# DEVELOPED TORQUE (Induction Motor)

2.4
The steady-state per-phase equivalent circuit in Figure 2.14 allows calculation of the stator current and torque developed in the induction motor under steady-state operating conditions. Balanced voltages and currents in individual phases of the stator winding are assumed, so that from the point of view of total power and torque the equivalent circuit represents one-third of the motor. The average developed torque is given by

where Pout denotes the output (mechanical) power of the motor, which is the difference between the input power, Pin, and power losses, Ploss, incurred in the resistances of stator and rotor.
The output power can conveniently be determined from the equivalent circuit using the concept of equivalent load resistance, Rh. Because the ohmic (copper) losses in the rotor part of the circuit occur in the rotor resistance, Rv the RT/s resistance appearing in this circuit can be split into RT and

as illustrated in Figure 2.15. Clearly, the power consumed in the rotor after subtracting the ohmic losses constitutes the output power transferred to the load. Thus,

and

which describes the equivalent circuit in Figure 2.14. Reactances Xs and Xp appearing in the impedance matrix, are called stator reactance and rotor reactance, respectively, and given by

and

An approximate expression for the developed torque can be obtained from the approximate equivalent circuit of the induction motor, shown

FIGURE 2.15 Per-phase equivalent circuit of the induction motor showing the equivalent load resistance.
in Figure 2.16. Except for very low supply frequencies, the magnetizing reactance is much higher than the stator resistance and leakage reactance. Thus, shifting the magnetizing reactance to the stator terminals of the equivalent circuit does not significantly change distribution of currents in the circuit. Now, the rms value, Ip of rotor current can be calculated as

where

denotes the total leakage reactance. When 7P given by Eq. (2.16), is substituted in Eq. (2.12), after some rearrangements based on Eqs. (2.4) and (2.6), the steady-state torque can be expressed as

The quadratic relation between the stator voltage and developed torque is the only serious weakness of induction motors. Voltage sags in power lines, quite a common occurrence, may cause such reduction in the torque that the motor stalls. The torque-slip relation (2.18) is illustrated in Figure

FIGURE 2.16 Approximate per-phase equivalent circuit of the induction motor.
2.17 for various values of the rotor resistance, RT (in squirrel-cage motors, selection of the rotor resistance occurs in the design stage, while the wound-rotor machines allow adjustment of the effective rotor resistance by connecting external rheostats to the rotor winding). Generally, low values of RT are typical for high-efficiency motors whose mechanical characteristic, that is the torque-speed relation, in the vicinity of rated speed is “stiff,” meaning a weak dependence of the speed on the load torque. On the other hand, motors with a high rotor resistance have a higher zero-speed torque, that is, the starting torque, which can be necessary in certain applications. A formula for the starting torque, TMst, is obtained from Eq. (2.18) by substituting 5=1, which yields

The maximum torque, TMmax, called a pull-out torque, corresponds to a critical slip, sCT, which can be determined by differentiating TM with respect to s and equalling the derivative to zero. That gives

and

FIGURE 2.17 Torque-slip characteristics of induction motors with various values of the rotor resistance.
It must be reminded that Eqs. (2.16) through (2.21) are based on the approximate equivalent circuit of the induction motor and, as such, they yield only approximate values of the respective quantities.

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