Synchronous generator principle and EMF equation

A synchronous generator (alternator) converts mechanical energy into AC electrical energy by electromagnetic induction, at a frequency fixed by its speed.

Working principle

1. The rotor carries the field winding, excited by DC through slip rings (or a brushless exciter); it produces alternate N and S poles.
2. The prime mover (water/steam turbine, engine) drives the rotor at synchronous speed $N_s$.
3. The rotating flux cuts the stationary three-phase armature conductors on the stator. By Faraday's law, an emf $e = Blv$ is induced; its direction is given by Fleming's right-hand rule.
4. Under a N pole the emf is in one direction, under a S pole in the other, so an alternating emf results. One cycle is produced per pair of poles: $f = \dfrac{PN_s}{120}$.
5. The three phase windings are displaced by 120° electrical, giving three balanced emfs 120° apart.
6. When load is connected, current flows and power is delivered; the field current is adjusted to keep voltage constant.

   N  ->  rotor turns  ->  S
   |  flux cuts stator     |
   v  conductors           v
  e = B l v  (sinusoidal, f = P Ns / 120)

EMF equation

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

$$\begin{aligned} \text{Flux cut per revolution} &= P\Phi \\ \text{Time for one revolution} &= \frac{60}{N_s}\ \text{s} \\ \text{Average emf per conductor} &= \frac{P\Phi N_s}{60} \\ \text{With } f = \frac{PN_s}{120}:\quad &= 2f\Phi\ \text{V} \\ \text{Average emf per phase} &= 2f\Phi\times Z = 4f\Phi T\ \text{V} \\ \text{RMS emf per phase} &= 1.11\times4f\Phi T = 4.44\,f\Phi T\ \text{V} \end{aligned}$$

using form factor 1.11 for a sine wave. For practical windings, which are short-pitched and distributed:

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

where

• pitch factor $k_p = \cos(\alpha/2)$, $\alpha$ = angle by which the coil is short of 180° electrical;
• distribution factor $k_d = \dfrac{\sin(m\beta/2)}{m\sin(\beta/2)}$, $m$ = slots per pole per phase, $\beta$ = slot angle.

Line voltage: $E_L = \sqrt3 E_{ph}$ for star, $E_L = E_{ph}$ for delta.