842 lines
34 KiB
TeX
842 lines
34 KiB
TeX
%%
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%% Make the revision and doc number macro's then they are defined in one place
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\ifthenelse {\boolean{paper}}
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{
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\begin{abstract}
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%
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%
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% do not ever try to put a paragraph in an abstract. Give incomprehensible
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% error messages at the wrong line number. Just like old fortran.
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%\paragraph{NOT WRITTEN YET USES PT100 DOC AS FRAME WORK: DO NOT READ}
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This paper analyses the example ciruit with an added safety component, given in the introduction chapter.
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The analysis is performed using Propositional Logic
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diagrams to assist the reasoning process.
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This chapter describes taking
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the failure modes of the components, analysing the circuit using FMEA
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and producing a failure mode model for the circuit as a whole.
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Thus after the analysis the Milli Volt Amplifier circuit, may be viewed
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from an FMEA perspective as a component itself, with a set of known failure modes.
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\end{abstract}
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}
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{
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\section{Overview}
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%\paragraph{NOT WRITTEN YET USES PT100 DOC AS FRAME WORK: DO NOT READ}
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%The analysis is performed using Propositional Logic
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%diagrams to assist the reasoning process.
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%This chapter describes taking
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%the failure modes of the components, analysing the circuit using FMEA
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%and producing a failure mode model for the circuit as a whole.
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%Thus after the analysis the Milli Volt Amplifier circuit, may be viewed
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%from an FMEA perspective as a component itself, with a set of known failure modes.
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}
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%
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%%\begin{figure}[h]
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%% \centering
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%% \includegraphics[width=400pt,bb=0 0 714 180,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier.jpg}
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%% % milli volt amplifier.jpg: 714x180 pixel, 72dpi, 25.19x6.35 cm, bb=0 0 714 180
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%% \caption{Milli Volt Amplifier four wire circuit}
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%% \label{fig:milli volt amplifier}
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%%\end{figure}
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%%
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%
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%\section{General Description of Milli Volt Amplifier four wire circuit}
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%
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%The Milli Volt Amplifier four wire circuit uses two wires to supply small electrical current,
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%and returns two sense volages by the other two.
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%By measuring voltages
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%from sections of this circuit forming potential dividers, we can determine the
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%resistance of the platinum wire sensor. The resistance
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%of this is directly related to temperature, and may be determined by
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%look-up tables or a suitable polynomial expression.
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%
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%
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%%\begin{figure}[h]
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%% \centering
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%% \includegraphics[width=150pt,bb=0 0 273 483,keepaspectratio=true]{./milli volt amplifier/vrange.jpg}
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%% % milli volt amplifier.jpg: 714x180 pixel, 72dpi, 25.19x6.35 cm, bb=0 0 714 180
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%% \caption{Milli Volt Amplifier expected voltage ranges}
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%% \label{fig:milli volt amplifiervrange}
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%%\end{figure}
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%%
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%
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%The voltage ranges we expect from this three stage potential divider\footnote{
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%two stages are required for validation, a third stage is used to measure the current flowing
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%through the circuit to obtain accurate temperature readings}
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%are shown in figure \ref{fig:milli volt amplifiervrange}. Note that there is
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%an expected range for each reading, for a given temperature span.
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%Note that the low reading goes down as temperature increases, and the higher reading goes up.
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%For this reason the low reading will be referred to as {\em sense-}
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%and the higher as {\em sense+}.
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%
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%\subsection{Accuracy despite variable \\ resistance in cables}
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%
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%For electronic and accuracy reasons a four wire circuit is preferred
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%because of resistance in the cables. Resistance from the supply
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% causes a slight voltage
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%drop in the supply to the Milli Volt Amplifier. As no significant current
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%is carried by the two `sense' lines, the resistance back to the ADC
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%causes only a negligible voltage drop, and thus the four wire
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%configuration is more accurate\footnote{The increased accuracy is because the voltage measured, is the voltage across
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%the thermistor and not the voltage across the thermistor and current supply wire resistance.}.
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%
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%\subsection{Calculating Temperature from \\ the sense line voltages}
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%
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%The current flowing though the
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%whole circuit can be measured on the PCB by reading a third
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%sense voltage from one of the load resistors. Knowing the current flowing
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%through the circuit
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%and knowing the voltage drop over the Milli Volt Amplifier, we can calculate its
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%resistance by Ohms law $V=I.R$, $R=\frac{V}{I}$.
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%Thus a little loss of supply current due to resistance in the cables
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%does not impinge on accuracy.
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%The resistance to temperature conversion is achieved
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%through the published Milli Volt Amplifier tables\cite{eurothermtables}.
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%The standard voltage divider equations (see figure \ref{fig:vd} and
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%equation \ref{eqn:vd}) can be used to calculate
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%expected voltages for failure mode and temperature reading purposes.
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%
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%%\begin{figure}[h]
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%% \centering
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%% \includegraphics[width=100pt,bb=0 0 183 170,keepaspectratio=true]{./milli volt amplifier/voltage_divider.png}
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%% % voltage_divider.png: 183x170 pixel, 72dpi, 6.46x6.00 cm, bb=0 0 183 170
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%% \caption{Voltage Divider}
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%% \label{fig:vd}
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%%\end{figure}
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%%%The looking at figure \ref{fig:vd} the standard voltage divider formula (equation \ref{eqn:vd}) is used.
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%
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%\begin{equation}
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%\label{eqn:vd}
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% V_{out} = V_{in}.\frac{Z2}{Z2+Z1}
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%\end{equation}
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%
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%\section{Safety case for 4 wire circuit}
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%
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%This sub-section looks at the behaviour of the Milli Volt Amplifier four wire circuit
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%for the effects of component failures.
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%All components have a set of known `failure modes'.
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%In other words we know that a given component can fail in several distinct ways.
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%Studies have been published which list common component types
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%and their sets of failure modes, often with MTTF statistics \cite{mil1991}.
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%Thus for each component, an analysis is made for each of its failure modes,
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%with respect to its effect on the
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%circuit. Each one of these scenarios is termed a `test case'.
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%The resultant circuit behaviour for each of these test cases is noted.
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%The worst case for this type of
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%analysis would be a fault that we cannot detect.
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%Where this occurs a circuit re-design is probably the only sensible course of action.
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%
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%
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%
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%\subsection{Single Fault FMEA Analysis \\ of Milli Volt Amplifier Four wire circuit}
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%
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%\label{fmea}
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%This circuit simply consists of three resistors.
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%Resistors according to the DOD Electronic component fault handbook
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%1991, fail by either going OPEN or SHORT circuit \cite{mil1991}.
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%%Should wires become disconnected these will have the same effect as
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%%given resistors going open.
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%For the purpose of this analyis;
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%$R_{1}$ is the \ohms{2k2} from 5V to the thermistor,
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%$R_3$ is the Milli Volt Amplifier thermistor and $R_{2}$ connects the thermistor to ground.
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%
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%We can define the terms `High Fault' and `Low Fault' here, with reference to figure
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%\ref{fig:milli volt amplifiervrange}. Should we get a reading outside the safe green zone
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%in the diagram we can consider this a fault.
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%Should the reading be above its expected range this is a `High Fault'
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%and if below a `Low Fault'.
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%
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%Table \ref{ptfmea} plays through the scenarios of each of the resistors failing
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%in both SHORT and OPEN failure modes, and hypothesises an error condition in the readings.
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%The range {0\oc} to {300\oc} will be analysed using potential divider equations to
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%determine out of range voltage limits in section \ref{ptbounds}.
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%
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%\begin{table}[ht]
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%\caption{Milli Volt Amplifier FMEA Single Faults} % title of Table
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%\centering % used for centering table
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%\begin{tabular}{||l|c|c|l|l||}
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%\hline \hline
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% \textbf{Test} & \textbf{Result} & \textbf{Result } & \textbf{General} \\
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% \textbf{Case} & \textbf{sense +} & \textbf{sense -} & \textbf{Symtom Description} \\
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%% R & wire & res + & res - & description
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%\hline
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%\hline
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% $R_1$ SHORT & High Fault & - & Value Out of Range Value \\ \hline
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%$R_1$ OPEN & Low Fault & Low Fault & Both values out of range \\ \hline
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% \hline
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%$R_3$ SHORT & Low Fault & High Fault & Both values out of range \\ \hline
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% $R_3$ OPEN & High Fault & Low Fault & Both values out of range \\ \hline
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%\hline
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%$R_2$ SHORT & - & Low Fault & Value Out of Range Value \\
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% $R_2$ OPEN & High Fault & High Fault & Both values out of range \\ \hline
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%\hline
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%\end{tabular}
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%\label{ptfmea}
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%\end{table}
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%
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%From table \ref{ptfmea} it can be seen that any component failure in the circuit
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%should cause a common symptom, that of one or more of the values being `out of range'.
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%Temperature range calculations and detailed calculations
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%on the effects of each test case are found in section \ref{milli volt amplifierrange}
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%and \ref{milli volt amplifiertemp}.
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%
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%
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%
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%\subsection{Range and Amplifier Calculations}
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%\label{milli volt amplifiertemp}
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%Milli Volt Amplifier resistors are designed to
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%have a resistance of \ohms{100} at {0\oc} \cite{aoe},\cite{eurothermtables}.
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%A suitable `wider than to be expected range' was considered to be {0\oc} to {300\oc}
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%for a given application.
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%According to the Eurotherm Milli Volt Amplifier
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%tables \cite{eurothermtables}, this corresponded to the resistances \ohms{100}
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%and \ohms{212.02} respectively. From this the potential divider circuit can be
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%analysed and the maximum and minimum acceptable voltages determined.
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%These can be used as bounds results to apply the findings from the
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%Milli Volt Amplifier FMEA analysis in section \ref{fmea}.
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%
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%As the Milli Volt Amplifier forms a potential divider with the \ohms{2k2} load resistors,
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%the upper and lower readings can be calculated thus:
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%
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%
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%$$ highreading = 5V.\frac{2k2+milli volt amplifier}{2k2+2k2+milli volt amplifier} $$
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%$$ lowreading = 5V.\frac{2k2}{2k2+2k2+milli volt amplifier} $$
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%So by defining an acceptable measurement/temperature range,
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%and ensuring the
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%values are always within these bounds we can be confident that none of the
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%resistors in this circuit has failed.
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%
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%To convert these to twelve bit ADC (\adctw) counts:
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%
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%$$ highreading = 2^{12}.\frac{2k2+milli volt amplifier}{2k2+2k2+milli volt amplifier} $$
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%$$ lowreading = 2^{12}.\frac{2k2}{2k2+2k2+milli volt amplifier} $$
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%
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%
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%\begin{table}[ht]
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%\caption{Milli Volt Amplifier Maximum and Minimum Values} % title of Table
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%\centering % used for centering table
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%\begin{tabular}{||c|c|c|l|l||}
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%\hline \hline
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% \textbf{Temperature} & \textbf{Milli Volt Amplifier resistance} &
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%\textbf{Lower} & \textbf{Higher} & \textbf{Description} \\
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%\hline
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%% {-100 \oc} & {\ohms{68.28}} & 2.46V & 2.53V & Boundary of \\
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%% & & 2017\adctw & 2079\adctw & out of range LOW \\ \hline
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% {0 \oc} & {\ohms{100}} & 2.44V & 2.56V & Boundary of \\
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% & & 2002\adctw & 2094\adctw & out of range LOW \\ \hline
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% {+300 \oc} & {\ohms{212.02}} & 2.38V & 2.62V & Boundary of \\
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% & & 1954\adctw & 2142\adctw & out of range HIGH \\ \hline
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%\hline
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%\end{tabular}
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%\label{ptbounds}
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%\end{table}
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%
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%Table \ref{ptbounds} gives ranges that determine correct operation. In fact it can be shown that
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%for any single error (short or opening of any resistor) this bounds check
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%will detect it.
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%
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%
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%\section{Single Fault FMEA Analysis \\ of Milli Volt Amplifier Four wire circuit}
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%
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%\subsection{Single Fault Modes as PLD}
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%
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%The component~failure~modes in table \ref{ptfmea} can be represented as contours
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%on a PLD diagram.
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%Each test case, is defined by the contours that enclose
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%it. The test cases here deal with single faults only
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%and are thus enclosed by one contour each.
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%
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%
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%%\begin{figure}[h]
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%% \centering
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%% \includegraphics[width=400pt,bb=0 0 518 365,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier_tc.jpg}
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%% % milli volt amplifier_tc.jpg: 518x365 pixel, 72dpi, 18.27x12.88 cm, bb=0 0 518 365
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%% \caption{Milli Volt Amplifier Component Failure Modes}
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%% \label{fig:milli volt amplifier_tc}
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%%\end{figure}
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%%
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%%ating input Fault
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%This circuit supplies two results, the {\em sense+} and {\em sense-} voltage readings.
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%To establish the valid voltage ranges for these, and knowing our
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%valid temperature range for this example ({0\oc} .. {300\oc}) we can calculate
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%valid voltage reading ranges by using the standard voltage divider equation \ref{eqn:vd}
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%for the circuit shown in figure \ref{fig:vd}.
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%
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%%
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%%\begin{figure}[h]
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%% \centering
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%% \includegraphics[width=100pt,bb=0 0 183 170,keepaspectratio=true]{./milli volt amplifier/voltage_divider.png}
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%% % voltage_divider.png: 183x170 pixel, 72dpi, 6.46x6.00 cm, bb=0 0 183 170
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%% \caption{Voltage Divider}
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%% \label{fig:vd}
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%%\end{figure}
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%%%The looking at figure \ref{fig:vd} the standard voltage divider formula (equation \ref{eqn:vd}) is used.
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%%
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%%\begin{equation}
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%%\label{eqn:vd}
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%% V_{out} = V_{in}.\frac{Z2}{Z2+Z1}
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%%\end{equation}
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%%
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%
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%
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%\subsection{Proof of Out of Range \\ Values for Failures}
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%\label{pt110range}
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%Using the temperature ranges defined above we can compare the voltages
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%we would get from the resistor failures to prove that they are
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%`out of range'. There are six test cases and each will be examined in turn.
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%
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%\subsubsection{ TC 1 : Voltages $R_1$ SHORT }
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%With milli volt amplifier at 0\oc
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%$$ highreading = 5V $$
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%Since the highreading or sense+ is directly connected to the 5V rail,
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%both temperature readings will be 5V..
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%$$ lowreading = 5V.\frac{2k2}{2k2+100\Omega} = 4.78V$$
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%With milli volt amplifier at the high end of the temperature range 300\oc.
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%$$ highreading = 5V $$
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%$$ lowreading = 5V.\frac{2k2}{2k2+212.02\Omega} = 4.56V$$
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%
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%Thus with $R_1$ shorted both readings are outside the
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%proscribed range in table \ref{ptbounds}.
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%
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%\subsubsection{ TC 2 : Voltages $R_1$ OPEN }
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%
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%In this case the 5V rail is disconnected. All voltages read are 0V, and
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%therefore both readings are outside the
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%proscribed range in table \ref{ptbounds}.
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%
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%
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%\subsubsection{ TC 3 : Voltages $R_2$ SHORT }
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%
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%With milli volt amplifier at 0\oc
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%$$ lowreading = 0V $$
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%Since the lowreading or sense- is directly connected to the 0V rail,
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%both temperature readings will be 0V.
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%$$ lowreading = 5V.\frac{100\Omega}{2k2+100\Omega} = 0.218V$$
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%With milli volt amplifier at the high end of the temperature range 300\oc.
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%$$ highreading = 5V.\frac{212.02\Omega}{2k2+212.02\Omega} = 0.44V$$
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%
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%Thus with $R_2$ shorted both readings are outside the
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%proscribed range in table \ref{ptbounds}.
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%
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%\subsubsection{ TC 4 : Voltages $R_2$ OPEN }
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%Here there is no potential divider operating and both sense lines
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%will read 5V, outside of the proscribed range.
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%
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%
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%\subsubsection{ TC 5 : Voltages $R_3$ SHORT }
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%
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%Here the potential divider is simply between
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%the two 2k2 load resistors. Thus it will read a nominal;
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%2.5V.
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%
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%Assuming the load resistors are
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%precision components, and then taking an absolute worst case of 1\% either way.
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%
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%$$ 5V.\frac{2k2*0.99}{2k2*1.01+2k2*0.99} = 2.475V $$
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%
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%$$ 5V.\frac{2k2*1.01}{2k2*1.01+2k2*0.99} = 2.525V $$
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%
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%These readings both lie outside the proscribed range.
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%Also the sense+ and sense- readings would have the same value.
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%
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%\subsubsection{ TC 6 : Voltages $R_3$ OPEN }
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%
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%Here the potential divider is broken. The sense- will read 0V and the sense+ will
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%read 5V. Both readings are outside the proscribed range.
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%
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%\subsection{Summary of Analysis}
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%
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%All six test cases have been analysed and the results agree with the hypothesis
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%put in Table \ref{ptfmea}. The PLD diagram, can now be used to collect the
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%symptoms. In this case there is a common and easily detected symptom for all these single
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%resistor faults : Voltage out of range.
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%
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%A spider can be drawn on the PLD diagram to this effect.
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%
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%In practical use, by defining an acceptable measurement/temperature range,
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%and ensuring the
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%values are always within these bounds we can be confident that none of the
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%resistors in this circuit has failed.
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%
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%
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%%\begin{figure}[h]
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%% \centering
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%% \includegraphics[width=400pt,bb=0 0 518 365,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier_tc_sp.jpg}
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%% % milli volt amplifier_tc.jpg: 518x365 pixel, 72dpi, 18.27x12.88 cm, bb=0 0 518 365
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%% \caption{Milli Volt Amplifier Component Failure Modes}
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%% \label{fig:milli volt amplifier_tc_sp}
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%%\end{figure}
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%%
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%
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%\subsection{Derived Component : The Milli Volt Amplifier Circuit}
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%The Milli Volt Amplifier circuit can now be treated as a component in its own right, and has one failure mode,
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%{\textbf OUT\_OF\_RANGE}. It can now be represnted as a PLD see figure \ref{fig:milli volt amplifier_singlef}.
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%
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%%\begin{figure}[h]
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%% \centering
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%% \includegraphics[width=100pt,bb=0 0 167 194,keepaspectratio=true]{./milli volt amplifier/milli volt amplifier_singlef.jpg}
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%% % milli volt amplifier_singlef.jpg: 167x194 pixel, 72dpi, 5.89x6.84 cm, bb=0 0 167 194
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%% \caption{Milli Volt Amplifier Circuit Failure Modes : From Single Faults Analysis}
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%% \label{fig:milli volt amplifier_singlef}
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%%\end{figure}
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%%
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%
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%%From the single faults (cardinality constrained powerset of 1) analysis, we can now create
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%%a new derived component, the {\emmilli volt amplifiercircuit}. This has only \{ OUT\_OF\_RANGE \}
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%%as its single failure mode.
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%
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%
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%%Interestingly we can calculate the failure statistics for this circuit now.
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%%Mill 1991 gives resistor stats of ${10}^{11}$ times 6 (can we get special stats for milli volt amplifier) ???
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%\clearpage
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%\subsection{Mean Time to Failure}
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%
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%Now that we have a model for the failure mode behaviour of the milli volt amplifier circuit
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%we can look at the statistics associated with each of the failure modes.
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%
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%The DOD electronic reliability of components
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%document MIL-HDBK-217F\cite{mil1992} gives formulae for calculating
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%the
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%%$\frac{failures}{{10}^6}$
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%${failures}/{{10}^6}$ % looks better
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%in hours for a wide range of generic components
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%\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}
|
|
%
|
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%\subsection{Statistics}
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%
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%%%
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%%% Need to talk abou the `detection time'
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%%% or `Safety Relevant Validation Time' ref can book
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%%% EN61508 gives detection calculations to reduce
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%%% statistical impacts of failures.
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%%%
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%
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%If we consider the failure modes to be statistically independent we can calculate
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%the FIT values for all the failures. The failure mode of concern, the undetectable {\textbf{FLOATING}} condition
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%requires that resistors $R_1$ and $R_2$ fail. We can multiply the MTTF
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%together and find an MTTF for both failing. The FIT value of 12.42 corresponds to
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%$12.42 \times {10}^{-9}$ failures per hour. Squaring this gives $ 154.3 \times {10}^{-18} $.
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%This is an astronomically small MTTF, and so small that it would
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%probably fall below a threshold to sensibly consider.
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%However, it is very interesting from a failure analysis perspective,
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%because here we have found a fault that we cannot detect at this
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%level. This means that should we wish to cope with
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%this fault, we need to devise a way of detecting this
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%condition in higher levels of the system.
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%
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%
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%
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%\vspace{20pt}
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%
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%%typeset in {\Huge \LaTeX} \today
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