Trying to calculate normal modes of nearly infinite network LC circuits

• slantz
In summary, the student is trying to solve a differential equation to find the normal modes in a circuit. They are unfamiliar with the equation and need help understanding it.
slantz

Homework Statement

The first circuit has a capacitor with capacitance c and an inductor with inductance L. In series with this is another capacitor which is connected to the next loop in the circuit.

Sorry for the crude drawing.

Homework Equations

The first part of the problem was to prove the equation could be written as dI2/dt2=w02(Ii-1-2Ii+Ii+1)

So that is a relevant equation and I have managed to do that just fine with Kirchoff's laws.

The Attempt at a Solution

The second part is to find the normal frequencies. Now I understand that this is very similar to a beaded string problem, or a discrete wave, however in the wave equation there is a Sin(k*xi) and I cannot for the life of me figure out what the equivalent equation is for a circuit.
I know that the current will repeat both with the individual loops and each loop will repeat with time, but I cannot figure out how to represent the space part of the wave equation in a circuit setting. This will be necessary for me to solve the differential equation to get the normal modes...

Help?

Last edited by a moderator:
Unless I'm missing something, the equation you have there can be written as
$$\frac{\mathrm{d}^2}{\mathrm{d}t^2}\begin{pmatrix}I_1 \\ I_2 \\ \vdots \\ I_{n-1} \\ I_{n}\end{pmatrix} = \omega_0^2\begin{pmatrix}-2 & 1 & 0 & 0 & \ddots \\ 1 & -2 & 1 & \ddots & 0 \\ 0 & 1 & \ddots & 1 & 0 \\ 0 & \ddots & 1 & -2 & 1 \\ \ddots & 0 & 0 & 1 & -2\end{pmatrix}\begin{pmatrix}I_1 \\ I_2 \\ \vdots \\ I_{n-1} \\ I_{n}\end{pmatrix}$$
Are you familiar with equations of this type, i.e. would you know how to solve it? It's the same mathematical procedure that is used in the discussion of coupled oscillators.

I feel like to find the resonant frequency in the matrix fashion I would just take the determinant of the matrix and set it to zero revealing the eigenvalues.

However to answer your question, no, I wouldn't know how to solve it, but am willing to do some reading if you send me in the correct direction.

I have taken linear algebra so it shouldn't be too difficult to learn, this material just has not been presented to me in that fashion just yet.

1. What are normal modes in a network LC circuit?

Normal modes in a network LC circuit refer to the different patterns of oscillation that the circuit can exhibit when excited by an external energy source. These modes are determined by the values of the inductance (L) and capacitance (C) in the circuit, and can be calculated using mathematical equations.

2. How do you calculate normal modes in a network LC circuit?

To calculate normal modes in a network LC circuit, you can use the equations for natural frequencies and mode shapes. The natural frequency equation is given by ω=1/√(LC), where ω is the angular frequency, L is the inductance, and C is the capacitance. The mode shape equation is given by ψ(x)=Asin(ωt), where ψ is the amplitude of the mode, x is the position, A is the initial amplitude, and ω is the natural frequency.

3. What is the significance of calculating normal modes in a network LC circuit?

Calculating normal modes in a network LC circuit has several important applications. It can help in understanding the behavior of the circuit and predicting its response to different input signals. It is also useful for designing and optimizing the circuit for specific purposes, such as filtering or amplification.

4. Can normal modes be calculated for a nearly infinite network LC circuit?

Yes, normal modes can be calculated for nearly infinite network LC circuits. In this case, the circuit is considered to have a large number of elements, but not truly infinite. The calculations for normal modes in this scenario may be more complex, but the same principles and equations can be applied.

5. Are there any limitations to calculating normal modes in a network LC circuit?

There are a few limitations to calculating normal modes in a network LC circuit. The equations used assume ideal circuit elements, which may not always be the case in real-world circuits. Additionally, the calculations may become more complicated for circuits with non-linear elements or multiple sources of energy. In these cases, numerical methods or approximations may be used to calculate the normal modes.

Replies
4
Views
2K
Replies
3
Views
3K
Replies
2
Views
7K