Band-Reject Filter Design: Notch Frequencies, Q-Factor, and Component Sizing

A band-reject filter is an electronic circuit that attenuates a specific range of frequencies while allowing all frequencies below and above that band to pass through unchanged. Often referred to as a "notch filter" when the rejected band is extremely narrow, this topology is the surgical tool of signal processing. Instead of rolling off high or low ends like standard low-pass or high-pass filters, a band-reject filter targets a specific interference spike—like 60 Hz mains hum or a switching regulator ripple—and carves it out of your signal path without degrading the surrounding bandwidth.

The Core Mechanics: Impedance Profiles and Signal Routing

To understand what a band-reject filter changes in a real circuit, you have to look at impedance. The filter alters the frequency-dependent impedance profile of the network. At the target center frequency ($f_c$), the circuit either creates a massive series impedance barrier (blocking the signal from reaching the load) or a near-zero shunt impedance path (diverting the unwanted frequency directly to ground). Frequencies outside this rejection band see a nominal impedance and pass through to the load with minimal insertion loss.

Think of it like a highway system where a specific weight class of trucks (the target frequency) is forced onto a decommissioned, high-friction gravel bypass, while sports cars (high frequencies) and sedans (low frequencies) take the smooth, direct mainline routes. The traffic still flows, but the specific nuisance vehicles are stalled out.

Design Data: Topologies, Q-Factors, and Component Realities

Selecting the right topology depends entirely on your required Q-factor, available board space, and whether you are dealing with passive audio lines or active sensor conditioning. The table below breaks down the four most common band-reject topologies you will encounter on the bench.

A critical reality of passive RC notch filters (like the Twin-T) is component tolerance and parasitic Equivalent Series Resistance (ESR). If your capacitors have a 5% tolerance, your notch depth might only be 15 dB instead of the theoretical 60 dB. The "leakage" through the mismatched arms prevents complete destructive interference at $f_c$. For deep notches, you must use 1% or 0.1% metal film resistors and C0G/NP0 ceramic or polypropylene film capacitors.

Worked Example: Designing a 60 Hz Mains Hum Notch Filter

Let’s design a passive Twin-T notch filter to strip 60 Hz mains hum from an audio preamplifier output. The Twin-T network consists of two parallel T-networks: one low-pass (two resistors, one capacitor to ground) and one high-pass (two capacitors, one resistor to ground).

The formula for the center frequency of a symmetrical Twin-T filter is:

$$f_c = \frac{1}{2 \pi R C}$$

First, we select a standard capacitor value. Let’s use C = 120 nF (0.12 µF) for the high-pass arms ($C_1$ and $C_2$). Now we solve for R:

$$R = \frac{1}{2 \pi \times 60 \times 120 \times 10^{-9}}$$

$$R = \frac{1}{4.523 \times 10^{-5}} \approx 22,104 \, \Omega$$

We will use 22.1 kΩ 1% metal film resistors for the low-pass arms ($R_1$ and $R_2$). The Twin-T topology requires the ground-leg components to be exactly double the capacitance and half the resistance of the series arms. Therefore:

• $C_3$ (Ground leg): Needs to be $2 \times 120$ nF = 240 nF. (Achieved by placing two 120 nF capacitors in parallel).
• $R_3$ (Ground leg): Needs to be $22.1 \text{ k}\Omega / 2$ = 11.05 kΩ. (Achieved by placing two 22.1 kΩ resistors in parallel).

The Q-Factor Problem: A purely passive Twin-T filter has an inherent Q-factor of exactly 0.25. This means the "notch" is actually a very wide, shallow valley that will severely attenuate your 40 Hz and 90 Hz audio bass frequencies alongside the 60 Hz hum. To fix this, we inject a small amount of positive feedback using an op-amp buffer (like the TI OPA2134). By feeding a portion of the output signal back into the $R_3/C_3$ ground node, we can artificially boost the Q-factor to 10 or 15, narrowing the rejection band to just a few Hertz and leaving your audio bass completely intact.

Where You Meet This in Practice (And Common Pitfalls)

You will encounter band-reject filters across vastly different voltage and frequency domains. Here is where they earn their keep on the jobsite and the bench:

• Biomedical Instrumentation (ECG/EEG): Biological signals are in the microvolt range and easily swamped by 50/60 Hz electromagnetic interference. Designers use active MFB notch filters or switched-capacitor ICs (like the Analog Devices LTC1068) to notch out mains hum without distorting the critical ST-segment of the heartbeat waveform.
• Audio Engineering: Ground loops cause 60 Hz (or 120 Hz full-wave rectified) hum. While a high-pass filter at 80 Hz would kill the hum, it also kills the kick drum. A high-Q notch filter surgically removes the hum while preserving the sub-bass.
• Power Electronics & VFDs: Variable Frequency Drives generate massive harmonic currents. Passive LC band-reject traps (tuned to the 5th, 7th, 11th, and 13th harmonics) are installed on the busbars to shunt these specific frequencies to ground, preventing them from overheating upstream transformers.