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Comparison complexity results.

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Very good for Bubba Oscillator
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17
mybib.bib
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@ -924,6 +924,23 @@ strength of materials, the causes of boiler explosions",
year = "2000" year = "2000"
} }
@Manual{lm358,
title = {Datasheet: Low-Power dual operation amplifiers LM158,LM258,LM358: Doc ID 2163 Rev 10},
key = {Doc ID 2163 Rev 10},
author = {ST Microelecronics.},
OPTorganization = {},
address = {http://www.st.com/},
OPTedition = {},
OPTmonth = {},
year = {2012},
OPTnote = {},
OPTannote = {},
OPTurl = {},
OPTdoi = {},
OPTissn = {},
OPTlocalfile = {},
OPTabstract = {},
}
@Manual{tlp181, @Manual{tlp181,
title = {TLP 181 Datasheet}, title = {TLP 181 Datasheet},

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43
submission_thesis/CH5_Examples/copy.tex
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@ -901,10 +901,10 @@ when it becomes a V2 follower).
The {\fm} $DiffAMPIncorrect$ may seem like a vague {\fm}---however, this {\fm} is impossible to detect in this circuit--- The {\fm} $DiffAMPIncorrect$ may seem like a vague {\fm}---however, this {\fm} is impossible to detect in this circuit---
in fault finding terminology~\cite{garrett}~\cite{maikowski} this {\fm} is said to be unobservable, and in EN61508~\cite{en61508} in fault finding terminology~\cite{garrett}~\cite{maikowski} this {\fm} is said to be unobservable, and in EN61508~\cite{en61508}
terminology is called an undetectable fault. terminology is an `undetectable~fault'.
% %
Were this failure to have safety implications, this FMMD analysis will have revealed Were this failure to have safety implications, this FMMD analysis will have revealed
the un-observability and would likely prompt re-design of this this un-observability condition; this would likely prompt re-design of this
circuit. A typical way to solve an un-observability such as this is circuit. A typical way to solve an un-observability such as this is
to periodically switch in test signals in place of the input signal. to periodically switch in test signals in place of the input signal.
%\footnote{A typical way to solve an un-observability such as this is %\footnote{A typical way to solve an un-observability such as this is
@ -1211,9 +1211,19 @@ could be easily detected; the failure symptom $FilterIncorrect$ may be less obs
This example shows the analysis of a linear signal path circuit with three easily identifiable This example shows the analysis of a linear signal path circuit with three easily identifiable
{\fgs} and re-use of the Sallen-Key {\dc}. {\fgs} and re-use of the Sallen-Key {\dc}.
\clearpage
\clearpage
%
% BUBBAOSC
%
\section{Quad Op-Amp Oscillator} \section{Quad Op-Amp Oscillator}
\label{sec:bubba} \label{sec:bubba}
@ -1221,7 +1231,7 @@ This example shows the analysis of a linear signal path circuit with three easil
\centering \centering
\includegraphics[width=200pt]{CH5_Examples/circuit3003.png} \includegraphics[width=200pt]{CH5_Examples/circuit3003.png}
% circuit3003.png: 503x326 pixel, 72dpi, 17.74x11.50 cm, bb=0 0 503 326 % circuit3003.png: 503x326 pixel, 72dpi, 17.74x11.50 cm, bb=0 0 503 326
\caption{Circuit 3} \caption{Circuit diagram for the Quad Op-Amp `Bubba' Oscillator}
\label{fig:circuit3} \label{fig:circuit3}
\end{figure} \end{figure}
@ -1233,8 +1243,8 @@ The circuit implements an oscillator using four 45 degree phase shifts, and an i
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. The circuit provides two outputs with a quadrature phase relationship.
% %
From a fault finding perspective this circuit cannot be decomposed From a fault finding perspective this circuit cannot be decomposed,
because the whole circuit is enclosed within a feedback loop, as the whole circuit is enclosed within a feedback loop,
hence a fault anywhere in the loop is likely to affect all stages. hence a fault anywhere in the loop is likely to affect all stages.
% %
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.
@ -1314,7 +1324,7 @@ 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}
\label{sec:bubba1}
Our {\fg} for this analysis can be expressed thus: Our {\fg} for this analysis can be expressed thus:
% %
%$$ G^1_0 = \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 PHS45^1_4, INVAMP^1_0 \} ,$$ %$$ G^1_0 = \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 PHS45^1_4, INVAMP^1_0 \} ,$$
@ -1368,7 +1378,7 @@ we may also discover new derived components that may be of use for other analyse
\clearpage \clearpage
\subsection{FMMD Analysis of Bubba Oscillator using a finer grained modular approach (i.e. more hierarchical stages)} \subsection{FMMD Analysis of Bubba Oscillator using a finer grained modular approach (i.e. more hierarchical stages)}
\label{sec:bubba2}
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.
We should be able to determine smaller {\fgs} and refine the model further. We should be able to determine smaller {\fgs} and refine the model further.
@ -1415,6 +1425,10 @@ A PHS45 {\dc} and an inverting amplifier\footnote{Inverting amplifiers apply a
form a {\fg} form a {\fg}
providing an amplified $225^{\circ}$ phase shift, analysed in table~\ref{tbl:phs225amp} providing an amplified $225^{\circ}$ phase shift, analysed in table~\ref{tbl:phs225amp}
resulting in the {\dc} $PHS225AMP$. resulting in the {\dc} $PHS225AMP$.
Applying FMMD we create a derived component $PHS225AMP$ which has the following failure modes:
$$
fm (PHS225AMP) = \{ 180\_phaseshift, NO\_signal .\} % 270\_phaseshift,
$$
% %
%---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 form a final {\fg} with $PHS135BUFFERED$ and $PHS225AMP$. Finally we form a final {\fg} with $PHS135BUFFERED$ and $PHS225AMP$.
@ -1485,9 +1499,12 @@ $$
% Our total comparison complexity is $58$, this contrasts with $468$ for traditional `flat' FMEA, % Our total comparison complexity is $58$, this contrasts with $468$ for traditional `flat' FMEA,
% and $250$ for our first stage functional groups analysis. % 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. % This has meant a drastic reduction in the number of failure-modes to check against components.
It has %also %It has %also
given us five {\dcs}, building blocks, which could potentially be re-used for similar circuitry This more de-composed approach has
to analyse in the future. given us five {\dcs}, building blocks, which could %
be re-used in other projects.
%potentially be re-used for similar circuitry
%to analyse in the future.
% %
% %
\subsection{Comparing both approaches} \subsection{Comparing both approaches}
@ -1512,6 +1529,8 @@ However, it involves a large reasoning distance, the final stage
having 24 failure modes to consider against each of the other seven {\dcs}. having 24 failure modes to consider against each of the other seven {\dcs}.
A finer grained approach produces more potentially re-usable {\dcs} and A finer grained approach produces more potentially re-usable {\dcs} and
involves several stages with lower reasoning distances. involves several stages with lower reasoning distances.
The lower reasoning distances, or complexity comparision figures are given in the metrics chapter~\ref{sec:chap7}
at section~\ref{sec:bubbaCC}.
@ -1552,7 +1571,7 @@ The following example is used to demonstrate FMMD analysis of a mixed analogue a
\paragraph{How the circuit works.} \paragraph{How the circuit works.}
A detailed description of \sd may be found in~\cite{mixedsignaldsp}[pp.69-80]. A detailed description of {\sd} may be found in~\cite{mixedsignaldsp}[pp.69-80].
The diagram in~\ref{fig:sigmadeltablock} shows the signal path used The diagram in~\ref{fig:sigmadeltablock} shows the signal path used
by this configuration for a \sd. by this configuration for a \sd.
% %

155
submission_thesis/CH6_Evaluation/copy.tex
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@ -365,6 +365,8 @@ $$
\sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720 \sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720
%\end{equation} %\end{equation}
$$ $$
%\clearpage
\subsection{Complexity Comparison applied to previous FMMD Examples} \subsection{Complexity Comparison applied to previous FMMD Examples}
All the FMMD examples in chapters \ref{sec:chap5} and \ref{sec:chap6} showed a marked reduction in comparison All the FMMD examples in chapters \ref{sec:chap5} and \ref{sec:chap6} showed a marked reduction in comparison
@ -372,14 +374,18 @@ complexity compared to the RFMEA worst case figures.
% %
% %
A table of complexity comparison vs. RFMEA is presented below. Complexity comparison vs. RFMEA for the first three examples
are presented in table~\ref{tbl:firstcc}.
%\usepackage{multirow} %\usepackage{multirow}
\begin{table}
\label{tbl:firstcc}
\begin{tabular}{ |l|l|l| } \begin{tabular}{ |c|l|l|c| }
\hline \hline
\textbf{Hierarchy} & \textbf{Analysis object} & \textbf{Complexity} \\ \textbf{Hierarchy} & \textbf{Derived} & \textbf{Complexity} & $|fm(c)|$: \textbf{number} \\
\textbf{Level} & \textbf{Description} & \textbf{Comparison} \\ \textbf{Level} & \textbf{Component} & \textbf{Comparison} & \textbf{of derived} \\
& & & \textbf{failure modes} \\
%\hline \hline %\hline \hline
%\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\ %\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\
%\hline \hline %\hline \hline
@ -391,37 +397,134 @@ A table of complexity comparison vs. RFMEA is presented below.
\multicolumn{3}{ |c| }{Inverting Amplifier Two stage FMMD Hierarchy: section~\ref{sec:invamp}} \\ \hline \multicolumn{3}{ |c| }{Inverting Amplifier Two stage FMMD Hierarchy: section~\ref{sec:invamp}} \\ \hline
%\multirow{3}{*} {Inverting Amplifier Two stage FMMD Hierarchy: section~\ref{sec:invamp}} & & \\ %\multirow{3}{*} {Inverting Amplifier Two stage FMMD Hierarchy: section~\ref{sec:invamp}} & & \\
0 & Potential Divider & 4 \\ \hline
1 & PD + Opamp & 8 \\ 0 & PD & 4 & 2 \\
& Inverting Amplifier: & FMMD 10 \\ 1 & INVAMP & 8 & 3 \\
& Inverting Amplifier: & RFMEA 16 \\ 2 & Total for INVAMP: & 10 (FMMD) & \\
\hline 0 & Total for INVAMP: & 16 (RFMEA) & \\
% & $(3-1) \times (4 + 2 +2)$ & & \\
\hline \hline
\multicolumn{3}{ |c| } {Inverting Amplifier One stage FMMD Hierarchy: section~\ref{sec:invamp}} \\ \hline \multicolumn{3}{ |c| } {Inverting Amplifier One stage FMMD Hierarchy: section~\ref{sec:invamp}} \\ \hline
0 & Resistors + Opamp & 16 \\ 0 & INVAMP & 16 & 3 \\
& Inverting Amplifier: & FMMD 16 \\ 1 & Total for INVAMP: & 16 (FMMD) & \\
& Inverting Amplifier: & RFMEA 16 \\ 0 & Total for INVAMP: & 16 (RFMEA) & \\
\hline \hline
\hline
\multicolumn{3}{ |c| } {Differencing Amplifier One stage FMMD Hierarchy: section~\ref{sec:invamp}} \\ \hline \multicolumn{3}{ |c| } {Differencing Amplifier Three stage FMMD Hierarchy: section~\ref{sec:diffamp}} \\ \hline
%\multirow{4}{*} {Differencing Amplifier FMMD Hierarchy: section~\ref{sec:diffamp}} & & \\ %\multirow{4}{*} {Differencing Amplifier FMMD Hierarchy: section~\ref{sec:diffamp}} & & \\
2 & Non inv Amp reused (see section~\ref{sec:noninvamp}) & 10 \\ 2 & NonInvAMP reused~\footnote{Reused analysed of NonInvAMP: see section~\ref{sec:invamp}.} & 10 & 3 \\
0 & Inverting amplifier & 16 \\ 0 & SEC\_AMP & 16 & 4 \\
& Differencing Amplifier: & FMMD 26 \\ 3 & DiffAMP & 7 & 4 \\
& Differencing Amplifier: & RFMEA 80 \\ \hline 3 & Total for DiffAMP & 33 (FMMD)& \\
0 & Total for DiffAMP: & 80 (RFMEA) & \\
% & Differencing Amplifier: & RFMEA 80-16 = 74 & \\
% & & & \\
\hline \hline
\hline
% \footnote{if we discount the comparison complexity for the pre-analysed INVAMP.}\hline
\multicolumn{3}{ |c| } {Five Pole Sallen Key Low Pass Filter: Three stage FMMD Hierarchy: section~\ref{sec:fivepolelp}} \\ \hline
%\multirow{4}{*} {Differencing Amplifier FMMD Hierarchy: section~\ref{sec:diffamp}} & & \\
0 & FirstOrderLP & 4 & 2 \\
1 & LP1 & 10 & 4 \\
\hline \hline 2 & SKLP & 48 & 4 \\
3 & FivePoleLP & 20 & 4 \\
3 & Total for FivePoleLP & 82 (FMMD)& \\
% & 20+48+10+4 & & \\
0 & Total for FivePoleLP & 384 (RFMEA) & \\
% & $(13-1) \times (3 \times 4 + 10 \times 2)$ & & \\ \hline
\hline
\end{tabular} \end{tabular}
\caption{Comparison Complexity figures for the first three examples in Chapter~\ref{sec:chap5}.}
\end{table}
% end table
The complexity comparison figures for the example circuits in chapter~\ref{sec:chap5} show The complexity comparison figures for the example circuits in chapter~\ref{sec:chap5} show
that for increasing complexity the performance benefits from FMMD become apparent. that for increasing complexity the performance benefits from FMMD are apparent.
\clearpage
\subsection{Comparison Complexity for the Bubba Oscillator Example.}
The Bubba oscillator example (see section~\ref{sec:bubba}) was chosen because it had a circular
signal path. It was also analysed twice, once by
{na\"{\i}vely} using the first {\fgs} identified, and secondly be de-composing
the circuit further.
We use these two analyses to compare the effect on comparison complexity (see table~\ref{tbl:bubbacc}) with that of RFMEA.
%
\begin{table}
\label{tbl:bubbacc}
\begin{tabular}{ |c|l|l|c| }
\hline
\textbf{Hierarchy} & \textbf{Derived} & \textbf{Complexity} & $|fm(c)|$: \textbf{number} \\
\textbf{Level} & \textbf{Component} & \textbf{Comparison} & \textbf{of derived} \\
& & & \textbf{failure modes} \\
%\hline \hline
%\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\
%\hline \hline
%Goalkeeper & GK & Paul Robinson \\ \hline
\hline
\multicolumn{3}{ |c| }{Bubba Oscillator one stage ({na\"{\i}ve}) FMMD Hierarchy: section~\ref{sec:bubba1}} \\ \hline
%\multirow{3}{*} {Inverting Amplifier Two stage FMMD Hierarchy: section~\ref{sec:invamp}} & & \\
\hline
1 & PHS45 & 4 & 2 \\
1 & INVAMP & 16 & 3 \\
0 & NIBUFF & 0 & 4 \\
%
% NIBUFF PHS45
% 8 components so LEVEL 2 (8-1) \times ( (3*4) + (4*2) + 3 ) + LEVEL 0 16 for the INVAMP
2 & Total for BUBBA: & 177 (FMMD) & \\
% R&C OPAMPS
% 14 components so 13 \times ( (10*2) (4*4) )
0 & Total for BUBBA: & 468 (RFMEA) & \\
% & $(3-1) \times (4 + 2 +2)$ & & \\
\hline \hline
\multicolumn{3}{ |c| } {Inverting Amplifier Multiple stage FMMD Hierarchy: section~\ref{sec:bubba2}} \\ \hline
1 & PHS45 & 4 & 2 \\
1 & INVAMP & 16 & 3 \\
0 & NIBUFF & 0 & 4 \\
2 & BUFF45 & 6 & 2 \\
3 & PHS135BUFFERED & 4 & 2 \\
2 & PHS225AMP & 5 & 2 \\
4 & BUBBA & 2 & 2 \\
%
%Level 1: 16 + 4 == 20
%Level 2: 6 + 5 == 11
%Level 3: 4 == 4
%Level 4: 2 == 2
%
1 & Total for BUBBA: & 37 (FMMD) & \\
0 & Total for BUBBA: & 468 (RFMEA) & \\
\hline
\hline
\end{tabular}
\caption{Complexity Comparison figures for the Bubba Oscillator FMMD example (see section~\ref{sec:bubba}).}
\end{table}
%
The initial {na\"{\i}ve} FMMD analysis reduces the number of checks by over half, the more de-composed analysis
by more than a factor of ten.
\subsection{Sigma delta Example: Comparison Complexity Results}
\label{sec:bubbaCC}
% \subsection{Exponential squared to Exponential} % \subsection{Exponential squared to Exponential}
% %
% can I say that ? % can I say that ?
@ -430,7 +533,7 @@ that for increasing complexity the performance benefits from FMMD become apparen
\section{Unitary State Component Failure Mode sets} \section{Unitary State Component Failure Mode sets}
\label{sec:unitarystate} \label{sec:unitarystate}
\paragraph{Design Descision/Constraint} \paragraph{Design Decision/Constraint}
An important factor in defining a set of failure modes is that they An important factor in defining a set of failure modes is that they
should represent the failure modes as simply and minimally as possible. should represent the failure modes as simply and minimally as possible.
It should not be possible, for instance, for It should not be possible, for instance, for