Active Band Reject Filter Design: How to Eliminate Target Frequencies

An active band reject filter is an op-amp-based circuit that sharply attenuates a specific target frequency band while amplifying or buffering the frequencies above and below it. If you are trying to strip 60 Hz mains hum out of an audio preamp or remove 50 Hz interference from an ECG sensor without killing the high-frequency transients, this is the exact tool for the job.

What this changes in a real circuit is profound: unlike passive LC or RC notch filters, which suffer from insertion loss and get detuned by the load impedance of the next stage, an active design uses operational amplifiers to provide high input impedance, low output impedance, and even passband gain. It allows you to surgically remove a narrow interference spike without degrading the overall signal-to-noise ratio or requiring bulky, expensive inductors.

What people commonly confuse it with is a band-pass filter (which does the exact opposite, rejecting everything except the target band) or a simple low-pass filter. A low-pass filter will indeed remove 60 Hz hum if your signal is strictly DC or sub-10Hz, but if you are processing audio or fast sensor edges, a low-pass filter will destroy your high-frequency data entirely.

The Core Mechanics and Common Confusions

At its heart, an active band reject filter (often called a notch filter) combines a frequency-selective passive network with an active gain element. The passive network creates a null at the target center frequency ($f_0$). The op-amp buffers the signal, preventing the source impedance from interacting with the filter network, and isolates the filter from the load impedance of whatever stage comes next.

The critical metric for any notch filter is the Q factor (Quality factor), which defines how narrow the rejection band is. A low Q (e.g., Q = 1) creates a wide, shallow dip that might attenuate a 20 Hz band around your target. A high Q (e.g., Q = 20) creates a razor-sharp spike of attenuation that removes only the exact 60.0 Hz fundamental, leaving 58 Hz and 62 Hz completely untouched. In active designs, we use positive feedback loops to artificially boost the Q factor far beyond what passive components alone can achieve.

Component Selection and Topology Data

Choosing the right topology and op-amp depends entirely on your application's noise floor and bandwidth requirements. Below is a baseline specification table for common real-world implementations.

For deeper design theory on how these topologies interact with op-amp gain-bandwidth products, the Analog Devices active filter design guide provides excellent transfer function derivations, while All About Circuits offers a solid primer on the underlying semiconductor behavior.

Worked Numeric Design: A 60 Hz Active Twin-T Notch

Let's design an active band reject filter to eliminate 60 Hz mains hum from a high-impedance audio sensor line. We will use the classic Twin-T network with an op-amp Q-enhancement loop. We'll select the OPA1612 for its ultra-low voltage noise density ($1.1 \text{ nV}/\sqrt{\text{Hz}}$) and high slew rate.

Step 1: Calculate the Base R and C Values
The center frequency formula for a Twin-T network is:

$$f_0 = \frac{1}{2 \pi R C}$$

We need $f_0 = 60\text{ Hz}$. Let's select a standard capacitor value to avoid massive, expensive film caps. We'll choose $C = 100\text{ nF}$ ($0.1 \mu\text{F}$). Solving for R:

$$R = \frac{1}{2 \pi \times 60 \times 100 \times 10^{-9}} = 26,525 \Omega$$

We will use standard 1% tolerance resistors. The closest E96 series value is $26.7\text{ k}\Omega$. This shifts our actual center frequency to $59.6\text{ Hz}$, which is well within the null bandwidth for 60 Hz hum.

Step 2: Map the Twin-T Component Values
The Twin-T requires specific ratios:

• $R_1 = R_2 = R = \mathbf{26.7\text{ k}\Omega}$
• $R_3 = R / 2 = \mathbf{13.3\text{ k}\Omega}$ (Use a $10\text{ k}\Omega$ fixed resistor in series with a $5\text{ k}\Omega$ trimmer pot for exact tuning)
• $C_1 = C_2 = C = \mathbf{100\text{ nF}}$
• $C_3 = 2C = \mathbf{200\text{ nF}}$ (Use two 100nF caps in parallel)

Step 3: Calculate the Q-Enhancement Feedback
A passive Twin-T has a dismal Q of 0.25, meaning the notch is incredibly wide and shallow. To sharpen it, we buffer the output with the first half of the OPA1612, then use the second half as a voltage divider to feed a fraction of the output signal ($\beta$) back into the ground leg of the Twin-T ($R_3$ and $C_3$).

The relationship between Q and the feedback factor $\beta$ is:

$$Q = \frac{0.5}{1 - \beta}$$

If we want a sharp notch with $Q = 10$, we solve for $\beta$:

$$10 = \frac{0.5}{1 - \beta} \implies \beta = 0.95$$

This means we need to feed exactly 95% of the output signal back to the Twin-T ground node. We achieve this by placing a $10\text{ k}\Omega$ multi-turn trimmer potentiometer between the op-amp output and ground, with the wiper connected to the Twin-T ground leg. Adjusting this pot tunes the depth and sharpness of the notch in real-time on the bench.

Where You Meet This in Practice

You will encounter active band reject filters in several critical domains where signal integrity is non-negotiable:

• Audio Mixing Consoles and DI Boxes: High-end analog mixers use active notch filters on channel inserts or master buses to surgically remove ground loop hum (50/60 Hz and its 120/180 Hz harmonics) without dulling the cymbal crashes or vocal sibilance that a low-pass filter would destroy.
• Biomedical Instrumentation (ECG/EEG): The human body acts as a massive antenna for mains electric fields. ECG amplifiers use ultra-high-Q active notch filters (often state-variable topologies using chopper-stabilized op-amps like the ADA4522) to strip out 50/60 Hz interference while preserving the sub-1 Hz to 100 Hz bandwidth of the heart's electrical activity.
• Industrial Vibration Analysis: When monitoring bearing health on a large AC motor, the fundamental rotational frequency (e.g., 1780 RPM = 29.6 Hz) will completely mask the high-frequency harmonics caused by a failing bearing race. An active band reject filter tuned to the motor's fundamental RPM removes the overwhelming baseline vibration, allowing the FFT analyzer to see the micro-volt bearing fault signatures.

Frequently Asked Questions

Can I just use a passive LC notch filter instead?
You can, but you will pay for it in size, cost, and loading effects. Inductors required for 50/60 Hz rejection are physically large, expensive, and prone to picking up magnetic interference from nearby transformers. Furthermore, a passive LC filter's notch depth will collapse if the load impedance of your next circuit stage drops below a few kilo-ohms. Active filters isolate the network from the load entirely.

Why does my active notch filter oscillate when I connect it to my ADC?
This is almost always a phase-margin issue caused by capacitive loading. If your ADC has a high input capacitance (or you have a long coaxial cable between the filter and the ADC), that capacitance interacts with the op-amp's output impedance, creating a pole that destroys phase margin. Fix it by adding a small series isolation resistor (typically $22 \Omega$ to $47 \Omega$) directly at the op-amp output pin, before the cable or ADC capacitor.

Should I use an active analog notch or just do it in DSP?
If your ADC has sufficient bit-depth (24-bit) and your interference isn't clipping the analog front-end, a digital FIR notch filter in software is vastly superior—it requires no component matching and never drifts with temperature. However, if the 60 Hz hum is so large that it consumes your ADC's dynamic range or causes rail-to-rail clipping in the analog gain stages, you must use an active analog band reject filter before the ADC to preserve the signal-to-noise ratio.