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INVOPAMP needs re-looking at.

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two ways of analysing it.

figures wrong for BUBBA CC, because INVAMP has CC values not accounted
for
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Robin Clark 2012-02-06 21:53:16 +00:00
parent 2f123ddc73
commit 0ed94c8b45
1 changed files with 154 additions and 49 deletions
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old_thesis/opamp_circuits_C_GARRETT/opamps.tex
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@ -779,9 +779,9 @@ We can now examine what effect each of these failures will have on the {\fg} (se
\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
FS1: R1 SHORT & & $LOW$ & & $PDLow$ \\
FS2: R1 OPEN & & $HIGH$ & & $PDHigh$ \\ \hline
FS3: R2 SHORT & & $HIGH$ & & $PDHigh$ \\
FS4: R2 OPEN & & $LOW$ & & $PDLow$ \\ \hline
\hline
\end{tabular}
@ -812,14 +812,34 @@ We can now form a {\fg} with $PD$ and $OPAMP$.
\end{figure}
We can collect symptoms from the analysis and cretae a derived component
\begin{table}[h+]
\caption{NIAMP: Single failure analysis}
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Non In Amp Effect} & & \textbf{Symptom} \\
\hline
FS1: PD HIGH & & $LOW$ & & $Low$ \\
FS2: PD LOW & & $HIGH$ & & $High$ \\ \hline
FS3: OPAMP $L_{UP}$ & & $HIGH$ & & $High$ \\
FS4: OPAMP $L_{DOWN}$ & & $LOW$ & & $Low$ \\
FS5: OPAMP $Noop$ & & $LOW$ & & $Low$ \\
FS5: OPAMP $Low slew$ & & $LOW$ & & $Lowpass$ \\ \hline
\hline
\end{tabular}
\label{tbl:pd}
\end{table}
We can collect symptoms from the analysis and create 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:
$$ fm(NI\_AMP) = \{ {lowpass}, {high}, {low} \}.$$
$$ fm(NIAMP) = \{ {lowpass}, {high}, {low} \}.$$
With this two stage analysis we have a comparison complexity (see equation~\ref{eqn:rd2}) of
$4.(2-1)=4$ for the potential divider and $6.(2-1)=6$, giving a total of $10$ for the $NIAMP$.
For this simple example, traditional flat/non-modular FMEA would have a CC of $(3-1).(4+2+2)=16$.
\section{Inverting OPAMP}
@ -835,7 +855,7 @@ $$ fm(NI\_AMP) = \{ {lowpass}, {high}, {low} \}.$$
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
One is to do this in two stages, by considering 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.
@ -844,7 +864,7 @@ Both approaches are followed in the next two sub-sections.
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
If we can refine the operational states of the functional group, we can obtain clearer
symptoms.
If we consider the input will only be positive, we can invert the potential divider (see table~\ref{tbl:pdneg}).
@ -866,11 +886,35 @@ 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).
\begin{table}[h+]
\caption{Inverting Amplifier: Single failure analysis}
\begin{tabular}{|| l | l | c | c | l ||} \hline
\textbf{Failure Scenario} & & \textbf{Inverted Amp Effect} & & \textbf{Symptom} \\ \hline
\hline
FS1: INVPD LOW & & NEGATIVE - input & & $ HIGH $ \\
FS2: INVPD HIGH & & Positive - input & & $ LOW $
FS5: AMP L\_DN & & $ INVAMP_{low} $ & & $ LOW $ \\ \hline
$$ fm(NI\_AMP) = \{ {lowpass}, {high}, {low} \}.$$
FS6: AMP L\_UP & & $INVAMP_{high} $ & & $ HIGH $ \\ \hline
FS7: AMP NOOP & & $INVAMP_{nogain} $ & & $ LOW $ \\ \hline
FS8: AMP LowSlew & & $ slow output \frac{\delta V}{\delta t} $ & & $ LOW PASS $ \\ \hline
\hline
\end{tabular}
\label{tbl:invamppd}
\end{table}
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).
$$ fm(NIAMP) = \{ {lowpass}, {high}, {low} \}.$$
\subsection{Inverting OPAMP using three components}
@ -903,11 +947,11 @@ $HIGH$ or $LOW$ output.
FS5: AMP L\_DN & & $ INVAMP_{low} $ & & $ OUT OF RANGE $ \\ \hline
FS2: AMP L\_UP & & $INVAMP_{high} $ & & $ OUT OF RANGE $ \\ \hline
FS6: AMP L\_UP & & $INVAMP_{high} $ & & $ OUT OF RANGE $ \\ \hline
FS3: AMP NOOP & & $INVAMP_{nogain} $ & & $ NO GAIN $ \\ \hline
FS7: AMP NOOP & & $INVAMP_{nogain} $ & & $ NO GAIN $ \\ \hline
FS4: AMP LowSlew & & $ slow output \frac{\delta V}{\delta t} $ & & $ LOW PASS $ \\ \hline
FS8: AMP LowSlew & & $ slow output \frac{\delta V}{\delta t} $ & & $ LOW PASS $ \\ \hline
\hline
\end{tabular}
\label{tbl:invamp}
@ -923,12 +967,15 @@ $$ fm(INVAMP) = \{ OUT OF RANGE, ZERO OUTPUT, NO GAIN, LOW PASS \} $$
%Could further refine this if MTTF stats available for each component failure.
\clearpage
\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; or in other words
the observability of the causes of failure are reduced.
the observability of the causes of failure are reduced. Instead of the more specific symptoms $HIGH$ or $LOW$ we
obtain $OUT OF RANGE$ instead.
\clearpage
\section{Op-Amp circuit 1}
@ -1442,17 +1489,18 @@ gain and the final 180 degrees of phase shift (making a total of 360 degrees of
From a fault finding perspective this circuit is less than ideal.
The signal path is circular (its a positive feedback circuit) and most failures would simply cause the output to stop oscillating.
The top level failure modes for the FMMD hierarchy bear this out.
However, FMMD is a bottom -up analysis methodology and we can therefore still identify
{\fgs} and apply analysis from a failure mode perspective.
%The top level failure modes for the FMMD hierarchy bear this out.
%However, FMMD is a bottom -up analysis methodology and we can therefore still identify
%{\fgs} and apply analysis from a failure mode perspective.
%
If we were to analyse this circuit using traditional FMEA (i.e. without modularisation) we observe 14 components with
($4.4 +10.2 = 36$) failure modes.
Applying equation~\ref{eqn:rd2} gives a complexity comparison figure of $13.36=468$.
We now create FMMD models and compare the complexity of FMMD and FMEA.
If we were to analyse this circuit without modularisation, we have 14 components with
($4.4 +10.2 = 36$) failure modes . Applying equation~\ref{eqnrd2} gives a complexity comparison figure of $13.36=468$.
The reduce the complexity required to analyse this circuit we apply FMMD and start by determining {\fgs}.
We identify three types functional groups, an inverting amplifier (analysed in section~\ref{fig:invamp}),
We apply FMMD and start by determining {\fgs}.
We initially identify three types functional groups, an inverting amplifier (analysed in section~\ref{fig:invamp}),
a 45 degree phase shifter (a {$10k\Omega$} resistor and a $10nF$ capacitor) and a non-inverting buffer
amplifier. We can name these $INVAMP$, $PHS45$ and $NIBUFF$ respectively.
We can use these {\fgs} to describe the circuit in block diagram form with arrows indicating the signal path, in figure~\ref{fig:bubbablock}.
@ -1506,6 +1554,8 @@ We have to analyse this circuit from the perspective of it being a {\em phase~sh
$$ fm (PHS45) = \{ 90\_phaseshift, nosignal, 0\_phaseshift \} $$
$$ CC(PHS45) = 4.1 = 4 $$
\subsection{Non Inverting Buffer: NIBUFF.}
The non-inverting buffer functional group, is comprised of one component, an op-amp.
@ -1513,6 +1563,9 @@ We use the failure modes for an op-amp~\cite{fmd91}[p.3-116] to represent this g
% GARK
$$ fm(NIBUFF) = fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$
Because we obtain the failure modes for $NIBUFF$ from the literature
its comparison complexity is zero.
$$ CC(NIBUFF) = 0 $$
%\subsection{Forming a functional group from the PHS45 and NIBUFF.}
% describe what we are doing, a buffered 45 degree phase shift element
@ -1522,18 +1575,11 @@ $$ fm(NIBUFF) = fm(OPAMP) = \{L\_{up}, L\_{dn}, Noop, L\_slew \} $$
We could at this point bring all the {\dcs} together into one large functional
group (see figure~\ref{fig:poss1finalbubba})
or we could try to merge smaller stages.
A PHS45 {\dc} and an inverting amplifier (which always gives $180^{\circ}$ phase shift), form a {\fg}
providing an amplified $225^{\circ}$ phase shift, which we can call $PHS225AMP$.
%
We could also merge the $NIBUFF$ and $PHS45$
{\dcs} into a {\fg} and the resultant derived component from this we could call a $BUFF45$,
and then with those three, form a $PHS135BUFFERED$
functional group---with the remaining $PHS45$ and the $INVAMP$ (re-used from section~\ref{sec:invamp})in a second group $PHS225AMP$---
and then merge $PHS135BUFFERED$ and $PHS225AMP$ in a final stage (see figure~\ref{fig:poss2finalbubba})
Initially we use the first identified {\fgs} to create our model without further stages of refinement/hierarchy.
\subsection{FMMD Analysis using one large functional group}
\subsection{FMMD Analysis using initially identified functional groups}
\begin{figure}[h+]
\centering
@ -1605,10 +1651,31 @@ returns three failure modes,
$$ fm(BubbaOscillator) = \{ NO_{osc}, HI_{fosc}, LO_{fosc} \} . $$
For the final stage of this FMMD model, we can calculate the complexity using equation~\ref{eqn:rd2}.
$$ CC = 28.8 = 224$$
To obtain the total comparison complexity $TCC$, we need to add the complexity from the
{\dcs} that $BubbaOscillator$ was built from.
$$ TCC = 28.8 + 4.4 + 4.0 = 240$$
%As we have re-used the analysis for BUFF45 we could even reasonably remove
%$3.4=12$ from this result, because the results from $BUFF45$ have been used four times.
Traditional FMEA would have lead us to a much higher comparison complexity
of $468$ failure modes to check against components.
The analysis here appears top-heavy; we should be able to refine the model more
and break this down into smaller functional groups, by allowing more stages of hierarchy and hopefully
this should lead a further reduction in the complexity comparison figure.
\clearpage
\subsection{FMMD Analysis using smaller functional groups}
\subsection{FMMD Analysis using more hierarchical stages}
The example above---from the initial {\fgs}---used one very large functional group to model the circuit.
This mean a quite large comparison complexity for this final stage.
We should be able to determine smaller {\fgs} and refine the model further.
\begin{figure}[h+]
\centering
@ -1618,17 +1685,35 @@ $$ fm(BubbaOscillator) = \{ NO_{osc}, HI_{fosc}, LO_{fosc} \} . $$
\label{fig:poss2finalbubba}
\end{figure}
We can take a more modular approach by creating two intermediate functional groups, a buffered $45^{\circ}$ phase shifter (BUFF45)
we can combine three $BUFF45$'s to make
a $135^{\circ}$ buffer phase shifter (PHS135BUFFERED).
We can combine a $PHS45$ and a $NIBUFF$ to create
and an amplifying $225^{\circ}$ phase shifter (PHS225AMP).
By combining PHS225AMP and PHS135BUFFERED we can create a more modularised hierarchical
model of the bubba oscillator.
The proposed hierarchy is shown in figure~\ref{fig:poss2finalbubba}.
BUFF45 will comprise of a $PHS45$ {\dc} and a $NIBUFF$.
%
We take the $NIBUFF$ and $PHS45$
{\dcs} into a {\fg} giving the {\dc} $BUFF45$.
$BUFF45$ is a {\dc} representing an actively buffered $45^{\circ}$ phase shifter.
and with those three, form a $PHS135BUFFERED$
functional group.
$PHS135BUFFERED$ is a {\dc} representing an actively buffered $135^{\circ}$ phase shifter.
A PHS45 {\dc} and an inverting amplifier\footnote{Inverting amplifiers always apply a $180^{\circ}$ phase shift.}, form a {\fg}
providing an amplified $225^{\circ}$ phase shift, which we can call $PHS225AMP$.
%---with the remaining $PHS45$ and the $INVAMP$ (re-used from section~\ref{sec:invamp})in a second group $PHS225AMP$---
Finally we can merge $PHS135BUFFERED$ and $PHS225AMP$ in a final stage (see figure~\ref{fig:poss2finalbubba})
%We can take a more modular approach by creating two intermediate functional groups, a buffered $45^{\circ}$ phase shifter (BUFF45)
%we can combine three $BUFF45$'s to make
%a $135^{\circ}$ buffer phase shifter (PHS135BUFFERED).
%We can combine a $PHS45$ and a $NIBUFF$ to create
%and an amplifying $225^{\circ}$ phase shifter (PHS225AMP).
% By combining PHS225AMP and PHS135BUFFERED we can create a more modularised hierarchical
% model of the bubba oscillator.
% The proposed hierarchy is shown in figure~\ref{fig:poss2finalbubba}.
\begin{table}[h+]
\caption{BUFF45: Failure Mode Effects Analysis} % title of Table
\label{tbl:buff45}
@ -1658,6 +1743,7 @@ $$
fm (BUFF45) = \{ 90\_phaseshift, 0\_phaseshift, NO\_signal .\}
$$
$$ CC(BUFF45) = 7.1 = 7 $$
We can now combine three $BUFF45$ {\dcs} and create a $PHS135BUFFERED$ {\dc}.
@ -1696,6 +1782,7 @@ fm (PHS135BUFFERED) = \{ 90\_phaseshift, 180\_phaseshift, NO\_signal .\}
$$
$$ CC (PHS135BUFFERED) = 3.2 = 6 $$
@ -1728,6 +1815,7 @@ $$
fm (PHS225AMP) = \{ 270\_phaseshift, 180\_phaseshift, NO\_signal .\}
$$
$$ CC(PHS225AMP) = 7.1 $$
The $PHS225AMP$ consists of a $PHS45$ and an $INVAMP$ (which provides $180^{\circ}$ of phase shift).
@ -1764,15 +1852,32 @@ $$
fm (BUBBAOSC) = \{ LO_{fosc}, HI_{osc}, NO\_signal .\}
$$
We could trace the DAGs here and ensure that both analysis strategies worked ok.....
%We could trace the DAGs here and ensure that both analysis strategies worked ok.....
$$ CC(BUBBAOSC) = 6.(2-1) = 6 $$
We can now add the comparison complexities for all levels of the analysis represented in figure~\ref{fig:poss2finalbubba}.
We have at the lowest level two $PHS45$ {\dcs} giving a CC of 8, at the next level four $BUFF45$ {\dcs} giving $(4-1).7=21$,
and penultimately $PHS135BUFFERED$ with 6 and $PHS225AMP$ with 7. The final top stage of the hierarchy, $BUBBAOSC$ has a CC of 6.
Our total comparison complexity is $48$, this contrasts with 468 for traditional `flat' FMEA,
and 240 for our first stage functional groups analysis.
This has meant a drastic reduction in the number of failure-modes to check against components.
It has also given us five {\dcs}, building blocks, which may be re-used for similar circuitry
to analyse in the future.
\subsection{Comparing both approaches}
Large FG and less hierarchy, and more hierarchy and smaller fgs.
In general with large functional groups the comparison complexity
is higher, by an order of $O(N^2)$.
Smaller functional groups mean less by-hand checks are required.
It also means a more finely grained model. This means that
there are more {\dcs} and this increases the possibility of re-use.
The more we can modularise, the more we decimate the $O(N^2)$ effect
of complexity comparison.
More re-use-able fgs with smaller groups. Less chance of making a mistake (lower CC)
%More re-use-able fgs with smaller groups. Less chance of making a mistake (lower CC)