A Criterion on Existence and Uniqueness of Behavior in Electric Circuit

Transcription

1 Institute Institute of of Advnced Advnced Engineering Engineering nd nd Science Science Interntionl Journl of Electricl nd Computer Engineering (IJECE) Vol 6, No 4, August 2016, pp ISSN: , DOI: /ijecev6i A Criterion on Existence nd Uniqueness of Behvior in Electric Circuit Tkuy Hirt *, Eko Setiwn *, Kzuy Ymguchi **, nd Ichijo Hodk *** * Interdisciplinry Grdute School of Agriculture nd Engineering, University of Miyzki ** Deprtment of Control Engineering, Ntionl Institute of Technology Nr College *** Deprtment of Environmentl Robotics, Fculty of Engineering, University of Miyzki Article Info Article history: Received Feb 5, 2016 Revised My 22, 2016 Accepted Jun 8, 2016 Keyword: behvior of electric circuit switching circuit circuit nlysis ABSTRACT Behvior of electric circuits cn be observed by solving circuit equtions symboliclly s well s numericlly In generl, symbolic computtion for circuits with certin number of circuit elements needs much more time thn numericl computtion It is resonble to check the existence nd uniqueness of the solution to circuit equtions beforehnd in order to void computtion for the cse of no solution Indeed, some circuits hve no solution; in tht cse, one should notice it nd void to wit meningless computtion This pper proposes new theorem to check whether given circuit equtions hve solution nd their voltges nd currents of ll circuit elements re uniquely determined or not The theorem is suitble for developing computer lgorithm nd helps quick symbolic computtion for electric circuits Copyright c 2016 Institute of Advnced Engineering nd Science All rights reserved Corresponding Author: Ichijo Hodk Deprtment of Environmentl Robotics, Fculty of Engineering, University of Miyzki 1-1, Gkuen Kibndi Nishi, Miyzki, , Jpn hijhodk@ccmiyzki-ucjp 1 INTRODUCTION Any simultion of behvior of n ctul circuit is bsed on modelling of the circuit with ideliztion The modelling is crucil step of nlysis nd design; it depicts the ctul circuit s circuit digrm, determines working point, nd linerizes chrcteristics of electric components in the ctul circuit with working frequency Behvior of the ctul circuit is represented by solution to circuit equtions derived from the circuit digrm nd the fundmentl lws such s Kirchhoff s lws, Ohm s lw, nd the electric chrcteristics of inductor nd cpcitor SPICE1, defcto stndrd circuit simultor, simultes nd plots the behvior by numericl clcultion Tht is, we there ssume existence nd uniqueness of behvior We lso ssume them if we mesure behvior of n ctul circuit Hence, modelling process should not scrifice existence nd uniqueness of behvior; voltges nd currents t components in the ctul circuit or in the simultion model should be single-vlued functions defined on ll time Some studies ddress circuit digrms which hve lost existence nd uniqueness of behvior For exmple, some of inductor currents nd cpcitor voltges in the circuit digrm re not eligible for member of stte vribles2-5 This cn be trnslted into problem to find spnning tree in the circuit by grph theory3-5 However, little is known bout direct procedure to check given circuit digrm In this pper, we propose quick criterion to check whether solution to liner circuit equtions exists nd is uniquely determined or not The criterion enbles us to vlidte complicted circuits such s switching circuit with computers, nd thus drw proper circuit digrm - digrm which hs existence nd uniqueness of behvior - with help of computers Notice tht the criterion is written in symbolic eqution Tht mens our result would be fundmentl result to contribute the field of symbolic clcultion67 of electric circuits Journl Homepge: w w w i i e e s s j j o u r r n l l c c o m

2 1530 ISSN: LAWS OF ELECTRIC CIRCUITS AND CIRCUIT EQUATIONS For given circuit digrm, circuit equtions re defined by the fundmentl lws, Kirchhoff s voltge lw(kvl), Kirchhoff s current lw(kcl), Ohm s lw, nd the electric chrcteristics of inductor nd cpcitor, where ll vribles in the lws nd the chrcteristics re ssumed to be single-vlued functions of time t In generl, we hve Aw + By + Cz = 0 (1) H(x(t) x(t 0 )) = t t 0 z 1 (p)dp (2) where, A, B, C, nd H re constnt mtrices whose entries re determined by the lws nd the chrcteristics of electric circuit nd resistnce of resistors, x = il, u = vv i I, w = u, y : the vector of node voltges, z x 1 = vl, z = i V v I v R i R z 1 v S i S, (3) nd v V, v I,, v L, v R, nd v S re voltges of voltge sources, current sources, cpcitors, inductors, resistors nd switches, nd i V, i I,, i L, i R, nd i S re currents of voltge sources, current sources, cpcitors, inductors, resistors nd switches, nd v N is potentil of nodes, respectively This pper ssumes tht ll the vribles re Riemnn integrble, nd thus the nottion in the eqution (2) is Riemnn integrl (for definition of integrl, see eg 8) N1 N0 N2 N3 Figure 1 A boost converter - CCM (S1: ON, S2: OFF) Exmple Figure 1 nd 2 represent so-clled continuous conduction mode (CCM) nd discontinuous conduction mode (DCM) of boost converter 9, respectively Circuit equtions of the circuit digrm in Figure 1 re represented IJECE Vol 6, No 4, August 2016:

3 IJECE ISSN: in the form (1) nd (2), where A = 0 1 0, B = 0, C = 1 0 0, (4) R H = L 0, x = 0 C il, z 1 = vl, w = v V i L, y = v N1 v N3 v N0 v N2, nd z = i V v R i R v L v S1 i S1 v S2 i S2 (5) N1 N0 N2 N3 Figure 2 A boost converter - DCM (S1: OFF, S2: OFF) Circuit equtions of Figure 2 re represented in the form (1) nd (2), where A = 0 1 0, B = 0, C = R (6) A Criterion on Existence nd Uniqueness of Behvior in Electric Circuit (Tkuy Hirt)

4 1532 ISSN: nd the other nottions re lredy defined s (5) The vrible v L cnnot be uniquely determined on the lgebric circuit equtions (6) of Figure 2 for ny v V, i L, nd, which is understood by looking t the circuit digrm We will propose theorem to cpture the bove sitution by rnk clcultions of mtrices A, B nd C in the lgebric eqution (1) 3 THE EXISTENCE AND UNIQUENESS OF SOLUTION OF ALGEBRAIC CIRCUIT EQUATIONS If for ny w, solution (y, z) to the lgebric circuit eqution (1) of circuit exists nd z is unique, then the circuit is sid to be proper in this pper A circuit is sid to be improper if it is not proper Theorem 1 A circuit is proper if nd only if the coefficient mtrices of its lgebric eqution in the form (1) stisfy rnk A B C = rnk B C = rnk B + m where m is the number of column of the mtrix C The proof of Theorem 1 is given in the ppendix A Rnk clcultion in Theorem 1 is performed on generl purpose symbolic computtion system widely used This helps us check whether circuit is proper or not, even if it is lrge nd complicted, nd then difficult to check by hnd Exmple (A proper circuit) We recll the circuit in Figure 1 nd obtin for (4) Therefore the circuit is proper by Theorem 1 rnk A B C = 12, rnk B C = 12, rnk B = 3, m = 9 Exmple (An improper circuit) We recll the circuit in Figure 2 nd obtin rnk A B C = 12, rnk B C = 11, rnk B = 3, m = 9 for (6) Therefore the circuit is improper by Theorem 1 The whole behvior of proper circuit is given by Corollry 1 in the ppendix A 4 CONCLUSION Behvior of n ctul electric circuit, set of voltges nd currents of ll circuit components, is ssumed to be expressed s single-vlued function defined on ll time, in generl Designers depict the ctul circuit s schemtic digrm to simulte behvior of the ctul circuit Creful designers model the circuit s digrm with decision of including prsitic elements or not nd reflect working point nd working frequency of the circuit on the digrm They write circuit equtions equivlent to the digrm, nd then, solve the equtions to simulte behvior of the ctul circuit However, existence nd uniqueness which re ssumed to behvior of the ctul circuit re not necessrily inherited to the solution becuse the digrm nd the equtions re reduction of the ctul circuit This pper hs proposed criterion for the existence nd uniqueness to be gurnteed The proposed criterion is expressed s equlity between rnks of coefficient mtrices in the circuit equtions Effective nd quick clcultion of mtrix rnk is vilble in generl purpose symbolic computing tools Therefore our result contributes to the field of symbolic computtion of electric circuits A LEMMA, COROLLARY AND PROOF Corollry 1 If circuit is proper, the equtions (1) nd (2) re uniquely solved s: z = K 1 x + K 2 u (7) x(t) = e F (t t0) x(t 0 ) + e F t t t 0 e F p Gu(p)dp (8) Proof 1 We immeditely obtin the expression (7) by Theorem 1 This includes z 1 = K 11 x + K 21 u Notice tht the mtrix H is block-digonl whose blocks hve cpcitnces C s nd self or mutul inductnces (L s nd M s respectively) in their entries nd is invertible If we put F = H 1 K 11 nd G = H 1 K 21 nd pply integrtion by prts8 with mtrix exponentil function, we hve (8) We remrk tht the solution (8) is obtined without ssuming x(t) to be differentible IJECE Vol 6, No 4, August 2016:

5 IJECE ISSN: Lemm 1 Let B nd C be mtrices with common number of rows The eqution By + Cz = 0 hs solution z for ny y if nd only if rnk B C = rnk C Proof 2 (only if) There is mtrix Z such tht B = CZ Then B C = CZ C = CZ U m, where U m is n identity mtrix rnk B C = rnk C, becuse Z U m is row full rnk (if) Since rnk B C = rnk C nd Im B C Im C, Im B C = Im C Hence Im B Im B C = Im C This mens tht for every fixed y, there is z such tht By + Cz = 0 Proof 3 (For Theorem 1) (only if) By the ssumption nd Lemm 1, rnk A B C = rnk B C Let w = 0 The solution spce Q of the y1 y2 eqution (1) is Q = kerb C Let Q nd y z 2 ker B Since Q, z = 0 Hence, dim(kerb C) = dim(ker B) By rnk-nullity theorem, we obtin rnk B C = rnk B + m y (if) Let Q(w) = { Aw+By+Cz = 0} By Lemm 1, for every fixed w, Q(w) φ Since rnk B C rnk B+ z rnk C nd rnk C m in generl, rnk B C = rnk B + m follows m = rnk B C rnk B rnk C m So, rnk C = m = rnk B C rnk B (9) Use generl equlity rnk B C = rnk B + rnk C dim(im B Im C)10 nd (9) show dim(im B Im C) = 0 (10) y1 y2 Let nd Q(w) Then By z 1 z 1 + Cz 1 = By 2 + Cz 2 By (10), C(z 1 z 2 ) = 0 The first equlity in (9) 2 implies z 1 = z 2 REFERENCES 1 L W Ngel nd D O Pederson, SPICE (Simultion Progrm with Integrted Circuit Emphsis), Memorndum No ERLM382 Electronic Reserch Lbortory, T Bshkow, The A Mtrix, New Network Description, IRE Trnsctions on Circuit Theory, Vol 4, No 3, pp , E S Kuh nd RA Rohrer, The Stte-Vrible Approch to Network Anlysis, Proceedings of the IEEE, Vol 53, No 7, pp , R A Rohrer, Circuit Theory: An Introduction to the Stte Vrible Approch, McGrw-Hill, D A Clhn, Computer-Aided Network Design, Revised Edition, McGrw-Hill, A Luchett, S Mnetti, nd A Retti, SAPWIN- symbolic simultor s support in electricl engineering eduction, IEEE Trnsctions on Eduction, Vol 44, No 2, T Hirt, K Ymguchi, nd I Hodk, A Symbolic Eqution Modeler for Electric Circuits, ACM Communictions in Computer Algebr, Vol 49, No 3, Issue 193, K Knopp, Theory nd Appliction of Infinite Series, Blckie, R W Erickson nd D Mksimović, Fundmentls of Power Electronics, Second Edition, Kluwer, P Lncster nd M Tismenetsky, The Theory of Mtrices, Second Edition, Acdemic Press, 1985 A Criterion on Existence nd Uniqueness of Behvior in Electric Circuit (Tkuy Hirt)