Circuit Theory/Introduction to Filtering

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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 ciruits

[edit] RC Circuit

A circuit with one resistor in series with the load 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 ciruit is more stable at frequencies from zero upto the response frequency : $\frac{1}{RC}$ . This circuit is ideal for Low Pass Frequency Filter .

[edit] CR Circuit

A circuit with one capacitor in serires with the load 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 ciruit is more stable at frequencies from the response frequency $\frac{1}{RC}$ upto infinity. This circuit is ideal for High Pass Frequency Filter .

[edit] RL Circuit

A circuit with one resistor in serires with the load 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 $\frac{R}{L}$ upto infinity. This circuit is ideal for High Pass Frequency Filter .

[edit] LR Circuit

A circuit with one inductor in serires with the load 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 ciruit is more stable at frequencies from zero upto 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

[edit] Conclusion

Filter Types	High Pass Filter	Low Pass Filter	Low Pass Filter	High Pass Filter
Circuit	RL	LR	RC	CR
ω_ο	$\frac{R}{L}$	$\frac{R}{L}$	$\frac{1}{RC}$	$\frac{1}{RC}$
T	$\frac{L}{R}$	$\frac{L}{R}$	CR	CR
Z	$R + j ω L$	$R + j ω L$	$R + \frac{1}{j\omega C}$	$R + \frac{1}{j\omega C}$
$\frac{V_o}{V_i}$	$\frac{j\omega L}{R + j\omega L}$	$\frac{R}{R + j\omega L}$	$\frac{\frac{1}{j\omega C}}{R + \frac{1}{j\omega C}}$	$\frac{j\omega CR}{1 + j\omega CR}$
Frequency Response	ω = 0 V_o = 0 ω = ω_ο V_o = V_i ω = 0 V_o = V_i	ω = 0 V_o = V_i ω = ω_ο V_o = V_i ω = 0 V_o = 0	ω = 0 V_o = V_i ω = ω_ο V_o = V_i ω = 0 V_o = 0	ω = 0 V_o = 0 ω = ω_ο V_o = V_i ω = 0 V_o = V_i
Stability	Circuit is stable at Frequencies ω = ω_ο→Infinity	Circuit is stable at Frequencies ω = 0→ω_ο	Circuit is stable at Frequencies ω = 0→ω_ο	Circuit is stable at Frequencies ω = ω_ο→Infinity