Second-order Systems Peak Time

Second-order systems Peak Time:

Here the peak time is obtained in terms of . The system is assumed to be underdamped.

 

Considering the equation,

 

 

we obtain the peak time by differentiating c(t) with respect to time and letting this derivative equal zero. Since

 

 

and the cosine terms in this last equation cancel each other, dc/dt, evaluated at t = t_p , can be simplified to

 

 

This last equation yields the following equation:

 

 

Since the peak time corresponds to the first peak overshoot, ω_dt_p = π. Hence

 

 

The peak time t_p corresponds to one-half cycle of the frequency of damped oscillation.