# Practical Electronics/High Pass Filter

## Low Pass Filter

### RL Network

${\displaystyle {\frac {V_{o}}{V_{i}}}={\frac {Z_{L}}{Z_{R}+Z_{L}}}={\frac {j\omega L}{R+j\omega L}}={\frac {1}{1+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}}$
${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {R}{L}}}$
${\displaystyle \omega =0V_{o}=0}$
${\displaystyle \omega _{o}={\sqrt {\frac {1}{RC}}}V_{o}={\frac {V_{i}}{2}}}$
${\displaystyle \omega =00V_{o}=00}$
Plot three points above we have a graph ${\displaystyle Vo-\omega }$  . From graph, we see voltage does not change with frequency on High Frequency therefore RL network can be used as High Pass Filter

### CR Network

${\displaystyle {\frac {V_{o}}{V_{i}}}={\frac {Z_{R}}{Z_{R}+Z_{C}}}={\frac {R}{R+{\frac {1}{j\omega C}}}}={\frac {j\omega T}{1+j\omega T}}}$
${\displaystyle T=RC}$
${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {1}{RC}}}$
${\displaystyle \omega =0V_{o}=0}$
${\displaystyle \omega _{o}={\sqrt {\frac {1}{LC}}}V_{o}={\frac {V_{i}}{2}}}$
${\displaystyle \omega =00V_{o}=00}$
Plot three points above we have a graph ${\displaystyle Vo-\omega }$  . From graph, we see voltage does not change with frequency on High Frequency therefore LR network can be used as High Pass Filter

## Summary

In general

1. High Pass Filter can be constructed from the two networks RL or CR network .
2. High Pass Filter has stable voltage does not change with frequency on Low Frequency .
3. High pass filter can be expressed in a mathematical form of
${\displaystyle {\frac {V_{o}}{V_{i}}}={\frac {j\omega T}{1+j\omega T}}}$
T = RC for RC network
${\displaystyle T={\frac {L}{R}}}$  for RL network