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On a mathematical model of squirrel-cage induction motors

Authors:	Dr Eng P Drozdowski Dr-Habil Eng T J Sobczyk

Affiliation:	(1) Institute of Electrotechnics and Electronics, Technical University of Cracow, Warszawska 24, PL 31-155 Cracow;(2) Institute of Electrical Machines and Power System Control, University of Mining and Metallurgy, Mickiewicza 30, PL 30-056 Cracow, Poland

Abstract:	Contents This paper presents a mathematical model of 3-phase squirrel-cage induction motor accounting for a very high number of space harmonics in which, due to application of a special transformation of voltages and currents, a differential equation system with constant coefficients is obtained. The number of space harmonics is so high that it is possible to perform a direct computation of the alternating component of the electromagnetic torque which decides on the parasitic synchronous torque.An application example of this model for the analysis of steady and dynamic states of a concrete squirrel-cage motor is given. Über ein mathematisches Modell des Käfigläufermotors Übersicht Im Artikel wird ein mathematisches Modell dreiphasiger Käfigläufermotoren mit Berücksichtigung einer großen Zahl räumlicher Harmonischer vorgestellt. Unter Anwendung einer speziellen Transformation der elektrischen Spannungen und Ströme erhält man ein Gleichungssystem mit konstanten Koeffizienten Die Zahl der räumlichen Harmonischen ist so groß, daß die Berechnung der Wechselkomponenten des elektromagnetischen Moments und damit des parasitären synchronen Moments möglich ist. Die Anwendung des mathematischen Modells wird anhand eines Beispiels für dynamischen und statischen Betrieb vorgestellt. List of symbols and abbreviations $a = e^{j\frac{{2\pi }}{3}} ,b = e^{j\frac{{2\pi }}{N}}$ N number of bars of the rotor cage - L ₈ total self inductance of stator phase winding - M ₈ mutual inductance of stator windings - R _rg,L _rg resistance and leakage inductance of a rotor end ring segment-respectively - R _b,L _b resistance and leakage inductance of a rotor cage bar-respectively $k_{rk} = \sin \left( {k\frac{\pi }{N}} \right)$ - k=0,1, ... N—1 number of the rotor currents symmetric component and theM _sr matrix column corresponding to it $\mu _0 = 4 \cdot 10^{ - 7} \frac{H}{m}$ - l equivalent axial length of stator core - equivalent width of air-gap - p pole pair numer - z number of turns per phase - v order of harmonic - k _sv stator winding coefficient for thev-th harmonic - k _rv rotor winding coefficient for thev-th harmonic - k _skv skewness coefficient for thev-th harmonic - U _s stator voltage vector in symmetrical components - U _r rotor voltage vector in symmetrical components - i _s stator current vector in symmetrical components - O zero matrix - R _s stator resistance matrix - R _r rotor resistance matrix - LL _s stator inductance matrix - L _rr rotor inductance matrix - M _sr matrix of stator-rotor mutual inductances - T _e electromagnetic torque - T _m motor load torque - rotor position angle - ₀ initial rotor position angle - $\omega = \frac{{d\varphi }}{{dt}}$ rotor angular velocity - (*) conjugation index of a complex number - Re {y} real part of a complex number - Im {y} imaginary part of a complex number - T transposition index - X]^x highest integer not greater thanX and of the same sign asX $v(\bmod N) = v - \left {\frac{v}{N}} \right]^x \cdot N$ - U r.m.s. value of the phase voltage - _u pulsation of supply voltage

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