Modeling of Induction Machines

Abstract

This chapter introduces and explains mathematical model of induction machines. This model represents transient and steady-state behavior in electrical and mechanical subsystems of the machine. Analysis and discussion introduces and explains Clarke and Park coordinate transforms. The model includes differential equations that express the voltage balance in stator and rotor windings, inductance matrix which relates flux linkages and currents, Newton differential equation of motion, expression for the air-gap power, and expression for the electromagnetic torque. The model development process starts with replacing the three-phase machine with two-phase equivalent. Namely, the three-phase voltages, currents, and flux linkages are transformed in two-phase variables by appropriate transformation matrix which implements 3$Φ$/2$Φ$ transform, also called Clarke coordinate transform. Two-phase model is formulated in stationary coordinate frame. The drawbacks and difficulties in using this model are the rationale for introducing and applying Park coordinate transform, which results in the machine model in synchronous dq coordinate frame. Necessary techniques and procedures of applying and using coordinate transforms are explained in detail, including representation of machine vectors by complex numbers. The operable model of induction machines is obtained in dq coordinate frame which revolves synchronously with the stator field. The merits and practical uses of the model in dq frame are explained at the end of the chapter.