# EMF Equation of a Transformer, Transformation & turns ratio

A Transformer has primary side and secondary side. One is high voltage(HV) side and other is low voltage(LV) side depending upon the application. If someone buy a transformer from the market. He or she doesn’t get to know, which is HV side and which is LV side by naked eyes. So we need to know EMF Equation of a transformer to define the primary and secondary winding.

## Mutual flux inside a Transformer

The mutual flux in the core is proportional to the sinusoidal applied Alternating current source. Also illustrated in the figure, between points ‘a’ and ‘b’, the total change of flux is 2ϕm Webers.

This change of flux occurs in half cycle, or in a time T/2 sec, where T is the time period, or the time required for the one wave to complete one cycle. The time T/2 and 1/2f seconds are equally related.

## What is EMF Equation of a transformer ?

Whenever primary side of the winding is connected to the source. A current flows in the transformer primary side. As you know there is no electrical connection between primary and secondary winding. Hence a medium is needed for the transfer of the power. Which is core of the Transformer. Now, this power travels as the mutual flux transfers. Accordingly we are going to derive emf equation of a transformer.

As of the Faraday’s law of electromagnetic induction. Mutual flux induces the EMF in the secondary winding. This mutual flux enduces EMF in both, primary and secondary windings. Whereas In primary winding it is known as back EMF, which is very less in practical.

Finally the total EMF induced in each winding must be proportional to the number of turns in that winding.

E_1\propto N_1
\frac{E_1}{E_2} = \frac{N_1}{N_2}

Where both E1 and E2 are the primary and secondary induced EMFs respectively and N1 and N2 are the number of turns in primary and secondary winding.

## Derivation of the EMF Equation of the transformer

As in the following figure, mutual flux is sinusoidal. Further we are going to derive the emf equation of transformer.

ϕ=ϕ_msin\omega t ---------(1)
e_1=-N_1 \frac{d\phi}{{dt}} --------(2)

Further Put equation (1) in (2).

e_1=-N_1 \phi_m \omega. cos\omega t

Basically, the maximum EMF is at (Coswt = 1)

E_m=-N_1 \phi_m \omega ~~~~~~ volts

Put w = 2πf.

E_m=-2\pi f . N_1 \phi_m ~ volt
E_{rms}=-\frac{2 \pi}{\sqrt 2} f.N_1 \phi_m ~ volt
E_{rms}=-\sqrt 2 \pi f . N_1 \phi_m ~ volts

OR

E_{rms}=-4.44 f . N_1 \phi_m ~ volts

Also it(E1 = -4.44 f N1 ϕm) is called as the EMF Equation of a transformer. Here negative sign is due to the Lenz’s law.

Similarly for secondary side induced EMF is calculated by

\frac{E_1}{E_2} = \frac{N_1}{N_2}

Further, by Putting a value on the E1, we get

\frac{4.44 f . N_1 \phi_m}{E_2} = \frac{N_1}{N_2}

N1 cancels both sides.

4.44 f .\phi_m N_2 = E_2

Similarly this is the formula for induced EMF Equation of the transformer in secondary side.

Basically in terms of the maximum flux density Bm (WB/m2) in area of the cross section of the core.

E_{rms}=-4.44 f . N B_m A ~ volts

The applied voltage must balance the induced primary voltage E1 and hence,

V_1=4.44 f . N_1 \phi_m ~ volts

Thus we see that the applied voltage V1 and secondary voltage -E2 are π radians out of the phase.

The ratio of the induced voltage in the two winding is called voltage transformation ratio K. Now this helps to calculate other parameters of the transformer.

K=\frac{E_1}{E_2} = \frac{N_1}{N_2}

Especially as per IEEE consideration, a practitioner should take below formula.

K=\frac{E_{H.V}}{E_{L.V}} = \frac{N_{H.V}}{N_{L.V}}

The EMF equation of the Transformer shows, that the EMF per number of turns is the same for the primary as well as the secondary winding.

## What is Turn Ratio?

Turn ratio is defined as the ratio of the number of turns of the primary to the secondary winding. This is a unitless quantity because it is a ratio of the number of turns.

\bf Turns~Ratio= \bf \frac{Number~of ~Primary~turns~(N_1)}{Number~of ~Secondary~turns~(N_2)}
Also,~\bf Turns~Ratio= \bf \frac{N_1}{N_2} = \bf \frac{V_1}{V_2} = \bf \frac{I_2}{I_1}

## What is the Voltage Transformation Ratio?

The Voltage transformation ratio is the ratio of the output voltage to the input voltage of the transformer. Similarly, this gives the idea of the transformer’s ability of voltage transformation. It is denoted by K.

\bf Voltage~Transformation~Ratio,~K=\frac{V_2}{V_1}

Further, For the ideal transformer, volt-ampere is equal to the output volt-ampere. So it will be as below:

\bf V_1 I_1= V_2 I_2

Therefore Voltage transformation ratio may be written as:

Also,\bf Voltage~Transformation~Ratio,~K=\frac{V_2}{V_1}=\frac{I_1}{I_2}

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