Question 9.17: A 50 Hz, 3-phase induction motor has a rated voltage V1. The...

(a)                                   $s_{\max , T}=\frac{R_{2}^{\prime}}{X_{2}^{\prime}}=0.2$

Where  $X_{2}^{\prime}=$ = standstill 50 Hz rotor reactance

At rated voltage ($V_{1}$ ) and frequency (50 Hz)

$I_{s}(1)=I_{2}^{\prime}(1)=\frac{V_{1}}{\sqrt{R_{2}^{\prime 2}+X_{2}^{\prime 2}}}$                                   (i)

$T_{s}(1)=\frac{V_{1}^{2} R_{2}^{\prime}}{R_{2}^{\prime 2}+X_{2}^{\prime 2}}$                                    (ii)

$T_{\max }(1)=\frac{3}{\omega_{s}} \cdot \frac{0.5 V_{1}^{2}}{X_{2}^{\prime}}$                                            (iii)

At voltage $V_{2}$  and frequency 60 Hz

$I_{2}(2)=I_{2}^{\prime}(2)=\frac{V_{2}}{\sqrt{R_{2}^{\prime 2}+\left(\frac{6}{5}\right)^{2} X_{2}^{\prime 2}}}$                                              (iv)

$T_{s}(2)=\frac{V_{2}^{2} R_{2}^{\prime}}{R_{2}^{\prime 2}+\left(\frac{6}{5}\right)^{2} X_{2}^{\prime 2}}$                                              (v)

$T_{\max }(2)=\frac{3}{\left(\frac{6}{5}\right) \omega_{s}} \frac{0.5 V_{2}^{2}}{\left(\frac{6}{5}\right) X_{2}^{\prime}}$                                                    (vi)

Dividing Eqs (iv), (v) and (vi) respectively by Eqs (i), (ii) and (iii)

(b)                                      $\frac{V_{2}^{2}}{V_{1}^{2}} \cdot \sqrt{\frac{(0.2)^{2}+1}{(0.2)^{2}+\left(\frac{6}{5}\right)^{2}}}=1$

This ratio will also give equal staring torques.