Sun's Blog

模拟集成电路期末复习：用漫画串起运放、反馈、滤波和限制

Thu, 28 May 2026 00:00:00 GMT

这篇是博客发布流程的第一篇测试文：不用重新生成图片，只复用已经整理好的模拟集成电路漫画页，再配一份期末复习用的文字和公式。目标不是把课件完整搬一遍，而是把考试里最容易连成一条线的概念先钉住：

理想运放规则 -> 负反馈建立闭环 -> RC 进入反馈变成滤波器 -> 开关电容把电阻变成时钟控制 -> 静态误差和动态限制决定真实边界。

1. 理想运放：先记住两条“分析规则”

理想运放在线性负反馈里最常用的两条近似是：

$$ v_+ \approx v_- $$

$$ i_+ \approx i_- \approx 0 $$

第一条常被叫作“虚短”：两个输入端电位近似相等。第二条常被叫作“虚断”：输入端电流近似为零。

但考试里真正要小心的是条件：这两条规则只在负反馈成立、运放未饱和、开环增益足够大时好用。它不是物理短路，也不是任何电路里都成立。

一个更稳的分析动作是：

先判断是不是负反馈。
再判断输出有没有可能饱和。
最后才用 $v_+ \approx v_-$ 和 $i_\pm \approx 0$ 列方程。

2. 负反馈：把“大而不准”的开环增益变成“准而可控”的闭环增益

运放本身开环增益 $A$ 很大，但不稳定、离散性强、频率一高还会下降。负反馈的意义是牺牲一部分增益，换来闭环传输的可控性。

最常见的反馈形式可以写成：

$$ A_f=\frac{A}{1+A\beta} $$

当环路增益足够大：

$$ A\beta \gg 1 $$

闭环增益近似为：

$$ A_f \approx \frac{1}{\beta} $$

也就是说，闭环结果主要由反馈网络 $\beta$ 决定，而不是由运放开环增益 $A$ 决定。这就是反相、同相、仪表放大器、有源滤波器能稳定工作的底层原因。

复习时可以把负反馈理解成一个自动纠错系统：输出偏离目标，反馈就把误差送回输入端，让差模输入变小。

3. 二阶滤波器：$\omega_n$ 定位置，$Q$ 定尖锐程度

有源滤波器的核心不是背每张电路图，而是读懂传递函数。标准二阶分母通常写成：

$$ s^2+\frac{\omega_n}{Q}s+\omega_n^2 $$

这里两个参数最重要：

$\omega_n$：自然角频率，决定响应大概发生在哪个频率附近。
$Q$：品质因数，决定响应尖不尖、峰值强不强、阶跃后振不振。

低通、高通、带通、带阻、全通常常共用类似的二阶分母，只是分子不同。判断滤波器类型时，先看两个极限：

$$ \omega \to 0 $$

$$ \omega \to \infty $$

如果低频通过、高频衰减，是低通；如果低频被压掉、高频通过，是高通；如果两边都压掉、中间通过，是带通。

考试里 $Q$ 的直觉很关键：$Q$ 大，选择性更强，但也更容易峰化、振铃，对元件误差更敏感。

4. 开关电容：用时钟把电阻“合成”出来

集成电路里，大电阻往往面积大、精度差。开关电容电路用一个电容和两相不交叠时钟来搬运电荷，等效出一个电阻。

每个时钟周期转移的电荷近似为：

$$ \Delta Q=C\Delta V $$

若时钟频率为 $f_s$，平均电流为：

$$ I=Cf_s\Delta V $$

和电阻电流：

$$ I=\frac{\Delta V}{R} $$

对比得到等效电阻：

$$ R_{\text{eq}}=\frac{1}{Cf_s} $$

这个公式很适合期末记忆：开关电容的“阻值”由电容和时钟频率决定。$f_s$ 越高，单位时间搬运电荷越多，等效电阻越小。

它的工程意义也很直接：片上电容比例和时钟频率通常比绝对电阻更好控制，所以开关电容滤波器适合做集成的精确时间常数。

5. 静态限制：输入偏置电流会把“虚断”打破成误差

理想运放假设输入电流为零，但真实运放输入级总会有偏置电流。若输入端等效电阻较大，偏置电流会在电阻上形成电压误差：

$$ V_{\text{err}} \approx I_B R $$

这类误差在积分器、传感器前端、跨阻放大器里尤其明显，因为这些电路常常有高阻节点或电容长期积累误差。

一个常见补偿思路是让两输入端看到的等效电阻尽量匹配，使两个偏置电流造成的压降相互抵消。若偏置电流不相等，还要考虑输入失调电流：

$$ I_{OS}=|I_{B+}-I_{B-}| $$

期末做题时不要只写“输入电流约为零”。如果题目给了 $I_B$、$I_{OS}$ 或高阻反馈网络，就要立刻想到静态误差。

6. 动态限制：压摆率决定大信号最快能变多快

增益带宽积描述小信号频率响应，但压摆率描述大信号变化速度上限。运放输出电压最快变化率定义为：

$$ SR=\max\left|\frac{dV_o}{dt}\right| $$

对正弦输出：

$$ v_o(t)=V_p\sin(2\pi ft) $$

最大斜率为：

$$ \max\left|\frac{dv_o}{dt}\right|=2\pi fV_p $$

为了不发生压摆率失真，需要：

$$ 2\pi fV_p \le SR $$

因此满功率带宽为：

$$ f_{\text{PB}}=\frac{SR}{2\pi V_p} $$

这个式子很适合区分两个概念：

小信号带宽：由开环频响、闭环增益和补偿决定。
大信号带宽：受 $SR$ 和输出幅度限制。

如果题目说“频率不高但波形变三角”，优先想到压摆率；如果题目说“小幅正弦幅度滚降”，优先想到增益带宽积。

7. 一张期末复习地图

主题	先抓住什么	高频考法
理想运放	负反馈下 $v_+\approx v_-$，$i_\pm\approx 0$	判断虚短虚断是否能用
负反馈	$A_f=A/(1+A\beta)$	环路增益大时闭环由反馈网络决定
有源滤波	看 $H(s)$ 的极点、零点和极限	判断滤波类型、Bode 斜率、稳定性
二阶响应	$\omega_n$ 定频率，$Q$ 定峰化和带宽	Butterworth、带宽、振铃、灵敏度
开关电容	$R_{\text{eq}}=1/(Cf_s)$	用电荷守恒推等效电阻
静态误差	$I_B R$、$V_{OS}$、CMRR、PSRR	判断误差来源和补偿方法
动态限制	GBP 小信号，SR 大信号	区分带宽滚降和压摆率失真

最后复习建议：不要把每章当孤立知识点。模拟集成电路的期末题通常是在问同一件事：

这个电路在理想模型下怎么工作？真实运放的哪些非理想性会让它偏离？应该用什么电路或参数去补救？

能按这三个问题组织答案，很多推导就不会散。

Markdown Extended Features

Wed, 01 May 2024 00:00:00 GMT

GitHub Repository Cards

You can add dynamic cards that link to GitHub repositories, on page load, the repository information is pulled from the GitHub API.

::github{repo="Fabrizz/MMM-OnSpotify"}

Create a GitHub repository card with the code ::github{repo="<owner>/<repo>"}.

::github{repo="saicaca/fuwari"}

Admonitions

Following types of admonitions are supported: note tip important warning caution

:::note Highlights information that users should take into account, even when skimming. :::

:::tip Optional information to help a user be more successful. :::

:::important Crucial information necessary for users to succeed. :::

:::warning Critical content demanding immediate user attention due to potential risks. :::

:::caution Negative potential consequences of an action. :::

Basic Syntax

:::note
Highlights information that users should take into account, even when skimming.
:::

:::tip
Optional information to help a user be more successful.
:::

Custom Titles

The title of the admonition can be customized.

:::note[MY CUSTOM TITLE] This is a note with a custom title. :::

:::note[MY CUSTOM TITLE]
This is a note with a custom title.
:::

GitHub Syntax

[!TIP] The GitHub syntax is also supported.

> [!NOTE]
> The GitHub syntax is also supported.

> [!TIP]
> The GitHub syntax is also supported.

Spoiler

You can add spoilers to your text. The text also supports Markdown syntax.

The content :spoiler[is hidden ayyy]!

The content :spoiler[is hidden **ayyy**]!

Expressive Code Example

Wed, 10 Apr 2024 00:00:00 GMT

Here, we'll explore how code blocks look using Expressive Code. The provided examples are based on the official documentation, which you can refer to for further details.

Expressive Code

Syntax Highlighting

Regular syntax highlighting

console.log('This code is syntax highlighted!')

Rendering ANSI escape sequences

ANSI colors:
- Regular: [31mRed[0m [32mGreen[0m [33mYellow[0m [34mBlue[0m [35mMagenta[0m [36mCyan[0m
- Bold:    [1;31mRed[0m [1;32mGreen[0m [1;33mYellow[0m [1;34mBlue[0m [1;35mMagenta[0m [1;36mCyan[0m
- Dimmed:  [2;31mRed[0m [2;32mGreen[0m [2;33mYellow[0m [2;34mBlue[0m [2;35mMagenta[0m [2;36mCyan[0m

256 colors (showing colors 160-177):
[38;5;160m160 [38;5;161m161 [38;5;162m162 [38;5;163m163 [38;5;164m164 [38;5;165m165[0m
[38;5;166m166 [38;5;167m167 [38;5;168m168 [38;5;169m169 [38;5;170m170 [38;5;171m171[0m
[38;5;172m172 [38;5;173m173 [38;5;174m174 [38;5;175m175 [38;5;176m176 [38;5;177m177[0m

Full RGB colors:
[38;2;34;139;34mForestGreen - RGB(34, 139, 34)[0m

Text formatting: [1mBold[0m [2mDimmed[0m [3mItalic[0m [4mUnderline[0m

Editor & Terminal Frames

Code editor frames

console.log('Title attribute example')

<!-- src/content/index.html -->
<div>File name comment example</div>

Terminal frames

echo "This terminal frame has no title"

Write-Output "This one has a title!"

Overriding frame types

echo "Look ma, no frame!"

# Without overriding, this would be a terminal frame
function Watch-Tail { Get-Content -Tail 20 -Wait $args }
New-Alias tail Watch-Tail

Text & Line Markers

Marking full lines & line ranges

// Line 1 - targeted by line number
// Line 2
// Line 3
// Line 4 - targeted by line number
// Line 5
// Line 6
// Line 7 - targeted by range "7-8"
// Line 8 - targeted by range "7-8"

Selecting line marker types (mark, ins, del)

function demo() {
  console.log('this line is marked as deleted')
  // This line and the next one are marked as inserted
  console.log('this is the second inserted line')

  return 'this line uses the neutral default marker type'
}

Adding labels to line markers

// labeled-line-markers.jsx
<button
  role="button"
  {...props}
  value={value}
  className={buttonClassName}
  disabled={disabled}
  active={active}
>
  {children &&
    !active &&
    (typeof children === 'string' ? <span>{children}</span> : children)}
</button>

Adding long labels on their own lines

// labeled-line-markers.jsx
<button
  role="button"
  {...props}

  value={value}
  className={buttonClassName}

  disabled={disabled}
  active={active}
>

  {children &&
    !active &&
    (typeof children === 'string' ? <span>{children}</span> : children)}
</button>

Using diff-like syntax

+this line will be marked as inserted
-this line will be marked as deleted
this is a regular line

--- a/README.md
+++ b/README.md
@@ -1,3 +1,4 @@
+this is an actual diff file
-all contents will remain unmodified
 no whitespace will be removed either

Combining syntax highlighting with diff-like syntax

  function thisIsJavaScript() {
    // This entire block gets highlighted as JavaScript,
    // and we can still add diff markers to it!
-   console.log('Old code to be removed')
+   console.log('New and shiny code!')
  }

Marking individual text inside lines

function demo() {
  // Mark any given text inside lines
  return 'Multiple matches of the given text are supported';
}

Regular expressions

console.log('The words yes and yep will be marked.')

Escaping forward slashes

echo "Test" > /home/test.txt

Selecting inline marker types (mark, ins, del)

function demo() {
  console.log('These are inserted and deleted marker types');
  // The return statement uses the default marker type
  return true;
}

Word Wrap

Configuring word wrap per block

// Example with wrap
function getLongString() {
  return 'This is a very long string that will most probably not fit into the available space unless the container is extremely wide'
}

// Example with wrap=false
function getLongString() {
  return 'This is a very long string that will most probably not fit into the available space unless the container is extremely wide'
}

Configuring indentation of wrapped lines

// Example with preserveIndent (enabled by default)
function getLongString() {
  return 'This is a very long string that will most probably not fit into the available space unless the container is extremely wide'
}

// Example with preserveIndent=false
function getLongString() {
  return 'This is a very long string that will most probably not fit into the available space unless the container is extremely wide'
}

Collapsible Sections

// All this boilerplate setup code will be collapsed
import { someBoilerplateEngine } from '@example/some-boilerplate'
import { evenMoreBoilerplate } from '@example/even-more-boilerplate'

const engine = someBoilerplateEngine(evenMoreBoilerplate())

// This part of the code will be visible by default
engine.doSomething(1, 2, 3, calcFn)

function calcFn() {
  // You can have multiple collapsed sections
  const a = 1
  const b = 2
  const c = a + b

  // This will remain visible
  console.log(`Calculation result: ${a} + ${b} = ${c}`)
  return c
}

// All this code until the end of the block will be collapsed again
engine.closeConnection()
engine.freeMemory()
engine.shutdown({ reason: 'End of example boilerplate code' })

Line Numbers

Displaying line numbers per block

// This code block will show line numbers
console.log('Greetings from line 2!')
console.log('I am on line 3')

// Line numbers are disabled for this block
console.log('Hello?')
console.log('Sorry, do you know what line I am on?')

Changing the starting line number

console.log('Greetings from line 5!')
console.log('I am on line 6')

Simple Guides for Fuwari

Mon, 01 Apr 2024 00:00:00 GMT

Cover image source: Source

This blog template is built with Astro. For the things that are not mentioned in this guide, you may find the answers in the Astro Docs.

Front-matter of Posts

---
title: My First Blog Post
published: 2023-09-09
description: This is the first post of my new Astro blog.
image: ./cover.jpg
tags: [Foo, Bar]
category: Front-end
draft: false
---

Attribute	Description
`title`	The title of the post.
`published`	The date the post was published.
`description`	A short description of the post. Displayed on index page.
`image`	The cover image path of the post.<br/>1. Start with `http://` or `https://`: Use web image<br/>2. Start with `/`: For image in `public` dir<br/>3. With none of the prefixes: Relative to the markdown file
`tags`	The tags of the post.
`category`	The category of the post.
`draft`	If this post is still a draft, which won't be displayed.

Where to Place the Post Files

Your post files should be placed in src/content/posts/ directory. You can also create sub-directories to better organize your posts and assets.

src/content/posts/
├── post-1.md
└── post-2/
    ├── cover.png
    └── index.md

Markdown Example

Sun, 01 Oct 2023 00:00:00 GMT

An h1 header

Paragraphs are separated by a blank line.

2nd paragraph. Italic, bold, and monospace. Itemized lists look like:

this one
that one
the other one

Note that --- not considering the asterisk --- the actual text content starts at 4-columns in.

Block quotes are written like so.

They can span multiple paragraphs, if you like.

Use 3 dashes for an em-dash. Use 2 dashes for ranges (ex., "it's all in chapters 12--14"). Three dots ... will be converted to an ellipsis. Unicode is supported. ☺

An h2 header

Here's a numbered list:

first item
second item
third item

Note again how the actual text starts at 4 columns in (4 characters from the left side). Here's a code sample:

# Let me re-iterate ...
for i in 1 .. 10 { do-something(i) }

As you probably guessed, indented 4 spaces. By the way, instead of indenting the block, you can use delimited blocks, if you like:

define foobar() {
    print "Welcome to flavor country!";
}

(which makes copying & pasting easier). You can optionally mark the delimited block for Pandoc to syntax highlight it:

import time
# Quick, count to ten!
for i in range(10):
    # (but not *too* quick)
    time.sleep(0.5)
    print i

An h3 header

Now a nested list:

First, get these ingredients:
- carrots
- celery
- lentils
Boil some water.

Dump everything in the pot and follow this algorithm:

 find wooden spoon
 uncover pot
 stir
 cover pot
 balance wooden spoon precariously on pot handle
 wait 10 minutes
 goto first step (or shut off burner when done)

Do not bump wooden spoon or it will fall.

Notice again how text always lines up on 4-space indents (including that last line which continues item 3 above).

Here's a link to a website, to a local doc, and to a section heading in the current doc. Here's a footnote [^1].

[^1]: Footnote text goes here.

Tables can look like this:

size material color

9 leather brown 10 hemp canvas natural 11 glass transparent

Table: Shoes, their sizes, and what they're made of

(The above is the caption for the table.) Pandoc also supports multi-line tables:

keyword text

red Sunsets, apples, and other red or reddish things.

green Leaves, grass, frogs and other things it's not easy being.

A horizontal rule follows.

Here's a definition list:

apples : Good for making applesauce. oranges : Citrus! tomatoes : There's no "e" in tomatoe.

Again, text is indented 4 spaces. (Put a blank line between each term/definition pair to spread things out more.)

Here's a "line block":

| Line one | Line too | Line tree

and images can be specified like so:

Inline math equations go in like so: $\omega = d\phi / dt$. Display math should get its own line and be put in in double-dollarsigns:

$$I = \int \rho R^{2} dV$$

$$ \begin{equation*} \pi =3.1415926535 ;8979323846;2643383279;5028841971;6939937510;5820974944 ;5923078164;0628620899;8628034825;3421170679;\ldots \end{equation*} $$

And note that you can backslash-escape any punctuation characters which you wish to be displayed literally, ex.: `foo`, *bar*, etc.

Include Video in the Posts

Tue, 01 Aug 2023 00:00:00 GMT

Just copy the embed code from YouTube or other platforms, and paste it in the markdown file.

---
title: Include Video in the Post
published: 2023-10-19
// ...
---

<iframe width="100%" height="468" src="https://www.youtube.com/embed/5gIf0_xpFPI?si=N1WTorLKL0uwLsU_" title="YouTube video player" frameborder="0" allowfullscreen></iframe>

YouTube

Bilibili