# Source Follower

\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*}

# Source Follower

common-source stage requires a large load impedance to achieve a high voltage gain, if the load is low-impedance, we need a source follower be placed after the amplifier to act as a buffer.

From Fig. 3.35, we have (apply KVL to output)

V_\tx{out} = (g_m V_1 + g_{mb} V_{bs})R_S\\ \tx{where } \begin{aligned} V_1 &= V_\tx{in}-V_\tx{out}\\ V_{bs} &= -V_\tx{out} \end{aligned}

$V_\tx{out} = (g_m V_\tx{in} - g_m V_\tx{out}-g_{mb}V_\tx{out})R_S\\ A_v=\frac{V_\tx{out}}{V_\tx{in}}=\frac{g_m R_S}{1+(g_m+g_{mb})R_S}\\ =\frac{g_m}{1/R_S+g_m+g_{mb}}$

$\frac{1}{2}\mu_n C_{ox}\frac{W}{L}(V_\tx{in}-V_\tx{out}-V_\tx{TH})^2 R_S = V_\tx{out}$

$A_v=\dfrac{R_S}{1/g_m+(1+g_{mb}/g_m)R_S}\\ \approx \dfrac{R_S}{(1+g_{mb}/g_m)R_S}\\ \approx \dfrac{1}{1+\eta}$

$I_X-g_m V_X-g_{mb}V_X=0\\ R_\tx{out}=\frac {1}{g_m+g_{mb}}=\frac{1}{g_m}\Vert \frac{1}{g_{mb}}$

\begin{aligned} A_v &= g_m R_\tx{out}\\ &=\frac{g_m}{g_m+g_{mb}}\\ &= \frac{1}{1+\eta} \end{aligned}

\begin{aligned} A_v &= \frac{R_\tx{eq}}{R_\tx{eq}+\dfrac{1}{g_m}} \\ &= \frac{g_m}{g_m + \dfrac{1}{R_\tx{eq}}} \end{aligned} \tag{3.96}

• nonlinearity
• The nonlinear dependence of $V_\tx{TH}$ upon the source potential
• In submicron technologies, $r_O$ of the transistor also changes substantially with $V_{DS}$ (Chapter 14)
• body effect (PFETs’ body effect can be eliminated by tying bulk to souce)
• Source followers shift the dc level of the signal by $V_{GS}$, consider the example in Figure 3.45
• Without the source follower, the minimum value of $V_X$ is $V_{GS1}-V_\tx{TH1}$
• With the source follower, the minimum value of $V_X$ is $V_{GS2}+(V_{GS1}-V_\tx{TH1})$

# Summary

$A_v = \frac{g_m}{g_m+g_{mb}+\dfrac{1}{r_O}+\dfrac{1}{R_S}+\dfrac{1}{R_L}}$

$$A_v = \frac{g_{m1}}{g_{m1}+g_{m2}+g_{mb2}+1/r_{O2}+1/r_{O1}}$$