Formula of EMF Equation | Transformers | Electrical Engineering

In this article we will discuss about the formula of emf equation.

When an alternating (sinusoidal) voltage is applied to the primary winding of a transformer, an alternating (sinusoidal) flux, as shown is Fig. 10.2., is set up in the iron core which links both the windings (primary and secondary windings).

Let ɸ_max = Maximum value of flux in webers

And f = Supply frequency in hertz.

As illustrated in Fig. 10.2, the magnetic flux increases from zero to its maximum value ɸ_max in one-fourth of a cycle i.e., in 1/4f second.

So average rate of change of flux, = dɸ/dt = ɸ_max/1/4 f = 4 f ɸ_max

Since average emf induced per turn in volts is equal to the average rate of change of flux.

So average emf induced per turn = 4 f ɸ_max volts

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

.^.. RMS value of emf induced per turn = 1.11 × 4 f ɸ_max volts … (10.1)

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

RMS value of emf induced in primary, E₁ = EMF induced per turn × number of primary turns

= 4.44 f ɸ_max × N₁ = 4.44 f N₁ ɸ_max volts … (19.2)

Similarly rms value of emf induced in secondary, E₂ = 4.44 f ɸ_max × N₂ volts … (10.3)

The above relations for emf induced in primary and secondary windings can be derived alternatively as below:

The instantaneous value of sinusoidally varying flux may be given as:

ɸ = ɸ_max sin ω

.^.. Instantaneous value of emf induced per turn = -dɸ/dt volts = – ω ɸ_max cos ωt = ωɸ_max sin (ωt – π/2) volts

It is clear from the above equation that the maximum value of emf induced per turn

= ω ɸ_max = 2 π f ɸ_max volts

^..^. ω = 2 π f

And rms value of emf induced per turn = 1/√2 × 2 π f ɸ_max = 4.44 f ɸ_max volts

Hence rms value of emf induced in primary, E₁ = 4.44 f N₁ ɸ_max volts

And rms value of emf induced in secondary, E₂, = 4.44 f N₂ ɸ_max volts

In an ideal transformer the voltage drops in primary and secondary windings are negligible, so

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

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

Note:

If B_max is the maximum allowable flux density in Wb/m² (or T) and a is the area of x-section of iron core in square metres, then in Eqs. (10.1), (10.2) and (10.3), ɸ_max is given as-

ɸ_max = B_max a webers

Resistance and Leakage Reactance:

In preceding discussions we considered an ideal transformer, which according to our assumptions, has got no resistance in the windings and no leakage flux but in actual practice it is impossible to obtain such an ideal transformer.

In actual transformer both the windings, primary and secondary windings have finite resistances R₁ and R₂ which cause copper losses and voltage drop in them.

The result is that:

(i) The secondary terminal voltage V₂ is less than the secondary induced emf E₂ and is equal to phasor difference of secondary induced emf E₂ and voltage drop in the secondary winding I₂ R₂, if magnetic leakage is negligible, i.e.

V₂ = E₂ – I₂ R₂

Where I₂ is the secondary current and R₂ is the secondary winding resistance.

(ii) Similarly the counter emf of primary – E₁ is equal to phasor difference of voltage applied to the primary winding V₁ and voltage drop in the primary winding I₁ R₁, provided magnetic leakage is negligible, i.e.,

– E₁ = V₁ – I₁ R₁

Where I₁ is the primary current and R₁ is the primary resistance.

It was previously assumed that the entire flux φ, developed by the primary winding, links with and cut every turn of both the primary and secondary windings. In practice, it is impossible to realize this condition.

However, part of the flux set up by the primary winding links only the primary turns, as illustrated in Fig. 10.12 by flux φ_L1. Also, some of the flux-set up by the secondary winding links only the secondary turns, as illustrated in Fig. 10.12 by φ_L2. These two fluxes φ_L1 and φ_L2 are known as leakage flux i.e., that flux which leaks out of the core and does not link both windings. The flux which does pass completely through the core and links both the windings is known as the mutual flux and is illustrated as φ.

The primary leakage flux φ_L1 linking with the primary winding is produced by primary ampere-turns only, therefore, it is proportional to primary current, number of primary turns being fixed.

On no load primary current is so small that leakage flux φ_L1 produced by it can be neglected but on load primary current increases resulting in increase in ampere-turns and hence leakage flux φ_L1 increases. The primary leakage flux φ_L1 is in phase with I₁ and produces self-induced emf E_L1 given by E_L1 = 2 π f L₁ I₁ in primary winding, where L₁ is the self-inductance of the primary winding produced by primary leakage flux φ_L1.

The self-induced emf E_L1, due to primary leakage flux, in the primary winding must lag leakage flux φ_L1 and primary current I by 90°. The emf necessary to balance this counter emf is opposite and equal to it and, therefore, leads the primary current I₁ by 90°. As this emf, induced by the primary leakage flux, is proportional to the current and lags it by 90°, it is nothing more than a reactance voltage and is denoted by – I₁ X₁.

The component of line voltage that balances this emf is + I₁ X₁. The effect of the primary leakage flux, therefore, is to induce an emf that opposes the flow of current to the transformer.

The reactance of the primary winding, X₁ can be obtained by dividing self-induced emf E_L1 by the primary current I₁ i.e.:

Similarly secondary leakage flux φ_L2 is set up by secondary ampere-turns and is proportional to secondary current I₂. On no load there is no current in secondary winding and, therefore, no leakage flux exists across the secondary winding on no load.

On load leakage flux φ_L2, in phase with secondary current I₂ and produces self-induced emf E_L2 = 2 π f L₂I₂ in the secondary winding where L₂ is self-inductance of secondary winding due to leakage flux φ_L2. This is also a reactance voltage, and the component that balances it leads the secondary current by 90°.

The secondary reactance X₂ opposes the current flowing out of the transformer and can be obtained by dividing self-induced emf in secondary winding, E_L2 by the secondary current I₂ i.e.:

The effect of magnetic leakage is, thus to produce in their respective windings emfs of self-inductance which are proportional to the current, and are, therefore, equivalent in effect to the addition of an inductive coil in series with each winding, the reactance of which is called the leakage reactance.

A transformer with magnetic leakage and winding resistance is equivalent to an ideal transformer (having no resistance and leakage reactance) having inductive and resistive coils connected in series with each winding as shown in Fig. 10.13.

Few important points to be kept in mind are given below:

1. The leakage flux links one or the other winding but not both, hence it in no way contributes to the transfer of energy from the primary winding to the secondary winding.

2. The applied voltage to the primary winding, V₁ will have to meet the reactive drop I₁ X₁ in addition to I₁ R₁. Similarly induced emf in the secondary winding E₂ will have to meet the resistive and reactive drops I₂ R₂, and I₂ X₂ respectively.

3. Transformation ratio is reduced due to resistance and magnetic leakage, since these reduce the secondary terminal voltage V₂ on load for a given primary applied voltage V₁.

4. The main useful flux φ decreases slightly with the increase in load but the leakage fluxes are practically proportional to the currents in the respective winding.

5. In actual transformers, the primary and secondary windings are not placed on separate legs, as shown in Fig. 10.1, as due to their being widely separate, large primary and secondary leakage fluxes would result. The leakage fluxes are reduced to a minimum by sectionalizing and inter­leaving the primary and secondary windings.