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?

enter image source here

1 Answer
Jan 2, 2017

Answer:

#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#