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Half way through re-org of sigma delta

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Robin P. Clark 2012-09-26 19:45:44 +01:00
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submission_thesis/CH5_Examples/Makefile
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@ -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 \ 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 \ 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 \ tree_abstraction_levels.png vrange.png sigma_delta_block.png ftcontext.png ct1.png hd.png \
sigdel1.png sdadc.png bubba_euler_1.png bubba_euler_2.png sigdel1.png sdadc.png bubba_euler_1.png bubba_euler_2.png eulersd.png

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submission_thesis/CH5_Examples/bubba_euler_2.dia
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submission_thesis/CH5_Examples/copy.tex
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@ -1457,8 +1457,9 @@ could be easily detected; the failure symptom $FilterIncorrect$ may be less obs
This circuit is described in the Analog Applications Journal~\cite{bubba}[p.37]. This circuit is described in the Analog Applications Journal~\cite{bubba}[p.37].
The circuit implements an oscillator using four 45 degree phase shifts, and an inverting amplifier to provide The circuit implements an oscillator using 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). gain and the final 180 degrees of phase shift (making a total of 360). % degrees of phase shift).
The circuit provides two outputs with a quadrature phase relationship.
%
From a fault finding perspective this circuit cannot be de-composed because the whole circuit is enclosed within a feedback loop. From a fault finding perspective this circuit cannot be de-composed because the whole circuit is enclosed within a feedback loop.
However, this is not a problem for FMMD, as {\fgs} are readily identifiable. However, this is not a problem for FMMD, as {\fgs} are readily identifiable.
The signal path is circular (its a positive feedback circuit) and most failures would simply cause the output to stop oscillating. The signal path is circular (its a positive feedback circuit) and most failures would simply cause the output to stop oscillating.
@ -1546,7 +1547,7 @@ $$ CC(NIBUFF) = 0 $$
\subsection{Bringing the functional Groups Together: FMMD model of 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 We could at this point bring all the {\dcs} together into one large functional
group (see figure~\ref{fig:poss1finalbubba}) group (see figure~\ref{fig:bubbaeuler1}) %{fig:poss1finalbubba})
or we could try to merge smaller stages. or we could try to merge smaller stages.
Initially we use the first identified {\fgs} to create our model without further stages of refinement/hierarchy. Initially we use the first identified {\fgs} to create our model without further stages of refinement/hierarchy.
@ -1555,9 +1556,9 @@ Initially we use the first identified {\fgs} to create our model without further
\subsection{FMMD Analysis using initially identified functional groups} \subsection{FMMD Analysis using initially identified functional groups}
Our functional group for this analysis can be expressed thus: Our functional group for this analysis can be expressed thus:
or in Euler diagram format as in figure~\ref{fig:bubbaeuler1}. %
$$ G^1_0 = \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 PHS45^1_4, INVAMP^1_0 \} ,$$ $$ G^1_0 = \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 PHS45^1_4, INVAMP^1_0 \} ,$$
or in Euler diagram format as in figure~\ref{fig:bubbaeuler1}.
% HTR 23SEP2012 \begin{figure}[h+] % HTR 23SEP2012 \begin{figure}[h+]
% HTR 23SEP2012 \centering % HTR 23SEP2012 \centering
% HTR 23SEP2012 \includegraphics[width=300pt,keepaspectratio=true]{CH5_Examples/poss1finalbubba.png} % HTR 23SEP2012 \includegraphics[width=300pt,keepaspectratio=true]{CH5_Examples/poss1finalbubba.png}
@ -1565,7 +1566,7 @@ $$ G^1_0 = \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0
% HTR 23SEP2012 \caption{Bubba Oscillator: One large functional group using the initial functional groups to model oscillator.} % HTR 23SEP2012 \caption{Bubba Oscillator: One large functional group using the initial functional groups to model oscillator.}
% HTR 23SEP2012 \label{fig:poss1finalbubba} % HTR 23SEP2012 \label{fig:poss1finalbubba}
% HTR 23SEP2012 \end{figure} % HTR 23SEP2012 \end{figure}
%
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\includegraphics[width=400pt]{./CH5_Examples/bubba_euler_1.png} \includegraphics[width=400pt]{./CH5_Examples/bubba_euler_1.png}
@ -1573,7 +1574,7 @@ $$ G^1_0 = \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0
\caption{Euler diagram showing the hierarchy of the initial FMMD analysis performed on the Bubba Oscillator circuit.} \caption{Euler diagram showing the hierarchy of the initial FMMD analysis performed on the Bubba Oscillator circuit.}
\label{fig:bubbaeuler1} \label{fig:bubbaeuler1}
\end{figure} \end{figure}
%
\begin{table}[h+] \begin{table}[h+]
\caption{Bubba Oscillator: Failure Mode Effects Analysis: One Large Functional Group} % title of Table \caption{Bubba Oscillator: Failure Mode Effects Analysis: One Large Functional Group} % title of Table
\label{tbl:bubbalargefg} \label{tbl:bubbalargefg}
@ -1632,17 +1633,17 @@ $$ G^1_0 = \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0
Collecting symptoms from table~\ref{tbl:bubbalargefg} we can show that for single failure modes, applying $fm$ to the bubba oscillator Collecting symptoms from table~\ref{tbl:bubbalargefg} we can show that for single failure modes, applying $fm$ to the bubba oscillator
returns three failure modes, returns three failure modes,
%
$$ fm(BubbaOscillator) = \{ NO_{osc}, HI_{fosc}\} . $$ %, LO_{fosc} \} . $$ $$ fm(BubbaOscillator) = \{ NO_{osc}, HI_{fosc}\} . $$ %, LO_{fosc} \} . $$
%
%For the final stage of this FMMD model, we can calculate the complexity using equation~\ref{eqn:rd2}. %For the final stage of this FMMD model, we can calculate the complexity using equation~\ref{eqn:rd2}.
%$$ CC = 28 \times 8 = 224$$ %$$ CC = 28 \times 8 = 224$$
%
%To obtain the total comparison complexity ($TCC$), we need to add the complexity from the %To obtain the total comparison complexity ($TCC$), we need to add the complexity from the
%{\dcs} that $BubbaOscillator$ was built from. %{\dcs} that $BubbaOscillator$ was built from.
%
%$$ TCC = 28 \times 8 + 4 \times 4 + 4 \times 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 %As we have re-used the analysis for BUFF45 we could even reasonably remove
%$3 \times 4=12$ from this result, because the results from $BUFF45$ have been used four times. %$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 %Traditional FMEA would have lead us to a much higher comparison complexity
@ -1659,7 +1660,7 @@ we may also discover new derived components that may be of use for other analyse
\clearpage \clearpage
\subsection{FMMD Analysis of Bubba Oscillator using more hierarchical stages} \subsection{FMMD Analysis of Bubba Oscillator using a finer grained modular approach (i.e. more hierarchical stages)}
The example above---from the initial {\fgs}---used one very large functional group to model the circuit. 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.
@ -1688,28 +1689,25 @@ We take the $NIBUFF$ and $PHS45$
and with those three, form a $PHS135BUFFERED$ and with those three, form a $PHS135BUFFERED$
functional group. functional group.
$PHS135BUFFERED$ is a {\dc} representing an actively buffered $135^{\circ}$ phase shifter. $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.}, A PHS45 {\dc} and an inverting amplifier\footnote{Inverting amplifiers always apply a $180^{\circ}$ phase shift.},
form a {\fg} form a {\fg}
providing an amplified $225^{\circ}$ phase shift, which we can call $PHS225AMP$. 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$--- %---with the remaining $PHS45$ and the $INVAMP$ (re-used from section~\ref{sec:invamp})in a second group $PHS225AMP$---
Finally we can merge $PHS135BUFFERED$ and $PHS225AMP$ in a final stage (see figure~\ref{fig:poss2finalbubba}) Finally we can merge $PHS135BUFFERED$ and $PHS225AMP$ in a final stage (see figure~{fig:bubbaeuler2}) % \ref{fig:poss2finalbubba})
%
%We can take a more modular approach by creating two intermediate functional groups, a buffered $45^{\circ}$ phase shifter (BUFF45) %We can take a more modular approach by creating two intermediate functional groups, a buffered $45^{\circ}$ phase shifter (BUFF45)
%we can combine three $BUFF45$'s to make %we can combine three $BUFF45$'s to make
%a $135^{\circ}$ buffer phase shifter (PHS135BUFFERED). %a $135^{\circ}$ buffer phase shifter (PHS135BUFFERED).
%
%We can combine a $PHS45$ and a $NIBUFF$ to create %We can combine a $PHS45$ and a $NIBUFF$ to create
%and an amplifying $225^{\circ}$ phase shifter (PHS225AMP). %and an amplifying $225^{\circ}$ phase shifter (PHS225AMP).
%
% By combining PHS225AMP and PHS135BUFFERED we can create a more modularised hierarchical % By combining PHS225AMP and PHS135BUFFERED we can create a more modularised hierarchical
% model of the bubba oscillator. % model of the bubba oscillator.
% The proposed hierarchy is shown in figure~\ref{fig:poss2finalbubba}. % The proposed hierarchy is shown in figure~\ref{fig:poss2finalbubba}.
%
\begin{table}[h+] \begin{table}[h+]
\caption{BUFF45: Failure Mode Effects Analysis} % title of Table \caption{BUFF45: Failure Mode Effects Analysis} % title of Table
\label{tbl:buff45} \label{tbl:buff45}
@ -1732,18 +1730,16 @@ Finally we can merge $PHS135BUFFERED$ and $PHS225AMP$ in a final stage (see fig
\end{tabular} \end{tabular}
\end{table} \end{table}
%
Collecting symptoms from table~\ref{tbl:buff45}, we can create a derived component $BUFF45$ which has the following failure modes: Collecting symptoms from table~\ref{tbl:buff45}, we can create a derived component $BUFF45$ which has the following failure modes:
$$ $$
fm (BUFF45) = \{ 0\_phaseshift, NO\_signal .\} % 90\_phaseshift, fm (BUFF45) = \{ 0\_phaseshift, NO\_signal .\} % 90\_phaseshift,
$$ $$
%
%$$ CC(BUFF45) = 7 \times 1 = 7 $$ %$$ CC(BUFF45) = 7 \times 1 = 7 $$
%
We can now combine three $BUFF45$ {\dcs} and create a $PHS135BUFFERED$ {\dc}. We can now combine three $BUFF45$ {\dcs} and create a $PHS135BUFFERED$ {\dc}.
%
\begin{table}[h+] \begin{table}[h+]
\caption{PHS135BUFFERED: Failure Mode Effects Analysis} % title of Table \caption{PHS135BUFFERED: Failure Mode Effects Analysis} % title of Table
\label{tbl:phs135buffered} \label{tbl:phs135buffered}
@ -1770,20 +1766,20 @@ We can now combine three $BUFF45$ {\dcs} and create a $PHS135BUFFERED$ {\dc}.
\end{tabular} \end{tabular}
\end{table} \end{table}
%
%
Collecting symptoms from table~\ref{tbl:phs135buffered}, we can create a derived component $PHS135BUFFERED$ which has the following failure modes: Collecting symptoms from table~\ref{tbl:phs135buffered}, we can create a derived component $PHS135BUFFERED$ which has the following failure modes:
$$ $$
fm (PHS135BUFFERED) = \{ 90\_phaseshift, NO\_signal .\} % 180\_phaseshift, fm (PHS135BUFFERED) = \{ 90\_phaseshift, NO\_signal .\} % 180\_phaseshift,
$$ $$
%
%
%$$ CC (PHS135BUFFERED) = 3 \times 2 = 6 $$ %$$ CC (PHS135BUFFERED) = 3 \times 2 = 6 $$
%
%
%
The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\circ}$ of phase shift). The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\circ}$ of phase shift).
%
\begin{table}[h+] \begin{table}[h+]
\caption{PHS225AMP: Failure Mode Effects Analysis} % title of Table \caption{PHS225AMP: Failure Mode Effects Analysis} % title of Table
\label{tbl:phs225amp} \label{tbl:phs225amp}
@ -1805,21 +1801,21 @@ The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\cir
\end{tabular} \end{tabular}
\end{table} \end{table}
%
Collecting symptoms from table~\ref{tbl:phs225amp}, we can create a derived component $PHS225AMP$ which has the following failure modes: Collecting symptoms from table~\ref{tbl:phs225amp}, we can create a derived component $PHS225AMP$ which has the following failure modes:
$$ $$
fm (PHS225AMP) = \{ 180\_phaseshift, NO\_signal .\} % 270\_phaseshift, fm (PHS225AMP) = \{ 180\_phaseshift, NO\_signal .\} % 270\_phaseshift,
$$ $$
%
%$$ CC(PHS225AMP) = 7 \times 1 $$ %$$ CC(PHS225AMP) = 7 \times 1 $$
%
The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\circ}$ of phase shift). The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\circ}$ of phase shift).
%
%
%
To complete the analysis we now bring the derived components $PHS135BUFFERED$ and $PHS225AMP$ together To complete the analysis we now bring the derived components $PHS135BUFFERED$ and $PHS225AMP$ together
and perform FMEA with these. and perform FMEA with these.
%
\begin{table}[h+] \begin{table}[h+]
\caption{BUBBAOSC: Failure Mode Effects Analysis} % title of Table \caption{BUBBAOSC: Failure Mode Effects Analysis} % title of Table
\label{tbl:bubba2} \label{tbl:bubba2}
@ -1841,18 +1837,17 @@ and perform FMEA with these.
\end{tabular} \end{tabular}
\end{table} \end{table}
%
Collecting symptoms from table~\ref{tbl:bubba2}, we can create a derived component $BUBBAOSC$ which has the following failure modes: Collecting symptoms from table~\ref{tbl:bubba2}, we can create a derived component $BUBBAOSC$ which has the following failure modes:
$$ $$
fm (BUBBAOSC) = \{ HI_{osc}, NO\_signal .\} % LO_{fosc}, fm (BUBBAOSC) = \{ HI_{osc}, NO\_signal .\} % LO_{fosc},
$$ $$
%
%We could trace the DAGs here and ensure that both analysis strategies worked ok..... %We could trace the DAGs here and ensure that both analysis strategies worked ok.....
%
%$$ CC(BUBBAOSC) = 6 \times (2-1) = 6 $$ %$$ CC(BUBBAOSC) = 6 \times (2-1) = 6 $$
%
%
% We can now add the comparison complexities for all levels of the analysis represented in figure~\ref{fig:poss2finalbubba}. % We can now add the comparison complexities for all levels of the analysis represented in figure~\ref{fig:poss2finalbubba}.
% We have at the lowest level two $PHS45$ {\dcs} giving a CC of 8 and $INVAMP$ with a CC of 10, % We have at the lowest level two $PHS45$ {\dcs} giving a CC of 8 and $INVAMP$ with a CC of 10,
% at the next level four $BUFF45$ {\dcs} giving $(4-1).7=21$, % at the next level four $BUFF45$ {\dcs} giving $(4-1).7=21$,
@ -1864,10 +1859,10 @@ $$
It has %also It has %also
given us five {\dcs}, building blocks, which could potentially be re-used for similar circuitry given us five {\dcs}, building blocks, which could potentially be re-used for similar circuitry
to analyse in the future. to analyse in the future.
%
%
\subsection{Comparing both approaches} \subsection{Comparing both approaches}
%
%In general with large functional groups the comparison complexity %In general with large functional groups the comparison complexity
%is higher, by an order of $O(N^2)$. %is higher, by an order of $O(N^2)$.
Smaller functional groups mean less by-hand checks are required. Smaller functional groups mean less by-hand checks are required.
@ -1875,9 +1870,21 @@ It also means a more finely grained model. This means that
there are more {\dcs} and this increases the possibility of re-use. there are more {\dcs} and this increases the possibility of re-use.
% HTR The more we can modularise, the more we decimate the $O(N^2)$ effect % HTR The more we can modularise, the more we decimate the $O(N^2)$ effect
% HTR of complexity comparison. % HTR of complexity comparison.
%
\clearpage
\section{Sigma Delta Analogue to Digital Converter.} %($\Sigma \Delta ADC$)}
\section{Sigma Delta Analogue to Digital Converter ($\Sigma \Delta ADC$).} %($\Sigma \Delta ADC$)}
\label{sec:sigmadelta} \label{sec:sigmadelta}
The following example is used to demonstrate FMMD analysis of a mixed analogue and digital circuit (see figure~\ref{fig:sigmadelta}). The following example is used to demonstrate FMMD analysis of a mixed analogue and digital circuit (see figure~\ref{fig:sigmadelta}).
\begin{figure}[h] \begin{figure}[h]
@ -1887,18 +1894,16 @@ The following example is used to demonstrate FMMD analysis of a mixed analogue a
\caption{Sigma Delta Analogue to Digital Converter} \caption{Sigma Delta Analogue to Digital Converter}
\label{fig:sigmadelta} \label{fig:sigmadelta}
\end{figure} \end{figure}
%
\nocite{f77} \nocite{f77}
\nocite{sccs} \nocite{sccs}
\nocite{electronicssysapproach} \nocite{electronicssysapproach}
%
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\includegraphics[width=200pt,keepaspectratio=true]{./CH5_Examples/sigma_delta_block.png} \includegraphics[width=200pt,keepaspectratio=true]{./CH5_Examples/sigma_delta_block.png}
% sigma_delta_block.png: 828x367 pixel, 72dpi, 29.21x12.95 cm, bb=0 0 828 367 % sigma_delta_block.png: 828x367 pixel, 72dpi, 29.21x12.95 cm, bb=0 0 828 367
\caption{Sigma Delta ADC signal path} \caption{Electrical signal path Block diagram: $\Sigma \Delta ADC$} % Analogue to Digital Converter }
\label{fig:sigmadeltablock} \label{fig:sigmadeltablock}
\end{figure} \end{figure}
@ -1929,36 +1934,40 @@ and fed into the summing integrator completing the negative feedback loop.
\subsection{FMMD analysis of \sd } \subsection{FMMD analysis of \sd }
The partslist for the \sd : The partslist for the \sd :
%
$$\{ IC1, IC2, IC3, IC4, R1, R2, R3, R4, C1 \} $$. $$\{ IC1, IC2, IC3, IC4, R1, R2, R3, R4, C1 \} $$.
%
IC1,2 and 3 are all Op-amps and we have failure modes from section~\ref{sec:opampfm}. IC1,2 and 3 are all Op-amps and we have failure modes from section~\ref{sec:opampfm}.
%
$$ fm(OPAMP) = \{ HIGH, LOW, NOOP, LOW\_SLEW \} $$ $$ fm(OPAMP) = \{ HIGH, LOW, NOOP, LOW\_SLEW \} $$
%
We examine the literature for a failure model for the D-type flip flop~\cite{fmd91}[3-105], the CD4013B~\cite{cd4013Bds}, We examine the literature for a failure model for the D-type flip flop~\cite{fmd91}[3-105], the CD4013B~\cite{cd4013Bds},
and obtain its failure modes, which we can express using the $fm$ function: and obtain its failure modes, which we can express using the $fm$ function:
%%
$$ fm ( CD4013B) = \{ HIGH, LOW, NOOP \} $$ $$ fm ( CD4013B) = \{ HIGH, LOW, NOOP \} $$
%
The resistors and capacitor failure modes we take from EN298~\cite{en298}[An.A] The resistors and capacitor failure modes we take from EN298~\cite{en298}[An.A]
%
$$ fm ( R ) = \{OPEN, SHORT\} $$ $$ fm ( R ) = \{OPEN, SHORT\} $$
%
$$ fm ( C ) = \{OPEN, SHORT\} $$ $$ fm ( C ) = \{OPEN, SHORT\} $$
%
We are also given a CLOCK. For the purpose of example we shall attribute
one failure mode to this, that it might stop.
%
$$ fm ( CLOCK ) = \{ STOPPED \} $$
\subsection{Identifying initial {\fgs}} \subsection{Identifying initial {\fgs}}
\subsubsection{Summing Junction} \subsubsection{Summing Junction Integrator (SUMJINT)}
We now need to choose {\fgs}. The signal path is circular, but we can start We now need to choose {\fgs}. The signal path is circular, but we can start
with the input voltage, which is applied via $R2$, we term this voltage $V_{in}$. with the input voltage, which is applied via $R2$, we term this voltage $V_{in}$.
% %
The feedback voltage for the ADC is supplied via $R1$, we term this voltage as $V_{fb}$. The feedback voltage for the ADC is supplied via $R1$, we term this voltage as $V_{fb}$.
%The input voltage is supplied via $R2$ and we term this voltage as $V_{in}$. %The input voltage is supplied via $R2$ and we term this voltage as $V_{in}$.
$R2$ and $R1$ form a summing junction to IC1: they thus work to fulfil this specific function. $R2$ and $R1$ form a summing junction to IC1: they balance the integrator provided
This can be our first {\fg} and we analyse it in table~\ref{tbl:suml=j}. by the capacitor C1 and the opamp IC1.
This can be our first {\fg} and we analyse it in table~\ref{tbl:sumjint}.
%For the symptoms, we have to think in terms of the effect %For the symptoms, we have to think in terms of the effect
%on its performance as a summing junction and not be %on its performance as a summing junction and not be
%distracted by the integrator formed by $C_1$ and $IC1$. %distracted by the integrator formed by $C_1$ and $IC1$.
@ -1967,8 +1976,8 @@ $$G^0_1 = \{R1, R2 \}$$
\begin{table}[h+] \begin{table}[h+]
\center \center
\caption{R1,R2 Summing Junction: Failure Mode Effects Analysis} % title of Table \caption{ Summing Junction Integrator: Failure Mode Effects Analysis} % title of Table
\label{tbl:sumj} \label{tbl:sumjint}
\begin{tabular}{|| l | l | c | c | l ||} \hline \begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{failure result} & & \textbf{Symptom} \\ \textbf{Failure Scenario} & & \textbf{failure result} & & \textbf{Symptom} \\
@ -1978,74 +1987,111 @@ $$G^0_1 = \{R1, R2 \}$$
FS2: $R1$ $SHORT$ & & $V_{fb}$ dominates input & & $V_{fb} DOM$ \\ \hline FS2: $R1$ $SHORT$ & & $V_{fb}$ dominates input & & $V_{fb} DOM$ \\ \hline
FS3: $R2$ $OPEN$ & & $V_{fb}$ dominates input & & $V_{fb} DOM$ \\ FS3: $R2$ $OPEN$ & & $V_{fb}$ dominates input & & $V_{fb} DOM$ \\
FS4: $R2$ $SHORT$ & & $V_{in}$ dominates input & & $V_{in} DOM$ \\ \hline FS4: $R2$ $SHORT$ & & $V_{in}$ dominates input & & $V_{in} DOM$ \\ \hline
FS5: $IC1$ $HIGH$ & & output perm. high & & HIGH \\
FS6: $IC1$ $LOW$ & & output perm. low & & LOW \\ \hline
FS7: $IC1$ $NOOP$ & & no current to drive C1 & & NO\_INTEGRATION \\
FS8: $IC1$ $LOW\_SLEW$ & & signal delay to C1 & & NO\_INTEGRATION \\ \hline
FS9: $C1$ $OPEN$ & & no capacitance & & NO\_INTEGRATION \\
FS10: $C1$ $SHORT$ & & no capacitance & & NO\_INTEGRATION \\ \hline
% \hline
% FS1: $IC2$ $HIGH$ & & output perm. high & & HIGH \\
% FS2: $IC2$ $LOW$ & & output perm. low & & LOW \\ \hline
% FS3: $IC2$ $NOOP$ & & no current drive & & LOW \\
% FS4: $IC2$ $LOW\_SLEW$ & & delayed signal & & LOW\_SLEW \\ \hline
% \hline
\hline \hline
\end{tabular} \end{tabular}
\end{table} \end{table}
%
%
% \end{tabular}
% \end{table}
From the analysis in table~\ref{tbl:sumj} we collect symptoms. From the analysis in table~\ref{tbl:sumj} we collect symptoms.
We can create the derived component We can create the derived component
$SUMJ$.% which has the failure modes from collecting its symptoms. $SUMJINT$.% which has the failure modes from collecting its symptoms.
We now state: We now state:
$$ fm(SUMJ) = \{ V_{in} DOM, V_{fb} DOM \} .$$ $$ fm(SUMJUINT) = \{ V_{in} DOM, V_{fb} DOM, NO\_INTEGRATION, HIGH, LOW \} .$$
\subsubsection{Buffered Integrator} % \subsubsection{Buffered Integrator}
%
% Following the signal path, the next functional group is the integrator.
% %
% This integrator is formed by placing $C1$ in the negative feedback loop of $IC2$\cite{aoe}[p.222].
% The output of the integrator is fed into IC2, which acts as a buffer,
% %performing the function of
% isolating the integrator from any load on its output.
% These three components work together to form a buffered integrator,
% and nicely form a {\fg}.
%
% $$G^0_2 = \{IC1, C1, IC2\}.$$
%
% The buffered integrator is analysed in table~\ref{tbl:intg}.
%
%
% \begin{table}[h+]
% \center
% \caption{IC1,C1,IC2 Buffered Integrator: Failure Mode Effects Analysis} % title of Table
% \label{tbl:intg}
%
% \begin{tabular}{|| l | l | c | c | l ||} \hline
% \textbf{Failure Scenario} & & \textbf{failure result } & & \textbf{Symptom} \\
% & & & & \\
% \hline \hline
%
%
%
% From the analysis in table~\ref{tbl:intg}, we can now create a derived component
% $BFINT$ which has the failure modes from collecting symptoms from the analysis in table~\ref{tbl:intg}.
% We can state
%
% $$ fm (BFINT) = \{ HIGH, LOW, NO\_INTEGRATION , LOW\_SLEW \} $$
%
Following the signal path, the next functional group is the integrator.
%
This integrator is formed by placing $C1$ in the negative feedback loop of $IC2$\cite{aoe}[p.222].
The output of the integrator is fed into IC2, which acts as a buffer,
%performing the function of
isolating the integrator from any load on its output.
These three components work together to form a buffered integrator,
and nicely form a {\fg}.
$$G^0_2 = \{IC1, C1, IC2\}.$$ \subsubsection{High Impedance Signal Buffer (HISB)}
The buffered integrator is analysed in table~\ref{tbl:intg}.
Next in the signal path (see figure~\ref{fig:sigmadeltablock}) is a signal buffer.
This presents a high impedance to the circuit driving it.
This prevents electrical loading, and thus interference with, the SUMJINT stage.
This is simply an op-amp
with the input connected to the +ve input and the -ve input grounded.
It therefore has the failure modes of an Op-amp.
\begin{table}[h+] \begin{table}[h+]
\center \center
\caption{IC1,C1,IC2 Buffered Integrator: Failure Mode Effects Analysis} % title of Table
\label{tbl:intg} % \center
\caption{ High Impedance Signal Buffer (HISB) : Failure Mode Effects Analysis} % title of Table
\begin{tabular}{|| l | l | c | c | l ||} \hline \begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{failure result } & & \textbf{Symptom} \\
\textbf{Failure Scenario} & & \textbf{failure result} & & \textbf{Symptom} \\
& & & & \\ & & & & \\
\hline \hline \hline\hline
FS1: $IC1$ $HIGH$ & & output perm. high & & HIGH \\
FS2: $IC1$ $LOW$ & & output perm. low & & LOW \\ \hline FS5: $IC2$ $HIGH$ & & output perm. high & & HIGH \\
FS3: $IC1$ $NOOP$ & & no current to drive C1 & & NO\_INTEGRATION \\ FS6: $IC2$ $LOW$ & & output perm. low & & LOW \\ \hline
FS4: $IC1$ $LOW\_SLEW$ & & signal delay to C1 & & NO\_INTEGRATION \\ \hline FS7: $IC2$ $NOOP$ & & no current to output & & $NOOP$ \\
FS8: $IC2$ $LOW\_SLEW$ & & delay signal & & $LOW\_SLEW$ \\ \hline
FS3: $C1$ $OPEN$ & & no capacitance & & NO\_INTEGRATION \\
FS4: $C1$ $SHORT$ & & no capacitance & & NO\_INTEGRATION \\ \hline % FS1: $IC2$ $HIGH$ & & output perm. high & & HIGH \\
% FS2: $IC2$ $LOW$ & & output perm. low & & LOW \\ \hline
\hline % FS3: $IC2$ $NOOP$ & & no current drive & & LOW \\
FS1: $IC2$ $HIGH$ & & output perm. high & & HIGH \\ % FS4: $IC2$ $LOW\_SLEW$ & & delayed signal & & LOW\_SLEW \\ \hline
FS2: $IC2$ $LOW$ & & output perm. low & & LOW \\ \hline % \hline
FS3: $IC2$ $NOOP$ & & no current drive & & LOW \\
FS4: $IC2$ $LOW\_SLEW$ & & delayed signal & & LOW\_SLEW \\ \hline
\hline
\end{tabular} \end{tabular}
\end{table} \end{table}
% \hline
%
From the analysis in table~\ref{tbl:intg}, we can now create a derived component % \end{tabular}
$BFINT$ which has the failure modes from collecting symptoms from the analysis in table~\ref{tbl:intg}. % \end{table}
We can state
$$ fm (BFINT) = \{ HIGH, LOW, NO\_INTEGRATION , LOW\_SLEW \} $$
\subsubsection{Digital level to analogue level conversion ($DL2AL$).} \subsubsection{Digital level to analogue level conversion ($DL2AL$).}
Digital level to analogue level conversion is performed by IC3 in conjunction with a potential divider formed by R3,R4. Digital level to analogue level conversion is performed by IC3 in conjunction with a potential divider formed by R3,R4.
@ -2113,19 +2159,41 @@ $$ fm (DL2AL^2) = \{ LOW, HIGH, LOW\_SLEW \} $$
\subsection{First {\fgs} analysed} \subsection{First {\fgs} analysed}
We have analysed the initial {\fgs} and can now take stock of the situation We have analysed the initial {\fgs} and
and see what is now required. Figure~\ref{fig:sigdel1} shows how far the have created our first {\dcs}. %and can now take stock of the situation
hierarchy has been built. %and see what is now required.
%Figure~\ref{fig:sigdel1} shows which {\fgs} we have analysed so far.
%hierarchy has been built.
These are:
\begin{itemize}
\item SUMJINT --- A summing junction and integrator,
\item HISB --- A High impedance buffer,
\item DIGITALBUFF --- A one bit digital buffer,
\item DL2AL --- A digital to analog level converter.
\end{itemize}
These {\dcs} follow to signal path shown in figure~\ref{fig:sigmadeltablock}.
We now use these {\dcs} to create a final {\fg} to represent the failure mode
behaviour of the $\Sigma \Delta ADC$. We represent this
in the Euler diagram in figure~\ref{fig:eulersd}.
\begin{figure}[h+] \begin{figure}[h]
\centering \centering
\includegraphics[width=400pt]{./CH5_Examples/sigdel1.png} \includegraphics[width=400pt]{./CH5_Examples/eulersd.png}
% sigdel1.png: 766x618 pixel, 72dpi, 27.02x21.80 cm, bb=0 0 766 618 % eulersd.png: 1018x334 pixel, 72dpi, 35.91x11.78 cm, bb=0 0 1018 334
\caption{First stage of FMMD analysis: Sigma delta Converter} \caption{Euler diagram showing the functional grouping used to model the $\Sigma \Delta ADC$}
\label{fig:sigdel1} \label{fig:eulersd}
\end{figure} \end{figure}
%
% \begin{figure}[h+]
% \centering
% \includegraphics[width=400pt]{./CH5_Examples/sigdel1.png}
% % sigdel1.png: 766x618 pixel, 72dpi, 27.02x21.80 cm, bb=0 0 766 618
% \caption{First stage of FMMD analysis: Sigma delta Converter}
% \label{fig:sigdel1}
% \end{figure}
IC4 is as yet unused, the signal path connects IC4 and DL2AL. These seem natural candidates IC4 is as yet unused, the signal path connects IC4 and DL2AL. These seem natural candidates
for the next {\fg}. for the next {\fg}.
@ -2137,7 +2205,7 @@ BFINT and SUMJ are adjacent in the signal path and these are chosen as a {\fg} a
\subsubsection{{\fg} $BFINT^1$ and $SUMJ^1$} \subsubsection{{\fg} $BFINT^1$ and $SUMJ^1$}
We now form a {\fg} with the two derived components $BFINT^1$ and $SUMJ^1$. We now form a {\fg} with the two derived components $BFINT^1$ and $SUMJINT^1$.
This forms a buffered integrating summing junction which we analyse in table~\ref{tbl:BISJ}. This forms a buffered integrating summing junction which we analyse in table~\ref{tbl:BISJ}.
$$ G^1_0 = \{ BFINT^1, SUMJ^1 \} $$ $$ G^1_0 = \{ BFINT^1, SUMJ^1 \} $$
@ -2188,7 +2256,7 @@ called $BISJ^2$.
The functional group formed by $IC4$ and $DL2AL$ takes the flip flop clocked and buffered The functional group formed by $IC4$ and $DL2AL$ takes the flip flop clocked and buffered
value, and outputs it at analogue voltage levels for the summing junction. value, and outputs it at analogue voltage levels for the summing junction.
$ G^2_1 = \{ IC4^0, DL2AL^2 \} $ $ G^2_1 = \{ IC4^0, DL2AL^2, CLOCK\} $
We analyse the buffered flip flop circuitry in table~\ref{tbl:FFB}. We analyse the buffered flip flop circuitry in table~\ref{tbl:FFB}.
@ -2208,8 +2276,9 @@ We analyse the buffered flip flop circuitry in table~\ref{tbl:FFB}.
FS5: $DL2AL^2$ $HIGH$ & & output perm. low & & $OUTPUT STUCK$ \\ \hline FS5: $DL2AL^2$ $HIGH$ & & output perm. low & & $OUTPUT STUCK$ \\ \hline
FS6: $DL2AL^2$ $LOW\_SLEW$ & & no current drive & & $LOW\_SLEW$ \\ FS6: $DL2AL^2$ $LOW\_SLEW$ & & no current drive & & $LOW\_SLEW$ \\
FS7: $CLOCK^0$ $STOPPED$ & & output stuck & & $OUTPUT STUCK$ \\
\hline
\hline \hline
\end{tabular} \end{tabular}
\end{table} \end{table}
@ -2325,6 +2394,27 @@ We now show the final hierarchy in figure~\ref{fig:sdadc}.
\section{Applying FMMD to Software} \section{Applying FMMD to Software}
\label{sec:elecsw} \label{sec:elecsw}
FMMD can be applied to software, and thus we can build complete failure models FMMD can be applied to software, and thus we can build complete failure models

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