Ch5 Operational Amplifiers

Term	Definition
Operational amplifiers (op amps)	运算放大器
Gain	增益
Transistor	晶体管
Capacitor	电容
Diode	（电子）二极管
Pin	针
Dual In-line Package (DIP)	双列直插式封装
Saturate	饱和
Open-loop / Close-loop	开环 / 闭环
Feedback	反馈
Inverting amplifier	反相放大器
Noninverting amplifier	同相放大器
Voltage follower	电压跟随器; 电压输出器
Cascade	级联

Introduction

Dependent Element (有源)

运放是一个特性与电压控制电压源类似的电子元件。

Active elements behaving like a VCVS, and can be used in making VCCS or CCCS.

Perform mathematical operations (addition, subtraction, differentiation, integration etc.)

Use nodal method for analyzing op amp circuits.

Operational Amplifiers

An active circuit element designed to perform mathematical operations of addition, subtraction, multiplication, division, differentiation, and integration.

Inverting Input 反向 Noninverting Input 正向 (same polarity)

需要外接电源 $V^-,V^+$ 。

Offset Null 平衡（调零）

Two inputs.

Applying to inverting terminal 2 opposite polarity at output.
Applying to noninverting terminal 3 same polarity at output.

One output at terminal 6.

Highlight: The Equivalent Circuit of Op Amp. It consists of $v_d$ (input voltage), $R_i$ input resistance, $v_o$ output voltage, $R_o$ output resistance. $A$ , the voltage gain.

Convesion to dB $A_{\mathrm{dB}}=20\log A$ .

![image-20230320102002635](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320102002635.png)

$v_d$ : difference $v_o$ : output.

$A$ : 开环电压增益。闭环：从输出到输入。

![image-20230320102425526](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320102425526.png)

和运放的设计有关。

Op amp is power supplied ( $i_o$ is not necessarily zero)
Op amp is assumed to operate in linear range
Possibility of saturation must be borne in mind when designing with op amps.

The open-loop gain of an op amp is 100,000. Calculate the output voltage when there are inputs of +10 μV on the inverting terminal and +20 μV on the noninverting terminal.
$v_o=A(v_2-v_1)=A(20-10) \times 10^{-6}=1 \mathrm{~V}$
The output voltage of an op amp is −4 V when the noninverting input is 1 mV. If the open-loop gain of the op amp is $2 × 10^6$ , what is the inverting input? $1.002 \mathrm{~mV}$ .

![image-20230320103744242](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320103744242.png)

a. Input Resistance = $2 \mathrm{~M\Omega}$

b. Output Resistance = $50 \mathrm{~\Omega}$

c. Voltage Gain = $2 \times 10^5$ .

Very stiff question. Do approximation.

![image-20230320104846253](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320104846253.png)

Working with nonideal model is rather boring and not necessary for a high quality (ideal) op amp.

\left\{ \begin{matrix} \frac{v_s -v_2}{R_i}=\frac{v_2}{R_1}+\frac{v_2-v_o}{R_2}\\ \frac{v_o}{R_3}=\frac{A(v_2-v_s)-v_o}{R_o}+\frac{v_2-v_o}{R_2} \end{matrix} \right.

Where $R_i : 10^5$ to $10^{13}$

![image-20230320110536348](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320110536348.png)

Ideal Op Amp

$A \approx \infin,R_i \approx \infin,R_o \approx 0$ .

“虚短”，“虚断”

![image-20230320111352602](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320111352602.png)

v_{-}=0,v_s/10 k=(0-v_0)/20k \Rightarrow v_o/v_s \approx -2

![image-20230320111724270](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320111724270.png)

1mA=\frac{-v_o}{2k\Omega} \Rightarrow v_o=-2V \quad -1V

![image-20230320112133629](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320112133629.png)

No current.

\frac{0.9}{50k}+\frac{0.9-v_o}{100k}=0

Take full advantage of $i_1=0,i_2=0,v_1=v_2$ .
$i_o$ is not necessarily zero
Nodal method is preferred.

Inverting Amplifier

![image-20230320112827052](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320112827052.png)

![](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320113003488.png)

![image-20230320113607850](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320113607850.png)

-v_o/R=i_s \quad

![image-20230320114036574](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230320114036574.png)

Noninverting Amplifier

![image-20230322082010524](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230322082010524.png)

v_o=\left(1+\frac{R_f}{R_1}\right)v_i=A_vv_i

![image-20230322082414388](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230322082414388.png)

Summing Amplifier

![image-20230322084041972](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230322084041972.png)

\boxed{v_o=-\left(\frac{R_f}{R_1}v_1 + \frac{R_f}{R_2} v_2 + \frac{R_f}{R_3}v_3\right)}

![image-20230322084533090](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230322084533090.png)

![image-20230322090349896](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230322090349896.png)

![image-20230322090537570](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230322090537570.png)

![image-20230322090848134](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230322090848134.png)

Cascaded Op Amp Circuits

![image-20230322091412288](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230322091412288.png)

Problems

Chapter 5, Problem 1

![image-20230402153922070](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402153922070.png)

Chapter 5, Problem 5.

![image-20230402154336672](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402154336672.png)

![image-20230402154411301](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402154411301.png)

Chapter 5, Problem 8

![image-20230402154516168](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402154516168.png)

Chapter 5, Problem 9

Chapter 5, Problem 15

![image-20230402160110435](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402160110435.png)

![image-20230402160120774](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402160120774.png)

Chapter 5, Problem 19

![image-20230402160726347](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402160726347.png)

Chapter 5, Problem 24

![image-20230402161336142](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402161336142.png)

Chapter 5, Problem 36

![image-20230402172804245](C:\Users\Steven Meng\AppData\Roaming\Typora\typora-user-images\image-20230402172804245.png)

To get $R_{\mathrm{Th}}$ , turn off all voltage source and apply current source $I_o$ at terminals a-b.