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Robin_PHD/opamp_circuits_C_GARRETT/opamps.tex
Robin Clark 9c46c5e2de de-coupling caps example pp15-16 
unintended side effects
2011-10-28 20:51:39 +01:00

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\documentclass[a4paper,10pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{tikz}
\usetikzlibrary{shapes,snakes}
\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows}
\usepackage{subfigure}
\usepackage{amsfonts,amsmath,amsthm}
\usepackage{algorithm}
\usepackage{algorithmic}
\usepackage{lastpage}
\newcommand{\fg}{\em functional~group}
\newcommand{\fgs}{\em functional~groups}
\newcommand{\dc}{\em derived~component}
\newcommand{\dcs}{\em derived~components}
\newcommand{\bc}{\em base~component}
\newcommand{\bcs}{\em base~components}
\newcommand{\irl}{in~real~life}
%\usepackage{glossary}
%opening
\title{Example OPAMP circuits}
\author{Robin}
\begin{document}
\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}
\maketitle
\tableofcontents
\listoffigures
\section{Non-Inverting OPAMP}
Consider a non inverting op-amp designed to amplify
a small positive voltage (typical use would be a thermocouple amplifier
taking a range from 0 to 25mV and amplifiying it to the useful range of an ADC, approx 0 to 4 volts).
\begin{figure}[h+]
\centering
\includegraphics[width=100pt]{./mvampcircuit.png}
% mvampcircuit.png: 243x143 pixel, 72dpi, 8.57x5.04 cm, bb=0 0 243 143
\label{fig:mvampcircuit}
\caption{positive mV amplifier circuit}
\end{figure}
We can begin by looking for functional groups.
The resistors $ R1, R2 $ perform a fairly common function in electronics, that of the potential divider.
So we can examine $\{ R1, R2 \}$ as a {\fg}.
\subsection{The Resistor in terms of failure modes}
We can now determine how the resistors can fail.
According to GAS standard EN298 the failure modes to consider for resistors are OPEN and SHORT.
We can express the failure modes of a component using the function $fm$, thus for the resistor, $ fm(R) = \{ OPEN, SHORT \}$.
We have two resistors in this circuit and therefore four component failure modes to consider for the potential divider.
We can now examine what effect each of these failures will have on the {\fg}.
\subsection{Analysing a potential divider in terms of failure modes}
\begin{figure}[h+]
\centering
\includegraphics[width=100pt,keepaspectratio=true]{./pd.png}
% pd.png: 361x241 pixel, 72dpi, 12.74x8.50 cm, bb=0 0 361 241
\label{fig:pdcircuit}
\caption{Potential Divider Circuit}
\end{figure}
\begin{table}[h+]
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Pot Div Effect} & & \textbf{Symptom} \\
\hline
FS1: R1 SHORT & & $LOW$ & & $PDLow$ \\ \hline
FS2: R1 OPEN & & $HIGH$ & & $PDHigh$ \\ \hline
FS3: R2 SHORT & & $HIGH$ & & $PDHigh$ \\ \hline
FS4: R2 OPEN & & $LOW$ & & $PDLow$ \\ \hline
\hline
\end{tabular}
\end{table}
We can now create a {\dc} for the potential divider, $PD$.
$$ fm(PD) = \{ PDLow, PDHigh \}$$
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\%).
\subsection{Analysing the non-inverting amplifier in terms of failure modes}
$$ fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$
We can now form a {\fg} with $PD$ and $OPAMP$.
\begin{figure}
\centering
\includegraphics[width=300pt]{./non_inv_amp_fmea.png}
% non_inv_amp_fmea.png: 964x492 pixel, 96dpi, 25.50x13.02 cm, bb=0 0 723 369
\label{fig:invampanalysis}
\end{figure}
We can collect symptoms from the analysis and cretae a derived component
to represent the non-inverting amplifier $NI\_AMP$.
We now have can express the failure mode behaviour of this type of amplifier thus:
$$ fm(NI\_AMP) = \{ {lowpass}, {high}, {low} \}.$$
\section{Inverting OPAMP}
\begin{figure}[h]
\centering
\includegraphics[width=200pt]{./invamp.png}
% invamp.png: 378x207 pixel, 72dpi, 13.34x7.30 cm, bb=0 0 378 207
\caption{Inverting Amplifier Configuration}
\label{fig:invamp}
\end{figure}
This configuration is interesting from methodology perspective.
There are two ways in which we can tackle this.
One is to do this in two stages, by considing the gain resistors to be a potential divider
and then combining it with the OPAMP failure mode model.
The other way is to place all three components in a {\fg}.
Both approaches are followed in the next two sub-sections.
\subsection{Inverting OPAMP using a Potential Divider {\dc}}
Re-using the $PD$ - potential divider works only if the input voltage is negative.
We want if possible to have detectable errors, HIGH and LOW are better than OUTOFRANGE.
If we can refine the operational states of the fungional group, we can obtain clearer
symptoms.
If we consider the input will only be positive, we can invert the potential divider.
\begin{table}[h+]
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Inverted Pot Div Effect} & & \textbf{Symptom} \\
\hline
FS1: R1 SHORT & & $HIGH$ & & $PDHigh$ \\ \hline
FS2: R1 OPEN & & $LOW$ & & $PDLow$ \\ \hline
FS3: R2 SHORT & & $LOW$ & & $PDLow$ \\ \hline
FS4: R2 OPEN & & $HIGH$ & & $PDHigh$ \\ \hline
\hline
\end{tabular}
\end{table}
We can form a {\dc} from this, and call it an inverted potential divider $INVPD$.
We can now form a {\fg} from the OPAMP and the $INVPD$
This gives the same results as the analysis from figure~\ref{fig:invampanalysis}.
The differences are the root causes or component failure modes that
lead to the symptoms (i.e. the symptoms are the same but causation tree will be different).
$$ fm(NI\_AMP) = \{ {lowpass}, {high}, {low} \}.$$
\subsection{Inverting OPAMP using three components}
We can use this for a more general case, because we can examine the
effects on the circuit for each operational case (i.e. input +ve
or input -ve). Because symptom collection is defined as surjective (from component failure modes
to symptoms) we cannot have a component failure mode that maps to two different symptoms (within a functional group).
Note that here we have a more general symptom $ OUT OF RANGE $ which could mean either
$HIGH$ or $LOW$ output.
\begin{table}[h+]
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Inverted Amp Effect} & & \textbf{Symptom} \\ \hline
\hline
FS1: R1 SHORT +ve in & & NEGATIVE out of range & & $ OUT OF RANGE $ \\
FS1: R1 SHORT -ve in & & POSITIVE out of range & & $ OUT OF RANGE $ \\ \hline
FS2: R1 OPEN +ve in & & zero output & & $ ZERO OUTPUT $ \\
FS2: R1 OPEN -ve in & & zero output & & $ ZERO OUTPUT $ \\ \hline
FS3: R2 SHORT +ve in & & $INVAMP_{nogain} $ & & $ NO GAIN $ \\
FS3: R2 SHORT -ve in & & $INVAMP_{nogain} $ & & $ NO GAIN $ \\ \hline
FS4: R2 OPEN +ve in & & NEGATIVE out of range $ $ & & $ OUT OF RANGE$ \\
FS4: R2 OPEN -ve in & & POSITIVE out of range $ $ & & $OUT OF RANGE $ \\ \hline
FS5: AMP L\_DN & & $ INVAMP_{low} $ & & $ OUT OF RANGE $ \\ \hline
FS2: AMP L\_UP & & $INVAMP_{high} $ & & $ OUT OF RANGE $ \\ \hline
FS3: AMP NOOP & & $INVAMP_{nogain} $ & & $ NO GAIN $ \\ \hline
FS4: AMP LowSlew & & $ slow output \frac{\delta V}{\delta t} $ & & $ LOW PASS $ \\ \hline
\hline
\end{tabular}
\end{table}
$$ fm(INVAMP) = \{ OUT OF RANGE, ZERO OUTPUT, NO GAIN, LOW PASS \} $$
Much more general. OUT OF RANGE symptom maps to many component failure modes.
Observability problem... system. In fact can we get a metric of how observable
a system is using the ratio of component failure modes X op states to a symptom ????
Could further refine this if MTTF stats available for each component failure.
\subsection{Comparison between the two approaches}
If the input voltage can be negative the potential divider
becomes reversed in polarity.
This means that detecting which failure mode has occurred from knowing the symptom, has become a more difficult task.
\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.
\pagebreak[3]
\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, but it is being fed by the first amplifier.
The first amplifier was grounded and received as input `+V1' (presumably
a positive voltage).
This means the junction of R1 R3 is always +ve.
This means the input voltage `+V2' could be lower than this.
This means R3 R4 is not a potential divider with R4 being on the positive side.
It could be on either polarity (i.e. the other way around R4 could be the negative side).
Here it is more intuitive to model the resistors not as a potential divider, but individually.
%This means we are either going to
%get a high or low reading if R3 or R4 fail.
\begin{table}[ht]
\caption{Second Amplifier $SEC\_AMP$: 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: $R3\_open$ & +V2 follower & & AMPIncorrectOutput\\ \hline
TC6: $R3\_short$ & Undefined & & AMPIncorrectOutput \\
& (impedance of IC1 vs +V2) & & \\ \hline
TC5: $R4\_open$ & High or Low output & & AMPIncorrectOutput \\
& +V2$>$+V1 $\mapsto$ High & & \\
& +V1$>$+V2 $\mapsto$ Low & & \\ \hline
TC6: $R4\_short$ & +V2 follower & & AMPIncorrectOutput \\ \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 4 ways $\{ AMPHigh, AMPLow, LowPass, AMPIncorrectOutput\}$.
We can now create a derived component, $SEC\_AMP$, to represent it.
$$ fm(SEC\_AMP) = \{ AMPHigh, AMPLow, LowPass, AMPIncorrectOutput \} $$
%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 $NI\_AMP$ and $SEC\_AMP$.
\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: $NI\_AMP$ AMPHigh & opamp 2 driven high & & DiffAMPLow \\
TC2: $NI\_AMP$ AMPLow & opamp 2 fdriven low & & DiffAMPHigh \\
TC3: $NI\_AMP$ LowPass & opamp 2 driven with lag & & DiffAMP\_LP \\ \hline
TC4: $SEC\_AMP$ AMPHigh & Diff amplifier high & & DiffAMPHigh\\
TC5: $SEC\_AMP$ AMPLow & Diff amplifier low & & DiffAMPLow \\
TC6: $SEC\_AMP$ LowPass & Diff amplifier lag/lowpass & & DiffAMP\_LP \\ \hline
TC7: $SEC\_AMP$ IncorrectOutput & Output voltage & & DiffAMPIncorrect \\
TC7: $SEC\_AMP$ & $ \neg (V2 - V1) $ & & \\ \hline
\hline
\end{tabular}
\label{ampfmea}
\end{table}
Collecting the symptoms, we can determine the failure modes for this circuit, $\{DiffAMPLow, DiffAMPHigh, DiffAMP\_LP, DiffAMPIncorrect \}$.
We now create a derived component to represent the circuit in figure~\ref{fig:circuit1}.
$$ fm (DiffAMP) = \{DiffAMPLow, DiffAMPHigh, DiffAMP\_LP DiffAMPIncorrect\} $$
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 $XOR$ gates.
This circuit performs poorly from a safety point of view.
Its failure modes could be indistinguishable from valid readings (especially
wihen it becomes a V2 follower).
\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}
\clearpage
\section{Standard Non-inverting OP AMP}
\clearpage
\section{Unintended Side Effects: A Problem for FMMD analysis}
A problem with modularising according to functionality is that we can have component failures that would
intuitively be associated with one {\fg} that may cause unintended side effects in other
{\fgs}.
For instance were we to have a component that that on failing $SHORT$ could bring down
a voltage supply rail, this could have drastic consequences for other functional groups in the system we are examining.
\pagebreak[3]
\subsection{Example de-coupling capacitors in logic circuits}
A good example of this are de-coupling capacitors, often used over the power supply pins of all chips in a digital logic circuit.
Were any of these capacitors to fail $SHORT$ they could bring down the supply voltage to the other logic chips.
To a power-supply, shorted capacitors on the supply rails are a potential source of the symptom, $SUPPLY\_SHORT$.
In a logic chip/digital circuit {\fg} open capacitors are a potential source of symptoms caused by the failure mode $INTERFERENCE$.
So we have a `symptom' of the power-supply, and a `failure~mode' of the logic chip to consider.
The FMMD solution to this is to include the de-coupling capacitors
in the power-supply {\fg}.
% decision, could they be included in both places ????
% I think so
Because the capacitor has two potential failure modes (EN298)
this raises another issue for FMMD. A de-coupling capacitor going $OPEN$ might not be considered relevant to
a power-supply module (but there might be additional noise on its output rails).
But in {\fg} terms the power supply, now has a new symptom that of $INTERFERENCE$.
Some logic chips are more susceptible to $INTERFERENCE$ than others.
A logic chip with de-coupling capacitor failing, may operate correctly
but interfere with other chips in the circuit.
There is no reason why the de-coupling capacitors could not be included {\em in the {\fg} they would intuitively be associated with as well}.
This allows for the general principle of a component failure affecting more than one {\fg} in a circuit.
This allows functional groups to share components where necessary.
\pagebreak[3]
\subsection{{\fgs} Sharing components and Hierarchy}
With electronics we need to follow the signal path to make sense of failure modes
effects on other parts of the circuit further down that path.
%{\fgs} will naturally have to be in the position of starter
A power-supply is naturally first in a signal path (or failure reasoning path).
That is to say, if the power-supply is faulty, its failure modes are likely to affect
the {\fgs} that have to use it.
This means that most electronic components should be placed higher in an FMMD
hierarchy than the power-supply.
A shorted de-coupling capactitor caused a `symptom' of the power-supply,
and an open de-coupling capactitor can be considered a `failure~mode' of the logic chip to consider.
If components can be shared between functional groups, this means that components
must be shareable between {\fgs} at different levels in the FMMD hierarchy.
This hierarchy and an optionally shared de-coupling capacitor (with line highlighted in red and dashed) are shown
in figure~\ref{fig:shared_component}.
\begin{figure}
\centering
\includegraphics[width=250pt,keepaspectratio=true]{./shared_component.png}
% shared_component.png: 729x670 pixel, 72dpi, 25.72x23.64 cm, bb=0 0 729 670
\caption{Optionally shared Component}
\label{fig:shared_component}
\end{figure}
\subsection{Hierarchy and structure}
By having this structure, the logic circuit element, can accept failure modes from the
power-supply (for instance these might, for the sake of example include: $NO\_POWER$, $LOW\_VOLTAGE$, $HIGH\_VOLTAGE$, $NOISE\_HF$, $NOISE\_LF$.
Our logic circuit may be able to cope with $LOW\_VOLTAGE$ and $NOISE\_LF$, but react with a serious symptom to $NOISE\_HF$ say.
But in order to process these failure modes it must be at a higher stage in the FMMD hierarchy.
\end{document}
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