0195136047.pdf

514 ELECTROMECHANICS

f=m

du

dt

=fe−fd−fk=fe−du−kx=Bli−du−kx

wheredrepresents the damping coefficient,krepresents the spring constant, andfeis the magnetic

force due to current flow in the coil. Using phasor techniques, we have

V(jω)=j ωLI (j ω)+RI (jω)+BlU(jω)

and

(j ωm+d)U(jω)+

k

jω

U(jω)=BlI(jω)

Neglecting the coil inductanceL,weget

I(jω)=

V(jω)−BlU(jω)

R

The frequency response of the loudspeaker is then given by

U(jω

V(jω)

=

Bl

Rm

jω

(j ω)^2 +jω(d/m+B^2 l^2 /Rm)+k/m

12.2 EMF Produced by Windings

The time variation of emf for a single conductor corresponds to the spatial variation of air-gap flux

density. By suitable winding design, the harmonics can be reduced appreciably, and the waveform

of the generated emf can be made to approach a pure sine shape.

Figure 12.2.1 shows an elementary single-phase, two-pole synchronous machine. In al-

most all cases, the armature winding of a synchronous machine is on the stator and the field

winding is on the rotor, because it is constructionally advantageous to have the low-power

field winding on the rotating member. The field winding is excited by direct current, which

is supplied by a dc source connected to carbon brushes bearing on slip rings (or collector

rings). The armature windings, though distributed in the slots around the inner periphery of

the stator in an actual machine, are shown in Figure 12.2.1(a) for simplicity as consisting

of a single coil ofNturns, indicated in cross section by the two sidesaand−aplaced in

diametrically opposite narrow slots. The conductors forming these coil sides are placed in

slots parallel to the machine shaft and connected in series by means of the end connections.

The coil in Figure 12.2.1(a) spans 180° (or a completepole pitch, which is the peripheral

distance from the centerline of a north pole to the centerline of an adjacent south pole) and

is hence known as afull-pitchcoil. For simplicity and convenience, Figure 12.2.1(a) shows

only a two-pole synchronous machine with salient-pole construction; the flux paths are shown

by dashed lines. Figure 12.2.1(b) illustrates a nonsalient-pole or cylindrical-rotor construc-

tion. The stator winding details are not shown and the flux paths are indicated by dashed

lines.

The space distribution of the radial air-gap flux density around the air-gap periphery can be

made to approximate a sinusoidal distribution by properly shaping the rotor pole faces facing the

air gap,

B=Bmcosβ (12.2.1)

whereBmis the peak value at the rotor pole center andβis measured in electrical radians from

the rotor pole axis (or the magnetic axis of the rotor), as shown in Figure 12.2.1. The air-gap flux

per pole is the integral of the flux density over the pole area. For a two-pole machine,