Pitch factor, distribution factor and EMF equation

Pitch factor (coil span factor), $k_p$

The pitch factor is the ratio of the emf induced in a short-pitched (chorded) coil to the emf that would be induced in a full-pitched coil.

If the coil is short of full pitch (180° electrical) by an angle $\alpha$, the emfs in the two coil sides are not in phase and add as phasors:

$$k_p = \frac{\text{phasor sum}}{\text{arithmetic sum}} = \cos\frac{\alpha}{2}$$

Significance: short pitching slightly reduces the emf ($k_p < 1$) but it eliminates or reduces harmonics (e.g. $\alpha = 36^\circ$ removes the 5th harmonic, since $k_{p5} = \cos(5\alpha/2) = 0$), saves copper in end connections and reduces leakage reactance.

Distribution (breadth) factor, $k_d$

The distribution factor is the ratio of the phasor sum to the arithmetic sum of the emfs of coils distributed in several slots of a phase group.

With $m$ slots per pole per phase and slot angle $\beta = \dfrac{180^\circ}{\text{slots per pole}}$:

$$k_d = \frac{\sin(m\beta/2)}{m\sin(\beta/2)}$$

Significance: distributing the winding gives a more sinusoidal emf, better use of the core and better cooling; $k_d < 1$ only slightly reduces the emf.

EMF equation of an alternator

Let

• $P$ = number of poles, $N_s$ = speed in rpm,
• $\Phi$ = flux per pole (Wb), $Z$ = conductors in series per phase, $T = Z/2$ turns per phase.

In one revolution, each conductor cuts flux $P\Phi$ in time $60/N_s$ seconds.

$$\begin{aligned} \text{Average emf per conductor} &= \frac{d\Phi}{dt} = \frac{P\Phi}{60/N_s} = \frac{P\Phi N_s}{60} \\ \text{Since } f = \frac{PN_s}{120},\quad \frac{PN_s}{60} &= 2f \\ \text{Average emf per conductor} &= 2f\Phi\ \text{V} \\ \text{Average emf per turn} &= 2 \times 2f\Phi = 4f\Phi\ \text{V} \\ \text{Average emf per phase} &= 4f\Phi T \\ \text{RMS emf per phase} &= 1.11 \times 4f\Phi T = 4.44\, f\Phi T\ \text{V} \end{aligned}$$

(form factor of a sine wave = 1.11). Including the winding factors for a short-pitched, distributed winding:

$$E_{ph} = 4.44\, k_p\, k_d\, f\, \Phi\, T\ \text{volts}$$

For a star-connected machine, line voltage $E_L = \sqrt{3}\,E_{ph}$; for delta, $E_L = E_{ph}$. The product $k_w = k_p k_d$ is called the winding factor.