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\chapter{Theory}
\label{cha:theory}

This chapter introduces the theory and facts that are related to this project. It describes the necessary parts of the ISO standards, measurement theory and methods to analyse  acquired data.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Previous research}
No previous research directly relevant to the reuse of test equipment was found. Though research that has been made on topics relevant to project, such as measurement techniques and curve fitting, are presented in this theory chapter.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ISO standards}
The ISO organisation, International Organization for Standardization, was founded in 1947 and has since published more than 22,500 International Standards. ISO standards does not only cover the electronic industry, but almost every industry. The purpose of the standards is to ensure safety, reliability and quality of products in a unified way, making international trade easier. The name ISO comes from the Greek word \emph{isos}, which means \emph{equal}.
\cite{site:iso_about}

A standard is developed and maintained by a Technical Committee, TC. The TC consists of, amongst others, experts in the area that the standard concerns \cite{site:iso_who_develops_standards}. A new standard is only developed when there exists a need for this from the industry or other groups that may require it \cite{site:iso_developing_standards}. Existing standards are automatically scheduled for review five years after its last publication, but can manually be reviewed before that time by the committee \cite{iso_guidance_review}. During the review process it will be decided if the standard is still valid, need to be updated or if it should be removed \cite{iso_guidance_review}.

The naming convention used for ISO standards is in the format \emph{number-part:year}, where the \emph{number} is the identifier to the unique ISO standard, \emph{part} denotes the part of the standard if it is divided into several parts and \emph{year} is the publishing year. For example; the name \emph{ISO~7637-2:2011} refers to part 2 of the ISO~7637 standard published in 2011, whilst \emph{ISO~7637-2:2004} would refer to an earlier version of the exact same document published in 2004.

To get hold of a copy of a standard, one need to buy it from ISO store or from a national ISO member. \cite{site:iso_shopping_faqs}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ISO~7637 and ISO~16750}

\textbf{The ISO~7637 standard}, \emph{Road vehicles — Electrical disturbances from
conduction and coupling}, concerns the electrical environment in road vehicles. The standard consists of four parts, as of August 2019.

Part 1, \emph{Definitions and general considerations}, defines some abbreviations and technical terms that are used throughout the standard. It also intended use of the standard. \cite{iso_7637_1}

Part 2, \emph{Electrical transient conduction along supply lines only}, defines the test procedures related to disturbances that are carried along the supply lines of a product. Both emission, disturbances created by the DUT, and immunity, the DUT's capability to withstand disturbances, are covered. This part defines the test pulses that are of interest for this project, and the verification of them. \cite{iso_7637_2}

Part 3, \emph{Electrical transient transmission by capacitive and inductive coupling via lines other than supply lines}, defines immunity tests against disturbances on other interfaces that the power supply. It focuses on test setups and different ways of coupling the signals. \cite{iso_7637_3}

Part 5, \emph{Enhanced definitions and verification methods for harmonization of pulse generators according to ISO~7637}, proposes an alternative verification method of the test pulses defined in ISO~7637-2. The main difference from the method described in ISO~7637-2 is that the DC voltage, $U_A$, should not only be 0~V during the verification, but also be set to the nominal voltage, $U_N$. This will not be considered deeply in this report, since it is only a proposal and makes the verification equipment more difficult. \cite{iso_7637_5}

The ISO~16750, \emph{Road vehicles -- Environmental conditions and testing for electrical and electronic equipment}, concerns different environmental factors that a product might face in a vehicle, such as mechanical shocks, temperature changes and acids. Part 2 of the standard, \emph{Electrical Loads}, deals with some electrical aspects that was previously part of the ISO~7637 standard. This is the only part of ISO~16750 that will be considered.
\cite{iso_16750_1, iso_16750_2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Test pulses}

All test pulses defined in ISO~7637 and ISO~16750 are supposed to simulate events that can occur in a real vehicle's electrical environment, that equipment must be able to withstand. The properties of these test pulses are well defined, to allow for unified testing regardless of which test lab that performs the test. In the real world, however, the disturbances might of course differ from the test pulses since a real case environment is not controlled. \cite{iso_7637_2,iso_16750_2, comparison_iso_7637_real_world}

The test pulses of interest defined in ISO~7637 are denoted \emph{Test pulse 1}, \emph{Test pulse 2a}, \emph{Test pulse 3a} and \emph{Test pulse 3b}. The test pulse of interest defined in ISO~16750 is denoted \emph{Load dump Test A}. There are more pulses and tests defined in these standards, but those are not in the scope of this project.

The general characteristics in common for all pulses are the DC voltage $U_A$, the surge voltage $U_s$, the rise time $t_r$, the pulse duration $t_d$ and the internal resistance $R_i$. The property \emph{internal resistance} is only in series with the generated pulse, not in series with the DC power source. For pulses that are supposed to be applied several times, $t_1$ usually denotes the time between the start of two consecutive pulses. The timings are illustrated in  \autoref{fig:doubleexp}.

\begin{figure}[H]
	\centering
	\begin{subfigure}[t]{0.45\textwidth}
	    \includegraphics[width=\textwidth]{doubleexpfunc}
	    \caption{The surge voltage $U_S$ is the puse maximum voltage disregarding the offset voltage $U_A$. The rise $t_r$ time is defined as the time elapsed from 0.1 to 0.9 times the surge voltage on the rising edge of the pulse. The duration $t_d$  is defined as the time from 0.1 times the maximum voltage on the rising edge, back to the same level of the falling edge.}
   	    \label{fig:doubleexp}
	\end{subfigure}\hfill
	\begin{subfigure}[t]{0.45\textwidth}
	    \includegraphics[width=\textwidth]{doubleexpfuncrep}    
	    \caption{The repetition time is defined as the time between two adjacent rising edges.}
   	    \label{fig:doubleexprep}
	\end{subfigure}
	\caption{The common properties of the pulses, as defined by ISO~7637.}
\end{figure}

An important observation is that the definition of the surge voltage, $U_s$, differs in ISO~7637 and ISO~16750 as depicted in \autoref{fig:loadDumpTestA}.

%%%%%%%%%%%%%%%%%%%
\subsection{Test pulse 1}
This pulse simulates the event of the power supply being disconnected while the DUT is connected to other inductive loads. The other inductive loads will generate a voltage transient of reversed polarity onto the DUT's supply lines.

In the standard there are two additional timings associated to this pulse, $t_2$ and $t_3$, which are defining the disconnection time for the power supply during the voltage transient. In practice $t_3$ can be very short, specified to less than 100 µs, and the step seen in \autoref{fig:pulse1} might be too short to be clearly distinguishable when seen on a oscilloscope.

\begin{figure}[H]
    %\captionsetup{width=.5\linewidth}
    \centering
    \includegraphics[width=\textwidth]{pulse1}    
    \caption{Illustration of test pulse 1.}
    \label{fig:pulse1}
\end{figure}

\begin{table}[H]
    \centering
    \caption{Parameter values for pulse 1}
    \begin{tabularx}{0.7\textwidth}{|X|c|c|}
        \hline
        \textbf{Parameter}	&\textbf{\SI{12}{\volt} system}	&\textbf{\SI{24}{\volt} system}	\\
        \hline
        $U_A$			& \SIrange{13.8}{14.2}{\volt}	& \SIrange{27.8}{28.2}{\volt}	\\
        \hline
        $U_S$			& \SIrange{-75}{-150}{\volt}	& \SIrange{-300}{-600}{\volt}	\\
        \hline
        $R_i$				& \SI{10}{\ohm}				& \SI{50}{\ohm}				\\
        \hline
        $t_d$			& \SI{2}{\milli\second}			& \SI{1}{\milli\second}			\\
        \hline
        $t_r$				& \SIrange{0.5}{1}{\micro\second}	& \SIrange{1.5}{3}{\micro\second} \\
        \hline
        $t_1$ 			& \multicolumn{2}{c|}{$\geq$\SI{0.5}{\second}}				\\
        \hline
        $t_2$ 			& \multicolumn{2}{c|}{\SI{200}{\milli\second}} 				\\
        \hline
        $t_3$ 			& \multicolumn{2}{c|}{$<$\SI{100}{\micro\second}} 			\\
        \hline
    \end{tabularx}    
    \label{tab:pulse1}
\end{table}

%%%%%%%%%%%%%%%%%%%
\subsection{Test pulse 2a}
This pulse simulates the event of a load, parallel to the DUT, being disconnected. The inductance in the wiring harness will then generate a positive voltage transient on the DUT's supply lines. distinguishable when seen on a oscilloscope.

\begin{figure}[H]
    %\captionsetup{width=.5\linewidth}
    \centering
    \includegraphics[width=\textwidth]{pulse2a}    
    \caption{Illustration of test pulse 2a.}
    \label{fig:pulse2a}
\end{figure}

\begin{table}[H]
    \centering
    \caption{Parameter values for pulse 2a}
    \begin{tabularx}{0.7\textwidth}{|X|c|c|}
        \hline
        \textbf{Parameter}	&\textbf{\SI{12}{\volt} system}	&\textbf{\SI{24}{\volt} system}	\\
        \hline
        $U_A$			& \SIrange{13.8}{14.2}{\volt}	& \SIrange{27.8}{28.2}{\volt}	\\
        \hline
        $U_S$			&  \multicolumn{2}{c|}{\SIrange{37}{112}{\volt}}				\\
        \hline
        $R_i$				&  \multicolumn{2}{c|}{\SI{2}{\ohm}}						\\
        \hline
        $t_d$			&  \multicolumn{2}{c|}{\SI{0.05}{\milli\second}}				\\
        \hline
        $t_r$				&  \multicolumn{2}{c|}{\SIrange{0.5}{1}{\micro\second}}		\\
        \hline
        $t_1$ 			&  \multicolumn{2}{c|}{\SIrange{0.2}{5}{\second}}			\\
        \hline
    \end{tabularx}    
    \label{tab:pulse2a}
\end{table}

%%%%%%%%%%%%%%%%%%%
\subsection{Test pulse 3a and 3b}
Test pulse 3a and 3b simulates transients ``which occur as a result of the switching process'' as stated in the standard \cite{iso_7637_2}. The formulation is not very clear, but is interperted and explained by Frazier and Alles \cite{comparison_iso_7637_real_world} to be the result of a mechanical switch breaking an inductive load. These transients are very short, compared to the other pulses, and the repetition time is very short. The pulses are sent in bursts, grouping a number of pulses together and separating groups by a fixed time.

These pulses contain high frequency components, up to 100~MHz, and special care must be taken when running tests with them as well as when verifying them.


\begin{figure}[H]
	\centering
	\begin{subfigure}[t]{0.45\textwidth}
	    \includegraphics[width=\textwidth]{pulse3a}
	    \caption{Pulse 3a}
   	    \label{fig:pulse3a}
	\end{subfigure}\hfill
	\begin{subfigure}[t]{0.45\textwidth}
	    \includegraphics[width=\textwidth]{pulse3b}    
	    \caption{Pulse 3b}
   	    \label{fig:pulse3b}
	\end{subfigure}
	\caption{Pulse 3a and 3b are applied in bursts. Each individual pulse is a double exponential curve with the same properties, $t_r$ and $t_d$, as e.g. pulse 2a}
	\label{fig:pulse3}
\end{figure}

\begin{table}[H]
    \centering
    \caption{Parameter values for pulse 3a and 3b}
    \begin{tabularx}{0.7\textwidth}{|X|c|c|}
        \hline
        \textbf{Parameter}	&\textbf{\SI{12}{\volt} system}	&\textbf{\SI{24}{\volt} system}	\\
        \hline
        $U_A$			& \SIrange{13.8}{14.2}{\volt}	& \SIrange{27.8}{28.2}{\volt}	\\
        \hline
        Pulse 3a $U_S$		& \SIrange{-112}{-220}{\volt}	& \SIrange{-150}{-300}{\volt}	\\
        \hline
        Pulse 3b $U_S$		& \SIrange{75}{150}{\volt}		& \SIrange{150}{300}{\volt}	\\
        \hline
        $R_i$				&  \multicolumn{2}{c|}{\SI{50}{\ohm}}						\\
        \hline
        $t_d$			&  \multicolumn{2}{c|}{\SIrange{105}{195}{\nano\second}}		\\
        \hline
        $t_r$				&  \multicolumn{2}{c|}{\SIrange{3.5}{6.5}{\nano\second}}		\\
        \hline
        $t_1$			&  \multicolumn{2}{c|}{\SI{100}{\micro\second}}				\\
        \hline
        $t_4$			&  \multicolumn{2}{c|}{\SI{10}{\milli\second}}				\\
        \hline
        $t_5$			&  \multicolumn{2}{c|}{\SI{90}{\milli\second}}				\\
        \hline
    \end{tabularx}    
    \label{tab:pulse3}
\end{table}

%%%%%%%%%%%%%%%%%%%
\subsection{Load dump Test A}
The Load dump Test A simulates the event of disconnecting a battery that is charged by the vehicles alternator, the current that the alternator is driving will give rise to a long voltage transient.

This pulse has the longest duration, $t_d$, of all the test pulses. It also has the lowest internal resistance. These properties makes it capable of transferring high energies into a low impedance DUT or dummy load.

Prior to 2011, the Load dump Test A was part of the ISO~7637-2 standard under the name \emph{Test pulse 5a}. The surge voltage $U_s$ was in the older standard, \mbox{ISO~7637-2:2004}, defined as the voltage between the DC offset voltage $U_A$ and the maximum voltage. In the newer standard, \mbox{ISO~16750-2:2012}, $U_s$ is defined as the absolute peak voltage. Only the former definition is used in this paper, $U_s = \hat{U} - U_A$.

\begin{figure}[H]
    %\captionsetup{width=.5\linewidth}
    \centering
    \includegraphics[width=\textwidth]{load dump a}
    \caption{Illustration of load dump Test A. Note the different definition of $U_S$ compared to the other pulses.}
    \label{fig:loadDumpTestA}
\end{figure}

\begin{table}[H]
    \centering
    \caption{Parameter values for load dump Test A}
    \begin{tabularx}{0.7\textwidth}{|X|c|c|}
        \hline
        \textbf{Parameter}	&\textbf{\SI{12}{\volt} system}		&\textbf{\SI{24}{\volt} system}		\\
        \hline
        $U_A$			& \SIrange{13.8}{14.2}{\volt}		& \SIrange{27.8}{28.2}{\volt}		\\
        \hline
        $U_S$ ISO~16750	& \SIrange{79}{101}{\volt}			& \SIrange{151}{202}{\volt}		\\	
        \hline
        $U_S$ ISO~7637	& \SIrange{64.8}{87.2}{\volt}		& \SIrange{122.8}{174.2}{\volt}		\\
        \hline
        $R_i$				& \SIrange{0.5}{4}{\ohm}			& \SIrange{1}{8}{\ohm}			\\
        \hline
        $t_d$			& \SIrange{40}{400}{\milli\second}	& \SIrange{100}{350}{\milli\second}	\\
        \hline
        $t_r$				& \multicolumn{2}{c|}{\SIrange{5}{10}{\milli\second}}					\\
        \hline
    \end{tabularx}    
    \label{tab:loadDumpTestA}
\end{table}

%%%%%%%%%%%%%%%%%%%
\subsection{Application of test pulses}
During a test, the nominal voltage is first applied between the plus and minus terminal of the DUT's power supply input by the test equipment. Then a series of test pulses are applied between the same terminals. The pulses are repeated at specified intervals, $t_1$, as depicted in \autoref{fig:doubleexprep}.

An example of how a test pulse can be applied by the test equipment is depicted in \autoref{fig:test_equipment_setup}.

\begin{figure}[H]
    %\captionsetup{width=.5\linewidth}
    \centering
    \includegraphics[width=\textwidth]{test_equipment_setup}
    \caption{Illustration of how the test equipment can apply a test pulse to the DUT whilst also providing the DC supply throuht an external PSU.}
    \label{fig:test_equipment_setup}
\end{figure}

%%%%%%%%%%%%%%%%%%%
\subsection{Verification}
The test pulses are to be verified before they are applied to the DUT. The voltage levels and the timings are to be measured both without any load, and with a matched load, $R_L$ = $R_i$, attached. The standard omits the rise time constraint when the load is attached, except for pulse 3a and 3b. \cite{iso_7637_2}

The verification is to be conducted with $U_A$ set to 0. There is, however, a proposal to set $U_A$ equal to the nominal voltage during the verification process, as the behaviour of the pulse generators has proven differ in this case \cite{iso_7637_5}. In this project $U_A = 0$ will be used.

The limits, and tolerances, for the pulses are summarised in \autoref{tab:verification-list}. The matched loads are to be within 1\% of the nominal value.

The instruments used for measuring the pulses must have at least \SI{400}{\mega\hertz}, since pulse 3a and 3b contains frequency components of up to \SI{200}{\mega\hertz}.

\begin{table}[H]
    \caption{These are all of the verifications that needs to be made before each use of the equipment, along with the limits for each case.}
\begin{adjustbox}{width=\columnwidth,center}
    %\centering
    \begin{tabular}{|l|r|r|r|r|} 
        \hline
         & & \multicolumn{3}{c|}{Limits}\\
        Pulse & Match resistor & $U_S$ & $t_d$ & $t_r$ \\
        \hline
        Pulse 1, 12 V, Open        &              & \SIrange{-110}{ -90}{\volt} & \SIrange{1.6}{2.4}{\milli\second} & \SIrange{0.5}{  1}{\micro\second} \\
        Pulse 1, 12 V, Matched     & 10 \si{\ohm} & \SIrange{-110}{ -90}{\volt} & \SIrange{1.6}{2.4}{\milli\second} & \SIrange{0.5}{  1}{\micro\second} \\
        Pulse 1, 24 V, Open        &              & \SIrange{-660}{-540}{\volt} & \SIrange{0.8}{1.2}{\milli\second} & \SIrange{1.5}{  3}{\micro\second} \\
        Pulse 1, 24 V, Matched     & 50 \si{\ohm} & \SIrange{-660}{-540}{\volt} & \SIrange{0.8}{1.2}{\milli\second} & \SIrange{1.5}{  3}{\micro\second} \\
        Pulse 2a, Open             &              & \SIrange{67.5}{82.5}{\volt} & \SIrange{ 40}{ 60}{\micro\second} & \SIrange{0.5}{  1}{\micro\second} \\
        Pulse 2a, Matched          &  2 \si{\ohm} & \SIrange{67.5}{82.5}{\volt} & \SIrange{ 40}{ 60}{\micro\second} & \SIrange{0.5}{  1}{\micro\second} \\
        Pulse 3a, Open (1k)        &              & \SIrange{-220}{-180}{\volt} & \SIrange{105}{195}{\nano\second}  & \SIrange{3.5}{6.5}{\nano\second}  \\
        Pulse 3a, Match            & 50 \si{\ohm} & \SIrange{-120}{ -80}{\volt} & \SIrange{105}{195}{\nano\second}  & \SIrange{3.5}{6.5}{\nano\second}  \\
        Pulse 3b, Open (1k)        &              & \SIrange{ 180}{ 220}{\volt} & \SIrange{105}{195}{\nano\second}  & \SIrange{3.5}{6.5}{\nano\second}  \\
        Pulse 3b, Match            & 50 \si{\ohm} & \SIrange{  80}{ 120}{\volt} & \SIrange{105}{195}{\nano\second}  & \SIrange{3.5}{6.5}{\nano\second}  \\
        Load dump A, 12 V, Open    &              & \SIrange{  90}{ 110}{\volt} & \SIrange{320}{480}{\milli\second} & \SIrange{  5}{ 10}{\milli\second} \\
        Load dump A, 12 V, Matched &  2 \si{\ohm} & \SIrange{  90}{ 110}{\volt} & \SIrange{320}{480}{\milli\second} & \SIrange{  5}{ 10}{\milli\second} \\
        Load dump A, 24 V, Open    &              & \SIrange{ 180}{ 220}{\volt} & \SIrange{280}{420}{\milli\second} & \SIrange{  5}{ 10}{\milli\second} \\
        Load dump A, 24 V, Matched &  2 \si{\ohm} & \SIrange{ 180}{ 220}{\volt} & \SIrange{280}{420}{\milli\second} & \SIrange{  5}{ 10}{\milli\second} \\
        \hline
    \end{tabular}
\end{adjustbox}
    \label{tab:verification-list}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Resistors at high frequencies}
\label{sec:theory:resistors_at_high_frequencies}

When working with resistors at high frequencies, one must consider the parasitc properties of the resistor. Vishay presents a model which consists of the resistance $R$, internal inductance $L$, internal capacitance $C$, external lead inductance $L_C$ and external ground capacitance $C_G$. Since the external ground capacitance is very small in comparison to the other parasitics, it has been neglected in this thesis. The model used for the simulations is depicted in \autoref{fig:nonIdealResistor}, with the values $L = \SI{0.1}{\nano\henry}$, $C = \SI{1}{\pico\farad}$ and $L_C = \SI{1}{\nano\henry}$. This is a bit higher than the values in Vishays paper, but those are also for smaller packages. \cite{vishay_hf_resistor} An approximation of the combined inductance of more than \SI{1}{\nano\henry} for the 1206 package is also in line with the values in a technical information note from AVX for capacitors, the package lead inductance should be similar for capacitors and resistors\cite{avx_cap_parasitic}.

\begin{figure}[H]
    \includegraphics[width=0.5\textwidth]{nonIdealResistor}    
    \caption{At high frequencies a resistors parasitic inductance and capacitance will affect the behavior of the circuit. This is the model used in this thesis when simulating circuits.}
    \label{fig:nonIdealResistor}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Measurement}
There are several measurement methods needed during the project. To verify the test pulses, voltage has to be measured over time. To verify the dummy loads, resistance has to be measured. To verify the attenuators, their magnitude response has to be measured. This chapter describes the necessary measurement theory required for this project.

%%%%%%%%%%%%%%%%%%%
\subsection{Resistance}
\label{sec:measurement:resistance}
To measure resistance, a current is fed through the resistor and the resulting voltage is measured to calculate the resistance using ohms law. This is typically carried out using a multimeter and two probe wires connecting to each terminal of the resistor. When measuring very low valued resistors, however, the resistance in the probe wires can be significant in relation to the resistor measured and will affect the accuracy. One way of overcoming this is to perform a 4-wire measurement using a so called \emph{Kelvin connection}. In this method the current that is fed through the resistor using one pair of wire, and the resulting voltage is measured at the desired point using another pair according to \autoref{fig:kelvin_measurement}.\cite{theCircuitDesignersCompanion}

\begin{figure}[H]
    %\captionsetup{width=.5\linewidth}
    \includegraphics[width=0.5\textwidth]{kelvin_measurement}    
    \caption{When measuring a low value resistor, the \emph{Kelvin connection} can be used to determine the resistance at the point where the voltmeter is connected without the resistance in the probe leads affecting the result.}
    \label{fig:kelvin_measurement}
\end{figure}

%%%%%%%%%%%%%%%%%%%
\subsection{Oscilloscopes, bandwidth, rise time and probes}
When using an oscilloscope to measure voltage over time, there are several limiting factors to how fast signals one can measure. The oscilloscope itself has a specified bandwidth, as do the probe and any attenuators used. All of these combined determine how short rise times that can be measured accurately. The rise time of the measured will be affected by these properties and the rise time displayed on the oscilloscope screen will be approximately according to \autoref{equ:riseComposite}, where $T_N$ is the \SIrange{10}{90}{\percent} rise time limit for each part in the chain. \cite{highSpeedDigitalDesign}

\begin{equation}
\label{equ:riseComposite}
T_{rise~composite} = \sqrt{ T_1^2 + T_2^2 + ... + T_N^2}
\end{equation}

Since \autoref{equ:riseComposite} is based on the rise time limitation but the specification usually tells the \SI{3}{\deci\bel} bandwidth, a conversion can be made according to \autoref{equ:bwToRise}. \cite{highSpeedDigitalDesign}

\begin{equation}
\label{equ:bwToRise}
T_{10-90} = \frac{0.338}{F_{ \SI{3}{\deci\bel}}}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Analysis}
The data points from the measurement must be processed and evaluated to determine if the measured pulse is within the specified limits.

%%%%%%%%%%%%%%%%%%%
\subsection{Mathematical description}
All of the test pulses applied to the vehicle equipment can individually be described mathematically by variations of the double exponential function shown in \autoref{eq:doubleexp}. The properties of interest, the ones which are specified in the standards, are the surge voltage $ U_s $, the rise time $ t_r $, the duration $ t_d $ and the repetition time $ t_1 $. \cite{iso_7637_2}

\begin{equation}
    u(t)=k(e^{\alpha t} - e^{\beta t}) + U_{A}
    \label{eq:doubleexp}
\end{equation}

It is not in the scope of this report to actually fit this function to the measured pulse, and further analyze it.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Instrumentation and control}
The following chapter describes the different instruments that were used, and their control interfaces.

%%%%%%%%%%%%%%%%%%%
\subsection{GPIB}
IEEE-488, or GPIB which it is often called, is a parallel bus interface. It is mainly used to interconnect lab instrumentation such as multimeters, signal generators and spectrum analyzers.

%%%%%%%%%%%%%%%%%%%
\subsection{Tektronix TDS7104 Oscilloscope}
The oscilloscope that is available is a Tektronix TDS7104, with specifications as seen in \autoref{tab:tds7104}. It has GPIB interface and TekVISA GPIB, an API for sending GPIB commands over ethernet, available for remote control. \todo[Lägg till specs eller ta bort helt]

%%%%%%%%%%%%%%%%%%%
\subsection{xxxxx Isolated differential probe}
\todo[Peta in kort beskrivning av prob och dess data]
\label{sec:hv-diff-probe}

%%%%%%%%%%%%%%%%%%%
\subsection{EM Test MPG 200 Micropulse generator}
The MPG~200 is used to generate \emph{Test pulse 1} and \emph{2a}. MPG is an abbreviation for \emph{MicroPulse Generator}. The instrument is designed to generate test pulses according to the older ISO~7637-2:1990 version, but the parameters can be adjusted to comply with the new ISO~7637:1990 standard. The adjustable parameter ranges are shown in \autoref{tab:mpg200_specs}. The instrumentation panels can be seen in \autoref{fig:mpg200}.

\begin{table}[H]
    \caption{Adjustable parameters in the MPG 200}
\begin{adjustbox}{center}
    %\centering
    \begin{tabular}{|l|r|} 
        \hline
        Parameter & Range \\
        \hline
        $U_S$  & \SIrange{20}{600}{\volt} \\
        $U_S$ polarity & +, - \\
        $R_s$  & \SIlist{2;4;10;20;30;50}{\ohm}  \\
        $t_1$  & \SIrange{0.2}{99.0}{\second} \\
        $t_2$  & \SIrange{0}{10}{\second} \\
        \hline
    \end{tabular}
\end{adjustbox}
    \label{tab:mpg200_specs}
\end{table}


\begin{figure}
	\centering
	\begin{subfigure}{0.7\textwidth}
		\includegraphics[width=\textwidth]{mpg200-front}
		\caption{Front.}
   	    \label{fig:mpg200-front}
	\end{subfigure}
	
	\begin{subfigure}{0.7\textwidth}
		\includegraphics[width=\textwidth]{mpg200-back}
		\caption{Back.}
   	    \label{fig:mpg200-back}
	\end{subfigure}
	\caption{The MPG~200 is used to generate test pulse 1 and 2a.}
   	\label{fig:mpg200}
\end{figure}

%%%%%%%%%%%%%%%%%%%
\subsection{EM Test EFT 200 Burst generator}
The EFT~200 is used to generate \emph{Test pulse 3a} and \emph{3b}. EFT is an abbreviation for \emph{Electrical Fast Transient}. The instrument is designed to generate test pulses according to the older ISO~7637-2:1990 version, but the parameters can be adjusted to comply with the new ISO~7637:1990 standard. The adjustable parameter ranges are shown in \autoref{tab:eft200_specs}. The instrumentation panels can be seen in \autoref{fig:eft200}.

\begin{table}[H]
    \caption{Adjustable parameters in the EFT 200}
\begin{adjustbox}{center}
    %\centering
    \begin{tabular}{|l|r|} 
        \hline
        Parameter & Range \\
        \hline
        $U_S$  & \SIrange{25}{1500}{\volt} \\
        $U_S$ polarity & +, - \\
        Coupling & any combination of +, - and GND \\
        \hline
    \end{tabular}
\end{adjustbox}
    \label{tab:eft200_specs}
\end{table}


\begin{figure}
	\centering
	\begin{subfigure}{0.7\textwidth}
		\includegraphics[width=\textwidth]{eft200-front}
		\caption{Front.}
   	    \label{fig:eft200-front}
	\end{subfigure}
	
	\begin{subfigure}{0.7\textwidth}
		\includegraphics[width=\textwidth]{eft200-back}
		\caption{Back.}
   	    \label{fig:eft200-back}
	\end{subfigure}
	\caption{The EFT~200 is used to generate test pulse 3a and 3b.}
   	\label{fig:eft200}
\end{figure}

%%%%%%%%%%%%%%%%%%%
\subsection{EM Test LD 200 Load dump}
The LD~200 is used to generate \emph{Load dump Test A}. LD is an abbreviation for \emph{Load Dump}. The instrument is designed to generate test pulses according to the older ISO~7637-2:1990 version, but the parameters can be adjusted to comply with the new ISO~16750:2012 standard. The adjustable parameter ranges are shown in \autoref{tab:ld200_specs}. The instrumentation panels can be seen in \autoref{fig:ld200}.

\begin{table}[H]
    \caption{Adjustable parameters in the LD 200}
\begin{adjustbox}{center}
    %\centering
    \begin{tabular}{|l|r|} 
        \hline
        Parameter & Range \\
        \hline
        $U_S$  & \SIrange{20}{200}{\volt} \\
        $R_s$  & \SIlist{0.5;1;2;10}{\ohm}  \\
        $t_d$  & \SIrange{50}{400}{\milli\second} \\
        \hline
    \end{tabular}
\end{adjustbox}
    \label{tab:ld200_specs}
\end{table}


\begin{figure}
	\centering
	\begin{subfigure}{0.7\textwidth}
		\includegraphics[width=\textwidth]{ld200-front}
		\caption{Front.}
   	    \label{fig:ld200-front}
	\end{subfigure}
	
	\begin{subfigure}{0.7\textwidth}
		\includegraphics[width=\textwidth]{ld200-back}
		\caption{Back.}
   	    \label{fig:ld200-back}
	\end{subfigure}
	\caption{The LD~200 is used to generate load dump test a.}
    \label{fig:ld200}
\end{figure}

%%%%%%%%%%%%%%%%%%%
\subsection{EM Test CNA 200 Coupling Network}
The SNA~200 is a coupling network used to multiplex the pulse generators outputs. It contains several relays to select the appropriate generator output. The SNA~200 has one interface for each pulse generator, but no interface for a computer. It is automatically controlled by the pulse generators. This allows the DUT to be connected only to the CNA~200 and not to each individual pulse generator. \autoref{fig:test_setup_cna_dut} shows the connections between the instruments in this setup. There is also a coaxial connection for calibration of pulse 3a and pulse 3b on the front panel. The instrumentation panels can be seen in \autoref{fig:cna200}.

\begin{figure}[H]
    %\captionsetup{width=.5\linewidth}
    \includegraphics[width=0.5\textwidth]{test setup pulse injection}    
    \caption{The CNA~200 allows each pusle generator to output their pulses through a common interface towards the DUT.}
    \label{fig:test_setup_cna_dut}
\end{figure}


\begin{figure}
	\centering
	\begin{subfigure}{0.7\textwidth}
		\includegraphics[width=\textwidth]{cna200-front}
		\caption{Front.}
   	    \label{fig:cna200-front}
	\end{subfigure}
	
	\begin{subfigure}{0.7\textwidth}
		\includegraphics[width=\textwidth]{cna200-back}
		\caption{Back.}
   	    \label{fig:cna200-back}
	\end{subfigure}
	\caption{The CNA~200 is used to couple all of the other pulse generators outputs to a common output.}
   	\label{fig:cna200}
\end{figure}


%%%%%%%%%%%%%%%%%%%
\subsection{Rohde \& Schwarz ZVL13}
\label{sec:rohde_schwarz_zvl}
The ZVL13 is a vector network analyzer that operates in the frequency range \SI{9}{\kilo\hertz} to \SI{13.6}{\giga\hertz}. It is, in this project, used to measure the magnitude and phase response between its two ports.

%%%%%%%%%%%%%%%%%%%
\subsection{PAT 50 and PAT 1000}
\label{sec:hv-attenuators}
\todo[Skriv någonting bättre här!]



\begin{figure}[H]
	\centering
    %\captionsetup{width=.5\linewidth}
    \includegraphics[width=0.5\textwidth]{pat_attenuators}    
    \caption{The two attenuators that were used in the project.}
    \label{fig:pat_attenuators}
\end{figure}

