# 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 ${\displaystyle Z_{1}}$ at some voltage ${\displaystyle V_{1}}$ to another voltage ${\displaystyle V_{2}}$ by the following calculation:

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

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

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

## Referring Impedances across Transformers

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

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

Where ${\displaystyle n\,}$ is the transformer winding ratio

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

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

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

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

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

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