\frametitle{A tiny bit of electronics: the op-amp}
\centering
\begin{figure}
\epsfig{file=opamp.ps}\\
\putbox{-24mm}{18mm}{$v_p$}%
\putbox{-24mm}{5mm}{$v_n$}%
\putbox{35mm}{11.5mm}{$v_o$}%
\end{figure}
\pause
\[
v_o = G(v_p - v_n)
\]
\end{frame}
\begin{frame}
\frametitle{The two key properties}
\begin{itemize}[<+->]
\item infinite input gain ($G \approx\infty$)
\item zero input current
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Inside the box}
\note<1>{\vspace{10em} In spite of the idealization, the op-amp is quite easy to approximate with only a few transistors. It requires a dual power supply, whose value will determine the actual limit the maximum and minimum output voltage }
\begin{figure}
\epsfig{file=opampSch.ps}\\
\putbox{-36mm}{32mm}{$v_p$}%
\putbox{28mm}{32mm}{$v_n$}%
\putbox{33mm}{41mm}{$v_o$}%
\putbox{-5mm}{66mm}{$+V_{cc}$}%
\putbox{-5mm}{1mm}{$-V_{cc}$}%
\end{figure}
\end{frame}
\begin{frame}
\frametitle{The op-amp in open loop: comparator}
\note<1>{\vspace{10em} Output is gain times the difference. Ideally, since gain is infinite, output should be either $+\infty$ or $-\infty$ according to the sign of the difference. Of course it's maxed out at the power supply level}
\begin{figure}
\epsfig{file=opamp.ps}\\
\putbox{-23mm}{17.5mm}{$x$}%
\putbox{-24mm}{5mm}{$V_T$}%
\putbox{35mm}{11.5mm}{$y$}%
\end{figure}
\pause
\[
y =\begin{cases}
+V_{cc} & \mbox{if $x > V_T$} \\
-V_{cc} & \mbox{if $x < V_T$}
\end{cases}
\]
\end{frame}
\begin{frame}
\frametitle{The op-amp in closed loop: buffer}
\note<1>{\vspace{10em} output is gain times difference. If output was less than $x$, output would increase; similarly, if output was larger than $x$, it would decrease; the only stable point is for $y=x$. We use this circuit because, since there's no input current, we can decouple the input from the output and we can measure voltage values without affecting the source}
\centering
\begin{figure}
\epsfig{file=closedLoop.ps}\\
\putbox{-23mm}{24mm}{$x$}%
\putbox{35mm}{17.5mm}{$y$}%
\end{figure}
\pause
\[
y = x
\]
\end{frame}
\begin{frame}
\frametitle{The op-amp in closed loop: inverting amplifier}
\note<1>{\vspace{10em} like before, output will stabilize once the input difference is zero. For that to happen the voltage at the inverting input should be equal to zero. At that point the current flowing into $R_1$ is $i=x/R_1$; this current cannot flow into the op-amp (rule 2) so it must flow into $R_2$ giving an output voltage of $-R_2i =-(R_2/R_1)x$}
\centering
\begin{figure}
\epsfig{file=iAmp.ps}\\
\putbox{-35mm}{28mm}{$x$}%
\putbox{34mm}{21.5mm}{$y$}%
\putbox{-22mm}{32mm}{$R_1$}%
\putbox{5mm}{43mm}{$R_2$}%
\end{figure}
\pause
\vspace{-1em}
$y =-(R_2/R_1)x$
\end{frame}
\begin{frame}
\frametitle{A/D Converter: Sample \& Hold}
\note<1>{\vspace{10em} The first buffer decouples the input from the rest. The MOSFET is just like a switch, opening in sync with the pulse train generated by a xtal oscillator. The output of the mosfet drives the capacitor that acts as a memory for the value between pulses (the "hold" part). The second buffer isolates the capacitor, so that it doesn't discharge during the hold phase}
\begin{figure}
\centering
\begin{minipage}{11.5cm}
\raggedright
\epsfig{file=sahL.ps}\\
% translate x=1365 y=233 scale 0.28
\putbox{2.07in}{1.48in}{\em T1}%
\putbox{2.87in}{0.99in}{\em C1}%
\putbox{0.05in}{1.09in}{$x(t)$}%
\putbox{0.05in}{0.28in}{$k(t)$}%
\putbox{0.85in}{0.07in}{$F_s$}%
\putbox{4.22in}{1.42in}{$x[n]$}%
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{A/D Converter: 2-Bit Quantizer}
\note<1>{\vspace{10em} The multi-resistor voltage divider generates equally spaced boundary points (the ``$i_k$'' in our previous discussion). Each op-amp is a comparator that goes to $+V_{cc}$ when the input is greater than the relative boundary point. So the output of the battery of comparators gives the interval number the input belongs to as a series of ``1'' (unary notation). The XOR and diodes form a simple logic network for binary encoding. work out an example with $x =0.2V$, output is 10}
\begin{figure}
\centering
\begin{minipage}{11.5cm}
\raggedright
\epsfig{file=adL.ps}\\
% translate x=1024 y=944 scale 0.20
\putbox{0.62in}{2.10in}{\em R}%
\putbox{0.66in}{1.36in}{\em R}%
\putbox{0.66in}{0.80in}{\em R}%
\putbox{0.62in}{0.06in}{\em R}%
\putbox{0.06in}{1.97in}{$+V_0$}%
\putbox{0.93in}{2.32in}{$x[n]$}%
\putbox{2.93in}{0.93in}{LSB}%
\putbox{2.93in}{1.49in}{MSB}%
\putbox{2.14in}{2.23in}{11}%
\putbox{2.14in}{1.66in}{10}%
\putbox{2.19in}{1.10in}{01}%
\putbox{0.32in}{1.62in}{$+0.5V_0$}%
\putbox{0.66in}{1.10in}{$0$}%
\putbox{0.32in}{0.58in}{$-0.5V_0$}%
\putbox{0.10in}{0.23in}{$-V_0$}%
\end{minipage}
\end{figure}
\end{frame}
\section{D/A Conversion}
\begin{frame}
\frametitle{D/A Converter}
\note<1>{\vspace{10em} Each voltage source is switched on or off according to the associated bit $b_k$ in the input $x =0.b_0b_1\ldots b_{N-1}$ ($b_0$ is the MSB). By applying Thevenin's theorem the R-2R resistor ladder is designed to appear as a resistor of value $R$ connected to a voltage source of value $R =\sum_k b_k V_0/2^k$ so that the output voltage is $y =-V_0 x$}