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

Question

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.

Answer 1

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}]\$/π

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

\$I_{av}\$ = \$[2*√2*I_{rms}]/π\$

Answer 2

The nominal value of the machine flux can be determined based on the stator current \$\hat{I}_s\$ in the no-load condition i.e. \$s\rightarrow 0\$ i.e. \$\frac{R_r}{s}\rightarrow\infty\$ which means \$\hat{I}_r\sim 0\$. Based on the first equation in the no-load condition we have:

$$\hat{V}_s = \left[R_s + j(X_{ls} + X_m)\right]\cdot\hat{I}_s$$

So for the stator current (which is basically magnetizing current) we can write

$$\hat{I}_m = \hat{I}_s = \frac{\hat{V}_s}{R_s + j(X_{ls} + X_{m})}$$

and for its magnitude we have

$$ I_m = I_s = \frac{V_s}{\sqrt{R_s^2 + (X_{ls} + X_m)^2}} = \frac{230}{\sqrt{0.294^2 + (2\pi\cdot 60\cdot(0.00139 + 0.041))^2}}\approx 14.4\,A$$.

The machine flux is then

$$\lambda = L_m\cdot i_m = 0.0.41\cdot 14.4 \approx 0.6\,Wb$$

Answer 3

Your calculations are correct, but it's possible to calculate the nominal rotor flux linkage by first calculating the RMS magnitude of the voltage drop across the Rs/s resistor and then dividing this voltage by the angular frequency omega = 2PIf. The nominal slip, s, can be obtained from the nominal speed and number of pole pairs