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Section 14-23 S 1103 TISCHENDORF, C. Topological index-calculat ion of DAEs in circuit simulation In electric circuit simulation we are confronted with highly nonlinear DAEs with low smoothness properties. They may have index 2 but they do not belong to the class of Hessenberg form systems that are well understood. The classic and the charge-oriented modified analysis are shown to lead to the same DAE-index if the circuit models satisfy some natural assumptions. W e present a topological criteria f o r calculating the index. This makes it possible to determine the index also for high-dimensional circuit equation systems. 1. M a t h e m a t i c a l network modelling The development of integrated circuits requires tests in order to see whether the circuits have the desired properties. The behaviour of circuits presently used and developped is influenced by a number of parasitic terms like scattered capacitances and lines. Therefore, experimental measurements are not accurate enough and the simulation of electronic circuits gets an important role. The circuit elements are described by mathematical models, numerical methods replace typical measurements like the determination of time behaviour. A number of frequently used simulation programs base on the modified node analysis (MNA), e.g. SPICE and TITAN. One reason for that is the possibility t o generate the network equations automatically. We studied the equation systems obtained by this simulation technique in more detail. Our results are restricted t o lumped circuits satisfying the following assumption: All circuit elements of the circuits are modelled as a combination of the following network elements: resistances, inductances, capacitances, independent voltage sources, independent current sources and voltage-controlled current sources. The resistances, inductances and capacitances may be linear or nonlinear but passive and time-invariant. Furthermore, if circuit elements are modelled with help of voltage-controlled current sources then all branch voltages controlling the sources must be defined by a capacitance. Note, this assumption is less a restriction to real circuits but more a restriction to the models describing the circuit. Example 1. The circuit represents a model for a field effect transistor with the terminals Gate (G), Source (S) and Drain (0). The current source between node s and node d is controlled by the branch voltages U G d , U G ~ a nd '1Lds. Obviously, the branch voltages U G d and U G ~ a re defined by the capacitances C G ~ and C G ~ . Additionally, the branch voltage U d s is defined by the capacitance CGd + C G ~ . The diodes represent nonlinear time-invariant resistors. Hence, the assumption given above is satisfied. 0 2. DAE index of the network equations Formulating the network equations by the modified nodal analysis (MNA) we obtain a quasilinear DAE A ( x ( t ) ) 4 t ) + g ( x ( t ) ) = r ( t ) . (1) The vector x contains the node potentials e (without the datum node), the branch currents ir, of inductances and the branch currents iv of the voltage sources. The equation system (1) consist of node equations (obtained by the Kirchhoffs nodal law), the voltage-current relations for the inductances and the voltage-time relations for the voltage
s 1104 ZAMM . Z. Anrrew. Math. Mech. 78 (1998) S3 sources. In more detail, system (1) has the following structure: AcC(e)Aze + A c b ( A g e ) + ARij(Aze) + A L ~ L + Avzv = -A sjs(.), L ( i L ) a L - A L e T = 0, AFe = vs(.). Here, A c , AL, AR, Av and A, describe the (reduced) incidence matrices for the capacitive branches, the inductive branches, the resistant branches, the voltage source branches and the current source branches, respectively. There- fore, AEe, A z e , A',e and A;e describe the branch voltages for the capacitances, the inductances, the resistances and the voltage sources, respectively. The matrices C ( e ) and L(iL) reflect the capacitance and the inductance matrix, respectively. The voltage-current relation for the resistances is given by the voltage-dependent function i j . The voltage-current relation for the voltage controlled current sources is represented by the voltage-dependent function b. The time-dependent functions jS(.) a nd u s ( , ) r epresent the waveforms of current and voltage sources, respectively. Note, the matrices C(e), L(iL) and $,(y) are positive definit, since capacitances, inductances and resistances are assumed to be passive. This implies the following Lemma. L e m m a 1. The nullspace as well as the image space of A(x) of the system (1) ,are constant. An adapted charge oriented formulation of the system (1) is given by the following system Acq + A c b ( A g e ) + ARB(A;e) + A L ~ L + Avzv = -ASjs(.), Ace = us(.), 4- A z e = 0, 4 = J,(A%), $ = J~(ZL). The charge-waveform f, and the flux-waveform f b satisfy the relations The assumptions t o the model given above guarantee the following theorem which makes it possible t o determine the index for circuit-DAEs by the topology of the network. T h e o r e m 1. The classical MNA (system (2)-(4)) and the charge oriented MNA (system (5)-(9)) lead to DAEs of the same index. The DAEs are of index 2 if a n d only if the network contains a loop of capacitances and voltage sources or a cutset of inductances and/or current sources. Otherwise, the DAEs have index 1. Acknowledgements The author thanks to Prof. A . Reibiger from the Technical University in Dresden and Dr. R . Lamour f r om the Humholdt- University of Berlin for helpful discussions in circuit theory. Particularly, I want to appreciate the encouragement given by/ Dr. U. Feldrnann from the SIEMENS A G in Munich. 3. References 1 BIJLIRSCH, R. A N D GILG, A .: Effiziente numerische Verfahren fur die Simulation elektrischer Schaltungen; In H. Schwiirtzel, 2 C H U A , L .O., DESOER, C H . A . A N D KUH, €3.: Linear and nonlinear circuits. McGraw-Hill Book Company, 1987. 3 DESOER, C H.A. AND K U H , E .: Basic circuit theory; McGraw-Hill Book Company, Singapore, 1969 4 GUNTHER, M ., DENK, G . A N D FELDMANN, U.: H ow models for MOS transistors reflect charge distribution effects; 5 R. MARZ A ND c. TISCHENDORF: Recent results in solving index 2 differential algebraic equations in circuit simulatioik; 6 TISCHENDORF, c.: S olution of index-2 differential algebraic equations and its application in circuit simulation; Berlin, editor; Informatik in der Praxis; Springer Verlag 1986, 3-2. Preprint 1745, T H-Darmstadt 1975. SIAM J. Sci. Stat. Co mput., 18 No. 1 (1997) 139-159 Humboldt-Univ., Diss. 1996, appeared in Logos-Verlag 1996. Addresses: DR. CAREN T lSCHENDORF,Humboldt-Universitat z u Berlin, Math-Nat. Fak. 11, Institut fur Angewandte Mathematik, Unter den Linden 6, 10099 Berlin, Germany.