The meaning of time constant [1st order model]

First-order process with different time-constant.

Transfer function: G = K/(tao.s+1) [K = 1 in this case]
and unit step input = 1/s at time t = 1

For time domain, the solution of 1^st order process is as follow:
output(t) = K.input(t).[1-exp(-t/tao)]

If t = tao (at time-constant), then output(tao) = K.input(tao).[1-exp(-1)] = 0.623.K.input(tao). That means at time-constant output have been changed 62.3% of the total change which it is going to make [K.input(infinite)].

In this example, total change is equal to 1 [K = 1 and input step-size = 1] then time-constant is time at output is equal to 0.623.

Time-constant	Output	Time	Time-constant estimate
1 2 5	0.6247 0.62283 0.62318	1.98 2.95 5.88	0.98 1.95 4.88

Estimate of time-constant from the output data is close to real time-constant of the process, the truncation error is occurred because of the software (MATLAB). Then this example just used to illustrate the concept of time-constant. If time-constant is large then process dynamic will slow (over-damped).

Time-constant of the process describes how fast of process dynamic to reach new steady-state with respect to input change.