# Fundamental Physics/Electronics/Electronics Filter

## Electronics Filter

Filter is an electronics device that provide a constant voltage , does not change with frequency , over frequency . It's reponse can be characterized from its Transfer function ${\displaystyle H(j\omega )}$

${\displaystyle {\frac {v_{o}}{v_{i}}}=H(j\omega )}$

There are 5 types of filters Low pass filter , High pass filter , Band pass filter , Resonance tuned bandpass filter . Resonance tuned bandreject filter

## Low pass filter

Low pass filter is an electronics device that has a constant voltage over low frequency . Low pass filter can be constructed from LR and RC as shown .

L-R ,

Low pass filter has a transfer function

${\displaystyle H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {1}{1+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}=RC}$

Frequency response of Low pass filter

${\displaystyle \omega =0.v_{o}=v_{i}}$
${\displaystyle \omega =\omega _{o}.v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega =00.v_{o}=0}$
${\displaystyle \omega ={\frac {1}{T}}}$

Cut off frequency, ${\displaystyle \omega _{o}}$ , frequency at which ${\displaystyle v_{o}={\frac {1}{2}}v_{i}}$

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

## High pass filter

High pass filter is an electronics that has a constant voltage over high frequency . High pass filter can be constructed from LR and RC as shown .

,

High pass filter has a transfer function

${\displaystyle H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {j\omega T}{1+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}=RC}$

Frequency response of High pass filter

${\displaystyle \omega =0.v_{o}=0}$
${\displaystyle \omega =\omega _{o}.v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega =00.v_{o}=v_{i}}$

Cut off frequency, ${\displaystyle \omega _{o}}$ , frequency at which ${\displaystyle v_{o}={\frac {1}{2}}v_{i}}$

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

## Band pass filter

Low pass filter has a constant voltage over a band of frequencies . Band pass filter can be constructed from LPF and HPF

Band pass filter has a transfer function

${\displaystyle H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {1}{1+j\omega T_{L}}}{\frac {j\omega T_{H}}{1+j\omega T_{H}}}}$

Frequency band pass that has constant voltage ${\displaystyle v_{o}=v_{i}}$

${\displaystyle {\frac {R}{L}}-{\frac {1}{RC}}}$
${\displaystyle {\frac {1}{RC}}-{\frac {L}{R}}}$

## Resonance tuned band pass filter

Transfer function

${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R}{j\omega L+{\frac {1}{j\omega C}}+R}}={\frac {j\omega RC}{j\omega ^{2}LC+j\omega RC+1}}}$

Frequency response

${\displaystyle v_{o}(\omega =0)=0}$ Capacitor opens circuit
${\displaystyle v_{o}(\omega =0)=v_{i}}$ ${\displaystyle Z_{L}=-Z_{C}}$
${\displaystyle v_{o}(\omega =00)=0}$ Inductor opens circuit

## Resonance tuned band rejected filter

Transfer function

${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R}{R+j\omega C+{\frac {1}{j\omega L}}}}={\frac {j\omega RL}{j\omega ^{2}LC+j\omega LR+1}}}$

Frequency response

${\displaystyle v_{o}(\omega =0)=v_{i}}$ Capacitor opens circuit, inductor shorts circuit
${\displaystyle v_{o}(\omega =0)=0}$ ${\displaystyle Z_{L}=-Z_{C}}$
${\displaystyle v_{o}(\omega =00)=v_{i}}$ Inductor opens circuit, capacitor shorts circuit