Consistent initial values for DAE systems in circuit simulation

Consistent initial values for DAE systems in circuit simulation

D. Estevez Schwarz

Abstract

One of the di culties of the numerical integration methods for dieren- tial-algebraic equations (DAEs) is computing consistent initial values be- fore starting the integration, i.e. , calculating values that satisfy the given algebraic constraints as well as the hidden constraints if higher index prob- lems are considered.

This paper presents an approach to calculate consistent initial values for index-2 DAEs starting up from possibly inconsistent ones. Firstly, the idea is exposed for linear DAEs and then it is shown how the results can be applied to those systems arising from modied nodal analysis (MNA) in circuit simulation. This article starts up from 8] and 6]. Several denotations and results we use were introduced there in more detail.

Key words:

Consistent initial values consistent initialization dierential- algebraic equation DAE index circuit simulation modied nodal analysis MNA structural properties.

AMS Subject Classication:

94C05, 65L05.

1 Introduction

Roughly speaking, the problem of determining consistent initial values for die- rential-algebraic equations can be described as follows. For ordinary dierential equations, initial values have to be prescribed for all variables to determine a unique solution. However, dierential-algebraic equations consist of dierential equations coupled with derivative-free equations, i.e., not all components appear in dynamic form. Indeed, some of them are determined by algebraic constraints.

In Section 2 a convenient characterization of consistent initial values is intro- duced.

One approach to determine consistent initial values is to locate a selection of variables for which we may prescribe initial values and to construct a nonsingu- lar system that provides the values for the remaining ones. In this context, we have to consider two problems:

1. The selection of variables is not arbitrary.

2. The values that are assigned to the selected variables have to be chosen in such a way that the nonlinear system is solvable.

In 6] it was analyzed how to construct a nonsingular system and how to treat (1) for systems arising from circuit simulation by means of modied nodal anal- ysis, while (2) was not discussed. Nevertheless, the practical realization of that approach leads to new insights, provided that the values from (2) are suitably chosen. These results will be presented in the present paper.

To illustrate the approach before considering the special systems arising from circuit simulation, we describe the basic ideas for linear DAEs in Section 3.

Then, in Section 4 we introduce the equations of the modied nodal analysis.

The structural properties of these systems, which were pointed out in 8] and 6], are summarized in Section 5. The new results are presented in Section 6.

Since, in circuit simulation, the operating point is frequently used for starting the integration, it is separately analyzed in Section 6.1. In Section 6.2 the approach is described for a more general case. In practice, the values obtained in the Sections 6.1 and 6.2 can be calculated by solving relatively small linear systems as described in Section 6.3.

Finally, in Section 7 it is shown how this approach may be combined with an initialization strategy that takes into account possible initialization preferences of the user of a simulation package.

2 About consistent initial values for DAEs

We consider dierential-algebraic equations, i.e. , equations of the form

f(^{x0 x t}) = 0 (2.1)

where_dx^df0 is singular. In this article,_dx^df0 is assumed to have a constant nullspace.

Note that, if we dene a projector^Qonto^ker_dx^df0 and^P :=^I;Q, then equation (2.1) may be written as

f(^{Px0 x t}) = 0^:

Denition 2.1

A vector ^{x0 2Rm} is a consistent initial value of (2.1) if there exists a solution of (2.1) that fulls^x(^t0) =^x0.

Taking into account that the singularity of _dx^df0 implies that (2.1) contains some algebraic equations, a consistent initial value has to full precisely those alge- braic equations. Moreover, the dierentiation of these algebraic equations may lead to further algebraic equations, called hidden constraints, which a consistent initial value has to full, too. This fact is closely related to the index concept.

Actually, we are also interested in the corresponding values of the derivatives appearing in the DAE, i.e., in the values of^Px0 if ^P is dened as^P :=^I;Q for a projector^Qonto^ker_dx^df0.

Denition 2.2

A vector(^{x0 Py0}) is a consistent initialization of (2.1) if^x0 is a consistent initial value and (^{x0 Py0}) fulls the equation ^f(^{Py0 x0 t0}) = 0.

For simplicity, we will rst present the approach for linear dierential-algebraic equations (DAEs). In the course of the article, it will be shown how it can be extended to those quasi-linear systems obtained by modied nodal analysis (MNA).

3 An overview of the approach for linear DAEs

For a short outline of the main ideas presented in this article, the tractability index and the spaces related to its denition are introduced.

Consider a linear DAE of the form:

Ax

0+^Bx=^q(^t) (3.1)

where^Ais singular.

For the tractability index we dene^N := ker^Aand^S:=^fz: ^Bz2im^Ag.

Denition 3.1

The DAE (3.1) is called index-1-tractable if the matrix ^A1:=

A+^BQis nonsingular for a constant projector ^Qonto^N.¹

Remarks:

1. The matrix^A1 is nonsingular if and only if^N\S =^f0^g. 2. The denition does not depend on the choice of the projector^Q.

For the denition of the index two we dene^N1:= ker^A1and^S1:=^fz: ^BPz2 im^A1gfor^P := (^I;Q).

Denition 3.2

The DAE (3.1) is called index-2-tractable if 1. it is not index-1-tractable,

2. ^A2:=^A1+^BPQ1 is nonsingular for a projector^Q1 onto^N1.²

Remarks:

1. The matrix^A2is nonsingular if and only if^N1\S1=^f0^g. 2. The denition does not depend on the choice of the projector^Q1.

Denition 3.3

In the following, the canonical projector^Q1:=^{Q1A;12 BP} onto

N

1 along^S1 is considered.

1cf. 10].

2cf. 10].

In the index-2 case the space^N\Srepresents all the components that are deter- mined neither by a dierential nor by an algebraic equation. These components can be determined only by inherent dierentiation.

Furthermore, for index-2 equations hidden constraints appear if we derive a part of the system's equations. This implies that an initial value has to be chosen in such a way that not only the system's equations, but, additionally, the hidden constraints have to be fullled.

An ecient approach³ to calculate a consistent initialization for index-2 DAEs at^tl consists in solving the system⁴:

Ayl+^Bxl = ^q(^tl) (3.2)

PP

1(^xl^;) +^PQ1yl^{;PQ1A;12 q0}(^tl) +^Qyl = 0 (3.3) for an arbitraryand^P1:=^I;Q1.

Note that^PP1(^xl^;) = 0 xes the dynamic components,^PQ1yl=^{PQ1A;12 q0}(^tl) describes the hidden constraints, and^Qyl= 0 precisely xes the values we are not interested in, obtaining a nonsingular system.

The results presented in 8] imply that, for nonlinear circuits, this approach has the disadvantage that the projectors^PP1and^PQ1depend on the solution.

Nevertheless, in 6] it was pointed out how the method could be reformulated in terms of other constant projectors.

In this article we will suppose that we already know values (^{xl yl}) that full the equations of the system (3.1) at^tlthat are not necessarily consistent⁵, i.e., which probably do not full the hidden constraints.⁶ Without loss of generality, we also assume that^Qyl= 0 is fullled.

We denote by (^xl ^yl) the consistent value we obtain from (3.2) - (3.3) by setting

:=^xl and dene:

x

+l :=^xl^;xl y

l+=^yl^;yl: (3.4)

From (3.2) - (3.3) it follows that:

Ay

+l +^Bx+_l = 0 (3.5)

PP

1 x

+l +^PQ1y+_l +^{PQ1yl;PQ1A;12 q0}(^tl) +^Qyl = 0 (3.6) i. e., we can calculate the value (^x+_l ^y_l⁺) from that system. Note that (3.6) implies

PP

1 x

+l = 0 (3.7)

PQ

1 y

l+ = ^{PQ1A;12 q0}(^tl)^;PQ1yl (3.8)

Qy

l+ = 0^: (3.9)

3cf. 7]. Note that this approach can also be extended to some nonlinear cases.

4We introduce the index^lto distinguish the values at an arbitrary time^tlfrom^t0, which represents the time we start the integration process.

5These values may be known from an integration process.

6In the following we consider only the index-2 case, because in the index-1 case a value that ful ls the system's equations is automatically consistent.

The task of determining values for (^x+_l ^y_l⁺) by making use of (3.5) - (3.6) may look very similar to the direct computation of (^xl ^yl) 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 calculation costs can be reduced considerably.

From the equations (3.5) - (3.9) it can be deduced that^x+_l ^2N\S. This follows because (3.5) implies^x+_l ^2S and, if we multiply (3.5) by^PQ1A;12 , we obtain

PQ

1 x

+l = 0. Taking into account (3.7), ^x+_l ^2N has to be given.

This property will be of special interest with regard to circuit simulation.

Remarks

1. At rst glance, this approach seems not to be very helpful, because a value that fulls the system's equations has to be given a priori. Fortunately, in circuit simulation we can take advantage of the structural properties that guarantee the existence of the DC operating point to compute such a value.

2. The circuit simulation by means of MNA leads to quasi-linear DAEs. The described projectors, spaces and index denitions of the tractability in- dex can be extended to nonlinear systems (cf. 10]). The denitions for the equations arising from circuit simulation by means of modied nodal analysis were discussed in detail in 8]. There it was proved that, under certain restrictions on the controlled sources (see Tables 5.1 and 5.2), the space^N\S is constant and the^N\S-component appears only in linear relations of the DAE. Therefore, the above approach can be successfully extended.

3. In 6] it was already pointed out how to transcribe topologically the equa- tions that describe the hidden constraints analogously to (3.8) with the aid of constant projectors. Nevertheless, the topological initialization pre- sented there is nally based on the idea of xing only the dynamic com- ponents and calculate the values for the remaining variables by means of the system's equations. This approach has the advantage that no specic (^{xl yl}) has to be given, but the disadvantage that the obtained values depend on the choice of the variables for which initial values have been prescribed.

4. In the course of this article it will be shown that an electrical explanation for the rearrangement from (^{xl yl}) to (^xl ^yl) can be given.

4 The MNA equations

Let us analyze the DAE system obtained by the application of the MNA from lumped networks containing nonlinear and possibly time-variant resistances, capacitances, inductances, independent voltage and current sources, and some specic controlled sources.

We denote by^qandthe charge associated with the capacitances and the uxes associated with the inductances, by^jLand^jV the current vector of inductances and voltage sources and by^ethe vector of node potentials.

On the other hand, ⁱ(), and ^v() represent functions of current and voltage sources. In this paper, we will assume special prerequisites for the controlled sources.

Analogously to 8], n-terminal resistances, capacitances and inductances are completely described by (^n;1) currents entering the (^n;1) terminals and then (^n;1) branch voltages across each of these (^n;1) terminals and the reference terminalⁿ.

To write down the MNA ⁷ equations, we split the reduced incidence Matrix A into the element-related incidence matrices ^A = (^AC^AL^AR^AV^AI), where

AC, ^AL, ^AR, ^AV, and ^AI describe the branch-current relation for capacitive branches, inductive branches, resistive branches, branches of voltage sources and branches of current sources, respectively.

If we dene

C(^{u t}) := ^@q(^{u t})

@u q

0t(^{u t}) := ^@q(^{u t})

@t

L(^{j t}) := ^@(^{j t})

@j

0t(^{j t}) :=^@(^{j t})

@t

the DAE system we obtain from networks by the conventional MNA reads

AC^C(^A_TC^{e t})^A_TC^de

dt

+^AC^qt⁰(^A_TC^{e t}) +^AR^r(^A_TR^{e t})

+^AL^jL+^AV^jV +^AIⁱ() = 0 (4.1)

L(^jL ^t)^djL

dt

+⁰_t(^jL ^t)^;A_TL^e = 0 (4.2)

ATV^e;v() = 0^: (4.3) Later on we will also need^G(^{u t}) :=^@r_@u^(ut).

We rst analyze the network with respect to the conventional MNA and, af- terwards, extend the results to the systems obtained by charge-oriented MNA.

These systems are⁸:

AC^dq

dt

+^AR^r(^A_TR^{e t}) +^AL^jL+^AV^jV +^AIⁱ() = 0 (4.4)

d

dt

;ATL^e= 0 (4.5)

ATV^e;v() = 0 (4.6)

q;qC(^A_TC^{e t}) = 0 (4.7)

;L(^jL ^t) = 0^: (4.8)

7A detailed discussion on how we set up the system's equations can be found in 8] and 12].

8cf. again 8] and 12].

Analogously to 8] and 6], we suppose that the capacitance matrix^C(^A_TC^{e t}), inductance matrix ^L(^jL ^t), and conductance matrix ^G(^A_TR^{e t}) of all capaci- tances, inductances and resistances, respectively, are positive denite⁹.

We will also make use of the fact that the reduced incidence matrix (^AC^AL^AR^AV) has full row rank and that^AV has full column rank, because cutsets of current sources only and loops of voltage sources only are forbidden (cf. 18], 8]).

5 Index analysis and consistent initialization for circuit simulation

5.1 Some denitions and results

In this section we repeat some of the results 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 following denitions and results.

Denition 5.1

To characterize the topological properties of the network, we dene the projectors ^QC,^QV^;C, ^QV^;C, and ^QR^;CV onto ker^A_TC, ker^A_TV^QC, ker^Q_TC^AV, and ker^A_TR^QC^QV^;C, respectively.

Note that^QCRV :=^QC^QV^;C^QR^;CV is a projector onto ker(^AC^AR^AV)^T. The complementary projectors will be denoted by^P :=^I;Q, with the cor- responding subindices.

Denition 5.2

1. An L-I cutset is a cutset consisting of inductances and/or current sources only.

2. A C-V loop is a loop consisting of capacitances and voltage sources.

In 8] , 18] the following was shown to hold:

Lemma 5.3

1. If the network does not contain L-I cutsets, then ^QCRV = 0.

2. If the network does not contain C-V loops, then ^QV^;C= 0.

In this article, we suppose that the controlled sources that form part of the network full the conditions exposed below in the Tables 5.1 and 5.2.

Regarding equations (5.3), (5.5), and (5.7) from Table 5.2, the assumptions made for the controlled current sources imply that

QTCRV^AIⁱ((^AC^AV^AR)^{Te j}L ^jV ^t) =^Q_TCRV^AItⁱt (5.9)

9Of course, the same restriction on the positive de niteness of the conductance matrix from Corollary 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 voltage sources, no positive de niteness of the corresponding conductance matrix has to be assumed.

If we consider the element-related splitting of ^QV^;C, i. e.,

QV^;C= ( ^QV^;C)t

( ^QV^;C)contr:

then we can summarize the prerequisites we assume for the controlled volt- age sources as follows:

QTV^;C^v(^{ATe dq}(^A_TC^{e t})

dt

jL ^jV ^t) = ^Q_TV^;_C^vt(^t) (5.1)

v(^{ATe dq}(^A_TC^{e t})

dt

jL ^jV ^t) = ^v(^A_TC^{e j}L ^t) (5.2) for a suitable function^vand for a vector^vt(^t) that contains the functions of independent voltage sources and zeros instead of the functions of controlled voltage sources. Analogously as in 8] and 6], in the following we will drop the index *.

Table 5.1: Condition for controlled voltage sources

is always fullled. Thus, we generally assume that the controlled sources do not form part of the C-V loops or L-I cutsets.

To shorten denotations we write

i((^AC^AV^AR)^{Te j}L ^P_V;C^jV ^t) (5.10) when we do not distinguish between (5.4), (5.6), and (5.8).¹⁰

Lemma 5.4

The matrices

H

1(^A_TC^{e t}) := ^AC^C(^A_TC^{e t})^A_TC+^Q_TC^QC

H

2 := ^Q_TC^AV^ATV^QC+^Q_TV^;_C^QV^;C

H

3 := ^A_TV^QC^QTC^AV + ^Q_TV^;_C^Q_V^;_C

H

4(^A_TC^{e t}) := ^Q_TV^;_C^A_TV^H1;1(^A_TC^{e t})^AV^Q_V^;_C+ ^P_TV^;_C^P_V^;_C

H

5(^jL ^t) := ^Q_TCRV^AL^L;1(^jL ^t)^A_TL^QCRV +^P_TCRV^PCRV

H

6 := ^Q_TV^;_C^A_TV^AV^Q_V^;_C+ ^P_TV^;_C^P_V^;_C

H

7 := ^Q_TCRV^AL^ATL^QCRV +^P_TCRV^PCRV

are nonsingular.¹¹

10These assumptions can be transcribed into topological criteria analogously as it was done in 8]. The result would be similar, with the only dierence that now the branch potentials of branches that form part of L-I cutsets would not be allowed to control controlled current sources.

11cf. 8], 6].

For controlled current sources we suppose that at least one of the following characterizations holds:

(a)

QTCRV^AIⁱ(^{ATe dq}(^A_TC^{e t})

dt

jL ^jV ^t) = ^Q_TCRV^AItⁱt (5.3)

i(^{ATe dq}(^A_TC^{e t})

dt

jL ^jV ^t) = ⁱa(^A_TC^{e A}_TV^{e j}L ^t) (5.4) for a suitable functionⁱa.

(b)

QTC^AIb = 0 (5.5)

i(^{ATe dq}(^A_TC^{e t})

dt

jL ^jV ^t) = ⁱb((^AC^AV^AR)^{Te j}L ^P_V;C^jV ^t)(5.6) for a suitable functionⁱb.

(c)

QTV^;C^QTC^AIc = 0 (5.7)

i(^{ATe dq}(^A_TC^{e t})

dt

jL ^jV ^t) = ⁱc((^AC^AV^AR)^{Te j}L ^t) (5.8) for a suitable functionⁱc.

Table 5.2: Conditions for controlled current sources In 8] the following result was obtained:

Theorem 5.5

Consider lumped electric circuits satisfying the assumptions of the Tables 5.1 and 5.2. Then it holds:

1. If the network contains neither L-I cutsets nor controlled C-V loops, then the conventional MNA leads to a DAE system with index-1 and the con- straints are only the explicit ones:

QTC^AR^r(^A_TR^{e t}) +^AL^jL+^AV^jV

+^AIacⁱac((^AC^AR^AV)^{Te j}L ^t) = 0 (5.11)

ATV^e;v(^A_TC^{e j}L ^t) = 0^: (5.12) 2. If the network contains L-I cutsets or C-V loops, then the conventional MNA leads to a DAE system with index-2. With regard to the constraints, we distinguish the following three possibilities.

(a) If the network does not contain an L-I cutset (but contains controlled C-V loops), then the constraints are the explicit ones, (5.11) and

(5.12), and, additionally, the hidden constraint:

QTV^;C^ATV^H1;1(^A_TC^{e t})^P_TC

AC^qt⁰(^A_TC^{e t}) +^AR^r(^A_TR^{e t}) +^AL^jL

+^AV^jV +^AIⁱ((^AC^AR^AV)^{Te j}L ^P_V;C^jV ^t)

+ ^Q_TV^;_C^dvt

dt

= 0^:(5.13) (b) If the network does not contain controlled C-V loops, but contains

L-I cutsets, the constraints are the explicit ones, (5.11) and (5.12), and, additionally, the hidden constraint:

QTCRV ^AL^L;1(^jL ^t)^;A_TL^e;0_t(^jL ^t)+^AIt^dit

dt

= 0^: (5.14) (c) If the network contains L-I cutsets and C-V loops, then the con- straints are the explicit ones, (5.11) and (5.12), and the hidden ones (5.13) and (5.14).

Remark:

In 8] it was proved that the hidden constraint (5.13) resulted from

QTV^;C^ATV^dedt

= ^Q_TV^;_C^dvt

dt

(5.15) and that (5.14) arose from

QTCRV ^AL^djL

dt

+^AIt^dit

dt

= 0^: (5.16)

For the sake of simplicity, we will sometimes drop the arguments of the ^H matrices in the following and write a dot if they are not constant.

In 6] it was shown that a splitting of the system can be performed in such a way that consistent initial values can be calculated successively as described below.

Corollary 5.6

Let the values(^PC^{e0 j}L⁰) be given. If the network contains con- trolled sources that full the conditions of the Tables 5.1 and 5.2, we can deter- mine consistent initial values for the system (4.1)-(4.3) gradually.

We split ^e0 =^PC^e0+^QC^PV^;C^e0+^QC^QV^;C^PR^;CV^e0+^QC^QV^;C^QR^;CV^e0

and^jV⁰= ^QV^;C^jV⁰+ ^PV^;C^jV⁰, and obtain the corresponding consistent values from

PC^e0 := ^PC^e0+^PC^AV^Q_V^;_C^H6^;1Q_TV;C^;vt(^t0)^;A_TV^PC^e0

jL⁰ := ^j_L⁰+^A_TL^QCRV^H7;1QTCRV ^;;AItⁱt(^t0)^;AL^j0L

QC^PV^;C^e0 := ^QC^H2;1QTC^AV^P_TV;C^;;ATV^PC^e0+^v(^A_TC^{e0 j}L^{0 t0}) whereas the value of^QC^QV^;C^PR^;CV^e0 can be obtained by solving the equation

PTR^;CV^QTV^;C^QTC^AR^r(^A_TR(^PC+^QC^PV^;C+^QC^QV^;C^PCRV)^e0) +^AL^jL⁰+^AIaⁱa(^A_TC^{e0 A}_TV^{e0 j}L^{0 t0})

= 0^:

Then we calculate

PV^;C^jV⁰ := ^;H3;1A_TV^QC^PTV^;C^QTC^AR^r(^A_TR^{e0 t0}) +^AL^jL⁰+^AIa cⁱac((^AC^AV^AR)^{Te j}L ^t)

:

Next, we can determine the remaining values by means of

QCRV^e0 := ^;;QCRV^H5;1()^Q_TCRV

AL^L;1(^jL^{0 t0})^A_TL(^PC+^QC^PV^;C+^QC^QV^;C^PR^;CV)^e0

;AL^L;1(^jL^{0 t0})⁰_t(^jL^{0 t0}) +^AIt^dit

dt

(^t0)

QV^;C^jV⁰ := ^;H4;1() ^Q_TV^;_C^A_TV^H1;1()^P_TC ^AC^qt⁰(^A_TC^{e0 t0}) +^AR^r(^A_TR^{e0 t0}) +^AL^jL⁰+^AV^P_V^;_C^j_V0+^AIⁱ((^AC^AV^AR)^{Te0 j}L^{0 P}_V^;_C^j_V0

t

0)

;H

;1

4 () ^Q_TV^;_C^dvt

dt

(^t0)^:

If the charge-oriented MNA is considered, we set additionally:

q

0 := ^qC(^A_TC^{e0 t0})

0 := L(^jL^{0 t0})^:

Remarks:

Note that each time the matrices ^H1;1() = ^H1;1(^A_TC^{e0 t0}), ^H4;1() =

H

;1

4 (^A_TC^{e0 t0}) or ^H5;1() = ^H5;1(^jL^{0 t0}) appear, we already know the corresponding values^A_TC^e0or^jL⁰and, therefore, can insert them. On the other hand, the conditions of the Tables 5.1 and 5.2 imply precisely that this holds analogously for the controlled sources.¹².

Of course, if the network contains no C-V-loops and no L-I cutsets, then the corresponding equations dened with the aid of the projectors ^QV^;C

and^QCRV, respectively, do not appear.

Corollary 5.6 implies that the choice of (^{e0 j}_L^{0 j}_V⁰) is arbitrary as long as the nonlinear equation that leads to the expression for^QC^QV^;C^PR^;CV^e0

is solvable.

From the results presented in 8] it follows directly that, for the controlled sources we consider here, the space^N\S() is constant and can be described by

N\S() = im^QCRV ^f0^gim ^QV^;C

12Note that the controlled sources we permit do not change the spaces associated with the DAEs and, in this context, imply that we do not need to alter the order, as it was done in 6].

for the conventional MNA and

N\S() =^f0^gf0^gim^QCRV ^f0^gim ^QV^;C

for the charge-oriented MNA. Observe that ^QCRV^e and ^QV^;C^jV appear only in linear expressions of the equations (4.1) - (4.3) and (4.4) - (4.8).

5.2 Topological analysis of the network

Let us analyze the topology of a given circuit to locate the equations corre- sponding to (5.15) and (5.16), i. e., to the equations that lead to the hidden constraints¹³. In 6] it was shown that these equations can be obtained directly from the network by making use of the following two procedures. They pre- cisely determine the linearly independent equations that describe the hidden constraints arising from C-V loops and L-I cutsets, respectively.

PROCEDURE 1

1. Search a C-V loop in the given network graph. If no loop is found, then end.

2. Write the equation resulting from the sum of the derivatives of the charac- teristic equations of the voltage sources contained in the C-V loop, taking into account the orientation of the loop and the reference direction of the considered branches.

For instance, if the voltage sources^{v1 ::: v}k form a part of the C-V loop and we dene

i:=

+1 if the orientation of the loop coincides with that of^vi

;1 else,

then the equation we write in this step is^Pk_i⁼¹i((^A_TV^e)⁰_i^;v0_i) = 0.

3. Form a new network graph by deleting the branch of one voltage source that forms a part of the loop, leaving the nodes unchanged.

4. Return to 1, considering the new network graph.

The following procedure starts again from the initial graph.

PROCEDURE 2

1. Search an L-I cutset. If one is found, then pick an arbitrary inductance of this cutset. Realize that we can always nd such an inductance because cutsets of current sources only are forbidden. If no cutset is found, then end.

13A similar topological analysis of the network can be found in 3].

2. Write a new equation resulting by derivation of the cutset equation arising from 1.

For instance, if the current sources^{i1 ::: i}kand the inductances^jL^{1 ::: j}L^~k

form a part of the L-I cutset and we dene

j :=

+1 if the orientation of the cutset coincides with that ofⁱj

;1 else,

~

j :=

+1 if the orientation of the cutset coincides with that of^j_Lj

;1 else,

then the equation obtained in this step reads^Pk_j⁼¹jⁱ⁰j+^P~k_i⁼¹~j^jLi⁰ = 0.

3. Delete the chosen inductance from the network contracting its incident nodes.

4. Return to 1, considering the new network graph.

In 6], an extension of these procedures determines how to x the dynamic components for calculating consistent initial values. The result permits to assign specic values to a selection of variables, and to determine the remaining by making use of the system's equation¹⁴. Nevertheless, the values obtained in that way depended on the choice of variables selected.

6 Calculating a consistent value starting up from a value fullling the system's equations

The approach presented in this section distributes the index-2 property along all aected elements. It results from the approach presented in Section 3 and can be summarized as follows:

Theorem 6.1

For networks that contain only controlled sources as specied in the Tables 5.1 and 5.2 we obtain consistent initial values starting up from possibly inconsistent ones that full the system's equations in the following way:

1. Add additional currents that ow through the C-V-loops as a consequence of the hidden constraints described by PROCEDURE 1 to the values of the currents through the branches that form a part of C-V-loops.

2. Add convenient values to the node potentials to full the additional volt- age across the L-I cutsets dened by the hidden constraints described by PROCEDURE 2.

The meaning of this theorem becomes clear in the course of the article. The Theorems 6.5 and 6.6 describe the statement properly.

In this chapter, we rst give an overview of our aim, explaining the approach

14Therefore, this result is similar to the one obtained in 16].

for a special case, the DC operating point, and then we generalize the results to apply them for arbitrary points. The proofs will be pointed out for the general case only.

In Section 6.4 we illustrate why the class of controlled sources for which this approach holds cannot be extended if no further topological considerations are made.

6.1 An initialization related to the DC operating point

A common way for obtaining values to start the numerical integration in circuit simulation is to calculate the DC operating point. Therefore, we consider this point separately. We calculate the DC operating point by setting the current through the capacitances and the voltages across the inductances equal to zero.

Note that for calculating the DC operating point for the charge-oriented MNA, the matrix _dx^df has to be nonsingular.

Topologically, this implies that:

1. Cutsets of capacitances and/or current sources are forbidden, i.e., that (^AL ^AR ^AV) has full row rank.

2. Loops of inductances and/or voltage sources are forbidden, i.e., (^AL ^AV) has full column rank.

Furthermore, the non-singularity of _dx^df implies assumptions on the resistances and on the controlled sources.

Similar considerations hold for the conventional MNA, whereas^q_t⁰() and ⁰_t() have to be considered separately . Observe that, if time-dependent capacitors or inductors appear in the network, then the existence of the DC operating point and the non-singularity of the corresponding matrix _dx^df are not equivalent for the conventional MNA.

In the index-2 case the DC operating point has not to be consistent. Neverthe- less, the values obtained for the DC operating point, (^{e0 j}_L^{0 j}_V⁰), full Kircho's laws, and precisely the equations

AR^r(^A_TR^{e0 t0}) +^AL^jL⁰+^AV^jV⁰ +^AIⁱ(^PCRV^{e0 j}L^{0 P}_V;C^jV^{0 t0}) = 0(6.1)

;ATL^e0 = 0(6.2)

ATV^e0;v(^A_TC^{e0 j}L^{0 t0}) = 0(6.3) for the conventional MNA, and, additionally, the equations

q;qC(^A_TC^{e0 t0}) = 0 (6.4)

;L(^j_L^{0 t0}) = 0 (6.5)

if the charge-oriented MNA is considered.

Theorem 6.2

For the conventional MNA we obtain a consistent initialization (^{e0 j}L^{0 j}V^{0 P}C^{e00 j}L⁰⁰) related to the DC operating point (^{e0 j}_L^{0 j}_V⁰) in the fol- lowing way: