Topic 5: Dynamic Simulation of

Induction Motor

Spring 2004

ECE 8830 - Electric Drives

Stationary Reference Frame Modeling of the Induction Motor

We now consider how the model of the induction motor that we have developed can be used to simulate the dynamic performance of the induction motor.

We will consider the model of the motor in the stationary reference frame.

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Consider a 3, P-pole, symmetrical induction motor in the stationary reference frame with windings as shown below:

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Consider first the input voltages for the given neutral connections of the stator and rotor windings shown.

The three applied voltages to the stator terminals vag, vbg, and vcg need not be balanced or sinusoidal. In general, we can write:

cs cg sgv v v bs bg sgv v v

as ag sgv v v

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Therefore,

In simulation, the voltage vsg can be determined from the flow of phase currents into the neutral connection by:

where Rsg and Lsg are the resistance and inductance between the two neutral points. Of course, if s and g are shorted, vsg=0.

3 ( ) ( )sg as bs cs ag bg cgv v v v v v v

0( ) ( ) 3sg sg as bs cs sg as bs cs sg sg s

d dv R i i i L i i i R L i

dt dt

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Now consider transformation of stator abc phase voltages to qd0 stationary voltages.

With the q-axis aligned with the stator a-phase axis, the following equations apply:

2 1 1 2 1 1

3 3 3 3 3 3sqs as bs cs ag bg cg sgv v v v v v v v

1 1( ) ( )3 3

sds cs bs cg bgv v v v v

0

1 1( ) ( )3 3s as bs cs ag bg cg sgv v v v v v v v

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Transformation of the abc rotor winding voltages to the qd0 stationary reference frame can be done in two steps.

First transform the “referred” rotor abc phase voltages to a qd0 reference frame attached to the rotor with the q-axis aligned to the axis of the rotor’s a-phase winding.

In the second step, transform the qd0 rotor quantities to the stationary qd0 stator reference frame.

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Step 1 ->

where vrn’ = voltage between points r and n and the primes indicate voltages referred to the stator side.

' ' ' ' ' ' ' '2 1 1 2 1 1

3 3 3 3 3 3rqr ar br cr an bn cn rnv v v v v v v v

' ' ' ' '1 1( ) ( )3 3

rdr cr br cn bnv v v v v

' ' ' ' ' ' ' '0

1 1( ) ( )3 3

rr ar br cr an bn cn rnv v v v v v v v

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Step 2 ->

where r(t) = rotor angle

at time t, r(0)= rotor angle

at time t=0, and r(t) =

angular velocity of rotor.

' ' 'cos ( ) sin ( )s r rqr qr r dr rv v t v t

' ' 'sin ( ) cos ( )s r rdr qr r dr rv v t v t

0

( ) ( ) (0)t

r r rt t dt

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

The qd0 voltages at both the stator and rotor terminals, referred to the same stationary qd0 reference frame, can be used as inputs along with the load torque to obtain the qd0 currents in the stationary reference frame. These can then be transformed to obtain the phase currents in the stator and rotor windings.

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

The inverse transformation to obtain the stator abc phase currents from the qd0 currents is given by:

0s

as qs si i i

0

1 3

2 2s s

bs qs ds si i i i

0

1 3

2 2s s

cs qs ds si i i i

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

The abc rotor currents are obtained by a two-step inverse transformation process. Step 1 transforms the stationary qd0 currents back to the qd frame attached to the rotor. Step 2 resolves the qd rotor currents back to the abc rotor phase currents.

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Step 1 ->

Step 2 ->

' ' 'cos ( ) sin ( )r s sqr qr r dr ri i t i t ' ' 'sin ( ) cos ( )r s sdr qr r dr ri i t i t

' ' '0

r sar qr ri i i

' ' ' '0

1 3

2 2r r rbr qr dr ri i i i

' ' ' '0

1 3

2 2r r rcr qr dr ri i i i

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

The model equations can be rearranged into the form of equations (6.112) to (6.117) in Ong’s book (provided in separate handout).

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

The torque equation is:

(eq. 6.118)

The equation of motion of the rotor is given by:

where Tmech is the externally-applied mechanical torque in the direction of the rotor speed and Tdamp is the damping torque in the opposite direction of rotation.

3( )

2 2s s s s

em ds qs qs dsb

PT i i

rmem mech damp

dJ T T Tdt

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Normalized to the base (or rated speed) of the rotor b is given by:

(eq. 6.120)2 ( / )b r b

em mech damp

J dT T T

P dt

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Stationary Reference Frame Modeling of the Induction Motor (cont’d)

Saturation of Mutual Flux

See Ong text.

Linearized Model

Solving the nonlinear equations by numerical integration allows visualization of the dynamic performance of a motor. However, in designing a control system, we would like to use linear control techniques. For this application we need to develop a linearized model of the induction motor.

Linearized Model (cont’d)

To develop a linearized model for the

induction motor, we select an operating point and perturb the system with small perturbations over a linear regime.

Linearized Model (cont’d)

The general form of the behavior of the induction motor may be described by the function:

f( , x, u, y) =0 where x is a vector of state variables ( ); u is the vector of

input variables ( ); and y is the vector of desired outputs, such as

.

x

' ', , , , /qs qr ds dr r b , ,e e

qs ds mechv v T

, ,e eqs ds emi i T

Linearized Model (cont’d)

When a small perturbation is applied to each of the components of the x, u, and y variables, the perturbed variables will satisfy the equation:

where the 0 subscript denotes the steady state value about which the perturbation is applied.

f( xx=x0+x , x0+ x , u0+u, y0+y) =0

Linearized Model (cont’d)

In steady state,

xx=x0

Neglecting higher order terms and regrouping some of the terms in the earlier equations, the linear equations including perturbations can be re-written as:

x A x B u

y C x D u

Linearized Model (cont’d)

See Ong text to learn how to use Matlab/Simulink to solve for the [A B C D] matrix.

See handout from Krishnan’s book for more detailed description of small signal analysis of induction motor.