0195136047.pdf

522 ELECTROMECHANICS

EXAMPLE 12.2.2

The armature of a four-pole dc machine has a simplex lap wound commutator winding (which has

the number of parallel paths equal to the number of poles) with 120 two-turn coils. If the flux per

pole is 0.02 Wb, calculate the dc voltage appearing across the brushes located on the quadrature

axis when the machine is running at 1800 r/min.

Solution

This example can be solved by the direct application of Equation (12.2.15),

Ea=

PφnZ

α× 60

For our example,P= 4 ,φ= 0. 02 ,n=1800, the number of conductorsZ= 120 × 2 × 2 = 480

and the number of parallel pathsα=4 for the simplex lap winding (the same as the number of

poles). Therefore,

Ea=

4 × 0. 02 × 1800 × 480

4 × 60

=288 V dc

12.3 Rotating Magnetic Fields

When a machine has more than two poles, only a single pair of poles needs to be considered

because the electric, magnetic, and mechanical conditions associated with every other pole pair

are repetitions of those for the pole pair under consideration. The angle subtended by one pair

of poles in aP-pole machine (or one cycle of flux distribution) is defined to be 360electrical

degrees,or2πelectrical radians.So the relationship between the mechanical anglemand the

anglein electrical units is given by

θ=

P

2

θm (12.3.1)

because one complete revolution hasP/2 complete wavelengths (or cycles). In view of this

relationship, for a two-pole machine, electrical degrees (or radians) will be the same as mechanical

degrees (or radians).

In this section we set out to show that a rotating field of constant amplitude and sinusoidal

space distribution of mmf around a periphery of the stator is produced by a three-phase winding

located on the stator and excited by balanced three-phase currents when the respective phase

windings are wound 2π/3 electrical radians (or 120 electrical degrees) apart in space. Let us

consider the two-pole, three-phase winding arrangement on the stator shown in Figure 12.3.1.

The windings of the individual phases are displaced by 120 electrical degrees from each other in

space around the air-gap periphery. The reference directions are given for positive phase currents.

The concentrated full-pitch coils, shown here for simplicity and convenience, do in fact represent

the actual distributed windings producing sinusoidal mmf waves centered on the magnetic axes

of the respective phases. Thus, these three sinusoidal mmf waves are displaced by 120 electrical

degrees from each other in space. Let a balanced three-phase excitation be applied with phase

sequencea–b–c,

ia=Icosωst; ib=I cos(ωst−120°); ic=Icos(ωst−240°) (12.3.2)

whereIis the maximum value of the current, and the timet=0 is chosen arbitrarily when the

a-phase current is a positive maximum. Each phase current is an ac wave varying in magnitude