Robin Clark
2078211ae0
Use and contrast two valid functional groups for the inverting amplifier r# Please enter the commit message for your changes. Lines starting
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15 KiB
Plaintext
\def\layersep{3.5cm}
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\ifthenelse {\boolean{paper}}
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{
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\abstract{
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This paper analyses a non-inverting op-amp
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configuration, with the opamp and gain resistors using the FMMD
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methodology.
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%
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It has three base components, two resistors
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and one op-amp.
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The two resistors are used as a potential divider to program the gain
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of the amplifier. We consider the two resistors as a functional group
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where their function is to operate as a potential divider.
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%
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The base component error modes of the
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resistors are used to model the potential divider from
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a failure mode perspective.
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%
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We determine the failure symptoms of the potential divider and
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consider these as failure modes of a new derived component.
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We can now create a functional group representing the non-inverting amplifier,
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by bringing the failure modes from the potential divider and
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the op-amp into a functional group.
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%
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This can be analysed and a derived component to represent the non inverting
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amplifier determined.
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}
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\section{Introduction}
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}
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{
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This chapter analyses a non-inverting op-amp
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configuration, with the opamp and gain resistors using the FMMD
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methodology.
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%
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It has three base components, two resistors
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and one op-amp.\section{Introduction}
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The two resistors are used as a potential divider to program the gain
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of the amplifier. We consider the two resistors as a functional group
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where their function is to operate as a potential divider.
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%
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The base component error modes of the
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resistors are used to model the potential divider from
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a failure mode perspective.
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%
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We determine the failure symptoms of the potential divider and
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consider these as failure modes of a new derived component.
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We can create a functional group representing the non-inverting amplifier,
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by bringing the failure modes from the potential divider and
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the op-amp into a functional group.
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%
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This can now be analysed and a derived component to represent the non inverting
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amplifier determined.
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\section{Introduction: The non-inverting amplifier}
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}
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A standard non inverting op amp (from ``The Art of Electronics'' ~\cite{aoe}[pp.234]) is shown in figure \ref{fig:noninvamp}.
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./noninv.png}
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% noninv.jpg: 341x186 pixel, 72dpi, 12.03x6.56 cm, bb=0 0 341 186
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\caption{Standard non inverting amplifier configuration}
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\label{fig:noninvamp}
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\end{figure}
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The function of the resistors in this circuit is to set the amplifier gain.
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They operate as a potential divider and program the minus input on the op-amp
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to balance them against the positive input, giving the voltage gain ($G_v$)
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defined by $ G_v = 1 + \frac{R2}{R1} $ at the output.
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A functional group, is an ideally small in number collection of components,
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that interact to provide
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a function or task within a system.
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As the resistors work to provide a specific function, that of a potential divider,
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we can treat them as a functional group. This functional group has two members, $R1$ and $R2$.
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Using the EN298 specification for resistor failure ~\cite{en298}[App.A]
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we can assign failure modes of $OPEN$ and $SHORT$ to the resistors.
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Thus $R1$ has failure modes $\{R1\_OPEN, R1\_SHORT\}$ and $R2$ has failure modes $\{R2\_OPEN, R2\_SHORT\}$.
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\clearpage
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\section{Failure Mode Analysis of the Potential Divider}
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Modelling this as a functional group, we can draw a simple closed curve
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to represent each failure mode, taken from the components R1 and R2,
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in the potential divider, shown in figure \ref{fig:fg1}.
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./fg1.png}
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% fg1.jpg: 430x271 pixel, 72dpi, 15.17x9.56 cm, bb=0 0 430 271
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\caption{potential divider `functional group' failure modes}
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\label{fig:fg1}
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\end{figure}
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We can now look at each of these base component failure modes,
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and determine how they will affect the operation of the potential divider.
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%Each failure mode scenario we look at will be given a test case number,
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%which is represented on the diagram, with an asterisk marking
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%which failure modes is modelling (see figure \ref{fig:fg1a}).
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Each labelled asterisk in the diagram represents a failure mode scenario.
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The failure mode scenarios are given test case numbers, and an example to clarify this follows
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in table~\ref{pdfmea}.
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\begin{figure}[h+]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./fg1a.png}
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% fg1a.jpg: 430x271 pixel, 72dpi, 15.17x9.56 cm, bb=0 0 430 271
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\caption{potential divider with test cases}
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\label{fig:fg1a}
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\end{figure}
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\begin{table}[ht]
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\caption{Potential Divider: 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{pdfmea}
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\end{table}
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We can now collect the symptoms of failure. From the four base component failure modes, we now
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have two symptoms, where the potential divider will give an incorrect low voltage (which we can term $LowPD$)
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or an incorrect high voltage (which we can term $HighPD$).
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We can represent the collection of these symptoms by drawing connecting lines between
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the test cases and naming them (see figure \ref{fig:fg1b}).
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\begin{figure}[h+]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./fg1b.png}
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% fg1b.jpg: 430x271 pixel, 72dpi, 15.17x9.56 cm, bb=0 0 430 271
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\caption{Collection of potential divider failure mode symptoms}
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\label{fig:fg1b}
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\end{figure}
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%\clearpage
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We can now make a `derived component' to represent this potential divider.
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This {\dc} will have two failure modes.
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We can use the symbol $\bowtie$ to represent taking the analysed
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{\fg} and creating from it, a {\dc}.
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%We could represent it algebraically thus: $ \bowtie(PotDiv) =
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\begin{figure}[h+]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./dc1.png}
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% dc1.jpg: 430x619 pixel, 72dpi, 15.17x21.84 cm, bb=0 0 430 619
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\caption{From functional group to derived component}
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\label{fig:dc1}
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\end{figure}
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Because the derived component is defined by its failure modes and
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the functional group used to derive it, we can use it
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as a building block for other {\fgs} in the same way as we used the resistors $R1$ and $R2$.
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\clearpage
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\section{Failure Mode Analysis of the OP-AMP}
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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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We can represent these failure modes on a diagram (see figure~\ref{fig:op1}).
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\begin{figure}[h+]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./op1.png}
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% op1.jpg: 406x221 pixel, 72dpi, 14.32x7.80 cm, bb=0 0 406 221
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\caption{Op Amp failure modes}
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\label{fig:op1}
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\end{figure}
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%\clearpage
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\section{Bringing the OP amp and the potential divider together}
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We can now consider bringing the OP amp and the potential divider together to
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for an amplifier. We have the failure modes of the functional group for the potential divider,
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so we do not need to consider the individual resistor failure modes that define its behaviour.
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We can make a new functional group to represent the amplifier, by bringing the component \textbf{opamp}
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and the component potential divider into a new functional group.
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This functional group has the failure modes from the op-amp component, and the failure modes
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from the potential divider {\dc} to analyse represented by figure~\ref{fig:fgamp}.
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\begin{figure}[h+]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./fgamp.png}
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% fgamp.jpg: 430x330 pixel, 72dpi, 15.17x11.64 cm, bb=0 0 430 330
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\caption{Amplifier Functional Group}
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\label{fig:fgamp}
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\end{figure}
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We can now place test cases on this (note this analysis considers single failure modes only
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where we want to model multiple failures, we can over lap contours, and place the test cases in overlapping
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regions) see figure~\ref{fig:fgampa}.
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\begin{figure}[h+]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./fgampa.png}
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% fgampa.jpg: 430x330 pixel, 72dpi, 15.17x11.64 cm, bb=0 0 430 330 hno
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\caption{Amplifier Functional Group with Test Cases}
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\label{fig:fgampa}
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\end{figure}
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\clearpage
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\begin{table}[ht]
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\caption{Non Inverting Amplifier: 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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For this amplifier configuration we have three failure modes, $AMPHigh, AMPLow, LowPass$.%see figure~\ref{fig:fgampb}.
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We can now derive a `component' to represent this amplifier configuration (see figure ~\ref{fig:noninvampa}).
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\begin{figure}[h+]
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\centering
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\includegraphics[width=200pt,keepaspectratio=true]{./noninvampa.png}
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% noninvampa.jpg: 436x720 pixel, 72dpi, 15.38x25.40 cm, bb=0 0 436 720
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\caption{Non Inverting Amplifier Derived Component}
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\label{fig:noninvampa}
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\end{figure}
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%failure mode contours).
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\clearpage
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\section{Directed Acyclic Failure Mode Graph}
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\begin{figure}
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\centering
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\begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep]
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\tikzstyle{every pin edge}=[<-,shorten <=1pt]
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\tikzstyle{fmmde}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt]
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\tikzstyle{component}=[fmmde, fill=green!50];
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\tikzstyle{failure}=[fmmde, fill=red!50];
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\tikzstyle{symptom}=[fmmde, fill=blue!50];
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\tikzstyle{annot} = [text width=4em, text centered]
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% Draw the input layer nodes
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%\foreach \name / \y in {1,...,4}
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% This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4}
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% \node[component, pin=left:Input \#\y] (I-\name) at (0,-\y) {};
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\node[component] (OPAMP) at (0,-6) {$OPAMP$};
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\node[component] (R1) at (0,-11) {$R_1$};
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\node[component] (R2) at (0,-15) {$R_2$};
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%\node[component] (C-3) at (0,-5) {$C^0_3$};
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%\node[component] (K-4) at (0,-8) {$K^0_4$};
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%\node[component] (C-5) at (0,-10) {$C^0_5$};
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%\node[component] (C-6) at (0,-12) {$C^0_6$};
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%\node[component] (K-7) at (0,-15) {$K^0_7$};
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% Draw the hidden layer nodes
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%\foreach \name / \y in {1,...,5}
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% \path[yshift=0.5cm]
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\node[failure] (OPAMPLU) at (\layersep*2,-3) {latchup};
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\node[failure] (OPAMPLD) at (\layersep*2,-6) {latchdown};
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\node[failure] (OPAMPNP) at (\layersep*2,-9) {noop};
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\node[failure] (OPAMPLS) at (\layersep*2,-11) {lowslew};
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\node[failure] (R1SHORT) at (\layersep,-11) {$R1_{SHORT}$};
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\node[failure] (R1OPEN) at (\layersep,-13) {$R1_{OPEN}$};
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\node[failure] (R2SHORT) at (\layersep,-15) {$R2_{SHORT}$};
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\node[failure] (R2OPEN) at (\layersep,-17) {$R2_{OPEN}$};
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% Draw the output layer node
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% % Connect every node in the input layer with every node in the
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% % hidden layer.
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% %\foreach \source in {1,...,4}
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% % \foreach \dest in {1,...,5}
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\path (OPAMP) edge (OPAMPLU);
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\path (OPAMP) edge (OPAMPLD);
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\path (OPAMP) edge (OPAMPNP);
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\path (R1) edge (R1SHORT);
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\path (R1) edge (R1OPEN);
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\path (R2) edge (R2SHORT);
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\path (R2) edge (R2OPEN);
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% Potential divider failure modes
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%
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\node[symptom] (PDHIGH) at (\layersep*2,-13) {$PD_{HIGH}$};
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\node[symptom] (PDLOW) at (\layersep*2,-15) {$PD_{LOW}$};
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\path (R1OPEN) edge (PDHIGH);
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\path (R2SHORT) edge (PDHIGH);
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\path (R2OPEN) edge (PDLOW);
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\path (R1SHORT) edge (PDLOW);
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\node[symptom] (AMPHIGH) at (\layersep*3,-13) {$AMP_{HIGH}$};
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\node[symptom] (AMPLOW) at (\layersep*3,-15) {$AMP_{LOW}$};
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\node[symptom] (AMPLP) at (\layersep*3,-15) {$LOWPASS$};
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\path (PDLOW) edge (AMPHIGH);
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\path (OPAMPLU) edge (AMPHIGH);
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\path (PDHIGH) edge (AMPLOW);
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\path (OPAMPNP) edge (AMPLOW);
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\path (OPAMPLD) edge (AMPLOW);
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% %\node[symptom,pin={[pin edge={->}]right:Output}, right of=C-1a] (O) {};
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% \node[symptom, right of=C-1a] (s1) {s1};
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% \node[symptom, right of=C-2a] (s2) {s2};
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%
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%
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%
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% \path (C-2b) edge (s1);
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% \path (C-1a) edge (s1);
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%
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% \path (C-2a) edge (s2);
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% \path (C-1b) edge (s2);
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%
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% %\node[component, right of=s1] (DC) {$C^1_1$};
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%
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% %\path (s1) edge (DC);
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% %\path (s2) edge (DC);
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%
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%
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%
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% % Connect every node in the hidden layer with the output layer
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% %\foreach \source in {1,...,5}
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% % \path (H-\source) edge (O);
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%
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% % Annotate the layers
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% \node[annot,above of=C-1a, node distance=1cm] (hl) {Failure modes};
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% \node[annot,left of=hl] {Base Components};
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% \node[annot,right of=hl](s) {Symptoms};
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%\node[annot,right of=s](dcl) {Derived Component};
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\end{tikzpicture}
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% End of code
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\caption{DAG representing failure modes and symptoms of $FG^0_1$}
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\label{fig:dag0}
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\end{figure}
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\section{Conclusion}
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We now have a derived component that represents the failure modes of a non-inverting
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op-amp based amplifier. We can now use this to model higher level designs, where we have systems
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that use this type of amplifier.
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If failure mode/reliability statistics were required these could be derived
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from the model, as each failure mode of the derived component
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is traceable to one or more base component failure mode causes, for which established
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statistical literature is available ~\cite{mil1991}~\cite{fmd91}.
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Software used to edit these diagrams, keeps the model in a directed acyclic graph data structure
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for this purpose.
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%\clearpage % refs etc come next
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%\vspace{60pt}
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%$$ \int_{0\-}^{\infty} f(t).e^{-s.t}.dt \; | \; s \in \mathcal{C}$$
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%\today
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% $$\frac{-b\pm\sqrt{ {b^2-4ac}}}{2a}$$
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%\today
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