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1 %$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 $
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 \vfill
179 \begin{figure}[b]
180 \noindent\rule[-0.25in]{\textwidth}{1pt}
181 \begin{tiny}
182 \begin{verbatim}
183 $RCSfile: c_rcs0.tex,v $
184 $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_rcs0/c_rcs0.tex,v $
185 $Revision: 1.2 $
186 $Author: dtashley $
187 $Date: 2001/07/01 19:46:09 $
188 \end{verbatim}
189 \end{tiny}
190 \noindent\rule[0.25in]{\textwidth}{1pt}
191 \end{figure}
192
193 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
194 % $Log: c_rcs0.tex,v $
195 % Revision 1.2 2001/07/01 19:46:09 dtashley
196 % Move out of binary mode for use with CVS.
197 %
198 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
199 % $History: c_rcs0.tex $
200 %
201 % ***************** Version 5 *****************
202 % User: Dashley1 Date: 12/22/00 Time: 12:56a
203 % Updated in $/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems
204 % Tcl automated method of build refined.
205 %
206 % ***************** Version 4 *****************
207 % User: Dashley1 Date: 6/28/00 Time: 12:09p
208 % Updated in $/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems
209 % Substantial edits, forming thoughts.
210 %
211 % ***************** Version 3 *****************
212 % User: David T. Ashley Date: 6/27/00 Time: 11:51p
213 % Updated in $/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems
214 % Initial check-in.
215 %
216 % ***************** Version 2 *****************
217 % User: Dashley1 Date: 6/27/00 Time: 7:36p
218 % Updated in $/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems
219 % Edits for rationmetric conversion systems.
220 %End of file C_RCS0.TEX

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