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opamp_circuits_C_GARRETT/Makefile
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PNG_DIA = circuit1_dag.png mvampcircuit.png pd.png invamp.png shared_component.png tree_abstraction_levels.png three_tree.png blockdiagramcircuit2.png circuit2h.png

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opamp_circuits_C_GARRETT/opamps.tex
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@ -70,11 +70,16 @@ We can express the failure modes of a component using the function $fm$, thus fo
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}.
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}
@ -85,6 +90,7 @@ We can now examine what effect each of these failures will have on the {\fg}.
\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
@ -94,6 +100,7 @@ We can now examine what effect each of these failures will have on the {\fg}.
FS4: R2 OPEN & & $LOW$ & & $PDLow$ \\ \hline
\hline
\end{tabular}
\label{tbl:pd}
\end{table}
We can now create a {\dc} for the potential divider, $PD$.
@ -152,9 +159,10 @@ Re-using the $PD$ - potential divider works only if the input voltage is negativ
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.
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
@ -164,6 +172,7 @@ If we consider the input will only be positive, we can invert the potential divi
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 $INVPD$.
@ -181,7 +190,7 @@ lead to the symptoms (i.e. the symptoms are the same but causation tree will be
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). Because symptom collection is defined as surjective (from component failure modes
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.
@ -189,6 +198,7 @@ $HIGH$ 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
@ -213,23 +223,25 @@ $HIGH$ or $LOW$ output.
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.
%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.
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}
@ -250,35 +262,36 @@ 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).
$$ fm(R) = \{ OPEN, SHORT \}$$
%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.
% \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 $fm(PD) = \{PDHigh,PDLow\}$.
@ -406,7 +419,7 @@ two derived components of the type $NI\_AMP$ and $SEC\_AMP$.
\begin{tabular}{||l|c|c|l|l||}
\hline \hline
\textbf{Test} & \textbf{Dual Amplifier} & \textbf{ } & \textbf{General} \\
\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symtom Description} \\
\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symptom Description} \\
% R & wire & res + & res - & description
\hline
\hline
@ -463,11 +476,282 @@ wihen it becomes a V2 follower).
\centering
\includegraphics[width=200pt]{./circuit2002.png}
% circuit2002.png: 575x331 pixel, 72dpi, 20.28x11.68 cm, bb=0 0 575 331
\caption{circuit2}
\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}
@ -478,8 +762,9 @@ wihen it becomes a V2 follower).
\caption{Circuit 3}
\label{fig:circuit3}
\end{figure}
\clearpage
\section{Standard Non-inverting OP AMP}
%\clearpage
%\section{Standard Non-inverting OP AMP}
\clearpage
@ -501,6 +786,8 @@ The main concept of FMMD is to build a hierarchy of failure behaviour from the {
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.
@ -679,6 +966,17 @@ 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
\section{Side Effects: A Problem for FMMD analysis}
A problem with modularising according to functionality is that we can have component failures that would
@ -844,7 +1142,7 @@ Rigorous FMEA (RFMEA).
\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 $(|fg| = 3) \wedge (|fm(c)| = 3)$}
\caption{FMMD Hierarchy with $(|fg| = 3)$ } % \wedge (|fm(c)| = 3)$}
\label{fig:three_tree}
\end{figure}
@ -942,9 +1240,11 @@ $$
%\end{equation}
$$
\subsection{Exponential squared to Exponential}
can I say that ?
% \subsection{Exponential squared to Exponential}
%
% can I say that ?
\bibliographystyle{plain}
\bibliography{../vmgbibliography,../mybib}
\end{document}

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@ -657,7 +657,7 @@ safety, as it can miss unexpected effects due to `unexpected' component interact
The Statistical Analysis methodology is the core philosophy
of the Safety Integrity Levels (SIL) embodied in EN61508 \cite{en61508}
and its international analog standard IOC5108.
and its international analog is standard IOC5108.
@ -669,6 +669,7 @@ and its international analog standard IOC5108.
\item No possibility to model base component level double failure modes.
\item As with all failure mode methodologies based on FMEA, does not model component failure modes
that may cause more than one type of SYSTEM failure.
\item Because FMEDA is based on one entry per component failure mode, top level symptoms are not grouped, and will be listed in a fragmented way, and may not have the same description.
\end{itemize}
%AND then how we can solve all there problems