The Physics of Frequency-Selective Passive Filters
In analog signal processing, a filter is a linear two-port circuit designed to transmit certain frequency bands unattenuated (the passband) while suppressing unwanted frequencies (the stopband). Passive filters utilize combinations of resistors ($R$), inductors ($L$), and capacitors ($C$) without active amplification devices like op-amps or transistors.
Filtering behavior arises from the fundamental frequency dependence of electrical reactive components:
- Resistors ($R$): Exhibit purely real impedance ($Z_R = R$) independent of frequency. Dissipate electrical energy as thermal heat.
- Capacitors ($C$): Store energy in an electric field. Complex impedance is $Z_C = frac{1}{j omega C} = -j frac{1}{2pi f C}$. Reactance is inversely proportional to frequency, acting as an open circuit at DC ($omega to 0$) and a short circuit at infinite frequency ($omega to infty$).
- Inductors ($L$): Store energy in a magnetic field. Complex impedance is $Z_L = j omega L = j 2pi f L$. Reactance is directly proportional to frequency, presenting zero impedance at DC and infinite impedance at high frequency.
First-Order RC and RL Filter Formulations
1. RC Low-Pass Filter
Output taken across capacitor. High frequencies shunt to ground:
$$H(s) = frac{V_{text{out}}}{V_{text{in}}} = frac{1}{1 + s R C} implies f_c = frac{1}{2pi R C}$$
Magnitude Response: $|H(f)| = frac{1}{sqrt{1 + (f / f_c)^2}}, quad text{Phase: } theta(f) = -arctanleft(frac{f}{f_c}right)$$
2. RC High-Pass Filter
Capacitor blocks DC, output taken across resistor:
$$H(s) = frac{s R C}{1 + s R C} implies f_c = frac{1}{2pi R C}$$
Magnitude Response: $|H(f)| = frac{f / f_c}{sqrt{1 + (f / f_c)^2}}, quad text{Phase: } theta(f) = +90^circ - arctanleft(frac{f}{f_c}right)$$
3. RL Filter Cutoff
Because $Z_L = omega L$, the time constant is $tau = L / R$:
$$f_c = frac{R}{2pi L}$$
Second-Order RLC Resonance & Quality Factor ($Q$)
When both inductive and capacitive energy storage elements are present, the circuit possesses a natural second-order oscillation frequency where inductive and capacitive reactances cancel out:
The Thomson Resonant Frequency Formula
$$X_L = X_C implies 2pi f_0 L = frac{1}{2pi f_0 C} implies f_0 = frac{1}{2pi sqrt{L C}}$$
The sharpness and energy dissipation of the resonant peak is characterized by the Quality Factor ($Q$) and Damping Ratio ($zeta$):
| Topological Model | Quality Factor ($Q$) | Damping Ratio ($zeta = frac{1}{2Q}$) | -3dB Bandwidth ($Delta f$) |
|---|---|---|---|
| Series RLC Circuit | $$Q = frac{1}{R} sqrt{frac{L}{C}} = frac{omega_0 L}{R}$$ | $$zeta = frac{R}{2} sqrt{frac{C}{L}}$$ | $$Delta f = frac{f_0}{Q} = frac{R}{2pi L}$$ |
| Parallel RLC Tank | $$Q = R sqrt{frac{C}{L}} = frac{R}{omega_0 L}$$ | $$zeta = frac{1}{2R} sqrt{frac{L}{C}}$$ | $$Delta f = frac{f_0}{Q} = frac{1}{2pi R C}$$ |
Damping Classifications:
- Overdamped ($zeta > 1, Q < 0.5$): No resonant peaking. Slow transient step response.
- Critically Damped ($zeta = 1, Q = 0.5$): Fastest return to equilibrium without overshoot.
- Underdamped ($zeta < 1, Q > 0.5$): Sharp frequency peaking and transient ringing. When $Q > 1/sqrt{2} approx 0.707$, the frequency response displays a resonant peak above $0text{ dB}$.