# 负反馈放大器

## 概述

• 能根据需要增加或减少输入阻抗（根据反馈形式）
• 能根据需要增加或减少输出阻抗（根据反馈形式）
• 降低信号失真（增加线性）
• 展宽频带
• 是增益对参数扰动的灵敏度降低
• 能够控制放大器的阶跃响应

• 如果设计不周，可能会导致工作不稳定
• 放大器的实际增益降低
• 接入反馈的放大器（也称闭环放大器）的输入、输出阻抗与未接入反馈的放大器（也称开环放大器）的增益有关。

## 经典反馈

### 降低增益

${\displaystyle V_{out}=A_{OL}\cdot V'_{in}}$

${\displaystyle V'_{in}=V_{in}-\beta \cdot V_{out}}$

${\displaystyle V_{out}=A_{OL}(V_{in}-\beta \cdot V_{out})}$

${\displaystyle V_{out}(1+\beta \cdot A_{OL})=V_{in}\cdot A_{OL}}$

${\displaystyle A_{\mathrm {fb} }={\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}={\frac {A_{OL}}{1+\beta \cdot A_{OL}}}}$

L = β AOL在反馈分析中十分常见，有时也被称作是环路增益（loop gain）。(1 + β AOL)在分析中也较为常见，被称为倒灵敏度因数（desensitivity factor）或改善因数（improvement factor）。

### 展宽频带

${\displaystyle A_{OL}(f)={\frac {A_{0}}{1+jf/f_{C}}}\ ,}$

${\displaystyle A_{fb}(f)={\frac {A_{OL}}{1+\beta A_{OL}}}}$
${\displaystyle ={\frac {A_{0}/(1+jf/f_{C})}{1+\beta A_{0}/(1+jf/f_{C})}}}$
${\displaystyle ={\frac {A_{0}}{1+jf/f_{C}+\beta A_{0}}}}$
${\displaystyle ={\frac {A_{0}}{(1+\beta A_{0})\left(1+j{\frac {f}{(1+\beta A_{0})f_{C}}}\right)}}\ .}$

## 信号流分析

"In contrast to block diagram and two-port approaches to the feedback network analysis problem, signal flow methods mandate no a priori assumptions as to the unilateral or bilateral properties of the open loop and feedback subcircuits. Moreover, they are not predicated on mutually independent open loop and feedback subcircuit transfer functions, and they do not require that feedback be implemented only globally. Indeed signal flow techniques do not even require explicit identification of the open loop and feedback subcircuits. Signal flow thus removes the detriments pervasive of conventional feedback network analyses but additionally, it proves to be computationally efficient as well."

"变量xS, xO表示输入和输出信号，此外准确示出了，其他两个变量xi, xj通过控制变量P联系起来。变量aij为权重分支。变量xi, xj和控制变量P模拟了一个受控的发电机，或者说是电路两个结点间电压和电流的关系。

${\displaystyle x_{O}=a_{11}x_{S}+a_{12}x_{j}\ ,}$
${\displaystyle x_{i}=a_{21}x_{S}+a_{22}x_{j}\ ,}$

${\displaystyle x_{j}=Px_{i}\ .}$

${\displaystyle {\frac {x_{O}}{x_{S}}}=a_{11}+{\frac {a_{12}a_{21}P}{1-Pa_{22}}}\ .}$

## 用二端口网络分析反馈

### 用二端口替代反馈网络

g11 g12 g21 g22
${\displaystyle {\frac {1}{R_{f}+R_{2}}}}$  ${\displaystyle -{\frac {R_{2}}{R_{2}+R_{f}}}}$  ${\displaystyle {\frac {R_{2}}{R_{2}+R_{f}}}}$  ${\displaystyle R_{2}//R_{f}\ }$

### 有负载的开环增益

${\displaystyle A_{OL}={\frac {\beta i_{B}}{i_{S}}}=g_{m}R_{C}\left({\frac {\beta }{\beta +1}}\right)\left({\frac {R_{1}}{R_{22}+{\frac {r_{\pi 2}+R_{C}}{\beta +1}}}}\right)\ .}$

### 反馈增益

${\displaystyle A_{FB}={\frac {A_{OL}}{1+{\beta }_{FB}A_{OL}}}}$
${\displaystyle A_{FB}={\frac {A_{OL}}{1+{\frac {R_{2}}{R_{2}+R_{f}}}A_{OL}}}\ ,}$

### 输入和输出电阻

#### 电阻定义的背景

${\displaystyle V_{x}=I_{x}R_{in}+\beta v_{out}\ ,}$

${\displaystyle R_{in}(fb)={\frac {V_{x}}{I_{x}}}=\left(1+\beta A_{v}\right)R_{in}\ .}$

## 参考文献及注释

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2. ^ Palumbo, Gaetano & Salvatore Pennisi. Feedback amplifiers: theory and design. Boston/Dordrecht/London: Kluwer Academic. 2002: 64. ISBN 0792376439.
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4. ^ Currently on display at Bell Laboratories in Mountainside, New Jersey
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6. ^ Black, Harold. Inventing the negative feedback amplifier. IEEE Spectrum. Dec 1977.
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8. ^ Willy M. C. Sansen. Analog design essentials. New York; Berlin: Springer. 2006: §0513–§0533, p. 155–165. ISBN 0-387-25746-2.
9. ^ Partha Pratim Sahu. §8.2 Partitioning. VLSI Design. McGraw Hill Education. 2013: 253. ISBN 9781259029844. dividing a circuit into smaller parts ...[so]...the number of connections between parts is minimized
10. ^ Gaetano Palumbo, Salvatore Pennisi. Feedback Amplifiers: Theory and Design. Springer Science & Business Media. 2002. ISBN 9780792376439. In real cases, unfortunately, blocks...cannot be assumed to be unidirectional.
11. ^ Wai-Kai Chen. §1.2 Methods of analysis. Feedback, Nonlinear, and Distributed Circuits. CRC Press. 2009: 1-3. ISBN 9781420058826.
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14. ^ Gaetano Palumbo & Salvatore Pennisi. Feedback amplifiers: theory and design. Springer Science & Business Media. 2002: 66. ISBN 9780792376439.
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16. ^ For an introduction, see Rahul Sarpeshkar. Chapter 10: Return ratio analysis. Ultra Low Power Bioelectronics: Fundamentals, Biomedical Applications, and Bio-Inspired Systems. Cambridge University Press. 2010: 240 ff. ISBN 9781139485234.
17. ^ Wai-Kai Chen. §11.2 Methods of analysis. Circuit Analysis and Feedback Amplifier Theory. CRC Press. 2005: 11–2 ff. ISBN 9781420037272.
18. ^ Gaetano Palumbo, Salvatore Pennisi. §3.3 The Rosenstark Method and §3.4 The Choma Method. Feedback Amplifiers: Theory and Design. Springer Science & Business Media. 2002: 69 ff. ISBN 9780792376439.
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21. ^ Ashok K. Goel Feedback topologies 互联网档案馆存檔，存档日期2008-02-29.
22. ^ Zimmer T & Geoffroy D: Feedback amplifier. [2013-04-28]. （原始内容存档于2013-07-02）.
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25. ^ A. S. Sedra and K.C. Smith. Microelectronic Circuits Fifth Edition. New York: Oxford. 2004. Example 8.4, pp. 825–829 and PSpice simulation pp. 855–859 [2015-02-10]. ISBN 0-19-514251-9. （原始内容存档于2009-02-04）.
26. ^ 如果考虑前馈的话，它会使开环增益发生变化，通常来说这个量相对于开环增益本身是相当小的，所以可以舍弃。还要注意到主放大器模块是单向的。