Formula Sheet: Three-Phase Induction Motors
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 (\(\Omega\)) |
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
\[P_{out} = P_{mech} - P_{friction} -
P_{windage}\]
\[P_{in} =
\sqrt{3}V_sI_s\cos\phi\]
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
\[P_{nl} = 3 V_s I_{nl}
\cos\phi_{nl}\]
22. Power
During Blocked Rotor Test
\[P_{sc} = 3 V_s I_{sc}
\cos\phi_{sc}\]
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
\[I_{sc} =
\frac{V_s}{Z_{block}}\]