281 lines
11 KiB
TeX
281 lines
11 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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We will be providing multiple variations on three different types of
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AC signals at $V_{in}$.
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We selected a resistor ${R=\SI{7.5}{\kohm}}$, with a $5\%$ precision
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band; note that the exact resistance value of the resistor really
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isn't important, since we are not studying its characteristics; we
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only need its impedance to be much larger than the 50-ohm internal
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resistance of our function generator.
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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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\subsection{Signal type ~\ref{type:ac} (sinusoidal AC)}
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Given frequency $\omega$ (rad./s) and magnitude (peak amplitude) $V_m$, signal
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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} (sinusoidal AC with DC offset)}
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Given frequency $\omega$ (rad./s), peak amplitude $V_m$ and DC offset $V_b$, we
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can model signal type ~\ref{type:acoffset} 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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\begin{quote}
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Note: I won't show it here, but general, for a periodic signal made
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of two components $v = v_a + v_b$, where the two components are of
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different frequencies, it can be shown from Equation ~\ref{eq:rms}
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that the RMS voltage $V_{RMS}$ of the combined signal is equal to
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the sum of the squares of the RMS voltages of its additive
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components, i.e.
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\begin{equation*}
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\label{eq:rms_sq_comp}
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V_{RMS} = \sqrt{{V_{aRMS}}^{2} + {V_{bRMS}}^{2}}.
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\end{equation*}
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\end{quote}
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\subsection{Signal type ~\ref{type:squarewave} (square wave)}
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Given magnitude $V_{m}$, frequency $\frac{1}{T}$ (Hz), and duty cycle
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$D$, our square wave (type ~\ref{type:squarewave}) can be modeled as:
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\begin{equation}
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0 < t \leq T, \quad v_{\ref{type:squarewave}}(t) = \\
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\begin{cases}
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V_{m} & 0 \leq t < D T \\
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-V_{m} & D T \leq t < T
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\end{cases}
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\end{equation}
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This time, the integration is trivial: at any time,
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$v_{\ref{type:squarewave}}(t)$ is either $V_{m}$ or $-V_{m}$, so
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$(v_{\ref{type:squarewave}}(t))^{2}={V_{m}}^{2}$:
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\begin{equation*}
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V_{\ref{type:squarewave}RMS} = \sqrt{\frac{1}{T} \int_{0}^{T}{(V_{m})^{2}dt} } = \sqrt{\frac{1}{\cancel{T}} \cancel{T} {V_{m}}^{2}} = V_{m}. \
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\numberthis \label{deriv:squarewave}
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\end{equation*}
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We will reference these three derivations in our ``Experimental Results'' section below.
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\section{Numerical Modeling Results}
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\section{Experimental Results}
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\begin{figure}[h]
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\caption{Breadboard, with resistor $R$ connected to our DMM, scope and function generator}
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\label{fig:breadboard}
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\centering
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\includegraphics[width=\textwidth]{lab1breadboard}
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\end{figure}
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The circuit we implemented can be seen in
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Figure~\ref{fig:breadboard}. This configuration connects one resistor
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leg with the signal lead of the oscilloscope, the positive lead each
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of the function generator and DMM; repeat the same with the other
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resistor leg, the ground lead of the scope, and the negative leads of
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each the DMM and function generator. In effect, all pieces of
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equipment are placed in parallel, consistent with any other procedure
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for measuring the facets of a signal's voltage.
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(Our circuit builder was Peyton; our checker was Will; I was grouped
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with these two as there were an odd number of students.)
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We used this configuration for the entire lab procedure, and adjusted
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both our function generator and oscilloscope through the variations of
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the three different signal types; in all three cases, we first used
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the oscilloscope to read the period and magnitude of the signal, and
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then used the DMM to measure the signal's RMS voltage.
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\subsection{Experiment ~\ref{type:ac} (sinusoidal AC)}
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Given a read period of $T$ seconds, we calculate the frequency as
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$\frac{1}{T}$ Hz. For the RMS voltage, we use the formula derived at
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Equation ~\ref{deriv:ac}:
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\begin{equation*}
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V_{\ref{type:ac}RMS} = \frac{V_{m}}{\sqrt{2}}
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\end{equation*}
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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 & 100.1 Hz & 1.48 V & 1.4236 V \\
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2V & 50 kHz & 2.05 V & 19.95 us & 50.13 kHz & 1.45 V & 1.4112 V \\
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5V & 100 Hz & 5.11 V & 10.01 ms & 99.90 Hz & 3.61 V & 3.5522 V \\
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5V & 50 kHz & 5.11 V & 20.01 us & 49.98 kHz & 3.61 V & 3.5451 V \\
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\end{longtable}
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\subsection{Experiment ~\ref{type:acoffset} (sinusoidal AC with DC offset)}
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Given a read period of $T$ seconds, we calculate the frequency as
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$\frac{1}{T}$ Hz. For this signal's RMS voltage, we use the formula derived at
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Equation ~\ref{deriv:acoffset}:
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\begin{equation*}
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V_{\ref{type:acoffset}RMS} = \sqrt{\frac{V_{m}^{2}}{2} + V_{b}^{2}}
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\end{equation*}
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TODO NOTE ERROR could not check dc offset voltage bias
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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. & DC bias & Read Mag. & Read Period & Calc. Freq. & Calc. RMS & Meas. RMS \\
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2V & 100 Hz & 2V & 2.13 V & 10.00 ms & 100.0 Hz & 2.50 V & 2.44 V \\
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2V & 100 Hz & -5V & 2.11 V & 9.996 ms & 100.0 Hz & 5.22 V & 5.19 V \\
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5V & 100 Hz & 2V & 5.15 V & 9.998 ms & 100.0 Hz & 4.15 V & 4.05 V \\
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5V & 100 Hz & -5V & 5.20 V & 9.997 ms & 100.0 Hz & 6.21 V & 6.16 V \\
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\end{longtable}
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\subsection{Experiment ~\ref{type:squarewave} (square wave)}
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Given a read period of $T$ seconds, we calculate the frequency as
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$\frac{1}{T}$ Hz. As we saw when deriving Equation
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~\ref{deriv:squarewave}, this signal's RMS voltage is the same as its
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magnitude.
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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. & Duty & Read Mag. & Read Period & Calc. Freq. & Calc. RMS & Meas. RMS \\
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2V & 100 Hz & 25\% & 2.11 V & 10.00ms & 100.0 Hz & 2.11 V & 2.02 V \\
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2V & 100 Hz & 50\% & 2.13 V & 10.00ms & 100.0 Hz & 2.13 V & 2.01 V \\
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5V & 100 Hz & 25\% & 5.20 V & 9.998ms & 100.0 Hz & 5.20 V & 5.04 V \\
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5V & 100 Hz & 50\% & 5.20 V & 9.999ms & 100.0 Hz & 5.20 V & 5.01 V \\
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\end{longtable}
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\section{Data Comparison}
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\section{Conclusions}
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\nocite{*}
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\printbibliography
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\end{document}
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