First-Order System Response Calculator
A first-order process with gain 2 and time constant 3 reaches 8.64665 at t = 6 after a step of 5: 86.5 % of its final value.
8.64664716763387
Full answer, then cut to fewer decimals (not rounded):
- 8.64664716763387
- 8.6466471676338
- 8.646647167633
- 8.64664716763
- 8.6466471676
- 8.646647167
- 8.64664716
- 8.6466471
- 8.646647
- 8.64664
- 8.6466
- 8.646
- 8.64
- 8.6
- Fraction of the final change
- 86.46647168 %
- Final value K M
- 10
- Initial slope K M / τ
- 3.333333333
- Time to 63.2 % (= τ)
- 3
- Time to 95 % (≈ 3τ)
- 8.987196821
- Time to 99.3 % (5τ)
- 15
Step-by-step solution
- y(t) = K M (1 − e^(−t/τ)) = 2 × 5 × (1 − e^(−6/3))
- y = 8.646647168
First-Order System Response Formula
y(t) = K M (1 − e^(−t/τ))
G(s) = K/(τs + 1); 63.2 % at t = τ
Worked Example
A process with K = 2 and τ = 3 after a step of 5, at t = 6:
- y(t) = K M (1 − e^(−t/τ)) = 2 × 5 × (1 − e^(−6/3))
- y = 8.646647168
Final value K M: 10 · Initial slope K M / τ: 3.333333333 · Time to 63.2 % (= τ): 3 · Time to 95 % (≈ 3τ): 8.987196821 · Time to 99.3 % (5τ): 15
GATE tip
A thermometer, a stirred tank and an RC circuit are all first order: identify K and τ from the physics and reuse this result.
More Instrumentation and Process Control Calculators
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Ziegler–Nichols Tuning Calculator
PID: K_c = 0.6K_u, τ_I = P_u/2, τ_D = P_u/8
reaction curve: K_c = 1.2τ/(Kθ), τ_I = 2θ, τ_D = 0.5θ
Example: A loop with ultimate gain 4 and ultimate period 2 min: PID controller gain K_c = 2.4.
Frequently asked questions
How long does a first-order system take to settle?
About 4τ to come within 2 % and 5τ within 1 %.
What is the initial slope?
KM/τ: if it kept that rate, the response would reach its final value in exactly one time constant.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). First-Order System Response Calculator. Retrieved October 29, 2026, from https://scientificconverters.com/calculators/first-order-response-calculator/
IEEE
Scientific Converters, "First-Order System Response Calculator," scientificconverters.com. Accessed: Oct. 29, 2026. [Online]. Available: https://scientificconverters.com/calculators/first-order-response-calculator/