Electrical M/C’s and Power Utilization 3(2+1)

LESSON 26. Polyphase motor - Phase diagram, torque equation, starting and speed control methods

The induced emf in any single conductor, l centimeters in length, in a field having a density of B gausses, the conductor moving at a velocity of v cm/s with respect to the field, is e = Blv10^-8 volts, the flux, the conductor and the velocity being mutually perpendicular Therefore, when a conductor is cutting flux at a uniform velocity, the flux being sinusoidally or otherwise dis­tributed in space, the emf in the conductor is zero when it is moving in a region where B, the flux density, is zero; the emf is a maximum when the conductor is moving in a region where B the flux density, is a maximum.

Alternating-current torque when current and flux are in space-phase. It may be said that e, the emf per conductor, is in phase with the flux. It further follows that

Fig. 11.1 Torque developed

the wave shape of the emf. in a single con­ductor is the same as the shape of the space-distribution curve of the flux. At small values of slip, the angle a, between the induced emf in each conductor and the current in the conductor, is small and therefore the current in each of the conductors, is practically in phase with its induced emf.    As the induced emf is a maximum when the conductor is in the field of greatest flux density, the current will be a maximum at practically the same instant. The current is then in time-phase with the emf and hence in space-phase with the flux. Under these conditions the current in the particular conductor which is under the center of the pole, is a maximum, and that in the other conductors is less, decreasing sinusoidally as indicated.

The force acting on each conductor is proportional to its current and to the flux density of that part of the field in which the conductor finds itself. The torque curve (Fig. 11.1) is obtained by taking the product of the current and flux at each point at each point, multiplied by a constant. It will be noted that this torque curve is of double frequency, it is always positive and reaches zero twice every cycle.

As the value of the slip increases, the reactance of the rotor increases, the reactance being proportional to the rotor frequency and hence to the slip, and the angle a by which the current lags its induced emf increases, since tan a = 2π fsL₂/R₂. The current in any conductor will not reach its maximum value until a time-degrees after the induced emf has reached its maximum value.

Operating Characteristics of the Squirrel-cage Motor

The squirrel-cage motor, like the direct-current shunt motor, operates at substantially constant speed. As the rotor cannot reach the speed of the rotating magnetic field, it must at all times operate with a certain amount of slip. At no load the slip is very small. As load is applied to the rotor, more rotor current is required to develop the necessary torque in order to carry the increased load. Consequently, the rotating magnetic field must cut the rotor conductors at an increased rate, in order to produce the necessary increase of current. The slip of the rotor must accordingly increase, so that the rotor speed drops. The ratio of the slip to the total power delivered to the rotor is proportional to the I²R loss in the rotor. As the resistance of the squirrel cage is very low, the I²R loss is low and, therefore, the slip for ordinary loads is small. In large motors, 50 hp. or greater, the slip is of the order of 1 to 2 per cent, at full load. In the smaller sizes of motor, the slip may be as high as 8 to 10 per cent at full load.

Figure 11.2 shows the characteristic curves of a 10-hp. squirrel-cage motor. It will be noted that the torque, speed and efficiency curves are very similar to those of a shunt motor.

Fig. 11.2 Operating characteristics of a squirrel-cage induction motor.

Fig. 11.3 Increase of  power factor with load

The power factor increases with the load for the following reason: At no load the motor takes a current I₀, in figure 11.3, which is   mostly magnetizing   current,   although  there   is   a  small energy component necessary to supply the no-load losses. The power factor at no load is cos q₀, the value of which may be as low as 0.10 to 0.15. The back emf of the motor remains nearly constant from no load to full load. Therefore, the flux must re­main substantially constant, just as it does in the transformer, so that the magnetizing current changes but slightly from no load to full load.   As load is applied to the motor, an energy current I₁’ is required to carry the load.    This current, when combined with  I₀, gives the total current I₁ at this load, and the resulting power-factor is cos q_i. As the load increases, an energy current I₂’ is required. The total current then becomes I₂ and the corresponding power-factor becomes cos q₂. It will be observed that the power factor angle decreases and therefore the power factor increases as the load on the motor increases. The increased reactance drops in the stator and in the rotor with increase of load tend to oppose this increase of power-factor and when the load exceeds a certain value may even bring about a decrease of power-factor.

As the power-factor increases, a smaller increase of current is required for a given increase of load than would be necessary if the power factor were constant. Therefore, the current in­creases more slowly than the load as shown. At first the efficiency increases rapidly and reaches a maximum value for the same reason that it does in other electrical apparatus. At all loads there are certain fixed losses, such as core loss and fric­tion. In addition there are the load losses (I²R) which increase nearly as the square of the load. Therefore, at light loads the efficiency is low because the fixed losses are large as compared with the input. As the load increases, the efficiency increases to a maximum, the fixed and variable losses being equal at this point. Beyond this point the I²R losses become rela­tively large, causing the efficiency to decrease.

One disadvantage of the squirrel-cage motor lies in the fact that it takes a very large current at low power-factor on starting, and in spite of this large current it develops  little torque. When the motor is at standstill, the squirrel cage acts as the short-circuited secondary of a transformer, causing the motor to take an excessive current on starting, if full voltage is applied.

Figure 11.4 shows the variation of torque with slip for two different values of line voltage. For small values of slip up to and beyond full load, which is the ordinary range of operation, the torque is substantially proportional to the slip. At higher values of slip, however, the torque curve bends over and finally reaches a maximum. This maximum is called the break-dawn torque.

Fig. 11.4 Slip-torque curves for squirrel-cage motor.

It is also true that the torque of an induction motor for a given slip is proportional to the square of the line voltage. For this reason a 10 per cent drop in voltage (Fig. 11.4), may cause a 19 per cent reduction in the break-down and starting torques.

The stator impedance also reduces the break-down torque. A high stator impedance means a comparatively large impedance drop in the stator for a given current. This decreases the back emf, E, hence the air-gap flux becomes less, and therefore the value of the rotor current at any given slip is reduced. This results in a reduction of torque for each value of slip.

The effect of each of these various factors upon the break-down torque is shown in the following equation: The break-down torque

where K is constant, V is the terminal voltage, r₁  is the stator resistance, x₁  is the stator reactance, and x₂ is the rotor reactance at standstill. As the squirrel-cage motor is ordinarily started at low voltage, it develops but little starting torque, because the flux is small.

Small induction motors are directly connected through simple 3 pole switches and the higher starting current is not minded much. Because the magnitude of current is relatively less. Howver for motors of more than 3 hp, the higher starting current should reduced to save the motor. As the squirrel cage motor at starting is equivalent to a short-circuited   transformer, it is necessary to reduce the starting current  in the larger sizes.    One simple method as shown in figure 11.5 is to use a delta-connected motor.    By means of a triple-pole, double-throw   (T.P.D.T) switch the windings are first thrown in star across  the line, thus applying only l/3 of the normal voltage to each coil. This makes the line current one-third the value if the motor were directly across the line. When the motor has attained sufficient speed, the switch is thrown over, connecting the motor in delta across the line.

Fig. 11.5 Star delta method of starting an induction motor.