First-Order + Dead Time

System Parameters

K — Process Gain 2.0

τ — Time Constant (s) 3.0

θ — Dead Time (s) 1.0

Input Settings

Step Amplitude 1.0

Presets

Transfer Function G(s) = K · e^−θs / (τs + 1)

FOPDT is the most widely used model in process control. It captures gain (K), speed of response (τ), and transport delay (θ).

Response Metrics

Key Relationships 63.2% at t = θ + τ
95% at t ≈ θ + 3τ
98% at t ≈ θ + 4τ
Final value = K × Amplitude

Final Value

2.00

63.2% Time

4.0 s

95% Time

10.0 s

98% Time

13.0 s

Initial Slope

0.67

Parameters & Equations

The FOPDT model and its exact time-domain responses.

Model

Transfer function G(s) = K · e^−θs / (τs + 1) K — process gain · τ — time constant (s) · θ — dead time (s).

Differential form τ · dy/dt + y(t) = K · u(t − θ) u is the input, delayed by θ before it affects y.

Exact Responses

Step response (amplitude A) y(t) = 0 (t < θ)
y(t) = K·A·(1 − e^{−(t−θ)/τ}) (t ≥ θ)

Ramp response (slope A) y(t) = 0 (t < θ)
y(t) = K·A·[(t−θ) − τ(1 − e^{−(t−θ)/τ})] (t ≥ θ)

Parameters

Symbol	Meaning	Default
K	Process gain (steady output / input)	2.0
τ	Time constant (63.2% speed)	3.0 s
θ	Dead time (transport delay)	1.0 s
A	Input step amplitude	1.0

Key Relationships

Response landmarks Final value = K·A
63.2% at t = θ + τ
95% at t ≈ θ + 3τ
98% at t ≈ θ + 4τ
Initial slope = K·A/τ θ shifts the curve in time; τ sets how fast it approaches the final value; K scales it.

First-Order Plus Dead Time