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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,

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.

- Related Questions & Answers
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