0195136047.pdf

524 ELECTROMECHANICS

for that instant when thea-phase current is a positive maximum; Figure 12.3.2(b) refers to that

instant when theb-phase current is a positive maximum; the intervening time corresponds to

120 electrical degrees. It can be seen from Figure 12.3.2 that during this time interval, the

resultant sinusoidal mmf waveform has traveled (or rotated through) 120 electrical degrees

of the periphery of the stator structure carrying the three-phase winding. That is to say, the

resultant mmf is rotating in synchronism with time variations in current, with its peak amplitude

remaining constant at^3 / 2 times that of the maximum phase value. Note that the peak value of

the resultant stator mmf wave coincides with the axis of a particular phase winding when that

phase winding carries its peak current. The graphical process can be continued for different

instants of time to show that the resultant mmf is in fact rotating in synchronism with the supply

frequency.

Although the analysis here is carried out only for a three-phase case, it holds good for anyq-

phase (q>1; i.e., polyphase) winding excited by balancedq-phase currents when the respective

phases are wound 2π/qelectrical radians apart in space. However, in a balanced two-phase case,

note that the two phase windings are displaced 90 electrical degrees in space, and the phase

currents in the two windings are phase-displaced by 90 electrical degrees in time. The constant

amplitude of the resultant rotating mmf can be shown to beq/2 times the maximum contribution

of any one phase. Neglecting the reluctance of the magnetic circuit, the corresponding flux density

in the air gap of the machine is then given by

Bg=

μ 0 F

g

(12.3.8)

wheregis the length of the air gap.

Production of Rotating Fields from Single-Phase Windings

In this subsection we show that a single-phase winding carrying alternating current produces a

stationary pulsating fluxthat can be represented by two counterrotating fluxes of constant and

equal magnitude.

Let us consider a single-phase winding, as shown in Figure 12.3.3(a), carrying alternating

currenti=I cosωt. This winding will produce a flux-density distribution whose axis is fixed

along the axis of the winding and that pulsates sinusoidally in magnitude. The flux density along

the coil axis is proportional to the current and is given byBmcosωt, whereBmis the peak flux

density along the coil axis.

Let the winding be on the stator of a rotating machine with uniform air gap, and let the flux

density be distributed sinusoidally around the air gap. Then the instantaneous flux density at any

positionθfrom the coil axis can be expressed as

B(θ)=(Bmcosωt)cosθ (12.3.9)

which may be rewritten by the use of the trigonometric identity following Equation (12.3.5) as

B(θ)=

Bm

2

cos(θ−ωt)+

Bm

2

cos(θ+ωt) (12.3.10)

The sinusoidal flux-density distribution given by Equation (12.3.9) can be represented by a

vectorBmcosωtof pulsating magnitude on the axis of the coil, as shown in Figure 12.3.3(a).

Alternatively, as suggested by Equation (12.3.10), this stationary pulsating flux-density vector can

be represented by two counterrotating vectors of constant magnitudeBm/2, as shown in Figure

12.3.3(b). While Equation (12.3.9) represents a standing space wave varying sinusoidally with

time, Equation (12.3.10) represents the two rotating components of constant and equal magnitude,