\def\rload{R_{\rm load}}
\input ../format
\titlesub{Part 2: Progress Report}{Topic: Mystery Circuit}

\section{Project Topic}

Our group will be working on the \link{Mystery Circuit Modelling
Scenario from SIMIODE}{https://simiode.org/resources/3187/download/4-23%
-S-MysteryCircuit-StudentVersion.pdf}. This applies Kirchhoff's Voltage
and Current Laws to the given circuit, which describe, respectively,
that the sum of all voltages in a closed loop is zero and the sum of all
currents at a node is zero. The circuit we're examining is an RLC
(resistor, inductor, capacitor) circuit, with zeroed initial conditions.
The specific circuit has two linked loops of resistors and capacitors,
in which ``gain,'' the ratio between chosen voltage differentials in the
circuit can be modeled mathematically. Because there are two connected
loops, there are three different currents. There is the current coming
off of the battery $x(t)$, the current split at the middle node becoming
$y(t)$ and $z(t)$. We are examining ${E(t)\over z(t)\rload}$ as the
``gain'' in the system. The first part uses $\omega = 100$ and the
entire problem uses $E(t) = \sin(\omega t)$.

\section{Progress}

From Kirchhoff's Voltage law over the first (xy) loop,
$$E(t) = \sin(\omega t) = x(t)R_1 + {1\over C_1}\int y(t)dt.$$
Kirchhoff's Voltage law also applies to the second yz-loop:
$${1\over C_1}\int y(t)dt = {1\over C_2}\int z(t)dt + z(t)\rload.$$
Differentiating and rearranging gives:
$$x'(t) = -{y(t) \over R_1C_1} + {\omega\cos(\omega t) \over R_1},$$
$$z'(t) = {y(t) \over C_1\rload} - {z(t) \over C_2\rload}$$

Kirchhoff's current law tells us that $y(t) + z(t) = x(t)$, so
$$y'(t) = x'(t) - z'(t) = -{y(t)\over R_1C_1} +
{\omega\cos(\omega t) \over R_1} - {y(t)\over C_1\rload}
+ {z(t) \over C_2\rload},$$
giving a system of differential equations to solve.

\section{Further Exploration}

We will reduce the system to a linear homogenous system and use the
Eigenvalues and Eigenvectors to find a similar solution to the system
of differential equations we have developed, and accordingly 
model the circuit. Using this, we can use either the method of
undetermined coefficients or variation of parameters to find a solution
to this system in general for various resistances and capacitances.
Furthermore, we will graphically represent various parameters,
investigate sensitivity to initial conditions, and attempt to generalize
our findings.

\bye
