When you're putting together a closed-loop servo system, you'll want to nail down how fast your loop actually responds before you run into instability or end up with a system that's just too slow for the job. This Servo Loop Bandwidth Calculator uses your natural frequency (ωn) and damping ratio (ζ) to compute bandwidth, rise time, settling time, and peak overshoot—the core stats you need for tuning in motion control, robotics, or any precision application. If your bandwidth is off, the system might either overshoot and oscillate, or just lag behind. Below are the formulas, an example calculation, a look at control theory, and a FAQ for common questions.
What is servo loop bandwidth?
Servo loop bandwidth tells you over what frequency range your closed-loop system can follow changes at the input before the response starts to trail off. It's a good indicator of how quickly your servo reacts. Aim for higher bandwidth if you want speed, but keep in mind you'll also be picking up more noise and may be closer to instability if you're not careful.
Simple Explanation
Think of bandwidth as your system’s reflexes: high bandwidth is like a sharp sports car—quick on the uptake but can get twitchy. Low bandwidth is your work truck—reliable, not easily upset, but a bit sluggish to commands. It all comes down to weighing how quick you need it to be against how much stability you can afford to give up.
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Closed-Loop Servo System Diagram
Servo Loop Bandwidth Calculator
Servo Loop Bandwidth Interactive Visualizer
Adjust natural frequency and damping ratio to see directly how they shape closed-loop responses—bandwidth, rise/settling time, and overshoot are all shown in real time. Change parameters and watch the system's step and frequency response update live.
BANDWIDTH (-3dB)
15.8 rad/s
RISE TIME
0.22 s
SETTLING TIME
0.38 s
OVERSHOOT
4.6%
FIRGELLI Automations — Interactive Engineering Calculators
How to Use This Calculator
- Type in your system's natural frequency (ωn) in rad/s.
- Type in your damping ratio (ζ); usually a value between 0 and 2.
- Check your input values are what you intended.
- Click Calculate for immediate results.
This calculator is intended for education, concept evaluation, and preliminary design. Results are based on the equations and assumptions described on this page, but cannot account for every real-world load case, tolerance, material property, environmental condition, installation detail, safety factor, code, or regulatory requirement. Verify all inputs, assumptions, units, and results independently before selecting components or using the result in a real application. Safety-critical, structural, medical, lifting, transportation, or regulated applications must be reviewed by a qualified engineer.
📹 Video Walkthrough — How to Use This Calculator
Mathematical Equations
Bandwidth Calculation
Here’s the core bandwidth formula for a standard second-order system.
BW ≈ ωn√(1-2ζ²+√(4ζ⁴-4ζ²+2))
Time Domain Parameters
Rise Time (10%-90%):
tr = (π - tan⁻¹(√(1-ζ²)/ζ)) / (ωn√(1-ζ²))
Settling Time (2% criterion):
ts = 4 / (ζωn)
Peak Overshoot:
Mp = 100 × e(-πζ/√(1-ζ²)) %
Peak Time:
tp = π / (ωn√(1-ζ²))
Simple Example
Given: ωn = 10 rad/s, ζ = 0.7 (underdamped, near-optimal damping)
- Bandwidth: ≈ 10.5 rad/s
- Rise time: ≈ 0.22 s
- Settling time: ≈ 0.57 s
- Peak overshoot: ≈ 4.6%
Control Theory & Applications
Bandwith is a shortcut for seeing how quickly and accurately your closed-loop control can track commands or reject disturbances. The bandwidth number is practical—it’s your “real world” speed limit. Too low, and your system lags. Too high, and you’re open to noise or instability; bandwidth is always a tradeoff against plant realities.
Understanding Closed-Loop Response
The second-order response of a typical servo boils down to these two numbers: natural frequency (ωn) and damping ratio (ζ). Both affect not only bandwidth but also rise and settling time, and whether you get overshoot.
Natural frequency is how fast your system would oscillate with no damping. Damping ratio tells you how quickly those oscillations die out, or if they hang around at all. Together, they let you predict the basic feel of your transient and frequency response—before you ever power up your actuator.
Bandwidth and System Performance
The -3dB bandwidth is the point where your system’s response starts dropping off quickly—it’s about 70% of its steady-state gain. This is the cutoff for practical tracking and disturbance rejection. Here’s what bandwidth impacts in real designs:
- Command Tracking: Higher bandwidth equals quicker following of command signals
- Disturbance Rejection: You need enough bandwidth to knock down disturbances before they propagates
- Stability Margins: Increasing bandwidth pares down stability; the further you go, the tighter the margins
- Noise Sensitivity: As bandwidth goes up, you invite in more sensor and EMI noise
Applications in Motion Control
Servo loop bandwidth is central for sizing and tuning automated systems. For FIRGELLI linear actuators, you want enough for fast moves but not so much that you introduce chatter. Use these calculations for:
- CNC Machine Tools: Need higher bandwidth to keep the tool path tight and surface finishes clean
- Robotic Systems: Fine-tune the bandwidth for responsive, precise trajectory control
- Pick-and-Place Operations: The right bandwidth trims your cycle time without sacrificing accuracy
- Medical Devices: Smooth, slow precision generally benefits from moderate, well-damped bandwidth
Design Trade-offs
There’s no perfect setting—you’re always balancing:
Speed vs. Stability: Push for higher bandwidth if you need speed, but keep an eye out for instability, especially if your system includes transport lags or the plant is not fully modeled. The calculator's a blunt tool to help you find a starting point to iterate from.
Accuracy vs. Noise: Increase bandwidth for better accuracy (faster following of references), but plan for more noise and potential actuator dither, especially with “noisy” feedback like incremental encoders.
Overshoot vs. Rise Time: Damping ratio is your main lever here. Under 0.7 typically means the response is quick but will overshoot, while higher values slow the response but keep things smooth. There’s no substitute for plotting both before making a choice.
Design Examples
Example 1: Linear Actuator Positioning System
System Requirements:
- Position accuracy: ±0.1mm
- Maximum settling time: 0.5 seconds
- Minimal overshoot preferred
Design Process:
Start with settling time: ts = 4/(ζωn) ≤ 0.5s
If you want almost no overshoot, use ζ = 0.8 (mildly underdamped)
So: ωn ≥ 4/(0.8 × 0.5) = 10 rad/s
With ωn = 10 rad/s and ζ = 0.8:
- Bandwidth: 11.6 rad/s (1.85 Hz)
- Rise time: 0.178 s
- Settling time: 0.5 s
- Overshoot: 1.5%
This approach delivers tight response with almost no overshoot. Always check if the actuator and sensor hardware can keep up.
Example 2: High-Speed Pick-and-Place Robot
System Requirements:
- Fast response time for high throughput
- Rise time < 0.1 seconds
- Some overshoot acceptable (< 20%)
Design Process:
For a nimble system, try ζ = 0.5 (lightly underdamped)
For the rise time requirement: tr ≈ 1.8/ωn < 0.1s
So: ωn > 18 rad/s
With ωn = 20 rad/s and ζ = 0.5:
- Bandwidth: 19.3 rad/s (3.07 Hz)
- Rise time: 0.075 s
- Settling time: 0.4 s
- Overshoot: 16.3%
This is typical for a quick pick-and-place. You can trade some overshoot for high throughput, but it depends on your downstream process tolerance.
Practical Implementation Considerations
Actuator Limitations: Calculated bandwidth needs to match what the actuator can do in the real world. Electric actuators are bandwidth-limited by their own speed and acceleration profiles—don’t rely on calculations alone.
Sensor Resolution: Feedback source must resolve fast changes cleanly at your target bandwidth. Low-resolution or slow sensors can add noise or quantization steps that will kill performance above a certain frequency.
Controller Implementation: Watch the controller sampling rate—digital implementation lag needs to be comfortably below your lowest settling time. As a general rule, use a controller rate at least 10x your bandwidth.
Mechanical Resonances: Only ideal second-order models match the calculator results. Real-world mechanicals add resonance modes that need attention. If you’re near a structural resonance, extra filters or notch controls may be mandatory.
When projects use multiple axes or actuators, run the numbers for each one. For coordinated moves, align bandwidth and damping across axes so that timing and response are predictable.
Frequently Asked Questions
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About the Author
Robbie Dickson
Chief Engineer & Founder, FIRGELLI Automations
Robbie Dickson brings over two decades of engineering expertise to FIRGELLI Automations. With a distinguished career at Rolls-Royce, BMW, and Ford, he has deep expertise in mechanical systems, actuator technology, and precision engineering.
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