Non-Smooth Modelling of Electrical Systems
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
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.
Electrical model of the extended DC-DC buck converter.
Mechanical model associated with the extended DC-DC buck converter.
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