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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
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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@ -250,35 +250,36 @@ We begin by identifying functional groups from the components in the circuit.
\subsection{Functional Group: Potential Divider}
Here 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 +407,7 @@ 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
@ -468,6 +469,164 @@ wihen it becomes a V2 follower).
\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 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.
\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.
Capacitors generally fail OPEN but some types fail OPEN and SHORT.
We will consider the latter type for this analysis.
\begin{table}[h+]
\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
\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{Symptom Description} \\
% R & wire & res + & res - & description
\hline
\hline
TC1: $OPAMP$ LatchUP & Output High & & LP1High \\
TC2: $OPAMP$ LatchDown & Output Low & & LP1Low \\ \hline
TC3: $OPAMP$ No Operation & Output Low & & LP1Low \\
TC4: $OPAMP$ Low Slew & Unwanted Low pass filtering & & LP1ExtraLowPass \\ \hline
TC5: $LPnofilter $ & No low pass filtering & & LP1NoLowPass \\ \hline
TC6: $LPnosignal $ & No input signal & & LP1low \\
\hline
\hline
\end{tabular}
\label{tbl:firststage}
\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 $LP1$.
$$ fm(LP1) = \{ LP1High, LP1Low, LP1ExtraLowPass, LP1NoLowPass \} $$
\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 one and re-use the 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{Amplifier} & \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 \\ \hline
TC3: $OPAMP$ No Operation & Output Low & & SKLPLow \\
TC4: $OPAMP$ Low Slew & Unwanted Low pass filtering & & SKLPIncorrect \\ \hline
TC5: $R1 OPEN$ & No input signal & & SKLPIncorrect \\ \hline
TC6: $R1 SHORT$ & incorrect low pass filtering & & SKLPIncorrect \\
TC7: $R2 OPEN$ & No input signal & & SKLPnosignal \\ \hline
TC8: $R2 SHORT$ & incorrect low pass filtering & & SKLPIncorrect \\
TC9: $C1 OPEN$ & reduced/incorrect low pass filtering & & SKLPIncorrect\\ \hline
TC10: $C1 SHORT$ & reduced/incorrect low pass filtering & & SKLPIncorrect \\
TC11: $C2 OPEN$ & reduced/incorrect low pass filtering & & SKLPIncorrect \\ \hline
TC12: $C2 SHORT$ & No input signal, low signal & & SKLPnosignal \\
\hline
\hline
\end{tabular}
\label{tbl:firststage}
\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.
We represent this as a block diagram to represent the signal flow, in figure~\ref{fig:blockdiagramcircuit2}.
\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 five pole low pass filter}
\label{fig:blockdiagramcircuit2}
\end{figure}
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 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}
So out final {\fg} will consist of the derived components
$\{ LP1, SKLP_1, SKLP_2 \}$.
\clearpage
\section{Op-Amp circuit 3}