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 circuits

RC Circuit[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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