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Since the first edition of Analysis of Electric Machinery was published, the reference frame theory that was detailed in the book has become the universally accepted approach for the analysis of both electric machines and electric drive systems. Now in its second edition, Analysis of Electric Machinery and Drive Systems presents, in one resource, the application of this theory to the analysis, simulation, and design of the complete drive system including the machine, converter, and control. | Chapter 8 LINEARIZED MACHINE EQUATIONS 8.1 INTRODUCTION The equations that describe the behavior of the induction and synchronous machines are nonlinear and can be solved only with the aid of a computer. However considerable insight can be gained regarding the small-excursion behavior from the linearized version of these equations. Although standard computer algorithms may be used to automatically linearize these equations it is important to be aware of the steps necessary to perform linearization. This procedure is set forth by applying Taylor s expansion about an operating point. The resulting set of linear differential equations describes the dynamic behavior of small displacements or small excursions about this operating point. The machines can then be treated as linear systems with regard to small disturbances whereupon basic linear system theory can be used to calculate eigenvalues and to establish transfer functions for use in the design of controls associated with the machines. In this chapter the nonlinear equations of the induction and synchronous machines are linearized. Although these equations are valid for operation with stator voltages of any frequency as in variable-frequency drive systems only rated frequency operation is considered in detail. Eigenvalues are determined for both induction and synchronous machines and transfer functions are formulated for the purpose of establishing the basis from which control analysis and design may begin. With present-day computer languages linearization calculation of eigenvalues and formulation of transfer functions can be done automatically from the nonlinear equations that describe these machines 1-3 . The material presented in this chapter serves to inform the reader of the steps necessary in order to automatically characterize small-excursion operation of induction and synchronous machines. 311 312 LINEARIZED MACHINE EQUATIONS 8.2 MACHINE EQUATIONS TO BE LINEARIZED The linearized machine equations are .
