164 lines
6.3 KiB
TeX
164 lines
6.3 KiB
TeX
\documentclass[paper=A4]{article}
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\usepackage[utf8]{inputenc}
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\usepackage[a4paper, left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}
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\usepackage{siunitx}
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\sisetup{
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input-decimal-markers={.},
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output-decimal-marker = {.},
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\usepackage{graphicx}
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\usepackage{fancyhdr}
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\usepackage{lastpage}
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\usepackage{subfigure}
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\usepackage{float}
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\usepackage{wrapfig}
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\usepackage{multicol}
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\usepackage{amsmath, amssymb}
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\usepackage{tikz}
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\usepackage{hyperref}
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\usepackage{listings}
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\usepackage{xcolor}
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\usepackage{eso-pic}
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\lstdefinestyle{mystyle}{
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backgroundcolor=\color{backcolour},
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commentstyle=\color{codegreen},
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keywordstyle=\color{orange},
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basicstyle=\ttfamily\footnotesize,
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tabsize=2
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}
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\lstset{style=mystyle}
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\hyphenpenalty=10000 %to stop cutting words in a paragraph
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\pagestyle{fancy}
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\fancyhf{}
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\rhead{\includegraphics*[scale=0.013]{./Pictures/FaSTTUBe_Logo_ohneAuto.png}}
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\rfoot{Page \thepage \hspace{1pt} of \pageref{LastPage}}
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\lhead{Car 313, 01.05, Rev. 1}
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\chead{\large TS Discharge Circuit Schematic}
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\begin{document}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.98\textwidth]{./Documents/DC.pdf}
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\caption{Schematic of the Discharge Circuit PCB}
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\end{figure}
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% DC Highlighting
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\AddToShipoutPicture*{
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\begin{tikzpicture}[remember picture, overlay]
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\fill[yellow, opacity=0.4] ([xshift=4.02cm,yshift=6.37cm]current page.center) rectangle ++(1.39cm,1.99cm);
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\fill[yellow, opacity=0.4] ([xshift=5.85cm,yshift=6.37cm]current page.center) rectangle ++(1.34cm,1.99cm);
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\end{tikzpicture}
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}
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\section*{Discharge Time}
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As seen in the schematic, for our discharge circuitry a PTC (PTCEL13R251NxE) is used.\cite{ptc_datasheet}
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The total capacitance of the DC-link capacitor from the two inverters (Emsiso emDrive H100) that we are using is about \SI{200}{\micro\farad},
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and the maximum voltage of the accumulator is 403.2V.\cite{emdriver_datasheet} Using the RC discharging circuit equation, we obtain the highest resistance that the PTC can have so that we are still within the 5s discharge limit.
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\begin{align}
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V_C &= V_0 \cdot e^{-t/RC} \\
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\SI{60}{\volt} &= \SI{403.2}{\volt} \cdot e^{-\SI{5}{\second}/(R_{PTC} \cdot \SI{200}{\micro\farad})} \\
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R_{PTC} &\approx \SI{13123}{\ohm}
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\end{align}
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\begin{wrapfigure}{r}{0.4\textwidth}
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\includegraphics[width=\linewidth]{./Pictures/PTC-R-T.png}
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\caption{Resistance vs. Temperature for PTCEL13 (typical)}
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\label{fig:PTC_T_R}
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\end{wrapfigure}
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To calculate how many discharge attempts can be made before the discharge time exceeds \SI{5}{\second},
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we first determine the temperature at which the PTC has a resistance of \SI{13123}{\ohm}. This value can be obtained from the PTC's datasheet (see Fig.~\ref{fig:PTC_T_R}).\cite{ptc_datasheet}
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From the graph, the corresponding temperature is approximately \SI{165}{\celsius}.
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We assume the PTC reaches this temperature instantly after each discharge and that heat dissipation is negligible (since the thermal time constant $\tau_{th}$ is \SI{130}{\second}).
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To determine the maximum allowable thermal energy before the PTC cools down, we assume an ambient temperature of \SI{45}{\celsius}.
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Given the thermal capacity $C_{th} = \SI{1.45}{\joule\per\kelvin}$, the maximum thermal energy that can be absorbed is:
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\[
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E = \Delta T \cdot C_{th} = (\SI{165}{\celsius} - \SI{45}{\celsius}) \cdot \SI{1.45}{\joule\per\kelvin} = \SI{174}{\joule}
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\]
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\newpage
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Next, we calculate the energy dissipated in one discharge:
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\begin{align}
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E &= \frac{1}{2} \cdot C \cdot V^2 \\
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&= \frac{1}{2} \cdot \SI{200}{\micro\farad} \cdot (\SI{403.2}{\volt})^2 = \SI{16.26}{\joule}
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\end{align}
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Therefore, the number of discharges possible before the discharge time exceeds \SI{5}{\second} is:
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\[
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\frac{\SI{174}{\joule}}{\SI{16.26}{\joule}} \approx 10.7 \Rightarrow \textbf{10 discharges}
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\]
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\section*{Permanent TS Voltage}
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We can find the equilibrium temperature by finding the temperature at which the heat loss is equal to the power emitted.
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To find that, we first convert the graph provided in the datasheet (fig. \ref{fig:PTC_T_R}) to a Look Up Table (LUT), a \hyperref[py_script]{python script} is then created
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with the two function listed below to find the equilibrium point. \\
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($DF$: dissipation factor. For the PTC used: \SI{19.5}{\milli\watt/\kelvin}).
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\begin{align}
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(T_{eq} - T_{amb}) \cdot DF = & P_{dissipated} \\
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V_{TS} ^ 2 / R_{PTC} = & P_{created}
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\end{align}
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After the execution of the script, we can see that the power dissipation at equilibrium is about \SI{1.84}{\watt}.
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The equilibrium temperature and the corresponding resistance calculated is then \SI{139}{\celsius} and \SI{88.5}{\kilo\ohm} accordingly.
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We can see that this is smaller then the maximum temperature rated at \SI{165}{\celsius}.
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To find whether the MOSFET STB10LN80K5 can survive the permanent TS voltage, we first have to calculate the current going through it. \cite{mosfet_datasheet}
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\begin{align}
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I & = V/R = \SI{403.2}{\volt}/\SI{88.5}{\kilo\ohm} \\
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& = \SI{4.56}{\milli\ampere}
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\end{align}
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Since the MOSFET drain current $I_D$ is rated for 8A, it will work under permanent TS voltage.
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\newpage
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\section*{Python script}
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\label{py_script}
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\lstinputlisting[language=python]{./Documents/ptc.py}
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\bibliographystyle{plain}
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\newpage
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\renewcommand\refname{Reference}
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\begin{thebibliography}{00}
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\bibitem{emdriver_datasheet} \textit{emDrive HXXX Datasheet}. \href{https://www.emdrive-mobility.com/portfolio/emdrive-h100/}{www.emdrive-mobility.com}
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\bibitem{ptc_datasheet} \textit{Vishay PTCEL13R251NxE Datasheet}. \href{https://www.vishay.com/docs/29165/ptcel_series.pdf}{www.vishay.com}, 09.2024
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\bibitem{mosfet_datasheet} \textit{ST STB10LN80K5 Datasheet}. \href{https://www.st.com/resource/en/datasheet/stb10ln80k5.pdf}{www.st.com}, 02.2016
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\end{thebibliography}
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\end{document}
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