A single-phase ac motor is sketched in Fig. 15.30. Very small synchronous motors of this kind are occasionally used for constant-speed applications, but most single-phase motors are induction motors. Single-phase motors are often encountered in daily life, but their use is confined to low-power applications, where three-phase power is often unavailable.
The magnetic field produced by the stator in Fig. 15.30 always points in the x direction, oscillating between upward and downward. Since the field is oscillating rather than rotating, it seems remarkable that such a motor can work at all. Insight can be gained by “decomposing” the oscillating field into two rotating fields turning in opposite directions. Let B(t) be given by
where ex is the unit vector in the x direction. This expression is identically equal to
A moment’s study reveals that B1(t) is a rotating field rotating in the counterclockwise direction, while B2(t) is a rotating field rotating clockwise. (For example, at t = 0, B2 points up; at t = π/2ω it points to the right; at t = π/ ω it points down; and so on.) It is possible for the rotor to lock onto one or the other of these two component fields and rotate in response to it. For the case of an induction motor, the torque-speed characteristic is found by adding to Fig. 15.29, which represents the response to one rotating field, a similar inverted curve representing response to the oppositely rotating field. This is shown in Fig. 15.31. The curve marked “sum” passes through the origin, indicating that the motor has no starting torque. (This is of course what one would expect from symmetry.) Once started in either direction, however, the motor generates torque that tends to keep it rotating in that direction.
In order for single-phase motors to be self-starting, the stator poles can be shaded. A shaded pole is designed to make the magnetic field on one side lag the field on the other side of the same pole. This causes the field to move
Torque-speed characteristic of a single-phase induction motor.
(a) Torque from the clockwise-rotating component of the stator field.
(b) Torque from the counterclockwise component.
sideways slightly across the p~le, providing a small component of rotating field that allows the motor to start. Shaded poles are used in small motors up to about 1/20 hp. Another approach, used up to about 1/2 hp, is to provide a split-phase stator winding. In this method the main stator winding is supplemented by a starting winding. The latter has high inductance and thus carries . a current perhaps 30° to 45° out of phase with that of the main winding. This is equivalent to excitation by a two-phase power source and provides a component of rotating field. When the motor has gained speed the starting winding is cut out of the circuit by a centrifugal switch on the shaft.
Capacitor-start motors are similar, except that a capacitor is placed in series with a starting winding to increase the phase difference between the windings. This results in higher starting torque. In a capacitor motor the starting winding is not switched out, and its series capacitance is used to correct the power factor, thus increasing the efficiency.
The motors used in analog control systems are usually servo motors, small reversible dc motors with permanent-magnet stators. In modern work, however, control functions are increasingly performed by digital systems. An appropriate output device for such systems is the digital stepping motor. Stepping motors are similar to polyphase synchronous motors. However, they are intended not to turn continuously but rather to move to desired positions under digital control.
The design of a four-pole, four-phase stepping motor is illustrated in Fig. 15.32(a), and a typical sequence of the four magnet currents is shown in Fig. 15.32(b). (Note that each current passes through two coils-current 1 through the coils at 9 o’clock and 3 o’clock, and so on.) Each current can have only two values, designated (+) and (-), and we shall assume that when the current is ( +) the pole is north and when the current is ( – ) the pole is south. In Fig. 15.32(a) the rotor is shown at time t1 [first column of Fig. 15.32(b)]. Poles 1 and 2 are south, and a north pole of the rotor rests between them. At time t2 pole 1 changes to north and pole 3 changes to south, and the rotor moves one-eighth revolution clockwise; thus its north pole now rests between poles 2 and 3. We note that there is no possibility of the rotor moving in the wrong direction; since at t2 poles 1 and 4 are both north, the north pole of the rotor cannot possibly move counterclockwise. For continuous clockwise rotation, the coil voltages continue as shown in Fig. 15.32(b). We note that the four coil currents are four square waves in four different phases, 90° apart; hence this is called a four-phase motor. The number of degrees through which a four-phase p-pole motor turns in each step is 180/p, 45° in this example. Real motors are made with a larger number of poles in order to give finer position control. A typical motor might have 24 poles and turn 7.5° per step.
In fact only two currents are needed to drive the motor because I1 = – I3 and I2 = – I4, Thus we need pass only a single current through coils 1 and 3, with coil 3 connected backward. A simple digital circuit known as a
A four-phase four-pole digital stepping motor.
(a) Machine design.
four-phase square-wave generator is used to produce I1 and 12 in response to inputs DIRECTION and CLOCK, as shown in Fig. 15.33(a). When DIRECTION = 1 each clock pulse advances the square wave by one time unit, following the pattern of Fig. 15.32(b). The clock pulses do not have to arrive at regular intervals; they are the output of a control system, and the motor can start and stop as necessary. When DIRECTION = 0 the square wave is generated in reverse phase sequence, and the motor moves 45° counterclockwise at each pulse. A typical sequence of operations is shown in Fig. 15.33(b). During the interval between the second and third clock pulses the motor does not move. When DIRECTION changes from 1 to 0 the direction of motion is reversed.
• Transducers are analog devices intended to produce mechanical output proportional to electrical input, or vice versa. Two important classes are movingcoil transducers and variable reluctance transducers.
• Actuators are electromechanical output devices intended to perform mechanical operations in response to electrical inputs. Examples are solenoids and relays.
• Dc rotating machines are usually distinguished by the use of a shaft-mounted rotary switch called a commutator. The commutator switches the currents in the armature so that the torque is constantly in one direction.
• Transducers and motors can be represented by circuit models. Each of these models contains a voltage source representing the back emf generated by mechanical motion.
• Dc rotating machines can be shunt connected, series connected, compound, or separately excited. These connections are used in order to obtain suitable torque speed characteristics.
• Two important classes of ac machines are synchronous machines and induction machines. The former is most common as a generator, the latter as a motor.
• Ac machines are based on the use of a rotating magnetic field.
• In synchronous machines the rotor turns at the same speed as the rotating field. (Thus synchronous motors can operate at only a single fixed speed determined by the power-line frequency.)
• A synchronous motor running without load can be used as an adjustable reactance for power factor correction. When used as a capacitor in this way the machine is called a synchronous condenser.
• Large ac machines are usually used in connection with three-phase power. Small motors, however, are often single-phase machines. Shaded poles, splitphase stator windings, or capacitor starting are used to obtain starting torque in single-phase motors.
• Stepping motors are small actuators that resemble ac motors. They are used as output devices in digital control systems.