General Second Order Systems

Sources

1. General Second Order EK2. (Lecture Slides)

General Second Order Transfer Function

Where

Type of Response

Determining the Poles and Step Response Given the Damping Ratio

Damping Ratio	Poles	Step Response
		Undamped
		Underdamped
	(repeated)	Critically Damped
		Overdamped

Natural Frequency

The natural frequency indicates how much the system oscillates given that there is no damping.

Where

• is a system pole element

Damping Ratio

Characteristics of a General Second-Order System Transient Response

Rise Time

The rise time refers to the time needed for the response to go from 10% to 90% of its steady state value.

Damping Ratio	Normalized Rise Time
0.1	1.104
0.2	1.203
0.3	1.321
0.4	1.463
0.5	1.638
0.6	1.854
0.7	2.126
0.8	2.467
0.9	2.883

Peak Time

The peak time refers to the length in time for the waveform to either reach the first or the maximum peak.

Percent Overshoot

The percent overshoot measures how much the waveform go beyond the steady state (expressed as a percentage of the steady state). It could also refer to the peak time’s final value.

Settling Time

The settling time refers to the time it takes for the response to reach and settle to 2% of its steady state value.

It can be approximated using the formula