Simulation of an Induction motor in different frame of references

As the application of induction motor in the industry is huge, it is important to study its steady-state along with the transient behaviour. Various model has been developed for this, and the d-q model or two-axis model has been tested good and proven to be accurate.

Equations that govern the performances of an induction motor are

• 3 phase to two phase conversion
• flux equation
• equivalent circuit equation
• torque euation
• motor equation

The matrix below represents the transformation from three phase to two phase

`[F_{0dq}]=[P_theta][F_{ABC}]`

where, `[F_{0dq}]`is the zero, d and q axis voltages, `[F_{ABC}]`is the three phase votage matrix and `[P_theta]`is the transformation matrix

`[P_theta]=[[0.5,0.5,0.5],[costheta,cos(theta-2pi/3),cos(theta+2pi/3)],[sin theta,sin(theta-2pi/3),sin(theta+2pi/3)]]`

Sufficient flux is generated by the flux generator equations i.e.

`V_{ds}=R_si_{ds}-omega_rlamda_{qs}+Slamda_{ds}`

`V_{qs}=R_si_{qs}+omega_rlamda_{ds}+Slamda_{qs}`

equivalent circuit equation: Based on the equivalent circuit discussed in the lecture, we can write it as

`V_{rd}=0=R_ri_{rd}-omega_{da}lamda_{rq}+S{L_{lr}i_{rd}+L_mi_{sd}+L_mi_{rd}}`

`V_{rq}=0=R_ri_{rq}+omega_{da}lamda_{rd}+S{L_{lr}i_{rq}+L_mi_{sq}+L_mi_{rq}}`

torque equation gives the expression for the electromagnetic torque i.e.

`T_e = 3/2 P/2(lamda_{ds}i_{qs}-lamda_{qs}i_{ds})`

`lamda_{qs}=L_mi_{qr}+(L_{ls}+L_m)i_{qs}`

`lamda_{ds}=L_mi_{dr}+(L_{ls}+L_m)i_{ds}`

Finally, the motor equation gives the relationship between the electrical and mechnical parameter

`T_e-T_l=Jd omega_m/dt`, where `omega_m = 2/P omega_e`

Now, the study is done on different reference frame

1. stationary d-q reference frame
2. synchronous reference frame
3. rotor reference frame
4. rotor flux reference frame

In stationary d-q reference frame, `theta=0`, because d-q axis do not rotate in addition to that d-axis is chosen to coincide with the stator phase A axis. Hence, `omega=d theta/dt = 0`

Simulation results of `iqs,ids,iqr,idr` at stationary d,q- axis reference frame

In synchronously rotating reference frame `theta = omega_st`. Here, the d and q-axis are rotating at synchronous speed. Hence, `omega=d theta/dt = omega_s`. Similarly, for rotor reference frame, `omega = omega_r`. Here, d and q-axis are rotating at rotor speed

Simulation results of `iqs,ids,iqr,idr` at synchronous reference frame

Simulation results of `iqs,ids,iqr,idr` at rotor reference frame

The frequency of the mmf will be `Sf`where,`f`is the supply frequency i.e. 60Hz. slip will be calculated as `(omega_s-omega_r)/omega_s`

Simulation results of `iqs,ids,iqr,idr` at rotor flux reference frame