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The 3-phase input power fed to the stator of a 3-phase induction motor is given by,

$$\mathrm{π_{ππ } = \sqrt{3} π_πΏπΌ_πΏ cos \varphi_i = 3 π_{π πβ} πΌ_{π πβ} cos \varphi_i}$$

Where,

- V
_{L}= Line voltage - I
_{L}= Line current - V
_{sph}= Stator phase voltage - I
_{sph}= Stator phase current - cos \varphi
_{i}= input power factor

*Stator copper losses*or*I*losses in the stator winding resistances, which are given as follows −^{2}R

$$\mathrm{π_{π ππ’} = 3 πΌ_{π πβ}^{2} π _{π πβ}}$$

- Hysteresis and eddy current losses in the stator core, which are known as
*stator-core losses*and are given by −

$$\mathrm{π_{π πΆ} = π_{π β} + π_{π π}}$$

Therefore, the power output of the stator will be,

$$\mathrm{π_{ππ } = π_{ππ }− π_{π ππ’} − π_{π πΆ}}$$

This output power of the stator (Pos) is transferred to the rotor of the machine across the air-gap between the stator and the rotor. It is also known as air-gap power (P_{g}) of the machine. Hence,

$$\mathrm{Power\:output\:of\:stator (π_{ππ })\:=\: Air\:gap\:power (π_π)\:=\:Input\:power\:to\:rotor (π_{ππ})}$$

- Rotor copper losses or I^2R losses in the rotor resistance, which are given by,
$$\mathrm{π_{πππ’} = 3 πΌ_2^2 π _2}$$

- Rotor core losses or hysteresis and eddy current losses, which are given as follows −
$$\mathrm{π_{ππΆ} = π_{πβ} + π_{ππ}}$$

- Friction and windage losses (P
_{fw}) - Stray load losses (P
_{misc}) consisting of all the losses which are not included in the above losses like losses due to harmonic fields.

If the rotor copper losses are subtracted from the rotor input power (P_{g} or P_{ir}), then the remaining power is converted from electrical power to mechanical power. It is known as *developed mechanical power *(P_{m}).

$$\mathrm{Developed\:mechanical\:power,\: π_π = π_{ππ} − π_{πππ’}}$$

$$\mathrm{⇒ π_π = π_π − π_{πππ’} = π_π − (3πΌ_2^2π _2)}$$

Therefore, the output power of the motor is given by,

$$\mathrm{π_π = π_π − π_{ππ€} − π_{πππ π}}$$

The power P_{o} is known as useful power or shaft power.

At the starting and during the acceleration, the rotor core losses are high and decreases with the increase in the speed of the motor. The friction and windage losses are zero at start and increases with the increases in the speed.

Consequently, the sum of core losses and friction and windage losses is approximately constant with varying speed of the motor. Hence, these losses may lumped together and are called as rotational losses and given as follows −

$$\mathrm{π{πππ‘ππ‘πππππ\:πππ π ππ }= π_πΆ + π_{ππ€} + π_{πππ π}}$$

Then, the output power of the motor is given by,

$$\mathrm{π_π = π_π − π{πππ‘ππ‘πππππ\:πππ π ππ }= π_π = π_π − π_πΆ − π_{ππ€} − π_{πππ π}}$$

The power diagram of the induction motor is shown in the figure below.

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