Referring Impedances

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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 [math]Z_{1}[/math] at some voltage [math]V_{1}[/math] to another voltage [math]V_{2}[/math] by the following calculation:

[math]Z_{2} = Z_{1} \left( \frac{V_{2}}{V_{1}} \right)^{2} [/math]

Where [math] Z_{1} \, [/math] is the impedance at voltage [math]V_{1}[/math] ([math] \Omega [/math])

[math] Z_{2} \, [/math] is the impedance at voltage [math]V_{2}[/math] ([math] \Omega [/math])

Referring Impedances across Transformers

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

[math] n = \frac{V_{t2} \left( 1 + t_{p} \right)}{V_{t1}} \, [/math]

Where [math] n \, [/math] is the transformer winding ratio

[math] V_{t2} \, [/math] is the transformer nominal secondary voltage at the principal tap (Vac)
[math] V_{t1} \, [/math] is the transformer nominal primary voltage (Vac)
[math] t_{p} \, [/math] 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:

[math] Z_{HV} = \frac{Z_{LV}}{n^{2}} \, [/math]

Where [math] Z_{HV} \, [/math] is the impedance referred to the primary (HV) side ([math] \Omega [/math])

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

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

[math] Z_{LV} = Z_{HV} \times n^{2} \, [/math]