reasoning for diff dual op-amp example circuit

opamp_circuits_C_GARRETT/opamps.tex

@@ -23,6 +23,8 @@
 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}
@@ -36,6 +38,157 @@ Proof of analysis concept.
 \end{figure}
 
 
+The amplifier in figure~\ref{fig:circuit1} amplifies the difference between 
+the 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.
+$$ fm(R) = \{ OPEN, SHORT \}$$
+
+
+
+\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
+$fm(PD) = \{PDHigh,PDLow\}$.
+
+
+The potential divider is used to program the gain of IC1.
+IC1 and PD1 provide the function of buffering
+/amplifying the signal $+V1$.
+We can treat IC1 and PD1 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: 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, $NONINVAMP$, 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 mode are therefore the same.
+
+\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        &  dual amplifier high           &      &   DiffAMPHIGH \\  
+ TC5: $NI\_AMP2$ AMPLow         &  dual amplifier low              &      &  DiffAMPLow \\  
+ TC6: $NI\_AMP2$ LowPass        &  dual 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
+of the failure modes and derived components here.
+By doing this we can trace any top level fault back to 
+a component failure mode that could have caused it.
+
+
+
+
+
 
 \clearpage
 \section{Op-Amp circuit 2}