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

$H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {1}{1+j\omega T}}$ $T={\frac {L}{R}}=RC$ Frequency response of Low pass filter

$\omega =0.v_{o}=v_{i}$ $\omega =\omega _{o}.v_{o}={\frac {v_{i}}{2}}$ $\omega =00.v_{o}=0$ $\omega ={\frac {1}{T}}$ Cut off frequency, $\omega _{o}$ , frequency at which $v_{o}={\frac {1}{2}}v_{i}$ $\omega _{o}={\frac {1}{T}}$ ## 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

$H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {j\omega T}{1+j\omega T}}$ $T={\frac {L}{R}}=RC$ Frequency response of High pass filter

$\omega =0.v_{o}=0$ $\omega =\omega _{o}.v_{o}={\frac {v_{i}}{2}}$ $\omega =00.v_{o}=v_{i}$ Cut off frequency, $\omega _{o}$ , frequency at which $v_{o}={\frac {1}{2}}v_{i}$ $\omega _{o}={\frac {1}{T}}$ ## 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

$H(j\omega )={\frac {v_{o}}{v_{i}}}={\frac {1}{1+j\omega T_{L}}}{\frac {j\omega T_{H}}{1+j\omega T_{H}}}$ Frequency band pass that has constant voltage $v_{o}=v_{i}$ ${\frac {R}{L}}-{\frac {1}{RC}}$ ${\frac {1}{RC}}-{\frac {L}{R}}$ ## Resonance tuned band pass filter

Transfer function

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

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

## Resonance tuned band rejected filter

Transfer function

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

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