A 50 Hz, 3-phase induction motor has a rated voltage V1. The motor’s breakdown torque at rated voltage and frequency occurs at a slip of 0.2. The motor is instead run from a 60 Hz supply of voltage V2. The stator impedance can be neglected. (a) If V2 = V1, find the ratio of currents and torques

Question

A 50 Hz, 3-phase induction motor has a rated voltage [latex] V_{1}[/latex]. The motor’s breakdown torque at rated voltage and frequency occurs at a slip of 0.2. The motor is instead run from a 60 Hz supply of voltage [latex] V_{2}[/latex]. The stator impedance can be neglected.

(a) If [latex] V_{2}=V_{1}[/latex], find the ratio of currents and torques at starting. Also find the ratio of maximum torques.

(b) Find the ratio V2 /V1 such that the motor has the same values of starting current and torque at 50 and 60 Hz.

Accepted Answer

(a) [latex] s_{\max , T}=\frac{R_{2}^{\prime}}{X_{2}^{\prime}}=0.2[/latex]

Where [latex] X_{2}^{\prime}=[/latex] = standstill 50 Hz rotor reactance

At rated voltage ([latex] V_{1}[/latex] ) and frequency (50 Hz)

[latex] I_{s}(1)=I_{2}^{\prime}(1)=\frac{V_{1}}{\sqrt{R_{2}^{\prime 2}+X_{2}^{\prime 2}}}[/latex] (i)

[latex] T_{s}(1)=\frac{V_{1}^{2} R_{2}^{\prime}}{R_{2}^{\prime 2}+X_{2}^{\prime 2}}[/latex] (ii)

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

At voltage [latex] V_{2}[/latex] and frequency 60 Hz

[latex] I_{2}(2)=I_{2}^{\prime}(2)=\frac{V_{2}}{\sqrt{R_{2}^{\prime 2}+\left(\frac{6}{5} ight)^{2} X_{2}^{\prime 2}}}[/latex] (iv)

[latex] T_{s}(2)=\frac{V_{2}^{2} R_{2}^{\prime}}{R_{2}^{\prime 2}+\left(\frac{6}{5} ight)^{2} X_{2}^{\prime 2}}[/latex] (v)

[latex] T_{\max }(2)=\frac{3}{\left(\frac{6}{5} ight) \omega_{s}} \frac{0.5 V_{2}^{2}}{\left(\frac{6}{5} ight) X_{2}^{\prime}}[/latex] (vi)

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

[latex] \frac{I_{s}(2)}{I_{s}(1)}=\frac{V_{2}}{V_{1}} \cdot \sqrt{\frac{R_{2}^{\prime 2}+X_{2}^{\prime 2}}{R_{2}^{\prime 2}+\left(\frac{6}{5} ight)^{2} X_{2}^{\prime 2}}}[/latex]

[latex] =\frac{V_{2}}{V_{1}} \cdot \sqrt{\frac{s_{\max , T}^{2}+1}{s_{\max , T}^{2}+\left(\frac{6}{5} ight)^{2}}}[/latex]

[latex] =1 imes \sqrt{\frac{(0.2)^{2}+1}{(0.2)^{2}+\left(\frac{6}{5} ight)^{2}}}=0.838[/latex]

[latex] \frac{T_{s}(2)}{T_{s}(1)}=\frac{V_{2}^{2}}{V_{1}^{2}} \cdot \frac{s_{\max , T}^{2}+1}{s_{\max , T}^{2}+\left(\frac{6}{5} ight)^{2}}[/latex]

[latex] =1 imes \frac{(0.2)^{2}+1}{(0.2)^{2}+\left(\frac{6}{5} ight)^{2}}=0.703[/latex]

[latex] \frac{T_{\max }(2)}{T_{\max }(1)}=\frac{V_{2}^{2}}{V_{1}^{2}}\left(\frac{5}{6} ight)^{2}=0.694[/latex]

(b) [latex] \frac{V_{2}^{2}}{V_{1}^{2}} \cdot \sqrt{\frac{(0.2)^{2}+1}{(0.2)^{2}+\left(\frac{6}{5} ight)^{2}}}=1[/latex]

[latex] \frac{V_{2}}{V_{1}}=1.19[/latex]

This ratio will also give equal staring torques.

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

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