diff --git a/opamp_circuits_C_GARRETT/Makefile b/opamp_circuits_C_GARRETT/Makefile index 506bb95..933e7ca 100644 --- a/opamp_circuits_C_GARRETT/Makefile +++ b/opamp_circuits_C_GARRETT/Makefile @@ -1,6 +1,6 @@ -PNG_DIA = circuit1_dag.png mvampcircuit.png pd.png invamp.png shared_component.png tree_abstraction_levels.png +PNG_DIA = circuit1_dag.png mvampcircuit.png pd.png invamp.png shared_component.png tree_abstraction_levels.png three_tree.png diff --git a/opamp_circuits_C_GARRETT/opamps.tex b/opamp_circuits_C_GARRETT/opamps.tex index f0b889c..13ece3e 100644 --- a/opamp_circuits_C_GARRETT/opamps.tex +++ b/opamp_circuits_C_GARRETT/opamps.tex @@ -813,7 +813,7 @@ This can be simplified if we can determine the total number of failure modes in equation~\ref{eqn:rd} becomes $$ RD(fg) = fT.(|fg|-1).$$ \pagebreak[4] -\subsection{Reasoning Distance Examples} +\subsection{Reasoning Distance Examples}(c-1) The potential divider discussed in section~\ref{potdivfmmd} has a four failure modes and two components and therefore has an $RD$ of 4. $$RD(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$ @@ -823,5 +823,96 @@ having 3 failure modes each, we would have an $RD$ of $$RD(fictitious) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$ +This would be the polynomial ($O(N^2)$) result of applying FMEA rigorously (we could term this +Rigorous FMEA (RFMEA). + + +\pagebreak[4] +\subsection{Using the concept of Reasoning Distance to compare RFMEA with FMMD} + +\begin{figure} + \centering + \includegraphics[width=400pt,keepaspectratio=true]{./three_tree.png} + % three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385 + \caption{FMMD Hierarchy with $(|fg| = 3) \wedge (|fm(c)| = 3)$} + \label{fig:three_tree} +\end{figure} + +Because components have variable numbers of failure modes, + and {\fgs} have variable numbers of components it is difficult to +come up with a general formula for comparing the number of checks to make for +RFMEA and FMMMD. +If we were to create an example by fixing the number of components in a {\fg} +and the number of failure modes per component, we can derive formulae +to represent the number of checks to make. + +Consider $k$ to be the number of components in a {\fg} (i.e. $k=|fg|$), +$f$ is the number of failure modes per component (i.e. $f=|fm(c)|$), and +$L$ to be the number of levels in the hierarchy of an FMMD analysis. +We can represent the number of failure scenarios to check in an FMMD +with equation~\ref{eqn:anscen}. + +\begin{equation} + \label{eqn:anscen} + \sum_{n=0}^{L} {k}^{n}.k.f.(k-1) +\end{equation} + +The thinking behind equation~\ref{eqn:anscen}, is that for each level of analysis -- counting down from the top -- +there are ${k}^{n}$ {\fgs} within each level; we need to apply RFMEA to each {\fg} on the level. +The number of checks to make for RFMEA is number of components $k$ multiplied by the number of failure modes $f$ +checked against the remaining components in the {\fg} $(k-1)$. + +If, for the sake of example we fix the number of components in a {\fg} to three and +the number of failure modes per component to three, an FMMD hierarchy +would look like figure~\ref{fig:three_tree}. + +\subsection{Worked 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}, $3^2.3.2.3=168$. +Adding these together gives $242$ checks to make to perform RFMEA \textbf{within} +{\fgs}. + +If we were to take the system represented in figure~\ref{fig:three_tree}, and +apply RFMEA on it as a whole system, we can use equation~\ref{eqn:rd}, +$ RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1)$, where $|fg|$ is 27, $fm(c_n)$ is 3 +and $(|fg|-1)$ is 26. +This gives: +$ RD(fg) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$ + +In order to get general equations with which to compare RFMEA with FMMD +we can re-write equation~\ref{eqn:rd} in terms of the number of levels +in an FMMD hierarchy. The number of components in is number of components +in a {\fg} raised to the power of the level plus one. +Thus we re-write equation~\ref{eqn:rd} as: + + +\begin{equation} + \label{eqn:fmea_state_exp21} + \sum_{n=1}^{k^{L+1}}.(k^{L+1}-1).f \; , % \\ + %(N^2 - N).f +\end{equation} + +or + +\begin{equation} + \label{eqn:fmea_state_exp22} + k^{L+1}.(k^{L+1}-1).f \;. % \\ + %(N^2 - N).f +\end{equation} + +We can now use equation~\ref{eqn:anscen} and \ref{eqn:fmea_state_exp22} to compare (for fixed sizes of $|fg|$ and $|fm(c)|$) +the two approaches, for the work required to perform rigorous checking. + + +\subsection{Exponential squared to Exponential} + +can I say that ? + \end{document} diff --git a/opamp_circuits_C_GARRETT/three_tree.dia b/opamp_circuits_C_GARRETT/three_tree.dia index ce105bd..716fdbe 100644 Binary files a/opamp_circuits_C_GARRETT/three_tree.dia and b/opamp_circuits_C_GARRETT/three_tree.dia differ diff --git a/presentations/fmea/SIL.png b/presentations/fmea/SIL.png deleted file mode 100644 index e40cb2d..0000000 Binary files a/presentations/fmea/SIL.png and /dev/null differ diff --git a/presentations/fmea/burntoutpinto.png b/presentations/fmea/burntoutpinto.png deleted file mode 100644 index b5d0baf..0000000 Binary files a/presentations/fmea/burntoutpinto.png and /dev/null differ diff --git a/presentations/fmea/non_inv_amp_fmea.png b/presentations/fmea/non_inv_amp_fmea.png deleted file mode 100644 index 7d2cb25..0000000 Binary files a/presentations/fmea/non_inv_amp_fmea.png and /dev/null differ diff --git a/presentations/fmea/tech_meeting.png b/presentations/fmea/tech_meeting.png deleted file mode 100644 index c25606b..0000000 Binary files a/presentations/fmea/tech_meeting.png and /dev/null differ