Mathematical model of small induction motors

Mathematical Model of Small Induction Motors with Magnetic Asymmetry
Vladimir A. Lavrinenko Department of Electrical Engineering, Chuvash State University, Cheboksary, 4280 17, Russia Phone: 07-8352-443830 E-mail: vodol@chuvsu.ru
Abstract - The paper presents d-q model for study the steady-state performances of a capacitor induction motor, in which took into account mutual interaction betweenspace harmonics of magnetic conductivity and magnetomotive forces. The model base upon crossfield theory and method of the harmonic analysis for the air gap permeance. Performance characteristics for capacitor induction motor are computed. The mathematical model has suflcient accuracy of calculation. Key words: capacitor induction motor, space harmonics, cross-field theory, harmonic analysis

1.Introduction
Capacitor motors (Fig.1) are widely used to drive fans in household appliances. The popularity of these motors is mainly due to their low cost. The demerit of the motor is a deep in a torque-speed curve because of influence of additional torques. Mutual interactions between space harmonics of the air gap permeances and magnetomotive forces are the reasons of making of additionalasynchronous torques. This defect can become the cause starting torque difficulties.

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Fig. 1. Cross-section of a salient-pole capacitor motor: 1 - yoke of stator core; 2 - main winding; 3 - squirrel cage winding; 4 auxiliary winding; 5 magnetic shunt

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There are many parameters influencing the shape of the flux density distribution curve. The increase of the air gap is thesimplest method. Optimal using of asymmetrical air gap enables to improve the shape of the flux density distribution curve and starting characteristics easily [14,15]. The main feature of the motor is the stepped air gap. The air gap under magnetic shunts has major value than under magnetic poles. Although construction of these motors is relatively simple, theoretical analysis is difficult suchthat much design work is accomplished empirically. The pioneer of the theory of the two phase motor was Morill [18], who studied the capacitor motor, and Trickey who gave before World War I1 the paper [22] where he considered a starting performances of shaded-pole motors. In the late fifties, discussions were not achieved conceming the validity and physical significance of the revolving field andcross field theories of the purely single-phase squirrel cage machine. In the mid sixties, Alger [l] still used the expressions ((the dilemma of single-phase induction machine theory.)) One of the most preeminent specialists in the world, Veiinott, writes about belt leakage reactance in squirrel cage motors [30]. The use of the generalized machine concept might contribute to enlighten the problem[6,7,10,2 1,271. Mishkin [17] has presented a model based upon direct solution of the air gap field equations. Williamson and Begg used for the analysis of cage induction motors the combine’d approach based on a field theory and circuit theory [33]. The analysis developed by Pdoujadoff [21] is based on electric circuits with time varying coefficients. When evaluating the air gap inductancesinvolving the rotor meshes, a rather complicated double integration process is required. Vas has investigated transient and steady-state operation of induction motors with electrical asymmetry [25,26]. He has presented a method in which took into account saturation in nonsalient-pole machines [27,28]. The equations in axes revolving with arbitrary speed conceming a stator are received. Principalperformance equations are derived in [34] by applying the symmetrical components method to an equivalent circuit. Advanced modeling techniques for the design of an innovative induction machine is created in [35].

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In 1998, I presented original model of the shadedpole induction motor with a stepped air gap [13]. This was a combined field and circuit theory as the one presented below. From...