control//step response//first-order system//time constant

The time constant is the time a first-order system takes to cover 63 % of the change after a step in its input, and it is the single number that says how fast a process responds: it decides how often a controller has to measure and act, how long a step test must run, and how slow a filter can be before it hides the dynamics. After one \(\tau\) the response has done \(1-e^{-1}\approx63\,\%\) of its way, after \(3\tau\) about 95 %, after \(5\tau\) more than 99 %; a rule of thumb follows that a first-order process has settled after four or five time constants.

The time constant is the time a first-order system takes to cover 63 % of the change after a step in its input, and it is the single number that says how fast a process responds: it decides how often a controller has to measure and act, how long a step test must run, and how slow a filter can be before it hides the dynamics. After one τ\tauτ the response has done 1−e−1≈63 %1-e^{-1}\approx63,%1−e−1≈63% of its way, after 3τ3\tau3τ about 95 %, after 5τ5\tau5τ more than 99 %; a rule of thumb follows that a first-order process has settled after four or five time constants.

The scales spread over five orders of magnitude, and knowing which one you face is most of the design. A drone's motor and propeller answer in tens of milliseconds, a linearized tank of half a square metre in a couple of hundred seconds, the housing of an industrial motor heats over tens of minutes. A loop on the first needs a kilohertz controller on the drone itself; the last could be closed by an operator with a clipboard.

Knowing the order of τ\tauτ is knowing how often to look and act. Sample several times per time constant of the fastest dynamics that matters (ten is the usual margin), wait four or five of the slowest before reading a step, and size any filter so its own time constant stays well below the one you want to see.

It is read graphically. The tangent to the step response at its start reaches the final value after exactly τ\tauτ, and the 63 % crossing gives the same number, so a pen and a trend chart identify a plant (FOPDT model). Rise time from 10 % to 90 % is τln⁡9≈2.2 τ\tau\ln 9\approx2.2,\tauτln9≈2.2τ; half-life is τln⁡2≈0.69 τ\tau\ln2\approx0.69,\tauτln2≈0.69τ. The three are often confused in specifications.

A sensor has one too. A thermocouple in a thick sheath or a heavily filtered pressure transmitter adds its own lag to the loop, and a first-order low-pass filter is exactly a time constant written in code, τ=1/(2πfc)\tau=1/(2\pi f_c)τ=1/(2πfc​): about 8 ms at a 20 Hz cut-off (first-order low-pass filter).

For a linear system with many modes each mode has its own, τi=−1/σi\tau_i=-1/\sigma_iτi​=−1/σi​, with σi\sigma_iσi​ the real part of its eigenvalue, and the slowest one dominates what is seen in the long run (modes). A pure delay has no time constant at all: nothing moves before the dead time (delay and lag).