%%
%% Make the revision and doc number macro's then they are defined in one place
\ifthenelse {\boolean{paper}}
{
\begin{abstract}

%
%
%  do not ever try to put a paragraph in an abstract. Give incomprehensible
% error messages at the wrong line number. Just like old fortran.

%\paragraph{NOT WRITTEN YET USES PT100 DOC AS FRAME WORK: DO NOT READ}
This paper analyses the example ciruit with an added safety component, given in the introduction chapter.

The analysis is performed using Propositional Logic
diagrams to assist the reasoning process.
This chapter describes taking
the failure modes of the components, analysing the circuit using FMEA
and producing a failure mode model for the circuit as a whole.
Thus after the analysis the Milli Volt Amplifier circuit, may be viewed 
from an FMEA perspective as a component itself, with a set of known failure modes.

\end{abstract}
}
{
\section{Overview}

%\paragraph{NOT WRITTEN YET USES PT100 DOC AS FRAME WORK: DO NOT READ}
%The analysis is performed using Propositional Logic
%diagrams to assist the reasoning process.
%This chapter describes taking
%the failure modes of the components, analysing the circuit using FMEA
%and producing a failure mode model for the circuit as a whole.
%Thus after the analysis the Milli Volt Amplifier circuit, may be viewed 
%from an FMEA perspective as a component itself, with a set of known failure modes.
}


%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=400pt,bb=0 0 714 180,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier.jpg}
%% % milli volt amplifier.jpg: 714x180 pixel, 72dpi, 25.19x6.35 cm, bb=0 0 714 180
%% \caption{Milli Volt Amplifier four wire circuit}
%% \label{fig:milli volt amplifier}
%%\end{figure}
%%
%
%\section{General Description of Milli Volt Amplifier four wire circuit}
%
%The Milli Volt Amplifier four wire circuit uses two wires to supply small electrical current,
%and returns two sense volages by the other two.
%By measuring voltages
%from sections of this circuit forming potential dividers, we can determine the 
%resistance of the platinum wire sensor. The resistance
%of this is directly related to temperature, and may be determined by
%look-up tables or a suitable polynomial expression.
%
%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=150pt,bb=0 0 273 483,keepaspectratio=true]{./milli volt amplifier/vrange.jpg}
%% % milli volt amplifier.jpg: 714x180 pixel, 72dpi, 25.19x6.35 cm, bb=0 0 714 180
%% \caption{Milli Volt Amplifier expected voltage ranges}
%% \label{fig:milli volt amplifiervrange}
%%\end{figure}
%%
%
%The voltage ranges we expect from this three stage potential divider\footnote{
%two stages are required for validation, a third stage is used to measure the current flowing
%through the circuit to obtain accurate temperature readings}
%are shown in figure \ref{fig:milli volt amplifiervrange}. Note that there is
%an expected range for each reading, for a given temperature span.
%Note that the low reading goes down as temperature increases, and the higher reading goes up.
%For this reason the low reading will be referred to as {\em sense-}
%and the higher as {\em sense+}.
%
%\subsection{Accuracy despite variable \\ resistance in cables}
%
%For electronic and accuracy reasons a four wire circuit is preferred
%because of resistance in the cables. Resistance from the supply
% causes a slight voltage 
%drop in the supply to the Milli Volt Amplifier. As no significant current
%is carried by the two `sense' lines, the resistance back to the ADC
%causes only a negligible voltage drop, and thus the four wire 
%configuration is more accurate\footnote{The increased accuracy is because the voltage measured, is the voltage across
%the thermistor and not the voltage across the thermistor and current supply wire resistance.}.
%
%\subsection{Calculating Temperature from \\ the sense line voltages}
%
%The current flowing though the 
%whole circuit can be measured on the PCB by reading a third
%sense voltage from one of the load resistors. Knowing the current flowing
%through the circuit   
%and knowing the voltage drop over the Milli Volt Amplifier, we can calculate its 
%resistance  by Ohms law $V=I.R$, $R=\frac{V}{I}$. 
%Thus a little loss of supply current due to resistance in the cables
%does not impinge on accuracy.
%The resistance to temperature conversion is achieved 
%through the published Milli Volt Amplifier tables\cite{eurothermtables}.
%The standard voltage divider equations (see figure \ref{fig:vd} and 
%equation \ref{eqn:vd}) can be used to  calculate
%expected voltages for failure mode and temperature reading purposes.
%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=100pt,bb=0 0 183 170,keepaspectratio=true]{./milli volt amplifier/voltage_divider.png}
%% % voltage_divider.png: 183x170 pixel, 72dpi, 6.46x6.00 cm, bb=0 0 183 170
%% \caption{Voltage Divider}
%% \label{fig:vd}
%%\end{figure}
%%%The looking at figure \ref{fig:vd} the standard voltage divider formula (equation \ref{eqn:vd}) is used.
%
%\begin{equation}
%\label{eqn:vd}
% V_{out} = V_{in}.\frac{Z2}{Z2+Z1}
%\end{equation}
%
%\section{Safety case for 4 wire circuit}
%
%This sub-section looks at the behaviour of the Milli Volt Amplifier four wire circuit 
%for the effects of component failures.
%All components have a set of known `failure modes'.
%In other words we know that a given component can fail in several distinct ways.
%Studies have been published which list common component types 
%and their sets of failure modes, often with MTTF statistics \cite{mil1991}.
%Thus for each component, an analysis is made for each of its failure modes,
%with respect to its effect on the 
%circuit. Each one of these scenarios is termed a `test case'.
%The resultant circuit behaviour for each of these test cases is noted. 
%The worst case for this type of
%analysis would be a fault that we cannot detect.
%Where this occurs a circuit re-design is probably the only sensible course of action.
%
%
%
%\subsection{Single Fault FMEA Analysis \\ of Milli Volt Amplifier Four wire circuit}
%
%\label{fmea}
%This circuit simply consists of three resistors.
%Resistors according to the DOD Electronic component fault handbook
%1991, fail by either going OPEN or SHORT circuit \cite{mil1991}.
%%Should wires become disconnected these will have the same effect as
%%given resistors going open. 
%For the purpose of this analyis;
%$R_{1}$ is the \ohms{2k2} from 5V to the thermistor,
%$R_3$ is the Milli Volt Amplifier thermistor and $R_{2}$ connects the thermistor to ground.
%
%We can define the terms `High Fault' and `Low Fault' here, with reference to figure
%\ref{fig:milli volt amplifiervrange}. Should we get a reading outside the safe green zone
%in the diagram we can consider this a fault.
%Should the reading be above its expected range this is a `High Fault'
%and if below a `Low Fault'.
%
%Table \ref{ptfmea} plays through the scenarios of each of the resistors failing
%in both SHORT and OPEN failure modes, and hypothesises an error condition in the readings.
%The range {0\oc} to {300\oc} will be analysed using potential divider equations to 
%determine out of range voltage limits in section \ref{ptbounds}.
%
%\begin{table}[ht]
%\caption{Milli Volt Amplifier FMEA Single Faults} % title of Table
%\centering % used for centering table
%\begin{tabular}{||l|c|c|l|l||}
%\hline \hline
% \textbf{Test} & \textbf{Result} & \textbf{Result } & \textbf{General}   \\
% \textbf{Case} &  \textbf{sense +} & \textbf{sense -} & \textbf{Symtom Description}   \\
%%   R         &    wire        & res +         & res -    & description
%\hline
%\hline
% $R_1$ SHORT    &  High Fault        &  -       &  Value Out of Range Value \\ \hline
%$R_1$  OPEN     &  Low Fault         & Low Fault     &  Both values out of range \\ \hline
% \hline
%$R_3$  SHORT    &  Low Fault          & High Fault      &   Both values out of range \\ \hline
% $R_3$  OPEN     &  High Fault         & Low  Fault    &  Both values out of range \\ \hline
%\hline
%$R_2$ SHORT     &  -         &  Low Fault   &  Value Out of Range Value \\  
% $R_2$ OPEN     &  High Fault         & High Fault     &  Both values out of range \\ \hline
%\hline
%\end{tabular} 
%\label{ptfmea}
%\end{table}
%
%From table \ref{ptfmea} it can be seen that any component failure in the circuit
%should cause a common symptom, that of one or more of the values being `out of range'.
%Temperature range calculations and detailed calculations
%on the effects of each test case are found in section \ref{milli volt amplifierrange}
%and \ref{milli volt amplifiertemp}. 
%
%
%
%\subsection{Range and Amplifier Calculations}
%\label{milli volt amplifiertemp}
%Milli Volt Amplifier resistors are designed to 
%have a resistance of \ohms{100}  at {0\oc} \cite{aoe},\cite{eurothermtables}.
%A suitable `wider than to be expected range' was considered to be {0\oc} to {300\oc}
%for a given application. 
%According to the Eurotherm Milli Volt Amplifier
%tables \cite{eurothermtables}, this corresponded to the resistances \ohms{100}
%and \ohms{212.02} respectively. From this the potential divider circuit can be
%analysed and the maximum and minimum acceptable voltages determined.
%These can be used as bounds results to apply the findings from the 
%Milli Volt Amplifier FMEA analysis in section \ref{fmea}.
%
%As the Milli Volt Amplifier forms a potential divider with the \ohms{2k2}  load resistors,
%the upper and lower readings can be calculated thus:
%
%
%$$ highreading = 5V.\frac{2k2+milli volt amplifier}{2k2+2k2+milli volt amplifier} $$
%$$ lowreading = 5V.\frac{2k2}{2k2+2k2+milli volt amplifier} $$
%So by defining an acceptable measurement/temperature range,
%and ensuring the 
%values are always within these bounds we can be confident that none of the
%resistors in this circuit has failed.
%
%To convert these to twelve bit ADC (\adctw) counts:
%
%$$ highreading = 2^{12}.\frac{2k2+milli volt amplifier}{2k2+2k2+milli volt amplifier} $$
%$$ lowreading = 2^{12}.\frac{2k2}{2k2+2k2+milli volt amplifier} $$
%
%
%\begin{table}[ht]
%\caption{Milli Volt Amplifier Maximum and Minimum Values} % title of Table
%\centering % used for centering table
%\begin{tabular}{||c|c|c|l|l||}
%\hline \hline
% \textbf{Temperature} & \textbf{Milli Volt Amplifier resistance} & 
%\textbf{Lower} & \textbf{Higher} & \textbf{Description}   \\ 
%\hline
%% {-100 \oc} &  {\ohms{68.28}} &  2.46V         &   2.53V      & Boundary of   \\ 
%%            &                 &  2017\adctw   &   2079\adctw  &   out of range LOW \\ \hline
%  {0 \oc}   & {\ohms{100}}    &  2.44V         &   2.56V       & Boundary of         \\
%            &                 &  2002\adctw   &   2094\adctw   &  out of range LOW        \\ \hline
% {+300 \oc} & {\ohms{212.02}}  &  2.38V         &   2.62V       & Boundary of                       \\
%            &                  &  1954\adctw   &   2142\adctw   &  out of range HIGH  \\ \hline
%\hline
%\end{tabular} 
%\label{ptbounds}
%\end{table}
%
%Table \ref{ptbounds} gives ranges that determine correct operation. In fact it can be shown that
%for any single error (short or opening of any resistor) this bounds check
%will detect it.
%
%
%\section{Single Fault FMEA Analysis \\ of Milli Volt Amplifier Four wire circuit}
%
%\subsection{Single Fault Modes as PLD}
%
%The component~failure~modes in table \ref{ptfmea} can be represented as contours
%on a PLD diagram. 
%Each test case, is defined by the contours that enclose
%it. The test cases here deal with single faults only
%and are thus enclosed by one contour each.
%
%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=400pt,bb=0 0 518 365,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier_tc.jpg}
%% % milli volt amplifier_tc.jpg: 518x365 pixel, 72dpi, 18.27x12.88 cm, bb=0 0 518 365
%% \caption{Milli Volt Amplifier Component Failure Modes}
%% \label{fig:milli volt amplifier_tc}
%%\end{figure}
%%
%%ating input Fault  
%This circuit supplies two results, the {\em sense+} and {\em sense-} voltage readings.
%To establish the valid voltage ranges for these, and knowing our
%valid temperature range for this example ({0\oc} .. {300\oc}) we can calculate
%valid voltage reading ranges by using the standard voltage divider equation \ref{eqn:vd}
%for the circuit shown in figure \ref{fig:vd}.
%
%%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=100pt,bb=0 0 183 170,keepaspectratio=true]{./milli volt amplifier/voltage_divider.png}
%% % voltage_divider.png: 183x170 pixel, 72dpi, 6.46x6.00 cm, bb=0 0 183 170
%% \caption{Voltage Divider}
%% \label{fig:vd}
%%\end{figure}
%%%The looking at figure \ref{fig:vd} the standard voltage divider formula (equation \ref{eqn:vd}) is used.
%%
%%\begin{equation}
%%\label{eqn:vd}
%% V_{out} = V_{in}.\frac{Z2}{Z2+Z1}
%%\end{equation}
%%
%
%
%\subsection{Proof of Out of Range \\ Values for Failures}
%\label{pt110range}
%Using the temperature ranges defined above we can compare the voltages
%we would get from the resistor failures to prove that they are
%`out of range'. There are six test cases and each will be examined in turn.
%
%\subsubsection{ TC 1 : Voltages $R_1$ SHORT } 
%With milli volt amplifier at 0\oc 
%$$ highreading = 5V $$
%Since the highreading or sense+ is directly connected to the 5V rail,
%both temperature readings will be 5V..
%$$ lowreading = 5V.\frac{2k2}{2k2+100\Omega} = 4.78V$$
%With milli volt amplifier at the high end of the temperature range 300\oc.
%$$ highreading = 5V $$
%$$ lowreading = 5V.\frac{2k2}{2k2+212.02\Omega} = 4.56V$$
%
%Thus with $R_1$ shorted both readings are outside the 
%proscribed range in table \ref{ptbounds}.
%
%\subsubsection{ TC 2 : Voltages $R_1$ OPEN } 
%
%In this case the 5V rail is disconnected. All voltages read are 0V, and 
%therefore both readings are outside the
%proscribed range in table \ref{ptbounds}.
%
%
%\subsubsection{ TC 3 : Voltages $R_2$ SHORT }
%
%With milli volt amplifier at 0\oc 
%$$ lowreading = 0V $$
%Since the lowreading or sense- is directly connected to the 0V rail,
%both temperature readings will be 0V.
%$$ lowreading = 5V.\frac{100\Omega}{2k2+100\Omega} = 0.218V$$
%With milli volt amplifier at the high end of the temperature range 300\oc.
%$$ highreading = 5V.\frac{212.02\Omega}{2k2+212.02\Omega} = 0.44V$$
%
%Thus with $R_2$ shorted both readings are outside the 
%proscribed range in table \ref{ptbounds}.
%
%\subsubsection{ TC 4 : Voltages $R_2$ OPEN }
%Here there is no potential divider operating and both sense lines
%will read 5V, outside of the proscribed range.
%
%
%\subsubsection{ TC 5 : Voltages $R_3$ SHORT } 
%
%Here the potential divider is simply between
%the two 2k2 load resistors. Thus it will read a nominal;
%2.5V.
%
%Assuming the load resistors are 
%precision components, and then taking an absolute worst case of 1\% either way.
%
%$$ 5V.\frac{2k2*0.99}{2k2*1.01+2k2*0.99} = 2.475V $$
%
%$$ 5V.\frac{2k2*1.01}{2k2*1.01+2k2*0.99} = 2.525V $$
%
%These readings both lie outside the proscribed range.
%Also the sense+ and sense- readings would have the same value.
%
%\subsubsection{ TC 6 : Voltages $R_3$ OPEN } 
%
%Here the potential divider is broken. The sense- will read 0V and the sense+ will
%read 5V. Both readings are outside the proscribed range.
%
%\subsection{Summary of Analysis}
%
%All six test cases have been analysed and the results agree with the hypothesis
%put in  Table \ref{ptfmea}. The PLD diagram, can now be used to collect the
%symptoms. In this case there is a common and easily detected symptom for all these single 
%resistor faults :  Voltage out of range.
%
%A spider can be drawn on the PLD diagram to this effect.
%
%In practical use, by defining an acceptable measurement/temperature range,
%and ensuring the 
%values are always within these bounds we can be confident that none of the
%resistors in this circuit has failed.
%
%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=400pt,bb=0 0 518 365,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier_tc_sp.jpg}
%% % milli volt amplifier_tc.jpg: 518x365 pixel, 72dpi, 18.27x12.88 cm, bb=0 0 518 365
%% \caption{Milli Volt Amplifier Component Failure Modes}
%% \label{fig:milli volt amplifier_tc_sp}
%%\end{figure}
%%
%
%\subsection{Derived Component : The Milli Volt Amplifier Circuit}
%The Milli Volt Amplifier circuit can now be treated as a component in its own right, and has one failure mode, 
%{\textbf OUT\_OF\_RANGE}. It can now be represnted as a PLD see figure \ref{fig:milli volt amplifier_singlef}.
%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=100pt,bb=0 0 167 194,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier_singlef.jpg}
%% % milli volt amplifier_singlef.jpg: 167x194 pixel, 72dpi, 5.89x6.84 cm, bb=0 0 167 194
%% \caption{Milli Volt Amplifier Circuit Failure Modes : From Single Faults Analysis}
%% \label{fig:milli volt amplifier_singlef}
%%\end{figure}
%%
%
%%From the single faults (cardinality constrained powerset of 1) analysis, we can now create
%%a new derived component, the {\emmilli volt amplifiercircuit}. This has only \{ OUT\_OF\_RANGE \}
%%as its single failure mode.
%
%
%%Interestingly we can calculate the failure statistics for this circuit now.
%%Mill 1991 gives resistor stats of ${10}^{11}$ times 6 (can we get special stats for milli volt amplifier) ???
%\clearpage
%\subsection{Mean Time to Failure}
%
%Now that we have a model for the failure mode behaviour of the milli volt amplifier circuit
%we can look at the statistics associated with each of the failure modes.
%
%The DOD electronic reliability of components
%document  MIL-HDBK-217F\cite{mil1992} gives formulae for calculating
%the
%%$\frac{failures}{{10}^6}$
%${failures}/{{10}^6}$ % looks better
%in hours for a wide range of generic components
%\footnote{These figures are based on components from the 1980's and MIL-HDBK-217F
%can give conservative reliability figures when applied to
%modern components}. 
%
%Using the MIL-HDBK-217F\cite{mil1992}  specifications for resistor and thermistor
%failure statistics we calculate the reliability of this circuit.
%
%
%\subsubsection{Resistor FIT Calculations}
%
%The formula for given in MIL-HDBK-217F\cite{mil1992}[9.2] for a generic fixed film non-power resistor
%is reproduced in equation \ref{resistorfit}. The meanings 
%and values assigned to its co-efficients are described in table \ref{tab:resistor}.
%
%\begin{equation}
%% fixed comp resistor{\lambda}_p = {\lambda}_{b}{\pi}_{R}{\pi}_Q{\pi}_E
%resistor{\lambda}_p = {\lambda}_{b}{\pi}_{R}{\pi}_Q{\pi}_E
% \label{resistorfit}
%\end{equation}
%
%\begin{table}[ht]
%\caption{Fixed film resistor Failure in time assessment} % title of Table
%\centering % used for centering table
%\begin{tabular}{||c|c|l||}
%\hline \hline
% \em{Parameter}      &  \em{Value}    &   \em{Comments} \\
%                     &                &      \\ \hline \hline
% ${\lambda}_{b}$ &  0.00092        & stress/temp base failure rate  $60^o$ C  \\   \hline
% %${\pi}_T$ &  4.2         & max temp of $60^o$ C\\ \hline
% ${\pi}_R$ &  1.0         & Resistance range $< 0.1M\Omega$\\ \hline
% ${\pi}_Q$ &  15.0         & Non-Mil spec component\\ \hline
% ${\pi}_E$ &  1.0         & benign ground environment\\ \hline
%
%\hline \hline
%\end{tabular}
%\label{tab:resistor}
%\end{table}
%
%Applying equation \ref{resistorfit} with the parameters from table \ref{tab:resistor}
%give the following failures in ${10}^6$ hours:
%
%\begin{equation}
% 0.00092 \times 1.0 \times 15.0 \times 1.0 = 0.0138  \;{failures}/{{10}^{6} Hours}
% \label{eqn:resistor}
%\end{equation}
%
%While MIL-HDBK-217F gives MTTF for a wide range of common components,
%it does not specify how the components will fail (in this case OPEN or SHORT). {Some standards, notably EN298 only consider resistors failing in OPEN mode}.
%FMD-97 gives 27\% OPEN and 3\% SHORTED, for resistors under certain electrical and environmental stresses. This example 
%compromises and uses a 90:10 ratio, for resistor failure. 
%Thus for this example resistors are expected to fail OPEN in 90\% of cases and SHORTED 
%in the other 10\%.
%A standard fixed film resistor, for use in a benign environment, non military spec at
%temperatures up to 60\oc is given a probability of 13.8 failures per billion ($10^9$)
%hours of operation (see equation \ref{eqn:resistor}). 
%This figure is referred to as a FIT\footnote{FIT values are measured as the number of 
%failures per Billion (${10}^9$) hours of operation, (roughly 114,000 years). The smaller the 
%FIT number the more reliable the fault~mode} Failure in time.
%
%The formula given for a thermistor in  MIL-HDBK-217F\cite{mil1992}[9.8] is reproduced in
%equation \ref{thermistorfit}. The variable meanings and values are described in table \ref{tab:thermistor}.
%
%\begin{equation}
%% fixed comp resistor{\lambda}_p = {\lambda}_{b}{\pi}_{R}{\pi}_Q{\pi}_E
%resistor{\lambda}_p = {\lambda}_{b}{\pi}_Q{\pi}_E
% \label{thermistorfit}
%\end{equation}
%
%\begin{table}[ht]
%\caption{Bead type Thermistor Failure in time assessment} % title of Table
%\centering % used for centering table
%\begin{tabular}{||c|c|l||}
%\hline \hline
% \em{Parameter}      &  \em{Value}    &   \em{Comments} \\
%                     &                &      \\ \hline \hline
% ${\lambda}_{b}$ &  0.021        & stress/temp base failure rate bead thermistor \\   \hline
% %${\pi}_T$ &  4.2         & max temp of $60^o$ C\\ \hline
% %${\pi}_R$ &  1.0         & Resistance range $< 0.1M\Omega$\\ \hline
% ${\pi}_Q$ &  15.0         & Non-Mil spec component\\ \hline
% ${\pi}_E$ &  1.0         & benign ground environment\\ \hline
%
%\hline \hline
%\end{tabular}
%\label{tab:thermistor}
%\end{table}
%
%
%\begin{equation}
% 0.021 \times 1.0 \times 15.0 \times 1.0 = 0.315 \; {failures}/{{10}^{6} Hours}
% \label{eqn:thermistor}
%\end{equation}
%
%
%Thus thermistor, bead type, non military spec is given a FIT of 315.0
%
%Using the RIAC finding we can draw up the following table (table \ref{tab:stat_single}),
%showing the FIT values for all faults considered.
%
%
%
%\begin{table}[h+]
%\caption{Milli Volt Amplifier FMEA Single // Fault Statistics} % title of Table
%\centering % used for centering table
%\begin{tabular}{||l|c|c|l|l||}
%\hline \hline
% \textbf{Test} & \textbf{Result} & \textbf{Result } & \textbf{MTTF}   \\
% \textbf{Case} &  \textbf{sense +} & \textbf{sense -} & \textbf{per $10^9$ hours of operation}   \\
%%   R         &    wire        & res +         & res -    & description
%\hline
%\hline
%TC:1 $R_1$ SHORT    &  High Fault        &  -       &  1.38  \\ \hline
%TC:2 $R_1$  OPEN     &  Low Fault         & Low Fault     &  12.42\\ \hline
% \hline
%TC:3 $R_3$  SHORT    &  Low Fault          & High Fault      & 31.5  \\ \hline
%TC:4 $R_3$  OPEN     &  High Fault         & Low  Fault    &   283.5 \\ \hline
%\hline
%TC:5 $R_2$ SHORT     &  -         &  Low Fault   &  1.38  \\  
%TC:6 $R_2$ OPEN     &  High Fault         & High Fault     & 12.42 \\ \hline
%\hline
%\end{tabular} 
%\label{tab:stat_single}
%\end{table}
%
%The FIT for the circuit as a whole is the sum of MTTF values for all the 
%test cases. The Milli Volt Amplifier circuit here has a  FIT of 342.6. This is a MTTF of
%about 360 years per circuit.
%
%A Probablistic tree can now be drawn, with a FIT value for the Milli Volt Amplifier
%circuit and FIT values for all the component fault modes that it was calculated from.
%We can see from this that that the most likely fault is the thermistor going OPEN.
%This circuit is around 10 times more likely to fail in this way than in any other.
%Were we to need a more reliable temperature sensor this would probably 
%be the fault~mode we would scrutinise first.
%
%
%%\begin{figure}[h+]
%% \centering
%% \includegraphics[width=400pt,bb=0 0 856 327,keepaspectratio=true]{./milli volt amplifier/stat_single.jpg}
%% % stat_single.jpg: 856x327 pixel, 72dpi, 30.20x11.54 cm, bb=0 0 856 327
%% \caption{Probablistic Fault Tree : Milli Volt Amplifier Single Faults}
%% \label{fig:stat_single}
%%\end{figure}
%
%
%The Milli Volt Amplifier analysis presents a simple result for single faults. 
%The next analysis phase looks at how the circuit will behave under double simultaneous failure
%conditions.
%
%\clearpage
%\section{ Milli Volt Amplifier Double Simultaneous \\ Fault Analysis}
%
%In this section we examine the failure mode behaviour for all single 
%faults and double simultaneous faults.
%This corresponds to the cardinality constrained powerset of
%the failure modes in the functional group.
%All the single faults have already been proved in the last section.
%For the next set of test cases, let us again hypothesise
%the failure modes, and then examine each one in detail with
%potential divider equation proofs.
%
%Table \ref{tab:ptfmea2} lists all the combinations of double
%faults and then hypothesises how the functional~group will react
%under those conditions.
%
%\begin{table}[ht]
%\caption{Milli Volt Amplifier FMEA Double Faults} % title of Table
%\centering % used for centering table
%\begin{tabular}{||l|l|c|c|l|l||}
%\hline \hline
% \textbf{TC}      &\textbf{Test} & \textbf{Result} & \textbf{Result } & \textbf{General}   \\
% \textbf{number} &\textbf{Case} &  \textbf{sense +} & \textbf{sense -} & \textbf{Symtom Description}   \\
%%   R         &    wire        & res +         & res -    & description
%\hline
%\hline
% TC 7: & $R_1$ OPEN $R_2$ OPEN    &  Floating input Fault        &   Floating input Fault    &  Unknown value readings \\ \hline
% TC 8: & $R_1$ OPEN $R_2$ SHORT   &  low        &   low    &  Both out of range  \\ \hline
%\hline
% TC 9: & $R_1$ OPEN $R_3$ OPEN   &  high        &   low     &  Both out of Range \\ \hline
% TC 10: & $R_1$ OPEN $R_3$ SHORT   &  low        &   low    &  Both out of range \\ \hline
%\hline
%
% TC 11: & $R_1$ SHORT $R_2$ OPEN    &  high        &   high    &  Both out of range \\ \hline
%TC 12: &  $R_1$ SHORT $R_2$ SHORT   &  high        &   low    &  Both out of range  \\ \hline
%\hline
% TC 13: & $R_1$ SHORT $R_3$ OPEN   &  high        &   low     &  Both out of Range \\ \hline
%TC 14: &  $R_1$ SHORT $R_3$ SHORT   &  high        &   high    &  Both out of range \\ \hline
%
%\hline
% TC 15: & $R_2$ OPEN $R_3$ SHORT   &  high        &   Floating input Fault    &  sense+ out of range \\ \hline
%TC 16: &  $R_2$ OPEN $R_3$ SHORT   &  high        &   high    &  Both out of Range \\ \hline
%TC 17: &  $R_2$ SHORT $R_3$ OPEN   &  high        &   low    &  Both out of Range \\ \hline
%TC 18: &  $R_2$ SHORT $R_3$ SHORT   &  low        &   low    &  Both out of Range \\ \hline
%\hline
%\end{tabular}
%\label{tab:ptfmea2}
%\end{table}
%
%\subsection{Verifying complete coverage for a \\ cardinality constrained powerset of 2}
%
%
%
%It is important to check that we have covered all possible double fault combinations.
%We can use the equation \ref{eqn:correctedccps2} 
%\ifthenelse {\boolean{paper}}
%{
%from the definitions paper
%\ref{pap:compdef}
%,
%reproduced below to verify this.
%
%\indent{
% where:
% \begin{itemize}
%  \item The set $SU$ represents the components in the functional~group, where all components are guaranteed to have unitary state failure modes.
%  \item The indexed set $C_j$ represents all components in set $SU$.
%  \item The function $FM$ takes a component as an argument and returns its set of failure modes.
%  \item $cc$ is the cardinality constraint, here 2 as we are interested in double and single faults.
% \end{itemize}
%}
%\begin{equation}
% |{\mathcal{P}_{cc}SU}| = {\sum^{k}_{1..cc}  \frac{|{SU}|!}{k!(|{SU}| - k)!}}   
%- \sum^{p}_{2..cc}{{\sum^{j}_{j \in J}   \frac{|FM({C_j})|!}{p!(|FM({C_j})| - p)!}}  }
% \label{eqn:correctedccps2}
%\end{equation}
%
%}
%{
%\begin{equation}
% |{\mathcal{P}_{cc}SU}| = {\sum^{k}_{1..cc}  \frac{|{SU}|!}{k!(|{SU}| - k)!}}
%- \sum^{p}_{2..cc}{{\sum^{j}_{j \in J}   \frac{|FM({C_j})|!}{p!(|FM({C_j})| - p)!}}  }
% %\label{eqn:correctedccps2}
%\end{equation}
%}
%
%
%$|FM(C_j)|$ will always be 2 here,  as all the components are resistors and have two failure modes.
%
%%
%% Factorial of zero is one ! You can only arrange an empty set one way !
%
%Populating this equation with $|SU| = 6$ and $|FM(C_j)|$ = 2.
%%is always 2 for this circuit, as all the components are resistors and have two failure modes.
%
%\begin{equation}
% |{\mathcal{P}_{2}SU}| = {\sum^{k}_{1..2}  \frac{6!}{k!(6 - k)!}}   
%- \sum^{p}_{2..2}{{\sum^{j}_{1..3}   \frac{2!}{p!(2 - p)!}}  }
% %\label{eqn:correctedccps2}
%\end{equation}
%
%$|{\mathcal{P}_{2}SU}|$ is the number of valid combinations of faults to check
%under the conditions of unitary state failure modes for the components (a resistor cannot fail by being shorted and open at the same time).
%
%Expanding the sumations
%
%
%$$ NoOfTestCasesToCheck = \frac{6!}{1!(6-1)!} + \frac{6!}{2!(6-2)!} - \Big( \frac{2!}{2!(2 - 2)!} + \frac{2!}{2!(2 - 2)!} + \frac{2!}{2!(2 - 2)!} \Big) $$
%
%$$ NoOfTestCasesToCheck = 6 + 15 - ( 1 + 1 + 1 )  = 18 $$
%
%As the test case are all different and are of the correct cardinalities (6 single faults and (15-3) double)
%we can be confident that we have looked at all `double combinations', of the possible faults
%in the milli volt amplifier circuit. The next task is to investigate
%these test cases in more detail to prove the failure mode hypothesis set out in table \ref{tab:ptfmea2}.
%
%
%\subsection{Proof of Double Faults Hypothesis }
%
%\subsubsection{ TC 7 : Voltages $R_1$ OPEN $R_2$ OPEN } 
%\label{milli volt amplifier:bothfloating}
%This double fault mode produces an interesting symptom.
%Both sense lines are floating.
%We cannot know what the {\adctw} readings on them will be.
%In practise these would probably float to  low values
%but for the purpose of a safety critical analysis
%all we can say is the values are `floating' and `unknown'.
%This is an interesting case, because it is, at this stage an undetectable
%fault that must be handled.
%
%
%\subsubsection{ TC 8 : Voltages $R_1$ OPEN $R_2$ SHORT } 
%
%This cuts the supply from Vcc. Both sense lines will be at zero.
%Thus both values will be out of range.
%
%
%\subsubsection{ TC 9 : Voltages $R_1$ OPEN $R_3$ OPEN } 
%
%Sense- will be floating.
%Sense+ will be tied to Vcc and will thus be out of range.
%
%\subsubsection{ TC 10 : Voltages $R_1$ OPEN $R_3$ SHORT } 
%
%This shorts ground to the 
%both of the sense lines.
%Both values thuis out of range.
%
%\subsubsection{ TC 11 : Voltages $R_1$ SHORT $R_2$ OPEN } 
%
%This shorts both sense lines to Vcc.
%Both values will be out of range.
%
%
%\subsubsection{ TC 12 : Voltages $R_1$ SHORT $R_2$ SHORT }
%
%This shorts the sense+ to Vcc and the sense- to ground.
%Both values will be out of range.
%
%
%
%
%
%
%
%
%
%\subsubsection{ TC 13 : Voltages $R_1$ SHORT $R_3$ OPEN }
%
%This shorts the sense+ to Vcc and the sense- to ground.
%Both values will be out of range.
%
%\subsubsection{ TC 14 : Voltages $R_1$ SHORT $R_3$ SHORT }
%
%This shorts the sense+ and sense- to Vcc.
%Both values will be out of range.
%
%\subsubsection{ TC 15 : Voltages $R_2$ OPEN $R_3$ OPEN }
%
%This shorts the sense+ to Vcc and causes sense- to float.
%The sense+ value will be out of range.
%
%
%\subsubsection{ TC 16 : Voltages $R_2$ OPEN $R_3$ SHORT }
%
%This shorts the sense+ and sense- to Vcc.
%Both values will be out of range.
%
%
%
%
%
%\subsubsection{ TC 17 : Voltages $R_2$ SHORT $R_3$ OPEN }
%
%This shorts the sense- to Ground.
%The sense- value will be out of range.
%
%
%\subsubsection{ TC 18 : Voltages $R_2$ SHORT $R_3$ SHORT }
%
%This shorts the sense+ and sense- to Vcc.
%Both values will be out of range.
%
%\clearpage
%\subsection{Double Faults Represented on a PLD Diagram}
%
%We can show the test cases on a diagram with the double faults residing on regions 
%corresponding to overlapping contours see figure \ref{fig:plddouble}.
%Thus $TC\_18$ will be enclosed by the $R2\_SHORT$ contour and the $R3\_SHORT$ contour.
%
%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=450pt,bb=0 0 730 641,keepaspectratio=true]{milli volt amplifier/plddouble.jpg}
%% % plddouble.jpg: 730x641 pixel, 72dpi, 25.75x22.61 cm, bb=0 0 730 641
%% \caption{Milli Volt Amplifier Double Simultaneous Faults}
%% \label{fig:plddouble}
%%\end{figure}
%
%The usefulnes of equation \ref{eqn:correctedccps2} is apparent. From the diagram it is easy to verify
%the number of failure modes considered for each test case, but complete coverage for
%a given cardinality constraint is not visually obvious.
%
%\subsubsection{Symptom Extraction}
%
%We can now examine the results of the test case analysis and apply symptom abstraction.
%In all the test case results we have at least one an out of range value, except for
%$TC\_7$  
%which has two unknown values/floating readings. We can collect all the faults, except $TC\_7$, 
%into the symptom $OUT\_OF\_RANGE$. 
%As a symptom $TC\_7$ could be described as $FLOATING$. We can thus draw a PLD diagram representing the
%failure modes of this functional~group, the milli volt amplifier circuit from the perspective of double simultaneous failures,
%in figure \ref{fig:dubsim}.
%
%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=450pt,bb=0 0 730 641,keepaspectratio=true]{milli volt amplifier/plddoublesymptom.jpg}
%% % plddouble.jpg: 730x641 pixel, 72dpi, 25.75x22.61 cm, bb=0 0 730 641
%% \caption{Milli Volt Amplifier Double Simultaneous Faults}
%% \label{fig:plddoublesymptom}
%%\end{figure}
%
%
%\clearpage
%\subsection{Derived Component : The Milli Volt Amplifier Circuit}
%The Milli Volt Amplifier circuit again, can now be treated as a component in its own right, and has two failure modes, 
%{\textbf{OUT\_OF\_RANGE}} and {\textbf{FLOATING}}. 
%It can now be represented as a PLD see figure \ref{fig:milli volt amplifier_doublef}.
%
%%\begin{figure}[h]
%% \centering
%% \includegraphics[width=100pt,bb=0 0 167 194,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier_doublef.jpg}
%% % milli volt amplifier_singlef.jpg: 167x194 pixel, 72dpi, 5.89x6.84 cm, bb=0 0 167 194
%% \caption{Milli Volt Amplifier Circuit Failure Modes : From Double Faults Analysis}
%% \label{fig:milli volt amplifier_doublef}
%%\end{figure}
%
%\subsection{Statistics}
%
%%%
%%% Need to talk abou the `detection time'
%%% or `Safety Relevant Validation Time' ref can book
%%% EN61508 gives detection calculations to reduce
%%% statistical impacts of failures.
%%%
%
%If we consider the failure modes to be statistically independent we can calculate
%the FIT values for all the failures. The failure mode of concern, the undetectable {\textbf{FLOATING}} condition
%requires that resistors $R_1$ and $R_2$ fail. We can multiply the MTTF
%together and find an MTTF for both failing. The FIT value of 12.42 corresponds to
%$12.42 \times {10}^{-9}$ failures per hour. Squaring this gives $ 154.3 \times {10}^{-18} $.
%This is an astronomically small MTTF, and so small that it would
%probably fall below a threshold to sensibly consider.
%However, it is very interesting from a failure analysis perspective,
%because here we have found a fault that we cannot detect at this
%level. This means that should we wish to cope with
%this fault, we need to devise a way of detecting this
%condition in higher levels of the system.
%
%
%
%\vspace{20pt}
%
%%typeset in {\Huge \LaTeX} \today
