Modeling of Induction Motor using dq0 Transformations First Semester 1431/1432

Introduction Steady state model developed in previous studies of induction motor neglects electrical transients due to load changes and stator frequency variations. Such variations arise in applications involving variable-speed drives. Variable-speed drives are converter-fed from finite sources, which unlike the utility supply, are limited by switch ratings and filter sizes, i.e. they cannot supply large transient power.

Introduction (cont’d) Thus, we need to evaluate dynamics of converter-fed variable-speed drives to assess the adequacy of the converter switches and the converters for a given motor and their interaction to determine the excursions of currents and torque in the converter and motor. Thus, the dynamic model considers the instantaneous effects of varying voltages/currents, stator frequency and torque disturbance.

Circuit Model of a Three-Phase IM Assumptions and Definitions: Space mmf and flux waves are considered to be sinusoidally distributed, thereby neglecting the effect of teeth and slots. The machine is regarded as group of linear coupled circuits, permitting superposition to be applied, while neglecting saturation, hysteresis, and eddy currents. Ls : self inductance per phase of the stator windings. Ms: mutual inductance per phase of the stator windings. rs: resistance per phase of the stator windings. Lr : self inductance per phase of the rotor windings. Mr: mutual inductance per phase of the rotor windings rr: resistance per phase of the rotor windings. Msr: maximum value of mutual inductance between any stator phase and any rotor phase.

Circuit Model of a Three-Phase IM

Voltage Equations Stator Voltage Equations:

Voltage Equations (cont’d) Rotor Voltage Equations:

Flux Linkage Equations

Flux Linkage Equations In general, we can assume: Let:

Flux Linkage Equations In general, we can assume: Let:

Flux Linkage Equations Stator: Rotor:

Flux Linkage Equations

Model of Induction Motor To build up our simulation equations, we could just differentiate each expression for , e.g. But since Lsr depends on position , which will generally be a function of time, the trigonometric terms will lead to a mess!

Park’s Transformation The Park’s transformation is a three-phase to two-phase transformation for synchronous machine analysis. It is used to transform the stator variables of a synchronous machine onto a dq reference frame that is fixed to the rotor. The +ve d-axis is aligned with the magnetic axis of the field winding and the +ve q-axis is defined as leading the +ve d-axis by /2.

Park’s Transformation (cont’d)  d-axis q-axis In induction machine, the d-axis is assumed to align on a-axis at t = 0 and rotate with synchronous speed () The result of this transformation is that all time-varying inductances in the voltage equations of an induction machine due to electric circuits in relative motion can be eliminated.

Park’s Transformation (cont’d) The Park’s transformation equation is of the form: where f can be i, v, or .

Park’s Transformation (cont’d) where K is a convenient constant. The current id and iq are proportional to the components of mmf in the direct and quadrature axes, respectively, produced by the resultant of all three armature currents, ia, ib, and ic. For balanced phase currents of a given maximum magnitude, the maximum value of id and iq can be of the same magnitude. Under balanced conditions, the maximum magnitude of any one of the phase currents is given by . To achieve this relationship, a value of 2/3 is assigned to the constant K.

Park’s Transformation (cont’d) The inverse transform is given by: Of course, [T][T]-1=[I]

Park’s Transformation (cont’d) Thus, and

Induction Motor Model in dq0 q-axis   d-axis

Induction Motor Model in dq0 (cont’d) Lets us define new “dq0” variables. Our induction motor has two subsystems - the rotor and the stator - to transform to our orthogonal coordinates: So, on the stator, where [Ts]= [T()], ( =  t) and on the rotor, where [Tr]= [T()], ( =  - r = ( r) t )

Induction Motor Model in dq0 (cont’d)

Induction Motor Model in dq0 (cont’d) Now: Just constants!! Our double reference frame transformation eliminates the trigonometric terms found in our original equations.

Induction Motor Model in dq0 (cont’d) Let us look at our new dq0 constitutive law and work out simulation equations.

Induction Motor Model in dq0 (cont’d) Using the differentiation product rule:

Induction Motor Model in dq0 (cont’d) For the stator this matrix is: For the rotor the terminal equation is essentially identical but the matrix is:

Induction Motor Model in dq0 (cont’d) Simulation model; Stator Equations:

Induction Motor Model in dq0 (cont’d) Simulation model; Rotor Equations:

Induction Motor Model in dq0 (cont’d) Zero-sequence equations (v0s and v0r) may be ignored for balanced operation. For a squirrel cage rotor machine, vdr= vqr= 0.

Induction Motor Model in dq0 (cont’d) We can also write down the flux linkages:

Induction Motor Model in dq0 (cont’d) The torque of the motor in qd0 frame is given by: where P= # of poles F=ma, so: where = load torque