Abstract
A simulation of an electro-mechanical drive including simultaneously the torsional behaviour of a mechanical load, a power electronic supply for soft-starting and a numerical transient field calculation requires the solution of a system of differential and algebraic equations. The Crank-Nicholson method for PDEs of parabolic type implies an application of the trapezoidal rule of integration with respect to time. This method of integration applied to a more general system of differential and algebraic equations can cause stability problems by numerically conditioned hazardous oscillations. These disturbances are suppressed by a modification of the trapezoidal rule sometimes referred to as θ-method with weighting-factors θ and 1 — θ for the integration. However, the damping of natural oscillations of the system sometimes increases unacceptably. Best results concerning the damping are achieved, if θ remains at 0.5. Numerically critical eigenvalues of the system and their stimulation can be avoided by taking into account additional properties of the system in most practical cases.
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References
1. Krawczyk A., Tegopoulos J. A.: Numerical Modelling of Eddy Currents. Clarendon Press, Oxford, 1993.
2. Krawczyk A.: The application of FEM to transient problems of electromagnetic engineering. Rozprawy Elektrotechniczne 29, (2), pp. 331–344, 1983.
3. Tsukerman, I.: A Stability Paradox for Time-Stepping Schemes in Coupled Field-Circuit Problems. IEEE Transactions on Magnetics, Vol. 31, No. 3, pp. 1857–1860, May 1995.
4. Belmans, R., Hameyer K. and Mertens, R.: Combined Time-Harmonic-Transient Approach to Calculate the Steady-State Behaviour of Induction Machines. IEEE Transactions on Magnetics, 1999.
5. Spiro, H.: Simulation integrierter Schaltungen, 2. Auflage. Oldenbourg Verlag München, Wien, 1990.
6. Klocke, M.: Zur Berechnung dynamischer Vorgänge bei von einem Drehstromsteller gespeisten Antrieben mit Asynchronmaschinen und mehreren gekoppelten Massen mittels Finite-Differenzen-Zeitschrittrechnung. Dissertation, University of Dortmund, 1999.
7. Feldmann, U., Günther, M.: Some Remarks on Regularization of Circuit Equations. Conference Proceedings of X. ISTET, 1999.
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© 2001 Springer-Verlag Berlin Heidelberg
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Klocke, M. (2001). The “Modified Trapezoidal Rule” (θ-Method) for the Integration of DAEs Modelling an Electro-Mechanical Drive Including External Circuitry and Magnetic Field-Distribution. In: van Rienen, U., Günther, M., Hecht, D. (eds) Scientific Computing in Electrical Engineering. Lecture Notes in Computational Science and Engineering, vol 18. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56470-3_40
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DOI: https://doi.org/10.1007/978-3-642-56470-3_40
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