AC Machine Modeling

In this chapter, analytical models of induction machines are first described. Analytical models of DFIG are then presented. The models are related to electric machine only. Converter controls will be discussed in Chapter 3. Two examples are given to demonstrate the usage of the analytical models to simulate free acceleration of an induction machine or analyze the consequences of DFIG stator voltage dip.

2.1 Space Vector and Complex Vector Explanation

The type of induction machines to be discussed is three-phase induction machines where both the stator and the rotor have abc windings. The key dynamics of the magnetic field can be described by Faraday’s law that the change of flux field induces electromagnetomotive force (EMF). To model a three-phase system, a direct approach is to express the EMF phase by phase. However, this approach will lead to the use of phase-coupled and time-varied inductances. The contribution of Park’s transformation is to develop analytical models based on a dq rotating reference frame. When the rotating speed of the reference frame is the same as the speed of the rotating flux, the models will have a much simpler form and the inductances of the dq-axis are decoupled and constant.

In this chapter, however, we will not use Park’s transformation to derive the analytical models. Instead, we rely on the concept of space vector to derive the analytical models due to its insightful and simple explanation of physics. In this section, let us first define and explain space vector as well as complex vector.

With the assumptions such as sinusoidal distribution of air gap flux and a uniform air gap, three-phase balanced stator currents in a nonsalient two-pole AC machine (Fig. 2.1) will lead to an magnetomotive force (MMF) which can be viewed as a traveling waveform with a constant magnitude in the air gap (Fig. 2.2) or a rotating MMF—a space vector. This MMF leads to a rotating flux in the air gap (Fig. 2.1). The rotating flux in the air gap then induces sinusoidal EMFs in the three-phase stator and rotor windings.

The concept of space vector comes from this physical mechanism: three-phase currents lead to a rotating MMF. For each phase current, at a random position (defined as angle α referring to the a-axis) in the air gap, the resulting MMF can be found based on Ampere’s law. The mathematical expressions are listed as follows:

a(α)=NiacosαFb(α)=Nibcosα−2π3Fc(α)=Niccosα+2π3

where α is the general angle in the air gap referring to the a-axis, F is MMF and N is the number of windings in each phase.

From the above equations, we can find that Fa reaches maximum when α = 0. Accordingly, Fb is maximum when =2π3; Fc is maximum when =4π3. Consider the current i as a sinusoidal function of time t:

a(t)=Imcos(ωst+θa)ib(t)=Imcos(ωst+θa−2π3)ic(t)=Imcos(ωst+θa+2π3)

where Im is the amplitude of the current, ωs is the angular frequency of the current and θa is the initial angle at t = 0.

Then the total MMF can be found as

(α,t)=NImcos(ωst+θa)cosα+cosωst+θa−2π3cosα−2π3+cosωst+θa+2π3cosα+2π3=32NImcosα−ωst−θa

It can be observed that three-phase balanced currents result in a rotating MMF space vector with a constant magnitude and a constant speed. Based on the physical mechanism, we can now define a space vector through a similar procedure.

The space vector is defined as

⃗(t)=23ej0fa(t)+ej2π3fb(t)+ej4π3fc(t)

Should fa(t), fb(t), and fc(t) be balanced sinusoidal waveforms with an angular speed ω and initial angle θ0, and an amplitude of fm, the space vector can be further expressed as follows:

⃗(t)=|F|ej(ωt+θ0)=|F|ejθ0ejωt=fα−jfβ

where |F| = fm (the amplitude of the waveforms).

The space vector can be further expressed in real and imaginary components fα and fβ, where

αfβ=231cos2π3cos2π30−sin2π3sin2π3fafbfc

2.1.1 Examples of Space Vector