Settling Time Calculator Formula
Understand the math behind the settling time calculator. Each variable explained with a worked example.
Formulas Used
Settling Time
ts = -log(percent / 100) / (zeta * wn)Settling Time (4/sigma approx)
ts_approx = 4 / (zeta * wn)Decay Rate (sigma)
sigma = zeta * wnVariables
| Variable | Description | Default |
|---|---|---|
wn | Natural Frequency (omega_n)(rad/s) | 10 |
zeta | Damping Ratio (zeta) | 0.5 |
percent | Settling Criterion(%) | 2 |
How It Works
Second-Order Settling Time
Settling time is the time required for the system response to remain within a specified percentage band (typically 2% or 5%) of the final value.
Formula
ts ≈ -ln(criterion) / (zeta × omega_n)
For the common 2% criterion: ts ≈ 4 / (zeta × omega_n). For 5%: ts ≈ 3 / (zeta × omega_n). The decay rate sigma = zeta × omega_n determines how quickly oscillations die out.
Worked Example
A second-order system with omega_n = 10 rad/s, zeta = 0.5, 2% criterion.
- 01sigma = 0.5 × 10 = 5 s⁻¹
- 02ts = -ln(0.02) / 5 = 3.912 / 5 = 0.782 s
- 03Approximate: 4 / 5 = 0.800 s
Frequently Asked Questions
What is the difference between 2% and 5% settling time?
The 2% settling time is when the response stays within ±2% of the final value. The 5% settling time is less strict and therefore shorter. The 2% criterion is more common in practice.
How does damping affect settling time?
Increasing damping ratio reduces oscillations but makes the system slower (higher tau). The settling time is minimized at zeta ≈ 0.7 for a given natural frequency. Very low or very high damping both increase settling time.
Does this formula work for overdamped systems?
The exponential envelope approximation works best for underdamped systems (zeta < 1). For overdamped systems (zeta > 1), the response has no oscillations and the settling time depends on the slower of two real poles.
Ready to run the numbers?
Open Settling Time Calculator