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In three - phase electrical circuits, there is often the need to transform from a Delta configuration to a Star configuration and vice versa. The formulas for making these conversions are detailed below:

$Z_{12} = Z_1 + Z_2 + \frac{Z_1 Z_2}{Z_3}$

$Z_{23} = Z_2 + Z_3 + \frac{Z_2 Z_3}{Z_1}$

$Z_{13} = Z_3 + Z_1 + \frac{Z_3 Z_1}{Z_2}$

$Z_1 = \frac{Z_{12} Z_{13}}{Z_{12}+Z_{13}+ Z_{23}}$

$Z_2 = \frac{Z_{12} Z_{23}}{Z_{12}+Z_{13}+ Z_{23}}$

$Z_3 = \frac{Z_{23} Z_{13}}{Z_{12}+Z_{13}+ Z_{23}}$

With the above formulas, you can easily transform from a Delta configuration to a Star configuration or vice versa.

**For a Star to Delta Transformation**, as shown in the formula above:$Z_{12} = Z_1 + Z_2 + \frac{Z_1 Z_2}{Z_3}$

$Z_{23} = Z_2 + Z_3 + \frac{Z_2 Z_3}{Z_1}$

$Z_{13} = Z_3 + Z_1 + \frac{Z_3 Z_1}{Z_2}$

**For a Delta to Star Transformation, we have**:$Z_1 = \frac{Z_{12} Z_{13}}{Z_{12}+Z_{13}+ Z_{23}}$

$Z_2 = \frac{Z_{12} Z_{23}}{Z_{12}+Z_{13}+ Z_{23}}$

$Z_3 = \frac{Z_{23} Z_{13}}{Z_{12}+Z_{13}+ Z_{23}}$