diff --git a/opamp_circuits_C_GARRETT/opamps.tex b/opamp_circuits_C_GARRETT/opamps.tex index d65cfe8..3a62cfe 100644 --- a/opamp_circuits_C_GARRETT/opamps.tex +++ b/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 -paths between failure modes and components, necessary to achieve RFMEA. +We could term this comkparison~complexity, as it is the number of +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 -having 3 failure modes each, we would have an $CC$ of +Even considering a $fictitious$ system with just 81 components (with these components +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.