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Published June 1, 2013 | Submitted
Journal Article Open

Variational integrators for electric circuits


In this contribution, we develop a variational integrator for the simulation of (stochastic and multiscale) electric circuits. When considering the dynamics of an electric circuit, one is faced with three special situations: 1. The system involves external (control) forcing through external (controlled) voltage sources and resistors. 2. The system is constrained via the Kirchhoff current (KCL) and voltage laws (KVL). 3. The Lagrangian is degenerate. Based on a geometric setting, an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. A time-discrete variational formulation provides an iteration scheme for the simulation of the electric circuit. Dependent on the discretization, the intrinsic degeneracy of the system can be canceled for the discrete variational scheme. In this way, a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits; in particular, a comparison to BDF methods (which are usually the method of choice for the simulation of electric circuits) shows that even for simple LCR circuits, a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator.

Additional Information

© 2013 Elsevier Inc. Received 9 March 2011. Received in revised form 19 November 2012. Accepted 5 February 2013. Available online 20 February 2013. This contribution was partly developed and published in the course of the Collaborative Research Centre 614 "Self-Optimizing Concepts and Structures in Mechanical Engineering" funded by the German Research Foundation (DFG) under Grant No. SFB 614. The authors acknowledge partial support from NSF grant CMMI-092600. The authors gratefully acknowledge Henry Jacobs, Melvin Leok, and Hiroaki Yoshimura for delightful discussions about variational mechanics for degenerate systems. Furthermore, the authors thank Stefan Klus, Sujit Nair, Olivier Verdier, and Hua Wang for helpful discussions regarding circuit theory. Finally, we thank Sydney Garstang and Carmen Sirois for proofreading the document.

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