Unlocking the Math of Power: Equation of Induced EMF in Alternators

Demonstrative Video

Equation of Induced EMF

\[\begin{array}{cl} Z= & \mbox{No. of conductors}=2T=2\times \mbox{No. of turns per phase}\\ P= & \mbox{No. of poles}\\ f= & \mbox{frequency in Hz}\\ \Phi= & \mbox{flux/pole in webers}\\ K_{d}= & \mbox{distribution factor}=\dfrac{\sin\left(m\beta/2\right)}{m\sin\left(\beta/2\right)}\\ K_{p}= & \mbox{pitch factor}=\cos\left(\alpha/2\right)\\ N= & \mbox{rotor rpm} \end{array}\]

In one revolution of the rotor (i.e. 60/N second) each stator conductor is cut by a flux of \(\Phi P\)
\[\begin{aligned} d\Phi & =\Phi P\\ dt & =60/N \end{aligned}\]
Then,
\[\begin{aligned} & =\dfrac{d\Phi}{dt}\\ &=\dfrac{\Phi PN}{60}\\ & =\dfrac{\Phi P}{60}\times\dfrac{120f}{P}\\ & =2f\Phi \end{aligned}\]
Avg. emf induced per conductor
\[=2f\Phi Z=4f\Phi T\]
For Z-conductors in series/phase, Avg. emf/phase

\[=1.11\times4f\Phi T =4.44f\Phi T\]
R.M.S value of emf/phase
Equation is for full-pitched and concentrated coils.
\[= \boxed{4.44K_pK_df\Phi T}\]
Actually available voltage/phase

\[\boxed{K_p = \cos \left(n\alpha/2\right)}\]
\(K_p = \cos \left(5\alpha/2\right)\)\(5^{th}\) degrees (electrical) for the fundamental flux wave, then for different harmonics If the short-pitch angle or chording angle is
\[\boxed{K_{d}=\dfrac{\sin \left(n \cdot m\beta/2\right)}{m\sin \left(n\beta/2\right)}}\]
Similarly, distribution factor for different harmonics
\[\boxed{f_n=nf}\]
Frequency is also changed.