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
emf-eq-of-transformer-figureAs shown in the above figure that the flux changes from + ϕm to – ϕm in half a cycle of 1/2f seconds.

By Faraday’s Law

Let E1 is the emf induced in the primary winding
emf-eq-1

Where Ψ = N1ϕ
emf-eq-2

Since ϕ is due to AC supply ϕ = ϕm Sinwt
emf-eq-3

So the induced emf lags flux by 90 degrees.

Maximum valve of emf
emf-eq-4

But w = 2πf
emf-eq-5

Root mean square RMS value is
emf-eq-6

Putting the value of E1max in equation (6) we get
emf-eq-7

Putting the value of π = 3.14 in the equation (7) we will get the value of E1 as
emf-eq-8

Similarly
emf-eq-9

Now, equating the equation (8) and (9) we get
emf-eq-10

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.
emf-eq-11

For a sinusoidal wave
emf-eq-12

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