# Time constant of 2nd order system

Why there is no general definition of time constant for 2nd or higher order systems , while 1st order systems have a proper definition of time constant.

is Time constant defined for every systems irrespective of their orders or it's only defined for 1st order systems ?

Why there is no general definition of time constant for 2nd or higher order systems , while 1st order systems have a proper definition of time constant.

Only the over-damped 2nd order filter has a useful time constant. For the underdamped case (when given a step input) it produces a decaying sinewave hence, its time-domain response is best defined by the damped natural frequency of the oscillations ($$\\omega_d\$$) and zeta (the damping ratio, $$\\zeta\$$).

The low-pass filter formulas for a normalized frequency of 1 radian per second are: -

For each category, the first formula is the frequency domain transfer function and how it transfers to the time domain via laplace transform tables.

Note that only the over-damped scenario has time constants associated with it.

The properties of transfer functions are best described and characterized by the locations of the poles and zeroes in the frequency domain. This applies primarily for filter applications. In control systems, very often we make also use from the characteristics in the time domain (step response).

For a 1st-order system, there is only one real pole which - in the time domain - corresponds to an exponential step response. Only for such a function we can define a single time constant which describes how fast the step response is approaching its final value.

For 2nd-order systems, there are several different transfer functions which allow to define two different factors (dimension: time). Such an interpretation in the time domain (step response) is important, in particular, for control systems (and less important for filters, for example). These factors (time constants) describe (a) the form and (b) the time which is needed to reach the final state of the step response.

• Examples (controllers): P-T2, D-T2, I-T1, PD-T1, PI, PID,....

• Selected example (PD-T1): H(s)=K(1+sT2)/(1+sT1).... with T2>T1.

Step response: The asymptote at t=0 crosses the time axis at t=T1. The value at t=0 is g(t=0)=K*T2/T1.

• Thanks for answer! "There are several different transfer function ", is it related to different conditions related to 2nd order system like under damped, over damped , critically damped etc? I didn't get that point – user215805 Aug 18 '20 at 10:01
• Yes - as mentioned by you already (classical filter functions, different poles, different damping) - however, there are other functions with pole-zero combinations which determine the step response (controllers like PD-T1, PI,..)). – LvW Aug 18 '20 at 10:25