Symbolic Analysis of Analog Circuits: Techniques and Applications: A Special Issue of Analog Integrated Circuits and Signal Processing

81 Pages·1993·6.031 MB·English

by  Marco Amadori, Roberto Guerrieri (auth.), Lawrence P. Huelsman, Georges G. E. Gielen (eds.)

SYMBOLIC ANALYSIS OF
ANALOG CIRCUITS:
TECHNIQUES AND APPLlCATIONS
edited by
Lawrence P. Huelsman
University of Arizona, Tuscon
and
Georges G.E. Gielen
Katholieke Universiteit Leuven Belgium
A Special Issue of
ANALOG INTEGRATED CIRCUITS AND SIGNAL PROCESSING
Reprinted from ANALOG INTEGRATED CIRCUITS AND SIGNAL PROCESSING
Val. 3, No. 1 (1993)
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
ANALOG CIRCUITS AND SIGNAL PROCESSING
Consulting Editor
Mohammed Ismail
Ohio State University
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Contents
Special Issue on Symbolic Analysis ofAnalog Circuits:
Techniques and Applications
Guest Editors: Lawrence P. Huelsman and Georges G.£. Gielen
Editorial Mohammed Ismail. David G. Haigh and Nobuo Fuji 5
Guest Editors Introduction Lawrence P. Huelsman and Georges G.E. Gielen 7
Symbolic Analysis of Simplified Transfer Functions .
· Marco Amadori, Roberto Guerrieri and Enrico Malavasi 9
Symbolic Analysis of Large-Scale Networks Using a Hierarchical Signal Flowgraph Approach .....
·........................................... Marwan M. Hassoun and Kevin S. McCarville 31
Formula Approximation for Flat and Hierarchical Symbolic Analysis .
·......................... FV Fernandez. A. ROdriguez-Vazquez, J.D. Martin and J.L. Huertas 43
Symbolic Simulators for the Fault Diagnosis of Nonlinear Analog Circuits .
·......................................................... S. Manetti and M.e. Piccirilli 59
More Efficient Algorithms for Symbolic Network Analysis: Supernodes and Reduced Loop Analysis
· RalfSommer, Dirk Ammermann and Eckhard Hemig 73
Llbrary of Congrcss Cataloging-in-Publication Data
Symbolic analysis of analog circuits : techniques and applications I
edited by Lawrence P. Huelsman, Georges G. E. Gielen.
p. cm. --(The Kluwer international series in engineering and
computer science : SECS 219. Analog circuits and signal processing)
Issued also as a special issue of Analog integrated circuils and
signal processing, volume 3, no. 1, January 1993.
ISBN 978-1-4613-6424-5 ISBN 978-1-4615-3240-8 (eBook)
DOI 10.1007/978-1-4615-3240-8
1. Electric circuits, Linear. 2. Symbolic circuit analysis.
3. Electronic circuit design-oData processing. 1. Huelsman,
Lawrence P. II. Gielen, Georges. III. Series: Kluwer international
series in engineering and computer science ; SECS 219. IV. Series:
Kluwer international ser ies in engineering and computer science.
Analog circuits and signal processing.
TK454 . 15 . LS6S96 1993
621 . 3815--dc20 92-40211
CIP
Copyright © 1993 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1993
Softcover reprint ofthe hardcover Ist edition 1993
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, mechanical, photo-copying, record ing,
or otherwise, without the prior written permission of the publisher, Springer Science+
Business Media, LLC.
Printed on acid-free paper.
Analog Integrated Circuits and Signal Processing 3, 5 (1993)
© 1993 Kluwer Academic Publishers, Boston. Manufactured inThe Netherlands.
Editorial
Wearepleasedtoannouncethatstartingwiththis issueofAnalogIntegratedCircuitsandSignalProcessing(Volume
3, 1993), the Journal will appear six times a years instead offour. This increase in issues per year is intended
to keep up with the increased number of high quality papers being submitted for publication.
WearealsoverypleasedtowelcomeasmembersoftheEditorialBoard, Drs. JohnChoma,Jr.,JohanHuijsing,
EdgarSanchez-Sinencio, TrondSretherandGaborTernes, andwelookforward tothevaluablecontributionsthey
will make to our Journal.
Mohammed Ismail
David G. Haigh
Nobuo Fujii
Editors-in-Chief
Analog Integrated Circuits and Signal Processing 3, 7 (1993)
© 1993 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Guest Editorial
This Special Issue is dedicated to the techniques and applications of symbolic analysis for analog circuits. The
generaltopicofanalogcircuitanalysismay bedivided intotwo maincategories. Thefirstofthese isusuallycalled
numericanalysis. Inthis, numericvaluesofohms, henries, farads, gain, andsoforth, areassigned totheappropri
ate circuit elements. The interconnection ofthe elements in the circuit is specified by topological information,
typicallygivenas nodenumbers. Thegoaloftheanalysisisthegenerationofnumeric informationgivingsinusoidal
steady-stateor time-domain response information, which is presented either in tabular form or, more usually, as
plotted information. Examples of numeric analysis are readily seen as the output from the SPICE program or
itsPCcounterpartPSpice. Thesecondgeneralcategoryofanalog circuitanalysis istheoneaddressed in thisSpecial
Issue, namelysymbolicanalysis. Inthis, literalnames(symbolic values)areassignedtotheelementsofthecircuit.
These literal names represent the symbolic (nonnumeric) values ofthe ohms, henries, farads, gain, and so forth
ofthecircuitelements. Thesenames, togetherwith the topological interconnection information, areusedtocreate
anetworkfunction inthecomplexvariablesorzwhichgivesadescriptionoftherelationbetweenthetransformed
output and input variables ofthe circuit. In such a network function, the coefficients ofvarious powers ofs or
zappear as explicit functions ofthe literal names of the circuit elements.
Symbolicanalysisand thecomputertechniques for automatedsymbolicanalysis, i.e., theautomaticgeneration
of analytic equations describing a circuit's electrical behavior, have reattracted much attention in recent years.
They represent a natural way ofanalyzing a circuit, a way taught in all basic engineering courses and practiced
by real-lifedesigners. Symbolicanalysis isfarmoregeneral than numericanalysis, sinceifofferscompletefreedom
inthechoiceofapplications, and includes sinusoidal steady-stateandtime-domainstudiesasspecialcases. Numeric
simulators such as SPICEhave becomemuch more popularthan symbolic ones as design supporting CAD tools,
because they can rapidly and accurately simulate a circuit's behavior, including its transient response. They are
also able to simulate larger-sizecircuits. Incontrast, however, symbolic analyses can provide many results which
are simply not available from numeric simulation methods. Most importantly, they can provide explicit insight
into the dominantbehaviorand propertiesofacircuit. Among the useful applications ofthis insightarethedeter
mination of derivatives of the network function with respect to one or more elements. Such literal information
providesdirectapplicationto sensitivitydetermination. Anotherapplicationofthe insightobtainedfrom symbolic
analysis is the development ofthe equations which are required in the use ofoptimization techniques to provide
solutions to specific design specifications. With SPICE-like numerical simulators, the same insight can only be
obtained after combining and often extrapolating the results ofnumerous simulation runs. In addition, symbolic
analysis can also be used in many other applications, such as in compiled-codeevaluation for statistical analysis,
and automated synthesis orfailure diagnosis ofanalog circuits, much the sameway as symbolic Boolean analysis
is used for synthesis and verification of logic circuits.
Foralongtime, symboliccircuitanalysishas beenregarded as anacademic topic. Itis truethatithascomputa
tionalcomplexitylimits whichhaveprevented itfrom beingfeasible for large-sizecircuits. In recentyearshowever,
enormous progress has been made in developing more advanced techniques and algorithms for symbolic circuit
analysis. This has resulted in an extension ofthe functionality ofsymbolic simulators, including for instance the
automatic generation ofsimplified symbolic expressions or the automatic generation of symbolic distortion for
mulas. At the same time, the capabilities ofsymbolic analysis have been extended toward larger circuits by the
introductionofhierarchical methods. All these advancements have resulted in the recent developmentofseveral
successful symbolic simulators such as ISAAC, ASAP, SC, and SSPICE. As a result, symbolic analysis is finally
becoming an attractive tool to assist designers in real-life circuit design.
ThisSpecialIssuecontainsfiveselectedpapersthatpresentrecentdevelopmentsinthefieldofsymbolicanalysis
for analog circuits. The first paper, by Amadori et aI., presents original algorithms for the direct generation 0;'
simplified symbolic transfer functions based on the relative magnitudes ofthe circuit elements. These simplified
expressions, which show the dominant contributions only, provide a good approximation for the overall circuit
behavior. Also, an algorithm for thesimplified symboliccomputationofthepolesand zeroesofthe transfer func
tions is described. Hassoun and McCarville, in the second paper, describean approach to the symbolic analysis
8 Huelsman and Gielen
oflarge-scale networks based on hierarchical decomposition. The total network is recursively decomposed into
smallersubblocks, whichareanalyzedseparately. Theexpressionforthetotalnetworkisthenobainedbycombin
ing bottom-up the expressions for the subblocks. This tremendously reduces the CPU time and the number of
symbolic terms for large circuits. In the third paper, Fernandez et al. describe new criteria and algorithms for
thegenerationofsimplifiedexpressions, bothforflatandhierarchicalsymbolicanalysis. Amajordifferencebetween
this approachand thatgivenbythefirst paper, is thatthesimplificationiscarriedouttaking into accountarange
ofelement values instead ofa single nominal value for the magnitude ofeach circuit element. The technique of
simplificationisalsoextendedtothehierarchical formulas, whichwouldbetheresultofthedecompositionmethod
ofthesecondpaper. Thiscombinationopensnew perspectives for thefastgeneration ofbothexactandsimplified
symbolicexpressions for largecircuits. In the fourth paper, Manettiand Piccirilli show how dedicated simulators
basedoncompiledsymbolicformulascanboosttheefficiencyofapplicationsrequiringrepetitivecircuitevaluation.
Nonlinear circuits are handled with piecewise linear approximation. The application ofthe method to nonlinear
circuitfaultdiagnosisispresented,inwhichtheactualelementvaluesandhencealsofaultycomponentsareextracted
by fitting the simulated to the measured response. Finally, in the short paper by Sommer et al., two alternative
network equation formulations are highlighted: supernode and reduced loop analysis. Compared to the classical
node, loop, and MNAformulations, thedescribedvariants resultinsimplerequations. Thisis advantageous both
for manual analysis as well as for computerized symbolic analysis.
The editors would like to thank all the authors who submitted papers, all the reviewers who participated in
the final selection ofthe papers, and the Kluwer Editorial Staff for their efforts in creating this Special Issue.
We hope that this Issue will provide you, the reader, with a useful introduction to the potential and powerofthe
use of symbolic analysis techniques in analog design.
Lawrence P. Huelsman
Georges G.E. Gielen
LawrenceP. HuelsmanreceivedtheBSEEdegreefrom CaseInstituteofTechnologyandthe MSEEand Ph.D.
degreesfromthe UniversityofCaliforniaatBerkeley. HeisaFellowoftheInstituteofElectricalandElectronic
Engineers. Hecurrently holds anappointmentas ProfessorEmeritusofElectricaland ComputerEngineeringat
the University ofArizona.
Dr. Huelsmanistheauthororcoauthorofsixteenbooksincluding: BasicCircuit Theory-3rdEd. published
byPrentice-Hall,Inc.;andOperationalAmplifiers:DesignandApplications.IntroductiontoOperationalAmplifier
TheoryandApplications,andIntroductiontotheTheoryandDesignofActiveFilters, publishedbytheMcGraw
HillBookCompany.Japanese,German,SpanishandRussiantranslationshavebeenmadeofseveralofhisbooks.
He has also published many papersin the areaofactivecircuittheory.
HehasservedasAssociateEditoroftheIEEETransactionsonCircuitandSystemTheoryandtheIEEETrans
actions onEducation and was technicalchairman ofthe IEEERegion SixAnnualConference. He isamember
ofthesteeringcommitteefortheMidwestSymposiumonCircuitsandSystems. Heisamemberofseveralscien
tific,engineering,andhonorarysocieties,includingTauBetaPi,PhiBetaKappa,EtaKappaNu,andSigmaXi. HehasreceivedtheAnderson
Prizeofthe CollegeofEngineeringand Mines ofthe University ofArizona for his contributions to education.
GeorgesG.E. Gielen was born in Heist-op-den-Berg, Belgium, on August25, 1963. He received the E.E. and
Ph.D.degreesinelectricalengineeringfromtheKatholiekeUniversiteitLeuven, Heverlee,Belgium,in 1986and
1990, respectively. From 1986until 1990, hewasappointedby theBelgianNational FundofScientificResearch
asaResearchAssistantattheESATlaboratoryoftheKatholiekeUniversiteitLeuven,workingonsymbolicanalysis
andanalogdesignautomation. From1990until 1991,hewasconnectedtothe UniversityofCalifornia,Berkeley,
asaVisitingLecturerandVisitingResearchEngineer,workingonbehavioralmodelsforanalogintegratedcircuits.
InOctober1991,hewasagainappointedbytheBelgianNationalFundofScientificResearchasaSeniorResearch
Assistantatthe ESAT-MICASlaboratoryoftheKatholieke UniversiteitLeuven, Heverlee, Belgium, wherehe is
currentlyheadingtheanalogdesignautomationgroup. Hisresearchinterestsareinthedesignofanalogandmixed
analog-digitalintegratedcircuitsandinanalogdesignautomation(modelingandsimulation, synthesis,optimiza-
tion,layoutandtesting).Hehasauthoredorco-authoredmorethan30papers,includingseveralchaptersforedited
books.In1991,healsopublishedabookonsymbolicanalysisanddesignautomationofanalog integratedcircuits.
Analog Integrated Circuits and Signal Processing 3, 9-29 (1993)
© 1993 Kluwer Academic Publishers, Boston. Manufactured inThe Netherlands.
Symbolic Analysis of Simplified Transfer Functions
MARCO AMADORI, ROBERlO GUERRIERI, AND ENRICO MALAVASI
Dipartimentodi Elettronica, Informatica e Sistemistica, Universita di Bologna, viale Risorgimento, 2-40/36Bologna, Italy
Abstract. Thiscontributionpresentsnew algorithms for theautomatic simplifiedcomputationofsymbolictransfer
functions oflinear circuits. The problem ofsymbolic simplification oftransfer functions is defined and a set of
algorithms able to cope with this problem and the simplified computation ofpoles and zeroes is developed and
discussed. Results are reported with examples ofcircuits analyzed by our algorithms, showing good accuracy in
theirapproximation, whencomparedtothecorrespondingSPICEsimulations. Ourtechniquemergesthesimplifica
tion procedure and the evaluation ofthe transfer functions, thus achieving significant improvements in terms of
CPU time, compared with direct evaluation of the functions themselves.
1. Introduction methodology have been presented in [10], [11], and
[13]. Thegoalofthisapproach istwofold. Firstly, the
Thepresenttrendintherealizationofcomplexsystems, availability of simple expressions provides a better
whereanaloganddigitalpartsoftencoexistonthesame understandingofthebehaviorofthe circuitandofthe
chip, requiresCADtoolstoautomatethevariousdesign relativeimportanceofeachparameter, avoidingatthe
phases. Although the design ofthe digital circuitry is same time a tedious and error-pronemanual elabora
extensivelycarriedoutbyautomatictools, itisnoteasy tion. Secondly, compactanalyticexpressionssimplify
tofindappropriateCADtoolsfortheanalogsynthesis; theworkofnumericoptimizersusedforcircuitsynthe
therefore a significant effort is required by expert sis, since the problem is considerably reduced in its
designersto implementtheanalogpartsby hand. This complexityaswellasinthenumberofparameters[14].
introducesdelayandcostintothechipdesignprocess. This feature would allow the use ofcircuit synthesis
Aphaserequiringmuchtimeand effortisthecom toolssuchasIDAC [15]andOPASYN [16] forarbitrary
putationofthetransferfunctions thatdescribethe cir architecturecells. In[10], simplificationisachievedby
cuitbehavior. Theresearchintheautomaticderivation exploiting the knowledge ofthe circuit structure. The
ofsymbolictransferfunctions ofelectroniccircuitswas retrieval of peculiar components and architectures in
developedduring theseventies[1-3]. Duetothecom the schematic (such as differential pairs and current
plexity ofthe symbolic expressions and the difficulty mirrors) and the recognitionofnonsignificantcapaci
ofdealingwithnonlinearproblems,thisfieldwaslater tancesallow thetool to simplifythecircuitneglecting
abandoned,althoughithadproducedseveralinteresting nonrelevantparameters. InISAAC [13] a heuristicap
contributions, suchasefficientalgorithmsforthecom proach based on the order of magnitude of circuit
putationoftransferfunctions [4-6]andaccurateestima parametersisdescribed. Thisallowsthetooltoneglect
tionsofthecomputationalcomplexityoftheseproblems theparameterswhichprovidesmallcontributionstothe
[7, 8]. Inthefollowing years, numerical simulationof overall transfer function.
linear and nonlinear circuits [9] received a much Unfortunately, neitherapproachaddressestheprob
strongeremphasissinceitremovedtheabovelimitations. lem of formal definition of simplification. Both use
Inrecenttimes, thesymbolicanalysisapproachhas heuristicswhendealingwiththereductionofcomputa
gainednew interest[10-12], aimingattheachievement tionand, finally, donotanalyzethestabilityproperties
ofautomatic computation ofsimplifiedtransfer func ofthesimplifiedexpressions, whenworking inthefre
tions. Abroadoverviewofthetechniquesforsymbolic quency domain.
analysiscan befound in[13], whileapplicationsofthis In this paper we propose a novel approach to the
simplificationoftransferfunctions. Fouroriginalalgo
Thisworkhasbeenpartiallysupportedbythe NationalCouncilof
rithmsarepresented. Thefirstreducesthefull rational
Research(CNR)underProgettoFinalizzatoMADESSandbyagrant
ofSGS-Thomson, Agrate Brianza. expressionofatransfer function by meansofa global
10 Amadori, Guerrieri and Malavasi
strategy based on the orders ofmagnitude ofeach of r
2..:
itsterms. Itcanbeappliedtothefully expandedexpres P = aisi (1)
sion of the transfer function. The second and third i=oO
algorithmsperformthesimplificationduringthecom
where r, maximum degree in s of the polynomial,
putation ofthe determinant ofthe admittance matrix
depends upon the numberofcircutcapacitors and the
atzerofrequency and in thefrequency domain respec
topology of the network.
tively. Thefourth algorithmfinds the locationsofpoles
andzeroesofatransferfunction byexploitingthepole
PROPERTY 2. Ifthe circuit is stable, polynomial (I) is
splittingand poleclusteringhypotheses. All thesealgo
complete, that is,
rithms are proved to fulfill a consistent definition of
optimal symbolic simplification. The problems raised ai ~ 0, i = 0, ..., r
by symmetricand antisymmetric architectures, due to
Furthermore all its coefficients have the same sign.
thefactthattheirtransferfunctions arenot in minimal
form, areinvestigatedandsuitableapproachesaresug
gested. The requiredparameterclassificationdoesnot PROPERTY 3. The coefficients ai are given by
relyontheuser'sexpertisebutisautomatic andbased
Ki
on simulation. a = L"".."J Ak (2)
Theorganizationofthispaperisasfollows. InSec I I
k=l
tion2thebasicpropertiesoftheadmittancematrixare
recalledand thesymbolic structureofatransfer func whereKiindicates the numberofmonomials inai and
At
tion is described. In Section 3 two algorithms for the is the monomial given by the product ofn factors
automatic simplification ofa transfer function at zero of the form
frequency areillustrated. Theirextensiontothegeneral Aik = (g\k ... gnk-i)(CIk'.. Cik) (3)
case in the frequency domain is described in Section
gf
4. In Section 5 the implementation ofthe algorithms where are conductances or transconductances and
Cf
is illustrated and results showing the suitabilityofour are capacitances. Ifi = 0,
symbolicsimplificationarereported. Finally, inSection
n
6 some conclusions are drawn. A~ = II (gt) (4)
i=l
2. Basic Definitions and Properties is the product ofexactly n conductances.
The transfer functions of an n-port linear circuit are Unfortunatelythesymboliccomputationoftheabove
given by the ratios ofdeterminants derived from the polynomials leads to very complex expressions even
nodaldefiniteadmittance matrix(DAM)ofthecircuit. in simple cases [7, 8]. Such expressions are difficult
In what follows we will recall some properties ofthe to evaluate and to use for design purposes. Consider,
DAM. Proofs ofthese properties can be found in [17, for instance,thetwo-stageoperationalamplifiershown
pp. 197 ff], along witha comprehensivesurvey ofthis infigure 1, whoseequivalentcircuitis showninfigure
subject. Forsakeofsimplicityandsinceouremphasis 2. Itsadmittancematrix isofthefifth orderand, even
isonproblemsstemmingfrom thelinearizationofcir keeping the differential-stage transistor matching into
cuits implemented in integrated technology, the only account, itcontains 14differentparameters: the exact
circuitelementsweconsiderareconductances,capaci expressionofitsdifferentialgain has 390monomials,
tances and voltage-controlled current sources. 48 in the numerator, and 342 in the denominator.
Giv~nalinearcircuitwithn + 1distinctnodes, the Sinceevenforsmallnetworksthecomputedexpres
DAM Yassociated withitisan invertiblesquarematrix sions are very complex, it is essential to simplify the
of size n, with the following properties. above polynomials, removing some monomials so as
to minimizetheperturbationinducedintothefunction.
PROPERTY 1. The polynomial P given by P = det(Y) Thesimplificationprocedureisbasedonthefollowing
can be expressed as definitions.
Symbolic Analysis of Simplified Transfer Functions 11
Vdd
M6
~ M1 M2 ~ '--_~
Vin1 I4-____-.--___--t-.l Vin2 Cc
Out
Cl
I
M5 M7
Vbias --l.----H...._:..:..:,
Vss
Fig. 1. Schematicofatwo-stage operational amplifier.
Gm4 v3
,f',.---=":"----l 1-----1---,--- Out
Viol -
Gm2 ( Viol .. v5 ) .. v5
C1
Go?
Fig. 2. Small-signal circuit for the two-stage op amp.