Automatic analogue circuit synthesis using genetic algorithms

J,B,Grimbleby

Ahstract: Most analoguc systcn1s mc designcd manually bccausc automatic circuit synthesis tools arc availahlc Cor only a lillliled range or design problems, ;\ new approach to cirellil synthesis hased on genetic alg()1'ithllls is prescnted, Using this I11ctllOd it is possible ill principle to synthcsise circuits to t11eel any linear or I1lJlllincdr, ['requency-doillain or tillIe-domain, ,�peci[icatioll, Whcn applied to cxisting liltct, desigtl problems this circuit synthesis tm:thod prodLlccs design solutions that arc mme el'liciellt liliin thosc resulting I'rolll ['(mmd design ltlc(hods or created manually hy an cxpcriclll:cd analoguc circuit designcr. ----- ---- ------- ------------- --- ----

Introduction

Digital systcms Gtn !lOW bc synthesiscd ;tuloillatically on a computer, hut (his is not tlte Glse Il)\' analogue systems, I ,'ormal design solutions do exist, but only for limitcd classes 01' analogue design prohlems (1'01' example linear ['1'1';­qucncy-doillain lillers) and most analoguc eirellit design is still peri'l)J'\\led manually by s kilhl engineers,

Thcse obscrvations Oil <Imt\oguc circuit synthesis do not extend to circuit analysis, and excellenl analogue circuit ,tmtlys is tools slieh as SPIC'l' havc heell ,tv,tilahle 1'01' tllany years, It might be thought that ill order to synlhesise cir­cuits it is simply ncecss,lry to 'close the 100[,)' around circuit amtiysis, Indeed this is exactly what Ilappens wilen a circuit is subjcctcd to nUlllerieal opt illlisalioll, LJ lll'(lttunutely, ,tlthough llUtllCrieal optim is,ltioll is a trlle dcsign technique, its applicat ion is sevcrely limiLed by tlIe lilcL that it operates only on [Ixed circuit topologics,

1[' an oplimisation tcehnique could be [(lund that Illodi­lied both circuit topology and cOlllponcnt values, tllen this cOlild lim11 the basis or an (\l\(tlog:uc circuit synthesis method, C,etletic algorithms «( ,As) arc jllst such ,Ill ol')timi­saLion technique, [)cvelopcd in the 1970s ami 19?-;()s [I, 21, CiAs havc heen succcssl'ully applied to a wide range 0[' tlwnerieal and nOIH1ulileric,t\ optimisatiol\ problems,

2 Genetic algorithms

(;As operate on the principle or 'survival 0[' thc [lttesL', gCIl­eraLing new design solutions rmm H popLtlatioll 01' cxisting solutions, and discarding those deslgll solutions which 11Hve all ill[crior perl'()t'lnanee or Iitness, I':aeh Illeillber 0[' tlte population has u 'CIU'(JIllOSOllle' which cOllsists or a number of' 'gcnes'; each gcne represents olle pan oj' the design so[u­tioll, New dcsign solutions mc crc;tted by 'hreeding' rrom a pair 0[' existing solutions, Pareltts arc 1',lndomly selected

�) [LI':, 2()()O Ih'F /'f'()(,(Y'liillgs (lnlilll' no, �()()OO7711

1)0!; 111,1 040iip-ctb;;'()[)0Il770

Paper l irsl received IOlh Scpleillber 11)1)1) anri in revi",,1 [illin 7.5th .I lily 201111 'I he aLilhor is wilh the Flectl"Ollie I:n�illcerillg (il"OlIl', Titc l !nivClsily oC IZead� illg, 1'0 Ilox �?'i, I{catling R(;(, (1/\ Y, I!K

i'mlll the present population, hut with a bi,1s towmds the rittest individuals, and their ehl'OlllOSOllWS al'c lllcrged to produce lhc chromosulllc or the olE,Pt inL'., A pl'llportion or o[E;pring is thcll suhjeetcd to randotl l tllulatiotl,

A relatcd tecitniquc is genelie pIl,grllmming «(JI'), in which the chromosome rcpresents not tlte dcsign solution direeLly bLlt a vari,tble-kllglh expn.:ssiOIl cont,linillg tenni n,tis ;lIlt! operatOt-s wltich, whctt cvalultted, yields the design solutioll [J), III G],s it is litis eX[lresSiol1, rather [hall the design solution itselr, that is slll1Fcl to crossovcr and mLlUI­lion,

3 Application of genetic algorithms to circuit synthesis

One approach to cil"CLtit synthesis is to ,IHow �I (;A or (II' 10 determinc both tilc topology ;\Ild the cOlllponent values, This approach has bccn adopted hy [(t1/a, IknnclL Andre and Ke,lllc [4, 5), who Lise (;1' with a variuhlc-Iength eApression con[,lining topo[ogy-Illodifying operalors, COI11-ponent-ereaLing operators and (tl'ith11lelie-pelforming opel'" ,Ilors, In Koza's ,tpproach the operators that modi!,y the circuit topology ami select component valucs me inscpara­hle, and all arc Linder the eont\'()l o!' the eVDlutiomtry proe­csses opc m ling on the exprcssilln, Circuit littless is cvaluated using a SPICE sillluiatol', the code lor which has been incorporated into the synthesis program,

('ircuit synthesis involves hoth the selection or a sLlitahle topology and choice or cotnponcnt values, and as Koza has shown, these Illay be opLimised simLlILalleoLlsly, Ilowcver, there is no reason why these operations should not be per­rormed scparatcly, with diiTercnl optimisation methods being uscd, Clearly the circuit topology must be chosen [lrst, ,tIld thc appropriatc algorithm ['or this is a CA, hlr e,\eh eil'euit topology gCllcr;tled the eomponenl values call thetl be optimised, and the pel'1'(Jrlmtnee 0[' the eit'Cllit returned as the [iltlcss I'ullction to thc (iA,

The COlllpOIll;nt values could also hc optimiscd Llsillg (\ (fA but, 1'01' j)mblcllls involving well-hehaved ohjcctive fUllctiolls dependcnt Oil the values 0[' a [ixed tllll11hCI' or vmiab1cs, it is well esulblislled that nU11lerical optimisation lllethods eOtlVerge IlILlCh 1�lster and involve ICwer objcctive I'ullction cvaillations, No optimisation method can be guar­anteed to [ind the global optimu\ll, but it has been [(llllld

16J that numcrical optimisatioll oj' componcllt valucs achicves a high jlropmtiol1 or I'csults closc to the global optimum. This hybrid ,lpprO<leh using a GA to sclect ,I suitablc topology and numerical optimisation to ehoosc componcnt values is likcly to be more enicient thm1 using ,I Ci/\ or GP to perrorm both rUl1ctions.

4 Hybrid genetic algorithm

The most important decision in applying (,As to <In opti­misation problem is how to represent the design solutiol1 ill the chromosome . A circuit topology CWl be spccilied by a list or com poncnt types togethcr with their termina l nodes, and it is naluml, lhcrci()rc, to makc cach gcne of the chro­mosomc represent a single component. Chromosomes arc normally or Jlxed length and contain a IIxed 1I11mber or genes, whereas circuits can clJnt,lil1 any number or compo­nents, This problem ean bc avoidcd by having, in ,Idditioll to (he standard component types or resistor, capacitor and inductor. an 'el11ply' component type. Since the CiA oJler­ates on circllit topology only, thc gcnes do l10t contain component va Illes. In c++ the dat,1 memhers 01' the gCJII' class arc delincd: onum component_type (rasis Lor, capacitor, inullcto r', empty}; t ypedef short int node;

c las s gene {

};

pri vate: component;_type type; nodo ni, n2;

publi.c:

Thc data member of the c/imlJIIISOIlW class IS simply <In array 0[' gcncs:

class chromosome {

};

private:

gena genes [16] ; public:

The number of' genes determines Lhe maximum circuit com­plexity so that this parameter should be Cit least cqual to thc anticipated complexity or the optimull1 circuit conligLl­ration. In i�lct, it should be somewhat larger than this to allow the circuit to evolve towmds the oplill1l1m by compo­nent deletion. Choosing too large a valuc , hm'l'ever. bids to inemciency hee<lllse unrealistically complcx circuits will he generated, and will need to be analysed.

output ,

Fig.1 /iJidl'('(i-ljt'ill'J'

The 1l1Pllt 01 the cirClilt is ,dways node I and the output is node 2: 110de 0 is the rc/cre11cc (or ground) node. hg. I shows a bridgcd-T l11ier with 11umbcred nodes. A possible chromosome (containing only six. gcnes Jllr clarity) reprc­senting this circuit is shown in Table L 120

'fable 1: Chromosome representing bridged-T filter ------------------------Gene 1 (jell(};'> C,OIIO 3 Gone 4 Gnnn" Genn (i

lype msistor ("nply capacitol losistor CClp<lcilor omply 3 7

3 o

Ncw individual, arc gcncratcd by uniJl)J'[n genetic crosso­ver from (wo parents (witl1 pro[xlbilily typically 70';;,) or by mutating a single parenl. In either case, i'urLher l11utation (with prolxlbility (ypically�()%) may thel l be applied. The eJ'Jiciency oj' the CiA docs not depcnd critically on either oj' these prolxlililitics. P,lI'l:n(s arc selected by tourmllllenl between two randomly-choscn individuab.

The population size i,� norl11ally KO 20(); a 1,Irger POj1l1hl­lion would prohably rcsult in <Ill increil,ed rohustness 01' the OA, bUl would lead to longcr sYllthc,is ti ll lCS . Lacl1 individual or the initial poplilatiol1 is crcatcd hy repeatcdly generating random chrumosoilles ulltil OliC is Illllnd that COITcsponcis to a viable (colll1cctcd) circuit. Sleady�stalc replacemcnt 01' lhe population is used: a new inclividlwl rcplaccs lhc Icast�Jil mCll1ber 01' tlie ex.isting PO[llll,llioli pro­vided thaL it has a superio l Illness.

The cruelest form oj' nilitiOI1l mutation would he (0 selcct r,lllclol11ly <I gene in the ollsprillg\; cilronHlSOlne, and replace it with a IICW gellc or ralldolll type ,l1ld 1',lildoli1 tcr­minal nodcs. UnJ'ortllmltcly tlii� process gCl1er<iles a high proporlion oj' Icthal Illutations ill whiel1 a viable circui! is transJ'olllled into a non-viable circui!. The SllCCl',S r,lte or mut,ltioll call he improved hy IlHldil)'illg the cirCliit only ill ways thal arc likely (but Ilot gllanllllccd) [0 Icad lo ,I viable rcsult. One oj' the rollowing fUlIr circlli l tramJonnatiuns, selected at randol1l, is applied to the olLpling\ chromo­SOl1le:

(i) Replace an ex isting compo\1cnt by an open circui!.

(ii) Replace an cxisting componel lt by ,I sbon. circllil.

(iii) Conncct a nclV r,lllllul1l eomjlOnClll ill par,tl il'! with an existing componcnt.

(iv) ('O\1l1cet a new random cOl11ponent ill scnes with <Ill exisl illg com pOlleli t. These operations alone would be sumeiL�nl to trans[(JrI11 any givcn circuit illto ,II1Y other circllil, and it might be thought that crossover is lherel(lrc slljWrJ1uous. This is in lilcl not the case, ;IS the use or crossover we,ltly improves thc cmcicncy.

Oncc a circuit topology has been gencrated its Jilncss is ev,illiated al'tcr I1lll11eriGtily optimising its cOlnpollcn( val­lies lIsing ,I quasi-Newtoll algorithm based Oil the Davi­dOll I'lclcher Powcll (DIT) l11ethod. This Gills all objective I'Lll1clion thal retUnlS thc sI11l1-()I�slJu;lres oj' thc dirtcrcrlccs hetween the circuit', response and the targel response al. a scq uenec of logarithmically-spaccd rreq ucncics. To rcd ucc the amoLint or eOl11putalilln involvcd a sYl11bolic analysis or each IlCW circuit iopology is pcrrol'llwd hel(m� nUl11crical optimisation, This illvolvl's const rUl'ling vol/age ,lI1d cur­rcnt graphs (cOiTespondin!', to thc vol l age and current inci dcncc matricc,) ['rom the circuit, and al'tcr coalescing appropria!c pairs oj' nudes, finding ,ill the COllll110n sp:Il1'

nin).', trees oj' the two gmplls [71. The restl ll is two linked lists 0[' symbolic tcrms, corrcsponding to thc llUl1ler,ltor ,lI1d dcnominator of' the VO[(,lgc i'rellllcncy-rcspollse rUIlC­(i011. Subs titul ing colllponcnt values into the symholic l(lI'ITl givcs the 11L1111el'ie<il l'rCL[L1l'llcy-rl'sponsc l'unctioI1; substitut­ing l'rcquency into Ihc nUll1cric;il I'reqlll,ncy-rcsponse I'unc­lion givcs thc volt'lgc gain.

As tllc response ;IPJll"Oaehl:s the [al'get responsc, [he nbjlx:live rllnetiol1 lends 10 ICW ,Inti the rccipmcal 01' [i1c objective runelioll I'or Ihe optililised circuit is rclllrilcd as the litncss, In Ihc nhsencc 0[' ,IllY othcl' conslminls, thc (J;\ will gem:ralc succcssively more Clll11l)lcx circliits, hculllsC a coll1plcx circuit will in general pJ'()vide ,I beller lit III a lar­gel rcspOI1SC [han ;1 simple cil'Cllil. To prevent this [he lil­IlCSS is Illldtiplied I,y ,I penally ['lI11ctioll p(lI) thaI is unity ['or simple eirellits, IJ\lI whicll rapidly hl:eoilies smallcr as the complexily (11l1:asured hy the I1Ullllll:r 0[' llodcs II) cxceeds somc prede1l:l'Inined kvell! II !!I,\':

I I lIif/ f/ ,/I/ O ,,' ) when: II is a ('onst;\lll, IVJlica\ly �, <Il1d /J IIn\' is sci III tile ,Inticipalcd Clllllplcxity,

.

This hybrid .. (iA cil'cuil synlhcsis Illclhod is relTlHrkcthle 1'01' inemporatillg 110 design IUle, or expert knowledge; it simply works lowanls satisl'ying Ihe de�,ign goals, It is thc (lntitllesis 01' the tnldition,Ji 't:XPI:rt system' "P[1I'O,lcl1 [0 illl,Jioglle cin:uit lie,iV,Il,

!:i Simple frequency"domain filter' benchmark

An ohviollS way Ill' (estill)', Ihe elfeeti\,elless oj' the hyhrid­(iA is to sYlltl1esi,e a cil'l:uil lo a spccilicatioll rill' which ;1 lill'mal design method cxists, ('ollsider the l1()nnaliscd l()w­pass liller speeiliccltilln:

p,l,s-band edge:

stop,band edge: maximulTl pass .. hand gain:

I,Orad!s

1,5 rad/s

Odl�

Illinilllllill pass-hand g;Lin: I dl�

rnaXillll11l1 slOp-h,111d gain: 4(,dll

hlllcnving lhe tl'mliliol1al lilter Liesi)'11 proCl:dLll'e, the 1II'st stage is lo clloose ,I suitable Illler "pproxilllalillll, 1'J'()vided tlml p,lss-lxlI1c! rippll:s ,11ll1 (\ 110lH110IHJ\Ollic sto[)-b,lIld an� acecptahle, the Illost l:meicill lilter approxilllatioll is the elliptic, Tile lowcsl-ordcr clliptic ;lppmxilll,ltion l1leelint', the spccij'icatioll is 01' 5tll order, and tile I'iIler call bt.: illlille­Illellted as an equally-terminated ladder rilter as shown ill lig,2.

I input> I (' o,�ru"

(y y y-\ -" Cf i '

lllIiJ2 !

ell I,mao r

r (outpull

II HI 10

I

To allow Ill\' the Elct lhat Ihe lllaXilllUIll pass-band 1�;lin 0[' ,111 eqwilly-tel'lliindled liller is (, dB, all 01' the gains in lhe ,peei[lcatioI1 must he redllccd by (, clll, The impedance levcls in Ihi� liltcI' ,II'C (J\'ound I �2 a III I ,lilY practic;Ji imple­IllcI1lation would relplire tliell] to hc sealed to a IllO\'(; re,il" istic level. h1l' l;Xcllllple, tile I'csislor ,1I1e! illductor v,ilues could be Illulliplicd hy [0000, and tile C,lPllCitor vailies divided by IOUOO, 1'\"" :1 shows the COITCSPOllclillg I're-1I liency response,

Tlie hybricl"C;\ circuil synlhesis pmgmlll .I',cncrilted the circuit shown ill Fig, 4, l lsill!', a [lopu\;l[ion sile 01' f.:() circuits and 100 gcneratiolls the time taken 10 sYlllhcsise lhis Ilher Oil a :\()() [YIII:;, 1'1 I [lCI'solnJi C(lllllllitcr was lIllclcr Ihree hours, The respon,c 01' the IlyhriLi-C i;\ desil',n is shown ill hg, 'i. (Illcl it is clear tlial the design is fully cOll1plialll with the spccilication.

(]

( "llj UJ -'1-0

-!SO

-70 I I I I l_1

o o,� O,f; LO fruqllunc:y, rml/t;

Fig,3 ""('(//lI'I/(T I'!',I/JOI/SI' 01 i'1ii/lliljilil'l'

Fig,4 ill-iJril/.(i;ll/niglled !iller

o

·10

III -�o ·0

L

��I -flO

-!;O

-GO

\(\ > , ( -

( I

?,O [i,n

-/0 0,1 ILl. Il,[i '1,0 �,O

frequency, r<:\c!!:.;

Fig.5 hl'I/W'/II:I' rcsl'"II.\(' 01 /n/Jrid-(,'/I dcs(�lIcdfif/!'1'

Surprisingly, tile hybrid-GA design is not simply as good as the elliptic lilter, hut with (lilly six rellelive compOl1ents is actually a more cl'lieiellt dcsign, Although the hybrid-CiA lksign ll1ecLs the specilk,ltioll, Ilcverthcless it is probably suhoptilmil with respect to performance Etctors (such as component valul: sensitivity) tlmt arc not included in the design goals, Tilis dirlicul1y c<ln in principlc bc overeomc silllply by llllldiryillg the ohjective I'Llnction 10 include these pcrilJrl1l<lnCC [�lc[ors, hut Ihis will certainly ,ldverscly aCieel lile specd Dr tbe sYl1thesis program,

As Jill' as cOll1ponent valuc sensitivity is cOllccrned thc synthcsis progl'<llll e,lll he ll10dilied to include an item or 'expert knowledge': equal1y"lcrminatecl ladder configura" lions me: known 10 haw low scnsitivilY, Till.: synthesis PJ'O­l'-ram was thcrcillJ'l'; constrained to gel1l;rate only LC circuils hetWl.:ell equal-value tcrmilUltion resistances and the result is ShOWll in Fig, (l, The response 01' this circuit is shown in Fig, 7, Again thc synthcsise:d circuit meeis the spccilicalion, while llsing kwer compollcnts thall the rilll:r l'cslLitinp, 1'n11l1 the traditional l 'oJ'llml design process hased 011 <Ill elliptic respollsc,

Fig.6 I frhl'icl-(,A "(,s(�lIcci <'![III1I1I'-/crllliIlUII'I/jil/('1'

o

-20

m ·30 ' '0 c, 'l1.40 -

-so

·60

·'/0 0.1

r<� -L ___ I ; _ _ I

D.? 0.5 1.0 frequency. rad/s

2.0 5.0

Fig.7 HC![I/WIT IFSJIOIISI' ,{/ir/mil-(ill dC.I'(!!,II('r/ C/fIl{///),-ll'rllli(/{(/crl/i//('}

6 Nielsen's filter design problem

In a papcr dcscribing a ClJntinuolis-timc IIIter compiler. Nielsen [R] uscs as all cxamplc a highly aSYlllmclric band­pass Illlcr speeilicatioll 1'01' a modclll application:

pass-band: 31.2 45.6 k 1-17 maximum pass-lxll1d ripple:

lower slop-bund edgc:

lower Slop-band gain:

upper stop-band:

uppcr stop-band gain:

O.6dB

20k[ I/.

<: 3gdB

()l).D -g4.0 k I I/.

< 73dB

gain abovc slop-hand: < - 55 dB

LJnrorlunatcly Nielscn's !iller compiler is bascd on lmdi­tional IIllcr dcsign melhods (lL1d can only gcnct',llc iii lc I' responscs that arc or the standard types (low-pass, high­pass, sY11llllctrical band-pass and sY11lmclrical band-stop). Although standard riller responses call be I'()lllld to satisfy aSYlllmetrie spcciliealiolls slIch as that given above, they arc likcly to be 01' unnecessarily high order. For example, to mcet Niclsen's specilication a IOth-()t'(ler elliptic Ii Iter is required., the response 0[- which is shown in I'-ig. g.

o

-20

-40 m ,j c '«j d)'60 -

·80

-100 -� 10 20

. � .

- -'--' (,\� .. 50 100

frequency. kHz

Fig.8 /'i'C![III'IIt!· ),('.11)01l.\(' 0/ 10th-order ('Iii/IiiI' /J1ll1i/'PllsljiIIC)'

;'>00

This clearly achieves an unnccessarily high degt-ee of altclluation in the lowcr stop·band. Niclsen WiIS thet'C/'orc forced to design by hand a suitable passive prototype Ii Iter. Ilis design consisls of an Rth-oreler dOLLbly-lenninated Elter containing ten reactive componcnts as well as the two tcr­mination resistanccs.

Koza r4] uses lhis lilter spccilieation as a delll0l1s1mtioll 01' lhe c1kctivellcss of his C; l' circuit-synthesis program. Arter Il)l) gcnerations a circuit emcl'ged that meets the spec­illealio1l, hut which e01ltains it total or]R componcnts and is dearly nol a cost-elTcetive design.

The hybrid-CIA synthesis progl'illll described here L1ses randomly chosen componenl wliucs around unity as it slarling point 1'01' numerical optilllis�ltion, and optil11iS�ltion is most likely to succeed il' the v;liucs corresponding to thc global optimu lll eire also ,tround unity_ This C(!I1 be ,Ichieved by selling the impedance levels to I Q (illel by scal­ing the Illler cut-oj'1' li'equencics to around I md/s. In the case or Niclsen's liller, a frequency sc;t!ing lilctor or h -2n x 45000 is lIsed, and the scaled speeiliea tion is used as lhe target responsc filr the hybrid-CiA circuit-synthesis program.

lJsing Nielsen's speeillcalion, and constrained to gencratc an equally-tcrminaled conliguration, the hybrid-CiA gcner­<tIcs thc circuit shown in Fig. 9. It contains one ICwer COt11� ponenl than Nielsen's manually-designed filtcr. This lith­ordcr circuit would not have resultcd li'om any traditional design procedure. Although it superlicially resembles a tra­ditional eqLtally-lcrlllinaled Iaddct· !llter, the series combina­tion: I.d/eh and tile scries/parallcl colllhinaliotL Lg/Cj/1 J arc non-sl<ll1da I·d.

o

"20

-40 ru T)

-BO '

· 10G 10 ?U 5G

frequcncy. IcHy Fig.10 hnll((,II(l' re.\pmm' ({h.l'hrirl-C;/f de, .... �!!.IIf.'rI ,:\l;ej\'cnjiilt'l'

200

01' COUI'SC thc response or (his circuit is eentetwl on I md/s ntther than the 45 kl J/. oj' the specillcation. To convert thc Ii/tel' to a filrm that meet, the original speeillea­tion the capacitor und ind uclor values a re divided by the nortlUllising filetOt· fl. The ill1pedances ,liso need to he bl'OlIgll1 lip to it more praetica I lcvel. Fig. 10 shows the fllter response.

It is clear that this 1iIler is fully eOlllpliant with the speci­lieation. while at [he same time bcing more el'licicnl (ill

tl:rlllS or thl: 11 1I1ll bl:t· 01' COtllpOIll:nh) than till: tntclltlomtl clli[ltil:-derivcd Illter, I(o/a's C;P circuit synthl:SlS hllcr and Nielsen's tmtl1ually-deslgned 111ter.

7 Time-domain design problem

Therc is no reason why the hybrid-U;\ synlhesis method should be limited to I'requelll:y-domain lilLers. (;iven a cir­cuit topology and component values, it is only slightly more complicated to Gllculate lhe impulse or step response than the ['requeney response.

Consider the f'ollowing lInil.-step response specilkatioll 1'01' an approximation to the ideal ,tveraging iii tel':

.iJ(!') � t =L (l.()?' I'm II s:: I <I

(J(t) � I =L 0.0:2 1'0), ] �! This speciliGttion is compared with the circuit's t'esponsc at intervals 01' (l,OS s Irol11 () s 10 4.0 s ,111(\ is the only dGsign inrortnalion supplied to the synthesis program. The aetu,d Ltllll-slep response or the cireuil under consideraliotl lS cal­cuiatl:d by thl: tllatrix l:Xpolll:lltial method. hlch citntil topology is ,tmllysl:d to givl: the symbolic transil;r ['unction, To calculatl: the rCSpllnSl: the eOll1pollClll values arc suhsti­tuted into thl: translcr fUIll:tiotl, lhl: stalc cquation� dcrivl:d, and thc transition tmltt'ix calculatcd 1'01' thc requin:d time step, Only a single matrix multiplication is then required to caleulatG the rcsponse atl:Hch ti\lle [loint.

Ie 1.801[1[ Il1pLif>--"ryy,

r -- . I Cl> (I.UU609 t �I

�Lc O.Ot51ll I .

. --r <outilUl I

Lj lomo( til :1:1[)30II � Ct O.O?7/G

t(I·LO � 1 L --�r L :

1.2

'1.0

<1J � (J.n <) � UJ �_ O.G ill 10 .�

O.?

tilllC, S Fig,12 (jllil·.III'/) 1'1'''1'01/.1(, 0/ hI II/ill·(;,j (/c.I(�/",II (i1'('mgill,� jil/n'

3

Using a population sit:e 01' flO circuils and 200 gencra­liolls, the rilter shmvn in I jig. II W�tS synthesised in around 10 hours, rig. I� shows lhc unit�step response ol'this liller,

ami it lS cblr thaI the synthcsised liltl:r lIleels lhc t'Gl[uired time·domain specilication.

8 Conclusion

The design cxamples presented abovc indicate lImt the hybrid-CiA method or citntit synthesis is pral:tical on widely availablc pcrsonal computers, and generales circuits that arc el'licienl and I'ully mcel thl: design goals. Driven purely by thl: SI'Jl:ciJicatioll, the clreuit synthl:sis program incorporates nl) expcrt knowledge or circuit design (except in some cases wherc the topology io constrained to be 01' equally-termimttGd 1'000m).

CircLiits gcnerated by thc synthesis progntm itrC ollen novel, and would not have bccn gl:nl:l'Clted by any estab­lisli(�d design LGd1l1iquG, Indeed, thc synthl:sis program appGars to be creative in a way lhat is not ollen associated with computet·s. [11 sOllle eaSl:S lhl: circuits gcncraled me more cl'l1cient than lhose rl:sulting rrom tntditional design methods (as lhe example 01' the sllilrp cut-olT filler demon­strated).

III principle, the hybrid-CiA itpprmlch Gill be applied 10 any citutit design problem rOt· which (\ means or evaluating potential cireuils against thc design gmlis is available, Com­putalional clrort is the only limiling I�let()r.

Almost all lhG computational elTort goes towards calcu­lating circuit responses, I �ach new eircLlit Itlust bl: optimisl:d numeril:ally lo obtain the eomponelll values, and this will lnvolve many thoLisands 01' responsc evaluations. Linew­cit'cui! analysis is 1�lst ,tnd errlcienl, and synthesising circuits with up to 16 compollGnls is possihll: Oil standard personal l:Olllputers, NOlllinear amtlysis, by contrast, involves thl: solutiOIl or nonlincar dilkt'Gnli,,1 eqLt<ltions and absorbs far more eomputatiomti d'll)rt lhan linear analysis. At prl:senl it is impnLctical to synthesise nonlinear cirGuils or any com­plexity on rcs using thl: hybrid· CIA method.

9 References

1101 ,I ANI), .1.1 L: 'AdaJliatioll ill mlliral and artilicial ,ystellls' (l:ni· vcrsily or Michif!dn I'r�", Ann r\ri1or, Micli .. 1l)75) (;01,1 )111 :IH ,. D.I:.: .( ienelic ,i1gmillHllS in ,e,,,,:Il. oplillli/alion. ami maehille lc:1rl1illg' (Adclisnn·Wesicy, I<eadillg. Mass .. Il)�')) ]( )/l\ . .I, R.: '( le:l1ctic programming: Oil lhe progratllllling or compuL­ers by mealls Ormllllml sdection' (YIlT I'rl'ss, 1'.1'12)

4 1(0/;\ .I.R, HLNNI'.'IT, 1:.11.. A Nl ll< L, D., and KLANI':, MA: 'Aullllllaled WYWIWY( i desi,,11 for both topology and millponcnl 'vL\lue:.; or electrical circuits L1�illg gelletlc pl'ognlllllllillg'. (icl1dic PI'O­[''''limning I ')l)(,: Proccc<i illes oC Ihe l 'ir,t Allnllal conlercnce . .I lIlv S I'lll[iml '( Jnivcr,iLy, (""1l1xidge, MA. 19%, J'jl. 123 1.11

. . 1(0//\, .IK. IILNNI:Tr. 1'.11., r\NDRL. I)., ami KI'-"\1\I:, MA. 'I,'our pi"ohkms ror which a computer pr(lgl"�il1l evolved hy geneLic ]l!"llgnllllinillg is l"ompclitivc with human pcrf'onmIlH . .'c'. Procccdiligs or the 19')(, II '.LI·: intellLalion,,1 conlCrcncc Oil 1':m/llli(lI/w), ['III II/m la/ioll , I\agoya, ,hllXlll. M'lY 1<)1)(" PI'. I· ·ID

() (,RI M III.I·,IIY, .I.B.: '1lyhrid genetic ,dg,)rith11ls ror amlogltC nelwork synlhesis'. I'mceeliill!'s oJ' 11':1-:/[1'1:1: inlcrrmlion:t1 congress 011 IO'l'Illll· limllilT COllljJlllilliml (('I;.e ''!'!J, W,dlinglon I)C, .Iuly I <)<)1), PI'. 17K I 17k7

7 (il{ II'vlllI.EIIY. lB.: '/\lgorith11l Il)r Jlmlillg Lhe COllllllOI) Slxllllling trees 01' I\VOI�I'''JlI1S', 1c'I1'1'//'()1I 1.1'1/, 10SI, 17. pp. 470471 Nil :ISI :N, I.R.: .;\ ('.'1' l'tlter compile!' • J't'Olll 'pcci['tcatioll 10 layoul'. Analog lillI',':/" CiI'Clii/I' S(�/",I f'I'IJi'I'.I'.I .. I ')ll'i. 7. PI'. 21 :U

. 121