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Robin_PHD/opamp_circuits_C_GARRETT/opamps.tex
Robin Clark d27bbcfdf5 .
2011-11-23 09:41:41 +00:00

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\documentclass[a4paper,10pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{tikz}
\usetikzlibrary{shapes,snakes}
\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows}
\usepackage{subfigure}
\usepackage{amsfonts,amsmath,amsthm}
\usepackage{algorithm}
\usepackage{algorithmic}
\usepackage{lastpage}
\newcommand{\fg}{\em functional~group}
\newcommand{\fgs}{\em functional~groups}
\newcommand{\dc}{\em derived~component}
\newcommand{\dcs}{\em derived~components}
\newcommand{\bc}{\em base~component}
\newcommand{\bcs}{\em base~components}
\newcommand{\irl}{in~real~life}
\newcommand{\abslevel}{\ensuremath{\psi}}
\newcommand{\FG}{\ensuremath{G}}
%\usepackage{glossary}
%opening
\title{Example OPAMP circuits}
\author{Robin}
\begin{document}
\begin{abstract}
Circuits from email conversation.
Not a document to be proof read.
Proof of analysis concept.
Function $fm$ applied to a component returns its failure modes.
\end{abstract}
\maketitle
\tableofcontents
\listoffigures
\section{Non-Inverting OPAMP}
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 amplifiying it to the useful range of an ADC, approx 0 to 4 volts).
\begin{figure}[h+]
\centering
\includegraphics[width=100pt]{./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.
According to GAS standard EN298 the failure modes to consider for resistors are OPEN and 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]{./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: Sinlge failure analysis}
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Pot Div Effect} & & \textbf{Symptom} \\
\hline
FS1: R1 SHORT & & $LOW$ & & $PDLow$ \\ \hline
FS2: R1 OPEN & & $HIGH$ & & $PDHigh$ \\ \hline
FS3: R2 SHORT & & $HIGH$ & & $PDHigh$ \\ \hline
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 use 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}
$$ fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$
We can now form a {\fg} with $PD$ and $OPAMP$.
\begin{figure}
\centering
\includegraphics[width=300pt]{./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}
We can collect symptoms from the analysis and cretae a derived component
to represent the non-inverting amplifier $NI\_AMP$.
We now have can express the failure mode behaviour of this type of amplifier thus:
$$ fm(NI\_AMP) = \{ {lowpass}, {high}, {low} \}.$$
\section{Inverting OPAMP}
\begin{figure}[h]
\centering
\includegraphics[width=200pt]{./invamp.png}
% invamp.png: 378x207 pixel, 72dpi, 13.34x7.30 cm, bb=0 0 378 207
\caption{Inverting Amplifier Configuration}
\label{fig:invamp}
\end{figure}
This configuration is interesting from methodology perspective.
There are two ways in which we can tackle this.
One is to do this in two stages, by considing the gain resistors to be a potential divider
and then combining it with the OPAMP failure mode model.
The other way is to place all three components in a {\fg}.
Both approaches are followed in the next two sub-sections.
\subsection{Inverting OPAMP using a Potential Divider {\dc}}
Re-using the $PD$ - potential divider works only if the input voltage is negative.
We want if possible to have detectable errors, HIGH and LOW are better than OUTOFRANGE.
If we can refine the operational states of the fungional group, we can obtain clearer
symptoms.
If we consider the input will only be positive, we can invert the potential divider (see table~\ref{tbl:pdneg}).
\begin{table}[h+]
\caption{Inverted Potential divider: Single failure analysis}
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Inverted Pot Div Effect} & & \textbf{Symptom} \\
\hline
FS1: R1 SHORT & & $HIGH$ & & $PDHigh$ \\ \hline
FS2: R1 OPEN & & $LOW$ & & $PDLow$ \\ \hline
FS3: R2 SHORT & & $LOW$ & & $PDLow$ \\ \hline
FS4: R2 OPEN & & $HIGH$ & & $PDHigh$ \\ \hline
\hline
\end{tabular}
\label{tbl:pdneg}
\end{table}
We can form a {\dc} from this, and call it an inverted potential divider $INVPD$.
We can now form a {\fg} from the OPAMP and the $INVPD$
This gives the same results as the analysis from figure~\ref{fig:invampanalysis}.
The differences are the root causes or component failure modes that
lead to the symptoms (i.e. the symptoms are the same but causation tree will be different).
$$ fm(NI\_AMP) = \{ {lowpass}, {high}, {low} \}.$$
\subsection{Inverting OPAMP using three components}
We can use this for a more general case, because we can examine the
effects on the circuit for each operational case (i.e. input +ve
or input -ve), see table~\ref{tbl:invamp}. Because symptom collection is defined as surjective (from component failure modes
to symptoms) we cannot have a component failure mode that maps to two different symptoms (within a functional group).
Note that here we have a more general symptom $ OUT OF RANGE $ which could mean either
$HIGH$ or $LOW$ output.
\begin{table}[h+]
\caption{Inverting Amplifier: Single failure analysis}
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Inverted Amp Effect} & & \textbf{Symptom} \\ \hline
\hline
FS1: R1 SHORT +ve in & & NEGATIVE out of range & & $ OUT OF RANGE $ \\
FS1: R1 SHORT -ve in & & POSITIVE out of range & & $ OUT OF RANGE $ \\ \hline
FS2: R1 OPEN +ve in & & zero output & & $ ZERO OUTPUT $ \\
FS2: R1 OPEN -ve in & & zero output & & $ ZERO OUTPUT $ \\ \hline
FS3: R2 SHORT +ve in & & $INVAMP_{nogain} $ & & $ NO GAIN $ \\
FS3: R2 SHORT -ve in & & $INVAMP_{nogain} $ & & $ NO GAIN $ \\ \hline
FS4: R2 OPEN +ve in & & NEGATIVE out of range $ $ & & $ OUT OF RANGE$ \\
FS4: R2 OPEN -ve in & & POSITIVE out of range $ $ & & $OUT OF RANGE $ \\ \hline
FS5: AMP L\_DN & & $ INVAMP_{low} $ & & $ OUT OF RANGE $ \\ \hline
FS2: AMP L\_UP & & $INVAMP_{high} $ & & $ OUT OF RANGE $ \\ \hline
FS3: AMP NOOP & & $INVAMP_{nogain} $ & & $ NO GAIN $ \\ \hline
FS4: AMP LowSlew & & $ slow output \frac{\delta V}{\delta t} $ & & $ LOW PASS $ \\ \hline
\hline
\end{tabular}
\label{tbl:invamp}
\end{table}
$$ fm(INVAMP) = \{ OUT OF RANGE, ZERO OUTPUT, NO GAIN, LOW PASS \} $$
%Much more general. OUT OF RANGE symptom maps to many component failure modes.
%Observability problem... system. In fact can we get a metric of how observable
%a system is using the ratio of component failure modes X op states to a symptom ????
%Could further refine this if MTTF stats available for each component failure.
\subsection{Comparison between the two approaches}
If the input voltage can be negative the potential divider
becomes reversed in polarity.
This means that detecting which failure mode has occurred from knowing the symptom, has become a more difficult task; or in other words
the observability of the causes of failure are reduced.
\clearpage
\section{Op-Amp circuit 1}
\begin{figure}[h]
\centering
\includegraphics[width=200pt]{/home/robin/projects/thesis/opamp_circuits_C_GARRETT/circuit1001.png}
% circuit1001.png: 420x300 pixel, 72dpi, 14.82x10.58 cm, bb=0 0 420 300
\caption{Circuit 1}
\label{fig:circuit1}
\end{figure}
The amplifier in figure~\ref{fig:circuit1} amplifies the difference between
the input voltages $+V1$ and $+V2$.
It would be desirable to represent this circuit as a derived component called say $DiffAMP$.
We begin by identifying functional groups from the components in the circuit.
\subsection{Functional Group: Potential Divider}
For the gain setting resistors R1,R2 -- we can re-use the potential divider from section~\ref{potdivfmmd}.
%R1 and R2 perform as a potential divider.
%Resistors can fail OPEN and SHORT (according to GAS burner standard EN298 Appendix A).
%$$ fm(R) = \{ OPEN, SHORT \}$$
% \begin{table}[ht]
% \caption{Potential Divider $PD$: Failure Mode Effects Analysis: Single Faults} % title of Table
% \centering % used for centering table
% \begin{tabular}{||l|c|c|l|l||}
% \hline \hline
% \textbf{Test} & \textbf{Pot.Div} & \textbf{ } & \textbf{General} \\
% \textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symtom Description} \\
% % R & wire & res + & res - & description
% \hline
% \hline
% TC1: $R_1$ SHORT & LOW & & LowPD \\
% TC2: $R_1$ OPEN & HIGH & & HighPD \\ \hline
% TC3: $R_2$ SHORT & HIGH & & HighPD \\
% TC4: $R_2$ OPEN & LOW & & LowPD \\ \hline
% \hline
% \end{tabular}
% \label{tbl:pdfmea}
% \end{table}
%
% By collecting the symptoms in table~\ref{tbl:pdfmea} we can create a derived
% component $PD$ to represent the failure mode behaviour
% of a potential divider.
Thus for single failure modes, a potential divider can fail
with $fm(PD) = \{PDHigh,PDLow\}$.
The potential divider is used to program the gain of IC1.
IC1 and PD provide the function of buffering
/amplifying the signal $+V1$.
We can now examine IC1 and PD as a functional group.
\pagebreak[3]
\subsection{Functional Group: Amplifier}
Let use 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\%).
$$ fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$
By bringing the $PD$ derived component and the $OPAMP$ into
a functional group we can analyse its failure mode behaviour.
\begin{table}[ht]
\caption{Non Inverting Amplifier $NI\_AMP$: Failure Mode Effects Analysis: Single Faults} % title of Table
\centering % used for centering table
\begin{tabular}{||l|c|c|l|l||}
\hline \hline
\textbf{Test} & \textbf{Amplifier} & \textbf{ } & \textbf{General} \\
\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symtom Description} \\
% R & wire & res + & res - & description
\hline
\hline
TC1: $OPAMP$ LatchUP & Output High & & AMPHigh \\
TC2: $OPAMP$ LatchDown & Output Low : Low gain& & AMPLow \\ \hline
TC3: $OPAMP$ No Operation & Output Low & & AMPLow \\
TC4: $OPAMP$ Low Slew & Low pass filtering & & LowPass \\ \hline
TC5: $PD$ LowPD & Output High & & AMPHigh \\ \hline
TC6: $PD$ HighPD & Output Low : Low Gain& & AMPLow \\ \hline
%TC7: $R_2$ OPEN & LOW & & LowPD \\ \hline
\hline
\end{tabular}
\label{ampfmea}
\end{table}
Collecting the symptoms we can see that this amplifier fails
in 3 ways $\{ AMPHigh, AMPLow, LowPass \}$.
We can now create a derived component, $NI\_AMP$, to represent it.
$$ fm(NI\_AMP) = \{ AMPHigh, AMPLow, LowPass \} $$
\subsection{The second Stage of the amplifier}
The second stage of this amplifier, following the signal path, is the amplifier
consisting of $R3,R4,IC2$.
This is in exactly the same configuration as the first amplifier, but it is being fed by the first amplifier.
The first amplifier was grounded and received as input `+V1' (presumably
a positive voltage).
This means the junction of R1 R3 is always +ve.
This means the input voltage `+V2' could be lower than this.
This means R3 R4 is not a potential divider with R4 being on the positive side.
It could be on either polarity (i.e. the other way around R4 could be the negative side).
Here it is more intuitive to model the resistors not as a potential divider, but individually.
%This means we are either going to
%get a high or low reading if R3 or R4 fail.
\begin{table}[ht]
\caption{Second Amplifier $SEC\_AMP$: Failure Mode Effects Analysis: Single Faults} % title of Table
\centering % used for centering table
\begin{tabular}{||l|c|c|l|l||}
\hline \hline
\textbf{Test} & \textbf{Amplifier} & \textbf{ } & \textbf{General} \\
\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symtom Description} \\
% R & wire & res + & res - & description
\hline
\hline
TC1: $OPAMP$ LatchUP & Output High & & AMPHigh \\
TC2: $OPAMP$ LatchDown & Output Low : Low gain & & AMPLow \\ \hline
TC3: $OPAMP$ No Operation & Output Low & & AMPLow \\
TC4: $OPAMP$ Low Slew & Low pass filtering & & LowPass \\ \hline
TC5: $R3\_open$ & +V2 follower & & AMPIncorrectOutput\\ \hline
TC6: $R3\_short$ & Undefined & & AMPIncorrectOutput \\
& (impedance of IC1 vs +V2) & & \\ \hline
TC5: $R4\_open$ & High or Low output & & AMPIncorrectOutput \\
& +V2$>$+V1 $\mapsto$ High & & \\
& +V1$>$+V2 $\mapsto$ Low & & \\ \hline
TC6: $R4\_short$ & +V2 follower & & AMPIncorrectOutput \\ \hline
%TC7: $R_2$ OPEN & LOW & & LowPD \\ \hline
\hline
\end{tabular}
\label{ampfmea}
\end{table}
Collecting the symptoms we can see that this amplifier fails
in 4 ways $\{ AMPHigh, AMPLow, LowPass, AMPIncorrectOutput\}$.
We can now create a derived component, $SEC\_AMP$, to represent it.
$$ fm(SEC\_AMP) = \{ AMPHigh, AMPLow, LowPass, AMPIncorrectOutput \} $$
%Its failure modes are therefore the same. We can therefore re-use
%the derived component for $NI\_AMP$
\pagebreak[4]
\subsection{Modelling the circuit}
For the final stage of this we can create a functional group consisting of
two derived components of the type $NI\_AMP$ and $SEC\_AMP$.
\begin{table}[ht]
\caption{Difference Amplifier $DiffAMP$ : Failure Mode Effects Analysis: Single Faults} % title of Table
\centering % used for centering table
\begin{tabular}{||l|c|c|l|l||}
\hline \hline
\textbf{Test} & \textbf{Dual Amplifier} & \textbf{ } & \textbf{General} \\
\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symptom Description} \\
% R & wire & res + & res - & description
\hline
\hline
TC1: $NI\_AMP$ AMPHigh & opamp 2 driven high & & DiffAMPLow \\
TC2: $NI\_AMP$ AMPLow & opamp 2 fdriven low & & DiffAMPHigh \\
TC3: $NI\_AMP$ LowPass & opamp 2 driven with lag & & DiffAMP\_LP \\ \hline
TC4: $SEC\_AMP$ AMPHigh & Diff amplifier high & & DiffAMPHigh\\
TC5: $SEC\_AMP$ AMPLow & Diff amplifier low & & DiffAMPLow \\
TC6: $SEC\_AMP$ LowPass & Diff amplifier lag/lowpass & & DiffAMP\_LP \\ \hline
TC7: $SEC\_AMP$ IncorrectOutput & Output voltage & & DiffAMPIncorrect \\
TC7: $SEC\_AMP$ & $ \neg (V2 - V1) $ & & \\ \hline
\hline
\end{tabular}
\label{ampfmea}
\end{table}
Collecting the symptoms, we can determine the failure modes for this circuit, $\{DiffAMPLow, DiffAMPHigh, DiffAMP\_LP, DiffAMPIncorrect \}$.
We now create a derived component to represent the circuit in figure~\ref{fig:circuit1}.
$$ fm (DiffAMP) = \{DiffAMPLow, DiffAMPHigh, DiffAMP\_LP DiffAMPIncorrect\} $$
Its interesting here to note that we can draw a directed graph (figure~\ref{fig:circuit1_dag})
of the failure modes and derived components.
Using this we can trace any top level fault back to
a component failure mode that could have caused it.
In fact we can re-construct an FTA diagram from the information in this graph.
We merely have to choose a top level event and work down using $XOR$ gates.
This circuit performs poorly from a safety point of view.
Its failure modes could be indistinguishable from valid readings (especially
wihen it becomes a V2 follower).
\begin{figure}[h]
\centering
\includegraphics[width=400pt]{./circuit1_dag.png}
% circuit1_dag.png: 797x1145 pixel, 72dpi, 28.12x40.39 cm, bb=0 0 797 1145
\caption{Directed Acyclic Graph of Circuit1 failure modes}
\label{fig:circuit1_dag}
\end{figure}
\clearpage
\section{Op-Amp circuit 2}
\begin{figure}[h]
\centering
\includegraphics[width=200pt]{./circuit2002.png}
% circuit2002.png: 575x331 pixel, 72dpi, 20.28x11.68 cm, bb=0 0 575 331
\caption{circuit 2}
\label{fig:circuit2}
\end{figure}
The circuit in figure~\ref{fig:circuit2} shows a five pole low pass filter.
Starting at the input, we have a first order low pass filter buffered by an op-amp,
the output of this is passed to a Sallen~Key~\cite{aoe}[p.267] second order lowpass filter.
The output of this is passed into another Sallen~Key filter -- which although it may have different values
for its resistors/capacitors and thus have a different frequency response -- is idential from a failure mode perspective.
Thus we can analyse the first Sallen~Key low pass filter and re-use the results.
\begin{figure}[h]
\centering
\includegraphics[width=400pt,keepaspectratio=true]{./blockdiagramcircuit2.png}
% blockdiagramcircuit2.png: 689x83 pixel, 72dpi, 24.31x2.93 cm, bb=0 0 689 83
\caption{Signal Flow though the five pole low pass filter}
\label{fig:blockdiagramcircuit2}
\end{figure}
\paragraph{First Order Low Pass Filter.}
We begin with the first order low pass filter formed by $R10$ and $C10$.
%
This configuration (or {\fg}) is very commonly
used in electronics to remove unwanted high frequencies/interference
form a signal; Here it is being used as a first stage of
a more sophisticated low pass filter.
%
R10 and C10 act as a potential divider, with the crucial difference between a purely resistive potential divider being
that the impedance of the capacitor is lower for higher frequencies.
Thus higher frquencies are attenuated at the point that we
read its output signal.
However, from a failure mode perspective we can analyse it in a very similar way
to a potential divider (see section~\ref{potdivfmmd}).
Capacitors generally fail OPEN but some types fail OPEN and SHORT.
We will consider the latter type for this analysis.
We analyse the first order low pass filter in table~\ref{tbl:firstorderlp}.\\
\begin{table}[h+]
\caption{FirstOrderLP: Failure Mode Effects Analysis: Single Faults} % title of Table
\label{tbl:firstorderlp}
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{First Order} & & \textbf{Symptom} \\
& & \textbf{Low Pass Filter} & & \\
\hline
FS1: R10 SHORT & & $No Filtering$ & & $LPnofilter$ \\ \hline
FS2: R10 OPEN & & $No Signal$ & & $LPnosignal$ \\ \hline
FS3: C10 SHORT & & $No Signal$ & & $LPnosignal$ \\ \hline
FS4: C10 OPEN & & $No Filtering$ & & $LPnofilter$ \\ \hline
\hline
\end{tabular}
\end{table}
We can collect the symptoms $\{ LPnofilter,LPnosignal \}$ and create a derived component
called $FirstOrderLP$. Applying the $fm$ function yields $$ fm(FirstOrderLP) = \{ LPnofilter,LPnosignal \}.$$
\paragraph{Addition of Buffer Amplifier: First stage.}
The opamp IC1 is being used simply as a buffer. By placing it between the next stages
on the signal path we remove the possibility of unwanted signal feedback.
The buffer is one of the simplest op-amp configurations.
It has no other components, and so we can now form a {\fg}
from the $FirstOrderLP$ and the OPAMP component.
\begin{table}[ht]
\caption{First Stage LP1: Failure Mode Effects Analysis: Single Faults} % title of Table
\label{tbl:firststage}
\centering % used for centering table
\begin{tabular}{||l|c|c|l|l||}
\hline \hline
\textbf{Test} & \textbf{Circuit} & \textbf{ } & \textbf{General} \\
\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symptom Description} \\
% R & wire & res + & res - & description
\hline
\hline
TC1: $OPAMP$ LatchUP & Output High & & LP1High \\
TC2: $OPAMP$ LatchDown & Output Low & & LP1Low \\
TC3: $OPAMP$ No Operation & Output Low & & LP1Low \\
TC4: $OPAMP$ Low Slew & Unwanted Low pass filtering & & LP1filterincorrect \\ \hline
TC5: $LPnofilter $ & No low pass filtering & & LP1filterincorrect \\
TC6: $LPnosignal $ & No input signal & & LP1nosignal \\ \hline
\hline
\hline
\end{tabular}
\end{table}
From the table~\ref{tbl:firststage} we can see three symptoms of failure of
the first stage of this circuit (i.e. R10,C10,IC1).
We can create a derived component for it, lets call it $LP1$.
$$ fm(LP1) = \{ LP1High, LP1Low, LP1filterincorrect, LP1nosignal \} $$
In terms terms of the circuit we have modelled the functional groups $FirstOrderLP$, and
$LP1$. We can represent these on the circuit diagram by drawing contours around the components
on the schematic as in figure~\ref{fig:circuit2002_LP1}.
\begin{figure}[h]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./circuit2002_LP1.png}
% circuit2002_LP1.png: 575x331 pixel, 72dpi, 20.28x11.68 cm, bb=0 0 575 331
\caption{Circuit showing functional groups modelled so far.}
\label{fig:circuit2002_LP1}
\end{figure}
\paragraph{Second order Sallen Key Low Pass Filter.}
The next two filters in the signal path are R1,R2,C2,C1,IC2 and R3,R4,C4,C3,IC3.
From a failure mode perspective these are identical.
We can analyse the first one and then re-use these results for the second.
\begin{table}[ht]
\caption{Sallen Key Low Pass Filter SKLP: Failure Mode Effects Analysis: Single Faults} % title of Table
\centering % used for centering table
\begin{tabular}{||l|c|c|l|l||}
\hline \hline
\textbf{Test} & \textbf{Circuit} & \textbf{ } & \textbf{General} \\
\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symptom Description} \\
% R & wire & res + & res - & description
\hline
\hline
TC1: $OPAMP$ LatchUP & Output High & & SKLPHigh \\
TC2: $OPAMP$ LatchDown & Output Low & & SKLPLow \\
TC3: $OPAMP$ No Operation & Output Low & & SKLPLow \\
TC4: $OPAMP$ Low Slew & Unwanted Low pass filtering & & SKLPfilterIncorrect \\ \hline
TC5: R1 OPEN & No input signal & & SKLPfilterIncorrect \\
TC6: R1 SHORT & incorrect low pass filtering & & SKLPfilterIncorrect \\ \hline
TC7: R2 OPEN & No input signal & & SKLPnosignal \\
TC8: R2 SHORT & incorrect low pass filtering & & SKLPfilterIncorrect \\ \hline
TC9: C1 OPEN & reduced/incorrect low pass filtering & & SKLPfilterIncorrect\\
TC10: C1 SHORT & reduced/incorrect low pass filtering & & SKLPfilterIncorrect \\ \hline
TC11: C2 OPEN & reduced/incorrect low pass filtering & & SKLPfilterIncorrect \\
TC12: C2 SHORT & No input signal, low signal & & SKLPnosignal \\ \hline
\hline
\hline
\end{tabular}
\label{tbl:sallenkeylp}
\end{table}
We now can create a derived component to represent the Sallen Key low pass filter, which we can call $SKLP$.
$$ fm ( SKLP ) = \{ SKLPHigh, SKLPLow, SKLPIncorrect, SKLPnosignal \} $$
\paragraph{A failure mode model of Op-Amp Circuit 2.}
We now have {\dcs} representing the three stages of this filter
and this follows the signal flow in the filter circuit (see figure~\ref{fig:blockdiagramcircuit2}).
As the signal has to pass though each block/stage
in order to be `five~pole' filtered, we need to bring these three blocks together into a {\fg}
in order to get a failure mode model for the whole circuit.
We can index the Sallen Key stages, and these are marked on the ciruit schematic in figure~\ref{fig:circuit2002_FIVEPOLE}.
\begin{figure}[h]+
\centering
\includegraphics[width=200pt]{./circuit2002_FIVEPOLE.png}
% circuit2002_FIVEPOLE.png: 575x331 pixel, 72dpi, 20.28x11.68 cm, bb=0 0 575 331
\caption{Functional Groups in Five Pole Low Pass Filter on schematic}
\label{fig:circuit2002_FIVEPOLE}
\end{figure}
\pagebreak[4]
So our final {\fg} will consist of the derived components $\{ LP1, SKLP_1, SKLP_2 \}$.
We represent the desired FMMD hierarchy in figure~\ref{fig:circuit2h}.
\begin{figure}[h]+
\centering
\includegraphics[width=300pt]{./circuit2h.png}
% circuit2h.png: 676x603 pixel, 72dpi, 23.85x21.27 cm, bb=0 0 676 603
\caption{FMMD Hierarchy for five pole Low Pass Filter}
\label{fig:circuit2h}
\end{figure}
%\pagebreak[4]
%$$ fm ( SKLP ) = \{ SKLPHigh, SKLPLow, SKLPIncorrect, SKLPnosignal \} $$
%$$ fm(LP1) = \{ LP1High, LP1Low, LP1ExtraLowPass, LP1NoLowPass \} $$
\begin{table}[ht]+
\caption{Five Pole Low Pass Filter: Failure Mode Effects Analysis: Single Faults} % title of Table
\centering % used for centering table
\begin{tabular}{||l|c|l|l|l||}
\hline \hline
\textbf{Test} & \textbf{Circuit} & \textbf{ } & \textbf{General} \\
\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symptom Description} \\
% R & wire & res + & res - & description
\hline
\hline
TC1: $LP1$ LP1High & signal HIGH & & HIGH \\
TC2: $LP1$ SKLPLow & signal LOW & & LOW \\
TC3: $LP1$ LP1filterIncorrect & filtering incorrect & & FilterIncorrect \\
TC4: $LP1$ LP1nosignal & no signal propogated & & NO\_SIGNAL \\ \hline
TC5: $SKLP_1$ High & signal HIGH & & HIGH \\
TC6: $SKLP_1$ Low & signal LOW & & LOW \\
TC7: $SKLP_1$ filterIncorrect & filtering incorrect & & FilterIncorrect \\
TC8: $SKLP_1$ nosignal & no signal propogated & & NO\_SIGNAL \\ \hline
TC9: $SKLP_2$ High & signal HIGH & & HIGH \\
TC10: $SKLP_2$ Low & signal LOW & & LOW \\
TC11: $SKLP_2$ filterIncorrect & filtering incorrect & & FilterIncorrect \\
TC12: $SKLP_2$ nosignal & no signal propogated & & NO\_SIGNAL \\ \hline
\hline
\hline
\end{tabular}
\label{tbl:fivepole}
\end{table}
We now can create a {\dc} to represent the circuit in figure~\ref{fig:circuit2}, we can call it
$FivePoleLP$ and applying the $fm$ function to it (see table~\ref{tbl:fivepole}) yields $fm(FivePoleLP) = \{ HIGH, LOW, FilterIncorrect, NO\_SIGNAL \}$.
\pagebreak[4]
The failure modes for the low pass filters are very similar, and the propogation of the signal
is simple (as it is never inverted). The circuit under analysis is -- as shown in the block diagram (see figure~\ref{fig:blockdiagramcircuit2}) --
three opamp driven non-inverting low pass filter elements; It is not suprising therefore that they have very similar failure modes.
From a safety point of view, the failure modes $LOW$, $HIGH$ and $NO\_SIGNAL$
could be easily detected; the failure symptom $FilterIncorrect$ may be less observable.
\clearpage
\section{Op-Amp circuit 3}
\begin{figure}[h]
\centering
\includegraphics[width=200pt]{/home/robin/projects/thesis/opamp_circuits_C_GARRETT/circuit3003.png}
% circuit3003.png: 503x326 pixel, 72dpi, 17.74x11.50 cm, bb=0 0 503 326
\caption{Circuit 3}
\label{fig:circuit3}
\end{figure}
%\clearpage
%\section{Standard Non-inverting OP AMP}
\clearpage
\section{Basic Concepts Of FMMD}
\paragraph {Definitions}
\begin{itemize}
\item {\bc} - a component with a known set of unitary state failure modes. Base here mean a starting or `bought~in' component.
\item {\fg} - a collection of components chosen to perform a particular task
\item {\em symptom} - a failure mode of a functional group caused by one or more of its component failure modes.
\item {\dc} - a new component derived from an analysed {\fg}
\end{itemize}
\paragraph{ Creating a fault hierarchy.}
The main concept of FMMD is to build a hierarchy of failure behaviour from the {\bc}
level up to the top, or system level, with analysis stages between each
transition to a higher level in the hierarchy.
The first stage is to choose
{\bcs} that interact and naturally form {\fgs}. The initial {\fgs} are collections of base components.
%These parts all have associated fault modes. A module is a set fault~modes.
From the point of view of fault analysis, we are not interested in the components themselves, but in the ways in which they can fail.
A {\fg} is a collection of components that perform some simple task or function.
%
In order to determine how a {\fg} can fail,
we need to consider all failure modes of its components.
%
By analysing the fault behaviour of a `{\fg}' with respect to all its components failure modes,
we can determine its symptoms of failure.
%In fact we can call these
%the symptoms of failure for the {\fg}.
With these symptoms (a set of derived faults from the perspective of the {\fg})
we can now state that the {\fg} (as an entity in its own right) can fail in a number of well defined ways.
%
In other words we have taken a {\fg}, and analysed how
\textbf{it} can fail according to the failure modes of its components, and then
determined the {\fg} failure modes.
\paragraph{Creating a derived component.}
We create a new `{\dc}' which has
the failure symptoms of the {\fg} from which it was derived, as its set of failure modes.
This new {\dc} is at a higher `failure~mode~abstraction~level' than {\bcs}.
%
\paragraph{An example of a {\dc}.}
To give an example of this, we could look at the components that
form, say an amplifier. We look at how all the components within it
could fail and how that would affect the amplifier.
%
The ways in which the amplifier can be affected are its symptoms.
%
When we have determined the symptoms, we can
create a {\dc} (called say AMP1) which has a {\em known set of failure modes} (i.e. its symptoms).
We can now treat $AMP1$ as a pre-analysed, higher level component.
The amplifier is an abstract concept, in terms of the components.
The components brought together in a specific way make it an amplifier !
%What this means is the `fault~symptoms' of the module have been derived.
%
%When we have determined the fault~modes at the module level these can become a set of derived faults.
%By taking sets of derived faults (module level faults) we can combine these to form modules
%at a higher level of fault abstraction. An entire hierarchy of fault modes can now be built in this way,
%to represent the fault behaviour of the entire system. This can be seen as using the modules we have analysed
%as parts, parts which may now be combined to create new functional groups,
%but as parts at a higher level of fault abstraction.
\paragraph{Building the Hierarchy.}
Applying the same process with {\dcs} we can bring {\dcs}
together to form functional groups and create new {\dcs}
at even higher abstraction levels. Eventually we will have a hierarchy
that converges to one top level {\dc}. At this stage we have a complete failure
mode model of the system under investigation.
\begin{figure}[h]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./tree_abstraction_levels.png}
% tree_abstraction_levels.png: 495x292 pixel, 72dpi, 17.46x10.30 cm, bb=0 0 495 292
\caption{FMMD Hierarchy showing ascending abstraction levels}
\label{fig:treeabslev}
\end{figure}
Figure~\ref{fig:treeabslev} shows an FMMD hierarchy, where the process of creating a {\dc} from a {\fg}
is shown as a `$\bowtie$' symbol.
\subsection{An algebraic notation for identifying FMMD enitities}
Consider all `components' to exist as
members of a set $\mathcal{C}$.
%
Each component $c$ has an associated set of failure modes.
We can define a function $fm$ that returns a
set of failure modes $F$, for the component $c$.
Let the set of all possible components be $\mathcal{C}$
and let the set of all possible failure modes be $\mathcal{F}$.
We now define the function $fm$
as
\begin{equation}
\label{eqn:fm}
fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}.
\end{equation}
This is defined by, where $c$ is a component and $F$ is a set of failure modes,
$ fm ( c ) = F. $
We can use the variable name $FG$ to represent a {\fg}. A {\fg} is a collection
of components.
%We thus define $FG$ as a set of chosen components defining
%a {\fg}; all functional groups
We can state that
{\FG} is a member of the power set of all components, $ \FG \in \mathcal{P} \mathcal{C}. $
We can overload the $fm$ function for a functional group {\FG}
where it will return all the failure modes of the components in {\FG}
given by
$$ fm (FG) = F. $$
Generally, where $\mathcal{{\FG}}$ is the set of all functional groups,
\begin{equation}
fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}.
\end{equation}
%$$ \mathcal{fm}(C) \rightarrow S $$
%$$ {fm}(C) \rightarrow S $$
\paragraph{Abstraction Levels of {\fgs} and {\dcs}}
\label{sec:indexsub}
We can indicate the abstraction level of a component by using a superscript.
Thus for the component $c$, where it is a `base component' we can assign it
the abstraction level zero, $c^0$. Should we wish to index the components
(for example as in a product parts-list) we can use a sub-script.
Our base component (if first in the parts-list) could now be uniquely identified as
$c^0_1$.
We can further define the abstraction level of a {\fg}.
We can say that it is the maximum abstraction level of any of its
components. Thus a functional group containing only base components
would have an abstraction level zero and could be represented with a superscript of zero thus
`${\FG}^0$'. % The functional group set may also be indexed.
We can apply symptom abstraction to a {\fg} to find
its symptoms.
%We are interested in the failure modes
%of all the components in the {\fg}. An analysis process
We define the symptom abstraction process with the symbol `$\bowtie$'.% is applied to the {\fg}.
%
The $\bowtie$ function takes a {\fg}
as an argument and returns a newly created {\dc}.
%
%The $\bowtie$ analysis, a symptom extraction process, is described in chapter \ref{chap:sympex}.
The symptom abstraction process must always raise the abstraction level
for the newly created {\dc}.
Using $\abslevel$ to symbolise the fault abstraction level, we can now state:
$$ \bowtie({\FG}^{\abslevel}) \rightarrow c^{{\abslevel}+N} | N \ge 1. $$
\paragraph{Functional Groups may be indexed}
We will typically have more than one {\fg} on each level of FMMD hierarchy ( expect the top level where there will only be one)
we could index the {\fgs} with a sub-script, and can then uniquely identify them using their level and their index.
For example ${\FG}^{3}_{2}$ would be the second {\fg} at the third level of abstraction in an FMMD hierarchy.
\paragraph{The symptom abstraction process in outline.}
The $\bowtie$ function processes each component in the {\fg} and
extracts all the component failure modes.
With all the failure modes, an analyst can
determine how each failure mode will affect the {\fg}, and then collect common symptoms.
A new {\dc} is created
where its failure modes, are the symptoms from {\fg}.
Note that the component must have a higher abstraction level than the {\fg}
it was derived from.
\paragraph{Surjective constraint applied to symptom collection.}
We can stipulate that symptom collection process is surjective.
% i.e. $ \forall f in F $
By stipulating surjection for symptom collection, we ensure
that each component failure mode maps to at least one symptom.
We also ensure that all symptoms have at least one component failure
mode (i.e. one or more failure modes that caused it).
%
\subsection{FMMD Hierarchy}
By applying stages of analysis to higher and higher abstraction
levels, we can converge to a complete failure mode model of the system under analysis.
Because the symptom abstraction process is defined as surjective (from component failure modes to symptoms)
the number of symptoms is guaranteed to be less than or equal to
the number of component failure modes.
In practise however, the number of symptoms greatly reduces as we traverse
up the hierarchy.
This is a natural process. When we have complicated systems
they always have a small number of system failure modes in comparison to
the number of failure modes in its sub-systems/components..
\section{Examples of Derived Component like concepts in safety literature}
Idea stage on this section
\begin{itemize}
\item Look at OPAMP circuits, pick one (say $\mu$741)
\item examine number of components and failure modes
\item outline a proposed FMMD analysis
\item Show FMD-91 OPAMP failure modes -- compare with FMMD
\end{itemize}
\clearpage
Two areas that cannot be automated. Choosing {\fgs} and the analysis/symptom collection process itself.
\section{Side Effects: A Problem for FMMD analysis}
A problem with modularising according to functionality is that we can have component failures that would
intuitively be associated with one {\fg} that may cause unintended side effects in other
{\fgs}.
For instance were we to have a component that on failing $SHORT$ could bring down
a voltage supply rail, this could have drastic consequences for other
functional groups in the system we are examining.
\pagebreak[3]
\subsection{Example de-coupling capacitors in logic circuits}
A good example of this, are de-coupling capacitors, often used
over the power supply pins of all chips in a digital logic circuit.
Were any of these capacitors to fail $SHORT$ they could bring down
the supply voltage to the other logic chips.
To a power-supply, shorted capacitors on the supply rails
are a potential source of the symptom, $SUPPLY\_SHORT$.
In a logic chip/digital circuit {\fg} open capacitors are a potential
source of symptoms caused by the failure mode $INTERFERENCE$.
So we have a `symptom' of the power-supply, and a `failure~mode' of
the logic chip to consider.
A possible solution to this is to include the de-coupling capacitors
in the power-supply {\fg}.
% decision, could they be included in both places ????
% I think so
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
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 $INTERFERENCE$.
Some logic chips are more susceptible to $INTERFERENCE$ than others.
A logic chip with de-coupling capacitor failing, may operate correctly
but interfere with other chips in the circuit.
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}.
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 does not break the modularity of the FMMD technique, because, as {\irl}
one component failure may affect more than one sub-system.
It does uncover a weakness in the FMMD methodology though.
It could be very easy to miss the side effect and include
the component causing the side effect into the wrong {\fg}, or only one germane {\fg}.
\pagebreak[3]
\subsection{{\fgs} Sharing components and Hierarchy}
With electronics we need to follow the signal path to make sense of failure modes
effects on other parts of the circuit further down that path.
%{\fgs} will naturally have to be in the position of starter
A power-supply is naturally first in a signal path (or failure reasoning path).
That is to say, if the power-supply is faulty, its failure modes are likely to affect
the {\fgs} that have to use it.
This means that most electronic components should be placed higher in an FMMD
hierarchy than the power-supply.
A shorted de-coupling capactitor caused a `symptom' of the power-supply,
and an open de-coupling capactitor should be considered a `failure~mode' relevant to the logic chip.
% to consider.
If components can be shared between functional groups, this means that components
must be shareable between {\fgs} at different levels in the FMMD hierarchy.
This hierarchy and an optionally shared de-coupling capacitor (with line highlighted in red and dashed) are shown
in figure~\ref{fig:shared_component}.
\begin{figure}
\centering
\includegraphics[width=250pt,keepaspectratio=true]{./shared_component.png}
% shared_component.png: 729x670 pixel, 72dpi, 25.72x23.64 cm, bb=0 0 729 670
\caption{Optionally shared Component}
\label{fig:shared_component}
\end{figure}
\subsection{Hierarchy and structure}
By having this structure, the logic circuit element, can accept failure modes from the
power-supply (for instance these might, for the sake of example include: $NO\_POWER$, $LOW\_VOLTAGE$, $HIGH\_VOLTAGE$, $NOISE\_HF$, $NOISE\_LF$.
Our logic circuit may be able to cope with $LOW\_VOLTAGE$ and $NOISE\_LF$, but react with a serious symptom to $NOISE\_HF$ say.
But in order to process these failure modes it must be at a higher stage in the FMMD hierarchy.
\pagebreak[4]
\section{Defining the concept of `comparison~complexity' in FMEA}
%
% DOMAIN == INPUTS
% RANGE == OUTPUTS
%
When performing FMEA we have a system under investigation, which will
comprise of a collection of components which have associated failure modes.
The object of FMEA is to determine cause and effect:
from the failure modes (the causes) to the effects (or symptoms of failure).
%
To perform FMEA rigorously
we could stipulate that every failure mode must be checked for effects
against all the components in the system.
We could term this `rigorous~FMEA'~(RFMEA).
The number of checks we have to make to achieve this gives an indication of the complexity of the task.
%
We could term this comkparison~complexity, as it is the number of
paths between failure modes and components, necessary to achieve RFMEA, for a given system/functional~group.
% (except its self of course, that component is already considered to be in a failed state!).
%
Obviously, for a small number of components and failure modes we have a smaller number
of checks to make than for a complicated larger system.
%
We can consider the system as a large {\fg} of components.
We represent the number of components in the {\fg} $G$, by
$ | G | .$
An indexing and sub-scripting notation to identify particular {\fgs}
within an FMMD hierarchy is given in section~\ref{sec:indexsub}.
The function $fm$ has a component as its domain and the components failure modes as its range (see equation~\ref{eqn:fm}).
We can represent the number of failure modes in a component $c$, to be $ | fm(c) | .$
If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|\FG|} $ we can express
the number of checks required to rigorously examine every
failure mode against all the other components in the system.
We can define this as a function, Comparison Complexity, $CC$, with its domain as the system
or {\fg}, $\FG$, and
its range as the number of checks to perform to satisfy a rigorous FMEA inspection.
Where $\mathcal{\FG}$ represents the set of all {\fgs}, and $ \mathbb{N} $ any natural integer, $CC$ is defined by,
\begin{equation}
%$$
CC(G \in \mathcal{\FG}) \rightarrow \mathbb{N},
%$$
\end{equation}
and, where n is the number of components in the system/{\fg}, $|fm(c_i)|$ is the number of failure modes
in component ${c_i}$, is given by
\begin{equation}
\label{eqn:CC}
%$$
%%% when it was called reasoning distance -- 19NOV2011 -- RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1)
CC(\FG) = (n-1) \sum_{1 \le i \le n} fm(c_i).
%$$
\end{equation}
This can be simplified if we can determine the total number of failure modes in the system $K$, (i.e. $ K = \sum_{n=1}^{|G|} {|fm(c_n)|}$);
equation~\ref{eqn:CC} becomes $$ CC(\FG) = K.(|\FG|-1).$$
%Equation~\ref{eqn:rd} can also be expressed as
%
% \begin{equation}
% \label{eqn:rd2}
% %$$
% CC(G) = {|G|}.{|fm(c_n)|}.{(|fg|-1)} .
% %$$
% \end{equation}
\subsection{A general formula for counting Comparison Complexity in an FMMD hierarchy}
An FMMD Hierarchy will have reducing numbers of functional groups as we progress up the hierarchy.
In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to
all {\fgs} on each level.
We define a helper function $g$ with a domain of the level $i$ in an FMMD hierarchy $H$, and a co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level),
defined by
\begin{equation}
%$$
g(H, i) \rightarrow \forall {\FG}^{\xi} \;where\; ({\xi} = {i}) \wedge ({\FG}^{\xi} \in H) .
%$$
\end{equation}
Where $L$ represents the number of levels in the FMMD hierarchy,
$|g(\xi)|$ represents the number of functional groups on the level
and $H$ represents an FMMD hierarchy,
we overload the comparison complexity thus:
%$$
\begin{equation}
\label{eqn:gf}
CC(H) = \sum_{\xi=0}^{L} \sum_{j=1}^{|g(H,\xi)|} CC({\FG}_{j}^{\xi}).
%$$
\end{equation}
\pagebreak[4]
\subsection{Complexity Comparison Examples}
The potential divider discussed in section~\ref{potdivfmmd} has four failure modes and two components and therefore has $CC$ of 4.
$$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
Even considering a $fictitious$ system with just 81 components (with these components
having 3 failure modes each) we would have an $CC$ of
$$CC(fictitious) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$
Ensuring all component failure modes are checked against all other components in a system
-- applying FMEA rigorously -- could be termed
Rigorous FMEA (RFMEA).
The computational order for RFMEA would be polynomial ($O(N^2.K)$) (where $K$ is the variable number of failure modes).
This order may be acceptable in a computational environment: However, the choosing of {\fgs} and the analysis
process are human activities. It can be seen that it is practically impossible to achieve
RFMEA for anything but trival systems.
%
% Next statement needs alot of justification
%
FMMD reduces the comparison complexity enough to make
rigorous checking feasible.
\pagebreak[4]
%\subsection{Using the concept of Complexity Comparison to compare RFMEA with FMMD}
\begin{figure}
\centering
\includegraphics[width=400pt,keepaspectratio=true]{./three_tree.png}
% three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385
\caption{FMMD Hierarchy with number of components in {\fg} fixed to 3 $(|G| = 3)$ } % \wedge (|fm(c)| = 3)$}
\label{fig:three_tree}
\end{figure}
\subsection{Comparing FMMD and RFMEA comparison complexity}
Because components have variable numbers of failure modes,
and {\fgs} have variable numbers of components it is difficult to
use the general formula for comparing the number of checks to make for
RFMEA and FMMMD.
If we were to create an example by fixing the number of components in a {\fg}
and the number of failure modes per component, we can derive formulae
to represent the number of checks to make.
Consider $k$ to be the number of components in a {\fg} (i.e. $k=|{\FG}|$),
$f$ is the number of failure modes per component (i.e. $f=|fm(c)|$), and
$L$ to be the number of levels in the hierarchy of an FMMD analysis.
We can represent the number of failure scenarios to check in a (fixed parameter for $|{\FG}|$ and $|fm(c_i)|$) FMMD hierarchy
with equation~\ref{eqn:anscen}.
\begin{equation}
\label{eqn:anscen}
\sum_{n=0}^{L} {k}^{n}.k.f.(k-1)
\end{equation}
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.
The number of checks to make for RFMEA is number of components $k$ multiplied by the number of failure modes $f$
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
the number of failure modes per component to three, an FMMD hierarchy
would look like figure~\ref{fig:three_tree}.
\subsection{Worked Example}
Using the diagram in figure~\ref{fig:three_tree}, we have three levels of analysis.
Starting at the top, we have a {\fg} with three derived components, each of which has
three failure modes.
Thus the number of checks to make in the top level is $3^0.3.2.3=18$.
On the level below that, we have three {\fgs} each with a
an identical number of checks, $3^1.3.2.3=56$.%{\fg}
On the level below that we have nine {\fgs}, $3^2.3.2.3=168$.
Adding these together gives $242$ checks to make to perform FMMD (i.e. RFMEA \textbf{within the}
{\fgs}).
If we were to take the system represented in figure~\ref{fig:three_tree}, and
apply RFMEA on it as a whole system, we can use equation~\ref{eqn:CC},
$CC(G) = \sum_{n=1}^{|G|} |fm(c_n)|.(|G|-1)$, where $|G|$ is 27, $fm(c_n)$ is 3
and $(|G|-1)$ is 26.
This gives:
$CC(G) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$.
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
in an FMMD hierarchy.
%
The number of components in the system, is number of components
in a {\fg} raised to the power of the level plus one.
Thus we re-write equation~\ref{eqn:CC} as:
\begin{equation}
\label{eqn:fmea_state_exp21}
\sum_{n=1}^{k^{L+1}}.(k^{L+1}-1).f \; , % \\
%(N^2 - N).f
\end{equation}
or
\begin{equation}
\label{eqn:fmea_state_exp22}
k^{L+1}.(k^{L+1}-1).f \;. % \\
%(N^2 - N).f
\end{equation}
We can now use equation~\ref{eqn:anscen} and \ref{eqn:fmea_state_exp22} to compare (for fixed sizes of $|G|$ and $|fm(c)|$)
the two approaches, for the work required to perform rigorous checking.
For instance, having four levels
of FMMD analysis, with these fixed numbers,
%(in addition to the top zeroth level)
will require 81 base level components.
$$
%\begin{equation}
\label{eqn:fmea_state_exp22}
3^4.(3^4-1).3 = 81.(81-1).3 = 19440 % \\
%(N^2 - N).f
%\end{equation}
$$
$$
%\begin{equation}
% \label{eqn:anscen}
\sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720
%\end{equation}
$$
% \subsection{Exponential squared to Exponential}
%
% can I say that ?
\bibliographystyle{plain}
\bibliography{../vmgbibliography,../mybib}
\end{document}
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