Equivalent circuit of the transformer

What is the Equivalent Circuit of Transformer?

Before discussing the equivalent circuit of the transformer need to understand the practical transformer. A practical transformer always possesses winding resistances which play the major and important role in copper losses. As the primary winding resistance is R1 while secondary winding possesses R2. Basically, the part of the primary winding flux as well the secondary winding one complete the path through air and link with the respective winding. Such flux is called leakage flux.

The leakage flux through primary winding is produced due to primary current I1 which is in phase with the I1 and links with the primary winding only while the leakage flux through the secondary winding is produced due to the current I2 which is in phase with the I2 and link with the secondary winding also. Due to these leakage fluxes, the corresponding self-induced emf’s are present there and due to these self-induced emf’s, the primary voltage V1 has to overcome primary self-induced emf to produce the E1 on the primary winding while the E2 induced on the secondary winding has to overcome the secondary self-induced emf to produce V2 on load, due to which these self-induced emf’s are treated as fictitious voltage drops across the reactances X1 and X2 playing their major role along with primary and secondary winding resistances R1 and R2 in series, respectively.

We know that if resistance and reactance of the system are given, we can find the impedance as we know that impedance is the combined effect of reactances and resistances of the system. As we know that the primary and secondary resistances and reactances are R1, R2, and X1, X2 respectively.

Calculations -Equivalent circuit of the transformer

Therefore, primary and secondary winding impedance is

Z1 = R1 + jX1

Z2 = R2 + jX2

And their corresponding magnitudes are,

 Z1 = (R12 + X12)1/2

Z2 = (R22 + X22)1/2

The combination of fixed and variable resistances and reactances, which exactly simulate the performance and working of the machine is known as the equivalent circuit of the machine.

For the transformer, the no-load primary current has two components,

                                  Magnetizing component = Im = Iosinϕo

                                             Active component = Ic = Iocosϕo     

Magnetizing component of no-load current produces the flux and is assumed to flow through reactance Xo while an active component of no-load current representing the core losses is assumed to flow through the resistance Ro. So, this circuit containing both Ro and Xo in parallel is called the exciting circuit. So, we can write

Ro = V1/Ic

Xo = V1/Im      

We know that when the load is connected to the secondary winding of the transformer, current I­2

flows through load due to which voltage drop across R2 and X2 occurs and due to this current I2, primary winding draws an additional current which is

I2 = K I2                 where K is the voltage transformation constant.

Now, we can say that I1 is the phasor addition of Io and I2 and this primary winding current causes voltage drop across R1 and X1.

It can be written as

I1 = Io + I2

The general procedure for finding equivalent circuit parameters of the transformer is to refer secondary side parameters to the primary one or primary side parameters to the secondary side.

So, transferring secondary winding parameters to the primary side we get,

R2 = R2 / K2

X2 = X2 / K2

Z2 = Z2 / K2


E2 = E2 / K

I2 = KI2


K = N2 / N1

Similarly, the primary winding parameters can be transferred to the secondary side and we can obtain the equivalent circuit parameters referred to the secondary.

R1 = R1*K2

X1 = X1*K2

Z1 = Z1*K2


E1 = K*E1

I1 = I1 / K

Io = Io / K


K = N2 / N1

Remembering rules of Equivalent Circuit of Transformer

While transferring the parameters remember the rule that,

Low voltage winding => High Current => Low Impedance

High voltage winding => Low Current => High Impedance

Now as long as the no-load branch which is also known as the exciting circuit is in between Z1 and Z2 or Z2 and Z1the impedances cannot be combined and further simplification of the circuit cannot be done. So, the process of approximation takes place due to which further simplification or equivalency of the circuit can be found.

To get approximation done further, the first step is to shift the exciting circuit behind the R1 and X1 due which Io gets neglected due to which such an equivalent circuit is known as the approximate equivalent circuit. Now, the circuit parameters can be combined such as R1 and R2 to equivalent resistance referred to the primary R1e and reactances can be combined such as X1 and X2 to equivalent reactance referred to the primary X1e so the total impedance referred to the primary side becomes Z1e.

So, we can write their relations to be

R1e = R1 + R2 = R1 + (R2/K2)

X1e = X1 + X2 = X1 + (X2/K2)

Z1e = Re + jX1e

And its magnitude is              Z1e = (R1e2 + X1e2) 1/2


Ro = V1/Ic

Xo = V1/Im   


Im = Iosinϕo

Ic = Iocosϕo

Similarly, the relations for the secondary approximate equivalent circuit are

R2e = R2 + R1 = R2 + (R1*K2)

X2e = X2 + X1 = X2 + (X1*K2)

Z2e = R2e + jX2e

And its magnitude is              Z2e = (R2e2 + X2e2) 1/2


Ro = V1/Ic

Xo = V1/Im

The secondary winding parameters for the approximate equivalent circuit are used to find the approximate voltage drop in the transformer.

Related Topics;

  1. All About Transformer
  2. Efficiency and losses of a transformer
  3. Characteristics of an ideal transformer
  4. 3-phase transformer
  5. Open Circuit and Short Circuit Test of Transformer