 WAZIPOINT Engineering Science & Technology: Transformer E.M.F. Equation and Derivation

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

The article on Transformer emf equation derivation helps us to calculate the primary and secondary winding-induced voltage and its r.m.s value.

The derivate relation between primary and secondary windings turns the 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 detailed calculation of the E.M.F. equation for the primary winding, the secondary can get in a similar way.

Let us consider a transformer where
N1 = No. of turns in the primary;
N2 = No. of turns in secondary;
Φm= Maximum flux in core in Webers
= Bm×A;
f = Frequency of a.c. input in Hz;

As shown in the Figure below-

The flux increases from its zero value to maximum value Φm in one-quarter of the cycle i.e. in 1/4f second.

∴ The average rate of change of flux =
Φm/(1/4f)
= 4fΦm 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 = 4fΦm 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 = 1.11×4fΦm = 4.44fΦm 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
E1 = 4.44fN1Φm = 4.44fN1BmA

So, the derivated equation-1 for induced e.m.f in the primary winding is as below:
E1 = 4.44fN1Φm = 4.44fN1BmA

Similarly, r.m.s. value of the e.m.f. induced in secondary is,
E2 = 4.44fN2Φm = 4.44fN2BmA

So, the derivated equation-2 for induced e.m.f in the secondary winding is as below:
E2 = 4.44fN2Φm = 4.44fN2BmA

It is seen from equations 1 and 2 that E1/N1 =E2/N2 =4.44fΦm

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, V1=E1 and E2=V2 where V2 is the terminal voltage.

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