finish off detailed double analysis

pt100/pt100.tex

@@ -24,7 +24,6 @@ from an FMEA persepective as a component itself, with a set of known failure mod
 \end{abstract}
 }
 {
-
 \section{Overview}
 The PT100, or platinum wire \ohms{100} sensor is 
 a widely used industrial temperature sensor that is
@@ -43,8 +42,6 @@ the failure modes of the components, analysing the circuit using FMEA
 and producing a failure mode model for the circuit as a whole.
 Thus after the analysis the PT100 temperature sensing circuit, may be veiwed 
 from an FMEA persepective as a component itself, with a set of known failure modes.
-
-
 }
 
 \begin{figure}[h]
@@ -271,6 +268,7 @@ and are thus enclosed by one contour each.
  \label{fig:pt100_tc}
 \end{figure}
 
+ating input Fault  
 This circuit supplies two results, sense+ and sense- voltage readings.
 To establish the valid voltage ranges for these, and knowing our
 valid temperature range for this example ({0\oc} .. {300\oc}) we can calculate
@@ -566,8 +564,143 @@ conditions.
 \clearpage
 \section{ PT100 Double Simultaneous \\ Fault Analysis}
 
+In this section we examine the failure mode behaviour for all single 
+faults and double simultaneous faults.
+This corresponds to the cardinality contstrained powerset of
+the failure modes in the functional group.
+All the single faults have already be proved in the last section.
+For the next set of test cases, let us again hypothesise
+the failure modes, and then examine each one in detail with
+potential divider equation proofs.
+
+Table \ref{tab:ptfmea2} lists all the combinations of double
+faults and then hypothesises how the functional~group will react
+under those conditions.
+
+\begin{table}[ht]
+\caption{PT100 FMEA Double Faults} % title of Table
+\centering % used for centering table
+\begin{tabular}{||l|l|c|c|l|l||}
+\hline \hline
+ \textbf{TC}      &\textbf{Test} & \textbf{Result} & \textbf{Result } & \textbf{General}   \\
+ \textbf{number} &\textbf{Case} &  \textbf{sense +} & \textbf{sense -} & \textbf{Symtom Description}   \\
+%   R         &    wire        & res +         & res -    & description
+\hline
+\hline
+ TC 7: & $R_1$ OPEN $R_2$ OPEN    &  Floating input Fault        &   Floating input Fault    &  Unknown value readings \\ \hline
+ TC 8: & $R_1$ OPEN $R_2$ SHORT   &  low        &   low    &  Both out of range  \\ \hline
+\hline
+ TC 9: & $R_1$ OPEN $R_3$ OPEN   &  high        &   low     &  Both out of Range \\ \hline
+ TC 10: & $R_1$ OPEN $R_3$ SHORT   &  low        &   low    &  Both out of range \\ \hline
+\hline
+
+ TC 11: & $R_1$ SHORT $R_2$ OPEN    &  high        &   high    &  Both out of range \\ \hline
+TC 12: &  $R_1$ SHORT $R_2$ SHORT   &  high        &   low    &  Both out of range  \\ \hline
+\hline
+ TC 13: & $R_1$ SHORT $R_3$ OPEN   &  high        &   low     &  Both out of Range \\ \hline
+TC 14: &  $R_1$ SHORT $R_3$ SHORT   &  high        &   high    &  Both out of range \\ \hline
+
+\hline
+ TC 15: & $R_2$ OPEN $R_3$ SHORT   &  high        &   Floating input Fault    &  sense+ out of range \\ \hline
+TC 16: &  $R_2$ OPEN $R_3$ SHORT   &  high        &   high    &  Both out of Range \\ \hline
+TC 17: &  $R_2$ SHORT $R_3$ OPEN   &  high        &   low    &  Both out of Range \\ \hline
+TC 18: &  $R_2$ SHORT $R_3$ SHORT   &  low        &   low    &  Both out of Range \\ \hline
+\hline
+\end{tabular}
+\label{tab:ptfmea2}
+\end{table}
+
+\subsection{Verifying complete coverage for a \\ cardinality constrained powerset of 2}
+
+It is important to check that we have covered all possible double fault combinations.
+We can use the equation \ref{eqn:correctedccps2}, reproduced below to verify this.
 
 
-DO THE DOUBLE
+
-% typeset in {\Huge \LaTeX} \today
+\begin{equation}
+ |{\mathcal{P}_{cc}SU}| = {\sum^{k}_{1..cc}  \frac{|{SU}|!}{k!(|{SU}| - k)!}}   
+- \sum^{p}_{2..cc}{{\sum^{j}_{j \in J}   \frac{|FM({C_j})|!}{p!(|FM({C_j})| - p)!}}  }
+ \label{eqn:correctedccps2}
+\end{equation}
+
+
+$|FM(C_j)|$ is always 2 here,  as all the components are resistors and have two failure modes.
+
+%
+% Factorial of zero is one ! You can only arrange an empty set one way !
+
+Populating this equation with $|SU| = 6$ and $|FM(C_j)|$ is always 2 here as all the components are resistors and have two failure modes.
+
+\begin{equation}
+ |{\mathcal{P}_{2}SU}| = {\sum^{k}_{1..2}  \frac{6!}{k!(6 - k)!}}   
+- \sum^{p}_{2..2}{{\sum^{j}_{1..3}   \frac{2!}{p!(2 - p)!}}  }
+ \label{eqn:correctedccps2}
+\end{equation}
+
+$|{\mathcal{P}_{2}SU}|$ is the number of valid combinations of faults to check
+under the conditions of unitary state failure modes for the components (a resistor cannot fail by being shorted and open at the same time).
+
+Expanding the sumations
+
+
+$$ NoOfTestCasesToCheck = \frac{6!}{1!(6-1)!} + \frac{6!}{2!(6-2)!} - \Big( \frac{2!}{2!(2 - 2)!} + \frac{2!}{2!(2 - 2)!} + \frac{2!}{2!(2 - 2)!} \Big) $$
+
+$$ NoOfTestCasesToCheck = 6 + 15 - ( 1 + 1 + 1 )  = 18 $$
+
+As the test case are all different and are of the correct cardinalities (6 single faults and (15-3) double)
+we can be confident that we have looked at all combinations up to two, of the possible faults
+in the pt100 circuit. The next task is to investigate
+these test cases in more detail to prove the failure mode hypothese set out in table \ref{tab:ptfmea2}.
+
+
+\subsection{Proof of Double Faults Hypothese }
+
+\subsubsection{ TC 7 : Voltages $R_1$ OPEN $R_2$ OPEN } 
+
+This double fault mode produces an interesting symptom.
+Both sense lines are floating.
+We cannot know what the {\adctw} readings on them will be.
+In practise these would probably float to  low values
+but for the purpose of a safety critical analysis
+all we can say is the values are `floating' and `unknown'.
+This is an interesting case, because it is, at this stage an undetectable
+fault that must be handled.
+
+
+\subsubsection{ TC 8 : Voltages $R_1$ OPEN $R_2$ SHORT } 
+
+This cuts the supply from Vcc. Both sense lines will be at zero.
+Thus both values will be out of range.
+
+
+\subsubsection{ TC 9 : Voltages $R_1$ OPEN $R_3$ OPEN } 
+
+Sense- will be floating.
+Sense+ will be tied to Vcc and will thus be out of range.
+
+\subsubsection{ TC 10 : Voltages $R_1$ OPEN $R_3$ SHORT } 
+
+This shorts ground to the 
+both of the sense lines.
+Both values thuis out of range.
+
+\subsubsection{ TC 11 : Voltages $R_1$ SHORT $R_2$ OPEN } 
+
+This shorts both sense lines to Vcc.
+Both values will be out of range.
+
+
+\subsubsection{ TC 12 : Voltages $R_1$ SHORT $R_2$ SHORT }
+
+This shorts the sense+ to Vcc and the sense- to ground.
+Both values will be out of range.
+
+
+
+
+
+
+
+
+typeset in {\Huge \LaTeX} \today
 

standards/standards.tex

@@ -4,29 +4,37 @@
 \ifthenelse {\boolean{paper}}
 {
 \begin{abstract}
-This chapter describes the legal frameworks and standards organisations
+This paper describes the legal frameworks and standards organisations
 that exist in Europe and North America.
 Some specific standards (that the author has experience with directly)
 are reviewed.
 \end{abstract}
 }
-{}
+{
-
+This chapter describes the legal frameworks and standards organisations
+that exist in Europe and North America.
+Some specific standards (that the author has experience with directly)
+are reviewed.
+}
 
 \section{Introduction}
 
 \subsection{Product Life Cycle}
-i
+
 difffernent areas
 EN61508 REQ to SPEC to DESIGN
 
 
 EN298
-DESIGN TO PRODUCT
+DESIGN TO 
+TESTING (EMC PRODUCT
 
 FM 
 PRODUCT VERIFICATION MONITORING
 
+
+NEW A PRODUCT LIFE CYCLE IMAGE WITH AN EULER DIAGRMA FOR THE DIFFERENT STANDARDS
+
 Different agencies - approval is testing of new product
 and verification to standard - manufacturing overwatch / supervision
 word on tip of tounge -