0195136047.pdf

516 ELECTROMECHANICS

e=−

dλ

dt

=ωNφsinωt−N

dφ

dt

cosωt (12.2.5)

The minus sign associated with Faraday’s law in Equation (12.2.5) implies generator reference

directions, as explained earlier. Considering the right-hand side of Equation (12.2.5), the first term

is a speed voltage caused by the relative motion of the field and the stator coil. The second term

is a transformer voltage, which is negligible in most rotating machines under normal steady-state

operation because the amplitude of the air-gap flux wave is fairly constant. The induced voltage

is then given by the speed voltage itself,

e=ωNφsinωt (12.2.6)

Equation (12.2.6) may alternatively be obtained by the application of the cutting-of-flux concept

given by Equation (12.1.3), from which the motional emf is given by the product ofBcoiltimes

the total active length of the conductorsleffin the two coil sides times the linear velocity of the

conductor relative to the field, provided that these three are mutually perpendicular. For the case

under consideration, then,

e=Bcoilleffv=(Bmsinωt)( 2 lN )(rωm)

or

e=(Bmsinωt)( 2 lN)

(

r 2 ω

P

)

=ωN

2

P

2 Bmlrsinωt (12.2.7)

which is the same as Equation (12.2.6) when the expression forφ, given by Equation (12.2.3), is

substituted.

The resulting coil voltage is thus a time function having the same sinusoidal waveform as

the spatial distributionB. The coil voltage passes through a complete cycle for each revolution

of the two-pole machine of Figure 12.2.1. So its frequency in hertz is the same as the speed of

the rotor in revolutions per second (r/s); that is, the electrical frequency is synchronized with the

mechanical speed of rotation. Thus, a two-pole synchronous machine, under normal steady-state

conditions of operation, revolves at 60 r/s, or 3600 r/min, in order to produce 60-Hz voltage. For

aP-pole machine in general, however, the coil voltage passes through a complete cycle every

time a pair of poles sweeps, i.e.,P/2 times in each revolution. The frequency of the voltage wave

is then given by

f=

P

2

·

n

60

Hz (12.2.8)

wherenis the mechanical speed of rotation in r/min. Thesynchronous speedin terms of the

frequency and the number of poles is given by

n=

120 f

P

r/min (12.2.9)

The radian frequencyωof the voltage wave in terms ofωm, the mechanical speed in radians per

second (rad/s), is given by

ω=

P

2

ωm (12.2.10)

Figure 12.2.2 shows an elementary single-phase synchronous machine with four salient poles;

the flux paths are shown by dashed lines. Two complete wavelengths (or cycles) exist in the

flux distribution around the periphery, since the field coils are connected so as to form poles of

alternate north and south polarities. The armature winding now consists of two coils(a 1 ,−a 1 )