Page 15 - Circuit Analysis II with MATLAB Applications

The Series RLC Circuit with DC Excitation

or
1
2
---s +
s + R ------- = 0
L LC
from which
2
R
R
1
s s 2 = – ------ r -------- – ------- (1.3)
1
2L 4L 2 LC
We will use the following notations:
R 1
2
D = ------ Z = ----------- E = D – Z 2 Z = Z – D 2
2
0
S
2L LC S S 0 nS 0 S
­ ° ® ° ¯ ­ ° ° ® ° ° ¯ ­ ° ° ® ° ° ¯ ­ ° ° ® ° ° ¯ (1.4)
D or Damping Resonant Beta Damped Natural
Coefficient Frequency Coefficient Frequency
s
where the subscript stands for series circuit. Then, we can express (1.3) as
2
2
2
s s 1 2 = – D S r D – Z = – D S r E S if D ! Z 2 0 (1.5)
S
0
S
or
2
2
2
s s = – D S r Z – D = – D S r Z nS if Z ! D 2 S (1.6)

0
S
1
2
0
2
Case I: If D ! Z 2 0 , the roots s 1 and s 2 are real, negative, and unequal. This results in the over-
S
damped natural response and has the form
s t s t
1
2
i t = k e + k e (1.7)
2
1
n
2
Case II: If D = Z 2 0 , the roots and are real, negative, and equal. This results in the criticallys 1 s 2
S
damped natural response and has the form
– D t
S
i t = e k + k t (1.8)
2
1
n
2 2
Case III: If Z ! D S , the roots and are complex conjugates. This is known as the underdampeds 1 s 2
0
or oscillatory natural response and has the form
– D t – D t
S
S
i t = e k cos Z t + k sin Z t = k e cos Z t + M (1.9)

n
2
3
1
nS
nS
nS
A typical overdamped response is shown in Figure 1.2 where it is assumed that i 0 = 0 . This plot
n
was created with the following MATLAB code:
1-3 Circuit Analysis II with MATLAB Applications
Orchard Publications