Electrical Machines and Power Electronics - 3 phase, 6 pole induction motor

[VinnyCee]

Homework Statement

A three phase, 6-pole, $\Delta$ connected induction motor is rated 208V, 60Hz. It has $R_R\,=\,5.2\,\Omega$, $R_S\,=\,2.2\,\Omega$, $X_m\,=\,220\,\Omega$, $X_{lr}\,=\,7.5\,\Omega$, $X_{ls}\,=\,2.3\,\Omega$.

Calculate the starting torque and both phase and line currents when the motor is started in $\Delta$ and Y.

Homework Equations

Various equations from the notes about induction machines and multiple pole pairs.

$$I_{S,\,Y}\,=\,\frac{\frac{V_{rated}}{\sqrt{3}}}{(R_S\,+\,j\,X_{ls})+(j\,X_m\,||\,R_R\,+\,j\,X_{lr})}$$

$$T_Y\,=\,3\,\cdot\,\frac{P_{gap}}{\omega_s}\,\cdot\,\frac{p}{2}$$

$$I_{R,\,Y}\,=\,I_{S,\,Y}\,\cdot\,\frac{j\,X_m}{R_R\,+\,j\,X_{lr}\,+\,j\,X_m}$$

$$P_{gap}\,=\,|I_{R,\,Y}|\,\cdot\,R_R$$

The Attempt at a Solution

$$I_{S,\,Y}\,=\,\frac{\frac{208}{\sqrt{3}}}{(2.2\,+\,j\,2.3)+(j\,220\,||\,5.2\,+\,j\,7.5)}\,=\,10\,\angle\,-53.8^{\circ}$$

In Y, the line current is same as phase current above, right?

I get the torque in Y as $T_Y\,=\,11.68\,Nm$.

Does that look right?

 

[VinnyCee]

This is correct. The solutions also state...

$$I_{ph,\,Y}\,=\,|I_{S,\,Y}|\,=\,10\,A$$

$$I_{l,\,Y}\,=\,I_{ph,\,Y}\,=\,10\,A$$

And for $\Delta$:

$$T_{\Delta}\,=\,3\,\cdot\,T_Y\,=\,35\,A$$

$$I_{ph,\,\Delta}\,=\,\frac{208}{120}\,\cdot\,10\,=\,17.34\,A$$

$$I_{l,\,\Delta}\,=\,\sqrt{3}\,\cdot\,I_{ph,\,\Delta}\,=\,30\,A$$