%%% Appendix for detailed workings out from CH5
\chapter{Detailed FMMD analyses}
\section{Bubba Oscillator FMMD analyses}

For clarity the detailed workings of the FMMD analysis stages in many of the examples
in chapter 5 have been moved here for reference.

\subsection{PHS45 Detailed Analysis}
\label{detail:PHS45}

\begin{table}[h+]
\caption{PhaseShift: Failure Mode Effects Analysis: Single Faults} % title of Table
\label{tbl:firstorderlp}

\begin{tabular}{|| l | c |  l ||} \hline
% \textbf{Failure Scenario} & &  \textbf{First Order}        &   & \textbf{Symptom}          \\ 
 %                          & &  \textbf{Low Pass Filter}    &   &                           \\
       \textbf{Failure} &  \textbf{$PHS45$ }         & \textbf{Derived Component}          \\ 
 \textbf{cause}   &   \textbf{Effect}                & \textbf{Failure Mode}          \\
                      
               \hline
     FS1: R  SHORT    &   0 degree's of phase shift      &    $0\_phaseshift$  \\        
     %                      90 degree's of phase shift   &  &    $90\_phaseshift$         
     FS2: R  OPEN     &    No Signal                     &    $nosignal$           \\ \hline
     FS3: C  SHORT    &    Grounded,No Signal            &    $nosignal$           \\ 
     FS4: C  OPEN     &    0 degree's of phase shift     &    $0\_phaseshift$           \\ \hline

\hline
\end{tabular}
\end{table}  
% PHS45 



\subsection{Bubba Oscillator: One Large Functional Group: Detailed Analysis}
\label{detail:BUBOSC1}



\begin{table}[h+]
\caption{Bubba Oscillator: Failure Mode Effects Analysis: One Large Functional Group} % title of Table
\label{tbl:bubbalargefg}

\begin{tabular}{|| l | l | c | c | l ||} \hline
% \textbf{Failure Scenario} & &  \textbf{Bubba}        &   & \textbf{Symptom}          \\ 
%                           & &  \textbf{Oscillator}    &   &                           \\
                           
 \textbf{Failure} & & \textbf{$BubbaOscillator$ }      &   & \textbf{Derived Component}          \\ 
 \textbf{cause}   & &  \textbf{Effect}             &   & \textbf{Failure Mode}          \\
 
               \hline


     FS1: $PHS45_1$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
     FS2: $PHS45_1$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  \hline 
    % FS3: $PHS45_1$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 

     FS3: $NIBUFF_1$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
     FS4: $NIBUFF_1$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
     FS5: $NIBUFF_1$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
     FS6: $NIBUFF_1$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 

     FS7: $PHS45_2$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
     FS8: $PHS45_2$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
     %FS10: $PHS45_2$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 


     FS9: $NIBUFF_2$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
     FS10: $NIBUFF_2$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
     FS11: $NIBUFF_2$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
     FS12: $NIBUFF_2$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 

    FS13: $PHS45_3$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
     FS14: $PHS45_3$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  \hline 
   %  FS17: $PHS45_3$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 

     FS15: $NIBUFF_3$  $L_{up}$      &  &  output high No Oscillation &  &    $NO_{osc}$           \\  
     FS16: $NIBUFF_3$  $L_{dn}$      &  &  output low No Oscillation  &  &     $NO_{osc}$              \\       
     FS17: $NIBUFF_3$  $N_{oop}$     &  &  output low No Oscillation  &  &     $NO_{osc}$                \\  
     FS18: $NIBUFF_3$  $L_{slew}$    &  &  signal lost                &  &   $NO_{osc}$          \\  \hline 

    FS19: $PHS45_4$  $0\_phaseshift$        &  &  osc frequency high               &  &   $HI_{fosc}$          \\  
     FS20: $PHS45_4$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  \hline 
  %   FS24: $PHS45_4$  $90\_phaseshift$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 

     FS21: $INVAMP$  $OUTOFRANGE$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
    FS22: $INVAMP$  $ZEROOUTPUT$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
    FS23: $INVAMP$  $NOGAIN$             &  &  signal lost                      &  &   $NO_{osc}$          \\  
    FS24: $INVAMP$  $LOWPASS$             &  &  signal lost                      &  &   $NO_{osc}$          \\  \hline 
  

 % FS1: $CAP_{10nF}$  $OPEN$          &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
 % FS1: $CAP_{10nF}$  $SHORT$         &  &  osc frequency low                &  &   $LO_{fosc}$          \\  \hline 
\hline

\end{tabular}
\end{table}  
Collecting symptoms from table~\ref{tbl:bubbalargefg} we can show that for single failure modes, applying $fm$ to the bubba oscillator 
returns three failure modes,
%
$$ fm(BubbaOscillator) = \{ NO_{osc}, HI_{fosc}\} . $$ %, LO_{fosc} \} . $$





\subsection{BUFF45: Detailed Analysis}
\label{detail:BUFF45}



\begin{table}[h+]
\caption{BUFF45: Failure Mode Effects Analysis} % title of Table
\label{tbl:buff45}

\begin{tabular}{|| l | l | c | c | l ||} \hline
 %\textbf{Failure Scenario} & &  \textbf{BUFF45}         &   & \textbf{Symptom}          \\ 
 %                          & &                          &   &                           \\
\textbf{Failure} & & \textbf{$BUFF45$ }      &   & \textbf{Derived Component}          \\ 
 \textbf{cause}   & &  \textbf{Effect}             &   & \textbf{Failure Mode}          \\
 
               \hline
     FS1: $PHS45_1$  $0\_phaseshift$        &  &  phase shift low              &  &   $0\_phaseshift$       \\  
     FS2: $PHS45_1$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  \hline 
     %FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $90\_phaseshift$          \\  \hline 

     FS3: $NIBUFF_1$  $L_{up}$      &  &  output high   &  &    $NO_{signal}$           \\  
     FS4: $NIBUFF_1$  $L_{dn}$      &  &  output low   &  &     $NO_{signal}$              \\       
     FS5: $NIBUFF_1$  $N_{oop}$     &  &  output low   &  &     $NO_{signal}$                \\  
     FS6: $NIBUFF_1$  $L_{slew}$    &  &  signal lost                &  &   $NO_{signal}$          \\  \hline 


\hline

\end{tabular}
\end{table} 


collecting symptoms from table~\ref{tbl:buff45}, we can create a derived component $BUFF45$ which has the following failure modes:
$$
fm (BUFF45) = \{  0\_phaseshift, NO\_signal .\} % 90\_phaseshift,
$$
%




\subsection{PHS135BUFFERED: Failure Mode Effects Analysis} % title of Table
\label{detail:PHS135BUFFERED}


\begin{table}[h+]
\caption{PHS135BUFFERED: Failure Mode Effects Analysis} % title of Table
\label{tbl:phs135buffered}

\begin{tabular}{|| l | l | c | c | l ||} \hline
 %\textbf{Failure Scenario} & &  \textbf{PHS135 Buffered}         &   & \textbf{Symptom}          \\ 
  %                         & &                          &   &                           \\
 \textbf{Failure} & & \textbf{$PHS135BUFFERED$ }      &   & \textbf{Derived Component}          \\ 
 \textbf{cause}   & &  \textbf{Effect}             &   & \textbf{Failure Mode}          \\
                      
                           
               \hline
     FS1: $PHS45_1$  $0\_phaseshift$        &  &  phase shift low              &  &   $90\_phaseshift$       \\  
     FS2: $PHS45_1$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  \hline 
    %FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 

    FS3: $PHS45_2$  $0\_phaseshift$        &  &  phase shift low              &  &   $90\_phaseshift$       \\  
     FS4: $PHS45_2$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  \hline 
    % FS6: $PHS45_2$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 

        FS5: $PHS45_3$  $0\_phaseshift$        &  &  phase shift low              &  &   $90\_phaseshift$       \\  
     FS6: $PHS45_3$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  \hline 
    % FS9: $PHS45_3$  $90\_phaseshift$         &  &  phase shift high                &  &    $180\_phaseshift$          \\  \hline 



\hline

\end{tabular}
\end{table} 
%
%
Collecting symptoms from table~\ref{tbl:phs135buffered}, we can create a derived component $PHS135BUFFERED$ which has the following failure modes:
$$
fm (PHS135BUFFERED) = \{ 90\_phaseshift,  NO\_signal .\} % 180\_phaseshift,
$$
%

\subsection{PHS225AMP: Failure Mode Effects Analysis} % title of Table
\label{detail:PHS225AMP}
\begin{table}[h+]
\caption{PHS225AMP: Failure Mode Effects Analysis} % title of Table
\label{tbl:phs225amp}

\begin{tabular}{|| l | l | c | c | l ||} \hline
 %\textbf{Failure Scenario} & &  \textbf{PHS225AMP}         &   & \textbf{Symptom}          \\ 
 %                          & &  \textbf{Oscillator}    &   &                           \\
 \textbf{Failure} & & \textbf{$PHS225AMP$ }      &   & \textbf{Derived Component}          \\ 
 \textbf{cause}   & &  \textbf{Effect}             &   & \textbf{Failure Mode}          \\
                               
               \hline
     FS1: $PHS45_1$  $0\_phaseshift$        &  &  phase shift low               &  &   $180\_phaseshift$       \\  
     FS2: $PHS45_1$  $no\_signal$            &  &  signal lost                      &  &   $NO_{signal}$          \\  \hline
    % FS3: $PHS45_1$  $90\_phaseshift$         &  &  phase shift high                &  &    $270\_phaseshift$          \\  \hline 

     FS3: $INVAMP$  $L_{up}$      &  &  output high   &  &    $NO_{signal}$           \\  
     FS4: $INVAMP$  $L_{dn}$      &  &  output low   &  &     $NO_{signal}$              \\       
     FS5: $INVAMP$  $N_{oop}$     &  &  output low   &  &     $NO_{signal}$                \\  
     FS6: $INVAMP$  $L_{slew}$    &  &  signal lost                &  &   $NO_{signal}$          \\  \hline 

\hline

\end{tabular}
\end{table} 
%
Applying FMMD we create a derived component $PHS225AMP$ which has the following failure modes:
$$
fm (PHS225AMP) = \{ 180\_phaseshift,  NO\_signal .\} % 270\_phaseshift,
$$



\subsection{BUBBAOSC: Failure Mode Effects Analysis} % title of Table
\label{detail:BUBBAOSC}


\begin{table}[h+]
\caption{BUBBAOSC: Failure Mode Effects Analysis} % title of Table
\label{tbl:bubba2}

\begin{tabular}{|| l | l | c | c | l ||} \hline
 %\textbf{Failure Scenario} & &  \textbf{BUBBAOSC}         &   & \textbf{Symptom}          \\ 
 %                          & &                          &   &                             \\
                           
\textbf{Failure} & & \textbf{$BUBBAOSC$ }      &   & \textbf{Derived Component}          \\ 
 \textbf{cause}   & &  \textbf{Effect}             &   & \textbf{Failure Mode}          \\
    
               \hline
     %FS1: $PHS135BUFFERED$  $180\_phaseshift$        &  &  phase shift high              &  &     $LO_{fosc}$                     \\  
     FS1: $PHS135BUFFERED$  $no\_signal$            &  &  signal lost                      &  &   $NO_{osc}$          \\  
     FS2: $PHS135BUFFERED$  $90\_phaseshift$         &  &  phase shift low                &  &    $HI_{osc}$       \\  \hline 

    % FS4: $PHS225AMP$  $270\_phaseshift$      &  &  phase shift high   &  &     $LO_{fosc}$                               \\  
     FS4: $PHS225AMP$  $180\_phaseshift$      &  &  phase shift low   &  &     $HI_{osc}$                               \\       
     FS5: $PHS225AMP$  $NO\_signal$     &  &  lost signal             &  &     $NO_{signal}$                              \\  \hline 
   

\hline

\end{tabular}
\end{table} 
%
Collecting symptoms from table~\ref{tbl:bubba2}, we can create a derived component $BUBBAOSC$ which has the following failure modes:
$$
fm (BUBBAOSC) = \{  HI_{osc}, NO\_signal .\} %  LO_{fosc},
$$
\clearpage
\section{Sigma Delta Detailed FMMD Analyses}



