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Robin_PHD/burner/pt100.tex~

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%
% 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 assist the reasoning process.
This chapter describes taking
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.
\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 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
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 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, 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
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}
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,
$R_3$ 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
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 hypothesises an error condition in the readings.
The range 0\oc to 300\oc will be analysed using potential divider equations to
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
\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
$R_3$ SHORT & Low Fault & High 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
\hline
\end{tabular}
\label{ptfmea}
\end{table}
From table \ref{ptfmea} it can be seen that any component failure in the circuit
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}.
\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}
\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 {0\oc} to {300\oc}
for a given application.
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:
$$ highreading = 5V.\frac{2k2+pt100}{2k2+2k2+pt100} $$
$$ lowreading = 5V.\frac{2k2}{2k2+2k2+pt100} $$
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:
$$ highreading = 2^{12}.\frac{2k2+pt100}{2k2+2k2+pt100} $$
$$ lowreading = 2^{12}.\frac{2k2}{2k2+2k2+pt100} $$
\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 & Boundary of \\
& & 2002\adctw & 2094\adctw & out of range LOW \\ \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 vo
\section{Single Fault FMEA Analysis of PT100 Four wire circuit}
\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.
% OK we can look at stats here.
% the probabilities of the faults occurring in failures per billion hours
% of operation.
\subsection{Statistical MTTF for the PT100 circuit}
Mil1991\cite{mil1991} gives a mean time to failure for a fixed film resistor
at up to 60oC at a low stress (current) level as
$$0.00092 . 1.0 . 15 . 1.0 . 1000$$
13.8 failure per billion hours of operation.
RAC \cite{rac} states that a resistor will fail 9/10 OPEN and 1/10 SHORT.
So 13.8 - 1.38 OPEN
1.38 SHORT
MILL 1991 gives Thermisitors, bead $$0.21 15 1.0 1000$$
3150 failures per billion hours of operation.
Again we can apply the RAC division of resistor errors.
We can now see the six error types and see a statistical
prediction of which will occur. We can also
determine the reliability of the circuit as a whole.
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
%
%\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 & Boundary of \\
% & & 2002\adctw & 2094\adctw & out of range LOW \\ \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}
%
\begin{table}[ht]
\caption{PT100 FMEA Single Fault Statistics} % title of Table
\centering % used for centering table
\begin{tabular}{||l|c|c|l|c||}
\hline \hline
\textbf{Test} & \textbf{Result} & \textbf{Result } & \textbf{MTTF} \\
\textbf{Case} & \textbf{sense +} & \textbf{sense -} & \textbf{per $10^9$ hours of operation} \\
% R & wire & res + & res - & description
\hline
\hline
$R_1$ SHORT & High Fault & - & 12.42 \\ \hline
$R_1$ OPEN & Low Fault & Low Fault & 1.38 \\ \hline
\hline
$R_3$ SHORT & Low Fault & High Fault & 2835 \\ \hline
$R_3$ OPEN & High Fault & Low Fault & 315 \\ \hline
\hline
$R_2$ SHORT & - & Low Fault & 12.42 \\
$R_2$ OPEN & High Fault & High Fault & 1.38 \\ \hline
\hline
\end{tabular}
\label{pt100_single_stats}
\end{table}
The ciruit overall has a MTTF of (13.8*2 + 3150) 3177.6
per billion ($10^9$) hours of operation.
This gives an individual circuit a MTTF of around 39 years.
Interestingly though we can now look at the results of our analysis
as a probablistioc tree. We can see the overall reliability of the circuit
and we can see the most likely fault (the thermisitor going OPEN circuit).
The circuit is 8 times more likely to fail in this way than in any other.
Were we to need a more reliable temperature sensor this would probably
be the component area we would scrutinise first.
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