# Step and Impulse Response of a Second-Order System

Step and Impulse Response of a Second-Order System

Consider a second-order process, where the transfer function is given by , where is the process time constant and is the damping coefficient.

H(s)=1+2γτs+1

2

τ

2

s

τ

γ

This Demonstration shows the response of this process when subject to a step input of amplitude (i.e., , where is the unit step function) or an impulse input of amplitude (i.e., , where is the Dirac delta function). The response is obtained by Laplace inversion using the Mathematica built-in function, InverseLaplaceTransform.

A

AU(t)

U(t)

A

Aδ(t)

δ(t)

When , one gets an underdamped response with oscillatory behavior. Critically damped and overdamped systems result when and , respectively. For these last two cases, the response does not exhibit oscillations. A critically damped response returns to steady state faster than an overdamped response when the system is subject to an impulse input.

0<γ<1

γ=1

γ>1