# Common-Gate Stage

\begin{align*} \newcommand{\dif}{\mathop{}\!\mathrm{d}} \newcommand{\belowarrow}[1]{\mathop{#1}\limits_{\uparrow}} \newcommand{\bd}{\boldsymbol} \newcommand{\tx}{\text} \newcommand{\L}{\mathscr{L}} \newcommand{\p}{\partial\,} \end{align*}

# Common-Gate Stage

Fig 3.48(a) 可以转换成 Fig 3.48(b)。We first study the large-signal behavior.

• $V_b-V_\tx{in}<V_\tx{TH}$, $M_1$ is off, $V_\tx{out}=V_{DD}$
• $V_b-V_\tx{in}>V_\tx{TH}$, $M_1$ is in saturation, $I_D$ is
$I_D = \frac{1}{2} \mu_n C_\tx{ox} \frac{W}{L} (V_b - V_\tx{in}-V_\tx{TH})^2$

and $V_\tx{out}$ is

$V_\tx{out}=V_{DD} - I_D R_D$

we can obtain a small-signal gain of

\begin{aligned} \frac{\p V_\tx{out}}{\p V_\tx{in}}&=-R_D \frac{\p I_D}{\p V_\tx{in}}\\ &=-R_D \mu_n C_\tx{ox} \frac{W}{L}(V_b-V_\tx{in}-V_\tx{TH}) \\ &\quad (-1-\frac{\p V_\tx{TH}}{\p V_\tx{in}})\\ \tx{with }&\frac{\p V_\tx{TH}}{\p V_\tx{in}}=\frac{\p V_\tx{TH}}{\p V_{SB}}=\eta\\ \frac{\p V_\tx{out}}{\p V_\tx{in}}&=g_m(1+\eta)R_D \end{aligned}

$\eta$ 是那个体效应的参数，所以和 $\p V_{TH}$ 有关。注意到 $\eta$ 的存在增大了增益。

• $V_b-V_\tx{TH}>V_\tx{out}$, $M_1$ is in triode region

The small-signal equivalent circuit of CG stage is

$r_O(\frac{-V_\tx{out}}{R_D}-g_m V_1 - g_{mb}V_1) - \frac{V_\tx{out}}{R_D}R_S+V_\tx{in} = V_\tx{out}\\ \tx{where } V_1 - \frac{V_\tx{out}}{R_D}R_S+V_\tx{in}=0\\ \Downarrow\\ A_v=\frac{V_\tx{out}}{V_\tx{in}}=\frac{(g_m+g_{mb})r_O+1}{r_O+(g_m+g_{mb})r_OR_S+R_S+R_D}R_D\\ =\frac{[1+(g_m+g_{mb})r_O]R_D}{r_O+[1+(g_m+g_{mb})r_O]R_S+R_D}$

R_DI_X+r_O[I_X-(g_m+g_{mb})V_X]=V_X\\ \begin{aligned} \Rightarrow R_\tx{in}&=\frac{V_X}{I_X}=\frac{R_D+r_O}{1+(g_m+g_{mb})r_O}\\ &=\frac{R_D}{1+(g_m+g_{mb})r_O}+\frac{1}{1/r_O+g_m+g_{mb}}\\ &\approx \frac{R_D}{(g_m+g_{mb})r_O}+\frac{1}{g_m+g_{mb}} \end{aligned}

$R_\tx{out}=\left\{ [1+(g_m+g_{mb})r_O]R_S+r_O \right\} \Vert R_D\\ 或=\left\{ [1+(g_m+g_{mb})R_S]r_O+R_S \right\} \Vert R_D$

# 总结

• $R_\tx{out}=\left\{ [1+(g_m+g_{mb})r_O]R_S+r_O \right\} \Vert R_D$ 和 Common-source 一样，这个简单
• $R_\tx{in}=\dfrac{R_D+r_O}{1+(g_m+g_{mb})r_O}$，前面的 Degenerated Factor 是乘以 $R_S$，这里恰好相反（注意，这里没有考虑 $R_S$，实际要加上 $R_S$）
• $A_v=\dfrac{(g_m+g_{mb})r_O+1}{r_O+(g_m+g_{mb})r_OR_S+R_S+R_D}R_D$，分母是所以电阻串联（且 $R_S$ 乘上 Degenerated Factor），而分母是 Degenerated Factor 乘上 $R_D$

\begin{aligned} A_v &= \frac{R_D}{R_S+\dfrac{(R_D+r_O)}{1+(g_m+g_{mb})r_O}}\\ &=\frac{i \cdot R_{D}}{i\cdot R_{\tx{in}}}\\ &=\frac{V_\tx{out}}{V_\tx{in}} \end{aligned}