Notch Filter for Resonance Suppression Calculator
Design a 2nd-order notch filter to suppress a specific resonance frequency, with computed depth, -3dB bandwidth, and a verified frequency response.
🔇 What is a Notch Filter for Resonance Suppression?
A notch filter is a second-order band-stop network that attenuates a narrow band of frequencies around a chosen center frequency while leaving frequencies outside that band essentially unaffected. In control systems and motion control, notch filters are inserted directly into a servo or feedback loop to suppress a specific mechanical resonance, a frequency at which a physical structure (a motor shaft, a machine frame, a robotic arm link) naturally wants to vibrate, so the loop can be tuned for faster, higher-gain response without that resonance ringing or destabilizing the system.
Motion-control engineers use notch filters constantly when tuning servo drives for CNC machines, robotic arms, semiconductor wafer stages, and camera gimbals, any system where a lightly-damped mechanical resonance would otherwise limit how aggressively the position or velocity loop can be tuned. Audio and signal-processing engineers use the identical filter structure to remove 50/60 Hz electrical hum from a recording, or to null out a single problematic frequency in a parametric equalizer, while power system engineers use notch filters to reject specific harmonic frequencies from a measurement or control signal.
A common misconception is that a notch filter is a single fixed shape, one depth for one width. In reality, depth and width are two independent design parameters, a notch filter can be made extremely deep but very narrow, moderately deep but wide, or anywhere in between, by choosing the zero Q (Qz, depth) and pole Q (Qp, bandwidth) separately. An ideal, infinitely deep notch corresponds to Qz approaching infinity, while any finite Qz gives a real, achievable filter with a finite (though often still very large) attenuation at the target frequency.
This calculator takes the target resonance frequency fn along with Qp and Qz, computes the resulting notch depth in dB, the -3 dB bandwidth and its edge frequencies, and verifies the full frequency response directly, at the notch center, at both bandwidth edges, and far away from resonance, so you can confirm the filter behaves exactly as designed before implementing it in a real control loop.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — 50 Hz Motor Resonance, Moderate Depth
fn = 50 Hz, Qp = 5, Qz = 50
Example 2 — 120 Hz Resonance, Shallower Notch
fn = 120 Hz, Qp = 8, Qz = 40
Example 3 — 200 Hz Resonance, Deep Narrow Notch
fn = 200 Hz, Qp = 4, Qz = 100
❓ Frequently Asked Questions
🔗 Related Calculators
What is a notch filter used for in control systems?
A notch filter is inserted into a control loop to suppress a specific troublesome frequency, most commonly a mechanical resonance in a servo or motion-control system, without significantly affecting the loop's response at other frequencies. This lets the loop run at higher gain and bandwidth than the resonance would otherwise allow before it starts ringing or becoming unstable.
What is the transfer function of a finite-depth notch filter?
H(s) = (s^2 + (wn/Qz)s + wn^2) / (s^2 + (wn/Qp)s + wn^2), where wn = 2*pi*fn is the notch center frequency in rad/s, Qz sets the depth of the notch, and Qp sets its bandwidth. An ideal, infinitely deep notch is the limit as Qz approaches infinity.
How do you calculate notch filter depth?
At the exact center frequency fn, |H(jwn)| simplifies to Qp/Qz, so the depth in dB is 20*log10(Qp/Qz). For example, Qp = 5 and Qz = 50 gives a depth of 20*log10(0.1) = -20 dB.
How do you calculate notch filter bandwidth?
The -3 dB bandwidth follows the standard Q-factor relationship BW = fn/Qp, using the pole Q, Qp. For fn = 50 Hz and Qp = 5, BW = 10 Hz, spanning approximately 45 Hz to 55 Hz around the center frequency.
Why do I need two separate Q values, Qz and Qp?
Qz (the zero's Q) controls how deep the notch is, while Qp (the pole's Q) controls how wide it is. These are independent design choices, real digital notch filter design commonly separates them so you can make a notch very deep without also making it impractically narrow, or vice versa.
What happens if Qz is less than or equal to Qp?
The filter stops being a notch. At Qz = Qp, |H(jw)| = 1 at every frequency (the zero and pole cancel exactly, an all-pass response). At Qz less than Qp, the ratio Qp/Qz at the center frequency exceeds 1, meaning the filter boosts rather than suppresses that frequency, the opposite of what a resonance suppression filter is for.
Is the BW = fn/Qp formula exact?
It is the standard, widely used approximation for the -3 dB points of a 2nd-order notch, and it is quite accurate for moderate to high Qp. Direct computation of |H(jw)| at the estimated edge frequencies typically lands within a few tenths of a dB of the ideal -3 dB point, close enough for practical filter design and tuning.
Does the notch filter affect frequencies far from the resonance?
Very little. As frequency moves away from the center frequency fn in either direction, the (wn^2 - w^2)^2 term dominates both the numerator and denominator of |H(jw)|^2 and they converge to the same value, so the gain returns to 1 (0 dB). This calculator verifies this directly by evaluating |H(jw)| at one-tenth and ten times the notch frequency.
Can I use this notch filter design for an audio or electrical application?
Yes, the same finite-depth notch biquad structure is used in parametric EQ notch filters, hum (50/60 Hz) rejection filters, and general band-stop filter design, not just mechanical resonance suppression. The math is identical, only the interpretation of what fn represents changes.
How narrow should I make a notch filter for a servo resonance?
Narrow enough (high Qp) to avoid attenuating frequencies the control loop actually needs for good response, but wide enough (moderate Qp) to still suppress the resonance if its exact frequency drifts slightly with temperature, wear, or load changes. A Qp in the range of 3 to 10 is a common practical starting point before fine-tuning against a real measured resonance.
Why is the notch depth independent of the center frequency?
At w = wn, the (wn^2 - w^2) term in both the numerator and denominator of |H(jw)| is exactly zero, leaving only the (w*wn/Q) terms, whose ratio simplifies to Qp/Qz. Since fn (through wn) appears identically in both the surviving numerator and denominator terms, it cancels out of the ratio entirely, leaving depth as a function of Qp and Qz alone.