__The Power Balance of an Induction Motor __

The total (apparent) power of an induction motor is

* S* =

*P*

_{1}+

*j*Q

_{l}= 3V

_{1}

*I*

_{1}cos φ

_{1}+

*j*3V

_{1}

*I*

_{1}sin φ

_{1}

where *P*_{1} is its active power and Q_{1} is it reactive power.

The active power *P*_{1} defines the average power associated with the irreversible conversion of the electric energy drawn by the motor from the supply line into mechanical, thermal and other forms of energy.

The reactive power Q_{1} defines the maximum time rate of energy exchange between the supply source and the magnetic field of the motor, Eq. (2.52).

The active power and efficiency of an induction motor. The way in which electric power is converted in an induction motor is illustrated in Fig. 20. In this diagram, *P*_{1} = 3*V*_{1}*I*_{1} cos φ_{1} is the power drawn by the motor from a three phase supply line . *P*_{w1} is the power lost as heat in the stator conductors. The remaining power, *P*_{rf} is converted to the power of a rotating magnetic field. *P*_{c} is the core loss which includes hysteresis and eddy-current losses in the stator core. The rotor core loss is of no practical importance because at the rotor frequency *f*_{2} (1-3 Hz) hysteresis and eddy-current losses in the rotor core are negligible.

The remainder of the rotating magnetic field power is the electromagnetic power of the rotor

*P*_{em} = *P*_{rf} - *P*_{c}

On substracting the power lost as heat in the rotor conductors, *P*_{w2} from *P*_{em} we obtain the mechanical power developed by the rotor, *P*_{m }. Hence

*P*_{m} = *P*_{1} - *P*_{W1} - *P*_{c} - *P*_{W2}

However, the useful mechanical power available at the motor shaft, *P*_{2} is less than the developed mechanical power *P*_{m} by the power lost through windage and friction, *P*_{wf }, so

*P*_{2} = *P*_{m} - *P*_{wf}

The ratio of the available mechanical power *P*_{2} to the active power drawn from the supply line, *P*_{1} gives the efficiency of an induction motor

η = *P*_{2}/*P*_{1}

Under rated operating conditions the efficiency of present-day induction motors is 80%-95%.

The reactive power and power factor of an induction motor. The reactive power Q_{l} is associated with the reversible exchange of energy between the magnetic field of the motor and the supply line. The existence of a magnetic field is essential to the operation of an induction motor, so reactive power is inevitable.

In the design and operation of induction motors, it is important to know the relation between active and reactive power. This relation is defined in terms of the power factor

cos φ_{1} = *P*_{1}/ √(*P*^{2}_{1}+ Q^{2}_{1}) (14.21)

As follows from an analysis of Eq. (14.11b), when the stator-phase terminal voltage *V*_{l} is constant and *I*_{l}≤*I*_{l,rtd }the magnetic flux Φ_{r} associated with the rotating magnetic field of the machine is likewise constant and independent of load. This implies that the energy stored in the magnetic field of an induction motor and the reactive power are constant, too, and independent of load. However, as the load is increased, the active power of an induction motor goes up. Therefore, it follows from Eq. (14.21) that an increase in load causes the power factor of the machine to increase too. If, at no load the power factor of an induction motor is 0.1-0.15, it usually goes up to 0,8-0,9 at rated load.