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 1 dashley 140 %$Header$ 2 3 \chapter[\crcszeroshorttitle{}]{\crcszerolongtitle{}} 4 5 \label{crcs0} 6 7 8 \section{Introduction} 9 10 \index{ratiometric conversion and calculation} 11 This chapter describes the construction and analysis of ratiometric conversion and 12 measurement systems. By \emph{ratiometric}, we mean that the system requires input 13 from multiple A/D channels to infer the data of interest, typically a potentiometer 14 position. Ratiometric conversion and calculation systems are most often used in 15 small microcontroller work because they can reduce cost by eliminating regulated 16 voltage supplies. Successive sections in the chapter describe the analysis of progressively 17 more complex ratiometric conversion and calculation systems. 18 19 20 \section{Ratiometric Conversion In Hardware Versus Ratiometric Calculation In Software} 21 22 Need to include a differentiation between conversion in hardware and 23 calculation in software. 24 25 26 %Section tag: srsy1 27 % 28 \section{Potentiometer With $V_{+}$ Reference And Hardware Ratiometric Conversion} 29 30 The simplest ratiometric potentiometer system 31 that would be constructed in practice 32 is shown in Fig. \ref{crcs0:srsy1:smplsys0}. 33 In this system, microcontroller software must sense 34 the potentiometer position $R_{P1}/R_P$\footnote{We hope that 35 all of our readers have a background that allows them to 36 analyze resistor networks. For readers without this background, 37 we recommend reading and working through the exercises in an 38 undergraduate circuit analysis text.} even as 39 $V_{+}$ varies within the interval 40 $V_{+} \in [V_{+MIN}, V_{+MAX}]$. Such systems, with 41 additional filtering and current-limiting components, 42 are commonly used in automobiles to allow a microcontroller 43 software load to sense seat or 44 mirror position. 45 \index{seat position} 46 \index{mirror position} 47 \index{battery voltage} 48 Using automobile battery voltage as $V_{+}$ 49 has the advantage that a regulated voltage is not 50 required, thus saving the component cost and circuit board 51 area of a voltage regulator. 52 53 \begin{figure}[!tb] 54 \centering 55 \includegraphics[width=4.6in]{c_rcs0/s_rsy1/smplsys0.eps} 56 \caption{Simple Ratiometric Measurement System With Hardware Ratiometric Conversion} 57 \label{crcs0:srsy1:smplsys0} 58 \end{figure} 59 60 In the circuit of Fig. \ref{crcs0:srsy1:smplsys0}, the microcontroller 61 \index{A/D converter}A/D converter will convert $V_P$ using $V_R$ as a voltage 62 reference according to the relationship in (\ref{crcs0:srsy1:eq000}), where $N_{MAX}$ 63 is the maximum count of the A/D converter. The \index{floor function}$floor(\cdot{})$ 64 function in (\ref{crcs0:srsy1:eq000}) is used to model the effect of 65 \index{quantization}quantization---the 66 A/D count $N$ is required to be $\in \vworkintsetnonneg$. 67 68 \begin{equation} 69 \label{crcs0:srsy1:eq000} 70 N = \left\lfloor { \frac{N_{MAX} V_P}{V_R} } \right\rfloor 71 \end{equation} 72 73 74 75 %Section tag: srsy0 76 % 77 \section{Fixed $r_{1}$, Fixed $r_{2}$ System} 78 The simplest ratiometric system that would be constructed in practice 79 is shown in Fig. \ref{crcs0:srsy0:fr1fr2a}. 80 In Fig. \ref{crcs0:srsy0:fr1fr2a}, 81 assume that the potentiometer is positioned so that 82 $R_{P1}$ is the resistance from the potentiometer wiper 83 to ground, and $R_{P2}$ is the resistance from the potentiometer 84 wiper to $V_{+}$. By definition, $R_{P} = R_{P1} + R_{P2}$. $z_R$ and 85 $z_P$ are the transfer coefficients which relate voltage to A/D counts. 86 These transfer coefficients are an analysis convenience, and correspond to 87 A/D converter characteristics. 88 89 \begin{figure}[!tb] 90 \centering 91 \includegraphics[height=2.5in]{c_rcs0/s_rsy0/smplsys0.eps} 92 \caption{Simple Ratiometric Measurement System With Software Ratiometric Calculation} 93 \label{crcs0:srsy0:fr1fr2a} 94 \end{figure} 95 96 The circuit is designed to allow 97 estimation of $R_{P1}$ (effectively, the potentiometer position) 98 under conditions of varying $V_{+}$. The economy of such a circuit 99 comes from the characteristic that $V_{+}$ need not be regulated, 100 thus allowing less expensive lower-capacity voltage regulators or 101 fewer voltage regulators to be used in an embedded system. 102 In an vehicle, for example, $V_{+}$ may be the battery voltage of 103 the vehicle, which will vary substantially based on which 104 electrical loads are turned on, whether the starter motor is 105 engaged, etc. 106 107 The critical analysis question is, 108 how accurately can $R_{P1}/R_P$ be estimated under conditions 109 of varying $V_{+} \in [V_{+MIN}, V_{+MAX}]$? Or, equivalently, 110 given measured values of $y_R, y_P \in \vworkintsetnonneg$ 111 and given $V_{+} \in [V_{+MIN}, V_{+MAX}]$, 112 what inequality describes the possible values of $R_{P1}/R_P$ 113 (i.e. how much can be inferred or implied from the observation)? 114 115 116 From analysis of the circuit of Fig. \ref{crcs0:srsy0:fr1fr2a}, 117 it can be shown that (\ref{crcs0:srsy0:eq000}) applies. 118 However, because an A/D 119 count is necessarily $\in \vworkintsetnonneg$, (\ref{crcs0:srsy0:eq000b}) must be 120 used for analysis. 121 122 \begin{equation} 123 \label{crcs0:srsy0:eq000} 124 y_R = \frac{R_1 z_R V_{+}}{R_1 + R_2} 125 \end{equation} 126 127 \begin{equation} 128 \label{crcs0:srsy0:eq000b} 129 y_R = \left\lfloor\frac{R_1 z_R V_{+}}{R_1 + R_2}\right\rfloor 130 \end{equation} 131 132 Similarly, (\ref{crcs0:srsy0:eq000c}) describes $y_P$ for analysis. 133 134 \begin{equation} 135 \label{crcs0:srsy0:eq000c} 136 y_P = \left\lfloor\frac{R_{P1} z_R V_{+}}{R_P}\right\rfloor 137 \end{equation} 138 139 140 \section{Unplaced Equations} 141 142 This section is a holding place for equations until can get my 143 thoughts together. 144 145 \begin{equation} 146 y_P = \frac{R_{P1}}{R_P} V_{+} 147 \end{equation} 148 149 \begin{equation} 150 V_{+} = y_P \left( {\frac{R_P}{R_{P1}}} \right) 151 \end{equation} 152 153 \begin{equation} 154 y_R = \frac{R_1}{R_1 + R_2} V_{+} 155 \end{equation} 156 157 \begin{equation} 158 V_{+} = \frac{y_R ( R_1 + R_2)}{R_1} 159 \end{equation} 160 161 \begin{equation} 162 y_P \left( {\frac{R_P}{R_{P1}}} \right) = y_R \left( {\frac{R1 + R2}{R1}} \right) 163 \end{equation} 164 165 \begin{equation} 166 \frac{R_P}{R_{P1}} = \frac{y_R}{y_P} \left( {\frac{R_1 + R_2}{R_1}} \right) 167 \end{equation} 168 169 \begin{equation} 170 \frac{R_{P1}}{R_P} = \frac{y_P}{y_R} \left( {\frac{R_1}{R_1 + R_2}} \right) 171 \end{equation} 172 173 \begin{equation} 174 \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} 175 \end{equation} 176 177 178 dashley 278 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 179 dashley 140 \vfill 180 \begin{figure}[b] 181 \noindent\rule[-0.25in]{\textwidth}{1pt} 182 \begin{tiny} 183 \begin{verbatim} 184 dashley 278 $HeadURL$ 185 $Revision$ 186 $Date$ 187 $Author$ 188 dashley 140 \end{verbatim} 189 \end{tiny} 190 \noindent\rule[0.25in]{\textwidth}{1pt} 191 \end{figure} 192 dashley 278 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 193 dashley 140 % 194 %End of file C_RCS0.TEX

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