Step Response Time Calculator Formula

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

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Formulas Used

Rise Time (10% to 90%)

rise_time = 2.197 * tau

Time to 63.2% of Final Value

time_to_63 = tau

Time to 95% of Final Value

time_to_95 = 3 * tau

Time to 99% of Final Value

time_to_99 = 4.605 * tau

Steady-State Output

final_value = gain * step_size

Variables

VariableDescriptionDefault
tauTime Constant (tau)(s)1
gainDC Gain (K)5
step_sizeStep Input Magnitude1

How It Works

First-Order Step Response

A first-order system responds to a step input as an exponential approach to the final value. The speed of response is governed by the time constant tau.

Key Relationships

y(t) = K × u × (1 - e^(-t/tau))

  • At t = tau: output reaches 63.2% of final value
  • Rise time (10-90%): t_r = 2.197 × tau
  • Time to 95%: t = 3 × tau
  • Time to 99%: t = 4.605 × tau

    • First-order systems have no overshoot; the output monotonically approaches the steady state.

    Worked Example

    A temperature control system with tau = 1 s, gain = 5, step input = 1.

    tau = 1gain = 5step_size = 1
    1. 01Final value = 5 × 1 = 5
    2. 02Rise time = 2.197 × 1 = 2.197 s
    3. 03Time to 63.2% = 1 s (output = 3.16)
    4. 04Time to 95% = 3 s (output = 4.75)
    5. 05Time to 99% = 4.605 s (output = 4.95)

    Ready to run the numbers?

    Open Step Response Time Calculator