Settling Time Calculator Formula

Understand the math behind the settling time calculator. Each variable explained with a worked example.

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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 * wn

Variables

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.

wn = 10zeta = 0.5percent = 2

01sigma = 0.5 × 10 = 5 s⁻¹
02ts = -ln(0.02) / 5 = 3.912 / 5 = 0.782 s
03Approximate: 4 / 5 = 0.800 s

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