STABILITY ANALYSIS OF INDUCTION MOTORS UNDER VARIOUS MODES OF OPERATION

Abstract

The prediction of open-loop stability of an induction motor is important for considering its suitability for use under variable frequency or phase control mode of operation, or to ensure its stability at the design stage through control of para meters. The available literature deals only with the stability analysis of symmetrical three phase induction motor under variable frequency mode of operation, with balanced sinusoidal applied voltage. This dissertation presents a comprehensive stability analysis and study using an approximate analytical technique of nonlinear system analysis, viz. the harmonic balance method, for variable frequency mode of operation, and direct digital simula tion for phase angle control mode of operation of polyphase and single phase induction motors. For stability analysis of induction motors the steady state solution of the dynamic system equations at the operating point under consideration is required to be obtained in analytical form. The nonlinear system equations for the induction motors cannot be solved by exact analytical methods except in the case of symmetrical induction motor for which the system equations become linear differential equations with constant coefficients at steady state, as the speed become constant. The conventional methods of electrical machines analysis for obtaining steady state solutions are inadequate for the dynamic problem of stability analysis of other types of induction motors, because they do not take into account the harmonic variations of speed. ii As a first step towards the determination of stability under variable frequency mode of operation the harmonic balance method is used for obtaining steady state solution of the non linear system equations. This method of obtaining solution of nonlinear system equations is suitable under the following situa tion- (i) if the existence and the nature of steady state periodic solution of the nonlinear system can be assumed ; and/or (ii) if the nonlinear system contains periodic forcing functions These conditions are fulfilled in the case of all types of induction motors with continuous variables and forcing functions. This method is therefore used for the stability analysis of sym metrical as well as asymmetrical induction motors rnder variable frequency mode of operation. This method is applied as follows. (1) A suitable periodic solution in Fourier series expansion with unknown coefficients is assumed for all the variables in the system equations. These variables are assumed to be varying slowly near the steady state. (2) The assumed solution is substituted in the system equation?, and the terms containing sine and cosine components are separated out from each equation. This gives autonomous system equations. (3) For determining the values of the unknown coefficients of the Fourier series at the steady state operating point, the nonlinear algebraic equations obtained from (2) are solved for the variables involved. For this purpose the iii procedure is indicated below* (a) Any constant speed is considered on the normal operating region of the torque versus speed charac teristic of the motor. With constant speed, the nonlinear algebraic equations are linearised. These linear algebraic equations are solved by Gaussion elimination method or any other suitable method. (b) For solving the complete set of nonlinear algebraic equations, the values of the variables obtained in (a), together with the constant speed, are taken as the initial guess. The nonlinear algebraic equations are then solved by Newton-Raphson method. This procedure gives a very fast convergence - in three or four iterations. The steady state solution thus obtained gives the har monic speed variation of the induction motor as an integral part of the solution. The solutions are more accurate, through computer oriented, compared to those obtained by conventional methods of electrical machines analysis, as the assumption of constant speed is only an intermediate step, and is not used in the final solution. (*f) For stability analysis the nonlinear autonomous system equations obtained in (2) are linearised about the steady state operating point. The characteristic equation is tested by Routh-Hurwitz criterion for checking the stabi lity of the operating point.

The following cases of induction motors under variable frequency mode of control are studied for steady state solution and stability analysis of the operating point by the harmonic balance method. Symmetrical induction motor with balanced sinusoidal applied voltages Symmetrical induction motor with unbalanced sinusoidal applied voltages Single phase induction motor Permanent split-phase motor Permanent-split capacitor motor. The results obtained for symmetrical induction motor are verified against those reported by previous authors. In other cases the results have been verified by direct digital simulation of the system equations. It has been observed that the method gives sufficient accuracy in all cases, even with the first order approximation in the harmonic balance method. Using the above method the effects of variations of para meters on the instability regions are studied in detail. The qualitative theory of differential equations is applied to obtain a better understanding of the nature of the steady state solutions of the system equations thereby bringing into sharper focus the meaning of the term "STABILITY" as applied to induction motors. The conventional definition of instability of induction motors, viz. a state of sustained oscillations, is inadequate when applied to single phase and asymmetrical polyphase induction motors. Amore comprehensive definition of stability has been formulated in the present work. The existence of the phenomena of "subharmonic oscillations" and "almost periodic oscillations" has been verified in all the cases of induction motors. A wide variety of such oscillations is possible with certain combinations of machine parameters and forcing functions. For phase angle control of induction motors the applied voltages are and currents may be discontinuous, and direct digital simulation of the system equations is unavoidable. This method has been used here for the stability study of single phase induc tion motor and symmetrical two phase induction motor under phase angle control mode of operation. In order to overcome the inherent drawback of direct digital simulation in terms of excessive computer time, fast converging techniques have been presented which substantially reduce the computer time. In brief, a complete investigation of the stability problem of symmetrical and asymmetrical induction motors under variable frequency and phase-controlled modes of operation is presented in this dissertation.