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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 internal resistance is only in series with the pulse generator, not 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{doubleexpfunc}.


\begin{figure}
	\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, measured at 0.1 times the maximum pulse voltage.}
   	    \label{fig:doubleexprep}
	\end{subfigure}
	\caption{The common properties of the pulses.}
\end{figure}

\todo[fixa referens till rätt kapitel]
An important observation is that the definition of the surge voltage, $U_s$, differs in ISO~7637 and ISO~16750 as seen in \autoref{sec:us_difference}. \hl{In this report, only the definition from ISO~7637 is used.}

\subsection{Test pulse 1}
This pulse simulates the event of the power supply being disconnected while the DUT is connected to other inductive loads. This leads to the other inductive loads generating a voltage transient of reversed polarity to 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.

\todo[Två bilder, en på kurvan och en på kretsen som orsakar den. En tabell med parametervärden.]

\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.

\todo[Två bilder, en på kurvan och en på kretsen som orsakar den. En tabell med parametervärden.]

\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.

\todo[Fyra bilder, två på kurvorna, en på burst och en på kretsen som orsakar den? En tabell med parametervärden.]

\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, $U_s = \hat{U} - U_A$.

\todo[Två bilder, en på kurvan och en på kretsen som orsakar den. En tabell med parametervärden.]

\subsection{Test setup}
During a test, the nominal voltage is first applied between the plus and minus terminal of the DUT's power supply input. 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}.
\todo[Snygg bild på uppkoppling och kopplingsnätverk.].

\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, with and without a matched load attached $R_L$ = $R_i$. 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 $U_A$ to the actual 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. \cite{iso_7637_2}

\begin{table}[h]
\begin{adjustbox}{width=\columnwidth,center}
    %\centering
    \begin{tabular}{|l|r|r|r|r|} 
        \hline
         & & \multicolumn{3}{c|}{Limits}\\
        Pulse & Match resistor (\si{\ohm}) & $U_S$ (\si{\volt}) & $t_d$ (\si{\second}) & $t_r$ (\si{\second}) \\
        \hline
        Pulse 1, 12 V, Open         &    & $[ -110, -90 ]$   & $[1.6,2.4]$ \si{\milli}   & $[0.5,1]$ \si{\micro}     \\
        Pulse 1, 12 V, Matched      & 10 & $[ -110, -90 ]$   & $[1.6,2.4]$ \si{\milli}   & $[0.5,1]$ \si{\micro}     \\
        Pulse 1, 24 V, Open         &    & $[ -660, -540 ]$  & $[0.8,1.2]$ \si{\milli}   & $[1.5,3]$ \si{\micro}     \\
        Pulse 1, 24 V, Matched      & 50 & $[ -660, -540 ]$  & $[0.8,1.2]$ \si{\milli}   & $[1.5,3]$ \si{\micro}     \\
        Pulse 2a, Open              &    & $[ 67.5, 82.5 ]$  & $[40,60]$ \si{\micro}     & $[0.5,1]$ \si{\micro}     \\
        Pulse 2a, Matched           &  2 & $[ 67.5, 82.5 ]$  & $[40,60]$ \si{\micro}     & $[0.5,1]$ \si{\micro}     \\
        Pulse 3a, Open (1k)         &    & $[ -220, -180 ]$  & $[105,195]$ \si{\nano}    & $[3.5,6.5]$ \si{\nano}    \\
        Pulse 3a, Match             & 50 & $[ -120, -80 ]$   & $[105,195]$ \si{\nano}    & $[3.5,6.5]$ \si{\nano}    \\
        Pulse 3b, Open (1k)         &    & $[ 180, 220 ]$    & $[105,195]$ \si{\nano}    & $[3.5,6.5]$ \si{\nano}    \\
        Pulse 3b, Match             & 50 & $[ 80, 120 ]$     & $[105,195]$ \si{\nano}    & $[3.5,6.5]$ \si{\nano}    \\
        Load dump A, 12 V, Open     &    & $[ 90, 110 ]$     & $[320,480]$ \si{\milli}   & $[5,10]$ \si{\milli}      \\
        Load dump A, 12 V, Matched  &  2 & $[ 90, 110 ]$     & $[320,480]$ \si{\milli}   & $[5,10]$ \si{\milli}      \\
        Load dump A, 24 V, Open     &    & $[ 180, 220 ]$    & $[280,420]$ \si{\milli}   & $[5,10]$ \si{\milli}      \\
        Load dump A, 24 V, Matched  &  2 & $[ 180, 220 ]$    & $[280,420]$ \si{\milli}   & $[5,10]$ \si{\milli}      \\
        \hline
    \end{tabular}
\end{adjustbox}
    \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.}
    \label{tab:verification-list}
\end{table}

\section{Differences between the new and the old standard}
Since the equipment used the project is designed for the older version of the standard, ISO~7637-2:2004 and possibly even ISO~7637-1:1990 together with ISO~7637-2:1990, the differences of importance between these will be presented in this chapter to see what parameters might be a problem for the older equipment to fulfil.

One of the most notable differences is the removal of a test pulse from ISO~7637-2 that was called \emph{Pulse 5a} and \emph{Pulse 5b}, this was instead introduced to the ISO~16750-2 under the name \emph{Load dump A} and \emph{Load dump B}.

\subsection{Supply voltage}

\textbf{ISO 7637-2:2004}

12V: $U_A = 13.5 \pm 0.5$

12V: $U_B = 12.5 \pm 0.2$

24V: $U_A = 27 \pm 1$

24V: $U_B = 24 \pm 0.4$

\textbf{ISO 7637-1:2015}

12V: $U_A = 13 \pm 1$

24V: $U_A = 26 \pm 2$

\textbf{ISO 16750-1}

12V: $U_A = 14 \pm 0.2$

12V: $U_B = 12.5 \pm 0.2$

24V: $U_A = 28 \pm 0.2$

24V: $U_B = 24 \pm 0.2$

\subsection{Definitions}
ISO~7637 and ISO~16750

The definitions are stated in the first part of the standards, ISO~7637-1 and ISO~16750-1.


\todo[Snygga till presentationen av definitionerna]

\bm{$U_S$}

For pulse 1, 2a, 3a and 3b: $U_{TOP} - U_N$

For Load dump A: $U_{TOP}$


\bm{$U_A$}

For pulse 1, 2a, 3a and 3b: $13.5 \pm 0.5$ V, $27 \pm 1$ V \cite{iso-7637-}

For Load dump A: $14 \pm 0.2$ V, $28 \pm 0.2$ V \cite{iso-16750-1}


\subsection{Tolerances}
Some of the allowed ranges on the pulses parameters has changed compared to the older version of the standard.

\todo[Gör snygga tabeller]

\textbf{Pulse 1}
Old, 12V: $U_S -75 -- -100$ V
Old, 24V: $U_S -450 -- -600$ V

New, 12V: $U_S -75 -- -150$ V
New, 24V: $U_S -300 -- -600$ V

\textbf{Pulse 2a}
Old: $U_S 37 -- 50$ V

New: $U_S 37 -- 112$ V

\textbf{Pulse 3a and Pulse 3b}
Old, 12V: $\lvert U_S \rvert 112 -- 150$ V
Old, 24V: $\lvert U_S \rvert 150 -- 200$ V

New, 12V: $\lvert U_S \rvert 112 -- 220$ V
New, 24V: $\lvert U_S \rvert 150 -- 300$ V

Old: $t_r 100 -- 200$ µs

New: $t_r 105 -- 195$ µs

\textbf{Load dump A}
No change, other that the definitions.

\subsection{Tolerances for verification}
\textbf{Pulse 1}

Old, 24V, matched: $U_S -300 \pm 30$ V

New, 24V, matched: $U_S -300 \pm 60$ V

\textbf{Pulse 2a}

Old, open: $U_S 50 \pm 5$ V, matched  $U_S 25 \pm 5$ V

New, open: $U_S 75 \pm 7.5$ V, matched  $U_S 35.5 \pm 7.5$ V

\textbf{Pulse 3a and Pulse 3b}

No change.

\textbf{Load dump A}

No change.

\section{Resistors at high frequencies}
\label{theory_parasitic_properties}
\todo[Ta upp teori kring parasitiska effekter och icke-ideala modeller]

\url{https://www.edn.com/design/components-and-packaging/4423492/Resistors-aren-t-resistors}

\url{https://www.vishay.com/docs/60107/freqresp.pdf}

Källan \cite{vishay_hf_resistor}.

When working with resistors at high frequencies, one must care for the parasitc properties of the resistor.

Chapter 3.1.6 \cite{theCircuitDesignersCompanion}

\autoref{fig:nonIdealResistor}. \cite{theElectricalEngineeringHandbook}

\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.}
    \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}
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}
    %\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{High Voltage}
The highest voltage that can be generated by the pulse generators is 1500~V, which is higher than any of the standards require but will serve as the design goal for the verification equipment. This is a higher voltage than most acquisition devices can measure without the use of external attenuators \cite{source}.

Resistive attenuators.. \todo[fyll på]

\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}

\subsection{Measurement errors}
\todo[Put good theory here]

\subsection{RF Attenuators}
Linearity, tolerances, power, combinations of resistors, impedances

\subsection{Tolerances and maximum ratings}
Resistors, Power, Voltages, surges,
Relays, isolation, dielectric strength

\section{Analysis}
The data points from the measurement must be processed and evaluated to determine if the measured pulse is within the specified limits. There are several techniques to accomplish this, which have different advantages.\cite{source}

\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} and \autoref{fig: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}

\subsection{What is good}
\label{sec:goodness}
\todo[Någonting om vad som anses bra]

\subsection{Curve fitting?}
\todo[Läs på om ämnet och se ifall det kan vara rimligt]

\subsection{Max/min limits?}
\todo[Användandet av max/min-fönster]

\subsection{Parameter extraction?}
\todo[Detta är nog ett påhittat ord, kanske menar jag curve fitting?]

\subsection{Evaluation/simulation/robustness}
\todo[Jämför och evaluera de två eller tre metoderna med hänseende till vad som står i ``goodness'']

----

\squareit{
Essentially an embedded high voltage probe.

Bra böcker

High frequency content

Measurement \& Instrumentation Principles, Alan S Morris

High-Voltage Engineering and Testing, 3rd Edition, 
Chapter 15 - Basic Measuring Techniques

Handbook of Measuring System Design, Peter H Sydenham and Richard Thorn


\textbf{Artiklar om högspänningsmätning}

High Voltage Measurements Using Slab Coupled 
Optical Sensors (SCOS) 
LeGrand Shumway, Nikola Stan, Freddy Seng, Rex King, 
Richard Selfridge, Stephen Schultz  
Department of Electrical \& Computer Engineering, Brigham Young University, Provo, Utah 84602, USA 

High frequency high voltage probe embedded 
into an extremely low inductivity high voltage 
supply connection 
D. Ketel, H. Hirsch, M. Malek 
University of Duisburg-Essen, Bi
smarckstr. 81, 47057 Duisburg, Germany, http://www.ets.uni-due.de 
Tel: +49(0) 203 379 3789, email: daniel.ketel@uni-due.de 

Optical measurement of current and voltage on power systems
AJ. Rogers

Application of the Pockels Effect to High Voltage 
Measurement
Fei Long Jianhuan  Zhang   Chunrong  Xie      Zhiwei  Yuan    
Mechanical Electrical Engineering Department, Xiamen University, Xiamen, 361005 China

% konstruktion: https://www.venkel.com/media/wysiwyg/technical/docs/technical-papers/the_definitive_guide_to_smt_resistor_selection.pdf


\textbf{Curve fitting och Double exponential pulse function}

Shape Properties of Pulses Described by Double Exponential Function and Its Modified Forms

On the  Design  and  Generation of the Double  Exponential Function S. C. Dutta  Roy and D. K. Bhargava 
}

\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. 
\todo[fyll på och hitta källor, lägg in bild på interface]

\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 in bild på utrustning, och tabell med data]

\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.

\todo[Lägg in bild på utrustning, och tabell med data]

\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.

\todo[Lägg in bild på utrustning, och tabell med data]

\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.

\todo[Lägg in bild på utrustning, och tabell med data]


\subsection{EM Test CNA 200 Coupling Network}
The SNA~200 is a coupling network, used to multiplex the pulse generators into one box. 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.

\todo[Lägg in bild på utrustning, och tabell med data]

\subsection{Rohde \& Schwarz ZVL13}
\label{sec:rohde_schwarz_zvl}
The ZVL13 is a vector network analyzer. It is, in this project, used to measure the magnitude and phase response between its two ports.

\todo[Lägg in bild på utrustning, och tabell med data]

\subsection{PAT 50 and PAT 1000}
\label{theory_pat_attenuators}






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