模拟集成电路(6)----单级放大器（共源共栅级 Cascode Stage）

大信号分析

对M1
$V_{x}\geq V_{in}-V_{TH1}\quad V_{x}=V_{B}-V_{GS2}\\V_{B}\geq V_{in}-V_{TH1}+V_{GS2}$
对M2
$\begin{aligned}&V_{out}\geq V_{B}-V_{TH2}\\&V_{out}\geq V_{in}-V_{TH1}+V_{GS2}-V_{TH2}\\&V_{out}\geq V_{OD1}+V_{OD2}\end{aligned}$

输入输出电阻

$\begin{aligned}&\mathrm{When~}V_{in}<V_{TH1},\mathrm{M}_{1}\mathrm{~and~}\mathrm{M}_{2}\mathrm{~are~off},V_{out}=V_{DD},V_{x}=V_{B}-V_{TH2}\end{aligned}$

$\begin{aligned}&\mathrm{When~}V_{\mathrm{in}}>V_{\mathrm{TH}1},V_{\mathrm{out}}\mathrm{drops},\\&-V_{GS2}\text{increases, resulting in a drop of }V_{X}\end{aligned}$

$R_{out}=[1+(g_{m2}+g_{mb2})r_{O2}]r_{O1}+r_{O2}\\=r_{O1}+r_{O2}+(g_{m2}+g_{mb2})r_{O2}r_{O1}$
$\ g_{m}r_{O}>>1,\quad\mathrm{then}\quad R_{out}\approx(g_{m2}+g_{mb2})r_{O2}r_{O1}\approx g_{m2}r_{O2}r_{O1}$

$\begin{aligned}&R_{out}\approx(g_{m3}+g_{mb3})r_{O3}R_{S}\approx\underline{g_{m3}r_{O3}R_{S}}\\&R_{S}\approx g_{m2}r_{O2}r_{O1}\\&\to R_{out}\approx g_{m3}r_{O3}\cdot g_{m2}r_{O2}\cdot r_{o1}\end{aligned}$

小信号分析

增益

$A_{\nu}=-g_{m1}R_{D}$
如若是电流源负载

$A_v=-G_mr_{out}$

$G_m{\approx}g_{m1}$

$A_{\nu}\approx-g_{m1}\cdot g_{m2}r_{o2}r_{o1}=-g_{m1}r_{o1}\cdot g_{m2}r_{o2}$

可以结合课本3.21去理解

$g_{m}=\sqrt{2I_{\mathrm{D}}\mu_{\mathrm{n}}C_{\mathrm{ox}}\frac{W}{L}}\\r_{\mathrm{o}}=\frac{1}{\lambda I_{\mathrm{D}}}\\A_{\mathrm{int}}=g_{m}r_{\mathrm{o}}$

$g_{m1}=\frac{g_{\mathrm{m}}}{2}\\r_{\mathrm{ol}}=4r_{\mathrm{o}}\\A_{\mathrm{intl}}=2A_{\mathrm{int}}$

$g_{m2}=g_{m}\\r_{\mathrm{o2}}=A_{\mathrm{int}}r_{\mathrm{o}}\\A_{\operatorname{int}2}=A_{\operatorname{int}}^2$
用PMOS cascode电流源作负载

$\begin{aligned}r_{outn}&\approx g_{m2}r_{o2}\cdot r_{o1}\\r_{outp}&\approx g_{m3}r_{o3}\cdot r_{o4}\end{aligned}$

$\begin{aligned}A_{\nu}&\approx-g_{m1}\cdot(r_{outn}\parallel r_{outp})\\&\approx-g_{m1}\cdot(g_{m3}r_{o3}\cdot r_{o4}\parallel g_{m2}r_{o2}\cdot r_{o1})\end{aligned}$