Variational integrators for electric circuits

Abstract

In this contribution, we develop a variational integrator for the simulation of (stochastic and multiscale) electric circuits. When considering the dynamics of an electrical 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.

A Fokker-Planck model for a non-linear switched system

“…This paper is the first of its kind in the sense that 'introducing the notion of Fokker-Planck modeling into switched electrical networks'. Equations (9), (13) and (14) are main analytic findings for the stochastic switched system of the paper.…”

“…Variational methods completely hinge on the Hamiltonian and Lagrangian principles for the network analysis, refer [12], [13]. Extensive literature is available on variational methods for non-conservative networks using deterministic Hamiltonian and Lagrangian approaches.…”

Optimal control of a slot car racer

“…The task to describe the system dynamics with discrete variational calculus includes to formulate the electrical, magnetic and mechanical energy of the system and to apply the discrete Lagrange-d'Alembert principle. This is less common in electrical engineering but leads to a structure preserving time stepping scheme which serves as equality constraints for the nonlinear programming problem, resulting from the discretization of the optimal control problem by the DMOC method [3][4][5][6]. The computed optimal profiles can be embedded to an experimental setup for a slot car racer with an underling camera tracking system which allows to correct the vehicle towards the desired state via computer.…”

Variational integrators for nonlinear electric circuits

“…The resistors (with relation u R = f R (v R )) and voltage sources u S = f S (v S , t) are incorporated in the Lagrangian force given byTo reduce the system's dimension, we define the Lagrangian and the forces on the space of meshes (following [1, 2]) asf Cj (y) dy with the Legendre transform FL M (q,ṽ) = (q, K T 2,L f L (K 2,Lṽ )) = (q,φ). Accordingly, we define the reduced Lagrangian force asDefining the mesh fluxesφ = K T 2 ϕ ∈ R n−m we can derive the equations of motion for the nonlinear circuit with the Lagrange-d'Alembert principle [1,3], i.e. we are searching for curvesq(t),ṽ(t) andφ(t) satisfyingFixing the initial and final points forq(0),q(T ) and taking variations δq, δṽ, δφ gives the implicit Euler-Lagrange equations on the mesh space ∂L M ∂q + f M L −φ = 0,q =ṽ, ∂L M ∂ṽ −φ = 0 which read for the nonlinear circuit aṡWhile the Lagrangian L with its Legendre transform FL is degenerate if any other components beside inductors are involved in the circuit, the reduction to L M can cancel out this degeneracy iff K T 2,L f L (K 2,L ·) : R n−m → R n−m is (locally) invertible.…”

“…Defining the mesh fluxesφ = K T 2 ϕ ∈ R n−m we can derive the equations of motion for the nonlinear circuit with the Lagrange-d'Alembert principle [1,3], i.e. we are searching for curvesq(t),ṽ(t) andφ(t) satisfying…”