8.2 First-order and second-order passive filters

Passive Filter Types

Passive filters shape the frequency content of signals using only resistors, capacitors, and inductors. By choosing different combinations of these components, you can build filters that pass or block specific frequency ranges. This section covers first-order filters (RC and RL) and second-order filters (LC and RLC), along with the key parameters that define their performance.

RC and RL Filters

RC and RL filters are first-order filters, meaning they each contain a single reactive component. That single reactive element gives them a roll-off of -20 dB/decade.

• An RC low-pass filter places the capacitor across the output. At low frequencies the capacitor's impedance is high, so most of the voltage appears across it. At high frequencies the capacitor's impedance drops, shorting the output to ground and attenuating the signal.
• An RC high-pass filter swaps the positions of the resistor and capacitor. Now the capacitor blocks low frequencies (high impedance) and passes high frequencies (low impedance) to the output.
• RL filters work on the same principle but use an inductor instead of a capacitor. Because an inductor's impedance increases with frequency, the component placement for low-pass and high-pass is reversed compared to RC circuits.

The cutoff frequency for a first-order RC filter is:

$f_c = \frac{1}{2\pi RC}$

For a first-order RL filter:

$f_c = \frac{R}{2\pi L}$

At the cutoff frequency, the output is attenuated by -3 dB (about 70.7% of the input voltage).

LC and RLC Filters

Adding a second reactive component creates a second-order filter with a roll-off of -40 dB/decade, twice as steep as a first-order filter.

• An LC filter pairs an inductor and capacitor to form a resonant circuit. Depending on how you connect them (series or parallel), you get either a band-pass configuration (passes a narrow range of frequencies around resonance) or a band-stop configuration (rejects that same range).
• An RLC filter adds a resistor to the LC pair. The resistor controls energy dissipation, which directly sets the sharpness of the frequency response. Without it, an ideal LC circuit would have infinite gain at resonance, which isn't physically realizable.

RLC filters can also be configured as second-order low-pass or high-pass filters, not just band-pass and band-stop. The topology (series vs. parallel, and where you take the output) determines the filter type.

Filter Characteristics

Filter Order and Roll-off Rate

Filter order equals the number of independent reactive components (capacitors or inductors) in the circuit. It directly controls how aggressively the filter attenuates signals outside the passband.

Q Factor and Bandwidth

The Q factor (quality factor) describes how selective a filter is around its center frequency. It applies mainly to second-order (and higher) filters, especially band-pass and band-stop types.

$Q = \frac{f_0}{BW}$

where $f_0$ is the center frequency and $BW$ is the -3 dB bandwidth.

• A high Q (say, Q = 50) means a very narrow passband. The filter is highly selective but passes only a tight range of frequencies.
• A low Q (say, Q = 1) means a wide passband with a gentle transition. Less selective, but it captures a broader range of signals.

There's a trade-off here: narrower bandwidth gives better selectivity, but if the bandwidth is too narrow, you risk attenuating parts of the signal you actually want to keep.

RLC Filter Parameters

Damping Ratio and System Response

The damping ratio $\zeta$ (zeta) describes how the filter behaves during transients and at resonance. For a series RLC circuit:

$\zeta = \frac{R}{2} \sqrt{\frac{C}{L}}$

The three regimes of damping are:

1. Underdamped ($\zeta < 1$): The response oscillates before settling. In the frequency domain, this produces a resonance peak, meaning gain near $f_0$ actually exceeds the passband level. Lower $\zeta$ values create taller, sharper peaks.
2. Critically damped ($\zeta = 1$): The fastest return to steady state with no overshoot or oscillation. This is the boundary condition and corresponds to a maximally flat response with no resonance peak.
3. Overdamped ($\zeta > 1$): The response is sluggish with no oscillation. The frequency response rolls off more gradually, and the filter becomes less selective.

The damping ratio and Q factor are related:

$Q = \frac{1}{2\zeta}$

So a high-Q filter is underdamped, and a low-Q filter is overdamped. Critically damped corresponds to $Q = 0.5$.

Natural Frequency and Resonance

The natural frequency $\omega_n$ is the frequency at which the LC portion of the circuit would oscillate if there were no resistance:

$\omega_n = \frac{1}{\sqrt{LC}}$

Or in hertz:

$f_n = \frac{1}{2\pi\sqrt{LC}}$

Resonance occurs when the driving frequency equals the natural frequency. At that point, the inductor's and capacitor's impedances are equal in magnitude but opposite in sign, so they cancel. In a series RLC circuit, this means the impedance drops to just $R$, and current reaches its maximum.

For band-pass and band-stop filters, $f_n$ sets the center frequency of the pass or reject band. You can tune the filter by changing $L$ or $C$. For instance, doubling $C$ lowers the center frequency by a factor of $\sqrt{2}$ (about 1.41), shifting the filter's response downward.