Calculator · Dynamics & Control
First-Order System Calculator
Enter the DC gain K and the time constant τ of the standard first-order lag G(s) = K/(τs + 1). Every value below is closed-form and uses exactly the definitions of the First-Order Lab.
Input
τ must be strictly positive.
Results
| Pole s = −1/τ | -1 |
|---|---|
| DC gain G(0) = K | 1 |
| Time constant τ | 1 s |
| Time to 63.2% of the step change | 1 s |
| Rise time, 10%–90% (τ·ln 9) | 2.1972 s |
| Settling time, 2% band (4τ) | 4 s |
| Settling time, 5% band (3τ) | 3 s |
| Break (corner) frequency 1/τ | 1 rad/s |
| Magnitude at the break frequency | |G(j/τ)| = |K|/√2 (−3.0103 dB relative to the DC magnitude |K|); the pole contributes −45° of phase |
| Stability | Asymptotically stable for every τ > 0 |
Technical reference
G(s) = K / (τs + 1), pole at s = −1/τ
y(t) = K(1 − e^(−t/τ)) for a unit step, t63% = τ
Exact rise time (10%–90%) = τ·ln 9; reported settling estimates: 2% ≈ 4τ, 5% ≈ 3τ
- K
- DC gain, G(0)[–]
- τ
- time constant; time to reach 63.2% of the total step change[s]
- 1/τ
- break (corner) frequency, where |G| = |K|/√2 and the pole contributes −45° of phase[rad/s]
τ is the time at which the step response has covered 63.2% (1 − e⁻¹) of the way from its initial value to K; it is not a rise or settling time itself.
The 10%–90% rise time τ·ln 9 is exact for this canonical model. Anemodia reports the conventional rounded settling estimates 4τ and 3τ; the exact 2% and 5% threshold times are −τ·ln(0.02) ≈ 3.912τ and −τ·ln(0.05) ≈ 2.996τ.
For K ≠ 0, the magnitude at ω = 1/τ is |K|/√2, or −3.0103 dB relative to |K|. The first-order pole contributes −45° of phase. For K = 0 the response is identically zero, so relative magnitude in dB and phase are undefined.
Assumptions
- · Continuous-time, single-input single-output, linear time-invariant.
- · Rise time is stated with the EXPLICIT 10%–90% convention; no unlabelled rise time is reported.
- · Settling times use the conventional rounded exponential-envelope estimates 4τ and 3τ for the 2% and 5% bands.
- · τ ≤ 0 is refused rather than reinterpreted: it is not a standard stable first-order lag.
Continue
K and τ travel with you; the Lab adds responses to arbitrary excitations.