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submission_thesis/CH6_Evaluation/copy.tex
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@ -6,7 +6,7 @@
%
This chapter begins by defining a metric for the complexity of an FMEA analysis task.
%
This concept is called `comparisson~complexity' and is a means to assess
This concept is called `comparison~complexity' and is a means to assess
the performance of FMMD against current FMEA methodologies.
%
This metric is developed using set threory % formally
@ -54,8 +54,8 @@ We can view FMEA as a process, taking each component in the system and for each
applying analysis with respect to the whole system.
%
This however entails a problem: which other components in the system must we
check against the %current failure mode.
each particular failure mode.
check, against %current failure mode.
each particular failure mode?
%
Often a component failing will have obvious effects on functionally adjacent components.
Sometimes %though, perhaps in the case of de-coupling capacitors in a digital ciruit,
@ -115,7 +115,7 @@ $ | G | $. %,
%within an FMMD hierarchy is given in section~\ref{sec:indexsub}).
\paragraph{Defining Components}
We define the set of all components as $\mathcal{C}$. Individiual components are denoted as $c$
We define the set of all components as $\mathcal{C}$. Individual components are denoted as $c$
with additional indexing when appropriate.
\paragraph{Defining a function that returns failure modes given a component.}
@ -281,12 +281,15 @@ would look like figure~\ref{fig:three_tree}.
\subsection{RFMEA FMMD Comparison Example}
Using the diagram in figure~\ref{fig:three_tree}, we have three levels of analysis.
%
Starting at the top, we have a {\fg} with three derived components, each of which has
three failure modes.
Thus the number of checks to make in the top level is $3^0.3.2.3=18$.
On the level below that, we have three {\fgs} each with a
an identical number of checks, $3^1.3.2.3=56$.%{\fg}
On the level below that we have nine {\fgs}, $3^2.3.2.3=168$.
%
Thus the number of checks to make in the top level is $3^0\times3\times2\times3 = 18$.
On the level below that, we have three {\fgs} each with
an identical number of checks, $3^1 \times 3 \times 2 \times 3 = 56$.%{\fg}
%
On the level below that we have nine {\fgs}, $3^2 \times 3\times2\times3=168$.
Adding these together gives $242$ checks to make to perform FMMD (i.e. RFMEA {\em{within the}}
{\fgs}).
@ -497,10 +500,9 @@ Where this occurs a circuit re-design is probably the only sensible course of ac
\paragraph{Single Fault FMEA Analysis of $Pt100$ Four wire circuit.}
\label{fmea}
The Pt00 circuit consists of three resistors, two `current~supply'
The Pt100 circuit consists of three resistors, two `current~supply'
wires and two `sensor' wires.
Resistors %according to the European Standard EN298:2003~\cite{en298}[App.A]
, are considered to fail by either going OPEN or SHORT (see section~\ref{sec:res_fms}). %circuit\footnote{EN298:2003~\cite{en298} also requires that components are downrated,
Resistors, are considered to fail by either going OPEN or SHORT (see section~\ref{sec:res_fms}). %circuit\footnote{EN298:2003~\cite{en298} also requires that components are downrated,
%and so in the case of resistors the parameter change failure mode~\cite{fmd-91}[2-23] can be ommitted.}.
%Should wires become disconnected these will have the same effect as
%given resistors going open.
@ -646,15 +648,15 @@ The \ohms{2k2} loading resistors should have a good temperature co-effecient
To calculate the resistance of the Pt100 element % (and thus derive its temperature),
knowing $V_{R3}$ we now need the current flowing in the temperature sensor loop.
%
Lets use, for the sake of example $R_2$ to measure the current.
Lets use, for the sake of example, $R_2$ to measure the current.
%
We can calculate the current $I$, by reading
the voltage over the known resistor $R_2$ and using ohms law\footnote{To calculate the resistance of the Pt100 we need the current flowing though it.
We can determine this via ohms law applied to $R_2$, $V=IR$, $I=\frac{V}{R_2}$,
and then using $I$, we can calculate $R_{3} = \frac{V_{3}}{I}$.} and then use ohms law again to calculate
the voltage over the known resistor $R_2$ and using Ohms law\footnote{To calculate the resistance of the Pt100 we need the current flowing though it.
We can determine this via Ohms law applied to $R_2$, $V=IR$, $I=\frac{V}{R_2}$,
and then using $I$, we can calculate $R_{3} = \frac{V_{3}}{I}$.} and then use Ohms law again to calculate
the resistance of $R_3$.
%
As ohms law is linear, the accuracy of the reading
As Ohms law is linear, the accuracy of the reading
will be determined by the accuracy of $R_2$ and $R_{3}$. It is reasonable to
take the mean square error of these accuracy figures~\cite{probstat}.
@ -785,7 +787,7 @@ resistors in this circuit has failed.
\subsection{Derived Component : The Pt100 Circuit}
The Pt100 circuit can now be treated as a component in its own right, and has one failure mode,
{\textbf OUT\_OF\_RANGE}. This is a single, detectable failure mode. The observability of a
fault condition is very good with this circuit.This should not be a surprise, as the four wire $Pt100$
fault condition is very good with this circuit. This should not be a surprise, as the four wire $Pt100$
has been developed for safety critical temperature measurement.
%
\ifthenelse{\boolean{pld}}
@ -1289,7 +1291,7 @@ condition in higher levels of the system.
\subsubsection{Side Effects: A Problem for FMMD analysis}
\label{sec:sideeffects}
A problem with modularising according to functionality is that we can have component failures that would
A problem with modularising according to functionality is that we can have component failures that would % poss split infinitive
intuitively be associated with one {\fg} that may cause unintended side effects in other
{\fgs}.
For instance were we to have a component that on failing $SHORT$ could bring down
@ -1327,7 +1329,7 @@ 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}.
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}.% poss split infinitive
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.