2
Induction Machine Equations in Phase Quantities: Assisted by Space Vectors

2-1    Introduction

In ac machines, the stator windings are intended to have a sinusoidally distributed conductor density in order to produce a sinusoidally distributed field distribution in the air gap. In the squirrel-cage rotor of induction machines, the bar density is uniform. Yet the currents in the rotor bars produce a magnetomotive force (mmf) that is sinsuoidally distributed. Therefore, it is possible to replace the squirrel-cage with an equivalent wound rotor with three sinsuoidally distributed windings.

In this chapter, we will briefly review the sinusoidally distributed windings and then calculate their inductances for developing equations for induction machines in phase (a-b-c) quantities. The development of these equations is assisted by space vectors, which are briefly reviewed. The analysis in this chapter establishes the framework for the dq winding-based analysis of induction machines under dynamic conditions carried out in the next chapter.

2-2    Sinusoidally Distributed Stator Windings

In the following analysis, we will also assume that the magnetic material in the stator and the rotor is operated in its linear region and has an infinite permeability.

In ac machines of Fig. 2-1a, windings for each phase ideally should produce a sinusoidally distributed radial field (F, H, and B) in the air gap. Theoretically, this requires a sinusoidally distributed winding in each phase. If each phase winding has a total of Ns turns (i.e., 2Ns conductors), the conductor density ns(θ) in phase-a of Fig. 2-1b can be defined as

The angle θ is measured in the counter-clockwise direction with respect to the phase-a magnetic axis. Rather than restricting the conductor density expression to a region 0 < θ < π, we can interpret the negative of the conductor density in the region π < θ < 2π in Eq. (2-1) as being associated with carrying the current in the opposite direction, as indicated in Fig. 2-1b.

In a multi-pole machine (with p > 2), the peak conductor density remains Ns/2, as in Eq. (2-1) for a two-pole machine, but the angle θ is expressed in electrical radians. Therefore, we will always express angles in all equations throughout this book by θ in electrical radians, thus making the expressions for field distributions and space vectors applicable to two-pole as well as multi-pole machines. For further discussion on this, please refer to example 9-2 in Reference [1].

The current ia through this sinusoidally distributed winding results in the air gap a magnetic field (mmf, flux density, and field intensity) that is co-sinusoidally distributed with respect to the position θ away from the magnetic axis of the phase

and

The radial field distribution in the air gap peaks along the phase-a magnetic axis, and at any instant of time, the amplitude is linearly proportional to the value of ia at that time. Notice that regardless of the positive or the negative current in phase-a, the flux-density distribution produced by it in the air gap always has its peak (positive or negative) along the phase-a magnetic axis.

2-2-1    Three-Phase, Sinusoidally Distributed Stator Windings

In the previous section, we focused only on phase-a, which has its magnetic axis along θ = 0°. There are two more identical sinusoidally distributed windings for phases b and c, with magnetic axes along θ = 120° and θ = 240°, respectively, as represented in Fig. 2-2a. These three windings are generally connected in a wye-arrangement by joining terminals a′, b′, and c′ together, as shown in Fig. 2-2b. A positive current into a winding terminal is assumed to produce flux in the radially outward direction. Field distributions in the air gap due to currents ib and ic are identical in sinusoidal shape to those due to ia, but they peak along their respective phase-b and phase-c magnetic axes.

2-3    Stator Inductances (Rotor Open-Circuited)

The stator windings are assumed to be wye-connected as shown in Fig. 2-2b where the neutral is not accessible. Therefore, at any time

For defining stator-winding inductances, we will assume that the rotor is present but it is electrically inert, that is “somehow” hypothetically of-course, it is electrically open-circuited.

2-3-1    Stator Single-Phase Magnetizing Inductance Lm,1-phase

As shown in Fig. 2-3a, hypothetically exciting only phase-a (made possible only if the neutral is accessible) by a current ia results in two equivalent flux components represented in Fig. 2-3b: (1) magnetizing flux which crosses the air gap and links with other stator phases and the rotor, and (2) the leakage flux which links phase-a only. Therefore, the self-inductance of a stator phase winding is

Therefore,

It requires no-load and blocked-rotor tests to estimate the leakage inductance Lℓs, but the single-phase magnetizing inductance Lm,1-phase can be easily calculated by equating the energy storage in the air gap to :

where r is the mean radius at the air gap, ℓ is the length of the rotor along its shaft axis, Ns equals the number of turns per phase, and p equals the number of poles.

2-3-2    Stator Mutual-Inductance Lmutual

As shown in Fig. 2-4, the mutual-inductance Lmutual between two stator phases can be calculated by hypothetically exciting phase-a by ia and calculating the flux linkage with phase-b

Note that only the magnetizing flux (not the leakage flux) produced by ia links the phase-b winding. The current ia produces a sinusoidal flux-density distribution in the air gap, and the two windings are sinusoidally distributed. Therefore, the flux linking phase-b winding due to ia can be shown to be the magnetic flux linkage of phase-a winding times the cosine of the angle between the two windings (which in this case is 120°):

Therefore, in Eq. (2-8), using Eq. (2-6a) and Eq. (2-9b),

The same mutual inductance exists between phase-a and phase-c, and between phase-b and phase-c.

The expression for the mutual inductance can also be derived from energy storage considerations (see Problem 2-2).

2-3-3    Per-Phase Magnetizing-Inductance Lm

Under the condition that the rotor is open-circuited, and all three phases are excited in Fig. 2-2b such that the sum of the three phase currents is zero as given by Eq. (2-5),

Using Eq. (2-10) for Lmutual, and from Eq. (2-5) replacing (−ib − ic) by ia in Eq. (2-11),

Using Eq. (2-7),

Note that the single-phase magnetizing inductance Lm,1-phase does not include the effect of mutual coupling from the other two phases, whereas the per-phase magnetizing-inductance Lm in Eq. (2-13) does. Hence, Lm is 3/2 times Lm,1-phase.

2-3-4    Stator-Inductance Ls

Due to all three stator currents (not including the flux linkage due to the rotor currents), the total flux linkage of phase-a can be expressed as

where the stator-inductance Ls is

2-4    Equivalent Windings in a Squirrel-Cage Rotor

For developing equations for dynamic analysis, we will replace the squirrel cage on the rotor by a set of three sinusoidally distributed phase windings. The number of turns in each phase of these equivalent rotor windings can be selected arbitrarily. However, the simplest, hence an obvious choice, is to assume that each rotor phase has Ns turns (similar to the stator windings), as shown in Fig. 2-5a. The voltages and currents in these windings are defined in Fig. 2-5b, where the dotted connection to the rotor-neutral is redundant for...