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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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