Derivation of Formula for Calculation of Turns per Volts of Transformer Design

For designing a transformer, we need certain number of turns on each side for a specific rating transformer. Voltages on each side have direct relation with number of turns. So, we are interested in finding  voltage per turn and for designing, turns per voltage. This is obtained from basic voltage equation of transformer:

$E=4.44fB_mA_i$

This equation is derived from basic equations. Here, we are going to derive this.

As we know that emf induced is given by rate of change of flux:

$e=N\frac{d\phi}{dt}$

RMS value is linked with $\sqrt{2}$ with peak value i.e. $e=\sqrt{2} E$. So, putting this we get:

$E=\frac {N}{ \sqrt {2} } \frac {d \phi }{dt}$

$\phi (t) = AB\sin(\omega t)$

Taking derivative w.r.t t of above equation

$\frac{d \phi (t)}{dt} =\omega AB \cos (\omega t)$

$\frac{d \phi (t)}{dt}=2 \pi f A B_m$

And we know that $\omega =2 \pi f$

$E= N \frac{2 \pi f AB_m}{\sqrt{2}}$

This is total emf induced. But we are interested in voltage per turn. So, dividing both sides by total number of turns (N).

$E_T=4.44fAB_m$

We are also interested in finding turns per volts So,

$T_E=\frac{1}{4.44fAB_m}$

P.S: As for 3 phase transformers, this derivation is different so the factor 4.44 will not present there.

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