*SYNCHRONOUS SPEED*

The speed at which the magnetic field rotates is known as the *synchronous speed*. The synchronous speed of a three-phase motor is determined by two factors:

1. The number of stator poles

2. The frequency of the ac line

Because 60 Hz is a standard frequency throughout the United States and Canada, the following gives the synchronous speeds for motors with different numbers of poles:

**Speed Performance**

The field set up by the stator windings cuts the copper bars of the rotor. Voltages induced in the squirrel-cage winding set up currents in the rotor bars. As a result, a field is created on the rotor core. The attraction between the stator field and the rotor field causes the rotor to follow the stator field. The rotor always turns at a speed that is slightly less than that of the stator field (less than the synchronous speed). In this way, the stator field cuts the rotor bars and induces the necessary rotor voltages and currents to create the rotor field.

The torque produced by an induction motor results from the interaction between the *stator flux *and the *r**otor flux*. If the rotor is turned at the same speed as the stator field, there will be no relative motion between the rotor bars and the stator field. This means that no torque can be produced. A torque is produced *only *when the rotor turns at a speed that is less than synchronous speed. At no load, the mechanical losses of the motor can be over- come by a small torque. The rotor speed will be slightly less than the synchronous speed of the stator field.

As a mechanical load is applied to the motor shaft, the rotor speed will decrease. The stator field turns at a constant synchronous speed and cuts the rotor bars at a faster rate per second. The voltages and currents induced in the rotor bars increase accordingly, causing a greater induced rotor voltage. The resulting increase in the rotor current causes a large torque at a slightly lower speed.

The squirrel-cage winding was described as consisting of heavy copper bars welded to two end rings. The impedance of this winding is relatively low. Therefore, a slight decrease in the speed causes a large increase in the currents in the rotor bars. Because the rotor circuit of a squirrel-cage induction motor has a low impedance, the speed regulation of this motor is very good.

*PERCENT SLIP*

The speed performance of squirrel-cage induction motors is measured in terms of *percent slip*. In determining percent slip, the synchronous speed of the stator field is used as a reference point. The synchronous speed for a particular motor is constant, because the number of poles and the frequency remain the same. *Slip *is the number of revolutions per minute by which the rotor falls behind the speed of the rotating field of the stator. Slip is determined by subtracting the speed of the rotor from the synchronous speed of the stator field. For example, a three-phase, two-pole induction motor has a full-load speed of 3480 r/min. The synchronous speed of the stator field is

Smaller values of percent slip mean that the motor has better speed regulation. When determined at the rated load, the percent slip of most squirrel-cage induction motors varies from 2% to 5%. This type of motor is considered to be a constant-speed motor because there is a small decrease in the speed between the no-load and full-load points.

__ROTOR FREQUENCY__

In the previous example, the rotor slips behind the speed of the stator field by 120 revolutions per minute. The flux of the two stator poles passes a given rotor bar of the squirrel-cage winding only 120 times every minute. Thus, the voltages and currents induced in the rotor will have a very low frequency. The rotor frequency is

Unit 1 showed that when a conductor passes a pair of unlike poles, one cycle (Hz) of voltage is induced in the conductor. In this example, a pair of stator poles passes a given bar in the squirrel-cage rotor 120 times per minute or twice per second. Thus, the frequency must be 2 Hz. If the slip is increased, the rotor frequency will increase, because the flux of the revolving field will cut a given bar in the squirrel-cage winding more times per second. This relationship can be expressed as a formula

Note that methods 1 and 2 both give the same frequency for the rotor. This frequency is an important factor in the operation of the motor. A change in the rotor frequency causes a change in the inductive reactance component (X= 2Tif L) of the rotor impedance. Thus, a change in the frequency will affect the starting and running characteristics of the motor.

**PROBLEM 1**

**Statement of the Problem**

A 5-hp, 220-V, three-phase, 60-Hz, squirrel-cage induction motor has eight poles. At the rated load, it has a speed of 870 r/min. Determine

1. the synchronous speed.

2. the percent slip, at the rated load.

3. the rotor frequency, at the instant of start-up.

4. ssthe rotor frequency, at the rated load.

**Solution**

1. The synchronous speed of the stator field is found using the frequency formula trans- posed to solve for S:

2. At the rated load, the percent slip is

3. At start-up, the rotor is not turning. The slip at this instant is unity, or 100%. There- fore, the rotor frequency and the stator frequency are both 60 Hz.

4. The rotor turns at 870 r/min at the rated load. The rotor frequency at this speed can be determined as follows:

Method 1:

Slip, in r/min = synchronous speed - rotor speed

= 900 - 870 = 30 r/min

__TORQUE AND SPEED CHARACTERISTICS__

The torque produced by an induction motor depends on the strengths of the stator and rotor fields and the phase relationship between the fields:

This torque formula is similar to the formula for the torque of a dc motor: T = k X 8f X IA. The difference between the formulas is in the cos 8R function. The equivalent dia- gram of the rotor is an inductor and resistor in series. Because rotor frequency is a function of slip, it follows that cos 8R varies with slip.

__STARTING CHARACTERISTICS__

At the instant the motor is started, the rotor is not turning and there is 100% slip. The rotor frequency at this moment is equal to the stator frequency. The inductive reactance of the rotor is very large compared to the effective resistance component. Also, the rotor has a very low lagging power factor. This means that the rotor flux lags the stator flux by a large phase angle. As a result, the interaction between the two fields is small and the starting torque is low.

As the speed of the motor increases, the percent slip and the frequency of the rotor decrease. The decrease in the rotor frequency causes the inductive reactance and the impedance of the rotor to decrease. Thus, the phase angle between the stator and rotor fluxes is reduced. The torque then increases to its maximum value at about 20% slip. As the rotor continues to accelerate, the torque decreases until it reaches the value required to turn the mechanical load applied to the motor shaft. The slip at this point is between 2% and 5%.

**Starting Current**

At start-up the stator field cuts the rotor bars at a faster rate than when the rotor is turning. The large voltage induced in the rotor causes a large rotor current. As a result, the stator current will also be high at start-up. The squirrel-cage induction motor resembles a static transformer during this brief instant. That is, the stator may be viewed as the primary or input winding, and the squirrel-cage rotor winding as the secondary winding.

Most three-phase, squirrel-cage induction motors are started with the rated line voltage applied directly to the motor terminals. This means that the starting surge of current reaches a value as high as three to five times the full-load current rating of the motor. This high starting current requires induction motors to have starting protection. This protection may be rated as high as three times the full-load current rating of the motor. In some instances, very large induction motors are started with auxiliary starters. These devices reduce the motor voltage at start-up to limit the starting surge of current. As a result, there is less voltage disturbance on the feeder circuit supplying the motor load.

**Starting with Reduced Voltage**

There are problems in starting a large induction motor with a reduced voltage. For example, assume that the voltage applied to the motor terminals at start-up is reduced to 50% of the rated nameplate voltage. The magnetizing flux of the stator is also reduced to half of the normal value. The voltages and currents induced in the rotor are similarly reduced by half. The resulting torque output of the motor is reduced to one-fourth of its original value. Figure 16–4 shows that a 50% reduction in voltage causes the torque to decrease to 25% of its normal value.

The torque formula given previously shows why the large reduction in the torque out- put occurs. Both the stator flux (<P S ) and the rotor current (IR ) are reduced to half of their original values. This means that the product (torque) of these terms is only one-fourth of

its original value. For a given value of slip, the torque varies as the square of the impressed voltage.

As explained previously, an increase in slip increases the rotor frequency and the inductive reactance of the rotor. In the normal operating range of the motor from no load to full load, the rotor frequency seldom is greater than 2 to 3 Hz. Therefore, a change in the frequency has negligible effects on the impedance of the rotor at full load, and even at 125% of the rated load.

**Motor with Overload**

When a motor has a heavy overload, the percent slip will increase, causing an increase in the rotor frequency. The increased frequency causes an increase in the inductive reactance and the impedance of the rotor circuit. Two effects result from the increase in the inductive reactance of the rotor circuit. First, the power factor of the rotor decreases, causing the rotor current to lag the induced rotor voltage. The rotor field flux will not reach its maximum value until the peak value of the stator flux wave has passed it. Although the cur- rents in the stator and rotor circuits increase because of the overload, the fluxes of the stator and the rotor fields are out of phase with each other. Therefore, there is less interaction

between the fields and the torque decreases. The second effect is that the increase in the inductive reactance and the impedance of the rotor decrease the rate at which the rotor current increases with an increase in slip. Because of these two effects, the torque increase will be less rapid. The torque reaches its maximum value at approximately 20% slip in the typical squirrel-cage induction motor.

**Breakdown Torque**

In Figure 16–4, note that the torque curve increases as a straight line well beyond the rated load. As the percent slip increases between 10% and 20%, there is a reduction in the rate at which the torque increases. Finally, at approximately 20% slip, the torque reaches its maximum value. The point of the maximum torque output is called the *breakdown point*. An increase in the load beyond this point results in less torque being developed by the motor and the rotor stops. As shown in the figure, this breakdown point is reached between 200% and 300% of the rated torque.

The following example shows that for a given value of slip, the torque varies as the square of the impressed voltage. Assume that a 240-V, squirrel-cage motor is operated on a 208-V, three-phase circuit. The value of 208 V is 87% of the rated voltage of the motor. The torque output is 0.872 = 0.75. This means that the breakdown torque is reduced to 75% of its rated value.