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Figure 1730 Two-pole synchronous machine
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Figure 1731 Four-pole three-phase alternator
three-phase wye connection has been divided into two coils, wound in different locations, according to the schematic stator diagram of Figure 1731 One could then envision analogous con gurations with greater numbers of poles, obtained in the same fashion, that is, by dividing each arm of a wye connection into more windings The arrangement shown in Figure 1731 requires that a further distinction be made between mechanical degrees, m , and electrical degrees, e In the four-pole alternator, the ux will see two complete cycles during one rotation of the rotor, and therefore the voltage that is generated in the coils will also oscillate at twice the frequency of rotation In general, the electrical degrees (or radians) are related to the mechanical degrees by the expression e = p m 2 (1756)
where p is the number of poles In effect, the voltage across a coil of the machine goes through one cycle every time a pair of poles moves past the coil Thus, the
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17
Introduction to Electric Machines
Figure 1732 Automotive alternator (Courtesy: Delphi Automotive Systems)
frequency of the voltage generated by a synchronous generator is p n f = Hz (1757) 2 60 where n is the mechanical speed in rev/min Alternatively, if the speed is expressed in rad/s, we have p e = m (1758) 2 where m is the mechanical speed of rotation in rad/s The number of poles employed in a synchronous generator is then determined by two factors: the frequency desired of the generated voltage (eg, 60 Hz, if the generator is used to produce AC power), and the speed of rotation of the prime mover In the latter respect, there is a signi cant difference, for example, between the speed of rotation of a steam turbine generator and that of a hydroelectric generator, the former being much greater A common application of the alternator is in automotive battery-charging systems in which, however, the generated AC voltage is recti ed to provide the DC current required for charging the battery Figure 1732 depicts an automotive alternator
THE SYNCHRONOUS MOTOR
Synchronous motors are virtually identical to synchronous generators with regard to their construction, except for an additional winding for helping start the motor and minimizing motor speed over- and undershoots The principle of operation is, of course, the opposite: an AC excitation provided to the armature generates a magnetic eld in the air gap between stator and rotor, resulting in a mechanical torque To generate the rotor magnetic eld, some DC current must be provided to the eld windings; this is often accomplished by means of an exciter, which consists of a small DC generator propelled by the motor itself, and therefore mechanically connected to it It was mentioned earlier that to obtain a constant torque in an electric motor, it is necessary to keep the rotor and stator magnetic elds constant relative to each other This means that the electromagnetically rotating eld in the stator and the mechanically rotating rotor eld should be aligned at all times The only condition for which this can occur is if both elds are rotating at the synchronous speed, ns = 120f/p Thus, synchronous motors are by their very nature constant-speed motors For a non salient pole (cylindrical-rotor) synchronous machine, the torque can be written in terms of the AC stator current, iS (t), and of the DC rotor current, If : T = kiS (t)If sin( ) (1759)
where is the angle between the stator and rotor elds (see Figure 177) Let the angular speed of rotation be d m rad/s (1760) dt where m = 2 n/60, and let e be the electrical frequency of iS (t), where iS (t) = 2IS sin( e t) Then the torque may be expressed as follows: T = k 2IS sin( e t)If sin( ) (1761) m =
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