single failures analysis

pt100/pt100.tex

@@ -13,9 +13,10 @@ Once considering single faults (cardinality constrained powerset of 1) and then
 possibility of double simultaneous faults (cardinality constrained powerset of 2).
 
 The analysis is performed using Propositional Logic
-diagrasms to aid in the reasoning process, which takes
+diagrasms to assist the reasoning process.
-the failure modes of the components, and produces a 
+This chapter describes taking
-failure mode model for the circuit as a whole.
+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 itsself, with a set of know failure modes.
 
@@ -33,7 +34,7 @@ from an FMEA persepective as a component itsself, with a set of know failure mod
 
 \section{Overview of PT100 four wire circuit}
 
-The PT100 four wire circuit consists of teo resistors supplying 
+The PT100 four wire circuit consists of two resistors supplying 
 a current to a third, the thermistor or PT100. By measuring volatges
 from sections of this circuit forming potential dividers, we can determine the 
 current resistance of the platinum wire sensor. The resistance
@@ -61,12 +62,17 @@ and the higher as {\em sense+}.
 
 \subsection{Accuracy despite variable resistance in cables}
 
-For electronic and accuracy reasons the four wire circiut is used
+For electronic and accuracy reasons a four wire circuit is preffered
 because of resistance in the cables. Resitance from the supply
  causes a slight voltage 
 drop in the supply to the PT100. As no significant current
 is carried by the two `sense' lines the resistance back to the ADC
-causes only a negligible voltage drop. The current flowing though the 
+causes only a negligible voltage drop, and thus the four wire 
+configuration is more accurate. 
+
+\subsection{Calculating Temperature from the sense line voltages}
+
+The current flowing though the 
 whole circuit can be measured on the PCB by reading a third
 sense voltage from one of the load resistors. Knowing the current flowing
 through the circuit   
@@ -98,14 +104,14 @@ Where this occurs a circuit re-design is probably the only sensible course of ac
 \subsection{Single Fault FMEA Analysis of PT100 Four wire circuit}
 
 \label{fmea}
-Looking at this circuit, it simply consists of three resistors.
+This circuit simply consists of three resistors.
 Resistors according to the DOD Electronic component fault handbook
 1991, fail by either going OPEN or SHORT circuit \cite{mil1991}.
 %Should wires become disconnected these will have the same effect as
 %given resistors going open. 
 For the purpose of his analyis;
 $R_{1}$ is the \ohms{2k2} from 5V to the thermistor,
-$R_p$ is the PT100 thermistor and $R_{2}$ connects the thermistor to ground.
+$R_3$ is the PT100 thermistor and $R_{2}$ connects the thermistor to ground.
 
 We can define the terms `High Fault' and `Low Fault' here, with reference to figure
 \ref{fig:pt100vrange}. Should we get a reading outside the safe green zone
@@ -114,9 +120,10 @@ Should the reading be above its expected range this is a `High Fault'
 and if below a `Low Fault'.
 
 The Table \ref{ptfmea} plays through the scenarios of each of the resistors failing
-in both SHORT and OPEN failure modes, and predicts an error condition in the readings.
+in both SHORT and OPEN failure modes, and hypothesises an error condition in the readings.
 The range 0\oc to 300\oc will be analysed using potential divider equations to 
-to the out of range voltage limits in section \ref{ptbounds}.
+determine out of range voltage limits in section \ref{ptbounds}.
+
 \begin{table}[ht]
 \caption{PT100 FMEA Single Faults} % title of Table
 \centering % used for centering table
@@ -130,8 +137,8 @@ to the out of range voltage limits in section \ref{ptbounds}.
  $R_1$ SHORT    &  High Fault        &  -       &  Value Out of Range Value \\ \hline
 $R_1$  OPEN     &  Low Fault         & Low Fault     &  Both values out of range \\ \hline
  \hline
-$R_p$  SHORT    &  Low Fault          & High Fault      &   Both values out of range \\ \hline
+$R_3$  SHORT    &  Low Fault          & High Fault      &   Both values out of range \\ \hline
- $R_p$  OPEN     &  High Fault         & Low  Fault    &  Both values out of range \\ \hline
+ $R_3$  OPEN     &  High Fault         & Low  Fault    &  Both values out of range \\ \hline
 \hline
 $R_2$ SHORT     &  -         &  Low Fault   &  Value Out of Range Value \\  
  $R_2$ OPEN     &  High Fault         & High Fault     &  Both values out of range \\ \hline
@@ -141,17 +148,49 @@ $R_2$ SHORT     &  -         &  Low Fault   &  Value Out of Range Value \\
 \end{table}
 
 From table \ref{ptfmea} it can be seen that any component failure in the circuit
-will cause a common symptom, that of one or more of the values being out of range.
+should cause a common symptom, that of one or more of the values being `out of range'.
 Temperature range calculations and detailed calculations
 on the effects of each test case are found in section \ref{pt100range}
-and \ref{pt100temp}. So by defining an acceptable measurement/temperature range, and ensuring the 
+and \ref{pt100temp}. 
-values are always within these bounds we can be confident that none of the
-resistors in this circuit has failed.
 
 
-\subsection{Single Fault Modes as PLD}
+\pagebreak
+% \subsection{Single Fault Modes as PLD}
+% 
+% The component~failure~modes in table \ref{ptfmea} can be represented as contours 
+% on a PLD diagram. Each test case, or analysis into the effects of the component failure
+% caused by the component~failure is represented by an labelled asterisk.
+% 
+% 
+% \begin{figure}[h]
+%  \centering
+%  \includegraphics[width=400pt,bb=0 0 518 365,keepaspectratio=true]{./pt100/pt100_tc.jpg}
+%  % pt100_tc.jpg: 518x365 pixel, 72dpi, 18.27x12.88 cm, bb=0 0 518 365
+%  \caption{PT100 Component Failure Modes}
+%  \label{fig:pt100_tc}
+% \end{figure}
+% 
+% This circuit supplies two results, sense+ and sense- voltage readings.
+% To establish the valid voltage ranges for these, and knowing our
+% valid tempperature range for this example ({0\oc} .. {300\oc}) we can calculate
+% valid voltage reading ranges by using the standard voltage divider equation \ref{eqn:vd}
+% for the circuit shown in figure \ref{fig:vd}.
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=100pt,bb=0 0 183 170,keepaspectratio=true]{./pt100/voltage_divider.png}
+ % voltage_divider.png: 183x170 pixel, 72dpi, 6.46x6.00 cm, bb=0 0 183 170
+ \caption{Voltage Divider}
+ \label{fig:vd}
+\end{figure}
+%The looking at figure \ref{fig:vd} the standard voltage divider formula (equation \ref{eqn:vd}) is used.
+
+\begin{equation}
+\label{eqn:vd}
+ V_{out} = V_{in}.\frac{Z2}{Z2+Z1}
+\end{equation}
 
-% Place in PLD diagram
 
 \subsection{Range and PT100 Calculations}
 \label{pt100temp}
@@ -169,8 +208,13 @@ PT100 FMEA analysis in section \ref{fmea}.
 As the PT100 forms a potential divider with the \ohms{2k2}  load resistors,
 the upper and lower readings can be calculated thus:
 
+
 $$ highreading = 5V.\frac{2k2+pt100}{2k2+2k2+pt100} $$
 $$ lowreading = 5V.\frac{2k2}{2k2+2k2+pt100} $$
+So by defining an acceptable measurement/temperature range,
+and ensuring the 
+values are always within these bounds we can be confident that none of the
+resistors in this circuit has failed.
 
 To convert these to twelve bit ADC (\adctw) counts:
 
@@ -201,69 +245,6 @@ Table \ref{ptbounds} gives ranges that determine correct operation. In fact it c
 for any single error (short or opening of any resistor) this bounds check
 will detect it.
 
-\subsection{Proof of Out of Range Values for Failures}
-\label{pt110range}
-Using the temperature ranges defined above we can compare the voltages
-we would get from the resistor failures to prove that they are
-`out of range'. There are six cases and each will be examined in turn.
-
-\subsubsection{ Voltages $R_1$ SHORT } 
-With pt100 at 0\oc 
-$$ highreading = 5V $$
-Since the highreading or sense+ is directly connected to the 5V rail,
-both temperature readings will be 5V..
-$$ lowreading = 5V.\frac{2k2}{2k2+68\Omega} = 4.85V$$
-With pt100 at the high end of the temperature range 300\oc.
-$$ highreading = 5V $$
-$$ lowreading = 5V.\frac{2k2}{2k2+212.02\Omega} = 4.56V$$
-
-Thus with $R_1$ shorted both readingare outside the 
-proscribed range in table \ref{ptbounds}.
-
-\subsubsection{ Voltages $R_1$ OPEN } 
-
-In this case the 5V rail is disconnected. All voltages read are 0V, and 
-therefore both readings are outside the
-proscribed range in table \ref{ptbounds}.
-
-\subsubsection{ Voltages $R_p$ SHORT } 
-
-Here the potential divider is simply between
-the two 2k2 load resistors. Thus it will read a nominal;
-2.5V.
-
-Assuming the load resistors are 
-precision components, and then taking an absolute worst case of 1\% either way.
-
-$$ 5V.\frac{2k2*0.99}{2k2*1.01+2k2*0.99} = 2.475V $$
-
-$$ 5V.\frac{2k2*1.01}{2k2*1.01+2k2*0.99} = 2.525V $$
-
-These readings both lie outside the proscribed range.
-Also the sense+ and sense- readings would have the same value.
-
-\subsubsection{ Voltages $R_p$ OPEN } 
-
-Here the potential divider is broken. The sense- will read 0V and the sense+ will
-read 5V. Both readings are outside the proscribed range.
-
-\subsubsection{ Voltages $R_2$ SHORT }
-
-With pt100 at -100\oc 
-$$ lowreading = 0V $$
-Since the lowreading or sense- is directly connected to the 0V rail,
-both temperature readings will be 0V.
-$$ lowreading = 5V.\frac{68\Omega}{2k2+68\Omega} = 0.15V$$
-With pt100 at the high end of the temperature range 300\oc.
-$$ highreading = 5V $$
-$$ lowreading = 5V.\frac{212.02\Omega}{2k2+212.02\Omega} = 0.44V$$
-
-Thus with $R_2$ shorted both readingare outside the 
-proscribed range in table \ref{ptbounds}.
-\subsubsection{ Voltages $R_2$ OPEN }
-Here there is no potential divider operating and both sense lines
-will read 5V, outside of the proscibed range.
-
 %\vbox{
 %\subsubsection{Calculating Bounds: High Value : HP48 RPL}
 %
@@ -305,22 +286,146 @@ will read 5V, outside of the proscibed range.
 %Three readings are taken. A reading to confirm the voltage level
 %over $R_2$ is taken,
 %from which the current can be determined.
-%The two sense lines then give the voltage over the PT100 thermistor.
+%The two sense lines then give the vo
-%As we know the current flowing through it we can determine the 
-%resistance.
-%
-%After verification (PT100 voltages/readings in range etc) the temperature
-%value is determined by interpolation via the PT100 tables \cite{eurothermtables}.
-%First order low pass filtering is then applied to smooth the value.
-%\section{Water Level Readings - \ft Inputs}
-%\label{wl}
-%After h/w revision 0.4, water level sensor \ft connections are wired to the TDS daughterboard,
-%but are passed to the main unit via a multiplexer, and connect to the
-%14 pin harwin (to PIN 13 of JP1 \cite{pcbAI222562}).
-%
-%The safety critical \ft water~level readings are thus handled in the \wlc.
-%
 
-\subsection{Single Fault FMEA Analysis of PT100 Four wire circuit}
+\section{Single Fault FMEA Analysis of PT100 Four wire circuit}
-typeset in {\Huge \LaTeX} \today
+
+\subsection{Single Fault Modes as PLD}
+
+The component~failure~modes in table \ref{ptfmea} can be represented as contours
+on a PLD diagram. Each test case, or analysis into the effects of the component failure
+caused by the component~failure is represented by an labelled asterisk.
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt,bb=0 0 518 365,keepaspectratio=true]{./pt100/pt100_tc.jpg}
+ % pt100_tc.jpg: 518x365 pixel, 72dpi, 18.27x12.88 cm, bb=0 0 518 365
+ \caption{PT100 Component Failure Modes}
+ \label{fig:pt100_tc}
+\end{figure}
+
+This circuit supplies two results, sense+ and sense- voltage readings.
+To establish the valid voltage ranges for these, and knowing our
+valid tempperature range for this example ({0\oc} .. {300\oc}) we can calculate
+valid voltage reading ranges by using the standard voltage divider equation \ref{eqn:vd}
+for the circuit shown in .
+
+
+
+\subsection{Proof of Out of Range Values for Failures}
+\label{pt110range}
+Using the temperature ranges defined above we can compare the voltages
+we would get from the resistor failures to prove that they are
+`out of range'. There are six test cases and each will be examined in turn.
+
+\subsubsection{ TC1 : Voltages $R_1$ SHORT } 
+With pt100 at 0\oc 
+$$ highreading = 5V $$
+Since the highreading or sense+ is directly connected to the 5V rail,
+both temperature readings will be 5V..
+$$ lowreading = 5V.\frac{2k2}{2k2+68\Omega} = 4.85V$$
+With pt100 at the high end of the temperature range 300\oc.
+$$ highreading = 5V $$
+$$ lowreading = 5V.\frac{2k2}{2k2+212.02\Omega} = 4.56V$$
+
+Thus with $R_1$ shorted both readingare outside the 
+proscribed range in table \ref{ptbounds}.
+
+\subsubsection{ TC2 : Voltages $R_1$ OPEN } 
+
+In this case the 5V rail is disconnected. All voltages read are 0V, and 
+therefore both readings are outside the
+proscribed range in table \ref{ptbounds}.
+
+
+\subsubsection{ TC 4 : Voltages $R_2$ SHORT }
+
+With pt100 at -100\oc 
+$$ lowreading = 0V $$
+Since the lowreading or sense- is directly connected to the 0V rail,
+both temperature readings will be 0V.
+$$ lowreading = 5V.\frac{68\Omega}{2k2+68\Omega} = 0.15V$$
+With pt100 at the high end of the temperature range 300\oc.
+$$ highreading = 5V $$
+$$ lowreading = 5V.\frac{212.02\Omega}{2k2+212.02\Omega} = 0.44V$$
+
+Thus with $R_2$ shorted both readingare outside the 
+proscribed range in table \ref{ptbounds}.
+
+\subsubsection{ TC : 5 Voltages $R_2$ OPEN }
+Here there is no potential divider operating and both sense lines
+will read 5V, outside of the proscibed range.
+
+
+\subsubsection{ TC 5 : Voltages $R_3$ SHORT } 
+
+Here the potential divider is simply between
+the two 2k2 load resistors. Thus it will read a nominal;
+2.5V.
+
+Assuming the load resistors are 
+precision components, and then taking an absolute worst case of 1\% either way.
+
+$$ 5V.\frac{2k2*0.99}{2k2*1.01+2k2*0.99} = 2.475V $$
+
+$$ 5V.\frac{2k2*1.01}{2k2*1.01+2k2*0.99} = 2.525V $$
+
+These readings both lie outside the proscribed range.
+Also the sense+ and sense- readings would have the same value.
+
+\subsubsection{ TC 6 : Voltages $R_3$ OPEN } 
+
+Here the potential divider is broken. The sense- will read 0V and the sense+ will
+read 5V. Both readings are outside the proscribed range.
+
+\subsection{Summary of Analysis}
+
+All six test cases have been analysed and the results agree with the hypothesis
+put in  Table \ref{ptfmea}. The PLD diagram, can now be used to collect the
+symptoms. In this case there is a common and easily detected symptom for all these single 
+resistor faults :  Voltage out of range.
+
+A spider can be drawn on the PLD diagram to this effect.
+
+In practical use, by defining an acceptable measurement/temperature range,
+and ensuring the 
+values are always within these bounds we can be confident that none of the
+resistors in this circuit has failed.
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt,bb=0 0 518 365,keepaspectratio=true]{./pt100/pt100_tc_sp.jpg}
+ % pt100_tc.jpg: 518x365 pixel, 72dpi, 18.27x12.88 cm, bb=0 0 518 365
+ \caption{PT100 Component Failure Modes}
+ \label{fig:pt100_tc_sp}
+\end{figure}
+
+
+The PT100 circuit can now be treated as a component in its own right, and has one failure mode, 
+{\textbf OUT\_OF\_RANGE}. It can now be represnted as a PLD see figure \ref{fig:pt100_singlef}.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=100pt,bb=0 0 167 194,keepaspectratio=true]{./pt100/pt100_singlef.jpg}
+ % pt100_singlef.jpg: 167x194 pixel, 72dpi, 5.89x6.84 cm, bb=0 0 167 194
+ \caption{PT100 Circuit Failure Modes : From Single Faults Analysis}
+ \label{fig:pt100_singlef}
+\end{figure}
+
+
+
+
+%Interestingly we can calculate the failure statistics for this circuit now.
+%Mill 1991 gives resistor stats of ${10}^{11}$ times 6 (can we get special stats for pt100) ???
+
+The PT100 analysis presents a simple result for single faults. 
+The next analysis phase looks at how the circuit will behave under double simultaneous failure
+conditions.
+
+
+\section{ PT100 Double Simultaneous Fault Analysis}
+
+% typeset in {\Huge \LaTeX} \today
 

pt100/pt100.tex.backup

@@ -0,0 +1,312 @@
+%
+% Make the revision and doc number macro's then they are defined in one place
+
+\begin{abstract}
+The PT100, or platinum wire \ohms{100} sensor is 
+a wisely used industrial temperature sensor that is
+are slowly replacing the use of thermocouples in many 
+industrial applications below 600\oc, due to high accuracy\cite{aoe}.
+
+This chapter looks at the most common configuration, the 
+four wire circuit, and analyses it from an FMEA perspective twice.
+Once considering single faults (cardinality constrained powerset of 1) and then again, considering the 
+possibility of double simultaneous faults (cardinality constrained powerset of 2).
+
+The analysis is performed using Propositional Logic
+diagrasms to aid in the reasoning process, which takes
+the failure modes of the components, and produces 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 itsself, with a set of know failure modes.
+
+\end{abstract}
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt,bb=0 0 714 180,keepaspectratio=true]{./pt100/pt100.jpg}
+ % pt100.jpg: 714x180 pixel, 72dpi, 25.19x6.35 cm, bb=0 0 714 180
+ \caption{PT100 four wire circuit}
+ \label{fig:pt100}
+\end{figure}
+
+
+\section{Overview of PT100 four wire circuit}
+
+The PT100 four wire circuit consists of teo resistors supplying 
+a current to a third, the thermistor or PT100. By measuring volatges
+from sections of this circuit forming potential dividers, we can determine the 
+current resistance of the platinum wire sensor. The resistance
+of this is directly related to temperature, and may be determined by
+look-up tables or a suitable polynomial expression.
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=150pt,bb=0 0 273 483,keepaspectratio=true]{./pt100/vrange.jpg}
+ % pt100.jpg: 714x180 pixel, 72dpi, 25.19x6.35 cm, bb=0 0 714 180
+ \caption{PT100 expected voltage range}
+ \label{fig:pt100vrange}
+\end{figure}
+
+
+
+
+The voltage ranges we expect from from this three stage potential divider
+are shown in figure \ref{fig:pt100vrange}. Note that there is
+an expected range for each reading for a given temperature span.
+
+\subsection{Accuracy despite variable resistance in cables}
+
+For electronic and accuracy reasons the four wire circiut is used
+because of resistance in the cables. Resitance from the supply
+ causes a slight voltage 
+drop in the supply to the PT100. As no significant current
+is carried by the two `sense' lines the resistance back to the ADC
+causes only a negligible voltage drop. The current flowing though the 
+whole circuit can be measured on the PCB by reading a third
+sense voltage from one of the load resistors. Knowing the current flowing
+through the circuit   
+and knowing the voltage drop over the PT100, we can calculate its 
+resistance  by ohms law $V=I.R$, $R=\frac{I}{V}$. 
+Thus a little loss of supply current due to resistance in the cables
+does not impinge on accuracy.
+The resistance to temperature conversion is achieved 
+through the published PT100 tables\cite{eurothermtables}.
+
+\section{Safety case for 4 wire circuit}
+
+This sub-section looks at the behaviour of the PT100 four wire circuit 
+for the effects of component failures.
+All components have a set of known `failure modes'.
+In other words we know that a given component can fail in several distict ways.
+Studies have been published which list common component types 
+and their sets of failure modes, often with MTTF statistics \cite{mil1991}.
+Thus for each component, an analysis is made for each of it failure modes,
+with respect to its effect on the 
+circuit. Each one of these scenarios is termed a `test case'.
+The resultant circuit behaviour for each of these test cases is noted. 
+The worst case for this type of
+analysis would be a fault that we cannot detect.
+Where this occurs a circuit re-design is probably the only sensible course of action.
+
+
+
+\subsection{Single Fault FMEA Analysis of PT100 Four wire circuit}
+
+\label{fmea}
+Looking at this circuit, it simply consists of three resistors.
+Resistors according to the DOD Electronic component fault handbook
+1991, fail by either going OPEN or SHORT circuit \cite{mil1991}.
+%Should wires become disconnected these will have the same effect as
+%given resistors going open. 
+For the purpose of his analyis;
+$R_{1}$ is the \ohms{2k2} from 5V to the thermistor,
+$R_p$ is the PT100 thermistor and $R_{2}$ connects the thermistor to ground.
+
+\begin{table}[ht]
+\caption{PT100 FMEA Single Faults} % title of Table
+\centering % used for centering table
+\begin{tabular}{||l|c|c|l|l||}
+\hline \hline
+ \textbf{Test} & \textbf{Result} & \textbf{Result } & \textbf{General}   \\
+ \textbf{Case} &  \textbf{sense +} & \textbf{sense -} & \textbf{Symtom Description}   \\
+%   R         &    wire        & res +         & res -    & description
+\hline
+\hline
+ $R_1$ SHORT    &  High         &  -       &  Value Out of Range Value \\ \hline
+$R_1$  OPEN     &  Low          & Low      &  Both values out of range \\ \hline
+ \hline
+$R_p$  SHORT    &  Low           & High       &   Both values out of range \\ \hline
+ $R_p$  OPEN     &  High          & Low      &  Both values out of range \\ \hline
+\hline
+$R_2$ SHORT     &  -         &  Low    &  Value Out of Range Value \\  
+ $R_2$ OPEN     &  High          & High      &  Both values out of range \\ \hline
+\hline
+\end{tabular} 
+\label{ptfmea}
+\end{table}
+
+From table \ref{ptfmea} it can be seen that any component failure in the circuit
+will cause a common symptom, that of one or more of the values being out of range.
+Temperature range calculations and detailed calculations
+on the effects of each test case are found in section \ref{pt100range}
+and \ref{pt100temp}. So by defining an acceptable measurement/temperature range, and ensuring the 
+values are always within these bounds we can be confident that none of the
+resistors in this circuit has failed.
+
+
+\subsection{Single Fault Modes as PLD}
+
+% Place in PLD diagram
+
+\subsection{Range and PT100 Calculations}
+\label{pt100temp}
+PT100 resistors are designed to 
+have a resistance of ohms{100}  at 0 \oc \cite{eurothermtables}.
+A suitable `wider than to be expected range' was considered to be {-100\oc} to {300\oc}. 
+According to the Eurotherm PT100
+tables \cite{eurothermtables}, this corresponded to the resistances \ohms{60.28}
+and \ohms{212.02} respectively. From this the potential divider circuit can be
+analysed and the maximum and minimum acceptable voltages determined.
+These can be used as bounds results to apply the findings from the 
+PT100 FMEA analysis in section \ref{fmea}.
+
+As the PT100 forms a potential divider with the \ohms{2k2}  load resistors,
+the upper and lower readings can be calculated thus:
+
+$$ highreading = 5V.\frac{2k2+pt100}{2k2+2k2+pt100} $$
+$$ lowreading = 5V.\frac{2k2}{2k2+2k2+pt100} $$
+
+To convert these to twelve bit ADC (\adctw) counts:
+
+$$ highreading = 2^{12}.\frac{2k2+pt100}{2k2+2k2+pt100} $$
+$$ lowreading = 2^{12}.\frac{2k2}{2k2+2k2+pt100} $$
+
+
+\begin{table}[ht]
+\caption{PT100 Maximum and Minimum Values} % title of Table
+\centering % used for centering table
+\begin{tabular}{||c|c|c|l|l||}
+\hline \hline
+ \textbf{Temperature} & \textbf{PT100 resistance} & 
+\textbf{Lower} & \textbf{Higher} & \textbf{Description}   \\ 
+\hline
+ {-100 \oc} &  {\ohms{68.28}} &  2.46V         &   2.53V      & Boundary of   \\ 
+            &                 &  2017\adctw   &   2079\adctw  &   out of range LOW \\ \hline
+  {0 \oc}   & {\ohms{100}}    &  2.44V         &   2.56V       & Mid Range         \\
+            &                 &  2002\adctw   &   2094\adctw   &          \\ \hline
+ {+300 \oc} & {\ohms{212.02}}  &  2.38V         &   2.62V       & Boundary of                       \\
+            &                  &  1954\adctw   &   2142\adctw   &  out of range HIGH  \\ \hline
+\hline
+\end{tabular} 
+\label{ptbounds}
+\end{table}
+
+Table \ref{ptbounds} gives ranges that determine correct operation. In fact it can be shown that
+for any single error (short or opening of any resistor) this bounds check
+will detect it.
+
+\subsection{Proof of Out of Range Values for Failures}
+\label{pt110range}
+Using the temperature ranges defined above we can compare the voltages
+we would get from the resistor failures to prove that they are
+`out of range'. There are six cases and each will be examined in turn.
+
+\subsubsection{ Voltages $R_1$ SHORT } 
+With pt100 at -100\oc 
+$$ highreading = 5V $$
+Since the highreading or sense+ is directly connected to the 5V rail,
+both temperature readings will be 5V..
+$$ lowreading = 5V.\frac{2k2}{2k2+68\Omega} = 4.85V$$
+With pt100 at the high end of the temperature range 300\oc.
+$$ highreading = 5V $$
+$$ lowreading = 5V.\frac{2k2}{2k2+212.02\Omega} = 4.56V$$
+
+Thus with $R_1$ shorted both readingare outside the 
+proscribed range in table \ref{ptbounds}.
+
+\subsubsection{ Voltages $R_1$ OPEN } 
+
+In this case the 5V rail is disconnected. All voltages read are 0V, and 
+therefore both readings are outside the
+proscribed range in table \ref{ptbounds}.
+
+\subsubsection{ Voltages $R_p$ SHORT } 
+
+Here the potential divider is simply between
+the two 2k2 load resistors. Thus it will read a nominal;
+2.5V.
+
+Assuming the load resistors are 
+precision components, and then taking an absolute worst case of 1\% either way.
+
+$$ 5V.\frac{2k2*0.99}{2k2*1.01+2k2*0.99} = 2.475V $$
+
+$$ 5V.\frac{2k2*1.01}{2k2*1.01+2k2*0.99} = 2.525V $$
+
+These readings both lie outside the proscribed range.
+Also the sense+ and sense- readings would have the same value.
+
+\subsubsection{ Voltages $R_p$ OPEN } 
+
+Here the potential divider is broken. The sense- will read 0V and the sense+ will
+read 5V. Both readings are outside the proscribed range.
+
+\subsubsection{ Voltages $R_2$ SHORT }
+
+With pt100 at -100\oc 
+$$ lowreading = 0V $$
+Since the lowreading or sense- is directly connected to the 0V rail,
+both temperature readings will be 0V.
+$$ lowreading = 5V.\frac{68\Omega}{2k2+68\Omega} = 0.15V$$
+With pt100 at the high end of the temperature range 300\oc.
+$$ highreading = 5V $$
+$$ lowreading = 5V.\frac{212.02\Omega}{2k2+212.02\Omega} = 0.44V$$
+
+Thus with $R_2$ shorted both readingare outside the 
+proscribed range in table \ref{ptbounds}.
+\subsubsection{ Voltages $R_2$ OPEN }
+Here there is no potential divider operating and both sense lines
+will read 5V, outside of the proscibed range.
+
+%\vbox{
+%\subsubsection{Calculating Bounds: High Value : HP48 RPL}
+%
+%
+%HP RPL calculator program to take pt100 resistance
+%and convert to voltage and {\adctw} values.
+%
+%\begin{verbatim}
+%<< -> p 
+%        <<
+%            p 2200 + 2200 2200 + p + / 5 * DUP 5
+%            / 4096 *
+%        >>
+%>>
+%\end{verbatim}
+%}
+%
+%\vbox{
+%\subsubsection{Calculating Bounds: LOW Value : HP48 RPL}
+%
+%
+%HP RPL calculator program to take pt100 resistance
+%and convert to voltage and {\adctw} values.
+%
+%\begin{verbatim}
+%<< -> p 
+%        <<
+%            p 2200 2200  p 2200 + + / 5 * DUP 5
+%            / 4096 *
+%        >>
+%>>
+%\end{verbatim}
+%}
+%
+%\subsection{Implementation of Four Wire Circuit}
+%
+%A standard 4 wire PT100\cite[pp 992]{aoe} circuit is read by 
+%ports on the 12 bit ADC of the PIC18F2523\cite{pic18f2523}.
+%Three readings are taken. A reading to confirm the voltage level
+%over $R_2$ is taken,
+%from which the current can be determined.
+%The two sense lines then give the voltage over the PT100 thermistor.
+%As we know the current flowing through it we can determine the 
+%resistance.
+%
+%After verification (PT100 voltages/readings in range etc) the temperature
+%value is determined by interpolation via the PT100 tables \cite{eurothermtables}.
+%First order low pass filtering is then applied to smooth the value.
+%\section{Water Level Readings - \ft Inputs}
+%\label{wl}
+%After h/w revision 0.4, water level sensor \ft connections are wired to the TDS daughterboard,
+%but are passed to the main unit via a multiplexer, and connect to the
+%14 pin harwin (to PIN 13 of JP1 \cite{pcbAI222562}).
+%
+%The safety critical \ft water~level readings are thus handled in the \wlc.
+%
+
+\subsection{Single Fault FMEA Analysis of PT100 Four wire circuit}
+typeset in {\Huge \LaTeX} \today
+

pt100/pt100.tex~

@@ -0,0 +1,325 @@
+%
+% Make the revision and doc number macro's then they are defined in one place
+
+\begin{abstract}
+The PT100, or platinum wire \ohms{100} sensor is 
+a wisely used industrial temperature sensor that is
+are slowly replacing the use of thermocouples in many 
+industrial applications below 600\oc, due to high accuracy\cite{aoe}.
+
+This chapter looks at the most common configuration, the 
+four wire circuit, and analyses it from an FMEA perspective twice.
+Once considering single faults (cardinality constrained powerset of 1) and then again, considering the 
+possibility of double simultaneous faults (cardinality constrained powerset of 2).
+
+The analysis is performed using Propositional Logic
+diagrasms to aid in the reasoning process, which takes
+the failure modes of the components, and produces 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 itsself, with a set of know failure modes.
+
+\end{abstract}
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt,bb=0 0 714 180,keepaspectratio=true]{./pt100/pt100.jpg}
+ % pt100.jpg: 714x180 pixel, 72dpi, 25.19x6.35 cm, bb=0 0 714 180
+ \caption{PT100 four wire circuit}
+ \label{fig:pt100}
+\end{figure}
+
+
+\section{Overview of PT100 four wire circuit}
+
+The PT100 four wire circuit consists of teo resistors supplying 
+a current to a third, the thermistor or PT100. By measuring volatges
+from sections of this circuit forming potential dividers, we can determine the 
+current resistance of the platinum wire sensor. The resistance
+of this is directly related to temperature, and may be determined by
+look-up tables or a suitable polynomial expression.
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=150pt,bb=0 0 273 483,keepaspectratio=true]{./pt100/vrange.jpg}
+ % pt100.jpg: 714x180 pixel, 72dpi, 25.19x6.35 cm, bb=0 0 714 180
+ \caption{PT100 expected voltage ranges}
+ \label{fig:pt100vrange}
+\end{figure}
+
+
+
+
+The voltage ranges we expect from from this three stage potential divider
+are shown in figure \ref{fig:pt100vrange}. Note that there is
+an expected range for each reading for a given temperature span.
+Note that the low reading goes down as temperature increases, and the higher reading goes up.
+For this reason the low reading will be reffered to as {\em sense-}
+and the higher as {\em sense+}.
+
+\subsection{Accuracy despite variable resistance in cables}
+
+For electronic and accuracy reasons the four wire circiut is used
+because of resistance in the cables. Resitance from the supply
+ causes a slight voltage 
+drop in the supply to the PT100. As no significant current
+is carried by the two `sense' lines the resistance back to the ADC
+causes only a negligible voltage drop. The current flowing though the 
+whole circuit can be measured on the PCB by reading a third
+sense voltage from one of the load resistors. Knowing the current flowing
+through the circuit   
+and knowing the voltage drop over the PT100, we can calculate its 
+resistance  by ohms law $V=I.R$, $R=\frac{I}{V}$. 
+Thus a little loss of supply current due to resistance in the cables
+does not impinge on accuracy.
+The resistance to temperature conversion is achieved 
+through the published PT100 tables\cite{eurothermtables}.
+
+\section{Safety case for 4 wire circuit}
+
+This sub-section looks at the behaviour of the PT100 four wire circuit 
+for the effects of component failures.
+All components have a set of known `failure modes'.
+In other words we know that a given component can fail in several distict ways.
+Studies have been published which list common component types 
+and their sets of failure modes, often with MTTF statistics \cite{mil1991}.
+Thus for each component, an analysis is made for each of it failure modes,
+with respect to its effect on the 
+circuit. Each one of these scenarios is termed a `test case'.
+The resultant circuit behaviour for each of these test cases is noted. 
+The worst case for this type of
+analysis would be a fault that we cannot detect.
+Where this occurs a circuit re-design is probably the only sensible course of action.
+
+
+
+\subsection{Single Fault FMEA Analysis of PT100 Four wire circuit}
+
+\label{fmea}
+Looking at this circuit, it simply consists of three resistors.
+Resistors according to the DOD Electronic component fault handbook
+1991, fail by either going OPEN or SHORT circuit \cite{mil1991}.
+%Should wires become disconnected these will have the same effect as
+%given resistors going open. 
+For the purpose of his analyis;
+$R_{1}$ is the \ohms{2k2} from 5V to the thermistor,
+$R_p$ is the PT100 thermistor and $R_{2}$ connects the thermistor to ground.
+
+We can define the terms `High Fault' and `Low Fault' here, with reference to figure
+\ref{fig:pt100vrange}. Should we get a reading outside the safe green zone
+in the diagram we can consider this a fault.
+Should the reading be above its expected range this is a `High Fault'
+and if below a `Low Fault'.
+
+The Table \ref{ptfmea} plays through the scenarios of each of the resistors failing
+in both SHORT and OPEN failure modes, and predicts an error condition in the readings.
+The range 0\oc to 300\oc will be analysed using potential divider equations to 
+to the out of range voltage limits in section \ref{ptbounds}.
+\begin{table}[ht]
+\caption{PT100 FMEA Single Faults} % title of Table
+\centering % used for centering table
+\begin{tabular}{||l|c|c|l|l||}
+\hline \hline
+ \textbf{Test} & \textbf{Result} & \textbf{Result } & \textbf{General}   \\
+ \textbf{Case} &  \textbf{sense +} & \textbf{sense -} & \textbf{Symtom Description}   \\
+%   R         &    wire        & res +         & res -    & description
+\hline
+\hline
+ $R_1$ SHORT    &  High Fault        &  -       &  Value Out of Range Value \\ \hline
+$R_1$  OPEN     &  Low Fault         & Low Fault     &  Both values out of range \\ \hline
+ \hline
+$R_p$  SHORT    &  Low Fault          & High Fault      &   Both values out of range \\ \hline
+ $R_p$  OPEN     &  High Fault         & Low  Fault    &  Both values out of range \\ \hline
+\hline
+$R_2$ SHORT     &  -         &  Low Fault   &  Value Out of Range Value \\  
+ $R_2$ OPEN     &  High Fault         & High Fault     &  Both values out of range \\ \hline
+\hline
+\end{tabular} 
+\label{ptfmea}
+\end{table}
+
+From table \ref{ptfmea} it can be seen that any component failure in the circuit
+will cause a common symptom, that of one or more of the values being out of range.
+Temperature range calculations and detailed calculations
+on the effects of each test case are found in section \ref{pt100range}
+and \ref{pt100temp}. So by defining an acceptable measurement/temperature range, and ensuring the 
+values are always within these bounds we can be confident that none of the
+resistors in this circuit has failed.
+
+
+\subsection{Single Fault Modes as PLD}
+
+% Place in PLD diagram
+
+\subsection{Range and PT100 Calculations}
+\label{pt100temp}
+PT100 resistors are designed to 
+have a resistance of ohms{100}  at 0 \oc \cite{eurothermtables}.
+A suitable `wider than to be expected range' was considered to be {-100\oc} to {300\oc}. 
+According to the Eurotherm PT100
+tables \cite{eurothermtables}, this corresponded to the resistances \ohms{60.28}
+and \ohms{212.02} respectively. From this the potential divider circuit can be
+analysed and the maximum and minimum acceptable voltages determined.
+These can be used as bounds results to apply the findings from the 
+PT100 FMEA analysis in section \ref{fmea}.
+
+As the PT100 forms a potential divider with the \ohms{2k2}  load resistors,
+the upper and lower readings can be calculated thus:
+
+$$ highreading = 5V.\frac{2k2+pt100}{2k2+2k2+pt100} $$
+$$ lowreading = 5V.\frac{2k2}{2k2+2k2+pt100} $$
+
+To convert these to twelve bit ADC (\adctw) counts:
+
+$$ highreading = 2^{12}.\frac{2k2+pt100}{2k2+2k2+pt100} $$
+$$ lowreading = 2^{12}.\frac{2k2}{2k2+2k2+pt100} $$
+
+
+\begin{table}[ht]
+\caption{PT100 Maximum and Minimum Values} % title of Table
+\centering % used for centering table
+\begin{tabular}{||c|c|c|l|l||}
+\hline \hline
+ \textbf{Temperature} & \textbf{PT100 resistance} & 
+\textbf{Lower} & \textbf{Higher} & \textbf{Description}   \\ 
+\hline
+ {-100 \oc} &  {\ohms{68.28}} &  2.46V         &   2.53V      & Boundary of   \\ 
+            &                 &  2017\adctw   &   2079\adctw  &   out of range LOW \\ \hline
+  {0 \oc}   & {\ohms{100}}    &  2.44V         &   2.56V       & Mid Range         \\
+            &                 &  2002\adctw   &   2094\adctw   &          \\ \hline
+ {+300 \oc} & {\ohms{212.02}}  &  2.38V         &   2.62V       & Boundary of                       \\
+            &                  &  1954\adctw   &   2142\adctw   &  out of range HIGH  \\ \hline
+\hline
+\end{tabular} 
+\label{ptbounds}
+\end{table}
+
+Table \ref{ptbounds} gives ranges that determine correct operation. In fact it can be shown that
+for any single error (short or opening of any resistor) this bounds check
+will detect it.
+
+\subsection{Proof of Out of Range Values for Failures}
+\label{pt110range}
+Using the temperature ranges defined above we can compare the voltages
+we would get from the resistor failures to prove that they are
+`out of range'. There are six cases and each will be examined in turn.
+
+\subsubsection{ Voltages $R_1$ SHORT } 
+With pt100 at -100\oc 
+$$ highreading = 5V $$
+Since the highreading or sense+ is directly connected to the 5V rail,
+both temperature readings will be 5V..
+$$ lowreading = 5V.\frac{2k2}{2k2+68\Omega} = 4.85V$$
+With pt100 at the high end of the temperature range 300\oc.
+$$ highreading = 5V $$
+$$ lowreading = 5V.\frac{2k2}{2k2+212.02\Omega} = 4.56V$$
+
+Thus with $R_1$ shorted both readingare outside the 
+proscribed range in table \ref{ptbounds}.
+
+\subsubsection{ Voltages $R_1$ OPEN } 
+
+In this case the 5V rail is disconnected. All voltages read are 0V, and 
+therefore both readings are outside the
+proscribed range in table \ref{ptbounds}.
+
+\subsubsection{ Voltages $R_p$ SHORT } 
+
+Here the potential divider is simply between
+the two 2k2 load resistors. Thus it will read a nominal;
+2.5V.
+
+Assuming the load resistors are 
+precision components, and then taking an absolute worst case of 1\% either way.
+
+$$ 5V.\frac{2k2*0.99}{2k2*1.01+2k2*0.99} = 2.475V $$
+
+$$ 5V.\frac{2k2*1.01}{2k2*1.01+2k2*0.99} = 2.525V $$
+
+These readings both lie outside the proscribed range.
+Also the sense+ and sense- readings would have the same value.
+
+\subsubsection{ Voltages $R_p$ OPEN } 
+
+Here the potential divider is broken. The sense- will read 0V and the sense+ will
+read 5V. Both readings are outside the proscribed range.
+
+\subsubsection{ Voltages $R_2$ SHORT }
+
+With pt100 at -100\oc 
+$$ lowreading = 0V $$
+Since the lowreading or sense- is directly connected to the 0V rail,
+both temperature readings will be 0V.
+$$ lowreading = 5V.\frac{68\Omega}{2k2+68\Omega} = 0.15V$$
+With pt100 at the high end of the temperature range 300\oc.
+$$ highreading = 5V $$
+$$ lowreading = 5V.\frac{212.02\Omega}{2k2+212.02\Omega} = 0.44V$$
+
+Thus with $R_2$ shorted both readingare outside the 
+proscribed range in table \ref{ptbounds}.
+\subsubsection{ Voltages $R_2$ OPEN }
+Here there is no potential divider operating and both sense lines
+will read 5V, outside of the proscibed range.
+
+%\vbox{
+%\subsubsection{Calculating Bounds: High Value : HP48 RPL}
+%
+%
+%HP RPL calculator program to take pt100 resistance
+%and convert to voltage and {\adctw} values.
+%
+%\begin{verbatim}
+%<< -> p 
+%        <<
+%            p 2200 + 2200 2200 + p + / 5 * DUP 5
+%            / 4096 *
+%        >>
+%>>
+%\end{verbatim}
+%}
+%
+%\vbox{
+%\subsubsection{Calculating Bounds: LOW Value : HP48 RPL}
+%
+%
+%HP RPL calculator program to take pt100 resistance
+%and convert to voltage and {\adctw} values.
+%
+%\begin{verbatim}
+%<< -> p 
+%        <<
+%            p 2200 2200  p 2200 + + / 5 * DUP 5
+%            / 4096 *
+%        >>
+%>>
+%\end{verbatim}
+%}
+%
+%\subsection{Implementation of Four Wire Circuit}
+%
+%A standard 4 wire PT100\cite[pp 992]{aoe} circuit is read by 
+%ports on the 12 bit ADC of the PIC18F2523\cite{pic18f2523}.
+%Three readings are taken. A reading to confirm the voltage level
+%over $R_2$ is taken,
+%from which the current can be determined.
+%The two sense lines then give the voltage over the PT100 thermistor.
+%As we know the current flowing through it we can determine the 
+%resistance.
+%
+%After verification (PT100 voltages/readings in range etc) the temperature
+%value is determined by interpolation via the PT100 tables \cite{eurothermtables}.
+%First order low pass filtering is then applied to smooth the value.
+%\section{Water Level Readings - \ft Inputs}
+%\label{wl}
+%After h/w revision 0.4, water level sensor \ft connections are wired to the TDS daughterboard,
+%but are passed to the main unit via a multiplexer, and connect to the
+%14 pin harwin (to PIN 13 of JP1 \cite{pcbAI222562}).
+%
+%The safety critical \ft water~level readings are thus handled in the \wlc.
+%
+
+\subsection{Single Fault FMEA Analysis of PT100 Four wire circuit}
+typeset in {\Huge \LaTeX} \today
+