# Practical Electronics/Low Pass Filter

## Low Pass Filter

### LR Network

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

### RC Network

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

## Summary

In general

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