From 52f512b38c7121c4e6f069f11665eb1037b8a671 Mon Sep 17 00:00:00 2001 From: Aidan Sharpe <33038791+Adog64@users.noreply.github.com> Date: Thu, 12 Sep 2024 15:31:27 -0400 Subject: [PATCH] VLSI homework 1, and IoT reading assignment 1 done --- .../ECOMMS/notes/ecomms-notes.aux | 11 ++- .../ECOMMS/notes/ecomms-notes.fdb_latexmk | 31 +++++- .../ECOMMS/notes/ecomms-notes.fls | 54 +++++++++++ .../ECOMMS/notes/ecomms-notes.log | 42 ++++++--- .../ECOMMS/notes/ecomms-notes.pdf | Bin 33399 -> 113032 bytes .../ECOMMS/notes/ecomms-notes.synctex.gz | Bin 3770 -> 7216 bytes .../ECOMMS/notes/ecomms-notes.tex | 22 +++++ .../ECOMMS/notes/ecomms-notes.toc | 2 + .../assignments/reading-assignment-1.md | 43 +++++++-- .../VLSI/homework/homework-1/Figure_1.png | Bin 0 -> 49702 bytes .../VLSI/homework/homework-1/homework-1.md | 88 ++++++++++++++++++ .../VLSI/homework/homework-1/homework-1.pdf | Bin 0 -> 254102 bytes .../VLSI/homework/homework-1/problem_1.py | 42 +++++++++ .../VLSI/homework/homework-1/problem_2.py | 29 ++++++ .../.~lock.Lecture01-intromos.pptx# | 1 - .../lecture-slides/Lecture03-nonideal.pptx | Bin 0 -> 698342 bytes 16 files changed, 336 insertions(+), 29 deletions(-) create mode 100644 7th-Semester-Fall-2024/VLSI/homework/homework-1/Figure_1.png create mode 100644 7th-Semester-Fall-2024/VLSI/homework/homework-1/homework-1.md create mode 100644 7th-Semester-Fall-2024/VLSI/homework/homework-1/homework-1.pdf create mode 100644 7th-Semester-Fall-2024/VLSI/homework/homework-1/problem_1.py create mode 100644 7th-Semester-Fall-2024/VLSI/homework/homework-1/problem_2.py delete mode 100644 7th-Semester-Fall-2024/VLSI/lecture-slides/.~lock.Lecture01-intromos.pptx# create mode 100644 7th-Semester-Fall-2024/VLSI/lecture-slides/Lecture03-nonideal.pptx diff --git a/7th-Semester-Fall-2024/ECOMMS/notes/ecomms-notes.aux b/7th-Semester-Fall-2024/ECOMMS/notes/ecomms-notes.aux index 58a7876..4e1fba3 100644 --- a/7th-Semester-Fall-2024/ECOMMS/notes/ecomms-notes.aux +++ b/7th-Semester-Fall-2024/ECOMMS/notes/ecomms-notes.aux @@ -18,5 +18,14 @@ \providecommand \oddpage@label [2]{} \BKM@entry{id=1,open,dest={746F632E73656374696F6E},srcline={16}}{5C3337365C3337375C303030435C3030306F5C3030306E5C303030745C303030655C3030306E5C303030745C30303073} \pgfsyspdfmark {pgfid1}{4025872}{43485761} +\pgfsyspdfmark {pgfid2}{6935668}{37980737} +\pgfsyspdfmark {pgfid3}{36232565}{37980737} +\BKM@entry{id=2,open,dest={636861707465722E31},srcline={20}}{5C3337365C3337375C303030315C3030305C303430} 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@@ -0,0 +1,88 @@ +# VLSI Homework 1 - Aidan Sharpe + +## Problem 1 +```python +import numpy as np +import matplotlib.pyplot as plt + +e_0 = 8.85E-14 +k_ox = 3.9 +e_ox = e_0 * k_ox + +mu = 350 # 350 cm^2 / (V*s) +t_ox = 100E-8 # 100A + +C_ox = e_ox / t_ox +W = 1.2 +L = 0.6 + +beta = mu * C_ox * W/L + +V_g_range = np.arange(6) +V_t = 0.7 +V_dd = np.linspace(0, 5, 50) + +def I_ds(V_gs, V_ds, V_t, beta): + if V_gs < V_t: + return np.zeros_like(V_ds) + + V_dsat = V_gs - V_t + + return np.where(V_ds < V_dsat, + beta * (V_gs - V_t - V_ds/2) * V_ds, + beta * (V_gs - V_t)**2 / 2) + +for V_s in (0, 2): + for V_g in V_g_range: + V_gs = V_g - V_s + V_ds = V_dd - V_s + plt.plot(V_ds, + 1000*I_ds(V_gs, V_ds, V_t, beta), + label="$V_{gs}$ = "+str(V_gs)) + + plt.xlabel("$V_{ds}$[V]") + plt.ylabel("$I_{ds}$[mA]") + plt.legend() + plt.show() +``` +![](Figure_1.png) + +## Problem 2 +Show that a MOSFET with channel length $2L$, when operating in the linear region, has the same current $I_{DS1}$ as two MOSFETs in series each with channel length $L$. Assume the same gate voltage $V_{DD}$ for all MOSFETs in the design and the same supply voltage $V_{DS}$ for both designs. + +### Known +The current through a MOSFET in the linear region is: +$$I_{DS} = \beta\left(V_{GS} - V_t - \frac{V_{DS}}{2}\right)V_{DS}$$ +The parameter $\beta$ is defined as: +$$\beta = \mu C_\text{ox} \frac{W}{L}$$ + +### Defining Variables +The factor $\beta_1$ for the singular MOSFET is $\mu C_\text{ox} \frac{W}{2L}$, and the factor $\beta_2$ for each of the MOSFETs in series is $\mu C_\text{ox} \frac{W}{L}$. Therefore, $\beta_2 = 2\beta_1$. + +The current $I_{DS1}$ through the singular MOSFET is: +$$I_{DS1} = \beta_1\left(V_{DD} - V_t - \frac{V_{DS}}{2}\right)V_{DS}$$ +The current $I_{DS2}$ through both MOSFETs in series is both: +$$I_{DS2} = \beta_2\left(V_{DD} - V_1 - V_t - \frac{V_{DS} - V_1}{2}\right)(V_{DS} - V_1)$$ +$$I_{DS2} = \beta_2\left(V_{DD} - V_t - \frac{V_1}{2}\right)V_1$$ + +### Expanding $I_{DS2}$ +The first equation for $I_{DS2}$ can be rewritten as the difference of two terms: +$$I_{DS2} = \beta_2\left(V_{DD} - V_t - \frac{V_{DS}}{2} - \frac{V_1}{2}\right)(V_{DS} - V_1)$$ +$$I_{DS2} = \beta_2\left(V_{DD} - V_t - \frac{V_{DS}}{2} - \frac{V_1}{2}\right)V_{DS} - \beta_2\left( V_{DD} - V_t - \frac{V_{DS}}{2} - \frac{V_1}{2}\right)V_1$$ + +The term $-\frac{V_1 V_{DS}}{2}$ can be pulled out of the first term, and its complement can be pulled out of the second term. Therefore, $I_{DS2}$ can be simplified to the difference of two terms each in the form of the current through a MOSFET in the linear region. +$$I_{DS2} = \beta_2\left(V_{DD} - V_t - \frac{V_{DS}}{2}\right)V_{DS} - \beta_2\left( V_{DD} - V_t - \frac{V_1}{2}\right)V_1$$ + +### Reducing $I_{DS2}$ +Notice that the second term is the inverse of the second equation for $I_{DS2}$ under "Defining Variables". Setting them equal: +$$I_{DS2} = \beta_2\left(V_{DD} - V_t - \frac{V_{DS}}{2}\right)V_{DS} - \beta_2\left( V_{DD} - V_t - \frac{V_1}{2}\right)V_1 = \beta_2\left(V_{DD} - V_t - \frac{V_1}{2}\right)V_1$$ + +Therefore: +$$\beta_2\left(V_{DD} -V_t - \frac{V_{DS}}{2}\right)V_{DS} = 2\beta_2\left(V_{DD} - V_t - \frac{V_1}{2}\right)V_1$$ + +Substituting $\beta_2$ on the left for $2\beta_1$ gives: +$$2\beta_1\left(V_{DD} - V_t - \frac{V_{DS}}{2}\right)V_{DS} = 2\beta_2\left(V_{DD} - V_t - \frac{V_1}{2}\right)V_1$$ + +Therefore: +$$2I_{DS1} = 2I_{DS2}$$ +$$\boxed{I_{DS1} = I_{DS2}}$$ diff --git a/7th-Semester-Fall-2024/VLSI/homework/homework-1/homework-1.pdf b/7th-Semester-Fall-2024/VLSI/homework/homework-1/homework-1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4837593afc26cb2be12ba271c6817f3e32d2e08a GIT binary patch literal 254102 zcmeFYbySsK*DegAponY%Q3M1;LK>uFgGfnBOLup72oiz_f=Gjaba#We>28pe?(Y89 z-u(1=&UxPPp7;CrjKdfX95?s9Vy-!_dCgelZv=(unCMy2$$JNzM$y?x7)dO3Owf6F 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