Merge branch 'master' of dev:/home/robin/git/thesis

opamp_circuits_C_GARRETT/opamps.tex

@@ -825,7 +825,7 @@ 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)
@@ -970,9 +970,18 @@ its range as the number of checks to perform to satisfy a rigorous FMEA inspecti
 
 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 $$ RD(fg) = fT.(|fg|-1).$$
+Equation~\ref{eqn:rd} can also be expressed as
+
+\begin{equation}
+\label{eqn:rd2}
+%$$
+ RD(fg) = {|fg|}.{|fm(c_n)|}.{(|fg|-1)} .
+%$$
+\end{equation}
+
 
 \pagebreak[4]
-\subsection{Reasoning Distance Examples}(c-1)
+\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.
 $$RD(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
@@ -1032,17 +1041,17 @@ Starting at the top, we have a {\fg} with three derived components, each of whic
 three failure modes.
 Thus the number of checks to make in the top level is $3^0.3.2.3=18$.
 On the level below that, we have three {\fgs} each with a 
-an identical number of checks, $3^1.3.2.3=56$.{\fg}
+an identical number of checks, $3^1.3.2.3=56$.%{\fg}
 On the level below that we have nine {\fgs}, $3^2.3.2.3=168$.
-Adding these together gives $242$ checks to make to perform RFMEA \textbf{within}
-{\fgs}.
+Adding these together gives $242$ checks to make to perform FMMD (i.e. RFMEA \textbf{within the}
+{\fgs}).
 
 If we were to take the system represented in  figure~\ref{fig:three_tree}, and 
 apply RFMEA on it as a whole system, we can use equation~\ref{eqn:rd},
 $ RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1)$, where $|fg|$ is 27, $fm(c_n)$ is 3
 and $(|fg|-1)$ is 26.
 This gives:
-$ RD(fg) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$
+$RD(fg) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$.
 
 In order to get general equations with which to compare RFMEA with FMMD
 we can re-write equation~\ref{eqn:rd} in terms of the number of levels 
@@ -1069,6 +1078,26 @@ We can now use  equation~\ref{eqn:anscen} and \ref{eqn:fmea_state_exp22} to comp
 the two approaches, for the work required to perform rigorous checking.
 
 
+For instance, having four levels 
+of FMMD analysis, with these fixed numbers,
+%(in addition to the top zeroth level)
+will require 81 base level components.
+
+$$
+%\begin{equation}
+  \label{eqn:fmea_state_exp22}
+  3^4.(3^4-1).3 = 81.(81-1).3 = 19440 % \\
+  %(N^2 - N).f 
+%\end{equation}
+$$
+
+$$
+%\begin{equation}
+ % \label{eqn:anscen}
+  \sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720
+%\end{equation}
+$$
+
 \subsection{Exponential squared to Exponential}
 
 can I say that ?