0195136047.pdf

12.4 FORCES AND TORQUES IN MAGNETIC-FIELD SYSTEMS 531

δr

δs

δ

+

+

++

+ +++

+

++

+

Magnetic axis

of stator

Magnetic axis of rotor

(a) (b)

Rotor

winding

Stator winding

+

− +−

Stator coil

Rotor coil

θ

(c)

Fs sin δ = F sin δr

Fr sin δ = F sin δs

Fr

Fs

F

θ

Figure E12.4.1Elementary two-pole rotating machine with uniform air gap.(a)Winding distribution.
(b)Schematic representation.(c)Vector diagram of mmf waves.

Using trigonometric identities, the product of the three trigonometric terms in this

equation may be expressed to yield

Te=

−LIsIr

4

[sin{[ωm+(ωs+ωr)]t+α+θ 0 }

+sin{[ωm−(ωs+ωr)]t−α+θ 0 }

+sin{[ωm+(ωs−ωr)]t−α+θ 0 }

+sin{[ωm−(ωs−ωr)]t+α+θ 0 }]

(b) The average value of each of the sinusoidal terms in the previous equation is zero, unless

the coefficient oftis zero in that term. That is, the average torque (Te)avdeveloped by

the machine is zero unless

ωm=±(ωs±ωr)

which may also be expressed as

|ωm|=|ωs±ωr|