Three Loop RL Circuit

Find the current in each resistor as a function of time.

Solution:

Taking loop currents clockwise, we get the equations:

15I₁ - 5I₂ - 6I₃ = 12 (1)

-5I₁ + 18I₂ - 3I₃ = -2 dI₂/dt (2)

-6I₁ - 3I₂ + 16I₃ = 0 (3)

Eliminating I₁ from equations (1) and (3), we get

I₃ = (72 + 75I₂)/204 (4)

Eliminating I₃ from equation (1) by substituting (4) into it, we get

Substituting this equation and equation (4) into equation (2), we get

where a and b are being used temporarily to make the integration easier to comprehend:

Substituting back into the equations for I₁ and I₃, we get

The current in the 5 ohm resistor is I₁ - I₂. The current in the 6 ohm resistor is I₁ - I₃. The current in the 3 ohm resistor is I₂ - I₃.