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On a mathematical model of squirrel-cage induction motors

Über ein mathematisches Modell des Käfigläufermotors

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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.

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.

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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

δ:

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

ϕ:

rotor position angle

ϕ0 :

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

ω u :

pulsation of supply voltage

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Authors and Affiliations

  1. Institute of Electrotechnics and Electronics, Technical University of Cracow, Warszawska 24, PL 31-155, Cracow

    P. Drozdowski

  2. Institute of Electrical Machines and Power System Control, University of Mining and Metallurgy, Mickiewicza 30, PL 30-056, Cracow, Poland

    T. J. Sobczyk (Associate Professor)

Authors
  1. P. Drozdowski
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  2. T. J. Sobczyk
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Drozdowski, P., Sobczyk, T.J. On a mathematical model of squirrel-cage induction motors. Archiv f. Elektrotechnik 70, 371–382 (1987). https://doi.org/10.1007/BF01574005

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