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\title[Current Mode Interconnect]
{Current Mode Interconnect}

%\subtitle
%{Include Only If Paper Has a Subtitle}

\author[Marshnil Dave, Dinesh Sharma]
{Marshnil Dave, Dinesh Sharma}
\institute[Department Of Electrical Engineering, IIT Bombay]
{
Department Of Electrical Engineering\\
Indian Institute Of Technology, Bombay
}
\date[\today] % (optional, should be abbreviation of conference name)
{\today}
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\begin{document}

\begin{frame}
\frametitle{The Beta Multiplier}
\begin{columns}
\column{4cm}
\includegraphics[width=4cm]{./Images/traditional-beta.eps}
\column{8cm}
We have $I_2 = m I_1$ with $m = \frac{(W/L)_{p2}}{(W/L)_{p1}}$\\
Equating voltages at the gates of n transistors:
\[V_{GS1}  + R I_1 = V_{GS2}\]
\[V_{Tn} + \sqrt{\frac{2 I_1}{K_{n1}}} + R I_1
= V_{Tn} + \sqrt{\frac{2 I_2}{K_{n2}}} \]
\[\mbox{so } \sqrt{\frac{2 I_1}{K_{n1}}} + R I_1
=\sqrt{\frac{2 m I_1}{K_{n2}}} \]
\end{columns}
Therefore, \vspace{1ex}\\
\[ R I_1
= \sqrt{\frac{2 I_1}{K_{n1}}}
  \left(\sqrt{\frac{m K_{n1}}{K_{n2}}} -1 \right)\]
\end{frame}

\begin{frame}
\frametitle{The Beta Multiplier}
\begin{columns}
\column{4cm}
\includegraphics[width=4cm]{./Images/traditional-beta.eps}
\column{8cm}
\[R I_1
= \sqrt{\frac{2 I_1}{K_{n1}}}
  \left(\sqrt{\frac{m K_{n1}}{K_{n2}}} -1 \right)\]
\[\mbox{so }\sqrt{I_1} = \frac{1}{R} \sqrt{\frac{2}{K_{n1}}}
  \left(\sqrt{\frac{m K_{n1}}{K_{n2}}} -1 \right)\]
\[\mbox{or } \sqrt{2 K_{n1} I_1} =
\frac{2}{R} \left(\sqrt{\frac{m K_{n1}}{K_{n2}}} -1 \right)\]
\end{columns}
\[\mbox{But } \sqrt{2 K_{n1} I_1} = g_{mn1},\mbox{\hspace{4em}} m = \frac{(W/L)_{p2}}{(W/L)_{p1}}\]
\[\mbox{Therefore } g_{mn1} = \frac{2}{R}\left(
\sqrt{\frac{(W/L)_{p2}/(W/L)_{p1}}{(W/L)_{n2}/(W/L)_{11}}} -1 \right)
= \mbox{ constant}\]
\end{frame}

\begin{frame}
\frametitle{The Beta Multiplier}
\begin{columns}
\column{4cm}
\includegraphics[width=4cm]{./Images/traditional-beta.eps}
\column{8cm}
\[g_{mn1} = \frac{2}{R}\left(
\sqrt{\frac{(W/L)_{p2}/(W/L)_{p1}}{(W/L)_{n2}/(W/L)_{11}}} -1 \right)\]
Thus $g_{mn1}$ depends only on R and geometry.
\end{columns}
Also, {\em any} nMOS transistor on the same chip
whose current is proportional to $I_1$ or $I_2$
will have a $g_{mn}$ which is
unaffected by temperature, supply variation etc.\vspace{1ex}\\
Thus the gate voltage of the n channel transistors ($V_{biasn}$)
can be used to bias n channel transistors on the chip,
whose \gm \ is to be kept constant.
\end{frame}

\begin{frame}
\frametitle{Zero input impedance circuit}
Low $r_{in}$ amps are used for photo-detectors.
{(\small C.-K. Kim et al,
``High Injection Efficiency Readout Circuit for Low Resistance Infrared Detector'',
IEE Electronic Letters, 35, 1507, 1999).}
\begin{columns}
\column{4cm}
\includegraphics[width=4cm]{./Images/beta-mult.eps}
\column{8cm}
$
\begin{array}{lclcl}
i_1 & = & g_{mn1}v_1 & = & g_{mp1}(v - v_2)\\
i_2 & = & g_{mn2}v_1 & = & - g_{mp2} v_2\\
v_2 & = & -\frac{g_{mn2}}{g_{mp2}} v_1 &
= & -\frac{g_{mn2}}{g_{mp2}} \frac{i_1}{g_{mn1}}\\
\end{array}
$
\[ i_1 = g_{mp1} v + \frac{g_{mn2}/g_{mn1}}{g_{mp2}/g_{mp1}} i_1\]
\end{columns}
\[\mbox{define } \Gamma \equiv \frac{g_{mn2}/g_{mn1}}{g_{mp2}/g_{mp1}}
\mbox{\hspace{3em} then, } i_1 (1 - \Gamma) = g_{mp1} v\]
\[\mbox{This gives  } r_{in} = (1 - \Gamma)/g_{mp1} \]
By making $\Gamma$ = 1, we can make $r_{in} = 0$.
\end{frame}

\begin{frame}
  \frametitle{Robustness of design}
In saturation,
\[ I_d = \frac{1}{2} \mu C_{ox}\frac{W}{L} (V_g - V_T)^2 \]
\[\mbox{So, } g_m = \mu C_{ox}\frac{W}{L} (V_g - V_T) =
\sqrt{2 \mu C_{ox}\frac{W}{L} I_d} \]
\[g_{mn2}/g_{mn1} = \sqrt{\frac{(W/L)_{n2}}{(W/L)_{n1}} \frac{I_2}{I_1}}\]
\[g_{mp2}/g_{mp1} = \sqrt{\frac{(W/L)_{p2}}{(W/L)_{p1}} \frac{I_2}{I_1}}\]
\[\mbox{Therefore } \Gamma \equiv \frac{g_{mn2}/g_{mn1}}{g_{mp2}/g_{mp1}}
= \sqrt{\frac{(W/L)_{n2}/(W/L)_{n1}}{(W/L)_{p2}/(W/L)_{p1}} }\]

Thus $\Gamma$ depends only on geometry and is independent of temperature
and device parameter variation.
\end{frame}
\end{document}

