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