Transformer equivalent circuit, referring parameters

An equivalent circuit replaces the magnetically coupled windings with an electric network that has the same terminal behaviour. Winding resistance and leakage reactance are put in series, and the core is represented by a shunt branch.

Equivalent circuit (ideal transformer $N_1:N_2$ in the centre)

   R1     X1                R2      X2
 o-/\/\--UUU--+---+----+--/\/\--UUU--o
              |   |    |  ideal      
  V1        Rc   Xm    ) N1:N2 (       V2   Load
              |   |    |             
 o------------+---+----+-------------o

• $R_1, X_1$: primary resistance and leakage reactance. $R_2, X_2$: the secondary values.
• $R_c$: core-loss resistance. $X_m$: magnetising reactance, with $I_c = E_1/R_c$ and $I_\mu = E_1/X_m$ (where $E_1 \approx V_1$). Together they form $I_0$.
• $a = N_1/N_2$.

Referring all parameters to the primary side

The power in each element must remain the same after referring. Secondary quantities are referred to the primary as follows:

$$V_2' = aV_2,\qquad I_2' = \frac{I_2}{a}$$

For the secondary impedance, equate the power (and reactive power) before and after referring:

$$I_2'^{\,2}R_2' = I_2^2R_2 \Rightarrow R_2' = R_2\left(\frac{I_2}{I_2'}\right)^2 = a^2R_2$$ $$X_2' = a^2X_2,\qquad Z_L' = a^2Z_L$$

The referred circuit has all the elements on the primary side:

   R1    X1     R2'     X2'
 o-/\/\--UUU--+--/\/\--UUU--+---o
              |            |
  V1        Rc   Xm       ZL' (= a^2 ZL)
              |            |
 o------------+------------+---o

The total referred values are then

$$R_{01} = R_1 + a^2R_2,\qquad X_{01} = X_1 + a^2X_2,\qquad Z_{01} = \sqrt{R_{01}^2 + X_{01}^2}$$

The shunt branch ($R_c, X_m$) is often moved to the input terminals (approximate circuit) because $I_0$ is small. With this approximation the circuit is $V_1$ across $R_c \parallel X_m$, followed by $R_{01} + jX_{01}$ in series with $Z_L'$.

Referring to the secondary side

Primary quantities are referred to the secondary by dividing by $a$ (voltage) and $a^2$ (impedance):

$$R_1' = \frac{R_1}{a^2},\quad X_1' = \frac{X_1}{a^2},\quad R_{02} = R_2 + \frac{R_1}{a^2},\quad X_{02} = X_2 + \frac{X_1}{a^2}$$

Similarly $R_c' = R_c/a^2$, $X_m' = X_m/a^2$, and the primary current becomes $aI_1$.

Rule: an impedance is multiplied by $a^2$ when moved from the secondary to the primary, and divided by $a^2$ when moved from the primary to the secondary, where $a = N_1/N_2$.