Non-Smooth Modelling of Electrical Systems

Authors: Michael Möller

The well-developed non-smooth modelling framework used for mechanical systems can be adopted for electrical systems by extending the classical electromechanical analogy to non-smooth systems. A non-smooth modelling approach to electrical circuits allows for idealized diodes and switching components. To formulate the model of a circuit there are basically three approaches called the charge approach, the flux approach and mixed approaches. The charge approach uses the charges and associated currents as variables, while the voltages are balanced. In the flux approach the fluxes with associated voltages constitute the variables while balancing the currents. Both approaches can be used to describe non-smooth electrical systems. In the charge approach the voltages can be impulsive and the currents are finite, while the flux approach allows the currents to be impulsive while the voltages are always finite.

For each non-smooth electrical component a set-valued branch relation is formulated. The set-valued branch relation for a diode is analogous to the set-valued force law of a sprag clutch. Switches are modelled as spark gap with variable break through voltage. The force law of dry friction is analogous to the branch relation of a spark gap in the charge approach. With the set-valued branch relations, the dynamics of the circuit is then described as a differential inclusion. For the numerical solution, the differential inclusion can be formulated as complementarity system and discretised with a difference scheme based on Moreau's midpoint rule.

Publications:

Glocker, Ch., Models of non-smooth switches in electrical systems. Int. J. Circuit Theory Appl. 30, 205-234, 2005

Möller, M., Glocker, Ch., Non-smooth modelling of electrical systems using the flux approach. Nonlinear Dynamics 50, pp. 273-295, 2007