ANALOG CIRCUIT SIMULATION BY STATE VARIABLE METHOD

Transcription

1 U.P.B. Sc. Bull., Seres C, Vol. 77, Iss., 25 ISSN ANAOG CIRCUIT SIMUATION BY STATE VARIABE METHOD Rodc VOICUESCU, Mh IORDACHE 22 An nlog crcut smulton method, bsed on the stte euton pproch, s presented. For lrge tme-nvrnt nlog crcuts ths method utomtclly formultes the stte eutons n symbolc, symbolc-numerc nd numerc norml form. Accordng to the proposed crcut nlyss method n effcent lgorthm tht genertes the stte eutons n norml full symbolc, prtlly symbolc or numerc norml form s descrbed. The SYSEG SYmbolc Stte Euton Generton progrm ws mproved by the mplemented of new routnes s: the generton of the stte eutons n opertonl form, the reducton of the number of the stte eutons nd the utomtc generton of the trnsfer networks for MIMO crcuts. Usng the stte eutons n opertonl form, the crcut functons n full symbolc, prtlly symbolc or numerc form re generted. Keywords: stte vrbles, stte eutons, symbolc nlyss, crcut nlyss, crcut functons. Introducton Stte vrble method hs the dvntge of remrkble systemtzton n wrtng the crcut eutons. Ths s essentlly mtrx method nd t s the only vlble method tht llows the ulty nlyss of the nlog crcuts. The stte euton method s used by commercl nd professonl progrms both for lner nd nonlner crcut smulton. Ths method s wde spred nd cn be used to solve ll the types of crcuts ncludng the ones wth excess elements. It uses mnmum number of ndependent vrbles (stte vrbles). If these vrbles re known t the ntl tme moment t =, the evoluton of the system for t > t cn be determned. [-]. Ech complete crcut nvrble n tme stsfes two bsc types of eutons: Krchhoff's lws (KC nd KV) nd the consttutve eutons of the crcut elements. The frst order dfferentl eutons system, obtned by mnpulton of these reltons, cn be expressed s: PhD student, Fculty of Electrcl Engneerng, Unverstte POITEHNICA dn Buchrest, Romn, e-ml: voculescu.rodc@gml.com 2 Prof., Fculty of Electrcl Engneerng, Unverstte POITEHNICA dn Buchrest, Romn

2 2 Rodc Voculescu, Mh Iordche ( x, t ), y g( x, t) x = f = () nd represent the stte eutons of the crcut n norml form, where: x = [ x ] t,x2,...,xn - s the stte vector ( x,x2,...,xn - re the crcut stte vrbles, n s the crcut complexty order); y = [ y ] t, y2,..., y p - s the output vector (p s the number of output vrbles). For lner nd pecewse-lner nonlner systems the norml mtrx form of the stte eutons s: where: x = Ax By Cy (2) A - s the coeffcent mtrx of the stte vrbles (stte mtrx) (n x n); B - s the coeffcent mtrx of the nput vector (n x m). C - s the coeffcent mtrx of the output vector frst order dervtves. A, B,C mtrces contns prmeters ssocted to the nonlner crcut elements: dfferentl resstnces for current controlled nonlner resstors; dfferentl conductnce for voltge controlled nonlner cpctors; dfferentl nductnces for current controlled nonlner nductors, nd dfferentl cpctnces for voltge controlled nonlner cpctors. A crcut whch contns loops mde up only by cpctors or/nd ndependent voltge sources or/nd voltge controlled sources or cut-sets mde up only by nductors or/nd ndependent current sources or/nd current controlled sources s known s crcut wth excess elements of the frst order. A crcut whch contns loops mde up only by nductors or/nd ndependent voltge sources or/nd voltge controlled sources or cut-sets mde up only by cpctors or/nd ndependent current sources or/nd current controlled sources s known s crcut wth excess elements of the second order, [, 2, ]. There re two common methods used to formulte the stte eutons: the euvlent source method nd the topologcl method of the norml tree. The euvlent source method for crcuts wthout excess elements conssts n replcement of the nductors wth del ndependent current sources nd of the cpctors wth del ndependent voltge sources. For crcuts wth excess elements n excess nductor s replced by n del ndependent voltge source wth the voltge eul to the nductor voltge. An excess cpctor s replced by n del ndependent current source wth current eul to the cpctor current. The topologcl method of the norml tree cn be esy mplemented on the computer.

3 Anlog crcut smulton by stte vrble method The norml tree topologcl method In order to generte the stte eutons specl tree nmed norml tree s chosen. A norml tree must contn: All the ndependent nd ll the controlled voltge sources; All the controlled brnches of the voltge controlled sources; All the controllng brnches of the current controllng sources; As mny cpctors s possble (the cpctors tht re not n the norml tree re excess elements); As mny voltge controlled nonlner resstors; As mny lner resstors s possble; A mnmum number of nductors (the nductors tht re n the norml tree re excess elements); A norml tree wll not contn ny ndependent nd controlled current source or ny nonlner current controlled resstor. The norml tree wll hve n- crcut brnches nd the correspondng cotree wll hve l-n crcut brnches (n the number of the crcut nodes; l the number of the crcut brnches). The number of the stte vrbles wll be the sum between the cpctors from the norml tree nd the nductors from the co-tree. The co-tree brnches re clled - lnks nd the norml tree brnches re clled norml tree brnches (twgs). To ech norml tree brnch (lnk) t s ttched cut-set ( loop) whch contns no other tree brnch (lnk) but lnks (tree brnches). Ths cut-set (loop) hs the orentton dentclly to the orentton of the ttched norml tree brnch (ttched lnk), [2]. After choosng ll the brnches nd the ssocted cut-sets, respectvely ll the lnks nd the ssocted loops, step by step lgorthm s followed. Step : Expressng the currents of the current-controlled sources n respect of ther controllng vrbles (ccordng to the defnton reltons), nd tkng nto ccount thtc = C dvc / dt, the KC s ppled on the cut-sets ssocted to the tree brnch cpctors; Step 2: Expressng the voltges of the voltge-controlled sources n respect of ther controllng vrbles, nd tkng nto ccount tht: vr = R R, v = d / dt, the KV s ppled on the loops ssocted to the lnk cpctors; Step : Apply the KV on the loops ssocted to the lnk nductors; Step : Apply the KC on the cut-sets ssocted to the tree brnch nductors; Step 5: Wrte the KC on the cut-sets ssocted to the tree brnch resstors; Step 6: Wrte the KV on the loops ssocted to the lnk resstors; Step 7: Consderng s ndependent vrbles the current vectors Rt, nd Rl, the eutons from step 5 nd 6 re solved;

4 26 Rodc Voculescu, Mh Iordche Step : The symbolc/numerc expressons obtned t the step 7 re ntroduced nto the eutons from the step 2 nd. The eutons generted n ths wy, together wth the expressons from the step 7 re ntroduced nto the eutons from the steps nd. Solvng these eutons n respect of the dfferentls of the stte vrbles, the stte eutons n norml symbolc/numerc form re obtned.. Descrpton of the SYSEG Progrm SYSEG SYmbolc Stte Euton Generton progrm, [6], (Fg. ) ws mproved by the mplemented of new routnes s: the generton of the stte eutons n opertonl form, the reducton of the number of the stte eutons nd the utomtc generton of the trnsfer networks for MIMO crcuts. These crcuts my contn both lner nd nonlner resstors, nductors, nd cpctors, ndependent voltge nd current sources, nd the four types of lner controlled sources. SYSEG progrm uses the followng mn wndow: Fg.. SYSEG progrm wndow. After openng the mn wndow, the followng steps re performed: Open the crcut nput fle pressng Open button; Fg. 2. SYSEG progrm second wndow.

5 Anlog crcut smulton by stte vrble method 27 Run the nput crcut fles by pressng Execute Syseg button descrbed n Fg.; In order to generte the stte eutons, the progrm uses severl subroutnes shown n Fg.: Fg.. Progrm subroutnes After selectng the subroutne, by pressng the Input vrbles button, the exctton unttes wll be ntroduced n the wndow descrbed n Fg.; Fg.. Input dt. By pressng Input fle generton button, the nput fle s generted nd the messge from Fg. 5 wll pper; Fg. 5. Fle generton progrm wndow

6 2 Rodc Voculescu, Mh Iordche Next Strt Mple button s pressed nd the stte eutons wll be obtned by runnng ths fle n Mple, [].. Exmple A nonlner crcut wth excess elements s presented n Fg. 6. A norml tree tht stsfes the reurements presented bove s chosen. The crcut n Fg. 6 contns two chrge-controlled (.c.) nonlner cpctors C du nd C du, whch hve the followng chrcterstcs: u = () u = () The crcut hs nodes nd brnches. Fg. 6. Dgrm of the nlyzed nonlner crcut. To pply the stte vrble method, both chrge-controlled nonlner cpctors re substtuted by euvlent ndependent voltge sources. The euvlent crcut s shown n Fg. 7.

7 Anlog crcut smulton by stte vrble method 2 Fg. 7. Dgrm of the nlyzed crcut norml tree. The norml tree (mrked by bold lnes) nd the correspondng co-tree contn the followng brnches: AN = { e,e 2,e,e,R 5,R 6,R 7 } (5) C AN = {,,} (6) The number of the stte vrbles results s the sum between the number of cpctors from the norml tree nd the number of nductors from the co-tree: n = n C nc = 5. The stte vector x wll be: x = (7) Followng the bove lgorthm for the crcut n Fg 7, n whch the two nonlner cpctors were substtuted by euvlent ndependent voltge sources, we get the eutons: = () = () u = u = e = () e = () KV ppled on the loops ssocted to the lnk nductors gves:

8 22 Rodc Voculescu, Mh Iordche d ( b ): = R55 e e2 e dt (2) d ( b ): = R66Z e e dt () d ( b ): = R77 e dt () KC ppled on the cut-sets ssocted to the tree brnch resstors gves: R ): 5 = (5) ( 5 ( 6 ( 7 R ): 6 = (6) R ): 7 = (7) Substtutng the eutons (5) - (7) nto the eutons (2) (), the symbolc norml-form of the stte eutons of the crcut n Fg. 7 re obtned: d R5 = e e2 e dt d R6 = e e. () dt d R7 = e dt From () A, B, C mtrces wll be determned. R5 - R6 A = -, R 7 C =. B =, () Now we use the Syseg progrm to nlyze the euvlent crcut represented n Fg. 7, whch s descrbed by the followng nput fle:

9 Anlog crcut smulton by stte vrble method 22 6 e 7 6 e2 2 e e 2 R5 R6 5 R Usng mbsb symbolc generton routne we wll obtn the followng results: Number of the stte vrbles n_stv =, Number of the nput (exctton) unttes n_ex=, Number of the dfferentl nput (exctton) unttes n_dex=, Number of the ouput vrbles n_outvr =, Stte vector Stte_vector=, [ I, I, I ] Input (exctton) vector y Input_vector y =, [ e, e2, e, e ] A = stte mtrx R7 A=, R5 R6 B = b mtrx of the nput (exctton) untty coeffcents

10 222 Rodc Voculescu, Mh Iordche Stte mtrces A, B, C obtned by Syseg progrm re dfferent from the stte mtrces () becuse the rrngement of the stte vrble s dfferent. Substtutng the eutons (), () nd the eutons (5) (7) nto the eutons (2) (), the symbolc norml-form of the stte eutons of the crcut n Fg. 6 re obtned: = = = = = R dt d R dt d e e R dt d (2) The euton system (2) cn by wrtten n the followng mtrx form: Cy By A x Ax x =, (2) where: = =, R R R A A, B=, C = c mtrx of the dfferentl nput (exctton) untty coeffcents C=,

11 Anlog crcut smulton by stte vrble method 22 B =, C =. (22) If the two chrge-controlled nonlner cpctors re substtuted by the two lner cpctors, hvng the cpctnces C nd C, nd the two del ndependent voltges e nd e 2 re substtuted by one ndependent voltge source denoted by e, we cn generted the nput dmttnce I7 _() s Y () s = Y7 _,7 _() s =, cllng the generte stte spce vector mtrx E () s numercl res lp routne. Usng Symbolc generton (generte stte spce vector mtrx numercl res lp) routne we obtn the followng symbolc expresson for the nput dmttnce: Y( s) :=( s C C ( C R6 C C C R7) s ( C C C C R6 C R7) s 2 ( C R7 C R7 C R6) s) ( s 5 C C ( R5 C C C C R7 C R6 C ) s ( C C R5 C C R7 C C R5 C R6 C C R6 C R7) s ( R5 C R5 C R6 C R7 R5 C R5 C C R7 R6 C C R6 C R7 CR7) s 2 ( R5CR7 R5CR7 R6CR7 R5CR6) s R7 R5 R6) For the numerc vlues: C = C =. F, R 5 =. Ω, R 6 =.2 Ω, R 7 =.2 Ω, = H, = 2.22 H nd =.6 H we obtned the followng results: Y_n :=. ( s s.2 ) ( s s.6 ) ( s.27 ) ( s s 66.7 ) ( s 2.72 s ) Poles_Y_n := { -.27, I, I, I, I } Egen_Vlues := { -.27, I, I, I, I } Zerous_Y_n := { I, I, I, I } We cn denote tht the poles of the nput dmttnces ( s) = Y ( s) re dentclly to the egenvlue vlues. Y 7 _,7 _

12 22 Rodc Voculescu, Mh Iordche 6. Conclusons The stte vrble method hs hgh degree of generlty nd cn be ppled to ny crcut structure, ncludng excess elements. The SYSEG SYmbolc Stte Euton Generton progrm ws mproved by the mplemented of new routnes s: the generton of the stte eutons n opertonl form, the reducton of the number of the stte eutons nd the utomtc generton of the trnsfer networks for MIMO crcuts n symbolc, prtlly-symbolc, or numerc form. The progrm ws lso enhnced by usng some new subroutnes nd frendly nterfce. The crcuts to be nlyzed my contn both lner nd nonlner elements such s resstors, nductors or cpctors, ndependent voltge nd current sources, nd ny lner two-ports controlled sources. Strtng from descrpton of the crcut usng netlst fle type, the progrm genertes the norml tree, then followng smple lgorthm wrte the crcut eutons nd fnlly get the stte eutons n the desre form. Usng n pproprte subroutne, the progrm cn offer lso ny trnsfer functon n one of the bove specfed forms. R E F E R E N C E S [] M. Iordche, ucn Mndche, Computer-ded nlyss of nlogc nonlner crcuts, Poltehnc Press, Buchrest, 2 [2] M. Iordche, uc Dumtru, The modern theory of electrc crcuts vol. II, A Eductonl Press, Buchrest, 2 [] M. Bde, Computer-ded nlyss of electrcl crcuts, Infomed Crov Press, 2 [] R. Voculescu, Generton progrm by stte eutons, The Interntonl Conference on Appled nd Theoretcl Electrcty ICATE 22, Crov, pp. -5. [5] P. Crste, Anlyss of nonlner crcuts, U.P.B. Press, Buchrest, 7 [6] M. Iordche, uc Dumtru, Computer ded smulton of nlog crcuts, Poltehnc Press, Buchrest, 22 [7] M. Pred, P. Crste, Electrcl engneerng fundments, Vol. 2, E.D.P. Press, Buchrest, [] C.I. Mocnu, Electrcl crcuts theory, D.P. Press, Buchrest, 7 [] P. Young, Prmeter estmton for contnuous-tme models - survey, Automtc, Vol. 7, No., pp. 2-,.