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# 3Dynamic Analysis of Induction Machines in Terms of dq Windings

## 3-1    Introduction

In this chapter, we will develop equations to analyze induction machine operation under dynamic conditions. We will make use of space vectors as intermediary in transforming a-b-c phase winding quantities into equivalent dq-winding quantities that we will use for dynamic (nonsteady state) analysis. We will see in later chapters the benefits of d- and q-axis analysis in controlling ac machines.

## 3-2    dq Winding Representation

We studied in the previous chapter that the stator and the rotor flux linkages and depend on the rotor angle θm because the mutual inductances between the stator and the rotor windings are position dependent. The main reason for the d- and q-axis analysis in machines like the induction machines is to control them properly, for example, using vector control principles. In most textbooks, this analysis is discussed as a mathematical transformation called the Park's transformation. In this chapter, we will take a physical approach to this transformation, which is much easier to visualize and arrive at identical results.

### 3-2-1    Stator dq Winding Representation

In Fig. 3-1a at time t, phase currents ia (t), ib (t), and ic (t) are represented by a stator current space vector

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