EMF equation of an alternator

Derivation of the emf equation

Let

• $P$ = number of poles, $\phi$ = flux per pole (Wb)
• $N_s$ = speed in rpm, $f = \frac{PN_s}{120}$ = frequency
• $Z$ = total conductors in series per phase, $T_{ph} = Z/2$ turns per phase

Average emf per conductor. In one revolution a conductor cuts $P\phi$ weber in time $\frac{60}{N_s}$ s. So the average emf per conductor is

$$e_{avg} = \frac{P\phi}{60/N_s} = \frac{P\phi N_s}{60} = 2f\phi\ \text{V}$$

since $\frac{PN_s}{60} = 2f$.

Per turn (two conductors): $e_{turn} = 4f\phi$.

Per phase with $T_{ph}$ turns in series:

$$E_{avg} = 4 f \phi T_{ph}$$

Form factor for a sinusoidal emf $= \frac{\text{rms}}{\text{avg}} = 1.11$:

$$E_{ph} = 1.11\times 4f\phi T_{ph} = 4.44\, f\,\phi\, T_{ph}\ \text{V}$$

Winding factors. In a practical winding, the coils are short-pitched and the phase belt is distributed over several slots. This reduces the emf by two factors:

• Pitch factor $K_p = \cos(\alpha/2)$, where $\alpha$ is the angle by which the coil is short-pitched (electrical degrees).
• Distribution factor $K_d = \dfrac{\sin(m\gamma/2)}{m\sin(\gamma/2)}$, where $m$ is slots per pole per phase and $\gamma$ the slot angle.

$$\boxed{E_{ph} = 4.44\,K_pK_d\, f\,\phi\, T_{ph}\ \text{V}}$$

For a star connection, line emf $E_L = \sqrt3E_{ph}$.