From d0cfa0651b85d86e6c5e5a9e62e0e3d718ceac50 Mon Sep 17 00:00:00 2001
From: Robin Clark <R.P.Clark@brighton.ac.uk>
Date: Sun, 23 Sep 2012 18:41:07 +0100
Subject: [PATCH] Need to check the Bubba OSC. Chris said the 90_degree symptom
 of the PHS45 was actually 0_degrees. So that ripples through. And then do the
 SUMJ, mix it with the opamp

---
 submission_thesis/CH4_FMMD/Makefile           |   2 +-
 submission_thesis/CH4_FMMD/cfg.dia            | Bin 2351 -> 2379 bytes
 submission_thesis/CH4_FMMD/copy.tex           |   7 +-
 submission_thesis/CH5_Examples/Makefile       |   2 +-
 .../CH5_Examples/bubba_euler_1.dia            | Bin 0 -> 2759 bytes
 .../CH5_Examples/bubba_euler_2.dia            | Bin 0 -> 2433 bytes
 submission_thesis/CH5_Examples/copy.tex       | 370 +++++++-----------
 7 files changed, 152 insertions(+), 229 deletions(-)
 create mode 100644 submission_thesis/CH5_Examples/bubba_euler_1.dia
 create mode 100644 submission_thesis/CH5_Examples/bubba_euler_2.dia

diff --git a/submission_thesis/CH4_FMMD/Makefile b/submission_thesis/CH4_FMMD/Makefile
index b92da04..2ade291 100644
--- a/submission_thesis/CH4_FMMD/Makefile
+++ b/submission_thesis/CH4_FMMD/Makefile
@@ -1,6 +1,6 @@
 
 
-PNG_DIA = cfg2.png  cfg.png  compco2.png  compco3.png  compco.png  component.png  componentpl.png  fmmd_uml2.png  fmmd_uml.png  partitioncfm.png master_uml.png top_down_de_comp.png dc1.png dc2.png eulerfmmd.png
+PNG_DIA = cfg2.png  cfg.png  compco2.png  compco3.png  compco.png  component.png  componentpl.png  fmmd_uml2.png  fmmd_uml.png  partitioncfm.png master_uml.png top_down_de_comp.png dc1.png dc2.png eulerfmmd.png 
 
 
 %.png:%.dia
diff --git a/submission_thesis/CH4_FMMD/cfg.dia b/submission_thesis/CH4_FMMD/cfg.dia
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diff --git a/submission_thesis/CH4_FMMD/copy.tex b/submission_thesis/CH4_FMMD/copy.tex
index cf58070..ffaa685 100644
--- a/submission_thesis/CH4_FMMD/copy.tex
+++ b/submission_thesis/CH4_FMMD/copy.tex
@@ -830,6 +830,7 @@ as {\fcs} in table~\ref{tbl:ampfmea1}.
 %
 %
 % 
+\label{sec:invamp}
 %
 \begin{figure}[h+]
  \centering
@@ -1476,11 +1477,11 @@ that it inherits a set of failure modes.
 %We thus have a `new' component, %or system building block, but 
 %with a known and traceable
 %fault behaviour.
-A {\fg} must comprise of two or more components, and the UML diagram shows this
-with the two to many relationship.
+A {\fg} must comprise of at least one component, and the UML diagram shows this
+with the one to many relationship.
 Under exceptional circumstances a component may need to be a member of more than 
 one {\fg} (this is looked at in section~\ref{sec:sideeffects}). The relationship between
-the {\fg} and component is therefore $ \star \leftrightarrow 2..\star$.
+the {\fg} and component is therefore $ \star \leftrightarrow 1..\star$.
 %
 A {\fg} will only be associated with one {\dc} and is given a one to one relationship in the UML diagram.
 %
diff --git a/submission_thesis/CH5_Examples/Makefile b/submission_thesis/CH5_Examples/Makefile
index 5b89d49..db91feb 100644
--- a/submission_thesis/CH5_Examples/Makefile
+++ b/submission_thesis/CH5_Examples/Makefile
@@ -5,7 +5,7 @@ PNG_DIA = blockdiagramcircuit2.png  bubba_oscillator_block_diagram.png  circuit1
 	poss1finalbubba.png   poss2finalbubba.png   pt100.png   pt100_doublef.png   pt100_singlef.png       \
 	pt100_tc.png         pt100_tc_sp.png      shared_component.png   stat_single.png    three_tree.png  \
 	tree_abstraction_levels.png   vrange.png sigma_delta_block.png ftcontext.png ct1.png hd.png         \
-        sigdel1.png sdadc.png
+        sigdel1.png sdadc.png bubba_euler_1.png bubba_euler_2.png
 
 
 
diff --git a/submission_thesis/CH5_Examples/bubba_euler_1.dia b/submission_thesis/CH5_Examples/bubba_euler_1.dia
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diff --git a/submission_thesis/CH5_Examples/bubba_euler_2.dia b/submission_thesis/CH5_Examples/bubba_euler_2.dia
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diff --git a/submission_thesis/CH5_Examples/copy.tex b/submission_thesis/CH5_Examples/copy.tex
index 34dbe37..cac68f0 100644
--- a/submission_thesis/CH5_Examples/copy.tex
+++ b/submission_thesis/CH5_Examples/copy.tex
@@ -7,7 +7,16 @@
 % 
 % * OPERATIONAL STATE (perhaps a self test on an ADC where it is set to output and driven high and low and read)
 
-
+% to do: 23SEP2012
+%
+% 90_degrees is an incorrect failure mode in bubba and must be purged
+%
+% summing junction in sigma delta is not a valid fg, prob have to include
+% the op-amp....
+%
+% very annoying to have to pull out the comparison complexity.
+% makes the comparisons between approaches have less meaning.
+% have to discuss this.
 
 \label{sec:chap5}
 
@@ -25,15 +34,18 @@ we are conforming to for our particular project).
 This is followed by several example FMMD analyses, 
 the first analysing a common configuration of
 the inverting amplifier (see section~\ref{sec:invamp}) using 
-an op-amp and two resistors, which demonstrates how the potential divider from section~\ref{subsec:potdiv}
-~\ref{sec:chap4} 
-can be re-used. %, but with provisos.
+an op-amp and two resistors, which demonstrates how the re-use of the potential divider from section~\ref{subsec:potdiv}.
+The inverting amplifier is analysed again, but this time with different
+{\fgs}. The two approaches, i.e. choice of membership for {\fgs},  are then discussed.
+%~\ref{sec:chap4} 
+%can be re-used. %, but with provisos.
 %
 %The first
 %(see section~\ref{sec:diffamp}) 
 Section~\ref{sec:diffamp} analyses a circuit where two op-amps are used 
 to create a differencing amplifier. 
-Re-use of the potential divider model is discussed in the context of this circuit, 
+Building on the two approaches section~\ref{sec:invamp}, re-use of the potential divider {\dc}
+is discussed in the context of this circuit, 
 where its re-use is appropriate in the first stage and 
 not in the second.
 %
@@ -48,6 +60,7 @@ Section~\ref{sec:sigmadelta} shows FMMD analysing the sigma delta analogue to di
 analogue and digital signals.
 %
 % Moving Pt100 to metrics
+%
 %Sections~\ref{sec:Pt100}~and~\ref{sec:Pt100d} demonstrate both statistical 
 %failure mode classification %  analysis for top level events traced back to {\bc} failure modes
 %and the analysis of double simultaneous failure modes.
@@ -222,13 +235,14 @@ as shown below.
  \item Shorted 3.9\% $\mapsto$  SHORT
  \item Lead damage 1.9\% $\mapsto$ OPEN.
 \end{itemize}
+%
 The main causes of drift are overloading of components.
 This is borne out in in the FMD-91~\cite{fmd91}[232] entry for a resistor network where the failure
 modes do not include drift.
 %
 If we can ensure that our resistors will not be exposed to overload conditions, the 
 probability of drift (sometimes called parameter change) occurring
-is significantly reduced, enough for some standards to exclude it~\cite{en298}.
+is significantly reduced, enough for some standards to exclude it~\cite{en298}~\cite{en230}.
 
 \paragraph{Resistor failure modes according to EN298.}
 
@@ -554,121 +568,7 @@ component {\fms} in FMEA or FMMD and require interpretation.
 % is shown as a `$\derivec$' symbol.
 % 
 % 
-% 
-% \section{Example Analysis: Non-Inverting OPAMP}
-% \label{sec:noninvamp}
-% Consider a non inverting op-amp designed to amplify
-% a small positive voltage (typical use would be a thermocouple amplifier
-% taking a range from 0 to 25mV and amplifying it to the useful range of an ADC, approx 0 to 4 volts).
-%  
-%  
-% \begin{figure}[h+]
-%  \centering
-%  \includegraphics[width=100pt]{CH5_Examples/mvampcircuit.png}
-%  % mvampcircuit.png: 243x143 pixel, 72dpi, 8.57x5.04 cm, bb=0 0 243 143
-%  \label{fig:mvampcircuit}
-%  \caption{positive mV amplifier circuit}
-% \end{figure}
-% 
-% We can begin by looking for functional groups. 
-% The resistors $ R1, R2 $  perform a fairly common function in electronics, that of the potential divider.
-% So we can examine  $\{ R1, R2 \}$  as a {\fg}. 
-% 
-% 
-% \subsection{The Resistor in terms of failure modes}
-% 
-% We can now determine how the resistors can fail.
-% We consider the {\fms} for resistors to be OPEN and SHORT (see section~\ref{ros}).
-% %, i.e.
-% %$ fm(R) = \{ OPEN, SHORT \} . $
-% 
-% We can express the failure modes of a component using the function $fm$, thus for the resistor,  $ fm(R) = \{ OPEN, SHORT \}$.
-% 
-% 
-% We have two resistors in this circuit and therefore four component failure modes to consider for the potential divider.
-% We can now examine what effect each of these failures will have on the {\fg} (see table~\ref{tbl:pd}).
-% 
-% 
-% \subsection{Analysing a potential divider in terms of failure modes}
-% 
-% 
-% \label{potdivfmmd}
-% 
-% 
-% 
-% \begin{figure}[h+]
-%  \centering
-%  \includegraphics[width=100pt,keepaspectratio=true]{CH5_Examples/pd.png}
-%  % pd.png: 361x241 pixel, 72dpi, 12.74x8.50 cm, bb=0 0 361 241
-%  \label{fig:pdcircuit}
-%  \caption{Potential Divider Circuit}
-% \end{figure}
-%  
-% 
-% \begin{table}[h+]
-% \caption{Potential Divider: Single failure analysis}
-% \begin{tabular}{|| l | l | c | c | l ||} \hline
-%  \textbf{Failure Scenario} & &  \textbf{Pot Div Effect}  &   & \textbf{Symptom}          \\ 
-%                \hline
-%      FS1: R1 SHORT    &  &  $LOW$   &  &  $PDLow$              \\  
-%      FS2: R1 OPEN     &  &  $HIGH$  &  &  $PDHigh$             \\  \hline
-%      FS3: R2 SHORT    &  &  $HIGH$  &  &  $PDHigh$             \\ 
-%      FS4: R2 OPEN     &  &  $LOW$   &  &  $PDLow$              \\ \hline
-% \hline
-% \end{tabular}
-% \label{tbl:pd}
-% \end{table}
-% 
-% We can now create a {\dc} for the potential divider, $PD$.
-% 
-%  $$ fm(PD) = \{ PDLow, PDHigh \}$$
-% 
-% %Let us now consider the op-amp. According to 
-% %FMD-91~\cite{fmd91}[3-116] an op-amp may have the following failure modes:
-% %latchup(12.5\%), latchdown(6\%), nooperation(31.3\%), lowslewrate(50\%).
-% 
-% 
-% \subsection{Analysing the non-inverting amplifier in terms of failure modes}
-% 
-% From section~\ref{sec:opamp_fms}
-% $$ fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$
-% 
-% 
-% We can now form a {\fg} with $PD$ and $OPAMP$.
-% 
-% \begin{figure}
-%  \centering
-%  \includegraphics[width=300pt]{CH5_Examples/non_inv_amp_fmea.png}
-%  % non_inv_amp_fmea.png: 964x492 pixel, 96dpi, 25.50x13.02 cm, bb=0 0 723 369
-%  \label{fig:invampanalysis}
-% \end{figure}
-% 
-% 
-% 
-% 
-% \begin{table}[h+]
-% \caption{NIAMP: Single failure analysis}
-% \begin{tabular}{|| l | l | c | c | l ||} \hline
-%  \textbf{Failure Scenario} & &  \textbf{Non In Amp Effect}  &   & \textbf{Symptom}          \\ 
-%                \hline
-%      FS1: PD HIGH              &  &  $LOW$   &  &  $Low$              \\  
-%      FS2: PD LOW               &  &  $HIGH$  &  &  $High$             \\  \hline
-%      FS3: OPAMP $L_{UP}$       &  &  $HIGH$  &  &  $High$             \\ 
-%      FS4: OPAMP $L_{DOWN}$     &  &  $LOW$   &  &  $Low$              \\ 
-%      FS5: OPAMP $Noop$         &  &  $LOW$   &  &  $Low$              \\  
-%      FS5: OPAMP $Low slew$     &  &  $LOW$   &  &  $Lowpass$              \\ \hline
-% 
-% \hline
-% \end{tabular}
-% \label{tbl:pd}
-% \end{table}
-% 
-% We can collect symptoms from the analysis and create a derived component 
-% to represent the non-inverting amplifier $NI\_AMP$.
-% We can now express the failure mode behaviour of this type of amplifier thus:
-% 
-% $$ fm(NIAMP) = \{  {lowpass}, {high}, {low} \}.$$
-% 
+%
 % 
 
 \clearpage
@@ -1591,7 +1491,7 @@ determine {\dcs}.
 This has been analysed in section~\ref{sec:invamp}.
 The inverting amplifier, as a {\dc}, has the following failure modes:
 
-$$ fm(INVAMP) = \{ HIGH, LOW, LOW PASS \}. $$
+$$ fm(INVAMP) = \{ AMP\_High, AMP\_Low, LowPass \}. $$ % \{ HIGH, LOW, LOW PASS \}. $$
 
 % METRICS and has a CC of 10.
 
@@ -1612,7 +1512,8 @@ Our functional group for the phase shifter consists of a resistor and a capacito
  \textbf{Failure Scenario} & &  \textbf{First Order}        &   & \textbf{Symptom}          \\ 
                            & &  \textbf{Low Pass Filter}    &   &                           \\
                \hline
-     FS1: R  SHORT    &  &  90 degree's of phase shift   &  &    $90\_phaseshift$           \\ \hline
+     FS1: R  SHORT    &  &  0 degree's of phase shift    &  &    $0\_phaseshift$           \\ \hline
+     %                      90 degree's of phase shift   &  &    $90\_phaseshift$           \\ \hline
      FS2: R  OPEN     &  &  No Signal                    &  &    $nosignal$           \\ \hline
      FS3: C  SHORT    &  &  Grounded,No Signal           &  &    $nosignal$           \\ \hline
      FS4: C  OPEN     &  &  0 degree's of phase shift    &  &    $0\_phaseshift$           \\ \hline
@@ -1624,10 +1525,10 @@ Our functional group for the phase shifter consists of a resistor and a capacito
 % PHS45 
 
 
-$$ fm (G_0) = \{ 90\_phaseshift, nosignal, 0\_phaseshift \} $$
-
-$$ CC(G_0) = 4.1 = 4 $$
+$$ fm (G_0) = \{  nosignal, 0\_phaseshift \} $$
 
+%$$ CC(G_0) = 4 \times 1 = 4 $$
+%23SEP2012
 \subsection{Non Inverting Buffer: NIBUFF.}
 
 The non-inverting buffer functional group, is comprised of one component, an op-amp.
@@ -1654,17 +1555,25 @@ Initially we use the first identified {\fgs} to create our model without further
 \subsection{FMMD Analysis using initially identified functional groups}
 
 Our functional group for this analysis can be expressed thus:
-$$ G^1_0 = PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 PHS45^1_4, INVAMP^1_0 \} .$$
+or in Euler diagram format as in figure~\ref{fig:bubbaeuler1}.
+$$ G^1_0 =  \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 PHS45^1_4, INVAMP^1_0 \} ,$$
 
-\begin{figure}[h+]
+% HTR 23SEP2012 \begin{figure}[h+]
+% HTR 23SEP2012  \centering
+% HTR 23SEP2012  \includegraphics[width=300pt,keepaspectratio=true]{CH5_Examples/poss1finalbubba.png}
+% HTR 23SEP2012  % largeosc.png: 916x390 pixel, 72dpi, 32.31x13.76 cm, bb=0 0 916 390
+% HTR 23SEP2012  \caption{Bubba Oscillator: One large functional group using the initial functional groups to model oscillator.}
+% HTR 23SEP2012  \label{fig:poss1finalbubba}
+% HTR 23SEP2012 \end{figure}
+
+\begin{figure}[h]
  \centering
- \includegraphics[width=300pt,keepaspectratio=true]{CH5_Examples/poss1finalbubba.png}
- % largeosc.png: 916x390 pixel, 72dpi, 32.31x13.76 cm, bb=0 0 916 390
- \caption{Bubba Oscillator: One large functional group using the initial functional groups to model oscillator.}
- \label{fig:poss1finalbubba}
+ \includegraphics[width=400pt]{./CH5_Examples/bubba_euler_1.png}
+ % bubba_euler_1.png: 946x404 pixel, 72dpi, 33.37x14.25 cm, bb=0 0 946 404
+ \caption{Euler diagram showing the hierarchy of the initial FMMD analysis performed on the  Bubba Oscillator circuit.}
+ \label{fig:bubbaeuler1}
 \end{figure}
 
-
 \begin{table}[h+]
 \caption{Bubba Oscillator: Failure Mode Effects Analysis: One Large Functional Group} % title of Table
 \label{tbl:bubbalargefg}
@@ -1677,40 +1586,40 @@ $$ G^1_0 = PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 P
 
      FS1: $PHS45_1$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
      FS2: $PHS45_1$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
-     FS3: $PHS45_1$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
+    % FS3: $PHS45_1$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
 
-     FS4: $NIBUFF_1$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
-     FS5: $NIBUFF_1$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
-     FS6: $NIBUFF_1$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
-     FS7: $NIBUFF_1$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 
+     FS3: $NIBUFF_1$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
+     FS4: $NIBUFF_1$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
+     FS5: $NIBUFF_1$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
+     FS6: $NIBUFF_1$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 
 
-     FS8: $PHS45_2$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
-     FS9: $PHS45_2$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
-     FS10: $PHS45_2$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
+     FS7: $PHS45_2$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
+     FS8: $PHS45_2$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
+     %FS10: $PHS45_2$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
 
 
-     FS11: $NIBUFF_2$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
-     FS12: $NIBUFF_2$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
-     FS13: $NIBUFF_2$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
-     FS14: $NIBUFF_2$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 
+     FS9: $NIBUFF_2$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
+     FS10: $NIBUFF_2$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
+     FS11: $NIBUFF_2$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
+     FS12: $NIBUFF_2$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 
 
-    FS15: $PHS45_3$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
-     FS16: $PHS45_3$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
-     FS17: $PHS45_3$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
+    FS13: $PHS45_3$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
+     FS14: $PHS45_3$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
+   %  FS17: $PHS45_3$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
 
-     FS18: $NIBUFF_3$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
-     FS19: $NIBUFF_3$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
-     FS20: $NIBUFF_3$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
-     FS21: $NIBUFF_3$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 
+     FS15: $NIBUFF_3$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
+     FS16: $NIBUFF_3$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
+     FS17: $NIBUFF_3$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
+     FS18: $NIBUFF_3$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 
 
-    FS22: $PHS45_4$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
-     FS23: $PHS45_4$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
-     FS24: $PHS45_4$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
+    FS19: $PHS45_4$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
+     FS20: $PHS45_4$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
+  %   FS24: $PHS45_4$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
 
-     FS25: $INVAMP$  $OUTOFRANGE$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
-    FS26: $INVAMP$  $ZEROOUTPUT$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
-    FS27: $INVAMP$  $NOGAIN$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
-    FS28: $INVAMP$  $LOWPASS$             &  &  signal lost                      &  &   $NO_{osc}$          \\  \hline 
+     FS21: $INVAMP$  $OUTOFRANGE$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
+    FS22: $INVAMP$  $ZEROOUTPUT$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
+    FS23: $INVAMP$  $NOGAIN$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
+    FS24: $INVAMP$  $LOWPASS$             &  &  signal lost                      &  &   $NO_{osc}$          \\  \hline 
   
 
  % FS1: $CAP_{10nF}$  $OPEN$          &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
@@ -1724,23 +1633,25 @@ $$ G^1_0 = PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 P
 Collecting symptoms from table~\ref{tbl:bubbalargefg} we can show that for single failure modes, applying $fm$ to the bubba oscillator 
 returns three failure modes,
 
-$$ fm(BubbaOscillator) = \{ NO_{osc}, HI_{fosc}, LO_{fosc} \} . $$
+$$ fm(BubbaOscillator) = \{ NO_{osc}, HI_{fosc}\} . $$ %, LO_{fosc} \} . $$
 
-For the final stage of this FMMD model, we can calculate the complexity using equation~\ref{eqn:rd2}.
-$$ CC = 28.8 = 224$$
+%For the final stage of this FMMD model, we can calculate the complexity using equation~\ref{eqn:rd2}.
+%$$ CC = 28 \times 8 = 224$$
 
-To obtain the total comparison complexity ($TCC$), we need to add the complexity from the 
-{\dcs} that $BubbaOscillator$ was built from.
+%To obtain the total comparison complexity ($TCC$), we need to add the complexity from the 
+%{\dcs} that $BubbaOscillator$ was built from.
 
-$$ TCC = 28.8 + 4.4 + 4.0 + 10 = 250$$
+%$$ TCC = 28 \times 8 + 4 \times 4 + 4 \times 0 + 10 = 250$$
 
 %As we have re-used the analysis for BUFF45 we could even reasonably remove
-%$3.4=12$ from this result, because the results from $BUFF45$ have been used four times. 
-Traditional FMEA would have lead us to a much higher comparison complexity
-of $468$ failure modes to check against components.
-However, the analysis here appears top-heavy; we should be able to refine the model more
-and break this down into smaller functional groups, by allowing more stages of hierarchy and hopefully
-this should lead a further reduction in the complexity comparison figure.
+%$3 \times 4=12$ from this result, because the results from $BUFF45$ have been used four times. 
+%Traditional FMEA would have lead us to a much higher comparison complexity
+%of $468$ failure modes to check against components.
+%However, 
+The analysis here appears top-heavy; we should be able to refine the model more
+and break this down into smaller functional groups, by allowing more stages of hierarchy.
+%and hopefully
+%this should lead a further reduction in the complexity comparison figure.
 By decreasing the size of the modules with further refinement,
 we may also discover new derived components that may be of use for other analyses in the future.
 
@@ -1748,20 +1659,27 @@ we may also discover new derived components that may be of use for other analyse
 
 \clearpage
 
-\subsection{FMMD Analysis using more hierarchical stages}
+\subsection{FMMD Analysis of Bubba Oscillator using more hierarchical stages}
 
 The example above---from the initial {\fgs}---used one very large functional group to model the circuit.
-This mean a quite large comparison complexity for this final stage.
+%This mean a quite large comparison complexity for this final stage.
 We should be able to determine smaller {\fgs} and refine the model further.
 
-\begin{figure}[h+]
- \centering
- \includegraphics[width=300pt,keepaspectratio=true]{CH5_Examples/poss2finalbubba.png}
- % largeosc.png: 916x390 pixel, 72dpi, 32.31x13.76 cm, bb=0 0 916 390
- \caption{Bubba Oscillator: Smaller Functional Groups, One more FMMD hierarchy stage.}
- \label{fig:poss2finalbubba}
-\end{figure}
+% HTR 23SEP2012 \begin{figure}[h+]
+% HTR 23SEP2012  \centering
+% HTR 23SEP2012  \includegraphics[width=300pt,keepaspectratio=true]{CH5_Examples/poss2finalbubba.png}
+% HTR 23SEP2012  % largeosc.png: 916x390 pixel, 72dpi, 32.31x13.76 cm, bb=0 0 916 390
+% HTR 23SEP2012  \caption{Bubba Oscillator: Smaller Functional Groups, One more FMMD hierarchy stage.}
+% HTR 23SEP2012  \label{fig:poss2finalbubba}
+% HTR 23SEP2012 \end{figure}
 
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt]{./CH5_Examples/bubba_euler_2.png}
+ % bubba_euler_2.png: 1241x617 pixel, 72dpi, 43.78x21.77 cm, bb=0 0 1241 617
+ \caption{Euler diagram showing functional groupings for the Bubba oscillator using a more de-composed approach.}
+ \label{fig:bubbaeuler2}
+\end{figure}
 
 %
 We take the $NIBUFF$ and $PHS45$
@@ -1771,7 +1689,8 @@ and with those three, form a $PHS135BUFFERED$
 functional group.
 $PHS135BUFFERED$ is a {\dc} representing an actively buffered $135^{\circ}$ phase shifter.
 
-A PHS45 {\dc} and an  inverting amplifier\footnote{Inverting amplifiers always apply a $180^{\circ}$ phase shift.}, form a {\fg}
+A PHS45 {\dc} and an  inverting amplifier\footnote{Inverting amplifiers always apply a $180^{\circ}$ phase shift.}, 
+form a {\fg}
 providing an amplified $225^{\circ}$ phase shift, which we can call $PHS225AMP$.
 
 %---with the remaining $PHS45$ and the $INVAMP$ (re-used from section~\ref{sec:invamp})in a second group $PHS225AMP$---
@@ -1801,12 +1720,12 @@ Finally we can merge $PHS135BUFFERED$ and  $PHS225AMP$ in a final stage (see fig
                \hline
      FS1: $PHS45_1$  $0\_phaseshift$        &  &  phase shift low              &  &   $0\_phaseshift$       \\  
      FS2: $PHS45_1$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  
-     FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $90\_phaseshift$          \\  \hline 
+     %FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $90\_phaseshift$          \\  \hline 
 
-     FS4: $NIBUFF_1$  $L_{up}$      &  &  output high   &  &    $NO_{signal}$           \\  
-     FS5: $NIBUFF_1$  $L_{dn}$      &  &  output low   &  &     $NO_{signal}$              \\       
-     FS6: $NIBUFF_1$  $N_{oop}$     &  &  output low   &  &     $NO_{signal}$                \\  
-     FS7: $NIBUFF_1$  $L_{slew}$    &  &  signal lost                &  &   $NO_{signal}$          \\  \hline 
+     FS3: $NIBUFF_1$  $L_{up}$      &  &  output high   &  &    $NO_{signal}$           \\  
+     FS4: $NIBUFF_1$  $L_{dn}$      &  &  output low   &  &     $NO_{signal}$              \\       
+     FS5: $NIBUFF_1$  $N_{oop}$     &  &  output low   &  &     $NO_{signal}$                \\  
+     FS6: $NIBUFF_1$  $L_{slew}$    &  &  signal lost                &  &   $NO_{signal}$          \\  \hline 
 
 
 \hline
@@ -1817,10 +1736,10 @@ Finally we can merge $PHS135BUFFERED$ and  $PHS225AMP$ in a final stage (see fig
 
 Collecting symptoms from table~\ref{tbl:buff45}, we can create a derived component $BUFF45$ which has the following failure modes:
 $$
-fm (BUFF45) = \{ 90\_phaseshift, 0\_phaseshift, NO\_signal .\}
+fm (BUFF45) = \{  0\_phaseshift, NO\_signal .\} % 90\_phaseshift,
 $$
 
-$$ CC(BUFF45) = 7.1 = 7 $$
+%$$ CC(BUFF45) = 7 \times 1 = 7 $$
 
 We can now combine three $BUFF45$ {\dcs} and create a $PHS135BUFFERED$ {\dc}.
 
@@ -1835,15 +1754,15 @@ We can now combine three $BUFF45$ {\dcs} and create a $PHS135BUFFERED$ {\dc}.
                \hline
      FS1: $PHS45_1$  $0\_phaseshift$        &  &  phase shift low              &  &   $90\_phaseshift$       \\  
      FS2: $PHS45_1$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  
-     FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 
+    %FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 
 
-    FS4: $PHS45_2$  $0\_phaseshift$        &  &  phase shift low              &  &   $90\_phaseshift$       \\  
-     FS5: $PHS45_2$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  
-     FS6: $PHS45_2$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 
+    FS3: $PHS45_2$  $0\_phaseshift$        &  &  phase shift low              &  &   $90\_phaseshift$       \\  
+     FS4: $PHS45_2$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  
+    % FS6: $PHS45_2$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 
 
-        FS7: $PHS45_3$  $0\_phaseshift$        &  &  phase shift low              &  &   $90\_phaseshift$       \\  
-     FS8: $PHS45_3$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  
-     FS9: $PHS45_3$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 
+        FS5: $PHS45_3$  $0\_phaseshift$        &  &  phase shift low              &  &   $90\_phaseshift$       \\  
+     FS6: $PHS45_3$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  
+    % FS9: $PHS45_3$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 
 
 
 
@@ -1855,11 +1774,11 @@ We can now combine three $BUFF45$ {\dcs} and create a $PHS135BUFFERED$ {\dc}.
 
 Collecting symptoms from table~\ref{tbl:phs135buffered}, we can create a derived component $PHS135BUFFERED$ which has the following failure modes:
 $$
-fm (PHS135BUFFERED) = \{ 90\_phaseshift, 180\_phaseshift, NO\_signal .\}
+fm (PHS135BUFFERED) = \{ 90\_phaseshift,  NO\_signal .\} % 180\_phaseshift,
 $$
 
 
-$$ CC (PHS135BUFFERED) = 3.2 = 6 $$
+%$$ CC (PHS135BUFFERED) = 3 \times 2 = 6 $$
 
 
 
@@ -1873,14 +1792,14 @@ The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\cir
  \textbf{Failure Scenario} & &  \textbf{PHS225AMP}         &   & \textbf{Symptom}          \\ 
                            & &  \textbf{Oscillator}    &   &                           \\
                \hline
-     FS1: $PHS45_1$  $0\_phaseshift$        &  &  phase shift low               &  &   $270\_phaseshift$       \\  
+     FS1: $PHS45_1$  $0\_phaseshift$        &  &  phase shift low               &  &   $180\_phaseshift$       \\  
      FS2: $PHS45_1$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  
-     FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 
+    % FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $270\_phaseshift$          \\  \hline 
 
-     FS4: $INVAMP$  $L_{up}$      &  &  output high   &  &    $NO_{signal}$           \\  
-     FS5: $INVAMP$  $L_{dn}$      &  &  output low   &  &     $NO_{signal}$              \\       
-     FS6: $INVAMP$  $N_{oop}$     &  &  output low   &  &     $NO_{signal}$                \\  
-     FS7: $INVAMP$  $L_{slew}$    &  &  signal lost                &  &   $NO_{signal}$          \\  \hline 
+     FS3: $INVAMP$  $L_{up}$      &  &  output high   &  &    $NO_{signal}$           \\  
+     FS4: $INVAMP$  $L_{dn}$      &  &  output low   &  &     $NO_{signal}$              \\       
+     FS5: $INVAMP$  $N_{oop}$     &  &  output low   &  &     $NO_{signal}$                \\  
+     FS6: $INVAMP$  $L_{slew}$    &  &  signal lost                &  &   $NO_{signal}$          \\  \hline 
 
 \hline
 
@@ -1889,10 +1808,10 @@ The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\cir
 
 Collecting symptoms from table~\ref{tbl:phs225amp}, we can create a derived component $PHS225AMP$ which has the following failure modes:
 $$
-fm (PHS225AMP) = \{ 270\_phaseshift, 180\_phaseshift, NO\_signal .\}
+fm (PHS225AMP) = \{ 180\_phaseshift,  NO\_signal .\} % 270\_phaseshift,
 $$
 
-$$ CC(PHS225AMP) = 7.1 $$
+%$$ CC(PHS225AMP) = 7 \times 1 $$
 
 The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\circ}$ of phase shift).
 
@@ -1909,13 +1828,13 @@ and perform FMEA with these.
  \textbf{Failure Scenario} & &  \textbf{BUBBAOSC}         &   & \textbf{Symptom}          \\ 
                            & &                          &   &                             \\
                \hline
-     FS1: $PHS135BUFFERED$  $180\_phaseshift$        &  &  phase shift high              &  &     $LO_{fosc}$                     \\  
-     FS2: $PHS135BUFFERED$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
-     FS3: $PHS135BUFFERED$  $90\_phaseshift$         &  &  phase shift low                &  &    $HI_{osc}$       \\  \hline 
+     %FS1: $PHS135BUFFERED$  $180\_phaseshift$        &  &  phase shift high              &  &     $LO_{fosc}$                     \\  
+     FS1: $PHS135BUFFERED$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
+     FS2: $PHS135BUFFERED$  $90\_phaseshift$         &  &  phase shift low                &  &    $HI_{osc}$       \\  \hline 
 
-     FS4: $PHS225AMP$  $270\_phaseshift$      &  &  phase shift high   &  &     $LO_{fosc}$                               \\  
-     FS5: $PHS225AMP$  $180\_phaseshift$      &  &  phase shift low   &  &     $HI_{osc}$                               \\       
-     FS6: $PHS225AMP$  $NO\_signal$     &  &  lost signal             &  &     $NO_{signal}$                              \\  \hline 
+    % FS4: $PHS225AMP$  $270\_phaseshift$      &  &  phase shift high   &  &     $LO_{fosc}$                               \\  
+     FS4: $PHS225AMP$  $180\_phaseshift$      &  &  phase shift low   &  &     $HI_{osc}$                               \\       
+     FS5: $PHS225AMP$  $NO\_signal$     &  &  lost signal             &  &     $NO_{signal}$                              \\  \hline 
    
 
 \hline
@@ -1926,33 +1845,36 @@ and perform FMEA with these.
 
 Collecting symptoms from table~\ref{tbl:bubba2}, we can create a derived component $BUBBAOSC$ which has the following failure modes:
 $$
-fm (BUBBAOSC) = \{  LO_{fosc}, HI_{osc}, NO\_signal .\}
+fm (BUBBAOSC) = \{  HI_{osc}, NO\_signal .\} %  LO_{fosc},
 $$
 
 %We could trace the DAGs here and ensure that both analysis strategies worked ok.....
 
-$$ CC(BUBBAOSC) = 6.(2-1) = 6 $$
+%$$ CC(BUBBAOSC) = 6 \times (2-1) = 6 $$
 
 
-We can now add the comparison complexities for all levels of the analysis represented in figure~\ref{fig:poss2finalbubba}.
-We have at the lowest level two $PHS45$ {\dcs} giving a CC of 8 and $INVAMP$ with a CC of 10, at the next level four $BUFF45$ {\dcs} giving $(4-1).7=21$,
-and penultimately $PHS135BUFFERED$ with 6 and $PHS225AMP$ with 7. The final top stage of the hierarchy, $BUBBAOSC$ has a CC of 6.
-Our total comparison complexity is $58$, this contrasts with $468$ for traditional `flat' FMEA,
-and $250$ for our first stage functional groups analysis.
-This has meant a drastic reduction in the number of failure-modes to check against components.
-It has also given us five {\dcs}, building blocks, which may be re-used for similar circuitry
+% We can now add the comparison complexities for all levels of the analysis represented in figure~\ref{fig:poss2finalbubba}.
+% We have at the lowest level two $PHS45$ {\dcs} giving a CC of 8 and $INVAMP$ with a CC of 10, 
+% at the next level four $BUFF45$ {\dcs} giving $(4-1).7=21$,
+% and penultimately $PHS135BUFFERED$ with 6 and $PHS225AMP$ with 7. 
+% The final top stage of the hierarchy, $BUBBAOSC$ has a CC of 6.
+% Our total comparison complexity is $58$, this contrasts with $468$ for traditional `flat' FMEA,
+% and $250$ for our first stage functional groups analysis.
+% This has meant a drastic reduction in the number of failure-modes to check against components.
+It has %also 
+given us five {\dcs}, building blocks, which could potentially be re-used for similar circuitry
 to analyse in the future.
 
   
 \subsection{Comparing both approaches}
 
-In general with large functional groups the comparison complexity
-is higher, by an order of $O(N^2)$.
+%In general with large functional groups the comparison complexity
+%is higher, by an order of $O(N^2)$.
 Smaller functional groups mean less by-hand checks are required.
 It also means a more finely grained model. This means that
 there are more {\dcs} and this increases the possibility of re-use.
-The more we can modularise, the more we decimate the $O(N^2)$ effect
-of complexity comparison.
+% HTR The more we can modularise, the more we decimate the $O(N^2)$ effect
+% HTR of complexity comparison.
 
 
 \section{Sigma Delta Analogue to Digital Converter.} %($\Sigma \Delta ADC$)}