Direct Current and Transient Analysis 133
and s 1 and s 2 are referred as the complex frequencies given by
sw 12 , ^202
R.2.110 As mentioned, the natural response of a parallel RLC circuit results in one of the
following three cases:
a. Overdamped
b. Critical damped
c. Underdamped
Observe that the elements defi ne the specifi c case.
R.2.111 Let us analyze each case, starting with the overdamped parallel confi guration that
occurs for the following condition:
^22 w 0
and since
sw 12 , ^2 02
then
^2 ww^022 ^20 ^0
Note that both s 1 and s 2 are real, distinct, and negative. Then the solution of the
differential equation of R.2.109 is of the form
vt Ae Ae
()st s t
1212
where A 1 and A 2 are constants that can be evaluated from the network initial
conditions.
R.2.112 The critical-damped parallel case occurs for the following condition:
^2 w 02 0
Therefore both s 1 and s 2 are equal to −α, and α is real and negative. The solution
of the differential equation of R.2.109 is then of the form
vt Ae Ate
()tt
12
where A 1 and A 2 are constants that can be evaluated from the network initial
conditions.
R.2.113 The underdamped parallel case occurs for the following condition:
(^22) w 0
Then s 1 and s 2 become complex conjugate frequencies, and the response of the
differential equation of R.2.109 is of the following form:
vt e A wt A wt
t
()[ cos(dd) sin( )]
(^12)