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,

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}

ϕ = 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,

$$\mathrm{𝐸_𝑠 = 4.44\: 𝑘_{𝑐𝑠} \:𝑘_{𝑑𝑠}\: 𝑓_𝑠 \:\varphi\: 𝑁_𝑠 … (1)}$$

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

$$\mathrm{𝐸_{𝑟0} = 4.44 \:𝑘_{𝑐𝑟}\: 𝑘_{𝑑𝑟}\: 𝑓_𝑠\: \varphi\: 𝑁_𝑟 … (2)}$$

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

$$\mathrm{𝐸_{𝑟𝑠} = 𝑠 𝐸_{𝑟0}}$$

$$\mathrm{\therefore 𝐸_{𝑟𝑠} = 4.44 𝑘_{𝑐𝑟} 𝑘_{𝑑𝑟}\: 𝑠 \:𝑓_𝑠 \:\varphi \:𝑁_𝑟 … (3)}$$

Now, let,

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

Then,

$$\mathrm{𝐸_𝑠 = 4.44 𝑘_{𝑤𝑠}\: 𝑓_𝑠 \:\varphi \:𝑁_𝑠 … (4)}$$

And

$$\mathrm{𝐸_{𝑟𝑠} = 4.44 𝑘_{𝑤𝑟} 𝑠 𝑓_𝑠 \varphi 𝑁_𝑟 … (5)}$$

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

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

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,

$$\mathrm{\frac{𝐼′_𝑟}{𝐼_𝑟}=\frac{𝑁_{𝑒𝑟}}{𝑁_{𝑒𝑠}}=\frac{1}{𝑎_{𝑒𝑓𝑓}}… (7)}$$

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.