0195136047.pdf

12.3 ROTATING MAGNETIC FIELDS 523

sinusoidally with time. Hence, the corresponding component mmf waves vary sinusoidally with
time. The sum of these components yields the resultant mmf.
Analytically, the resultant mmf at any point at an angleθfrom the axis of phaseais given by
F(θ)=Facosθ+Fbcos(θ−120°)+Fccos(θ−240°) (12.3.3)

But the mmf amplitudes vary with time according to the current variations,

Fa=Fmcosωst; Fb=Fmcos(ωst−120°); Fc=Fmcos(ωst−240°) (12.3.4)

Then, on substitution, it follows that

F(θ,t)=Fmcosθ cosωst+Fmcos(θ−120°)cos(ωst−120°)

+Fmcos(θ−240°)cos(ωst−240°) (12.3.5)

By the use of the trigonometric identity

cosαcosβ=

1

2

cos(α−β)+

1

2

cos(α+β)

and noting that the sum of three equal sinusoids displaced in phase by 120° is equal to zero,
Equation (12.3.5) can be simplified as

F(θ,t)=

3

2

Fmcos(θ−ωst) (12.3.6)

which is the expression for the resultant mmf wave. It has a constant amplitude^3 / 2 Fm,isa
sinusoidal function of the angleθ, and rotates in synchronism with the supply frequency; hence it
is called arotating field.The constant amplitude is^3 / 2 times the maximum contributionFmof any
one phase. The angular velocity of the wave isωs= 2 πfselectrical radians per second, where
fsis the frequency of the electric supply in hertz. For aP-pole machine, the rotational speed is
given by

ωm=

2

P

ωsrad/sorn=

120 fs

P

r/min (12.3.7)

which is the synchronous speed.
The same result may be obtained graphically, as shown in Figure 12.3.2, which shows
the spatial distribution of the mmf of each phase and that of the resultant mmf (given by the
algebraic sum of the three components at any given instant of time). Figure 12.3.2(a) applies

Axis of phase a

Axis of phase c

Axis of phase b

+

+ +

θ

−a

a

−b

b

−c

c

Figure 12.3.1Simple two-pole, three-phase wind-

ing arrangement on a stator.