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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"
}
@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,
title = {TLP 181 Datasheet},

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related_papers_books/LM358_CD00000464.pdf Normal file
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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---
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
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
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
@ -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
{\fgs} and re-use of the Sallen-Key {\dc}.
\clearpage
\clearpage
%
% BUBBAOSC
%
\section{Quad Op-Amp Oscillator}
\label{sec:bubba}
@ -1221,7 +1231,7 @@ This example shows the analysis of a linear signal path circuit with three easil
\centering
\includegraphics[width=200pt]{CH5_Examples/circuit3003.png}
% 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}
\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).
The circuit provides two outputs with a quadrature phase relationship.
%
From a fault finding perspective this circuit cannot be decomposed
because the whole circuit is enclosed within a feedback loop,
From a fault finding perspective this circuit cannot be decomposed,
as the whole circuit is enclosed within a feedback loop,
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.
@ -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}
\label{sec:bubba1}
Our {\fg} for this analysis can be expressed thus:
%
%$$ G^1_0 = \{ PHS45^1_1, NIBUFF^0_1, PHS45^1_2, NIBUFF^0_2, PHS45^1_3, NIBUFF^0_3 PHS45^1_4, INVAMP^1_0 \} ,$$
@ -1368,7 +1378,7 @@ we may also discover new derived components that may be of use for other analyse
\clearpage
\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.
%This mean a quite large comparison complexity for this final stage.
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}
providing an amplified $225^{\circ}$ phase shift, analysed in table~\ref{tbl: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 $PHS225AMP$---
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,
% 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.
%It has %also
This more de-composed approach has
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}
@ -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}.
A finer grained approach produces more potentially re-usable {\dcs} and
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.}
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
by this configuration for a \sd.
%

153
submission_thesis/CH6_Evaluation/copy.tex
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@ -365,6 +365,8 @@ $$
\sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720
%\end{equation}
$$
%\clearpage
\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
@ -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}
\begin{table}
\label{tbl:firstcc}
\begin{tabular}{ |l|l|l| }
\begin{tabular}{ |c|l|l|c| }
\hline
\textbf{Hierarchy} & \textbf{Analysis object} & \textbf{Complexity} \\
\textbf{Level} & \textbf{Description} & \textbf{Comparison} \\
\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
@ -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
%\multirow{3}{*} {Inverting Amplifier Two stage FMMD Hierarchy: section~\ref{sec:invamp}} & & \\
0 & Potential Divider & 4 \\
1 & PD + Opamp & 8 \\
& Inverting Amplifier: & FMMD 10 \\
& Inverting Amplifier: & RFMEA 16 \\
\hline
0 & PD & 4 & 2 \\
1 & INVAMP & 8 & 3 \\
2 & Total for INVAMP: & 10 (FMMD) & \\
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
0 & Resistors + Opamp & 16 \\
& Inverting Amplifier: & FMMD 16 \\
& Inverting Amplifier: & RFMEA 16 \\
0 & INVAMP & 16 & 3 \\
1 & Total for INVAMP: & 16 (FMMD) & \\
0 & Total for INVAMP: & 16 (RFMEA) & \\
\hline
\multicolumn{3}{ |c| } {Differencing Amplifier One stage FMMD Hierarchy: section~\ref{sec:invamp}} \\ \hline
\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}} & & \\
2 & Non inv Amp reused (see section~\ref{sec:noninvamp}) & 10 \\
0 & Inverting amplifier & 16 \\
& Differencing Amplifier: & FMMD 26 \\
& Differencing Amplifier: & RFMEA 80 \\ \hline
2 & NonInvAMP reused~\footnote{Reused analysed of NonInvAMP: see section~\ref{sec:invamp}.} & 10 & 3 \\
0 & SEC\_AMP & 16 & 4 \\
3 & DiffAMP & 7 & 4 \\
3 & Total for DiffAMP & 33 (FMMD)& \\
0 & Total for DiffAMP: & 80 (RFMEA) & \\
% & Differencing Amplifier: & RFMEA 80-16 = 74 & \\
% & & & \\
\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 \\
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
\hline \hline
\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
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}
%
% 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}
\label{sec:unitarystate}
\paragraph{Design Descision/Constraint}
\paragraph{Design Decision/Constraint}
An important factor in defining a set of failure modes is that they
should represent the failure modes as simply and minimally as possible.
It should not be possible, for instance, for