## E.M.F. Equation of a Transformer and derivation the Primary and Secondary Winding Induced Voltage

The article of Transformer emf equation derivation helps us to calculate the primary and secondary winding induced voltage and it's r.m.s value.

The derivated relation between primary and secondary windings turns number and the magnetic flux in the transformer core.

The r.m.s. value of the e.m.f. induced in primary and secondary windings are similar to calculate.

Here, we will show the detail calculation of E.M.F. equation for the primary winding, the secondary can get in a similar way.

Let us consider a transformer where

N

N

Φ

= B

f = Frequency of a.c. input in Hz;

As shown in Figure below-

_{1}= No. of turns in the primary;N

_{2}= No. of turns in secondary;Φ

_{m}= Maximum flux in core in webers= B

_{m}×A;f = Frequency of a.c. input in Hz;

As shown in Figure below-

The flux increases from its zero value to maximum value Φ

∴ The average rate of change of flux = Φ

_{m}in one-quarter of the cycle i.e. in 1/4f second.∴ The average rate of change of flux = Φ

_{m/(}1/4f) = 4fΦ

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

∴ Average e.m.f./turn = 4fΦ

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

Form factor = (r.m.s. value)/(average value) = 1.11

_{m}Wb/s or volt.Now, the rate of change of flux per turn means induced e.m.f. in volts.

∴ Average e.m.f./turn = 4fΦ

_{m}volt.If flux Φ varies sinusoidally, then r.m.s. value of induced e.m.f. is obtained by multiplying the average value with the form factor.

Form factor = (r.m.s. value)/(average value) = 1.11

∴ r.m.s. value of e.m.f./turn = 1.11×4fΦ

_{m}= 4.44fΦ_{m}volt.Now, r.m.s. value of the induced e.m.f. in the whole of the primary winding

= (induced e.m.f/turn)×No. of primary turns

E

_{1}= 4.44fN

_{1}Φ

_{m}= 4.44fN

_{1}B

_{m}A

So, the derivated equation-1 for induced e.m.f in the primary winding is as below:

E_{1}= 4.44fN_{1}Φ_{m}= 4.44fN_{1}B_{m}A

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

E

_{2}= 4.44fN_{2}Φ_{m}= 4.44fN_{2}B_{m}ASo, the derivated equation-2 for induced e.m.f in the secondary winding is as below:

E_{2}= 4.44fN_{2}Φ_{m}= 4.44fN_{2}B_{m}A

It is seen from equations 1 and 2 that E

_{1}/N_{1}=E_{2}/N_{2}=4.44fΦ_{m}.It means that e.m.f./turn is the same in both the primary and the secondary windings.

In an ideal transformer on no-load, V

_{1}=E_{1}and E_{2}=V_{2}where V_{2}is the terminal voltage.
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