%$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_rcs0/c_rcs0.tex,v 1.2 2001/07/01 19:46:09 dtashley Exp$ \chapter[\crcszeroshorttitle{}]{\crcszerolongtitle{}} \label{crcs0} \section{Introduction} \index{ratiometric conversion and calculation} This chapter describes the construction and analysis of ratiometric conversion and measurement systems. By \emph{ratiometric}, we mean that the system requires input from multiple A/D channels to infer the data of interest, typically a potentiometer position. Ratiometric conversion and calculation systems are most often used in small microcontroller work because they can reduce cost by eliminating regulated voltage supplies. Successive sections in the chapter describe the analysis of progressively more complex ratiometric conversion and calculation systems. \section{Ratiometric Conversion In Hardware Versus Ratiometric Calculation In Software} Need to include a differentiation between conversion in hardware and calculation in software. %Section tag: srsy1 % \section{Potentiometer With $V_{+}$ Reference And Hardware Ratiometric Conversion} The simplest ratiometric potentiometer system that would be constructed in practice is shown in Fig. \ref{crcs0:srsy1:smplsys0}. In this system, microcontroller software must sense the potentiometer position $R_{P1}/R_P$\footnote{We hope that all of our readers have a background that allows them to analyze resistor networks. For readers without this background, we recommend reading and working through the exercises in an undergraduate circuit analysis text.} even as $V_{+}$ varies within the interval $V_{+} \in [V_{+MIN}, V_{+MAX}]$. Such systems, with additional filtering and current-limiting components, are commonly used in automobiles to allow a microcontroller software load to sense seat or mirror position. \index{seat position} \index{mirror position} \index{battery voltage} Using automobile battery voltage as $V_{+}$ has the advantage that a regulated voltage is not required, thus saving the component cost and circuit board area of a voltage regulator. \begin{figure}[!tb] \centering \includegraphics[width=4.6in]{c_rcs0/s_rsy1/smplsys0.eps} \caption{Simple Ratiometric Measurement System With Hardware Ratiometric Conversion} \label{crcs0:srsy1:smplsys0} \end{figure} In the circuit of Fig. \ref{crcs0:srsy1:smplsys0}, the microcontroller \index{A/D converter}A/D converter will convert $V_P$ using $V_R$ as a voltage reference according to the relationship in (\ref{crcs0:srsy1:eq000}), where $N_{MAX}$ is the maximum count of the A/D converter. The \index{floor function}$floor(\cdot{})$ function in (\ref{crcs0:srsy1:eq000}) is used to model the effect of \index{quantization}quantization---the A/D count $N$ is required to be $\in \vworkintsetnonneg$. $$\label{crcs0:srsy1:eq000} N = \left\lfloor { \frac{N_{MAX} V_P}{V_R} } \right\rfloor$$ %Section tag: srsy0 % \section{Fixed $r_{1}$, Fixed $r_{2}$ System} The simplest ratiometric system that would be constructed in practice is shown in Fig. \ref{crcs0:srsy0:fr1fr2a}. In Fig. \ref{crcs0:srsy0:fr1fr2a}, assume that the potentiometer is positioned so that $R_{P1}$ is the resistance from the potentiometer wiper to ground, and $R_{P2}$ is the resistance from the potentiometer wiper to $V_{+}$. By definition, $R_{P} = R_{P1} + R_{P2}$. $z_R$ and $z_P$ are the transfer coefficients which relate voltage to A/D counts. These transfer coefficients are an analysis convenience, and correspond to A/D converter characteristics. \begin{figure}[!tb] \centering \includegraphics[height=2.5in]{c_rcs0/s_rsy0/smplsys0.eps} \caption{Simple Ratiometric Measurement System With Software Ratiometric Calculation} \label{crcs0:srsy0:fr1fr2a} \end{figure} The circuit is designed to allow estimation of $R_{P1}$ (effectively, the potentiometer position) under conditions of varying $V_{+}$. The economy of such a circuit comes from the characteristic that $V_{+}$ need not be regulated, thus allowing less expensive lower-capacity voltage regulators or fewer voltage regulators to be used in an embedded system. In an vehicle, for example, $V_{+}$ may be the battery voltage of the vehicle, which will vary substantially based on which electrical loads are turned on, whether the starter motor is engaged, etc. The critical analysis question is, how accurately can $R_{P1}/R_P$ be estimated under conditions of varying $V_{+} \in [V_{+MIN}, V_{+MAX}]$? Or, equivalently, given measured values of $y_R, y_P \in \vworkintsetnonneg$ and given $V_{+} \in [V_{+MIN}, V_{+MAX}]$, what inequality describes the possible values of $R_{P1}/R_P$ (i.e. how much can be inferred or implied from the observation)? From analysis of the circuit of Fig. \ref{crcs0:srsy0:fr1fr2a}, it can be shown that (\ref{crcs0:srsy0:eq000}) applies. However, because an A/D count is necessarily $\in \vworkintsetnonneg$, (\ref{crcs0:srsy0:eq000b}) must be used for analysis. $$\label{crcs0:srsy0:eq000} y_R = \frac{R_1 z_R V_{+}}{R_1 + R_2}$$ $$\label{crcs0:srsy0:eq000b} y_R = \left\lfloor\frac{R_1 z_R V_{+}}{R_1 + R_2}\right\rfloor$$ Similarly, (\ref{crcs0:srsy0:eq000c}) describes $y_P$ for analysis. $$\label{crcs0:srsy0:eq000c} y_P = \left\lfloor\frac{R_{P1} z_R V_{+}}{R_P}\right\rfloor$$ \section{Unplaced Equations} This section is a holding place for equations until can get my thoughts together. $$y_P = \frac{R_{P1}}{R_P} V_{+}$$ $$V_{+} = y_P \left( {\frac{R_P}{R_{P1}}} \right)$$ $$y_R = \frac{R_1}{R_1 + R_2} V_{+}$$ $$V_{+} = \frac{y_R ( R_1 + R_2)}{R_1}$$ $$y_P \left( {\frac{R_P}{R_{P1}}} \right) = y_R \left( {\frac{R1 + R2}{R1}} \right)$$ $$\frac{R_P}{R_{P1}} = \frac{y_R}{y_P} \left( {\frac{R_1 + R_2}{R_1}} \right)$$ $$\frac{R_{P1}}{R_P} = \frac{y_P}{y_R} \left( {\frac{R_1}{R_1 + R_2}} \right)$$ $$\frac{R_P V}{R_P V + 1} < \frac{\lfloor R_P V \rfloor}{\lfloor R_R V \rfloor} < \frac{R_P V + 1}{R_R V}$$ \vfill \begin{figure}[b] \noindent\rule[-0.25in]{\textwidth}{1pt} \begin{tiny} \begin{verbatim} $RCSfile: c_rcs0.tex,v$ $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_rcs0/c_rcs0.tex,v$ $Revision: 1.2$ $Author: dtashley$ $Date: 2001/07/01 19:46:09$ \end{verbatim} \end{tiny} \noindent\rule[0.25in]{\textwidth}{1pt} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % $Log: c_rcs0.tex,v$ % Revision 1.2 2001/07/01 19:46:09 dtashley % Move out of binary mode for use with CVS. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % $History: c_rcs0.tex$ % % ***************** Version 5 ***************** % User: Dashley1 Date: 12/22/00 Time: 12:56a % Updated in $/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems % Tcl automated method of build refined. % % ***************** Version 4 ***************** % User: Dashley1 Date: 6/28/00 Time: 12:09p % Updated in$/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems % Substantial edits, forming thoughts. % % ***************** Version 3 ***************** % User: David T. Ashley Date: 6/27/00 Time: 11:51p % Updated in $/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems % Initial check-in. % % ***************** Version 2 ***************** % User: Dashley1 Date: 6/27/00 Time: 7:36p % Updated in$/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems % Edits for rationmetric conversion systems. %End of file C_RCS0.TEX