# How to calculate rotor flux of the three phase squirrel cage induction motor?

I have a three phase squirrel cage induction motor with given parameters (stator winding connected in delta)

• nominal power: $$\P_n = 22.4\,\mathrm{kW}\$$
• nominal stator voltage: $$\V_{sn} = 230\,\mathrm{V}\$$
• nominal stator current: $$\I_{sn} = 39.5\,\mathrm{A}\$$
• nominal stator frequency: $$\f_{sn} = 60\,\mathrm{Hz}\$$
• nominal speed: $$\n_n = 1168\,\mathrm{min}^{-1}\$$
• number of pole pairs: 3
• stator resistance per phase (T equivalent circuit): $$\R_s = 0.294\,\Omega\$$
• stator leakage inductance per phase (T equivalent circuit): $$\L_{sl} = 0.00139\,\mathrm{H}\$$
• rotor resistance per phase (T equivalent circuit): $$\R_r = 0.156\,\Omega\$$
• rotor leakage inductance per phase (T equivalent circuit): $$\L_{rl} = 0.0007401\,\mathrm{H}\$$
• magnetizing inductance per phase (T equivalent circuit): $$\L_{m} = 0.041\,\mathrm{H}\$$

I have been struggling with calculation of the nominal value of the rotor flux. My idea was that I will use the T equivalent circuit for that purpose

and I will use nominal values of the motor quantities i.e. I set the motor operating point into the nominal operation point (nominal slip, nominal stator voltage etc.). Then I calculate phasor of the stator current ($$\\hat{I}_s\$$) and phasor of the rotor current ($$\\hat{I}_r\$$) according to the below given set of equations

$$\begin{bmatrix} \hat{V}_s \\ 0 \end{bmatrix} = \begin{bmatrix} R_s + j\cdot(X_{sl} + X_m) & j\cdot X_m \\ j\cdot X_m & \frac{R_r}{s} + j\cdot(X_{rl} + X_m) \end{bmatrix} \cdot \begin{bmatrix} \hat{I}_s \\ \hat{I}_r \end{bmatrix}$$

For the motor parameters mentioned above the Scilab command linsolve gave me

$$\hat{I}_s = - 34.946619 + 17.574273j$$ $$\hat{I}_r = 35.797115 - 3.954462j$$

Based on the known phasors of the stator and rotor current I used the below given equation for calculation of the phasor of the rotor flux

$$\hat{\lambda}_r = (L_{rl} + L_m)\cdot\hat{I}_r + L_m\cdot\hat{I}_s$$

which gives $$\\lambda_r = 0.5554856 - 0.0613638j\$$ i.e. $$\|\lambda_r|=0.5588647\,\mathrm{V}\cdot\mathrm{s}\$$. This value seems to me to be too low. So I have doubts regarding the way I have used for its calculation. Unfortunatelly I don't know any other way for its calculation which I can use for verification. Can anybody tell me whether the applied procedure is correct or not? I would also appreciate any idea how to verify my results. Thanks in advance.

I don't think the leakage rotor magnetic field it has to be included (since the stator and rotor main magnetic field has to be the same).

However, it can be calculated using T equivalent diagram, the current $$\I_{s}\$$. At first calculate the equivalent impedance of parallel $$\X_m\$$ with $$\X_{rl}\$$ and $$\R_r/s\$$ and then divide $$\V_{sn}\$$ by total impedance $$\R_{s}+jX_s+jX_m\$$||($$\R_{r}/s+jX_{lr}\$$).

The result will be the same. $$\I_r\$$ you can calculate by dividing the $$\I_s\$$ voltage drop on the equivalent impedance of $$\jX_m\$$ and $$\R_r/s + jX_{lr}\$$ by $$\Z_r = R_r/s+jX_{lr}\$$.

I think the flux units are average values but $$\I_s\$$ and $$\I_r\$$ are RMS.

$$\I_{av} = 1/π\int_{ωt = 0}^{ωt = π} I_{max}*sin(ωt) \,dt = 2*[I_{max}]\$$

Given that $$\I_{rms}\$$ = $$\I_{max}/√2\$$

$$\I_{av}\$$ = $$\[2*√2*I_{rms}]/π\$$