The DC Loop gain and Servo Function are:
\begin{equation} L_{GDC}=- 72.93 g_{mo} \end{equation} \begin{equation} S_{DC}=\frac{4153.0 g_{mo}}{4153.0 g_{mo} + 56.94} \end{equation}SLiCAP's Servo Function is:
\begin{equation} S_{DC}=\frac{4153.0 g_{mo}}{4153.0 g_{mo} + 56.94} \end{equation}The Loop Gain Error required is:
\begin{equation} LG_{err}=10\,\left[ \mathrm{percent}\right] \end{equation} \begin{equation} g_{mo max}=1.358 \end{equation} \begin{equation} g_{mo min}=0.1234 \end{equation} \begin{equation} g_{mo min}=\frac{\left(A_{y} - \Delta_{A y}\right) \left(0.5 R_{a} R_{c} g_{o XMi} + R_{a} + 0.25 R_{b} R_{c} g_{m XMi} + 0.25 R_{b} R_{c} g_{o XMi} + 0.5 R_{b} - 0.25 R_{c} Z_{i} g_{m XMi} + 0.25 R_{c} Z_{i} g_{o XMi} + R_{c} + 0.5 Z_{i}\right)}{\Delta_{A y} R_{c} \left(0.25 R_{a} R_{b} g_{m XMi} + R_{a} + 0.25 R_{b} Z_{i} g_{m XMi} + 0.5 Z_{i}\right)} \end{equation}The loop Gain can be written in terms of just $C_{gso}$
\begin{equation} L_{g}=\frac{N_{1}}{C_{gso} D_{1} + D_{2}} \end{equation}The Servo Function becomes:
\begin{equation} S_{func}=- \frac{N_{1}}{C_{gso} D_{1} + D_{2} - N_{1}} \end{equation}This can be written as two components, one where $C_{gso}=0$ the other is a function of $C_{gso}$:
\begin{equation} S_{func}=- \frac{L_{cgso} S_{0cgso}}{1 - L_{cgso}} \end{equation} \begin{equation} S_{0cgso}=- \frac{N_{1}}{D_{2} - N_{1}} \end{equation} \begin{equation} L_{cgso}=\frac{D_{2} - N_{1}}{C_{gso} D_{1}} \end{equation} \begin{equation} S_{func}=\frac{2.248 \cdot 10^{4} s \left(1.87 \cdot 10^{-13} \pi + 1.979 \cdot 10^{-12}\right) + 512.5}{C_{gso} \left(1.842 \cdot 10^{5} s^{2} \cdot \left(1.87 \cdot 10^{-13} \pi + 1.979 \cdot 10^{-12}\right) + 2633.0 s\right) + 2.818 \cdot 10^{4} s \left(1.87 \cdot 10^{-13} \pi + 1.979 \cdot 10^{-12}\right) + 569.4} \end{equation}This is similar to determining the bandwidth for a system that has a loop gain and zeros in the feedback loop
That means the bandwidth is determined by the minimum between two functions.
So I can write the Servo function bandwidth as a function of $C_{gso}$ where:
\begin{equation} \frac{D_{2} - N_{1}}{C_{gso} D_{1}}=\frac{2.818 \cdot 10^{4} s \left(1.87 \cdot 10^{-13} \pi + 1.979 \cdot 10^{-12}\right) + 569.4}{C_{gso} \left(1.842 \cdot 10^{5} s^{2} \cdot \left(1.87 \cdot 10^{-13} \pi + 1.979 \cdot 10^{-12}\right) + 2633.0 s\right)} \end{equation}Now I can solve for the maximum value of the gate capacitance:
\begin{equation} C_{gso}=9.741 \cdot 10^{-11} \end{equation} \begin{equation} C_{gso}=\frac{0.5 \left|{0.5 R_{a} R_{b} R_{c} g_{mo} + 0.25 R_{a} R_{b} R_{c} g_{o XMi} + 0.5 R_{a} R_{b} + 0.5 R_{a} R_{c} Z_{i} g_{mo} + 0.25 R_{a} R_{c} Z_{i} g_{o XMi} + 0.5 R_{a} Z_{i} + 0.5 R_{b} R_{c} Z_{i} g_{mo} + 0.5 R_{b} R_{c} Z_{i} g_{o XMi} + 0.5 R_{b} R_{c} + R_{b} Z_{i} + 0.5 R_{c} Z_{i}}\right|}{\pi \left|{R_{c}}\right| \left|{f_{max}}\right| \left|{0.5 R_{a} R_{b} + 0.5 R_{a} Z_{i} + R_{b} Z_{i}}\right|} \end{equation}The maximum possible bandwidth of the servo function occurs when $C_{gso}=0$:
\begin{equation} f_{S max}=1.253 \cdot 10^{9} \end{equation}
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SLiCAP: Symbolic Linear Circuit Analysis Program, Version 2.0.1 © 2009-2023 SLiCAP development team
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Last project update: 2023-11-25 20:52:48