added double fault diagrams

pt100/pt100.tex

@@ -268,7 +268,7 @@ and are thus enclosed by one contour each.
  \label{fig:pt100_tc}
 \end{figure}
 
-ating input Fault  
+%ating input Fault  
 This circuit supplies two results, sense+ and 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
@@ -612,19 +612,42 @@ TC 18: &  $R_2$ SHORT $R_3$ SHORT   &  low        &   low    &  Both out of Rang
 
 \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}, reproduced below to verify this.
+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.
+  \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}
+ \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}
+}
 
-$|FM(C_j)|$ is always 2 here,  as all the components are resistors and have two failure modes.
+
+$|FM(C_j)|$ 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 !
@@ -740,13 +763,77 @@ The sense- value will be out of range.
 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]{pt100/plddouble.jpg}
+ % plddouble.jpg: 730x641 pixel, 72dpi, 25.75x22.61 cm, bb=0 0 730 641
+ \caption{PT100 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 pt100 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]{pt100/plddoublesymptom.jpg}
+ % plddouble.jpg: 730x641 pixel, 72dpi, 25.75x22.61 cm, bb=0 0 730 641
+ \caption{PT100 Double Simultaneous Faults}
+ \label{fig:plddoublesymptom}
+\end{figure}
+
+
+\clearpage
+\subsection{Derived Component : The PT100 Circuit}
+The PT100 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:pt100_doublef}.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=100pt,bb=0 0 167 194,keepaspectratio=true]{./pt100/pt100_doublef.jpg}
+ % pt100_singlef.jpg: 167x194 pixel, 72dpi, 5.89x6.84 cm, bb=0 0 167 194
+ \caption{PT100 Circuit Failure Modes : From Double Faults Analysis}
+ \label{fig:pt100_doublef}
+\end{figure}
+
+\subsection{Statistics}
+
+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
+%typeset in {\Huge \LaTeX} \today
-