- #1

Yoni V

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## Homework Statement

Given the circuit in the picture, I need to:

a. write a system of diff. equations that could solve the charges on each cap. and current on each resistor, and write them in a vector/matrix form.

b. solve the equations for t>>τ

## Homework Equations

Kirchoff's laws, perhaps Thevenin's theorem, although this wasn't my attempt at it.

## The Attempt at a Solution

I wrote the equations for voltage loops.

First was the outer loop, yielding

##V=I_{1}R_{1}+I_{2}R_{2}+I_{3}R_{3}=I_{1}R_{1}+\frac{1}{2}I_{1}R_{2}+\frac{1}{4}I_{1}R_{3}=R_{1}\dot{Q_{1}}+R_{2}\dot{Q_{2}}+I_{3}R_{3}##

Second was

##V=R_{1}\dot{Q_{1}}+R_{2}\dot{Q_{2}}+Q_{2}/C_{2}##

Third was

##V=R_{1}\dot{Q}_{1}+Q_{1}/C_{1}+Q_{2}/C_{2}##

And lastly, although obviously one of these is redundant, the loop containing both capacitors yields

##I_{2}R_{2}=R_{2}\dot{Q}_{2}=Q_{2}/C_{2}-Q_{1}/C_{1}\Rightarrow Q_{2}/C_{2}=Q_{1}/C_{1}+R_{2}\dot{Q_{2}}##

I think I'm supposed to have enough info here to be able to write a decent set of diff. equations, but somehow meshing the equations together didn't result in anything. I mean, first I'm not sure if the equations are indeed enough for a theoretical solution, and second I'm a little puzzled at how to transform all of this into vector form.

Any leads?

Thanks!

Edit: Trying to force an answer from this resulted in:

##

\begin{bmatrix}\dot{Q_{1}}\\

\dot{Q_{2}}

\end{bmatrix}=\begin{bmatrix}\frac{-1}{R_{1}C_{1}} & \frac{-1}{R_{1}C_{2}}\\

\frac{-1}{R_{2}C_{1}} & \frac{1}{R_{2}C_{2}}

\end{bmatrix}\begin{bmatrix}Q_{1}\\

Q_{2}

\end{bmatrix}+\begin{bmatrix}\frac{V}{R_{1}}\\

0

\end{bmatrix}

##

And then for part b, it results (as for all other equation combos) that Q1=Q2=0 (equating the derivatives to 0). This doesn't make much sense to me, but I can't figure out what's wrong.

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