Added reasoning distance essay

opamp_circuits_C_GARRETT/opamps.tex

@@ -74,7 +74,7 @@ We can now examine what effect each of these failures will have on the {\fg}.
 
 
 \subsection{Analysing a potential divider in terms of failure modes}
-
+\label{potdivfmmd}
 \begin{figure}[h+]
  \centering
  \includegraphics[width=100pt,keepaspectratio=true]{./pd.png}
@@ -553,4 +553,64 @@ power-supply (for instance these might, for the sake of example  include: $NO\_P
 Our logic circuit may be able to cope with $LOW\_VOLTAGE$ and $NOISE\_LF$, but react with a serious symptom to $NOISE\_HF$ say.
 But in order to process these failure modes it must be at a higher stage in the FMMD hierarchy.
 
+\pagebreak[4]
+\section{Defining the concept of `reasoning distance' in FMEA}
+
+%
+% DOMAIN == INPUTS
+% RANGE == OUTPUTS
+%
+
+When performing FMEA we have system under investigation, which will comprise of a collection of components which have associated failure modes.
+The object of FMEA is to determine cause and effect: from the failure modes (the causes) to the effects (or symptoms of failure).
+%
+To perform FMEA rigorously
+we could stipulate that every failure mode must be checked for effects
+against all the components in the system.
+We could term this `rigorous~FMEA'~(RFMEA).
+The number of checks we have to make to achieve this gives an indication of the complexity of the task.
+%
+We could term this complexity a reasoning distance, as it is the sum of 
+all the paths between failure modes and components, necessary to achieve RFMEA.
+
+% (except its self of course, that component is already considered to be in a failed state!).
+%
+Obviously, for a small number of components and failure modes we have a smaller number
+of checks to make than for a complicated larger system.
+%
+We can consider the system as a large {\fg} of components.
+We represent the number of components in the {\fg} by 
+$$ | fg | .$$
+
+The function $fm$ has a component as its domain and the components failure modes as its range.
+We can represent the number of failure modes in a component $c$, to be  $$ | fm(c) | .$$
+
+If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|fg|} $ we can express 
+the number of checks required to rigorously check every
+failure mode against all the other components in the system.
+We can define this as a function, $RD$, with its domain as the system or {\fg}, $fg$, and
+its range as the number of checks to perform to satisfy a rigorous FMEA inspection.
+
+\begin{equation}
+\label{eqn:rd}
+%$$
+ RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1) 
+%$$
+\end{equation}
+
+This can be simplified if we can determine the total number of failure modes in the system $fT$, (i.e. $ fT = \sum_{n=1}^{|fg|} {|fm(c_n)|}$);
+equation~\ref{eqn:rd} becomes $$ RD(fg) = fT.(|fg|-1).$$
+
+\pagebreak[4]
+\subsection{Reasoning Distance Examples}
+
+The potential divider discussed in section~\ref{potdivfmmd} has a four failure modes and two components and therefore has an $RD$ of 4.
+$$RD(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
+
+Were we to consider a $fictitious$ system with 81 components, with these components
+having 3 failure modes each, we would have an $RD$ of 
+
+$$RD(fictitious) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$
+
+
 \end{document}