In Electrical Machines, transformer is a cornerstone. It serves as an vital link in the transmission and distribution of electrical energy. We all know that transformer is a device which transfers electrical energy from one circuit to another by keeping its frequency constant.

Understanding the equivalent circuit helps engineers design transformers for specific applications, optimize their performance, and predict how they will behave under different load conditions. It’s a fundamental concept for anyone studying electrical engineering or working with Electrical Machines.

In this article, we are going to delve the detailed information about the equivalent circuit of the transformer.

You can watch a quick YouTube video or read along.

## What is Equivalent Circuit of transformer?

● Equivalent Circuit of transformer is a compact and efficient means of describing or predicting that how well the transformer will work.

● The knowledge about equivalent circuit is very important in estimating and checking the performance of transformer under all load conditions i.e. from no load to full load including short-circuited secondary.

● It makes calculations and analysis easier because we are working in one winding only, which is more convenient.

● The accurate efficiency and voltage regulation calculations can be easily done and can be compared with requirements. It ensures that there have been no errors in design and construction of transformers.

● Its important to know the value of percentage impedance during installing a transformer in existing power system because the value of %Z of all the transformers should be matched properly during the parallel operation of power transformer.

Let us see the equivalent circuits of transformer one by one.

**i) Basic Magnetically coupled circuit **

The equivalent circuit of basic magnetically coupled circuit is shown below.

Primary and secondary parameters are listed below.

I_{o} → no load current , \overline{I_{c}}+\overline{I_{m}}

I_{c} → core loss current , I_{m} → magnetising current

R_{c }→ core loss resistance , X_{m} → magnetising reactance

I_{1} → primary current , I_{2} → Secondary current

V_{1} → Primary voltage , V_{2} → Secondary voltage

R_{1} → Primary resistance , R_{2} → Secondary resistance

X_{L1} → Primary leakage reactance , X_{L2 }→ Secondary leakage reactance

E_{1} and E_{2} → Primary and secondary induced EMf

N_{1} and N_{2} → No. of turns of primary and secondary

The impedance is the sum of resistance and reactance. Therefore, primary and secondary impedances will become,

Z_{1} = R_{1}+ jX_{L1} , Z_{2} = R_{2} + jX_{L2 }

Exact equivalent circuit is difficult to analyze due to no-load branch of R_{c} and X_{m}. Therefore, we shift this branch to supply side without losing much accuracy.

Now, to make calculations simpler, we will shift all the circuit parameters from one side to other.

The equivalent circuit will now become

In this we have transferred the secondary parameters to primary side for making calculations simple.

When we shift the parameters like R_{2} and X_{2} to primary side it will change into

R_{2 }→ R_{2}‘

X_{2} → X_{2}‘

Cu loss by R_{2} in sec. = Cu loss produced carrying current I_{1} in pri

I_{2}^{2}R_{2} = I_{1}^{2} R_{2}‘

R_{2}^{'}= \frac{R_{2}}{K^{2}} [ Secondary resistance referred to primary)

Similarly when we transfer reactance to primary side it will be X_{L2}‘ to the primary side

The equivalent resistance referred to primary is given as,

R_{o1} = R_{1} + R_{2}‘

= R_{1}+\frac{R_{2}}{K^{2}}

Equivalent leakage reactance referred to primary

X_{o1} = X_{L1 }+ X_{L2}‘

= X_{L1}+\frac{X_{L2}}{K^{2}}

Total Impedance referred to primary side

\overline{Z_{o1}}=R_{o1}+jX_{o1}** |\overline{Z_{o1}}|=\sqrt{R^{2}{o1}+X^{2}{o1}} **

The total resistance referred to primary side

R_{o1} = R_{1}+\frac{R_{2}}{K^{2}}

The total leakage reactance referred to primary is given as,

X_{o1} = X_{L1}+\frac{X_{L2}}{K^{2}}

Total impedance Primary Side

\overline{Z_{o1}}=R_{o1}+jX_{o1}** Approximate Equivalent Circuit Referred to primary** **is **

**c) Exact Equivalent Circuit Referred to secondary side**

Transferring R_{1} to the secondary side

Cu loss by R_{1} in pri. = Cu loss produced carrying current I_{2} in sec

I_{1}^{2}R_{1} = I_{2}^{2}R_{1}‘

**R _{1}‘ = K^{2}R_{1 }**

Primary resistance referred to secondary side

Similarly transferring XL1 to the secondary side

**X _{L1}‘ = K^{2} X_{L1}**

This is the primary leakage reactance referred to secondary side.

Equivalent Resistance referred to secondary

**R _{o2} = R_{2} + R_{1}‘ = R_{2} +K^{2}R_{1} **

Equivalent leakage reactance referred to secondary

X_{o2 }= X_{L2} + X_{L1}‘

**X _{o2} = X_{L2} +K^{2}X_{L1}**

Total impedance secondary side is

\overline{Z_{o2}}=R_{o2}+jX_{o2}** |\overline{Z_{o2}}|=\sqrt{R^{2}{o2}+X^{2}{o2}}**

Copper losses is given by

P_{cu} = P_{cu1} + P_{cu2 }

Pcu = I_{1}^{2}R_{1} + I_{2}^{2}R_{2}

= I_{1}^{2}R_{1 }+ I_{1}^{2}R_{2}‘

= I_{1}^{2}(R_{1}+R_{2}‘)

P_{cu} = I_{1}^{2} R_{o1} —(a)

P_{cu }= I_{2}^{2}R_{2} + I_{2}^{2}R_{1}‘ = I_{2}^{2}(R_{2}+R_{1}‘)

P_{cu} = I_{2}^{2} R_{o2} —(b)

Equating eq.(i) and (ii) we get,

\frac{R_{o1}}{R_{o2}}=\left ( \frac{I_{2}}{I_{1}} \right )^{2} \frac{R_{o1}}{R_{o2}}=\left ( \frac{I_{2}}{I_{1}} \right )^{2}=\frac{1}{K^{2}} R_{o1}=\frac{R_{o2}}{K^{2}}R_{o2} = K^{2} R_{o1}

**Other Equivalent Parameters **

i) Secondary parameter referred to primary

V_{2}^{'}= \frac{V_{2}}{K} E_{2}^{'}= \frac{E_{2}}{K}I_{2}‘ = KI_{2}

**2) primary parameter referred to Secondary**

V_{1}‘ = KV_{1}

E_{1}‘ = KE_{1}

and I_{1^{'}}=\frac{I_{1}}{K}

R_{c}‘ = K^{2}R_{c}

X_{m}‘ = K^{2}X_{m}

The approximate Equivalent Circuit Referred to secondary is

In this way, we can draw the equivalent circuits for the primary and secondary side.

Using the equivalent circuit, engineers can perform calculations and simulations to understand how a transformer responds to changes in load, voltage, and other parameters.

This allows for efficient design, optimization, and troubleshooting of transformer-based systems.

Hope, so this article helped you to understand the equivalent circuit of transformers thoroughly.