Understanding Long LC Circuits

[channel1]

Homework Statement

I'm curious about infinitely(?) long LC Circuits. Say you have a circuit (and I am going to describe this sort of like a matrix) with capacitors on the top row, inductors on the bottom row, and one inductor in each column. Assume a current with value "i" is going up through the columns. Can someone please help me understand what is going on in the system? I understand how to solve a basic LC circuit but I can't find any good examples with multiple capacitors and inductors within the same system.

----C---------C---------C----
l l
L L
l l
----L---------L---------L----

Homework Equations

C(series) = (1/C + 1/C +...)^-1
C(parallel) = C+C+...
L(series) = L+L+...
L(parallel) = (1/L + 1/L +...)^-1

The Attempt at a Solution

I tried looking at this as as 2 repeating series: (a capacitor and inductor in series) + (an inductor in parallel with (a capacitor and inductor in series)) but i don't think that's right...

 

[channel1]

ugh this website auto "corrected" my diagram -_- the columns go in the spaces between the rows (so not directly under the capacitors but directly under the "-----" gaps)

 

[channel1]

actually this is a much better diagram that i found, unfortunately the couldn't figure it out either: http://i.imgur.com/YJDaD.png

 

[gneill]

Usually with these sorts of problems the idea is to identify a 'unit cell' of what comprises the ladder network and then assume that, since it is infinite in length, adding one more cell to the front end ( or back end) won't change the impedance.

You end up with an equation that goes something like Z = Z + Zcell, or Z = Z || Zcell, or something similar. Of course you might have to deal with series or parallel bits when the cell is added, but you get the idea.

 

[asteriatic]

I can offer some insight into the behavior of long LC circuits. First, it's important to understand that an infinitely long LC circuit is a theoretical concept and not something that can be practically implemented. However, it can serve as a useful model for understanding the behavior of real-world LC circuits.

In the circuit you have described, there are multiple capacitors and inductors arranged in a repeating pattern. This can be thought of as a series of smaller LC circuits connected in parallel. Each individual LC circuit will have its own resonant frequency, determined by the values of the capacitor and inductor in that circuit.

When a current with value "i" is applied to the circuit, it will flow through each individual LC circuit, causing them to oscillate at their respective resonant frequencies. The presence of multiple LC circuits in the system can result in complex interactions and interference patterns, making it difficult to predict the behavior of the overall circuit.

To solve this type of circuit, you can use the equations you have listed in your post, combined with the principles of Kirchhoff's laws and Ohm's law. It may also be helpful to break the circuit down into smaller sections and solve each section separately before combining the results.

In summary, understanding long LC circuits involves understanding the behavior of individual LC circuits and how they interact with each other in a larger system. It may also require some mathematical analysis and problem-solving skills. I hope this helps clarify the concept for you.