### Torque-Speed Characteristic Of INDUCTION MOTOR

In order to estimate the speed torque characteristic let us suppose that a sinusoidal voltage is impressed on the machine. Recalling that the equivalent circuit is the per-phase representation of the machine, the current drawn by the circuit is given by

where Vs is the phase voltage phasor and Is is the current phasor. The  magnetizing current is neglected. Since this current is flowing through      (R′ r/s). , the air-gap power is given by

The mechanical power output was shown to be (1−s)Pg (power dissipated in (R′ r/s). The torque is obtained by dividing this by the shaft speed !m.Thus we have,

The torque may be plotted as a function of ‘s’ and is called the torque-slip (or torque- speed, since slip indicates speed) characteristic — a very important characteristic of the induction machine. Eqn. 17 is valid for a two-pole (one pole pair) machine. In general, this expression should be ultiplied by p, the number of pole-pairs. A typical torque-speed characteristic is  shown in fig. 22. This plot corresponds to a 3 kW, 4 pole, 60 Hz machine. The rated operating speed is 1780 rpm.

We must note that the approximate equivalent circuit was used in deriving this relation. Readers with access to MATLAB or suitable equivalents (octave, scilab available free under GNU at the time of this writing) may find out the difference caused by using the ‘exact’ equivalent circuit by using the script found here. A comparison between the two is found in the plot of fig. 23. The plots correspond to a 3 kW, 4 pole, 50 Hz machine, with a rated speed of 1440 rpm. It can be seen that the approximate equivalent circuit is a good approximation in the operating speed range of the machine. Comparing fig. 22 with fig. 23,we can see that the slope and shape of the characteristics are dependent intimately on the machine parameters.

Further, this curve is obtained by varying slip with the applied voltage being held constant. Coupled with the fact that this is an equivalent circuit valid under steady state, it implies that if this characteristic is to be measured  experimentally, we need to look at the torque for a given speed after all transients have died down. One cannot, for example, try to obtain this curve by directly starting the motor with full voltage applied to the terminals

negative and measuring the torque and speed dynamically as it runs up to steady speed.

Another point to note is that the equivalent circuit and the values of torque predicted is valid when the applied voltage waveform is sinusoidal. With non-sinusoidal voltage wave- forms, the procedure is not as straightforward.

Respect to the direction of rotation of the air-gap flux, the rotor maybe driven to higher speeds by a prime mover or may also be rotated in the reverse direction. The torque-speed relation for the machine under the entire speed range is called the complete speed-torque characteristic. A typical curve is shown in fig. 7.1 for a four-pole machine, the synchronous speed being 1500 rpm. Note that speeds correspond to slip values greater than 1, and speeds greater than 1500 rpm correspond to negative slip. The plot also shows the operating modes of the induction machine in various regions. The slip axis is also shown for convenience.

Restricting ourselves to positive values of slip, we see that the curve has a peak point. This is the maximum torque that the machine can produce, and is called as stalling torque. If the load torque is more than this value, the machine stops rotating or stalls. It occurs at a slip ˆs, which for the machine of fig. 7.1 is 0.38. At values of slip lower than ˆs, the curve falls steeply down to zero at s = 0. The torque at synchronous speed is therefore zero. At values of slip higher than s = ˆs, the curve falls slowly to a minimum value at s = 1. The torque at s = 1 (speed = 0) is called the starting torque.

The value of the stalling torque may be obtained by differentiating the expression fortorque with respect to zero and setting it to zero to find the value of ˆs. Using this method,

the negative sign being valid for negative slip.

The expression shows that Te is the independent of R′ while ˆs is directly proportional to R′ r. This fact can be made use of conveniently to alter ˆs. If it is possible to change R′ r, then we can get a whole series of torque-speed characteristics, the maximum torque remaining constant all the while. But this is a subject to be discussed later.

We may note that if R′r is chosen equal to  , ˆs, becomes unity, which means that the maximum torque occurs at starting.

Thus changing of R′ r, wherever possible   can serve as a means to control the starting torque. While considering the negative slip range, (generator mode) we note that the maximum torque is higher than in the positive slip region (motoring mode).

Efficiency

The efficiency of induction machines at nominal operation, with neglection of stator copper losses (R1 = 0), computes to:

The nominal slip sn is supposed to be kept as small as possible, in order to achieve proper nominal efficiency. Usual amounts for nominal slips are:

When taking stator copper losses and hysteresis losses into account, real applications actually show lower efficiency amounts between approx. 0,8 and 0,95