Practical Electronics/Circuits Analysis/Two Port Network

RC

$\frac{V_o}{V_i}=\frac{\frac{1}{j\omega C}}{R+\frac{1}{j\omega C}}=\frac{1}{1+j\omega T}$

$T=RC$

${\omega }_o=\frac{1}{T}=\frac{1}{RC}$

This circuit has Stable voltage at low frequency therefore suitable for filtering low frequency therefore the name Low Pass Filter

CR

$\frac{V_o}{V_i}=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega T}{1+j\omega T}$

$T=RC$

${\omega }_o=\frac{1}{T}=\frac{1}{RC}$

This circuit has Stable voltage at high frequency therefore suitable for filtering high frequency therefore the name High Pass Filter

LR

$\frac{V_o}{V_i}=\frac{R}{R+j\omega L}=\frac{1}{1+j\omega T}$

$T=RC$

${\omega }_o=\frac{j\omega L}{R+j\omega L}=\frac{1}{1+j\omega T}$

This circuit has Stable voltage at low frequency therefore suitable for filtering low frequency therefore the name Low Pass Filter

RL

$\frac{V_o}{V_i}=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega T}{1+j\omega T}$

$T=\frac{L}{R}$

${\omega }_o=\frac{1}{T}=\frac{1}{RC}$

This circuit has Stable voltage at high frequency therefore suitable for filtering high frequency therefore the name High Pass Filter

LC - R

$\frac{V_o}{V_i}=\frac{R}{R+j\omega L+\frac{1}{j\omega C}}$

Tuned Resonance Selected Band Pass Filter

R - LC

$\frac{V_o}{V_i}=\frac{j\omega L+\frac{1}{j\omega C}}{R+j\omega L+\frac{1}{j\omega C}}$

$T=RC$

${\omega }_o=\frac{1}{T}=\frac{1}{RC}$

Tuned Resonance Selected Band Reject Filter

LC// - R

$\frac{V_o}{V_i}=\frac{R}{R+j\omega C+\frac{1}{j\omega L}}$

Tuned Resonance Selected Band Pass Filter

R - LC//

$\frac{V_o}{V_i}=\frac{j\omega C+\frac{1}{j\omega L}}{R+j\omega C+\frac{1}{j\omega L}}$

Tuned Resonance Selected Band Pass Filter

Tuned Resonance Selected Band Pass Filter

LR + CR

Transfer Function

$\frac{V_o}{V_i}=(\frac{1}{1+j\omega \frac{L}{R}})(\frac{j\omega RC}{1+j\omega RC})$

Band Pass or band of frequencies that has a stable voltage

$\frac{R}{L}-\frac{1}{RC}$ provided that $\frac{1}{RC}>\frac{R}{L}$

Band Pass Filter

RC - RL

Transfer Function

$\frac{V_o}{V_i}=()()$

Band Pass or band of frequencies that has a stable voltage

$\frac{1}{RC}-\frac{R}{L}$ provided that $\frac{R}{L}>\frac{1}{RC}$

Band Pass Filter

Summary

Low Pass Filter

$\frac{V_o}{V_i}=\frac{\frac{1}{j\omega C}}{R+\frac{1}{j\omega C}}=\frac{1}{1+j\omega T}$

$T=RC$

${\omega }_o=\frac{1}{T}=\frac{1}{RC}$

High Pass Filter

$\frac{V_o}{V_i}=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega T}{1+j\omega T}$

$T=RC$

${\omega }_o=\frac{1}{T}=\frac{1}{RC}$