For the motor of Problem 8.A.2, assume that the stator impedance is given by
$Z_{1}=1.86+j2.56\ \Omega$
Find the internal mechanical power, output power, power factor, input power, developed torque, and efficiency, assuming that friction losses are 15 W.

Question

For the motor of Problem 8.A.2, assume that the stator impedance is given by

[latex]Z_{1}=1.86+j2.56\ \Omega[/latex]

Find the internal mechanical power, output power, power factor, input power, developed torque, and efficiency, assuming that friction losses are 15 W.

Accepted Answer

We need the backward-field impedance

[latex]Z_{b}=\frac{j0.5X_{m}^{\prime}[0.5(R_{2}^{\prime}/(2-s))+j0.5X_{2}^{\prime}]}{0.5(R_{2}^{\prime}/(2-s))+j0.5(X_{2}^{\prime}+X_{m}^{\prime})} \ ={\frac{j26.75[(1.78/1.95)+j1.3]}{(1.78/1.95)+j(1.3+26.75)}} \ =0.8293+j1.2667\ \Omega[/latex]

As a result,

[latex]Z_{i}=Z_{1}+Z_{f}+Z_{b} \ =\;15.089+j20.8067=25.7023\angle54.05\;\mathrm{deg}\;\Omega[/latex]

The input current is thus

[latex]I_{1}={\cfrac{V}{Z_{i}} =\frac{110\angle0}{25.7023\angle54.05}}=4.2798\angle-54.05~\mathrm{deg}{\mathrm{A}}[/latex]

The forward-gap power is

[latex]P_{g f}=|I_{1}|^{2}R_{f}=227.125\,\mathrm{W}[/latex]

The backward-gap power is

[latex]P_{g b}=|I_{1}|^{2}R_{b}=15.1899\,\mathrm{W}[/latex]

As a result, the internal mechanical power is

[latex]{{P_{m}=(1-s)(P_{g f}-P_{g b})}}\ {{=201.338\,\mathrm{W}}}[/latex]

The output power is thus

[latex]P_{o}=201.338-15=186.338\,\mathrm{W}[/latex]

The power factor is given by

[latex]\mathrm{PF}=\cos54.05\,\mathrm{deg}=0.5871[/latex]

The input power is calculated as

[latex]P_{\mathrm{in}}=110(4.2798)(0.5871)=276.383\,\mathrm{W}[/latex]

As a result, the efficiency is

[latex]\eta={\frac{P_{\mathrm{o}}}{P_{\mathrm{in}}}}={\frac{186.338}{276.383}}=0.6742[/latex]

The synchronous speed is

[latex]n_{s}={\frac{120f}{P}}=1800\,\mathrm{rpm}[/latex]

The rotor speed in radls is thus

[latex]\omega_{r}={\frac{2\pi n_{s}}{60}}(1-s)=57\pi[/latex]

The torque output is given by

[latex]T_{o}={\frac{P_{o}}{\omega_{r}}}={\frac{186.338}{57\pi}}=1.0406\,\mathrm{N}\cdot\mathrm{m}[/latex]

Question 8.A.3: For the motor of Problem 8.A.2, assume that the stator imped......

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