%
% 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.

\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.
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}

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.

%\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

