Question: coupled oscillating circuits

[Israfil]

Dear readers :)

I've tried to figure this out for quite some time now, I hope anyone can help me on this:

I'm looking for the differential equations for 2 PARALLEL oscillating circuits coupled by a capacitor. I've tried to start similar as in http://www.ruhr.de/home/leser/mathe/355.pdf
for a serial oscillating circuit, but I failed :(

I'll insert the circuit as an image. I've also been searching the net and my books for hours... Please help me...

Thank You!

Israhttp://www-lehre.inf.uos.de/~dklose/osc_circuit.jpg

 

[berkeman]

Have you been using KVL or KCL equations? Can you post your equations so far?

 

[Israfil]

Hi :)

I've e.g. tried
1: 2*U_c + U_C3 = 0
2: U_c + U_L = 0

=>
1: 2/C \int I_C dt + 1/C_3 \int I_L - I_C dt = 0
2: 1/C \int I_C dt + L * d/dt I_L = 0

=>
1: 2/C* I_C + 1/C_3 * (I_L - I_C) = 0
2: L* d^2/dt^2 I_L + 1/C * I_C = 0

the problem I have here is - in my opinion that I can't make an equation of a harmonic oscillator out of them. I mean like m*d^2/dt^2 x + k*x = 0 because I can't get 2 proper equations with d^2/dt^2 in it...
Do you know what I mean?

p.s. How do I type formulas properly / where is the howto?

Thank You!

 

[berkeman]

p.s. How do I type formulas properly / where is the howto?

There's an Intro to LaTex tutorial sticky in the Tutorials forum:

https://www.physicsforums.com/forumdisplay.php?f=151

Very helpful. I printed out one of the first documents mentioned in the thread, and keep it pinned to the wall next to my PC. It really helps the readability of posts.

As to your question, the circuit you've shown is more of a resonant filter circuit, not an oscillator circuit. Is that part of the confusion?

 

[Israfil]

thank you so far :)

I know its a filter ... it's more a matter of language because I'm German and not that much used to writing about such topics in English. I hope you can help me anyway.

I'm familliar with TeX, I just didn't know how to use it in here...

 

[Israfil]

by the way: What's KVL and KCL?

 

[berkeman]

by the way: What's KVL and KCL?

Kirchoff's voltage and current laws. That's the usual way to solve a circuit like this, write the loop voltage equations, or write the node current sum equations. I usually prefer the node current sum equations (KCL).

So write an equation for each node in the circuit that shows that the sum of all the currents into that node must equal zero. Use the complex impedance for each component and the voltage across it to represent the current. You end up with a set of simultaneous equations that you then solve for the node voltages. Make sense?

 

[dfx]

by the way: What's KVL and KCL?

Kirchoff Voltage Law and Kirchoff Current Law, respectively

 

[berkeman]

BTW, I looked at the paper you attached to your original post (OP), and it looks like it uses KVL loop equations. You can solve it either way.

 

[Israfil]

hey berkeman,

I really tried ... (with KVL)

I started with
$$\begin{eqnarray*}<br /> \begin{split}<br /> U_0 &=& R_1 I_a + L \dot{I}_a - L \dot{I}_b\\<br /> 0 &=& \frac{1}{C} (I_b - I_c) - L \ddot{I}_b + L \ddot{I}_c\\<br /> 0 &=& \frac{1}{C_3} I_c + \frac{1}{C}(I_c - I_d) - \frac{1}{C}(I_b-I_c)\\<br /> 0 &=& L \ddot{I}_d - L \ddot{I}_e - \frac{1}{C}(I_c - I_d)\\<br /> 0 &=& R_2 I_e - L \dot{I}_d + L \dot{I}_e\\<br /> \end{split}<br /> \end{eqnarray*}$$
As far as I understand the aim to get the resonant frequncies is to find harmonic solutions for $$(\ddot{I}_b - \ddot{I}_c)$$ and for $$(\ddot{I}_b + \ddot{I}_c)$$.

I managed to find
$$(\ddot{I}_b - \ddot{I}_c) + \frac{1}{LC}(I_b - I_c) = 0 \\$$ so I have a harmonic solution here, but
the question is: How do I get to the other ($$C_3$$-dependent) solution?
I've been trying for hours now, I'd really appreciate some help or advice or... thank you!

 

[berkeman]

I don't know if I'll be able to help much more today (apologies), but I'm working on a deadline at the moment. Maybe try solving a simplified circuit first (like the first example in the paper you posted), and see if that helps in working with the equations. You might also try the complex impedance version of the loop equations, to see if they are easier to work with.

 

[Israfil]

Hey :)

I got as far that I know I have to use the total impedance, which I calculated.

Does anyone know, how I get the 2 resonant frequencies out of the total impedance? That'd be great!

Isra