# EMF Equation of a Transformer

When a sinusoidal voltage is applied to the primary winding of a transformer, alternating flux ϕm sets up in the iron core of the transformer. This sinusoidal flux links with both primary and secondary winding. The function of flux is a sine function. The rate of change of flux with respect to time is derived mathematically.

The derivation of EMF Equation of the transformer is shown below. Let

• ϕm be the maximum value of flux in Weber
• f be the supply frequency in Hz
• N1 is the number of turns in the primary winding
• N2 is the number of turns in the secondary winding

Φ is the flux per turn in Weber
As shown in the above figure that the flux changes from + ϕm to – ϕm in half a cycle of 1/2f seconds.

Let E1 is the emf induced in the primary winding

Where Ψ = N1ϕ

Since ϕ is due to AC supply ϕ = ϕm Sinwt

So the induced emf lags flux by 90 degrees.

Maximum valve of emf

But w = 2πf

Root mean square RMS value is

Putting the value of E1max in equation (6) we get

Putting the value of π = 3.14 in the equation (7) we will get the value of E1 as

Similarly

Now, equating the equation (8) and (9) we get

The above equation is called the turn ratio where K is known as transformation ratio.

The equation (8) and (9) can also be written as shown below using the relation

(ϕm = Bm x Ai) where Ai is the iron area and Bm is the maximum value of flux density.

For a sinusoidal wave