0195136047.pdf

526 ELECTROMECHANICS

ωt

(a) (b)

i = I cos ωt

B = Bm cos ωt Bm cos ωt

ωt

ω

ω

Bm/ 2

Bm/ 2

Figure 12.3.3Single-phase winding carrying alternating current, producing a stationary pulsating flux or

equivalent rotating flux components.

12.4 Forces and Torques in Magnetic-Field Systems

We mentioned earlier that the greater ease of storing energy in magnetic fields largely accounts

for the common use of electromagnetic devices for electromechanical energy conversion. In a

magnetic circuit containing an air-gap region, the energy stored in the air-gap space is several

times greater than that stored in the iron portion, even though the volume of the air gap is only a

small fraction of that of the iron.

The energy-conversion process involves an interchange between electric and mechanical

energy via the stored energy in the magnetic field. This stored energy, which can be determined

for any configuration of the system, is astate functiondefined solely by functional relationships

between variables and the final values of these variables. Thus, theenergy methodis a powerful

tool for determining the coupling forces of electromechanics.

In asingly excited system,a change in flux density from a value of zero initial flux density to

Brequires an energy input to the field occupying a given volume,

Wm=vol

∫B 1

0

HdB (12.4.1)

which can also be expressed by

Wm=

∫λ 1

0

i(λ) dλ=

∫φ 1

0

F(φ) dφ (12.4.2)

Note that the currentiis a function of the flux linkagesλand that the mmfFis a function of the

fluxφ; their relations depend on the geometry of the coil, the magnetic circuit, and the magnetic

l Figure 12.4.1Graphical interpretation of energy

and coenergy in a singly excited nonlinear system.