Practical Electronics/Low Pass Filter

Low Pass Filter

LR Network

${\displaystyle {\frac {V_{o}}{V_{i}}}={\frac {Z_{R}}{Z_{R}+Z_{L}}}={\frac {R}{R+j\omega L}}={\frac {1}{1+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}}$

${\displaystyle \omega =0V_{o}=V_{i}}$
${\displaystyle \omega _{o}={\sqrt {\frac {1}{LC}}}V_{o}={\frac {V_{i}}{2}}}$
${\displaystyle \omega =00V_{o}=0}$
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

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

${\displaystyle \omega =0V_{o}=V_{i}}$
${\displaystyle \omega _{o}={\sqrt {\frac {1}{LC}}}V_{o}={\frac {V_{i}}{2}}}$
${\displaystyle \omega =00V_{o}=0}$
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