Solution strategies for transient, field-circuit coupled systems

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Abstract
Transient simulation times for field-circuit coupled models of realistic electromagnetic devices become unacceptably high. A magnetodynamic formulation is coupled to an electric circuit analysis, yielding a sparse, symmetric and indefinite matrix. The unknown circuit currents correspond to negative eigenvalues in the matrix spectrum. The Quasi-Minimal Residual method performs better than the Minimal Residual approach that is restricted to positive definite preconditioners. The positive definite variant is solved by the Conjugate Gradient method without explicitly building the dense coupled matrix. As an example, both approaches are applied to an induction motor. Finite element simulation techniques are commonly used in the design and optimisation of electromagnetic devices. The computation of the dynamic behaviour of magnetic fields involves the simulation of the electric network that excites or is excited by the magnetic field. As the differential equations representing both phenomena are linearly dependent upon each other, simulation by means of one coupled system matrix is particularly attractive (1). For a large range of technical devices operating at low frequencies, a clear distinction can be made between electrically conducting and non-conducting media. As a consequence, a description of the electric behaviour of the device in terms of a lumped parameter model may reach a sufficient accuracy. The relative difference in permeability, however, is much lower. Moreover, the permeability may be non-linear and the paths followed by the magnetic flux are usely rather irregular. As a consequence, the magnetic models requires a finer discretisation, e.g. by means of finite elements. Both the coupling of two physical phenomena and the hybrid nature of the discretisation methods assign specific properties to the coupled system matrix. Magnetodynamic models of transformers, induction machines and induction furnaces are relatively small but have to be simulated many times. In the case of transient simulation, a huge number of sequential solutions is required. This fact justifies a detailed study of the influence of the field-circuit coupling mechanism on the efficiency of the system solution. In this paper, coupling schemes are selected from the viewpoint of the resulting system properties. Appropriate iterative solving techniques for the coupled system matrix are developed. The coupling techniques are judged upon the efficiency of the applicable iterative solvers.

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