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28
submission_thesis/CH5_Examples/copy.tex
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@ -33,7 +33,7 @@ a variety of typical embedded system components including analogue/digital and e
% %
%This is followed by several example FMMD analyses, %This is followed by several example FMMD analyses,
\begin{itemize} \begin{itemize}
\item The first example applies FMMD to an operational amplifier inverting amplifier (see section~\ref{sec:invamp}); \item The first example applies FMMD to an operational-amplifier inverting amplifier (see section~\ref{sec:invamp});
%using an op-amp and two resistors; %using an op-amp and two resistors;
this demonstrates re-use of a potential divider {\dc} from section~\ref{subsec:potdiv}. this demonstrates re-use of a potential divider {\dc} from section~\ref{subsec:potdiv}.
This amplifier is analysed twice, using different compositions of {\fgs}. This amplifier is analysed twice, using different compositions of {\fgs}.
@ -64,7 +64,7 @@ by applying FMMD to a sigma delta ADC.
%shows FMMD analysing the sigma delta %shows FMMD analysing the sigma delta
%analogue to digital converter---again with a circular signal path---which operates on both %analogue to digital converter---again with a circular signal path---which operates on both
%analogue and digital signals. %analogue and digital signals.
\item Section~\ref{sec:Pt100} demonstrates FMMD being applied to commonly used Pt100 \item Section~\ref{sec:Pt100} demonstrates FMMD being applied to a commonly used Pt100
safety critical temperature sensor circuit, this is analysed for single and double failure modes. safety critical temperature sensor circuit, this is analysed for single and double failure modes.
@ -257,7 +257,7 @@ safety critical temperature sensor circuit, this is analysed for single and doub
\end{figure} \end{figure}
%This configuration is interesting from methodology pers. %This configuration is interesting from methodology pers.
There are two obvious ways in which we can model this circuit: There are two obvious ways in which we can model this circuit.
One is to do this in two stages, by considering the gain resistors to be a potential divider One is to do this in two stages, by considering the gain resistors to be a potential divider
and then combining it with the OPAMP failure mode model. and then combining it with the OPAMP failure mode model.
The second is to place all three components in one {\fg}. The second is to place all three components in one {\fg}.
@ -269,7 +269,7 @@ Ideally we would like to re-use {\dcs} from the $PD$ from section~\ref{subsec:po
looks a good candidate for this. looks a good candidate for this.
% %
However, However,
we cannot directly re-use $PD$ , and not just because we cannot directly re-use $PD$, and not just because
the potential divider is floating i.e. that the polarity of the potential divider is floating i.e. that the polarity of
the R2 side of the potential divider is determined by the output from the op-amp. the R2 side of the potential divider is determined by the output from the op-amp.
% %
@ -777,7 +777,7 @@ $$ fm(NI\_AMP) = \{ AMPHigh, AMPLow, LowPass \} .$$
\subsection{The second Stage of the amplifier} \subsection{The second stage of the amplifier}
The second stage of this amplifier, following the signal path, is the amplifier The second stage of this amplifier, following the signal path, is the amplifier
consisting of $R3,R4$ and $IC2$. consisting of $R3,R4$ and $IC2$.
@ -1301,7 +1301,7 @@ $$ fm (G_0) = \{ nosignal, 0\_phaseshift \} $$
%23SEP2012 %23SEP2012
\subsection{Non Inverting Buffer: NIBUFF.} \subsection{Non Inverting Buffer: NIBUFF.}
The non-inverting buffer {\fg}, is comprised of one component, an op-amp. The non-inverting buffer {\fg} is comprised of one component, an op-amp.
We use the failure modes for an op-amp~\cite{fmd91}[p.3-116] to represent this group. We use the failure modes for an op-amp~\cite{fmd91}[p.3-116] to represent this group.
% GARK % GARK
We can express the failure modes for the non-inverting buffer ($NIBUFF$) thus: We can express the failure modes for the non-inverting buffer ($NIBUFF$) thus:
@ -1464,7 +1464,7 @@ we create a $PHS135BUFFERED$ {\dc}. The FMMD analysis may be viewed at section~\
% %
% %
The $PHS225AMP$ consists of a $PHS45$, providing $45^{\circ}$ of phase shift, and an The $PHS225AMP$ consists of a $PHS45$, providing $45^{\circ}$ of phase shift, and an
$INVAMP$, providing $180^{\circ}$ giving a total of $225^{\circ}$. $INVAMP$, providing $180^{\circ}$ giving a total of $225^{\circ}$.
Detailed FMMD analysis may be found in section~\ref{detail:PHS225AMP}. Detailed FMMD analysis may be found in section~\ref{detail:PHS225AMP}.
% %
@ -1617,7 +1617,7 @@ 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].
We express the failure modes for the resistors (R) and Capacitors (C) thus: We express the failure modes for the resistors (R) and capacitors (C) thus:
% %
$$ fm ( R ) = \{OPEN, SHORT\},$$ $$ fm ( R ) = \{OPEN, SHORT\},$$
% %
@ -1647,7 +1647,7 @@ This can be our first {\fg} and we analyse it in table~\ref{tbl:sumjint}.
% %
$$FG = \{R1, R2, IC1, C1 \}$$ $$FG = \{R1, R2, IC1, C1 \}$$
That is the failure modes (see FMMD analysis at~\ref{detail:SUMJINT})of our new {\dc} That is, the failure modes (see FMMD analysis at~\ref{detail:SUMJINT}) of our new {\dc}
$SUMJINT$ are $$\{ V_{in} DOM, V_{fb} DOM, NO\_INTEGRATION, HIGH, LOW \} .$$ $SUMJINT$ are $$\{ V_{in} DOM, V_{fb} DOM, NO\_INTEGRATION, HIGH, LOW \} .$$
%\clearpage %\clearpage
@ -1885,9 +1885,9 @@ The \sd example, shows that FMMD can be applied to mixed digital and analogue ci
%% STATS MOVED TO FUTURE WORK %% STATS MOVED TO FUTURE WORK
%% %%
For this example we look at an industry standard temperature measurement circuit, For this example we look at an industry standard temperature measurement circuit,
the Pt100. The four wire Pt100 configuration commonly used well known safety critical circuit. the Pt100. The four wire Pt100 configuration is a commonly used and well known safety critical circuit.
Applying FMMD lets us look at this circuit in a fresh light. Applying FMMD lets us look at this circuit in a fresh light.
we analyse this for both single and double failures, We analyse this for both single and double failures,
in addition it demonstrates FMMD coping with component parameter tolerances. in addition it demonstrates FMMD coping with component parameter tolerances.
The circuit is described traditionally and then analysed using the FMMD methodology. The circuit is described traditionally and then analysed using the FMMD methodology.
@ -1905,7 +1905,7 @@ industrial applications below 600\oc, due to high accuracy\cite{aoe}.
FMMD is performed twice on this circuit FMMD is performed twice on this circuit
firstly considering single faults only firstly considering single faults only
%(cardinality constrained powerset of 1) %(cardinality constrained powerset of 1)
and again, considering the and secondly, considering the
possibility of double faults. % (cardinality constrained powerset of 2). possibility of double faults. % (cardinality constrained powerset of 2).
% %
% \ifthenelse {\boolean{pld}} % \ifthenelse {\boolean{pld}}
@ -2287,7 +2287,7 @@ All six test cases have been analysed and the results agree with the hypothesis
put in table~\ref{ptfmea}. put in table~\ref{ptfmea}.
%The PLD diagram, can now be used to collect the symptoms. %The PLD diagram, can now be used to collect the symptoms.
In this case there is a common and easily detected symptom for all these single In this case there is a common and easily detected symptom for all these single
resistor faults : Voltage out of range. resistor faults---that of---`voltage~out~of~range'.
% %
% A spider can be drawn on the PLD diagram to this effect. % A spider can be drawn on the PLD diagram to this effect.
% %
@ -2469,7 +2469,7 @@ Both values will be out of range.
\paragraph{ TC 17 : Voltages $R_2$ SHORT $R_3$ OPEN } \paragraph{ TC 17 : Voltages $R_2$ SHORT $R_3$ OPEN }
This shorts the sense- to Ground. This shorts the sense- to ground.
The sense- value will be out of range. The sense- value will be out of range.

41
submission_thesis/CH6_Evaluation/copy.tex
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@ -9,7 +9,7 @@ This chapter begins by defining a metric for the complexity of an FMEA analysis
This concept is called `comparison~complexity' and is a means to assess This concept is called `comparison~complexity' and is a means to assess
the performance of FMMD against current FMEA methodologies. the performance of FMMD against current FMEA methodologies.
% %
This metric is developed using set threory % formally This metric is developed using set theory % formally
and then formulae are presented for calculating the and then formulae are presented for calculating the
complexity of applying FMEA to a group of components. complexity of applying FMEA to a group of components.
% %
@ -218,7 +218,7 @@ we overload the comparison complexity thus:
The potential divider discussed in section~\ref{subsec:potdiv} has four failure modes and two components and therefore has $CC$ of 4. The potential divider discussed in section~\ref{subsec:potdiv} has four failure modes and two components and therefore has $CC$ of 4.
$$CC(potdiv) = \sum_{n=1}^{2} |2| \times (|1|) = 4 $$ $$CC(potdiv) = \sum_{n=1}^{2} |2| \times (|1|) = 4 $$
We combine the potential divider with an op-amp which has four failure modes We combine the potential divider with an op-amp which has four failure modes
to form a {\fg} with two components one with four failure modes and the other (the potential divider) with two. to form a {\fg} with two components, one with four failure modes and the other (the potential divider) with two.
$$CC(invamp) = 2 \times 1 + 4 \times 1 = 6 $$ $$CC(invamp) = 2 \times 1 + 4 \times 1 = 6 $$
To analyse the inverting amplifier with FMMD we required 10 reasoning stages. To analyse the inverting amplifier with FMMD we required 10 reasoning stages.
Using RFMEA we obtain $ 2 \times (3-1) + 2 \times (3-1) + 4 \times (3-1)$ = 16. Using RFMEA we obtain $ 2 \times (3-1) + 2 \times (3-1) + 4 \times (3-1)$ = 16.
@ -290,9 +290,10 @@ with equation~\ref{eqn:anscen}.
The thinking behind equation~\ref{eqn:anscen}, is that for each level of analysis -- counting down from the top -- The thinking behind equation~\ref{eqn:anscen}, is that for each level of analysis -- counting down from the top --
there are ${k}^{n}$ {\fgs} within each level; we need to apply RFMEA to each {\fg} on the level. there are ${k}^{n}$ {\fgs} within each level; we need to apply RFMEA to each {\fg} on the level.
The number of checks to make for RFMEA is number of components $k$ multiplied by the number of failure modes $f$ %
The number of checks to make for RFMEA, is the number of components $k$ multiplied by the number of failure modes $f$
checked against the remaining components in the {\fg} $(k-1)$. checked against the remaining components in the {\fg} $(k-1)$.
%
If, for the sake of example, we fix the number of components in a {\fg} to three and If, for the sake of example, we fix the number of components in a {\fg} to three and
the number of failure modes per component to three, an FMMD hierarchy the number of failure modes per component to three, an FMMD hierarchy
would look like figure~\ref{fig:three_tree}. would look like figure~\ref{fig:three_tree}.
@ -304,7 +305,8 @@ Using the diagram in figure~\ref{fig:three_tree}, we have three levels of analys
Starting at the top, we have a {\fg} with three derived components, each of which has Starting at the top, we have a {\fg} with three derived components, each of which has
three failure modes. three failure modes.
% %
Thus the number of checks to make in the top level is $3^0\times3\times2\times3 = 18$. Thus the number of checks to make in the top level is $3^0\times3\times2\times3 = 18$.
%
On the level below that, we have three {\fgs} each with On the level below that, we have three {\fgs} each with
an identical number of checks, $3^1 \times 3 \times 2 \times 3 = 56$.%{\fg} an identical number of checks, $3^1 \times 3 \times 2 \times 3 = 56$.%{\fg}
% %
@ -323,7 +325,7 @@ In order to get general equations with which to compare RFMEA with FMMD,
we can re-write equation~\ref{eqn:CC} in terms of the number of levels we can re-write equation~\ref{eqn:CC} in terms of the number of levels
in an FMMD hierarchy. in an FMMD hierarchy.
% %
The number of components in the system, is number of components The number of components in the system, is the number of components
in a {\fg} raised to the power of the level plus one. in a {\fg} raised to the power of the level plus one.
Thus we re-write equation~\ref{eqn:CC} as: Thus we re-write equation~\ref{eqn:CC} as:
@ -372,7 +374,7 @@ $$
All the FMMD examples in chapters \ref{sec:chap5} All the FMMD examples in chapters \ref{sec:chap5}
and \ref{sec:chap6} showed a marked reduction in comparison and \ref{sec:chap6} showed a marked reduction in comparison
complexity compared to the RFMEA worst case figures. complexity compared to the RFMEA worst case figures.
To calculate RFMEA Comparison complexity equation~\ref{eqn:CC} is used. To calculate RFMEA comparison complexity equation~\ref{eqn:CC} is used.
% %
% %
Complexity comparison vs. RFMEA for the first three examples Complexity comparison vs. RFMEA for the first three examples
@ -652,7 +654,7 @@ $ fm(R) \in \mathcal{U} $.
We can make this a general case by taking a set $F$ (with $f_1, f_2 \in F$) representing a collection We can make this a general case by taking a set $F$ (with $f_1, f_2 \in F$) representing a collection
of component failure modes. of component failure modes.
We can define a boolean function {\ensuremath{\mathcal{ACTIVE}}} that returns We can define a Boolean function {\ensuremath{\mathcal{ACTIVE}}} that returns
whether a fault mode is active (true) or dormant (false). whether a fault mode is active (true) or dormant (false).
We can say that if any pair of fault modes is active at the same time, then the failure mode set is not We can say that if any pair of fault modes is active at the same time, then the failure mode set is not
@ -703,8 +705,9 @@ is then applied to it.}.
\paragraph{Reason for Constraint.} Were this constraint to not be applied \paragraph{Reason for Constraint.} Were this constraint not to be applied
each component would not contribute $N$ failure modes to consider but potentially each component would not contribute $N$ failure modes, % to consider
but potentially
$2^N$. $2^N$.
% %
This would make the job of analysing the failure modes This would make the job of analysing the failure modes
@ -715,7 +718,7 @@ in a {\fg} impractical due to the sheer size of the task.
\section{Handling Simultaneous Component Faults} \section{Handling Simultaneous Component Faults}
For some integrity levels of static analysis, there is a need to consider not only single For some integrity levels of static analysis, there is a need to consider not only single
failure modes in isolation, but cases where more then one failure mode may occur failure modes in isolation, but cases where more than one failure mode may occur
simultaneously. simultaneously.
% %
Note that the `unitary state' conditions apply to failure modes within a component. Note that the `unitary state' conditions apply to failure modes within a component.
@ -1057,7 +1060,7 @@ $ \Omega(C) = fm(C) \cup \{OK\} $).
The $OK$ statistical case is the (usually) largest in probability, and is therefore The $OK$ statistical case is the (usually) largest in probability, and is therefore
of interest when analysing systems from a statistical perspective. of interest when analysing systems from a statistical perspective.
For these examples the OK state is not represented area proportionately, but included For these examples, the OK state is not represented area proportionately, but included
in the diagrams. in the diagrams.
This is of interest for the application of conditional probability calculations This is of interest for the application of conditional probability calculations
such as Bayes theorem~\cite{probstat}. such as Bayes theorem~\cite{probstat}.
@ -1072,7 +1075,7 @@ Another way to view this is to consider the failure modes of a
component, with the $OK$ state, as a universal set $\Omega$, where component, with the $OK$ state, as a universal set $\Omega$, where
all sets within $\Omega$ are partitioned. all sets within $\Omega$ are partitioned.
Figure \ref{fig:partitioncfm} shows a partitioned set representing Figure \ref{fig:partitioncfm} shows a partitioned set representing
component failure modes $\{ B_1 ... B_8, OK \}$ : partitioned sets component failure modes $\{ B_1 ... B_8, OK \}$: partitioned sets
where the OK or empty set condition is included, obey unitary state conditions. where the OK or empty set condition is included, obey unitary state conditions.
Because the subsets of $\Omega$ are partitioned, we can say these Because the subsets of $\Omega$ are partitioned, we can say these
failure modes are unitary state. failure modes are unitary state.
@ -1119,7 +1122,7 @@ of the failure modes as new failure modes.
We can model this using an Euler diagram representation of We can model this using an Euler diagram representation of
an example component with three failure modes\footnote{OK is really the empty set, but the term OK is more meaningful in an example component with three failure modes\footnote{OK is really the empty set, but the term OK is more meaningful in
the context of component failure modes} $\{ B_1, B_2, B_3, OK \}$ see figure \ref{fig:combco}. the context of component failure modes} $\{ B_1, B_2, B_3, OK \}$ see figure \ref{fig:combco}.
%
For the purpose of example let us consider $\{ B_2, B_3 \}$ For the purpose of example let us consider $\{ B_2, B_3 \}$
to be intrinsically mutually exclusive, but $B_1$ to be independent. to be intrinsically mutually exclusive, but $B_1$ to be independent.
This means the we have the possibility of two new combinations This means the we have the possibility of two new combinations
@ -1137,8 +1140,8 @@ as shaded sections of figure \ref{fig:combco2}.
We can calculate the probabilities for the shaded areas We can calculate the probabilities for the shaded areas,
assuming the failure modes are statistically independent assuming the failure modes are statistically independent,
by multiplying the probabilities of the members of the intersection. by multiplying the probabilities of the members of the intersection.
We can use the function $P$ to return the probability of a We can use the function $P$ to return the probability of a
failure mode, or combination thereof. failure mode, or combination thereof.
@ -1209,14 +1212,16 @@ in the power-supply {\fg}.
Because the capacitor has two potential failure modes (EN298), Because the capacitor has two potential failure modes (EN298),
this raises another issue for FMMD. A de-coupling capacitor going $OPEN$ might not be considered relevant to this raises another issue for FMMD. A de-coupling capacitor going $OPEN$ might not be considered relevant to
a power-supply module (but there might be additional noise on its output rails). a power-supply module (but there might be additional noise on its output rails).
But in {\fg} terms the power supply, now has a new symptom that of $INTERFERENCE$. %
But in {\fg} terms, the power supply now has a new symptom that of $INTERFERENCE$.
% %
Some logic chips are more susceptible to $INTERFERENCE$ than others. Some logic chips are more susceptible to $INTERFERENCE$ than others.
A logic chip with de-coupling capacitor failing, may operate correctly A logic chip with de-coupling capacitor failing, may operate correctly
but interfere with other chips in the circuit. but interfere with other chips in the circuit.
% %
There is no reason why the de-coupling capacitors There is no reason why the de-coupling capacitors
could not be included {\em in the {\fg} they would intuitively be associated with as well}.% poss split infinitive could not be included % {\em in the {\fg} they would intuitively be associated with as well}.% poss split infinitive
in {\fgs} that they would not intuitively be associated with.
% %
This allows for the general principle of a component failure affecting more than one {\fg} in a circuit. This allows for the general principle of a component failure affecting more than one {\fg} in a circuit.
This allows functional groups to share components where necessary. This allows functional groups to share components where necessary.

2
submission_thesis/appendixes/algorithmic.tex
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@ -4,7 +4,7 @@
\label{sec:algorithmfmmd} \label{sec:algorithmfmmd}
This section decribes the algorithm for performing one step of This section decribes the algorithm for performing one step of
FMMD analysis i.e. FMMD analysis
analysing a {\fg} and determining from it a {\dc}. analysing a {\fg} and determining from it a {\dc}.
Algorithms using set theory describe the process. Algorithms using set theory describe the process.
It begins with an overview of the FMMD process, and then contrasts and compares it It begins with an overview of the FMMD process, and then contrasts and compares it