The “Modified Trapezoidal Rule” (θ-Method) for the Integration of DAEs Modelling an Electro-Mechanical Drive Including External Circuitry and Magnetic Field-Distribution

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.