain responses such as the reflection terms o r S n for the cross structure. This is due to the fact that the reflection coefficient is negative which is an im proper transfer function. In other words, the program fails to find an analog filter that can produce a negative pulse. Further improvements is required to handle these special cases. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 CONCLUSIONS 6.1 Concluding Remarks In this research, results for the electromagnetic behavior of shielded multilayered geom etries w ere obtained based on a general com puter code developed using a full-wave numerical technique. The code was further enhanced in both versatility and efficiency, by im plementing non-uniform discretization o f interconnect geometries allowing the analysis of more com plicated circuits without resorting to the unnecessary increase of discretization density. The program w as also im proved by providing the ability to automatically recognize and eliminate com m on com putations o f basis functions through the use of links. Parallel processing techniques w ere developed and compared for efficiency and storage. It was found that a frequency-based partitioning o f the workload is superior to spatial partitioning o f the physical interconnect layout. Also, basis function integral sharing was found to further enhance the parallel code. To validate the code several geom etries were analyzed, chosen to test the various features o f the method. These included tw o-ports, N-ports, multilayered conducting strips, and non-uniform discretization. A time-domain analysis was also developed that could potentially be used to treat such problem s as high frequency interference which may occur in integrated circuits. A sim ple planar geometry, the cross junction, was analyzed to dem onstrate the effectiveness o f the method. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2 Improvements and Recommendations Although results presented in this thesis dem onstrate the accuracy o f this m ethod, certain issues ought to be addressed. For example, the structure o f the FORTRAN program m ing language allows for static m em ory allocation. This means that the number o f interconnect layers must be built into the program and not a user-specified option. An obvious improvement would be the conversion to “C ” programming language which would allow dynamic memory allocation as well as easier linked list handling of basis function integrals. Furthermore, to analyze realistic structures, the incorporation of conduction loss o f interconnects as well as dielectric loss should be im plemented, thereby adding greatly to the code’s generality. Dielectric losses can be incorporated into any structure since the dielectric primitivities of all layers are treated as complex numbers by the computer code. However, validation of the code’s ability to handle such lossy structures by comparison with published results has to be performed before applying the code to new lossy structures. It is also recommended to perform further convergence analysis due to introduction o f non-uniform discretization. This is to test the validity o f the assumption that discretization lim its o f uniform analysis holds for non-uniform analysis. A nother highly recommended study is that of the incorporation of lumped elements as well as vias into the impedance matrix, in order to analyze a m uch wider range of circuits. This is seen as a natural step towards the developm ent of a general purpose full-wave electromagnetic analysis tool for multilayered circuits. *"7 * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A INPUT GENERATION PROGRAM T he generation of input files describing complex structures can be quite cumbersome. Such structures may have to be discretized into hundreds of cells to ensure adequate analysis. As m entioned in Chapter 3, each of these cells is described by six variables: • coordinates of the test function • length o f the cell directly in front o f the above point • length o f the cell directly behind the above point • width o f both ceils • type o f cell To avoid manual entry of cell locations and the possibility o f human input error, a com puter program was written in the “C” language to automatically generate the input from a com puter draw ing. The program is executed as follows: M akelnput <deltaz> <deltay> <inputfi,ename> T he program initiates a bitmap editor which offers a Graphical U ser Interface (GUI) to draw the layout (Fig. A -l). The structure is first drawn in black and white, or any set o f two colors. The object may also be scanned from a drawn or photographed image. Once the drawing is completed and the file is saved, the program executes another system program which converts the bitmap file to an ASCn file. This simply converts the Hexadecimal image bit to T ’ if the bit in on (black) and to ‘O’ if the bit is off (white). T he execution of this step can be done manually as follows: bmtoa -char 01 <filename.bmp> <newfilename> 58 R e p ro du ced with permission o f the copyright owner. Further reproduction prohibited without permission. H^bttMpJSSBSBHHBBSSHSSi^BSIil^BSSSSSSSS (tof ile } (ElMiO ri l l —: Ohm) l « » w t : <boh> Si m : 30*3(1 Clear ( Set ( ( Invert I ttw u k ( c«Tr ( Point ) c C am ) ( B«ct<nql« t* r k ) r~ ii^"~ ) ) fFUlri Kactanqlg) (_ Clrcl* ) C Filled Circle") C Flood Fill ) ( Set Mot Spot ) C Clear Hot ^ o Q c t>^° ~t Figure A-l. The GUI drawing program used to create the input file. The program then analyzes the new ly formed matrix and creates all the y and z directed cells necessary to describe the layout. The input parameters, deltaz and d eltay are used for all the cells as the physical length, in meters, of the front and back o f each cell. The output is saved in the file <newfilename> which can then be appended to the original input file discussed in Table 3-1, which contains the complete geometry description. The procedure is repeated for each layer in the structure. Since the GUI bitmap editor handles constant bit sizes, the program can only create uniformly discretized grids. However, the program can be used to create a geom etry resembling that bemg analyzed as much as uniform discretization allows. The user m ay then manually edit the edge cells to m ake the input file conform exactly to the desired layout. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B TIME-DOMAIN ANALYSIS PROGRAM This analysis is made possible with the use o f the program T D A N A L Y S IS . It is invoked in the Matlab environm ent as follows: [z,p,y]= TD AN ALYSIS {H, w, na,nb, input, t) where H w na, njj An array o f complex frequency response data obtained with the frequency-domain method An array o f corresponding frequencies of H A "uess of the maximum order of the numerator and denominator of the transfer function which is: A „ s + A„ _ , s + . . . + A , s + A T( s) = input t z p The time-domain input signal to the system The discrete time array associated with the input signal An array o f the zeros o f the system An array o f the poles o f the system such that: ( s - z „ ) ( s - z „ _ l ) . . . ( s - z 0) T( s ) = where n, m y are the final numerator and denominator orders is the discrete time-domain output of the system w hen subjected with the signal input at times specified by t 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES 1. T. Itoh, Num erical Techniques fo r Microwave a n d Millimeter-Wave Passive Structures, New York: John W iley & Sons, 1989. 2. V. N, Rayapati, B. Kaminska, “Performance analysis o f multilayer interconnections for megabit static randor- access memory chip,” IE E E Trans, on Components, Hybrids, and M anufacturing Technology, vol. 16, pp. 4 6 9-4/4, August 1993. 3. Jesse Sheinwald. “MMIC compatible bandpass filter design: A survey o f applicable techniques,” M icrowave Journal, pp. 26-41, M arch 1994. 4. EEsof Inc., Circuit Element Catalog, vol. 1, W estlake Village, CA 1993. 5. K. C. G upta, M.D. Abouzahra. Analysis and Design o f I 'lanar M icrowave Components, IEEE Press, New York, 1994. 6. C. N. C hang, J. F. Cheng, “Hybrid quasistatic analysis o f multilayer m icrostrip lines,” IEE Proceedings-H, vol. 140, pp. 79-83, April 1993. 7. C. N. C hang, J. 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Zaky, T.E. van Deventer, “ Research consortium in electromagnetics and electrostatic discharge,” Report no. 5 fo r Bell Canada, pp. 19-23, 1994. 40. W . L. Stutzman, G. A. Thiele, A ntenna Theory and Design, New York, John Wiley & Sons Inc., 1981. 41. R. C. D aigle, G. W . Bull, D. J. Doyle, “M ultilayer microwave boards: manufacturing and d e s i g n M icrowave Journal, April 1993. 42. Personal communications with Xidong Wu, University o f Toronto, 1995. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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