284 lines
9.9 KiB
TeX
284 lines
9.9 KiB
TeX
\documentclass[a4paper,10pt]{article}
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\usepackage[utf8x]{inputenc}
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\usepackage{graphicx}
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\usepackage{fancyhdr}
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\usepackage{tikz}
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\usetikzlibrary{shapes,snakes}
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\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows}
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\usepackage{subfigure}
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\usepackage{amsfonts,amsmath,amsthm}
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\usepackage{algorithm}
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\usepackage{algorithmic}
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\usepackage{lastpage}
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%\usepackage{glossary}
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%opening
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\title{Example OPAMP circuits}
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\author{Robin}
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\begin{document}
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\begin{abstract}
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Circuits from email conversation.
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Not a document to be proof read.
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Proof of analysis concept.
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Function $fm$ applied to a component returns its failure modes.
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\end{abstract}
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\maketitle
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\tableofcontents
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\listoffigures
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\clearpage
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\section{Op-Amp circuit 1}
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt]{/home/robin/projects/thesis/opamp_circuits_C_GARRETT/circuit1001.png}
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% circuit1001.png: 420x300 pixel, 72dpi, 14.82x10.58 cm, bb=0 0 420 300
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\caption{Circuit 1}
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\label{fig:circuit1}
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\end{figure}
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The amplifier in figure~\ref{fig:circuit1} amplifies the difference between
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the input voltages $+V1$ and $+V2$.
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It would be desirable to represent this circuit as a derived component called say $DiffAMP$.
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We begin by identifying functional groups from the components in the circuit.
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\subsection{Functional Group: Potential Divider}
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R1 and R2 perform as a potential divider.
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Resistors can fail OPEN and SHORT (according to GAS burner standard EN298 Appendix A).
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$$ fm(R) = \{ OPEN, SHORT \}$$
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\begin{table}[ht]
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\caption{Potential Divider $PD$: Failure Mode Effects Analysis: Single Faults} % title of Table
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\centering % used for centering table
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\begin{tabular}{||l|c|c|l|l||}
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\hline \hline
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\textbf{Test} & \textbf{Pot.Div} & \textbf{ } & \textbf{General} \\
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\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symtom Description} \\
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% R & wire & res + & res - & description
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\hline
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\hline
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TC1: $R_1$ SHORT & LOW & & LowPD \\
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TC2: $R_1$ OPEN & HIGH & & HighPD \\ \hline
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TC3: $R_2$ SHORT & HIGH & & HighPD \\
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TC4: $R_2$ OPEN & LOW & & LowPD \\ \hline
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\hline
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\end{tabular}
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\label{tbl:pdfmea}
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\end{table}
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By collecting the symptoms in table~\ref{tbl:pdfmea} we can create a derived
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component $PD$ to represent the failure mode behaviour
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of a potential divider.
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Thus for single failure modes, a potential divider can fail
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with $fm(PD) = \{PDHigh,PDLow\}$.
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The potential divider is used to program the gain of IC1.
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IC1 and PD provide the function of buffering
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/amplifying the signal $+V1$.
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We can now examine IC1 and PD as a functional group.
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\pagebreak[3]
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\subsection{Functional Group: Amplifier}
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Let use now consider the op-amp. According to
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FMD-91~\cite{fmd91}[3-116] an op amp may have the following failure modes:
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latchup(12.5\%), latchdown(6\%), nooperation(31.3\%), lowslewrate(50\%).
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$$ fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$
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By bringing the $PD$ derived component and the $OPAMP$ into
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a functional group we can analyse its failure mode behaviour.
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\begin{table}[ht]
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\caption{Non Inverting Amplifier $NI\_AMP$: Failure Mode Effects Analysis: Single Faults} % title of Table
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\centering % used for centering table
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\begin{tabular}{||l|c|c|l|l||}
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\hline \hline
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\textbf{Test} & \textbf{Amplifier} & \textbf{ } & \textbf{General} \\
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\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symtom Description} \\
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% R & wire & res + & res - & description
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\hline
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\hline
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TC1: $OPAMP$ LatchUP & Output High & & AMPHigh \\
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TC2: $OPAMP$ LatchDown & Output Low : Low gain& & AMPLow \\ \hline
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TC3: $OPAMP$ No Operation & Output Low & & AMPLow \\
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TC4: $OPAMP$ Low Slew & Low pass filtering & & LowPass \\ \hline
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TC5: $PD$ LowPD & Output High & & AMPHigh \\ \hline
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TC6: $PD$ HighPD & Output Low : Low Gain& & AMPLow \\ \hline
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%TC7: $R_2$ OPEN & LOW & & LowPD \\ \hline
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\hline
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\end{tabular}
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\label{ampfmea}
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\end{table}
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Collecting the symptoms we can see that this amplifier fails
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in 3 ways $\{ AMPHigh, AMPLow, LowPass \}$.
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We can now create a derived component, $NI\_AMP$, to represent it.
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$$ fm(NI\_AMP) = \{ AMPHigh, AMPLow, LowPass \} $$
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\subsection{The second Stage of the amplifier}
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The second stage of this amplifier, following the signal path, is the amplifier
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consisting of $R3,R4,IC2$.
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This is in exactly the same configuration as the first amplifier, but it is being fed by the first amplifier.
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The first amplifier was grounded and received as input `+V1' (presumably
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a positive voltage).
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This means the junction of R1 R3 is always +ve.
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This means the input voltage `+V2' could be lower than this.
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This means R3 R4 is not a potential divider with R4 being on the positive side.
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It could be on either polarity (i.e. the other way around R4 could be the negative side).
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Here it is more intuitive to model the resistors not as a potential divider, but individually.
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%This means we are either going to
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%get a high or low reading if R3 or R4 fail.
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\begin{table}[ht]
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\caption{Differencing Amplifier $D\_AMP$: Failure Mode Effects Analysis: Single Faults} % title of Table
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\centering % used for centering table
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\begin{tabular}{||l|c|c|l|l||}
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\hline \hline
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\textbf{Test} & \textbf{Amplifier} & \textbf{ } & \textbf{General} \\
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\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symtom Description} \\
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% R & wire & res + & res - & description
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\hline
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\hline
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TC1: $OPAMP$ LatchUP & Output High & & AMPHigh \\
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TC2: $OPAMP$ LatchDown & Output Low : Low gain & & AMPLow \\ \hline
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TC3: $OPAMP$ No Operation & Output Low & & AMPLow \\
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TC4: $OPAMP$ Low Slew & Low pass filtering & & LowPass \\ \hline
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TC5: $R3\_open$ & +V2 follower & & AMPIncorrectOutput\\ \hline
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TC6: $R3\_short$ & Undefined & & AMPIncorrectOutput \\
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& (impedance of IC1 vs +V2) & & \\ \hline
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TC5: $R4\_open$ & High or Low output & & AMPIncorrectOutput \\
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& +V2$>$+V1 $\mapsto$ High & & \\
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& +V1$>$+V2 $\mapsto$ Low & & \\ \hline
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TC6: $R4\_short$ & +V2 follower & & AMPIncorrectOutput \\ \hline
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%TC7: $R_2$ OPEN & LOW & & LowPD \\ \hline
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\hline
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\end{tabular}
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\label{ampfmea}
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\end{table}
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Collecting the symptoms we can see that this amplifier fails
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in 4 ways $\{ AMPHigh, AMPLow, LowPass, AMPIncorrectOutput\}$.
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We can now create a derived component, $D\_AMP$, to represent it.
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$$ fm(D\_AMP) = \{ AMPHigh, AMPLow, LowPass, AMPIncorrectOutput \} $$
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%Its failure modes are therefore the same. We can therefore re-use
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%the derived component for $NI\_AMP$
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\pagebreak[4]
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\subsection{Modelling the circuit}
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For the final stage of this we can create a functional group consisting of
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two derived components of the type $NI\_AMP$ and $D\_AMP$.
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\begin{table}[ht]
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\caption{Difference Amplifier $DiffAMP$ : Failure Mode Effects Analysis: Single Faults} % title of Table
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\centering % used for centering table
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\begin{tabular}{||l|c|c|l|l||}
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\hline \hline
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\textbf{Test} & \textbf{Dual Amplifier} & \textbf{ } & \textbf{General} \\
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\textbf{Case} & \textbf{Effect} & \textbf{ } & \textbf{Symtom Description} \\
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% R & wire & res + & res - & description
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\hline
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\hline
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TC1: $NI\_AMP$ AMPHigh & opamp 2 driven high & & DiffAMPLow \\
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TC2: $NI\_AMP$ AMPLow & opamp 2 fdriven low & & DiffAMPHigh \\
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TC3: $NI\_AMP$ LowPass & opamp 2 driven with lag & & DiffAMP\_LP \\ \hline
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TC4: $D\_AMP$ AMPHigh & Diff amplifier high & & DiffAMPHigh\\
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TC5: $D\_AMP$ AMPLow & Diff amplifier low & & DiffAMPLow \\
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TC6: $D\_AMP$ LowPass & Diff amplifier lag/lowpass & & DiffAMP\_LP \\ \hline
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TC7: $D\_AMP$ IncorrectOutput & Output voltage & & DiffAMPIncorrect \\
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TC7: $D\_AMP$ & $ \neg (V2 - V1) $ & & \\ \hline
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\hline
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\end{tabular}
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\label{ampfmea}
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\end{table}
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Collecting the symptoms, we can determine the failure modes for this circuit, $\{DiffAMPLow, DiffAMPHigh, DiffAMP\_LP, DiffAMPIncorrect \}$.
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We now create a derived component to represent the circuit in figure~\ref{fig:circuit1}.
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$$ fm (DiffAMP) = \{DiffAMPLow, DiffAMPHigh, DiffAMP\_LP DiffAMPIncorrect\} $$
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Its interesting here to note that we can draw a directed graph (figure~\ref{fig:circuit1_dag})
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of the failure modes and derived components.
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Using this we can trace any top level fault back to
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a component failure mode that could have caused it.
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In fact we can re-construct an FTA diagram from the information in this graph.
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We merely have to choose a top level event and work down using or gates.
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This circuit performs poorly from a safety point of view.
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Its failure modes could be indistinguishable from valid readings (especially
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wihen it becomes a V2 follower).
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\begin{figure}[h]
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\centering
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\includegraphics[width=400pt]{./circuit1_dag.png}
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% circuit1_dag.png: 797x1145 pixel, 72dpi, 28.12x40.39 cm, bb=0 0 797 1145
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\caption{Directed Acyclic Graph of Circuit1 failure modes}
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\label{fig:circuit1_dag}
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\end{figure}
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\clearpage
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\section{Op-Amp circuit 2}
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt]{./circuit2002.png}
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% circuit2002.png: 575x331 pixel, 72dpi, 20.28x11.68 cm, bb=0 0 575 331
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\caption{circuit2}
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\label{fig:circuit2}
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\end{figure}
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\clearpage
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\section{Op-Amp circuit 3}
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt]{/home/robin/projects/thesis/opamp_circuits_C_GARRETT/circuit3003.png}
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% circuit3003.png: 503x326 pixel, 72dpi, 17.74x11.50 cm, bb=0 0 503 326
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\caption{Circuit 3}
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\label{fig:circuit3}
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\end{figure}
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\clearpage
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\section{Standard Non-inverting OP AMP}
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\end{document}
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