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:

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