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OK sending circuit 2 to C Garrett

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Robin Clark 2011-11-17 18:05:36 +00:00
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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 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} \subsection{Analysing a potential divider in terms of failure modes}
\label{potdivfmmd} \label{potdivfmmd}
\begin{figure}[h+] \begin{figure}[h+]
\centering \centering
\includegraphics[width=100pt,keepaspectratio=true]{./pd.png} \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+] \begin{table}[h+]
\caption{Potential Divider: Sinlge failure analysis}
\begin{tabular}{|| l | l | c | c | l ||} \hline \begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Pot Div Effect} & & \textbf{Symptom} \\ \textbf{Failure Scenario} & & \textbf{Pot Div Effect} & & \textbf{Symptom} \\
\hline \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 FS4: R2 OPEN & & $LOW$ & & $PDLow$ \\ \hline
\hline \hline
\end{tabular} \end{tabular}
\label{tbl:pd}
\end{table} \end{table}
We can now create a {\dc} for the potential divider, $PD$. 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. 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 If we can refine the operational states of the fungional group, we can obtain clearer
symptoms. 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+] \begin{table}[h+]
\caption{Inverted Potential divider: Single failure analysis}
\begin{tabular}{|| l | l | c | c | l ||} \hline \begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Inverted Pot Div Effect} & & \textbf{Symptom} \\ \textbf{Failure Scenario} & & \textbf{Inverted Pot Div Effect} & & \textbf{Symptom} \\
\hline \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 FS4: R2 OPEN & & $HIGH$ & & $PDHigh$ \\ \hline
\hline \hline
\end{tabular} \end{tabular}
\label{tbl:pdneg}
\end{table} \end{table}
We can form a {\dc} from this, and call it an inverted potential divider $INVPD$. 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 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 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). 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 Note that here we have a more general symptom $ OUT OF RANGE $ which could mean either
$HIGH$ or $LOW$ output. $HIGH$ or $LOW$ output.
@ -189,6 +198,7 @@ $HIGH$ or $LOW$ output.
\begin{table}[h+] \begin{table}[h+]
\caption{Inverting Amplifier: Single failure analysis}
\begin{tabular}{|| l | l | c | c | l ||} \hline \begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Inverted Amp Effect} & & \textbf{Symptom} \\ \hline \textbf{Failure Scenario} & & \textbf{Inverted Amp Effect} & & \textbf{Symptom} \\ \hline
\hline \hline
@ -213,23 +223,25 @@ $HIGH$ or $LOW$ output.
FS4: AMP LowSlew & & $ slow output \frac{\delta V}{\delta t} $ & & $ LOW PASS $ \\ \hline FS4: AMP LowSlew & & $ slow output \frac{\delta V}{\delta t} $ & & $ LOW PASS $ \\ \hline
\hline \hline
\end{tabular} \end{tabular}
\label{tbl:invamp}
\end{table} \end{table}
$$ fm(INVAMP) = \{ OUT OF RANGE, ZERO OUTPUT, NO GAIN, LOW PASS \} $$ $$ fm(INVAMP) = \{ OUT OF RANGE, ZERO OUTPUT, NO GAIN, LOW PASS \} $$
Much more general. OUT OF RANGE symptom maps to many component failure modes. %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 %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 ???? %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. %Could further refine this if MTTF stats available for each component failure.
\subsection{Comparison between the two approaches} \subsection{Comparison between the two approaches}
If the input voltage can be negative the potential divider If the input voltage can be negative the potential divider
becomes reversed in polarity. 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 \clearpage
\section{Op-Amp circuit 1} \section{Op-Amp circuit 1}
@ -474,8 +486,8 @@ wihen it becomes a V2 follower).
The circuit in figure~\ref{fig:circuit2} shows a five pole low pass filter. 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, 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 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 The output of this is passed into another Sallen~Key filter -- which although it may have different values
for its resistors/capacitors and thus a different frequency response) is idential from a failure mode perspective. 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. Thus we can analyse the first Sallen~Key low pass filter and re-use the results.
@ -501,13 +513,16 @@ that the impedance of the capacitor is lower for higher frequencies.
Thus higher frquencies are attenuated at the point that we Thus higher frquencies are attenuated at the point that we
read its output signal. read its output signal.
However, from a failure mode perspective we can analyse it in a very similar way However, from a failure mode perspective we can analyse it in a very similar way
to a potential divider. to a potential divider (see section~\ref{potdivfmmd}).
Capacitors generally fail OPEN but some types fail OPEN and SHORT. Capacitors generally fail OPEN but some types fail OPEN and SHORT.
We will consider the latter type for this analysis. 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+] \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 \begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{First Order} & & \textbf{Symptom} \\ \textbf{Failure Scenario} & & \textbf{First Order} & & \textbf{Symptom} \\
& & \textbf{Low Pass Filter} & & \\ & & \textbf{Low Pass Filter} & & \\
@ -526,7 +541,7 @@ We will consider the latter type for this analysis.
We can collect the symptoms $\{ LPnofilter,LPnosignal \}$ and create a derived component We can collect the symptoms $\{ LPnofilter,LPnosignal \}$ and create a derived component
called $FirstOrderLP$. Applying the $fm$ function yields $$ fm(FirstOrderLP) = \{ LPnofilter,LPnosignal \}.$$ called $FirstOrderLP$. Applying the $fm$ function yields $$ fm(FirstOrderLP) = \{ LPnofilter,LPnosignal \}.$$
\paragraph{Addition of Buffer Amplifier: first stage.} \paragraph{Addition of Buffer Amplifier: First stage.}
The opamp IC1 is being used simply as a buffer. By placing it between the next stages 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. on the signal path we remove the possibility of unwanted signal feedback.
@ -536,6 +551,7 @@ from the $FirstOrderLP$ and the OPAMP component.
\begin{table}[ht] \begin{table}[ht]
\caption{First Stage LP1: Failure Mode Effects Analysis: Single Faults} % title of Table \caption{First Stage LP1: Failure Mode Effects Analysis: Single Faults} % title of Table
\label{tbl:firststage}
\centering % used for centering table \centering % used for centering table
\begin{tabular}{||l|c|c|l|l||} \begin{tabular}{||l|c|c|l|l||}
\hline \hline \hline \hline
@ -554,7 +570,7 @@ from the $FirstOrderLP$ and the OPAMP component.
\hline \hline
\end{tabular} \end{tabular}
\label{tbl:firststage}
\end{table} \end{table}
From the table~\ref{tbl:firststage} we can see three symptoms of failure of From the table~\ref{tbl:firststage} we can see three symptoms of failure of
@ -580,7 +596,7 @@ on the schematic as in figure~\ref{fig:circuit2002_LP1}.
\paragraph{Second order Sallen Key Low Pass Filter.} \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. 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. From a failure mode perspective these are identical.
We can analyse one and re-use the results for the second. We can analyse the first one and then re-use these results for the second.
\begin{table}[ht] \begin{table}[ht]
\caption{Sallen Key Low Pass Filter SKLP: Failure Mode Effects Analysis: Single Faults} % title of Table \caption{Sallen Key Low Pass Filter SKLP: Failure Mode Effects Analysis: Single Faults} % title of Table
@ -613,6 +629,11 @@ We can analyse one and re-use the results for the second.
\label{tbl:sallenkeylp} \label{tbl:sallenkeylp}
\end{table} \end{table}
We now can create a derived component to represent the Sallen Key low pass filter, which we can call $SKLP$. We now can create a derived component to represent the Sallen Key low pass filter, which we can call $SKLP$.
@ -763,7 +784,9 @@ could be easily detected; the failure symptom $FilterIncorrect$ may be less obs
\paragraph{ Creating a fault hierarchy.} \paragraph{ Creating a fault hierarchy.}
The main concept of FMMD is to build a hierarchy of failure behaviour from the {\bc} 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 level up to the top, or system level, with analysis stages between each
transition to a higher level in the hierarchy. $$ fm(LP1) = \{ LP1High, LP1Low, LP1ExtraLowPass, LP1NoLowPass \} $$ transition to a higher level in the hierarchy.
The first stage is to choose The first stage is to choose
{\bcs} that interact and naturally form {\fgs}. The initial {\fgs} are collections of base components. {\bcs} that interact and naturally form {\fgs}. The initial {\fgs} are collections of base components.
@ -943,7 +966,7 @@ This is a natural process. When we have a complicated systems
they always have a small number of system failure modes. they always have a small number of system failure modes.
\section{The Case for Derived Components} \section{Examples of Derived Component like concepts in safety literature}
Idea stage on this section Idea stage on this section
\begin{itemize} \begin{itemize}
@ -1118,7 +1141,7 @@ Rigorous FMEA (RFMEA).
\centering \centering
\includegraphics[width=400pt,keepaspectratio=true]{./three_tree.png} \includegraphics[width=400pt,keepaspectratio=true]{./three_tree.png}
% three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385 % 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} \label{fig:three_tree}
\end{figure} \end{figure}
@ -1214,9 +1237,11 @@ $$
%\end{equation} %\end{equation}
$$ $$
\subsection{Exponential squared to Exponential} % \subsection{Exponential squared to Exponential}
%
can I say that ? % can I say that ?
\bibliographystyle{plain}
\bibliography{../vmgbibliography,../mybib}
\end{document} \end{document}

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