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finished up the TS-DC-schematic

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TS-DC-schematic/TS-DC-schematic.tex
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\documentclass[paper=A4]{article}
\usepackage[utf8]{inputenc}
\usepackage[a4paper, left=2cm, rightr2cm, top=2cm, bottom=2cm]{geometry}
\usepackage[a4paper, left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}
\usepackage{siunitx}
\sisetup{
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\usepackage{hyperref}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{eso-pic}
\definecolor{codegreen}{rgb}{0,0.6,0.4}
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
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\rhead{\includegraphics*[scale=0.013]{./Pictures/FaSTTUBe_Logo_ohneAuto.png}}
\rfoot{Page \thepage \hspace{1pt} of \pageref{LastPage}}
\lhead{Car 313, 23.04, Rev. 1}
\lhead{Car 313, 01.05, Rev. 1}
\chead{\large TS Discharge Circuit Schematic}
\begin{document}
@ -66,50 +67,73 @@
\caption{Schematic of the Discharge Circuit PCB}
\end{figure}
% DC Highlighting
\AddToShipoutPicture*{
\begin{tikzpicture}[remember picture, overlay]
% Draw a yellow semi-transparent rectangle
\fill[yellow, opacity=0.4] ([xshift=3.1cm,yshift=6.8cm]current page.center) rectangle ++(1.37cm,1.86cm);
\fill[yellow, opacity=0.4] ([xshift=4.88cm,yshift=6.8cm]current page.center) rectangle ++(1.25cm,1.86cm);
\fill[yellow, opacity=0.4] ([xshift=4.02cm,yshift=6.37cm]current page.center) rectangle ++(1.39cm,1.99cm);
\fill[yellow, opacity=0.4] ([xshift=5.85cm,yshift=6.37cm]current page.center) rectangle ++(1.34cm,1.99cm);
\end{tikzpicture}
}
\section*{Discharge Time}
As seen in the Schematic, for our discharge circuitry a PTC (PTCEL13R251NxE) is used. 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}, and the maximum voltage of the accumulator is 403.2V. 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.
As seen in the Schematic, for our discharge circuitry a PTC (PTCEL13R251NxE\cite{ptc_datasheet}) is used.
The total capacitance of the DC-link capacitor from the two inverters (Emsiso emDrive H100\cite{emdriver_datasheet}) that we are using is about \SI{200}{\micro\farad},
and the maximum voltage of the accumulator is 403.2V. 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.
\begin{align}
V_C = & V_0 \cdot e^{-t/RC} \\
\SI{60}{\volt} = & \SI{403.2}{\volt} \cdot e^{-\SI{5}{\second}/(R_{PTC} \cdot \SI{200}{\micro\farad})} \\
R_{PTC} \approx & \SI{13123}{\ohm}
V_C &= V_0 \cdot e^{-t/RC} \\
\SI{60}{\volt} &= \SI{403.2}{\volt} \cdot e^{-\SI{5}{\second}/(R_{PTC} \cdot \SI{200}{\micro\farad})} \\
R_{PTC} &\approx \SI{13123}{\ohm}
\end{align}
\begin{wrapfigure}{r}{0.4\textwidth}
\includegraphics[width=1\linewidth]{./Pictures/PTC-R-T.png}
\includegraphics[width=\linewidth]{./Pictures/PTC-R-T.png}
\caption{Resistance vs. Temperature for PTCEL13 (typical)}
\label{fig:PTC_T_R}
\end{wrapfigure}
To Calculate how much discharge attempts we can have before the discharge reaches 5s, we first see at what temperature the PTC the resistance of \SI{13123}{\ohm} have. We can get this from the datasheet \cite{ptc_datasheet} of the PTC (fig. \ref{fig:PTC_T_R}).
To calculate how many discharge attempts can be made before the discharge time exceeds \SI{5}{\second},
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}.
As seen in the datasheet, the temperature is about \SI{165}{\celsius}. Assuming that the temperature rises instantly after the discharge, and the heat dissipation is negligible (since the thermal time constant $\tau_{th}$ is \SI{130}{\second}). As we are calculating the lest amount of precharge allowed, we assume the temperature of the environment to be \SI{45}{\celsius}. Since the thermal capacity $C_{th}$ is \SI{1.45}{\joule/\kelvin}, we can see that the total energy that we can generate from discharge is $E = \SI{120}{\kelvin} \cdot \SI{1.45}{\joule/\kelvin} = \SI{174}{\joule}$.
From the graph, the corresponding temperature is approximately \SI{165}{\celsius}.
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}).
We then calculate how much energy in produced in one discharge:
To determine the maximum allowable thermal energy before the PTC cools down, we assume an ambient temperature of \SI{45}{\celsius}.
Given the thermal capacity $C_{th} = \SI{1.45}{\joule\per\kelvin}$, the maximum thermal energy that can be absorbed is:
\[
E = \Delta T \cdot C_{th} = (\SI{165}{\celsius} - \SI{45}{\celsius}) \cdot \SI{1.45}{\joule\per\kelvin} = \SI{174}{\joule}
\]
\newpage
Next, we calculate the energy dissipated in one discharge:
\begin{align}
E = & \frac{1}{2} \cdot C \cdot V^2 \\
= & \SI{16.26}{\joule}
E &= \frac{1}{2} \cdot C \cdot V^2 \\
&= \frac{1}{2} \cdot \SI{200}{\micro\farad} \cdot (\SI{403.2}{\volt})^2 = \SI{16.26}{\joule}
\end{align}
Therefore, the total amount of discharge we can carry out before the time exceeds \SI{5}{\second} is $\SI{174}{\joule} / \SI{16.26}{\joule} = 10.7$ --- ten times.
Therefore, the number of discharges possible before the discharge time exceeds \SI{5}{\second} is:
\[
\frac{\SI{174}{\joule}}{\SI{16.26}{\joule}} \approx 10.7 \Rightarrow \textbf{10 discharges}
\]
\section*{Permanent TS Voltage}
We can find the equilibrium temperature by finding the temperature at which the heat loss is equal to the power emitted. 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 python script (listed below) is then created with the two function listed below to find the equilibrium point. (Here is the $DF$ the dissipation factor, and with this PTC it is \SI{19.5}{\milli\watt/\kelvin}).
We can find the equilibrium temperature by finding the temperature at which the heat loss is equal to the power emitted.
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
with the two function listed below to find the equilibrium point. \\
($DF$: dissipation factor. For the used PTC: \SI{19.5}{\milli\watt/\kelvin}).
\begin{align}
(T_{eq} - T_{amb}) \cdot DF = & P_{dissipated} \\
V_{TS} ^ 2 / R_{PTC} = & P_{created}
\end{align}
After the execution of the script, we can see that the power dissipation at equilibrium is about \SI{1.84}{\watt}. The equilibrium temperature and the corresponding resistance calculated is then \SI{139}{\celsius} and \SI{88.5}{\kilo\ohm} accordingly. We can see that this is smaller then the maximum temperature rated at \SI{165}{\celsius}.
After the execution of the script, we can see that the power dissipation at equilibrium is about \SI{1.84}{\watt}.
The equilibrium temperature and the corresponding resistance calculated is then \SI{139}{\celsius} and \SI{88.5}{\kilo\ohm} accordingly.
We can see that this is smaller then the maximum temperature rated at \SI{165}{\celsius}.
To find whether the MOSFET STB10LN80K5 can survive the permanent TS voltage, we first have to calculate the current going through it.
@ -120,16 +144,20 @@ To find whether the MOSFET STB10LN80K5 can survive the permanent TS voltage, we
Since the MOSFET drain current $I_D$ is rated for 8A, it will work under permanent TS voltage. \cite{mosfet_datasheet}
\newpage
\section*{Python script}
\label{py_script}
\lstinputlisting[language=python]{./Documents/ptc.py}
\bibliographystyle{plain}
\newpage
\renewcommand\refname{Reference}
\begin{thebibliography}{00}
\bibitem{ptc_datasheet} \textit{Vishay PTCEL13R251NxE}. \href{https://www.vishay.com/docs/29165/ptcel_series.pdf}{https://www.vishay.com/docs/29165/ptcel\_series.pdf}, 09.2024
\bibitem{mosfet_datasheet} \textit{ST STB10LN80K5}. \href{https://www.st.com/resource/en/datasheet/stb10ln80k5.pdf}{https://www.st.com/resource/en/datasheet/stb10ln80k5.pdf}, 02.2016
\bibitem{emdriver_datasheet} \textit{emDrive HXXX Datasheet}. \href{https://emsisosi.sharepoint.com/sites/EMSISOWebsite/Shared%20Documents/Forms/AllItems.aspx?id=%2Fsites%2FEMSISOWebsite%2FShared%20Documents%2FGeneral%2FemDrive%20Controllers%20web%20data%2FUser%20Manual%2FemDrive%20User%20Manual%5Fv2%5F3%2Epdf&parent=%2Fsites%2FEMSISOWebsite%2FShared%20Documents%2FGeneral%2FemDrive%20Controllers%20web%20data%2FUser%20Manual&p=true&ga=1}{emsisosi.sharepoint.com}
\bibitem{ptc_datasheet} \textit{Vishay PTCEL13R251NxE Datasheet}. \href{https://www.vishay.com/docs/29165/ptcel_series.pdf}{www.vishay.com}, 09.2024
\bibitem{mosfet_datasheet} \textit{ST STB10LN80K5 Datasheet}. \href{https://www.st.com/resource/en/datasheet/stb10ln80k5.pdf}{www.st.com}, 02.2016
\end{thebibliography}