An arrangement of resistors which connect three nodes of a network in a ‘delta’ configuration is shown in Fig. (I) below. In circuit analysis it is sometimes convenient to replace the delta by the ‘star’ configuration illustrated in Fig. (II). Corresponding node voltages and node currents are identical in both circuits.

Prove that the resistance of each star branch equals (the product of adjacent delta resistances) ÷ (the sum of delta resistances). That is,

${R}_{\mathrm{A}}=\frac{{R}_{1}{R}_{2}}{R},\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{R}_{\mathrm{B}}=\frac{{R}_{1}{R}_{3}}{R},\phantom{\rule{0ex}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{0ex}{0ex}}{R}_{\mathrm{C}}=\frac{{R}_{2}{R}_{3}}{R},$RA=RR1R2,RB=RR1R3,andRC=RR2R3, (1)

where $R={R}_{1}+{R}_{2}+{R}_{3}$R=R1+R2+R3.

In the circuit diagram below, we assign branch currents as shown and then apply Kirchhoff’s junction rule (see p. 286) to the delta configuration. This gives:

${i}_{\mathrm{A}}={i}_{1}-{i}_{2},\phantom{\rule{0ex}{0ex}}{i}_{\mathrm{B}}={i}_{3}-{i}_{1}$, and ${i}_{\mathrm{C}}={i}_{2}-{i}_{3}\mathrm{.}$ (2)

Equivalence requires that corresponding voltage drops are the same in both configurations, and so

$\begin{array}{c}{v}_{\mathrm{A}\mathrm{B}}={R}_{1}{i}_{1}={R}_{\mathrm{A}}{i}_{\mathrm{A}}-{R}_{\mathrm{B}}{i}_{\mathrm{B}}\\ {v}_{\mathrm{C}\mathrm{A}}={R}_{2}{i}_{2}={R}_{\mathrm{C}}{i}_{\mathrm{C}}-{R}_{\mathrm{A}}{i}_{\mathrm{A}}\\ {v}_{\mathrm{B}\mathrm{C}}={R}_{3}{i}_{3}={R}_{\mathrm{B}}{i}_{\mathrm{B}}-{R}_{\mathrm{C}}{i}_{\mathrm{C}}\end{array}\}\mathrm{.}\phantom{\rule{0ex}{0ex}}(3)\mathrm{}$

Using (2) to eliminate ${i}_{\mathrm{A}},{i}_{\mathrm{B}}$iA,iB and ${i}_{\mathrm{C}}$iC from (3) yields

$\begin{array}{c}({R}_{1}-{R}_{\mathrm{A}}-{R}_{\mathrm{B}}){i}_{1}+{R}_{\mathrm{A}}{i}_{2}+{R}_{\mathrm{B}}{i}_{3}=0\\ {R}_{\mathrm{A}}{i}_{1}+\left({R}_{2}-{R}_{\mathrm{A}}-{R}_{\mathrm{C}}\right){i}_{2}+{R}_{\mathrm{C}}{i}_{3}=0\\ {R}_{\mathrm{B}}{i}_{1}+{R}_{\mathrm{C}}{i}_{2}+\left({R}_{3}-{R}_{\mathrm{B}}-{R}_{\mathrm{C}}\right){i}_{3}=0\end{array}\}\mathrm{.}\phantom{\rule{0ex}{0ex}}\left(4\right)$

The set of equations (4) has non-trivial solutions if the determinant of the coefficients is zero, that is