E.m.f Equation of Transformer

Suppose , N1 = No. of turns of primary coil &

N2 = No. of turns of secondary coil of a transformer.

Φ_m = Maximum flux in core ( webers)

= B_m x A

f= frequency of alternating current in Hz

From the figure , it has been seen that the flux Φ increases from its zero value to maximum value Φ_m in one quarter of the cycle i.e in 1/4 f second

=4 f Φ_m Wb/s or volt

Now, rate of change of flux per turn means induced e.m.f in volts.

∴ average e.m.f/ turn = 4 f Φ_m volt

If the magnitude of flux Φ varies sinusoidally, then the r.m.s value of induced e.m.f is obtained by multiplying the average value with from factor.

∴ r.m.s value of e.m.f./turn = 1.11 x 4fΦ_m = 4.44 fΦ_m volt

Now, r.m.s value of the induced e.m.f in the primary winding

∴ E₁ = (induced e.m.f/turn) x No. of primary turns

∴ E₁ = 4.44 f Φ_m N1 (As Φ_m = B_m x A )

∴ E₁ = 4.44 f N1B_m A .....................(i)

Similarly, r.m.s value of the e.m.f. induced in secondary is,

∴ E₂ = (induced e.m.f/turn) x No. of Secondary turns

= 4.44 f Φ_m N2 (As Φ_m = B_m x A )

⇒ E₂ = 4.44 f N2 B_m A .....................(ii)

It is seen from equation (i) and (ii) that E₁ / N1 = E₂ / N2 = 4.44 f Φ_m .

from the above equation it is seen that the e.m.f/ turn is the same in both primary and secondary windings.

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Voltage Transformation Ratio :-

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From equation (i) and (ii) ,we get

∴ E₁ / N1 = E₂ / N2 = 4.44 f Φ_m = K

Constant K is known as voltage transformation ratio.

i) If N2 > N1 i.e K > 1,then transformer is called step-up transformer.

ii) If N2 < N1 i.e K < 1,then transformer is called step-down transformer.