OK now need to do working out for the final two stages

opamp_circuits_C_GARRETT/opamps.tex

@@ -35,6 +35,11 @@ Not a document to be proof read.
 Proof of analysis concept.
 
 Function $fm$ applied to a component returns its failure modes.
+
+
+The circuits specified are not typical saftey critical circuitry which usually
+has both redundancy and self~checking and/or diagnostic features build in. 
+These are examples of the FMMD methodology being applied to some standard electronic circuits.
 \end{abstract}
 \maketitle
 \tableofcontents    
@@ -457,7 +462,7 @@ We merely have to choose a top level event and work down using $XOR$ gates.
 
 This circuit performs poorly from a safety point of view.
 Its failure modes could be indistinguishable from valid readings (especially
-wihen it becomes a V2 follower). 
+when it becomes a V2 follower). 
 
 \begin{figure}[h]
  \centering
@@ -773,10 +778,20 @@ This circuit is described in the Analog Applications Journal~\cite{bubba}.
 The circuit uses four 45 degree phase shifts, and an inverting amplifier to provide
 gain and the final 180 degrees of phase shift (making a total of 360 degrees of phase shift).
 
-We identifiy three functional groups, the inverting amplifer (analysed in section~\ref{fig:invamp}),
-a 45 degree phase shifter (a {$10k\Omega$} resistor and a $10nF$ capacitor) and a noninverting buffer
+From a fault finding perspective this circuit is less than ideal.
+The signal path is circular (its a positive feedback circuit) and most failures would simply cause the output to stop oscillating.
+However, FMMD is a bottom -up analysis methodology and we can therefore still identify
+{\fgs} and apply analysis from a failure mode perspective.
+
+If we were to analyse this circuit without modularisation, we have 14 components with
+($4.4 +10.2 = 36$) failure modes . Applying equation~\ref{eqnrd2} gives a complexity comparison figure of $13.36=468$.
+The reduce the complexity required to analyse this circuit we apply FMMD and start by determining {\fgs}.
+ 
+
+We identify three types functional groups, an inverting amplifier (analysed in section~\ref{fig:invamp}),
+a 45 degree phase shifter (a {$10k\Omega$} resistor and a $10nF$ capacitor) and a non-inverting buffer
 amplifier. We can name these $INVAMP$, $PHS45$ and $NIBUFF$ respectively.
-We can use these {\fgs} to describe the circuit in block diagram form, as in figure ~\ref{fig:bubbablock}.
+We can use these {\fgs} to describe the circuit in block diagram form, see figure ~\ref{fig:bubbablock}.
 
 \begin{figure}[h]
  \centering
@@ -838,7 +853,7 @@ $$ fm(NIBUFF) = fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$
 
 % describe what we are doing, a buffered 45 degree phase shift element
 
-\subsection{Bringing the functional Groups Together: The `Bubba' Oscillator.}
+\subsection{Bringing the functional Groups Together: FMMD model of the `Bubba' Oscillator.}
 
 We could at this point bring all the {\dcs} together into one large functional 
 group (see figure~\ref{fig:poss1finalbubba})
@@ -848,7 +863,9 @@ The capactior and 180 degree inverting amplifier, form a {\fg}
 providing an amplified 225 degree phase shift, which we can call $PHS225AMP$.
 %
 We could also merge the $NIBUFF$ and $PHS45$
-{\dcs} into a {\fg} and the resulant derived component from this we could call a $BUFF45$, and then with those three, form a $PHS135BUFFERED$ functional group -- with the remaining $PHS45$ and the $INVAMP$ in a second group $PHS225AMP$,
+{\dcs} into a {\fg} and the resulant derived component from this we could call a $BUFF45$, 
+and then with those three, form a $PHS135BUFFERED$ 
+functional group---with the remaining $PHS45$ and the $INVAMP$ (re-used from section~\ref{sec:invamp})in a second group $PHS225AMP$---
 and then merge $PHS135BUFFERED$ and  $PHS225AMP$ in a final stage (see figure~\ref{fig:poss2finalbubba})
 
 
@@ -1239,7 +1256,14 @@ in component ${c_i}$, is given by
 \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 $$ CC(\FG) = K.(|\FG|-1).$$
+equation~\ref{eqn:CC} becomes 
+
+%$$
+\begin{equation}
+\label{eqn:rd2}
+ CC(\FG) = K.(|\FG|-1).
+\end{equation}
+%$$
 %Equation~\ref{eqn:rd} can also be expressed as
 % 
 % \begin{equation}

thesis.tex

@@ -5,7 +5,7 @@
 \usepackage{tikz} 
 \usetikzlibrary{shapes,snakes}
 \usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows}
-\usepackage{subfigure}
+%\usepackage{subfigure}
 \usepackage{amsfonts,amsmath,amsthm}
 \usepackage{algorithm}
 \usepackage{algorithmic}