Efficient Design and Optimisation Techniques for Planar Microwave Filters by S0ren F. Peik A Thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Electrical Engineering Waterloo, Ontario, Canada, 1999 ©Spren F. Peik 1999 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 1 * 1 National Library of Canada Bibliotheque nationale du Canada Acquisitions and Bibliographic Services Acquisisitons et services bibliographiques 395 W ellington Street Ottawa ON K1A 0N4 Canada 395, rue W ellington Ottawa ON K1A 0N4 Canada Your file Votre reference ISBN: 0-612-92046-1 O ur file Notre reference ISBN: 0-612-92046-1 The author has granted a non exclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sell copies of this thesis in microform, paper or electronic formats. 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While these forms may be included in the document page count, their removal does not represent any loss of content from the dissertation. Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. Canada Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. I hereby declare that I am the sole author o f this thesis. I authorise the University o f W aterloo to lend this thesis to other institutions or individuals for the purpose o f scholarly research. I further authorise the University o f W aterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request o f other institutions or individuals for the purpose o f scholarly research. ii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. The University o f W aterloo requires the signatures o f all persons using or photocopying this thesis. Please sign below, and give address and date. iii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Abstract The topic o f this thesis is the efficient and accurate synthesis o f arbitrarily shaped m icrowave filters. Currently, filter circuits are synthesised using an iterative optim isation in conjunction w ith an electrom agnetic (EM ) simulator. This so-called direct EM optim isation has tw o m ajor drawbacks. It does not generally attain the optim al solution, but rather a sub-optim al solution. In addition, direct E M optim isation is com putationally very expensive. In this thesis, we propose four novel techniques to overcom e these drawbacks. First, we propose the application o f a genetic algorithm (GA) to guide the optim iser towards the optim al solution. GA’s identify the optim al filter layout even where gradient-based m ethods fail. We enhance a generic GA in tw o ways. O ur algorithm allows the user to im pose arbitrary con straints. O ur G A also provides the user w ith vital data about the correlation and the sensitivity of the param eters. W ith these enhancem ents, our im proved GA generally attains better-perform ing circuits than gradient search m ethods at sim ilar com putational expenses. Second, w e extend the Cauchy m ethod for fast frequency sweeps to a m ulti-dim ensional Cauchy method, w ith respect to both frequency and physical dim ensions. Hence, our new algorithm can be used as a substitute for neural networks. The proposed m ulti-dim ensional C auchy m ethod accurately and inexpensively com putes circuit m odels from arbitrary topologies. It m inim ises the com putational cost o f the optim isation process. Exam ples dem onstrate that— w ithout sacrificing accuracy— the optimisation tim e is reduced by several orders o f magnitude. Third, for tackling large filter circuits, we introduce a hybrid optimisation schem e based on cou pling m atrix alteration. This new technique com bines an accurate— but expensive— com plete electrom agnetic analysis with an inexpensive— but less accurate— decom posed analysis. First, we derive tw o coupling m atrices o f the filter: one from the response found by com plete anal ysis, and one from the response found by decom posed analysis. We then com pare the two coupling m atrices and apply this knowledge to elim inate inaccuracies in the decom posed circuit m odel. O ur hybrid technique attains the com putational speed o f the decom posed analysis, but the accuracy o f the full analysis. The technique can optim ise filter circuits for w hich direct EM optim isation w ould require excessive com puter resources. Fourth, we create a technique for optim ising non-resonant filter structures, such as transversal fil ters. Transversal filters m ay include up to several hundred design param eters, m aking direct EM optim isation im possible. An optim isation o f such filters becom es feasible using the tim e-dom ain response rather than the frequency-dom ain response. O ur technique isolates the fault locations o f the circuit one after the other by tim e-dom ain reflectometry. Hence, the optim al circuit paiv R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. ram eters can be found one at a time. The tim e-dom ain optim isation presented in this thesis can optimise a transversal filter w ithout a restrictive upper lim it on the num ber o f param eters. Several exam ples throughout this thesis illustrate that the proposed techniques result in a consid erable gain in design reliability and a significant reduction o f com putational cost when com pared to conventional m ethods. One chapter is exclusively dedicated to the design and the m easure m ent of various m icrowave devices for superconductive satellite systems using our new tech niques. The m easured results are in excellent agreem ent w ith the theory. v R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I would like to thank Professor Y. L. Chow, Professor R.R. M ansour and P rofessor A. SafaviNaeini who agreed to be my supervisors. Their encouragem ent and advice during the course o f this thesis are deeply appreciated. They also generously supported m e in overcom ing the countless adm inistrative hurdles that resulted from w ith m y unusual situation w ith one supervisor in Hong Kong, one in C am bridge, and one in W aterloo, and m yself working full-tim e at C O M DEV, Cambridge. I take special pleasure in thanking all m y colleagues at C O M D EV International and C O M D EV Space Group, for providing m e w ith an excellent environm ent to pursue my research. This thesis could not have been com pleted w ithout the technical help of the C O M D EV R& D staff. In particular, I would like to thank Raafat M ansour, Bill Jolley, Tony Rom ano, Shen Ye, and Ke-Li Wu for the countless stim ulating discussions on m y thesis project. Last, but not least, I thank m y wife, Gudrun, who often m issed me on the w eekends and late evenings during the course o f this study. The preparation o f this thesis was largely supported by C O M DEV International, Cam bridge, Ontario. The synthesis techniques and hardware described in this thesis are the result o f research program s supported in part by the D efence A dvanced Research Projects Agency (DARPA), the Canadian Space Agency (CSA), the Canadian D epartm ent o f National D efence (DN D), and C O M D EV under N A SA Co-operative A greem ent NCC3-517. vi R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. To my parents vii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Contents 1 Introduction 1 1.1 M otivation for This Thesis and Proposed S o l u t i o n s .................................................. 2 1.2 Thesis Organisation 5 .......................................................................................................... 2 Available Filter Design Techniques 2.1 6 Standard F ilter D esign T e c h n iq u e s .................................................................................. 2.1.1 .................................................................... 8 2.2 Enhanced Filter Optim isation A l g o r i t h m s .................................................................... 13 2.3 Lim itations o f Available M ethods and O ur Proposed S o l u t i o n s ............................. 15 2.3.1 R obustness and Cost o f G radient C a l c u l a t i o n ............................................... 16 2.3.2 D iscrete M eshes and Interpolation E r r o r s ...................................................... 17 2.3.3 Com putational E x p e n s e s ...................................................................................... 18 2.4 Application o f Standard M ethods 6 C onclusion ............................................................................................................................ 3 Genetic Algorithms in Microwave Circuit Design 19 21 3.1 B a c k g r o u n d ............................................................................................................................ 21 3.2 A lgorithm Outline and A daption to Circuit S y n t h e s i s ................................................ 22 3.3 M icrowave-Specific Extensions 25 ..................................................................................... 3.3.1 Im plem enting Physical C onstraints ................................................................. 26 3.3.2 Param eter Space V is u a lis a tio n ........................................................................... 27 viii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 3.4 E x a m p l e s .............................................................................................................................. 29 3.4.1 K a-B and Stripline R ejection Filter D e s ig n ...................................................... 29 3.4.2 W aveguide Cavity F ilter D e s i g n ........................................................................ 33 3.5 O ptim isation Perform ance o f GA’s .................................................................................. 34 3.6 Conclusion 36 .......................................................................................................................... 4 Multi-Dimensional Cauchy Method 4.1 37 M ulti-D im ensional In te r p o la tio n .................................................................................... 38 4.1.1 N eural N e tw o r k s .................................................................................................... 38 C auchy M e t h o d .................................................................................................................... 42 4.2.1 O ne-D im ensional Cauchy M e t h o d .................................................................... 43 4.3 Adaptive S a m p l i n g ............................................................................................................ 46 4.4 M ulti-dim ensional Cauchy M e th o d .................................................................................. 46 4.4.1 Recursive Cauchy M e t h o d ................................................................................... 46 4.4.2 M ulti-D im ensional Cauchy E x p a n sio n .............................................................. 52 Num erical R e s u lts ................................................................................................................ 53 4.5.1 M icrostrip Line Im p e d a n c e ................................................................................... 53 4.5.2 R ecessed Line Fed M icrostrip A n t e n n a .......................................................... 53 4.5.3 N arrow -Band 3-Pole F i l t e r ................................................................................... 56 4.2 4.5 4.6 C onclusion ........................................................................................................................... 5 Hybrid Optimisation Using Coupling Matrix Alteration 5.1 5.2 57 61 B a c k g r o u n d ........................................................................................................................... 62 5.1.1 Space M a p p in g ........................................................................................................ 66 M -Param eter Correction......... ............................................................................................. 67 5.2.1 C oupling M atrix R e p re s e n ta tio n ........................................................................ 68 5.2.2 Coupling M atrix o f Three-Pole F i l t e r .............................................................. 71 ix R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 5.2.3 M -Param eter C orrected O p tim is a tio n ............................................................. 72 5.3 N um erical E x a m p le ............................................................................................................ 74 5.4 C om parison o f Com putation T i m e ................................................................................ 83 5.5 Conclusion 84 .......................................................................................................................... 6 Optimisation by Time-Domain Reflectometry 6.1 Theory 6.1.1 ................................................................................................................................. 88 Transform ation into T im e -D o m a in .................................................................... 90 6.1.3 Pinpointing o f Fault L ocation ........................................................................... 91 6.1.4 Elim ination o f F a u l t s ............................................................................................. 93 6.1.5 N on-U niqueness o f Fault E lim in a tio n ............................................................. 95 6.1.6 Elim inating M ultiple Faults ............................................................................... 95 Pulse T r a c k i n g ........................................................................................................ 97 6 .1.2 .1.7 Ideal F ilter Synthesis and R eal Filter Perform ance 87 ..................................... 6 6.2 Tuning o f F ilter H ardw are ............................................................................................... 97 6.3 Optim isation and Tuning o f N arrow -Band R esonant Filter C i r c u i t s ..................... 99 6.4 N um erical R e s u lts ................................................................................................................ 101 6.4.1 103 6.5 7 86 C om parison o f Com putation T i m e s ................................................................. C onclusion ........................................................................................................................... 105 Hardware Verification 106 7.1 K a-B and R ejection F i l t e r .................................................................................................. 106 7.2 3-dB Pow er D iv id e r............................................................................................................. 107 7.3 Butler-M atrix B e a m f o r m e r ............................................................................................... 109 7.4 C ryogenic S w itc h e s ............................................................................................................. 115 7.4.1 HTS C - S w i t c h ........................................................................................................ 115 7.4.2 HTS H igh Isolation S w itc h ................................................................................... 118 7.5 Six-Pole N arrow -Band F i l t e r ........................................................................................... 118 7.6 C o n c lu s io n s ........................................................................................................................... 122 x R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 8 Summary and Future Work 124 8.1 R eco m m en d atio n s................................................................................................................. 126 8.2 Final R e m a rk s ........................................................................................................................ 127 8.3 Future W o r k ........................................................................................................................... 128 A Theory of Space Mapping 129 B Coupling Matrix Algebra 132 B .l C oupling C apacitor Representations for B and-Stop Filters ................................... 132 B .2 D uality o f R e p re s e n ta tio n s ............................................................................................... 134 B.3 Coupling Values are Pure R eal N u m b e r s ........................................................................ 136 C Validity of Rational Function Representation C .l D istributed E l e m e n t s .......................................................................................................... D Circuit Layouts 137 138 140 xi R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. List of Tables 3.1 Transfer Param eter S 21 S p e c ific a tio n s ........................................................................... 30 3.2 Starting Values for Optim isation o f Folded Stub Filter ........................................... 34 5.1 Specifications for Narrow B and HTS F ilter ................................................................ 74 5.2 C PU Tim e C om parison for D ifferent Optim isation S tr a te g ie s ................................. 84 6.1 C om parison o f C PU Tim es for Conventional and Tim e-dom ain Optim isation 7.1 C om parison o f B ranch-C oupler T e c h n o lo g ie s ............................................................. 107 8.1 Recom m ended Optim isation M ethods for D ifferent Filter Types 126 . .......................... xii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 103 List of Figures 1.1 Analysis Versus Synthesis ................................................................................................ 2 2.1 D esign Procedure for Iterative Optim isation o f M icrowave Circuits ................... 7 2.2 Coupling o f Circuit Sections in a C ascaded and Com plete M icrostrip Circuit 2.3 (Side V i e w ) ..................................................................................................................... 9 Folded Two-Pole F i l t e r ................................................................................................. 9 2.4 Schem atic Circuit for Two Pole Filter 2.5 ........................................................................... Layouts U sed for Electrom agnetic S im u la tio n ....................................................... 10 11 2.6 (a) Filter R esponse C alculated from Circuit Theory, (b) Full-EM Sim ulation, (c) C ascaded E M Sim ulation, and (d) Full-EM Sim ulation From M odified Circuit . 2.7 Space M a p p i n g ............................................................................................................... 2.8 Objective Function W ith Two Local M axim a 12 15 ............................................................. 16 2.9 Planar C apacitor on a D iscrete G eom etry G rid o f Sonnet e m ........................... 17 2.10 C om putation Tim e as a Function o f C ircuit Com plexity........................................ 18 2.11 Planar N on-U niform Transm ission Line Filters 3.1 .......................................................... 19 Planar Circuit W ith Three Param eters and Its R epresentation as Individual W ith Three Chrom osom es (Binary C o d e d )....................................................................... 23 3.2 Population o f Four Individuals W ith Three Chrom osom es E a c h ....................... 23 3.3 Outline o f Genetic A lg o rith m ...................................................................................... 24 3.4 O pen Stripline Stub, Sm ith Charts for 20 GHz and 30 G H z .............................. 27 xiii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 3.5 L ayout and Param eters o f Four-Pole D ual M ode F ilte r ............................................ 28 3.6 Population A fter 20 Generations. Num bers are in mils 28 3.7 L ayout o f Stripline R ejection Filter, G ray=B efore optim isation, Black=A fter O ptim isation ......................................... ........................................................................................................................ 30 3.8 R esponse o f Stripline F ilter Before and A fter GA O p tim is a tio n ........................... 32 3.9 L ayout o f Four-Pole R ectangular Iris-Coupled W ave-Guide Filter Show ing all Adjustable Param eters ...................................................................................................... 3.10 Optim ised R esponse o f W aveguide F ilter L ayout 33 .................................................... 33 L ayout o f M icrostrip L ine F i l t e r ................................................................................... 34 ......................................................................... 35 3.13 Decay o f E rror Function for Optim isation M e t h o d s ................................................ 35 4.1 Neural N etw ork M odel o f M icrowave C i r c u i t ........................................................... 39 4.2 Planar Three-Pole F ilter ................................................................................................. 40 4.3 Neural N etw ork Approxim ation and Exact EM Sim ulation for R andom Input 3.11 3.12 Return Loss o f O ptim ised Stub Filter V e c t o r ...................................................................................................................................... 4.4 41 Neural N etw ork A pproxim ation and E xact E M Sim ulation for an Optim ised Cir cuit L a y o u t ........................................................................................................................... 41 4.5 Parallel R esonator and Input I m p e d a n c e ..................................................................... 43 4.6 Interpolation o f Four-Pole Filter Response, a) Linear, b) Cubic Spline, c) C auchy 45 4.7 Adaptive Sam pling Procedure 47 4.8 Sam ple Locations for Two-dim ensional and Three D im ensional Param eter Space 48 4.9 Steps in Recursive C auchy M ethod in Two D im e n s io n s ......................................... 50 4.10 Inverted Tree Representation for Recursive Cauchy M e th o d .................................. 51 ....................................................................................... 4.11 Im pedance o f M icrostrip L ine as a Function o f e r and w / h using M ulti-D im ensional 4.12 R ational Function E x p a n s io n ........................................................................................... 54 M icrostrip A ntenna w ith R ecessed line f e e d .............................................................. 54 4.13 R eturn Loss o f A ntenna Geom etry p \= 24.5 m m , p 2 - 1.4 m m , p:>= 4.2 m m C om puted by M ulti-D Cauchy M odel and Full E M -S im u la tio n ............................ xiv R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 55 4.14 Effect o f Inset location on A ntenna’s R eturn L o s s .................................................... 56 4.15 L ayout o f Three-pole F ilter Show ing Param eters p i to p 4 ...................................... 56 4.16 Interpolated S n Param eter U sing M ulti-D im ensional Cauchy M e t h o d 58 4.17 S ii Param eter W ithout Using M ulti-D im ensional Cauchy M ethod Showing the Sam pled Points Above and B e l o w ................................................................................. 58 4.18 R esponse o f O ptim ised Three-Pole F ilte r ..................................................................... 59 4.19 M onte-C arlo A nalysis o f O ptim ised Three-Pole f i l t e r ............................................. 60 5.1 Training and A pplication Phase s ...................................................................................... 63 5.2 R esponse o f Circuit L ayout p \ and p 2 ............................................................................... 64 5.3 R eal Part o f A 5 n ( f ) for L ayout p i and L ayout p 2 (Left), and Im aginary Part of A S 'n (f) for L ayout p \ and L ayout P 2 (Right) ............................................................ 65 .............................. 65 ................................................ 66 5.4 A djusted S-Param eter S'* u ( / ) for L ayout p 2 U sing [AS 1](p i) 5.5 Training and Application U sing Space M apping 5.6 Three-Pole Filter, a) C ascaded (Decom posed) Layout, b) Com plete Layout . . 5.7 Cascaded and C om plete Sim ulation R esults o f Three-Pole Filter.C learly Seen 67 in the Com plete Sim ulation is a Transm ission Zero at f=3.96 G H z D ue to Stray C oupling Betw een R esonator O ne and T hree................................................................ 68 5.8 D irect C oupled R e s o n a to r s .............................................................................................. 68 5.9 Inductive C oupled R esonators ....................................................................................... 69 5.10 Training and Application o f Coupling M atrix C o r r e c t i o n ....................................... 72 5.11 O ptim isation Including Correction o f C ascaded R e s p o n s e ....................................... 73 5.12 L ayout o f Six-pole F i l t e r ................................................................................................. 74 5.13 C ascaded Six-pole F i l t e r .................................................................................................. 75 5.14 Definition of Variable Param eters in Single R e s o n a to r.............................................. 75 5.15 R esponse o f Initial F ilter L a y o u t .................................................................................... 76 5.16 R esponse o f Optim ised F i l t e r ........................................................................................... 77 5.17 A dditional Couplings in Full-EM Sim ulation 78 ............................................................ xv R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 5.18 R esponse o f (G rid-Snapped) Filter Layout p geo3 using Fine M odel and C oarse M o d e l ...................................................................................................................................... 78 5.19 R esponse o f F ilter L ayout p g e o 2 using C orrected C oarse M odel and U ncorrected Coarse M odel. M arked in Gray is the Forbidden A rea o f the Specifications. . . 80 5.20 R esponse o f Filter W ith L ayout p g e 0 4 O ptim ised U sing Corrected C oarse m odel 81 5.21 Com parison: R esponse o f Filter by C ascaded C orrected Sim ulation and Full-EM Sim ulation at N ext G rid-Snap C alled L ayout p geo5 82 6.1 N on-U niform Transm ission L i n e ................................................................................... 6.2 Circuit L ayout U sing Circuit Theory and Realisation in M icrostrip Topology and 88 Their R espective R e s p o n s e ............................................................................................... 89 6.3 Frequency R esponse o f Ideal and Real C i r c u i t ........................................................... 90 6.4 Com putation o f Tim e-dom ain From Frequency-D om ain Electrom agnetic Sim u lation 6.5 ...................................................................................................................................... 91 Excitation Pulses in Tim e-D om ain a) Gaussian b) Differential Gaussian c) G aus sian Step d) Sine M odulated G aussian Pulse ............................................................... 91 6 .6 D esired R esponse s f f s (f) and A ctual R esponse s f f ( i ) in Tim e-D om ain . . . . 92 6.7 Exciting Gaussian Pulse and D ifference o f D esired and Actual Response; <1 Denotes the Tim e E lapsed B etw een Excitation and Getting the Echo o f the Fault L o c a tio n .................................................................................................................................. 93 6 .8 Optim isation Param eters at Fault L o c a tio n .................................................................. 94 6.9 O ptim ised Circuit L ayout 94 .............................................................................................. 6.10 D esired Frequency R esponse and Frequency R esponse of M odified C ircuit . . . 95 6.11 Three D ifferent M odifications o f the L ayout Yielding the D esired Response: a) Extending the width at Fault; b) Shortening the Line with Im pedance Zo ; c) A djusting C ircuit | Away From F a u l t .......................................................................... 96 6.12 Tim e Sequence o f Pulse Travelling Through a C i r c u i t ............................................. 98 6.13 B and-pass W indow F u n c t i o n .......................................................................................... 99 6.14 Tim e-D om ain R esponse o f Tuned N arrow -Band Filter ......................................... 6.15 Tim e-D om ain R esponse o f N arrow -Band Filter W ith R esonator three D etuned xvi R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 100 100 6.16 Planar N onuniform Transm ission Line Low-Pass F i l t e r .......................................... 101 6.17 Frequency-D om ain response 5 n F rom Circuit Theory (Desired Response), FullEM Sim ulation (Actual R e s p o n s e ) ................................................................................. 102 6.18 Tim e-D om ain R esponse s n (< ) From Circuit Theory and raw F u l l - E M 102 6.19 Variation o f Frequency and Tim e-Dom ain R esponse by Changing ru4 ................. 103 6.20 Tim e-D om ain R esponse o f sn(f) from Circuit Theory and Corrected Circuit U sing Full EM and TD R O p tim is a tio n .......................................................................... 104 6.21 Frequency-D om ain R esponse S n From Circuit Theory, Full-EM , Corrected Full-EM Sim ulation ......................................................................................................... 104 7.1 B lock D iagram o f a Typical R elays S a t e l l it e .............................................................. 107 7.2 Sim ulated (Thin Line) and M easured R esponse (Fat Line) o f Stripline F ilter .At the Top 5 n [d B ] and at the B ottom 5 i 2 [ d B ] ............................................................... 108 7.3 Com parison o f L-B and Hybrids ................................................................................... 108 7.4 L ayout o f H y b r i d ............................................................................................................... 109 7.5 Photograph o f the Folded H ybrid in C om parison W ith One C ent C o i n ............. 110 7.6 Full E M Sim ulated R esponse o f O ptim ised H y b r i d ................................................ 110 7.7 M easured R esponse o f O ptim ised H y b r i d .................................................................. 110 7.8 B eam Generation by M ultiple Fixed B eam form ing N etw ork Ill 7.9 Schem atic B utler M atrix Beam form er, The X ’s Represent 90° Hybrids W ith a ............................... Pow er Split Ratio o f 3dB /3dB ............................................................................................ Ill 7.10 L ayout o f 8 x 8 M atrix in H o u s in g ................................................................................... 112 7.11 Photograph o f 8 x 8 B utler M atrix B eam form er 113 ....................................................... 7.12 R eturn Loss o f 8 x 8 M atrix for B eam Ports 1R, 2R, 3R, 4R Sim ulated (left), and M easured (right) 7.13 Insertion Loss o f ................................................................................................................. 8 x8 113 M atrix W ith B eam -Port Input 1R, sim ulated (left), and m easured (r ig h t).................................................................................................................... 114 7.14 Phase Taper o f 8 x 8 M atrix Sim ulated and M e a s u re d ................................................ 114 7.15 114 Phase Distribution o f 8 x 8 m atrix (Sim ulated and M easured).................................... xvii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 7.16 Radiation Pattern (Sim ulated from M easured D a t a ) ................................................ 115 7.17 Four-For-Two Redundancy Sw itch A rrangem ent o f A N IK D (see [141]) . . . . 115 7.18 C -sw itch Operation M o d e s .............................................................................................. 116 7.19 L ayout o f C -Sw itch . Inset Shows One o f the Eight D.C. Decoupled Stub Lines. 116 7.20 Photograph o f Conventional C o-A xial C -Switch, HTS C-Sw itch and One D ollar C o i n ......................................................................................................................................... 117 7.21 Subsection o f C -Sw itch Showing the four Optim isation Param eters ................. 117 7.22 Sim ulated (Left) and M easured (Right) Perform ance o f C - s w itc h ........................ 118 7.23 L ayout o f Four PIN D iode Sw itch and B low -U p o f the Diodes Including D.C. 7.24 B ias N etw ork and R F C h o k e s ........................................................................................... 119 Photograph o f H igh-Isolation Sw itch W ith L i d ........................................................... 119 7.25 Sim ulated and M easured R esponse o f The High-Isolation Integrated H TS/PIN D iode S w i t c h ........................................................................................................................ 120 7.26 L ayout o f Six-Pole Filter in H o u s i n g ............................................................................ 121 7.27 Photograph o f the six-Pole F i l t e r ................................................................................... 121 7.28 Sim ulated and M easured Frequency R esponse o f Six-Pole F ilter W ithout Tuning 123 A .l Illustration o f Space M apping 130 B .l C apacitive C oupled Resonators Z ................................................................................ 132 B.2 Lossy Parallel Resonator C i r c u i t ................................................................................... 134 B.3 Duality o f C oupling N etw orks, On the left C oupling by Lum ped E lem ent Induc ....................................................................................... tive T-Network, on the Right Coupling by Capacitive 7r-N etw ork ......................... 135 C .l T-N etwork and 7r - N e t w o r k .............................................................................................. 137 C.2 Tangents o f x and Pade A p p ro x im atio n ......................................................................... 139 D .l L ayout o f LC Low -Pass C ir c u it....................................................................................... 141 D.2 L ayout o f sub-section o f six-pole f i l t e r ......................................................................... 142 D.3 L ayout o f Six-Pole filter 143 ................................................................................................. xviii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction "In scien ce one tries to tell p eople, in such a w ay as to be un dersto o d by everyone, som ething that no one ever knew before. But in poetry, i t ’s the exact o p p o site.", Paul Dirac M icrowave filters are the backbone o f any m odem m icrowave com m unication system. D esign and m anufacturing o f small, lightw eight, high-perform ing filters is a m ulti-billion dollar industry nowadays. Consequently, a lot o f effort is put into the precise and fast synthesis o f microwave filter circuits. The approach o f filter synthesis has changed over the years. In the early days o f m icrowave engi neering in the 1930’s, filter design was perform ed exclusively by formal synthesis. This approach uses a ladder netw ork o f lum ped im pedances to com pose a filter from a prototype function that represents the desired response. W here lum ped elem ents are im practical, distributed elem ents can approxim ate them. The continued refinem ent o f form al synthesis gave rise to w hat is now com m only called Filter Theory. The progressive m iniaturisation o f filters that becam e possible w ith the invention o f planar struc tures in the 1950’s, has m ade form al synthesis increasingly difficult. Form al synthesis cannot take fringe fields, parasitic elem ent coupling, etc. into account. These effects can adversely affect a circuit’s perform ance, especially at very high frequencies beyond the X-band. Further m ore, form al synthesis is restricted to regularly shaped layouts. A dvanced filters, however, m ay dem and irregularly shaped circuit elements. An alternative m ethod to form al synthesis was developed in the 1960’s. The m ethod originates from the fact that synthesis m ay not be possible for a certain circuit layout, but a circuit analysis is feasible. Figure 1.1 shows the definitions o f analysis and synthesis. Synthesis derives a circuit 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 2 CHAPTER 1. IN TRO D U C TIO N structure from a given response function. Analysis, on the other hand, calculates the response o f a given filter structure. ^ ^ y n th e s i^ ^ Response , Analysis Circuit Structure Figure 1.1: Analysis Versus Synthesis If analysis is possible, but form al synthesis is not, w e can still synthesise the filter by an iter ative optim isation process, wherein a circuit layout is m odified and analysed repeatedly until the specifications are satisfied. T e m e s and C a l a h a n in their classical paper [11] in 1967 were am ong the first to advocate the use o f such an iterative optim isation process to design m i crowave circuits. The draw back o f this m ethod is that, in general, this iterative process is rather tim e-consum ing. W ith only very sim ple com puters available in the 1960’s, the analysis— and therefore the optim isation— were restricted to very small circuits. The situation changed rapidly with the advent o f powerful digital com puters in the early 1970’s. N um erical m icrowave circuit analysis reached undream t-of capabilities. U sing finite-elem ent m ethods, it becam e possible to derive an accurate response o f alm ost any irregularly shaped microwave structure. W ith these sophisticated analysis techniques on hand, the synthesis of m icrowave circuits using iterative optim isation gained m uch popularity in the m icrowave com munity. Today, num erical m ethods are the m ain tools for analysis o f m icrowave circuits. M any com m ercial program s on the m arket tackle any kind o f analysis problem that involves microwave structures. Optim isation tools are either part o f the package, or they can be purchased separately. 1.1 Motivation for This Thesis and Proposed Solutions The design by num erical iterative optim isation m ight be the m ost adopted tool in microwave circuit design, but it is not the panacea o f m icrowave engineering. An application o f the iterative optim isation synthesis technique is accom panied by the following problems: 1. Present optim isation schemes do not guarantee to find the optimal circuit layout subject to the circuit specifications. The search m ay get trapped in a sub-optim al, unsatisfactory solution. The optim isation process is then said to be non-robust. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CH APTER 1. INTRO D U CTIO N 3 2. The optim isation process requires repeated analysis o f the circuit for evaluating the cir cuit’s perform ance. This repetitive analysis step can com putationally be extrem ely expen sive. W ith today’s techniques, expenses can often be reduced only by sacrificing accuracy. 3. The m ore com plex the circuit, the m ore apparent the problem s described in 1. and 2. b e com e. M edium -sized synthesis problem s already put today’s m icrowave design software to the limit. Even larger circuits can only be analysed and optim ised by decom position, i.e. by splitting them into several sub-circuits. This approach has the draw back o f losing accuracy in the simulations. 4. C om m ercial circuit sim ulation tools do not allow an arbitrary scaling o f the circuit’s ge ometry. Rather, the dim ensions can only be changed in finite steps in order to fall on the boundaries o f a fixed mesh. This lim itation is necessary to achieve acceptable per form ance o f the simulator. The optimal circuit geom etry m ay not be representable by a layout snapped to this discrete grid. H ence, the true optim um cannot be found. This thesis proposes a num ber o f novel techniques to avoid the problem s outlined above. R o bustness is im proved by switching to a novel class o f optim isation algorithm s, the so-called genetic algorithm s. We are am ong the first to apply these algorithm s to filter synthesis prob lems. To yield better perform ance, w e m odify the standard genetic algorithm when applying it to filter synthesis problem s. Especially, we allow the user to include knowledge about the circuit’s physical constraints. As shown in this work, such constraints cannot be incorporated into conventional optim isation schemes. The m easures taken result in novel robust optim isa tion algorithm s for m ore reliable m icrowave circuit designs. O ur algorithm s are m eant to be a supplem ent to existing optim isers rather than a substitution. T he problem o f large com putational expenses and too m any optim isation variables is solved by several actions: A m ulti-dim ensional Cauchy m ethod, hybrid optim isation and tim e-dom ain optim isation. These techniques m ay be used individually or combined. O ur m ulti-dim ensional Cauchy m ethod is a novel extension o f the one-dim ensional Cauchy m ethod for frequency response interpolation. The m ulti-dim ensional m ethod interpolates with respect to both frequency and geom etrical param eters. The technique can precisely derive the response o f any arbitrary filter geom etry from a small set o f pre-calculated response samples. W hen applied to the optim isation o f filter circuits, the m ulti-dim ensional Cauchy m ethod makes the evaluation o f the circuit perform ance (the analysis step in the optim isation) very inexpensive. The new interpolation scheme also allows us to avoid the problem o f finite m esh sizes in the circuit sim ulation software, as response sam ples can be com puted from any circuit layout. A p R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. INTRO D U CTIO N 4 plying the new interpolation scheme, we can use a very rough m esh for fast sim ulation and still achieve the desired accuracy. For large optim isation problem s, the database concept becom es impractical, because it requires a vast num ber o f circuit samples. The circuit m ust be calculated from sm aller decom posed sub sections and the overall response found by network-theory. Unfortunately, this decom position introduces errors. Parasitic couplings, w hich always exist in the real circuits, are cut off. As a result, the response calculated from the decom posed circuit is different from the response cal culated from the com plete circuit. In the past, this error introduced by the decom posed analysis had to be accepted for optimisation, as there was no practical way to perform an optim isation on the com plete circuit. We develop here a novel technique which provides the speed o f the decom posed analysis with the accuracy o f the com plete circuit analysis. The m ethod, called hybrid optim isation, perform s one circuit sim ulation using both the decom posed circuit and the com plete circuit. From their respective responses the coupling param eters are derived. The differences betw een coupling param eters describe the parasitic couplings betw een the filter resonators, w hich are present in the com plete circuit but m issing in the decom posed circuit. It is shown that we can adjust the responses derived from subsequent decom posed circuit analyses to include the parasitic coupling if we use the difference coupling matrix. The result is an accurate m odel with the com putational cost o f the decom posed analysis. O ur hybrid optim isation is specialised to resonant band-pass and band-stop filters. The technique cannot optim ise other filters, such as low-pass filters and transversal, i.e. non-resonant, filters. This problem can be tackled with another innovation we introduce here, nam ely optimisation using time-dom ain reflectometry. The tim e-dom ain response reveals different characteristics of the circuit than the frequency-dom ain response. This well-known fact is exploited by timedom ain reflectom etry (TD R) m easurem ents. Here, we adapt the TDR concept for a tim e-dom ain based synthesis o f m icrowave circuits. W ith our new technique, we can pinpoint one fault after the other and fix (i.e. optim ise) them one at a time. D ue to non-uniqueness, w e often even have several choices for optim ising the circuit. Applying the new technique to filter optimisation, we can split an optim isation problem with N param eters into N problem s with one parameter, which m akes the overall optim isation m uch easier to handle. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. IN TRO D U C TIO N 1.2 5 Thesis Organisation C hapter 2 gives a review o f the currently available filter design techniques. Standard m ethods as w ell as enhanced algorithm s are briefly explained and many references given. The chapter pinpoints the problem s inherent in the standard techniques as well as enhanced m ethods, and outlines our ideas for overcom ing these problems. C hapter 3 proposes the im plem entation o f genetic algorithm s in the filter design process. The new approach im proves the robustness o f the optim isation process significantly. We outline how we m odify the generic genetic algorithm to boost perform ance. Genetic algorithm s are also com pared to traditional optim isers with respect to robustness and com putational costs. The concepts are dem onstrated by several examples. Chapter 4 introduces a novel, accurate concept o f response interpolation that uses a m ulti dim ensional Cauchy method. The novel interpolation schem e perm its the fast calculation o f any m odified circuit response from response sam ples calculated a-priori. As dem onstrated by various exam ples, the concept gives the designer a fast and hassle-free optim isation, param eter plots, and M onte-C arlo analyses. C hapter 5 develops the idea o f hybrid optimisation. By hybrid optim isation we understand the optim isation o f filter structures utilising two different circuit m odels, nam ely a coarse, fast m odel and a fine, accurate m odel. R epresenting both m odels by their respective coupling m atrices, we succeed in correcting the coarse m odel so that the accurate response for any geom etry can be derived w ithout further fine m odel simulations. The novel m ethod achieves the com putational speed of the coarse m odel w ith the accuracy o f the fine model. O ur next new contribution, the filter optim isation by tim e-dom ain reflectometry, is introduced in C hapter 6 . We recom m end tim e-dom ain optim isation particularly for the synthesis o f non- uniform transm ission line filters, e.g. transversal filters. The chapter gives a detailed explanation o f the theory underlying tim e-dom ain reflectom etry and applications. C hapter 7 lists and describes all hardware built throughout the course o f this project. We verify the perform ance o f m icrow ave circuits synthesised by our new techniques on several pieces o f hardware. Am ong others, w e test a superconductive B utler m atrix beam form er, several cryo genic active switch circuits, and a superconductive planar six-pole narrow-band filter. F our appendices provide m athem atical details and derivations. Furtherm ore, the appendices contain technical draw ings o f the m anufactured hardware. A list of references can be found at the end. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Available Filter Design Techniques "I sa w that it was great, greater, an d greatest, w ith prodigious p o ssib ilities in its power. I w as determ in ed to m aster the book... It took m e several years before I cou ld understand as much as I p o ssib le could. Then I set M axw ell aside an d fo llo w e d m y own course. A nd I progressed much m ore quickly.", Oliver H eaviside on M axw ell’s book "Treatise on Electricity and Magnetism" O ver the years, many techniques have been reported in the literature for designing m icrowave filters. In this chapter, w e briefly review currently used filter design techniques and the draw backs associated with them. A dvanced techniques that w ere developed to avoid these drawbacks are the topic o f this chapter’s second section. These m ethod, even though very powerful, still have some limitations. R esolving these lim itations is the topic o f this thesis. A t the end o f this chapter, we present a num ber o f new ly developed techniques that enable us to design arbitrarily shaped large filter circuits in an accurate, reliable, and inexpensive way. 2.1 Standard Filter Design Techniques The starting point o f microwave filter design is the form al synthesis o f lum ped and distributed elem ent netw orks using circuit theory. W e can derive filter circuits for alm ost any desired filter response using circuit theory as described in [27, 29, 12]. Various aspects o f form al filter design techniques are the topic o f a vast num ber o f publications, e.g. [14, 23, 25, 28], The standard w ork [12] o f M a t t h a e i, Y oung and Jo n e s presents an exhaustive discussion o f form al filter design with m any supporting design tables and charts. 6 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 7 O ver the years, m any specialised design techniques for specific filters have been developed. Filters with finite transm ission zeros for better perform ance are developed in [1 6 ,1 7 ,1 8 ], am ong others. C oupled resonator representations o f bandpass filters are discussed in [26, 30, 31, 32], The so called transversal filters, first proposed by K allm ann in [109], are presented in [19, 20] and others. On the other hand, m any papers have been published with a focus on either circuit topology (e.g. W aveguide filter design [21], planar filters [22], superconductive filters [140]) or filter purpose, e.g. space applications[15]. However, it has been w ell-recognised that pure form al filter synthesis techniques have their lim i tations. T he m ethod can only synthesise filters using the ideal, basic, regularly shaped elem ents, nam ely inductors, capacitors, and transm ission lines. Filter layouts including arbitrarily shaped elem ents are not feasible. Furtherm ore, not every desired filter response can be achieved using the form al synthesis. In order to liberate the design process from these restrictions, an iterative circuit design by opti m isation was suggested in [11, 39, 40]. The idea originates from the fact that any circuit perfor m ance can be analysed, but not necessarily synthesised. The solution is an iterative optim isation process. W e m odify and analyse the circuit layout repeatedly until the circuit’s response satisfies the specifications. The flow o f the process is outlined in the chart o f Figure 2.1. O ptim isation techniques are discussed in-depth in the literature. There are basically three different ways o f searching the param eter space: G radient search m ethods, direct search m ethods, and guided random m ethods. For m ore inform ation, the reader m ay refer to [61, 81, 62, 65, 66 ]. ^ Start ^ Initial D esig n Circuit T op ology Parameter M odifications Circuit A nalysis Requirements . Satisfied ? ; I yes Q rn T ) Figure 2.1: D esign Procedure for Iterative Optim isation o f M icrow ave Circuits The filter design using iterative optim isation is only lim ited by the ability to analyse the circuit. Initially, circuit analysis was done exclusively on netw orks o f lum ped and distributed elem ents w hose analytical responses are available. Later, arbitrary elem ents w ith responses derived from R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 8 m easurem ents or heuristic form ulas w ere included. The actual circuit analysis is carried out using netw ork theory. The technique is exhaustively studied in [39] and [44]. W ith the aid o f today’s com puters, we can also include full electrom agnetic (so-called fu ll E M ) sim ulated circuit elem ents, see [42]. F or m ore inform ation on full EM sim ulation m ethods, refer to [36, 34, 35, 33]. The ultim ate application o f netw ork theory based analysis is decom posed fu ll E M simulation (also called cascaded fu ll EM ). The circuit is split into several sub-sections which are com puted separately using full E M sim ulation m ethods, and the overall response is com puted from circuit theory. D ecom position is used for exam ple in [42], Versatility and speed o f the com bined netw ork analysis/optim isation technique have led to their widespread application in com puter aided design (CAD) software o f m icrowave circuits. The m ost prom inent software packages using this approach are Touchstone [145], O SA 90 [147], and H P-M DS [146]. The m ethod has its drawbacks: N o interaction betw een non-adjacent circuit ele m ents can be considered. In contrast to the real circuit, non-adjacent coupling, o f the sub-circuits is not considered. These effects include stray quasi-static field coupling as well as coupling by box-m odes and surface-m odes inside the housing. Figure 2.2 illustrates the possible couplings o f a decom posed m icrostrip circuit versus the com plete circuit in a housing. It becom es clear, that highest accuracy is achieved with an optim isation using com plete fu ll EM simulation. C om plete sim ulation m eans that w e com pute the response from a single full EM sim ulation o f the circuit as one block. The com plete full E M simulation m odels the actual cir cuit m ost closely. Consequently, an iterative optim isation using the com plete full E M sim ulation results in the m ost accurate design. The com plete EM (also called direct) optim isation is ap plied for exam ple in [43, 49, 41, 46]. The m ethod’s handicap lies in its extrem e com putational dem ands. C om plete full EM sim ulations take hours or even days, even for relatively sm all-size circuits. 2.1.1 Application of Standard Methods The differences betw een the analysis m ethods, nam ely heuristic m odels+network-theory, de composed EM+network theory, and complete EM, are illustrated by considering a two-pole filter. A typical two-pole line filter is shown in Figure 2.3. W e can sim ulate this filter with circuit com posed o f com ponents based on heuristic m odels from e.g. [39], The schematic diagram o f the circuit from elem entary com ponents is shown in Figure 2.4. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 9 C ascad ed C ircuit ’V/V/V/V///V/V//V/V<V/\V/V/VV/V/V/V ;\v S u b -C ircu it 1 S u b -C ircu it 2 A ir M etallisation 'i m tk im M m % R eal C ircuit Wave-Guide Modes S pace Waves o Ouasi-Static E/H Fields Surface Waves Figure 2.2: Coupling o f C ircuit Sections in a C ascaded and Com plete M icrostrip C ircuit (Side View) Figure 2.3: Folded Two-Pole Filter R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 10 OV22 H5SUB5TRATE 200 SUBST-LaAl ER-24 MUR-1 C0ND-1.0E+306 ROUGH-O un TANO-O.O nil H-20 n il EQUATION LJ101-(10<40. K45) H-7 n il L-L res n i I SUBST-LaAl EQUATION L_H 12 -( 10<45. 0<45) EQUATIONL res-(120<130.2KlB0) SUBST-LaAl H-7 ni QF9 HSCRNR W P10 H5TL SUBST-LaAL Ka"i!, avw CHP4 H-7 n iI SUBST-LaAl H 1 :S-7 n il T-LaAl 1 HSOC CHP43 HSOC H -7 n i 1 O F13 SUBST-LaAl L-L NI2 n il >{ SUBST-LaAl L-L.M01 n i l CHP17 N SOM H 5-7 n il an : H-7 n i 1 J02 iTH H-7 n i l L-18 n i l SUBST-LaAl HSOC KSCTL SUBST-LaAl L - O n il H-7 n i 1 HSOC H-7 n L-50 n i l L-0 n il H-7 n i l SUBST-LaAl f a SrflR PQRTNUM-1 R-50.0 OH JX-0.0 OH AGROUND i s m rs. H-7 mi 1 L-L res n i l SUBST-LaAl 3 i CHP23 «CS« CHP33 CHP32 cm? SUBST-LaAL H-7 m il L-18 n i l OF47 H-7 n i l HSCTL S-7 n il Q P36 CfWB H-7 n il S-7 n il H-7 n il ono H50C L-0 n il H-7 mil SUBST-LaAl OP24 H51L H-7 n iI L-100 n il SUBST-LaAl .SMR PORTNUM-2 R-50.0 OH JX-0.0 OH AGROUND F igure 2.4: Schem atic Circuit for Two Pole Filter R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 11 The schematic circuit is optim ised for 4 G H z centre frequency, -15 dB return loss, and a 0.5% bandw idth. The response o f the optim ised filter using circuit theory is shown in Figure 2.6 (a). The response is verified by a full EM analysis. Three layouts are used for the sim ulation as shown in Figure 2.5: a com plete layout (Layout 1), a cascaded circuit (Layout 2), and a layout with a flipped second resonator (Layout 3). Layout 1 Layout 2 Layout 3 Figure 2.5: Layouts U sed for Electrom agnetic Sim ulation The response param eters S u and £21 for the three sim ulations are shown in Figure 2.6 (b) to (d). T he circuit theory sim ulation is also shown in Figure 2.6 (a). W e clearly see that the response of the full E M sim ulation o f L ayout 1 does not m atch the circuit theory sim ulation. A com parison betw een the sim ulations o f the cascaded circuit o f L ayout 2 and the m odified circuit o f Layout 3 shows a m uch closer resem blance o f the response with the theoretical results. The response is shifted only slightly in frequency. All layouts are identical from the circuit theory point o f view. So why are the responses so different? The reason for this lies in the differences in the separation o f the resonators. M utual coupling betw een the resonators o f layout 1 is m uch stronger than in layout 2 or 3, because the resonators are facing each other. In layout 3, the resonators are spatially separated, whereas in layout 2, they are com pletely decoupled from parasitic coupling. R F energy flows m ainly through the ports from one resonator to the next. This condition is violated in layout 1. Energy flows also via electrom agnetic coupling from resonator to resonator. Consequently, the response from layout 1 differs dram atically from the theoretical response. Obviously, the optim al layout o f an iteratively optim ised circuit is only as good as the underlying analysis m ethod. A lthough the filter is optim ised for 4 GHz, the use o f the heuristic m odels have led to an actual design w ith a centre frequency o f 4.02 GHz due to the m odel errors. Further, bandw idth and return loss o f the actual circuit are slightly different from the response derived using circuit theory. This clearly dem onstrates, that we cannot blindly trust the circuit theory m odelling. Accurate synthesis is not always feasible. In order to gain accuracy, w e have to use full EM models. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 3.9 3.9 12 GHz freq freq GHz (b) L ayout 1 (a) C ircuit-Theory Trace2«dB(FULLEM_DESIGN_AFS.,S(1,1]) Tr ace4-dB(FULLEM_DESIGN_AFS. . SC1,2]) Tracel-dB (S C l,m Trace6-dB(Stt,2]) s < / \ / /' \ / \ 1=1 1 (c) L ayout 2 Trace3*dBCDATAFROMHALFS. .SCI,1 ]) Trace5*dB(DATAFR0MHALFS..Stl,2 ]) 3.9 3.9 s \ V freq freq S GHz GHz N . [_ _ ] 1 A freq freq 4.1 4.1 GHz A GHz B (d) L ayout 3 Tr ace7-dB(FULLEMJJESIGNjb i r ror _AFS. . S[1,11) Trace8*dB(FULLEM_DESIGN_mirror_AFS..Stl,2 ]) Figure 2.6: (a) Filter R esponse C alculated from Circuit Theory, (b) Full-EM Sim ulation, (c) C ascaded EM Sim ulation, and (d) Full-EM Sim ulation From M odified Circuit R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 13 To sum m arise, w e can classify the available techniques into four categories: 1. Form al synthesis using circuit theory 2. Iterative optim isation using heuristic m odel, circuit theory analysis, and netw ork theory 3. Iterative optim isation using circuit m odels based on a full electrom agnetic sim ulation or m easurem ents 4. Iterative optim isation using com plete full electrom agnetic sim ulation (direct optimisation) The boundaries betw een the m ethods are not rigid and the different m ethods can be com bined. A ll o f these m ethods are in widespread use by m icrowave engineers. W hich technique to choose is often determ ined by a trade-off betw een required accuracy and the perm itted com putational costs. Typically, the engineer derives an initial design using m ethod 1, and then switches to m ethods 2 and 3 for a refinem ent o f the optim al design. In the last step, the design is verified by a com plete E M sim ulation. However, even w ith today’s fast com puters, a direct optim isation of large circuits (i.e. m ethod 4) is not feasible because o f excessive com putational costs. W ithout question, the goal is an optim isation w ith the accuracy of the full E M sim ulation and speed o f the circuit theory approach. In the follow ing section, we sum m arise w hat has been reported in the literature to com e closer to this goal. 2.2 Enhanced Filter Optimisation Algorithms T he basic technique o f iterative optim isation can be im proved by two strategies. R ecalling the flow chart o f Figure 2.1, we can either im prove the perform ance o f the analysis step, or upgrade the param eter m odification algorithm. The general im provem ent of the analysis step is a big— if not the biggest— research field in m icrowave theory. These im proved techniques are beyond the scope of this thesis. It is sufficient to say that they indirectly accelerate the optim isation process. In the follow ing sections, we concentrate on the review o f m ethods whose im m ediate goal is to im prove the optim isation process. Response Database and Interpolation: D uring the optim isation process, identical or alm ost identical circuit layouts are sim ulated repeatedly. Consequently, it is useful to store pre vious sim ulation results in a response database, as proposed by B a n d l e r in [43]. W ith R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 14 CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES a sufficiently large database, we can also interpolate the response o f unknown circuit lay outs from the know n saved results. This requires a m ulti-dim ensional interpolation. In the literature linear and quadratic interpolation has been proposed, e.g. in [43]. Cauchy Method and Adaptive Sampling: M i l l e r et. al. suggested in [76, 77, 82] to apply the Cauchy m ethod to the frequency response. The Cauchy m ethod is a very effective interpolation schem e based on rational functions. U sing this m ethod, a handful o f fre quency sam ples can often predict the exact frequency response over a wide frequency range. D h a e n e et. al. showed in [82] that adaptive sam pling can reduce the required num ber o f frequency sam ples even further. In M iller’s work, however, the m ethod is re stricted to frequency response interpolation. W e give a discussion o f the Cauchy m ethod in C hapter 4. There, we also extend the one-dim ensional Cauchy m ethod to a m ulti dim ensional Cauchy method. Neural Networks: A very pow erful interpolation schem e is a technique known as neural net works as described in [120, 122, 119]. G u p t a in [126, 127] and Z h a n g in [130, 131] were am ong the first to apply neural networks to microwave analysis. Neural networks com pute a response from a known set o f response samples. Z H A N G and G UPTA im proved the standard netw ork perform ance by introducing so-called knowledge-based neural net works (K B -N N ) [132, 133, 134, 128, 127]. K B -N N use heuristic m icrowave m odels as activation functions (neurons) rather than the standard function. IN [136] W u et. al. pro posed a technique for input-output reversal o f neural networks. In doing so, the network can be used not only for analysis but also for synthesis. M ore inform ation on neural nets is given in C hapter 4. Space Mapping and Hybrid Optimisation: B a n d l e r at al. developed a technique called space m apping in [72]. The m ethod is a hybrid optim isation scheme. The idea behind hybrid optim isation is to use a coarse circuit m odel, e.g. decom posed m odel, within the optimisation loop, and a fine m odel, e.g. a com plete EM , for verification and adjustm ents of the coarse model. Space m apping establishes a relationship betw een the param eter spaces o f a coarse m odel and o f a line m odel by m apping the two param eter spaces as illustrated in Figure 2.7. We briefly outline the theory o f the m apping algorithm in A p pendix A. The algorithm has been updated w ith several im provem ents [73]. Circuit Extraction: YE and M a n s o u r in [45] developed a technique for including the stray coupling o f real circuits into the decom posed analysis. The sub-circuits are connected by additional tw o-port netw orks w hich em ulate the m issing stray coupling. The netw ork R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES Parameter Space of Coarse Model 15 Parameter Space of fine Model Figure 2.7: Space M apping elem ents are derived using circuit param eter extraction. Adjoint Network Method: M ost optim isation techniques are based on the knowledge o f the gradient o f the objective function, i.e. its derivatives with respect to the geom etrical pa rameters. W hen using full E M sim ulators, the derivatives m ust be calculated numerically, i.e. by finite-difference m ethods. A llesa n d r i et. al. showed in [48] that the gradient can be com puted by an adjoint netw ork m ethod that is well known in circuit theory ana lysis. However, the m ethod is limited. The optim isation is only applicable to simulation m ethods and topologies w here the analytical gradient o f the Y-matrix o f the sub-circuit can be found. Order Recursive Gaussian Elimination: W hen designing a decom posed planar circuit by the m ethod o f m om ents (see [33]), w e often sim ulate very sim ilar subsections. It can be shown that there is a considerable overlap o f data o f the various sub-circuit simulations. In [5 0 ] , M i s r a et al. proposed the order recursive Gaussian elim ination algorithm to exploit the data-overlap. The m ethod is very limited, because w e need access to the the matrix form ulation in the sim ulation software and we m ust reform ulate the algorithm for each specific layout. 2.3 Limitations of Available Methods and Our Proposed Solutions Section 2.2 has shown that m any efforts have been m ade to im prove the standard optim isation algorithm s. The suggested techniques are a big step towards an accurate and fast optimisation. Nevertheless, m ost o f these techniques are only applicable to specific circuit layouts or suffer from other severe restrictions. The follow ing sections discuss in m ore detail the known prob lem s with robustness, com putational expenses, discrete geom etry grids, and large num ber of R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 16 optim isation variables. W ithin each section, we outline our proposed solution to the problem discussed. 2.3.1 Robustness and Cost of Gradient Calculation The traditional gradient optim isers [63] are often unable to find the best solution, because their search gets trapped in a sub-optim al solution (local m inim um ). For a typical objective function as shown in Figure 2.8, the gradient-guided optim iser always follows the direction o f steepest ascent. Thus, the param eter vector p is always guided to the closest peak in the objective func tion. Once the optim iser com es close enough to the local peak, the gradient-based algorithm cannot jum p to another peak. The algorithm is called non robust, because it fails to find the global maxim um. G lobal Maximum Local Maximum Figure 2.8: Objective Function W ith Two Local M axim a N one of the available techniques addresses the problem s with robustness. It is left up to the user o f the optim isation software to w atch the optim isation process carefully and to intervene in case the optim iser gets stuck. The problem can be fixed by using a robust optim isation algorithm which searches the param eter space globally. Genetic algorithm s (GA ) are o f this type. As shown in [95], their search is not stalled by the presence o f local m axim a. To our knowledge, genetic algorithm s have never been applied to the synthesis o f microwave filter. In this thesis, we propose to em ploy genetic algorithm s in a robust m icrowave filter synthesis. C hapter 3 dem onstrates how we use genetic algorithm s in connection with an EM -sim ulator for optimisation. As shown there, w e add some extensions to the standard GA to boost its optim isation perform ance and to extract intriguing circuit insights. A nother advantage o f GA’s is that they do not require any derivative inform ation. W hen using R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 17 direct full E M optim isation, the com putation o f the derivatives is very expensive. These costs are avoided by switching to genetic algorithm s. 2.3.2 Discrete Meshes and Interpolation Errors F or better perform ance, com m ercially available circuit sim ulators like Sonnet em [150] use a discrete m esh for the circuit layout. All vertices o f the circuit layout m ust fall on equally spaced grid-points. Figure 2.9 shows a typical grid used by Sonnet em. Due to the restrictions to the grid, the capacitor gap can take only discrete values. Thus the capacitance cannot be freely chosen. The circuit response o f layouts w ith geom etries “off-the-grid” m ust be interpolated. HI—II lh Figure 2.9: Planar C apacitor on a D iscrete G eom etry Grid o f Sonnet em L inear interpolation, cubic spline interpolation, and neural networks have been recom m ended for such geom etries. These schem es w ork very reliably as long as the interpolated functions are only weakly non-linear. N onetheless, the available m ethods have difficulties handling highly non-linear functions, in particular, functions with finite poles. The Cauchy m ethod does not suffer this problem , but has the lim itation that it is applicable to the frequency interpolation, only. To overcom e these problem s, we propose the m ulti-dim ensional Cauchy method. This novel m ethod is capable o f handling several variables, as well as reliably interpolating highly-nonlinear functions. The concept and exam ples are discussed in detail in Chapter 4. Besides its interpolation accuracy, the Cauchy m ethod can significantly reduce the C PU tim e required during a circuit optimisation. Since the m ethod derives all responses from a-priori cal culated response sam ples, the com putational costs for the circuit analysis in the optim isation loop are m inim al. The com putation o f the sam ples is a solitary event, and could be carried out over-night. Once the response sam ples are available, circuits can be analysed alm ost instanta neously. N ote however, that in practice the m ulti-dim ensional Cauchy m ethod is in practice for m edium size problem s only. F or circuits w ith m any optim isation variables, the advantages o f quick analysis are counterbalanced by the cost o f generating the response database. In order to R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 18 address large scale circuits, w e have to decom pose the circuit first. The sub-sections can then be m odelled with the m ulti-dim ensional Cauchy method. 2.3.3 Computational Expenses The m assive need for com puter resources is the m ost serious disadvantage o f optim isation by com plete full E M sim ulations. The com putational expenses can be dramatic. F or example, a single full electrom agnetic sim ulation o f an 8 x 8 beam form er outlined in Section 7.3 takes already 10 CPU -hours on a state-of-the-art H ew lett-Packard K-class machine. For problem s o f that order, we cannot do m ore than one or tw o sim ulations a day. To optim ise this circuit we need many o f such sim ulations. H ence, direct optim isation is im possible in a reasonable tim e frame. The main problem is that with increasing com plexity o f the circuit, the required C PU tim e in creases exponentially. Figure 2.10 shows the typical relationship between circuit com plexity and com putation time. A t a certain complexity, the lim its o f the underlying software and hardw are are reached. The num ber o f calculations becom es im m ense, and com putation tim e increases alm ost boundless. Sonnet Software [150] refers to this lim it as The Wall. The Wall is represented by the dotted line in Figure 2.10. Computation Time Complexity Figure 2.10: C om putation Time as a Function o f Circuit Complexity. Consequently, it is very im portant to lim it the circuit com plexity to stay below the wall. To reduce the complexity, the circuit can be decom posed as discussed earlier. The decrease in com putational expenses from using decom position is paid for by a decrease in accuracy. For many applications, the errors thus introduced are unacceptably high. The exam ple o f the twopole filter responses given in Figure 2.6 dem onstrates this clearly. Space m apping, as suggested to overcom e this problem , reduces the costs o f the optim isation with a hybrid optim isation algorithm. As outlined in C hapter 5, however, space m apping fails R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E LILTER D E SIG N TECHNIQUES 19 on circuits where the coarse m odel, i.e. the decom posed analysis, cannot describe the stray couplings. In this thesis, w e introduce a new hybrid optim isation concept. The m ethod builds on the direct coupled resonator representation. W e can describe coupled resonator filters com pletely by their coupling matrix. The key o f our new technique is to use an additional coupling m atrix in the decom posed analysis to represent all stray couplings. This concept and exam ples are presented in C hapter 5. The m ethod, however, is lim ited to resonant band-pass filter structures. F or non-resonant and low-pass/high-pass filter structures, w e offer an alternative optim isation technique. Such filters are often realised as non-uniform transm ission line filters such as shown in Figure 2.11. A direct optim isation o f large non-uniform transm ission line filters is not feasible, no m atter w hich o f the available techniques is used. U p to now, these filters are designed exclu sively by form al synthesis. D irect optim isation o f the frequency response cannot be perform ed, due to the vast num ber o f circuit param eters. Figure 2.11: Planar N on-U niform Transm ission Line Filters We show in C hapter 6 that direct E M optim isation is feasible when w e utilise the tim e-dom ain response. The tim e-dom ain response o f a microwave circuit reveals other properties than the frequency-response does. These characteristics can be exploited for the circuit optimisation. C hapter 6 introduces a new technique for non-uniform transm ission line filters. This new tech nique is based on on tim e-dom ain reflectometry. 2.4 Conclusion This C hapter has reviewed the m ost com m only used filter synthesis techniques. The techniques can be classified into tw o m ajor groups: form al synthesis and iterative optimisation. Form al synthesis derives the filter layout analytically from filter theory. Iterative optim isation is usually perform ed with tw o steps in a loop: the analyses o f the circuit by num erical sim ulation and the m odification o f the circuit param eters. The num erical sim ulation is done by one o f circuit theory, cascaded electrom agnetic analysis, or com plete electrom agnetic analysis. The pros and cons o f the available sim ulation m ethods were discussed. To sum m arise, the m odels are either accurate or com putationally fast, but never R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. A V A IL A B L E FILTER D E SIG N TECHNIQUES 20 both. The decom posed full E M analysis o f circuits is a technique often applied to com prom ise betw een accuracy and speed. The second step o f the optim isation loop— the m odification o f the param eters— is done with the goal to m axim ise the circuit’s perform ance. W e can update the param eters with various available algorithm s. Nowadays, a gradient-based (steepest descent) search technique is applied m ost often. The draw back o f this m ethod is, however, that it is not generally robust, i.e. it m ight stop searching the param eter space after getting trapped in a sub-optim al solution. In the next chapter, w e propose the application o f genetic algorithm s to filter optim isation. Genetic algorithm s search the param eter space by sim ulating evolution. Besides the choice o f the right optim isation algorithm , the key to an efficient m icrowave circuit optim isation is the developm ent o f accurate and fast circuit models. A prom ising approach is to m ove the expensive evaluations o f the circuit m odel out o f the optim isation loop. A ll required circuit responses are com puted a priori and saved in a database. Since the database cannot include all possible param eter vectors, the response com putations in the optim isation loop are interpolated from values that do exist in the database. In C hapter 4 w e propose an accurate m ulti-dim ensional interpolation scheme, the m ulti-dim ensional Cauchy m ethod. A s m entioned in this chapter, circuit theory provides excellent com putation speed, whereas full electrom agnetic m odels achieve superior accuracy. The tw o m ethods can be com bined into an hybrid optim isation technique. The hybrid optim isation uses the fast circuit theory m odel in the optim isation loop. To increase the accuracy o f the circuit theory model, w e include a correction term . The correction term is derived from a com parison o f one full-EM sim ulation with its corresponding circuit theory simulation. In C hapter 5, we discuss available hybrid optimisation m ethods and derive a new technique for hybrid optim isation. O ur new technique derives the correction term from the differences in the the coupling m atrix o f the circuit theory m odel and the full E M model. Since our hybrid optim isation m ethod is lim ited to resonant band-pass filter structures, we advo cate an alternative optim isation algorithm for non-resonant and low-pass filter structures. Using the tim e-dom ain, rather than frequency dom ain response, w e can optim ise such circuits m ore efficiently. The theory and exam ples are given in C hapter 6 . All hardware built and m easured throughout this thesis is described separately in C hapter 7. We do not only present filter circuits, but also other m icrowave devices such as switches and antenna beamform er. As dem onstrated, the above m ethods are also applicable to the synthesis o f such devices. The m easured responses o f the devices agree with the predicted responses calculated w ith our new techniques. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Genetic Algorithms in Microwave Circuit Design "Research is w hat I am doing i f I don 7 know w hat 1 am doing ", W em her von Braun In this chapter, we bring the concept o f evolutional com putation [87] in context with the synthesis o f m icrowave circuits. The algorithm s and techniques presented provide an additional optim i sation m ethod when others fail or perform poorly. B y no m eans are the algorithm s intended to substitute for the conventional m ethods entirely. The background and history o f genetic algorithm s are outlined in Section 3.1. An algorithm itself is explained in Section 3.2, w ith strong em phasis on the application to m icrowave circuits. In Section 3.3, w e describe the features we add to the algorithm to im prove its applicability to the design o f m icrowave circuits. 3.1 Background Genetic algorithm s [95] search an optim um by sim ulating biological evolution. They take their m o tiv a tio n fr o m th e m e c h a n is m o f n a tu ra l s e le c tio n , a ls o k n o w n as “ su r v iv a l o f th e fitte st” . In natural selection, individuals com pete for the opportunity to reproduce. Individuals with new features com e into the population when existing individuals exchange genetic m aterial during reproduction and when m utations occur. Genetic algorithm s (GA’s) work differently from traditional optim isation m ethods in four ways: 21 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. G ENETIC ALG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N 22 • GA’s w ork w ith a coding o f the param eter set, not on the param eters them selves • GA’s search from a population o f points, not from a single point • GA’s use payoff (objective function) inform ation, instead o f derivatives • GA’s use probabilistic transition rules, instead o f determ inistic rules The technique o f evolution program m ing leads to the follow ing properties of GA’s: • GA’s are very robust, as shown in [95]. T hat m eans that they do not get trapped in local m inim a w hile searching for param eter values that m inim ise the perform ance function. • GA’s do not require derivatives o f the perform ance function as m ost other m ethods do. C alculating the approxim ations o f derivatives, usually by sensitivity analysis [39], is very tim e consum ing. • GA’s are very versatile. There exist virtually no restrictions o f the perform ance function and the constraints. The evaluation function m ay not be continuous. The constraints may be non-linear [90]. GA’s are already being used com m ercially for scheduling and vehicle routing problem s [8 8 ]. Since a few years, they have also been applied increasingly in engineering design problem s. S c h a f f e r et al. [89] design digital filters using GA’s. T a n a k a et al. [91] apply GA’s to inverse electrom agnetic problem s. All o f the above-listed properties are also o f value for the optimisation o f m icrowave circuits. We usually know little about the error functions. They are occasionally discontinuous and may have many local m inim a. Derivatives o f the error function are m ost often unknown and can only be found by a tedious num erical approxim ation. This is our motivation for applying GA’s to the optimisation o f m icrow ave circuits. 3.2 Algorithm Outline and Adaption to Circuit Synthesis The term inology w e use here is a m ix from the areas o f com puter science and genetics. Since the algorithm is to be applied in microwave circuit optim isation, the m eaning o f the term s in the microwave application is also explained. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. G ENETIC A LG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N 23 We refer to m icrowave circuits in evolutional program m ing as individuals. The variable param eters, e.g., line widths and lengths, form their data structure or genetic structure. We assum e that the genetic structure can be described by a bit string, i.e. list o f 0 ’s and l ’s. In genetics, these strings are called chromosomes, and the individual bits are called genes. For instance, consider a planar circuit with three param eters as seen in Figure 3.1. The three param eters are the length o f the stub, width o f the stub, and its distance from port one. b in a ry c o d in g M IC in d iv id u a l p2 K-M 1101 p i = 13 m m ■=> 1101 p2 = 8 mm 1000 p3 = 1 1 m m !=> 1011 1000 10111 chrom osom e 1 chrom osom e 2 chrom osom e 3 Figure 3.1: Planar Circuit W ith Three Param eters and Its R epresentation as Individual With Three C hrom osom es (Binary Coded) The three param eters are coded in three chrom osom es (bit strings) o f length 4 bits. The string length is chosen based on the required accuracy. Here, an individual with the chrom osom e set {1101,1000,1011} represents a circuit w ith p i = 13 mm , p 2 = 8 m m a n d p 3 = 11mm. Traditional optimisers simulate, evaluate the error function and adjust the layout on one circuit at a time. GA’s w ork differently. They evaluate the error functions for a group o f individuals (circuits) characterised by different values o f the chrom osom es. Such a group o f individuals is called a population. A population o f four individuals for our exam ple is illustrated in Figure 3.2. chromosomes = parameter l\ l.iifr* individual one specific circuit genes = binary digits coding parameter value population= set of several different circuits Figure 3.2: Population o f Four Individuals W ith Three C hrom osom es Each One simulation step o f the population is called a generation. During the sim ulation the algorithm assigns every individual a fitness value. The higher the fitness value o f an individual, the better the individual m atches the specifications. In our example, the fitness value is calculated from the reciprocal o f the m ean-square-error betw een the response o f any specific circuit and the desired R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. G ENETIC ALG O R ITH M S I N M ICROW AVE CIRCUIT D E SIG N 24 response. Given the fitness values o f a population o f individuals, the algorithm sim ulates natural selection and reproduction to obtain the next generation. Individuals in the next generation m ay have different chrom osom es. These are obtained by the GA’s three basic operations: reproduction, in w hich individuals from one generation are selected for the next generation, crossover, in which genetic m aterial from one individual is exchanged with genetic m aterial of another individual, and m u ta tio n , in w hich the genetic m aterial o f an individual is altered. C ircuit 1 Circuit 2 Circuit 3 p2 C ircuit 4 ■* K-N X .__ p i = 13 mm - 1101 p2 = p3 = 11 mm - 1011 oS § ? I iE 55 o Individual «55 pl = 11 m m = 1011 p2 = 3 mm = 0011 p3 = 10 mm —1010 Individual pl = 4 mm = 0100 p2= 13 mm = 1101 p3 = 0 mm = 0000 Individual Jh pl = 9 mm - 1001 p2 = 11 mm = 1011 p3 - 15 mm =1111 Individual Generation 1101 1011 Reproduction 1011 Crossover 111: 1001 1011 Vt VV VT V ▼ y 11 t 1001 1000 1001 Mutation r ▼ t ▼ Generation 1 1 1 1 n+1 % O F ig u r e 3 .3 : O u tlin e o f G e n e tic A lg o r ith m Figure 3.3 shows the three operations being used to obtain a new generation from an existing population. In this exam ple, the population o f any generation consists o f 4 individuals with 3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC A LG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N 25 param eters each. F or clarity, we show the m odification o f one param eter (chrom osom e) only. The rem aining param eters are changed correspondingly. The tilted gray num bers indicate the fitness values o f the individuals, obtained from an EM sim ulation o f each specific circuit. Arrows with sm ooth lines indicate copying; arrows with squiggly lines indicate mutation. The selection o f individuals that are allow ed to reproduce is done random ly w ith probabilities based on their respective fitness values. In Figure 3.3, we note that individual 3 with the chrom o som e ( 0 1 0 0 ) is not allow ed to reproduce and individual 2 ( 1 0 1 1 ) which has a very high fitness value is allow ed tw o offsprings. The crossover corresponds to tw o individuals exchanging genetic m aterial. Conventionally speaking, crossover introduces som e perturbation into the param eters. C rossover is controlled by a preset probability. In typical im plem entations, it occurs w ith prob ability 0.5 to 0.6. C rossover random ly chooses a pair o f individuals, and exchanges substrings o f their genetic m aterial. In our circuit optim isation, this corresponds to a slight change in the layouts o f tw o specific circuits. Mutation is carried out by creating a new chrom osom es in w hich each bit is copied from the old without change w ith probability 1 —p mut, and a bit is toggled w ith probability p mut • GOLDBERG [95] suggests a mutation rate p m ut o f 0.03, but often it is necessary to experim ent until one finds values that work w ell for a given application. To search for an optim al solution, w e repeat the evolutional sim ulation process for som e num ber o f generations and take the fittest individual in the final generation as our solution. W e can stop the process after a fixed num ber o f generations and pick the best solution, or execute the algorithm until a satisfactory solution by preset criteria is achieved. In this section, we studied a sim ple genetic algorithm governed by three operations: reproduc tion, crossover, and m utation. A dvanced operators have been tried with the goal o f im proving the perform ance o f this sim ple settings. In [9 5 ] G o l d b e r g exam ines the m ost com m only used advanced operators. H e finds that their use generally does im prove the efficiency, i.e. speed and robustness, o f genetic algorithm s. 3.3 Microwave-Specific Extensions The algorithm discussed in Section 3.2 is com m on to any genetic-algorithm -based optimiser. The algorithm works satisfactorily when applied to m icrowave circuit optim isation. N everthe R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC ALG O R ITH M S I N M ICROW AVE CIRCUIT D E SIG N 26 less, the m ethod can be significantly improved. In this section we dem onstrate the versatile possibilities to adapt GA’s to specific applications. 3.3.1 Implementing Physical Constraints It is rare to find an engineering problem w ithout constraints. W hen physical considerations indicate that the optim um lies within a certain region o f the param eter space, the algorithm should take advantage o f this knowledge. C onstraints in m icrowave circuit design can take a variety o f form s. They include upper and low er bounds on param eters and non-negativity of network elem ents. Constraints can also be m uch m ore com plex. For exam ple, there could be a specific region in the param eter space, or param eter com binations, w hich are favourable or not acceptable. We can often derive additional constraints w hen we include our knowledge about the specific circuit topology. Any knowledge can generally help to pinpoint the optimal circuit layout faster and m ore reliably. This knowledge can be, for instance, past experience or insights on the theoretical behaviour o f the circuit. Gradient search algorithm s are lim ited to the im plem entation o f fairly sim ple constraints, how ever. B ackground inform ation, such as experience, insights, and com plex constraints are diffi cult, if not im possible, to im plem ent in the rigid gradient-based optim isation schemes. Genetic algorithm s are different. They pick points from the param eter space guided by the law o f the survival o f the fittest, rather than following a tedious gradient calculation. Picking a point p that represents a particular circuit is com putationally alm ost free. The selected point can be checked for coherence before the circuit is simulated. M any tests can be run on p before its objective function is calculated. These tests include, but are not lim ited to, sim ple and com bined constraints, validation o f conform ance with rules set up for the specific circuit, and obsoleteness o f circuits known to be ill-perform ing. Once a param eter vector is identified to be ill-perform ing, it can be elim inated from the gene-pool w ithout further com putational expenses. The algorithm sim ply picks a new point from the rem aining population and goes on. L et us illustrate the procedure on the follow ing example. Suppose, w e want to optim ise an open strip-line stub as shown in Figure 3.4. The goal o f the optim isation is that the input im pedance Z rn is zero at 20 G H z (short) and approaches infinity (open) at 30 GHz. Transm ission line theory tells us that this goal is theoretically achieved for an im pedance ratio o f 3:1 for the narrow to the wide line, and an electrical total length o f one-half wavelength at 30 GHz. The sm ith charts in Figure 3.4 explain why: The open end at point A m oves to point B, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC ALG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N 27 due to the line length o f the narrow line. The point B is transform ed into B’ due to the step. The w ide line moves the point B’ to the point C. As seen, at 20 GHz, C ends up at Z = 0 (short) and at 30 GHz, the im pedance at C is infinite (open). Figure 3.4: Open Stripline Stub, Sm ith C harts for 20 GHz and 30 GHz A conventional optim iser can only im plem ent sim ple bounds on the param eters w \, W2 , h , and 12- The im plem entation o f a constraint prescribing that the im pedance ratio o f the lines be close to 3:1 is very difficult w ith conventional optimiser. For genetic algorithm s, this constraint is sim ple to include. W ith a single line o f code, w e can check the ratio for every individual and sort out the ones that do not conform with our constraint before optimisation. W hat constraints are acceptable in the algorithm is only lim ited by the ability to im plem ent them in com puter code, in our case FO R TRA N code. An exam ple o f an insight-guided search is given in Section 3.4.1. 3.3.2 Parameter Space Visualisation In the previous section, w e dem onstrated how the algorithm profits from inform ation that the user gives to the program . The opposite also applies: The user can gain inform ation provided by the algorithm. It is characteristic o f genetic algorithm s to search the param eter space m ostly around areas with high fitness values. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC A LG O R ITH M S I N M ICROW AVE CIRCUIT D E SIG N 28 W hen we plot all individuals ever generated in the param eter space, we can m ake out areas with high population density. Such areas indicate high fitness o f the individuals located within. W e can use this inform ation to gain insights into the circuit’s perform ance and to guide our optim isation further. To illustrate the technique, we take a look at a four-pole dual m ode filter with three param eters shown in Figure 3.5. The filter is optim ised for a 2% passband at 4 GHz. A population plot of all generated individuals is shown in Figure 3.6. Figure 3.5: L ayout and Param eters o f Four-Pole D ual M ode Filter Best Individual: pl=0.438 p2=8.9 p3=-9.2 -10 Pl F ig u r e 3 .6 : P o p u la tio n A fte r 2 0 G e n e r a tio n s. N u m b e r s are in m ils We can see that the filter perform s best at values o f param eter p i around 0.5 mil. Param eter p \ does not seem to be correlated w ith param eter p 2 - Param eter p 2 , by contrast, varies widely, but does show some crowding around p 2 = 9 m ils and p 2 = -2 mil. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC ALG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N 3.4 29 Examples In the following exam ples, several m icrowave circuits are optimised. The exam ples dem onstrate the capabilities o f genetic algorithm s in optim isation o f microwave filter circuits. The perfor m ance o f the GA is com pared with that o f gradient-search algorithm. 3.4.1 Ka-Band Stripline Rejection Filter Design The K a-band filter presented here is part o f a satellite system currently under developm ent at SPAR Canada. The satellite system receives signals from a ground station in one frequency band f up, am plifies this signal, and transm its it back to earth in another frequency band f downThe am plitude o f the received signal is typically about 100 dB sm aller than the am plitude of the transm itted signal. H ence, if the transm itted signal— even in very sm all am ounts-reach the am plifier input, it would disturb the system. The system m ight even start oscillating at the transm itting frequency. Thus, it is im portant to block any signals of the transm itting frequency at the am plifier input by a band stop filter. The given satellite system operates at the following frequencies: R eceiver centre frequency : R eceiver band width: Transm itter centre frequency: Transm itter band width: 29.5 GHz 1.0 GHz 20.0 GHz 1.0 GHz In order to ensure reliable operation o f the amplifier, the filter transfer function has to satisfy the following specifications: A ttenuation o f transm itter signal : | (19. 5 . . . 20.5G H z)| < -70 dB A ttenuation o f receiver signal : |*S,2 i ( 2 9 . 0 . . . 30.0G H z)| > -0.2 dB The substrate values are pre-determ ined as follows: Circuit structure : Stripline technology D ielectric constant er : 3.0 Substrate height : 2 x 10 m ils equiv. 0.254 m m R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. G ENETIC ALG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N The layout is first designed with the circuit-based program to u c h sto n e 30 [145]. In order to obtain the desired response, we propose the filter layout seen in Figure 3.7. Four stubs branch off a transm ission line. Each stub has a step to a higher im pedance at h alf o f its length. The stubs follow the open-short input im pedance behaviour discussed in Figure 3.4. An EM -sim ulation of the layout shows that the filter design needs further optimisation, because its actual response does not satisfy the specifications. In our filter optim isation process, eight param eters are adjustable. These are the lengths and widths o f the stub lines, and the lengths o f the transm ission lines betw een the stub lines. All param eters (pi to p§) are shown in the sketch o f Figure 3.7. 91 [mm] 90 89 88 87 86 85 84 [m m ] 2 2 Figure 3.7: L ayout o f Stripline R ejection Filter, G ray=B efore optim isation, B lack=A fter O pti m isation R ecalling the term inology o f Section 3.2, we translate this situation into the GA settings: The param eters are coded into 10 bit binary strings. The population is set to 5 individuals, and the GA com putes ten generations. The fitness is calculated from an error function based on six frequency points as shown in Table 3.1. band stop band pass band frequency / 1 = 19.5 GHz f 2 = 20.0 GHZ / 3 = 20.5 G H Z / 4 = 29.0 GHz f 5 = 29.5 GHz / 6 = 30.0 GHz S 21 spec. 0 0 0 1 1 1 Table 3.1: Transfer Param eter S 2 1 Specifications R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. G ENETIC ALG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N 31 As an objective function, w e use the m ean-square-error function 2 E ( p i , . . . ,ps) = \ei ( p i , •••, p s ) \ 2 , (3 .1 ) 2= 1 where e\ and e 2 are the errors in the stop and pass band, respectively, i.e. n= 3 el = E l ° - | ^ ( / n , P t , . . , P 8 )|| (3-2) n— 1 71 = 6 e2 = X ) I1 - \s 2 i ( f n , P i , - - - , P s ) \ \ ■ (3-3) n = 4 The optim isation starts w ith the layout shown in gray in Figure 3.7, , which also shows the param eters p l to p 8 . Analysing the filter geometry, one notices that the filter works best when the ratio o f the line im pedances o f the narrow and wide stub lines §£(see Figure 3.7) is around 3:1. This knowledge can be im plem ented into the genetic algorithm. The search in the param eter space is then biased towards this ratio. This kind o f non-linear objective is difficult to im plem ent in traditional search methods. Further, due to the available etching technology, the m inim al line w idth is constrained to be 1 mil. In the genetic algorithm approach, constraints like this can be easily considered. Invalid geom etries, as the ones with too narrow lines, sim ply “die” , even before they are analysed. F or the optim isation, a m icro-genetic algorithm [97] is used in connection with a m om entm ethod EM -sim ulator. The results o f our optim isation are shown in Figure 3.8. A fter optim ising the transfer param eter 621 yields exactly the desired response, as defined in Table 3.1. Since the search is guided by physical insights as described above, the optim iser adjusts the circuit layout within a m ere 350 function calls. By com parison, the sam e optimisation w ithout constraints takes about 600 function calls. The gradient search optim isation cam e to a similarly optim ised circuit within 600 function calls, too. O ne function call here refers to the calculation o f the response at a single frequency point. The optim ised layout is shown in black in Figure 3.7. Altogether, the optim isation takes approxim ately 30 m inutes on an H P-700 series U N IX work station. A fter optim isation, the transfer param eter S 21 satisfies the desired 60 dB isolation at 20 GHz as shown in Figure 3.8. M easured results for this filter are given in C hapter 7. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC A LG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N befo re o p tim izatio n after optim izatio n f[G H z ] [d B ] 19.5 - 20.5 G H z 2 9 .0 - 3 0 .0 G H z -10 -20 -30 befo re o p tim izatio n -40 after o p tim izatio n -50 -60 -70 f[G H z ] Figure 3.8: R esponse o f Stripline Filter Before and A fter GA O ptim isation R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC ALG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N 3.4.2 33 Waveguide Cavity Filter Design This exam ple dem onstrates the optim isation o f an K u-band wave guide cavity filter. The four cavities o f the filter are iris-coupled as seen in Figure 3.9. The optim isation param eters p i to p5 are the iris widths and the cavity lengths. The optim isation goal is a -25 dB pass band at 15 GHz with 3% bandw idth. The analysis is perform ed by the full electrom agnetic m ode-m atching program em _ m ux [153]. P. M P5 N M P5 H M P. MM H Figure 3.9: L ayout o f Four-Pole R ectangular Iris-Coupled W ave-Guide F ilter Showing all A d justable Param eters As seen from the frequency response in Figure 3.10 the genetic algorithm obtains an overall better solution after approxim ately 500 function calls. The gradient search needs 220 function calls in this exam ple, but does not find the global m inim um . The optim isation stops when the algorithm is trapped in a local m inim um. [dB] -10 -20 -25 -30 G ra d ie n t O p tim ise r -35 ------------- G A O p tim ise r -40 14 1 4.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16 f [G H z ] Figure 3.10: O ptim ised R esponse o f W aveguide Filter Layout The quality o f the optimal solution o f the gradient-search algorithm depends strongly on the starting point given. A nother optim al response is found, if another starting point is chosen. By R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC A LG O R ITH M S I N M ICROW AVE CIRCUIT D E SIG N 34 contrast, the GA always yields the optim al response shown in Figure 3.10. The initial param eters m ay affect the convergence speed but do not affect the final optimal solution. 3.5 Optimisation Performance of GA’s In order to com pare the GA’s convergence speed m ore precisely with that o f a traditional gradientsearch optimiser, a standard problem is tackled by both optimisers: A folded stub m icrostrip filter as shown in Figure 3.11 is optimised. m>[t2 ■ j Figure 3.11: L ayout o f M icrostrip Line Filter All line widths are fixed at 4.8 mil. The substrate is 5 m ils thick with a dielectric constant o f 9.9. The design param eters p i to p3 are chosen as shown in Figure 5. F or the gradient search two different sets o f start values as shown in Table 3.2 are used. The starting values are chosen the sam e as in [43]. p l= P2 = p3= Start Set 1 Start Set 2 74 mils 62 mils 73 m ils 61 m ils 1 2 m ils 13 mils Table 3.2: Starting Values for Optim isation o f Folded Stub Filter Figure 3.12 presents the results o f the optimisation. It is seen that the GA finds the best solution overall, w hereas the gradient search (starting point set 1 used) yields a sub-optim al solution. Figure 3.13 shows the decay o f the error with respect to the num ber o f function calls. The GA’s error function decays faster than that o f the gradient search, no m atter which o f the tw o starting points is chosen. The decay rate o f the gradient search strongly depends on the starting point chosen. M oreover, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GENETIC ALG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N S21 [dB] -10 -20 -30 G enetic A lgorithm -50 -60 G radientSearch -70 -80 f [GHz] Figure 3.12: R eturn Loss o f Optim ised Stub Filter 25 Gradient Search Starting Point #2 20 15 G ra d ie n t S e a rc h S ta rtin g P o in t #1 10 5 G e n e tic Algorithm o 100 120 140 function calls per frequency point Figure 3.13: D ecay o f E rror Function for Optim isation M ethods R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. G ENETIC A LG O R ITH M S IN M ICROW AVE CIRCUIT D E SIG N 36 the gradient search does not find the overall solution for either starting point. By contrast, the GA’s error function decays slowly to the overall m inim um. 3.6 Conclusion Genetic algorithm s have been shown to be effective in the optim isation o f m icrowave circuits. Robustness m akes them a good choice for many synthesis tasks. GA’s are particularly effec tive for finding optim al solutions in high-dim ensional param eter spaces, as they do not require derivatives. Although searching by evolution for the best solution is generally known to be slow, it has been shown that genetic algorithm s can som etim es be even faster than directed optim isation m ethods such as gradient search. Especially, when physical insights are included in the algorithm s, and/or when various con straints and objectives m ust considered, the optim isation by “survival o f the fittest” is often the best available optim isation method. However, the superior optim isation speed o f the GA over the gradient search cannot be gener alised. The decay rate o f the error function depends in both m ethods on many param eters, e.g. starting value o f the gradient search, param eter range. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Multi-Dimensional Cauchy Method "If it keeps up, man w ill atrophy all his lim bs but the push-button finger", Frank L loyd Wright A num ber o f software packages are now com m ercially available for Electro-M agnetic (EM ) sim ulation o f m icrowave circuits. These packages provide reasonably accurate results, allowing designs to be im plem ented in a cost effective way. These packages, however, are typically very com putation intensive. It has been well recognised that the CPU tim e and m em ory space required to sim ulate fully integrated m icrowave circuits, using these E M sim ulators, far exceed the capabilities o f today’s com puter workstations. O ver the past years, there has been a strong interest am ong researchers to avoid this problem by using neural netw orks [130], space m apping [72] and param eter extraction [45]. The use of C auchy’s m ethod has also been proposed in [76, 77]. The C auchy m ethod yields a surprisingly accurate m atch o f the com puted points, betw een and even exterior to the sam pled points with the exact solution. However, m ost o f the papers published on applications o f C auchy’s m ethod deal with one dim ensional interpolation, nam ely frequency response interpolation. Here, we extend the one dim ensional interpolation for frequency response interpolation to m ulti-dim ensional Cauchy in terpolation with respect to both frequency and geom etric dim ensions. Two different approaches are suggested to achieve a m ulti-dim ensional approach: A recursive one-dim ensional application o f the standard Cauchy m ethod and m ulti-dim ensional rational function expansion. We start the chapter w ith a short review o f interpolation schemes. In the second part o f this chapter, w e propose the new m ulti-dim ensional interpolation scheme, called multi-dim ensional 37 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 38 Cauchy method. We explain why this schem e is superior to linear, quadratic, and polynom ial interpolation. A t the end o f this chapter, som e exam ples o f applications o f m ulti-dim ensional Cauchy m ethod are given. 4.1 Multi-Dimensional Interpolation In general, the interpolation problem consists o f determ ining a function y o f p from 7 given pairs o f num bers (p1, y 1), (p2, y 2) , . . . , (p7 , y 1 ). These pairs are here called sam ple p oint pairs. The set o f all available sam ple points is denoted by TLThe num ber p m ay be real, com plex, or— for m ulti-dim ensional interpolation- vector-valued (p). The function value y is here considered as one specific response of the circuit, e.g. return loss, w hich is typically complex. The linear m ulti-dim ensional interpolation defines a linear function in the n-dim ensional param eter s p a c e p = ( p i , p 2, ■■• ,p n ) by y(Pl , P2, ■■■i P n ) = Ao + A 1P 1 + A2P 2 -I (4.1) A„ P n w here the A’s are unknown coefficients. The A’s can be derived from n + 1 sam ple points from a system o f linear equations. We can then interpolate any point in the param eter space using eqn. (4.1). The algorithm can be expanded to higher-order approxim ations o f the function. In order to include quadratic term s, the function in eqn. (4.1) can be rewritten as y ( P l , P 2 , ■■■,Pn) = A0 + A lP i + A2p \ + A 3 P 2 + A4 P 2 + A 5 P 1P 2 b M nPn (4-2) This technique requires 2 n + l response samples. The sam e algorithm can obviously be expanded to higher order approxim ations. 4.1.1 Neural Networks Recently, neural netw orks have been introduced into the m icrowave engineering com m unity [125, 126, 130] as a fast and flexible tool for m icrowave circuit response interpolation. The input and output param eters o f the circuit are related by a netw ork of activation functions usually R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 39 arranged in several layers. A typical three-layer feed-forw ard neural netw ork is shown in figure 4.1. F or m ore inform ation about neural netw orks refer to [119]-[134]. N e u ro n s W e ig h te d L in k s input l.a \ -ik k ten I .aver ( Juiput L ayer Figure 4.1: Neural N etw ork M odel o f M icrow ave Circuit Follow ing our definitions, the input and output layer o f the netw ork correspond to the vector p and the vector y, respectively. The input layer and output layer are connected by a hidden layer. In each layer, w e find neurons that are activated depending on the input from the layer on the left. The activation level is com puted from the so-called activation function. A typical activation function is the logistic function 1 ah 1 + e x p (— £ " = i aiWih - @h ) (4.3) w here Wih are the adjustable weights between ith and hth neuron, and ©h is a threshold value. The netw ork has to be trained by a num ber o f sam ple input-output vector pairs (p, y ) in order to determ ine the weight factors and thresholds 0 ^ . These data have to be com puted by an expensive EM simulation. The sam ple p-vectors are chosen randomly, and y is com puted by an EM -sim ulation. During the training process, the neural net autom atically adjusts its weights w iy and threshold values Oh such that the error E betw een predicted and sam pled outputs is m inim ised. The num ber o f hidden neurons m ust be determ ined carefully. Too few neurons m odel the system Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM ENSIO NAL C A U C H Y M ETHO D 40 imprecisely. Too m any neurons tend to over-train the network, such that the training data are m atched nicely, but subsequent testing for unknown points fails. The perform ance o f the netw ork is tested by a second set o f sample vector pairs. These sam ples are not included in the training data set. If these unknown sam ple pairs are m odelled correctly, the netw ork is likely to represent a valid m odel for the m odelled system. A fter training and testing, the netw ork is ready to be used as a substitute to the EM -sim ulator. The netw ork can predict the output y for any given input param eters p within the trained region. The calculation is very fast since only a few basic algebraic operations are required. The m odel once trained can be reused in the design process many tim es w ithout the cost of additional EM -sim ulations. 4.1.1.1 Neural Network Filter Model This exam ple dem onstrates a neural netw ork m odel o f a narrow band three pole filter as shown in Figure 4.2. The m odelling is difficult due to extrem ely non-linear dependence o f the circuit’s response. P3 h Pb Figure 4.2: Planar Three-Pole Filter The netw ork consists o f 5 input neurons, 20 hidden neuron and 4 output neurons and is m od elled using the SNNS [119] neural netw ork generator. The m odel im plem ents five input circuit param eters, including the frequency. The output neurons represent the com plex S-param eters. T he netw ork is trained on 5000 random ly taken sam ples and tested on 500 additional samples. T he com putation o f the sam ples took 80 h. Training o f the network required 20 h. Figures 4.3 and 4.4 show the correlation o f S 'a for a random ly generated layout geom etry and for an optim ised layout, respectively. 4.1.1.2 Limitations of Neural Network Applications We can see that the neural netw ork m odels the approxim ate behaviour o f the circuit’s response correctly. However, an exact m odelling, particularly for values o f the S-param eters close to zero, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETH O D 41 -10 -15 -20 N eural N e t E M -S im . -25 3.86 3.9 3.92 3.94 3.96 3.98 f [GH z] Figure 4.3: Neural N etw ork Approxim ation and Exact EM Sim ulation for R andom Input Vector -105 p—i i < N e u ra l N e t P re d ic tio n § “ -2 0 E M -S im u la tio n -30 -35 -40 -45 3.7 3.75 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2 f [GHz] Figure 4.4: N eural N etw ork A pproxim ation and Exact EM Sim ulation for an O ptim ised Circuit Layout R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 42 is not achieved. The accuracy o f the neural net can be im proved further by adjusting the netw ork size and activation function, and by taking m ore training data points. The latter com es with the penalty o f m ore EM simulations. The requirem ent for m any sam ples originates from the fact that the standard neural netw ork is a black box. Theoretically, it can m odel anything, from m edical diagnosis to stock m arket fluctuations. This apparent advantage is actually a draw back when it com es to m icrowave cir cuit modelling. The netw ork requires a vast num ber o f response sam ples to m odel the circuit response accurately, since it does not have any background knowledge about microwave circuits. In the last years, several researchers focused on the use o f neural networks for m icrowave cir cuit m odelling and synthesis. The general idea is to include knowledge about the general cir cuit behaviour into the netw ork structure. W a t s o n , G u p t a , and M a h a ja n [127] as well as Z h a n g et al. [131]-[134] use special neurons, so called knowledge-based neurons. Each of these neurons includes a coarse m odel o f the circuit’s response. The netw ork itself is only used for m inor adjustm ents o f the coarse m odel. A nother approach by BURRASCANO and MONGIArdo [128, 129] applies Kohonen self-organising maps. The netw ork is split into tw o sections, one for m odelling the general behaviour o f the circuit and another one for adjusting the general behaviour according to a specific response with respect to the input param eter vector. T he new knowledge-based neural networks are an exciting new paradigm for circuit modelling. Including knowledge seem s to be the key towards an efficient application o f neural networks. One drawback o f this idea is, however, that we have to develop a specific network for each type o f circuit. In the following section, we discuss the Cauchy method. The Cauchy m ethod allows a general description o f the response o f passive microwave circuits. N o specific knowledge o f the circuit is required. 4.2 Cauchy Method Linear, quadratic, and spline interpolation schemes lack in ability to m odel poles for finite argu ments of a function property. Due to their nature, there is no possibility for the function value y (p ) to approach infinity for a finite value o f p. The transfer function o f passive m icrowave circuits, however, often includes such poles for finite values o f the circuit variables. An obvious exam ple is the input im pedance with respect to the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 43 frequency u> o f an ideal parallel m icrowave resonator as shown in Figure 4.5. A t resonance frequency, the circuit im pedance approaches infinity. Zresi^res) = 1™ ( j U l^ -L O r e s — :C Z D x ) ----- > OO ^ (4.4) -jj- gL Figure 4.5: Parallel R esonator and Input Im pedance R esponses, like the one shown above, can be represented by rational functions. In A ppendix C we show, that any response from a m icrowave circuit can be approxim ated by a rational function. Consequently, rational polynom ials as interpolation functions yield a m uch closer representation o f the system ’s response than other schemes, e.g. splines. Below, w e develop the m athem at ical fram ew ork for a one-dim ensional rational function interpolation. This algorithm , known as Cauchy method or Pade approximation, is a frequently applied m ethod for interpolation of the frequency response. M any published papers show the alm ost m agical pow er o f frequency response interpolation. Today, the technique is incorporated into many com m ercial microwave sim ulation software packages as so-called f a s t f requency sweep. 4.2.1 One-Dimensional Cauchy Method The system response S ( f ) is described by a rational polynom ial function o f num erator order N and denom inator order D . F or one param eter, m ost often the frequency / , the function has the form n( a0 + °1 / + a 2f 2H Qp + H n = l an f n ^ l +6 1 / +6 2 /2+... - 1+££= !& */"' In order to determ ine a function in the form o f eqn.(4.5) we need k = N + D + 1 arbitrary sam pling points. W hen using k = N + D + 1 sam ple points w e can set up a system o f linear equations whose solution yields the values for the coefficients a o , . . . , ajv and b \ , . . . , bnIn order to avoid m atrix inversion problem s o f the likely ill-conditioned m atrix, another faster and m ore stable algorithm was found by S to er and B u r l is c h . This algorithm o f the Neville type perform s the interpolation on tabulated data in a recurrent manner. The B urlisch-Stoer R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 44 algorithm does not require the inversion o f a matrix. The algorithm is outlined in the following paragraph. L et R \ be the value at f \ o f the unique rational function o f degree zero (i.e. a constant) pass ing through the point ( f i , S ( f i ) ) . Likew ise, define R 2 , R 3 , . . . , Rk- Now let R n be the ra tional polynom ial o f degree one passing through both ( / i , 5 ( / i ) ) and (J 2 , S ( f 2 ))- Likewise -^ 2 3 ) -^ 2 3 ) • • •, R ( k - i ) ( k ) ) ■ Similarly, we proceed for higher order polynom ials up to R m . . . k The polynom ial R. m.. .k is the value o f the unique interpolating polynom ial through all k points, i.e. the desired answer. The various R ’s form a tableau w ith ancestors on the left leading to a single descendant at the extrem e right. F or example, with k — 4 we have S im u la ted S im u la ted Sim u la ted S im u la ted The B urlisch-Stoer algorithm is a recurrent way o f filling in the num bers in the tableau a colum n at a time. It is based on the relationship betw een a daughter and her parents. B y [81], we have R(i+l)...(i+m) R i . . . ( i + m —1 ) (4.6) This technique produces the so-called diagonal rational function, where the degrees o f the nu m erator and denom inator are equal (if k is odd) or where the degree o f the denom inator is larger by one (if k is even). For the derivation o f the algorithm , refer to Stoer and B urlisch [80], To dem onstrate the efficiency o f the one-dim ensional Cauchy m ethod, the results o f different interpolation schem es (linear, spline, and Cauchy) are shown in Figure 4.6. The response o f a four-pole narrow-band filter is sam pled at 20 frequency points only. The linear interpolation does not yield m uch inform ation about the behaviour o f the circuit’s response. The spline inter polation provides better correlation, due to the sm ooth curve fitting; however, the curve does not m atch with the exact solution. The Cauchy m ethod, on the other hand, shows the exact response, even though only a few sam ple points are used. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETH O D 5 S ll 45 Sll [dB] 0 ■5 -10 -10 -15 -15 -20 -20 -25 -3 0 3 .7 -25 ® E xact Solu tion X Sam p le P oints — L inear Interpolation 3.8 3.9 4 -3 0 f [G H z] 4.1 4 .2 4 .3 4 .4 3.7 * E xact Solution X Sam ple P oin ts Spline Interpolation 3.8 3.9 4 4.1 f [GHz] 4 .2 4 .3 4 .4 S ll A •»»>»>»»«•»>»>» -10 -15 -20 -25 -3 0 • E xact Solu tion X Sam p le P oints — C auchy Interpolation f [GHz] 3.7 3 .8 3.9 4 4.1 4 .2 4 .3 4 .4 Figure 4.6: Interpolation o f Four-Pole F ilter Response, a) Linear, b) Cubic Spline, c) Cauchy R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 4.3 46 Adaptive Sampling As seen from Figure 4.6, m any frequency sam ples are taken at frequency bands where the re sponse is totally linear. There is not m uch inform ation gained from these sam ples; rather, we should concentrate the sam pling in bands o f high non-linearity. D haene et al. show in [82] that an adaptive sam pling scheme can be applied in order to focus the sampling in those areas. The technique is illustrated by the follow ing example. First, a few samples, let’s say five, are taken. Using these samples, tw o different rational polynom ial approxim ations are com puted. These two m odels are then scanned for the frequency with biggest mismatch. A t this point, the next sam ple is taken, and the procedure is repeated until both m odels agree. The procedure is illustrated by an exam ple in Figure 4.7. The Cauchy m ethod starts with 5 samples. From these samples, the response is interpolated for a rational function o f num erator order 2 and denom inator order 2. A nother response is interpolated using a rational function o f num erator order 1 and denom inator order 3. The curves are com pared, and the next sam ple is taken at the frequency point with the biggest m ism atch, here / = 9.6 GHz. The com parison is repeated, and m ore sam ples are taken, as long as necessary. A t the end, w e observe, that the sam ples are concentrated in areas w here the response is highly non-linear. Only a few sam ples are taken in intervals o f the response with very linear behaviour. 4.4 Multi-dimensional Cauchy Method We now extend the one-dim ensional interpolation for frequency response interpolation to a m ulti-dim ensional Cauchy interpolation with respect to both frequency and geom etric dim en sions. Two different approaches are suggested to achieve a m ulti-dim ensional approach: a recur sive one-dim ensional application o f the standard C auchy m ethod, and m ulti-dim ensional rational function expansion. 4.4.1 Recursive Cauchy Method The recursive m ethod solves the m ulti-dim ensional interpolation problem by a recursive algo rithm. The algorithm itself perform s a one-dim ensional Cauchy interpolation as described in Section 4.2. From a given set H o f 7 sam ple points and an arbitrary point p*, the algorithm C calculates the interpolated function value S*(p*). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D Sll 5 points [d B ) -10 t y-v -1 5 • •••'* -20 order(num) = 2 ' order(denom)= 2 o rder(num ) = 1 o rd e r(d e n o m )s 3 -2 5 20 -10 -1 5 • o rder(num ) = 3 ' o rd er(d e n o m )= 2 ■, o rder(num ) = 2 *• o rd er(d e n o m )= 3 6 points -20 -2 5 f [G H z ] -3 0 20 -10 -15 7 points -20 « y-\ o rder(num ) = 3 ' order(d e n o m )= 3 ; .**•. o rder(num ) = 2 ’• o rd er(d e n o m )= 4 -2 5 -3 0 -35 flG H z ] -4 0 20 10 0 -10 -20 points -3 0 » '•S -4 0 •, y-\ o rder(num ) = 4 ' order(d e n o m )= 3 o rder(num ) = 3 *• o rd er(d e n o m )= 4 -5 0 f IG H z ] -6 0 10 4 14 20 18 -10 -1 5 -20 -2 5 -3 0 -3 5 -4 0 -4 5 9 points i C' y-\ o rder(num ) = 4 \ o rd er(d e n o m )= 4 o rder(num ) = 3 ’• o rd er(d e n o m )= 5 I IG H z ] -5 0 20 Figure 4.7: Adaptive Sam pling Procedure R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM ENSIO NAL C A U C H Y M ETHO D 48 The set TL is put together by the pairs o f sam pling points p ' to p 1 . The algorithm C can be defined as the function C (p*,TL) w hich yields the interpolated response S* for p* using the sam ples TL. c (p*,n) = (4.7) (p", W , S ' ) , ( p " , S " ) , ( p ' " , S ' " ) , . . . , (p“ , S ” )} ) — > S ' Using these definitions, the algorithm can now be extended to a m ulti-dim ensional interpolation. F or this purpose, the set o f sam ple points m ust be extended from the one-dim ensional sample point set TL to a m ulti-dim ensional sam ple point array. 4.4.1.1 Choice of Sample Points in Parameter Space The sam ple points in an n-dim ensional param eter space are now represented by vectors p = ( p i , P 2 , . . . , p n )- For the recursive algorithm , the set o f sam ple point pairs (p, S ) m ust fall on a com pletely filled grid o f points. The grid does not have to be equidistant. Typically sample locations in the param eter space are shown in Figure 4.8 for tw o and three param eters. P2 ’ Pi pl’ pl” pl pl Figure 4.8: Sam ple Locations for Two-dim ensional and Three D im ensional P aram eter Space E ach dim ension has its own subset o f param eter values, as seen from Figure 4.8, w hich do not change for different values o f the other param eter dim ensions. In the given exam ple the set of sam ples in the p \-dim ension has the four param eter values p[ to p'{". 4.4.1.2 Algorithm Implementation The goal is to interpolate the function value S* o f an arbitrarily located point p* = {p\ ,p*2, . . . , p n* ) The algorithm can be divided into three steps: Step 1: The root process starts by interpolating the point p* with constant p 2 = Ps = P% etc. parallel to the p \ -axis, shown as a dashed line in Figure 4.9(a). This is a one-dim ensional R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 49 interpolation, so the algorithm C defined in eqn. (4.7) can be used. This step yields the desired interpolated point S* = C ( \ p l , p * 2, . . . ] , A ) (4.8) w here A is the set o f sam pling points for p[ to p'{" and p 2 — p,\, • • .=const. These are the points m arked with © in Figure 4.9. They m ay not fall on the grid o f known sam ple points. If that is the case, the algorithm proceeds to step 2 in order to determ ine the points © . O therw ise (the points are known), the algorithm proceeds with Step 3. Step 2: The algorithm calls itself for each o f the unknown points © . In the exam ple, the algorithm starts four new child processes C ((p i,^ )^ t) ,C(( p '",P2),B3) ,C((p?",rf),B4) (4.9) as shown in Figure 4.9(b). The B ’s are sets o f sam ple points with a fixed value for p\ as seen from Figure 4.9. The interpolation is now perform ed along the p 2 -axis. The routine called is exactly the sam e routine as already used in Step 1. The algorithm is thus recursive. Again, each subprocess checks if the set B is from known samples. If not, the algorithm starts another instance o f subprocesses in order to interpolate the points included in B using the next higher dim ension. In the example, this would be p^. Step 3: In case the subprocess determ ines that all sam ple points are known, it calculates the interpolated point using C and hands it back to the parent process w hich requested that point. Finally, the answ er for the root process in eqn. (4.8) will be found. T he iterations o f the algorithm can be represented in an inverted tree diagram (Figure 4.10). Each process C(TL,p*) requests the unknown sam ple points in its dim ension from a num ber of child processes and hands the result o f its interpolation to its parent. The tree term inates with “leaves” -processes that interpolate from know n sam ple points. Adaptive sam pling can be applied to the m ulti-dim ensional Cauchy m ethod. U sing the recursive m ethod, the sam ples m ust lie on a— not necessarily uniform — grid and adaptive sam pling can only be used in one dim ension w ithout constraints. However, when com bining the approaches, the adaptive sam pling can be applied to m ore dimensions. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D CO N3 O’ <D . 50 —a— -fc-a-Q — a o. a. c o io X X »-l £ T3 IS £ s 'o 0- 3 cr o «3 ° -* x • X X X X a X a -0 --X -X ---X 0) 1 -4-* Ul es --© --X -X - (b) Step 2 X X --X - o) Da- X a a X X <N P . X X o. a Figure 4.9: Steps in Recursive C auchy M ethod in Two Dim ensions R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D Figure 4.10: Inverted Tree Representation for Recursive C auchy M ethod R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 51 CHAPTER 4. M U LTI-D IM ENSIO NAL C A U C H Y M ETHO D 52 W hen using the m ulti-dim ensional polynom ial expansion, the choice o f sam ple points is uncon strained in the com plete param eter space, and the sam ples can be picked by the sam e m ethods as in the one-dim ensional case. This m eans, w e can use adaptive sam pling in the sam e way as it was used in the one-dim ensional case. 4.4.2 Multi-Dimensional Cauchy Expansion The m ulti-dim ensional m ethod can also be im plem ented by a single m ulti-dim ensional rational polynom ial o f the form \ S (Pi,P 2,P3,--0 = where P nu m { P i , P 2 , P 3 , ■ ■ •) T>n u m { P l i P 2 , P 3 i ■■■) s — ------------------------ r P d e n ( P l , P 2 , P 3 , - ■■) and Pden i ‘P \ >P 2 ■,P s , ■• ,. l m (4 -1 0 ) ■) are arbitrary polynom ials o f the param eters p i to p n in the num erator and denom inator, respectively. U sing this approach, the coefficients of the function (4.10) are determ ined directly. S a k a ta [ 8 3 ] showed the extension into tw o dim en sions and the general scheme is briefly discussed here. Eqn. 4.10 can be w ritten as a( , QQ + E j i l d j P j ( p i , P2, • • • , Pn) 1 + £?=1 b j P j ( p i , P 2 , ■■■i P n ) w here P j ( p i , p 2, • ■■, p n ) are the m ixed term s o f the existing param eters. The general form of the m ixed term elem ents is Pj = t l P i U) i= 0 ( 44 2 ) where u ( j ) is an integer function o f j . M uch attention m ust be paid to the choice o f the functions u ( j ) . The m ethod used here is to get all m ixed term s o f all param eters p \ , . . . , p n up to a specified m axim um pow er and then sort by power-sum s u sum = Y a =\ u i- Starting w ith the lowest pow er sum, the polynom ial is built. U sing this scheme, the start o f the polynom ial for tw o dim ensions is P { p i , P 2 ) = a 0 + a i p i + a 2p 2 + a 3p i p 2 + a4p l + a 5 p | + a 6p \ p 2 + . . . (4.13) This approach yields a linear equation system. Solving this system determ ines directly the co efficients o f eqn.(4.11) and, hence, a closed-form and differentiable equation o f the system ’s response S ( p 1, p 2, p 3, •. ■)■ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 53 As there are no restrictions whatsoever, a m ulti-dim ensional adaptive sam pling o f the param eter space can be applied in analogy to the one-dim ensional case. It should be m entioned that for large dim ensional problem s, it would be m ore efficient to split the problem into several problem s o f low er order, which are then solved recursively as shown in Section 4.4.1. 4.5 Numerical Results Three exam ples are given here to dem onstrate the interpolation using m ulti-dim ensional Cauchy method. All EM -sim ulations are perform ed using the Sonnet em planar electrom agnetic solver. A ll CPU tim es given refer to com putations on a H ew lett Packard K-class machine. 4.5.1 Microstrip Line Impedance The m ulti-dim ensional rational function expansion is dem onstrated by m odelling the line im pedance o f a m icrostrip line with respect to the line’s width-to-height-ratio f and the relative dielectric constant er o f the substrate. The m odelling algorithm described in Section 4.4.2 returns the follow ing closed-form rational function: Z ( p x = er , p 2 = *jr) = 5 4 9 .6 2 3 2 + 1 1 7 . 0 0 7 8 p x + 9 7 9 9 . 3 8 3 9 p 2 1 .0 + 0 . 4 5 0 2 1 p ! + 3 6 .5 7 0 9 p 2 - (4 -14) 7 . 4 3 6 7 ^ + 1 5 3 6 .0 6 5 2 p | + 2 9 9 .3 8 4 8 /> ip 2 + 0 .1 6 1 2 3 p f - 0 .0 1 9 4 6 p 2 + 1 2 . 5 5 1 9 p 2 + 3 6 . 3 2 6 3 0 p i p 2 + 7 . 9 0 6 E - 5p f - 1 7 .9 0 0 3 2 5 p | + 3 6 3 .8 9 4 7 2 p fp 2 0 .1 1 6 0 5 p 3 + 1 7 .7 3 3 8 7 p 2 p 2 _ 4 . 6 6 4 5 p i p | The sam ples are determ ined by an adaptive sam pling technique sim ilar to the m ethod used in [82]. Figure 4.11 shows the m odel approxim ation and the 19 adaptively taken sam ples. The m odel provides an accuracy within 4.5.2 0 .1 % error. Recessed Line Fed Microstrip Antenna Heuristic m odels o f m icrowave circuits are restricted to very basic topologies such as lines, gaps, step etc.. The m ulti-dim ensional Cauchy m ethod allows the creation o f fast and accurate m odels for any kind o f topology, as long as a param eterised EM -sim ulation is feasible. Here, the building o f a generic m odel for an recessed line fed m icrostrip antenna in C -band on 10 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D Z[Q] 54 200 150 100 0 .9 0.8 0/7 S 0.6 0 .5 0.3 ^ w / h Samples points 0.2 20 25 Figure 4.11: Im pedance o f M icrostrip L ine as a Function o f er and w / h using M ulti-D im ensional Rational Function Expansion m il-Rogers RT/duroid™ 5870 substrate (Figure 4.12) is discussed. An accurate m odel for such elem ents does not exist, and up to now, the design is based on simplified cavity m odels [85] or an expensive search by a vast num ber o f E M -sim ulations, which m ust be repeated whenever a new slightly m odified design is requested. Figure 4.12: M icrostrip A ntenna with Recessed line feed The m ulti-dim ensional Cauchy M ethod overcom es these problems. A ll expensive EM -sim ulations for the m icrostrip antenna are com puted at once. The problem has four dim ensions, the param eters p i to pz and the frequency / . Five param eter values per dim ension are required. H ence, the responses o f 5 4 = 625 sam ples have to be com puted for a com pletely filled grid o f sample points. This data acquisition phase requires approxim ately 10 h o f C PU time. Even though this initial effort is quite high, it pays off, as the established m odel can be used for a large variety o f antenna layouts w ithout any additional expensive com putations. The sim ulated geom etries fall on the follow ing grid: R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETH O D 55 pi = 17.5 m m , 20.3 m m , P2 - 0.0 m m , 2.8 m m , 5.6 m m , 8.4 mm, 11.2 m m pz = 2.8 m m , 5.6 m m , 8.4 m m , 11.2 mm, 14 mm / = 3.4 GHz 3.7 GHz 4.0 GHz 4.3 GHz 4.6 GHz 2 3 .1 m m , 25.9 mm, 28.7 m m The frequency response for the m odel is verified at an unsam pled arbitrary geom etry with p \ = 24.5 mm, p 2 = 1.4 m m , pz= 4.2 mm. The response com puted by full E M sim ulation and by the m odel is shown in Figure 4.13. As seen, the m odel is in very good agreem ent with the EM sim ulated result. a o -J E M -Sim ulation — M ulti-D Cauchy E 3 (U Ot 3 .6 3 .7 3 .8 3 .9 4 4 .1 4 .2 f [GHZ] Figure 4.13: R eturn Loss o f A ntenna G eom etry p \ = 24.5 m m , p 2 = 1.4 m m , pz= 4.2 m m C om puted by M ulti-D C auchy M odel and Full EM -Sim ulation 4.5.2.1 Parameter Plot of Feed Location Using the m ulti-dim ensional Cauchy m ethod m odel, the antenna is optim ised for m inim al return loss at 4 GHz. The effects o f the feed location on the return loss is shown by a param eter plot of the return loss in Figure 4.14. The param eter plots contain the inform ation o f 2000 frequency points. To obtain the sam e in form ation using EM -sim ulators, one has to perform a sim ulation that takes 33 C PU hours. O ur Cauchy m odel com putes the sam e inform ation in less than 4 CPU seconds. It can clearly be seen, that the established m odel enables the designer to investigate the circuit’s perform ance with respect to m odifications w ithout any additional expensive calculation. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 56 CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D O' -5 ..__= 1 8 .0 6 m m p 3= 5 .1 8 m m -10 -1 0 W -1 5 2. 3?-20 P2 = \ o c -2 5 1 2 .6 m m S -3 0 11 .2 m m ^ / / 9 ‘ 15 “C/5 -2 0 / -40 -45 3 .9 p 2= 7 .0 m m 4 .2 m m 4 .3 4 m m 4 .4 8 m m 4 .6 2 m m 4 .7 6 m m 4 .9 m m 5 .0 4 m m V u X /// " 1.4 m m e ’25 § -3 0 9 .8 m m - '''" ”" ^ W —— —2 .8 m m “ -35 " 4 .2 m m -4 0 OS -35 Pj = 18.06 m 8.4 m m (M— " - 5 .6 m m 7 .0 m m 3 .9 2 3 .9 4 3 .9 6 3 .9 8 4 .0 2 4 .0 4 4 .0 6 4 .0 8 5 .4 6 m m 5 .3 2 m m 5 .1 8 m m -45 1 4 5 .6 m m 4.1 3 .9 3 .9 2 3 .9 4 3 .9 6 3 .9 8 4 4 .0 2 4 .0 4 4 .0 6 4 .0 8 4.1 f [G H z] f [G H z] Figure 4.14: E ffect o f Inset location on A ntenna’s R eturn Loss 4.5.3 Narrow-Band 3-Pole Filter In this example, the S-param eters o f the response o f a planar superconductive m icrostrip filter as shown in Figure 4.15 are interpolated. The five param eters are the four geom etrical param eters, nam ely the gap length and resonator lengths, and the frequency. A sam ple grid with 5 sample points per dim ension for the geom etrical param eter is used. Figure 4.15: L ayout o f Three-pole Filter Showing Param eters p l to p4 D ue to the underlying lim itation o f the EM -sim ulation software, the param eter values are forced to lie on a 1.75 mils grid. The frequency dependency is determ ined by adaptive sam pling, as the last level o f the recursive algorithm . The frequency is chosen because it shows by far the largest variation o f the S-param eter response. Shown in Figure 4.16 is the interpolated response for the param eter values G ap length p l = 6.125 G ap length p2 = 16.625 Line length p3 = 23 8.875 Line length p 4 = 238.875 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 51 In com parison w ith the exact solution— obtained by a finer m eshing and finer frequency stepping— one notices the very good agreem ent o f the interpolated response with the exact solution. In Figure 4.17, the response is shown when the m ulti-dim ensional Cauchy m ethod is not applied. D ue to the restrictions that all geom etrical values have to fall on the 1.75 mils grid, the param eter values m ust be snapped to the next grid-point. Figure 4.17 shows the response for both rounded up and rounded down values for the param eters p \ to p^. In addition, the frequency resolution is lost, as only the sam pled frequencies can be shown. A s a result, the tw o-ripple filter response degrades to a m eaningless polygon. 4.5.3.1 Optimisation, Monte-Carlo Analysis U sing the m odel, the filter is optim ised with respect to the specifications: S n < -20 dB for 3.9 GHz < / < 3.95 GHz S 21 < -20 dB for / = 3.85 GHz and / = 4.00 GHz The optim isation by gradient search requires 19 iterations, which corresponds to 1140 function calls. A full EM -sim ulation would require a C PU tim e o f at least several days, as the circuit’s re sponse m ust be repeatedly com puted on a very fine grid. The m ulti-dim ensional Cauchy M ethod solves the problem within 3 CPU-m in. Having the m odel on hand, the developer can check the sensitivity o f the m odel by a M onteC arlo analysis. Again, the analysis— known to be expensive or not feasible w hen EM -sim ulation is used— can be applied with little effort using the predefined model. In Figure 4.19, a random variation o f ± 0 .1 m ils is added to all geom etrical param eters. N either optim isation nor M onte-C arlo analysis can be perform ed using direct EM -sim ulation within reasonable com putation time. The num ber o f required points exceeds even the capabil ities of m odem m ulti-processor com puters. M oreover, the planar electrom agnetic solver is not capable o f handling geom etries w ith sm all variations, as all vertices m ust fall on a relatively rough grid. The m ulti-dim ensional Cauchy m ethod m odel overcom es these problem s. A fter one expensive generation o f an on-grid database, all follow ing com putations o f arbitrary circuit variations can be obtained by the inexpensive recursive algorithm. 4.6 Conclusion In the past, m any publications have shown the rem arkable reduction o f com putational cost when Cauchy m ethod and adaptive sam pling are applied to the frequency response interpolation. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM ENSIO NAL C A U C H Y M ETHOD 58 5 S ll [dB ] ° ■5 -10 -1 5 -20 -2 5 -3 0 M u ltid im en sion al C auchy ° E xact S olu tion -3 5 -4 0 3 .7 3 .7 5 3 .8 3 .8 5 3 .9 3.9 5 4 4 .0 5 4.1 4 .1 5 4 .2 f [G H z] Figure 4.16: Interpolated 5 n Param eter U sing M ulti-D im ensional Cauchy M ethod S ll [d B ] -10 -15 S am p le b e lo w -20 S am p le ab ove -25 -30 -35 3.7 3 .7 5 3 .8 3 .8 5 3 .9 3 .9 5 4 4 .0 5 4.1 4 .1 5 4 .2 f [G H z] F igure 4.17: S u Param eter W ithout U sing M ulti-D im ensional Cauchy M ethod Show ing the Sam pled Points Above and Below R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D 59 -10 -35 *■ E M -Sim ulation -40 — C auchy-M ethod -45 3.85 3.9 3.95 4 .05 f [G H z] Figure 4.18: R esponse o f O ptim ised Three-Pole Filter This C hapter has shown that the m ethods can be extended and applied to m ulti-dim ensional problem s. This can be either done by recursive application o f the Cauchy m ethod or by a m ulti dim ensional rational polynom ial expansion. In doing so, sim ilar savings o f com putational expenses for m ulti-dim ensional problem s can be achieved as for the one-dim ensional case. In the exam ples given, we showed that a com plete and accurate num erical m odel o f a three-pole filter w ith 4 geom etrical param eter plus frequency dependency can be obtained. A s shown, we can apply the derived m odel to com plex operations, such as optim isation or M onte-C arlo analysis. The operations are perform ed at very little com putational expense. The m odelling o f m icrowave circuits using the m ulti-dim ensional C auchy m ethod is restricted to a few (~ 5 ) param eters, though, because the num ber o f simulations increases exponentially with the num ber o f param eters. Optim isation o f large circuits is not feasible with the m ethod proposed in this chapter. In the next chapter, we develop an optim isation schem e for large circuits. The algorithm uses com plete and decom posed circuit sim ulations to determ ine the optim ised layout fast and accu rately. F or an extrem ely fast calculation o f the decom posed circuit response, the decom posed sim ulation uses block models based on the m ulti-dim ensional Cauchy method. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. M U LTI-D IM EN SIO N AL C A U C H Y M ETHO D -10 -3 0 -35 -4 0 -45 L -5 0 3.8 3 .85 3.9 3.95 f [G H z] 4 4 .05 4.1 Figure 4.19: M onte-C arlo Analysis o f O ptim ised Three-Pole filter R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Hybrid Optimisation Using Coupling Matrix Alteration "There are only tw o w ays to live y o u r life. One is as though nothing is a m iracle. The o th er is as though everything is a m iracle. ", Albert Einstein The quality o f an iterative optim isation o f m icrowave circuits is determ ined by the underlying circuit analysis m ethod. Full electrom agnetic sim ulation is the m ost desirable analysis m ethod in the optim isation process as it attains the highest possible accuracy. Full E M sim ulation, how ever, com es w ith very high com putational costs. It is w ell known that these costs can becom e prohibitive for large circuits. Consequently, direct full EM optim isation o f large circuit is not feasible with the current tools. Large circuits can be usually optim ised by a sim plified circuit analysis only. The simplification m ay decrease accuracy, but it m ay t be the only choice if optim isation is necessary. Using simplified circuit m odels for optim isation often yields an inaccurate but sufficient solution o f the problem. The use o f the exact m odel can be restricted to the final verification o f the solution. If w e require both the high accuracy o f the full E M m odel and the speed o f the sim plified m odel, w e can com bine them. We call this a hybrid optim isation technique. In this chapter, w e discuss existing hybrid optim isation techniques and their limitations. We also propose a new m ethod especially tailored to an efficient m icrowave filter design. O ur technique is based on the coupled resonator representation o f the filter. 61 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A T IO N 5.1 62 Background Any hybrid optim isation relies on tw o sim ulation models, a fine model and a coarse m odel. The fine m odel is accurate but com putationally expensive, w hereas the coarse m odel is fast but less accurate. The coarse m odel is used to optim ise the circuit, and the fine m odel is used to verify o f the results and to correct the m odel based on the fast calculations. B oth m odels are essential in the hybrid optim isation process. For planar circuit optim isation, as discussed here, the coarse m odel is usually represented by a decom posed E M analysis or circuit theory analysis. The fine m odel is norm ally represented by a com plete E M analysis. A detailed outline o f the tw o m ethods is given in C hapter 2. H ybrid techniques generally w ork in tw o phases: a training phase and an application phase. D uring the training phase, w e collect data about the coarse and the fine m odel for a few specific circuit geom etries. B ased on this data, we derive an algorithm which adjusts the coarse model. O ur goal in the training phase is to derive an adjusted coarse m odel w hich m im ics the fine m odel very closely. In the application phase, we use this corrected coarse m odel to inexpensively com pute new, unknown circuit geometries. In the next section w e discuss some hybrid optim isation techniques and their lim itations. Later in Section 5.2 we will introduce a new hybrid optim isation scheme. O ur technique is particularly tailored towards filter optimisation. The proposed approach uses a coupled resonator filter m odel for correcting the coarse m odel. We could apply the correction o f the coarse m odel directly to the m odel’s output param eter, i.e., the S-param eter response. This approach, however, is not successful since the corrected m odels do not obey the law o f conservation o f com plex power. In this section, w e outline why. L et us define the S-param eter response o f the coarse m odel as the m atrix [S]c . T he response of the fine m odel is defined as the m atrix [S]f. B oth m atrices are functions o f the circuit param eter vector p. The vector p represents all param eterised layout features, e.g. line width and substrate thickness. Now, w e define a difference S-m atrix [A S ^ p ) by [A S ](p) = [S]f (p) - [S]c(p). (5.1) T his m atrix describes the differences betw een the fine m odel’s response and coarse m odel’s response. Since w e investigate frequency responses, the difference m atrix is actually an array of m atrices corresponding to a list o f frequency points, i.e. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID OPTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N ([A S]) = ( [ A S K /i), [A S K / 2 ), [A S ]( /3) , . . . ) 63 (5.2) For convenience, w e refer to this array sim ply as the m atrix [A S], even though it is really a list o f matrices. Now, we perform one coarse m odel sim ulation and one fine m odel sim ulation each with the arbitrary param eter start vector po. Then, w e calculate [A S ](jJq). Next, w e w ant to optim ise [S]f with respect to p. Since the calculation o f [S}j(p) is expensive, the optim iser uses the coarse m odel. To im prove the coarse m odel’s accuracy, w e adjust its response [S]c(p) to obtain a new S-m atrix [S]*(p) = [5]c(p) + [ A S ] ( f t ) . (5.3) In other words, the difference determ ined at po is added to all subsequent calculations o f [S]c. W e can divide hybrid optim isation algorithm s into tw o phases: a training phase and an appli cation phase. D uring the training phase we determ ine a [AS] from a coarse and fine model simulation. D uring the application phase, w e use the coarse m odel sim ulation exclusively and adjust its response by [AS]. The process is sum m arised is in Figure 5.1. Training: Application: F in e M o d el S im u la tio n l.\S| Simulation [AS] Figure 5.1: Training and Application Phase s In the case o f p = po, i.e., the layout used for training, the corrected m atrix [S']* is identical to the fine m odel’s m atrix [S]j by definition. However, if p A Po the adjusted m atrix [S]* represents an approxim ation o f [ S] f only. The corrected m atrix [S]*.(-p) and the accurate response [S'] / (p) are equal only if the difference m atrix [AS] is not a function o f p. F or real problem s, [AS] is always a function o f p. The question is w hether it is a slowly changing function. If yes the the approxim ation is likely to approxim ate the fine m odel well. If no the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID OPTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 64 approxim ation is not valid. The answ er to this question depends on the problem. In general, however, [AS] changes rapidly with p. A typical exam ple is outlined next. L et us calculate [AS] for the tw o-pole filter given in Figure 2.3. The coarse m odel is given by the circuit theory sim ulation, and the fine m odel is given by the full-em simulation. The difference m atrix [AS] is calculated for tw o slightly different geom etries. The first layout, w ith the param eter vector p \ , is the layout discussed in Figure 2.3. The second layout, with the param eter vector p 2 , is identical except for a slightly longer resonator line. As a result, the response S ( p 2 ) is shifted downwards in frequency w ith respect to S (p \). The tw o responses in logarithm ic scale are shown in Figure 5.2. f r eq freq f r eq freq T racel-dBtSU .tJ) Tricel-dBCSC1,1]) Trace2»dB(FULLEM_DESIGN_AFS. . SC I, 13) Tr 8 ce 2 *d B( DE SI GN _s hi f te d _f ul lent AFS. . S C I , 1]) Figure 5.2: R esponse o f C ircuit L ayout p \ and j72■ Now, let us take a look at the difference m atrix [A S]. The real and im aginary parts o f A S n for both layouts are shown in Figure 5.3. It can be seen, that A 5 n (pi) and A S u (]>>) are quite different. Consequently, a correction as proposed by eqn. (5.3) fails because [AS] changes dram atically when we go from p \ to p ‘2 - Figure 5.4 shows [S*}(p 2 ) when we use [AS] ( p \ ) for correction, i.e. [S*}(P2) = [Sc](p2) + [ A S M ) • (5-4) The corrected coarse m odel response S * u (p 2 ) is obviously not a good approxim ation o f the a ccu ra te fin e m o d e l r e s p o n s e [S]^(p 2 )- E v e n w o r s e , S * 11( p 2 ) e x c e e d s 0 d B in d ic a tin g n e g a tiv e resistance, hence, violating the unitary condition. In conclusion, the fixed S-param eter correction does not w ork properly, even for a sim ple exam ple as shown above. The application o f this algorithm to the Y-, Z-, and A B C D -m atrix, generally results in the sam e failure. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A TIO N R eal-P art of Delta-511 of L ay o ut LI and L a y o ut L2 65 Imag-Part of D e l t a - S l l of Layouy LI and L ay o ut L2 TraceB Trace? 2.0 2.0 « BQ . * A ’ j / -2.0 -2.0 V_ fti t ■\ GHz freq freq 3.9 3.9 Trace5-Real(OESIGN_orig..SCI,m-Real(DESiGN_orig..SC3,3]) TraceG»real(design_shifted..SC I , U)-Real(design_shifted..SC3,33) GHz GHz freq freq 4.1 4.1 GHzB GHz A Trace8*Ii*ag(DESIGN_orig..SCI,13)-Inag(DESlGN_orig..S[3,31) Trace7*Imag(deiign_shifted..S[l,13)-Iinag(design_shifted..SC3,3]) Figure 5.3: R eal Part o f A 5 n ( f ) for L ayout p\ and L ayout p 2 (Left), and Im aginary Part o f A S ' n ( f ) for L ayout p i and L ayout p 2 (Right) freq diff=design_orig.siml.SP.S[3,3]-des adj_ shifte ds design_shifte6.siml.SP.! Trace3=dB(adj_shifted) Figure 5.4: A djusted S-Param eter S * n ( f ) for L ayout p 2 U sing [ A 5 ] (pi) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 5.1.1 66 Space Mapping From the responses shown in Figure 5.2 we observe that the responses are very similar, except for a slight frequency shift. It m ay be possible, to adjust the response o f the coarse model by simply rescaling the frequency axis. L et us assum e the frequency shift betw een [Sjc and [S]f is constant or nearly constant for different geom etry param eters. Then, we can correct the coarse m odel response by rescaling the frequency variable. This approach changes the correction focus from the output (S-param eter) to the input (frequency). We can show that the adjustm ents can also be applied to other input variables, nam ely the param eter vector p. The system atic application o f this m ethod, called space m apping, was first suggested by B a n d le r [72]. An outline o f the space m apping algorithm is given in A ppendix A. The training and application phase of the space m apping algorithm is discussed in Figure 5.5. Training: Application: C o arse M odel S im u latio n Figure 5.5: Training and Application U sing Space M apping Space m apping always yields results com pliant to the unitary condition. Since we change the input parameter, the derived response m ust conform with the unitary condition as long as the param eter values are physical m eaningful, e.g. non-negative line-width. Stability and robustness o f the space m apping algorithm is questionable . A global m apping as described in A ppendix A.3 cannot be found in all cases. The param eter correction is a m athe m atical construct only, with little or no physical m eaning. The m apping is always local in the param eter space. Thus, strong derivation betw een the fine and coarse m odel can yield to a m ap ping legitim ate at the m apped point 'p*c but not at the desired optimal point j7?.pt. This m apping is said to be local. These problem s can often be circum vented with m ore iterations and sophis ticated algorithm s for deriving the m apping function as suggested in [73] w ith the thrust-region R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CH APTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 67 algorithm . M ore im portant is the fact that the param eter adjustm ents— as done by space m apping— have fundam ental limitations. The control o f the coarse m odel response is restricted to m anipulations w hich can be controlled by the m odel input param eters. Certain electrom agnetic phenom ena de tected in the fine m odel only— and thus not accessible in the coarse m odel— cannot be integrated in a param eter space m apping. F or instance, m any coarse m odels neglect radiation effects and stray coupling etc. in their derivation. These effects cannot be adjusted or included by tw eaking the accessible param eter vector p. A n exam ple o f this effect is given by a typical three-pole filter described in Figure 5.6. We sim ulate the filter as a com plete circuit (fine m odel) and by decom position (coarse m odel). F ig ure 5.7 shows the sim ulation results. It can be clearly seen, that the filter response is quite different betw een com plete and cascaded sim ulation. M ost notable is the transm ission zero at / = 3.96 GHz. W e see it as a dip-down in the logarithm ic S 2 1 response. Finite transm ission zeros are only possible for elliptic filter transfer functions. Thus, the zero m ust be caused by a stray coupling betw een the first and third resonator. This coupling is sketched in Figure 5.6. 3 Figure 5.6: Three-Pole Filter, a) C ascaded (Decom posed) Layout, b) C om plete Layout The decom posed m odel does not allow param eter adjustm ents which can excite this stray cou pling, because there is no physical connection betw een the first and last resonator in the cascaded m odel. Consequently, space m apping is not feasible for resonator coupled structures o f this kind. 5.2 M-Parameter Correction In order to include all couplings into the coarse m odel, we switch to the coupled resonator filter representation. This representation describes a resonant-coupled filter com pletely including all cross-couplings. Cross couplings are defined as couplings between non-adjacent resonators. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 68 0 -10 -10 -20 c2o CM -20 -3 0 -4 0 -3 0 -5 0 <n -4 0 -6 0 -7 0 -5 0 -8 0 -6 0 -9 0 3 .9 3 .9 2 3 .9 4 3 .9 6 3 .9 8 4 4 .0 2 4 .0 4 4 .0 6 4 .0 8 4.1 f [GHz] 3 .9 3 .9 2 3 .9 4 3 .9 6 3 .9 8 4 4 .0 2 4 .0 4 4 .0 6 4 .0 8 4.1 f [GHz] Figure 5.7: C ascaded and C om plete Sim ulation R esults o f Three-Pole Filter. Clearly Seen in the Com plete Sim ulation is a Transm ission Zero at f=3.96 GHz Due to Stray C oupling B etween Resonator O ne and Three. Figure 5.8 illustrates the description o f a filter using direct coupled resonators. The theory of direct coupled resonant structures is developed in detail in [30], [31], and [32]. Resonator 1 Input Coupling Resonator 2 Coupling Resonator 3 Output Coupling Coupling Resonator 4 Resonator 5 Coupling lie Figure 5.8: D irect C oupled Resonators A brief review o f the coupled resonator filters is given in the next section. In Section 5.2.3, we will show how we can extend the theory to a new hybrid optimisation scheme. 5.2.1 Coupling Matrix Representation Figure 5.9 shows a m ultiple inductive coupled iV-resonator filter structure. The resonators are assum ed identical, i.e. all resonances are at the sam e frequency m alised im pedance Z ujq — and with the nor — 112. The inductive coupling between the ith and j t h resonator R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A TIO N is denoted 69 — w hich are real num bers1— and assum ed to be frequency independent2. The resonators can include loss denoted by an ohm ic resistor R n . Further we denote the source and load im pedance as i ^ a n d R j y respectively. M M 12 M 23 M 34 N -2 .N -1 M n _ i >n Figure 5.9: Inductive C oupled Resonators The loop equations can be w ritten as h (R i + j{uL —j i \ M \ 2 + *2 - j i 2u M i 2 ( R 2 + j(uL — jiNMiN = e% — j i2wM23 — ■■■ — j i N M 2N — 0 = 0 = 0 : : : —j h M N i — j i 2LoMN2 — - ' - + j i N M i N w here M mn = M nm for all couplings. U sing the above norm alisations for ujq = and Z q = = 1 we can use the following abbreviation j { u) L - - ^ ) + R — j Z o w here A = Z q cu u)o U)o U) + Rn — j A + Rn , (5.5) . N o w , o u r lo o p e q u a tio n s ca n b e w r itte n in m a trix n o ta tio n [[R\ + y(A[7] - u[M})} i = [Z] i = e 'Appendix B discusses w hy w e can use pure real coupling values 2Frequency dependence can be included by a dispersion factor R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (5.6) CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A TIO N 70 where ( Ri 0 0 R2 0 0 ••• ■■■ 0 0 0 0 \ (5.7) [ R ] V o Rn o Rn J and ( Mu M 2\ M\2 M\n 22 M 2n M 2n M nn M nN m ■ M in [M} = (5.8) M ni \ M n i M M n2 ■■■ M nti ■■■ M N n } and [/] is the identity m atrix and e is the voltage excitation vector, i.e. ( ei \ e = (5.9) V o T h e main diagonal elem ents M nn are zero in the tuned case, i.e. all resonators have the sam e resonant frequency loq. The coupling m atrix com pletely describes the electrical behaviour of the filter. Thus, it is possible to represent any resonant band-pass lum ped elem ent filter by its coupling m atrix. Since distributed elem ent filters can be approxim ated with lum ped elem ent resonators using R ichard’s transform (see[27]), we can also represent those filters by their cou pling matrix. R ichard’s transform , however, is only valid for relatively small band-w idths (10% to 20%). Thus, the coupling m atrix representation is, strictly speaking, only valid for lum ped elem ent band-pass filters and narrow -band distributed elem ent band-pass filter. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 71 We can com plem ent the coupling m atrix representation with a dual circuit. A detailed outline of this duality is discussed in A ppendix B .l. 5.2.2 Coupling Matrix o f Three-Pole Filter We can use the coupling m atrix representation to describe the three-pole filters from the last section in Figure 5.6. The coupling m atrix o f the cascaded filter is / [M\c = V .0 0 .82 .0 0 .82 .0 0 .82 .0 0 .82 .0 0 \ (5.10) The self couplings are zero, i.e. the resonators are tuned exactly to the frequency luq. The m utual coupling M \2 and M 2 3 are different from zero, i.e. adjacent resonators are coupled. There is no coupling betw een resonator one and three, as seen from M = M 31 = 13 0 . The coupling m atrix o f the com plete full EM model -.2 0 .83 .83 1 \ .0 0 (5.11) 0 .83 [M]f -.2 0 to .83 -.2 0 is different. We observe now a coupling betw een resonator one and three, corresponding to the stray coupling. D ue to the m utual inductance M 13 the self coupling o f M \ \ and M 33 changes as well. D etails about the reasons are given in A ppendix B. M ost notable is the difference [A M ] = [ M \ f - [M ]c = V .20 - .0 1 .20 - .0 1 .00 -.0 1 .20 - .0 1 (5.12) .20 / which contains all stray couplings not considered by the coarse m odel, but included in the fine model. H ence, we can use this difference m atrix [A M ] for correcting a coarse m odel in a hybrid optimisation. The detailed algorithm is presented in the next section. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 5.2.3 72 M-Parameter Corrected Optimisation A s illustrated by the exam ple above, the difference coupling m atrix representation can m odel additional couplings introduced by upgrading from a cascaded analysis towards a full-em anal ysis. Consequently, we can use the difference m atrix [A M ] for correction o f a cascaded circuit analysis. The exact procedure is outlined below. L et us define the response o f the cascaded analysis with the layout param eter vector p as [S}c (p) and the response o f the Full-EM analysis as [S]f(p). From the response w e can derive the respective coupling m atrices [M ] c(p) and [ M] f (p ). The difference m atrix o f M is then defined as [A M ] ( p ) = [M]f (p) - [M]c(p) (5.13) The m atrix [A M ] (p) contains all additional couplings not included in the cascaded analysis plus corrections o f all other couplings. We can use [A M ] (p) to adjust the coarse m odel response at any param eter vector p x such that the adjusted response [M]*c(Px) = [M}c{p) + [A M ] ( p ) (5.14) m odels the real (full-EM ) circuit m uch closer. Since [A M ] ( p ) is constant for small changes o f the geometry, the same [A M ] (p) can be used to correct the coarse m odel response for any layout containing sm all geom etrical changes. In the exam ple below, we will see that [A M ] is constant with respect to p. Sim ilar to the other hybrid m ethods, our m ethod requires a training phase to acquire [A M ]. The subsequent application phase em ploys the difference m atrix for correction o f the coarse model. The procedure is outlined by Figure 5.10. Training: Application: Fine — |S |P— » | M ] , ------ ;r - — |S ] C 1M] ,— < 3— (M i; — IS) • M |c ------ Figure 5.10: Training and Application o f Coupling M atrix Correction R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 73 T he proposed technique can be incorporated into a hybrid optimisation scheme. R ather than calculating the accurate response in the loop using direct EM simulation (the fine m odel), we derive the response from the adjusted coarse m odel. The m atrix [A M ], required for adjustm ent, is derived in the training phase from just one response o f one full EM sim ulation and the cor responding coarse m odel analysis. This techniques offers a high accuracy in the optim isation w ithout the costs associated with direct EM simulation. F or the coarse m odel w e usually em ploy a decom posed circuit analysis. F or even m ore effi ciency, w e m odel here the sub-circuits o f the decom posed circuit using our m ulti-dim ensional Cauchy m ethod presented in C hapter 4. In doing so, w e take all expensive calculations out o f the optim isation loop. T he circuit analysis within the optim isation becomes ju st a m atter o f database requests and m inim al algebraic com putations. The flow o f the proposed optim isation schem e is outlined in Figure 5.11. ( Start ^ Parameter M odifications C ascaded Circuit A n alysis by M -C auchy Interpolation C ascaded Circuit A n alysis Full-EM Circuit A n alysis A djustm ents using [AM] R equirem ents S atisfied ? / ,yes F ull-EM Circuit A n alysis R equirem ents . S atisfied ? / -yes F igure 5.11: Optim isation Including Correction o f Cascaded R esponse Note, that at the end o f the optim isation process the perform ance o f the optim ised structure is checked by another single Full-EM simulation. If the perform ance passes the specification the synthesis is com plete. If the check fails, the correction [A M ] is not valid for the new layout, i.e. [A M ] is a function o f p. In that case, a new correction [A M ] is calculated using the already known responses o f the full EM and cascaded sim ulations. The optim isation process is repeated R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A T IO N 74 using the new [A M ] . 5.3 Numerical Example O ur novel optim isation technique is dem onstrated in the design of a 6 -pole narrow -band filter. T he filter— part o f a superconductive satellite com m unication system [74]— is realised as a pla nar thin-film T B C C O 3 m icrostrip circuit on a L aA 1 0 3 -substrate. Specifications for the filter are given in Table 5.1. C entre Frequency 4.5 GHz 3dB -B andw idth Pass band return loss Stop bands Stop band insertion loss 100 M H z <-17 dB f<4.44 GHz and f>4.56 GHz <-30 dB Table 5.1: Specifications for Narrow B and HTS Filter The initial filter layout is shown in Figure 5.12. We em ploy a com plete full EM sim ulation using Sonnet em [150] as the fine m odel. A decom posed circuit analysis as shown in Figure 5.13 is used for the coarse model. We m odel each sub-section o f the circuit using the m ulti-dim ensional Cauchy method. Figure 5.12: Layout o f Six-pole Filter The optimisation o f the filter breaks down into 6 steps: 1. Param eterisation and sim ulation o f the sub-circuit 2. Optim isation o f the coarse m odel 3 Thallium-Barium-Calcium-Copper-Oxide, a high-temperature superconductor, see [139][140] R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 75 Figure 5.13: Cascaded Six-pole Filter 3. Com parison o f the coarse m odel’s response with the fine m odel’s response 4. Deriving the correction m atrix [A M ] 5. O ptim isation w ith corrected coarse m odel using [A M ] 6 . Verify validity o f optim ised circuit. Step 1: Parameterisation and Simulation of the Sub-Circuit Each resonator in the filter is described by one sub-circuit. The param eters o f each resonator are the resonator length and the width o f the tw o coupling gaps as seen in Figure 5.14. We build a database o f the responses o f a sub-circuit for the three geom etrical param eters plus the frequency dependence as described in Section 4. The param eter values given here are always relative to the initial layout. A draw ing o f the initial layout can be found in A ppendix D. The database covers a sufficiently large range o f all param eter values using 5 sam ples per di mension. The com putation o f the database takes 2.4 hours o f C PU tim e on an H P-700 Series workstation. Once the database is built the C auchy m ethod developed in C hapter 4 is then ex clusively used to m odel the circuit. Figure 5.14: Definition o f Variable Param eters in Single R esonator Step 2: Optimisation of Cascaded Circuit The com plete 6 -pole filter is com posed o f six sub-circuits. Since the filter is sym m etric about the y-axis, we deal with three different resonators with three param eters each. In total, the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CH APTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 76 optim isation o f the filter requires the optim isation o f nine param eters. The raw filter layout w ithout any optim isation, i.e. param eter vector p geoi = ( 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ), has a response as shown in Figure 5.15. [dB] -10 -1 5 -20 -2 5 -3 5 -4 0 ' 5 ^ .3 8 4 .4 4 .4 2 4 .4 4 f [GHz] Figure 5.15: R esponse o f Initial Filter Layout The goal o f the optim isation is to design the filter such that it satisfies the specification as defined in Table 5.1. The filter is optim ised using Genetic algorithm search and m inim ax gradient-search com bined. F or now, optim isation is only perform ed on the coarse m odel, i.e. the cascaded circuit. The optim ised filter response is shown in Figure 5.16. We define the param eter vector of the layout that generates the optim ised response shown in Figure 5.16 as p geo2 . The num erical value is Pgeo2 = (2.62,1.02, - 5 . 4 , - 3 .9 8 , - 4 .8 3 , - 2 .9 9 , - 1 2 .3 , - 8 .2 9 , - 2 .1 ) (5.15) A ll num bers are in mils. The optim ised layout is not exact, however, as we used the coarse m odel for the optimisation. In the next step, we will com pare the coarse m odel’s response with a response calculated by the fine model. Step 3: Comparison with Full-EM In order to com pare the coarse m odel’s results w ith fine m odel’s results we have to sim ulate the sam e layout using both models. D ue to lim itations o f the full E M simulator, however, we cannot com pare arbitrary layouts. The reason behind this is the rigid layout grid used by the full EM simulator. A ll vertices o f the layout m ust fall on a grid o f fixed pitch as explained in Section 2.3.2. Hence, w e cannot check the layout p g e o 2 derived above. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 11 [dB] -10 -15 -20 -25 -30 -35 -40 -45 -50, 4 .3 8 4.4 4 .6 4.5 f [GHz] Figure 5.16: R esponse o f O ptim ised Filter Instead, the circuit’s vertices m ust be snapped to the nearest grid point. The sim ulation is per form ed on this snapped-to-grid layout. In doing so, w e can perform the sim ulation for the coarse m odel as w ell as the fine m odel. The snapped layout is defined as p geo3 . Since the snap grid is 0.2875 m ils the param eter vector becom es pgeo-i = (2 .5 8 7 5 ,1 .1 5 , - 5 .4 6 2 5 , - 4 .0 2 5 , - 4 .8 8 7 5 , - 2 .8 7 5 , - 1 2 .3 6 2 5 , - 8 .3 3 7 5 , - 2 .0 1 2 5 ) (5.16) The responses [S ] c ( p geo3) and [S]r(P[iro:i) are shown in Figure 5.18. N ot surprising, the re sponses, for the cascaded and full-EM analysis respectively, are different, because the full-EM analysis includes additional couplings. The additional couplings are shown in Figure 5.17. The conclusion is, that the circuit needs further optim isation using a m ore accurate m odel in order to m eet the specifications. For an accurate and rapid optim isation we now switch to our hybrid technique. Step 4: Finding the Correction [AM] O ur hybrid optim isation technique calls for the difference coupling m atrix derived [A M ]. This m atrix is derived from the coupling m atrices o f the coarse m odel’s response and the fine m odel’s response. In Step 3 w e already derived a response o f the coarse and the fine m odel, that is the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A T IO N 78 M , Figure 5.17: A dditional Couplings in Full-EM Sim ulation 0 [dB] -10 -20 -30 -40 -50 I____________t ____________I____________ I____________I____________ I____________ I____________ I____________ I____________I____________ I____________ I 4 .3 8 4 .4 4 .4 2 4 .4 4 4 .4 6 4 .4 8 4 .5 4 .5 2 4 .5 4 4 .5 6 4 .5 8 4 .6 f [GHz] Figure 5.18: R esponse o f (Grid-Snapped) F ilter L ayout p g e o 3 using Fine M odel and C oarse M odel R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A T IO N 79 response o f layout p geo3 - From the responses [S]c(pgeo3 ) and [ S] j(pge„?,) w e can calculate the respective coupling m atrices [M]c(pgeo3 ) and [ M ] f ( p geo3 ). The two m atrices are / - .0 0 9 6 4 .78917 .00000 .78917 .06138 .58990 .00000 .00000 .00000 .00000 .00000 .00000 .58990 .04509 .56839 .00000 .00000 .00000 .00000 .00000 .00000 .56839 .04756 .59129 .00000 .00000 .59129 .05796 .79165 .00000 .79165 - .0 1 1 3 8 / V .00000 .00000 .00000 .00000 \ and / [M ] f ( p geo3 ) — V \ .13423 .84662 - .0 0 7 8 1 .00000 .00054 .00000 .84662 .25104 .65190 - .0 2 0 9 0 - .0 0 7 0 7 .00002 - .0 0 7 8 1 .65190 .25251 .63468 - .0 2 1 7 2 .00000 .00000 - .0 2 0 9 0 .63468 .26261 .65203 -.0 2 0 0 1 .00054 - .0 0 7 0 7 - .0 2 1 7 2 .65203 .26221 .85066 .00000 .00002 .00000 -.02001 .85066 .17549 / ( 5 . 18) The difference o f the above coupling m atrices is / [ / \ M ] ( p geo 3) — V .14387 .05745 - .0 0 7 8 1 .00000 .00054 .05745 .18966 .06199 - .0 2 0 9 0 - .0 0 7 0 7 - .0 0 7 8 1 .06199 .20742 .06629 - .0 2 1 7 2 .00000 - .0 2 0 9 0 .06629 .21505 .06074 .00000 .00002 .00000 -.02001 .00054 - .0 0 7 0 7 - .0 2 1 7 2 .06074 .20425 .05901 .00000 .00002 .00000 -.02001 .05901 .18687 / \ ( 5 . 19 ) This m atrix represents all additional couplings detected by the full-EM sim ulation, but not con sidered in the cascaded sim ulation. W e note non-zero couplings of non-adjacent resonators due to stray coupling. Further, the values o f direct couplings (i.e. M ig+\ ) and self couplings change significantly. These changes are the result o f m any effects, e.g. stray coupling, different de em bedding, and changes in the housing wall locations. We do not have to know the exact cause o f these extra couplings. It is im portant only, that these additional couplings are accounted for in the matrix [A M ], We use the m atrix [ A M ] ( p ge03 ) to include the additional couplings into the coarse m odel sim u lation. The corrected coarse m odel m atrix becom es [M]*c = [M]c + [A M ] (pgeoZ) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (5.20) CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A T IO N 80 O ur first application o f the corrected m odel is the verification o f layout p geo2 - This layout is the optim ised layout found by the coarse m odel optim isation w hich was not snapped to the grid. The response o f the corrected m odel in com parison with the uncorrected m odel, is shown in Figure 5.19. Clearly seen, the response w ith the correction— assum ed to be accurate— violates the specification. H ence, the circuit needs further optimisation. [dB] -10 -1 5 -20 -2 5 -3 0 -3 5 -4 0 C ascad ed resp o n se C orrected resp o n se -4 5 -5 0 4 .3 8 4 .4 4 .4 2 4 .4 4 4 .4 6 4 .4 8 4 .5 4 .5 2 4 .5 4 4 .5 6 4 .5 8 4 .6 f[GHz] Figure 5.19: R esponse o f Filter L ayout p g e o 2 using C orrected Coarse M odel and U ncorrected Coarse M odel. M arked in Gray is the Forbidden A rea o f the Specifications. Step 5: Repeat Optimisation using Cascaded Corrected Response The optim isation is repeated on the cascaded response including the correction algorithm . The layout o f the optim ised circuit using the corrected coarse m odel is called p geo4 . The response o f this geom etry is shown in Figure 5.20. This response is a very good approxim ation o f the real response, since we included all stray couplings in our m odel. Still, w e have to verify that the response calculated by the corrected coarse m odel is accurate. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A T IO N 81 0 [dB] -5 -10 -15 -20 -25 -30 -35 -40 -45 '5 ^ .3 8 4 .4 4 .4 2 4 .4 4 4 .4 6 4 .4 8 4 .5 4 .5 2 4 .5 4 4 .5 6 4 .5 8 4 .6 f[GHz] Figure 5.20: R esponse o f F ilter W ith L ayout p ge0A O ptim ised U sing C orrected C oarse m odel R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 82 Step 6: Verify Validity of Optimised Circuit The quality o f the corrected coarse m odel calculations for filter layout o f p g e o 4 is validated by a full-EM sim ulation. Again, due to the nature o f the discrete layout grid, the geom etry vertices have to be snapped to the next grid point, yielding layout p geo5 - The response o f the corrected coarse m odel using layout p g e o 5 is com pared with the full-EM counterpart. The com parison is shown in Figure 5.21. [dB] -1 0 -20 -25 -30 -35 F u ll-E M -4 0 Cascaded + Corrected -45 -50, 4 .3 8 4 .5 f[GHz] F igure 5.21: Com parison: R esponse o f Filter by C ascaded Corrected Sim ulation and Full-EM Sim ulation at N ext G rid-Snap C alled L ayout p g e o 5 The graph shows an excellent agreem ent o f the corrected coarse m odel’s response with the full E M m odel’s response. W hen w e assum e the full E M sim ulation is exact, the filter layout p g e o 4 m ust be very closely to the exact solution. The optim ised filter is verified by hardw are m easure ments. The reader m ay refer to C hapter 7 for the results o f the m easurem ent. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A L T E R A T IO N 5.4 83 Comparison of Computation Time T his section gives an overview on the com putation tim es required using different optim isation approaches. The benchm arks are perform ed on an H P-U X 733 workstation. All tim es given are CPU-tim es. A com plete electrom agnetic sim ulation (frequency sweep) o f the layout of Figure 5.12 requires 22,080 C PU -seconds (6 h, 8 min). The sim ulation o f a single resonator (Figure 5.14), on the other hand, requires only 70 s for a com plete frequency sweeps. The database for the sub-circuit response includes five values for each param eter dim ension. H ence for building the database, we require 5 3 sim ulations o f the resonator layouts. This deter m ines the database generation tim e to be 8750 s (2.4 h). A sim ulation o f the com plete cascaded circuit using the database and Cauchy m ethod takes 3 s. The optim isation o f the circuit requires 15 iterations using first-derivative gradient search. The tim e required for the com plete standard gradient optim isation process can be sum m arised in the following formula, I = tsingle ' (1 3“ ftp a r ) ’ f^freq ' ^ i t e (5.21) where t Singie is the sim ulation tim e for a single circuit at a single point, n par is the num ber o f pa rameters, n f req is the num ber o f frequency sam ples required, and n u e is the num ber o f iterations necessary to find an optim al solution. In this form ula, n lte is the only variable undeterm ined prior to the optim isation. Experience shows that an optim um result can be achieved with 10 to 20 iterations for problem s w ith around 5 to 10 variables. W hen using the com plete full EM sim ulation for analysis the optimisation w ould last for 920 h or 39 days. The exact tim e could not be determ ined for obvious reasons. The response o f the optim ised circuit w ould be close to the desired response but lim ited by the discrete grid-snap. Using the cascaded circuit and rough grid m eshing for analysis, the optim isation takes 31500 s or 8.75 h. The result is not satisfying. Firstly, the calculations are based on a very rough grid and, secondly, errors introduced by decom position are present. The accuracy can be enhanced by finer m eshing. However, the tim e penalty is severe. Im plem enting the Cauchy m ethod, the total optim isation tim e is reduced to 2.5 h. This includes the tim e required for generating the database. The final optim ised response is closer to the de sired response because w e interpolate off-the-grid points. The simulation, however, still suffers from inaccuracies introduced by the decom position. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A T IO N 84 W hen applying our hybrid M -m atrix optim isation as suggested in this chapter, the optim isation tim e is 12 h 16 min. The tim e includes database generation, tw o com plete EM sim ulations and 25 optim isation iterations. The num ber o f iterations is increased because we have to perform two optim isation runs as shown in Section 5.11. The error o f the m ethod is very sm all as seen from the com parison o f Figure 5.18. The benchm ark results are sum m arised in Table 5.2. Conventional C om plete Circuit Optim isation Conventional C ascaded Circuit Optim isation Case. Circuit Opti. from Database-i- Cauchy O ur H ybrid O ptim isation C PU Time E rror 920 h = 39 days 8 h 45 min 2 h 33 min 1 2 h 16 min defined as exact very large large very small Table 5.2: C PU Tim e Com parison for D ifferent O ptim isation Strategies 5.5 Conclusion H ybrid optim isation com bines the advantages o f coarse and fine m odelling in the analysis. On the one hand, we exploit the coarse m odel’s com putational speed in the optim isation loop for a rapid— but inaccurate— calculation o f the optim al circuit layout. On the other hand, we exploit the fine m odel’s accuracy for verifying and adjusting the coarse model. We discussed in this chapter three hybrid optim isation techniques: the m odel output correction, the space m apping algorithm and our new coupling m atrix correction. M odel output correction adjusts the output param eters (e.g. S-param eters) according to the difference o f the port param eters of one coarse m odel and fine m odel simulation. A s shown above, this technique is not successful. The correction o f the coarse m odel often yields non-physical responses. Space m apping corrects the input param eters— rather than the output param eters— o f the coarse model. There is no problem with non-physical behaviour, because the corrected m odel output is always derived from a real physical m odel. However, space m apping is limited. The m apping can suppress only faults correctable by the input param eters. Space m apping cannot correct physical phenom ena w hich are not m odelled by the coarse model. These restrictions prevent us from applying space m apping to bandpass filter optim isation. For bandpass filter design we often use cascaded circuits as the coarse m odel and com plete EM R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. H Y B R ID O PTIM ISATIO N USING COUPLING M A T R IX A LT E R A T IO N 85 sim ulations as the fine m odel. The stray coupling betw een circuits is m odelled by the fine model but not by the coarse m odel. As shown, w e cannot find a m apping o f the param eter spaces. Stray coupling is considered in the coupling m atrix representation o f a filter. H ence, we pro posed a correction scheme based on the coupling m atrix representation. The coupling m atrix is calculated for both the fine and the coarse m odel’s response. Then we derive the difference m a trix by subtracting the m atrices o f the tw o m odels. The difference matrix represents all couplings not included by the coarse m odel, but m odelled by the fine m odel. Subsequent sim ulations o f the coarse m odel are adjusted using the difference matrix. This m ethod derives a response which includes all stray couplings usually only m odelled by the fine model. O ur corrected coarse m odel, however, has the com putational costs o f the coarse model, only. The new optim isation technique is dem onstrated on the design of a planar six-pole filter. The m ethod incorporates a cascaded EM analysis m odel based on the m ulti-dim ensional Cauchy m ethod (see C hapter 4) for the coarse m odel, and a com plete EM simulation for the fine model. W ith only tw o com plete EM sim ulations, the filter could be very accurately sim ulated and opti mised. W hen com pared with conventional approaches, our technique is the only one providing a very accurate solution within reasonable com putational expenses. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Optimisation by Time-Domain Reflectometry "Time is natures w ay o f keeping everything from happening at the sam e tim e .", Unknown This C hapter focuses on the optim isation o f wide-band and low-pass filters. For these filters w e cannot apply the hybrid optimisation technique proposed in Chapter 5 because our hybrid technique is restricted to narrow -band band-pass filters. W ide-band and low-pass filters are often realised as non-uniform transm ission line filters. The design of such filters is described in [1 0 3 ,1 1 6 ,1 1 8 ,9 9 ,1 1 1 ,1 1 0 ,1 1 2 ]. A very interesting sub-set o f non-uniform transm ission line filters are the transversal filters. Transversal filters, also known as Kallm ann filters, are finite im pulse response (FIR) filters. These filters can— in theory— be synthesised for any type o f causal response. D etailed design instm ctions on transversal filters are given in [109, 108]. Sim ilar to the form al filter synthesis o f coupled resonator filters, the synthesis o f non-uniform transm ission line filters assum es ideal circuit elem ents. Since real transm ission lines and line discontinuities (steps) are not ideal the response o f the synthesised layout differs from the de sired response. D irect E M optim isation o f the com plete circuit is very difficult, as these filters typically include a large num ber o f optim isation param eters. Cascading is not feasible either, as the decom position generates significant errors. This chapter suggests a new m ethod o f perform ing an accurate optim isation o f transm ission line filters using full EM sim ulations even for a vast num ber o f design param eters. 86 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O P T IM ISA T IO N B Y TIM E -D O M A IN R E F LE C TO M E TR Y 87 O ur new technique perform s the optim isation within the tim e-domain. The advantage o f using the tim e-dom ain is that each section o f the tim e-dom ain response corresponds to a geom etrical section in the circuit. H ence, we can optim ise the circuit layout section by section watching the respective response section exclusively. This approach simplifies the optim isation process significantly. O ur m ethod is adapted from tim e-dom ain reflectometry. Tim e-dom ain reflectom etry (TD R) is a well known m easurem ent technique for detecting discontinuities in both R F and optical trans m ission paths. TD R detects type and m agnitude of the discontinuity and— m ost interesting— the location o f the discontinuities. The characterisation o f discontinuities by tim e-dom ain reflec tom etry is the topic o f m any publications, e.g. [100, 105, 107, 106, 115]. O ur m ethod uses the tim e-dom ain response from an ideal circuit as a tem plate and com pares it with the actual response calculated by full electrom agnetic sim ulation or m easurem ent. We m ake adjustm ents based on the differences in the tim e-dom ain response o f the desired and actual circuit. In the next section, w e will introduce the theory o f our new tim e-dom ain optim isation algorithm for non-uniform transm ission lines. A pplications o f the algorithm and num erical exam ples are given in Section 6.3 and 6.4. 6.1 Theory Figure 6.1 describes an W -section transm ission line, w here each section i is characterised by its characteristic im pedance Zi and propagation delay r* . The length o f the ith line section is li. D elay and length o f the fine section are associated via the ith section’s wave propagation constant /% by h — ( 6 . 1) For an incident wave a(t) at port one, the reflected wave b(t) at port one is a sum m ation o f internal single or m ultiple reflections caused by the line’s discontinuities[106] _ | I m,m+ 1 n L o (:Tk,k+iTk+i,k) a(t-£™ = 17*)+m ult.Reflections (6.2) where (6.3) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY C) a(t) Z, b(t) T i Z2 Zj Zn T i T n T 2 .......... L1 L2 L n Figure 6.1: N on-U niform Transm ission Line T he m ultiple reflections m entioned in eqn. (6.2) are discussed in detail in [107]. The m ulti ple reflections occur— following causality— always later in tim e than the single reflections at a discontinuity and do not have influence on the first reflections as defined above. One notices from eqn. (6.2), that contributions in the response with respect to the line section i, do not occur earlier in tim e than E L i 2-7*- other words, the tim e-dom ain response up to ti=Yj\=i 2rfc *s not altered by the line sections * + 1 to N . We can now associate a tim e-interval T i o f the response b(t) with a length interval Hj o f the circuit using the relation Y i = [ t i , t i + i] H = [ L i , L i + i] (6.4) w here Li = E L t h ■ The intervals are related by t j = 4f to each other follow ing eqn.( 6.1). Pi Reflections observed in tim e interval Yj are caused by discontinuities in the length interval S i to Hj. Consequently, when we observe tim e interval Yj only, we can concentrate our optim isation on the param eter adjustm ent o f the discontinuities in the geom etrical interval Hi to Hj only. W hen we have optim ised all param eters o f the interval Hi to H j_ i, w e can fully concentrate our optim isation on the param eters in interval Hj .The next section discusses the algorithm in m ore detail. 6.1.1 Ideal Filter Synthesis and Real Filter Performance The circuit theory design o f non-uniform transm ission line filters is described in [103, 116, 118, 99, 104, 111, 110, 112, 109]. All o f the suggested design techniques, however, assum e ideal circuit elem ents. Analogous to the problem s described for the synthesis in the frequencydom ain, the response o f the ideal circuit m ay differ significantly from the actual realised circuit. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 89 A n exam ple o f a sim ple three line section filter is given in Figure 6.2 (a). The desired response in frequency dom ain is a notch type filter response that resonates at /o = 3 .7 5 GHz. The circuit is synthesised using transm ission line theory [27], The m iddle line section needs an electrical length of w here A is the wavelength at / = 3.75 GHz along the m iddle line section. T he ideal TEM -line section is converted into a m icrostrip line section as shown in Figure 6.2 (b). D ue to the im perfect m icrostrip m odel the actual frequency response— com puted using full elec trom agnetic analysis— is different from the ideal response. The ideal (desired) and actual re sponse is depicted in Figure 6.3. [m m ] 0 4 0 -0 .4 - 0.8 -1 0 0 10 20 30 40 50 60 70 80 90 [mm] F igure 6.2: Circuit L ayout U sing Circuit Theory and R ealisation in M icrostrip Topology and T heir Respective R esponse The goal o f our new optim isation algorithm is to adjust the actual circuit such that its response m atches the response o f the ideal circuit, w hich is our desired response. Thus, w e define the response o f the ideal circuit as the desired response S des( f ) and the response o f the real circuit as the actual response S act( f ) . The technique requires both responses to be available in the time-dom ain. W hen using a timedom ain sim ulation tool, the tim e-dom ain responses are known. W hen using a frequency-dom ain sim ulation tool, the response has to be transform ed into tim e-dom ain using Fourier transform . Fourier transform ation o f frequency-dom ain signals (obtained by EM sim ulators) is known to be com putationally expensive. In the next section, w e will show that this problem can be overcom e using the Cauchy m ethod applied to the frequency sweep. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 90 -20 -25 -30 -35 -4 0 -45 d e s ire d a c tu al -5 0 -55 3 .5 8 3 .6 3.6 2 3 .6 4 3 .6 6 3 .6 8 3 .7 3 .7 2 3 .7 4 3 .7 6 3 .7 8 3.8 f [G H z] Figure 6.3: Frequency R esponse o f Ideal and Real Circuit 6.1.2 Transformation into Time-Domain The response has to be transform ed into the tim e-dom ain when a frequency-dom ain sim ula tor is used. The tim e-dom ain response is com m only attained using the inverse fast Fourier Transform (FFT). However, applying the inverse F FT [98] to a frequency response requires an abundant num ber o f frequency sam ple points. The calculation o f these sam ples is com putation ally expensive, thus, m aking the overall process ineffective. With the im plem entation o f the Cauchy m ethod [76] and adaptive sam pling [82], as described in Section 4.1, these drawbacks are overcome. The response o f the circuit is sam pled at a few salient frequency sam ples, and the in-between points are interpolated by rational function interpolation. The procedure o f obtaining the tim e-dom ain response is sum m arised in Figure 6.4. The m ini m um required num ber o f sam pling points is calculated by the E M -sim ulator im plem enting adap tive sam pling, as proposed in C hapter 4. U sing the exact samples, an abundant set o f sam ples is interpolated using C auchy’s method. Since the frequency data is band-lim ited, the response S ( f ) has to be m ultiplied by a window -function A ( f ) to avoid G ibb’s phenom enon[113]. The function S ( f ) ■A ( f ) represents the response o f the system excited by the pulse A ( f ) in the frequency-dom ain. From the product, the inverse fast Fourier transform (IFFT) com putes the tim e-dom ain response b(t). The output b(t) is the convolution of the tim e-dom ain system re- R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY S(f) EM S im u la to r P u lse fu n c tio n abundant s e t of s a m p le s s a lie n t s a m p le s A(f) S(f) C a u c h y ’s M e th o d 91 S(f)A(f) s(t) ® a(t) FFT A d a p tiv e S a m p lin g Figure 6.4: C om putation o f Tim e-dom ain From Frequency-D om ain Electrom agnetic Sim ulation sponse w ith the selected pulse function a ( t ): 6 (f) = s(f) <8 >a(t) d= f s(r)a(t —r ) d r (6.5) Jo D epending on the application four different functions for a(t) are used, nam ely a Gaussian pulse, a differential Gaussian pulse, a step function and a sine-m odulated Gaussian pulse (see Figure 6.5). 1 I 1 1 0 0 0 0 -2 0 2 4 6 3 •1 10 -2 0 2 4 6 3 10 -2 0 2 4 6 8 10 •1 Figure 6.5: Excitation Pulses in Tim e-D om ain a) Gaussian b) Differential Gaussian c) Gaussian Step d) Sine M odulated Gaussian Pulse O ur algorithm uses the B ase-2 FFT algorithm. The frequency response consisting o f 2" sample points is transform ed into a 2"-point tim e-dom ain response. A m ore detailed discussion o f the FFT, the convolution theorem , and the w indowing functions is given in [98]. 6.1.3 Pinpointing of Fault Location L e t’s recall the problem o f adjusting the resonator circuit such that the actual circuit response m atches the desired response. Before w e start the actual optim isation o f the real circuit we pinpoint the fault location. We define fault locations as sections in the filter w here differences betw een the desired and actual responses originate. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN R E F LEC TO M ETRY 92 Com paring the actual and desired frequency-dom ain responses o f our exam ple line filter, we cannot obtain any inform ation about the fault location from the frequency response (see Figure 6.3). R ecalling Section 6 .1, however, we know that any discontinuity in the line can be connected to a certain interval in the tim e-dom ain response. The tim e-dom ain responses s f f s (t) and s f f 4 (f), using a Gaussian pulse excitation, are depicted in Figure 6 .6 . 0.4 0.3 desired actual 0.2 0.1 - 0.1 - 0.2 - 0.3 100 300 200 500 400 Ups] Figure 6 .6 : D esired R esponse s f l s (t) and A ctual R esponse S i f ( i ) in Tim e-Dom ain W e observe that the first reflections in tim e are identical. Consequently, the length interval S i, corresponding to the tim e interval Y i = [0 n s , 300 ns], behaves identically in both the ideal circuit and the actual c irc u it. The section S* o f the filter does not need any further adjustment. Reflections after t = 300 ns are different due to differences betw een the ideal circuit and the actual real circuit. By taking the difference of the tim e-dom ain responses, i.e. A s ( t ) = s act(t) - s des( t ), ( 6 .6 ) all echos which originate from identical discontinuities are elim inated. Only echos o f additional or different discontinuities remain. The difference pulse is zero up to the tim e when the first o f these rem aining echos reaches the port. This can be seen in Figure 6.7. W e include the excitation pulse into the figure, so that w e can read off the tim e-delay t \ betw een exciting the circuit and getting the echo. From the elapsed tim e t \ betw een the excitation o f port one and getting the echo pulse (see Figure 6.7) w e can derive the distance l\ o f the discontinuity from port one by h = , (6.7) w here /3 is the velocity o f the waves in the m icrostrip line. This m eans that the fault location can R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 93 0.4 S„ 0.3 xciting G aussian pulse 0.2 - 0.1 - 0.2 -0.3 -0.4 100 200 300 400 500 t lp s l Figure 6.7: Exciting Gaussian Pulse and Difference o f D esired and A ctual R esponse; t \ Denotes the Time Elapsed B etw een Excitation and Getting the Echo o f the Fault Location be pinpointed at l\ units away from port one. T he pinpointing o f the fault location am ong other discontinuities, w hich are identical in both structures, is analogous to clutter 1 elim ination in R A D A R systems. N on-coherent clutter elim i nation, as explained in [101], extracts targets from the clutter in the sam e manner: First a radar echo is recorded w ith the fixed echos (due to clutter) only. Then this recorded radar response is subtracted from the actual response so that all echos o f fixed targets disappear. The target echos rem ain and stand out. In our exam ple, the tim e t.\ is found to be 325 ps from Figure 6.7. For simplicity, w e used a free-space sim ulation o f the circuit(i.e. (3 — v = 3 ■10 8 m/s). U sing equation (6.1) we finally get l\ = 48m m ( 6 .8 ) Looking at Figure 6.2, we pinpoint the fault location at the second discontinuity. In the next section, w e use this inform ation to adjust the actual circuit’s output to the desired output. 6.1.4 Elimination of Faults To elim inate the fault w e m odify the circuit section causing the fault. H ere com es the advantage o f tim e-dom ain optimisation: The circuit is optim ised in one specific area only. We deal with one or two optim isation variables. 'clutter: undesired echos in the radar response due to fixed ground targets, weather etc. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 94 We located the fault at the second step around I = 48 mm. Hence, we concentrate our optim i sation on adjustm ents within this area. L et us call the length interval around this area E p . We m odify the circuit by varying the m etallisation at discontinuity in the interval Figure 6 .8 As seen from , varying the param eters p \ and p 2 m odify the m etallisation at this point. Pi 48mm Figure 6 .8 : Optim isation Param eters at Fault Location T he optim isation process adjusts these param eters until the actual tim e-dom ain response m atches the desired response in the corresponding tim e interval T p. Form ally speaking, the optim iser m inim ises the least square error (6‘9> ( W )2 - 6*^)2) * E tf = L o f the time interval T p starting at t l ='^2'k7=\ 2Tk an<l ending at t2=^2 'k=12 Tk using the optim isation variables p \ and p 2 This optim isation process is very robust, as the param eter p i and p? effect the error function alm ost proportionally within the small param eter range. The optim isation m ay take three or four steps. A possible outcom e o f the optim isation is described in Figure 6.9. a) Modification 1 -1 0 0 10 20 30 40 50 60 70 80 90 [m m ] Figure 6.9: O ptim ised Circuit Layout A fter the optim isation, the actual response is m atched with the desired response as shown in Figure 6.10. This m eans that w e can m anufacture the circuit with the desired response, as we assum ed the EM -sim ulation to be correct. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M A IN REF LEC TO M ETRY 95 -10 [dB ] -20 -3 0 -4 0 -5 0 -6 0 a c tu a l d e s ir e d -7 0 3 .5 8 3 .6 3 .6 2 3 .6 4 3 .6 6 3 .6 8 3 .7 3 .7 2 3 .7 4 3 .7 6 3 .7 8 3 .8 f [G H z ] Figure 6.10: D esired Frequency R esponse and Frequency R esponse o f M odified Circuit 6.1.5 Non-Uniqueness o f Fault Elimination Evidently, there exists m ore than one m odification leading to the desired response. Three possi ble m odifications, bringing out the desired response, are shown in Figure 6.11. The final circuit layout is said to be non-unique. The actual m odification used is left up to the user. Note that the circuit could even be fixed at a different location (see M odification 3) in case the fault location is not accessible. An increase o f capacitance in a line at one location— as done here— is equivalent to an increase o f inductance j away from that location. H ence, we can increase (thinning the line) the inductance in the m iddle o f the resonator alternatively to increasing the capacitance at the end o f the resonator. This dem onstrates that non-uniqueness can w ork to our advantage. We do not have to find the m odification, since many different solutions provides us with the desired fix. The optim isation becom es even m ore efficient, as w e m ay have to optim ise only one param eter rather than two param eters. 6.1.6 Eliminating Multiple Faults So far we dem onstrated our algorithm on the elim ination o f one fault only. In practice, however, a circuit needs m odifications at m ultiple locations. Below w e will outline the iterative elim ination R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 96 a) Modification 1 [m m ] a 4 0 -0 .4 - 0.8 -1 0 0 10 20 30 40 50 60 70 80 90 [m m j 0.8 [mm] Q 4 b) Modification 2 0 -0 .4 - 0.8 -1 0 0 10 20 30 40 50 60 70 80 90 [m m ] c) Modification 3 [m m ] a 4 0 -0 .4 - 0.8 -1 0 0 10 20 30 40 50 60 70 80 90 [m m ] Figure 6.11: Three D ifferent M odifications o f the L ayout Yielding the D esired Response: a) Extending the w idth at Fault; b) Shortening the Line with Im pedance Zo ; c) A djusting Circuit Away F rom Fault R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN R E F LEC TO M ETRY 97 o f faults. L et us recall the line filter o f Figure 6.1. The algorithm works by detecting and optim ising discontinuities sequentially. The iterative process starts w ith interval one with the goal o f m in im ising the error E \ in the first tim e interval by optim ising Z \ and w \ . Then, section tw o is m odified by adjusting Z 2 and w 2 and so on. We repeat this process for all intervals up to interval N . U sing this technique the high-dim ensional optim isation problem o f 2N variables is split into N tw o-dim ensional optim isation problem s w hich can be solved m ore easily. We apply the sequential tim e-dom ain optim isation m ethod on a another exam ple in Section 6.4. 6.1.7 Pulse Tracking In the above exam ples, we assum ed that the propagation constant k and, consequently, the prop agation speed j3 inside the circuit is known. Furtherm ore, w e assum ed to know how the pulse is travelling through the circuit so w e could use the sim ple equation (6.7) to pinpoint the fault location. However, in som e circuit structures the exact propagation speed and propagation path o f the pulse is not obvious. Thus, we im plem ented a pulse tracking routine in our program . The routine generates a timesequence o f the pulse travelling through the structure. The output o f the routine provides us the m ost wanted inform ation, i.e. location w ith respect to the tim e t, o f the pulse propagation inside the circuit. The tim e-sequence representation can be used to obtain the fault location from the echo tim e 11 — instead o f equation 6.7— even if the propagation speed and exact path is unknown. The propagation o f the Gaussian pulse in the first exam ple is shown in Figure 6.12. Sum m ing up, we synthesised a circuit w ith a desired response by a com bination o f a circuitbased program and EM -sim ulation. The circuit-based program is used for a fast synthesis o f the circuit. Faults in the derived layout are detected by an accurate EM -sim ulation. A pinpointing o f the faults in tim e-dom ain and an optim isation aim ed at the fault location fixes the faults. In doing so, the use o f the slow optim isation in a high-dim ensional param eter space is avoided. 6.2 Tuning of Filter Hardware Som e of the challenge in tuning filter hardware is in identifying which circuit section needs to be tuned and determ ining how to tune the filter to obtain the desired response. O ur proposed R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O P TIM ISATIO N B Y T IM E -D O M AIN REF LEC TO M ETRY 98 t= ,014 ns t= ,027 ns t= ,057 ns t= ,071ns t= ,085 ns t= ,114ns t= ,127 ns t= ,143 ns t= ,157 ns t= ,171ns t= ,185 ns t= ,200 ns t= ,214 ns Figure 6.12: Tim e Sequence o f Pulse Travelling Through a Circuit R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. t= ,043 ns CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN R E F LEC TO M ETRY 99 tim e-dom ain technique can help the person at the test bench to tune the filter hardware. In that case, the m easured response becom es the actual response and the theoretical response becom es the desired response. The fault locations can be pinpointed in analogy to the m ethod described above. However, adjusting the fault location is different. W e cannot change the circuit shape as easily as in the com puter simulation. On planar circuits we can add m etallized patches or cut away m etallisation, as done e.g. with laser beam s. Alternatively, w e can m ove dielectric blocks on the substrate surface. In waveguide filters, we com m only introduce tuning screws. 6.3 Optimisation and Tuning of Narrow-Band Resonant Filter Cir cuits A direct application o f the tim e-dom ain optim isation technique to narrow-band resonant filter structures is not practical. The im pulse response o f such filters stretches out over a very long tim e period. The energy is reflected in very sm all portions by the filter discontinuities. Accurate adjustm ents o f the discontinuities from these m inim al reflections is not feasible. The problem can be circum vented using a band-pass weighting function in the frequency do m ain. The frequency-dom ain response is m ultiplied by the weighting function, and the timedom ain transform ation is applied to this weighted frequency-dom ain response. A typical w eight ing function (also known as window function) is shown in Figure 6.13. The introduction o f a band-pass weighting function m eans physically, that the system is excited by a pulse function m ixed with a carrier frequency at /o . The excitation pulse can be interpreted as an am plitude m odulated signal. H ere w e im plem ent the sine m odulated Gaussian pulse as depicted in Figure 6.5 (d). Response Band of Interest 32730836^6 Window Function Figure 6.13: Band-pass W indow Function W hen excited with the sine m odulated Gaussian pulse, a tuned band-pass filter’s response shows characteristic dips. The m agnitude envelope (in log-scale) o f the tim e-dom ain response o f a tuned 5-pole filter is shown in Figure in 6.14. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. OPTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 100 -10 dB -20 dB -30 dB -40 dB -40 ns 0 ns 260 ns F igure 6.14: Tim e-D om ain R esponse o f Tuned N arrow -Band Filter B eginning with the first dip near t=0 ns, the next N dips represent the response o f the resonators 1 through N in the filter. This relationship was recently identified in [117] by a research group from Hew lett-Packard Test & M easurem ent. T he tim e span betw een the dips is the group delay from one resonator to the next. W hen one o f the resonators is de-tuned, i.e. it does not resonate at the desired frequency, the dip disappears. Figure 6.15 shows the response with resonator three de-tuned. -10 dB -20 dB -30 dB -40 dB -40 ns 0 ns 2 6 0 ns Figure 6.15: Tim e-Dom ain R esponse o f N arrow -Band F ilter W ith R esonator three Detuned The geom etry o f this de-tuned resonator can now be changed such that the characteristic dip appears. M inim ising the tim e-dom ain m agnitude at the dip locations results in a properly tuned filter as outlined in [117]. In analogy to the unw eighted tim e-dom ain optim isation, w e fix one fault location— here a resonator— after another, starting from the excitation port. The same group o f people at H -P expanded the technique to allow tuning o f the filter couplings. Observing the peaks betw een the dips in the response, they can also tune the resonator couplings. The m ethod relies, exactly as our m ethod, on the existence o f an ideally tuned filter. The cou R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y T IM E -D O M AIN REF LEC TO M ETRY 101 plings are adjusted with the goal to m atch the actual with the ideal response. Once all peaks are m atched the filter is tuned. 6.4 Numerical Results This num erical exam ple illustrates the functionality o f the tim e-dom ain optim isation technique, on the designed o f a non-uniform line low-pass filter. O ne o f the filter specifications is a return loss o f less than 25 dB in the passband below / = 1.5 GHz. From filter and transm ission line theory [27] the initial layout as shown in Figure 6.16 is synthesised. L1 L2 L3 L4 L5 Figure 6.16: Planar N onuniform Transm ission Line Low-Pass Filter The ideal response o f the initial layout in frequency dom ain is shown in Figure 6.17 m arked as circuit theory. We also com pute the response o f the initial layout using the full EM sim ulation H P-M om entum [149]. This response is the actual response. A s seen in Figure 6.21, the actual response does not yield the desired 25 dB rejection in the passband. We convert the response into the tim e-dom ain by inverse FFT. The inverse FFT is perform ed on 64 frequency samples in a 5 GHz frequencyband. Using C auchy m ethod the 64 sam ples are com puted from 9 original frequency samples. Only these nine sam ples are actually derived using full EM -sim ulation. Not surprisingly, the tim e-dom ain response (Figure 6.18) shows a significant deviation between circuit theory and EM -sim ulation. The filter requires a full-EM optim isation in order to m eet the specifications. The filter is optim ised to the specifications using our new tim e-dom ain optim isation m ethod. First we divide the filter into five geom etrical intervals. Each interval covers one discontinuity, i.e. im pedance step. W hen changing the tw o param eters o f the respective im pedance step (line length and width), the tim e-dom ain response is only m odified in the corresponding and later time-intervals. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 102 S ll [dB] -10 -15 Full-EM -20 -25 -30 Circuit-Theory -35 -40 -45 f [GHz] -50 0.5 3.5 2.5 F igure 6.17: Frequency-D om ain response S n From Circuit Theory (D esired Response), FullE M Sim ulation (Actual R esponse) — T ra n sien tjS ll} Circuit Theory Full-EM 0 1 2 4 3 t [ns] 5 - Transient] S 2 2 } ! Circuit Theory Full-EM 1 0 1 2 3 4 t [ns] Figure 6.18: Tim e-D om ain R esponse s n ( f ) From Circuit Theory and raw Full-EM R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 5 CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN R E F LEC TO M ETRY 103 F or example, Figure 6.19 illustrates the effects o f layout variations on both the frequency and tim e-dom ain response. Varying w 4 the frequency-dom ain response is affected over the com plete frequency band, w hereas in tim e-dom ain the effects are lim ited in the interval [0.25ns, 00 ]. The interval[0ns, 0.25ns] rem ains untouched by w ,\, as predicted in Section 6.1.4. 0 T im e-D o m ain R esp o n se f F re q u e n c y -D o m a in 6 m il R esp o n se 8 mil 10 mil 12 mil 14 m il 16 m il -10 -20 -30 -40 -50 -60 -70 -80 -90 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 . 0 .5 0 0.5 1 1.5 2 Figure 6.19: Variation o f Frequency and Tim e-D om ain R esponse by Changing W4 An optim isation o f the response using the m ethod described in this C hapter m atches the EM sim ulation response w ith the theoretical response as shown in Figure 6.20. It can be seen from F igure 6.21, that the adjusted frequency-response o f the optim ised layout m eets the specifica tions. 6.4.1 Comparison of Computation Times The synthesis o f the low-pass filter is a ten-dim ensional optimisation problem . T he full EM sim ulation o f the circuits takes 110 C PU seconds. Hence, a frequency-dom ain optim isation requires about 300 C PU min. The tim e-dom ain technique optim ises tw o param eters in parallel. It needs five optim isation runs for the five intervals. In total the optim iser perform s 35 full EM sim ulations, w hich is equivalent to about 64 C PU min. The results are sum m arised in Table 6.1. As seen, the required C PU tim e is reduced by a factor o f five. Savings in com putation tim e are even bigger for larger circuits. Conventional direct full E M optim isation O ur tim e-dom ain optim isation m ethod C PU Time E rror 300 min 64 min very small very small Table 6 .1: Com parison o f C PU Tim es for Conventional and Tim e-dom ain Optim isation R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. O PTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 104 _ T ransient{S ll} Circuit Theory Full-EM 0 1 3 2 4 t [ns] 5 -T ransient[S22} Circuit Theory Full-EM 0 ■1 1 3 2 4 t [ns] 5 Figure 6.20: Tim e-D om ain R esponse o f s n (£ ) from Circuit Theory and C orrected Circuit Using Full EM and TD R Optim isation 0 Sl l ■5 [dB] -10 -15 -20 Full-EM -25 L Full-EM tim e-dom ain adjusted -30 Circuit-Theory -35 -40 -45 -50 f [G H z] 0.5 2.5 3.5 Figure 6.21: Frequency-D om ain R esponse 5 n From C ircuit Theory, Full-EM , C orrected FullE M Simulation R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. OPTIM ISATIO N B Y TIM E -D O M AIN REF LEC TO M ETRY 6.5 105 Conclusion In this chapter w e introduce a new optim isation technique for non-uniform transm ission line filters. O ur technique is based on the theory o f tim e-dom ain reflectometry. We perform the optim isation in tw o steps: First the fault locations are located by com paring the tim e-dom ain response o f the circuit, i.e. the actual response, with the desired response. Then, the faults are rem oved using an optim isation aim ed at the specific fault location. The large set o f optim isation variables is divided into small sets o f one or tw o variables, which are m uch easier to handle than the high-dim ensional problem . Hence, w hen dealing with m edium or large size circuits, the proposed optim isation technique is superior to the optim isation o f the frequency-response. We showed that for a circuit with ten degrees of freedom the com putational costs are already reduced by 80%. The m ethod has in principle no upper bound o f the num ber o f param eters. A pplication o f the m ethod lies in the com puter aided design and com puter aided tuning of filters. A lso in this chapter, w e suggest an algorithm that overcom es the problem s with the Fourier transform o f a very sm all set o f frequency-response samples. B y applying Cauchy m ethod and adaptive frequency sam pling, the num ber o f required sam ples is reduced to a m inim um . It has been shown that w e can apply the proposed technique to narrow-band resonator filters as well. For optim um perform ance, w e have to excite the filter ports with a sine-m odulated pulse. This technique reduces the sam pling acquisition for the FFT to the frequency band o f interest. The response o f the filter exhibits characteristic dips and peaks in the dem odulated tim e-dom ain response. This characteristic can guide the user in the optim isation or tuning o f the filter. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 Hardware Verification "The only theorem o f superconductivity that can be p ro v e d is that any theory o f superconductivity is refutable", F elix Bloch The design tools proposed in this thesis are pointless, if w e cannot use them for the developm ent o f m icrowave device hardware. Consequently, w e verified the usability o f these techniques on the design o f functional hardware. In this chapter, w e present m easurem ents for som e o f the hardware designed with the help o f our new synthesis techniques. The hardware is developed at C O M D EV Space Group. C O M D EV is a specialised supplier for m icrowave satellite equipm ent. All devices appearing here are part o f experim ental satellite payloads. The em phasis is on applications o f high-tem perature superconductors (H TS) in satel lite m icrowave systems. The superconducting com ponents presented here are part o f the current DARPA High-Tem perature Superconductor Technology D evelopm ent project. Inform ation on this project can be found in [144]. Form er H TS space experim ents projects are discussed in e.g. [142, 143]. The project is a joint C anadian/U S program directed by the N ational A eronautics and Space A dm inistration (NASA), and funded by DARPA, the Canadian space Agency (CSA) and the D epartm ent o f National D efence (DND). 7.1 Ka-Band Rejection Filter The rejection filter— designed and optim ised using our enhanced genetic algorithm s— is inte grated into a K a-B and relay satellite. The typical block diagram of a relay satellite is shown in Figure 7.1. The purpose o f the rejection filter is to suppress any feedback from the down-link antenna into the up-link antenna. This coupling, if not suppressed, can cause oscillations. 106 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATION 107 L ocal I O sc . L o w N o is e Pow er A m p lif ie r A m p lif ie r R e je c tio n D ow n F ilte r C o n v e r te r P a ra s itic F e e d b a c k Figure 7.1: B lock D iagram o f a Typical R elays Satellite The optim ised design for the rejection filter shown in Figure 3.7 is verified by m easurem ent. The m easured results com pared to the sim ulated results are shown in Figure 7.2. The m easurem ents show excellent agreem ent with the sim ulation of the optim ised circuit. 7.2 3-dB Power Divider For a high-tem perature superconducting (HTS) L-B and beam form ing netw ork (shown in the next section) we design a 90°-branch-line coupler. This device is com m only known as a hybrid. Superconducting planar circuits achieve the sam e perform ance while using less volum e and w eight com pared to conventional coaxial hybrids. Figure 7.3 shows a the size com parison shown for an conventional versus HTS 90°-H ybrid pow er divider a L-band. A num erical com parison is given in Table 7.1. Loss Volume W eight W aveguide Co-axial HTS HTS drop-in 0.3 dB '1 0 in3 '3 0 0 g 0.3 dB 0.75 in3 100 g 0.3 dB 0.75 in3 40 g 0.3 dB 0.75 in3 5g Table 7.1: Com parison o f B ranch-C oupler Technologies R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATION 108 1 9 .5 -2 0 .5 GHz l 2 9 .0 - 3 0 .0 GHz o -5 -10 -1 5 -20 -25 -3 0 -35 ~4 0 5 10 15 20 25 1 9 .5 -^2 0 .5 G H z 0 30 f[G Hz] 35 2 9 . 0 -J30.0 G H z -10 -20 -30 -40 -50 -60 '7° 10 15 20 25 30 f[G Hz] 35 Figure 7.2: Sim ulated (Thin Line) and M easured R esponse (Fat Line) o f Stripline F ilter .At the Top S'nfdB J and at the B ottom S'i 2 [dB] Conventional 0 .7 5 HTS 0 .5 " ( 1.5" HTS drop-in Figure 7.3: C om parison o f L-B and Hybrids R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 109 CHAPTER 7. H A R D W A R E VERIFICATIO N It can be seen from the table, that the volum e and w eight can be significantly reduced by replac ing coaxial hybrids with HTS technology. Especially when several hybrids can be integrated on one substrate, the w eight reduction can be enormous. Conventional pow er dividers are typically realised using a ring or square shaped layout. H ow ever, such layouts occupy a large area. The area o f the hybrid can be reduced by about 60% w hen using a folded structure as shown in Figure 7.4. 1 - |---------------1 - 2 1— i 4-1 i— ' ------U 3 Figure 7.4: F ayout o f Hybrid We m odel the hybrid using the m ulti-dim ensional Cauchy m ethod in conjunction with a full EM simulator. U sing this approach, w e obtain a fast and accurate m odel o f the circuit. We optim ise the circuit’s perform ance (using a gradient-search algorithm ) for best return loss and equal pow er distribution at the outputs with a 90° phase shift. The optim isation uses four param eters: length and width o f the through lines and branch lines. A fter optimisation, the hybrid exhibits a return loss o f less than 25 dB and an isolation o f 30 dB over a bandw idth o f 300 M Hz. Pow er split is very close to the desired 3dB/3dB, and the phase shift is at the desired 90°. T he perform ance o f the structure is not degraded by folding. The response o f the optim ised hybrid is shown in Figure 7.6. A photograph o f the hybrid is shown in Figure 7.5. The perform ance o f the filter is verified by m easurem ents. The m easured results are shown in Figure 7.7. As seen, the m easured response is in good agreem ent with the simulation. 7.3 Butler-Matrix Beamformer On satellites it is som etim es desirable to feed an antenna array through a feed system that will excite a num ber o f beams. This netw ork is called a beam form ing network or simply beamform er. Each input port o f the beam form er excites the array so that the it produces one particular beam. D ifferent input ports generate beam s pointing in a different direction. The concept is illustrated in Figure 7.8. The beam s can be used sim ultaneously w ithout cross-talk in the beam form ing networks. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATIO N 110 Figure 7.5: Photograph o f the Folded H ybrid in C om parison W ith O ne C ent Coin 200 -10 150 -15 100 lap ] -20 -2 5 -3 0 -50 -35 -100 -4 0 1.5 1.52 -45 1.54 1.56 frequency [GHz] 1.58 1.6 1.56 1.6 -5 0 1.5 1.52 1.5 4 1.5 6 1 .5 8 1.6 frequency [GHz] Figure 7.6: Full EM Sim ulated R esponse o f Optim ised H ybrid o 200 -s 150 •10 100 •15 50 -20 0 -25 •50 •30 -100 -35 •150 •40 -45 1.5 1.52 1.54 1 56 f [GHz] .56 1.6 -200 .5 1.52 f [GHz] Figure 7.7: M easured R esponse o f O ptim ised Hybrid R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATION P a tte r n g e n e ra ted 111 P a tte rn g en erated P a tte r n g en e ra ted B eam form er B eam form er B eam form er P a tte rn g en e ra ted B eam form er ♦ 4 4 4 F eed F eed F eed Feed Figure 7.8: B eam Generation by M ultiple Fixed B eam form ing N etw ork B eam form ing networks are n -to -m netw orks and their purpose is to interconnect the n input ports to the individual m antenna ports w ith the required am plitude and phase. They can be either passive or active networks. Probably the m ost widely known m ultiple-beam m atrix feed is the B utler m atrix. It is well docum ented in the literature [38], Figure 7.9 is a schematic representation o f an 8x8 butler matrix. Antenna Ports x 45° <t> IX 67.: X +5° -2 0 ° +10° -1 5 ° Beam Ports F ig u r e 7 .9 : S c h e m a t ic B u tle r M a tr ix B e a m fo r m e r , T h e X ’s R e p r e se n t 9 0 ° H y b r id s W ith a P o w e r Split Ratio o f 3dB/3dB. In this section, we describe the layout and operation o f an 8x8 B utler m atrix. Twelve o f the hybrids designed in the last section are necessary to build the 8x8 beam form ing network. The R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CH APTER 7. H A R D W A R E VERIFICATIO N 112 hybrids are connected by delay lines with a phase delay as seen in Figure 7.9. T he beam form er is optim ised using gradient optim isation in connection with a cascaded elec trom agnetic simulation. The cascaded sim ulation yields satisfying results because we deal with relatively w ide-band structures. The interested reader m ay be referred to [7] and [8] for detailed inform ation on the beam form er design. T he m atrix is built using T B C C O films supplied by DuPont. A detailed layout o f the m atrix is shown in Figure 7.10. The overall size o f the m atrix is only 3.0 x 2.0 x 0.2 in3. The beam ports (inputs for transm it m ode) are num bered respective to the beam generated. That is, 1R m eans the first beam generated to the right. The antenna ports are num bered according to the order o f the connected antenna elem ents. A photograph o f the here discussed 8x8 m atrix is shown in F igure 7.11. 1 5 2 6 3 7 4 8 Plug-In GPO-Connectors M ulti-Layer Cross-Over D elay Lines Folded 90°-Hybrids 1R 4 L 3 R 2L 2 R 3 L 4 R 1L — - - ------- 2.046 ------------------------------------------------------- Figure 7.10: L ayout o f 8x8 M atrix in Housing The m easured and sim ulated results are sum m arised in Figures 7.12 to 7.15. Figure 7.12 shows the return loss o f the beam ports 1R to 4R. Figure 7.13 shows the insertion loss for the path from one beam port (shown here port 1R only) to the various antenna ports 1 to 8. The beam form er R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATION 113 is designed for uniform am plitude taper, i.e., all ports should receive | o f the input pow er (9 dB on a logarithm ic scale). T he sim ulated and m easured phase tapers o f all ports for the centre frequency 1.55 G H z are sum m arised in Figure 7.15. As seen, all phases are within a 10° accuracy. The total device insertion loss o f the m atrix is less than 0.2 dB. Figure 7.11: Photograph o f 8x8 B utler M atrix B eam form er o •s •10 •15 -20 -25 -30 •40 -45 -50 f req 1.5 1.52 1.54 f(QHz) .56 1.56 1.6 Figure 7.12: R eturn Loss o f 8x8 M atrix for B eam Ports 1R, 2R, 3R, 4R Sim ulated (left), and M easured (right) The m easured am plitude and phase taper are used to calculate a beam -pattem generated by an im aginary linear antenna array. The generated beam s are shown in Figure 7.16. It can be clearly seen that the beam form er perform s its intended function. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 114 CHAPTER 7. H A R D W A R E VERIFICATION 1.6 GHz E Figure 7.13: Insertion Loss o f 8x8 M atrix W ith B eam -Port Input 1R, sim ulated (left), and m ea sured (right) 200 150 90 cn 0 •50 •200 1.5 GHz 1.6 f GH z H 1.5 1.52 f(GHz) 1.56 1.55 1.6 Figure 7.14: Phase Taper o f 8x8 M atrix Sim ulated and M easured 1500 1000 500 0 -500 •1000 •1500 0 2 3 4 5 Antenna Port 6 7 8 9 Figure 7.15: Phase Distribution o f 8x8 m atrix (Sim ulated and M easured). Each Line corresponds to O ne Input Port. The D otted Lines M ark the Theoretical Phase Taper, W hereas The D ots M ark The Sim ulated and M easured Taper, Respectively. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 115 CHAPTER 7. H A R D W A R E VERIFICATIO N 2L 2R 3R 4R 4L F igure 7.16: Radiation Pattern (Sim ulated from M easured Data) 7.4 Cryogenic Switches The pow er am plifiers aboard a satellites m ay be switched to provide redundancy as illustrated in Figure 7.17. The scheme shown is term ed a four-to-two redundancy, m eaning that four channels are provided with two redundant amplifiers. Using the m atrix shown in the diagram , any channel can use three different amplifiers. Ch. A Ch. A Switch Switch Ch. B Ch. B Switch Switch Ch. C Ch. C Switch Switch Ch. D Ch. D Switch Position I: & Switch Position II: X / & Figure 7.17: Four-For-Two Redundancy Sw itch A rrangem ent o f A N IK D (see [141]) F or the incorporation into a cryogenic payload, we develop a new class o f cryogenic switches. The switches use planar thin-film HTS technology with integrated sem iconductor devices. 7.4.1 HTS C-Switch In this section we describe the design o f an planar integrated HTS C-switch. The switch has been designed and optim ised using the m ulti-dim ensional Cauchy method. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATIO N 116 A C -switch is a four port network. The tw o operation m odes are defined as shown in Figure 7.18. We choose the layout o f the switch as shown in Figure 7.19. U sing a special gap coupling o f the stub line with the circuit, w e attain D.C. de-coupling betw een the switch branches. S w itc h M o d e I: S w itc h M o d e II: Figure 7.18: C -sw itch Operation M odes P IN D io d e s / F E T s/ H T S S w itc h L in e R F C hoke In d u c to rs O n e L a y e r C irc u it ( n o D .C . C o u p lin g ) K -C o n n e c to r D C -B ia s F e e d F igure 7.19: L ayout o f C -Sw itch . Inset Shows One o f the Eight D.C. D ecoupled Stub Lines. For an efficient sim ulation and optim isation w e decom pose the circuit into four sub-sections, which are identical. The subsection is shown in Figure 7.21. We m odel this subsection using m ulti-dim ensional Cauchy method. T he response analysis o f the switch is perform ed on the cascaded circuit o f four identical subsection. W e optim ise the switch for best perform ance in the frequency-band from 4.5 GHz to 5.5 GHz. B est perform ance m eans m inim ised return loss and insertion loss with m axim ised isolation. The optim isation results are sum m arised in Figure 7.22(left). M easurem ents on the device (Fig ure 7.22 right) show a good agreem ent o f the simulation. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATIO N 117 F igure 7.20: Photograph o f Conventional C o-A xial C -Switch, HTS C -Sw itch and O ne D ollar Coin P o rt 3 p4 P o rt 1 P o rt 4 p3 -► 4 pl P3 P4 ■P o rt 2 -► 4 p2 pi Figure 7.21: Subsection o f C -Sw itch Showing the four O ptim isation Param eters R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATIO N 118 Transmission (on-state) <L O Transmission (on-state) o •10 Return Lass Return Loss -15 -20 (u0 ra CD Transmission (off-state) L -30 Trans mi 5S10I1 ff-stite) O / Isolation -50 freq 4 4.2 4.4 4.6 4.8 5 5.2 5.4 56 5.8 6 f [GHz] Isolation Figure 7.22: Sim ulated (Left) and M easured (Right) Perform ance o f C -sw itch 7.4.2 HTS High Isolation Switch The C -switch assum ed an ideal switching elem ent, i.e. ideal FET or Diode. In reality these elem ents are non-ideal, i.e. they are lossy in the on state and not perfectly open in the off-state. T he m easurem ents o f Section 7.4.1 are obtained by opening and closing m etal strips at the end o f the line. U sing sem iconductors does not yield the desired response due to the losses in the device. A new design o f the switch is optim ised for best perform ance when operating with P IN diodes. W e obtain the exact m odel o f the PIN diode from de-em bedded m easurem ents o f the diode at cryogenic tem peratures. T he assem bly o f the first tested switch using PIN diodes is shown in Figure 7.23. M easurem ents show a very good agreem ent betw een our optim ised sim ulated response and the m easured response. The com parison is shown in Figure 7.25. 7.5 Six-Pole Narrow-Band Filter H ere presented are the m easurem ents o f the six-pole filter designed in C hapter 5. This type of narrow-band filter is used in satellite filter assem blies for channel splitting. The detailed design R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 119 CHAPTER 7. H A R D W A R E VERIFICATIO N XT Side-View: •DC-Feed Spiral Pi n lndoctorDi ode Ribbon Bond Wire Bond Figure 7.23: L ayout o f Four PIN D iode Sw itch and B low -U p o f the D iodes Including D.C. Bias Netw ork and R F Chokes Figure 7.24: Photograph o f High-Isolation Sw itch W ith Lid R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATION [dB] 120 S21 -10 -15 -20 Sll -25 -30 On State -35 -40 o Measured -45 -50 — Simulated 3.5 4.5 f [GHz] 5.5 0 [dB] Sll -10 -20 -30 -40 -50 Off State S21 -60 -70 » Measured -80 -90 — Simulated 3.5 4.5 f [GHz] 5.5 F igure 7.25: Sim ulated and M easured R esponse o f The High-Isolation Integrated H TS/PIN D iode Switch R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATION 121 techniques is described in C hapter 5. The com plete assem bled circuit in a housing with Kconnectors is shown in Figure 7.26. A photograph o f the filter without lid can be seen in Figure 7.27. 1,337 Figure 7.26: L ayout o f Six-Pole Filter in Housing Figure 7.27: Photograph o f the six-Pole Filter The results o f the m easurem ents are shown in Figure 7.28. The m easured response is in excellent agreem ent w ith the theoretical response, obtained by our hybrid full E M and cascaded sim ula tion. The m easured response shows a slightly higher centre frequency and slightly w ider band width. We can attribute these deviations to im perfect substrates and m anufacturing tolerances. Indeed, past m easurem ents on other HTS devices com m only exhibit m uch stronger m ism atch b e R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATION 122 tw een sim ulation and m easurem ent. This indicates, that we alm ost com pletely elim inated errors from the num erical sim ulation and optim isation when applying our new developed techniques. 7.6 Conclusions O ur new techniques have been tested on the developm ent o f various m icrowave devices. In this chapter we have presented the design and m easured results for an L-band hybrid, an 8x8 B utler m atrix beam form er, tw o C -Band switches, and a narrow -band six-pole filter. The devices are all developed with the aid o f the here proposed synthesis techniques. W hen using our m ethods, we reduced developm ent tim e significantly w ithout neglecting an accurate design. The rem aining differences betw een the theoretical responses and m easured responses o f the devices can be attributed to im perfect m aterials and m anufacturing processes. These problem s can norm ally be fixed with circuit tuning. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7. H A R D W A R E VERIFICATIO N 123 -10 -15 -20 -25 -30 -35 o M easured -40 — Sim ulated -45 -50 I— 4 .3 5 4 .4 4 .4 5 4 .5 4 .5 5 4 .6 4 .6 5 -10 S21 [dB] -20 -30 -40 -50 -60 -70 <po 0 M easured — Sim ulated -80 -90 I— 4.35 4.4 4.45 4.5 4.55 4.6 4.65 f [GHZ] Figure 7.28: Sim ulated and M easured Frequency R esponse o f Six-Pole F ilter W ithout Tuning R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 8 Summary and Future Work "If som ething is so com plicated that you c a n ’t explain it in 10 seconds, then i t ’s p robably not w orth knowing anyway", Calvin in Calvin and H obbes This thesis is m otivated by the lack o f fast, accurate, and reliable synthesis techniques for m i crowave filter design. As discussed, filter synthesis is currently perform ed by form al synthesis and iterative optim isation in conjunction w ith an EM simulator. The trend is towards iterative optim isation since it is m ore accurate and not lim ited to regularly shaped circuit layouts. H ow ever, this approach still has shortcomings. W e have identified the main problem s o f the available techniques as follows: • The search for the optim al solution is unreliable That is the algorithm s are non-robust. • The com putational costs are high. • W hen sim plifications are used the accuracy o f the sim ulation is low. • The spatial grid for the circuit vertices is discrete. To resolve these problem s, we have proposed four new techniques, that can be com bined or used separately. These techniques are • Genetic algorithm s in circuit design • M ulti-dim ensional Cauchy m ethod 124 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8. S U M M A R Y A N D FU TURE W O R K 125 • Hybrid optim isation • Tim e-dom ain based optim isation We have dem onstrated that genetic algorithm s can increase the robustness o f m icrowave circuit optimisation. Genetic algorithm s do not get trapped in sub-optim al solutions. Rather, they seek the optimal solution by applying the law o f the survival o f the fittest. Further, genetic algorithm s do not need derivative inform ation. H ence, the optim isation can be as fast or even faster than a gradient guided search. In several exam ples, w e have shown the excellent perform ance of GA’s com pared to gradient search. Genetic algorithm s are not m eant to be a replacem ent for gradient-based optim isers, but a supplement. For a faster analysis step, w e have developed a new m ulti-dim ensional Cauchy m ethod as an al ternative to neural networks. O ur technique can predict the response o f unknown circuit layouts based on the knowledge o f several sam ple layouts and responses. This approach takes the expen sive calculations out o f the optim isation loop. The optim isation can be perform ed exclusively on the very inexpensive m ulti-dim ensional Cauchy m odel. As shown, the m ethod can speed up the optim isation by several orders o f m agnitude. The m ulti-dim ensional Cauchy m ethod is now a vital part o f the num erical design package used for the synthesis o f high-tem perature superconductive circuits at C O M D E V ’s corporate R & D departm ent. O ur enhanced Cauchy m ethod is in practice lim ited by the m axim um num ber o f param eters. Large circuits, still m ust be decom posed. As w e have shown, decom position neglects stray coupling betw een the circuit sub-sections. To include this stray coupling, we have developed here a new hybrid technique. O ur m ethod is based on the coupled resonator representation of band-pass filters. W e have dem onstrated that our techniques reduce the optim isation tim e by a factor o f several m agnitudes w ithout sacrificing accuracy. W ith the new technique, we accurately designed large circuits, that previously seem ed im possible to optim ise with today’s com puters. This new optim isation paradigm is now also part o f C O M D E V ’s filter design package. A s m entioned above, the coupled resonator representation is lim ited to band-pass structures. To tackle the optim isation o f low-pass and non-resonant filters, w e propose an optim isation based on tim e-dom ain reflectometry. We have shown that we can split one iV-dim ensional optim isa tion problem into N one-dim ensional optim isation problem s. The latter arrangem ent is m uch easier to handle than the former. H ence, w e gain significantly in optim isation perform ance. We can exploit tim e-dom ain optim isation in the com puter sim ulated design process as well as in hardw are m easurem ents. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8. S U M M A R Y A N D FU TURE W O R K 126 T he new techniques have been verified by num erous hardware m easurem ents. The m easured hardware is designed for upcom ing superconductive satellite microwave systems. The m easure m ents have m et or even exceeded our expectations. Slight differences in the responses between theory and practice are largely due to m anufacturing tolerances. 8.1 Recommendations Table 8.1 sum m arises the recom m ended design techniques for various filter types. Sm all filter circuits with up to 5 optim isation variables can best be designed by our new m ulti-dim ensional Cauchy method. The actual optim isation algorithm could either use the genetic algorithm or the gradient search technique. It is often recom m ended to start the optim isation with the GA, and then refine the solution w ith a gradient search. Filter Type Param eter Recom m ended D esign Techniques Recom m ended Opt. A lgorithm Arbitrary Filters R esonant Filters Non-U niform Transm ission Line filters < 5 M ulti-Dim. Cauchy M ethod H ybrid optim isation with C oupling M atrix Tim e-Dom ain Optim isation GA and/or G radient Search GA and/or G radient Search G radient Search 5-40 5-100 Table 8.1: R ecom m ended Optim isation M ethods for D ifferent F ilter Types Large resonant filters should be designed with our hybrid optim isation technique using the cou pling adjustments. Again, w e favour searching the param eter space with both G A and gradient search. F or the design o f non-uniform transm ission line filters o f large order, we advocate our timedom ain based optim isation. We recom m end using gradient-search for the param eter adjustm ents only. GA’s are less favourable in this case. Since the param eter adjustm ents problem is very linear, gradient search is m ore effective. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8. S U M M A R Y A N D FU TU RE W O RK 8.2 127 Final Remarks One conclusion o f our work is that a universal design technique for all filter types does not exist. Rather, the synthesis is m ost efficient when geared to a specific filter type. The perform ance of the design technique is the better the m ore inform ation about the particular filter type is presented to the design routine. On the other hand, w e could also show that the choice o f the optim isation algorithm (e.g. GA or gradient search) itself does not have a large im pact on the perform ance o f the design process. We can state that not a specific optim isation algorithm but additional inform ation can im prove the design process. This statem ent is known as the N o Free Lunch Theorem o f Optimisation. A proof can be found in [64]. Consequently, w e m ust focus our attention not on the optim isation algorithm itself, but on how to incorporate the m ost inform ation into the design process. A good exam ple is the circuit design using neural networks. Standard “black-box” netw orks perform poorly for very specific optim i sation problem s. As shown in e.g. [131, 132, 133, 134], neural networks perform m uch better when they include some fundam ental current-voltage relations relevant to the device. Including this knowledge reduces the required data acquisition from several hundred to a few samples. The sam e principle applies to space m apping, adjoint-netw ork m atrix method, and so on. The four techniques introduced in this thesis follow this path. W hen we include background knowledge in the genetic algorithm , it out-perform s gradient-based m ethods. O ur m ulti-dim en sional Cauchy m ethod includes know ledge by im plying that the circuit’s response is a rational polynom ial. H ybrid optim isation provides additional knowledge from the fine m odel to the coarse model. O ur tim e-dom ain based optim isation m akes use of inform ation in the transient response. K nowledge about the relationship betw een the response interval and the geom etrical interval helps us guide the optimiser. Incorporation o f knowledge about the circuit structure is the key to an efficient m icrowave cir cuit design. Throughout this thesis, w e have dem onstrated increased perform ance when using know ledge-based algorithm s. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8. S U M M A R Y A N D FU TURE W O RK 8.3 128 Future Work Filter synthesis is a very broad field o f research. In this thesis, we could only concentrate on a few filter topologies and design techniques. The follow ing research topics can be considered for future work: • Som e applications require filter topologies that are not covered in this thesis. A ctive filters are an example. Future research m ight concentrate on developing efficient techniques for those filter topologies. In particular, w e w ould like to consider algorithm s that incorporate know ledge about these filter topologies. • As shown in C hapter 7, the theoretical results do not m atch the m easurem ents perfectly. This is m ainly due to lim itations in the m anufacturing process and applied m aterials. We will im prove this situation in tw o ways. First, we will concentrate on filter layouts de signed for easier m anufacturing. Second, w e w ill try to im prove the m anufacturing process itself and the material. • The software used here is still in the developm ent stage. At this time, the use o f tools requires thorough understanding o f com puter program m ing and filter theory. It is desirable to im prove the program s to the point o f integration into an actual engineering environment. It is planned to incorporate these algorithm s into user-friendly C A D packages as part of C O M DEV-proprietary m icrowave sim ulation tools. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Appendix A Theory of Space Mapping Space m apping, first suggested by B a n d l e r , establishes a relationship betw een the coarse m odel param eter vector and the fine m odel param eter. W e refer to the vector o f the param e ters for the fine and coarse m odel as p p and p c , respectively. The aim o f the space m apping algorithm is to establish a m apping p — { p c '— >7 p f} (see Figure 2.7) such that IIR f {p f ) — R c { p c )\\ < £ (A .l) w here Ftp and R c is the response o f the fine and coarse m odel, respectively, and || || is a suitable norm. L et p (p f ) be defined as a linear com bination o f some predefined fundam ental fixed functions flip), M P), fn(p) (A. 2) in the form n (A 3 ) A m apping is established when w e know the coefficients a. Finding the a ’s for the m apping function p (p f ) is an iterative process. In the first step o f the space m apping algorithm the circuit is optim ised for an optim al response using the coarse model. The param eter vector o f the optim ised layout is defined as p*c . The circuit is now sim ulated with the geom etric param eters P q using the fine m odel, i.e. R(p*F = P c ) is calculated. This step is illustrated in Figure (a). 129 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. APP EN D IX A . T H E O R Y OF SPACE M APPING -►* •*C (a) coarse model space fine model space (b) coarse model space Pi fine model space -►* (c) coarse model space Pi fine model space Pi (d) coarse model space Pi fine model space Pi Figure A. 1: Illustration o f Space M apping R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 131 APP EN D IX A . TH E O R Y OF SPACE M APPING T he response R f (Pf ) plus som e additional responses o f points in the vincity o f p lF is now com pared to corresponding coarse m odel responses. By param eter extraction a corresponding P c and surrounding points are derived such that the error \\Rc(pb) - R f ( P f )\\ (A.4) is m inim ised. From the tw o sets o f corresponding coarse and fine m odel points (i.e. their re sponses are alm ost identical in their respective spaces) a m apping as used in eqn.( A. 3) is derived. _ii) T he derivation o f the correct m apping function p is an iterative process. L et p p 'b e the /'th iteration to approach the solution. The next iterate is found by a quasi-N ewton iteration w here M ^is obtained from B - h ® = -E ( j$ ) (A . 6 ) w here B_ is an approxim ation to the Jacobian o f the vector E with respect to p at the ith iteration. The m atrix B is updated at each iteration using B royden’s update [75]. The error function E is then evaluated from p (p f ), w hich can be found by a param eter extraction as described in [72] and [73]. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Appendix B Coupling Matrix Algebra B .l Coupling Capacitor Representations for Band-Stop Filters This appendix outlines a coupled resonator m odel that is dual to the inductive coupled resonator m odel described in Section 5.2.1. L et us assum e a netw ork with N resonators. The coupling between the resonators is achieved using shunt capacitors as described in Figure B .l. 2 Figure B .l: Capacitive C oupled Resonators Z There are + ^ u n k n o w n currents in the netw ork as shown. Using K irchhoff’s rules we can find the N nodes equations: 132 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. APP EN D IX B. COUPLING M A T R IX A L G E B R A *1 + *21 * 12 + *2 133 + ■ ■• + *ijV = iE (B .l) + ' ' ’ + *2 iV = 0 (B.2) : = 0 (B.3) I n i + *1V2 + • ■■+ *n —0 (B.4) w here we define i ki = - i ki and C mn = C nrn. A nd also there will be ( ^ — y ) loop equations: Z m im, “ Z n in — 0 *mn (B.5) J iO L /m n w ith all com binations o f m = 1 . . . N and n = ( m + 1 ) . . . N . Eqn. B.5 can be rearranged to imn = j ^ C m nZ n i n . (B . 6 ) Now, we can substitute all i mnterm s in eqn.( B.4) *1 + j u C w Z i h — iw (7 i2 ^ 2 * 2 4------- + j u C i N Z \ i i — J u C i n Z n i n = juCwZih —jujCnZ2i2 + *2 4----- + iwCiiv^2*2 —J+’CWZjyijv = •■■ + ••■ + *jv = *£ (B.7) 0 (B . 8 ) 0 (B.9) 0 (B.10) using O hm ’s law i k Z k = Vk we get the adm ittance m atrix ( rn y 12 Y21 Y22 y 1n • ■■ \ Y2N / V\ \ ( iE ^ V2 0 (B .l 1) 0 \ Yn 1 Yiv2 • ■■ Y n n ) \ V N ) \ 0 / w here the m atrix elem ents are R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. APPEN D IX B. COUPLING M A T R IX A L G E B R A Lmn — 134 jw C n N Ymm — En "F ^ ] jwCmk k= l,k^n for bandpass filter structures we define Ymas a resonator structure as shown in Figure and nor m alise such that Figure B.2: Lossy Parallel R esonator Circuit co L ui — + R — Zo uC .Wq U>0 + R = A co (B.12) and then the full m atrix is ( x-jwE^iCik -j w C i 2 -jw C x i \ ( ) - j w C 2N ~ jw C 21 V -jw C iN -jw C m \ ( i E ) v2 0 vs 0 VN ) 10 ) (B.13) B.2 Duality of Representations The Z-representation o f the left circuit is: The Y-representation o f the right circuit is vi = (R + + j w ( L + M )) i i + M i 2 *! = (G ' + JUU + M C ' + £>')) «i + D ' V2 v 2 = M i \ + ( R + -Gjq + ju j(L + M ) ) i2 ^ = (G ' + ^ U + M C ' + D' ) ) v x + D ' v 2 M ore general for a filter o f n th order the coupling m atrix and capacitor m atrix is The Z-representation o f the left circuit is: R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. APPEN D IX B. COUPLING M A T R IX A L G E B R A C HI— M > H h =L+M 135 R czh ?L+M Y -P re s e n ta tio n Z - P re s e n ta tio n F igure B.3: Duality o f C oupling Netw orks, On the left Coupling by Lum ped Elem ent Inductive T-Network, on the R ight C oupling by Capacitive 7r-N etw ork using A = R + + juL ( A - jui Y ,k = -ju M \ 1 -ju> M i2 -J w M ijv A-ju;E£LiM2* —j u ) M 2N M lk 21 —j u M j y i —j w M N2 A - \ J 2 k = i M ? ik J ( h ( \ VE ^ *2 0 *3 0 10 V lN ) ) (B.14) The Y-representation o f the right circuit is using A = G ' + j + ju C ' ( A' - ju j Y ,k = i D i k - ju jD - n \ -ju )D Ni -jw D n —j u D i N A' - j u Y ,k= i D 2k —j u D 2N —ju )D N 2 A ' - E f = i D Nk \ J ^ ( \ ( IE ^ v2 0 Vi 0 VN ) 10 ) (B.15) This shows that the Z-presentation m odel is a dual to the Y-presentation m odel. In the dual m odel with have to swap all param eters by R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 136 APP EN D IX B. COUPLING M A T R IX A L G E B R A 1 < ^ = f> V R B.3 V i 4 = f- G' L C' C L' M D' Coupling Values are Pure Real Numbers W hen looking at arbitrary inductive or capacitive coupled resonators, w e m ay think the cou pling values m ust be com plex in order to reflect phase delays betw een the coupled resonators. Nevertheless, com plex valued M -m atrices are not necessary. The reasoning is outlined here. L et us recall the arrangem ent o f Figure B.3. The coupling network is an inductive T-network. First, we assum e the elem ent values to be positive real. Thus, the T-network can create trans m ission phase shifts o f 0° to 90° in the com plex plane depending on the values o f M an L . As seen from the equivalent circuit, the inductance L is absorbed in the resonator inductance and will show up in the m atrix o f eqn. (B.14) as a non-zero value in the m ain diagonal. H ence, a real valued positive coupling m atrix can represent a 0° to 90° phase shift in the coupling structure. Further, w e are not lim ited to positive real values. N othing prevents us from choosing L and M negative real. Then, we can create any phase shift with the proper choice o f L and M , while still m aintaining a real coupling m atrix. Som e M -values m ay be negative though. Negative coupling values can be observed in the num erical coupling m atrices throughout C hapter 5. The theory is lim ited to narrow -band signal, because the frequency dispersion o f the T-networks is not identical to the dispersion o f the real coupling networks, i.e. delay lines. In band-lim ited signals (bandw idth<10% ), however, T-networks are a very good approxim ation o f delay lines. The errors are usually less than 1%. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Appendix C Validity of Rational Function Representation The port im pedances o f any lum ped linear passive elem ent m icrowave circuit can be represented as a rational function o f its elem ent values Inductance L , capacitance C , and resistance R , as well as its harm onic operating frequency uj. T he reason for this is as follows: Any lum ped passive elem ents network can be represented by a com position o f T-networks or 7r-networks as shown in Figure C .l. The decom position and com position rules are described in detail in [39]. H hJLH b z3 T ---------- 0 Figure C .l: T-Network and 7r-Network The Y-parameters (Y-matrix) as defined in [39] for a T-network is PI 1 Z2 + Z3 Z \Z 2 + Z ‘i Z,\ + Z \ Z$ —Z 3 —Z 3 Z \ + z% A nd for the 7r-network 137 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (C .l) APPEN D IX C. V A L ID IT Y OF R A T IO N A L FU N C TIO N REPRESENTATIO N 138 w here Y and Z o f lum ped linear passive elem ents are generally proportional or inverse propor tional to the frequency to and their respective elem ent values C , L , and R , i.e. Z oc ju) or Z oc — or Z oc j u j C or Z oc L or Z oc R (C.3) Hence, w e can describe the m atrix elem ents o f [Y] or [Z] analytically by rational polynom ial functions. T he polynom ial variables can be either the frequency or the elem ent values R , C , and L. Com posing the com plete circuit from cascaded T-networks and 7r-networks includes only the basic algebraic operations o f addition, subtraction, m ultiplication, and division. Consequently, the resulting transfer function rem ains a rational polynom ial function. This explains, that all netw orks o f lum ped passive elem ents can be represented as rational poly nomials. C.l Distributed Elements W e can use the rational function approxim ation also for a band-lim ited response representation o f distributed elem ents. In that case the response o f the distributed elem ent m ust be approxi m ated using a Pade approxim ation. U sing this m ethod the response o f a band-lim ited signal can be arbitrarily closely approxim ated. For exam ple, w e can approxim ate ta n (x ) by the rational polynom ial ta n (x ) « 1 x + ^-x 3 r, - 21—=— - . + 3r * 2 + (C.4) This Pade approxim ation o f ta n (:r) is shown in Figure C.2. Similarly, w e can approxim ate m ost distributed circuit elem ent param eter, e.g. transm ission lines, stubs etc. The S-param eters of such elem ents are functions o f the line lengths and line im pedance proportional to trigonom etric functions, representable by Pade approxim ation. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX C. V A L ID IT Y O F R A T IO N A L F U N C T IO N R E P R E S E N T A T IO N 30 20 10 0 -10 -20 -30 •3 •2 -1 0 x 1 2 3 Figure C.2: Tangents o f x and Pade Approxim ation R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Appendix D Circuit Layouts 140 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. APP EN D IX D. CIRC U IT L A Y O U T S D cfUUO h 0.3/40 304 / O.bBP/ 3 3/b 4 3383 0. 8161 ? 8130 Figure D .l: L ayout o f LC Low -Pass Circuit R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. APP EN D IX D. CIRC U IT LA Y O U T S 142 1.1127 0.0069 0,0069- J.0069 1.00f?3 - 1 •- 0 0092 - 0 0 7 8 2 ------ 0,0644 1 .0 0 69 0.0069 1. 1334 Figure D.2: L ayout o f sub-section o f six-pole filter R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. APPEN D IX D. C IRC U IT L A Y O U T S ~r T -~ ^ — r F igure D.3: L ayout o f Six-Pole filter R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] S.F. Peik, Y.L. Chow, F ilter Synthesis with Genetic A lgorithm s and Constraints, Proceedings o f the P IERS Conference 1997, Vol I, p. 229, 1997 [2] S.F. Peik, Y.L. Chow, Genetic Algorithm s A pplied to M icrowave Circuit Optim ization, Pro ceedings o f the A sian Pacific M icrow ave Conference 1997 , pp. 857 861, 1997 [3] S.F. Peik, R.R. M ansour, Y.L. Chow, M ultidim ensional Cauchy M ethod and Adaptive Sam pling for an A ccurate M icrow ave Circuit Modeling, IE E E M TT-S Sym posium D igest 1998 Baltimore, pp. 21-24, June 1998 [4] S.F. Peik, R.R. M ansour, Y.L. 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