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tags: Microelectronic Circuits
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# Coursera Introduction to Electronics Week 2-Operational Amplifiers Part 1
## Introductions to Operational Amplifiers
1. Symbol
2. Behaviors
1. Operating in Linear Region
$A$ is the slope of the graph.
2. Operating in Saturation Region
3. Ideal Model
$\begin{cases}i_+=i_-=0\\v_+=v_-\end{cases}$
4. Phisical Op Amps
## Applications
### Buffer Circuits
We use buffer circuits to boost power without changing voltage waveform.
### Non-Inverting Amplifiers
Focus on the point P
$v_{in}=\frac{R_3}{R_2+R_3}v_o\\v_o=\frac{R_2+R_3}{R_3}v_{in}\\Let\ the\ gain\ G=\frac{R_2+R_3}{R_3}\\\Rightarrow v_o=Gv_{in}$
### Inverting Amplifier
Focus on the point P, since it is a ideal amplifier, $i_{in}=0$,
$i_1=i_2\\\frac{v_{in}}{R_1}=\frac{v_o}{R_f}\\\Rightarrow V_o=-\frac{R_f}{R_1}V_{in}$
### Difference Circuit
$\begin{cases}v_a=\frac{R_f}{R_f+R_1}v_2\\i_1+i_2=0\Rightarrow \frac{v_a-v_1}{R_1}+\frac{v_a-v_o}{R_f}=0\end{cases}\\\Rightarrow\frac{R_f}{R_f+R_1}v_2(\frac{R_1+R_f}{R_1R_f})=\frac{v_1}{R_1}+\frac{v_o}{R_f}\\\Rightarrow v_o=\frac{R_f}{R_1}(v_2-v_1)$
### Summing Amplifier
$i_1+i_2+i_f=0\\\Rightarrow \frac{v_1}{R_1}+\frac{v_2}{R_2}+\frac{v_o}{R_f}=0\\\Rightarrow v_o=-\frac{R_f}{R_1}v_1-\frac{R_f}{R_2}v_2\\Let\ \begin{cases}G_1=-\frac{R_f}{R_1}\\G_2=-\frac{R_f}{R_2}\end{cases}\\\Rightarrow v_o=G_1v_1+G_2v_2$
### Differentiator Circuit
$v_o=-RC\frac{dv_{in}}{dt}$
### Integrator Circuit
$\begin{split}(i)&For\ t<0\\&v_o=0\end{split}\\\begin{split}(ii)&For\ t>0\\&\begin{cases}-v_{in}+iR+v_C+v_o=0\\i=C\frac{dv_C}{dt}\\v_c=\frac{1}{C}\int_0^tidt\end{cases}\\&v_o=-\frac{1}{RC}\int_0^tv_{in}dt\end{split}$
## Filters
1. Analog Filters
2. The Difference between Analog Filter and Passive Filter
1. Analog Filter
!
Analog filters have isolation, which means the $v_o$ doesn't depends on the output resistances.
2. Passive Filter
Passive filters deplete power.
3. Active Filter
Active filters have their own power supply and also provide isolation.

**Reference**
Coursera Introduction to Electronics by Georgia Institute of Technology