Equivalent circuit of Transformer

1. Equivalent circuit of Transformer

       The term equivalent circuit of a machine means the combination of fixed and variable resistances and reactances, which exactly simulates performance and working of the machine.

       For a transformer, no load primary current has two components,

                   I_m = I_o sinΦ_o = Magnetizing component 

                   I_c = I_o cosΦ_o = Active component 

       I_m produces the flux and is assumed to flow through reactance X_o called no load reractance while I_c is active component representing core losses hence is assumed to flow through the reactance R_o. Hence equivalent circuit on no load can be shown as in the Fig. 1. This circuit consisting of R_o and X_o in parallel is called exciting circuit. From the equivalent circuit we can write,

R_o = V₁/I_c

and  X_o= V₁/I_m

Fig. 1 No load equivalent circuit

       When the is connected to the transformer then secondary current I₂ flows. This causes voltage drop across R₂ and R₂. Due to I₂, primary draws an additional current

       I₂' = I₂/ K. Now I₁ is the phasor addition of I_o and I₂'. This I₁ causes the voltage drop across primary resistance R₁ and reactance X₁.

       Hence the equivalent circuit can be shown as in the Fig. 2.

Fig. 2

       But in the equivalent circuit, windings are not shown and it is further simplified by transferring all the values to the primary or secondary. This makes the transformer calculation much easy.

       So transferring secondary parameters to primary we get,
                          R₂'= R₂/K² ,      X₂' = X₂/K²'  ,       Z₂' = Z₂/K²

While                  E₂' = E₂/K'               I₂' = K I₂

Where                 K = N₂ /N₁

      While transferring the values remember the rule that

                  Low voltage winding High current Low impedance 

                  High voltage winding Low current High impedance

        Thus the exact equivalent circuit referred to primary can be shown as in the Fig. 3.

Fig.  3   Exact equivalent circuit referred to primary

       Similarly all the primary value can be referred to secondary and we can obtain the equivalent circuit referred to secondary.

                     R₁' = K² R₁ ,        X₁' = K² X₁,       Z₁' = K² Z₁
                      E₁'= K E₁,             I_o' = I₁ /K'    I_o' = I_o /K

       Similarly the exciting circuit parameters also gets transferred to secondary as R_o'and X_o '. The circuit is shown in the Fig.4.

Fig. 4  Exact equivalent circuit referred to secondary

       Now as long as no load branch i.e. exciting branch is in between Z₁ and Z₂', the impedances can not be combined. So further simplification of the circuit can be done. Such circuit is called approximate equivalent circuit.

1.1 Approximate Equivalent Circuit

       To get approximate equivalent circuit, shift the no load branch containing R_o and X_o to the left of R₁ and X₁. By doing this we are creating an error that the drop across R₁ and X₁due to I_o is neglected. Hence such an equivalent circuit is called approximate equivalent circuit.

       So approximate equivalent circuit referred to primary can be as shown in the Fig. 5.

Fig. 5 Approximate equivalent circuit referred to primary

       In this circuit now R₁ and R₂' can be combined to get equivalent resistance referred to primary R_1e as discussed earlier. Similarly X₁and X₁' can be combined to get X_1e. And equivalent circuit can be simplified as shown in the Fig. 6.

Fig. 6

       We know that,  R_1e = R₁ + R₂'= R₁ + R₂/K²
                               X_1e = X₁ + X₂' = X₁ + X₂/K²
                                Z_1e = R_1e + j X_1e
                                 R_o = V₁ /I_c  and   X_o = V₁ /I_m
                                I_c  = I_o  cosΦ_o and   Im = I_o  sinΦ_o
       In the similar fashion, the approximate equivalent circuit referred to secondary also can be obtained.

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