Transformer equivalent circuit referred to primary

The equivalent circuit of a transformer is a circuit of resistances and reactances that behaves exactly like the real transformer, so its performance can be calculated by simple circuit analysis.

Exact equivalent circuit

  I1   R1    X1          I2'     R2'   X2'
 o--->-/\/\--mmm--+--------+-->--/\/\--mmm--o
 +                |   I0   |                +
               [ R0 ]   [ X0 ]
 V1   E1          |        |            V2'
 -                |        |                -
 o----------------+--------+----------------o
     primary side      |  secondary referred

Parameters:

• $R_1, X_1$: primary winding resistance and leakage reactance.
• $R_2, X_2$: secondary resistance and leakage reactance.
• $R_0$: core-loss resistance (carries $I_w$, the working component of $I_0$).
• $X_0$: magnetising reactance (carries $I_\mu$).

In the actual transformer, $R_1, X_1, R_0, X_0$ are on the primary side and $R_2, X_2$ and the load $Z_L$ are on the secondary side, joined by an ideal transformer of ratio $K = N_2/N_1 = E_2/E_1$.

Transferring secondary quantities to primary

The rule: the referred quantity must give the same power, losses and VA as the original.

Voltage: since $E_2 = KE_1$,

$$E_2' = \frac{E_2}{K} = E_1, \qquad V_2' = \frac{V_2}{K}$$

Current: by ampere-turn balance $N_1I_2' = N_2I_2$,

$$I_2' = KI_2$$

Resistance: copper loss must be unchanged:

$$I_2'^2R_2' = I_2^2R_2 \;\Rightarrow\; R_2' = R_2\left(\frac{I_2}{I_2'}\right)^2 = \frac{R_2}{K^2}$$

Reactance: reactive power $I^2X$ must be unchanged:

$$I_2'^2X_2' = I_2^2X_2 \;\Rightarrow\; X_2' = \frac{X_2}{K^2}$$

Load impedance: similarly $Z_L' = \frac{Z_L}{K^2}$.

Check with impedance: $Z_2' = \frac{V_2'}{I_2'} = \frac{V_2/K}{KI_2} = \frac{Z_2}{K^2}$.

Approximate equivalent circuit referred to primary

Since $I_0$ is only 2-5% of rated current, the shunt branch can be moved to the input terminals. Then the series parts combine:

$$\begin{aligned} R_{01} &= R_1 + R_2' = R_1 + \frac{R_2}{K^2} \\ X_{01} &= X_1 + X_2' = X_1 + \frac{X_2}{K^2} \\ Z_{01} &= \sqrt{R_{01}^2 + X_{01}^2} \end{aligned}$$

 o---+--------+----[ R01 ]----[ jX01 ]----o
     |        |                           +
  [ R0 ]   [ jX0 ]                   V2' = V2/K
     |        |                     Z_L' = Z_L/K^2
 o---+--------+---------------------------o

Similarly, referred to the secondary: $R_{02} = R_2 + K^2R_1$, $X_{02} = X_2 + K^2X_1$, and $R_0, X_0$ are multiplied by $K^2$.

Rule to remember: going from the low-voltage side to the high-voltage side, impedances are multiplied by the square of the voltage ratio (HV/LV)$^2$; voltages scale by the ratio, currents by its inverse.