DAE-Index and Convergence Analysis of Lumped Electric Circuits Refined by 3-D Magnetoquasistatic Conductor Models

Transcription

1 Bergische Universität Wuppertal Fachbereich Mathematik un Naturwissenschaften Lehrstuhl für Angewante Mathematik un Numerische Mathematik Preprint BUW-AMNA 8/6 Sebastian Schöps DAE-Inex an Convergence Analysis of Lumpe Electric Circuits Refine by 3-D Magnetoquasistatic Conuctor Moels October 28

2 DAE-Inex an Convergence Analysis of Lumpe Electric Circuits Refine by 3-D Magnetoquasistatic Conuctor Moels Sebastian Schöps 1, Anreas Bartel 1, Herbert De Gersem 2, an Michael Günther 1 Abstract In this paper the fiel/circuit coupling is reconsiere for (non-linear) lumpe electric circuits refine by 3-D magnetoquasistatic conuctor moels, where the circuit is escribe by moifie noal analysis an the fiel is iscretize in the terms of the finite integration technique. The coupling of the two systems of ifferential-algebraic equations is iscusse an two numerical approaches are propose, the weak (co-simulation) an strong coupling (monolithic). The DAE-inex of the subproblems an of the full problem are analyze an the convergence properties of the co-simulation are stuie by inspecting the epenencies of algebraic constraints on solutions of previous iterations. Finally computational results of a simple half rectifier circuit are exemplarily given to prove the concepts. 1 Introuction Many basic elements in circuit analysis are escribe by (non-)linear relations, isregaring fiel effects. Sometimes, they are replace by more complex but more realistic companion moels to meet reality. These give, however, only a partial insight into fiel effects. In contrast refine moels irectly rely upon Maxwell s equations. We restrict our analysis to the refinement of two conuctor types, which are embee into electric circuits an exhibit proximity an skin effects with ey currents. The couple problem is a system of ifferential-algebraic equations (DAEs) originating from Kirchhof s Laws an the iscrete Maxwell Equations. It can be irectly aresse by solving one monolithic system using a fiel- or circuit-oriente approach. In the fiel approach, commonly the circuit is escribe using loop/branch techniques an is solve within the fiel simulator. This approach is quite successful an well unerstoo [1], but it is neither efficient for coupling with very large circuits nor usable within moern circuit simulators that are base on moifie noal 1 Bergische Universität Wuppertal, {schoeps,bartel,guenther}@math.uni-wuppertal.e 2 Katholieke Universitat Leuven, Herbert.DeGersem@kuleuven-kortrijk.be 1

3 2 Sebastian Schöps, Anreas Bartel, Herbert De Gersem, an Michael Günther analysis (MNA). The circuit-oriente approach relies on MNA. Although intensive research has been carrie out [2], companion moels are still wiesprea. Obviously the strongly couple approaches above will not have the avantages of both simulators. Therefore co-simulation becomes beneficial [3]. It allows the use of ifferent simulators for each subproblem an it provies a natural support for iversifying integration methos an time-stepping rates (multirate) towars the subproblems. The coupling is given mathematically by a ynamic iteration scheme. The paper is organize as follows: In Sections 2 an 3 we recall the circuit an fiel settings, in Section 4 we analyze the DAE-inex of the fiel-system, in Section 5 we introuce the monolithic coupling an its co-simulation with inex an convergence analysis, in Sections 6 an 7 we give an example an final conclusions. 2 Lumpe Electric Circuit Electric circuits are given by elements connecte by Kirchhoff s Laws an their escription is given here in terms of MNA, which yiels DAEs since it is base on reunant coorinates. In the charge-flux oriente formulation, [4], the system reas A C t q+a R r(a T R e,t)+a Li L + A V i V + A I i(t)+a λ i λ (A T λ e,t) =, t Φ AT L e =, AT Ve v(t) =, q q C (A T Ce,t) =, Φ Φ L (i L,t) =, (1) with incience matrices A, noe potentials e, inepenent an controlle current an voltage sources i, i λ an v, currents through voltage an flux controlle elements i V an i L, charges an fluxes q an Φ, functions of charges, fluxes an resistances q C, Φ L an r (with positive efinite erivatives), respectively. The numerical properties of (1) are well known, e.g. the DAE-inex has been iscusse by ecomposing the unknown (e,i V,i L,q,Φ) into algebraic an ifferential parts using a projector Q C onto the kernel of A T C, i.e., Q C kera T C = kera T C an A T CQ C =, an its complementary projection P C = I Q C. We assume in the terms above C1 No loops of capacitors an voltage sources, i.e., ker(a C,A R,A V ) T = {}. C2 C3 No cutsets of inuctors an/or current sources, i.e., kerqc T A V = {}. Voltage controlle current sources parallel to capacitors, i.e., QC T A λ i λ =. This splits the unknown into y := (P C e, j L ) T an z := (Q C e, j V,q,φ) T, such that t y = f 1(y,z,i λ ), = g 1 (y,z), (2) is an inex-1 escription of (1) since the erivative z g 1 can be shown to be nonsingular assuming C1-C3. This motivates the following important property [5]:

4 Analysis of Lumpe Electric Circuits Refine by 3-D MQS Conuctor Moels 3 Theorem 1. Let us consier a lumpe electric circuit in form (1) that respects C3, then the flux/charge oriente MNA leas to an inex-1 DAE iff C1-C2 hol, it leas otherwise to an inex-2 DAE. 3 Electromagnetic Fiel The electromagnetic fiel is escribe by Maxwell s equations. We assume that they are spatially iscretize by either the finite integration technique or the finite elements metho using lowest-orer Whitney elements an staggere gris [6,7]. This gives their iscrete counterpart, also calle the Maxwell s Gri Equations (MGE) C e = t b, C h = t + j, S = q, Sb =, (3) with iscrete curl operators C an C, ivergence operators S an S, electric an magnetic fiel strength e an h, current ensity, iscrete magnetic an electric flux j, b an, respectively. For linear materials, MGE are accomplishe by b = Mµ h, = Mε e, j = M σ e, (4) with symmetric positive (semi-)efinite matrices M µ, M ε an M σ for the permeabilities, permittivities an conuctivities, respectively; only M σ is generally singular ue to non-conucting regions [8]. We obtain from (3,4) the curl-curl equation M ε 2 t 2 a + M σ t a + K ν a = j src, (5) with a enoting the magnetic vector potential (MVP) an K ν := CM ν C. 4 Fiel Moels as Refine Network Elements i sol i str Conuctor moels for connecting fiel an circuit parts are well-known. Most common are soli an strane conuctors (Fig. 1). We use the given symbol for a (mulv sol v str (a) Soli (b) Strane (c) Symbol Fig. 1: Conuctor moels (a), (b) an evice symbol (c) that embes both into the circuit.

5 4 Sebastian Schöps, Anreas Bartel, Herbert De Gersem, an Michael Günther tiport) evice that consists of (multiple) conuctors of both types which are tightly couple by the fiel. The latter is escribe by the curl-curl equation an excite by the source current ensity j src ue to the connecte circuit [9]. Typically voltages of solis (v sol ) an currents of strane conuctors (i str ) are consiere to be given an thus the excitation reas j src = M σ Q sol v sol + Q str i str. (6) Here Q = [Q sol,q str ] enotes the coupling matrix. Each column correspons to a conuctor moel an imposes currents/voltages onto eges of the gri. The unknown currents i sol an voltages v str are obtaine by the aitional equations i sol = G sol v sol Q T sol M σ t a, v str = R str i str + Q T str t a, (7) with the iagonal matrices of soli conuctances G sol an strane resistances R str = G 1 str. Let the moel above fulfill the following assumptions F1 The fiel is magnetoquasistatic, i.e., max t max j src. F2 The solution is unique, i.e., [M σ,k ν ] := et(λm σ + K ν ) for a λ. F3 The moels are spatially isjunct, i.e., Q T (i) Q ( j) =, for all i j. F4 The excitation is consistent, i.e., ker(cq sol ) = {}, ker(cm σ,aniso + Q str) = {}. The first assumption F1 implies the neglect of M ε 2 t 2 a in the curl-curl equation (5), such that (5) becomes a first orer DAE, which is generally not uniquely solvable ue to the non-trivial nullspaces of M σ an C. Thus the integration requires the selection of one solution within the equivalent class efine by b = C a, therefore we nee the gauging assumption F2, [1]. From F3 we obtain the equivalent form M σ,fillin t a + K ν a = M σ Q sol v sol + Q str G str v str := j src, Q T sol K ν a = i sol, G str Q T strm + σ,aniso K ν a = i str, (8a) (8b) (8c) where M σ,aniso + is the pseuoinverse of the anisotropic conuctivity matrix for strane conuctors an M σ,fillin := M σ + Q str G str Q T str is a combine (ense) matrix for both conuctor types. Lemma 1. Let the fiel problem consist of soli an strane conuctors which fulfill F1-F3, then the curl-curl equation (8a) is (algebraic) inex-1 for given voltages an has only non-trivial components in the ifferential part. Proof. The matrix pencil of (8a) is regular ue to F2, so the symmetric positive semi-efiniteness of M σ,fillin implies that (8a) is inex-1 an there is the Kronecker form t a 1 (t)+u 1 K ν V 1 a 1 (t) = U 1 j src, a 2 (t) = U 2 j src, (9a) (9b)

6 Analysis of Lumpe Electric Circuits Refine by 3-D MQS Conuctor Moels 5 that splits the MVP a = V 1 a 1 +V 2 a 2 into ifferential an algebraic components by using the regular matrices U T = ( U1 T,U 2 T ) an V = (V1, V 2 ). From U 2 M σ,fillin = U 2 ( Mσ + Q str G str Q T str) = follows that both U 2 M σ an U 2 Q str G str vanish because the images of M σ an Q str are istinct, since F3 is assume. Hence we finally conclue that a 2 = U 2 j =. Let us now stuy the full system (8) in the abstract form t a = f 2a ( a,v λ ), = f 2b ( a,v λ ), = g 2 ( a,i λ ), (1) where the voltages v λ = (v sol,v str ) T an the currents i λ = (i sol,i str ) T are combine in vectors. The algebraic evaluation f 2b is trivial in our case because of Lemma 1 an the algebraic function g 2 consists of = g sol ( a,i sol ), = g str ( a,i str ). System (1) establishes a relation between currents (i sol, i str ) an voltages (v sol, v str ) an we can choose which quantity is treate as unknown for each conuctor type in the fiel system, since then the other quantity is efine by the couple electric circuit. Therefore we will istinguish between the possible sets in the following. Theorem 2. Let the fiel problem consist of soli an strane conuctors which fulfill F1-F4, then the system (8) has (ifferential) inex-2 an only for the case of given voltages v sol an v str it has inex-1. Proof. In the case of given voltages the currents i λ are obtaine by evaluations of the algebraic equation g 2. Thus one ifferentiation with respect to time yiels an ODEs, hence we have (ifferential) inex-1. In all other cases the arguments are analogue to the case of given i sol an v str. Now the function f 2a in (1) epens on the unknown v sol an one time erivative yiels the aitional hien constraint: = t g sol( a,i sol ) = a g sol f 2 ( a,v sol )+ t i sol =: h sol ( a,v sol, t i sol), an since the conuctivity matrices M σ an M σ,aniso reflect F3 (M σ,aniso Q sol = ), another ifferentiation of this constraint gives v sol h sol = Q T sol K νm + σ,fillin M σ Q sol = Q T sol K νq sol = Q T sol CT M ν CQ sol, which is non-singular ue to F4; thus it is inex-2. We conclue, if some voltage is consiere unknown, then (8) is an inex-2 Hessenberg system (with inex-1 evaluations), which is less ill-conitione than the general case. Aitionally the inex-2 variables enter the computation linearly an without time-epenence. Therefore the ifferential components are not affecte by the erivatives of perturbations, [11], an thus the numerical ifficulties correspon to inex-1.

7 6 Sebastian Schöps, Anreas Bartel, Herbert De Gersem, an Michael Günther 5 Coupling We assign the voltages v λ to ifferences of noe potentials e an map the controlle current sources i λ to the currents through the conuctors in (1) v λ = A T λ e, i λ = (i sol,i str ) T. (11) We finally obtain the monolithic system consisting of (1), (8) an (11). Theorem 3. Let us consier an electric circuit in the form (1) with C1-C2, which is monolithically couple via (11) to a fiel moel (8) of soli an strane conuctors fulfilling F1-F4, then the full system is (ifferential) inex-1. Proof. The algebraic part of the MVP is insignificant for soli an strane conuctors accoring to Lemma 1. Hence after embeing the fiel into the circuit system the separate unknowns of the full system rea y := (P C e, j L, a 1 ) T, z := (Q C e, j V,q,φ,i λ ) T. (12) The critical partial erivative of the algebraic equation z g consisting of g 1 an g 2 is non-singular, since the first is regular ue to C1-C2 an the secon is just an evaluation of a ifferential variable ( a 1 ). Thus we have inex-1. The assumption C3 is not require in the monolithic coupling because the algebraic part of the MVP was shown to vanish for the excitement of soli an strane conuctors. Alternatively the subproblems coul be treate separately by a waveform relaxation scheme (of Jacobi or Gauß-Seiel type). When applying these schemes to DAEs one has to pay attention to algebraic constraints to avoi numerical instabilities, [12]. We suggest the following Gauß-Seiel scheme t a (1) = f 2 ( a (1),v () ), v () := A T λ (y() + z () ), t y (1) = f 1 (y (1),z (1),i (1) λ ), = g 2 ( a (1),i (1) λ ), = g 1(y (1),z (1),i (1) λ ), where the fiel is couple via the current sources i λ to the circuit. The scheme above shows irectly that there is no epenence in algebraic constraints (g 1, g 2 ) on ol algebraic iterates (i λ, z) an hence the convergence is guarantee [13] an we obtain: Lemma 2. Let us consier an electric circuit in the form (1) that respects C1-C2 is couple via (11) to a fiel moel (8) of soli an strane conuctors fulfilling F1-F4, then both subproblems are inex-1 an the waveform-relaxation above will converge. The aitional assumption C3 can eliminate the i λ -epenence of the algebraic equation g 1 an allows us to exchanges the computational orer of the subproblems an thus we may compute the circuit first without losing the convergence guarantee.

8 Analysis of Lumpe Electric Circuits Refine by 3-D MQS Conuctor Moels v(t) C R loa -1 (a) Half rectifier: v eff = 25V, f = 5Hz, R loa = 1Ω an Shockley ioe I s =1µA (b) Voltage in noes 1 an 3, obtaine by mono, H = 5µs Fig. 2: Refine half rectifier circuit an its input an compute output voltages. 6 Numerical Experiments The numerical experiments have been obtaine with coe that is implemente within the COMSON DP ( using fiel moels constructe by EM Stuio from CST ( It is capable of both, the monolithic an the co-simulation of non-linear circuits refine by (linear) conuctor moels. The iscusse schemes have been supplie to the DP, where the time integration is kept simple by using Backwar Euler. The example of Fig. 2 is a refine half rectifier with a transformer consisting of two strane conuctors an a soli core. The co-simulation without iterations of time frames (cosim1) is slightly faster than the monolithic simulation (mono) using the same time steps H an it yiels comparable results if the accuracy requirement is quite low. For ecreasing step sizes cosim1 oes not linearly improve its accuracy as mono an cosim3 (3 iterations) o (Fig. 3), but the latter has an increase computational effort ue to the aitional iterations. Aaptive time-integrators in the co-simulation apply the same step size to the circuit an the fiel, as long as they o not have multirate potential itself. This is in line with the fact that the fiel reflects the ynamics of the couple circuit noes. 7 Conclusions The fiel problem is essentially an inex-1 DAE, the monolithic couple system is still inex-1 an the convergence of the propose co-simulation is guarantee, as ocumente by the computation of a refine rectifier circuit. The co-simulation uses problem-specific software packages an exploits multirate potentials if available in the circuit, but oes not have all possible benefits. Applying a step size an iteration control will improve efficiency, an the use of more complex equivalent circuits (e.g. aitional inuctivities) might require fewer fiel upates [14, 15]. Acknowlegements This work was partially supporte by the European Commission within the framework of the COMSON RTN project.

9 8 Sebastian Schöps, Anreas Bartel, Herbert De Gersem, an Michael Günther (a) mono, H = 1µs (b) cosim3, H = 1µs (c) cosim1, H = 1µs () mono, H = 1µs (e) cosim3, H = 1µs (f) cosim1, H = 1µs Fig. 3: Errors in the voltages compare to the results of mono, H = from Fig. 2b References 1. Benerskaya, G. et al.: Transient Fiel-Circuit Couple Formulation Base on the FIT an a Mixe Circuit Formulation. COMPEL 23(4), (24) 2. Dreher, T., Meunier, G.: 3D Line Current Moel of Coils an External Circuits. IEEE Trans. Mag. 31(3), (1995) 3. Berosian, G.: A New Metho for Coupling Finite Element Fiel Solutions with External Circuits an Kinematics. IEEE Trans. Mag. 29(2), (1993) 4. Günther, M., Rentrop, P.: Numerical Simulation of Elec. Circuits. GAMM 1/2, (2) 5. Estévez Schwarz, D., Tischenorf, C.: Structural Analysis of Electric Circuits an Consequences for MNA. Int. J. Circ. Theor. Appl. 28(2), (2) 6. Bossavit, A.: Computational Electromagnetism. Acaemic Press, San Diego (1998) 7. Clemens, M.: Large Systems of Equations in a Discrete Electromagnetism: Formulations an Numerical Algorithms. IEE Proc Sci Meas Tech 152(2), 5 72 (25) 8. Nicolet, A., Delincé, F.: Implicit Runge-Kutta Methos for Transient Magnetic Fiel Computation. IEEE Trans. Mag. 32(3), (1996) 9. De Gersem, H. et al.: Fiel-circuit Couple Moels in Electromagnetic Simulation. JCAM 168(1-2), (24) 1. Kettunen, L. et al.: Gauging in Whitney Spaces. IEEE Trans. Mag. 35(3), (1999) 11. Arnol, M. et al.: Errors in the Numerical Solution of Nonlinear Differential-Algebraic Systems of Inex 2. Martin-Luther-University Halle (1995) 12. Arnol, M., Günther, M.: Preconitione Dynamic Iteration for Couple Differential- Algebraic Systems. BIT 41(1), 1 25 (21) 13. Bartel, A.: Partial Differential-Algebraic Moels in Chip Design - Thermal an Semiconuctor Problems. Ph.D. thesis, TU Karlsruhe, VDI Verlag 14. Lange, E. et al.: A Circuit Coupling Metho Base on a Temporary Linearization of the Energy Balance of the Finite Element Moel. IEEE Trans. Mag. 44(6), (28) 15. Zhou, P. et al.: A General Co-Simulation Approach for Couple Fiel-Circuit Problems. IEEE Trans. Mag. 42(4), (26)