more work on fit and mil1992

pt100/pt100.tex

@@ -340,6 +340,7 @@ resistors in this circuit has failed.
 \end{figure}
 
 
+\subsection{Derived Component : The PT100 Circuit}
 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}.
 
@@ -352,6 +353,9 @@ The PT100 circuit can now be treated as a component in its own right, and has on
 \end{figure}
 
 
+%From the single faults (cardinality constrained powerset of 1) analysis, we can now create
+%a new derived component, the {\empt100circuit}. This has only \{ OUT\_OF\_RANGE \}
+%as its single failure mode.
 
 
 %Interestingly we can calculate the failure statistics for this circuit now.
@@ -359,20 +363,109 @@ The PT100 circuit can now be treated as a component in its own right, and has on
 \clearpage
 \subsection{Mean Time to Failure}
 
-Using the MIL1991\cite{mil1991} specifications for resistor and thermistor
+Now that we have a model for the failure mode behaviour of the pt100 circuit
+we can look at the statistics associated with each of the failure modes.
+
+The DOD electronic reliability of components
+document  MIL-HDBK-217F\cite{mil1992} gives formulae for calculating
+the
+%$\frac{failures}{{10}^6}$
+${failures}/{{10}^6}$ % looks better
+in hours for a wide range of generic components
+\footnote{These figures are based on components from the 1980's and MIL-HDBK-217F
+can give conservative reliability figures when applied to
+modern components}. 
+
+Using the MIL-HDBK-217F\cite{mil1992}  specifications for resistor and thermistor
 failure statistics we calculate the reliability of this circuit.
-MIL1991 gives MTTF for a wide range of common components.
-It does not specify how the components will fail (in this case OPEN or SHORT). {Some standards, notably EN298 only consider resistors failing in OPEN mode}.
+
+
+\subsubsection{Resistor FIT Calculations}
+
+The formula for given in MIL-HDBK-217F\cite{mil1992}[9.2] for a generic fixed film non-power resistor
+is reproduced in equation \ref{resistorfit}. The meanings 
+and values assigned to its co-efficients are described in table \ref{tab:resistor}.
+
+\begin{equation}
+% fixed comp resistor{\lambda}_p = {\lambda}_{b}{\pi}_{R}{\pi}_Q{\pi}_E
+resistor{\lambda}_p = {\lambda}_{b}{\pi}_{R}{\pi}_Q{\pi}_E
+ \label{resistorfit}
+\end{equation}
+
+\begin{table}[ht]
+\caption{Fixed film resistor Failure in time assessment} % title of Table
+\centering % used for centering table
+\begin{tabular}{||c|c|l||}
+\hline \hline
+ \em{Parameter}      &  \em{Value}    &   \em{Comments} \\
+                     &                &      \\ \hline \hline
+ ${\lambda}_{b}$ &  0.00092        & stress/temp base failure rate  $60^o$ C  \\   \hline
+ %${\pi}_T$ &  4.2         & max temp of $60^o$ C\\ \hline
+ ${\pi}_R$ &  1.0         & Resistance range $< 0.1M\Omega$\\ \hline
+ ${\pi}_Q$ &  15.0         & Non-Mil spec component\\ \hline
+ ${\pi}_E$ &  1.0         & benign ground environment\\ \hline
+
+\hline \hline
+\end{tabular}
+\label{tab:resistor}
+\end{table}
+
+Applying equation \ref{resistorfit} with the parameters from table \ref{tab:resistor}
+give the following failures in ${10}^6$ hours:
+
+\begin{equation}
+ 0.00092 \times 1.0 \times 15.0 \times 1.0 = 0.0138  \;{failures}/{{10}^{6} Hours}
+ \label{eqn:resistor}
+\end{equation}
+
+While MIL-HDBK-217F gives MTTF for a wide range of common components,
+it does not specify how the components will fail (in this case OPEN or SHORT). {Some standards, notably EN298 only consider resistors failing in OPEN mode}.
 FMD-97 Gives 27\% OPEN and 3\% SHORTED, for resistors under certain electrical and environmental stresses. This example 
 compromises and uses a 90:10 ratio, for resistor failure. 
 Thus for this example resistors are expevcted to fail OPEN in 90\% of cases and SHORTED 
 in the other 10\%.
-
 A standard fixed film resistor, for use in a benign environment, non military spec at
 temperatures up to 60\oc is given a probability of 13.8 failures per billion ($10^9$)
-hours of operation. This figure is referred to as a FIT\footnote{FIT values are measured as the number of failures per billion hours of operation, (roughly 1.1 Million years). The smaller the FIT number the more reliable the fault~mode}, Failure in time.
+hours of operation (see equation \ref{eqn:resistor}). 
+This figure is referred to as a FIT\footnote{FIT values are measured as the number of 
+failures per Billion (${10}^9$) hours of operation, (roughly 114,000 years). The smaller the 
+FIT number the more reliable the fault~mode} Failure in time.
 
-A thermistor, bead type, non military spec is given a FIT of 3150.
+The formula given for a thermistor in  MIL-HDBK-217F\cite{mil1992}[9.8] is reporoduced in
+equation \ref{thermistorfit}. The variable meanings and values are described in table \ref{tab:thermistor}.
+
+\begin{equation}
+% fixed comp resistor{\lambda}_p = {\lambda}_{b}{\pi}_{R}{\pi}_Q{\pi}_E
+resistor{\lambda}_p = {\lambda}_{b}{\pi}_Q{\pi}_E
+ \label{thermistorfit}
+\end{equation}
+
+\begin{table}[ht]
+\caption{Bead type Thermistor Failure in time assessment} % title of Table
+\centering % used for centering table
+\begin{tabular}{||c|c|l||}
+\hline \hline
+ \em{Parameter}      &  \em{Value}    &   \em{Comments} \\
+                     &                &      \\ \hline \hline
+ ${\lambda}_{b}$ &  0.021        & stress/temp base failure rate bead thermistor \\   \hline
+ %${\pi}_T$ &  4.2         & max temp of $60^o$ C\\ \hline
+ %${\pi}_R$ &  1.0         & Resistance range $< 0.1M\Omega$\\ \hline
+ ${\pi}_Q$ &  15.0         & Non-Mil spec component\\ \hline
+ ${\pi}_E$ &  1.0         & benign ground environment\\ \hline
+
+\hline \hline
+\end{tabular}
+\label{tab:thermistor}
+\end{table}
+
+
+\begin{equation}
+ 0.021 \times 1.0 \times 15.0 \times 1.0 = 0.315 \; {failures}/{{10}^{6} Hours}
+ \label{eqn:thermistor}
+\end{equation}
+
+
+Thus thermistor, bead type, non military spec is given a FIT of 315.0
 
 Using the RIAC finding we can draw up the following table (table \ref{tab:stat_single}),
 showing the FIT values for all faults considered.
@@ -392,8 +485,8 @@ showing the FIT values for all faults considered.
 TC:1 $R_1$ SHORT    &  High Fault        &  -       &  12.42 \\ \hline
 TC:2 $R_1$  OPEN     &  Low Fault         & Low Fault     &  1.38 \\ \hline
  \hline
-TC:3 $R_3$  SHORT    &  Low Fault          & High Fault      &   2835 \\ \hline
-TC:4 $R_3$  OPEN     &  High Fault         & Low  Fault    &   315 \\ \hline
+TC:3 $R_3$  SHORT    &  Low Fault          & High Fault      &   283.5 \\ \hline
+TC:4 $R_3$  OPEN     &  High Fault         & Low  Fault    &   31.5 \\ \hline
 \hline
 TC:5 $R_2$ SHORT     &  -         &  Low Fault   &  12.42 \\  
 TC:6 $R_2$ OPEN     &  High Fault         & High Fault     &  1.38 \\ \hline
@@ -403,13 +496,13 @@ TC:6 $R_2$ OPEN     &  High Fault         & High Fault     &  1.38 \\ \hline
 \end{table}
 
 The FIT for the circuit as a whole is the sum of MTTF values for all the 
-test cases. The PT100 circuit here has a  FIT of 3177.6. This is a MTTF of
+test cases. The PT100 circuit here has a  FIT of 342.6. This is a MTTF of
 about 360 years per circuit.
 
 A Probablistic tree can now be drawn, with a FIT value for the PT100
 circuit and FIT values for all the component fault modes that it was calculated from.
 We can see from this that that the most likely fault is the thermistor going OPEN.
-This circuit is 8 times more likely to fail in this way than in any other.
+This circuit is around 10 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 fault~mode we would scrutinise first.