Engineering RLC Circuit problem using laplace transforms

[tkaz23]

Homework Statement

The switch at S is closed at time t=0 and a constant ... va−vh=v0 is applied. Use Kirchoff's laws to write 3 equations for i,i1,i2, and q(t). What is the relation between q(t) and i,i1,i2? Given that the charge on C and all currents are initially zero, find an ODE for i1(t), given v0=1 volt, L=1 henry, C=1 farads, and R= 1 ohm.

Here is a picture of the RLC circuit (Hopefully this helps):
R i i_2
A___/\/\/\/\/___>__B___>___D
|aaaaaaaaaaaaaaaa | aaaaaaa|
aaaaaaaaaaaaaaa L $aaaaaaa C = q(t)
0 aaaaaaaaaaaaaaa $aaaaaaaa|
|_________________|_________|
H G F

Disclaimer: Disregard the blank a's. I am new to this, and I don't know how to get this to read white space. Sorry.

1st part asks to set equations for i,i_1,i_2, and q(t),
2nd part asks for a single equation for i_1,
3rd part asks for solution of part 2 using laplace transforms as well as expansions (solutions to the others)

Any help is appreciated, and thank you in advance. I have stared at this for quite sometime, and I have gotten no where.

Homework Equations

Kirchoff's 2 laws and Laplace Transform table

The Attempt at a Solution

This is what I got so far, but trying to use Laplace transforms on the i_1 eqn seems wrong because of the dq/dt in the left hand side.
I have i=i_1+i_2, i_2=dq=dt, and that the equations or differential equations are as follows for:
i_1) Ri_1+L(di_1\dt)=v_0(t)-R(dq/dt)
q) (1/C)q(t)+R(dq/dt)=v_0(t)-Ri_1

 

[vela]

You need to use CODE tags to preserve spacing.

        R       i      i_2
A___/\/\/\/\/___>__B____>___D
|                  |        |
                 L $      C = q(t)
0                  $        |
|__________________|________|
H                  G        F

 

[tkaz23]

Thank you for the pointing that out.

 

[vela]

The Attempt at a Solution

This is what I got so far, but trying to use Laplace transforms on the i_1 eqn seems wrong because of the dq/dt in the left hand side.
I have i=i_1+i_2, i_2=dq=dt, and that the equations or differential equations are as follows for:
i_1) Ri_1+L(di_1\dt)=v_0(t)-R(dq/dt)
q) (1/C)q(t)+R(dq/dt)=v_0(t)-Ri_1

I take it i₁ is the current through the inductor. If so, the equation $i = i_1 + i_2$ is fine.

I'm not sure how you got the other two equations. Start by first expressing the voltages V_AB, V_BG, and V_DF in terms of i, i₁, and q. If you go around the left loop (in a clockwise direction), KVL tells you that V_AB+V_BG+V_GH+V_HA = 0. What do you get when you substitute in the various values and expressions for those voltages?

 

[tkaz23]

You get i+(di_1/dt)-1=0 for the left loop, q(t)-(di_1/dt)=0 for the right loop, and i+q(t)-1=0 for the entire loop, if you do what you say to do. Now, we pretty much have one equation. However when trying to use a laplace transform on this, I am unsure of what to do with i_2.

 

[tkaz23]

I'm sorry to bump this, but I need to get someone to reply to me soon. If I hadn't stared at this thing for 4-5 hours, I would not be asking for help, and it is due in about an hour and a half. Please help! I am trying my best to understand things in a field, that I have relatively little clue of what to do.

 

[vela]

You get i+(di_1/dt)-1=0 for the left loop, q(t)-(di_1/dt)=0 for the right loop, and i+q(t)-1=0 for the entire loop, if you do what you say to do. Now, we pretty much have one equation. However when trying to use a laplace transform on this, I am unsure of what to do with i_2.

Note you get your third equation if you add the first two equations together. It's not an independent equation, so you can toss it. Also, you should get in the habit of using variables and plugging the numbers in only at the end. So you have the equations
\begin{align*}
i_1 + i_2 &= i \\
Ri + L\frac{di_1}{dt} - V &= 0 \\
\frac{1}{C} q - L\frac{di_1}{dt} &= 0 \\
\end{align*}
To see how i₂ works into this, you need to realize that i₂ is the rate that charge flows into the capacitor. How do you express this mathematically?

I'm sorry to bump this, but I need to get someone to reply to me soon. If I hadn't stared at this thing for 4-5 hours, I would not be asking for help, and it is due in about an hour and a half. Please help! I am trying my best to understand things in a field, that I have relatively little clue of what to do.

As you've discovered, it's best not to leave questions until the last minute. People offer help here on a volunteer basis, so while you'll often get a timely reply — that is, within a day or so — there's no guarantee that you'll get a quick reply.