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\cohead[Exercise 6]{Exercise 6}
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\cofoot[Fundamentals of Analog VLSI Design]{Fundamentals of Analog VLSI Design}
\rofoot[29.10.2025]{29.10.2025}
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\title{Fundamentals of Analog VLSI Design\\
Exercise 6 - Problem}
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\makeatother
\subtitle{The Fully Differential G\textsubscript{m}-R Gain Stage}
\author{Christian Enz (christian.enz@epfl.ch)}
\date{29.10.2025}
\begin{document}
\maketitle

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\chapter{Introduction}\label{introduction}

\begin{figure}

\centering{

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

}

\caption{\label{fig-Diff_Pair_with_Rload}Schematic of the differential
pair with resistive load.}

\end{figure}%

Figure~\ref{fig-Diff_Pair_with_Rload} shows a differential pair composed
of two NMOS transistors \(M_1\) and \(M_2\), loaded with resistors
\(R_1\) and \(R_2\), respectively. Since there are two input terminals,
the output current or voltage depends on both the input voltages
\(V_{i1}\) and \(V_{i2}\). It is usually more interesting to express the
output current or voltage in terms of the differential and common mode
input voltages \(V_{id}\) and \(V_{ic}\) defined as \begin{align*}
  V_{id} &\triangleq V_{i1} - V_{i2}\\
  V_{ic} &\triangleq \frac{V_{i1}+V_{i2}}{2}.
\end{align*} The differential mode and common mode operations are
defined for \(V_{ic} = const.\) and for \(V_{id} = 0\), respectively.
The input terminals are set to an appropriate common mode voltage
\(V_{ic}\), to which a differential voltage \(V_{id}\) is superimposed
according to \begin{align*}
  V_{i1} &= V_{ic} + \frac{V_{id}}{2}\\
  V_{i2} &= V_{ic} - \frac{V_{id}}{2}.
\end{align*}

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

\begin{itemize}
\tightlist
\item
  Draw the small-signal equivalent schematic of the circuit assuming the
  transistors are biased in saturation.
\item
  Calculate the small-signal differential voltage gain
  \(A_{vd} \triangleq V_{od}/V_{id}\).
\item
  Calculate the small-signal common-mode input voltage to differential
  output voltage gain \(A_{vc} \triangleq V_{od}/V_{ic}\).
\end{itemize}

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

\begin{itemize}
\tightlist
\item
  Draw the small-signal equivalent schematic of the circuit assuming the
  transistors are biased in saturation and including all the noise
  sources.
\item
  Calculate the output noise power spectral density (PSD) or output
  noise resistance assuming that the transistors and resistors are
  perfectly matched.
\item
  Calculate the input-referred thermal noise PSD and the equivalent
  thermal noise resistance \(R_{nin,th}\).
\item
  Calculate the input-referred flicker noise PSD and the equivalent
  flicker noise resistance \(R_{nin,fl}\).
\item
  Calculate the total output thermal noise power assuming that there is
  an output capacitance \(C\) in parallel with each of the load
  resistance \(R_1\) and \(R_2\) (assume that transistors, resistors and
  capacitors are perfectly matched).
\end{itemize}

\chapter{Offset analysis}\label{offset-analysis}

Mismatch between the two transistors of the differential pair
\(M_1\)-\(M_2\) and of the resistors \(R_1\)-\(R_2\) cause some non-zero
differential output voltage even for a zero differential input voltage
\(V_{id} = 0\).

\begin{itemize}
\tightlist
\item
  Calculate the differential mode output mismatch voltage in terms of
  drain current mismatch \(\Delta I_D\) and resistance mismatch
  \(\Delta R\). Hint: use the above noise analysis where the noise
  currents are replaced by current mismatch.
\item
  Calculate the input-referred offset voltage in terms of resistor
  mismatch \(\Delta R\) and MOS transistor mismatch (\(\beta\)- and
  \(V_{T0}\)-mismatch).
\item
  Determine the variance of the input referred offset voltage. How can
  it be minimized?
\end{itemize}

\chapter{Common-mode input range analysis (CMIR) and differential-mode
output range analysis
(DMOR)}\label{common-mode-input-range-analysis-cmir-and-differential-mode-output-range-analysis-dmor}

\begin{itemize}
\tightlist
\item
  Calculate the minimum and maximum common-mode input voltages
  \(V_{ic,min}\) and \(V_{ic,max}\). For this analysis, \(V_{id}\) is
  set to 0 and the ideal current source is replaced by a transistor
  (\(M_2\)).
\item
  Calculate the minimum and maximum output voltages \(V_{o,min}\) and
  \(V_{o,max}\); deduce the differential output voltage swing
  \(\Delta V_{od,max}\).
\end{itemize}

\chapter{Example}\label{example}

\section{Design}\label{design}

\begin{itemize}
\tightlist
\item
  Design the amplifier, i.e.~size the transistors, determine the values
  of the resistors and the bias current, to meet the specifications
  given in Table~\ref{tbl-specs}. For the design we will use a generic
  180nm bulk CMOS process. The physical parameters are given in
  Table~\ref{tbl-physics_param}, the global process parameters in
  Table~\ref{tbl-process_param} and finally the MOSFET parameters in
  Table~\ref{tbl-mos_param}.
\end{itemize}

\begin{longtable}[]{@{}lccc@{}}

\caption{\label{tbl-specs}Specifications for the differential pair with
resistive loads.}

\tabularnewline

\toprule\noalign{}
Specification & Symbol & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Input common mode voltage & \(V_{{ic}}\) & 0.8 & \(V\) \\
DC gain & \(A_{{dc}}\) & 25 & \(dB\) \\
DC gain & \(A_{{dc}}\) & 18 & - \\
Bandwidth & \(B\) & 1 & \(MHz\) \\
Input-referred thermal noise resistance & \(R_{{nin,th}}\) & 10 &
\(k \Omega\) \\
Maximum input-referred offset voltage & \(V_{{os}}\) & 2 & \(mV\) \\

\end{longtable}

\begin{longtable}[]{@{}ccc@{}}

\caption{\label{tbl-physics_param}Physical parameters}

\tabularnewline

\toprule\noalign{}
Parameter & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(T\) & 300 & \(K\) \\
\(U_T\) & 25.875 & \(mV\) \\

\end{longtable}

\begin{longtable}[]{@{}ccc@{}}

\caption{\label{tbl-process_param}Global process parameters}

\tabularnewline

\toprule\noalign{}
Parameter & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(V_{DD}\) & 1.8 & \(V\) \\
\(C_{ox}\) & 8.443 & \(\frac{fF}{\mu m^2}\) \\
\(W_{min}\) & 200 & \(nm\) \\
\(L_{min}\) & 180 & \(nm\) \\

\end{longtable}

\begin{longtable}[]{@{}cccc@{}}

\caption{\label{tbl-mos_param}Transistor process parameters}

\tabularnewline

\toprule\noalign{}
Parameter & NMOS & PMOS & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
sEKV parameters & & & \\
\(n\) & 1.27 & 1.31 & - \\
\(I_{{spec\Box}}\) & 715 & 173 & \(nA\) \\
\(V_{{T0}}\) & 0.455 & 0.445 & \(V\) \\
\(L_{{sat}}\) & 26 & 36 & \(nm\) \\
\(\lambda\) & 15 & 20 & \(\frac{{V}}{{\mu m}}\) \\
Overlap capacitances parameters & & & \\
\(C_{{GDo}}\) & 0.366 & 0.329 & \(\frac{{fF}}{{\mu m}}\) \\
\(C_{{GSo}}\) & 0.366 & 0.329 & \(\frac{{fF}}{{\mu m}}\) \\
\(C_{{GBo}}\) & 0 & 0 & \(\frac{{fF}}{{\mu m}}\) \\
Junction capacitances parameters & & & \\
\(C_J\) & 1 & 1.121 & \(\frac{{fF}}{{\mu m^2}}\) \\
\(C_{{JSW}}\) & 0.2 & 0.248 & \(\frac{{fF}}{{\mu m}}\) \\
Flicker noise parameters & & & \\
\(K_F\) & 8.1e-24 & 6.8e-23 & \(J\) \\
\(AF\) & 1 & 1 & - \\
\(\rho\) & 0.05794 & 0.4828 & \(\frac{{V \cdot m^2}}{{A \cdot s}}\) \\
Matching parameters & & & \\
\(A_{{VT}}\) & 5 & 5 & \(mV \cdot \mu m\) \\
\(A_{{\beta}}\) & 1 & 1 & \(\% \cdot \mu m\) \\
Source and drain sheet resistance parameter & & & \\
\(R_{{sh}}\) & 600 & 2386 & \(\frac{{\Omega}}{{\mu m}}\) \\
Width and length parameters & & & \\
\(\Delta W\) & 39 & 54 & \(\,nm\) \\
\(\Delta L\) & -76 & -72 & \(\,nm\) \\

\end{longtable}

\section{Simulation}\label{simulation}

\begin{itemize}
\tightlist
\item
  Simulate your design using the available qucs-s schematic.
\end{itemize}

\section{Reducing the supply voltage}\label{reducing-the-supply-voltage}

\begin{itemize}
\tightlist
\item
  What happens if \(V_{DD}=1\,V\)? Can you use the same design as before
  and fulfill all the specifications? Using the same \(V_{ic}\), get the
  new \(\Delta V_{od,max}\) and propose an alternative design relaxing
  one of the specifications.
\end{itemize}




\end{document}
