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\cohead[Exercise 7]{Exercise 7}
\lofoot[$\copyright$ C. Enz]{$\copyright$ C. Enz}
\cofoot[Fundamentals of Analog VLSI Design]{Fundamentals of Analog VLSI Design}
\rofoot[05.11.2025]{05.11.2025}
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  pdftitle={Fundamentals of Analog VLSI Design Exercise 7 - Problem},
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\title{Fundamentals of Analog VLSI Design\\
Exercise 7 - Problem}
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\subtitle{Gain and Bandwidth Enhancement}
\author{Christian Enz (christian.enz@epfl.ch)}
\date{05.11.2025}
\begin{document}
\maketitle

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\chapter{Problem 1: Gain Enhancement}\label{problem-1-gain-enhancement}

\begin{figure}

\centering{

\pandocbounded{\includegraphics[keepaspectratio]{Figures/Gain_enhancement_schematic_with_CL.png}}

}

\caption{\label{fig-Gain_enhancement_schematic}Schematic of gain
enhancement amplifier.}

\end{figure}%

The amplifier shown in Figure~\ref{fig-Gain_enhancement_schematic}
allows to enhance the DC gain by reducing the load conductance seen by
the differential pair M\textsubscript{1a}-M\textsubscript{1b} {[}1{]}
{[}2{]}.

\section{Small-signal analysis}\label{small-signal-analysis}

\begin{figure}

\centering{

\pandocbounded{\includegraphics[keepaspectratio]{Figures/negative_gm.png}}

}

\caption{\label{fig-negative_gm}Negative conductance.}

\end{figure}%

The cross-coupled pair M\textsubscript{3a}-M\textsubscript{3b} which is
redrawn in Figure~\ref{fig-negative_gm}, allow to implement a negative
conductance. When combined with transistors
M\textsubscript{2a}-M\textsubscript{2b}, it allows to reduce the
conductance seen by the differential pair
M\textsubscript{1a}-M\textsubscript{1b}.

\begin{itemize}
\tightlist
\item
  Draw the small-signal schematic of the circuit shown in
  Figure~\ref{fig-negative_gm}.
\item
  Derive the small-signal differential conductance
  \(G = \Delta I_{in}/\Delta V_{in}\), where
  \(\Delta I_{in} \triangleq \Delta I_1-\Delta I_2\) assuming a perfect
  matching between M\textsubscript{3a} and M\textsubscript{3b}.
\item
  Draw the small-signal circuit of the amplifier shown in
  Figure~\ref{fig-Gain_enhancement_schematic} including the output
  conductances in differential mode assuming a perfect matching between
  all the transistors in one current branch and the other. Hint:
  simplify the circuit by using the fact that the voltage at the common
  source node of M\textsubscript{1a}-M\textsubscript{1b} does not change
  in differential mode with perfect matching.
\item
  From the above analysis, derive the small-signal differential transfer
  function \begin{equation}
  A_d(s) \triangleq \frac{V_{od}(s)}{V_{id}(s)},
  \end{equation} where \(V_{od} \triangleq V_{o1}-V_{o2}\) and
  \(V_{id} \triangleq V_{i1}-V_{i2}\) are the output and input
  differential voltages, respectively.
\item
  Deduce the expression of the DC gain and the dominant pole.
\item
  What is the condition on \(G_{m3}\) for the pole to remain in the left
  half-plane?
\item
  What is the gain-enhancement factor \(K\) compared to the voltage gain
  without the cross-coupled pair
  M\textsubscript{3a}-M\textsubscript{3b}?
\item
  What is the value of the DC gain for \(G_{m3}=G_{m2}\)? How does it
  compare to a common-stage stage?
\item
  Find the gain-bandwidth product \(\omega_u\). Is the gain-bandwidth
  product also enhanced like the DC gain?
\end{itemize}

\section{Noise analysis}\label{noise-analysis}

\begin{itemize}
\tightlist
\item
  Draw the small-signal schematic of
  Figure~\ref{fig-Gain_enhancement_schematic} including all the noise
  sources but with \(\Delta V_{id}=0\). Hint: simplify the circuit by
  using the fact that the voltage at the common source node of
  M\textsubscript{1a}-M\textsubscript{1b} does not change in
  differential mode with perfect matching.
\item
  Derive the output noise resistance \(R_{nout}\).
\item
  Deduce the input-referred noise resistance \(R_{nin}\).
\item
  Calculate the input-referred thermal noise resistance \(R_{nt}\). How
  does it compare to the noise of the same amplifier without
  M\textsubscript{3a}-M\textsubscript{3b}?
\item
  Calculate the input-referred flicker noise resistance \(R_{nf}\). How
  does it compare to the noise of the same amplifier without
  M\textsubscript{3a}-M\textsubscript{3b}?
\end{itemize}

\chapter{Problem 2: Bandwidth
enhancement}\label{problem-2-bandwidth-enhancement}

\begin{figure}

\centering{

\pandocbounded{\includegraphics[keepaspectratio]{Figures/Gm_enhanced_ota_schematic.png}}

}

\caption{\label{fig-Gm_enhanced_ota_schematic}Enhanced \(G_m\)
differential OTA}

\end{figure}%

The differential OTA shown in Figure~\ref{fig-Gm_enhanced_ota_schematic}
shows a larger gain-bandwidth product compared to the case without the
cross-coupled transistors M\textsubscript{2a}-M\textsubscript{2b}
{[}1{]} {[}3{]}.

\section{Small-signal analysis}\label{small-signal-analysis-1}

\begin{figure}

\centering{

\pandocbounded{\includegraphics[keepaspectratio]{Figures/Gm_enhancement_schematic.png}}

}

\caption{\label{fig-Gm_enhancement_schematic}Enhanced \(G_m\)
differential transconductor}

\end{figure}%

The OTA of Figure~\ref{fig-Gm_enhanced_ota_schematic} uses the
differential transconductor shown in
Figure~\ref{fig-Gm_enhancement_schematic} which allows to increase the
equivalent transconductance compared to the case of a simple
differential pair.

\begin{itemize}
\tightlist
\item
  Draw the small-signal schematic of
  Figure~\ref{fig-Gm_enhancement_schematic}. Hint: simplify the circuit
  by using the fact that the voltage at the common source node of
  M\textsubscript{2a}-M\textsubscript{2b} does not change in
  differential mode with perfect matching.
\item
  Derive the small-signal equivalent transconductance \begin{equation}
  G_{meq} \triangleq \frac{\Delta I_{od}}{\Delta V_{id}},
  \end{equation} where
  \(\Delta I_{od} \triangleq \Delta I_1-\Delta I_2\) is the small-signal
  differential output current and
  \(\Delta V_{id} \triangleq \Delta V_{i1}-\Delta V_{i2}\) is the
  small-signal differential input voltage. How does it compare to the
  transconductance of a simple differential pair?
\item
  Derive the small-signal equivalent transadmittance \begin{equation}
  Y_{meq}(s) \triangleq \frac{\Delta I_{od}(s)}{\Delta V_{id}(s)},
  \end{equation} accounting for the the parasitic capacitance \(C_p\) at
  the source node of M\textsubscript{1a}-M\textsubscript{1b}. Hint: add
  a parasitic capacitance \(C_p\) connected between the source node of
  M\textsubscript{1a} and ground to the small-signal half-circuit used
  above.
\item
  Deduce the expression of the dominant pole \(\omega_p\).
\item
  What is the condition on \(G_{m2}\) for the pole \(\omega_p\) to
  remain in the left half-plane? Hint: express the condition in terms of
  the transconductance ratio \(\alpha \triangleq G_{ms1}/G_{m2}\).
\item
  Express the \(G_m\)-enhancement factor \(K\) compared to the
  transconductance of the differential pair
  M\textsubscript{1a}-M\textsubscript{1b} without the cross-coupled
  transistors M\textsubscript{2a}-M\textsubscript{2b}. Hint: express
  \(K\) in terms of the transconductance ratio \(\alpha\).
\item
  What is the value of \(K\) when
  M\textsubscript{1a}-M\textsubscript{1b} and
  M\textsubscript{2a}-M\textsubscript{2b} are both biased in weak
  inversion? Hint: M\textsubscript{1a}-M\textsubscript{1b} and
  M\textsubscript{2a}-M\textsubscript{2b} share the same bias current
  \(I_b\).
\item
  Derive the small-signal differential transfer function
  \begin{equation}
  A_d \triangleq \frac{V_{od}}{V_{id}}.
  \end{equation}
\item
  What is the value of the DC gain for \(G_{ms1}=G_{m2}\)? How does it
  compare to a common-stage stage?
\item
  Find the gain-bandwidth product \(\omega_u\). Is the gain-bandwidth
  product also enhanced like the DC gain?
\end{itemize}

\section{Noise analysis}\label{noise-analysis-1}

\begin{itemize}
\tightlist
\item
  Draw the small-signal schematic of
  Figure~\ref{fig-Gm_enhancement_schematic} including all the noise
  sources and with \(V_{id}=0\) and neglecting all output conductances.
  Hint: simplify the circuit by using the fact that the voltage at the
  common source node of M\textsubscript{2a}-M\textsubscript{2b} does not
  change in differential mode with perfect matching.
\item
  Derive the output noise conductance \(G_{nout}\).
\item
  Deduce the input-referred noise resistance \(R_{nin}\).
\item
  Calculate the input-referred thermal noise resistance \(R_{nt}\). How
  does it compare to the noise of a simple differential pair?
\item
  Calculate the input-referred flicker noise resistance \(R_{nf}\). How
  does it compare to the noise of a simple differential pair?
\end{itemize}

\chapter*{References}\label{references}
\addcontentsline{toc}{chapter}{References}

\phantomsection\label{refs}
\begin{CSLReferences}{0}{0}
\bibitem[\citeproctext]{ref-bib:sansen:aacd:2016}
\CSLLeftMargin{{[}1{]} }%
\CSLRightInline{W. Sansen, {``Opamps, gm-blocks or inverters?''} in
\emph{Efficient sensor interfaces, advanced amplifiers and low power RF
systems: Advances in analog circuit design 2015}, Springer, 2016, pp.
125--138.}

\bibitem[\citeproctext]{ref-bib:ohri:jssc:14:1:1979}
\CSLLeftMargin{{[}2{]} }%
\CSLRightInline{K. B. Ohri and M. J. Callahan, {``Integrated PCM
codec,''} \emph{IEEE Journal of Solid-State Circuits}, vol. 14, no. 1,
pp. 38--46, 1979, doi:
\href{https://doi.org/10.1109/JSSC.1979.1051139}{10.1109/JSSC.1979.1051139}.}

\bibitem[\citeproctext]{ref-bib:castello:jssc:25:3:1990}
\CSLLeftMargin{{[}3{]} }%
\CSLRightInline{R. Castello, A. G. Grassi, and S. Donati, {``A 500-nA
sixth-order bandpass SC filter,''} \emph{IEEE Journal of Solid-State
Circuits}, vol. 25, no. 3, pp. 669--676, 1990, doi:
\href{https://doi.org/10.1109/4.102659}{10.1109/4.102659}.}

\end{CSLReferences}




\end{document}
