A 440 V, 50 Hz, 4-pole 3-phase, delta-connected motor has a leakage impedance of (0.3 + j 5.5 + 0.25/s) Ω /phase (delta phase) referred to the stator. The stator to rotor voltage ratio is 2.5. Determine the external resistance to be inserted in each star phase of the rotor winding such that the

Question

A 440 V, 50 Hz, 4-pole 3-phase, delta-connected motor has a leakage impedance of (0.3 + j 5.5 + 0.25/s) [latex] \Omega /phase[/latex] (delta phase) referred to the stator. The stator to rotor voltage ratio is 2.5. Determine the external resistance to be inserted in each star phase of the rotor winding such that the motor develops a gross torque of 150 Nm at a speed of 1250 rpm.

Accepted Answer

[latex] \omega_{s}=\frac{2 \pi imes 1500}{60}=157.1 rad / s[/latex]

[latex] s=\frac{1500-1250}{1500}=0.167[/latex]

Total leakage impedance referred to stator (equivalent star basis)

[latex] =\frac{1}{3}[(0.3+0.25 / s)+j 5.5]=\left(0.1+\frac{0.083}{s}+j 1.83 ight) \Omega / ext { phase }( ext { star })[/latex]

From Eq (9.22) [latex] T=\frac{3}{\omega_{s}} \cdot I_{2}^{\prime 2}\left(R_{2}^{\prime} / s ight) =\frac{3}{\omega_{s}} \cdot \frac{V_{T H}^{2}\left(R_{2}^{\prime} / s ight)}{\left(R_{T H}+R_{2}^{\prime} / s ight)^{2}+\left(X_{T H}+X_{2}^{\prime} ight)^{2}}[/latex] with

[latex] R_{2 t}^{\prime}=R_{2}^{\prime}+R_{ ext }^{\prime} ; \quad R_{2}^{\prime}=0.083 \Omega, R_{2 t}^{\prime}=R_{2}^{\prime}[/latex] (total)

[latex] T=\frac{3}{\omega_{s}} \cdot \frac{V^{2}\left(R_{2 t}^{\prime} / s ight)}{\left(0.1+R_{2 t}^{\prime} / s ight)^{2}+\left(X_{1}+X_{2}^{\prime} ight)^{2}}[/latex]

Substituting values

[latex] 150=\frac{3}{157.1} \cdot \frac{(440 / \sqrt{3})^{2}\left(R_{2 t}^{\prime} / 00.167 ight)}{\left(0.1+R_{2 t}^{\prime} / 0.167 ight)^{2}+(1.83)^{2}}[/latex]

[latex] R_{2 t}^{\prime}-1.34 R_{2 t}^{\prime}+0.093=0[/latex]

[latex] R_{2 t}^{\prime}=1.27 \Omega ; 0.073 \Omega[/latex]

(The second value is less than [latex] R_{2}^{\prime}=0.083 \Omega[/latex] )

[latex] R_{ ext }^{\prime}=1.27-0.083 \Omega[/latex]

[latex] R_{ ext {ext }}( ext { rotor })=\frac{1.19}{(2.5)^{2}}=1.19 \Omega / ext { phase }[/latex]

Question 9.14: A 440 V, 50 Hz, 4-pole 3-phase, delta-connected motor has a ...

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