Transformer with winding Resistance and Magnetic Leakage

Let us discuss in this topic what will be effective in a transformer due to the presence of resistance & leakage reactance. The resistance of the Transformer means the resistance of the primary and secondary winding of the transformer. And leakage reactance of Transformer means there is some magnetic leaking in both of the windings that should be equal to inductive coils connected in both primary & secondary winding which is shown in fig-1.

Fig-1

Where, the R₁= Resistance of primary winding,
R₂ = Resistance of Secondary winding,
X₁= Reactance of primary winding,
X₂= Reactance of secondary winding,

Therefore, Primary impedance of transformer is

Z₁= √(R₁² + X₁²)

Therefore, Secondary impedance of transformer is

Z₂= √(R₂² + X₂²),

Due to the presence of resistance and leakage reactance, some voltage drop will occur in the primary winding.

If, V₁ =Applied primary voltage,
E₁= Primary induced voltage,

Then, we can write , that

V₁ = E₁ + I₁( R₁+jX₁)
V₁ = E₁ + I₁Z₁,

Similarly, The voltage drop will occur in secondary winding,

If V₂ = secondary terminal voltage,
and E₂= Secondary induced voltage,
Then, we can write, that
E₂=V₂ + I₂( R₂+jX₂)
E₂ =V₂ +I₂Z₂

Now, we draw the vector diagram below for different kinds of the load is shown in figs.

Fig-3 ( When Load is Inductive)

In these diagrams, the vector for resistive drops is parallel to current vector whereas reactive drops are perpendicular to the current vectors. The angle φ₁ between V₁ & I₁ is the power factor angle of the transformer. And we can also transfer leakage reactance from one winding to another winding in the same way as resistance transfer.
Therefore, the equivalent secondary reactance as referred to primary

(X₂’) = X₂/K² and

The equivalent primary reactance as referred to secondary (X₁’)=K²X₁,
And The reactance of the transformer as referred to primary

(X₀₁)=X₁ +X₂’ = X₁ + X₂/K²,

Similarly, The reactance of the transformer as referred to secondary

(X₀₂)=X₂ +X₁’ = X₂ + X₁K²,

Fig-5

Fig-6

We have discussed in the topic of equivalent resistance of transformer,
That, Effective resistance of the transformer as referred to primary of transformer

(R₀₁) = R₁ + R₂’= R₁ +R₂/K².

And Effective resistance of the transformer as referred to secondary of transformer,

R₀₂= R₂ + R₁’= R₂ +R₁K².

Hence, The total impedance of the transformer as referred to primary is given by,

Z₀₁ = √(R₀₁)² + (X₀₁ )²

And, the total impedance of the transformer as referred to secondary is given by,

Z₀₂ = √(R₀₂)² + (X₀₂ )²