0.3 Chapter 4: introduction to rotating machines  (Page 6/9)

• Fig. 4.15(a) shows schematically a four-pole dc machine.
• The machine is shown in laid-out form in Fig. 4.15(b).

Figure 4.15 (a) Cross section of a four-pole dc machine; (b) development of current sheet and mmf wave.

• The peak value of the sawtooth armature mmf wave can be written as

$(F_{\text{ag}}{)}_{\text{peak}}=(\frac{C_a}{\mathrm{2m}\text{.}\text{poles}})i_a\text{A}\text{.}\text{turn/ pole}$ (4.9)

Ca = total number of conductors in armature winding

m = number of parallel paths through armature winding

ia = armature current, A

$(F_{\text{ag}}{)}_{\text{peak}}=(\frac{N_a}{\text{poles}})i_a,N_a=C_a/(\mathrm{2m}):\text{no}\text{.}\text{of series armature turns}$ (4.10)

$(F_{\text{ag}}{)}_{\text{peak}}=\frac{8}{{\pi }^2}(\frac{N_a}{\text{poles}})i_a$ (4.11)

§4.4 Magnetic Fields In Rotating Machinery

• The behavior of electric machinery is determined by the magnetic fields created by currents in the various windings of the machine.

• The investigations of both ac and dc machines are based on the assumption of sinusoidal spatial distribution of mmf.
• Results from examining a two-pole machine can immediately be extrapolated to a multipole machine.

§4.4.1 Magnetic with Uniform Air Gaps

• Consider machines with uniform air gaps.

• Fig. 4.16(a) shows a single full-pitch, N-turn coil in a high-permeability magnetic structure $\mu \to \infty$ , with a concentric, cylindrical rotor.

• In Fig. 4.16(b) the air-gap mmf $F_{\text{ag}}$ is plotted versus angle ${\theta }_a$ .
• Fig. 4.16(c) demonstrates the air-gap constant radial magnetic field $H_{\text{ag}}$ .

$H_{\text{ag}}=\frac{F_{\text{ag}}}{g}$ (4.12)

$(H_{\text{agl}})=\frac{F_{\text{agl}}}{g}=\frac{4}{\pi }(\frac{\text{Ni}}{\mathrm{2g}})\text{cos}{\theta }_a$ (4.13)

$(H_{\text{agl}}{)}_{\text{peak}}=\frac{4}{\pi }(\frac{\text{Ni}}{\mathrm{2g}})$ (4.14)

• For a distributed winding, the air-gap magnetic field intensity is

$H_{\text{agl}}=\frac{4}{\pi }(\frac{k_w{N}_{\text{ph}}}{g\text{.}\text{poles}})i_a\text{cos}(\frac{\text{poles}}{2}{\theta }_a)$ (4.15)

Figure 4.16 The air-gap mmf and radial component of $H_{\text{ag}}$ for a concentrated full-pitch winding.

§4.4.2 Machines with Nonuniform Air Gaps

• The air-gap magnetic-field distribution of machines with nonuniform air gaps is more complex than that of uniform-air-gap machines.

• Fig. 4.17(a) shows the structure of a typical dc machine and Fig. 4.17 (b) shows the structure of a typical salient-pole synchronous machine.

Figure 4.17 Structure of typical salient-pole machines:

(a) dc machine and (b) salient-pole synchronous machine.

• Detailed analysis of the magnetic field distributions requires complete solutions of the field problem.

• Fig. 4.18 shows the magnetic field distribution in a salient-pole dc generator (obtained by finite-element solution).

Figure 4.18 Finite-element solution of the magnetic field distribution in a salient-pole dc generator. Field coils excited; no current in armature coils. (General Electric Company.)

§4.5 Rotating MMF Waves in AC Machines

• To understand the theory and operation of polyphase ac machines, it is necessary to study the nature of the mmf wave produced by a polyphase winding.

§4.5.1 MMF Wave of a Single-Phase Winding

• Fig. 4.19(a) shows the space-fundamental mmf distribution of a single-phase winding.

• Note that from Eq. (4.5), $F_{\text{agl}}$ is

$F_{\text{agl}}=\frac{4}{\pi }(\frac{k_w{N}_{\text{ph}}}{\text{poles}})i_a\text{cos}(\frac{\text{poles}}{2}{\theta }_a)$ (4.16)

When the winding is exicted by a current

$i_a=I{}_a\text{cos}{\omega }_{e}t$ (4.17)

the mmf distribution is given by

$\begin{array}{}{F}_{\text{agl}}=F_{\text{max}}\text{cos}(\frac{\text{poles}}{2}{\theta }_a)\text{cos}{\omega }_{e}t\\ =F_{\text{max}}\text{cos}({\theta }_{\text{ae}})\text{cos}{\omega }_{e}t\end{array}$ (4.18)

$F_{\text{max}}=\frac{4}{\pi }(\frac{k_w{N}_{\text{ph}}}{\text{poles}})I_a$ (4.19)

• • This mmf distribution remains fixed in space with an amplitude that varies sinusoidally in time at frequency ${\omega }_c$ , as shown in Fig. 4.19(a).
• The air-gap mmf of a single-phase winding exicted by a source of ac current can be resolved into rotating traveling waves.
  • By the identity $\text{cos}\alpha \text{cos}\beta =\frac{1}{2}\text{cos}(\alpha -\beta )+\text{cos}(\alpha +\beta )$