Nomenclature

\(P\)	Number of poles
\(f\)	Supply frequency (Hz)
\(N_s\)	Synchronous speed (rpm)
\(N_r\)	Rotor speed (rpm)
\(s\)	Slip
\(T_e\)	Electromagnetic torque (Nm)
\(P_{in}\)	Input power (W)
\(P_{out}\)	Output power (W)
\(P_{mech}\)	Mechanical power developed (W)
\(P_{rot}\)	Rotor copper losses (W)
\(P_{core}\)	Core losses (W)
\(P_{stator}\)	Stator copper losses (W)
\(I_s\)	Stator current (A)
\(I_r\)	Rotor current referred to the stator (A)
\(V_s\)	Stator voltage (V)
\(R_s\)	Stator resistance (\(\Omega\))
\(R_r'\)	Rotor resistance referred to the stator (\(\Omega\))
\(X_s\)	Stator reactance (\(\Omega\))
\(X_r'\)	Rotor reactance referred to the stator (\(\Omega\))
\(X_m\)	Magnetizing reactance (\(\Omega\))
\(Z_{eq}\)	Equivalent impedance (\(\Omega\))
\(E_r\)	Rotor induced EMF (V)
\(\eta\)	Efficiency
\(PF\)	Power factor
\(k_w\)	Winding factor
\(R_{th}, X_{th}\)	Thevenin equivalent parameters (\(\Omega\))
\(T_{max}\)	Maximum torque (Nm)
\(T_{start}\)	Starting torque (Nm)
\(T_{full-load}\)	Full-load torque (Nm)
\(I_{start}\)	Starting current (A)
\(Z_{block}\)	Impedance during blocked rotor test (\(\Omega\))
\(s_{max}\)	Slip at maximum torque
\(s_{fl}\)	Full-load slip
\(P_{nl}\)	Power during no-load test (W)
\(P_{sc}\)	Power during blocked rotor test (W)
\(I_{nl}\)	No-load current (A)
\(I_{sc}\)	Short-circuit current (A)
\(R_{nl}, X_{nl}\)	No-load equivalent parameters (\(\Omega\))

Formulas

1. Synchronous Speed

\[N_s = \frac{120f}{P}\]

2. Slip

\[s = \frac{N_s - N_r}{N_s}\]

3. Rotor Frequency

\[f_r = sf\]

4. Rotor Induced EMF (Referred to Stator)

\[E_r = sE_s\]

5. Equivalent Impedance

\[Z_{eq} = R_s + jX_s + \frac{\left(\frac{R_r'}{s} + jX_r'\right)X_m}{\frac{R_r'}{s} + j\left(X_r' + X_m\right)}\]

6. Stator Copper Loss

\[P_{stator} = I_s^2 R_s\]

7. Rotor Copper Loss

\[P_{rot} = sP_{mech}\]

8. Core Loss

\[P_{core} = P_{in} - (P_{stator} + P_{rot} + P_{mech})\]

9. Mechanical Power Developed

\[P_{mech} = \frac{1-s}{s} P_{rot}\]

10. Electromagnetic Torque

\[T_e = \frac{P_{mech}}{\omega_s}, \quad \omega_s = \frac{2\pi N_s}{60}\]

11. Output Power

\[P_{out} = P_{mech} - P_{friction} - P_{windage}\]

12. Input Power

\[P_{in} = \sqrt{3}V_sI_s\cos\phi\]

13. Efficiency

\[\eta = \frac{P_{out}}{P_{in}} \times 100\%\]

14. Power Factor

\[PF = \cos\phi = \frac{P_{in}}{\sqrt{3}V_sI_s}\]

15. Torque-Speed Relation

\[T_e \propto \frac{s}{R_r' + \left(\frac{R_r'}{s}\right)^2}\]

16. Condition for Maximum Torque

\[s_{max} = \frac{R_r'}{\sqrt{R_s^2 + (X_s + X_r')^2}}\]

17. Maximum Torque

\[T_{max} = \frac{E_s^2}{2\omega_s(R_r' + \sqrt{R_s^2 + (X_s + X_r')^2})}\]

18. Starting Torque

\[T_{start} = \frac{3}{\omega_s}\frac{V_s^2 R_r'}{(R_s + R_r')^2 + (X_s + X_r')^2}\]

19. Full-Load Torque

\[T_{full-load} = T_e \text{ at } s = s_{fl}\]

20. Impedance During Blocked Rotor Test

\[Z_{block} = \sqrt{R_s^2 + (X_s + X_r')^2}\]

21. Power During No-Load Test

\[P_{nl} = 3 V_s I_{nl} \cos\phi_{nl}\] \[R_{nl} = \frac{P_{nl}}{I_{nl}^2}, \quad X_{nl} = \sqrt{\frac{V_s^2}{I_{nl}^2} - R_{nl}^2}\] \[P_{core} \approx P_{nl} - 3I_{nl}^2R_s\]

22. Power During Blocked Rotor Test

\[P_{sc} = 3 V_s I_{sc} \cos\phi_{sc}\] \[R_{block} = \frac{P_{sc}}{I_{sc}^2}, \quad X_{block} = \sqrt{\frac{V_s^2}{I_{sc}^2} - R_{block}^2}\]

23. Thevenin Equivalent Parameters

\[R_{th} = \frac{R_s}{1 + \left(\frac{X_s}{X_m}\right)^2}, \quad X_{th} = \frac{X_m X_s}{X_m + X_s}\]

24. Short-Circuit Current

\[I_{sc} = \frac{V_s}{Z_{block}}\]