12.5 BASIC ASPECTS OF ELECTROMECHANICAL ENERGY CONVERTERS 539
since the average value of the square of a sine wave is one-half of the square of its peak value.
The total coenergy for the air-gap region is then given by
W′=(w′)av(volume of air-gap region)=
μ 0
4
F^2
g^2
πDlg (12.4.29)
whereDis the average diameter at the air gap andlis the axial length of the machine. Equation
(12.4.29) may be rewritten as follows by using Equation (12.4.26):
W′=
πDlμ 0
4 g
(Fs^2 +Fr^2 + 2 FsFrcosδ) (12.4.30)
The torque in terms of the interacting magnetic fields is obtained by taking the partial derivative
of the field coenergy with respect to the angleδ. For a two-pole machine, such a torque is given by
Te=
∂W′
∂δ
=−
πDlμ 0
2 g
FsFrsinδ=−KFsFrsinδ (12.4.31)
in whichKis a constant determined by the dimensions of the machine. The torque for aP-pole
cylindrical machine with a uniform air gap is then
Te=−
P
2
KFsFrsinδ (12.4.32)
Equations (12.4.31) and (12.4.32) have shown that the torque is proportional to the peak values
of the interacting stator and rotor mmfs and also to the sine of the space-phase angleδbetween
them (expressed in electrical units). The interpretation of the negative sign is the same as before, in
that the fields tend to align themselves by decreasing the displacement angleδbetween the fields.
Equation (12.4.32) shows that it is possible to obtain a constant torque, varying neither with
time nor with rotor position, provided that the two mmf waves are of constant amplitude and
have constant angular displacement from each other. While it is easy to conceive of the two mmf
waves having constant amplitudes, the question would then be how to maintain a constant angle
between the stator and rotor mmf axes if one winding is stationary and the other is rotating. Three
possible answers arise:
- If the stator mmf axis is fixed in space, the rotor mmf must also be fixed in space, even
when the rotor winding is physically rotating, as is the case with a dc machine.
- If the rotor mmf axis is fixed relative to the rotor, the stator mmf axis must rotate at
the rotor speed relative to the stationary stator windings, as is the case with a polyphase
synchronous machine.
- The two mmf axes must rotate at such speeds relative to their windings that they remain
stationary with respect to each other, as is the case with a polyphase induction machine
(which we will explain later).
12.5 Basic Aspects of Electromechanical Energy Converters
Whereas detailed differences and particularly challenging problems emerge among various ma-
chine types, this section briefly touches on the interrelated problems that are common to all
machine types, such as losses and efficiency, ventilation and cooling, machine ratings, magnetic
saturation, leakage and harmonic fluxes, and machine applications. Variousstandards, developed
by IEEE (Institute of Electrical and Electronics Engineers), NEMA (National Electrical Man-
ufacturers Association), ANSI (American National Standards Institute), and IEC (International
Electrotechnical Commission), deal with machine ratings, insulation and allowable temperature
rise, testing methods, losses, and efficiency determination.
