Merge branch 'main' of https://arnweb.nl/gitea/arne/EV6_HW_Imp

doc/reflection/main.tex

@@ -13,6 +13,8 @@
 \usepackage{hyperref}
 \usepackage{listings}
 
+\newcommand{\eqline}{\noalign{\smallskip} \hline \noalign{\smallskip}}
+
 \title{
     Reflection experiment\linebreak
     \large{EV6 - Hardware Implementation}
@@ -135,36 +137,79 @@
         \caption{Pulse measurement, no cable}
         \label{fig:probe_only}
     \end{figure}
+    The resulting pulse is as expected and within tolerances. 
 
-    \subsection{Open termination}
+    \subsection{Termination results}
+        \subsubsection{Open termination}
         \begin{figure}[H]
             \includegraphics[width=\linewidth]{./img/Graph - Unterminated End.png}
             \caption{Result for open termination}
             \label{fig:actual_open}
         \end{figure}
+        The open termination results in a reflection of roughly equal amplitude to the original signal. There is 36 ns between the original pulse and the reflection.
 
-    \subsection{Short termination}
+        \subsubsection{Short termination}
         \begin{figure}[H]
             \includegraphics[width=\linewidth]{./img/Graph - Terminated 0 Ohm.png}
             \caption{Result for short termination}
             \label{fig:actual_short}
         \end{figure}        
+        The short termination results in a reflection of the original signal, but inverted. The reflection is also of lower amplitude than the original signal. Interestingly, the time between the original pulse and the reflection is 36 ns exactly, which is identical as the open termination result.
 
-    \subsection{Matched termination}
+        \subsubsection{Matched termination}
         \begin{figure}[H]
             \includegraphics[width=\linewidth]{./img/Graph - Terminated 50 Ohm.png}
             \caption{Result for matched termination}
             \label{fig:actual_matched}
         \end{figure}
+        As expected, the matched termination results in no reflection at all.
 
-    \subsection{Unknown cable length}
+
+    \subsection{Cable length determination}
+    By measuring the difference in time between the original pulse and the reflection, we can calculate the length of the cable using the speed formula:
+    \begin{align*}
+        v &= \frac{s}{t}\\
+        s &= v \cdot t
+    \end{align*}
+
+    The speed of light in a cable is roughly 2/3 of the speed of light in a vacuum. Good quality cables include this value exactly, but unfortunately, we do not have this information for the cables used in this experiment. 
+
+    \begin{align*}
+        c &= 299792458 & [ms^{-1}]\\
+        v &= \frac{2}{3} \cdot c\\
+        &= 199861638,\overline{66} & [ms^{-1}]\\
+        t &= 36 & [ns]\\
+        &= 36 \cdot 10^{-9} & [s]\\
+        \frac{1}{2}t &= 18 \cdot 10^{-9} & [s]
+    \end{align*}
+
+    Time has to be divided by 2 since the reflection has to travel the same distance twice, once to the end of the cable and once back.
+
+    \begin{align*}
+        s &= 1199861638,\overline{66} \cdot (18 \cdot 10^{-9})\\
+        &= 3,5975094959 & [m]\\
+        \noalign{\smallskip} \hline \noalign{\smallskip}
+        s &\approx 3,6 & [m]
+    \end{align*}
+
+    The actual length of the cable is 3 meters, so the result is somewhat off, there are several possible reasons for this:
+    \begin{enumerate}
+        \item The speed of light in the cable and the male to male connectors is not exactly 66\% of the speed of light in a vacuum
+        \item The cables are not exactly 1 meter long
+        \item The length of the male to male connectors is not taken into account
+    \end{enumerate}
+
+    \subsubsection{Unknown 75 Ohm cable}
     \begin{figure}[H]
         \includegraphics[width=\linewidth]{./img/Graph - 75 Ohm cable.png}
         \caption{Result for unknown cable}
         \label{fig:unknown_cable}
     \end{figure}
+    In this example, we did not change the setting of the function generator, resulting in an initial pulse of rougly 3.5 volts instead of 2.5 volts. However, this does not affect the results of the measurement since we only look at the time difference between the original pulse and the reflection.
+
+    Using the same method as before, we find that the cable is roughly 332482034 meters long.
 
     \section{Conclusion}
-    Reflection = bad, termination = good.
+    Reflection = bad, termination = good, depending on what you wish to achieve. 
 \end{multicols}
 \end{document}