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Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-215$16.00 2001, Vol. 7, No. 2, pp. 215±223 Swets & Zeitlinger 1 2 C. TISCHENDORF and D. ESTEVEZ SCHWARZ ABSTRACT Circuit simulation is a standard task for the computer-aided design of electronic circuits. The transient analysis is well understood and realized in powerful simulation packages for conventional circuits. However, further developments in production engineering lead to new classes of circuits, which cause dif®culties for the numerical integration. The dimension of circuit models can be quite large (10 equations). The complexity of the models demands a higher level of abstraction. Parasitic effects become dominant. The signal to noise ratio becomes smaller. In this paper, we want to draw attention to three essential problems from a mathematical point of view, the DAE-index, consistent initial values, and asymptotic stability. These topics have been extensively analyzed only recently. We shall illustrate them by some simple examples. Keywords: circuit simulation, consistent initial values, differential-algebraic equations, index, stability, AMS classi®cation: 94C05, 65L80. 1 STRUCTURE OF CAD-BASED SYSTEMS FOR INTEGRATED CIRCUITS The modi®ed nodal analysis (MNA) is a widely used modeling technique which enables an automatic generation of the network equations under conservation of the circuit structure. It leads to differential algebraic equations (DAEs) of the form C x; tx _ f x; t 0; 1 where C x; t is a singular matrix and x consists of all nodal potentials and the currents through voltage de®ning elements. In case of the charge oriented MNA, Institute of Mathematics, Humboldt-University of Berlin, Seat: Rudower Chaussee 25, 10099 Berlin, Germany. In®neon Technologies, MP TI CS ATS, MCH B, Balanstr. 73, D-81541 Munich, Germany. Â 216 C. TISCHENDORF AND D. ESTEVEZ SCHWARZ the charges of capacitive elements as well as the ¯uxes of inductive elements are included additionally. A closer look onto the systems reveals special structural properties that can and should be exploited by numerical integrators. The model description is based on ®ve basic network elements. The static behavior is described by voltage sources, current sources and resistances. The dynamic behavior is re¯ected by capacitances and inductances. Splitting the incidence matrix A of the network elements into the element- related incidence matrices A A ; A ; A ; A ; A , where A , A , A , A , C L R V I C L R V and A describe the branch-current relations for capacitive branches, inductive branches, resistive branches, branches of voltage sources and branches of current sources, respectively, we obtain a system of the form dq A e; t C T T A A r A e; t A j A j A i A e; j ; j ; t 0; 2 C R L L V V I L V dt d j ; t ÿ A e 0; 3 dt T T A eÿ v A e; j ; j ; t 0; 4 L V where e represents the node potentials and j re¯ect the current vectors of L=V inductances / voltage sources. The functions q and describe the voltage- charge and current-¯ux relations for the dynamic elements. The controlling functions of current sources and voltage sources are given by i and v. 2 DAE INDEX OF THE NETWORK EQUATIONS Numerical methods like the Backward Difference Formulae (BDF) method can be applied to DAEs of the form (1) directly. They are often used successfully in simulation packages. However, they may fail. Investigations of general DAEs [1, 6, 9] indicate that this is usually the case if the DAE has a higher index. A detailed analysis [7, 8] of numerous examples shows that the index may be high and that it depends on different aspects: on the formulation of the network equation; on the kind of network elements used; on the structure of the circuit; on parameter values; on operating conditions of the circuit. MATHEMATICAL PROBLEMS IN CIRCUIT SIMULATION 217 Assuming all (possibly multi-port) capacitances, inductances and resistances to be positive de®nite as well as certain structural conditions for the controlled sources to be satis®ed, the following locally veri®able index characterization is possible : Theorem 2.1 The conventional MNA as well as the charge oriented MNA lead to an index-1 DAE if and only if the network contains neither L-I cutsets nor C-V loops. Otherwise, they provide an index-2 DAE. Remark An L-I cutset represents a cutset consisting of inductances and / or current sources only. A C-V loop describes a loop consisting of capacitances and voltage sources only. 3 CONSISTENT INITIAL VALUES The nonlinear equations in DAE systems represent constraints. Under suf®ciently smoothness conditions, DAEs can be considered as differential equations on manifolds. This implies that initial values must belong to a certain manifold, i.e., they have to be consistent. Providing consistent initial values in practice is a nontrivial task, but very important for the reliability of the simulation results as the following example shows. Formulating the network equations for the circuit given in Figure 1 by the modi®ed nodal analysis (MNA), we obtain q e 2 sin t 4 j ÿ sin tÿ 2 0; 5 1 V q ÿ j 0; 6 e ÿ e 2 sin t; 7 1 2 q e ; 8 q e ; 9 2 2 where q and q describe the charges of the capacitances C and C , 1 2 1 2 respectively. The variables e and e represent the node potentials at the 1 2 nodes 1 and 2. Finally, j re¯ects the current of the voltage source. If we integrate this circuit with the trapezoidal rule and start from an incon- sistent initial value, we obtain a completely wrong numerical solution (see Fig. 2). Â 218 C. TISCHENDORF AND D. ESTEVEZ SCHWARZ Fig. 1. Circuit containing a controlled current source that is controlled by the current of a sinusoidal voltage source between node 1 and node 2. Fig. 2. Result for the trapezoidal rule for (5)±(9) starting from the inconsistent value (4, 2, 2, 2, 500). If we start from a consistent value, the numerical solution coincides with the exact solution. One can observe a similar behavior for other methods, e.g., the implicit midpoint rule. Therefore, it is of great interest to have consistent initial values for the integration. In , a cheap algorithm for calculating MATHEMATICAL PROBLEMS IN CIRCUIT SIMULATION 219 consistent initial values (using operating points) is presented. It exploits the special circuit structure. It has to be mentioned that a numerical integration, starting from inconsistent initial values, can also be successful, depending on the method we use. In fact, the implicit Euler method provides acceptable results for the above example if we start from the inconsistent initial value . 4 ASYMPTOTIC STABILITY Numerical methods for DAEs do not always have those stability properties that are well-known for regular ODEs. We want to illustrate this by the following simple (theoretically constructed) example. The network equations for the circuit given in Figure 3 by MNA are given by the linear time-varying DAE q _ G te j 0; 10 1 1 1 V q _ G te i j ; t 0; 11 2 2 2 V e v e ; t; 12 1 2 q C e ; 13 1 1 1 q C e : 14 2 2 2 Here, e and e describe the voltages of the nodes 1 and 2 with respect to the 1 2 datum node. The variable j re¯ects the branch current of the voltage source. The variables q and q represent the charges of the capacitances C and C . 1 2 1 2 For simplicity, the capacities are assumed to be 1, that means, C C 1. 1 2 The resistances G and G are time-varying and have the conductances 1 2 G t 1ÿ tÿ ; G t ÿ ÿ t; 1 2 1ÿ t depending on the parameters and . The voltage source is assumed to be a voltage controlled, time-varying one: v e ; tÿ e : 2 2 1ÿ t Finally, the controlled current source depends also on the parameter and is given by i j ; t tj : V V Â 220 C. TISCHENDORF AND D. ESTEVEZ SCHWARZ Fig. 3. Circuit containing a controlled current source that is controlled by the current of a controlled voltage source. Then, the exact solution for this example reads q t e t e t exp t; 15 1 1 1 0 q t e tÿ 1ÿ te t exp t; 16 2 2 1 0 j tÿ 1ÿ te t exp t: 17 V 1 0 This implies that the solution is asymptotically stable for < 0. If we apply the implicit Euler method with a stepsize h 0:02 to the circuit (with ÿ5 and ÿ20), then we obtain, as expected, an asymptotically stable numerical solution (see Fig. 4). Contrarily, for a slightly bigger stepsize h 0:06, the numerical solution is unstable ( Fig. 5). This is an unknown behavior for A-stable methods for regular ODEs and has to be considered when integrating DAEs by standard numerical methods. Only recently, it has been shown that this instability effect does not occur if a DAE satis®es certain structural conditions . However, these structural conditions are not given, in general, for systems resulting from the charge- oriented MNA. But assuming certain modeling criteria (see [2, 3, 4]), the MATHEMATICAL PROBLEMS IN CIRCUIT SIMULATION 221 Fig. 4. Exact and numerical solution (implicit Euler h 0.02) for the system (10)±(14). Â 222 C. TISCHENDORF AND D. ESTEVEZ SCHWARZ Fig. 5. Exact and numerical solution (implicit Euler with h 0.06) for the system (10)±(14). MATHEMATICAL PROBLEMS IN CIRCUIT SIMULATION 223 charge-oriented MNA provides DAE systems with slightly different structural properties, which exclude instability effects as described above. 5 CONCLUSIONS The reliability of the results of numerical integration of general DAEs depends strongly on the structural properties of the DAE. For the equations obtained by Modi®ed Nodal Analysis, the relation between the topology of the network and the structural properties of the equations provides the possibility to diagnose the critical parts of the circuit . However, the structural analysis of circuits containing VHDL-AMS ele- ments and / or extended semiconductor models and the extension to stochastic DAEs are still a current matter of research. REFERENCES 1. Brenan, K.E., Campbell, S.L. and Petzold, L.R.: The Numerical Solution of Initial Value Problems in Ordinary Differential-Algebraic Equations. North Holland Publishing Co., 1989. 2. Estevez Schwarz, D.: Consistent Initialization for Differential-algebraic Equations and its Application to Circuit Simulation. Fachbereich Mathematik, Humboldt-Univ. zu Berlin, PhD Thesis, 2000. Available at: http://dochost.rz.hu-berlin.de/dissertationen. 3. Este Âvez Schwarz, D., Rodrõ Âguez Santiesteban, A. and Tischendorf, C.: Asymptotic Stability in Circuit Simulation., in preparation. 4. Este Âvez Schwarz, D. and Tischendorf, C.: Structural Analysis of Electric Circuits and Consequences for MNA. Int. J. Circ. Theor. Appl. 28 (2000), pp. 131±162. 5. Estevez Schwarz, D., Feldmann, U., Ma Èrz, R., Sturtzel, S. and Tischendorf, C.: Finding Bene®cial DAE Strucures in circuit simulation. Fachbereich Mathematik, Humboldt-Univ. zu Berlin, preprint 00-7, 2000. 6. Griepentrog, E. and Ma Èrz, R.: Differential-Algebraic Equations and Their Numerical Treatment. BSB B.G. Teubner Verlagsgesellschaft, Leipzig, Teubner-Texte zur Mathematik No. 88, 1986. 7. Gunther, M. and Feldmann, U.: CAD Based Electric Modeling in Industry. Part I: Mathematical Structure and Index of Network Equations. Surv. Math. Ind. 8 (1999), pp. 97±129. 8. Gu È nther, M. and Feldmann, U.: CAD Based Electric Modeling in Industry. Part II: Impact of Circuit Con®gurations and Parameters Mathematical Structure and Index of Network Equations. Surv. Math. Ind. 8 (1999), pp. 131±157. 9. Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, Heidelberg, 1991. 10. Rodrõ Âguez Santiesteban, A. and Ma Èrz, R.: Analyzing the Stability Behaviour of DAE Solutions and Their Approximations. Fachbereich Mathematik, Humboldt-Univ. zu Berlin, preprint 99±2, 1999.
Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis
Published: Jun 1, 2001
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