PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

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)