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Spectral analysis of permanent magnet electric machines

Pawel Witczak (Institute of Mechatronics and Information Systems, Technical University of Lodz, Lodz, Poland)
Beata Wawrzyniak (Institute of Mechatronics and Information Systems, Technical University of Lodz, Lodz, Poland)



This paper aims to provide the mathematical background for representation of permanent magnet AC machines in terms of rotating magnetic field waves of any shape instead of being restricted to the sinusoidal. The general idea is to replace the inductances in the mathematical model of the machine by means of adequate time functions of the flux linkage.


On the basis of several 2D or 3D solutions obtained by the finite element (FE) approach, the set of basis functions is generated for further post‐processing. These functions enable fast and accurate computations of back EMF time shape at any load conditions, which in turn gives the instantaneous values of terminal quantities like torque or voltage, depending on the regime of interest.


The permanent magnet machine (PMM) has been represented by means of the traveling non‐dispersive waves of the flux density in the air gap rotating with specified group velocity. The conversion between distributions of the flux density in space and flux linkage in time is obtained through filtering in the spectral domain using 2D or 3D discrete Fourier transform. The change of magnetic saturation due to arbitrary value of the machine load is incorporated by the interpolation between known magnitudes of the basis functions at given a priori RMS values of phase currents. It has been proved that a sinusoidal field machine is particular to the presented theory.

Research limitations/implications

The paper deals with the steady state of PMM; however, the extension towards the transient analysis is possible.

Practical implications

The paper presents a fast and accurate model of PMM for the analysis of its basic electromagnetic quantities.


The analysis of terminal quantities being different in time from sinusoidal or constant distributions, both electrical and mechanical, is usually performed by means of a time stepping approach. The required computing effort is still too high for real time applications. The presented method starts from single FE solutions and converts their accuracy on the set of mutually orthogonal functions having the clear representation in the spectral, mode‐frequency domain. The magnitudes of these basic functions enable one to express the electromagnetic power in a form equivalent to classic dq representation, but not constrained by sinusoidal input quantities.



Witczak, P. and Wawrzyniak, B. (2008), "Spectral analysis of permanent magnet electric machines", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 27 No. 4, pp. 919-928.



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