Consistent initial values for DAE systems in circuit simulation D. Estevez Schwarz Abstract One of the diculties of the numerical integration methods

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1 Consistent initia vaues for DAE systems in circuit simuation D. Estevez Schwarz Abstract One of the dicuties of the numerica integration methods for dierentia-agebraic equations (DAEs) is computing consistent initia vaues before starting the integration, i.e., cacuating vaues that satisfy the given agebraic constraints as we as the hidden constraints if higher index probems are considered. This paper presents an approach to cacuate consistent initia vaues for index-2 DAEs starting up from possiby inconsistent ones. Firsty, the idea is exposed for inear DAEs and then it is shown how the resuts can be appied to those systems arising from modied noda anaysis (MNA) in circuit simuation. This artice starts up from [8] and [6]. Severa denotations and resuts we usewere introduced there in more detai. Key words: Consistent initia vaues consistent initiaization dierentiaagebraic equation DAE index circuit simuation modied noda anaysis MNA structura properties. AMS Subject Cassication: 94C5, 65L5. Introduction Roughy speaking, the probem of determining consistent initia vaues for dierentia-agebraic equations can be described as foows. For ordinary dierentia equations, initia vaues have to be prescribed for a variabes to determine a unique soution. However, dierentia-agebraic equations consist of dierentia equations couped with derivative-free equations, i.e., not a components appear in dynamic form. Indeed, some of them are determined by agebraic constraints. In Section 2 a convenient characterization of consistent initia vaues is introduced. One approach to determine consistent initia vaues is to ocate a seection of variabes for which we may prescribe initia vaues and to construct a nonsinguar system that provides the vaues for the remaining ones. In this context, we have to consider two probems:. The seection of variabes is not arbitrary.

2 2. The vaues that are assigned to the seected variabes have to be chosen in such away that the noninear system is sovabe. In [6] itwas anayzed how to construct a nonsinguar system and how to treat () for systems arising from circuit simuation by means of modied noda anaysis, whie (2) was not discussed. Nevertheess, the practica reaization of that approach eads to new insights, provided that the vaues from (2) are suitaby chosen. These resuts wi be presented in the present paper. To iustrate the approach before considering the specia systems arising from circuit simuation, we describe the basic ideas for inear DAEs in Section 3. Then, in Section 4weintroduce the equations of the modied noda anaysis. The structura properties of these systems, which were pointed out in [8] and [6], are summarized in Section 5. The new resuts are presented in Section 6. Since, in circuit simuation, the operating point is frequenty used for starting the integration, it is separatey anayzed in Section 6.. In Section 6.2 the approach is described for a more genera case. In practice, the vaues obtained in the Sections 6. and 6.2 can be cacuated by soving reativey sma inear systems as described in Section 6.3. Finay, in Section 7 it is shown how thisapproach maybecombined with an initiaization strategy that takes into account possibe initiaization preferences of the user of a simuation package. 2 About consistent initia vaues for DAEs We consider dierentia-agebraic equations, i.e., equations of the form f(x x t)= (2.) where dx df df is singuar. In this artice, dx is assumed to haveaconstantnuspace. Note that, if we dene a projector Q onto ker dx df and P := I ; Q, then equation (2.) may be written as f(px x t)=: Denition 2. Avector x 2 R m is a consistent initia vaue of (2.) if there exists a soution of (2.) that fus x(t )=x. df Taking into account that the singuarity of dx impies that (2.) contains some agebraic equations, a consistent initia vaue has to fu precisey those agebraic equations. Moreover, the dierentiation of these agebraic equations may ead to further agebraic equations, caed hidden constraints, which a consistent initia vaue has to fu, too. This fact is cosey reated to the index concept. Actuay, we are aso interested in the corresponding vaues of the derivatives appearing in the DAE, i.e., in the vaues of Px if P is dened as P := I ; Q for a projector Q onto ker df dx. 2

3 Denition 2.2 Avector (x Py ) is a consistent initiaization of (2.) if x is aconsistent initia vaue and (x Py ) fus the equation f(py x t )=. For simpicity, we wi rst present the approach for inear dierentia-agebraic equations (DAEs). In the course of the artice, it wi be shown how it can be extended to those quasi-inear systems obtained by modied noda anaysis (MNA). 3 An overview of the approach for inear DAEs For a short outine of the main ideas presented in this artice, the tractabiity index and the spaces reated to its denition are introduced. Consider a inear DAE of the form: Ax + Bx = q(t) (3.) where A is singuar. For the tractabiity index we dene N := ker A and S := fz : Bz 2 im Ag. Denition 3. The DAE (3.) is caed index--tractabe if the matrix A := A + BQ is nonsinguar for a constant projector Q onto N. Remarks:. The matrix A is nonsinguar if and ony if N \ S = fg. 2. The denition does not depend on the choice of the projector Q. For the denition of the index twowe dene N := ker A and S := fz : BPz 2 im A g for P := (I ; Q). Denition 3.2 The DAE (3.) is caed index-2-tractabe if. it is not index--tractabe, 2. A 2 := A + BPQ is nonsinguar for a projector Q onto N. 2 Remarks:. The matrix A 2 is nonsinguar if and ony if N \ S = fg. 2. The denition does not depend on the choice of the projector Q. Denition 3.3 In the foowing, the canonica projector Q := Q A ; 2 BP onto N aong S is considered. cf. []. 2 cf. []. 3

4 In the index-2 case the space N \S represents a the components that are determined neither by a dierentia nor by an agebraic equation. These components can be determined ony by inherent dierentiation. Furthermore, for index-2 equations hidden constraints appear if we deriveapart of the system's equations. This impies that an initia vaue has to be chosen in such a way that not ony the system's equations, but, additionay, the hidden constraints have to be fued. An ecient approach 3 to cacuate a consistent initiaization for index-2 DAEs at t consists in soving the system 4 : Ay + Bx = q(t ) (3.2) PP (x ; )+PQ y ; PQ A ; 2 q (t )+Qy = (3.3) for an arbitrary and P := I ; Q. Note that PP (x ;) = xes the dynamic components, PQ y = PQ A ; 2 q (t ) describes the hidden constraints, and Qy = precisey xes the vaues we are not interested in, obtaining a nonsinguar system. The resuts presented in [8] impy that, for noninear circuits, this approach has the disadvantage that the projectors PP and PQ depend on the soution. Nevertheess, in [6] itwas pointed out how the method coud be reformuated in terms of other constant projectors. In this artice we wi suppose that we aready know vaues (x y ) that fu the equations of the system (3.) att that are not necessariy consistent 5, i.e., which probaby do not fu the hidden constraints. 6 Without oss of generaity, we aso assume that Qy = is fued. We denote by (x y ) the consistent vaue we obtain from (3.2) - (3.3) by setting := x and dene: From (3.2) - (3.3) it foows that: x + := x ; x y + = y ; y : (3.4) Ay + + Bx + = (3.5) PP x + + PQ y + + PQ y ; PQ A ; 2 q (t )+Qy = (3.6) i. e., we can cacuate the vaue (x + y+ ) from that system. Note that (3.6) impies PP x + = (3.7) PQ y + = PQ A ; 2 q (t ) ; PQ y (3.8) Qy + = : (3.9) 3 cf. [7]. Note that this approach can aso be extended to some noninear cases. 4 Weintroduce the index to distinguish the vaues at an arbitrary time t from t,which represents the time we start the integration process. 5 These vaues may beknown from an integration process. 6 In the foowing we consider ony the index-2 case, because in the index- case a vaue that fus the system's equations is automaticay consistent. 4

5 The task of determining vaues for (x + y+ )by making use of (3.5) - (3.6) may ook very simiar to the direct computation of (x y )from(3.2) - (3.3), but, in fact, if we know a description of the N \ S-components and/or the expressions for the equations corresponding to (3.8) and (3.9), the cacuation costs can be reduced consideraby. From the equations (3.5) - (3.9) it can be deduced that x + because (3.5) impies x + 2 S and, if we mutipy (3.5) by PQ A ; 2 2 N \S. This foows,we obtain PQ x + =. Taking into account (3.7), x + 2 N has to be given. This property wi be of specia interest with regard to circuit simuation. Remarks. At rst gance, this approach seems not to be very hepfu, because a vaue that fus the system's equations has to be given a priori. Fortunatey, in circuit simuation we can take advantage of the structura properties that guarantee the existence of the DC operating point to compute such avaue. 2. The circuit simuation by means of MNA eads to quasi-inear DAEs. The described projectors, spaces and index denitions of the tractabiity index can be extended to noninear systems (cf. []). The denitions for the equations arising from circuit simuation by means of modied noda anaysis were discussed in detai in [8]. There it was proved that, under certain restrictions on the controed sources (see Tabes 5. and 5.2), the space N \ S is constant and the N \ S-component appears ony in inear reations of the DAE. Therefore, the above approach can be successfuy extended. 3. In [6] it was aready pointed out how to transcribe topoogicay the equations that describe the hidden constraints anaogousy to (3.8) with the aid of constant projectors. Nevertheess, the topoogica initiaization presented there is nay based on the idea of xing ony the dynamic components and cacuate the vaues for the remaining variabes by means of the system's equations. This approach has the advantage that no specic (x y ) has to be given, but the disadvantage that the obtained vaues depend on the choice of the variabes for which initia vaues have been prescribed. 4. In the course of this artice it wi be shown that an eectrica expanation for the rearrangement from(x y )to(x y ) can be given. 4 The MNA equations Let us anayze the DAE system obtained by the appication of the MNA from umped networks containing noninear and possiby time-variant resistances, capacitances, inductances, independent votage and current sources, and some specic controed sources. 5

6 Wedenoteby q and the charge associated with the capacitances and the uxes associated with the inductances, by j L and j V the current vector of inductances and votage sources and by e the vector of node potentias. On the other hand, i(), and v() represent functions of current and votage sources. In this paper, we wi assume specia prerequisites for the controed sources. Anaogousy to [8], n-termina resistances, capacitances and inductances are competey described by (n ; ) currents entering the (n ; ) terminas and then (n ; ) branch votages across each of these (n ; ) terminas and the reference termina n. To write down the MNA 7 equations, we spit the reduced incidence Matrix A into the eement-reated incidence matrices A = (A C A L A R A V A I ), where A C, A L, A R, A V, and A I describe the branch-current reation for capacitive branches, inductive branches, resistive branches, branches of votage sources and branches of current sources, respectivey. If we dene C(u t) q t(u t) t) L(j t) t) t(j t) the DAE system we obtain from networks by the conventiona MNA reads A C C(A T C e de t)at C dt + A Cqt(A T C e t)+a Rr(A T Re t) Later on we wi aso need G(u t) +A L j L + A V j V + A I i() = (4.) L(j L t) dj L dt + t(j L t) ; A T Le = (4.2) A T V e ; v() = : (4.3) We rst anayze the network with respect to the conventiona MNA and, afterwards, extend the resuts to the systems obtained by charge-oriented MNA. These systems are 8 : A C dq dt + A Rr(A T R e t)+a Lj L + A V j V + A I i() = (4.4) d dt ; AT Le = (4.5) e ; v() = (4.6) A T V q ; q C (A T Ce t) = (4.7) ; L (j L t)=: (4.8) 7 A detaied discussion on how we set up the system's equations can be found in [8] and [2]. 8 cf. again [8] and [2]. 6

7 Anaogousy to [8] and [6], we suppose that the capacitance matrix C(A T Ce t), inductance matrix L(j L t), and conductance matrix G(A T Re t) of a capacitances, inductances and resistances, respectivey, are positive denite 9. We wi aso make use of the fact that the reduced incidence matrix (A C A L A R A V ) has fu row rank and that A V has fu coumn rank, because cutsets of current sources ony and oops of votage sources ony are forbidden (cf. [8], [8]). 5 Index anaysis and consistent initiaization for circuit simuation 5. Some denitions and resuts In this section we repeat some of the resuts presented in [8] and [6] concerning the index of the DAE system and the expressions for the hidden constraints in terms of appropriate projectors. To this end, we need the foowing denitions and resuts. Denition 5. To characterize the topoogica properties of the network, we dene the projectors Q C, Q V ;C, QV ;C, and Q R;CV onto kera T C, kerat V Q C, kerq T C A V, and kera T R Q CQ V ;C,respectivey. Note that Q CRV := Q C Q V ;C Q R;CV is a projector onto ker(a C A R A V ) T. The compementary projectors wi be denoted byp := I ; Q, with the corresponding subindices. Denition 5.2. An L-I cutset is a cutset consisting of inductances and/or current sources ony. 2. A C-V oop is a oop consisting of capacitances and votage sources. In [8],[8] the foowing was shown to hod: Lemma 5.3. If the network does not contain L-I cutsets, then Q CRV =. 2. If the network does not contain C-V oops, then QV ;C =. In this artice, we suppose that the controed sources that form part of the network fu the conditions exposed beow in the Tabes 5. and 5.2. Regarding equations (5.3), (5.5), and (5.7) from Tabe 5.2, the assumptions made for the controed current sources impy that Q T CRV A Ii((A C A V A R ) T e j L j V t)=q T CRV A Iti t (5.9) 9 Of course, the same restriction on the positive deniteness of the conductance matrix from Coroary 2.2 of [8] can be made here. Therefore, for the resistances with incidence nodes that are connected to each other by capacitances and/or votage sources, no positive deniteness of the corresponding conductance matrix has to be assumed. 7

8 If we consider the eement-reated spitting of QV ;C,i.e., ( QV Q V ;C = ;C ) t ( QV ;C ) contr: then we can summarize the prerequisites we assume for the controed votage sources as foows: Q T V ;C v(at e dq(at Ce t) j L j V t) dt = Q T V ;C v t (t) (5.) v(a T e dq(at Ce t) j L j V t) dt = v (A T C e j L t) (5.2) for a suitabe function v and for a vector v t (t) that contains the functions of independent votage sources and zeros instead of the functions of controed votage sources. Anaogousy as in [8] and [6], in the foowing we wi drop the index *. Tabe 5.: Condition for controed votage sources is aways fued. Thus, we generay assume that the controed sources do not form part of the C-V oops or L-I cutsets. To shorten denotations we write i((a C A V A R ) T e j L PV ;C j V t) (5.) when we do not distinguish between (5.4), (5.6), and (5.8). Lemma 5.4 The matrices H (A T C e t) := A CC(A T C e t)at C + Q T C Q C H 2 := Q T C A V A T V Q C + Q T V ;C Q V ;C H 3 := A T V Q CQ T C A V + Q T V ;C QV ;C H 4 (A T C e t) := Q T V ;C A T V H ; (AT C e t)a V Q V ;C + P T V ;C PV ;C H 5 (j L t) := Q T CRV A LL ; (j L t)a T L Q CRV + P T CRV P CRV are nonsinguar. H 6 := Q T V ;C A T V A V Q V ;C + P T V ;C PV ;C H 7 := Q T CRV A LA T L Q CRV + P T CRV P CRV These assumptions can be transcribed into topoogica criteria anaogousy as it was done in [8]. The resut woud be simiar, with the ony dierence that now the branch potentias of branches that form part of L-I cutsets woud not be aowed to contro controed current sources. cf. [8], [6]. 8

9 For controed current sources we suppose that at east one of the foowing characterizations hods: (a) Q T CRV A I i(a T e dq(at Ce t) j L j V t) dt = Q T CRV A Iti t (5.3) i(a T e dq(at Ce t) j L j V t) dt = i a (A T C e AT V e j L t) (5.4) for a suitabe function i a. (b) Q T C A Ib = (5.5) i(a T e dq(at Ce t) j L j V t) = i b ((A C A V A R ) T e j L PV ;C j V t)(5.6) dt for a suitabe function i b. (c) Q T V ;C QT C A Ic = (5.7) i(a T e dq(at Ce t) j L j V t) = i c ((A C A V A R ) T e j L t) (5.8) dt for a suitabe function i c. Tabe 5.2: Conditions for controed current sources In [8] the foowing resut was obtained: Theorem 5.5 Consider umped eectric circuits satisfying the assumptions of the Tabes 5. and 5.2. Then it hods:. If the network contains neither L-I cutsets nor controed C-V oops, then the conventiona MNA eads to a DAE system with index- and the constraints are ony the expicit ones: Q T C AR r(a T R e t)+a Lj L + A V j V +A Ia c i a c ((A C A R A V ) T e j L t) = (5.) A T V e ; v(at C e j L t) = : (5.2) 2. If the network contains L-I cutsets or C-V oops, then the conventiona MNA eads to a DAE system with index-2. With regard to the constraints, we distinguish the foowing three possibiities. (a) If the network does not contain an L-I cutset (but contains controed C-V oops), then the constraints are the expicit ones, (5.) and 9

10 (5.2), and, additionay, the hidden constraint: Q T V ;C AT V H ; (AT C e t)p C T A C qt(a T C e t)+a Rr(A T R e t)+a Lj L +A V j V + A I i((a C A R A V ) T e j L PV ;C j V t) + Q T V ;C dv t dt =:(5.3) (b) If the network does not contain controed C-V oops, but contains L-I cutsets, the constraints are the expicit ones, (5.) and (5.2), and, additionay, the hidden constraint: Q T CRV A L L ; (j L t) ; A T L e ; t(j L t) + A It di t dt =: (5.4) (c) If the network contains L-I cutsets and C-V oops, then the constraints are the expicit ones, (5.) and (5.2), and the hidden ones (5.3) and (5.4). Remark: In [8] it was proved that the hidden constraint (5.3) resuted from and that (5.4) arose from Q T CRV Q T V ;C AT V de dt = Q T dv t V ;C dt (5.5) dj L A L dt + A di t It =: (5.6) dt For the sake of simpicity, we wi sometimes drop the arguments of the H matrices in the foowing and write a dot if they are not constant. In [6] itwas shown that a spitting of the system can be performed in such away that consistent initia vaues can be cacuated successivey as described beow. Coroary 5.6 Let the vaues (P C e jl ) be given. If the network contains controed sources that fu the conditions of the Tabes 5. and 5.2, we can determine consistent initia vaues for the system (4.)-(4.3) graduay. We spit e = P C e + Q C P V ;C e + Q C Q V ;C P R;CV e + Q C Q V ;C Q R;CV e and j V = QV ;C j V + PV ;C j V, and obtain the corresponding consistent vaues from P C e := P C e + P C A V QV ;C H ; 6 j L := j L + A T L Q CRV H ; 7 QT CRV Q C P V ;C e := Q C H ; 2 QT C A V P T V ;C ; Q T V ;C vt (t ) ; A T V P Ce ; ;A It i t (t ) ; A L jl ; ;A T V P Ce + v(a T C e j L t ) whereas the vaue of Q C Q V ;C P R;CV e can be obtained by soving the equation A R r(a T R(P C + Q C P V ;C + Q C Q V ;C P CRV )e ) P T R;CV QT V ;C QT C +A L j L + A Ia i a (A T C e A T V e j L t ) = :

11 Then we cacuate P V ;C j V := ;H ; 3 AT V Q CPV T ;C QT C A R r(a T R e t ) +A L j L + A Ia c i a c ((A C A V A R ) T e j L t) Next, we can determine the remaining vaues by means of Q CRV e := ;; QCRV H ; 5 ()QT CRV A L L ; (j L t )A T L (P C + Q C P V ;C + Q C Q V ;C P R;CV ) e ;A L L ; (j L t ) di t t(j L t )+A It dt (t ) Q V ;C j V := ;H ; 4 () Q T V ;C A T V H ; ()P T C A C qt(a T C e t )+A R r(a T R e t ) +A L j L + A V PV ;C j V + A I i((a C A V A R ) T e j L PV ;C j V t ) ;H ; 4 () Q T V ;C dv t dt (t ): If the charge-oriented MNA is considered, we set additionay: Remarks: q := q C (A T C e t ) := L (j L t ): Note that each time the matrices H ; ; () = H (AT C e t ), H ; () = 4 H ; 4 (AT C e t ) or H ; ; 5 () = H 5 (j L t ) appear, we aready know the corresponding vaues A T C e or j L and, therefore, can insert them. On the other hand, the conditions of the Tabes 5. and 5.2 impy precisey that this hods anaogousy for the controed sources. 2. Of course, if the network contains no C-V-oops and no L-I cutsets, then the corresponding equations dened with the aid of the projectors QV ;C and Q CRV, respectivey, donotappear. Coroary 5.6 impies that the choice of (e jl j V ) is arbitrary as ong as the noninear equation that eads to the expression for Q C Q V ;C P R;CV e is sovabe. From the resuts presented in [8] it foows directy that, for the controed sources we consider here, the space N \ S() is constant and can be described by N \ S() =imq CRV fg im QV ;C 2 Note that the controed sources we permit do not change the spaces associated with the DAEs and, in this context, impy that we do not need to ater the order, as it was done in [6]. :

12 for the conventiona MNA and N \ S() =fg fg im Q CRV fg im QV ;C for the charge-oriented MNA. Observe thatq CRV e and QV ;C j V appear ony in inear expressions of the equations (4.) - (4.3) and (4.4) - (4.8). 5.2 Topoogica anaysis of the network Let us anayze the topoogy of a given circuit to ocate the equations corresponding to (5.5) and (5.6), i. e., to the equations that ead to the hidden constraints 3. In [6] itwas shown that these equations can be obtained directy from the network by making use of the foowing two procedures. They precisey determine the ineary independent equations that describe the hidden constraints arising from C-V oops and L-I cutsets, respectivey. PROCEDURE. Search a C-V oop in the given network graph. If no oop is found, then end. 2. Write the equation resuting from the sum of the derivatives of the characteristic equations of the votage sources contained in the C-V oop, taking into account the orientation of the oop and the reference direction of the considered branches. For instance, if the votage sources v ::: v k form a part of the C-V oop and we dene i := + if the orientation of the oop coincides with that of vi ; ese, then the equation we write in this step is P k i= i((a T V e) i ; v i)=. 3. Form a new network graph by deeting the branch of one votage source that forms a part of the oop, eaving the nodes unchanged. 4. Return to, considering the new network graph. The foowing procedure starts again from the initia graph. PROCEDURE 2. Search an L-I cutset. If one is found, then pick an arbitrary inductance of this cutset. Reaize that we can aways nd such an inductance because cutsets of current sources ony are forbidden. If no cutset is found, then end. 3 A simiar topoogica anaysis of the network can be found in [3]. 2

13 2. Write a new equation resuting by derivation of the cutset equation arising from. For instance, if the current sources i ::: i k and the inductances j L ::: j L~ k form a part of the L-I cutset and we dene j := ~ j := + if the orientation of the cutset coincides with that of ij ; ese, + if the orientation of the cutset coincides with that of jlj ; ese, then the equation obtained in this step reads P k j= ji j + P~ k i= ~ jj Li =. 3. Deete the chosen inductance from the network contracting its incident nodes. 4. Return to, considering the new network graph. In [6], an extension of these procedures determines how to x the dynamic components for cacuating consistent initia vaues. The resut permits to assign specic vaues to a seection of variabes, and to determine the remaining by making use of the system's equation 4. Nevertheess, the vaues obtained in that way depended on the choice of variabes seected. 6 Cacuating a consistentvaue starting up from a vaue fuing the system's equations The approach presented in this section distributes the index-2 property aong a aected eements. It resuts from the approach presented in Section 3 and can be summarized as foows: Theorem 6. For networks that contain ony controed sources as specied in the Tabes 5. and 5.2 we obtain consistent initia vaues starting up from possiby inconsistent ones that fu the system's equations in the foowing way:. Add additiona currents that ow through the C-V-oops as a consequence of the hidden constraints described by PROCEDURE to the vaues of the currents through the branches that form a part of C-V-oops. 2. Add convenient vaues to the node potentias to fu the additiona votage across the L-I cutsets dened by the hidden constraints described by PROCEDURE 2. The meaning of this theorem becomes cear in the course of the artice. The Theorems 6.5 and 6.6 describe the statement propery. In this chapter, we rst give an overview of our aim, expaining the approach 4 Therefore, this resut is simiar to the one obtained in [6]. 3

14 for a specia case, the DC operating point, and then we generaize the resuts to appy them for arbitrary points. The proofs wi be pointed out for the genera case ony. In Section 6.4 we iustrate why the cass of controed sources for which this approach hods cannot be extended if no further topoogica considerations are made. 6. An initiaization reated to the DC operating point A common way for obtaining vaues to start the numerica integration in circuit simuation is to cacuate the DC operating point. Therefore, we consider this point separatey. We cacuate the DC operating point by setting the current through the capacitances and the votages across the inductances equa to zero. Note that for cacuating the DC operating point for the charge-oriented MNA, the matrix df dx has to be nonsinguar. Topoogicay, this impies that:. Cutsets of capacitances and/or current sources are forbidden, i.e., that (A L A R A V ) has fu row rank. 2. Loops of inductances and/or votage sources are forbidden, i.e., (A L A V ) has fu coumn rank. Furthermore, the non-singuarity of df dx impies assumptions on the resistances and on the controed sources. Simiar considerations hod for the conventiona MNA, whereas qt() and t() have to be considered separatey. Observe that, if time-dependent capacitors or inductors appear in the network, then the existence of the DC operating point and the non-singuarity of the corresponding matrix df dx are not equivaent for the conventiona MNA. In the index-2 case the DC operating point has not to be consistent. Nevertheess, the vaues obtained for the DC operating point, (e jl j V ), fu Kircho's aws, and precisey the equations A R r(a T R e t )+A L j L + A V j V + A I i(p CRV e j L PV ;C j V t ) = (6.) for the conventiona MNA, and, additionay, the equations if the charge-oriented MNA is considered. ;A T L e = (6.2) A T V e ; v(a T C e j L t ) = (6.3) q ; q C (A T C e t )= (6.4) ; L (j L t )= (6.5) Theorem 6.2 For the conventiona MNA we obtain a consistent initiaization (e j L j V P C e j ) reated tothedcoperating point L (e jl j V ) in the foowing way: 4

15 . Set P CRV e := P CRV e, j L := j L, PV ;C j V := PV ;C j V Cacuate Q CRV e making use of the above setting and soving the equation Q T CRV ; A L L ; (j L t ) A T L e ; di t t(j L t ) + A It dt (t ) =: (6.6) 3. Cacuate QV ;C j V making use of the above setting and soving the equation Q T V ;C AT V H ; (AT C e t )PC T A C qt(a T C e t )+A R r(a T R e t )+A L j L +A V j V + A I i(p CRV e j L PV ;C j V t ) + Q T V ;C dv t dt (t )=:(6.7) 4. Cacuate the vaues of the derivatives of the votages across the capacitances and of the derivatives of the currents through the inductances according to (4.) and (4.2). For the charge-oriented MNA the resut can be directy adapted by cacuating q = q and = with (4.7) and (4.8) and by making use of (4.4) and (4.5) to cacuate the derivatives of the charges and uxes associated with the capacitances and inductances, respectivey. Proof: This resut is a direct consequence of Coroary 5.6. The equations (6.)-(6.3) guarantee that Q T CRV [A L jl + A It i t (t )] = and Q T V ;C AT V e = Q T V ;C v t (t ) are fued. Therefore, the resuts from Coroary 5.6 ead to P CRV e = P CRV e, j L = jl, PV ;C j V = PV ;C jv, and to the equations for Q CRV e and QV ;C j V as described. q.e.d. Coroary 6.3 The consistent initia vaues dened intheorem 6.2 can be cacuated in the foowing way: e := e ; H ; 5 (j L t )Q T CRV j L := j L j V := jv ; H ; 4 (AT C e t ) Q T dvt V ;C dt (t ) +A T V H ; (AT C e t )A C q t(a T C e t ) di t A It dt (t ) ; A L L ; (jl t ) t(jl t ) 5 Note that this impies precisey x + 2 N \ S(), (cf. Sections 3 and 5.). : 5

16 Then, the corresponding vaues of the derivatives of votages across the capacitances and currents through the inductances can be computed by: P C e := ;H ; (AT C e t )A C q t(a T C e t ) +H ; (AT C e t )A V H ; 4 (AT C e t ) Q T dvt V ;C dt (t ) +A T V H ; (AT C e t )A C q t(a T C e t ) j L := ;L ; (j L t ) t(j L t ) ; L ; (j L t )A T L H ; 5 (j L t )Q T CRV ;A L L ; (j L t ) t(j L t ) : A It di t dt (t ) For the charge-oriented MNA we set q := q and :=, whereas the corresponding vaues of the nontrivia derivatives of charges associated with the capacitances and uxes associated with the inductances can be obtained by: P C q := H ; C A V H ; 4 (AT C e t ) Q T V ;C dv t dt (t ) + Q T V ;C A T V H ; (AT C e t )A C qt(a T C e t ) := ;A T L H ; 5 (j L t ) for HC := A T C A C + Q T Q. Q T CRV A I t di t dt (t ) ; Q T CRV A LL ; (j L t ) t(j L t ) Note that H 4 () andh 5 () were dened in Section 5.. An exampe is given in Figure 6.. Proof: The coroary is a specia case of the Theorems 6.5 and 6.6. Coroary 6.4 The vaues obtained intheorem 6.2 impy that, at time t, the sum of the additiona power deivered to the network by the C-V-oops and L-I cutsets is equa to the sum of the additiona power absorbed by the branches of the C-V-oops and L-I-cutsets. Proof: The coroary is a specia case of Coroary 6.7. In Figure 6.2 we can observe how this approach can be carried out for the NAND-Gate described e.g. in [9]. We consider the case that it contains inear capacitances. The resut shows that there is a current thatows through V, through the MOSFET that is incident with node 6, and through V BB. Note that inside the MOSFET the current is divided. 6

17 e v (t) e 2 DC operating point: C C 2 R e = v(t ) e 2 = j V =: Consistent initia vaue: Conventiona MNA: C e + j V = ;j V + C 2 e 2 + R e 2 = e ; e 2 = v(t ): e = v(t ) e 2 = j V = v (t C + ) C 2 e = ; C C + C 2 v (t ) e 2 = C 2 C + C 2 v (t ): Figure 6.: Circuit with C-V oop 6.2 Computing consistent vaues starting up from possiby inconsistent ones For any vaue (x Py ) fuing the DAE equations but being not necessariy consistent, a consistent (x Py ) can be computed in an anaogous way asfor the operating point 6. Instead of (6.) - (6.3), (or (6.) - (6.5), correspondingy) the possiby inconsistent vaues fu: A C C(A T C e t )A T C e + A C q t(a T C e t )+A R r(a T R e t ) +A L j L + A V j V + A I i((a C A V A R ) T e j L PV ;C j V t ) = (6.8) for the conventiona MNA and L(jL t )j L + t(jl t ) ; A T L e = (6.9) A T V e ; v(a T C e j L t ) = (6.) A C q + A R r(a T R e t )+A L jl + A V jv +A I i((a C A V A R ) T e j L PV ;C j V t )i = (6.) ; A T L e = (6.2) A T V e ; v(a T C e j L t ) = (6.3) q ; q C (A T C e t ) = (6.4) ; L (j L t ) = (6.5) for the charge-oriented MNA. Consequenty, for these vaues Coroary 5.6 impies, anaogousy as in Theorem 6 Weintroduce the index to distinguish the vaues at an arbitrary time t from t,which was introduced to start the integration process. The resuts presented in this chapter may be of specia interest to cacuate consistent vaues starting up from the possiby inconsistent vaues for the soution an integration method suppies at t. 7

18 2 VDD 3 2 Q C = ::: : : : : ::: B@ CA Q V ;C = B@ CA 6 V C C = c c c c c c c c c c c c c B@ CA : V 2 V BB j + V j + V 2 j + BB j + DD The additiona current for correcting the DC operating point is: C A = ; 2cc 2cc c+c c+c ; 2cc 2cc c+c c+c 2cc 2cc ; 2cc(4cc+3c(c+c)) c+c c+c (c+c)(2cc+c(c+c)) ; C A {z } ;H 4 ; V V 2 V BB C : A {z } QV ;C dv t dt For c =:5 ;3, c =:24 ;3, c =:6 ;3, V = 9, V2 = VBB =we obtain: q q gd q gs q db ;3: ;5 q sb q C A = B 2gd : ;5 3: ;5 A q2gs = : ;5 : q 2db : ;5 q2sb : ;5 q3gd Bq3gs C B q 3db A j + V j + V 2 j + BB j + DD q 3sb Figure 6.2: Computation of j + V and q + for the NAND-Gate from [9]. 8

19 6.2, that ony the vaues of Q CRV e and QV ;C j V have tobeproperychanged to obtain consistent vaues (e j L j V ). As a consequence, the vaues of the appearing derivatives have to be adapted to get consistent (x Py ) that fus the systems corresponding to (4.) - (4.2), or (4.4) - (4.5). Theorem 6.5 For the conventiona MNA we obtain consistent vaues (e j L j V ) starting up from the possiby inconsistent vaues (e jl j V P Ce j L ) that fu the DAE equations as foows: e := e ; H ; 5 (j L t )Q T CRV j L := j L di t A It dt (t )+A L j L j V := j V ; H ; 4 (AT C e t ) Q T V ;C dvt dt (t ) ; A T V e Furthermore, the corresponding vaues of the derivatives can be cacuated by means of P C e := P C e + H ; (AT C e t )A V H ; 4 (AT C e t ) Q T V ;C dvt j L := j L ; L; (j L t )A T L H ; 5 (j L t )Q T CRV Proof: If, for a xed, we dene e + := e ; e j V + := j V ; j V di t A It dt (t )+A L j L : dt (t ) ; A T V e : then it hods that e + = Q CRV e + + and j V = QV + ;C j V, because of P CRV e = P CRV e and PV ;C j V = PV ;C jv. If we insert e = e + e + and j V = jv + j V + into the hidden constraints (5.3) and (5.4), which precisey have to be fued additionay by a consistentvaue, we obtain Q T CRV A L L ; (j L t )A T L Q CRV e + + A L j di t L + A It dt (t ) = and Q T V ;C A T V H ; (AT C e t )A V QV + ;C j V ; A T V P Ce + dv dt (t ) = : The expressions for e and j V foow thenbymutipying these expressions by H ; ; 5 () and H () to cacuate 4 e+ + and j V. Dening now P C e + := P C e ; P Ce j L + := j L ; j L 9

20 and making use of the fact that (e j L j V P C e j L ) has aso to fu the system (4.) -(4.2), we obtain A C C(A T C e t )A T C e + + A V j V + = which eads to the presented expressions. L(j L t )j + L ; AT L e+ = q.e.d. Theorem 6.6 For the charge-oriented MNA we obtain consistent vaues (e j L j V q ) starting up from the possiby inconsistent vaues (e j L j V q PC q ) that fu the DAE equations as foows: e := e ; H ; 5 (j L t )Q T CRV j L := j L j V := j V ; H ; 4 (AT C e t ) Q T V ;C dvt dt (t ) q := q := : A It di t dt (t )+A L L ; (j L t ) ;A T V H ; (AT C e t )A C q ; q t (A T C e t ) ; t (jl t ) Furthermore, the vaues of the derivatives of the charges and the uxes associated with the capacitances and the inductances, respectivey, are determined by P C q := PC q + H ; C A V H ; 4 (AT C e t ) Q T dv V ;C dt (t ) := ; A T L H ; 5 (j L t )Q T CRV for HC = A T C A C + Q T C QC. ;A T V H ; (AT C e t )A PC C q ; q t (A T C e t ) di t A It +A L L ; (j L t ) ; t (j L t ) Proof: Theorem 6.6 foows simiary to Theorem 6.5. dt (t ) Proof of Coroary 6.3: Coroary 6.3 can be interpreted as a specia case of the Theorems 6.5 and 6.6. For k =and P C e = ;H ; (AT C e t )A C q t(a T C e t ) j L = ;L ; (j L t ) t(j L t ) 2

21 or, if the charge-oriented MNA is considered, q =and =. q.e.d. Coroary 6.7 The vaues obtained in Theorem 6.5 (or 6.6, correspondingy) impy that, at time t, the sum of the additiona power deivered by the C-V oops and L-I cutsets is equa to the sum of the additiona power absorbed by the branches of the C-V oops and L-I cutsets. Proof: First of a, et us consider the eements that are aected by the corrections of the vaues. Taking into account j + V = QV ;C j + V, it foows from the projector anaysis made in [6] that ony the currents through those votage sources that form a part of C-V oops are aected. Considering now that the expressions of Theorem 6.6 impy A C q + + A V j + V = for q + = q ; q, it foows that ony the currents through capacitances that form a part of C-V oops change. Taking into account that Kircho's aws are vaid for the (x Py ) and for (x Py ) because both fu the MNA equations, Teegen's theorem 7 impies bx k= u k j k = and bx k= u k j k = i.e., = bx k= u k j k ; bx k= u k j k if b is the number of branches, u k and j k are the votages and currents obtained for the network from the consistent vaue (x Py ), and u k and j k are the votages and currents obtained from (x Py ). Setting u + := e ; e +, j V := j V ; j V, P Ce + := P C e ; P Ce, and j + L := j L ; j L for a xed, for the conventiona MNA this eads to = X L s from L;I;cutsets + X V s from C;V ;oops L(j L t )j + k v tk (t )j V + L s from L;I;cutsets j k + X k + C s from C;V ;oops X I s from L;I;cutsets u k u + k i tk(t ): C(A T C e t )A T C e + k and, if q + := q ; q and + := ; is dened, the charge-oriented MNA impies X = X + k j k + u k q + k 7 cf. for instance [5]. + X V s from C;V ;oops v tk (t )j V + C s from C;V ;oops k + X I s from L;I;cutsets u + k i tk(t ): 2

22 q.e.d. Proof of Coroary 6.4: Coroary 6.4 can be interpreted as a specia case of Coroary 6.7 for k =. Remark: If x denotes a vaue that fus the equations of the system and x the corresponding consistent vaue, then the above resuts impy that, if we start the integration process with an impicit Euer method, then we wi obtain the same resuts when starting up from x and from x. This is due to the fact that, on the one hand, the dynamic components of x and x coincide, and, on the other hand, the same Jacobian for the Newton method resuts in both cases, because of the inear occurrence of the correction. Observe that this is not the case for integration methods that utiize the vaues of a variabes from the preceding steps, as for instance the trapezoida rue. 6.3 The reevant inear system to cacuate the consistent vaues The expressions from the Sections 6. and 6.2 can be transformed in a way that we ony have to sove a reativey sma inear system. We present the resuts for an arbitrary point. Reca that we have denoted the n-termina capacitances, inductances and resistances by capacitances, inductances and resistances. In this way, we do not have to distinguish if some of them contro others or are controed by others. In the foowing our aim is to dene equations that permit the cacuation of the vaues dened in the ast section, but that do not require the direct cacuation of the inverses of the compete matrices H (), H 4 (), L(), and H 5 (). Note that these matrices were constructed by compementing the reevant matrices that described the C-V oops and L-I cutsets to obtain the non-singuarity. In practice, consideraby smaer matrices can be considered. Denition 6.8 Denote by A C and A V the incidence matrices of the capacitances and the votage sources that form a part of C-V oops, respectivey, and denote by q and C (A T Ce t) the charges and the capacitance matrix corresponding to them. Further et A L denote the incidence matrix of the inductances that form a part of L-I cutsets, and, j L and L (j L t) the uxes, currents and the inductance matrix corresponding to them. Theorem 6.9 For the conventiona MNA the soution ( QV + ;C j V P Ce + of the inear system: A C C (A T C e t )A T C e + + A V QV + ;C j V = (6.6) Q T V ;C AT V e + + Q T V ;C A T V e ; Q T V ;C v t (t ) = (6.7) ) 22

23 and the soution (Q CRV e +, j + L ) of the inear system L (j L t )j L + ; A T L Q CRV e + = (6.8) Q T CRV A Lj L + + Q T CRV A Lj L + Q CRV A It i t(t ) = (6.9) provide the vaues that permit us to cacuate the consistent vaues from Theorem 6.5. Note that P C e and j L are considered tobeconstant vectors. Remarks:. Equations (6.7) and (6.9) can be obtained by making use of Procedures and 2 from Section A practicabe reaization of the cacuation of suitabe vaues can be carried out making use of the projectors QV ;C and Q CRV dened in [6]. Theorem 6. For the charge-oriented MNA the soution of the inear system 8 A C C (A T C e t )A T C e + + A V QV + ;C j V = (6.2) Q T V ;C AT V e + + Q T V ;C A T V e ; Q T V ;C v t (t ) = (6.2) and the soution of the inear system q ; C (A T C e t )A T C e ; q t(a T C e t ) = (6.22) + L (j L t )j L ; A T L Q CRV e + = (6.23) Q T CRV A Lj L + + Q T CRV A Lj L + Q CRV A I i t(t ) = (6.24) ; L (j L t )j + L ; t(j L t ) = (6.25) provide the vaues required to cacuate the consistent vaues e and j V from Theorem 6.6. The corresponding vaues of the concerned derivatives can be xed then by: q = q + C (A T C e t )A T C e + = + L (j L t )j + L : Note that now q and are considered tobeconstant vectors. Proof: The theorems foow by straight-forward computation. Remark: The remarks we made for the conventiona MNA hod for the chargeoriented MNA anaogousy. 8 Note that we expand the system considered in Theorem 6.9 to avoid the cacuation of the inverse matrices to estabish the reation between the given vaues q, and the vaues Q T V ;C AT V e, j L that appear in (6.2) and (6.24), respectivey. 23

24 e i(t) DC operating point: e = e 2 = v 2(t ) e 3 = v (t ) L i (e ) c e 2 j V = j V 2 = i(t ) ; i c() j L = ;i(t ) j L = C(e 2 ; e 3)= Consistent initia vaue: v (t) e 3 C e = ;Li (t ) e 2 = v 2(t ) e 3 = v (t ) j V = C(v2(t ) ; v(t )) j V 2 = i(t ) ; i c(;li (t )) ; C(v2(t ) ; v(t )) j L = ;i(t ) v (t) 2 j L = ;i (t ) C(e 2 ; e 3)=C(v 2(t ) ; v (t )): Figure 6.3: Circuit with VCCS 6.4 Considering more genera controed sources For arbitrary controed sources, the index maychange in various ways 9. Therefore, we restrict our considerations to those presented in [8] to notice that the resuts of the atter section cannot even be appied to a sources considered there. As it was proved in that paper, some of the controing sources change the structure of the spaces associated with the DAE-system. Indeed, it was proved there that the ateration of the order in which we sove the equations in Theorem 3.2 of [8] became aso recognizabe in terms of the space N \ S() of the tractabiity index. For the approach presented in the above section, we use that this space is constant. Furthermore, we utiize that precisey the N \ S()- components occur ony ineary in the equations (4.) -(4.3) (or (4.4) -(4.6), respectivey) and that, if we modify their vaues, this changes ony the vaues of the derivatives of the capacitances and inductances that form a part of C-V oops or L-I cutsets. Uness these assumptions are fued, the presented approach to cacuate a consistent vaue starting from a vaue fuing the system's equations fais. The exampe of Figure 6.3 iustrates the probem for a cass of sources described in [8]. Note that, if, in Figure 6.3, we repace v 2 by a resistance, then, even if there appears no C-V oops, the current through v is aected by a hidden constraint. Therefore, j V and C(e 2 ; e 3)have to be rearranged, too. Nevertheess, the sources described in the Tabes 5. and 5.2 incude, for instance, the controed sources contained in the MOSFET. 9 cf. for instance [3] and [7]. 24

25 7 A combined possibiity of initiaization In the foowing, we present an initiaization approach that mayaowthe user of a simuation package to prescribe the vaues of some votages across capacitances and currents through inductances, and then to cacuate an initia vaue that is sti reated to a DC operating point. This approach isony possibe if the seected eements fu some topoogica restriction described beow. In this chapter we wi present the approach successivey and summarize the resut nay. Reca that for muti-termina eements with n terminas, we consider each pair of terminas n, with n ; as branches, if n denotes the reference termina 2. In the foowing, we assume that, if n-termina capacitances and inductances appear in the network, then for each one the user either assigns vaues to a or to none of its branch potentias and currents, respectivey. 2 Denotation 7. Suppose that the user wants to prescribe the vaues of the branch votages across some capacitive branches and the vaues of the currents through some inductive branches of a network G. Denote by ^G the graph of the network we obtain when substituting, in G, a capacitances and inductances for which the user wants to prescribe an initia vaue by independent constant votage and current sources, respectivey. Let the constant vaues characterizing these sources be precisey the vaues the user wants to assign. If the user wants to prescribe the vaues of an n-termina, then, to construct ^G, we introduce votage or current sources that connect the reference node of that n-termina with each of the other terminas. Let the matrices (A C A R A L A V A I )and(a ^C A ^R A ^L A ^V A ^I ) denote the incidence matrices of the origina and the modied graphs, respectivey. We denote by: A C ; the incidence matrix of those capacitances across which the user wants to assign an initia branch votage. A L ; the incidence matrix of those inductances through which the user wants to assign a initia current. Note that then A ^C and A ^L denote the incidence matrices of capacitances and inductances for which the user does not assign an initia vaue. the incidence matrix of the votage sources we introduce into the network instead of capacitances. A ^V + the incidence matrix of the current sources we introduce into the network instead of inductances. A ^I + 2 cf. [8]. 2 If this restriction is not kept, then further topoogica considerations concerning controing and the controed eements have to be made. 25

26 With these denotations, the foowing reations foow directy if we suppose that the incidence matrices are dened considering the eements of the same shape successivey: A C ; = A ^V + A ^V =(A V A ^V + )=(A V A C ;) A L ; = A ^I + A ^I =(A IA ^I + )=(A I A L ;) A C =(A ^C A C;) =(A ^C A ^V + ) A L =(A ^L A L;) =(A ^L A ^I + ): To appy the approach presented in this section, the choice of capacitances and inductances made has to fu the topoogica restrictions of Tabe 7.. TOPOLOGICAL RESTRICTIONS In this section, a specic choice of inductances and capacitances for which the user wants to prescribe initia vaues has to fu the foowing topoogica conditions:. The matrix (A V A C ;)=A ^V has fu coumn rank. 2. The matrix (A C A L ;A R A V )=(A ^C A ^R A ^L A ^V ) has fu row rank. 3. The matrix (A L ;A V A C ;)=(A ^L A^V ) has fu coumn rank. 4. The matrix (A L ;A R A V A C ;)=(A ^R A^L A^V ) has fu row rank. If these conditions are fued, we goonookingat ^G. Remarks about Tabe 7.: Tabe 7.: Topoogica restrictions Note that the conditions () - (2) mean that ^G does not contain oops of votage sources ony nor cutsets of current sources ony. If these conditions are not fued, then the initiaization the user wants to perform is not possibe. On the one hand, if () is not fued, then it is not possibe to assign a vaue to a chosen capacitances. On the other hand, if (2) is not fued, we cannot prescribe vaues for a chosen inductances. (cf. [6]). Furthermore, the conditions (3) - (4) have to be fued if ^G is DCsovabe. If these conditions are not met, this approach cannot be carried out for the seected inductances and capacitances. In this case, some further or fewer inductances and/or capacitances have to be considered. Indeed, these conditions are necessary but not sucient to guarantee the DC-sovabiity of ^G. For the approach of this section we require that ^G is DC sovabe. 26

27 From now on, et us suppose that ^G is DC sovabe and that we cacuate the DC operating point for ^G. In the foowing we wi show how and why we can make use of the vaues computed in this waytoobtainavaue that fus the MNA equations corresponding to G. We denote by (^q ^ (A ^C A ^R A ^L A ^V )T e j ^L j^v ) the variabes corresponding to ^G v ^V + and i ^I + the votage and the current of the introduced independent constant votage and current sources, respectivey ^v and ^i the compete votage and currentvectors of the votage and current sources of ^G. Making use of the equations that are fued for the DC operating point of ^G, we dene initia vaues for G by the foowing possibe reations: A ^R e t)+a r(at^r ^L j ^L + A ^V j ^V {z } =A V j V +A ^V + j ^V + + A ^I^i() {z } =A Ii()+A ^I + i ^I + = (7.) ;A e = (7.2) T^L A e ; ^v() T^V {z } = (7.3) = A T V e;v()= AT V + e;v V + = ^q ; ^q ^C e t) = (7.4) (AT^C ^ ; ^ ^L (j ^L t) = : (7.5) Of course, if the conventiona MNA is considered, the equations (7.4) - (7.5) wi not appear. Assignment 7.2 Denote by (^q ^ (A ^C A ^R A ^L A ^V )T e j ) the operating point ^L j^v of ^G. Let us x then the foowing vaues for the variabes of G by means of: A A T T C e := ^C e A T^V = +e j L := A T ^C e (7.6) v ^V + j ^L : (7.7) i ^I + For the remaining vaues of A T e and for jv we appy the vaues obtained for ^G directy. If we denote by jc the vector that contains the vaues of j V for the capacitances + for which we prescribed initia branch votages and zero ese, then we can set: P C e := C ; (A T C e t )(j C ; q t(a T C e t )) (7.8) j L := L ; (j L t )(A T L e ; t(j L t )) (7.9) 27

28 for the conventiona MNA and dq dt := jc (7.) d dt := A T L e (7.) q := q C (A T C e t ) (7.2) := L (j L t ) (7.3) for the charge-oriented MNA. Note that the derivatives of the charges and uxes can ony be distinct from zero for the capacitances and inductances for which we prescribed an initia vaue. Lemma 7.3 The vaues determined by Assignment 7.2 fu the MNA equations for G. Proof: For the charge-oriented MNA the equations A C dq dt + A Rr(A T R e t)+a Lj L + A V j V + A I i() = (7.4) have to be fued. We discuss each of them. d dt ; AT Le = (7.5) A T V e ; v() = (7.6) q ; q C (A T Ce t) = (7.7) ; L (j L t)= (7.8) Equation (7.4) is fued because of (7.), taking into account that we obtain dq A C dt = A ^C j C = A C ;j ^V = A + ^V j ^V + + A L jl = A ^L j^l + A ^I + i I + making use of (7.), (7.7), and A R r(a T R e t)=a ^R r(at^r e t). Equation (7.5) is fued because of (7.2) and the setting (7.). Equation (7.6) is fued because of (7.3). Equation (7.7) and (7.8) are fued because of (7.2) and (7.3). Therefore, this vaue fus the system's equations. q.e.d. For the conventiona MNA, the proof is anaogous. 28

29 Consider now the foowing procedure to cacuate a consistent initia vaue that meets the demands of the user of the simuation package. PROCEDURE 3 Suppose that the user wants to prescribe vaues for the branch potentias across the capacitances C ::: C n and vaues for the currents through the inductances L ::: L m.. Check if the graph ^G resuting according to Denotation 7. for the seection of capacitances and inductances is DC-sovabe. If not, this procedure cannot be appied. 2. Compute the DC operating point for ^G. 3. Fix the vaues for G as described in Assignment Compute a consistent vaue reated to the vaue obtained in Step 3 as described in the Theorems 6.5 and 6.6 for the conventiona and the chargeoriented MNA, respectivey. We summarize the resuts of this section in the foowing theorem: Theorem 7.4 If a seection of capacitances and inductances is permissibe according to step, then Procedure 3 yieds consistent initia vaues with the properties: The branch potentias across the seected capacitances and the currents through the seected inductances are the prescribed ones. The vaues of the currents that ow through the capacitances that are neither seected nor form a part of C-V oops are zero. The vaues of the votages across the inductances that are neither seected nor form a partofl-icutsetsarezero. An exampe is given in Figure 7.. There we consider the exampe of Figure 6. and suppose the user wants to prescribe the vaue v C for the votage across the capacitance C. Observe thatthe vaue obtained for the current through the capacitive branchofc 2 depends aso on the vaue v C. 8 Concusion In this artice we have presented how to make use of the specia structure of the equations obtained by means of the MNA in eectric circuit simuation to compute consistent vaues. This may give new insight into how to dea with these structura properties with regard to numerica circuit simuation. Nevertheess, these resuts are ony vaid if we make restrictions on the controed sources that appear in the network. For arbitrary controed sources, no such genera resuts seem to be possibe. 29