# 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

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

The RC circuit is more stable at frequencies from zero up to the response frequency :${\frac {1}{RC}}$  . 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

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

The CR circuit is more stable at frequencies from the response frequency ${\frac {1}{RC}}$  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

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

The RL circuit is more stable at frequencies from the response frequency ${\frac {R}{L}}$  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

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

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

The LR circuit is more stable at frequencies from zero up to the response frequency ${\frac {R}{L}}$  . 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 ${\frac {R}{L}}$  ${\frac {R}{L}}$  ${\frac {1}{RC}}$  ${\frac {1}{RC}}$
T ${\frac {L}{R}}$  ${\frac {L}{R}}$  CR CR
Z $R+j\omega L$  $R+j\omega L$  $R+{\frac {1}{j\omega C}}$  $R+{\frac {1}{j\omega C}}$
${\frac {V_{o}}{V_{i}}}$  ${\frac {j\omega L}{R+j\omega L}}$  ${\frac {R}{R+j\omega L}}$  ${\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}$  ${\frac {j\omega CR}{1+j\omega CR}}$
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