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Introduction
This package presents computer models of electric machines leading to the assessment of the dynamic performance of open- and closed-loop ac and dc drives. The Simulink/Matlab implementation is adopted because of its inherent integration of vectorized system representations in block diagram form, of numerical analysis methods, of graphical portrayal of time evolutions of signals combined with the simple implementation of the functionality of controllers and power electronic excitations. The development of Simulink models of drive assemblies is a relatively simple task consisting of combining inputoutput block representation of the various components making up the system. This approach provides a powerful design tool because of the ease of observing the effects of parameter modifications and of changes in system configurations and control strategies.
Under the rubric Animations, a series of movie clips portrays the motion of electric machines, magnetic fields, and space vectors.

The approach
Electric machines
The starting step in the mathematical modeling of ac machines is to describe them as coupled stator and rotor polyphase circuits in terms of so-called phase variables, namely stator currents ias, ibs, ics; rotor currents iar, ibr, icr for an induction machine or if, ikd, ikq for a synchronous machine; the rotor speed ωm ; and the angular displacement θ between stator and rotor windings. The magnetic coupling is expressed in terms of an inductance matrix which is a function of position θ. The matrix expression of the machine equations are readily formulated in Matlab or Simulink language. A detailed example of this approach is given in a later section.
The next step is to transform the original stator and rotor abc frames of reference into a common k or dq frame in which the new variables for voltages, currents, and fluxes can be viewed as 2-D space vectors. In this common frame the inductances become constant independent of position. Figure 1 illustrates various reference frames (coordinate systems): the triplet [As Bs Cs] denotes a three-phase system attached to the stator while the pair [as bs] corresponds to an equivalent two-phase system (zerosequence components can be ignored in Y-connected ac machines in which the neutral is normally isolated). Among possible choices of dq frames are the following:
a) Stator frame where ωk = 0
b) Rotor frame where ωk = ωm
c) Synchronous frame associated with the frequency ωs (possibly time varying)
of the stator excitation.
d) Rotor flux frame in which the d-axis lines up with the direction of the rotor
flux vector.
The choice of the common dq frame is usually dictated by the symmetry constraints imposed by the construction and excitation of the machine. With the complete symmetry encountered in a three-phase induction machine with balanced sinusoidal excitation, any one of the five frames can be used, although the synchronous frame is more convenient in as much as all signals appear as constant dc in steady state. However, certain control strategies may require the adoption of a specific frame, as is the case of vector control where the reference frame is attached to the rotor flux vector. In the presence of asymmetry, the common frame is attached to the asymmetrical member: an induction motor with unbalanced excitation or asymmetrical stator windings (the case of a capacitor motor) will be modeled in the stator frame where as a synchronous machine is represented in the rotor frame. In the common dq frame, the machine dynamic equations appear as differential equations with constant coefficients (independent of rotor position) and nonlinearities confined to products of variables associated with speed voltages and torque components.