EMF Equation and Transformation Ratio of Transformer

EMF Equation and Transformation Ratio of Transformer

In this article, we will derive the EMF Equation and Transformation Ratio of Transformer.

EMF Equation of Transformer

A transformer is a device used to transfer power from one circuit to another; it is mainly used to change the voltage level of power without changing frequency.

As we have seen in the construction of the transformer, it has two windings; primary winding and secondary winding.

When an alternating power is supplied to the primary winding, an alternating flux is set up in the iron core. The alternating power is sinusoidal in shape, and the induced flux is also sinusoidal.

The flux set up in the transformer core is linked with both the windings. And it is illustrated as shown in the below waveform.

flux set up in transformer
flux set up in transformer

The magnetic flux increase from zero to maximum value фmax. The maximum flux value is obtained from the above waveform at one-fourth cycle.

Hence, the time taken by the flux is 1/4f seconds to reach maximum value from zero.

    \[ dt = \frac{1}{4f} \]

Therefore, average rate of change of flux is;

    \[ \frac{d\phi}{dt} = \frac{\phi_{max}}{\frac{1}{4f}} = 4f \phi_{max} \]

An average EMF induced per turn (in volts) is equal to the rate of change of flux.

    \[ E_{avg} = 4f \phi_{max} \]

The flux varies sinusoidally. Hence, induced EMF is sinusoidal. And the form factor for a sinusoidal wave is 1.11.

So, the RMS value of induced EMF is 1.11 times of the average value.

    \[ E_{RMS} = 1.11 \times 4f \phi_{max} \]

If the number of turns in primary winding is N1 and in secondary winding N2, then RMS value of induced EMF in primary is;

    \[ E_1 = Induced \, EMF \, per \, turns \times Number \, of \, Primary \, turns \]

    \[ E_1 = 4.44 f \phi_{max} \times N_1 \, Volts \]

And RMS value of induced EMF in secondary winding is;

    \[ E_2 = 4.44f \phi_{max} \times N_2 \, Volts \]

The instantaneous value of sinusoidal varying flux is;

    \[ \phi = \phi_{max} sin (\omega t) \]

Instantaneous value of induced EMF per turn

    \[ E = \frac{-d \phi}{dt} = \omega \phi_{max} cos (\omega t)\]

    \[ E = \omega \phi_{max} sin (\omega t - \frac{\pi}{2}) \, volts \]

From the above equation, we can see that the maximum value of induced EMF per turns is;

    \[ E_{max} = \omega \phi_{max} = 2 \pi f \phi_{max} \, volts \]

So, the RMS value of induced EMF per turn is;

    \[ E_{RMS} = \frac{1}{\sqrt(2)} \times 2 \pi f \phi_{max} = 4.44 f \phi_{max} \, Volts \]

Therefore,

RMS value of induced EMF in primary winding;

    \[ E_1 = 4.44 f N_1 \phi_{max} \, Volts \]

RMS value of induced EMF in secondary winding;

    \[ E_2 = 4.44 N_2 \phi_{max} \, Volts \]

If we consider an ideal transformer, the voltage drops in transformer winding are negligible. And the transformer losses are zero.

So, the induced EMF in primary winding E1 equals the supplied voltage in primary V1. And the induced EMF in secondary winding E2 is equal to the terminal voltage V2.

Transformation Ratio

Transformation ratio is also known as turns ratio. It is denoted as K. For Transformation ratio, we take the ratio of above equations;

    \[ \frac{E_2}{E_1} = \frac{4.44 N_2 \phi_{max}}{4.44 f N_1 \phi_{max}} = K \]

For ideal transformer;

    \[ \frac{E_2}{E_1} = \frac{V_2}{V_1} = \frac{N_2}{N_1} = K \]

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

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

In ideal transformer, the losses are negligible. Hence, the volt-ampere input to the primary and volt-ampere output from secondary will be same.

    \[ output \, VA = input \, VA \]

    \[ V_2 I_2 = V_1 I_1 \]

Therefore,

    \[ \frac{V_2}{V_1} = \frac{I_1}{I_2} = \frac{E_2}{E_1} = \frac{N_2}{N_1} = K \]

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