0.3 Chapter 4: introduction to rotating machines  (Page 5/9)

$(F_{\text{agl}}{)}_{\text{peak}}=\frac{4}{\pi }(\frac{\text{Ni}}{2})$ (4.4)

Consider a distributed winding, consisting of coils distributed in several slots.

• Fig. 4.11(a) shows phase a of the armature winding of a simplified two-pole, three-phase ac machine and phases b and c occupy the empty slots.
• The windings of the three phases are identical and are located with their magnetic axes 120 degrees apart.The winding is arranged in two layers, each full-pitch coil of $N_c$ turns having one side in the top of a slot and the other coil side in the bottom of a slot a pole pitch away.
• Fig. 4.11(b) shows that the mmf wave is a series of steps each of height ${\mathrm{2N}}_c{i}_a$ . It can be seen that the distributed winding produces a closer approximation to a sinusoidal mmf wave than the concentrated coil of Fig.4.10 does.

Figure 4.11 The mmf of one phase of a distributed two-pole,

three-phase winding with full-pitch coils.

• The modified form of (4.3) for a distributed multipole winding is

$F_{\text{agl}}=\frac{4}{\pi }(\frac{k_w{N}_{\text{ph}}}{\text{poles}})i_a\text{cos}(\frac{\text{poles}}{2}{\theta }_a)$ (4.5)

$N_{\text{ph}}$ : number of series turns per phase,

$k_w$ : winding factor, a reduction factor taking into account the distribution of the winding, typically in the range of 0.85 to 0.95, $k_w=k_b{k}_p(\text{or}{k}_d{k}_p)$ .

• The peak amplitude of this mmf wave is

$(F_{\text{agl}}{)}_{\text{peak}}=\frac{4}{\pi }(\frac{k_w{N}_{\text{ph}}}{\text{poles}})i_a$ (4.6)

• Eq. (4.5) describes the space-fundamental component of the mmf wave produced by current in phase a of a distributed winding.

• If $i_a=I_m\text{cos}\mathrm{\omega t}$ the result will be an mmf wave which is stationary in space and varies sinusoidally both with respect to ${\theta }_a$ and in time.
• The application of three-phase currents will produce a rotating mmf wave.

• Rotor windings are often distributed in slots to reduce the effects of space harmonics.

• Fig. 4.12(a) shows the rotor of a typical two-pole round-rotor generator.
• As shown in Fig. 4.12(b), there are fewer turns in the slots nearest the pole face.
• The fundamental air-gap mmf wave of a multipole rotor winding is

$F_{\text{agl}}=\frac{4}{\pi }(\frac{k_r{N}_r}{\text{poles}})I_r\text{cos}(\frac{\text{poles}}{2}{\theta }_r)$ (4.7)

$(F_{\text{agl}}{)}_{\text{peak}}=\frac{4}{\pi }(\frac{k_r{N}_r}{\text{poles}})I_r$ (4.8)

Figure 4.12 The air-gap mmf of a distributed winding on the rotor of a round-rotor generator.

§4.3.2 DC Machines

• Because of the restrictions imposed on the winding arrangement by the commutator, the mmf wave of a dc machine armature approximates a sawtooth waveform more nearly than the sine wave of ac machines.

• Fig. 4.13 shows diagrammatically in cross section the armature of a two-pole dc machine.

• The armature coil connections are such that the armature winding produces a magnetic field whose axis is vertical and thus is perpendicular to the axis of the field winding.
• As the armature rotates, the magnetic field of the armature remains vertical due to commutator action and a continuous unidirectional torque results.
• The mmf wave is illustrated and analyzed in Fig. 4.14.

Figure 4.13 Cross section of a two-pole dc machine.

Figure 4.14 (a) Developed sketch of the dc machine of Fig. 4.22; (b) mmf wave; (c) equivalent sawtooth mmf wave, its fundamental component, and equivalent rectangular current sheet.

DC machines often have a magnetic structure with more than two poles.