# Emf equation of transformer

## Emf equation

When an alternating(sinusoidal) voltage is applied to the primary winding of a Transformer an alternating(sinusoidal) flux is produced in the iron core which links both the windings i.e primary as well as secondary.The derivation of Emf equation of transformer is shown below:

## Method 1

Let Φmax=maximum value of flux in webers

f=supply frequency in hertz.

N1=Number of turns in primary winding.

N2=Number of turns in secondary winding.

The magnetic flux increases from zero to its maximum value Φmax in one fourth of a cycle  i.e in 1/4f second.

Average rate of change of flux,

since average emf induced per turn in volts is equal to average rate of change of flux,

average emf induced per turn =4fΦmax volts.

since the flux varies sinusoidally,emf induced will be sinusoidal and form factor for sinusoidal wave is 1.11 i.e rms or effective value is 1.11 times the average value.

RMS value of emf induced per turn=1.11*4fΦmax volts.

If the number of turns on primary and secondary windings are N1 and N2 respectively,then, we have,

RMS value of emf induced in primary,

E1=EMF induced per turn*number of primary turns

=4.44fΦmax*N1

=4.44fN1Φmax volts

similarly,

rms value of emf induced in secondary,

E2=4.44fΦmax*N2 volts.

## Method 2

The instantaneous value of sinusoidally varying flux is given by,

Φ=Φmax sinwt

Instantaneous value of emf induced per turn,

The maximum value of emf induced per turn

=wΦmax

=2πfΦmax volts       :(w=2πf)

rms value of emf induced per turn

Hence the rms value of emf induced in primary ,

E1=4.44fN1Φmax volts

Also,the rms value of emf induced in secondary,

E2=4.44fN2Φmax volts.

Ideal transformer

In an ideal transformer Voltage drop in the primary and secondary are negligible.so,

Emf induced in primary winding,E=Applied voltage to primary,V1

and terminal voltage,V2=EMF induced in secondary,E2

## Voltage and current transformation ratios

The volts per turn is exactly the same for both the primary and secondary windings. In any transformer the secondary and primary induced emfs are related to each other by the ratio of the number of secondary and primary turns.

The constant k is called the voltage transformation ratio.

For step-up transformer,V2>V1 or voltage transformation ratio,K>1.

For step-down transformer,V2<V1 or voltage transformation ratio,K<1.

For an ideal transformer,

output VA=Input VA

or V2I2=V1I1

[For design of transformer https://engineeringnotesonline.com/transformer-design/]

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