**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 R_{1} while secondary winding possesses R_{2}. 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 I_{1} which is in phase with the I_{1} and links with the primary winding only while the leakage flux through the secondary winding is produced due to the current I_{2} which is in phase with the I_{2} 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 V_{1} has to overcome primary self-induced emf to produce the E_{1} on the primary winding while the E_{2} induced on the secondary winding has to overcome the secondary self-induced emf to produce V_{2 }on load, due to which these self-induced emf’s are treated as fictitious voltage drops across the reactances X_{1} and X_{2} playing their major role along with primary and secondary winding resistances R_{1} and R_{2} 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 R_{1}, R_{2,} and X_{1}, X_{2} respectively.

**Calculations -Equivalent circuit of the transformer**

Therefore, primary and secondary winding impedance is

**Z _{1} = R_{1} + jX_{1}**

**Z _{2} = R_{2} + jX_{2}**

And their corresponding magnitudes are,

**Z _{1} = (R_{1}^{2} + X_{1}^{2})^{1/2}**

**Z _{2} = (R_{2}^{2} + X_{2}^{2})^{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 = I _{m} = I_{o}sinϕ_{o}**

_{ } **Active component = I _{c} = I_{o}cosϕ_{o} **

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

**R _{o} = V_{1}/I_{c}**

**X _{o} = V_{1}/I_{m} **

We know that when the load is connected to the secondary winding of the transformer, current I_{2}

flows through load due to which voltage drop across R_{2} and X_{2} occurs and due to this current I_{2}, primary winding draws an additional current which is

**I _{2}^{’} = K I_{2} where K is the voltage transformation constant.**

Now, we can say that I_{1} is the phasor addition of I_{o} and I_{2}^{’} and this primary winding current causes voltage drop across R_{1} and X_{1}.

It can be written as

**I _{1} = I_{o} + I_{2}^{’}**

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,

**R _{2}^{’} = R_{2} / K^{2}**

**X _{2}^{’} = X_{2} / K^{2}**

**Z _{2}^{’} = Z_{2} / K^{2}**

while

**E _{2}^{’} = E_{2} / K**

**I _{2}^{’} = KI_{2}**

Where,

**K = N _{2} / N_{1}**

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

**R _{1}^{’} = R_{1}*K^{2}**

**X _{1}^{’} = X_{1}*K^{2}**

**Z _{1}^{’} = Z_{1}*K^{2}**

while,

**E _{1}^{’} = K*E_{1}**

**I _{1}^{’} = I_{1 }/ K**

**I _{o}^{’} = I_{o }/ K**

Where

**K = N _{2} / N_{1}**

**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 Z_{1} and Z_{2}^{’} or Z_{2} and Z_{1}^{’ }the 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 R_{1} and X_{1} due which I_{o} 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 R_{1} and R_{2}^{’} to equivalent resistance referred to the primary R_{1e }and reactances can be combined such as X_{1} and X_{2}^{’} to equivalent reactance referred to the primary X_{1e }so the total impedance referred to the primary side becomes Z_{1e}.

So, we can write their relations to be

**R _{1e} = R_{1} + R_{2}^{’} = R_{1} + (R_{2}/K^{2})**

**X _{1e} = X_{1} + X_{2}^{’} = X_{1} + (X_{2}/K^{2})**

**Z _{1e} = R_{e} + jX_{1e}**

And its magnitude is **Z _{1e} = (R_{1e}^{2} + X_{1e}^{2}) ^{1/2}**

And

**R _{o} = V_{1}/I_{c}**

**Xo = V _{1}/I_{m} **

Where

**I _{m} = I_{o}sinϕ_{o}**

**I _{c} = I_{o}cosϕ_{o}**

Similarly, the relations for the secondary approximate equivalent circuit are

**R _{2e} = R_{2} + R_{1}^{’} = R_{2} + (R_{1}*K^{2})**

**X _{2e} = X_{2} + X_{1}^{’} = X_{2} + (X_{1}*K^{2})**

**Z _{2e} = R_{2e} + jX_{2e}**

And its magnitude is **Z _{2e} = (R_{2e}^{2} + X_{2e}^{2}) ^{1/2}**

And

**R _{o}^{’} = V_{1}^{’}/I_{c}^{’}**

**Xo ^{’} = V_{1}^{’}/I_{m}^{’}**

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

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