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A PDAE Model for Interconnected Linear RLC Networks

A PDAE Model for Interconnected Linear RLC Networks Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-189$16.00 2001, Vol. 7, No. 2, pp. 189±203 Swets & Zeitlinger A PDAE Model for Interconnected Linear RLC Networks M. GUNTHER ABSTRACT In electrical circuit simulation, a re®ned generalized network approach is used to describe secondary and parasitic effects of interconnected networks. Restricting our investigations to linear RLC circuits, this ansatz yields linear initial-boundary value problems of mixed partial- differential and differential-algebraic equations, so-called PDAE systems. If the network ful®ls some topological conditions, this system is well-posed and has perturbation index 1 only: the solution of a slightly perturbed system does not depend on derivatives of the perturbations. As method-of-lines applications are often used to embed PDAE models into time-domain network analysis packages, it is reasonable to demand that the analytical properties of the approximate DAE system obtained after semidiscretization are consistent with the original PDAE system. Especially, both should show the same sensitivity with respect to initial and boundary data. We will learn, however, that semidiscretization may act like a deregularization of an index-1 PDAE model, if an inappropriate type of semidiscretization is used. Keywords: re®ned modeling, generalized network models, differential-algebraic equations (DAEs), partial differential equations (PDEs), partial differential-algebraic equations (PDAEs), a-priori estimates, perturbation index, method-of-lines (MOL), approximate DAE systems (ADAEs). 1 INTRODUCTION In network simulation packages, real circuit elements and interconnections are commonly replaced by companion models of ideal and compact network elements, whose properties are determined uniquely by ®xing electrical para- È È Fakultatfu È r Mathematik und Wirtschaftswissenschaften, Universitat Ulm, D-89081 Ulm, Germany. ± Permanent address: Universita Èt Karlsruhe (TH), Institut fu È r Wissenschaftliches Rechnen und Mathmatische Modellbildung (IWRMM), Engesserstr.6, D-76128 Karlsruhe, Germany. È 190 M. GUNTHER meters like capacitances or inductances. This yields a unique modeling approach, which allows for including parasitic and second order effects into the differential-algebraic (DAE) network approach. Examples are transistor models which approximate the physical behavior of semiconductor devices by companion models of different modeling levels, or transmission line models, which consist of RLC elements and controlled sources. Mathematically, this approach corresponds to a spatial discretization of the governing partial differential equations (PDEs) already at the modeling level. Another short-coming is the frequent use of arbitrary coupled sources that may destroy the structure of the network equations and thus lead to high-index systems [5, 9]. As an alternative, the co-simulation approach makes use of already existing simulation software for single parts of the system: different parts of the systems are modeled independently of each other and simulated by two simulation packages for electrical networks and electromagnetic ®elds; coupling is ensured by coupling the simulators. In addition to convergence problems, dif®culties may arise, since coupled systems are often characterized by very different time constants. A third approach is the use of generalized network models [6]. Re®ned models are allowed for interconnects and semiconductor devices, whose characteristic equations de®ne PDE models. Hence numerical methods can be tailored exactly to the resulting mathematical models ± the spatial discretiza- tion is not yet made at modeling level. Mathematically spoken, this approach leads to a coupled system of DAEs and PDEs, with the boundary conditions for the PDEs linked to the DAEs at the boundary nodes. Such systems are called partial differential-algebraic equations, shortly PDAE systems. In this paper, we will concentrate on the last approach for linear RLC networks. This ansatz yields linear initial-boundary value problems of mixed partial-differential and differential-algebraic equations. The analysis of such PDAE systems and their numerical discretization is in the focus of actual research: one aim is to generalize the DAE index concept to PDAE systems to get some knowledge on structural properties before discretization [4, 7, 8, 10]: for example, the sensitivity of the solution to small perturbations in the initial data and / or input signals. On the other hand, estimates are required for the impact of semidiscretization on the index of the resulting approximate DAE (ADAE) system: does the ADAE system properly re¯ect the behavior of the original PDAE system? Or does one detect an arti®cial smoothing effect? Or even a coarsening one? PDAE MODEL FOR RLC NETWORKS 191 It is already known that the last but one question has to be answered in the af®rmative. Semidiscretization may act like a regularization [1, 3]: the ADAE system is less sensitive w.r.t. input data than the PDAE model, and may yield physically incorrect solutions. In this contribution we will af®rm the last question, too. The paper is organized as follows. We ®rst derive the generalized network approach for interconnected linear RLC networks and discuss the impact of topological index-1 conditions on the structure of the DAE part. In the analysis of the PDAE model in Section 3 we ®rst prove uniqueness and derive a-priori estimates of the solution to establish well-posedness of the system. We see that the solution of a slightly perturbed system does not depend on derivatives of the perturbations ± the PDAE has perturbation index 1. These properties should be re¯ected by the approximate DAE system obtained after semidiscretization. We will learn, however, in Section 4 that semidiscretiza- tion may act like a deregularization of an index-1 PDAE model, if an inappropriate type of semidiscretization is used. 2 GENERALIZED NETWORK APPROACH In the following, we consider two electrical networks which are coupled by a system of d uniform lossy transmission lines. To derive a mathematical model, we use a generalized network approach: the electrical circuits are described by DAE models, whereas the transmission lines shown in Figure 1 are governed by a PDE model. Both models are linked via boundary node voltages and currents. Fig. 1. PDE-network model for a system of d uniform lossy transmission lines. È 192 M. GUNTHER 2.1 PDE Network Model for Transmission Lines Assuming quasi stationary behavior transverse to the wave propagation, the signal propagation in the transmission lines can be characterized by the telegrapher's equation 0 ˆ V …z; t†‡ LJ …z; t†‡ RJ…z; t†; (1a) z t 0 ˆ J …z; t†‡ CV …z; t†‡ GV…z; t†; (1b) z t dd where R; L; G and C 2 R are the positive-de®nite symmetric resistance, inductance, conductance and capacitance matrices per unit length. V…z; t† is a d-dimensional vector of line voltages with respect to ground, and J…z; t† is a d-dimensional vector of line currents. For V; J2H :ˆ H …0; l† the telegrapher's equation (1) holds in the sense of distributions inV :ˆ L …0; l† . This ®rst order hyperbolic system of partial differential equations is initialized by a set of initial values V…z; t †ˆ V …z†; …2a† J…z; t †ˆ J …z†; …2b† for all z 2 I :ˆ‰0; lŠ at initial time point t . After introducing 2d virtual current sources k :ˆ… ; .. .; † and 1 1;1 1; d k :ˆ… ; .. .; † at the boundaries, the characteristic equation for the 2 2;1 2;d PDE model of a lossy transmission line system in admittance form reads J…0; t† k k ˆ with k ˆ ; …3† ÿJ…l; t† k where the line currents J are de®ned by the telegrapher's equation (1). The PDE model is completed by the boundary conditions V…0; t† u ˆ ˆ: A u; (4) V…l; t† u which couples the PDE network model with the node potentials u of the DAE model for both electrical networks via an incidence matrix A . 2.2 DAE Network Model for Linear Electrical Circuits A network model is used to describe the electrical behavior of the circuit: network equations for node potentials are derived using Kirchhoff's laws and characteristic equations for the elements. This results in a system of differen- PDAE MODEL FOR RLC NETWORKS 193 tial-algebraic equations since only topology and no spatial dimension is considered. Using classical Modi®ed Nodal Analysis (MNA), only node potentials u, currents through inductive and resistive branches and , and L V currents k at the boundaries of interconnects are unknowns. The DAE network equations with x :ˆ…u; ; † read for a linear RLC network consisting of L V only linear capacitors, inductors and resistors, as well as independent voltage and current sources, 0 1 0 1 T > A CA 00 A GA A A C R L V B C B C x _ ‡ x @ A @ ÿA 00 A 0 L 0 00 0 ÿA 00 0 1 0 1 A A …t† B C B C ‡ 0 k‡ 0 ˆ 0 …5† @ A @ A 0 v v v…t† with consistent initial values x…t †ˆ x : …6† 0 0 The element-related incidence matrices A , A , A , A , A and A describe C L R V I the branch-current relations for capacitive, inductive, resistive branches and branches for voltage sources, current sources and transmission line elements. e e e The capacitance, inductance and conductance matrices C, L and G are assumed to be positive-de®nite and symmetric [9]. 2.3 Topological Index-1 Conditions With Q projecting onto A and P :ˆ I ÿ Q , the network variables x can be c c split into the differential part y :ˆ…P u; † and the algebraic components c L z :ˆ…Q u; † . Correspondingly, the network equations read after pure algebraic transformations > > e e 0 ˆ H y _ ‡ P A GA … y ‡ z †‡ A y ‡ A z ‡ A …t†‡ A k 1 R 1 L V 2 I 1 1 2 c R 0 ˆ L ÿ A … y ‡ z † L 1 L 1 ! ! > > > > > > e e z y Q A GA Q Q A Q A GA Q A R V R L 1 c R c c c R c 0 ˆ ‡ > > z y A Q 0 2 A 0 V V Q …A …t†‡ A k† ÿv v v…t† È 194 M. GUNTHER > > e e with H :ˆ A CA ‡ Q Q positive-de®nite and symmetric. The topolog- 1 C C C c ical index-1 conditions T1 ker…A ; A ; A † ˆf0g; C R V T2 ker Q A ˆf0g guarantee > > > Q A GA Q Q A R V c R c ker ˆ ker Q f0g: > c A Q 0 Thus z…t† is given as a linear function in y…t†, …t†; v v v…t† and k…t†. If there is a capacitive path to ground for all coupling nodes, the condition > > T3 Q A ˆ 0 …ker P A ˆf0g† c c holds, and z…t† is given as a linear function in y…t†, …t† and u…t† alone. Assuming that T1±T3 holds, the equations (1±6) de®ne a uniquely solvable mixed initial-boundary value problem of PDEs and DAEs, or partial differential-algebraic equations (PDAEs), of perturbation index 1. This will be shown in the next section. 3 ANALYTICAL PROPERTIES OF THE PDAE MODEL To derive the analytical properties of the PDAE (1±6), we proceed in three steps: First, we prove uniqueness in an L -sense independent of the network topology by using energy estimates. Based on these technique, we are able to derive a-priori estimates for the solution depending only on the model data, i.e., initial values, input sources and matrices of the linear PDE and DAE models. If the topological index-1 conditions hold, these two ingredients establish well-posedness of the PDAE system. In a last step, we investigate the sensitivity of the model with respect to small perturbations. 3.1 Uniqueness Let DV; DJ; Du; D ; D be the difference of two (suf®ciently smooth) L V 0 0 0 solutions of the PDAE (1±6) for the same data, i.e., initial values V ; J ; x , e e input sources …t†; v v v…t†, and matrices L; C; R; G of the linear PDE and L; C; G PDAE MODEL FOR RLC NETWORKS 195 of the DAE model. Considering the inner product of (1a) and (1b) with D J and DV, resp., inV , one obtains after integration by parts due to C, L symmetrical and positive-de®nite n o 1 d 2 2 1=2 1=2 0 ˆ kC DV…; t†k ‡kL D J…; t†k V V 2 dt > > ‡hi GDV…; t†; DV…; t† ‡hi RDJ…; t†; DJ…; t† ÿ…A Du† Dk; …7† whereh;i denotes the inner product inV andkk the associated norm. The network equations (5) yield for the last term > > > > > e e e ÿ…A Du† Dk ˆ Du A CA D u‡ A GA Du†‡ D LD j : …8† C R C R L e e e As the matrices C, L and G are symmetrical and positive-de®nite, we get after inserting (8) into (7) and integration with respect to time due to R; G positive- semide®nite the estimate 2 2 2 1=2 1=2 1=2 kC DV…; t †k ‡kL DJ…; t †k ‡kL D …t †k e e L e V V 2 2 2 1=2 > 1=2 > e e ‡kC A Du…t †k ‡ kG A Du…t†k dt  0; …9† C 2 R 2 which is equivalent to > > DV  D J  0; D  0; A Du  0; A Du  0: C R > > in the L -sense. Since A Du  Dv v v…t† 0 and A Du  L D  0 hold, V L Du  0 follows from the fact that a physically reasonable network must not contain cut sets of current sources: …A ; A ; A ; A † ˆf0g. Finally, the C R V L identity A D  0 in (5) together with the topological condition A ˆf0g V V V (no loops of only voltage sources) yields D ˆ 0. Thus uniqueness is established independent of the network topology whether the topological conditions T1±T3 hold or not. 3.2 A-Priori Estimates Analogous to (7±8) we have for a solution of the PDAE (1±6) n o 1 d > > > e e 0 ˆ hi CV…; t†; V…; t† ‡hi LJ…; t†; J…; t† ‡ u A CA u‡ L C L C L 2 dt > > ‡hi GV…; t†; V…; t† ‡hi RJ…; t†; J…; t† ‡ u A GA u > > ‡ u A …t†‡ v v v …t† : I V È 196 M. GUNTHER e e e The symmetry of C, L, G, C and L yields the energy estimate 2 2 2 …t† …t †‡ c …†‡k z…†k d ‡k k ‡kv v vk 2 2 0 1 2 L …t ;t† L …t ;t† 0 0 …10† in the differential components V; J and y with 2 2 2 …t† :ˆkV…; t†k ‡kJ…; t†k ‡ky…t†k ; V V 2 s s Z Z t t 2 2 k k 2 :ˆ k …†k d; kv v vk 2 :ˆ kv v v…†k d L …t ;t† 2 L …t ;t† 2 0 0 t t 0 0 and the constant no ÿ1 ÿ1 ÿ1 ÿ1 e e c ˆ 2 max C ; L ; H ; L 2 2 2 2 1 1 max kGk ;kRk ;kGA k ; kA k ; ; 2 2 R 2 2 2 2 depending on the matrices of the linear models. Using the topological conditions T1±T3, the algebraic components z are given as a linear function in 2 2 2 2 y, and v v v. Thus we have with kz…t†k  c …ky…t†k ‡k …t†k ‡kv v v…t†k † and 2 2 2 2 c :ˆ c …1‡ c † 3 1 2 2 2 …t† …t †‡ c …† d ‡k k ‡kv v vk : 0 3 2 2 L …t ;t† L …t ;t† 0 0 By Gronwall's lemma this yields the a-priori estimate 2 2 …t† exp…c t† …t †‡ c k k 2 ‡kv v vk 2 ; …11† 3 0 3 L …t ;t† L …t ;t† 0 0 and ®nally for the algebraic components 2 2 2 kz…t†k  c …t†‡k …t†k ‡kv v v…t†k : …12† 2 2 2 If the topological conditions T1±T3 hold, the PDAE initial-boundary value problem (1±6) is well posed. The solution depends continuously on the data, i.e., on the initial values, on the input signals and via the constants c ; c on the 1 2 matrices of the linear models. In the DAE sense it is well conditioned, too, as only data enter the solution, but not its derivatives. PDAE MODEL FOR RLC NETWORKS 197 Based on the a-priori estimates, one can show the existence of a unique solution for the PDAE (1±6): Under the assumptions of T1±T3 the PDAE system (1±6) has a unique solution …V; J; u; ; ;†,if L V n n 0 0 1  1 v V ; J 2H; 2 H …t ; t † ; v v v 2 H …t ; t † 0 e 0 e holds for the initial values and input signals. Both V; J and V ; J are bounded t t on the ®nite time interval ‰t ; t Š with 0 e V; J2H; V ; J 2V: t t For the network variables one gets n n n 1 u 1 | 1 | L V u 2 H …t ; t † ; 2 H …t ; t † ; 2 H …t ; t † : 0 e L 0 e V 0 e 3.3 Perturbation Analysis Let …V; J; x† be the (reference) solution of the PDAE system (1±6) with 0 0 consistent initial values V ; J and x . To investigate the sensitivity of this solution, we apply perturbations d…z; t† and l…t† to the right-hand sides of (1) b b and (5). The corresponding perturbed solution is denoted by …V; J; x† with 0 0 b b consistent perturbed initial values V ; J and x . The aim is to obtain estimates for the sensitivity of the solution, i.e., for the difference b b …DV; D J; Dx† :ˆ…V; J; x†ÿ…V; J; x† between both solutions. Due to the linearity of the systems, we get an a-priori energy estimate for 2 2 2 e…t† :ˆkDV…; t†k ‡kD J…; t†k ‡kDy…t†k ; V V 2 and kDe z…t†k of the type (11, 12): e…t† exp…e c t† e…t †‡e c kk ‡kk ; …13a† 2 2 3 0 3 L …‰t ; tŠ;V† L …t ; t† 0 0 2 2 2 kDe z…t†k  c …e…t†‡k…; t†k ‡k…t†k†…13b† 2 V 2 with the norm kk :ˆ kd…;†k d L …‰t ; tŠ;V† 0 È 198 M. GUNTHER and the constants no ÿ1 ÿ1 ÿ1 ÿ1 e e e c ˆ max C ; L ; H ; L 2 2 2 2 no max 2kGk ‡ 1; 2kRk ‡ 1; 2kGA k ‡ 1; 1 2 2 R 2 and ce :ˆ e c …1‡ c †. 3 1 2 Thus the norm of the difference of reference and perturbed solution can be estimated by the norm of the perturbations itself, and does not depend on derivatives of the perturbations, neither with respect to space nor time. Hence the PDAE system (1±6) has perturbation index 1. For a precise de®nition of the perturbation index of linear PDAE systems of hyperbolic-type (1±6) we refer to [7, 10]. 4 IMPACT OF SEMIDISCRETIZATION: PERTURBATION INDEX OF PDAE AND ADAE Summing up the results of Section 3, the PDAE model (1±6) meeting the topological conditions T1±T3 is well-posed and has perturbation index 1. This analytical behavior should be re¯ected by any type of numerical discretiza- tion. As the numerical simulation of interconnected electrical networks has to be embedded into time-domain network analysis packages that are based on a DAE description of systems in time alone, the method-of-lines (MOL) approach is a natural candidate for numerical discretization of the PDAE model: ®rst semidiscretization of the PDAE model with respect to space, and secondly numerical integration of the originating approximate system of differential-algebraic equations (ADAEs) in time. To re¯ect the analytical properties of the PDAE properly, the equality of PDAE and ADAE perturbation index de®nes an inalienable demand on the respective type of semidiscretization. However, as we will see in the following, this demand will not always be met. As an example, we will use linear ®nite elements that are based on two analytically equivalent mixed weak / strong formulation. 4.1 Mixed Weak / Strong Formulation Under the regularity assumptions 1 x V; J2H; V ; J 2V; x 2 H …t ; t † t t 0 e PDAE MODEL FOR RLC NETWORKS 199 of Section 3.2 the strong formulation (1) is equivalent to the weak formulation w w w LJ …; t†‡ V …; t†‡ RJ…; t† t z ; w w w ˆ 0 8 w w w ˆ 2HH : CV …; t†‡ J …; t†‡ GV…; t† w w w t z By integrating the term V …; t† ; w w w J …; t† twice by parts and formulating the boundary conditions (4) weakly for the node voltages, the PDAE system (1±6) can be transformed into the mixed weak / strong formulation LJ …; t†‡ V …; t†‡ RJ…; t† t z ; w w w CV …; t†‡ J …; t†‡ GV…; t† t z …14a† V…0; t† ÿw w w …0† ‡ A uÿ ˆ 0 8w w w 2HH; V…l; t† w w w …l† J…0; t† kÿ ˆ 0 …14b† ÿJ…l; t† 0 1 0 1 A CA 00 A GA A A R L V C R B C B C e _ @ Ax‡ @ Ax ÿA 00 0 L 0 ÿA 00 00 0 0 1 0 1 A A …t† J…0; t† B C B C ‡ 0 ‡ 0 ˆ 0: …14c† @ A @ A ÿJ…l; t† 0 v v v…t† One notes that the boundary condition (3) for the algebraic component k is formulated in a strong sense. Correspondingly, if the coupling condition (3) for the coupling currents is formulated weakly, we get the system LJ …; t†‡ V …; t†‡ RJ…; t† t z ; w w w CV …; t†‡ J …; t†‡ GV…; t† t z …15a† J…0; t† w w w …0† ÿ kÿ ˆ 0 8w w w 2HH; ÿJ…l; t† w w w …l† 2 È 200 M. GUNTHER V…0; t† A uÿ ˆ 0 …15b† V…l; t† 0 1 0 1 A CA 00 A GA A A R L V C R B C B C e _ @ Ax‡ @ Ax ÿA 00 0 L 0 ÿA 00 00 0 …15c† 0 1 0 1 A A …t† B C B C ‡ @ 0 Ak‡ @ 0 A ˆ 0: 0 v v v…t† Now the boundary condition (4) for the differential components A u is formulated in a strong sense. These two analytically equivalent formulations may form the basis for a Ritz±Galerkin approach with linear ®nite elements. 4.2 Semidiscretization If we apply semidiscretization with respect to space, we seek for an approxi- mate solution V ; J ; x ; k of (14) and (15) resp., with m m m m J …z; t† ˆ …t† w w w …z† V …z; t† iˆ1 1 m and w w w ; .. . ; w w w elements of a ®nite subspace of HH. Using Galerkin's principle, this approach de®nes the approximated index-1 DAE (ADAE) system 0 ˆ M n‡…K ‡ K ÿ K †n‡ b A y …16a† 1 1 2 3 1 1; m 0 ˆ k‡ b n …16b† > > e e 0 ˆ H y ‡ P A GA …y ‡ z †‡ A y …16c† 1 R 1; m L 1; m 1; m 2; m c R ‡ A z ‡ A …t† ÿ…b A † n V 2;m I 1 0 ˆ Ly _ ÿ A … y ‡ z†…16d† 2; m 1 L PDAE MODEL FOR RLC NETWORKS 201 > > > Q A GA Q Q A 1; m R V C R C 0 ˆ …16e† A Q 0 2; m ! ! > > > A …t† Q A GA Q A I R L 1; m C R C ‡ ‡ ÿv v v…t† A 0 2; m with n :ˆ… ; .. .; † and the element matrices 1 m C 0 i j M ˆ w w w ; w w w ; 0 L i; j 0 I i j K ˆ w w w ; w w w ; I 0 i; j R 0 i j K ˆ w w w ; w w w ; 0 G i; j > j > j i i K ˆ‰w w w …l†Š w w w …l† ÿ‰w w w …0†Š w w w …0† ; 1 2 1 2 i; j i; j 0 1 1 1 ÿw w w …0† w w w …l† 1 1 B C . . B C . . b ˆ @ . . A m m ÿw w w …0† w w w …l† 1 1 for (14). Thus the index of the ADAE systems ®ts to the perturbation index for the original PDAE that has to be solved numerically. For (15) we get the ADAE system 0 ˆ M n‡…K ‡ K ‡ K †nÿ b k …17a† 1 1 2 4 2 > > 0 ˆ A y ÿ b n …17b† 1; m > > e e 0 ˆ H y ‡ P A GA … y ‡ z †‡ A y …17c† 1 R 1; m L 1; m 1; m 2; m c R ‡ A z ‡ A …t† ‡ A k V 2; m I  m 0 ˆ L y _ ÿ A …y ‡ z†…17d† 2; m 1 ! ! > > > Q A A Q Q A ~ 1; m C RG R C C 0 ˆ …17e† A Q 0 2; m ! ! > > > A …t† Q A GA Q A I R L 1; m C R C ‡ ‡ y ÿv v v…t† A 0 2; m V È 202 M. GUNTHER with the element matrices > j > j i i K ˆ‰w w w …l†Š w w w …l† ‡‰w w w …0†Š w w w …0† ; 2 1 2 1 i; j i; j 0 1 1 1 w w w …0† w w w …l† 2 2 B C . . B C . . b ˆ : @ A . . m m w w w …0† w w w …l† 2 2 One notes that the index is larger than 1: the algebraic equation (19) does only depend on the differential components y and n. After differentiation of 1; m (17b), the equations (17a±17c) can be solved for n; y _ and k as a linear function in n; x: the matrix 0 1 M 0 ÿb 1 2 B e C 0 H A @ A > > ÿb A 0 is regular, since M and H are symmetric positive-de®nite and A has full 1 1 column rank. Thus the index is two, independent of the particular Galerkin ansatz. How can one explain these different results for system (14) and (15)? In both cases, semidiscretization yields coupled systems of two DAEs [2]. The structure of the coupling is given by the way the boundary conditions are formulated in a weak or strong sense. In (14) the algebraic component is de®ned in a strong sense. Thus we get two index-1 DAE systems y _ …t†ˆ f…y; z†; i i i …i ˆ 1; 2†…18† 0 ˆ h…y; z† i i > > with y :ˆ n , z :ˆ k , and y :ˆ… y ; y † , z :ˆ…z ; z † , which are 1 1 m 2 2 1; m 2; m 1; m 2; m coupled only via the right-hand side ± thus the whole system (14) has index 1. In (15), however, the differential components A u are de®ned in a strong manner. Correspondingly, we get y _ …t†ˆ f…y; z ; u†; i i i …i ˆ 1; 2†…19a† 0 ˆ h…y; z† i i 0 ˆ g…y; z†…19b† > > with y :ˆ n, z :ˆfg; h :ˆfg; y :ˆ… y ; y † , z :ˆ…z ; z † and 1 1 1 2 2 1; m 2; m 1; m 2; m u :ˆ k , i.e., one ODE and one index-1 DAE coupled via right-hand sides and m PDAE MODEL FOR RLC NETWORKS 203 n algebraic equaitons. One obtains index 2 for the whole system (19) due to the algebraic coupling, since @g…y; z†=@z ˆ 0: 5 CONCLUSION Using generalized network models for interconnects, the DAE network equations for linear RLC circuits are generalized to a linear PDAE model. If the network ful®ls the topological index-1 conditions, the system is well-posed and has perturbation index 1. Using semidiscretization with respect to space, the method-of-lines approach converts this PDAE model into an approximate DAE system in time only. To re¯ect the physical properties of the original PDAE model, the ADAE system should neither be more nor less sensitive. Although the formulation of PDAE boundary conditions in a weak or strong sense does not affect the analytical solution, it may have an impact on the index of the ADAE systems, if one applies a semidiscretization scheme such as linear ®nite elements that does not re¯ect the hyperbolic type of the PDE system. REFERENCES 1. Arnold, M.: A Note on the Uniform Perturbation index. Rostock. Math. Kolloq. 52 (1998), pp. 33±46. 2. Arnold, M. and Gu È nther, M.: Preconditioned Dynamic Iteration for Coupled Differential- Algebraic System. BIT 41(1) (2001), pp. 1±25. 3. Campbell, S.L. and Marszalek, W.: ODE / DAE Integrators and MOL Problems. Z.f. Angew. Math. 76 (Suppl. 1) (1996), pp. 251±254. 4. Campbell, S.L. and Marszalek, W.: The Index of an In®nite Dimensional Implicit System. Math. Comput. Modeling Dyn. Syst. 5 (1999), pp. 18±42. 5. Gu È nther, M. and Feldmann, U.: CAD Based Electric Circuit Modeling in Industry. I: Mathematical Structure and Index of Network Equations. II: Impact of Circuit Con®- gurations and Parameters. Surv. Math. Ind. 8 (1999), pp. 97±157. 6. Gu È nther, M. and Rentrop, P.: PDAE-Netzwerkmodelle in der Elektrischen Schaltungssi- mulation. In: W. John (Ed.): Analog '99: 5.ITG / GMM-Diskussionssitzung. `Entwicklung von Analogschaltungen mit CAE-Methoden mit dem Schwerpunkt Entwurfsmethodik und Parasitare Effekte.' Frankfurt, 2001, pp. 31±38. 7. Gu È nther, M. and Wagner, Y.: Index Concepts for Linear Mixed Systems of Differential- Algebraic and Hyperbolic-Type Equations. SIAM J. Sci. Comp. 22 (5) (2000), pp. 1610±1629. 8. Lucht, W., Strehmel, K. and Eichler-Liebenow, C.: Indexes and Special Discretization Methods for Linear Partial Differential Algebraic Equations. BIT 39 (1999), pp. 484±512. 9. Tischendorf, C.: Topological Index Calculation of Differential-Algebraic Equations in Circuit Simulation. Surv. Math. Ind. 8 (1999), pp. 187±199. 10. Wagner, Y.: A further Index Concept for PDAEs of Hyperbolic Type. Mathematics and Computers in Simulation 53 (2000), pp. 287±291. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical and Computer Modelling of Dynamical Systems Taylor & Francis

A PDAE Model for Interconnected Linear RLC Networks

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Taylor & Francis
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1744-5051
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1387-3954
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10.1076/mcmd.7.2.189.3649
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Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-189$16.00 2001, Vol. 7, No. 2, pp. 189±203 Swets & Zeitlinger A PDAE Model for Interconnected Linear RLC Networks M. GUNTHER ABSTRACT In electrical circuit simulation, a re®ned generalized network approach is used to describe secondary and parasitic effects of interconnected networks. Restricting our investigations to linear RLC circuits, this ansatz yields linear initial-boundary value problems of mixed partial- differential and differential-algebraic equations, so-called PDAE systems. If the network ful®ls some topological conditions, this system is well-posed and has perturbation index 1 only: the solution of a slightly perturbed system does not depend on derivatives of the perturbations. As method-of-lines applications are often used to embed PDAE models into time-domain network analysis packages, it is reasonable to demand that the analytical properties of the approximate DAE system obtained after semidiscretization are consistent with the original PDAE system. Especially, both should show the same sensitivity with respect to initial and boundary data. We will learn, however, that semidiscretization may act like a deregularization of an index-1 PDAE model, if an inappropriate type of semidiscretization is used. Keywords: re®ned modeling, generalized network models, differential-algebraic equations (DAEs), partial differential equations (PDEs), partial differential-algebraic equations (PDAEs), a-priori estimates, perturbation index, method-of-lines (MOL), approximate DAE systems (ADAEs). 1 INTRODUCTION In network simulation packages, real circuit elements and interconnections are commonly replaced by companion models of ideal and compact network elements, whose properties are determined uniquely by ®xing electrical para- È È Fakultatfu È r Mathematik und Wirtschaftswissenschaften, Universitat Ulm, D-89081 Ulm, Germany. ± Permanent address: Universita Èt Karlsruhe (TH), Institut fu È r Wissenschaftliches Rechnen und Mathmatische Modellbildung (IWRMM), Engesserstr.6, D-76128 Karlsruhe, Germany. È 190 M. GUNTHER meters like capacitances or inductances. This yields a unique modeling approach, which allows for including parasitic and second order effects into the differential-algebraic (DAE) network approach. Examples are transistor models which approximate the physical behavior of semiconductor devices by companion models of different modeling levels, or transmission line models, which consist of RLC elements and controlled sources. Mathematically, this approach corresponds to a spatial discretization of the governing partial differential equations (PDEs) already at the modeling level. Another short-coming is the frequent use of arbitrary coupled sources that may destroy the structure of the network equations and thus lead to high-index systems [5, 9]. As an alternative, the co-simulation approach makes use of already existing simulation software for single parts of the system: different parts of the systems are modeled independently of each other and simulated by two simulation packages for electrical networks and electromagnetic ®elds; coupling is ensured by coupling the simulators. In addition to convergence problems, dif®culties may arise, since coupled systems are often characterized by very different time constants. A third approach is the use of generalized network models [6]. Re®ned models are allowed for interconnects and semiconductor devices, whose characteristic equations de®ne PDE models. Hence numerical methods can be tailored exactly to the resulting mathematical models ± the spatial discretiza- tion is not yet made at modeling level. Mathematically spoken, this approach leads to a coupled system of DAEs and PDEs, with the boundary conditions for the PDEs linked to the DAEs at the boundary nodes. Such systems are called partial differential-algebraic equations, shortly PDAE systems. In this paper, we will concentrate on the last approach for linear RLC networks. This ansatz yields linear initial-boundary value problems of mixed partial-differential and differential-algebraic equations. The analysis of such PDAE systems and their numerical discretization is in the focus of actual research: one aim is to generalize the DAE index concept to PDAE systems to get some knowledge on structural properties before discretization [4, 7, 8, 10]: for example, the sensitivity of the solution to small perturbations in the initial data and / or input signals. On the other hand, estimates are required for the impact of semidiscretization on the index of the resulting approximate DAE (ADAE) system: does the ADAE system properly re¯ect the behavior of the original PDAE system? Or does one detect an arti®cial smoothing effect? Or even a coarsening one? PDAE MODEL FOR RLC NETWORKS 191 It is already known that the last but one question has to be answered in the af®rmative. Semidiscretization may act like a regularization [1, 3]: the ADAE system is less sensitive w.r.t. input data than the PDAE model, and may yield physically incorrect solutions. In this contribution we will af®rm the last question, too. The paper is organized as follows. We ®rst derive the generalized network approach for interconnected linear RLC networks and discuss the impact of topological index-1 conditions on the structure of the DAE part. In the analysis of the PDAE model in Section 3 we ®rst prove uniqueness and derive a-priori estimates of the solution to establish well-posedness of the system. We see that the solution of a slightly perturbed system does not depend on derivatives of the perturbations ± the PDAE has perturbation index 1. These properties should be re¯ected by the approximate DAE system obtained after semidiscretization. We will learn, however, in Section 4 that semidiscretiza- tion may act like a deregularization of an index-1 PDAE model, if an inappropriate type of semidiscretization is used. 2 GENERALIZED NETWORK APPROACH In the following, we consider two electrical networks which are coupled by a system of d uniform lossy transmission lines. To derive a mathematical model, we use a generalized network approach: the electrical circuits are described by DAE models, whereas the transmission lines shown in Figure 1 are governed by a PDE model. Both models are linked via boundary node voltages and currents. Fig. 1. PDE-network model for a system of d uniform lossy transmission lines. È 192 M. GUNTHER 2.1 PDE Network Model for Transmission Lines Assuming quasi stationary behavior transverse to the wave propagation, the signal propagation in the transmission lines can be characterized by the telegrapher's equation 0 ˆ V …z; t†‡ LJ …z; t†‡ RJ…z; t†; (1a) z t 0 ˆ J …z; t†‡ CV …z; t†‡ GV…z; t†; (1b) z t dd where R; L; G and C 2 R are the positive-de®nite symmetric resistance, inductance, conductance and capacitance matrices per unit length. V…z; t† is a d-dimensional vector of line voltages with respect to ground, and J…z; t† is a d-dimensional vector of line currents. For V; J2H :ˆ H …0; l† the telegrapher's equation (1) holds in the sense of distributions inV :ˆ L …0; l† . This ®rst order hyperbolic system of partial differential equations is initialized by a set of initial values V…z; t †ˆ V …z†; …2a† J…z; t †ˆ J …z†; …2b† for all z 2 I :ˆ‰0; lŠ at initial time point t . After introducing 2d virtual current sources k :ˆ… ; .. .; † and 1 1;1 1; d k :ˆ… ; .. .; † at the boundaries, the characteristic equation for the 2 2;1 2;d PDE model of a lossy transmission line system in admittance form reads J…0; t† k k ˆ with k ˆ ; …3† ÿJ…l; t† k where the line currents J are de®ned by the telegrapher's equation (1). The PDE model is completed by the boundary conditions V…0; t† u ˆ ˆ: A u; (4) V…l; t† u which couples the PDE network model with the node potentials u of the DAE model for both electrical networks via an incidence matrix A . 2.2 DAE Network Model for Linear Electrical Circuits A network model is used to describe the electrical behavior of the circuit: network equations for node potentials are derived using Kirchhoff's laws and characteristic equations for the elements. This results in a system of differen- PDAE MODEL FOR RLC NETWORKS 193 tial-algebraic equations since only topology and no spatial dimension is considered. Using classical Modi®ed Nodal Analysis (MNA), only node potentials u, currents through inductive and resistive branches and , and L V currents k at the boundaries of interconnects are unknowns. The DAE network equations with x :ˆ…u; ; † read for a linear RLC network consisting of L V only linear capacitors, inductors and resistors, as well as independent voltage and current sources, 0 1 0 1 T > A CA 00 A GA A A C R L V B C B C x _ ‡ x @ A @ ÿA 00 A 0 L 0 00 0 ÿA 00 0 1 0 1 A A …t† B C B C ‡ 0 k‡ 0 ˆ 0 …5† @ A @ A 0 v v v…t† with consistent initial values x…t †ˆ x : …6† 0 0 The element-related incidence matrices A , A , A , A , A and A describe C L R V I the branch-current relations for capacitive, inductive, resistive branches and branches for voltage sources, current sources and transmission line elements. e e e The capacitance, inductance and conductance matrices C, L and G are assumed to be positive-de®nite and symmetric [9]. 2.3 Topological Index-1 Conditions With Q projecting onto A and P :ˆ I ÿ Q , the network variables x can be c c split into the differential part y :ˆ…P u; † and the algebraic components c L z :ˆ…Q u; † . Correspondingly, the network equations read after pure algebraic transformations > > e e 0 ˆ H y _ ‡ P A GA … y ‡ z †‡ A y ‡ A z ‡ A …t†‡ A k 1 R 1 L V 2 I 1 1 2 c R 0 ˆ L ÿ A … y ‡ z † L 1 L 1 ! ! > > > > > > e e z y Q A GA Q Q A Q A GA Q A R V R L 1 c R c c c R c 0 ˆ ‡ > > z y A Q 0 2 A 0 V V Q …A …t†‡ A k† ÿv v v…t† È 194 M. GUNTHER > > e e with H :ˆ A CA ‡ Q Q positive-de®nite and symmetric. The topolog- 1 C C C c ical index-1 conditions T1 ker…A ; A ; A † ˆf0g; C R V T2 ker Q A ˆf0g guarantee > > > Q A GA Q Q A R V c R c ker ˆ ker Q f0g: > c A Q 0 Thus z…t† is given as a linear function in y…t†, …t†; v v v…t† and k…t†. If there is a capacitive path to ground for all coupling nodes, the condition > > T3 Q A ˆ 0 …ker P A ˆf0g† c c holds, and z…t† is given as a linear function in y…t†, …t† and u…t† alone. Assuming that T1±T3 holds, the equations (1±6) de®ne a uniquely solvable mixed initial-boundary value problem of PDEs and DAEs, or partial differential-algebraic equations (PDAEs), of perturbation index 1. This will be shown in the next section. 3 ANALYTICAL PROPERTIES OF THE PDAE MODEL To derive the analytical properties of the PDAE (1±6), we proceed in three steps: First, we prove uniqueness in an L -sense independent of the network topology by using energy estimates. Based on these technique, we are able to derive a-priori estimates for the solution depending only on the model data, i.e., initial values, input sources and matrices of the linear PDE and DAE models. If the topological index-1 conditions hold, these two ingredients establish well-posedness of the PDAE system. In a last step, we investigate the sensitivity of the model with respect to small perturbations. 3.1 Uniqueness Let DV; DJ; Du; D ; D be the difference of two (suf®ciently smooth) L V 0 0 0 solutions of the PDAE (1±6) for the same data, i.e., initial values V ; J ; x , e e input sources …t†; v v v…t†, and matrices L; C; R; G of the linear PDE and L; C; G PDAE MODEL FOR RLC NETWORKS 195 of the DAE model. Considering the inner product of (1a) and (1b) with D J and DV, resp., inV , one obtains after integration by parts due to C, L symmetrical and positive-de®nite n o 1 d 2 2 1=2 1=2 0 ˆ kC DV…; t†k ‡kL D J…; t†k V V 2 dt > > ‡hi GDV…; t†; DV…; t† ‡hi RDJ…; t†; DJ…; t† ÿ…A Du† Dk; …7† whereh;i denotes the inner product inV andkk the associated norm. The network equations (5) yield for the last term > > > > > e e e ÿ…A Du† Dk ˆ Du A CA D u‡ A GA Du†‡ D LD j : …8† C R C R L e e e As the matrices C, L and G are symmetrical and positive-de®nite, we get after inserting (8) into (7) and integration with respect to time due to R; G positive- semide®nite the estimate 2 2 2 1=2 1=2 1=2 kC DV…; t †k ‡kL DJ…; t †k ‡kL D …t †k e e L e V V 2 2 2 1=2 > 1=2 > e e ‡kC A Du…t †k ‡ kG A Du…t†k dt  0; …9† C 2 R 2 which is equivalent to > > DV  D J  0; D  0; A Du  0; A Du  0: C R > > in the L -sense. Since A Du  Dv v v…t† 0 and A Du  L D  0 hold, V L Du  0 follows from the fact that a physically reasonable network must not contain cut sets of current sources: …A ; A ; A ; A † ˆf0g. Finally, the C R V L identity A D  0 in (5) together with the topological condition A ˆf0g V V V (no loops of only voltage sources) yields D ˆ 0. Thus uniqueness is established independent of the network topology whether the topological conditions T1±T3 hold or not. 3.2 A-Priori Estimates Analogous to (7±8) we have for a solution of the PDAE (1±6) n o 1 d > > > e e 0 ˆ hi CV…; t†; V…; t† ‡hi LJ…; t†; J…; t† ‡ u A CA u‡ L C L C L 2 dt > > ‡hi GV…; t†; V…; t† ‡hi RJ…; t†; J…; t† ‡ u A GA u > > ‡ u A …t†‡ v v v …t† : I V È 196 M. GUNTHER e e e The symmetry of C, L, G, C and L yields the energy estimate 2 2 2 …t† …t †‡ c …†‡k z…†k d ‡k k ‡kv v vk 2 2 0 1 2 L …t ;t† L …t ;t† 0 0 …10† in the differential components V; J and y with 2 2 2 …t† :ˆkV…; t†k ‡kJ…; t†k ‡ky…t†k ; V V 2 s s Z Z t t 2 2 k k 2 :ˆ k …†k d; kv v vk 2 :ˆ kv v v…†k d L …t ;t† 2 L …t ;t† 2 0 0 t t 0 0 and the constant no ÿ1 ÿ1 ÿ1 ÿ1 e e c ˆ 2 max C ; L ; H ; L 2 2 2 2 1 1 max kGk ;kRk ;kGA k ; kA k ; ; 2 2 R 2 2 2 2 depending on the matrices of the linear models. Using the topological conditions T1±T3, the algebraic components z are given as a linear function in 2 2 2 2 y, and v v v. Thus we have with kz…t†k  c …ky…t†k ‡k …t†k ‡kv v v…t†k † and 2 2 2 2 c :ˆ c …1‡ c † 3 1 2 2 2 …t† …t †‡ c …† d ‡k k ‡kv v vk : 0 3 2 2 L …t ;t† L …t ;t† 0 0 By Gronwall's lemma this yields the a-priori estimate 2 2 …t† exp…c t† …t †‡ c k k 2 ‡kv v vk 2 ; …11† 3 0 3 L …t ;t† L …t ;t† 0 0 and ®nally for the algebraic components 2 2 2 kz…t†k  c …t†‡k …t†k ‡kv v v…t†k : …12† 2 2 2 If the topological conditions T1±T3 hold, the PDAE initial-boundary value problem (1±6) is well posed. The solution depends continuously on the data, i.e., on the initial values, on the input signals and via the constants c ; c on the 1 2 matrices of the linear models. In the DAE sense it is well conditioned, too, as only data enter the solution, but not its derivatives. PDAE MODEL FOR RLC NETWORKS 197 Based on the a-priori estimates, one can show the existence of a unique solution for the PDAE (1±6): Under the assumptions of T1±T3 the PDAE system (1±6) has a unique solution …V; J; u; ; ;†,if L V n n 0 0 1  1 v V ; J 2H; 2 H …t ; t † ; v v v 2 H …t ; t † 0 e 0 e holds for the initial values and input signals. Both V; J and V ; J are bounded t t on the ®nite time interval ‰t ; t Š with 0 e V; J2H; V ; J 2V: t t For the network variables one gets n n n 1 u 1 | 1 | L V u 2 H …t ; t † ; 2 H …t ; t † ; 2 H …t ; t † : 0 e L 0 e V 0 e 3.3 Perturbation Analysis Let …V; J; x† be the (reference) solution of the PDAE system (1±6) with 0 0 consistent initial values V ; J and x . To investigate the sensitivity of this solution, we apply perturbations d…z; t† and l…t† to the right-hand sides of (1) b b and (5). The corresponding perturbed solution is denoted by …V; J; x† with 0 0 b b consistent perturbed initial values V ; J and x . The aim is to obtain estimates for the sensitivity of the solution, i.e., for the difference b b …DV; D J; Dx† :ˆ…V; J; x†ÿ…V; J; x† between both solutions. Due to the linearity of the systems, we get an a-priori energy estimate for 2 2 2 e…t† :ˆkDV…; t†k ‡kD J…; t†k ‡kDy…t†k ; V V 2 and kDe z…t†k of the type (11, 12): e…t† exp…e c t† e…t †‡e c kk ‡kk ; …13a† 2 2 3 0 3 L …‰t ; tŠ;V† L …t ; t† 0 0 2 2 2 kDe z…t†k  c …e…t†‡k…; t†k ‡k…t†k†…13b† 2 V 2 with the norm kk :ˆ kd…;†k d L …‰t ; tŠ;V† 0 È 198 M. GUNTHER and the constants no ÿ1 ÿ1 ÿ1 ÿ1 e e e c ˆ max C ; L ; H ; L 2 2 2 2 no max 2kGk ‡ 1; 2kRk ‡ 1; 2kGA k ‡ 1; 1 2 2 R 2 and ce :ˆ e c …1‡ c †. 3 1 2 Thus the norm of the difference of reference and perturbed solution can be estimated by the norm of the perturbations itself, and does not depend on derivatives of the perturbations, neither with respect to space nor time. Hence the PDAE system (1±6) has perturbation index 1. For a precise de®nition of the perturbation index of linear PDAE systems of hyperbolic-type (1±6) we refer to [7, 10]. 4 IMPACT OF SEMIDISCRETIZATION: PERTURBATION INDEX OF PDAE AND ADAE Summing up the results of Section 3, the PDAE model (1±6) meeting the topological conditions T1±T3 is well-posed and has perturbation index 1. This analytical behavior should be re¯ected by any type of numerical discretiza- tion. As the numerical simulation of interconnected electrical networks has to be embedded into time-domain network analysis packages that are based on a DAE description of systems in time alone, the method-of-lines (MOL) approach is a natural candidate for numerical discretization of the PDAE model: ®rst semidiscretization of the PDAE model with respect to space, and secondly numerical integration of the originating approximate system of differential-algebraic equations (ADAEs) in time. To re¯ect the analytical properties of the PDAE properly, the equality of PDAE and ADAE perturbation index de®nes an inalienable demand on the respective type of semidiscretization. However, as we will see in the following, this demand will not always be met. As an example, we will use linear ®nite elements that are based on two analytically equivalent mixed weak / strong formulation. 4.1 Mixed Weak / Strong Formulation Under the regularity assumptions 1 x V; J2H; V ; J 2V; x 2 H …t ; t † t t 0 e PDAE MODEL FOR RLC NETWORKS 199 of Section 3.2 the strong formulation (1) is equivalent to the weak formulation w w w LJ …; t†‡ V …; t†‡ RJ…; t† t z ; w w w ˆ 0 8 w w w ˆ 2HH : CV …; t†‡ J …; t†‡ GV…; t† w w w t z By integrating the term V …; t† ; w w w J …; t† twice by parts and formulating the boundary conditions (4) weakly for the node voltages, the PDAE system (1±6) can be transformed into the mixed weak / strong formulation LJ …; t†‡ V …; t†‡ RJ…; t† t z ; w w w CV …; t†‡ J …; t†‡ GV…; t† t z …14a† V…0; t† ÿw w w …0† ‡ A uÿ ˆ 0 8w w w 2HH; V…l; t† w w w …l† J…0; t† kÿ ˆ 0 …14b† ÿJ…l; t† 0 1 0 1 A CA 00 A GA A A R L V C R B C B C e _ @ Ax‡ @ Ax ÿA 00 0 L 0 ÿA 00 00 0 0 1 0 1 A A …t† J…0; t† B C B C ‡ 0 ‡ 0 ˆ 0: …14c† @ A @ A ÿJ…l; t† 0 v v v…t† One notes that the boundary condition (3) for the algebraic component k is formulated in a strong sense. Correspondingly, if the coupling condition (3) for the coupling currents is formulated weakly, we get the system LJ …; t†‡ V …; t†‡ RJ…; t† t z ; w w w CV …; t†‡ J …; t†‡ GV…; t† t z …15a† J…0; t† w w w …0† ÿ kÿ ˆ 0 8w w w 2HH; ÿJ…l; t† w w w …l† 2 È 200 M. GUNTHER V…0; t† A uÿ ˆ 0 …15b† V…l; t† 0 1 0 1 A CA 00 A GA A A R L V C R B C B C e _ @ Ax‡ @ Ax ÿA 00 0 L 0 ÿA 00 00 0 …15c† 0 1 0 1 A A …t† B C B C ‡ @ 0 Ak‡ @ 0 A ˆ 0: 0 v v v…t† Now the boundary condition (4) for the differential components A u is formulated in a strong sense. These two analytically equivalent formulations may form the basis for a Ritz±Galerkin approach with linear ®nite elements. 4.2 Semidiscretization If we apply semidiscretization with respect to space, we seek for an approxi- mate solution V ; J ; x ; k of (14) and (15) resp., with m m m m J …z; t† ˆ …t† w w w …z† V …z; t† iˆ1 1 m and w w w ; .. . ; w w w elements of a ®nite subspace of HH. Using Galerkin's principle, this approach de®nes the approximated index-1 DAE (ADAE) system 0 ˆ M n‡…K ‡ K ÿ K †n‡ b A y …16a† 1 1 2 3 1 1; m 0 ˆ k‡ b n …16b† > > e e 0 ˆ H y ‡ P A GA …y ‡ z †‡ A y …16c† 1 R 1; m L 1; m 1; m 2; m c R ‡ A z ‡ A …t† ÿ…b A † n V 2;m I 1 0 ˆ Ly _ ÿ A … y ‡ z†…16d† 2; m 1 L PDAE MODEL FOR RLC NETWORKS 201 > > > Q A GA Q Q A 1; m R V C R C 0 ˆ …16e† A Q 0 2; m ! ! > > > A …t† Q A GA Q A I R L 1; m C R C ‡ ‡ ÿv v v…t† A 0 2; m with n :ˆ… ; .. .; † and the element matrices 1 m C 0 i j M ˆ w w w ; w w w ; 0 L i; j 0 I i j K ˆ w w w ; w w w ; I 0 i; j R 0 i j K ˆ w w w ; w w w ; 0 G i; j > j > j i i K ˆ‰w w w …l†Š w w w …l† ÿ‰w w w …0†Š w w w …0† ; 1 2 1 2 i; j i; j 0 1 1 1 ÿw w w …0† w w w …l† 1 1 B C . . B C . . b ˆ @ . . A m m ÿw w w …0† w w w …l† 1 1 for (14). Thus the index of the ADAE systems ®ts to the perturbation index for the original PDAE that has to be solved numerically. For (15) we get the ADAE system 0 ˆ M n‡…K ‡ K ‡ K †nÿ b k …17a† 1 1 2 4 2 > > 0 ˆ A y ÿ b n …17b† 1; m > > e e 0 ˆ H y ‡ P A GA … y ‡ z †‡ A y …17c† 1 R 1; m L 1; m 1; m 2; m c R ‡ A z ‡ A …t† ‡ A k V 2; m I  m 0 ˆ L y _ ÿ A …y ‡ z†…17d† 2; m 1 ! ! > > > Q A A Q Q A ~ 1; m C RG R C C 0 ˆ …17e† A Q 0 2; m ! ! > > > A …t† Q A GA Q A I R L 1; m C R C ‡ ‡ y ÿv v v…t† A 0 2; m V È 202 M. GUNTHER with the element matrices > j > j i i K ˆ‰w w w …l†Š w w w …l† ‡‰w w w …0†Š w w w …0† ; 2 1 2 1 i; j i; j 0 1 1 1 w w w …0† w w w …l† 2 2 B C . . B C . . b ˆ : @ A . . m m w w w …0† w w w …l† 2 2 One notes that the index is larger than 1: the algebraic equation (19) does only depend on the differential components y and n. After differentiation of 1; m (17b), the equations (17a±17c) can be solved for n; y _ and k as a linear function in n; x: the matrix 0 1 M 0 ÿb 1 2 B e C 0 H A @ A > > ÿb A 0 is regular, since M and H are symmetric positive-de®nite and A has full 1 1 column rank. Thus the index is two, independent of the particular Galerkin ansatz. How can one explain these different results for system (14) and (15)? In both cases, semidiscretization yields coupled systems of two DAEs [2]. The structure of the coupling is given by the way the boundary conditions are formulated in a weak or strong sense. In (14) the algebraic component is de®ned in a strong sense. Thus we get two index-1 DAE systems y _ …t†ˆ f…y; z†; i i i …i ˆ 1; 2†…18† 0 ˆ h…y; z† i i > > with y :ˆ n , z :ˆ k , and y :ˆ… y ; y † , z :ˆ…z ; z † , which are 1 1 m 2 2 1; m 2; m 1; m 2; m coupled only via the right-hand side ± thus the whole system (14) has index 1. In (15), however, the differential components A u are de®ned in a strong manner. Correspondingly, we get y _ …t†ˆ f…y; z ; u†; i i i …i ˆ 1; 2†…19a† 0 ˆ h…y; z† i i 0 ˆ g…y; z†…19b† > > with y :ˆ n, z :ˆfg; h :ˆfg; y :ˆ… y ; y † , z :ˆ…z ; z † and 1 1 1 2 2 1; m 2; m 1; m 2; m u :ˆ k , i.e., one ODE and one index-1 DAE coupled via right-hand sides and m PDAE MODEL FOR RLC NETWORKS 203 n algebraic equaitons. One obtains index 2 for the whole system (19) due to the algebraic coupling, since @g…y; z†=@z ˆ 0: 5 CONCLUSION Using generalized network models for interconnects, the DAE network equations for linear RLC circuits are generalized to a linear PDAE model. If the network ful®ls the topological index-1 conditions, the system is well-posed and has perturbation index 1. Using semidiscretization with respect to space, the method-of-lines approach converts this PDAE model into an approximate DAE system in time only. To re¯ect the physical properties of the original PDAE model, the ADAE system should neither be more nor less sensitive. Although the formulation of PDAE boundary conditions in a weak or strong sense does not affect the analytical solution, it may have an impact on the index of the ADAE systems, if one applies a semidiscretization scheme such as linear ®nite elements that does not re¯ect the hyperbolic type of the PDE system. REFERENCES 1. Arnold, M.: A Note on the Uniform Perturbation index. Rostock. Math. Kolloq. 52 (1998), pp. 33±46. 2. Arnold, M. and Gu È nther, M.: Preconditioned Dynamic Iteration for Coupled Differential- Algebraic System. BIT 41(1) (2001), pp. 1±25. 3. Campbell, S.L. and Marszalek, W.: ODE / DAE Integrators and MOL Problems. Z.f. Angew. Math. 76 (Suppl. 1) (1996), pp. 251±254. 4. Campbell, S.L. and Marszalek, W.: The Index of an In®nite Dimensional Implicit System. Math. Comput. Modeling Dyn. Syst. 5 (1999), pp. 18±42. 5. Gu È nther, M. and Feldmann, U.: CAD Based Electric Circuit Modeling in Industry. I: Mathematical Structure and Index of Network Equations. II: Impact of Circuit Con®- gurations and Parameters. Surv. Math. Ind. 8 (1999), pp. 97±157. 6. Gu È nther, M. and Rentrop, P.: PDAE-Netzwerkmodelle in der Elektrischen Schaltungssi- mulation. In: W. John (Ed.): Analog '99: 5.ITG / GMM-Diskussionssitzung. `Entwicklung von Analogschaltungen mit CAE-Methoden mit dem Schwerpunkt Entwurfsmethodik und Parasitare Effekte.' Frankfurt, 2001, pp. 31±38. 7. Gu È nther, M. and Wagner, Y.: Index Concepts for Linear Mixed Systems of Differential- Algebraic and Hyperbolic-Type Equations. SIAM J. Sci. Comp. 22 (5) (2000), pp. 1610±1629. 8. Lucht, W., Strehmel, K. and Eichler-Liebenow, C.: Indexes and Special Discretization Methods for Linear Partial Differential Algebraic Equations. BIT 39 (1999), pp. 484±512. 9. Tischendorf, C.: Topological Index Calculation of Differential-Algebraic Equations in Circuit Simulation. Surv. Math. Ind. 8 (1999), pp. 187±199. 10. Wagner, Y.: A further Index Concept for PDAEs of Hyperbolic Type. Mathematics and Computers in Simulation 53 (2000), pp. 287±291.

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Mathematical and Computer Modelling of Dynamical SystemsTaylor & Francis

Published: Jun 1, 2001

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