Analysing a 3-Loop RLC Circuit and Finding State Equations

[Captain1024]

Homework Statement

Obtain the state equations in Matrix form for the two-input and one-output circuit shown in the figure below where the output is i_2.

Homework Equations

Ohm's Law, KVL, KCL

The Attempt at a Solution

Currents i1, i2, i3 assumed clockwise. Is there a better assumption I could have made?

Nodes:
a: $i_1=i_2+i_\mathrm{L}$
b: $i_3=i_2+C\frac{\mathrm{d}v_C(t)}{\mathrm{d}t}$

Loops:
l1: $v_1-\mathrm{R}_1i_1-\mathrm{L}\frac{\mathrm{d}i_1(t)}{\mathrm{d}t}=0$
l2: $v_C-\mathrm{?}-\mathrm{R}_2i_2=0$
l3: $v_2-v_C-\mathrm{R}_3i_3=0$

Before I start solving for unknowns, I need a little help writing the loop equations. After that, I'm not sure how to choose my state variables.

-Captain1024

 

[axmls]

Your first loop equation is incorrect. Notice that the current in the branch with the inductor is $i_1 - i_2$, so you would instead have $-L \frac{d}{dt} \left(i_1(t) - i_2(t)\right)$.

 

[Captain1024]

My professor has posted the solution now. This thread may be deleted.

 

[gneill]

My professor has posted the solution now. This thread may be deleted.

Threads are not deleted unless there is some extraordinary reason. You may mark it "Solved", however. See the "MARK SOLVED" icon at the top of the thread.

 

[Captain1024]

Threads are not deleted unless there is some extraordinary reason. You may mark it "Solved", however. See the "MARK SOLVED" icon at the top of the thread.

I don't think marking a thread "solved" is an option anymore. I looked at the top of the thread and didn't find anything. Then, I searched the site for "mark solved" and found a thread from last year that mentioned that feature is gone. Thanks anyway.

-Captain1024

 

[gneill]

The feature is back. Go to the top of this page and look to the right of the thread title (all the way to the right of the page). Find the black icon:

Just hit that and it marks the thread "Solved".

 

[Captain1024]

The feature is back. Go to the top of this page and look to the right of the thread title (all the way to the right of the page). Find the black icon:

View attachment 95097

Just hit that and it marks the thread "Solved".

I logged in with Chrome, FireFox, and IE and I did not see that button next to my thread title. Is it possible this is a moderator only feature?

 

[gneill]

I logged in with Chrome, FireFox, and IE and I did not see that button next to my thread title. Is it possible this is a moderator only feature?

No, it's supposed to be available to all.

Leave it with me, I'll report the issue. Thanks for pointing it out.