Why damping ratio 0.707

The resulting transfer function between the input and output is: This is the simplest second-order system - there are no zeroes, just poles.

The poles of this second order system are located at: The poles of the system give us information about how the system responds because the poles encode all of the information about the natural frequency and the damping ratio. The decaying exponential has a time constant equal to: And the damped natural frequency is equal to: The damped natural frequency is typically close to the natural frequency - and is the frequency of thedecaying sinusoid underdamped system.

The system is overdamped. The system is critically damped. The system is underdamped. The system is undamped. Note the following: The vertical location of the pole is the frequency of the oscillations in the response damped natural frequency.

The horizontal location of the pole is the reciprocal of the time constant of the exponential decay. Hence, the farther the pole is to the left in the s-plane, the faster the transient response dies out. See the simulation example above. The time-domain response will not oscillate for more than period.

See below the pole locations are just slightly off. Time-domain Behavior The poles of the system can be written in a slightly different form as: In time domain: Or in Laplace domain: Time domain solution can be easily obtained by using the Inverse Laplace Transform. Reference 1 - MIT contains the time-domain solution to underdamped, overdamped, and critically damped cases.

In short, the time domain solution of an underdamped system is a single-frequency sine function multiplied with a decaying exponential. Over-damped and critically damped systems only have real poles, whereas under-damped systems have complex poles, which results in oscillations in the impulse and step responses.

Note that only under-damped systems are useful for implementing frequency-selective filters, because the pole angles are related to the cut-off frequency, i. So for frequency-selective filters you only consider under-damped systems, and then you can find different categories of under-damped systems according to their behavior in the frequency domain. Sign up to join this community.

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