# Equivalent Circuit of Electrical Transformer

**What is the Equivalent Circuit of Transformer?**

An equivalent circuit of a transformer is a graphical representation of a transformer circuit in which the resistance and leakage reactance are imagined to be external to the winding. The exact equivalent circuit of a transformer can be referred to as the primary or secondary side.

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First, we derive the equivalent resistance and leakage reactance.

**Equivalent Resistance**

In a practical transformer, we need to consider winding resistance. The resistance of two windings can be transferred to any one of two windings. It can be transferred to either the primary or secondary side to make calculation easy and simple. The winding resistance of the transformer is imagined by adding resistance in series as shown in the figure below.

Where,

- R
_{1}= Primary winding resistance - R
_{2}= Secondary winding resistance

Now, we transfer the primary winding resistance to the secondary side by introducing an additional resistance R_{2}’ in the primary winding. This new resistance is added in such a way that the power absorbed by the resistance R_{2}’ when carrying primary current I_{1} is equal to the power absorbed by the secondary resistance R_{2} when carrying secondary current I_{2}.

Therefore,

Where, K = Transformation ratio = I_{1} / I_{2}

R_{2}’ is equivalent secondary resistance referred to the primary side and it is shown in the figure below.

As shown in the above figure, the effective resistance of transformer referred to primary sides is R_{e1};

Similarly, we can transfer the primary resistance referred to the secondary side.

The equivalent circuit referred to as secondary is as shown in the figure below.

The total resistance of the transformer referred to secondary side is R_{e2}.

*R _{e}*

_{2}

*= R*

_{2}

*+ R*

_{1}

*‘*

*R _{e}*

_{2}

*=R*

_{2}

*+ K*

^{2}

*R*

_{1}

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**Equivalent Leakage Reactance **

The leakage reactance of both windings can be transferred to any one of winding similar to the resistance. The imaginary reactance of both windings is added as shown in the figure below.

Where,

- X
_{1}= Leakage reactance of primary winding - X
_{2}= Leakage reactance of secondary winding

The equivalent secondary leakage reactance referred to the primary winding is X_{2}*‘*;

The total equivalent reactance of transformer referred to the primary side is X_{e1};

Similarly, equivalent primary leakage reactance referred to secondary is X_{1}’;

*X*_{1‘}* = K*^{2}* X*_{1}

The total equivalent reactance of transformer referred to secondary side is X_{e2}.

*X _{e}*

_{2}

*= X*

_{2}

*+X*

_{1‘}

*X _{e}*

_{2}

*= X*

_{2}

*+K*

^{2}

*X*

_{1}

Now, Let’s find the equivalent impedance of the transformer by adding equivalent resistance and reactance.

Equivalent impedance referred to the primary side is Z_{e1};

*Z _{e}*

_{1}

*= R*

_{e}_{1}

*+ X*

_{e}_{1}

Equivalent impedance referred to secondary side is Z_{e2}.

*Z _{e}*

_{2}

*= R*

_{e}_{2}

*+ X*

_{e}_{2}

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**Equivalent Circuit of Transformer**

In the exact equivalent circuit of a transformer based on ideal transformer, we need to consider the no-load current. No-load current is a vector summation of working component I_{W} and magnetizing component I_{μ}. The working component of the no-load current passes through the pure resistance R_{0} and magnetizing component passes through the pure inductance X_{0}.

We can find the exact equivalent circuit of a transformer by adding a no-load component with resistances and reactances as shown in the figure below.

Where,

- V
_{1}= Supply voltage to the primary winding - V
_{2}= Load voltage

*V*_{1} *= E*_{1}* + I*_{1}* Z*_{1}

*E*_{2}* = V*_{2}* + I*_{2}* Z*_{2}

No-load resistance R_{0} represents the iron and core losses and the working component IW supplies the core losses. No-load inductance X_{0} represents a loss-free coil and the magnetizing component I_{μ} passes through X_{0}.

Now, to simplify the above equivalent circuit, transfer the resistance, reactance, voltage, and current either to the primary or secondary side.

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**Equivalent Circuit Referred to Primary Side**

In an equivalent circuit referred to the primary side, we will transfer all elements of the secondary side to the primary side.

Secondary induced EMF E_{2} referred to primary side;

Secondary terminal voltage (load voltage) V_{2} referred to the primary side;

Secondary resistance R_{2} referred to the primary side;

Secondary reactance X_{2} referred to the primary side;

The simplified exact equivalent circuit of transformer referred to the primary side is shown in the figure below.

In the above circuit, the no-load component (resistance R_{0} and reactance X_{0}) can be transferred before the primary resistance and reactance. By doing this, a very small error will introduce. But it can be neglected. So, the equivalent circuit looks like the figure below. This circuit is also known as an approximate equivalent circuit of transformer referred to the primary side.

Now, for simplification, we can add primary and secondary resistance and reactance.

*R*_{01} *= R*_{1}* + R‘*_{2}

*X*_{01} *= X*_{1}* + X‘*_{2}

Similarly, we can find the approximate equivalent circuit referred to the secondary side and this circuit is shown in the figure below.

Where,

Primary resistance referred to the secondary side;

*R‘*_{1}* = K*^{2}* R*_{1}

Primary reactance referred to the secondary side;

*X‘*_{1}* = K*^{2}* X*_{1}

Hence, the total resistance is;

*R*_{0}_{2}* =**R‘*_{1} + *R*_{2}

And total reactance is;

*X*_{0}_{2}* =**X‘*_{1} + *X*_{2}

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