## Induction machine

The
synchronous rotational speed n_{s} and synchronous angular speed
w_{s}
of a machine with p pole pairs running on a supply of frequency f_{s}
are:

n_{s} = 60f_{s} / p

w_{s}
= 2pf_{s} / p = 2pn_{s}
/ 60

**
The per-unit slip s of an induction
machine of synchronous rotational speed n _{s} running at rotational
speed n_{m} is:**

s = (n_{s} - n_{m}) / n_{s}

Rearranging for rotational speed n_{m}:

n_{m} = (1 - s)n_{s}

Using angular speed w instead of rotational speed n:

w_{m} = (1 - s)w_{s}

**
The rated load torque T _{M}
for a rated output power P_{M} is:**

T_{M} = P_{M} / w_{m} = 60P_{M} / 2pn_{m}

For an
induction machine with N_{s} stator turns and N_{r} rotor
turns running at slip s on a supply of voltage E_{s} and frequency f_{s},
the rotor induced voltage and frequency E_{r} and f_{r} are:

E_{r} = sE_{s}N_{r} / N_{s}

f_{r} = sf_{s}

For a rotor current I_{r}, the equivalent stator current I_{rs}
is:

I_{rs} = I_{r}N_{r} / N_{s}

Note that the rotor / stator ratios are N_{s} / N_{r} for
current, sN_{r} / N_{s} for voltage and s for frequency.

**
For an
induction machine with rotor resistance R _{r} and locked rotor
leakage reactance X_{r}, the rotor impedance Z_{r} at slip s
is:**

Z_{r} = R_{r} + jsX_{r}

The stator circuit equivalent impedance Z_{rf} for a rotor / stator frequency ratio s is:

Z_{rf} = R_{rs} / s + jX_{rs}

**
For an induction motor with
synchronous angular speed
w _{s}
running at angular speed
w_{m}
and slip s, the airgap transfer power P_{t}, rotor copper loss P_{r}
and gross output power P_{m} for a gross output torque T_{m}
are related by:**

P_{t} = w_{s}T_{m} = P_{r} / s = P_{m} / (1 - s)

P_{r} = sP_{t} = sP_{m} / (1 - s)

P_{m} = w_{m}T_{m} = (1 - s)P_{t}

The power ratios are:

P_{t} : P_{r} : P_{m} = 1 : s : (1 - s)

The gross motor efficiency h_{m} (neglecting stator and mechanical losses) is:

h_{m} = P_{m} / P_{t} = 1 - s

An induction
machine can be operated as a generator, a motor or a brake:

- for negative slip (speed above synchronous) the machine is a generator,

- for positive slip between 0 and 1 (speed below synchronous) the machine is
a motor,

- for positive slip greater than 1 (speed negative) the machine is a brake,

In all cases the magnetizing current (at lagging power factor) is provided
by the supply system.

No Load Test

If an induction machine with its rotor unloaded is energised at rated
voltage, then the input power represents the sum of the iron loss and
mechanical loss of the machine.

Locked Rotor Test

If an induction machine with its rotor locked is energised at a reduced
voltage which causes rated current input, then the input power represents
the sum of the full load copper loss and stray loss of the machine.

Stator Resistance Test

The resistance of the stator winding is measured using a small direct
current.