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Welcome back! You've successfully signed in. # Lecture 4 Operational Amplifier

## Background

Operational amplifier (Op Amp)

### Functions

When combined with resistors, capacitors, and inductors, can perform various functions:

1. amplification/scaling
2. sign changing
4. integration
5. differentiation
6. analog filtering (滤波器)
7. nonlinear functions (exponential, log, sqrt)
8. Isolate input from output.

### History

• The Operational Amplifier (op amp) was invented in the 40’s.
• Bell Labs filed a patent in 1941.
• Many consider the first practical op amp to be the vacuum tube K2-W invented in 1952 by George Philbrick.
• Bob Widlar at Fairchild invented the uA702 op amp in 1963.
• Until uA741, released in 1968, op amps became relatively inexpensive and started on the road to ubiquity.

## Op Amp Terminals

Five important terminals

The inverting input

The noninverting input

The output

The positive (+) power supply

The negative (-) power supply

## practical Op Amps

$$V_o = AV_d = A(V_p-V_n)$$

practical实际值 ideal理想值
$A$ open loop gain 开环增益 $10^5,10^8$ $\infty$
$R_i$ input resistance $10^5,10^{13}$ $\infty$
$R_0$ output resistance $10, 100$ 0

In practice, we have output voltage limit : $\pm V_{cc}$
\begin{equation} \begin{aligned} V_o = \left\{ \begin{array}{lr} -V_{cc} ,(AV_d<-V_{cc}) \\AVd,(-V_{cc}\leq AV_d\leq V_{cc}) \\V_{cc} ,(AV_d>V_{cc}) \end{array} \right. \end{aligned} \end{equation}

How do we know whether the Op Amp is operation in linear region?

$-\frac{V_{cc}}{A}\leq V_d \leq \frac{V_{cc}}{A} \Rightarrow$ linear region

otherwise saturation

## Example

$$V_0 = \frac{R_1+R_2}{R_2}$$
tradeoff between circuit gain and linear dynamic range of $V_s$

1. (a) $10^6$ (b) $5$
2. (a) $[-10\mu V ,10 \mu V ]$ (b) $[-2V,2V]$
3. (a) not precise (b) precise
4. (b) negative feed back, stable voltage supply

## Ideal Op Amp

Ideal:

1. $A = \infty$
2. $R_i = \infty$
3. $R_o = 0$

### Two Golden Rules (Very important)

\begin{equation} \begin{aligned} \left. \begin{array}{lr} A = \infty\\ V_o = A(V_p-V_n) \\V_o ,finite\space number \\ Assumption \space linear \space region \end{array} \right\} \Rightarrow V_p = V_n \end{aligned} \end{equation}

This is called virtual short虚短

$R_i = \infty \Rightarrow i_p= i_n = 0$

This is called virtual open 虚断

### 反向比例方大器 inverting Amplifier

Find $\frac{V_o}{V_s}$
$$\frac{0=V_s}{R_s}+ \frac{0-V_o}{R_f} = 0 \\frac{V_o}{V_s} = -\frac{R_f}{R_s}$$