Chapter 2, Problem 2.49P

Figure 2.33 gives the general $\Delta$ -Y transformation. (a) Show that the general transformation reduces to that given in Figure 2.16 for a balanced three-phase load. (b) Determine the impedances of the equivalent Y for the following $\Delta$ impedances: $Z_{\text{AB}}=j10,Z_{\text{BC}}=j20$, and $Z_{\text{CA}}=-j25\Omega$.

$\begin{array}{l}{Z}_{\text{AB}}=\frac{Z_{\text{A}}{Z}_{\text{B}}+Z_{\text{B}}{A}_{\text{C}}+Z_{\text{C}}{Z}_{\text{A}}}{Z_{\text{C}}}{Z}_{\text{A}}=\frac{Z_{\text{AB}}{Z}_{\text{CA}}}{Z_{\text{AB}}+Z_{\text{BC}}+Z_{\text{CA}}}\\ Z_{\text{BC}}=\frac{Z_{\text{A}}{Z}_{\text{B}}+Z_{\text{B}}{A}_{\text{C}}+Z_{\text{C}}{Z}_{\text{A}}}{Z_{\text{A}}}{Z}_{\text{B}}=\frac{Z_{\text{AB}}{Z}_{\text{BC}}}{Z_{\text{AB}}+Z_{\text{BC}}+Z_{\text{CA}}}\\ Z_{\text{CA}}=\frac{Z_{\text{A}}{Z}_{\text{B}}+Z_{\text{B}}{A}_{\text{C}}+Z_{\text{C}}{Z}_{\text{A}}}{Z_{\text{B}}}{Z}_{\text{A}}=\frac{Z_{\text{CA}}{Z}_{\text{BC}}}{Z_{\text{AB}}+Z_{\text{BC}}+Z_{\text{CA}}}\end{array}$