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Lab1.tex

@@ -179,6 +179,53 @@ We will reference these three derivations in our ``Experimental Results'' sectio
 
 \section{Numerical Modeling Results}
 
+For the numerical modeling, I opted to simulate signals types
+~\ref{type:ac} (sinusoidal AC about 0V) and ~\ref{type:squarewave}
+(square wave).
+
+In LTSpice, I assembled the circuit shown in Figure
+~\ref{fig:ac_2v_100hz_diag}; for the first signal, I specified a
+transient simulation from 0.1s to 0.2s:
+
+\begin{figure}[h]
+  \caption{Our first sinusoidal signal circuit, simulated in LTSpice}
+  \label{fig:ac_2v_100hz_diag}
+  \centering
+  \includegraphics[width=0.6\textwidth]{lab1_ac_2v_100hz_diag}
+\end{figure}
+
+I then used Ctrl+click on the signal label \texttt{V(n001)} to pull up
+the Waveform dialog shown in Figure ~\ref{fig:ac_2v_100hz_num},
+yielding a numerically-derived RMS voltage:
+
+\begin{figure}[h!]
+  \caption{This LTSpice dialog shows us measurements of our interval; of most interest is RMS}
+  \label{fig:ac_2v_100hz_num}
+  \centering
+  \includegraphics[width=0.3\textwidth]{lab1_ac_2v_100hz_num}
+\end{figure}
+
+I had to adjust the size length of the transient simulation to get an
+easily-viewable result.
+
+For the square-wave values: I switched from the \texttt{SINE} command
+to \texttt{PULSE}; this command requires a rise-time and fall-time,
+which are set as low as possible to mimic a true square-wave; in
+addition, we specify the duty cycle and frequency indirectly, instead
+by specifying the on time and period, as seen in Figure
+~\ref{fig:pulse_ltspice}.
+
+
+\begin{figure}[h]
+  \caption{The LTSpice \texttt{PULSE} command menu}
+  \label{fig:pulse_ltspice}
+  \centering
+  \includegraphics[width=\textwidth]{lab1_ltspice_pulse}
+\end{figure}
+
+The results of the numerical modeling are compiled below, in Tables
+~\ref{table:comparison_ac} and ~\ref{table:comparison_squarewave}.
+
 \section{Experimental Results}
 
 \begin{figure}[h]
@@ -217,15 +264,31 @@ Equation ~\ref{deriv:ac}:
 \end{equation*}
 
 \begin{longtable}[]{@{}lllllllll@{}}
-\toprule
-\endhead
-\bottomrule
-\endlastfoot
-Set Mag. & Set Freq. & Read Mag. & Read Period & Calc. Freq. & Calc. RMS & Meas. RMS \\
-2V       & 100 Hz    & 2.10 V    & 9.994 ms    & 100.1 Hz    & 1.48 V    & 1.4236 V  \\
-2V       & 50 kHz    & 2.05 V    & 19.95 us    & 50.13 kHz   & 1.45 V    & 1.4112 V  \\
-5V       & 100 Hz    & 5.11 V    & 10.01 ms    & 99.90 Hz    & 3.61 V    & 3.5522 V  \\
-5V       & 50 kHz    & 5.11 V    & 20.01 us    & 49.98 kHz   & 3.61 V    & 3.5451 V  \\
+  \toprule
+  \caption {Voltage measurements and period for ~\ref{type:ac} (sinusoidal AC)}
+  \endhead
+  \bottomrule
+  \endlastfoot
+  Set Mag. & Set Freq. & Read Mag. & Read Period & Calc. Freq. & Calc. RMS & Meas. RMS \\
+  2V       & 100 Hz    & 2.10 V    & 9.994 ms    & 100.1 Hz    & 1.48 V    & 1.4236 V  \\
+  2V       & 50 kHz    & 2.05 V    & 19.95 us    & 50.13 kHz   & 1.45 V    & 1.4112 V  \\
+  5V       & 100 Hz    & 5.11 V    & 10.01 ms    & 99.90 Hz    & 3.61 V    & 3.5522 V  \\
+  5V       & 50 kHz    & 5.11 V    & 20.01 us    & 49.98 kHz   & 3.61 V    & 3.5451 V  \\
+\end{longtable}
+
+Comparing the calculated and measured RMS values:
+
+\begin{longtable}[]{@{}llll@{}}
+  \toprule\noalign{}
+  \caption {RMS Error for ~\ref{type:ac} (sinusoidal AC)}
+  \endhead
+  \bottomrule\noalign{}
+  \endlastfoot
+         & Calc. RMS & Meas. RMS & Error \% \\
+  Case 1 & 1.48 V    & 1.4236 V  & 3.81 \%  \\
+  Case 2 & 1.45 V    & 1.4112 V  & 2.68 \%  \\
+  Case 3 & 3.61 V    & 3.5522 V  & 1.60 \%  \\
+  Case 4 & 3.61 V    & 3.5451 V  & 1.80 \%  \\
 \end{longtable}
 
 \subsection{Experiment ~\ref{type:acoffset} (sinusoidal AC with DC offset)}
@@ -238,10 +301,9 @@ Equation ~\ref{deriv:acoffset}:
   V_{\ref{type:acoffset}RMS} = \sqrt{\frac{V_{m}^{2}}{2} + V_{b}^{2}}
 \end{equation*}
 
-TODO NOTE ERROR could not check dc offset voltage bias
-
 \begin{longtable}[]{@{}lllllllll@{}}
   \toprule
+  \caption {Voltage measurements and period for ~\ref{type:acoffset} (sinusoidal AC with DC offset)}
   \endhead
   \bottomrule
   \endlastfoot
@@ -252,6 +314,21 @@ TODO NOTE ERROR could not check dc offset voltage bias
   5V       & 100 Hz    & -5V     & 5.20 V    & 9.997 ms    & 100.0 Hz    & 6.21 V    & 6.16   V  \\
 \end{longtable}
 
+Comparing the calculated and measured RMS values:
+
+\begin{longtable}[]{@{}llll@{}}
+  \toprule\noalign{}
+  \caption {RMS Error for ~\ref{type:acoffset} (sinusoidal AC with DC offset)}
+  \endhead
+  \bottomrule\noalign{}
+  \endlastfoot
+         & Calc. RMS & Meas. RMS & Error \% \\
+  Case 1 & 2.50 V    & 2.44   V  & 2.40 \%  \\
+  Case 2 & 5.22 V    & 5.19   V  & 0.57 \%  \\
+  Case 3 & 4.15 V    & 4.05   V  & 2.41 \%  \\
+  Case 4 & 6.21 V    & 6.16   V  & 0.81 \%  \\
+\end{longtable}
+
 \subsection{Experiment ~\ref{type:squarewave} (square wave)}
 
 Given a read period of $T$ seconds, we calculate the frequency as
@@ -261,6 +338,7 @@ magnitude.
 
 \begin{longtable}[]{@{}lllllllll@{}}
   \toprule
+  \caption {Voltage measurements and period for ~\ref{type:squarewave} (square wave)}
   \endhead
   \bottomrule
   \endlastfoot
@@ -271,9 +349,124 @@ magnitude.
   5V       & 100 Hz    & 50\% & 5.20 V    & 9.999ms     & 100.0 Hz    & 5.20 V    & 5.01  V   \\
 \end{longtable}
 
+Comparing the calculated and measured RMS values:
+
+\begin{longtable}[]{@{}llll@{}}
+  \toprule\noalign{}
+  \caption {RMS Error for ~\ref{type:squarewave} (square wave)}
+  \endhead
+  \bottomrule\noalign{}
+  \endlastfoot
+         & Calc. RMS & Meas. RMS & Error \% \\
+  Case 1 & 2.11 V    & 2.02  V   & 4.27 \%  \\
+  Case 2 & 2.13 V    & 2.01  V   & 5.63 \%  \\
+  Case 3 & 5.20 V    & 5.04  V   & 3.08 \%  \\
+  Case 4 & 5.20 V    & 5.01  V   & 3.65 \%  \\
+\end{longtable}
+
+Notably, this is the first time we have a >5\% error value; we will
+review this item in TODO WHERE?
+
 \section{Data Comparison}
+
+Here, I publish the modeling results for ~\ref{type:ac} (sinusoidal AC
+about 0V).
+
+\begin{longtable}[]{@{}lllll@{}}
+  \toprule\noalign{}
+  \caption {RMS Voltage comparison for ~\ref{type:ac} (sinusoidal AC)}
+  \label {table:comparison_ac}
+  \endhead
+  \bottomrule\noalign{}
+  \endlastfoot
+  RMS              & Case 1   & Case 2   & Case 3   & Case 4   \\
+  Analytic (A)     & 1.48 V   & 1.45 V   & 3.61 V   & 3.61 V   \\
+  Numerical (N)    & 1.4125 V & 1.4124 V & 3.5311 V & 3.5311 V \\
+  Experimental (E) & 1.4236 V & 1.4112 V & 3.5522 V & 3.5451 V \\
+  A-N error        & 4.78 \%  & 2.66 \%  & 2.23 \%  & 2.23 \%  \\
+  A-E error        & 3.96 \%  & 2.75 \%  & 1.63 \%  & 1.83 \%  \\
+  N-E error        & 0.78 \%  & 0.09 \%  & 0.59 \%  & 0.39 \%  \\
+\end{longtable}
+
+Next, I publish the modeling results for ~\ref{type:squarewave}
+(square-wave AC).
+
+\begin{longtable}[]{@{}lllll@{}}
+  \toprule\noalign{}
+  \caption {RMS Voltage comparison for ~\ref{type:squarewave} (square-wave AC)}
+  \label {table:comparison_squarewave}
+  \endhead
+  \bottomrule\noalign{}
+  \endlastfoot
+  RMS              & Case 1   & Case 2   & Case 3   & Case 4   \\
+  Analytic (A)     & 2.11 V   & 2.13 V   & 5.20 V   & 5.01 V   \\
+  Numerical (N)    & 1.9999 V & 1.9999 V & 4.9997 V & 4.9997 V \\
+  Experimental (E) & 2.02 V   & 2.01 V   & 5.04 V   & 5.01 V   \\
+  A-N error        & 5.51 \%  & 6.50 \%  & 4.01 \%  & 4.01 \%  \\
+  A-E error        & 4.46 \%  & 5.97 \%  & 3.17 \%  & 1.96 \%  \\
+  N-E error        & 1.00 \%  & 0.50 \%  & 0.80 \%  & 1.97 \%  \\
+\end{longtable}
+
 \section{Conclusions}
 
+Before completing the lab report out and making firm conclusions, I'd
+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.
+\end{quote}
+
+Returning to our simulation of a 5V sinusoidal waveform with no DC
+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},
+~\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]
+  \caption{The LTSpice simulation with no defined ``Maximum Step Size''}
+  \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}
+
+
+
 \nocite{*}
 \printbibliography