Referred quantities in a transformer

Referred quantities are values of voltage, current and impedance of one winding expressed as equivalent values on the other side, chosen so that power, losses and per-unit drops are unchanged. Referring lets the two windings be joined into a single equivalent circuit without the ideal transformer.

Let $K = N_2/N_1$ (transformation ratio).

Referred voltage

Emf per turn is the same in both windings: $E_1/N_1 = E_2/N_2$. So a secondary voltage seen on the primary side is

$$V_2' = \frac{N_1}{N_2}V_2 = \frac{V_2}{K}$$

and a primary voltage referred to the secondary is $V_1' = K V_1$.

Referred current

The secondary load current is balanced by the primary load component: $N_1 I_2' = N_2 I_2$ (mmf balance). Hence

$$I_2' = \frac{N_2}{N_1}I_2 = K I_2$$

and $I_1' = I_1 / K$ when the primary current is referred to the secondary. Volt-amperes are unchanged: $V_2' I_2' = (V_2/K)(K I_2) = V_2 I_2$.

Referred resistance

The copper loss must be the same before and after referring:

$$\begin{aligned} I_2'^2 R_2' &= I_2^2 R_2 \\ R_2' &= R_2\left(\frac{I_2}{I_2'}\right)^2 = \frac{R_2}{K^2} \end{aligned}$$

Referred reactance and impedance

The reactive power (or the voltage drop as a fraction of voltage) must be the same:

$$\begin{aligned} I_2'^2 X_2' &= I_2^2 X_2 \Rightarrow X_2' = \frac{X_2}{K^2} \\ Z_2' &= \frac{V_2'}{I_2'} = \frac{V_2/K}{K I_2} = \frac{Z_2}{K^2} \end{aligned}$$

Likewise, primary impedance referred to the secondary: $Z_1' = K^2 Z_1$.

Summary

Rule of thumb: to move an impedance from LV to HV multiply by (turns ratio)², i.e. HV side always has the larger impedance.

Example: for 2000/200 V ($K = 0.1$), $R_2 = 0.02\ \Omega$ referred to HV is $0.02/0.01 = 2\ \Omega$, so $R_{01} = R_1 + 2\ \Omega$.