Circuits: What is wrong with my polarities?

[Saladsamurai]

Homework Statement

I am working on this transient problem. When the switch is open, current only flows in the right loop and so the initial current on the inductor is i_o+ = 4A flowinf downward through the inductor.

When the switch is thrown at t = 0, the other loop gets started. I reduced the circuit to a Thevenin equivalent as shown. Now, I know that I am supposed to be solving for the i(t) shown in the last diagram, but I know from the 6A current source in the second to last diagram that my the real current is flowing in the opposite direction. So instead, I will solve for the real current i_R(t) flowing counter clockwise and then just negate it to yield the i(t) asked for.


Since I know the direction of the real current i_R(t), I have labeled the polarities of my circuit elements accordingly. However, when I solve the loop equation:

1.

(12/5)*i_R + 10*i'_R = 72/5​

using the Laplace transform, I get

2.

(12/5)*I(s)_R + 10*sI_R(s) - 10*4 = 72/5​

3.

i_R(t) = 6[1 - (1/3)*e^-6t/25]​

so that

4.

i(t) = 2*e^-6t/25 - 6​

But the answer in the book is

5.

i(t) = 10*e^-6t/25 - 6​

which I can rectify by messing with the signs in my loop equation. What am I doing wrong here?

Perhaps it has to do with the fact that my initial condition is a negative current with respect to i_R, so instead of subtracting it in the Laplace domain (equation 2), I should have added it? Actually, I am quite confident that that is what it is. What do you think?


EDIT Solved: That was the mistake. Since the initial current through the inductor was 'downward,' it was directed opposite my assumed positive direction for i_R(t). So when the Laplace calls for the IC, the sign has to be consistent.

 

[member 553083]

The correct equation is then2.(12/5)*I(s)R + 10*sIR(s) + 10*4 = 72/5 which yields 3.iR(t) = 6[1 + (1/3)*e-6t/25] and 4.i(t) = 10*e-6t/25 - 6As expected. Homework Equations1.(12/5)*iR + 10*i'R = 72/5 2.(12/5)*I(s)R + 10*sIR(s) - 10*4 = 72/5 3.iR(t) = 6[1 - (1/3)*e-6t/25] The Attempt at a Solution I figured out that I was making a mistake in my Laplace transform. I assumed that the initial current through the inductor was in the same direction as iR, but it was actually in the opposite direction. So when I went to compute the Laplace transform, I had to add the 4 A instead of subtracting it. Once I made this change, the equation yielded the correct answer.