EMF equation of a DC generator

The emf equation gives the average emf generated in the armature of a DC machine.

Let

• $\Phi$ = flux per pole (Wb), $P$ = number of poles
• $Z$ = total armature conductors, $A$ = number of parallel paths
• $N$ = speed in rpm

In one revolution a conductor cuts the flux of all poles, $P\Phi$ weber, in time $60/N$ second.

$$\text{average emf per conductor} = \frac{d\phi}{dt} = \frac{P\Phi}{60/N} = \frac{P\Phi N}{60}\ \text{V}$$

The $Z$ conductors form $A$ parallel paths, so each path has $Z/A$ conductors in series. The generated emf equals the emf of one path:

$$E_g = \frac{P\Phi N}{60}\times\frac{Z}{A} = \frac{\Phi Z N P}{60A}\ \text{V}$$

• Lap winding: $A = P$, so $E_g = \dfrac{\Phi Z N}{60}$
• Wave winding: $A = 2$, so $E_g = \dfrac{\Phi Z N P}{120}$

In terms of angular speed, $E_g = \dfrac{ZP}{2\pi A}\Phi\omega = K_a\Phi\omega$.