Question

Given a circuit with resistors R1 = 10 W, R2 = 15 W, R3 = 20 and voltage, `V_b = 12V`, see figure, find the currents through all the resistors?

Answer 1

`i_1 =12/13, i_2 = 12/65, i_3 = 48/65`

Explanation:

By Kirchhoff

`i_1=i_2+i_3`

Along a loop

`V_b=R_1i_1+R_2i_2`

Along the other loop

`R_3i_3-V_b-R_2i_2=0`

Joining the equations

`{(i_1=i_2+i_3),(V_b=R_1i_1+R_2i_2),(R_3i_3-V_b-R_2i_2=0):}`

or also

`((1,-1,-1),(R_1,R_2,0),(0,-R_2,R_3))((i_1),(i_2),(i_3))=((0),(V_b),(V_b))`

Solving for `i_1,i_2,i_3` we obtain

`((i_1 = ((2 R_2 + R_3) V_b)/(R_2 R_3 + R_1 (R_2 + R_3))),(i_2 = ((R_3-R_1) V_b)/(R_2 R_3 + R_1 (R_2 + R_3))),(i_3 = ((R_1 + 2 R_2) V_b)/(R_2 R_3 + R_1 (R_2 + R_3))))`

NOTE:

Supposing that the figures for `R_1, R_2` are relative to maximum allowed dissipation, the problem would be formulated as:

solve for `i_1,i_2,R_1,R_2` the system of equations

#{ (i_1^2R_1=10), (i_2^2R_2=15), (i_1 = ((2 R_2 + R_3) V_b)/(R_2 R_3 + R_1 (R_2 + R_3))), (i_2 = ((R_3-R_1) V_b)/(R_2 R_3 + R_1 (R_2 + R_3))) :}#

obtaining

`i_1=2.8,i_2=1.78,R_1=1.27,R_2=4.74`