# EMF Equation of a Transformer

When a sinusoidal voltage is applied to the primary winding of a transformer, alternating flux ϕ_{m} 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

- ϕ
_{m}be the maximum value of flux in Weber - f be the supply frequency in Hz
- N
_{1}is the number of turns in the primary winding - N
_{2 }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 + ϕ_{m} to – ϕ_{m} in half a cycle of 1/2f seconds.

By Faraday’s Law

Let E_{1} is the emf induced in the primary winding

Since ϕ is due to AC supply ϕ = ϕ_{m }Sinwt

So the induced emf lags flux by 90 degrees.

Putting the value of E_{1}max in equation (6) we get

Putting the value of π = 3.14 in the equation (7) we will get the value of E_{1} as

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 = B_{m} x A_{i}) where A_{i }is the iron area and B_{m} is the maximum value of flux density.

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