EHBSIM: MATLAB-BASED NONLINEAR CIRCUIT SIMULATION PROGRAM (HARMONIC BALANCE AND NONLINEAR ENVELOPE METHODS)

Transcription

1 EHBSIM: MATLAB-BASED OLIEAR CIRCUIT SIMULATIO PROGRAM (HARMOIC BALACE AD OLIEAR EVELOPE METHODS) Lonardo da C. Brio and Paulo H. P. d Carvalho LEMOM Univrsiy of Brasilia Brazil paulo@n.unb.br Absrac This papr prsns a nonlinar circui simulaion program calld EHBSim implmnd in Malab. This program offrs as simulaion mhods h Harmonic Balanc mhod, and spcially h onlinar Envlop mhod. Equaions sysms ha allow h simulaion of any circui archicur wr also usd. Th program has original algorihms implmnd o solv h associad circui quaion. Th dscripion of h onlinar Envlop Mhod as wll as a prformanc comparison bwn hos wo simulaion mhods, hrough h analysis of mulion xcid circuis, ar also carrid ou. I - ITRODUCTIO In h simulaion of nonlinar lcric circuis xcid by modulad carrirs, hr ar wo opions for rprsning h modulaion/dmodulaion procss: passband and basband. In passband simulaion, h carrir signal is includd in h modl of h lcric lmns (linar and nonlinar lmns). Th frquncy of h carrir signals is usually much highr han h highs frquncy of h inpu mssag (basband) signal. According o h yquis sampling horm, h sampling frquncy of h simulaion mus b a las wic as high as h carrir frquncy in ordr o allow h rcovry of h mssag. Howvr, h simulaion of a high frquncy signal is vry slow and consqunly infficin whn radiional im domain ingraion mhods [,] ar usd. To spd up h simulaion procss, a basband simulaion can b usd. A basband circui simulaion mhod, nown as h onlinar Envlop Mhod (LEM) [,3], which is a low-pass mhod, handls h complx nvlops of h modulad signals. Th following scions prsn h mahmaical dfiniion of a complx nvlop, as wll as h modls for basic linar circui lmns and h way o characriz nonlinar lmns in h nvlop domain. A Malab-basd nonlinar lcric circui simulaion program calld EHBSim (Envlop & Harmonic Balanc Simulaor) is also prsnd in his papr. Th EHBSim program prsns as main simulaion mhod h onlinar Envlop Mhod. This mhod wors in boh im and frquncy domains, and allows h analysis of circuis xcid by modulad carrirs by xracing h complx nvlops of h signals, which can b narrow or widband, analog or digial. Sinc

2 his chniqu is limid o h analysis of complx nvlops, i dmands rasonably low compuaional coss. Thus i allows fficin analysis of RF lcommunicaion sysms xcid by arbirary modulad carrirs. EHBSim also implmns h Harmonic Balanc Mhod (HBM) [,4,5]. Anohr faur of EHBsim is ha h implmnd quaion sysm yilds h simulaion of any circui opology. EHBSim uss h won-raphson algorihm [,,4] associad wih coninuaion chniqus [,] as iraiv mhod for solving h circui quaions. II - EHBSim SIMULATIO METHODS II. - Harmonic Balanc Mhod Th svral simulaion chniqus currnly in us can b classifid as im-domain, frquncydomain, and im and frquncy-domain chniqus. Among hs chniqus, h wllsablishd Harmonic Balanc Mhod has shown grar fficincy rough h las dcad. Howvr, his chniqu prsns srious difficulis whn h circui is xcid by modulad signals, dmanding, in hs cass, high compuaional coss. EHBSim implmns his mhod. I wors in boh im and frquncy domains. In his mhod h circui quaion is formulad in h frquncy domain, bu, bcaus of h impossibiliy o obain h modls of h nonlinar lmns in h frquncy domain, h valuaion of h nonlinar funcions (currn, volag and charg) ar prformd in h im domain. To prform his valuaion, w firs a h invrs Fourir ransform of h command (currn or volag) of h nonlinar funcion, opra h nonlinar funcion, and hn us h forward Fourir ransform o convr bac his funcion o h frquncy domain. Sinc h sysm of quaions is rprsnd in h frquncy domain, using Fourir cofficins, i canno giv h ransin soluion of h circui. Thrfor, h HBM is limid o h sarch of h sady-sa soluion. Th firs sp owards h applicaion of h HBM is o build h circui quaion. I is rprsnd in h frquncy domain and is a vcor composd by h combinaion of h lmn modls (obying h lmn laws) and h opology of h circui (obying Kirchhoff s laws). This quaion conains informaion abou h modls of linar lmns (rsisors, capaciors, inducors, ransmission lins, c.), nonlinar lmns (volag, currn and capacianc), and sourcs (volag and currn). Th quaion sysm as h form: F ( C )) = C ( ) A G ( ) A G ( C ( )) A C ( ) 0 L ( L L L L 3 L = () Eq. () rprsns h sgmnd modifid nodal quaion [,4]. In his quaion, only h minimal s of unnowns of h sysm ar considrd (commands of h nonlinar funcions). In xprssion (), is a vcor composd of gnric angular frquncis, C L ( ), G L (C L ( )) and G L ( ) ar, rspcivly, h Fourir cofficins of h nonlinar funcions commands c ( L ), h nonlinar lmns characrizd by nonlinar funcions g L ( c L ( )), and h indpndn sourcs g L(). Th marics A, A and A 3 ar composd by h modls of h lcric lmns, obying h circui opology. Thy ar also calld ransfr funcions. Eq. () is gnrally solvd using h won-raphson algorihm associad wih coninuaion chniqus, basd on h conrol of h xciaion sourc lvls. Thus, h lvl of h inpu sourcs of h circui ar rducd unil convrgnc (soluion) is achivd. Thn h lvl is discrly incrasd, h soluion found o a lowr lvl is usd as iniial soluion for a highr lvl, and h sarch of h

3 3 nx soluion sars. This procss is rpad unil h lvls of h sourcs rach h dsird valu. To simula mulion lcric circuis, on can us h mulidimnsional Fourir ransforms, and mor spcifically, h mulidimnsional fas Fourir ransforms, in ordr o a advanag of h gain in im and compuaional ffor offrd by hm. In his cas, rprsns a s composd by h fundamnal frquncis of h circui, hir harmonics, and h summaions and diffrncs bwn frquncis of h fundamnal componns and hir harmonics. Th dimnsion of h mulidmnsional FFT [,4] usd is givn by h numbr of fundamnal frquncis of h lcric signals. Th HBM is mor accura and fficin whn h circui is nar linar and h lcric signals ar nar sinusoidal. If h circui o b simulad conains signals wih abrup ransiions or if hy ar srongly nonlinar, such as hos commonly prsn in lcommunicaion circuis, h HBM dos no shows fficincy bcaus: () many mor frquncis ar ndd in ordr o rprsn corrcly h signals of h lcric circui maing h simulaion of such circui highly im-consuming; () h coninuaion chniqus ma mor calls o h HBM as h nonlinariy incrass. Evn wih h us of h FFTs o lin h im and frquncy domains and Krylov subspac mhods [6] o solv h spars linar quaion (), h us of h HBM o solv srongly nonlinar circuis prsning digial signals, which hav abrup ransiions, is xpnsiv and almos prohibiiv. II. - onlinar Envlop Mhod Th onlinar Envlop Mhod (LEM) is capabl of providing boh ransin and sady-sa rsponss of an lcric circui. This mhod can b fficinly applid o h simulaion of mulixcid modulad sysms whn h modulaion signals ar complx (.g., QAM signals). This mhod as ino accoun only h slow variaions of h modulad signals. In fac, h LEM shows a much br prformanc han convnional ransin analysis algorihms. In his mhod, on only nds o considr a numbr of sampl poins ncssary o corrcly rprsn h basband componns of h signals. Thus, h fficincy of his mhod, whn i is compard wih convnional ransin mhods, can b roughly simad as h rlaionship bwn h numbr of sampl poins ncssary o rprsn h modulad signal and h numbr of sampl poins ncssary o rprsn h basband (or modulaing) signal. For xampl, a ypical cllular lphon ransmission has a 30 Hz modulaion bandwidh riding on a GHz carrir. Hnc, an sima for h gain achivd by using h LEM insad of using convnional ransin mhods is ovr 3,000 in his cas. Hnc, if h nvlops (or complx modulaing signals) chang slowly if compard o h priod of h carrir, h simulaion by h LEM will b vry fficin. Mahmaical dscripion of h complx nvlop A gnric signal prsning wo fundamnal frquncis as h form: x j mω + nω ) ( ) = X mn ( () n= m= whr ω and ω ar h fundamnal angular frquncis and X mn ar h Fourir cofficins of x () ha corrspond o h frquncis m ω + nω. This dscripion can b asily xndd o

4 4 h cas w hav an arbirary numbr of fundamnal frquncis. Hr only wo fundamnal frquncis ar prsnd for simpliciy, bu wihou loss of gnraliy. Exprssion () can b rarrangd, rsuling in: Equaion (3) can b rwrin as: jmω jnω x ) = X mn n= m= ( (3) jnω x ) = X n ( ) n= ( (4) whr m= jmω X ( ) = X (5) n mn In h cas of a modulad signal, if ω is h carrir frquncy and ω is h fundamnal frquncy of h mssag or basband signal, w hav ha ω >> ω. In his siuaion, h cofficins X n (), for n =,L,, can b sn as slow im-varying Fourir cofficins. Thy ar calld complx nvlops of x (), and h xprssion (4) is calld Envlop Fourir sris. If w spli x () in wo im axs, or dimnsions, and, is mahmaical rprsnaion can b rwrin as: jnω x, ) = X n ( ) n= ( (6) whr jmω X ( ) = X (7) n m= Th signal x (, ) is T priodic in and T priodic in, as shown in Fig.. Thrfor, h wo-dimnsional vrsion of x() has mporal dimnsions associad wih h im scals of ach of h fundamnal frquncis. This is h principl of h mulidimnsional Fourir sris (in his cas, h bidimnsional Fourir sris). Fig. shows a bidimnsional grid conaining h sampls of x () along h wo im dimnsions. mn

5 5 Fig. Bidimnsional grid of im sampls. To rprsn h signal x (, ) compuaionally, i is ncssary o runca h numbr of harmonics of h fundamnal frquncis. Thn, w can wri: and jnω x, ) = X n ( ) n= ( (8) M jmω X n ) = X mn m= M ( (9) Thus, quaions (8) and (9) prsn harmonics of h carrir frquncy and M harmonics of h basband signal. Th Fourir cofficins of x () ar shown in Fig.. Ths cofficins can b obaind by firsly applying unidimnsional forward Fourir ransforms along ach row of h grid and hn applying unidimnsional forward Fourir ransforms along h columns of h grid. This procdur corrsponds o h bidimnsional forward Fourir ransform. Howvr, whn only h firs par of h abov procdur is prformd, h im-varying cofficins ), for X n ( n =,K,, ar found. I can b sn ha h sampld signal X n ( ) is an a an insan = and varying along on cycl of h high-frquncy componns. Ths componns ar calld im-varying complx nvlops of x (). I can b noicd ha h rms jnω ar nown a any insan. Howvr, only h rms X n ) nd o b nown in ordr o rprsn hos signals. Ths considraions ar h (

6 6 ssnc of h onlinar Envlop Mhod. Hnc, any signal can b buil by is complx nvlops whn h signal is composd by low-frquncy and high-frquncy componns. As h high-frquncy im bhavior is nown, h signal x () can b rconsrucd aing ino accoun only h sampls of is complx nvlops (low-frquncy variaions). Thn, only a small numbr of sampls (givn by h sampling horm) ar ncssary in ordr o rprsn an lcric signal along on cycl of is low-frquncy componns. Fig. Bidimnsional grid of Fourir cofficins. Modls for linar lcrical lmns By now, w ar abl o obain som modls for h basic linar lmns of any lcric circui and also sablish a way for valuaing h nonlinar funcions of h nonlinar lmns of circuis. Indpndn sourcs Th sandard form of an AM (Ampliud Modulaion) and FM (Frquncy Modulaion) signal is: whr s( ) = f ( )cos( ω + ϕ( )) (0)

7 7 f () is an ampliud modulaing signal; ϕ () is a phas modulaing signal; d ϕ ( ) / d is a frquncy modulaing signal; ω is h carrir frquncy. Th funcion s () can b convninly wrin as: whr f ( ) f ( ) ( ) = () jϕ( ) jω jϕ( ) jω s + Using quaion (8) and (9) o rprsn (), w hav: (, ) = S n ( n= jnω s ) (7) M nm m= M jmω S n ( ) = S (8) I can b vrifid ha in h abov xprssions h complx nvlops S ( ), n =, K,0, K,, ar: f ( ) jϕ( ) ( ) = is h complx nvlop rprsning h frquncy ω ; f ( ) jϕ( ) S ( ) = rprsns h frquncy ω ; and h rs of h nvlops ar null. S Th funcion s () can rprsn any indpndn sourc wavform of an lcric circui. Hnc, any indpndn sourc can b rprsnd by wo slow im-varying linar sourcs. This rprsnaion is shown in Fig. 3. Fig. 3 Envlop Domain rprsnaion of an indpndn sourc.

8 8 a) Linar Rsisor Whn xprssions (4) and (5) ar runcad and applid o h consiuiv law of h linar rsisor (), and h rsuling xprssion is spli ino wo im dimnsions and, xprssion (3) can b obaind. n= vr ( ) = RiR ( ) () R n n= jnω jnω V R ( ) = R I ( ) (3) n Sinc h quaions of qualiy (3) ar linarly indpndn, w can wri for ach harmonic of h fundamnal frquncy of h carrir: V R ) = RI R ( ) (4) n ( n Exprssion (4) is ru for n =,K,. Thn, i can b vrifid ha, for ach harmonic frquncy of ω, hr is an lmn composd by a rsisanc R. Thrfor, h s of + of hos lmns rprsns h rsisor in h nvlop domain. Fig. 4 shows h rsisor modl in h nvlop domain. Fig. 4 Envlop Domain rprsnaion of a linar rsisor. b) Linar Capaciors By applying runcad vrsions of xprssions (4) and (5) o h linar capacior consiuiv law (5) and spliing h rsuling xprssion in wo im dimnsions, on can obain is quivaln rprsnaion in h nvlop domain, as shown in Fig. 5.

9 9 i C dvc ( ) ( ) = C (5) d d = C V d jnω jnω I C n ( ) C n( ) n= n= Dvloping h quaion abov, aing h rms from boh sids of his xprssion ha rprsn h sam frquncyω, and spliing h rsuling xprssion ino wo im dimnsions and, w obain: I dv C n ( ) ) = C + jnωc V C n ( ) (6) d C n ( for n =,K,. Thn, i can b vrifid ha, for ach harmonic frquncy of ω, hr is an lmn composd by a capacianc C in paralll wih a complx conducanc jnω C. Thrfor, h s of + of hos lmns rprsn h capacior C in h nvlop domain. Fig. 5 Envlop Domain rprsnaion of a linar capacior. c) Linar inducors Th procdur dscribd abov can b applid in ordr o obain h inducor modl. Doing so, h rsuling xprssion is: V L n d I L n ( ) ) = L + jnωl I L n ( ) (7) d (

10 0 for n =,K,. Thrfor, o rprsn an inducor by h onlinar Envlop Mhod, w hav a s of + lmns. Each lmn is composd by an inducanc L in sris wih a complx rsisanc jnω L. Th nvlop domain rprsnaion of an inducor is schd in Fig.6. Fig. 6 Envlop Domain rprsnaion of a linar inducor. Modls of nonlinar lcric lmns Th modls in h nvlop domain of lmns govrnd by nonlinar funcions can also b asily obaind. Th following scions prsn h modls of nonlinar rsisanc/conducanc and capacianc or charg in h nvlop domain. d) onlinar rsisanc/conducanc For nonlinar lmns, w hav a physical modl dscribd by a funcion wih h form: q(, ) = f L ( g(, )) (8) whr f ( ) is a nonlinar opraor. I is ncssary o calcula h complx nvlops of L q, ) using h im sampls of g, ). Thus, i bcoms ncssary o obain g, ) for ( ( ( = and varying along on cycl of h high-frquncy componns of g (). Thos sampls ar obaind by applying h unidimnsional invrs Fourir ransform along on of h lins of h bidimnsional grid of Fig. whr =. Doing so w obain h vcor of mporal sampls prsnd in (0) from h complx nvlop sampl vcor (9). [ G ( ) L G ( ) L G ( ) ] G ( ) = 0 (9)

11 [ g(,0) g(, ) g(, )] g ( ) = L (0) To calcula h sampls of q (, ), h nonlinar opraor f ( L ) is applid o h sampls vcor (0), lading o [ q(,0) q(, ) q(, )] q ( ) = L () Finally, h unidimnsional dirc Fourir ransform is applid o (), rsuling in: [ Q ( ) L Q ( ) L Q ( ) ] Q ( ) = () 0 Following his procdur, h complx nvlops of q (, ) ar obaind. This procdur can b summarizd by h following xprssion: { f ( { ( ) })} Q( ) = F F G (3) L whr Q ( ) is h vcor of Fourir cofficins q ( ) a h im insan =, G ( ) is h vcor of Fourir cofficins g ( ) a =, and F {} and F {} ar unidimnsional dirc and invrs Fourir ransforms, rspcivly. Through h procdur dscribd abov, h rlaionships bwn ach nvlop Q ) of q(, ) wih h nvlops G n ( ) of g(, ), for n =, K,0, K,, ar found. Hnc, h way o dscrib h bhavior of a nonlinar rsisanc/conducanc in h nvlop domain is nown. I can b noicd ha h lmn in h im domain corrsponds o + lmns in h nvlop domain and, as shown, h rlaionships bwn is lcric signals ar calculad using h unidimnsional dirc and invrs Fourir ransforms. ) onlinar capacianc (charg) n ( Th formas of lcric circui quaions commonly usd by numric simulaors ar xprssd aing in accoun only currns and volags [,,4]. Consqunly, nonlinar capaciancs (or chargs) nd spcial ramn o b includd in hos quaions. In h cas of q () bing a nonlinar funcion of a volag v (), w hav: whr and q ( ) = q( v( )) (4) jnωn ( ) = Qn( (5) n= q )

12 jnω v ) = V n ( ) ( (6) n= Bu, dq( ) i( ) = (7) d whr jnω i ) = I n ( ) ( (8) n= Subsiuing xprssion (5) and (8) in (7), w obain: Thn, n= d = Q d n= jnω jnω I n( ) n ( ) n= dq ( ) jnω n jnω n( ) = jnω Qn( ) + (9) n= d I Thrfor, whn h signals ar spli ino wo im dimnsions and, i can b sn ha: I dq ( ) n n ( ) = jnωqn ( ) + (30) d for n =, K,0, K,. Consqunly, i is possibl o rprsn a nonlinar lmn of h yp of charg (or capacianc) by h complx nvlops of h currn ha flows rough i. I allows us o insr his yp of lmn ino h quaions ha rprsn an lcric circui. I mus b sn ha h cofficins Q n ), for n =, K,0, K,, can b calculad using xprssion (3). ( { q( { ( ) })} Q( ) = F F V (3) Elcric Circui Equaion Applid o h Envlop Mhod In ordr o analyz h im bhavior of a circui, i is ncssary o dfin h discr modls of circui lmns using on of h xising numrical ingraion mhods [,]. Afr h discrizaion of h modls, h quaion sysm of xprssion (3) for = can b buil.

13 3 F ( CL )) = CL ( ) Aˆ G L ( ) Aˆ G L ( CL ( )) Aˆ CL ( ) = 0 ( 3 (3) In xprssion (3), C L ( ), G L ( CL ( )) and G L ( ) ar, rspcivly, h Fourir cofficins of h nonlinar funcion commands c L (, ), h nonlinar lmns characrizd by nonlinar funcions g ( c L L ( )), and h indpndn sourcs g ( L ). Thos cofficins ar prsn in h frquncis ha ar mulipl of h carrir frquncy. Th marics Â, Â Â3 ar composd by h modls of h lcric lmns, obying h circui opology. EHBSim solvs quaion (3) using h won-raphson algorihm associad wih a coninuaion chniqu, basd on h conrol of h im-sp siz [,]. III - EHBSim PROGRAM FUCIOALITIES AD SIMULATIO EXAMPLES Th EHBSim main scrn, shown in Fig. 7, prsns h circui dscripion as main componn. As mniond, EHBSim implmns, bsids h LEM, h HBM. I allows on o ma prformanc comparisons bwn boh simulaion mhods. Th linar and nonlinar lmn modls o b nrd in h circui dscripion lis follow a paricular synax. EHBSim provids, bsids linar lmns, nonlinar lmns of currn, charg, volag, and flux yps. Th commands of hos nonlinar lmns can b volags, dlayd volags, drivaiv volags, currns, dlayd currns, and drivaiv currns. Th nonlinar funcions ha govrn h rlaionships bwn h lcric signals of nonlinar lmns can b crad by h usr. Th simulaion rsuls, rquird by h us of h probs V(Volmr) and A (Ammr), ar shown jus afr h nd of h simulaion. Th frquncy spcra of h rsponss, obaind by dcomposiion in Fourir sris, as wll as h wavforms ar prsnd. Fig. 7 EHBSim main scrn.

14 4 III. - Applicaion xampls Th aim of h following xampls is o dmonsra h us of h EHBSim program. Th xampls also mphasiz h suprioriy inhrn o h LEM whn compard wih h HBM for nonlinar circuis wih mulion xciaion. Th compuaional coss compard ar h im o achiv h circui rspons and h amoun of floaing-poin opraions spn. FM Dmoduador Fig. 8 dpics a FM dmodulaor. Th xciaion sourc Vs provids a FSK (Frquncy Shif Kying) signal wih h following wavform: s ( ) A cos( ω + 0 θ s ( ) d) (33) = whr θ () is a squar wav wih ampliud ω, and T is h θ () priod. For his xampl, A C s = 5 V, ω 0 = π rad/s, ω C = π 0 3 rad/s, and T = ms. 0 Fig. 8 FM dmodulaor. Th sady-sa rspons V (volag prob), obaind hrough h applicaion of h LEM, is shown in Fig. 9. Fig. 0 shows h diffrnc bwn h soluions obaind by boh simulaion mhods, whr h prcision of h LEM is vidn. Th grar divrgncs ar jusifid by h runcaion of h frquncy soluion obaind by h HBM, which provids a mporal oscillaion.

15 5 Fig. 9 Rspons V. In Fig. 9, h poins mard by squars rprsn h simulaion insans prformd by h LEM. Thus, i can b sn ha fw simulaion poins ar ndd in ordr o obain h circui rsponss. Using radiional im domain ingraion chniqus, i would b ncssary many simulaion poins for ach carrir cycl. Th LEM dmands only h sampls ncssary o rprsn h complx nvlops of h modulad signals Diffrnc HBM/LEM Tim (s) x 0-3 Fig. 0 Diffrnc bwn HBM and LEM V rsponss (hic solid lin) and HBM and LEM V rsponss (suprposd dashd lins). Tabl I shows h compuaional coss rquird by h HBM (scond column), and by h LEM (hird column). In ordr o ma h comparison, h circui was simulad by LEM unil h sady-sa rspons was rachd. Each row of h las column of Tabl I shows h raio bwn h compuaional coss spn by h HBM and by h LEM. Th suprioriy of LEM ovr HBM is qui imprssiv

16 6 whn h circui is xcid by modulad carrirs. I is clar ha h LEM offrs a much highr prformanc han h HBM. Tabl I COMPARISO BETWEE HBM AD LEM PERFORMACES HBM LEM Gain (HBM/LEM) Tim (s), Floaing-poin opraions AM Modulaor/Dmodulaor This xampl prsns an AM modulaor in cascad wih an AM dmodulaor, as shown in Fig.. Fig. AM modulaor/dmodulaor. Th xciaion sourcs Vf and Vp and Vp hav h wavforms of h xprssions (34) and (35), rspcivly. v f v p ( ) = 0.5 cos(π0 3 ) V (34) ( ) = 0.5 cos(π0 5 ) V (35) Th circui rsponss, obaind by h us of h LEM, ar probd by h volmrs V (Fig. ) and V (Fig. 3).

17 7 Fig. Rspons V. Fig. 3 Rspons V. Tabl II shows h prformanc masurmns an by h HBM and LEM simulaions. Tabl II COMPARISO BETWEE HBM AD LEM PERFORMACES HBM LEM Gain (HBM/LEM) Tim (s), Floaing-poin opraions Th br prformanc of LEM ovr HBM, whn h circui is xcid by modulad carrirs, is again vidn.

18 8 Signal Amplifir Anohr imporan advanag of h LEM whn i is compard wih radiional mhods is ha i achivs h sady-sa rspons quicly, i.., prdics h long rm ransin bhavior of circuis fficinly. This faur is du h main characrisic of his mhod: i handls only h slow im-varying complx nvlops of h modulad signals. Thus, only fw sampls of h complx nvlops along hir cycls ar ndd. Simulaing modulad circuis using radiional im domain ingraion mhods would b highly compuaionally xpnsiv bcaus hs mhods us im sps calculad considring h highs frquncy of h lcrical signals and h simulaion as many cycls of h basband signal. Thus, housands of simulaion poins would b ncssary. For xampl, considr h amplifir shown in Fig. 4. Fig. 5 shows h xciaion signal, Fig. 6 shows h compl rspons of h amplifir for linar and sauraion rgions, and Fig. 7 shows h sady-sa rspons for boh opraion rgions. Thr cycls of h basband signal wr ncssary uil h ransin rspons vanishd. Fig. 4 Amplifir. Fig. 5 Exciaion sourc V.

19 9 (a) (b) Fig. 6 Compl rspons V: (a) linar cas, (b) sauraion cas. As shown if Figs. (5) and (6), h sady-sa rspons is rachd afr hr cycls of h basband signal. Ths graphs also dpic h rlvan abrup ransiion ha occurs in h firs cycl of h signals, h linariy of h rspons obaind in h linar cas, and h high nonlinariy ordr of h rspons obaind in h sauraion cas.

20 0 (a) (b) Fig. 7 Sady-sa rspons V: (a) linar cas, (b) sauraion cas. Using radiional ransin mhods, abou,000 poins would b ncssary in ordr o achiv h sady-sa rspons in h firs cas (linar) and 5,000 simulaion poins in h

21 scond cas (sauraion). Th LEM nds abou 30 simulaion poins in boh cass o achiv h sady-sa rspons. I provids a mdium gain of approximaly 00. Th grar h diffrnc bwn h highs frquncy of h modulad signals and h highs frquncy of h basband signals, h grar h gain providd by h LEM. IV - COCLUSIOS Du h sysmaizaion implmnd, h EHBSim program is capabl o simula circui wih any archicur of opology, srongly nonlinar, sabl or unsabl, and muli-xcid by analog and digial signals. Original im ingraion algorihms wr also implmnd. Ths algorihms mploy h coninuaion soluions concps in ordr o guaran scur and fas convrgnc. Sinc EHBSim implmns h HBM and h LEM algorihms, i was possibl o ma comparisons bwn h wo chniqus (numric prcision and compuaional ffor). From h givn xampls, i was dmonsrad h ponialiis of LEM for simulaion of lcric circuis xcid by arbirary modulad signals. LEM also handls fficinly h ransin rspons, which is xrmly imporan o h analysis of digial communicaion sysms. REFERECES [] Brio L.C., Méodos do Equilíbrio Harmônico da Envolória ão-linar para a simulção d circuios d RF muli-xciados por sinais modulados analógicos digiais, MsC. Dissraion, Univriy of Brasília, 00. [] Chua L.O., and Lin Pn-Min, Compur-aidd analysis of lcronic circuis, Prnic Hall, 975. [3] goya E., and Larchvèqu R., Envlop Transin Analysis: a nw mhod for h ransin and sady sa analysis of microwav communicaion circuis and sysms, IEEE MTT-S Digs, Vol., pp , 996. [4] Carvalho P.H.P, Approch orin obj d l analys ds circuis non-linairs hyprfrquncs. Archicurs d simulaurs r bibliohqus d composans logicils, Docoral Thsis, Univrsiy of Limogs, 993. [5] Rizolli V., ri A., Sa of h ar and prsn rnds in nonlinar microwav CAD chniqus, IEEE Transacions on Microwav Thory an Tchniqus, Vol. 36, n.º, pp , 988. [6] Barr R., al, Tmplas for h Soluion of Linar Sysms: Building Blocs for Iraiv Mhods, SIAM, Philadlphia/PA, 994.