Power Flow Diagram and Losses of Induction Motor

---

---

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

$$\mathrm{P_{is} \:=\: \sqrt{3} \: V_{L}I_{L} \: \cos \varphi_{i} \: = \: 3 V_{sph} I_{sph} \: \cos \varphi_{i}}$$

Where,

• V_L = Line voltage
• I_L = Line current
• V_sph = Stator phase voltage
• I_sph = Stator phase current
• cos ϕ_i = input power factor

Stator Losses

• Stator copper losses or I²R losses in the stator winding resistances, which are given as follows −

$$\mathrm{P_{scu} \: = \: 3 I_{sph}^{2} R_{sph}}$$

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

$$\mathrm{P_{sC} \:=\: P_{sh} \:+\: P_{se}}$$

Therefore, the power output of the stator will be,

$$\mathrm{P_{os} \:=\: P_{is} \: - \: P_{scu} \: - \: P_{sC}}$$

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,

$$\mathrm{\text{Power output of stator } \: (P_{os}) \:=\: \text{ Air gap power } \: (P_g) \:=\: \text{ Input\:power\:to\:rotor } \: (P_{ir})}$$

Rotor Losses

• Rotor copper losses or I²R losses in the rotor resistance, which are given by,

$$\mathrm{P_{rcu} \:=\: 3 \: I_{2}^{2} R_{2}}$$

• Rotor core losses or hysteresis and eddy current losses, which are given as follows −

$$\mathrm{P_{rC} \:=\: P_{rh} \:+\: P_{re}}$$

• 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.

Mechanical Power Developed (P_m)

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).

$$\mathrm{\text{Developed mechanical power, } \: P_{m} \:=\: P_{ir} \: - \: P_{rcu}}$$

$$\mathrm{\Rightarrow \: P_{m} \:=\: P_{g} \:-\: P_{rcu} \:=\: P_{g} \:-\: (3I_{2}^{2}R_{2})}$$

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

$$\mathrm{P_{o} \:=\: P_{m} \:-\: P_{fw} \:-\: P_{misc}}$$

The power P_o is known as useful power or shaft power.

Rotational Losses

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 −

$$\mathrm{P_{\text{rotational losses}} \:=\: P_{C} \:+\: P_{fw} \:+\: P_{misc}}$$

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

$$\mathrm{P_{o} \:=\: P_{m} \:-\: P_{\text{rotational losses}} \:=\: P_{o} \:=\: P_{m} \:-\: P_C \:-\: P_{fw} \:-\: P_{misc}}$$

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