---
tags: Microelectronic Circuits
---
# Coursera Introduction to Electronics Week 3-Operational Amplifiers Part 2
## First-Order Pass Filters
### First-Order Low Pass Filters
1. Transfer Function
1. $H(\omega)=K_{DC}\frac{1}{\tau j \omega+1}$
* ※$|H(\omega)|=K_{DC}\frac{1}{\sqrt{1+(\tau\omega)^2}}\\\Rightarrow \omega_B=\frac{1}{\tau}\\At\ \omega_=\frac{1}{\tau}\\|H(\omega)|\approx=0.707K_{DC}$
2. Bandwidth: $\omega_B=\frac{1}{\tau}$
3. DC Gain: $H(0)=K_{DC}$
2. From Passive to Active Lowpass Filters
3. Derivation
$\begin{cases}From\ the\ path\ 1,\ v_{in}=Z_1i\\From\ the\ path\ 2,\ v_{in}=Z_1i+Z_fi+v_o\\Z_f=R_f||Z_C\end{cases}\\\Rightarrow\begin{cases}i=\frac{v_{in}}{Z_1}\\v_o=-Z_fi\\Z_f=R_f||Z_C\end{cases}\\\Rightarrow v_o=-\frac{Z_f}{Z_1}v_{in}\\\Rightarrow v_o=-\frac{R_f}{R_1}\frac{1}{Cj\omega+1}v_{in}$
4. Frequency Characteristics
$\begin{cases}|H(\omega)|=\frac{R_f}{R_1}\frac{1}{\sqrt{(R_fC_f\omega)^2+1}}\\∠H(\omega)=180-arctan(R_fC_f\omega)\\DC\ Gain=-\frac{R_f}{R_1}\\Bandwidth=\omega_b=\frac{1}{R_fC_f}\end{cases}$
### First-Order High Pass Filters
1. Transfer Function
1. $H(\omega)=\frac{Jj\omega}{\tau j\omega+1}$
2. Corner Frequency: $\omega_c=\frac{1}{\tau}$
3. Passband Gain: $K_{PB}=\frac{K}{\tau}$
2. Derivation
$v_o=\frac{-R_fCj\omega}{R_1Cj\omega+1}v_{in}$
3. Frequency Characteristics
1. $|H(\omega)|=\frac{R_fC\omega}{\sqrt{(R_1C\omega)^2+1}}$
2. $∠H(\omega)=-90^\circ-acrtan(R_1C\omega)$
3. Passband Gain$(\omega\to\infty)=-\frac{R_f}{R_1}$
4. Corner Frequency: $\omega_c=\frac{1}{R_1C}$
### Cascaded First-Order Filters
1. Transfer Functions in Hertz
$\begin{cases}For\ a\ low\ pass\ filter,\\\begin{cases}H(\omega)=K_{DC}\frac{1}{\tau j \omega+1}\\\omega_B=\frac{1}{\tau}\\H(0)=K_{DC}\end{cases}\\For\ a\ high\ pass\ filter,\\\begin{cases}H(\omega)=\frac{Jj\omega}{\tau j\omega+1}\\\omega_c=\frac{1}{\tau}\\K_{PB}=\frac{K}{\tau}\\\end{cases}\end{cases}\\Let\ \begin{cases}\omega=2\pi f\\\omega_o=\omega_B\ or\ \omega_c=2\pi f_o\end{cases}\\\Rightarrow\begin{cases}For\ a\ low\ pass\ filter,\\H(f)=K_{DC}\frac{1}{\frac{jf}{f_o}+1}\\For\ a\ high\ pass\ filter,\\H(f)=K_{PB}\frac{\frac{jf}{f_o}}{\frac{jf}{f_o}+1}\end{cases}$
2. Transfer Function of Cascaded First-Order Filters
$H_c=\frac{v_{out}}{v_{in}}=\frac{v_{out1}}{v_{in}}\cdot\frac{v_{out}}{v_{in2}}=H_{LP}H_{HP}$
3. Bandpass Filter Characteristics
$Let\ \begin{cases}Bandwidh=f_u-f_l\\Center\ Frequency=f_o=\sqrt{f_uf_l}\\Quailty\ Factor=Q=\frac{f_o}{Bandwidth}\end{cases}\\H_{BP}(f)=H_{LP}(f)H_{HP}(f)\\\Rightarrow K=K_{DC}K_{PB}(\frac{f_{lp}}{f_{lp}+f_{hp}})\\\Rightarrow \begin{cases}f_o=\sqrt{f_{jp}f_{hp}}\\Q=\frac{\sqrt{f_{jp}f_{hp}}}{f_{jp}+f_{hp}}\\Bandwidth=f_{jp}+f_{hp}\end{cases}$
## Second-Order Pass Filters
1. Characteristics
1. Transfer Functions
1. Low Pass Filters
$H_{LP2}(f)=K\frac{1}{(\frac{if}{f_o})^2+\frac{if}{f_o}\frac{1}{Q}+1}$
2. High Pass Filters
$H_{HP2}(f)=K\frac{(\frac{if}{f_o})^2}{(\frac{if}{f_o})^2+\frac{if}{f_o}\frac{1}{Q}+1}$
3. Band Pass Filters
$H_{BP2}(f)=K\frac{\frac{if}{f_o}\frac{1}{Q}}{(\frac{if}{f_o})^2+\frac{if}{f_o}\frac{1}{Q}+1}$
4. Notch Filters
$H_{BR2}(f)=H_{HP2(f)}+H_{LP2}(f)=K\frac{(\frac{if}{f_o})^2+1}{(\frac{if}{f_o})^2+\frac{if}{f_o}\frac{1}{Q}+1}$
2. $Q$
1. Low Pass Filters
2. High Pass Filters
3. Band Pass Filters
4. Butterworth Transfer Function$(Q=\frac{1}{\sqrt{2}})$
5. Chebyshev Transfer Function$(Q>\frac{1}{\sqrt{2}})$
* ※Forth-Order Butterworth and Chebyshev Filters
2. Circuits
1. Sallen-Key Low-Pass Filter
$\begin{cases}H_{LP2}(f)=K\frac{1}{(\frac{if}{f_0})^2+\frac{if}{f_0}\frac{1}{Q}+1}\\K=1+\frac{R_4}{R_3}\\f_o=\frac{1}{2\pi\sqrt{R_1R_2C_1C_2}}\\Q=\frac{\sqrt{R_1R_2C_1C_2}}{(1-K)R_1C_1+(R_1C_1)C_2}\end{cases}$
* Low Pass Design Equations
1. Special Case 1: $K=1,\ Solve\ for\ C's$
$K=1\\\Rightarrow\begin{cases}\begin{cases}R_3=\infty\\R_4=0\end{cases}\\C_1=\frac{Q}{\omega_0}(\frac{1}{R_1}+\frac{1}{R_2})\\C_2=\frac{1}{Q\omega_0(R_1+R_2)}\end{cases}$
* ※We can simplify it with $R_1=R_2$.
2. Special Case 2: $K=1,\ Solve\ for\ R's$
$K=1\\\Rightarrow\begin{cases}\begin{cases}R_3=\infty\\R_4=0\end{cases}\\R_1R_2=\frac{1}{2Q\omega_0C_2}(1\pm\sqrt{1-4Q^2\frac{Q_2}{Q_1}})\\4Q^2\frac{C_2}{C_1}\leq1\end{cases}$
3. Special Case 3: $R's\ equal\ and \ C's\ equal$
$\begin{cases}\begin{cases}R_1=R_2=R\\C_1=C_2=C\\K=1+\frac{R_4}{R_3}\end{cases}\\K=3-\frac{1}{Q}\\R=\frac{1}{\omega_oC}\end{cases}$
2. Sallen-Key Highpass Filter
$\begin{cases}H_{HP2}(f)=H_{HP2}(f)=K\frac{(\frac{if}{f_o})^2}{(\frac{if}{f_o})^2+\frac{if}{f_o}\frac{1}{Q}+1}\\K=1+\frac{R_4}{R_3}\\f_o=\frac{1}{2\pi\sqrt{R_1R_2C_1C_2}}\\Q=\frac{\sqrt{R_1R_2C_1C_2}}{(1-K)R_1C_1+(R_1C_1)C_2}\end{cases}$
* High Pass Design Equatios
1. Special Case 1: $(K=1,\ C_1=C_2=C)$
$\begin{cases}K=1\\C_1=C_2=C\end{cases}\\\Rightarrow\begin{cases}\begin{cases}R_3=\infty\\R_4=0\end{cases}\\R_1=\frac{1}{2Q\omega_oC}\\R_2=\frac{2Q}{\omega_oC}\end{cases}$
2. Special Case 2: $R's\ equal\ and \ C's\ equal$
$\begin{cases}\begin{cases}R_1=R_2=R\\C_1=C_2=C\\K=1+\frac{R_4}{R_3}\end{cases}\\K=3-\frac{1}{Q}\\R=\frac{1}{\omega_oR}\end{cases}$
3. Sallen-Key Bandpass Filter
$\begin{cases}H_{BP2}(f)=K\frac{\frac{if}{f_o}\frac{1}{Q}}{(\frac{if}{f_o})^2+\frac{if}{f_o}\frac{1}{Q}+1}\\K=\frac{R_2}{R_1+R_2}\frac{K_oR_3C_2}{(R_1||R_2)(C_1+C_2)+R_3C_2[1-\frac{K_oR_1}{R_1+R_2}]}\\f_o=\frac{1}{2\pi\sqrt{(R_1||R_2)R_3C_1C_2}}\\Q=\frac{\sqrt{(R_1||R_2)R_3C_1C_2}}{(R_1||R_2)(C_1+C_2)+R_3C_2[1-\frac{K_oR_1}{R_1+R_2}]}\\K_o=1+\frac{R_3}{R_4}\end{cases}$
* Bandpass Design Equations
* Special Case: $R's\ equal\ and \ C's\ equal$
$\begin{cases}R_1=R_2=R_3=R\\C_1=C_2=C\\K_o=1+\frac{R_3}{R_4}\end{cases}\\\Rightarrow\begin{cases}R=\frac{\sqrt{2}}{2\pi f_oC}\\K_o=4-\frac{1}{Q^2}\\K=4Q^2-1\end{cases}$
## Filtering Demonstration
1. Spectrum of Sine Wave
2. Spectrum of Sum of Two Sine Waves
3. Spectrum of Square Wave

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