# 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