---
title: |
  Fundamentals of Analog VLSI Design \
  Exercise 7 - Problem
subtitle: Gain and Bandwidth Enhancement
author: Christian Enz (christian.enz@epfl.ch)
affiliation: EPFL
date: 11-05-2025
date-format: "DD.MM.YYYY"
execute:
  echo: false
  warning: false
toc: true
number-sections: true
number-depth: 3
highlight-style: pygments
bibliography: references.bib
csl: ieee.csl
format: 
  html:
    include-in-header: mathjax.html
    code-fold: true
    html-math-method: mathjax
    css: styles.css
  pdf:
    keep-tex: true
    pdf-engine: pdflatex
    documentclass: scrreprt
    papersize: a4
    geometry:
      - top=20mm
      - bottom=20mm
      - left=20mm
      - right=20mm
    include-in-header:
      text: |
        \usepackage{scrlayer-scrpage}
        \rohead[\thepage]{\thepage}
        \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}
---

```{python}
# Initialization

from ekv_functions import *

import pandas as pd
import matplotlib.pyplot as plt
import subprocess
import shutil
import os
import os.path as op
import sys
sys.path.append(".")
import re

from IPython.display import Markdown
from tabulate import tabulate

plt.rcParams['svg.fonttype'] = 'none'
plt.rcParams['pdf.fonttype'] = 42
plt.rcParams['font.family'] = 'Arial'
plt.rcParams['mathtext.fontset'] = 'custom'
plt.rcParams['mathtext.rm'] = 'Arial'
plt.rcParams['mathtext.it'] = 'Arial:italic'
plt.rcParams['mathtext.bf'] = 'Arial:bold'

inkscapePath = r"C:\\Program Files\\Inkscape\bin\\inkscape.exe"
savePlotPath = "./Plots/"
#savePlots=True
savePlots=False

if savePlots:
  lw=1
  curvelw=2
  msize=6
  textFontSize=12
  BodePlotParams = {'figure.figsize': (6,6)}
  pltStyle='plt_style_medium.mplstyle'
else:
  lw=1
  curvelw=1
  msize=5
  textFontSize=9
  BodePlotParams = {'figure.figsize': (5,5)}
  pltStyle='plt_style_small.mplstyle'

plt.style.use(pltStyle)

circuitName="diff_pair"
version = "_v1" # Allows you to save different versions
newSim=True
#newSim=False
refloat=r'.*?([+-]?\d+([.]\d*)?([eE][+-]?\d+)?|[.]\d+([eE][+-]?\d+)?)'

def saveFigures(savePath, plotName):
#  print("Patience, saving plot!")
  figFolder = savePath + r"\\{}.{}"
  pngFile = figFolder.format(plotName,"png")
  pdfFile = figFolder.format(plotName,"pdf")
  svgFile = figFolder.format(plotName,"svg")
  plt.savefig(pngFile, format="png", dpi=1200, bbox_inches="tight", transparent=True)
  plt.savefig(pdfFile, format="pdf", dpi=1200, bbox_inches="tight", transparent=True)
  plt.savefig(svgFile, format="svg", dpi=1200, bbox_inches="tight", transparent=True)
  cmd_emf = '"{}" "{}\\{}.svg" --export-filename="{}\\{}.emf" --export-overwrite'.format(inkscapePath, savePath, plotName, savePath, plotName)
#    cmd_emf = '"{}" "{}\{}.pdf" --export-filename="{}\{}.emf" --export-overwrite'.format(inkscapePath, savePath, plotName, savePath, plotName)
#    cmd_pdf = '"{}" "{}\{}.svg" --export-filename="{}\{}.pdf" --export-overwrite'.format(inkscapePath, savePath, plotName, savePath, plotName)
  subprocess.call(cmd_emf, shell=True)
#    subprocess.call(cmd_pdf, shell=True)
```

# Problem 1: Gain Enhancement

![Schematic of gain enhancement amplifier.](Figures/Gain_enhancement_schematic_with_CL.png){#fig-Gain_enhancement_schematic}

The amplifier shown in @fig-Gain_enhancement_schematic allows to enhance the DC gain by reducing the load conductance seen by the differential pair M~1a~-M~1b~ @bib:sansen:aacd:2016 @bib:ohri:jssc:14:1:1979.

## Small-signal analysis

![Negative conductance.](Figures/negative_gm.png){#fig-negative_gm}

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

* Draw the small-signal schematic of the circuit shown in @fig-negative_gm.
* 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~3a~ and M~3b~.
* Draw the small-signal circuit of the amplifier shown in @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~1a~-M~1b~ does not change in differential mode with perfect matching.
* 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.
* Deduce the expression of the DC gain and the dominant pole.
* What is the condition on $G_{m3}$ for the pole to remain in the left half-plane?
* What is the gain-enhancement factor $K$ compared to the voltage gain without the cross-coupled pair M~3a~-M~3b~?
* What is the value of the DC gain for $G_{m3}=G_{m2}$? How does it compare to a common-stage stage?
* Find the gain-bandwidth product $\omega_u$. Is the gain-bandwidth product also enhanced like the DC gain?

## Noise analysis

* Draw the small-signal schematic of @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~1a~-M~1b~ does not change in differential mode with perfect matching.
* Derive the output noise resistance $R_{nout}$.
* Deduce the input-referred noise resistance $R_{nin}$.
* Calculate the input-referred thermal noise resistance $R_{nt}$. How does it compare to the noise of the same amplifier without M~3a~-M~3b~?
* Calculate the input-referred flicker noise resistance $R_{nf}$. How does it compare to the noise of the same amplifier without M~3a~-M~3b~?

# Problem 2: Bandwidth enhancement

![Enhanced $G_m$ differential OTA](Figures/Gm_enhanced_ota_schematic.png){#fig-Gm_enhanced_ota_schematic}

The differential OTA shown in @fig-Gm_enhanced_ota_schematic shows a larger gain-bandwidth product compared to the case without the cross-coupled transistors M~2a~-M~2b~ @bib:sansen:aacd:2016 @bib:castello:jssc:25:3:1990.

## Small-signal analysis

![Enhanced $G_m$ differential transconductor](Figures/Gm_enhancement_schematic.png){#fig-Gm_enhancement_schematic}

The OTA of @fig-Gm_enhanced_ota_schematic uses the differential transconductor shown in @fig-Gm_enhancement_schematic which allows to increase the equivalent transconductance compared to the case of a simple differential pair.

* Draw the small-signal schematic of @fig-Gm_enhancement_schematic. Hint: simplify the circuit by using the fact that the voltage at the common source node of M~2a~-M~2b~ does not change in differential mode with perfect matching.
* 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?
* 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~1a~-M~1b~. Hint: add a parasitic capacitance $C_p$ connected between the source node of M~1a~ and ground to the small-signal half-circuit used above.
* Deduce the expression of the dominant pole $\omega_p$.
* 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 $\alpha \triangleq G_{ms1}/G_{m2}$.
* Express the $G_m$-enhancement factor $K$ compared to the transconductance of the differential pair M~1a~-M~1b~ without the cross-coupled transistors M~2a~-M~2b~. Hint: express $K$ in terms of the transconductance ratio $\alpha$.
* What is the value of $K$ when M~1a~-M~1b~ and M~2a~-M~2b~ are both biased in weak inversion? Hint: M~1a~-M~1b~ and M~2a~-M~2b~ share the same bias current $I_b$.
* Derive the small-signal differential transfer function
\begin{equation}
  A_d \triangleq \frac{V_{od}}{V_{id}}.
\end{equation}
* What is the value of the DC gain for $G_{ms1}=G_{m2}$? How does it compare to a common-stage stage?
* Find the gain-bandwidth product $\omega_u$. Is the gain-bandwidth product also enhanced like the DC gain?

## Noise analysis

* Draw the small-signal schematic of @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~2a~-M~2b~ does not change in differential mode with perfect matching.
* Derive the output noise conductance $G_{nout}$.
* Deduce the input-referred noise resistance $R_{nin}$.
* Calculate the input-referred thermal noise resistance $R_{nt}$. How does it compare to the noise of a simple differential pair?
* Calculate the input-referred flicker noise resistance $R_{nf}$. How does it compare to the noise of a simple differential pair?

# References