Find the current I_L in the circuit shown in Fig. Q2.50.1 using Millman’s theorem.
The circuit can be redrawn as shown in Fig. Q2.50.2 and then the parallel connected voltage sources can be converted into a single source as shown in Fig. Q2.50.3
{R}_{\mathrm{eq}}={\frac{1}{{\frac{1}{2}}+{\frac{1}{8}}}}=1.6~\Omega ; {E}_{\mathrm{eq}}=\left({\frac{12}{2}}+{\frac{0}{8}}\right)\times1.6=9.6V
{R}_{\mathrm{eq}2}=\frac{1}{\begin{array}{l}{{\frac{1}{2}}+{\frac{1}{3}}}\end{array}}=1.2\cdot\Omega ;
{E}_{\mathrm{eq}}=\;\left({\frac{0}{2}}+{\frac{6}{3}}\right)\times1.2=2.4VWith reference to fig Q2.50.3,
\mathrm{{I_{L}}}={\frac{\mathrm{{E_{eq1}}}-\mathrm{{E_{eq2}}}}{{\mathrm{{R_{eq1}}}}+0.8+{\mathrm{{R}}}_{e q2}}}\,=\,{\frac{9.6-2.4}{1.6+0.8+1.2}}\,=\,2\,A