Three-phase Induction Motor - Winding EMFs (Stator EMF and Rotor EMF)

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

V_s = Stator applied voltage per phase

N_s = Number of stator winding turns in series per phase

N_r = Number of rotor winding turns in series per phase

= Resultant flux in air gap

E_s = Stator induced EMF per phase

E_r0 = EMF induced in the rotor per phase when the rotor is at standstill

E_rs = EMF induced in the rotor per phase when the rotor is rotating at a slip s

R_s = Resistance of stator winding per phase

R_r = Resistance of rotor winding per phase

L_r0 = Rotor inductance per phase at standstill due to leakage flux

X_r0 = Leakage reactance of the rotor winding per phase when the rotor is at standstill

f_s = Supply frequency

f_r = Frequency of the induced EMF in the rotor at a slip s

X_rs = Leakage reactance of rotor winding per phase when the rotor is rotating at a slip s

k_ds = Distribution factor of stator winding

k_dr = Distribution factor of rotor winding

k_cs = Coil span factor of stator winding

k_cr = Coil span factor of rotor winding

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

$$\mathrm{E_{s} \:=\: 4.44 \: k_{cs} \: k_{ds} \: f_{s} \: \varphi \: N_{s}\:\: \dotso \: (1)}$$

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

$$\mathrm{E_{r0} \:=\: 4.44 \: k_{cr} \: k_{dr} \: f_{s} \: \varphi \: N_{r}\:\: \dotso \: (2)}$$

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

$$\mathrm{E_{rs} \:=\: s E_{r0}}$$

$$\mathrm{\therefore \: E_{rs} \:=\: 4.44 \:k_{cr} \:k_{dr} \: s \: f_{s} \: \varphi \: N_{r} \:\: \dotso \: (3)}$$

Now, let,

• k_cs k_ds = k_w = Winding factor of stator
• k_cr k_dr = k_wr = Winding factor of rotor

Then,

$$\mathrm{E_{s} \:=\: 4.44 \:k_{ws} \: f_{s} \: \varphi \: N_{s} \:\: \dotso \: (4)}$$

And

$$\mathrm{E_{rs} \:=\: 4.44 \: k_{wr} \: s f_{s} \:\varphi\: N_{r} \:\: \dotso \: (5)}$$

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

$$\mathrm{\frac{E_{s}}{E_{rs}} \:=\: \frac{k_{ws} \: N_{s}}{k_{wr} \: N_{r}} \:=\: \frac{N_{es}}{N_{er}} \:=\: a_{eff} \:\: \dotso \: (6)}$$

Where, N_es and N_er are known as effective stator and rotor turns per phase, respectively.

And a_eff is known as effective turns ratio of an induction motor.

Also,

$$\mathrm{\frac{I′_r}{I_{r}} \:=\: \frac{N_{er}}{N_{es}} \:=\: \frac{1}{a_{eff}} \:\: \dotso \: (7)}$$

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.