trunk/c_rcs0/c_rcs0.tex

%$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 $\in \vworkintsetnonneg$.

\begin{equation}
\label{crcs0:srsy1:eq000}
N = \left\lfloor { \frac{N_{MAX} V_P}{V_R} } \right\rfloor
\end{equation}


%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.

\begin{equation}
\label{crcs0:srsy0:eq000}
y_R = \frac{R_1 z_R V_{+}}{R_1 + R_2}
\end{equation}

\begin{equation}
\label{crcs0:srsy0:eq000b}
y_R = \left\lfloor\frac{R_1 z_R V_{+}}{R_1 + R_2}\right\rfloor
\end{equation}

Similarly, (\ref{crcs0:srsy0:eq000c}) describes $y_P$ for analysis.

\begin{equation}
\label{crcs0:srsy0:eq000c}
y_P = \left\lfloor\frac{R_{P1} z_R V_{+}}{R_P}\right\rfloor
\end{equation}


\section{Unplaced Equations}

This section is a holding place for equations until can get my
thoughts together.

\begin{equation}
y_P = \frac{R_{P1}}{R_P} V_{+}
\end{equation}

\begin{equation}
V_{+} = y_P \left( {\frac{R_P}{R_{P1}}} \right)
\end{equation}

\begin{equation}
y_R = \frac{R_1}{R_1 + R_2} V_{+}
\end{equation}

\begin{equation}
V_{+} = \frac{y_R ( R_1 + R_2)}{R_1}
\end{equation}

\begin{equation}
y_P \left( {\frac{R_P}{R_{P1}}} \right) = y_R \left( {\frac{R1 + R2}{R1}} \right)
\end{equation}

\begin{equation}
\frac{R_P}{R_{P1}} = \frac{y_R}{y_P} \left( {\frac{R_1 + R_2}{R_1}} \right)
\end{equation}

\begin{equation}
\frac{R_{P1}}{R_P} = \frac{y_P}{y_R} \left( {\frac{R_1}{R_1 + R_2}} \right)
\end{equation}

\begin{equation}
\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}
\end{equation}


\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}

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% $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 $/uC Software Multi-Volume Book (A)/Chapter, RCS0, Ratiometric Conversion And Measurement Systems
% Edits for rationmetric conversion systems.
%End of file C_RCS0.TEX
1	dashley	6	%$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