Question +Vx يع 8r {ar 42 GV ( 34 3 Cou Find Vx using: a.) Kirchhoff's Law b.) Millman's Theorem en 4.) 20 & Tix I 8V 3ix 4V Find VX using a.) Kirchhoff's Law b.) Millman's Theorem.

(3)(a) virchhoff's law aspling kol in \operatorname{loop}(1) we get\begin{array}{c}6-8 I_{1}-v_{x}-3 I_{2}+3 v_{x}=0 \\6-8 I_{1}-3 I_{2}+2 v_{x}=0\end{array}uipling hui in loop(2) we yet\begin{array}{l}-4 I_{3}-8-3 v_{x}+3 I_{2}=0 \\-4\left(I_{1}+I_{2}\right)-8-3 v_{x}+3 I_{2}=0 \\-4 I_{1}-4 I_{2}-8-3 v_{x}+3 I_{2}=0 \\-4 I_{1}-I_{2}-8-3 v_{x}=0 \\4 v_{x}=-6 I_{1}\end{array}so we sat 6-8 I_{1}-3 I_{2}+2\left(-6 I_{1}\right)=0\begin{array}{c}6-8 I_{1}-3 I_{2}-12 I_{1}=0 \\-20 I_{1}-3 I_{2}=-6 \\20 I_{1}+3 I_{2}=6 \\-4 I_{1}-I_{2}-8-3\left(-6 I_{1}\right)=0 \\-4 I_{1}-I_{2}-8+18 I_{1}=0 \\14 I_{1}-I_{2}=8\end{array}solviny eqn (5) & (6) we get\begin{array}{l}I_{1}=0.4838 \mathrm{~A} \quad I_{2}=-1.225 \mathrm{~A} \\U_{x}=-6\left(I_{1}\right)=-6(0.4838)=-2.9028001 \mathrm{~s}\end{array}(b) millman's theorem\begin{array}{l}v_{E q}=v_{x}=\frac{\sum \pm v_{i} b_{i}}{\sum q_{i}} \text { (millman's } \quad \text { equdion) } \\v_{x}=\frac{\sum v_{1} u_{1}+v_{2} u_{2}+v_{3} u_{3}}{u_{1}+4_{2}+u_{3}} \\v_{x}=\frac{(-6 \times 1 / 8)+\left(-\beta v_{x} \times \frac{1}{8}\right)+\left(8 \times \frac{1}{4}\right)}{1 / 8+\frac{1}{3}+y_{4}} \\v_{x}=\frac{-0.75+\left(-v_{x} x\right)+2}{0.7083} \\0.7083 v_{x}=-0.75-v_{x}+2 \\1.7083 v^{2}=1.25 \\v_{x}=\frac{1.2 r}{1.7088_{3}}=0.7317 \mathrm{~V} \\\end{array}(4)(a) kirchoff's law4 p i giny hut in loop (1)\begin{aligned}-4 & -2 I_{1}-4 I_{2}-8=0 \\& -2 I_{1}-4 I_{2}=12 \\& 2 y_{1}+4 I_{2}=-12 \Rightarrow q_{1}+2 I_{2}=-6\end{aligned}applying kNL in lap (2)\begin{array}{l}-6 I_{3}-3 I_{3}+3 u_{x}+8-4 I_{2}=0 \\-9 I_{3}+3 x+8-4 I_{2}=0 \\v_{x}=4 I_{2} \\-9 I_{3}+12 I_{2}+8-4 I_{2}=0 \\-9\left(I_{1}+I_{2}\right)-11 I_{2}+8-4 I_{2}=0 \\-9 I_{1}-I_{2}=-8 \\9 I_{1}+I_{2}=8\end{array}solving equn (1) & (2) we get\begin{array}{l}I_{1}=1.294 \mathrm{~A}, I_{2}=-3.647 \mathrm{~A} \\N_{x}=4 I_{2}=-4 \times 3.647=-14.581\end{array}(b) Linnan's theorein\begin{array}{l}v_{e q}=\nu_{\lambda}=\frac{\sum \geq v_{i} u_{i}}{\sum q_{i}} \\v_{x}=\frac{\sum_{i=1}^{3} v_{1} u_{1}+v_{2} u_{2}+v_{3} u_{3}}{\sum_{i=1} G_{1}+u_{2}+G_{3}} \\V_{x}=\frac{-\left(4 \times v_{2}\right)+\left(-8 \times \frac{1}{4}\right)+\left(/ 3 v_{x} \times \frac{v_{0}}{\frac{9}{3}}\right)}{1 / 2+\frac{1}{4}+1 / 4} \\v_{x}=\frac{-\not x+\frac{z x+3}{x+v_{x}}}{0.8611} \Rightarrow v_{x}=\frac{-4+v_{x x} / 3}{0.8611} \\0.86112^{2} x=-4+\frac{1}{3} \\v x\left[0.8611-\frac{1}{3}\right]=-4 \\0.5277 v x=-4 \\u_{x}=-7.58 \mathrm{v} \\\end{array} ...