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

${\frac {V_{o}}{V_{i}}}={\frac {R}{R+{\frac {1}{j\omega C}}}}={\frac {j\omega T}{1+j\omega T}}$

$T=RC$

$\omega _{o}={\frac {1}{T}}={\frac {1}{RC}}$