8/19/2019 A Unified Wavelet-based Approach to Electrical Machine Modeling http://slidepdf.com/reader/full/a-unified-wavelet-based-approach-to-electrical-machine-modeling 1/5 A unified wavelet-based approach to electrical machine modeling S. Fedrigo c), . Gandelli c), A. Monti < ), F. Ponci (') 7 ipartimento di Elettrotecnica Politecnico di Milano Piazza Leonard0 da Vinci 32 20133 Milano (Italy) epartment of Electrical Engineering University of South Carolina Columbia, SC 29208 USA) Abstract- The paper presents an original approach to electrical machine modeling based on wavelet transformation. The main purpose of the approach is to give a more detailed description of the field in the airgap removing the hypothesis of sinusoidal field distribution typical of traditional space-phasor approach. As result, more detailed closed-form analysis is possible including torque ripple evaluation. 1. INTRODUCTION The modeling of air-gap electromotive force for motor design or motor control has followed different ways. In particular both numerical, through finite elements computation and analytical ways have been followed. All the proposed methods are in general time consuming [ 11-[2] and are structured to supply high level of detail. On the other hand the traditional space phasor approach implies a rough approximation neglecting the presence of the slots and assuming ideal materials and ideal conductors distribution. The method here proposed allows the synthesis of a more realistic model of the machine still requiring a low computational effort. The traditional approach for describing magnetic inter action s within electrical machines air-gap is based on a simplified space representation of the magnetic field. In particular, under ideal material conditions, the magneto- motive force along the air gap i s square-shaped as reported in Figure 1. For calculation purpose only the first harmonic in space, corresponding to a s inusoidal winding distr ibuti on, is considered. Figure 1: electro-magnetic force along the air gap and its irst space harmonic within a rotating electrical machine In this paper we propose a new simplified method to analyze the magneto-motive force without these assumptions. This method will lead to satisfactory results compared with traditional approach both in term of torque and torque ripple analysis. Starting from this new approach, we will achieve an original and unified way to study rotating electrical machines without the limits related to the electric currents waveform and physical structures. To achieve this result the Haar wavelet approach is used. Haar wavelets proved to be particularly efficient in synthetically represent a piecewise constant function with compact support such as the magnetic field along the air gap in electrical rotating machines. 11. HAAR RANSFORM BASICS The basics of Haar wavelet approach and applications are widely presented in [l-71. In particular we recall here that the orthogonal set of Haar functions, har(k,t), is defined from the Haar mother wavelet 1 ifOlt<1/2 h ar (l ,t )= -1 if 1/21t<l 0 for any other interval by translation of the non-negative integer n = k - and by contraction of 2' where j is the scale parameter. The function har(O,t), equal to unity in the interval [0,1], closes the orthogonal set. In our studies we deal with finite sequences of N samples of the analysed signal, {x(n N)} , where N = 2' . A band- limited Haar-transformable function is gven whose maximum wavelet frequency equals to N/2. Uniform sampling in the interval [0,1] is carried out, yielding a sequence of N samples univocally related to the Haar- sequence of N wavelet-coeficie nts. By ordering these sequences respectively i n the column vectors of samples xNand coefficients X, , he matrix relation follows: where we have introduced the square matrix [H,] , whose N rows are made up of samples of ordered Haar-functions: 0 7803 7091 0/01/$10°2001 IEEE 765
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A Unified Wavelet-based Approach to Electrical Machine Modeling
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8/19/2019 A Unified Wavelet-based Approach to Electrical Machine Modeling