Trying to calculate normal modes of nearly infinite network LC circuits

slantz
Messages
2
Reaction score
0

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.

It look something like http://imgur.com/YJDaD.png"
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:
Physics news on Phys.org
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.
 

Thread 'Please tell me where I am going wrong in this integral'

##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
View full post »

Thread 'Direction of the magnetic field of an oscillating charge'

Hello everyone, I’m considering a point charge q that oscillates harmonically about the origin along the z-axis, e.g. $$z_{q}(t)= A\sin(wt)$$ In a strongly simplified / quasi-instantaneous approximation I ignore retardation and take the electric field at the position ##r=(x,y,z)## simply to be the “Coulomb field at the charge’s instantaneous position”: $$E(r,t)=\frac{q}{4\pi\varepsilon_{0}}\frac{r-r_{q}(t)}{||r-r_{q}(t)||^{3}}$$ with $$r_{q}(t)=(0,0,z_{q}(t))$$ (I’m aware this isn’t...
View full post »

Thread 'Initial conditions for orbits around a wormhole'

Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
View full post »

Similar threads

How Do Infinitely Long LC Circuits Behave?
Replies
3
Views
2K
Discharging capacitor in RC circuit, confusing path
Replies
3
Views
3K
How Do You Calculate Parameters and Analyze a Leaky Capacitor Circuit?
Replies
1
Views
7K

Hot Threads

Recent Insights

Back
Top