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