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@ -5,33 +5,59 @@
This paper analyses a non-inverting op-amp
configuration, with the opamp and gain resistors using the FMMD
methodology.
%
It has three base components, two resistors
and one op-amp.
The two resistors are used as a potential divider to program the gain
of the amplifier. We consider the two resistors as a functional group
where the function is provides is to operate as a potential divider.
where their function is to operate as a potential divider.
%
The base component error modes of the
resistors are used to model the potential divider from
a failure mode perspective.
%
We determine the failure symptoms of the potential divider and
consider these as failure modes of a derived component.
consider these as failure modes of a new derived component.
We can now create a functional group representing the amplifier,
We can now create a functional group representing the non-inverting amplifier,
by bringing the failure modes from the potential divider and
the op-amp into a functional group.
This can now be analysed and a derived component to represent the non inverting
%
This can be analysed and a derived component to represent the non inverting
amplifier determined.
}
\section{Introduction}
}
{
This chapter analyses a non-inverting op-amp
configuration, with the opamp and gain resistors using the FMMD
methodology.
%
It has three base components, two resistors
and one op-amp.\section{Introduction}
The two resistors are used as a potential divider to program the gain
of the amplifier. We consider the two resistors as a functional group
where their function is to operate as a potential divider.
%
The base component error modes of the
resistors are used to model the potential divider from
a failure mode perspective.
%
We determine the failure symptoms of the potential divider and
consider these as failure modes of a new derived component.
We can create a functional group representing the non-inverting amplifier,
by bringing the failure modes from the potential divider and
the op-amp into a functional group.
%
This can now be analysed and a derived component to represent the non inverting
amplifier determined.
\section{Introduction: The non-inverting amplifier}
}
\section{Introduction}
A standard non inverting op amp (from ``The Art of Electronics'' ~\cite{aoe}[pp.234]) is shown in figure \ref{fig:noninvamp}.
@ -45,20 +71,25 @@ A standard non inverting op amp (from ``The Art of Electronics'' ~\cite{aoe}[pp
The functional of the resistors in this circuit is to set the amplifier gain.
The function of the resistors in this circuit is to set the amplifier gain.
They operate as a potential divider and program the minus input on the op-amp
to balance them against the positive input, giving the voltage gain ($G_v$)
defined by $ G_v = 1 + \frac{R2}{R1} $ at the output.
A functional group, is an ideally small in number collection of components,
that interact to provide
a function or task within a system.
As the resistors work to provide a specific function, that of a potential divider,
we can treat them as a functional group. This functional group has two members, R1 and R2,.
we can treat them as a functional group. This functional group has two members, $R1$ and $R2$.
Using the EN298 specification for resistor failure ~\cite{en298}[App.A]
we can assign failure modes of $OPEN$ and $SHORT$ to the resistors.
Thus $R1$ has failure modes $\{R1\_OPEN, R1\_SHORT\}$ and $R2$ has failure modes $\{R2\_OPEN, R2\_SHORT\}$.
\clearpage
\section{Failure Mode Analysis of the Potential Divider}
Modelling this as a functional group, we can draw a circle to represent each failure mode
Modelling this as a functional group, we can draw a simple closed curve
to represent each failure mode, taken from the components R1 and R2,
in the potential divider, shown in figure \ref{fig:fg1}.
@ -72,11 +103,14 @@ in the potential divider, shown in figure \ref{fig:fg1}.
We can now look at each of these base component failure modes,
and determine how they will affect the operation of the potential divider.
Each failure mode scenario we look at will be given a teat case number,
which is represented on the diagram, with an asterisk marking
which failure modes is is modelling (see figure \ref{fig:fg1a}).
%Each failure mode scenario we look at will be given a test case number,
%which is represented on the diagram, with an asterisk marking
%which failure modes is modelling (see figure \ref{fig:fg1a}).
Each labelled asterisk in the diagram represents a failure mode scenario.
The failure mode scenarios are given test case numbers, and an example to clarify this follows
in table~\ref{pdfmea}.
\begin{figure}[h]
\begin{figure}[h+]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/fg1a.png}
% fg1a.jpg: 430x271 pixel, 72dpi, 15.17x9.56 cm, bb=0 0 430 271
@ -106,13 +140,14 @@ which failure modes is is modelling (see figure \ref{fig:fg1a}).
We can now collect the symptoms of failure. From the four base component failure modes, we now
have two symptoms, $LowPD, HighPD$.
have two symptoms, where the potential divider will give an incorrect low voltage (which we can term $LowPD$)
or an incorrect high voltage (which we can term $HighPD$).
We can represent the collection of these symptoms by drawing connecting lines between
the test cases and naming them (see figure \ref{fig:fg1b}).
\begin{figure}[h]
\begin{figure}[h+]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/fg1b.png}
% fg1b.jpg: 430x271 pixel, 72dpi, 15.17x9.56 cm, bb=0 0 430 271
@ -120,13 +155,15 @@ the test cases and naming them (see figure \ref{fig:fg1b}).
\label{fig:fg1b}
\end{figure}
%\clearpage
We can now make a `derived component' to represent this potential divider.
This {\dc} will have two failure failure modes.
This {\dc} will have two failure modes.
We can use the symbol $\bowtie$ to represent taking the analysed
{\fg} and creating from it, a {\dc}.
%We could represent it algebraically thus: $ \bowtie(PotDiv) =
\begin{figure}[h]
\begin{figure}[h+]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/dc1.png}
% dc1.jpg: 430x619 pixel, 72dpi, 15.17x21.84 cm, bb=0 0 430 619
@ -134,29 +171,24 @@ We can use the symbol $\bowtie$ to represent taking the analysed
\label{fig:dc1}
\end{figure}
Because the derived component is defined by its failure modes, we can use it
as a building block for other {\fgs} in the same way as we used the resistors R1 and R2.
Because the derived component is defined by its failure modes and
the functional group used to derive it, we can use it
as a building block for other {\fgs} in the same way as we used the resistors $R1$ and $R2$.
\clearpage
\section{Failure Mode Analysis of the OP-AMP}
Let use now consider the op-amp. According to
FMD-91~\cite{fmd91}[3-116] an op amp may have the follow failure modes
FMD-91~\cite{fmd91}[3-116] an op amp may have the following failure modes:
latchup(12.5\%), latchdown(6\%), nooperation(31.3\%), lowslewrate(50\%).
We can represent these failure modes on a diagram (see figure~\ref{fig:op1}).
\clearpage
\section{Bringing the OP amp and the potential divider together}
We can now consider bringing the OP amp and the potential divider together to
for an amplifier. We have the failure modes of the functional group for the potential divider, so we do not need to consider the individual resistor failure modes that define its behaviour.
We can make a new functional group to represent the amplifier, by bringing the component opamp
and the component potential divider into a new functional group.
\begin{figure}[h]
\begin{figure}[h+]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/op1.png}
% op1.jpg: 406x221 pixel, 72dpi, 14.32x7.80 cm, bb=0 0 406 221
@ -164,12 +196,23 @@ and the component potential divider into a new functional group.
\label{fig:op1}
\end{figure}
%\clearpage
\section{Bringing the OP amp and the potential divider together}
We can now consider bringing the OP amp and the potential divider together to
for an amplifier. We have the failure modes of the functional group for the potential divider,
so we do not need to consider the individual resistor failure modes that define its behaviour.
We can make a new functional group to represent the amplifier, by bringing the component \textbf{opamp}
and the component potential divider into a new functional group.
This functional group has the failure modes from the op-amp component, and the failure modes
from the potential divider {\dc} to analyse represented by figure~\ref{fig:fgamp}.
\begin{figure}[h]
\begin{figure}[h+]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/fgamp.png}
% fgamp.jpg: 430x330 pixel, 72dpi, 15.17x11.64 cm, bb=0 0 430 330
@ -177,24 +220,19 @@ from the potential divider {\dc} to analyse represented by figure~\ref{fig:fgamp
\label{fig:fgamp}
\end{figure}
We can now place test cases on this (note this analysis considers single failure modes only
where we want to model multiple failures, we can over lap contours, and place the test cases in overlapping
regions) see figure~\ref{fig:fgampa}.
\begin{figure}[h]
\begin{figure}[h+]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/fgampa.png}
% fgampa.jpg: 430x330 pixel, 72dpi, 15.17x11.64 cm, bb=0 0 430 330
% fgampa.jpg: 430x330 pixel, 72dpi, 15.17x11.64 cm, bb=0 0 430 330 hno
\caption{Amplifier Functional Group with Test Cases}
\label{fig:fgampa.jpg}
\label{fig:fgampa}
\end{figure}
\clearpage
\begin{table}[ht]
\caption{Non Inverting Amplifier: Failure Mode Effects Analysis: Single Faults} % title of Table
@ -218,14 +256,13 @@ regions) see figure~\ref{fig:fgampa}.
\label{ampfmea}
\end{table}
For this amplifier configuration we have three failure modes, $AMPHigh, AMPLow, LowPass$ see figure~\ref{fig:fgampb}.
For this amplifier configuration we have three failure modes, $AMPHigh, AMPLow, LowPass$.%see figure~\ref{fig:fgampb}.
We can now derive a `component' to represent this amplifier configuration (see figure
and use it it to model higher level functional groups see figure~\ref{fig:noninvampa}.
We can now derive a `component' to represent this amplifier configuration (see figure ~\ref{fig:noninvampa}).
\begin{figure}[h]
\begin{figure}[h+]
\centering
\includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/noninvampa.png}
% noninvampa.jpg: 436x720 pixel, 72dpi, 15.38x25.40 cm, bb=0 0 436 720
@ -236,9 +273,24 @@ and use it it to model higher level functional groups see figure~\ref{fig:noninv
%failure mode contours).
\clearpage
\vspace{60pt}
$$ \int_{0\-}^{\infty} f(t).e^{-s.t}.dt \; | \; s \in \mathcal{C}$$
\today
\section{Conclusion}
$$\frac{-b\pm\sqrt{ {b^2-4ac}}}{2a}$$
\today
We now have a derived component that represents the failure modes of a non-inverting
op-amp based amplifier. We can now use this to model higher level designs, where we have systems
that use this type of amplifier.
If failure mode/reliability statistics were required these could be derived
from the model, as each failure mode of the derived component
is traceable to one or more base component failure mode causes, for which established
statistical literature is available ~\cite{mil1991}~\cite{fmd91}.
Software used to edit these diagrams, keeps the model in a directed acyclic graph data structure
for this purpose.
%\clearpage % refs etc come next
%\vspace{60pt}
%$$ \int_{0\-}^{\infty} f(t).e^{-s.t}.dt \; | \; s \in \mathcal{C}$$
%\today
% $$\frac{-b\pm\sqrt{ {b^2-4ac}}}{2a}$$
%\today

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