# Referring Impedances

In a system with multiple voltage levels, it is sometimes necessary convert impedances from one voltage to another, i.e. so that they can be used in a single equivalent circuit. Note that the whole process of referring impedances can be avoided by using the per-unit system.

## Referring Impedances in General

Generally, one can refer an impedance $Z_{1}$ at some voltage $V_{1}$ to another voltage $V_{2}$ by the following calculation:

$Z_{2} = Z_{1} \left( \frac{V_{2}}{V_{1}} \right)^{2}$

Where $Z_{1} \,$ is the impedance at voltage $V_{1}$ ($\Omega$)

$Z_{2} \,$ is the impedance at voltage $V_{2}$ ($\Omega$)

## Referring Impedances across Transformers

The winding ratio of a transformer can be calculated as follows:

$n = \frac{V_{t2} \left( 1 + t_{p} \right)}{V_{t1}} \,$

Where $n \,$ is the transformer winding ratio

$V_{t2} \,$ is the transformer nominal secondary voltage at the principal tap (Vac)
$V_{t1} \,$ is the transformer nominal primary voltage (Vac)
$t_{p} \,$ is the specified tap setting (%)

Using the winding ratio, impedances (as well as resistances and reactances) can be referred to the primary (HV) side of the transformer by the following relation:

$Z_{HV} = \frac{Z_{LV}}{n^{2}} \,$

Where $Z_{HV} \,$ is the impedance referred to the primary (HV) side ($\Omega$)

$Z_{LV} \,$ is the impedance at the secondary (LV) side ($\Omega$)
$n \,$ is the transformer winding ratio (pu)

Conversely, by re-arranging the equation above, impedances can be referred to the LV side:

$Z_{LV} = Z_{HV} \times n^{2} \,$