# 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,

𝑉𝑠 = Stator applied voltage per phase

𝑁𝑠 = Number of stator winding turns in series per phase

𝑁𝑟 = Number of rotor winding turns in series per phase

ϕ = Resultant flux in air gap

𝐸𝑠 = Stator induced EMF per phase

𝐸𝑟0 = EMF induced in the rotor per phase when the rotor is at standstill

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

𝑅𝑠 = Resistance of stator winding per phase

𝑅𝑟 = Resistance of rotor winding per phase

𝐿𝑟0 = Rotor inductance per phase at standstill due to leakage flux

𝑋𝑟0 = Leakage reactance of the rotor winding per phase when the rotor is at standstill

𝑓𝑠= Supply frequency

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

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

𝑘𝑑𝑠 = Distribution factor of stator winding

𝑘𝑑𝑟 = Distribution factor of rotor winding

𝑘𝑐𝑠 = Coil span factor of stator winding

𝑘𝑐𝑟 = 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,

• 𝑘𝑐𝑠 𝑘𝑑𝑠 = 𝑘𝑤 = Winding factor of stator
• 𝑘𝑐𝑟 𝑘𝑑𝑟 = 𝑘𝑤𝑟 = 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, Nes and Ner are known as effective stator and rotor turns per phase, respectively.

And 𝑎𝑒𝑓𝑓 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.