MODULE 5 — 3-Phase Induction Motor
5.1 Rotating Magnetic Field (RMF)
Synchronous Speed
$$ N_s = \frac{120f}{P} \quad \text{(RPM)} $$
Angular Velocity
$$ \omega_s = \frac{4\pi f}{P} \quad \text{(rad/s)} $$
Stator Currents
5.2 Slip & Frequencies
$$ s = \frac{N_s - N_r}{N_s} \quad N_r = N_s(1 - s) $$
$$ f_r = s \cdot f, \quad E_r = s \cdot E_{r0}, \quad X_r = s \cdot
X_{r0} $$
Standstill: $s=1$. Normal: $s \approx 0.01
- 0.05$.
5.3 Torque Equations
$$ T = \frac{3}{\omega_s} \cdot \frac{s E_2^2
R_2}{R_2^2
+ (sX_2)^2} $$
$$ T_{max} = \frac{3}{2\omega_s}
\frac{E_2^2}{X_2} \quad
\text{at } s_m = \frac{R_2}{X_2} $$
5.4 Power Flow & Efficiency
$$ P_{gap} = P_{in} - P_{stator} $$
$$ P_{cu(rotor)} = s \cdot P_{gap} $$
$$ P_{mech} = (1-s) P_{gap} $$
$$ \eta = \frac{P_{out}}{P_{in}} \times 100\% $$
5.5 Starting Methods
DOL vs Star-Delta
$$ I_{start(Y\Delta)} = \frac{1}{3} I_{start(DOL)} $$
$$ T_{start(Y\Delta)} = \frac{1}{3} T_{start(DOL)} $$
Auto-transformer: $I_{st} = x^2 I_{DOL}, T_{st} = x^2
T_{DOL}$
5.6 Speed Control (V/f)
$$ N_s = \frac{120f}{P}, \quad \frac{V}{f} = \text{constant} $$
Variable Frequency Drive keeps air-gap flux constant.