# Transformer with winding resistance and leakage reactance

## Resistance of transformer

An ideal transformer has no losses because it possesses no resistance. It is not possible to build an ideal transformer. But an actual transformer always has some resistance in the primary and secondary windings. Due to this resistance, there is some voltage drop in the two windings.

The result is that the EMF induced in the primary winding E_{1} is less than the primary input voltage. It is because of the resistance R_{1} in the primary winding. The equation becomes

The secondary terminal voltage V_{₂} is vectorially less than the secondary induced e.m.f. E_{2} because of the resistance R_{2} of the secondary winding. Hence, V_{2} becomes

## Leakage Flux in transformer

For an ideal transformer, it was assumed that the entire flux developed by the primary winding, links with every turn of both the primary and secondary windings. But in practice, it is not possible to realize this condition.

However, part of the flux Φ_{L1} set up by the primary winding links only with the primary turns, as illustrated in the figure below. Also, some of the flux Φ_{L2} set up by the secondary winding links only the secondary turns.

These two fluxes Φ_{L1} and Φ_{L2} are known as **leakage flux**. It is defined as the flux that leaks out of the core and does not link both windings.

The primary leakage flux Φ_{L1} is proportional to the primary current and the number of primary turns being fixed.

## Leakage Reactance of transformer

On no load, the primary current is so small that the leakage flux produced by it can be neglected. Under load conditions, primary current increases resulting in an increase in ampere-turns, and hence leakage flux increases.

The primary leakage flux Φ_{L1} is in phase with I_{1} and produces self-induced emf E_{L1} given by *E _{L1}=2πfL_{1}I_{1}* in the primary winding, where L

_{1}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 will lag the leakage flux Φ_{L1} and primary current I_{1} by 90°. This emf is nothing but the reactance voltage and is denoted by I_{1}X_{1}.

The effect of the primary leakage flux, therefore, is to induce an emf that opposes the flow of current through the transformer. The reactance of the primary winding X_{1} is written below as.

Similarly, when the transformer is operated on load, secondary current I_{2} flows through the secondary winding, which produces its own flux Φ_{2}. A part of this flux will leak out of the core, called secondary leakage flux Φ_{L2}, which is in phase with the secondary current I_{2}.

This leakage flux produces self-induced emf E_{L2}* = 2πfL _{2}I_{2}* in the secondary winding where L

_{2}is self-inductance of secondary winding due to leakage flux Φ

_{L2}.

This emf is also called a reactance voltage I_{2}X_{2} which lags the secondary current I_{2} by 90º, where X_{2} is the secondary reactance and is given as,

A transformer with winding resistance and leakage reactance is equivalent to an ideal transformer having inductive and resistive coils connected in series with each winding as shown below.

## Impedance of Transformer

As discussed above, both primary and secondary windings will have resistance and leakage reactance. The combination of both resistance and reactance is nothing but the **impedance of the transformer**. If R_{1} and R_{2} are primary and secondary resistances, X_{1} and X_{2} are leakage reactances of the transformer respectively, then Z_{1} and Z_{2} are the impedance of primary and secondary windings. The equations of impedances are given by,

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