Solve for the node voltage shown in Figure P5.53. 20 90 10

Step 1 of 3

Apply Kirchhoff’s Voltage Law in Loop 1.

\(20\angle 90^\circ  = 10{I_1} + j10 = j20\left( {{I_1} - {I_2}} \right)\)

\(\left( {10 + j10} \right){I_1} - j10\left( {{I_2}} \right) = j20.....\left( 1 \right)\)

Apply Kirchhoff’s Voltage Law in Loop 2.

\(\left( {{I_2} - {I_1}} \right)j10 - j5\left( {{I_2}} \right) + 10\angle 0^\circ  = 0\)

\( - 10 =  - j10{I_1} + j5\left( {{I_2}} \right).......\left( 2 \right)\)

Write the equations (1) and (2) in matrix form.

\(\left[ {\begin{array}{*{20}{c}}{10 + j10}&{ - j10}\\{ - j10}&{j5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{I_1}}\\{{I_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{j20}\\{ - 10}\end{array}} \right] \)