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# Induction Machines

### Formulas

The synchronous rotational speed ns and synchronous angular speed ws of a machine with p pole pairs running on a supply of frequency fs are:
ns = 60fs / p
ws = 2pfs / p = 2pns / 60

The per-unit slip s of an induction machine of synchronous rotational speed ns running at rotational speed nm is:
s = (ns – nm) / ns
Rearranging for rotational speed nm:
nm = (1 – s)ns
Using angular speed w instead of rotational speed n:
wm = (1 – s)ws

The rated load torque TM for a rated output power PM is:
TM = PM / wm = 60PM / 2pnm

For an induction machine with Ns stator turns and Nr rotor turns running at slip s on a supply of voltage Es and frequency fs, the rotor induced voltage and frequency Er and fr are:
Er = sEsNr / Ns
fr = sfs
For a rotor current Ir, the equivalent stator current Irs is:
Irs = IrNr / Ns
Note that the rotor / stator ratios are Ns / Nr for current, sNr / Ns for voltage and s for frequency.

For an induction machine with rotor resistance Rr and locked rotor leakage reactance Xr, the rotor impedance Zr at slip s is:
Zr = Rr + jsXr
The stator circuit equivalent impedance Zrf for a rotor / stator frequency ratio s is:
Zrf = Rrs / s + jXrs

For an induction motor with synchronous angular speed ws running at angular speed wm and slip s, the airgap transfer power Pt, rotor copper loss Pr and gross output power Pm for a gross output torque Tm are related by:
Pt = wsTm = Pr / s = Pm / (1 – s)
Pr = sPt = sPm / (1 – s)
Pm = wmTm = (1 – s)Pt
The power ratios are:
Pt : Pr : Pm = 1 : s : (1 – s)
The gross motor efficiency hm (neglecting stator and mechanical losses) is:
hm = Pm / Pt = 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.

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 The symbol font is used for some notation and formulae. If the Greek symbols for alpha beta delta do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae. B E 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] S s T V X Y Z d F f h q w 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]