A notch filter transfer function is a mathematical expression, typically denoted as $H(s)$ or $H(j\omega)$, that describes how a circuit attenuates a specific, narrow band of frequencies while passing all others with minimal loss. In a real circuit or installation, this function changes the frequency response by introducing a deep transmission zero at a target interference frequency—dropping the signal amplitude by 40dB to 60dB at that exact point without altering the phase or amplitude of the surrounding audio, RF, or data spectrum. Designers commonly confuse a true notch filter (high Q, narrow bandwidth) with a broad band-stop filter (low Q, wide rejection band) or a simple low-pass filter, which indiscriminately kills everything above a cutoff. Unlike a low-pass filter that acts like a physical height-restriction barrier blocking all tall vehicles, a notch filter is like a targeted toll booth that only stops vehicles with one specific license plate number while letting all other traffic flow freely.
The Core Math: What the Transfer Function Actually Does
To design an effective filter, you must understand the biquadratic (biquad) transfer function that governs second-order notch responses. The standard Laplace-domain equation is:
The Standard Notch Transfer Function:
$H(s) = \frac{s^2 + \omega_0^2}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2}$
Here, $s$ is the complex frequency variable ($j\omega$), $\omega_0$ is the center notch frequency in radians per second ($2\pi f_0$), and $Q$ is the quality factor. The numerator $(s^2 + \omega_0^2)$ creates the transmission zero; when the input frequency exactly matches $\omega_0$, the numerator evaluates to zero, theoretically yielding infinite attenuation. The denominator dictates the bandwidth and phase shift around the notch. A higher $Q$ results in a narrower, deeper notch, but demands tighter component tolerances and higher op-amp gain-bandwidth product (GBW).
According to the All About Circuits guide on band-stop filters, the physical realization of this math requires a network that sums a high-pass and a low-pass response, or utilizes a resonant trap to shunt the target frequency to ground while maintaining the passband.
Worked Numeric Example: Designing a 60Hz Active Twin-T
Let’s calculate the exact component values for a passive Twin-T network (which we will later buffer with an op-amp) targeting 60Hz mains hum.
Step 1: Define the target and pick a capacitor.
Target frequency $f_0 = 60$ Hz. We need high-stability capacitors, so we select $C = 100$ nF (0.1 µF) using 50V WIMA MKS polypropylene film capacitors. Avoid ceramics here for reasons detailed later.
Step 2: Calculate the base resistor value.
The Twin-T notch frequency is defined by $f_0 = \frac{1}{2 \pi R C}$. Rearranging for $R$:
$R = \frac{1}{2 \pi f_0 C} = \frac{1}{2 \pi \times 60 \times (100 \times 10^{-9})} = 26,525.8 \, \Omega$
Step 3: Select precision resistors.
Standard 1% E96 series resistors do not have an exact 26.5k value. To hit the 60Hz target precisely without relying on a trimmer potentiometer, we use a series combination: a 26.1 kΩ E96 resistor in series with a 430 Ω E96 resistor yields exactly 26,530 Ω, placing our notch at 59.99 Hz.
Step 4: Scale for the Twin-T branches.
The Twin-T topology requires two parallel branches. The high-pass branch requires $2R$ and $C/2$. The low-pass branch requires $R/2$ and $2C$ (though standard practice often mirrors the $R$ and $C$ values with a 2:1 ratio network).
- $2R = 53,060 \, \Omega$ (Use 52.3 kΩ + 750 Ω in series).
- $C/2 = 50$ nF (Wire two 100 nF film capacitors in series).
Where You Meet This in Practice
You will encounter the need for a notch filter transfer function in three primary domains:
- Audio and Instrumentation Preamps: Removing 50Hz or 60Hz ground loop hum from microphone preamps or turntable phono stages without muddying the 20Hz-20kHz audio spectrum.
- Biomedical Sensors (ECG/EEG):strong> Front-end analog filtering for ICs like the TI ADS1298. The biopotential signals are in the microvolt range; if 60Hz hum isn't notched analogously before the PGA (Programmable Gain Amplifier), it will clip the ADC input stage, causing digital aliasing that no DSP can fix.
- PLL and VCO Loops: Cleaning up specific reference spurs or switching converter ripple (e.g., a 500kHz spike from a buck regulator) in the control voltage path of a phase-locked loop.
Topology Decision Tree: Pick Your Circuit
Choosing the right active topology is where most designs fail. Use this decision matrix to select your circuit architecture based on your constraints.
| Topology | Q-Factor Capability | Tuning Ease | Component Count | Best Use Case |
|---|---|---|---|---|
| Twin-T (Active) | Medium (Q < 20) | Difficult (requires matched R/C pairs) | High (6+ passive parts) | Fixed-frequency, low-cost consumer audio. |
| Fliege | High (Q up to 50+) | Easy (tune Q with one resistor) | Medium (4 R, 2 C, 2 Op-amps) | Precision 50/60Hz hum removal, high-end sensors. |
| State-Variable | Very High (Q > 100) | Moderate | High (requires 3+ op-amps) | Parametric EQs, simultaneous LP/BP/HP outputs. |
| Passive LC Trap | Low (Limited by inductor ESR) | Difficult (requires variable inductor) | Low (1 L, 1 C) | RF paths, high-voltage power lines (no op-amp headroom limits). |
The Default Pick: For precision analog front-ends in 2026 requiring high-Q 50Hz/60Hz rejection, use the Fliege topology. It requires two op-amps but allows independent tuning of the notch frequency and the Q-factor. Pair it with a low-noise, high-GBW bipolar op-amp like the TI OPA211 (1.1 nV/√Hz noise, 80 MHz GBW) to ensure the active components do not degrade the noise floor of your sensor.
Component Selection and Real-World Parasitics
The math assumes ideal components. In reality, parasitics will destroy your notch depth if you ignore them.
The Capacitor Dielectric Trap
Never use X7R or Y5V ceramic capacitors in the frequency-determining network of a notch filter. X7R dielectrics exhibit high dielectric absorption (DA) and severe piezoelectric microphonic effects. When subjected to mechanical vibration or voltage bias, an X7R cap's value shifts, pulling your 60Hz notch off-target and limiting your maximum attenuation to roughly -30dB. Always specify C0G/NP0 ceramics (for values under 10nF) or Polypropylene/PET film capacitors (like WIMA MKS or Vishay Roederstein series) for values in the 100nF range. Film caps offer near-zero DA and excellent linearity, easily allowing notch depths of -60dB or greater.
Resistor Tolerance and Thermal Noise
A notch depth of -60dB requires component matching to at least 0.1%. Standard 1% thick-film resistors will limit your notch depth to roughly -40dB due to the imbalance in the bridge network. Specify 0.1% thin-film resistors (e.g., Susumu RG series or Vishay PAT series). Additionally, keep resistor values below 100 kΩ to minimize Johnson-Nyquist thermal noise, which becomes critical in high-gain biomedical and audio paths.
Frequently Asked Questions
Can I just skip the analog notch and use a digital IIR filter in my microcontroller?
You can, but only if the analog interference is small enough not to clip your ADC. If a 2V peak-to-peak 60Hz ground loop enters a 3.3V ADC input, it will consume your entire dynamic range. The ADC will clip, creating broadband harmonic distortion that a digital notch filter cannot remove. You must notch the hum in the analog domain before the PGA/ADC to preserve dynamic range.
Why is my active notch filter oscillating on the bench?
Oscillation in high-Q notch filters usually stems from three issues: (1) The op-amp's GBW is too low; for a 60Hz notch with Q=50, you need an op-amp with a GBW of at least $100 \times f_0 \times Q$ (roughly 300 kHz minimum, but 10 MHz is safer for phase margin). (2) The PCB layout has stray capacitance coupling the output back to the high-impedance summing nodes. (3) The power supply decoupling is inadequate, causing the op-amp to rail-to-rail oscillate. Ensure 100nF C0G caps are placed within 2mm of the op-amp VCC pins.
How do I tune the notch if my components drift?
This is why the Fliege topology wins for prototyping and production tuning. In a Fliege circuit, the Q-factor is set by a single resistor ratio, and the center frequency is set by a separate RC pair. You can place a 10kΩ multi-turn Bourns cermet trimmer in series with the frequency-setting resistor, inject a 60Hz sine wave from a function generator, and tweak the trimmer while watching the output on an oscilloscope until the waveform hits the absolute minimum millivolt reading.
For any high-precision analog front-end requiring 50Hz or 60Hz mains rejection, the default choice is the Fliege topology paired with a low-noise bipolar op-amp like the TI OPA2211. Do not default to the Twin-T unless your frequency is permanently fixed and you have 0.1% matched components on hand. Build the Fliege, specify C0G or film capacitors, tune the Q with a single resistor, and let the transfer function math do the heavy lifting.
