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*Wed Aug 14 23:10:36 2019 UTC*
(4 years, 11 months ago)
by *dashley*

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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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