
\ifthenelse {\boolean{paper}}
{
\abstract{ 
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.

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.

We can now create a functional group representing the 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}

A standard non inverting  op amp (from ``The Art of Electronics'' ~\cite{aoe}[pp.234]) is shown in figure \ref{fig:noninvamp}.

\begin{figure}[h]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/noninv.png}
 % noninv.jpg: 341x186 pixel, 72dpi, 12.03x6.56 cm, bb=0 0 341 186
 \caption{Standard non inverting amplifier configuration}
 \label{fig:noninvamp}
\end{figure}



The functional 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.

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,.
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 $\{R2\_OPEN, R2\_SHORT\}$.

\section{Failure Mode Analysis of the Potential Divider}

Modelling this as a functional group, we can draw a circle to represent each failure mode
in the potential divider, shown in figure \ref{fig:fg1}.


\begin{figure}[h]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/fg1.png}
 % fg1.jpg: 430x271 pixel, 72dpi, 15.17x9.56 cm, bb=0 0 430 271
 \caption{potential divider `functional group' failure modes}
 \label{fig:fg1}
\end{figure}

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}).

\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
 \caption{potential divider with test cases}
 \label{fig:fg1a}
\end{figure}


\begin{table}[ht]
\caption{Potential Divider: 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{pdfmea}
\end{table}


We can now collect the symptoms of failure. From the four base component failure modes, we now
have two symptoms, $LowPD, HighPD$. 

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]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/fg1b.png}
 % fg1b.jpg: 430x271 pixel, 72dpi, 15.17x9.56 cm, bb=0 0 430 271
 \caption{Collection of potential divider failure mode symptoms}
 \label{fig:fg1b}
\end{figure}

We can now make a `derived component' to represent this potential divider.
This {\dc} will have two failure 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]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/dc1.png}
 % dc1.jpg: 430x619 pixel, 72dpi, 15.17x21.84 cm, bb=0 0 430 619
 \caption{From functional group to derived component}
 \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.

\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
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]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/op1.png}
 % op1.jpg: 406x221 pixel, 72dpi, 14.32x7.80 cm, bb=0 0 406 221
 \caption{Op Amp failure modes}
 \label{fig:op1}
\end{figure}


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]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/fgamp.png}
 % fgamp.jpg: 430x330 pixel, 72dpi, 15.17x11.64 cm, bb=0 0 430 330
 \caption{Amplifier Functional Group}
 \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]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./noninvopamp/fgampa.png}
 % fgampa.jpg: 430x330 pixel, 72dpi, 15.17x11.64 cm, bb=0 0 430 330
 \caption{Amplifier Functional Group with Test Cases}
 \label{fig:fgampa.jpg}
\end{figure}



\begin{table}[ht]
\caption{Non Inverting Amplifier: 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{Symtom Description}   \\
%   R         &    wire        & res +         & res -    & description
\hline
\hline
 TC1: $OPAMP$ LatchUP    &  Output High          &      &  AMPHigh \\ 
 TC2: $OPAMP$ LatchDown  &  Output Low           &      &  AMPLow \\ \hline
 TC3: $OPAMP$ No Operation &  Output Low         &      &  AMPLow \\  
 TC4: $OPAMP$ Low Slew     &  Low pass filtering &      &  LowPass \\ \hline
 TC5: $PD$ LowPD         &   Output High          &      &  AMPHigh \\ \hline
 TC6: $PD$ HighPD        &  Output Low          &      &  AMPLow \\ \hline 
 %TC7: $R_2$ OPEN     &  LOW       &      &  LowPD \\ \hline
\hline
\end{tabular} 
\label{ampfmea}
\end{table}

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}.



\begin{figure}[h]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./noninvampa.png}
 % noninvampa.jpg: 436x720 pixel, 72dpi, 15.38x25.40 cm, bb=0 0 436 720
 \caption{Non Inverting Amplifier Derived Component}
 \label{fig:noninvampa}
\end{figure}


%failure mode contours).
\clearpage
\vspace{60pt}
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\today
