### Formulas

The synchronous rotational speed **n _{s}** and synchronous angular speed

**w**of a machine with

_{s}**p**pole pairs running on a supply of frequency

**f**are:

_{s}**n**

_{s}= 60f_{s}/ p**w**

_{s}= 2pf_{s}/ p = 2pn_{s}/ 60The per-unit slip **s** of an induction machine of synchronous rotational speed **n _{s}** running at rotational speed

**n**is:

_{m}**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**is:

_{M}**T**

_{M}= P_{M}/ w_{m}= 60P_{M}/ 2pn_{m}For an induction machine with **N _{s}** stator turns and

**N**rotor turns running at slip

_{r}**s**on a supply of voltage

**E**and frequency

_{s}**f**, the rotor induced voltage and frequency

_{s}**E**and

_{r}**f**are:

_{r}**E**

_{r}= sE_{s}N_{r}/ N_{s}**f**

_{r}= sf_{s}For a rotor current

**I**, the equivalent stator current

_{r}**I**is:

_{rs}**I**

_{rs}= I_{r}N_{r}/ N_{s}Note that the rotor / stator ratios are

**N**for current,

_{s}/ N_{r}**sN**for voltage and

_{r}/ N_{s}**s**for frequency.

For an induction machine with rotor resistance **R _{r}** and locked rotor leakage reactance

**X**, the rotor impedance

_{r}**Z**at slip

_{r}**s**is:

**Z**

_{r}= R_{r}+ jsX_{r}The stator circuit equivalent impedance

**Z**for a rotor / stator frequency ratio

_{rf}**s**is:

**Z**

_{rf}= R_{rs}/ s + jX_{rs}For an induction motor with synchronous angular speed **w _{s}** running at angular speed

**w**and slip

_{m}**s**, the airgap transfer power

**P**, rotor copper loss

_{t}**P**and gross output power

_{r}**P**for a gross output torque

_{m}**T**are related by:

_{m}**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**(neglecting stator and mechanical losses) is:

_{m}**h**

_{m}= P_{m}/ P_{t}= 1 – sAn 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.

### Formulas

NOTATION |
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BE f G I j k m N n P p R |
susceptance induced voltage frequency conductance current j-operator coefficient number of phases number of turns rotational speed power pole pairs resistance |
[siemens, S] [volts, V] [hertz, Hz] [siemens, S] [amps, A] [1Ð90°] [number] [number] [number] [revs/min] [watts, W] [number] [ohms, W] |
SsTVXYZdFfhqw |
voltamperes slip torque terminal voltage reactance admittance impedance loss angle magnetic flux phase angle efficiency temperature angular speed |
[volt-amps, VA] [per-unit] [newton-metres, Nm] [volts, V] [ohms, W] [siemens, S] [ohms, W] [degrees, °] [webers, Wb] [degrees, °] [per-unit] [centigrade, °C] [radians/sec] |