# 运算放大器

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

# 概论

• high voltage gain
• high input impedance
• low output impedance

• speed
• output voltage swings
• power dissipation

• Gain
• Small-signal Bandwidth
• unity-gain $f_u$
• 3-dB frequency $f_{3-\tx{dB}}$
• Large-Signal Behavior
• Output Swing
• Linearity
• Noise and Offset: determine the minimum signal level that can be processed
• Supply Rejection
• Slew Rate: $V_\tx{out}$ 对时间的比值（输入阶跃，输出的最大斜率）

# One-Stage Op Amps

## Simple OTA

$A_v = g_{m1} \frac{r_O}{2}$ $\tx{if } {\rm r_{O2}=r_{O4}=r_O} \tx{ then}\\ \tx{BW}=\frac{1}{2 \pi \dfrac{r_O}{2}(C_L+C_{n1})}$

## Telescopic Cascode Op Amps

 \begin{aligned} A_v &= (g_m r_{DS})_1\\ R_\tx{out} &= r_{DS1} \end{aligned} \begin{aligned} A_v &= (g_m r_{DS})_1(g_m r_{DS})_2\\ R_\tx{out} &= r_{DS1}(g_m r_{DS})_2 \end{aligned} $$\tx{BW} = \frac{1}{2\pi R_\tx{out}C_L}\\ \tx{GBW} = \frac{g_{m1}}{2\pi C_L}$$ 注意纵轴取了 log，$A_{v1}$ 和 $A_{v2}$ 实际是相乘而非相加。

$\begin{cases} V_\tx{out} \leq V_X + V_\tx{TH2}\\ V_\tx{out} \geq V_b-V_\tx{TH4}\\ V_X=V_b-V_{GS4} \end{cases}\\ \Rightarrow V_b-V_\tx{TH4} \leq V_\tx{out} \leq V_b-V_{GS4}+V_\tx{TH2}\\ \Rightarrow V_\max - V_\min = V_\tx{TH4}-(V_{GS4}-V_\tx{TH2})$

## Folded-Cascode Op Amps

the drawbacks of telescopic cascode op amps:

• limited output swings
• difficulty in choosing equal input and output CM levels

In order to alleviate the drawbacks, a “folded-cascode” op amp can be used.

1. (b) requires more bias current than (a)
• (a): $I_1,I_2$
• (b): $I_{SS},I_1,I_2$ which means that (b) consumes more power.
2. (b) has a wider voltage swing, especially when input and output shorted.
• (a): the input CM level cannot exceed $V_{b1}-V_{GS3}+V_\tx{TH1}$
• (b): the input CM level cannot exceed $V_{b1}-V_{GS3}-\vert V_\tx{THP}\vert$

$折叠式：\\ \vert A_v \vert \approx g_{m1} \{[g_{m3}r_{O3}(r_{O1}\Vert r_{O5})] \Vert [g_{m7} r_{O7}r_{O9}]\}\\ 套筒式：\\ A_v \approx g_{m1}[(g_{m3}r_{O3}r_{O1})\Vert (g_{m5}r_{O5}r_{O7})]$

• consumes more power 功耗大
• the gain is 2~3 times lower than telescopic cascode 增益小
• the pole is quite closer to the origin than that associated with the source of cascode devices in a telescopic topology 极点靠近原点，速度会变慢

# Two-Stage Op Amps

$A_v \approx {g_{m1,2}[(g_{m3,4}+g_{mb3,4})r_{O3,4}r_{O1,2}]\Vert (g_{m5,6}+g_{mb5,6})r_{O5,6}r_{O7,8}}\\ \times [g_{m9,10}(r_{O9,10}\Vert r_{O11,12})]$

# Gain Boosting

## Basic Idea

One-Stage Op 的增益不够，Two-Stage Op 的速度又太慢了（电容太大），为了解决这些问题，我们又提出了 Gain Boosting. 它的原理如下：在单个 CS Stage 前面再加一个放大器，并且将反相输入端和 S 是连在一起的

$G_m = \frac{I_\tx{out}}{V_\tx{in}}=\frac{A_1g_m}{1+(A_1+1)g_m R_S}$

$R_\tx{out}$ 的计算如下：$I_{r_O}=\dfrac{V_X-R_SI_X}{r_O}$，且 $I_0 = (-A_1R_SI_X-R_SI_X)g_m$，从而：

$I_X = (-A_1R_SI_X-R_SI_X)g_m + \frac{V_X-R_SI_X}{r_O}\\ \Downarrow\\ R_\tx{out} = r_O+(A_1+1)g_m r_O R_S+R_S$

![Figure 9.28](assets/images/Figure%209.28.jpg) 考虑 Fig. 9.28 从源端看进去的电阻，计算过程与上面类似，在此直接给出结果： $$R_X = \frac{R_D+r_O}{1+(A_1+1)g_m r_O}$$ 可以看出分母中的 $g_m$ 同样增大了 $(1+A_1)$ 倍。

\begin{aligned} \vert A_v \vert &\approx g_{m1}\left[ r_O+(A_1+1)g_m r_O R_S+R_S \right]\\ &\approx g_{m1}g_{m2}r_{O1}r_{O2}(A_1+1) \end{aligned}

# Common-Mode Feedback

## Basic Concept

• sensing the output CM level
• comparison with a reference
• returning the error to the amplifier’s bias network

• $I_{D3,4}>\frac{I_{SS}}{2}$, M3 and M4 enter the triode region so that their drain currents fall to $I_{SS}/2$
• $I_{D3,4}<\frac{I_{SS}}{2}$, M5 enters the triode region, thereby producing only $2I_{D3,4}$