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+% Created 2023-03-22 Wed 18:57
+% Intended LaTeX compiler: pdflatex
+\documentclass[11pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{graphicx}
+\usepackage{longtable}
+\usepackage{wrapfig}
+\usepackage{rotating}
+\usepackage[normalem]{ulem}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{capt-of}
+\usepackage{hyperref}
+\usepackage{ dsfont } \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
+\author{Lizzy Hunt}
+\date{\today}
+\title{Building an RC Circuit}
+\hypersetup{
+ pdfauthor={Lizzy Hunt},
+ pdftitle={Building an RC Circuit},
+ pdfkeywords={},
+ pdfsubject={},
+ pdfcreator={Emacs 28.2 (Org mode 9.6.1)}, 
+ pdflang={English}}
+\begin{document}
+
+\maketitle
+
+\section{The Experiment}
+\label{sec:orgd0521b4}
+The purpose of this experiment was to gain a better understanding of the effects on the voltage over a capacitor
+as a time-valued function when put in a circuit in series with a resistor. To achieve
+this goal - and to experimentally verify laws governing the total resistance and capacitance of
+configurations of resistors and capacitors using a multimeter - we built an "RC circuit" by combining
+resistors and capacitors from respective smaller-valued components.
+
+Additionally, this lab fulfilled the requirement allowing us to play with dangerously high temperature metal equipment \(\ddot\smile\).
+
+\subsection{Theory}
+\label{sec:orgcbdcb4a}
+Here we list some ideas the reader should be familiar with for reference later in the report.
+
+\subsubsection{Resistors}
+\label{sec:orga13127c}
+Consider \(n\) resistors, \(r_i \ni i \in [1, n]\) representing the total resistance of the i\textsuperscript{th} resistor,
+or sub-configuration of resistors, all in parallel. Then, the total resistance, R, of the group is:
+
+\begin{equation}
+R^{-1} = \sum_{i=1}^{n}(r_i)^{-1}
+\end{equation}
+
+Consider \(n\) resistors, \(r_i \ni i \in [0, n]\) representing the total resistance of the i\textsuperscript{th} resistor,
+or sub-configuration of resistors, all in series. Then, the total resistance, R, of the group is:
+
+\begin{equation}
+R = \sum_{i=1}^{n}(r_i)
+\end{equation}
+
+\subsubsection{Capacitors}
+\label{sec:org20ad8ce}
+Total capacitance of configurations of capacitors are similar to the inversion of the laws for
+resistors.
+
+Consider \(n\) capacitors, \(c_i \ni i \in [1, n]\) representing the total capacitor of the i\textsuperscript{th} capacitor,
+or sub-configuration of capacitors, all in series. Then, the total capacitance, C, of the group is:
+
+\begin{equation}
+C^{-1} = \sum_{i=1}^{n }(c_i)^{-1}
+\end{equation}
+
+Consider \(n\) capacitors, \(c_i \ni i \in [1, n]\) representing the total capacitor of the i\textsuperscript{th} capacitor,
+or sub-configuration of capacitors, all in parallel. Then, the total capacitance, C, of the group is:
+
+\begin{equation}
+C = \sum_{i=1}^{n }(c_i)
+\end{equation}
+
+\subsubsection{The RC Circuit}
+\label{sec:orgcf7b1ec}
+For a circuit with a resistor of resistance \(R\) and capacitor with capacitance \(C\) in series,
+we can model the voltage over the capacitor, \(V_C\), given an initial voltage \(V_0\) and final
+voltage \(V_f\), as a function of time:
+
+\begin{equation}
+V_C(t) = (V_0 - V_f)e^{-\frac{t}{RC}} + V_f
+\end{equation}
+
+\subsection{Procedure}
+\label{sec:org107e08f}
+
+The given procedure to exercise our knowledge of equations (1) - (4) if to build both a relatively higher-valued
+resistor, and capacitor, out of smaller-valued components - by soldering them in series / parallel configurations.
+
+Each pair of students is to produce a resistor and capacitor at a target value (and with a 10\% margin for error),
+determined by seating arrangement. By happenstance, our group was chosen to build:
+
+\begin{enumerate}
+\item A 22 kilo-ohm resistor (22 \(k \Omega\))
+\item A 1.67 micro-farad capacitor (\(\mu F\))
+\end{enumerate}
+
+out of only 10 \(k \Omega\) resistors, and 1 \(\mu F\) capacitors.
+
+I did not record the configuration we used for either. So, assume the following configurations throughout the rest of the lab (pretty sure
+these were pretty close to our monstrosities):
+
+\subsubsection{Building a Resistor}
+\label{sec:orga374597}
+Assume all resistors as \(10 k \Omega\)
+\begin{center}
+\includegraphics[width=240px]{./resistors.png}
+\end{center}
+
+In theory, the total resistance measured from the leftmost point to the rightmost is 22 \(k \Omega\):
+
+\begin{align*}
+R  &= 10^4 \text{ (leftmost resistor in series (2))} \\
+   &+ 10^4 \text{ (second leftmost resistor in series (2))} \\
+   &+ (\frac{5}{10^4})^{-1} \text{ (5 resistors in parallel (1))} \\
+   &= 2.20 * 10^4 \Omega
+\end{align*}
+
+\subsubsection{Building a Capacitor}
+\label{sec:orgc015c61}
+\begin{center}
+\includegraphics[width=200px]{./capacitors.png}
+\end{center}
+
+In theory, the total capacitance measured from the leftmost point to the rightmost is 1.67 \(\mu F\):
+
+\begin{align*}
+C  &= 2(\frac{3}{10^-6})^{-1} \text{ (two groups of 3 1-}\mu F \text{ capacitors in series (3) in parallel with) } \\
+   &+ 10^{-6} \text{ (another 1-} \mu F \text{ capacitor (4)) } \\
+   &\approx 1.67 * 10^{-6} F
+\end{align*}
+
+\subsubsection{Determining the \(RC\) constant}
+\label{sec:org1d0608b}
+
+To measure our \(RC\) constant, we connected two voltage probes over \(V_c\) (as shown in the diagram below) to a computer-generated
+positive square wave oscillating at 0.50 Hz with an amplitude of 5V. We then record for 1.5 seconds, polling at 1 kHz, from
+the time \(V_C\) is at 4.95 V (the capacitor has charged) - allowing us to record at least half a second of discharge
+from the capacitor.
+
+\begin{center}
+\includegraphics[width=200px]{./total_circuit.png}
+\end{center}
+
+We expect to see that as it discharges, the measured voltage over the capacitor would follow an exponentially decreasing fit,
+according to the \(-\frac{t}{RC}\) term in (5). To find the value of \(RC\) we measure the voltage at each discrete time step (\(\frac{1}{1000}\) of a second)
+from near the beginning of the exponential drop to where it reaches stability, and copy those values into a Magic Excel Sheet\textsuperscript{TM}. This
+region is somewhat shown in the figure below (some values are actually truncated):
+
+\begin{center}
+\includegraphics[width=200px]{./rc-discharge.png}
+\end{center}
+
+The Magic Excel Sheet\textsuperscript{TM} produces a good exponential fit to this data. But, it takes some manual fiddling with the \(RC\) value itself
+to determine the minimum sum of residuals (gradient descent inspired guess and check). The value of \(RC\)
+producing the lowest error by this measure, is our result.
+
+\section{Results}
+\label{sec:orgf43daaa}
+\subsection{Building a Resistor}
+\label{sec:org99028c5}
+
+The measured resistance (via multimeter) we obtained from our resistor was \(21.68 k \Omega\).
+
+\subsection{Building a Capacitor}
+\label{sec:org21807f6}
+
+The measured capacitance (via multimeter) we obtained from our resistor was \(1.78 \mu F\).
+
+\subsection{The Value Of \(RC\)}
+\label{sec:orgbc9c601}
+
+For our computer determined RC constant, we found it to be \(3.72 * 10^{-2}\) s.
+
+\section{Discussion}
+\label{sec:org8408366}
+\subsection{Building a Resistor}
+\label{sec:org9b4e64e}
+Our target value was \(22.00 k \Omega\), and we came out with \(21.68 k \Omega\) - an error of 1.45\%.
+
+\subsection{Building a Capacitor}
+\label{sec:orgf2e0c7e}
+Our target value was \(1.67 \mu F\), and we came out with \(1.78 \mu F\) - an error of 6.59\%.
+
+\subsection{The Value of \(RC\)}
+\label{sec:org7e41eff}
+If our resistor and capacitor were exactly on the target value, our \(RC\) constant would be \((2.20 * 10^4 \Omega)(1.67 * 10^{-6} F) = 3.67 * 10^{-2}\) s.
+
+The \(RC\) constant from the measured resistance and capacitance would be \((2.17 * 10^4 \Omega)(1.78 * 10^{-6} F) = 3.86 * 10^-2\) s.
+
+But, our human-gradient-descent-plus-excel-magic-thanks-computer told us it was \(3.72 * 10^{-2}\) s - a 3.62\% error from the
+theoretical measured value, and 1.36\% from the overall "target" value.
+\end{document}
+\ No newline at end of file