Using the parameters from the feedback design I can simplify my input referred noise to:
\begin{equation} S_{in}=\frac{6.273 \cdot 10^{-6} \Gamma_{XN} N_{s N18} T g_{m XN} k \left(\left(\frac{f_{\ell XN}}{f}\right)^{AF_{N18}} + 1\right) \left(1.595 \cdot 10^{10} f^{2} g_{m X}^{2} + 8.1 \cdot 10^{5} f_{T X}^{2}\right)}{f_{T X}^{2} g_{m X}^{2}} + \frac{6.273 \cdot 10^{-6} \Gamma_{XP} N_{s N18} T g_{m XP} k \left(\left(\frac{f_{\ell XP}}{f}\right)^{AF_{N18}} + 1\right) \left(1.595 \cdot 10^{10} f^{2} g_{m X}^{2} + 8.1 \cdot 10^{5} f_{T X}^{2}\right)}{f_{T X}^{2} g_{m X}^{2}} + 1200 T k\,\left[ \mathrm{\frac{V^{2}}{Hz}}\right] \end{equation}Also I am designing a balanced network so I want a solution where both input stages are identical; so I can again simplify:
\begin{equation} S_{in}=\frac{2.664 \cdot 10^{-20} T k \left(\Gamma_{XN} N_{s N18} \left(\left(\frac{f_{\ell XN}}{f}\right)^{AF_{N18}} + 1\right) \left(7.512 \cdot 10^{24} f^{2} g_{m XN}^{2} + 3.814 \cdot 10^{20} f_{T XN}^{2}\right) + 4.504 \cdot 10^{22} f_{T XN}^{2} g_{m XN}\right)}{f_{T XN}^{2} g_{m XN}}\,\left[ \mathrm{\frac{V^{2}}{Hz}}\right] \end{equation}Now I know that throughout the design $AF_{N18}$ will not change so I can simplify:
\begin{equation} AF_{N18}=1 \end{equation} \begin{equation} N_{s N18}=1.35 \end{equation} \begin{equation} S_{in}=\frac{3.597 \cdot 10^{-20} T k \left(\Gamma_{XN} \left(f + f_{\ell XN}\right) \left(7.512 \cdot 10^{24} f^{2} g_{m XN}^{2} + 3.814 \cdot 10^{20} f_{T XN}^{2}\right) + 3.336 \cdot 10^{22} f f_{T XN}^{2} g_{m XN}\right)}{f f_{T XN}^{2} g_{m XN}}\,\left[ \mathrm{\frac{V^{2}}{Hz}}\right] \end{equation}Here I have a function of 4 variables; $\Gamma_{XN}$, $g_{mXN}$, $f_{TXN}$ and $f_{\ell XN}$ which are functions of each other.
\begin{equation} NF_{eq}=\frac{8.419 \cdot 10^{-6} \cdot \left(0.009033 \Gamma_{XN} f_{T XN}^{2} f_{\ell XN} + 4.074 \cdot 10^{5} \Gamma_{XN} f_{T XN}^{2} + 1.007 \cdot 10^{18} \Gamma_{XN} f_{\ell XN} g_{m XN}^{2} + 1.678 \cdot 10^{26} \Gamma_{XN} g_{m XN}^{2} + 3.563 \cdot 10^{7} f_{T XN}^{2} g_{m XN}\right)}{R_{s} f_{T XN}^{2} g_{m XN}}\,\left[ \mathrm{1}\right] \end{equation}Replacing :
\begin{equation} f_{T XN}=\frac{0.5 g_{m XN}}{\pi c_{iss XN}} \end{equation} \begin{equation} NF_{eq}=\frac{3.368 \cdot 10^{-5} \pi^{2} c_{iss XN}^{2} \cdot \left(1.007 \cdot 10^{18} \Gamma_{XN} f_{\ell XN} g_{m XN}^{2} + 1.678 \cdot 10^{26} \Gamma_{XN} g_{m XN}^{2} + \frac{0.002258 \Gamma_{XN} f_{\ell XN} g_{m XN}^{2}}{\pi^{2} c_{iss XN}^{2}} + \frac{1.018 \cdot 10^{5} \Gamma_{XN} g_{m XN}^{2}}{\pi^{2} c_{iss XN}^{2}} + \frac{8.908 \cdot 10^{6} g_{m XN}^{3}}{\pi^{2} c_{iss XN}^{2}}\right)}{R_{s} g_{m XN}^{3}}\,\left[ \mathrm{1}\right] \end{equation}I have two design Parameters from my Noise Figure: $C_{iss}$ and $g_m$
By using the Maximum Temperature from the specs in the Noise Figure Equation I will find limits on $C_{iss}$ and $g_m$
\begin{equation} T_{max}=343.1\,\left[ \mathrm{K}\right] \end{equation}
From the Plot notice that Worst Case for the Noise Figure occurs when $\Gamma$ is the Highest, to model this I can set $\Gamma=2/3$
\begin{equation} NF_{eq}=\frac{2.261 \cdot 10^{13} \pi^{2} c_{iss XN}^{2} f_{\ell XN}}{R_{s} g_{m XN}^{1.0}} + \frac{3.768 \cdot 10^{21} \pi^{2} c_{iss XN}^{2}}{R_{s} g_{m XN}^{1.0}} + \frac{5.07 \cdot 10^{-8} f_{\ell XN}}{R_{s} g_{m XN}^{1.0}} + \frac{2.286}{R_{s} g_{m XN}^{1.0}} + \frac{300}{R_{s}}\,\left[ \mathrm{1}\right] \end{equation}
Here I would like to also say the minimum $c_{iss}=2/3W_iL_iC_{OX}$, doing this I can say the maximum $f_\ell=\alpha f_T$ where from the plot above I can see the maximum alpha is no greater than 1/8400
\begin{equation} NF_{eq}=\frac{1.256 \cdot 10^{19} \pi^{2} c_{iss XN}^{2}}{g_{m XN}^{1.0}} + 4.486 \cdot 10^{6} \pi c_{iss XN} + \frac{0.007621}{g_{m XN}^{1.0}} + 1 + \frac{1.006 \cdot 10^{-14}}{\pi c_{iss XN}}\,\left[ \mathrm{1}\right] \end{equation}Now I have a noise figure with two design variables: $c_{iss}$ and $g_m$.
More specifically, because I have taken values that maximize the Noise figure, I have a minimum $g_m$ since $NF\propto \frac{1}{g_m}$ and because I am using the smallest representation for $c_{iss}$ I am finding the maximum $c_{iss}$ that constrains the Noise figure.
Go to Balanced-Cross-Coupled-MOSFET-Noisy-Nullor-Analysis_index
SLiCAP: Symbolic Linear Circuit Analysis Program, Version 2.0.1 © 2009-2023 SLiCAP development team
For documentation, examples, support, updates and courses please visit: analog-electronics.tudelft.nl
Last project update: 2023-11-25 20:52:48