0.3 Chapter 4: introduction to rotating machines  (Page 8/9)

Figure 4.22 The production of a rotating magnetic field by means of three-phase currents.

§4.6 Generated Voltage

§4.6.1 AC Machines

Figure 4.23 Cross-sectional view of an elementary three-phase ac machine.

$B_{\text{peak}}=\frac{{\mathrm{4\mu }}_o}{\mathrm{\pi g}}(\frac{k_f{N}_f}{\text{poles}})I_f$ (4.41)

$B=B_{\text{peak}}\text{cos}(\frac{\text{poles}}{2}{\theta }_r)$ (4.42)

$\begin{array}{}\Phi =l{\int }_{-\pi /\text{poles}}^{+\pi /\text{poles}}{B}_{\text{peak}}\text{cos}(\frac{\text{poles}}{2}{\theta }_r)\text{rd}{\theta }_r\\ =(\frac{2}{\text{poles}}){\mathrm{2B}}_{\text{peak}}\text{lr}\end{array}$ (4.43)

$\begin{array}{}{\lambda }_a=k_w{N}_{\text{ph}}{\Phi }_p\text{cos}((\frac{\text{poles}}{2}){\omega }_{m}t)\\ =k_w{N}_{\text{ph}}{\Phi }_p\text{cos}{\omega }_{\text{me}}t\end{array}$ (4.44)

${\omega }_{\text{me}}=(\frac{\text{poles}}{2}){\omega }_m$ (4.45)

$e_a=\frac{{\mathrm{d\lambda }}_a}{\text{dt}}=k_w{N}_{\text{ph}}\frac{{\mathrm{d\Phi }}_p}{\text{dt}}\text{cos}{\omega }_{\text{me}}t-{\omega }_{\text{me}}{k}_w{N}_{\text{ph}}{\Phi }_p\text{sin}{\omega }_{\text{me}}t$ (4.46)

$e_a=-{\omega }_{\text{me}}{k}_w{N}_{\text{ph}}{\Phi }_p\text{sin}{\omega }_{\text{me}}t$ (4.47)

$E_{\text{max}}={\omega }_{\text{me}}{k}_w{N}_{\text{ph}}{\Phi }_p={\mathrm{2\pi f}}_{\text{me}}{k}_w{N}_{\text{ph}}{\Phi }_p$ (4.48)

$E_{\text{rms}}=\frac{\mathrm{2\pi }}{\sqrt{2}}{f}_{\text{me}}{k}_w{N}_{\text{ph}}{\Phi }_p=\sqrt{2}{f}_{\text{me}}{k}_w{N}_{\text{ph}}{\Phi }_p$ (4.49)

$E_a=\frac{1}{\pi }{\int }_0^{\pi }{\omega }_{\text{me}}{\mathrm{N\Phi }}_p\text{sin}({\omega }_{\text{me}}t)d({\omega }_{\text{me}}t)=\frac{2}{\pi }{\omega }_{\text{me}}{\mathrm{N\Phi }}_p$ (4.50)

$E_a=(\frac{\text{poles}}{\pi }){\mathrm{N\Phi }}_p{\omega }_m=\text{polesN}{\Phi }_p(\frac{n}{\text{30}})$ (4.51)

$E_a=(\frac{\text{poles}}{\mathrm{2\pi }})(\frac{C_a}{m}){\Phi }_p{\omega }_m=(\frac{\text{poles}}{\text{60}})(\frac{C_a}{m}){\Phi }_{p}n$ (4.52)

§4.7 Torque in Nonsalient-pole Machines

• Consider the elementary smooth-air-gap machine of Fig.4.24 with one winding on the stator and one on the rotor and with ${\theta }_m$ being the mechanical angle between the axes of the two windings. These windings are distributed over a number of slots so that their mmf waves can be approximated by space sinusoids. In Fig.4.24a the coil sides s, -s and r, -r mark the positions of the centers of the belts of conductors comprising the distributed windings. An alternative way of drawing these windings is shown in Fig.4.24b, which also shows reference directions for voltages and currents. Here it is assumed that current in the arrow direction produces a magnetic field in the air gap in the arrow direction, so that a single arrow defines reference directions for both current and flux.

Figure 4.24 Elementary two-pole machine with smooth air gap: (a) winding distribution and (b) schematic representation.

• The stator and rotor are concentric cylinders, and slot openings are neglected. Consequently, our elementary model does not include the effects of salient poles, which are investigated in later chapters. We also assume that the reluctances of the stator and rotor iron are negligible. Finally, although Fig.4.34 shows a two-pole machine, we will write the derivations that follow for the general case of a multipole machine, replacing ${\theta }_m$ by the electrical rotor angle.

${\theta }_{\text{me}}=\left[\frac{\text{poles}}{2}\right]{\theta }_m$ (4.53)

• Based upon these assumptions, the stator and rotor self-inductances $L_{\text{ss}}$ and $L_{\text{rr}}$ can be seen to be constant, but the stator-to-rotor mutual inductance depends on the electrical angle ${\theta }_{\text{me}}$ between the magnetic axes of the stator and rotor windings. The mutual inductance is at its positive maximum when ${\theta }_{\text{me}}$ =0 or 2 $\pi$ , is zero when ${\theta }_{\text{me}}=\pm \pi /2$ , and is at its negative maximum when ${\theta }_{\text{me}}=\pm \pi$ . On the assumption of sinusoidal mmf waves and a uniform air gap, the space distribution of the air-gap flux wave is sinusoidal, and the mutual inductance will be of the form

$L_{\text{sr}}({\theta }_{\text{me}})=L_{\text{sr}}\text{cos}({\theta }_{\text{me}})$ (4.54)

where the script letter E denotes an inductance which is a function of the electrical angle ${\theta }_{\text{me}}$ . The italic capital letter L denotes a constant value. Thus Lsr is the magnitude of the mutual inductance; its value when the magnetic axes of the stator and rotor are aligned ( ${\theta }_{\text{me}}$ = 0). In terms of the inductances, the stator and rotor flux linkages) ${\lambda }_s$ and ${\lambda }_r$ are