156 lines
5.5 KiB
TeX
156 lines
5.5 KiB
TeX
\begin{filecontents}[overwrite]{\jobname.bib}
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\end{filecontents}
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\documentclass{article}
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\usepackage[backend=biber]{biblatex}
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\addbibresource{\jobname.bib}
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\usepackage{circuitikz}
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\usepackage{siunitx}
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\usepackage[a4paper, total={6in, 8in}]{geometry}
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\usepackage{amsmath}
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\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}}
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\usepackage{cancel}
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\usepackage{graphicx}
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\usepackage[T1]{fontenc}
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\usepackage{framed}
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\usepackage{longtable,booktabs,array}
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\usepackage{caption}
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\title{Lab 1 \\\quad\\ \small performed 2025-01-27}
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\author{Martin Kennedy}
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\begin{document}
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\maketitle
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\section{Introduction}
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Unlike DC signals, AC signals are time-varying, posing unique
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challenges to recording, characterizing and otherwise studying them.
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In this lab, we will examine the circuit depicted in fig.~\ref{fig:circ}, and
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focus on comparing the measurement of one aspect of an AC signal --
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the RMS voltage -- as seen by two tools: the Digital Multimeter (DMM),
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and the oscillscope. We will use numerical and analytical methods to
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model this circuit and derive expected RMS values for comparison.
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Figure~\ref{fig:circ} depicts the circuit we are studying.
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\begin{figure}[h]
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\caption{Our simple circuit}
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\label{fig:circ}
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\centering
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\begin{circuitikz}[american voltages]
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\draw
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(0,0) to [sV,l=$V_{in}$] (0,2)
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to (3,2)
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to [ R, l_=$R$ ] (3,0)
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to (0,0)
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;
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\end{circuitikz}
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\end{figure}
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\section{Analytic Modeling Results}
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In the course of our lab procedure, three different types of AC signals are provided at $V_{in}$:
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\begin{enumerate}
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\item
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\label{type:ac}
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A sinusoidal signal with no DC offset, at two different frequencies and magnitudes
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\item
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\label{type:acoffset}
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The same sinusoidal signals as above, but with a small positive DC offset
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\item
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\label{type:squarewave}
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A square wave voltage, at $25\%$ and $50\%$ duty cycles, with two different AC magnitudes.
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\end{enumerate}
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For a signal $x(t)$, we are given the following formula for its RMS voltage:
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\begin{equation} \label{eq:rms}
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X_{RMS} = \sqrt{\frac{1}{T}\int{x(t)^{2}dt}}
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\end{equation}
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I chose to analyze signal types ~\ref{type:ac} and
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~\ref{type:acoffset}, to calculate the RMS AC voltages we expect to
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see.
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\subsection{Signal type ~\ref{type:ac}}
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Signal type ~\ref{type:ac} can be modeled as:
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\begin{equation}
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v_{\ref{type:ac}}(t) = V_m\cos{(\omega{t})}
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\end{equation}
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(We can arrive here from the general form,
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$V_m\cos{(\omega{t}+\theta)}$, by shifting the beginning and end of
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our measurement window by $-\theta$, as the oscilloscope will do when
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it measures RMS voltages from a peak-to-peak cycle.)
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Substituting $v_{\ref{type:ac}}$ into equation
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~\ref{eq:rms}, the RMS voltage $V_{\ref{type:ac}RMS}$ can be
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expressed as:
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\begin{align*}
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V_{\ref{type:ac}RMS} \
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&= \sqrt{\frac{1}{T}\int_{0}^{T}{V_{m}^{2}\cos^{2}{(\omega{t})}dt}} \\
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&= V_{m}\sqrt{\frac{1}{T} \int_{0}^{T}{\frac{1}{2} + \frac{\cos{(2\omega{t})}}{2}dt}} \\
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&= V_{m}\sqrt{\frac{1}{T} \left[ \frac{t}{2} + \frac{1}{4\omega} \sin{(2\omega t)} \right]_{0}^T} \\
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&= V_{m}\sqrt{\frac{1}{T} \left( \frac{T}{2} + \frac{1}{4\omega} \cancel{\sin{(2\omega T)}} - \frac{1}{4\omega} \cancel{\sin{(0)}} \right)} \\
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&= V_{m}\sqrt{\frac{1}{\cancel{T}} \frac{\cancel{T}}{2}} \\
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&= \frac{V_{m}}{\sqrt{2}}. \numberthis \label{deriv:ac}
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\end{align*}
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\subsection{Signal type ~\ref{type:acoffset}}
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Signal type ~\ref{type:acoffset} can be modeled as:
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\begin{equation}
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v_{\ref{type:acoffset}}(t) = V_m\cos{(\omega{t})}+V_{b}
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\end{equation}
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Thus, substituting $v_{\ref{type:acoffset}}$ into equation
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~\ref{eq:rms}, the RMS voltage $V_{\ref{type:acoffset}RMS}$ can be
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expressed as:
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\begin{align*}
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V_{\ref{type:acoffset}RMS} \
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&= \sqrt{\frac{1}{T}\int_{0}^{T}{ \left( V_{m} \cos{(\omega{t})} + V_{b} \right)^{2} dt}} \\
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&= \sqrt{\frac{1}{T} \left( V_{m} \right)^{2} \int_{0}^{T}{\cos^{2}{(\omega{t})} + 2 \frac{V_{b}}{V_{m}} \cos{(\omega{t})} + \left( \frac{V_{b}}{V_{m}} \right)^{2} dt}} \\
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&= V_{m}\sqrt{\frac{1}{T} \int_{0}^{T}{\frac{1}{2} + \frac{\cos{(2\omega{t})}}{2} + 2 \frac{V_{b}}{V_{m}} \cos{(\omega{t})} + \left( \frac{V_{b}}{V_{m}} \right)^{2} dt}} \\
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&= V_{m}\sqrt{\frac{1}{T} \left[ \frac{t}{2} + \frac{1}{4\omega} \sin{(2\omega t)} + \frac{2 V_{b}}{\omega V_{m}} \sin{(\omega t)} + t \left( \frac{V_{b}}{V_{m}} \right)^{2} \right]_{0}^T} \\
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&= V_{m}\sqrt{\frac{1}{T} \left( \frac{T}{2} + \frac{1}{4\omega} \cancel{\sin{(2\omega T)}} + \frac{2 V_{b}}{\omega V_{m}} \cancel{\sin{(\omega T)}} + T \left( \frac{V_{b}}{V_{m}} \right)^{2} - \frac{1}{4\omega} \cancel{\sin{(0)}} - \frac{2 V_{b}}{\omega V_{m}} \cancel{\sin{(0)}}\right)} \\
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&= V_{m}\sqrt{\frac{1}{\cancel{T}} \left( \cancel{T} \right) \left( \frac{1}{2} + \left( \frac{V_{b}}{V_{m}} \right)^{2} \right)} \\
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&= V_{m}\sqrt{\frac{1}{2} + \left( \frac{V_{b}}{V_{m}} \right)^{2}} \\
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&= \sqrt{\frac{V_{m}^{2}}{2} + V_{b}^{2}}. \numberthis \label{deriv:acoffset}
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\end{align*}
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\section{Numerical Modeling Results}
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\section{Experimental Results}
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\section{Data Comparison}
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\section{Conclusions}
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\begin{longtable}[]{@{}lllllllll@{}}
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\toprule
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\endhead
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\bottomrule
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\endlastfoot
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Set Mag. & Set Freq. & Read Mag. & Read Period & Calc. Freq. & Calc. RMS & Meas. RMS \\
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2V & 100 Hz & 2.10 V & 9.994 ms & XXXXXHz & .....V & 1.4236 V \\
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2V & 50 kHz & 2.05 V & 19.947 us & a & d & 1.4112 V \\
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5V & 100 Hz & 5.11 V & 10.007 ms & b & e & 3.5522 V \\
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5V & 50 kHz & 5.11 V & 20.005 us & c & f & 3.5451 V \\
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\end{longtable}
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\nocite{*}
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\printbibliography
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\end{document}
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