Signals and Systems

1. Sep 3, 2011

NHLspl09

Hey guys, I was wondering if I could receive any help on a homework problem I have. I need to find the differential equation relating the input to the output. I've began working on it but feel like I've hit a brick wall in my work, any input?

1. The problem statement, all variables and given/known data

Attachment - Problem
Working on part c

2. Relevant equations

Attachment - Work

3. The attempt at a solution

Attachment - Work

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2. Sep 3, 2011

yongs90

Take a look at my solution:
http://www.mypicx.com/09032011/Solution/

I leave the rearranging terms to you.. By the way how do you upload the picture like what is shown in this page. I don't know how to do it..

3. Sep 3, 2011

vela

Staff Emeritus
You solved for i1 in terms of i2 and its derivatives. Now plug that into the other equation to eliminate i1 completely and leave you with x(t) in terms of i2 and its derivatives. Then since y(t) = L di2/dt, you can write i2 and its derivatives in terms of y(t).

4. Sep 3, 2011

NHLspl09

The thing is I need to do it using differential equations and not in the s domain. All I do to upload pictures is scan a document to paint then save it as a jpeg file and upload it using the manage attachments tool here.

Ok, I understand what you mean until you say to plug i1 into the other equation.. do you mean plug it into my x(dot) equation?? Where it is (i1-i2)?

5. Sep 3, 2011

vela

Staff Emeritus
Yes, plug it into the x-dot equation anywhere i1 appears.

6. Sep 4, 2011

NHLspl09

I've done so and then solved for $\frac{di1}{dt}$ in equation 3 and also plugged that into the x(dot) equation. Do I now take the derivative of this equation to leave it in terms of x?

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7. Sep 4, 2011

vela

Staff Emeritus
Equation 3 should be y(t)=L di2/dt.

8. Sep 5, 2011

NHLspl09

If equation 3 is y(t)=L di2/dt, then I'm not quite sure where it is plugged into in the other formulas

9. Sep 5, 2011

NHLspl09

I've been trying to restart the problem and look at it from a different approach but still no luck, any idea if I'm doing something wrong here? I've followed the same steps as taken in my notes from class and I can't seem to grasp this problem.

Edit: Any thoughts as to if I should be using KCL rather than KVL since I can't seem to figure this problem out?

Last edited: Sep 5, 2011
10. Sep 5, 2011

vela

Staff Emeritus
\begin{align*}
x(t) - R_1 i_1 - \frac{1}{C}\int (i_1-i_2)\,dt &= 0 \\
\frac{1}{C}\int(i_2-i_1)\,dt - R_2 i_2 - L_1 \frac{di_2}{dt} &= 0
\end{align*}
(I think you had a sign error in the second equation.) The last term in the second equation is the voltage across the inductor, right? In other words, it's y(t). That's what your equation (3) should have been. I'm not sure why you used i1 there instead. In fact, when you differentiated equation (2), you correctly expressed the derivatives of i2 in terms of y(t).

You then differentiated each equation and got
\begin{align*}
\dot{x}(t) - R_1 \frac{di_1}{dt} - \frac{1}{C}(i_1-i_2) &= 0 \\
\frac{1}{C}(i_2-i_1) - R_2 \frac{di_2}{dt} - L_1 \frac{d^2i_2}{dt^2} &= 0
\end{align*}
You then solved the second equation for i1 and obtained$$i_1 = i_2 - R_2 C \frac{di_2}{dt} - L_1C \frac{d^2i_2}{dt^2}$$
Now just plug it into the first equation to get$$\dot{x}(t) - R_1 \frac{d}{dt} \left( i_2 - R_2 C \frac{di_2}{dt} - L_1C \frac{d^2i_2}{dt^2} \right) - \frac{1}{C} \left[ \left( i_2 - R_2 C \frac{di_2}{dt} - L_1C \frac{d^2i_2}{dt^2}\right) -i_2 \right] = 0$$
Note i1 is gone. Now simplify it and then use the fact that y(t) = L di2/dt to get rid of i2.

11. Sep 5, 2011

NHLspl09

Gotcha, my question is when you want to use y(t) = L di2/dt how in the world can you manipulate this to get rid of i2 throughout the xdot equation??

12. Sep 5, 2011

vela

Staff Emeritus
The same way you did it before after you solved for i1 after differentiating equation (2). What did you do back then?

13. Sep 6, 2011

NHLspl09

I ended up solving for a final answer of xdot = R1($\frac{y}{L1}$+R2C$\frac{ydot}{L1}$+y(2dot)C)+(R2$\frac{y}{L1}$+ydot). Thank you guys for all of your help and input on this problem!

14. Sep 6, 2011

vela

Staff Emeritus
Sorry, as you apparently realized, your original equations were right, and it was I who made the sign error in equation (2). So the loop equations are
\begin{align*}
\dot{x}(t) - R_1 \frac{di_1}{dt} - \frac{1}{C}(i_1-i_2) &= 0 \\
\frac{1}{C}(i_2-i_1) + R_2 \frac{di_2}{dt} + L_1 \frac{d^2i_2}{dt^2} &= 0
\end{align*}