Winding functions in transient magnetoquasistatic field-circuit coupled simulations

Sebastian Schöps (Graduate School Computational Engineering and Institut für Theorie Elektromagnetischer Felder, Technische Universitaet Darmstadt, Darmstadt, Germany)
Herbert De Gersem (Wave Propagation and Signal Processing Research Group, Department of Physics and Astronomy, KU Leuven – Kulak, Kortrijk, Belgium)
Thomas Weiland (Institut für Theorie Elektromagnetischer Felder, Technische Universitaet Darmstadt, Darmstadt, Germany)



The purpose of this paper is to review the mutual coupling of electromagnetic fields in the magnetic vector potential formulation with electric circuits in terms of (modified) nodal and loop analyses. It aims for an unified and generic notation.


The coupled formulation is derived rigorously using the concept of winding functions. Strong and weak coupling approaches are proposed and examples are given. Discretization methods of the partial differential equations and in particular the winding functions are discussed. Reasons for instabilities in the numerical time domain simulation of the coupled formulation are presented using results from differential-algebraic-index analysis.


This paper establishes a unified notation for different conductor models, e.g. solid, stranded and foil conductors and shows their structural equivalence. The structural information explains numerical instabilities in the case of current excitation.


The presentation of winding functions allows to generically describe the coupling, embed the circuit equations into the de Rham complex and visualize them by Tonti diagrams. This is of value for scientists interested in differential geometry and engineers that work in the field of numerical simulation of field-circuit coupled problems.



The work of S. Schöps is supported by the ”Excellence Initiative” of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt. The author thanks A. Bartel and M. Günther (Bergische Universität Wuppertal) for many fruitful discussions on the topic of coupled systems. The work of H. De Gersem is financially supported by grant KUL 3E100118 “Electromagnetic Field Simulation for Future Particle Accelerators” and grant KUL 3E080005 “Numerical field simulation, materials models and signal processing techniques for nonlinear ultrasonic non-destructive testing”.


Schöps, S., De Gersem, H. and Weiland, T. (2013), "Winding functions in transient magnetoquasistatic field-circuit coupled simulations", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 32 No. 6, pp. 2063-2083.

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