Electronics Handbook/Circuits/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

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

$T = RC$

$\omega_o = \frac{1}{T} = \frac{1}{RC}$