Andrew fish comments

opamp_circuits_C_GARRETT/opamps.tex

@@ -870,6 +870,7 @@ and let the set of all possible failure modes be $\mathcal{F}$.
 We now define the function $fm$
 as
 \begin{equation}
+\label{eqn:fm}
 fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}.
 \end{equation}
 This is defined by, where $c$ is a component and $F$ is a set of failure modes,
@@ -902,7 +903,7 @@ fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}.
 \paragraph{Abstraction Levels of {\fgs} and {\dcs}}
 
 
-
+\label{sec:indexsub}
 We can indicate the abstraction level of a component by using a superscript.
 Thus for the component $c$, where it is a `base component' we can assign it
 the abstraction level zero, $c^0$. Should we wish to index the components
@@ -958,7 +959,7 @@ mode (i.e. one or more failure modes that caused it).
 %
 
 \subsection{FMMD Hierarchy}
-\;
+
 By applying stages of analysis to higher and higher abstraction
 levels, we can converge to a complete failure mode model of the system under analysis.
 Because the symptom abstraction process is defined as surjective (from component failure modes to symptoms)
@@ -1085,8 +1086,8 @@ 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 number of 
+We could term this comkparison~complexity, as it is the number of 
-paths between failure modes and components, necessary to achieve RFMEA.
+paths between failure modes and components, necessary to achieve RFMEA, for a given system/functional~group.
 
 
 % (except its self of course, that component is already considered to be in a failed state!).
@@ -1097,27 +1098,39 @@ 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} $G$,  by 
 $ | G | .$
+An indexing and sub-scripting notation to identify particular {\fgs}
+within an FMMD hierarchy is given in section~\ref{sec:indexsub}.
 
-The function $fm$ has a component as its domain and the components failure modes as its range.
+The function $fm$ has a component as its domain and the components failure modes as its range (see equation~\ref{eqn:fm}).
 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_{|G|} $ we can express 
+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 examine every
 failure mode against all the other components in the system.
 We can define this as a function, Comparison Complexity, $CC$, with its domain as the system 
-or {\fg}, $G$, and
+or {\fg}, $\FG$, and
 its range as the number of checks to perform to satisfy a rigorous FMEA inspection.
 
+Where $\mathcal{\FG}$ represents the set of all {\fgs}, and $ \mathbb{N} $ any natural integer, $CC$ is defined by,
+\begin{equation}
+%$$
+ CC(G \in \mathcal{\FG}) \rightarrow \mathbb{N}, 
+%$$
+\end{equation}
+
+and, where n is the number of components in the system/{\fg},   $|fm(c_i)|$ is the number of failure modes
+in component ${c_i}$, is given by
+
 \begin{equation}
 \label{eqn:CC}
 %$$
  %%% when it was called reasoning distance -- 19NOV2011 -- RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1) 
- CC(G) = (n-1) \sum_{1 \le i \le n} fm(c_i)
+ CC(\FG) = (n-1) \sum_{1 \le i \le n} fm(c_i).
 %$$
 \end{equation}
 
 This can be simplified if we can determine the total number of failure modes in the system $K$, (i.e. $ K = \sum_{n=1}^{|G|} {|fm(c_n)|}$);
-equation~\ref{eqn:CC} becomes $$ CC(G) = K.(|G|-1).$$
+equation~\ref{eqn:CC} becomes $$ CC(\FG) = K.(|\FG|-1).$$
 %Equation~\ref{eqn:rd} can also be expressed as
 % 
 % \begin{equation}
@@ -1132,14 +1145,18 @@ An FMMD Hierarchy will have reducing numbers of functional groups as we progress
 In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to
 all {\fgs} on each level.
 
+We define a helper function $g$ that takes a level $\xi$ in an FMMD hierarchy $H$, and returns all the {\fgs} on that level,
+defined by 
+$$g(H, i) \rightarrow \forall {\FG}^{\xi} \;where\; ({\xi} = {i}) \wedge ({\FG}^{\xi} \in H) .$$
+
 Where $L$ represents the number of levels in the FMMD hierarchy,
-$|\xi|$ represents the number of functional groups on the level
+$|g(\xi)|$ represents the number of functional groups on the level
 and $H$ represents an FMMD hierarchy,
-we overload the comparison complexity equation thus:
+we overload the comparison complexity thus:
 %$$
 \begin{equation}
      \label{eqn:gf}
-      CC(H) = \sum_{\xi=0}^{L} \sum_{j=1}^{|\xi|} CC({G}_{j}^{\xi}).
+      CC(H) = \sum_{\xi=0}^{L} \sum_{j=1}^{|g(H,\xi)|} CC({\FG}_{j}^{\xi}).
 %$$
 \end{equation}
 
@@ -1147,11 +1164,11 @@ we overload the comparison complexity equation thus:
 \pagebreak[4]
 \subsection{Complexity Comparison Examples}
 
-The potential divider discussed in section~\ref{potdivfmmd} has a four failure modes and two components and therefore has an $CC$ of 4.
+The potential divider discussed in section~\ref{potdivfmmd} has  four failure modes and two components and therefore has   $CC$ of 4.
 $$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
 
-Were we to consider a $fictitious$ system with 81 components, with these components
+Even considering  a $fictitious$ system with just 81 components (with these components
-having 3 failure modes each, we would have an $CC$ of 
+having 3 failure modes each) we would have an $CC$ of 
 
 $$CC(fictitious) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$
 
@@ -1162,7 +1179,11 @@ The computational order for RFMEA would be polynomial ($O(N^2.K)$) (where $K$ is
 
 This order may be acceptable in a computational environment: However, the choosing of {\fgs} and the analysis
 process are human activities. It can be seen that it is practically impossible to achieve 
-RFMEA for anything but trival systems. FMMD reduces the comparison complexity enough to make
+RFMEA for anything but trival systems.
+%
+% Next statement needs alot of justification
+%
+FMMD reduces the comparison complexity enough to make
 rigorous checking feasible.