Question

Set up and solve the system of equations for the currents in the branches of the given network. ( explain using matricies ).

Set up and solve the system of equations for the currents in the branches of the given networlk (11.々, i3)-( 15 V 3s 362 5Ω 62 eBook

Answer 1

For the left hand loop, we have

For the left hand loop, we have 5i_3 - 3 i_1 + 15 = 0 = > 3i_1 - 5 i_3 = 15 For the right hand loop, we have -6i_2 - 5 i_3 + 35 = 0 = > 6i_2 + 5i_3 = 35 Adding these two equations, 3 i_1 + 6 i_2 = 50 Also we have i_2 = i_3 + i_1 Plugging (4) into (1), 3 i_1 + 5 i_1 - 5 i_2 = 15 = > 8 i_1 - 5 i_2 = 15 6 times (5) added to 5 times (3) gives 48 i_1 + 15 i_1 = 90 + 250 = 340 = > i_1 = 340/63 Plugging this back into (5) 5 i_2 = 8 times 340/63 - 15 = > i_2 = 8 times 68/63 - 3 = 544 - 189/93 = 355/63 Hence from (4) i_3 = 355/63 - 340/63 = 15/63

For the right hand loop, we have

$-6i_{2}-5i_{3}+35=0 \implies 6i_{2}+5i_{3}=35 \,\,\, \ldots \ldots (2)$

Adding these two equations,

Also we have

Plugging (4) into (1),

$3i_{1}+5i_{1}-5i_{2}=15 \implies 8i_{1}-5i_{2}=15 \,\,\, \ldots \ldots (5)$

6 times (5) added to 5 times (3) gives

Plugging this back into (5)

Hence from (4)