# Transformer Equivalent Circuit

Transformer Equivalent Circuit is the electrical circuit representation of equations describing the behavior of Transformer. In fact equivalent circuit of any electrical device is necessary for its performance analysis and to find any scope of further design modification. The equivalent circuit of a transformer consists of a combination of resistance, inductance, capacitance, voltage etc. This circuits can then be studied and analyzed by applying principles of circuit theory.

**Exact Equivalent Circuit of Transformer**

Let us now consider the equation which describes the behavior of a Transformer under load condition. To better understand, it is recommended to first read Transformer Phasor Diagram.

E_{2} = V_{2} + I_{2}(r_{2}+jx_{2})

V_{1} = V_{1}’ + I_{1}(r_{1} + jx_{1})

Where E_{2} = EMF induced in secondary winding

I_{2} = Load Current

V_{1} = Primary supply voltage

V_{1}’ = -E1, EMF induced in the primary winding.

By the above two equations, the transformer equivalent circuit can be drawn as shown below.

In the above equivalent circuit, (r_{1} + jx_{1}) and (r_{2}+jx_{2}) are the leakage impedances of primary and secondary windings respectively. The voltage V_{1}’ is treated as the voltage drop in the direction of primary current I_{1}.

Since the transformer primary current consists of two components: load component current and exciting current. The load component current counteracts the secondary mmf N_{2}I_{2}. Therefore this current is shown to flow in the primary winding in the equivalent circuit shown below. The exciting current I_{e} consists of core loss current I_{c} and magnetizing current I_{m}. Since I_{c} accounts for the core loss of transformer, a resistance R_{c} is shown in the transformer equivalent circuit such that

Core Loss P_{c} = R_{c}I_{c}^{2} = V_{1}’I_{c}

⇒R_{c} = V_{1}’ / I_{c}

Since the magnetizing current I_{m} lags V_{1}’ by 90°, this can be represented in the equivalent circuit by a reactance X_{m} such that

X_{m} = V_{1}’ / I_{m}

The resistance R_{c} and reactance X_{m} are called core loss resistance and magnetizing reactance respectively. X_{m} and R_{c} are shown in the below equivalent circuit. This is the exact transformer equivalent circuit.

**Transformer Equivalent Circuit Referred to Primary Side**

In transformer analysis, it is better to transfer different parameters to either primary side or secondary side. This is done to make the analysis and calculations easier. Therefore, transformer equivalent circuit should also be obtained when secondary parameters are referred to primary side. Let us obtain this.

Secondary resistance drop when referred to primary side must be multiplied with (N_{1}/N_{2}). Therefore,

Secondary Resistance drop I_{2}r_{2} referred to Primary

= (I_{2}r_{2})(N_{1}/N_{2})

But I_{2} = I_{1} (N_{1}/N_{2})

⇒ Secondary resistance drop I_{2}r_{2} referred to Primary

= I_{1} [(N_{1}/N_{2})^{2}r_{2}] = I_{1}r_{2}’

where r_{2}’ = (N_{1}/N_{2})^{2}r_{2}

This resistance r_{2}’ is called the *secondary resistance referred to primary side*. Therefore the total resistance of primary circuit

**r _{e1} = r_{1} + r_{2}’**

**r _{e1 }**is called the equivalent resistance of transformer referred to primary side.

Similarly,

The secondary leakage reactance drop when referred to primary side

= (I_{2}x_{2})(N_{1}/N_{2})

= I_{1} [(N_{1}/N_{2})^{2}x_{2}] = I_{1}x_{2}’

where x_{2}’ = (N_{1}/N_{2})^{2}x_{2}

This reactance x_{2}’ is called the *secondary reactance referred to primary side*. Therefore the total resistance of primary circuit

**x _{e1} = x_{1} + x_{2}’**

**x _{e1 }**is called the equivalent reactance of transformer referred to primary side.

Thus, the total leakage impedance of transformer referred to primary side is given as

**z _{e1} = r_{e1} + jx_{e1}**

Thus when the secondary resistance drop and reactance drop are referred to primary side, the transformer equivalent circuit will be as shown below.

**Transformer Equivalent Circuit Referred to Secondary Side**

Transformer equivalent circuit referred to secondary side can easily be obtained by transferring primary resistance and reactance drop to the secondary side. This is done exactly in the same way as discussed in arriving at equivalent circuit referred to primary. The equivalent circuit referred to secondary side is shown below.

The total leakage impedance of transformer referred to secondary side is given as

**z _{e2} = r_{e2} + jx_{e2}**

where r_{e2} = r_{2} + r_{1}’ and x_{e2} = x_{2} + x_{1}’

r_{1}’ = (N_{2}/N_{1})^{2}r_{1} and x_{1}’ = (N_{2}/N_{1})^{2}x_{1}

**Generalized Transformer Equivalent Circuit**

Generalized transformer equivalent circuit can either be obtained by referring all parameters to primary side or by secondary side and omitting the ideal transformer notation. The generalized form of equivalent circuit of transformer is shown below.

While using the above equivalent circuit, either all parameters should be referred to primary or secondary. One should not mix primary and secondary parameter in the generalized form.

**Approximate Transformer Equivalent Circuit**

The approximate equivalent circuit of transformer is obtained from the generalized equivalent circuit by moving the shunt branch to either primary side or secondary side. When the shunt branch is moved to primary side, the excitation current I_{e} don’t flow through the primary leakage impedance. But in actual transformer equivalent circuit, it does flow.

Thus in the above approximate circuit, voltage drop in primary leakage impedance due to exciting current I_{e} is excluded.

Similarly, when the shunt branch is moved to secondary side, the excitation current does flow through the secondary leakage impedance. But in actual or exact transformer equivalent circuit, it does not flow through the secondary leakage impedance. Thus this approximate circuit includes the voltage drop in secondary leakage impedance due to exciting current I_{e}.

Since the exciting current I_{e} is only about 2 to 6 % of the rated winding current, therefore exclusion or inclusion of voltage drop due to this current does not introduce significant error in the calculation. But these simplified equivalent circuits of transformer significantly reduce the computational labor.

A more simplified equivalent circuit can also be obtained by neglecting the shunt branch. This is justified as the excitation current is quite low as compared to winding current. The simplified circuit is shown below.

Again, for power transformer the winding resistance is quite low as compared to leakage reactance, therefore it will be quite convenient to neglect primary and secondary resistance. But this is only applicable for large power transformer having name plate rating more than 500 kVA. The circuit is shown below.

The generalized form of transformer equivalent circuit should be used when the exciting current is comparable to the rated current. Again, it should be bear in mind the side where all transformer quantities has been referred while dealing with generalized transformer equivalent circuit.