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Miller Compensation¶

\begin{matrix} \frac{V_{o}}{I_{i}} = \frac{(s C_{f} - g_{m}) R_{1} R_{2}}{1 + s X (s) + s^{2} Y (s)} \\ X (s) = R_{1} C_{1} + R_{2} C_{2} + R_{1} C_{f} (1 + g_{m} R_{2}) + R_{2} C_{f} \\ Y (s) = (C_{1} C_{2} + C_{1} C_{f} + C_{2} C_{f}) R_{1} R_{2} = (\binom{{C_{1}, C_{2}, C_{f}}}{2}) \cdot R_{1} R_{2} \end{matrix}

\begin{matrix} A_{0} = A (0) = - g_{m} R_{1} R_{2} \end{matrix}

\begin{matrix} A (s) |_{s = z} = A (z) = 0 \end{matrix}

implies there is no (AC)current go through $R_{D}$ , thus we can easily have

\begin{matrix} (V_{i} - V_{o}) s C_{f} = V_{i} g_{m} \\ ⟹ s C_{f} = g_{m} \end{matrix}

\begin{matrix} 1 + s X_{o} (s) + s^{2} Y_{o} (s) = (1 + s R_{1} C_{1}) (1 + s R_{2} C_{s}) \\ ⟹ {\begin{aligned} X_{o} (s) & = R_{1} C_{1} + R_{2} C_{2} \\ Y_{o} (s) & = R_{1} R_{2} C_{1} C_{2} \end{aligned} \end{matrix}

now consider $C_{f}$

⟹ {\begin{aligned} X (s) & = R_{1} C_{1}^{'} + R_{2} C_{2}^{'} = R_{1} (C_{1} + C_{f - m i l l e r}) + R_{2} (C_{2} + C_{f}) \\ Y (s) & = (C_{1} C_{2} + C_{1} C_{f} + C_{2} C_{f}) R_{1} R_{2} \end{aligned}

\begin{matrix} D (s) = (1 + \frac{s}{ω_{p 1}^{'}}) (1 + \frac{s}{ω_{p 2}^{'}}) = 1 + X (s) s + Y (s) s^{2} \end{matrix}

assume $ω_{p 1}^{'}$ dominates ( $ω_{p 1} ≪ ω_{p 2}$ ), then

\begin{matrix} D (s) \approx 1 + \frac{s}{ω_{p 1}} + \frac{s^{2}}{ω_{p 1} ω_{p 2}} \\ ⟹ {\begin{aligned} ω_{p 1} & = \frac{1}{R_{1} C_{1} + R_{2} C_{2} + R_{1} C_{f} (1 + g_{m} R_{2}) + R_{2} C_{f}} \approx \frac{1}{g_{m} R_{1} R_{2} C_{f}} \\ ω_{p 2} & = \frac{g_{m} C_{f}}{C_{1} C_{2} + C_{1} C_{f} + C_{2} C_{f}} \end{aligned} \end{matrix}