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# Winding EMFs in a 3-Phase Induction Motor; Stator EMF and Rotor EMF

Let suffixes "s" and "r" be used for stator and rotor quantities, respectively. Then,

π_{π } = Stator applied voltage per phase

π_{π } = Number of stator winding turns in series per phase

π_{π = Number of rotor winding turns in series per phase}

Ο = Resultant flux in air gap

πΈ_{π } = Stator induced EMF per phase

πΈ_{π0} = EMF induced in the rotor per phase when the rotor is at standstill

πΈ_{ππ } = EMF induced in the rotor per phase when the rotor is rotating at a slip π

π
_{π } = Resistance of stator winding per phase

π
_{π} = Resistance of rotor winding per phase

πΏ_{π0} = Rotor inductance per phase at standstill due to leakage flux

π_{π0} = Leakage reactance of the rotor winding per phase when the rotor is at standstill

π_{π }= Supply frequency

π_{π} = Frequency of the induced EMF in the rotor at a slip π

π_{ππ } = Leakage reactance of rotor winding per phase when the rotor is rotating at a slip π

π_{ππ } = Distribution factor of stator winding

π_{ππ} = Distribution factor of rotor winding

π_{ππ } = Coil span factor of stator winding

π_{ππ} = Coil span factor of rotor winding

Then, the induced EMF in the stator winding per phase is given by,

$$\mathrm{πΈ_π = 4.44\: π_{ππ } \:π_{ππ }\: π_π \:\varphi\: π_π … (1)}$$

The induced EMF per phase in the rotor when the rotor is at standstill is given by,

$$\mathrm{πΈ_{π0} = 4.44 \:π_{ππ}\: π_{ππ}\: π_π \: \varphi\: π_π … (2)}$$

The induced EMF per phase in the rotor when the rotor is rotating at a slip 's' is given by,

$$\mathrm{πΈ_{ππ } = π πΈ_{π0}}$$

$$\mathrm{\therefore πΈ_{ππ } = 4.44 π_{ππ} π_{ππ}\: π \:π_π \:\varphi \:π_π … (3)}$$

Now, let,

- π
_{ππ }π_{ππ }= π_{π€}= Winding factor of stator - π
_{ππ}π_{ππ}= π_{π€π}= Winding factor of rotor

Then,

$$\mathrm{πΈ_π = 4.44 π_{π€π }\: π_π \:\varphi \:π_π … (4)}$$

And

$$\mathrm{πΈ_{ππ } = 4.44 π_{π€π} π π_π \varphi π_π … (5)}$$

Now, taking the ratio of eqns. (4) and (5), we get,

$$\mathrm{\frac{πΈ_{π }}{πΈ_{ππ }}=\frac{π_{π€π } π_{π }}{π_{π€π} π_{π}}=\frac{π_{ππ }}{π_{ππ}}= π_{πππ}… (6)}$$

Where, N_{es} and N_{er} are known as effective stator and rotor turns per phase, respectively.

And π_{πππ} is known as effective turns ratio of an induction motor.

Also,

$$\mathrm{\frac{πΌ′_π}{πΌ_π}=\frac{π_{ππ}}{π_{ππ }}=\frac{1}{π_{πππ}}… (7)}$$

From equation (6), it is clear that the ratio between stator and rotor EMFs is constant at standstill. This ratio depends upon the turns ratio modified by the distribution and coil span factors of the windings. Hence, an induction motor behaves like a transformer. The number of slots in stator and rotor may be different, thus, the factors for the stator and rotor windings are not the same.

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