500 lines
19 KiB
TeX
500 lines
19 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[margin=1in]{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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This lab examines the circuit depicted in fig.~\ref{fig:circ}, and
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focuses 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. Numerical and analytical methods are used 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 simple circuit under study:
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\begin{figure}[h]
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\caption{The 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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Multiple variations on three different types of AC signals are
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provided at $V_{in}$.
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A resistor was selected with value ${R=\SI{7.5}{\kohm}}$, with a $5\%$
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precision band; note that the exact resistance value of the resistor
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really isn't important, since its characteristics are not under study;
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it is only important that its impedance be much larger than the 50-ohm
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internal resistance of the function generator in use.
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\section{Analytic Modeling Results}
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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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(One 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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the 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 root of the sum of the squares of the RMS voltages of its
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additive 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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These three derivations are re-referenced in the ``Experimental
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Results'' section below.
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\section{Numerical Modeling Results}
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For the numerical modeling, I opted to simulate signals types
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~\ref{type:ac} (sinusoidal AC about 0V) and ~\ref{type:squarewave}
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(square wave).
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In LTSpice, I assembled the circuit shown in Figure
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~\ref{fig:ac_2v_100hz_diag}; for the first signal, I specified a
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transient simulation from 0.1s to 0.2s:
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\begin{figure}[h]
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\caption{Our first sinusoidal signal circuit, simulated in LTSpice}
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\label{fig:ac_2v_100hz_diag}
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\centering
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\includegraphics[width=0.6\textwidth]{lab1_ac_2v_100hz_diag}
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\end{figure}
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I then used Ctrl+click on the signal label \texttt{V(n001)} to pull up
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the Waveform dialog shown in Figure ~\ref{fig:ac_2v_100hz_num},
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yielding a numerically-derived RMS voltage:
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\begin{figure}[h!]
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\caption{This LTSpice dialog shows us measurements of our interval; of most interest is RMS}
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\label{fig:ac_2v_100hz_num}
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\centering
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\includegraphics[width=0.3\textwidth]{lab1_ac_2v_100hz_num}
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\end{figure}
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I had to adjust the size length of the transient simulation to get an
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easily-viewable result.
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For the square-wave values: I switched from the \texttt{SINE} command
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to \texttt{PULSE}; this command requires a rise-time and fall-time,
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which are set as low as possible to mimic a true square-wave; in
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addition, we specify the duty cycle and frequency indirectly, instead
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by specifying the on time and period, as seen in Figure
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~\ref{fig:pulse_ltspice}.
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\begin{figure}[h]
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\caption{The LTSpice \texttt{PULSE} command menu}
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\label{fig:pulse_ltspice}
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\centering
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\includegraphics[width=\textwidth]{lab1_ltspice_pulse}
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\end{figure}
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The results of the numerical modeling are compiled below, in Tables
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~\ref{table:comparison_ac} and ~\ref{table:comparison_squarewave}.
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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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This configuration was used 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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Note: our use of the oscilloscope for magnitude measurements will
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later be identified as a key source of error.
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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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\caption {Voltage measurements and period for ~\ref{type:ac} (sinusoidal AC)}
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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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Comparing the calculated and measured RMS values:
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\begin{longtable}[]{@{}llll@{}}
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\toprule\noalign{}
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\caption {RMS Error for ~\ref{type:ac} (sinusoidal AC)}
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\endhead
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\bottomrule\noalign{}
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\endlastfoot
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& Calc. RMS & Meas. RMS & Error \% \\
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Case 1 & 1.48 V & 1.4236 V & 3.81 \% \\
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Case 2 & 1.45 V & 1.4112 V & 2.68 \% \\
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Case 3 & 3.61 V & 3.5522 V & 1.60 \% \\
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Case 4 & 3.61 V & 3.5451 V & 1.80 \% \\
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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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\begin{longtable}[]{@{}lllllllll@{}}
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\toprule
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\caption {Voltage measurements and period for ~\ref{type:acoffset} (sinusoidal AC with DC offset)}
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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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Comparing the calculated and measured RMS values:
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\begin{longtable}[]{@{}llll@{}}
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\toprule\noalign{}
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\caption {RMS Error for ~\ref{type:acoffset} (sinusoidal AC with DC offset)}
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\endhead
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\bottomrule\noalign{}
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\endlastfoot
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& Calc. RMS & Meas. RMS & Error \% \\
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Case 1 & 2.50 V & 2.44 V & 2.40 \% \\
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Case 2 & 5.22 V & 5.19 V & 0.57 \% \\
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Case 3 & 4.15 V & 4.05 V & 2.41 \% \\
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Case 4 & 6.21 V & 6.16 V & 0.81 \% \\
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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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\caption {Voltage measurements and period for ~\ref{type:squarewave} (square wave)}
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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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Comparing the calculated and measured RMS values:
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\begin{longtable}[]{@{}llll@{}}
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\toprule\noalign{}
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\caption {RMS Error for ~\ref{type:squarewave} (square wave)}
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\endhead
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\bottomrule\noalign{}
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\endlastfoot
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& Calc. RMS & Meas. RMS & Error \% \\
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Case 1 & 2.11 V & 2.02 V & 4.27 \% \\
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Case 2 & 2.13 V & 2.01 V & 5.63 \% \\
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Case 3 & 5.20 V & 5.04 V & 3.08 \% \\
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Case 4 & 5.20 V & 5.01 V & 3.65 \% \\
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\end{longtable}
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Notably, this is the first time we have a >5\% error value; we will
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review this item in TODO WHERE?
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\section{Data Comparison}
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Here, I publish the modeling results for ~\ref{type:ac} (sinusoidal AC
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about 0V).
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\begin{longtable}[]{@{}lllll@{}}
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\toprule\noalign{}
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\caption {RMS Voltage comparison for ~\ref{type:ac} (sinusoidal AC)}
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\label {table:comparison_ac}
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\endhead
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\bottomrule\noalign{}
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\endlastfoot
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RMS & Case 1 & Case 2 & Case 3 & Case 4 \\
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Analytic (A) & 1.48 V & 1.45 V & 3.61 V & 3.61 V \\
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Numerical (N) & 1.4125 V & 1.4124 V & 3.5311 V & 3.5311 V \\
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Experimental (E) & 1.4236 V & 1.4112 V & 3.5522 V & 3.5451 V \\
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A-N error & 4.78 \% & 2.66 \% & 2.23 \% & 2.23 \% \\
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A-E error & 3.96 \% & 2.75 \% & 1.63 \% & 1.83 \% \\
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N-E error & 0.78 \% & 0.09 \% & 0.59 \% & 0.39 \% \\
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\end{longtable}
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Next, I publish the modeling results for ~\ref{type:squarewave}
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(square-wave AC).
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\begin{longtable}[]{@{}lllll@{}}
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\toprule\noalign{}
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\caption {RMS Voltage comparison for ~\ref{type:squarewave} (square-wave AC)}
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\label {table:comparison_squarewave}
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\endhead
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\bottomrule\noalign{}
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\endlastfoot
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RMS & Case 1 & Case 2 & Case 3 & Case 4 \\
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Analytic (A) & 2.11 V & 2.13 V & 5.20 V & 5.01 V \\
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Numerical (N) & 1.9999 V & 1.9999 V & 4.9997 V & 4.9997 V \\
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Experimental (E) & 2.02 V & 2.01 V & 5.04 V & 5.01 V \\
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A-N error & 5.51 \% & 6.50 \% & 4.01 \% & 4.01 \% \\
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A-E error & 4.46 \% & 5.97 \% & 3.17 \% & 1.96 \% \\
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N-E error & 1.00 \% & 0.50 \% & 0.80 \% & 1.97 \% \\
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\end{longtable}
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\section{Conclusions}
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Before completing the lab report out and making firm conclusions, I'd
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like to address our two analysis questions:
|
|
|
|
\begin{quote}
|
|
PSpice: In the transient simulation profile: what is the role of
|
|
``Maximum Step Size''? Create an example and include waveform images
|
|
to illustrate your point.
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|
\end{quote}
|
|
|
|
Returning to our simulation of a 5V sinusoidal waveform with no DC
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|
bias at 50kHz: the larger we allow the ``Step Size'' to go, the fewer
|
|
timesteps LTSpice will take when performing its numerical simulation,
|
|
and thus the less precise our RMS value is / the less close it is to
|
|
$\frac{5}{\sqrt{2}} \approx 3.53553 ...$. In Figures
|
|
~\ref{fig:ltspice_large_timestep},
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|
~\ref{fig:ltspice_large_timestep_num},
|
|
~\ref{fig:ltspice_small_timestep}, and
|
|
~\ref{fig:ltspice_small_timestep_num}, we can see that restricting the
|
|
timestep size down to $\SI{1}{\ns}$ brings our RMS voltage much closer
|
|
to $\frac{5}{\sqrt{2}} \approx 3.53553$.
|
|
|
|
\begin{figure}[h]
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|
\caption{The LTSpice simulation with no defined ``Maximum Step Size''}
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|
\label{fig:ltspice_large_timestep}
|
|
\centering
|
|
\includegraphics[width=0.6\textwidth]{lab1_ltspice_large_timestep}
|
|
\end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\caption{The RMS value of the simulation with no defined ``Maximum Step Size''}
|
|
\label{fig:ltspice_large_timestep_num}
|
|
\centering
|
|
\includegraphics[width=0.3\textwidth]{lab1_ltspice_large_timestep_num}
|
|
\end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\caption{The LTSpice simulation with a ``Maximum Step Size'' of $\SI{1}{\ns}$}
|
|
\label{fig:ltspice_small_timestep}
|
|
\centering
|
|
\includegraphics[width=0.6\textwidth]{lab1_ltspice_small_timestep}
|
|
\end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\caption{The RMS value of the simulation under a ``Maximum Step Size'' of $\SI{1}{\ns}$}
|
|
\label{fig:ltspice_small_timestep_num}
|
|
\centering
|
|
\includegraphics[width=0.3\textwidth]{lab1_ltspice_small_timestep_num}
|
|
\end{figure}
|
|
|
|
As for the second question:
|
|
|
|
\begin{quote}
|
|
Considering your experimental data, explain why we can conclude that
|
|
the DMM is showing the true RMS regardless of the waveform.
|
|
\end{quote}
|
|
|
|
A value for RMS voltage was arrived at through multiple different
|
|
means; from this, it was discovered that no waveform yields a
|
|
particularly higher error for RMS voltage than any other; in fact,
|
|
more specifically, the highest error values are associated only with
|
|
the analytical derivation, which relies on an oscilloscope-derived
|
|
magnitude reading.
|
|
|
|
I'll review the oscilloscope-derived magnitude problem in a moment,
|
|
but I wanted to finish my point regarding the suitability of the DMM
|
|
to produce RMS voltage measurements: the lack of particularly high
|
|
error values when comparing experimental (DMM) RMS values for any of
|
|
the various waveforms indicates that the DMM's suitability does not,
|
|
within reason, depend on the waveform it's measuring.
|
|
|
|
As for the error values: the largest cases of error are when comparing
|
|
the experimental and numerical results to the analytical results for
|
|
2V square-wave AC. Despite setting the function generator to 2V, the
|
|
oscilloscope nevertheless reads a peak-to-peak magnitude of 4.22 and
|
|
4.26V (so, a peak-magnitude 2.11V of 2.13V, respectively). This likely
|
|
has to do with how the oscilloscope regisers peak-to-peak voltages,
|
|
relying on hazy extremes; but also, notably, because we had our
|
|
oscilloscope in 10X mode, reducing its signal sensitivity /
|
|
resolution.
|
|
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|
\nocite{*}
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|
\printbibliography
|
|
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|
\end{document}
|