The impedances at standstill of the inner and outer cages of a double-cage rotor are (0.01 + j 0.5) Ω and (0.05 + j 0.1) Ω respectively. The stator impedance may be assumed to be negligible. Calculate the ratio of the torques due to the two cages (i) at starting, and (ii) when running with a slip

Question

The impedances at standstill of the inner and outer cages of a double-cage rotor are (0.01 + j 0.5)[latex] \Omega[/latex] and (0.05 + j 0.1) [latex] \Omega[/latex] respectively. The stator impedance may be assumed to be negligible. Calculate the ratio of the torques due to the two cages (i) at starting, and (ii) when running with a slip of 5%.

Accepted Answer

From Eq. (9.31) [latex] T_{ start }=\frac{3}{\omega_{s}} \cdot \frac{V^{2} R_{2}^{\prime}}{R_{2}^{\prime 2}+X_{2}^{\prime 2}} [/latex] (as seen on the rotor side)

[latex] T_{s}=\frac{3}{\omega_{s}} \cdot \frac{V^{\prime 2} R_{2}^{2}}{R_{2}^{2}+X_{2}^{2}}[/latex]

Where [latex] V^{\prime}[/latex] = rotor induced emf

Substituting values

[latex] T_{s o}=\frac{3}{\omega_{s}} \cdot \frac{V^{\prime 2}(0.05)}{(0.05)^{2}+(0.1)^{2}}[/latex]

[latex] T_{s i}=\frac{3}{\omega_{s}} \cdot \frac{V^{\prime 2}(0.01)}{(0.01)^{2}+(0.5)^{2}}[/latex]

[latex] herefore[/latex] [latex] \frac{T_{s o}}{T_{s i}}=\frac{(0.01)^{2}+(0.5)^{2}}{(0.05)^{2}+(0.1)^{2}} imes\left(\frac{0.05}{0.01} ight)=100[/latex]

From Eq. (9.28) [latex] T=\frac{3}{\omega_{s}} \cdot \frac{V^{2}\left(R_{2}^{\prime} / s ight)}{\left(R_{2}^{\prime} / s ight)^{2}+X_{2}^{\prime 2}}[/latex] (as seen on the rotor side)

[latex] T=\frac{3}{\omega_{s}} \cdot \frac{V^{\prime 2}\left(R_{2} / s ight)}{\left(R_{2} / s ight)^{2}+X_{2}^{2}}[/latex]

Substituting values

[latex] T_{0}=\frac{3}{\omega_{s}} \cdot \frac{V^{\prime 2}(0.05 / 0.05)}{(0.05 / 0.05)^{2}+(0.1)^{2}}[/latex]

[latex] T_{i}=\frac{3}{\omega_{s}} \cdot \frac{V^{\prime 2}(0.01 / 0.05)}{(0.01 / 0.05)^{2}+(0.3)^{2}}[/latex]

[latex] herefore[/latex] [latex] \frac{T_{0}}{T_{i}}=\frac{(0.01 / 0.05)^{2}+(0.5)^{2}}{(0.05 / 0.05)^{2}+(0.1)^{2}} imes \frac{0.05}{0.01}=1.436[/latex]

Remark The outer cage contributes 100 times more torque than the inner cage at starting while it contributes only 1.436 times during running.

Question 9.18: The impedances at standstill of the inner and outer cages of...

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