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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{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}

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{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}

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