added calculations to support fmea

pt100/pt100.tex

@@ -111,16 +111,19 @@ $R_2$ SHORT     &  -         &  Low    &  Value Out of Range Value \\
 
 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 
+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}. 
@@ -166,6 +169,69 @@ 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 -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}
 %