# Circuit Theory/Introduction to Filtering

Filter is a circuit constructed from Resistor and Capacitor or Inductor in order to pass certain range of frequencies . The range of frequencies that make the circuit stable

Let examine the following circuits

## RC Circuit

A circuit with one resistor in series with the input and one capacitor parallel to the load

${\displaystyle V_{o}=V_{i}{\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}}$
ω = 0 ${\displaystyle \omega ={\frac {R}{L}}}$ ω = Infinity
${\displaystyle V_{o}=V_{i}}$ ${\displaystyle V_{o}=V_{i}}$ ${\displaystyle V_{o}=0}$

The RC circuit is more stable at frequencies from zero up to the response frequency :${\displaystyle {\frac {1}{RC}}}$ . This circuit is ideal for Low Pass Frequency Filter .

## CR Circuit

A circuit with one capacitor in series with the input and one resistor parallel to the load

${\displaystyle V_{o}=V_{i}{\frac {R}{R+{\frac {1}{j\omega C}}}}}$
ω = 0 ω = ${\displaystyle {\frac {R}{L}}}$ ω = Infinity
${\displaystyle V_{o}=0}$ ${\displaystyle V_{o}=V_{i}}$ ${\displaystyle V_{o}=V_{i}}$

The CR circuit is more stable at frequencies from the response frequency ${\displaystyle {\frac {1}{RC}}}$ up to infinity. This circuit is ideal for High Pass Frequency Filter .

## RL Circuit

A circuit with one resistor in serires with the input and one inductor parallel to the load

${\displaystyle V_{o}=V_{i}{\frac {j\omega L}{R+j\omega L}}}$
ω = 0 ${\displaystyle \omega ={\frac {R}{L}}}$ ω = Infinity
${\displaystyle V_{o}=0}$ ${\displaystyle V_{o}=V_{i}}$ ${\displaystyle V_{o}=V_{i}}$

The RL circuit is more stable at frequencies from the response frequency ${\displaystyle {\frac {R}{L}}}$ up to infinity. This circuit is ideal for High Pass Frequency Filter .

## LR Circuit

A circuit with one inductor in serires with the input and one resistor parallel to the load

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

ω = 0 ${\displaystyle \omega ={\frac {R}{L}}}$ ω = Infinity
${\displaystyle V_{o}=V_{i}}$ ${\displaystyle V_{o}=V_{i}}$ ${\displaystyle V_{o}=0}$

The LR circuit is more stable at frequencies from zero up to the response frequency ${\displaystyle {\frac {R}{L}}}$ . This circuit is ideal for Low Pass Frequency Filter .

In Conclusion, Resistor and Capacitor or Inductor can be used for constructing a Filter

• For Low Pass Filter use LR or RC
• For High Pass Filter use RL or CR

## Conclusion

Filter Types High Pass Filter Low Pass Filter Low Pass Filter High Pass Filter
Circuit RL LR RC CR
ωο ${\displaystyle {\frac {R}{L}}}$ ${\displaystyle {\frac {R}{L}}}$ ${\displaystyle {\frac {1}{RC}}}$ ${\displaystyle {\frac {1}{RC}}}$
T ${\displaystyle {\frac {L}{R}}}$ ${\displaystyle {\frac {L}{R}}}$ CR CR
Z ${\displaystyle R+j\omega L}$ ${\displaystyle R+j\omega L}$ ${\displaystyle R+{\frac {1}{j\omega C}}}$ ${\displaystyle R+{\frac {1}{j\omega C}}}$
${\displaystyle {\frac {V_{o}}{V_{i}}}}$ ${\displaystyle {\frac {j\omega L}{R+j\omega L}}}$ ${\displaystyle {\frac {R}{R+j\omega L}}}$ ${\displaystyle {\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}}$ ${\displaystyle {\frac {j\omega CR}{1+j\omega CR}}}$
Frequency Response
ω = 0 Vo = 0
ω = ωο Vo = Vi
ω = 0 Vo = Vi
ω = 0 Vo = Vi
ω = ωο Vo = Vi
ω = 0 Vo = 0
ω = 0 Vo = Vi
ω = ωο Vo = Vi
ω = 0 Vo = 0
ω = 0 Vo = 0
ω = ωο Vo = Vi
ω = 0 Vo = Vi
Stability Circuit is stable at Frequencies ω = ωο→Infinity Circuit is stable at Frequencies ω = 0→ωο Circuit is stable at Frequencies ω = 0→ωο Circuit is stable at Frequencies ω = ωο→Infinity