E12

Second Order Systems

Today we looked at second order systems of the form:

where ω0 is the natural frequency and ζ is the damping ratio.

The step response of such a function is given by

One thing that is apparent is that the time is always scaled by the natural frequency so we get terms of ω0t.  This means that as ω0 changes, the speed of the response changes, but not the shape.  In other words ω0 acts to dilate or contract the sequence.  This is shown below (the Matlab Script that generated these system is SecondOrder.m)

The effect of varying the natural frequency

ζ=0.25, ω0 varies.

Note that as ω0 varies it just changes the time scale of the response, the shape doesn't change.  Note also that the poles move away from the origin in a straight line as ω0 varies (below).   For underdamped systems the angle at which poles are located is determined by ζ, the distance from the origin is equal to ω0.

The effect of varying the damping factor

If ζ changes, the shape will change. For small ζ we get many oscillation.  As ζ increases we get fewer oscillation.  Critical damping (no oscillations) occurs when ζ=1.  A system is overdamped when ζ>1.  It is underdamped if 0<ζ<1.

ω0=1, ζ varies.

To reiterate, for underdamped systems the angle at which poles are located is determined by ζ, the distance from the origin is equal to ω0.   Therefore as ζ changes, the angle changes and a circular arc is described by the poles with a radius equal to ω0=1.  When ζ=1 the system is critically damped.  When ζ>1 the system is critically damped.   When ζ=0 the system is undamped.

Adding a Zero at the Origin to the Transfer Function

Adding a zero to the transfer function caused the final value of the step response to go to zero.  We developed this in terms of the Laplace Transform, and by making the argument that multiplication by s (a zero at the origin) is equivalent to differentiation.