# 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

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