General Considerations

Miller’s Law

$$ \begin{aligned} Z &= Z_1 + Z_2 \\ \frac{ V_X }{ Z_1 } &= \frac{ V_Y }{ Z_2 } = \frac{ V_X - V_Y }{ Z} \end{aligned} $$

$$ \begin{align} Z_1 &= \frac{ Z }{ 1 - A_v } \\ Z_2 &= \frac{ Z }{ 1- \dfrac{ 1 }{ A_v }} \end{align} $$

• $A_v = \dfrac{ V_{out} }{ V_{in} }$

• 注意: $Z$和main signal path之间的关系很重要

Association of Poles with Nodes

General Case

$$ \begin{aligned} H(s) &= \frac{V_{out}}{V_{in}}(s) \\ &= \frac{A_1}{1 + sR_SC_{in}} \frac{A_2}{1 + sR_1 C_N}\frac{1}{1 + sR_2C_P} \end{aligned} $$

Input Example

$$ \begin{aligned} Input \ Impedence \ Z_{in} &= (1+A) C_F \\ f_p &= \frac{1}{(1+A)R_SC_F} \end{aligned} $$

CG Stage With Parasitic Capacitances

• Neglecting CLM

$$ \begin{aligned} \omega_{in} &= [C_S (R_S || \frac{1}{g_{m1} + g_{mb1}})]^{-1} \\ &= [(C_{GS1} + C_{SB1})(R_S || \frac{1}{g_{m1} + g_{mb1}})]^{-1} \\ \omega_{out} &= [C_D R_D]^{-1} \\ &= [(C_{DG} + C_{DB}) R_D]^{-1} \\ \therefore A_v &= \frac{(g_m + g_{mb})R_D}{1 + (g_m + g_{mb})R_S} \frac{1}{1 + s/\omega_{in}}\frac{1}{1 + s/\omega_{out}} \end{aligned} $$

Common-Source Stage

High-frequency Model of CS Stage

电路图

Gain

• $C_{GD}$接在了输入和输出之间，因此根据Miller’s Law分别计算输出和输入阻抗

1. X to Ground

$$ \begin{aligned} Z_X &= C_{GS} + (1-A)C_{GD} \\ \omega_{in} &= \frac{1}{R_S [C_{GS} + (1 + g_mR_D)C_{GD}]} \end{aligned} $$

2. $V_{out}$ to Ground

$$ \begin{aligned} Z_X &= C_{DB} + (1-A^{-1})C_{GD} \\ &\approx C_{DB} + C_{GD} \\ \therefore \omega_{out} &= \frac{1}{R_S(C_{GS} + C_{GD})} \end{aligned} $$

3. when $R_S$ is large

$$ \begin{aligned} Z_X &= \frac{1}{sC_{eq}} || ( \frac{C_{GD} + C_{GS}}{C_{GD}} \frac{1}{g_m}) (???) \\ \therefore \omega_{out} &= \frac{1}{\Big[ R_D || \Big(\dfrac{C_{GD} + C_{GS}}{C_{GD}} \dfrac{1}{g_m}\Big) \Big] (C_{eq} + C_{DB})} \\ \therefore \frac{V_{out}}{V_{in}}(s) &= -\frac{g_mR_D}{(1 + \dfrac{s}{\omega_{in}}) (1 + \dfrac{s}{\omega_{out}})} \end{aligned} $$

Exact Transfer Function

1. High-Frequency Small Signal Equivalent Circuit

2. KCL方程

$$ \begin{aligned} 0 &= (V_{out} - V_X) sC_{GD} + V_{out} ( \frac{ 1 }{ R_D } + sC_{DB}) + g_mV_X\\ V_{X} &= \frac{ 1 }{ 1 + s R_S C_{GS} } V_{in} \end{aligned} $$

$$ \begin{equation} \begin{aligned} \therefore \ H(s) &= \frac{V_{out}}{V_{in}} \\ &= \frac{(C_{GD}s - g_m)R_D}{R_SR_D\xi s^2 + [R_S(1+g_mR_D)C_{GD} + R_SC_{GS} + R_D(C_{GD} + C_{DB})]s + 1} \\ \xi &= C_{GS}C_{GD} + C_{GS}C_{DB} + C_{GD}C_{DB} \end{aligned} \end{equation} $$

Two Pole Systems

• $D$: demoninator，分母

$$ \begin{equation} \begin{aligned} D &= (1 + \frac{s}{\omega_{p1}})(1 + \frac{s}{\omega_{p2}}) \\ &= \frac{s^2}{\omega_{p1}\omega_{p2}} + (\frac{1}{\omega_{p1}} + \frac{1}{\omega_{p2}})s + 1 \\ \end{aligned} \end{equation} $$

• 如果$\omega_{p2} \gg \omega_{p1}$，所以我们可以将s的系数近似看成$\dfrac{1}{\omega_{p1}}$

• 进而根据$s^2$的系数求出$\omega_{p2}$

3. 对于CS Stage, 有以下结论

$$ \begin{equation} \omega_{p1} = \frac{1}{R_S(1+g_mR_D)C_{GD} + R_SC_{GS} + R_D(C_{GD} + C_{DB})} \end{equation} $$

$$ \begin{equation} \begin{aligned} \omega_{p2} &= \frac{1}{\omega_{p1}} \frac{1}{R_S R_D \xi} \\ &= \frac{R_S(1+g_mR_D)C_{GD} + R_SC_{GS} + R_D(C_{GD} + C_{DB})}{R_SR_D\xi} \end{aligned} \end{equation} $$

• 若 $C_{GS} \gg (1+g_m R_D)C_{GD} + R_D(C_{GD} + C_{DB})/R_S$

$$ \begin{aligned} \omega_{p2} &\approx \frac{1}{R_D (C_{GD} + C_{DB})} \end{aligned} $$

• 与估算得到的结果相同

Summary of Approximation of Resistance

1. 一级估计

$$ \begin{aligned} Z_{in} &= \frac{ 1 }{ s[C_{GS} + (1 +g_m R_D) C_{GD}] } \end{aligned} $$

2. 高频估计(考虑输出结点的影响)

$$ \begin{aligned} Z_{in} &= \frac{ 1 }{ s C_{GS} } || \frac{ 1 + s R_D({C_{GD} +C_{DB}})}{ sC_{GD} ( 1 + g_m R_D + s R_D C_{DB})} \end{aligned} $$

Source Followers

Ciurcuit & Small Signal Model

Transfer Function and Poles

1. KCL

$$ \begin{aligned} V_1C_{GS}s + g_mV_1 &= V_{out}C_Ls \\ \therefore V_1 &= \frac{C_Ls}{g_m + C_{GS}s}V_{out} \end{aligned} $$

2. KVL

$$ \begin{aligned} V_{in} &= R_S[V_1C_{GS}s + (V_1 + V_{out})C_{GD}s] + V_1 + V_{out} \end{aligned} $$

3. Gain

$$ \begin{equation} \frac{V_{out}}{V_{in}}(s) = \frac{g_m + C_{GS}s}{R_S\xi s^2 + (g_mR_SC_{GD} + C_L + C_{GS})s + g_m} \end{equation} $$

4. Poles & Zeros

$$ \begin{aligned} \therefore \omega_{p1} &\approx \frac{g_m}{g_mR_SC_{GD} + C_L + C_{GS}} \\ &= \frac{1}{R_SC_{GD} + \dfrac{C_L + C_{GS}}{g_m}} \end{aligned} $$

• 如果 $R_S = 0$

$$ \begin{aligned} \omega_{p1} &= \frac{g_m}{C_L + C_{GS}} \end{aligned} $$

• Zeros

$$ \begin{equation} \omega_z = -\frac{ g_m }{ C_{GS}} \end{equation} $$

Impedence

Input Impedence

• $C_{GD}$ shunts the input, neglected.

$$ \begin{aligned} V_X &= \frac{I_X}{C_{GS}s} + (I_X + \frac{g_m I_X}{C_{GS}s}) (\frac{1}{g_{mb}} || \frac{1}{C_Ls}) \\ Z_{in} &= \frac{1}{C_{GS}s} + (1 + \frac{g_m}{C_{GS}s})\frac{1}{g_{mb} + C_L s} \\ Low \ Frequency: \\ g_{mb} &\gg |C_Ls| \\ \therefore Z_{in} &\approx \frac{1}{C_{GS}s}(1 + \frac{g_m}{g_{mb}}) + \frac{1}{g_{mb}} \end{aligned} $$

Output Impedence

Common-Gate Stage

$$ \begin{aligned} \frac{V_{out}}{V_{in}}(s) &= \frac{(g_m + g_{mb})R_D}{1 + (g_m + g_{mb})R_S} \frac{1}{\Big(1 + \dfrac{C_S}{g_m + g_{mb} + \dfrac{1}{R_S}}s \Big) (1 + R_DC_D s)} \\ \therefore Z_{in} &\approx \frac{Z_L}{(g_m + g_{mb})r_o} + \frac{1}{g_m + g_{mb}} \\ Z_L &= R_D || [1/(s C_D)] \end{aligned} $$

Cascode Stage

$$ \begin{aligned} \omega_{p,A} &= \frac{1}{R_S \Big[C_{GS1} + \Big( 1 + \dfrac{g_{m1}}{g_{m2} + g_{mb2}})C_{GD1}]} (???)\\ \omega_{p,X} &= \frac{g_{m2} + g_{mb2}}{2C_{GD1} + C_{DB1} + C_{SB1} + C_{GS2}} \\ \omega_{p,Y} &= \frac{1}{R_D(C_{DB2} + C_L + C_{GD2})} \end{aligned} $$

Appendix

Poles and Zeros