Saturday Morning Edit

fmmdset/fmmdset.tex

@@ -23,12 +23,16 @@ The FMMD process in outline is that,
 components are collected into functional groups, which are analysed from a failure mode perspective,
 and then a failure mode behaviour for each particular {\fg} is determined.
 From this failure mode behaviour we can now treat the {\fg}
-as a component or `black~box', with a known set of failure modes.
+as a component or `black~box', with a known set of failure symptoms.
+%
+%
+The failure symptoms of the {\fg} may be considered the failure modes of the 
+{\fg}, when viewed as a `black~box' or as a higher level `component'/sub-system. 
 We can thus  create a new component, a {\dc}, that we can use in place 
 of the functional group in our design.
 %
 By collecting {\dcs} into {\fgs} and analysing these into higher level {\dcs} a
-hierarchy is naturally formed. This hierarchy is termed an `FMMD failure mode tree'. 
+hierarchy is naturally formed. This hierarchy is termed an `FMMD~failure~mode~tree'. 
 
 From the FMMD failure mode trees, 
 modular re-usable sections of safety critical systems,
@@ -156,6 +160,14 @@ we need to consider all failure modes of its components.
 By analysing the fault behaviour of a `{\fg}' with respect to these component failure modes,
 we can derive a new set of possible failure modes. In fact we can call these
 the symptoms of failure for the {\fg}.
+
+
+We can stipulate that symptom collection process is surjective.
+% i.e. $ \forall f in F $ 
+By stipulating surjection for symptom collection, we ensure
+that each component failure mode maps to at least one one symptom.
+We also ensure that all symptoms have at least one component failure
+mode.
 %
 This new set of faults is the set of derived faults from the perspective of the {\fg}, and is thus at a higher level of 
 fault~mode abstraction. Thus we can say that the {\fg} as an entity, can fail in a number of well defined ways.
@@ -253,7 +265,7 @@ a set of derived failure modes. We are interested in the failure modes
 of all the components in the {\fg}. An analysis process
 defined by the symbol  `$\bowtie$' is applied to the {\fg}.
 
-iThe $\bowtie$ function takes a {\fg}
+The $\bowtie$ function takes a {\fg}
 as an argument and returns a newly created {\dc}.
 
 The $\bowtie$ analysis, a symptom extraction process, is described in chapter \ref{chap:sympex}.
@@ -272,6 +284,14 @@ it was derived from.
 
 By applying stages of analysis to higher and higher abstraction
 levels, we can converge to a complete failure mode model of the system under analysis.
+Because the symptom abstraction process is defined as surjective (from component failure modes to symptoms)
+the number of symptoms is guaranteed to the less than or equal to
+the number of component failure modes.
+
+In practice however, the number of symptoms greatly reduces as we traverse
+up the hierarchy.
+This is a natural process. When we have a complicated systems
+they always have a small number of system failure modes.
 
 An example of a simple system will illustrate this.
 

opamp_circuits_C_GARRETT/opamps.tex

@@ -41,10 +41,10 @@ Function $fm$ applied to a component returns its failure modes.
 \listoffigures  
 
 
-
 \section{Non-Inverting OPAMP}
 Consider a non inverting op-amp designed to amplify
-a small positive voltage, typical use would be a thermocouple.
+a small positive voltage (typical use would be a thermocouple amplifier
+taking a range from 0 to 25mV and amplifiying it to the range of an ADC approx 0 to 4 volts).
  
  
 \begin{figure}[h+]
@@ -115,6 +115,7 @@ We can now form a {\fg} with $PD$ and $OPAMP$.
  \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}
 
 
@@ -140,15 +141,89 @@ $$ fm(NI\_AMP) = \{  N\_INVAMP_{lowpass}, N\_INVAMP_{high}, N\_INVAMP_{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
-and then combining the poential divider with the OPAMP failure mode model.
+and then combining the potential divider with the OPAMP failure mode model.
 The other way is to place all three components in a {\fg}.
 
 \subsection{Inverting OPAMP using a Potential Divider {\dc}}
 
+Re-using the $PD$ - potential divider works only if the input voltage is negative.
+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.
+
+ $$ fm(NI\_AMP) = \{  N\_INVAMP_{lowpass}, N\_INVAMP_{high}, N\_INVAMP_{low} \}.$$
+
+
 \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 !
+
+
+
+\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 was essentially get an either situation with the error detection.
+
 
 \clearpage
 \section{Op-Amp circuit 1}