Designing a closed-loop servo system means you need to know exactly how fast your system can respond before it becomes unstable or too sluggish to be useful. Use this Servo Loop Bandwidth Calculator to calculate bandwidth, rise time, settling time, and peak overshoot using your system's natural frequency (ωn) and damping ratio (ζ). Getting these numbers right matters in motion control, robotics, and precision manufacturing — a poorly sized bandwidth means either a system that oscillates out of control or one that's too slow to track its reference. This page covers the formula, a worked example, the underlying control theory, and a full FAQ.
What is servo loop bandwidth?
Servo loop bandwidth is the frequency range over which a closed-loop control system can track an input signal without significant loss in response. It tells you how fast your servo system can react — higher bandwidth means faster response, but also greater sensitivity to noise and potential instability.
Simple Explanation
Think of bandwidth like the reflexes of your servo system — a high-bandwidth system reacts quickly to commands, like a sports car that responds instantly to steering input. A low-bandwidth system is more like a loaded truck: stable and smooth, but slow to change direction. You're picking the right balance between speed and stability for your specific job.
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Closed-Loop Servo System Diagram
Servo Loop Bandwidth Calculator
Servo Loop Bandwidth Interactive Visualizer
Explore how natural frequency and damping ratio affect closed-loop servo system performance including bandwidth, rise time, settling time, and overshoot. Adjust parameters to see real-time changes in the system's step response and frequency characteristics.
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
- Enter your system's natural frequency (ωn) in rad/s into the Natural Frequency field.
- Enter your damping ratio (ζ) — a value between 0 and 2 — into the Damping Ratio field.
- Review the input fields to confirm the values match your system's design parameters.
- Click Calculate to see your result.
Mathematical Equations
Bandwidth Calculation
Use the formula below to calculate servo loop bandwidth.
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
The servo loop bandwidth calculator is fundamental to understanding closed-loop control system performance. In servo systems, the bandwidth represents the frequency range over which the system can effectively track reference inputs with minimal attenuation. This characteristic directly impacts the system's ability to respond to commands and reject disturbances.
Understanding Closed-Loop Response
A closed-loop servo system's performance is characterized by its second-order response, which can be described by two key parameters: natural frequency (ωn) and damping ratio (ζ). These parameters determine not only the bandwidth but also critical time-domain characteristics like rise time, settling time, and overshoot.
The natural frequency represents the oscillation frequency of the system when undamped, while the damping ratio indicates how oscillations decay over time. Together, these parameters define the system's transient response and frequency domain characteristics.
Bandwidth and System Performance
The -3dB bandwidth is the frequency at which the closed-loop gain drops to 70.7% of its DC value. This frequency range is crucial for servo systems because:
- Command Tracking: Higher bandwidth allows the system to track faster-changing reference signals
- Disturbance Rejection: Adequate bandwidth ensures the system can respond to and correct for disturbances
- Stability Margins: Bandwidth affects both gain and phase margins, critical for robust operation
- Noise Sensitivity: Higher bandwidth may allow more high-frequency noise into the system
Applications in Motion Control
Servo loop bandwidth calculations are essential in various automation applications. For FIRGELLI linear actuators, proper bandwidth design ensures precise positioning with minimal settling time. Common applications include:
- CNC Machine Tools: High bandwidth ensures accurate tool positioning and smooth surface finishes
- Robotic Systems: Proper bandwidth design enables precise end-effector control and trajectory following
- Pick-and-Place Operations: Optimized bandwidth reduces cycle time while maintaining accuracy
- Medical Devices: Controlled bandwidth ensures smooth, precise movements in surgical robots
Design Trade-offs
When designing servo systems, engineers must balance several competing requirements:
Speed vs. Stability: Higher bandwidth generally means faster response, but excessive bandwidth can lead to instability, especially in systems with delays or unmodeled dynamics. The servo loop bandwidth calculator helps engineers find the optimal balance.
Accuracy vs. Noise: While higher bandwidth improves tracking accuracy for fast signals, it also allows more high-frequency noise to pass through the system. This is particularly important in applications using encoders or other feedback devices that may introduce noise.
Overshoot vs. Rise Time: The damping ratio significantly affects this trade-off. Lower damping ratios (ζ < 0.7) provide faster rise times but introduce overshoot, while higher damping ratios eliminate overshoot but slow the response.
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:
Starting with the settling time requirement: ts = 4/(ζωn) ≤ 0.5s
For minimal overshoot, choose ζ = 0.8 (slightly underdamped)
Therefore: ωn ≥ 4/(0.8 × 0.5) = 10 rad/s
Using our servo loop bandwidth calculator 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 design meets all requirements with excellent performance characteristics.
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 fast response, choose ζ = 0.5 (underdamped)
For rise time requirement: tr ≈ 1.8/ωn < 0.1s
Therefore: ωn > 18 rad/s
Using 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 aggressive design achieves the fast response needed for high-throughput applications.
Practical Implementation Considerations
Actuator Limitations: The calculated bandwidth must be achievable by your actuator. Electric linear actuators have maximum velocities and accelerations that may limit the practical bandwidth. Always verify that your actuator can provide the required dynamic response.
Sensor Resolution: The feedback sensor must have sufficient resolution and bandwidth to support the designed system bandwidth. Encoders with inadequate resolution can introduce quantization noise that degrades performance at higher bandwidths.
Controller Implementation: Digital controllers introduce sampling delays that can affect the achievable bandwidth. The sampling frequency should typically be at least 10 times the desired bandwidth to maintain good performance.
Mechanical Resonances: Real mechanical systems have resonant frequencies that can limit the practical bandwidth. The servo loop bandwidth calculator assumes an ideal second-order system, but mechanical resonances may require additional filtering or notch compensation.
For complex automation projects involving multiple actuators, using a systematic approach with bandwidth calculations ensures consistent performance across all axes of motion. This is particularly important in coordinated motion applications where timing between axes is critical.
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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