Page 201 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

An Underdamped Precess 115

These three voltages have to add up to match the applied
voltage.

V "R 1 It "( )d L di (6-20)
=' +Co' u u+ dt

Eq. (6-20) could be differentiated to get rid of the integral. Alter-
natively, the equation could be transformed to the Laplace domain
yielding

- - i -
V=iR+-+Lsi
Cs
The output/input transfer function is

1
i Y 1 Cs t 5
2
V = Q = G = R + _1_ + Ls = LCs + RCs + 1 = + R + _1_
52 5
Cs L LC
This expression looks similar to Eq. (6-5), which is repeated
here as

This suggests that

which further suggests

Therefore, the RLC process has the potential of behaving in an
underdamped manner similar to that of the mass/dashpot/spring
process. For example, with R, C, and L chosen such that~ < 1, the step
response will exhibit damped oscillations with a frequency of co,.

Question 6-2 Can you conceive of an electrical circuit that behaves similarly to
the first-order process introduced in Chap. 3?