8.2 Dynamic Model of an Induction Machine

A three-phase current system can be represented by a three-axis coordinate system, as shown in Figure 8.1a. Unfortunately, the three axes are linearly dependent, a fact that makes the mathematical description of a three-phase machine difficult. However, the linear dependence means that only two variables are necessary to describe three physical quantities. Hence, a complex, linearly independent coordinate system can be selected. Figure 8.1b shows the equivalent representation.

Figure 8.1 Coordinate transformation. (a) Currents expressed in a three-phase reference frame (a, b, c). (b) Currents expressed in a complex reference frame αβ

A three-phase stator currents system, with angular frequency ω₀, can be defined in a fixed three-phase coordinate frame:

8.1

8.2

8.3

For the stator currents, the transformation from a three- to a two-phase system is described by

8.4

8.5

8.6

The same coordinate transformation shown above is used for the electromagnetic variables. Thus, the equations of an induction machine [16] can be represented in any arbitrary reference frame rotating at an angular frequency ω_k. The variable ω denotes the rotor angular speed:

8.7

8.8

8.9

8.10

8.11

where:

• L_s, L_r, and L_m are the stator, rotor, and magnetizing inductances, respectively.
• R_s and R_r are the stator and rotor resistances.
• v_s and i_s are the stator voltage and current vectors.
• i_r is the rotor current vector.
• and are the stator and rotor flux vectors.
• T and p are electromagnetic torque and number of pole pairs, respectively.
• is the complex conjugate value of .

In (8.8), the rotor vector voltage v_r is equal to zero because a squirrel-cage motor is considered. Hence, the rotor winding is short-circuited.

If the mechanical equation of the rotor is considered in (8.12), it is possible to see that the torque affects the ratio of change in the mechanical rotor speed ω_m:

8.12

The coefficient J in (8.12) denotes the moment of inertia of the mechanical shaft, and T_l is the load torque connected to the machine; it corresponds to an external disturbance, which must be compensated by the control system. ω_m is the mechanical rotor speed, which is related to the electric rotor speed ω by the number of pole pairs p:

8.13

In order to develop an appropriate control strategy, it is convenient to write the equations of the machine in terms of state variables. The stator current i_s and the rotor flux vectors are selected as state variables. The stator current is especially selected because it is a variable that can be measured, and also undesired stator dynamics, like effects on the stator resistance, stator inductance, and back-emf, are avoided. Thus, according to [17] and [18], the equivalent equations of the stator and rotor dynamics of a squirrel-cage induction machine are obtained:

8.14

8.15

where

These equations will be used for estimating the stator and rotor flux, and for calculating predictions for the stator currents, stator flux, and electrical torque using the appropriate discrete-time version of the equations.