# Induction Machines

An induction motor consists of an external stator and an internal rotor. Windings embedded in the stator is excited by a 3-phase AC supply to produce a rotating magnetic field. The supply can be arranged in STAR or DELTA formation, where STAR formation has the voltage across each winding reduced by $$\sqrt{3}$$.

### Synchronous Speed

The rotational velocity of the magnetic field is the maximum velocity of the rotor and is referred to as the synchronous speed:

$$n_s = \frac{f_{electrical}}{PolePairs} (revolutions/second)$$
$$n_s = \frac{f_{electrical}}{PolePairs} \times 60 (rpm)$$
$$\omega _s = \frac{2 \pi f_{electrical}}{PolePairs}$$

### Slip

Due to various losses, the rotor will deviate slightly from synchronous speed.

$$s = \frac{n_s - n_r}{n_s}$$

Where $$n_s$$ is synchronous speed, $$n_r$$ is rotor speed.

## Parameter Tests

Using an ohm-meter to measure the resistance of each phase.

Measure $$P$$ and $$I$$ of each phase, use $$R_1$$ losses to determine friction and power/torque.

The total impedance can be found from voltage and current magnitudes, so the impedance of $$X_1$$ and $$X_M$$ can be found.

Measure $$P$$ and $$I$$ of each phase while the rotor is locked, since all power is lost in resistance and $$R_1$$ is known, $$R'_2$$ can be deduced. Assume $$X_1 = X'_2$$ so that the other parameters can be found.

## Simplified Equivalent Circuit

$$\omega _s = \frac{2 \pi f_{electrical}}{PolePairs}$$
$$SLIP\_s = \frac{n_s - n_r}{n_s} = \frac{\omega _s - \omega _r}{\omega _s }$$
$$|I'|_2 = \frac{V_1}{(R_1 + \frac{R'_2}{s})^2 + (X_1 + X'_2)^2}$$
$$P_{airgap} = 3 |I' _2|^2 \bigg( \frac{R'_2}{s} \bigg)$$
$$P_{mech} = 3 |I' _2|^2 \bigg( \frac{R'_2}{s} \bigg) (1-s)$$
$$P_{shaft} = P_{mech} - P_{friction,corelosses}$$
$$P_{shaft} = T_{shaft} \times \omega _{rotor}$$
$$T_{mech} = \frac{3}{\omega _s} |I'_2|^2 \bigg( \frac{R'_2}{s} \bigg)$$
$$T_{mech} = \frac{1}{\omega _s} P_{airgap}$$

### Maximum Torque

$$T_{mech\_max} = \frac{3}{2 \omega _s} \bigg( \frac{V_1 ^2}{ R_1 + \sqrt{ R_1 ^2 + (X_1 + X'_2)^2 } } \bigg)$$