0195136047.pdf

538 ELECTROMECHANICS

Forx= 0 .25 cm and|f|=100 N,

i^2 =

100 ( 0. 25 × 10 − 2 )^2

2. 5 π× 10 −^4

=

2. 5

π

= 0. 795

or

i= 0 .89 A

(b)Wm′(i, x)=

1

2

Li^2 =

1

2

5 π× 10 −^4

x

i^2

Sincef=∂Wm′(i, x)/∂x, the magnitude of the developed force is given by

|f|=

1

2

5 π× 10 −^4

x^2

i^2

Forx= 0 .25 cm and|f|=100 N,

i^2 =

100 × 2 ×( 0. 25 × 10 −^2 )^2

5 π× 10 −^4

=

2. 5

π

= 0. 795

or

i= 0 .89 A

Such relays find common application in industrial practice to remotely switch large

industrial loads. The student should recognize that a relatively low-level current can be

used to operate the relay, which in turn controls the opening and closing of a circuit that

carries large currents.

Starting with the flux linkages given by Equations (12.4.22) and (12.4.23), one can develop

the volt–ampere equations for the stator and rotor circuits. While the voltage and torque equations

for the idealized elementary machine of Example 12.4.1 with a uniform air gap are now obtained

from the coupled-circuit viewpoint, these can also be formulated from themagnetic-field viewpoint

based on the interaction of the magnetic fields of the stator and rotor windings in the air gap.

Since the mmf waves of the stator and rotor are considered spatial sine waves, they can be

represented by the space vectorsF ̄sandF ̄r, drawn along the magnetic axes of the stator and

rotor mmf waves, as in Figure E12.4.1, with the phase angleδ(in electrical units) between their

magnetic axes. The resultant mmfF ̄acting across the air gap is also a sine wave, given by the

vector sum ofF ̄sandF ̄r, so that

F^2 =Fs^2 +Fr^2 + 2 FsFrcosδ (12.4.26)

whereF’s are the peak values of the mmf waves. Assuming the air-gap field to be entirely radial,

the resultantH ̄-field is a sinusoidal space wave whose peak is given by

Hpeak=

F

g

(12.4.27)

wheregis the radial length of the air gap. Because of linearity, the coenergy is equal to the energy.

The average coenergy density obtained by averaging over the volume of the air-gap region is

(w′)av=

μ 0

2

(average value ofH^2 )=

μ 0

2

Hpeak^2

2

=

μ 0

4

F^2

g^2

(12.4.28)