单管放大

共源

	共源放大
电阻负载	$\begin{array}{}\text{(55)}& A_{\mathrm{v}}& =\frac{\partial V_{\text{ow }}}{\partial V_{\mathrm{in}}}\text{(56)}& & =-R_{\mathrm{D}}{\mu }_{\mathrm{n}}{C}_{\mathrm{ox}}\frac{W}{L}(V_{\mathrm{in}}-V_{\mathrm{TH}})\text{(57)}& & =-g_{\mathrm{m}}{R}_{\mathrm{D}}\end{array}$$\begin{align} A_{\mathrm{v}} &=\frac{\partial V_{\text {ow }}}{\partial V_{\mathrm{in}}} \\ &=-R_{\mathrm{D}} \mu_{\mathrm{n}} C_{\mathrm{ox}} \frac{W}{L}\left(V_{\mathrm{in}}-V_{\mathrm{TH}}\right) \\ &=-g_{\mathrm{m}} R_{\mathrm{D}} \end{align}$
考虑沟长调制	$A_v=-g_{\mathrm{m}}(r_0\parallel R_{\mathrm{D}})$$A_v=-g_{\mathrm{m}}\left(r_0 \|R_{\mathrm{D}}\right)$
二极管负载	$\begin{array}{}\text{(58)}& & \frac{V_x}{I_x}=\frac{1}{g_{\mathrm{m}}+g_{\mathrm{mb}}+r_{\mathrm{n}}^{-1}}\text{(59)}& & =\frac{1}{g_m+g_{mb}}\parallel r_0\text{(60)}& & \approx \frac{1}{g_m+g_{ml}}\end{array}$$\begin{align}&\frac{V_x}{I_x}=\frac{1}{g_{\mathrm{m}}+g_{\mathrm{mb}}+r_{\mathrm{n}}^{-1}}\\&=\frac{1}{g_m+g_{m b}} \|r_0\\&\approx \frac{1}{g_m+g_{m l}}\end{align}$
二极管负载增益	$\begin{array}{}\text{(61)}& A_{\mathrm{v}}& =-g_{\mathrm{m}1}\frac{1}{g_{\mathrm{m}2}+g_{\mathrm{mb}2}}\text{(62)}& & =-\frac{g_{\mathrm{m}1}}{g_{\mathrm{m}2}}\frac{1}{1+\eta }\text{(63)}& & =-\sqrt{\frac{(W/L{)}_1}{(W/L{)}_2}}\frac{1}{1+\eta }\end{array}$$\begin{align} A_{\mathrm{v}} &=-g_{\mathrm{m} 1} \frac{1}{g_{\mathrm{m} 2}+g_{\mathrm{mb} 2}} \\ &=-\frac{g_{\mathrm{m} 1}}{g_{\mathrm{m} 2}} \frac{1}{1+\eta} \\ &= -\sqrt{\frac{(W / L)_1}{(W / L)_2}} \frac{1}{1+\eta} \end{align}$
电流源负载	$A_v=-g_{\mathrm{m}}(r_{O1}\parallel r_{O2})$$A_v=-g_{\mathrm{m}}\left(r_{O1 }\|r_{O2}\right)$
带源极负反馈	$\begin{array}{}\text{(64)}& G_{\mathrm{m}}& =\frac{g_{\mathrm{m}}}{1+g_{\mathrm{m}}{R}_{\mathrm{s}}}={\frac{1}{\frac{1}{g_{\mathrm{m}}}+R_{\mathrm{s}}}}_{「\text{在}\text{源}\text{极}\text{通}\text{路}\text{上}\text{看}\text{到}\text{的}\text{电}\text{阻}」}\text{(65)}& A_{\mathrm{v}}& =-G_{\mathrm{m}}{R}_{\mathrm{D}}\text{(66)}& & =\frac{-g_m{R}_D}{1+g_m{R}_s}\end{array}$$\begin{align} G_{\mathrm{m}} &=\frac{g_{\mathrm{m}}}{1+g_{\mathrm{m}} R_{\mathrm{s}}} = \frac{1}{\frac{1}{g_{\mathrm{m}}}+ R_{\mathrm{s}}}_{「在源极通路上看到的电阻」} \\ A_{\mathrm{v}} &= -G_{\mathrm{m}} R_{\mathrm{D}} \\ &= \frac{-g_m R_D}{1+g_m R_s} \end{align}$
考虑沟长调制与体效应： 输出阻抗：	$\begin{array}{}\text{(67)}& G_{\mathrm{m}}& =\frac{I_{out}}{V_{in}}=\frac{V_{out}}{V_{in}}·\frac{I_{out}}{V_{out}}\text{(68)}& & =g_m{r}_o·\frac{1}{R_{out}}\text{(69)}& & =\frac{g_m{r}_o}{r_o+[1+(g_m+g_{mb})r_o]R_s}\text{(70)}& \text{(71)}& R_{\text{out }}& =[1+(g_{\mathrm{m}}+g_{\mathrm{mb}})R_{\mathrm{s}}]r_0+R_{\mathrm{s}}\text{(72)}& & =[1+(g_{\mathrm{m}}+g_{\mathrm{mb}})r_{\mathrm{b}}]R_{\mathrm{s}}+r_0\end{array}\to A_{\mathrm{v}}=-G_{\mathrm{m}}{R}_{\mathrm{out}}$$\begin{align} G_{\mathrm{m}} &= \frac{I_{out}}{V_{in }}=\frac{V_{out}}{V_{in }}·\frac{I_{out}}{V_{ou t }} \\ &=g_m r_o ·\frac{1}{R_{out }} \\ &= \frac{g_m r_o}{r_o + [1+(g_m +g_{mb})r_o]R_s} \\ \\ R_{\text {out }} &=\left[1+\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) R_{\mathrm{s}}\right] r_0+R_{\mathrm{s}} \\ &=\left[1+\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) r_{\mathrm{b}}\right] R_{\mathrm{s}}+r_0 \end{align} \to A_{\mathrm{v}} =-G_{\mathrm{m}} R_{\mathrm{out}}$
带负载	$\begin{array}{}\text{(73)}& A_{\mathrm{v}}& =\frac{g_m{r}_o}{r_o+[1+(g_m+g_{mb})r_o]R_s}(R_D//R_{out})\text{(74)}& & -\frac{g_{\mathrm{m}}{r}_0}{R_{\mathrm{s}}+r_0+(g_{\mathrm{m}}+g_{\mathrm{mb}})R_{\mathrm{s}}{r}_0)}\times \text{(75)}& & \frac{R_{\mathrm{D}}[R_{\mathrm{s}}+r_0+(g_{\mathrm{m}}+g_{\mathrm{ml}})R_{\mathrm{s}}{r}_0]}{R_{\mathrm{D}}+R_{\mathrm{s}}+r_0+(g_{\mathrm{m}}+g_{\mathrm{mb}})R_{\mathrm{s}}{r}_0}\end{array}$$\begin{align} A_{\mathrm{v}} &= \frac{g_m r_o}{r_o + [1+(g_m +g_{mb})r_o]R_s} (R_D //R_{out}) \\ &-\frac{g_{\mathrm{m}} r_0}{\left.R_{\mathrm{s}}+r_0+\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) R_{\mathrm{s}} r_0\right)} \times \\ & \space \space\space\space\space\frac{R_{\mathrm{D}}\left[R_{\mathrm{s}}+r_0+\left(g_{\mathrm{m}}+g_{\mathrm{ml}}\right) R_{\mathrm{s}} r_0\right]}{R_{\mathrm{D}}+R_{\mathrm{s}}+r_0+\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) R_{\mathrm{s}} r_0} \end{align}$

共漏

	源跟随器（共漏极）
忽略沟长调制	$\begin{array}{}\text{(76)}& A_{\mathrm{v}}& =\frac{g_{\mathrm{m}}{R}_{\mathrm{s}}}{1+(g_{\mathrm{m}}+g_{\mathrm{mb}})R_{\mathrm{s}}}=\frac{R_{\mathrm{s}}}{\frac{1}{g_{\mathrm{m}}}+\left(\frac{g_{\mathrm{m}}+g_{\mathrm{mb}}}{g_{\mathrm{m}}}\right)R_{\mathrm{s}}}\text{(77)}& g_{\mathrm{m}}& ={\mu }_{\mathrm{n}}{C}_{\mathrm{ox}}\frac{W}{L}(V_{\mathrm{m}}-V_{\mathrm{TH}}-V_{\mathrm{ott}})\end{array}$$\begin{align} A_{\mathrm{v}}&=\frac{g_{\mathrm{m}} R_{\mathrm{s}}}{1+\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) R_{\mathrm{s}}} = \frac{ R_{\mathrm{s}}}{\frac{1}{g_{\mathrm{m}}}+\left(\frac{ g_{\mathrm{m}}+g_{\mathrm{mb}}}{g_{\mathrm{m}}}\right) R_{\mathrm{s}}} \\ g_{\mathrm{m}} &=\mu_{\mathrm{n}} C_{\mathrm{ox}} \frac{W}{L}\left(V_{\mathrm{m}}-V_{\mathrm{TH}}-V_{\mathrm{ott}}\right)\\ \end{align}$ $\begin{array}{c}\\ \text{小}\text{信}\text{号}\text{方}\text{法}：(\begin{array}{}\text{(78)}& & V_{in}-V_1=V_{out},V_{bs}=-V_{out}\text{(79)}& & g_m{V}_1-g_{mb}{V}_{out}=V_{out}/R_s\text{(80)}& & A_v=\frac{V_{out}}{V_{in}}=g_m{R}_s/[1+(g_m+g_{mb})R_s]\end{array}\end{array}$$\\ 小信号方法： \left( \begin{align} &V_{in}-V_1=V_{out},V_{bs}=-V_{out}\\ &g_mV_1-g_{mb}V_{out}=V_{out}/R_s \\&A_v=\frac{V_{out}}{V_{in}}=g_mR_s/[1+(g_m+g_{mb})R_s] \end{align}\right.$	$A_v<1$$A_v<1$
电流源代替电阻	$\begin{array}{}\text{(81)}& R_{out}& =\frac{1}{g_m}\parallel \frac{1}{g_{mb}}=\frac{1}{g_m+g_{mb}}\text{(82)}& A_v& =\frac{\frac{1}{g_{mb}}}{\frac{1}{g_m}+\frac{1}{g_{mb}}}=\frac{g_m}{g_m+g_{mb}}\end{array}$$\begin{align} R_{out} &= \frac{1}{g_m}\|\frac{1}{g_{mb}} = \frac{1}{g_m+ g_{mb}} \\ A_v&=\frac{ \frac{1}{g_{mb}} }{\frac{1}{g_{m}}+\frac{1}{g_{mb}}}=\frac{g_m}{g_m+g_{mb}} \\ \end{align}$
驱动负载的源随器	$A_v=\frac{R_{eq}}{R_{eq}+\frac{1}{g_m}},R_{\mathrm{eq}}=(\frac{1}{g_{\mathrm{mb}}})\Vert r_{o1}\Vert r_{\mathrm{o}2}\parallel R_{\mathrm{L}}$$A_v=\frac{R_{eq}}{R_{eq}+\frac{1}{g_m}} , R_{\mathrm{eq}}=(\frac{1} { g_{\mathrm{mb}}})\left\|r_{o1}\right\|r_{\mathrm{o2}} \|R_{\mathrm{L}}$	源随器不一定是有效的驱动器。

共栅

	共栅极
不考虑沟长调制	$\begin{array}{}\text{(83)}& \frac{\partial V_{\text{out }}}{\partial V_{\text{in }}}& ={\mu }_{\mathrm{n}}{C}_{\mathrm{ox}}\frac{W}{L}{R}_{\mathrm{D}}(V_{\mathrm{b}}-V_{\mathrm{in}}-V_{\mathrm{TH}})(1+\eta )\text{(84)}& & =g_{\mathrm{m}}(1+\eta )R_{\mathrm{D}}\end{array}$$\begin{align} \frac{\partial V_{\text {out }}}{\partial V_{\text {in }}} &=\mu_{\mathrm{n}} C_{\mathrm{ox}} \frac{W}{L} R_{\mathrm{D}}\left(V_{\mathrm{b}}-V_{\mathrm{in}}-V_{\mathrm{TH}}\right)(1+\eta) \\ &=g_{\mathrm{m}}(1+\eta) R_{\mathrm{D}} \end{align}$	体效应使共栅级的等效跨导变大了
考虑MOS输出阻抗$R_0$$R_0$及$V_{in}$$V_{in}$阻抗$R_s$$R_s$	$\begin{array}{}\text{(85)}& r_{\mathrm{ov}}& {(\frac{-V_{\text{out }}}{R_{\mathrm{D}}}-g_{\mathrm{m}}{V}_1-g_{\text{inb }}{V}_1)}_{\text{流}\text{过}{r}_o\text{电}\text{流}}\text{(86)}& & -\frac{V_{\text{out }}}{R_{\mathrm{D}}}{R}_{\mathrm{S}}+V_{\text{in }}=V_{\text{out }}\text{(87)}& and:& V_1-\frac{V_{\text{out }}}{R_{\mathrm{D}}}{R}_{\mathrm{s}}+V_{\text{in }}=0\text{(88)}& \to :& \frac{V_{\text{out }}}{V_{\text{in }}}=\frac{(g_{\mathrm{m}}+g_{\mathrm{mb}})r_{\mathrm{o}}+1}{r_o+(g_{\mathrm{m}}+g_{\mathrm{mb}})r_o{R}_{\mathrm{s}}+R_{\mathrm{s}}+R_{\mathrm{D}}}{R}_{\mathrm{D}}\end{array}$$\begin{align} r_{\mathrm{ov}}&\left(\frac{-V_{\text {out }}}{R_{\mathrm{D}}}-g_{\mathrm{m}} V_1-g_{\text {inb }} V_1\right)_{流过r_o电流}\\ &-\frac{V_{\text {out }}}{R_{\mathrm{D}}} R_{\mathrm{S}}+V_{\text {in }}=V_{\text {out }} \\ and: &V_1-\frac{V_{\text {out }}}{R_{\mathrm{D}}} R_{\mathrm{s}}+V_{\text {in }}=0 \\ \to: &\frac{V_{\text {out }}}{V_{\text {in }}}=\frac{\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) r_{\mathrm{o}}+1}{r_o+\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) r_o R_{\mathrm{s}}+R_{\mathrm{s}}+R_{\mathrm{D}}} R_{\mathrm{D}}\end{align}$
	共源共栅级(cascade)
考虑沟长调制及体效应	$\begin{array}{}\text{(89)}& V_{\mathrm{in},\mathrm{eq}}& =\frac{r_{\mathrm{o}1}\parallel \frac{1}{g_{\mathrm{mb}1}}}{r_{\mathrm{o}1}\parallel \frac{1}{g_{\mathrm{mbl}}}+\frac{1}{g_{\mathrm{m}1}}}{V}_{\mathrm{in}},R_{\mathrm{eq}}=r_{on}\Vert \frac{1}{g_{\mathrm{mb}1}}\Vert \frac{1}{g_{\mathrm{m}1}}\text{(90)}& & Replaceitwith:\frac{V_{out}}{V_{in}}\end{array}$$\begin{align} V_{\mathrm{in},\mathrm{eq}}&=\frac{r_{\mathrm{o1}}\|\frac{1}{g_{\mathrm{mb1}}}}{r_{\mathrm{o1}}\|\frac{1}{g_{\mathrm{mbl}}}+\frac{1}{g_{\mathrm{m1} }}}V_{\mathrm{in}},R_{\mathrm{eq}}=r_{on}\left\|\frac{1}{g_{\mathrm{mb1}}}\right\|\frac{1}{g_{\mathrm{m} 1}}\\ &Replace\space it \space with:\frac{V_{out}}{V_{in}} \end{align}$
共栅极输入输出阻抗：	$\begin{array}{}\text{(91)}& R_x& =\frac{V_x}{I_x}=\frac{r_0}{1+(g_{\mathrm{m}}+g_{\mathrm{mb}})r_0}\text{(92)}& & =\frac{1}{\frac{1}{r_0}+g_{\mathrm{m}}+g_{\mathrm{mht}}}\text{(93)}& & =r_o\parallel \frac{1}{g_m}\parallel {\frac{1}{g_{mb}}}_{\text{理}\text{想}\text{电}\text{流}\text{源}\text{负}\text{载}\text{的}\text{共}\text{栅}\text{极}\text{输}\text{入}\text{电}\text{阻}}\text{(94)}& \text{(95)}& R_{\text{out }}& =\{[1+(g_{\mathrm{m}}+g_{\mathrm{mb}})r_o]R_{\mathrm{S}}+r_o\}\parallel R_{{\mathrm{D}}_{\text{带}\text{源}\text{级}\text{负}\text{反}\text{馈}\text{共}\text{栅}\text{极}\text{的}\text{输}\text{出}\text{电}\text{阻}}}\end{array}$$\begin{align} R_{x} &=\frac{V_x}{I_x} =\frac{r_0}{1+\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) r_0} \\ &=\frac{1}{\frac{1}{r_0}+g_{\mathrm{m}}+g_{\mathrm{mht}}} \\ &= r_o \| \frac{1}{g_m}\|\frac{1}{g_{mb}}_{理想电流源负载的共栅极输入电阻} \\ \\R_{\text {out }}&=\left\{\left[1+\left(g_{\mathrm{m}}+g_{\mathrm{mb}}\right) r_o\right] R_{\mathrm{S}}+r_o\right\} \|R_{\mathrm{D}_{带源级负反馈共栅极的输出电阻}}\end{align}$
输出电阻：	$\begin{array}{}\text{(96)}& R_{\mathrm{out}}& =[1+(g_{\mathrm{m}2}+g_{\mathrm{mb}2})r_{\mathrm{o}2}]r_{o1}+r_{o2}\text{(97)}& & \approx (g_{m2}+g_{mb2})\cdot r_{o1}{r}_{o2}\text{(98)}& & {}_{「\text{共}\text{源}\text{共}\text{栅}\text{输}\text{出}\text{阻}\text{抗}」}\end{array}$$\begin{align}R_{\mathrm{out}}&=\left[1+\left(g_{\mathrm{m} 2}+g_{\mathrm{mb} 2}\right) r_{\mathrm{o2}}\right] r_{o1}+r_{o2}\\ &\approx (g_{m2}+g_{mb2})\cdot r_{o1}r_{o2}\\&_{「共源共栅输出阻抗」}\end{align}$
带电流源负载的Cascade	$\begin{array}{}\text{(99)}& R_x& =r_o\parallel \frac{1}{g_m}\parallel {\frac{1}{g_{mb}}}_{M_1\text{漏}\text{端}\text{看}\text{进}\text{去}\text{的}\text{负}\text{载}}\text{(100)}& I_{out}& =g_{m1}{V}_{in}\frac{r_{o1}}{r_{o1}+r_o\parallel \frac{1}{g_m}\parallel \frac{1}{g_{mb}}}\text{(101)}& G_m& =\frac{I_{out}}{V_{in}}=\frac{g_{\mathrm{m}1}{r}_{\mathrm{o}1}[r_{\mathrm{o}2}(g_{\mathrm{m}2}+g_{\mathrm{mb}2})+1]}{r_{o1}{r}_{\mathrm{o}2}(g_{\mathrm{m}2}+g_{\mathrm{mb}2})+r_{\mathrm{o}1}+r_{\mathrm{o}2}}\text{(102)}& |A_v|& =G_m{R}_{out}=g_{\mathrm{m}1}{r}_{\mathrm{o}1}[r_{\mathrm{o}2}(g_{\mathrm{m}2}+g_{\mathrm{mb}2})+1]\end{array}$$\begin{align}R_x &= r_o \|\frac{1}{g_m}\|\frac{1}{g_{mb}}_{M_1漏端看进去的负载} \\I_{out} &= g_{m1}V_{in}\frac{r_{o1}}{r_{o1}+r_o\|\frac{1}{g_m}\|\frac{1}{g_{mb}} } \\ G_m &=\frac{I_{out}}{V_{in}}=\frac{g_{\mathrm{m} 1} r_{\mathrm{o1} }\left[r_{\mathrm{o2}}\left(g_{\mathrm{m} 2}+g_{\mathrm{mb} 2}\right)+1\right]}{r_{o1} r_{\mathrm{o2}}\left(g_{\mathrm{m} 2}+g_{\mathrm{mb} 2}\right)+r_{\mathrm{o1}}+r_{\mathrm{o2}}}\\|A_v|&=G_mR_{out}=g_{\mathrm{m} 1} r_{\mathrm{o1} }\left[r_{\mathrm{o2}}\left(g_{\mathrm{m} 2}+g_{\mathrm{mb} 2}\right)+1\right] \end{align}$	$\begin{array}{}\text{(103)}& if{G}_m& \approx g_m\text{(104)}& then|A_v|& =g_{m1}{R}_{out}\end{array}$$\begin{align}if \space G_{m}&\approx g_m\\ then\space|A_v|&= g_{m1}R_{out} \end{align}$
上图电流源用PMOS Cascade实现：	$\begin{array}{}\text{(105)}& R_{\text{out }}& =\{[1+(g_{\mathrm{m}2}+g_{\mathrm{mb}2})r_{o2}]r_{o1}+r_{02}\}\text{(106)}& & \parallel \{[1+(g_{\mathrm{m}3}+g_{\mathrm{mb}3})r_{o3}]r_{o4}+r_{o3}\}\text{(107)}& \text{(108)}& \left|A_v\right|& \approx g_{\mathrm{m}1}[\left(g_{\mathrm{m}2}{r}_{\mathrm{n}2}{r}_{\mathrm{m}1}\right)\parallel \left(g_{\mathrm{m}3}{r}_{03}{r}_{\mathrm{m}3}\right)]\end{array}$$\begin{align} R_{\text {out }}&=\left\{\left[1+\left(g_{\mathrm{m2}}+g_{\mathrm{mb2}}\right) r_{o2}\right] r_{o1}+r_{02}\right\} \\ &\space \space\space\space \|\left\{\left[1+\left(g_{\mathrm{m} 3}+g_{\mathrm{mb3}}\right) r_{o3}\right] r_{o4}+r_{o3}\right\}\\\\\left|A_v \right| &\approx g_{\mathrm{m} 1}\left[\left(g_{\mathrm{m} 2} r_{\mathrm{n} 2} r_{\mathrm{m} 1}\right)\|\left(g_{\mathrm{m} 3} r_{03} r_{\mathrm{m} 3}\right)\right] \end{align}$

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