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\cohead[Exercise 3]{Exercise 3}
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\cofoot[Fundamentals of Analog VLSI Design]{Fundamentals of Analog VLSI Design}
\rofoot[01.10.2025]{01.10.2025}
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\title{Fundamentals of Analog VLSI Design\\
Exercise 3 - Problem}
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\subtitle{Design and Optimization of the Common-source Gain Stage}
\author{Christian Enz (christian.enz@epfl.ch)}
\date{01.10.2025}
\begin{document}
\maketitle

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\chapter*{Introduction}\label{introduction}
\addcontentsline{toc}{chapter}{Introduction}

\begin{figure}

\centering{

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

}

\caption{\label{fig-CS_schematic}Common-source gain stage.}

\end{figure}%

In this exercise we want to apply the \(G_m/I_D\) design methodology
using the inversion coefficient \(IC\) to the simple common source
amplifier shown in Figure~\ref{fig-CS_schematic} for various
specifications. Most of these specifications are related to the
transistor transconductance \(G_m\). The exercise will show how to use
the EKV functions related to the transconductance, in particular the
normalized source transconductance which, for a long-channel transistor,
only depends on the inversion coefficient according to
\begin{equation}\label{eqn:gms}
  gms(IC) \triangleq \frac{G_{ms}}{G_{spec}} = \frac{G_m\,n U_T}{I_{spec}} = \frac{\sqrt{4 IC+1}-1}{2},
\end{equation} where \(G_{spec} = I_{spec}/U_T\) is the specific
conductance and \(I_{spec} = I_D/IC\) is the specific current also given
by \(I_{spec} = I_{spec\Box}\,W/L\).

In the case the drain current and inversion coefficient are imposed, you
can get the transconductance from the normalized \(G_m/I_D\) function
\begin{equation}
  gmid(IC) = \frac{G_m\,n U_T}{I_D} = \frac{G_{ms}\,U_T}{I_D} = \frac{gms(IC)}{IC} = \frac{\sqrt{4 IC+1}-1}{2 IC}
\end{equation} as \begin{equation}
  G_m = gmid(IC) \cdot \frac{I_D}{n U_T}
\end{equation}

In the case the normalized transconductance is imposed, you can get the
inversion coefficient by inverting \eqref{eqn:gms} resulting in
\begin{equation}
  IC = g_{ms} \cdot (g_{ms}+1).
\end{equation}

In the case the transconductance and the current are given you can get
the inversion coefficient from \begin{equation}
  IC = \frac{1-gmsid}{gmsid^2}
\end{equation} where \begin{equation}
  gmsid = \frac{G_m\,n U_T}{I_D} = \frac{G_{ms}\,U_T}{I_D}
\end{equation}

For short-channel transistors we need to account for velocity saturation
(VS). The normalized source transconductance and normalized \(G_m/I_D\)
functions then become \begin{align}
  gms(IC,\lambda_c) &\triangleq \frac{G_{ms}}{G_{spec}} = \frac{G_m\,n U_T}{I_{spec}} = \frac{\sqrt{4 IC+1+(\lambda_c\,IC)^2}-1}{2+\lambda_c^2\,IC},\\
  gmid(IC,\lambda_c) &= \frac{G_m\,n U_T}{I_D} = \frac{G_{ms}\,U_T}{I_D} = \frac{gms(IC)}{IC} = \frac{\sqrt{4 IC+1+(\lambda_c\,IC)^2}-1}{IC\;(2+\lambda_c^2\,IC)},
\end{align} where \(\lambda_c = L_{sat}/L_{eff}\) is the VS parameter.

All the above EKV functions, and actually many more, are coded in python
in the ekv\_functions.py file.

For the design we assume 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}.

\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 & \(\frac{fF}{\mu m^2}\) \\
\(H_{dif}\) & 200 & \(nm\) \\
\(W_{min}\) & 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}

Note that the effective channel width and length are defined as follows
\begin{align}
  W_{eff} &\triangleq W + \Delta W,\\
  L_{eff} & \triangleq L + \Delta L,
\end{align} where \(W\) and \(L\) are drawn width and length.

\chapter{Problem 1: Imposing the gain bandwidth product (unity gain
frequency)}\label{problem-1-imposing-the-gain-bandwidth-product-unity-gain-frequency}

Design the CS amplifier shown in Figure~\ref{fig-CS_schematic}, for the
specifications given in Table~\ref{tbl-specs_problem1} at room
temperature.

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

\caption{\label{tbl-specs_problem1}Specifications for problem 1.}

\tabularnewline

\toprule\noalign{}
Specification & Symbol & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Gain-bandwidth product & \(GBW\) & 10 & \(MHz\) \\
Inversion coefficient & \(IC\) & 1 & - \\
Load capacitance & \(C_L\) & 1 & \(pF\) \\

\end{longtable}

Carry out the design procedure both for long- and short-channel devices
in saturation, using the drawn lengths \(L_{long} = 1\,\mu m\) and
\(L_{short} = 0.18\,\mu m\), respectively. Note that velocity saturation
(VS) has to be accounted for in the short-channel case. We assume that
the self-loading capacitance at the drain can be neglected (i.e.~assume
a constant load capacitance equal to \(C_L\)).

Carry out the design procedure following these steps:

\begin{itemize}
\tightlist
\item
  Draw the equivalent small-signal schematic ignoring the transistor
  parasitic capacitances.
\item
  Derive the transfer function
  \(A_v(s) \triangleq \Delta V_{out}/\Delta V_{in}\) and extract the
  expressions of the dc gain, cut-off frequency and gain-bandwidth
  product (unity gain frequency) \(f_u\).
\item
  Find the transistor transconductance value \(G_m\).
\item
  Find the normalized source transconductance
  \(g_{ms}(IC) \triangleq G_{ms}/G_{spec} = n\,G_m/G_{spec} = n U_T\,G_m/I_{spec}\)
  from the imposed inversion coefficient \(IC\). Then, deduce the
  specific current \(I_{spec}\) and the bias current \(I_b\).
\item
  Find the \(W_{eff}/L_{eff}\) ratio and the effective width \(W_{eff}\)
  for both the long and short channel case using the given drawn length
  \(L\). Derive the drawn width \(W\).
\item
  Deduce the output conductance value \(G_{ds}\) and then calculate the
  dc gain value assuming the bias current source is ideal.
\item
  Simulate the circuit using the ngspice simulator and compare the
  theoretical results with the simulations.
\end{itemize}

\chapter{Problem 2: Imposing the power
consumption}\label{problem-2-imposing-the-power-consumption}

Design the CS amplifier shown in Figure~\ref{fig-CS_schematic}, for the
specifications given in Table~\ref{tbl-specs_problem2} at room
temperature.

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

\caption{\label{tbl-specs_problem2}Specifications for problem 2.}

\tabularnewline

\toprule\noalign{}
Specification & Symbol & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Power consumption & \(P\) & 20 & \(\mu W\) \\
Inversion coefficient & \(IC\) & 1 & - \\
Load capacitance & \(C_L\) & 1 & \(pF\) \\

\end{longtable}

Carry out the design procedure both for long- and short-channel devices
in saturation, using the drawn lengths \(L_{long} = 1\,\mu m\) and
\(L_{short} = 0.18\,\mu m\), respectively. Note that velocity saturation
(VS) has to be accounted for in the short-channel case. We assume that
the self-loading capacitance at the drain can be neglected (i.e.~assume
a constant load capacitance \(C_L\)).

Carry out the design procedure following these steps:

\begin{itemize}
\tightlist
\item
  Find the bias current \(I_b\) for the specified maximum power
  consumption.
\item
  Find the transconductance value of the transistor for the given
  inversion coefficient \(IC\) by means of the current efficiency.
\item
  Find the gain-bandwidth product (unity gain frequency) using the
  specific current \(I_{spec}\).
\item
  Find the \(W_{eff}/L_{eff}\) ratio and the effective width \(W_{eff}\)
  for both the long and short channel case using the given drawn length
  \(L\). Derive the drawn width \(W\).
\item
  Deduce the output conductance value \(G_{ds}\) and then calculate the
  dc gain value assuming the bias current source is ideal.
\item
  Simulate the circuit using the ngspice simulator and compare the
  theoretical results with the simulations.
\end{itemize}

\chapter{Problem 3: Imposing the gain bandwidth product and the current
budget}\label{problem-3-imposing-the-gain-bandwidth-product-and-the-current-budget}

Design the same CS amplifier shown in Figure~\ref{fig-CS_schematic}
imposing both the gain-bandwidth product and the current budget with the
specifications given in Table~\ref{tbl-specs_problem3} at room
temperature.

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

\caption{\label{tbl-specs_problem3}Specifications for problem 3.}

\tabularnewline

\toprule\noalign{}
Specification & Symbol & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Gain-bandwidth product & \(GBW\) & 1 & \(MHz\) \\
Bias current & \(I_b\) & 400 & \(nA\) \\
Load capacitance & \(C_L\) & 1 & \(pF\) \\
Transistor length & \(L\) & 1 & \(\mu m\) \\

\end{longtable}

Carry out the design procedure only for the long-channel case with
\(L=L_{long} = 1\,\mu m\) following these steps:

\begin{itemize}
\tightlist
\item
  Find the transistor transconductance value.
\item
  Find the inversion coefficient \(IC\) by means of the current
  efficiency and then the specific current \(I_{spec}\).
\item
  Find the \(W_{eff}/L_{eff}\) ratio and the effective width \(W_{eff}\)
  for the long channel case only using the given drawn length \(L\).
  Derive the drawn width \(W\).
\item
  Deduce the output conductance value \(G_{ds}\) and then calculate the
  dc gain value assuming the bias current source is ideal.
\item
  Simulate the circuit using the ngspice simulator and compare the
  theoretical results with the simulations.
\end{itemize}

\chapter{Problem 4: Imposing the input-referred
noise}\label{problem-4-imposing-the-input-referred-noise}

In this problem we want to design the CS amplifier shown in
Figure~\ref{fig-CS_schematic} imposing the input-referred noise
including both the thermal and flicker noise. The specifications at room
temperature are given in Table~\ref{tbl-specs_problem4}.

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

\caption{\label{tbl-specs_problem4}Specifications for problem 4.}

\tabularnewline

\toprule\noalign{}
Specification & Symbol & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Corner frequency & \(f_k\) & 100 & \(kHz\) \\
Input referred white PSD & \(\sqrt{{S_{{n,th}}}}\) & 30 &
\(\frac{{nV}}{{\sqrt{{Hz}}}}\) \\
& \(10\,\log(S_{{n,th}})\) & -150.46 &
\(\frac{{dBv}}{{\sqrt{{Hz}}}}\) \\
Inversion coefficient & \(IC\) & 1 & - \\
Load capacitance & \(C_L\) & 1 & \(pF\) \\

\end{longtable}

Carry out the design procedure following these steps:

\begin{itemize}
\tightlist
\item
  Using \(S_{nth}\) determine the transconductance \(G_m\).
\item
  Find the normalized transconductance \(g_{ms}(IC)\).
\item
  Determine the specific current \(I_{spec}\), the bias current \(I_b\)
  and the \(W_{eff}/L_{eff}\) ratio.
\item
  Using the noise corner frequency \(f_k\), determine the effective gate
  area \(W_{eff} \cdot L_{eff}\).
\item
  Finally, calculate the effective and drawn dimensions of the
  transistor.
\item
  Simulate the circuit using the ngspice simulator and compare the
  theoretical results with the simulations.
\end{itemize}

\chapter{Problem 5: Minimum current for a given gain-bandwidth product
with
self-loading}\label{problem-5-minimum-current-for-a-given-gain-bandwidth-product-with-self-loading}

In this problem we want to account for the self-loading effect which is
particularly important when the transistor is getting wide as it
typically happens in weak inversion. The specifications at room
temperature are given in Table~\ref{tbl-specs_problem5}.

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

\caption{\label{tbl-specs_problem5}Specifications for problem 5.}

\tabularnewline

\toprule\noalign{}
Specification & Symbol & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Gain-bandwidth product & \(GBW\) & 25 & \(MHz\) \\
Load capacitance & \(C_{L0}\) & 20 & \(fF\) \\
Transistor length & \(L\) & 1 & \(\mu m\) \\

\end{longtable}

Follow the procedure illustrated in the lecture notes.

Simulate your design and check whether it meets the specifications.

\chapter{Problem 6: Minimum current for a given DC gain and
gain-bandwidth product with
self-loading}\label{problem-6-minimum-current-for-a-given-dc-gain-and-gain-bandwidth-product-with-self-loading}

In this problem we want to account for the self-loading effect but use
the additional degree of freedom, namely the transistor gate length, to
set the dc gain. The specifications at room temperature are given in
Table~\ref{tbl-specs_problem6}.

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

\caption{\label{tbl-specs_problem6}Specifications for problem 6.}

\tabularnewline

\toprule\noalign{}
Specification & Symbol & Value & Unit \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
DC gain & \(A_{dc}\) & 50 & \(dB\) \\
Gain-bandwidth product & \(GBW\) & 25 & \(MHz\) \\
Load capacitance & \(C_{L0}\) & 20 & \(fF\) \\

\end{longtable}

Follow the procedure illustrated in the lecture notes.

Simulate your design and check whether it meets the specifications.




\end{document}
