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\title{Example OPAMP circuits}
\author{Robin}

\begin{document}

\maketitle

\begin{abstract}
Circuits from email conversation.

Not a document to be proof read.

Proof of analysis concept.

Function $fm$ applied to a component returns its failure modes.
\end{abstract}
\clearpage
\section{Op-Amp circuit 1}

\begin{figure}[h]
 \centering
 \includegraphics[width=200pt]{/home/robin/projects/thesis/opamp_circuits_C_GARRETT/circuit1001.png}
 % circuit1001.png: 420x300 pixel, 72dpi, 14.82x10.58 cm, bb=0 0 420 300
 \caption{Circuit 1}
 \label{fig:circuit1}
\end{figure}


The amplifier in figure~\ref{fig:circuit1} amplifies the difference between 
the input voltages $+V1$ and $+V2$.
It would be desirable to represent this circuit as a derived component called say $DiffAMP$.
We begin by identifying functional groups from the components in the circuit.


\subsection{Functional Group: Potential Divider}

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.

Thus for single failure modes, a potential divider can fail
with $fm(PD) = \{PDHigh,PDLow\}$.


The potential divider is used to program the gain of IC1.
IC1 and PD provide the function of buffering
/amplifying the signal $+V1$.
We can now examine IC1 and PD as a functional group.


\subsection{Functional Group: Amplifier}

Let use now consider the op-amp. According to 
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\%).


$$ fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$


By bringing the $PD$ derived component and the $OPAMP$ into 
a functional group we can analyse its failure mode behaviour.


\begin{table}[ht]
\caption{Non Inverting Amplifier $NI\_AMP$: 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 : Low gain&      &  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 : Low Gain&      &  AMPLow \\ \hline 
 %TC7: $R_2$ OPEN     &  LOW       &      &  LowPD \\ \hline
\hline
\end{tabular} 
\label{ampfmea}
\end{table}


Collecting the symptoms we can see that this amplifier fails
in 3 ways $\{ AMPHigh, AMPLow,  LowPass \}$.
We can now create a derived component, $NI\_AMP$, to represent it.

 
$$ fm(NI\_AMP) = \{ AMPHigh, AMPLow, LowPass \} $$
 



\subsection{The second Stage of the amplifier}

The second stage of this amplifier, following the signal path, is the amplifier
consisting of $R3,R4,IC2$.

This is in exactly the same configuration as the first amplifier.
Its failure modes are therefore the same. We can therefore re-use
the derived component for $NI\_AMP$

\pagebreak[4]
\subsection{Modelling the circuit}

For the final stage of this we can create a functional group consisting of
two derived components of the type $NI\_AMP$.



\begin{table}[ht]
\caption{Difference Amplifier $DiffAMP$ : 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{Dual Amplifier} & \textbf{  } & \textbf{General}   \\
 \textbf{Case} &  \textbf{Effect} & \textbf{  } & \textbf{Symtom Description}   \\
%   R         &    wire        & res +         & res -    & description
\hline
\hline
 TC1: $NI\_AMP1$ AMPHigh        &  opamp 2 driven high          &      & DiffAMPLow \\ 
 TC2: $NI\_AMP1$  AMPLow        &  opamp 2 fdriven low          &      &  DiffAMPHigh \\  
 TC3: $NI\_AMP1$ LowPass        &  opamp 2 driven with lag      &      &  DiffAMP\_LP \\  \hline 
 TC4: $NI\_AMP2$ AMPHigh        &  Diff amplifier high           &      &   DiffAMPHigh\\  
 TC5: $NI\_AMP2$ AMPLow         &  Diff amplifier low              &      &  DiffAMPLow \\  
 TC6: $NI\_AMP2$ LowPass        &  Diff amplifier lag/lowpass        &      &  DiffAMP\_LP \\ \hline 
 %TC7: $R_2$ OPEN     &  LOW       &      &  LowPD \\ \hline
\hline
\end{tabular} 
\label{ampfmea}
\end{table}



Collecting the symptoms, we can determine the failure modes for this circuit, $\{DiffAMPLow, DiffAMPHigh,  DiffAMP\_LP\}$.


We now create a derived component to represent the circuit in figure~\ref{fig:circuit1}.

$$ fm (DiffAMP) = \{DiffAMPLow, DiffAMPHigh,  DiffAMP\_LP\} $$


Its interesting here to note that we can draw a directed graph (figure~\ref{fig:circuit1_dag})
of the failure modes and derived components.
Using this we can trace any top level fault back to 
a component failure mode that could have caused it.
In fact we can re-construct an FTA diagram from the information in this graph.
We merely have to choose a top level event and work down using or gates.

\begin{figure}[h]
 \centering
 \includegraphics[width=400pt]{./circuit1_dag.png}
 % circuit1_dag.png: 797x1145 pixel, 72dpi, 28.12x40.39 cm, bb=0 0 797 1145
 \caption{Directed Acyclic Graph of Circuit1 failure modes}
 \label{fig:circuit1_dag}
\end{figure}





\clearpage
\section{Op-Amp circuit 2}


  \begin{figure}[h]
 \centering
 \includegraphics[width=200pt]{./circuit2002.png}
 % circuit2002.png: 575x331 pixel, 72dpi, 20.28x11.68 cm, bb=0 0 575 331
 \caption{circuit2}
 \label{fig:circuit2}
\end{figure}


\clearpage
\section{Op-Amp circuit 3}

 \begin{figure}[h]
 \centering
 \includegraphics[width=200pt]{/home/robin/projects/thesis/opamp_circuits_C_GARRETT/circuit3003.png}
 % circuit3003.png: 503x326 pixel, 72dpi, 17.74x11.50 cm, bb=0 0 503 326 
\caption{Circuit 3}
 \label{fig:circuit3}
\end{figure}


\end{document}