Page 28 - Circuit Analysis II with MATLAB Applications

Chapter 1 Second Order Circuits

2
2
2
s s = – D P r D – Z = – D P r E P if D ! Z 2 0 (1.44)

1
2
P
0
P
or
2
2
s s = – D P r Z – D 2 P = – D P r Z nP if Z ! D 2 P (1.45)

1
0
0
2
Note: From (1.4) and (1.43) we observe that D z D P
S
As in a series circuit, the natural response v t can be overdamped, critically damped, or under-
n
damped.
2 2
Case I: If D ! Z 0 , the roots s 1 and s 2 are real, negative, and unequal. This results in the over-
P
damped natural response and has the form
s t s t
v t = k e 1 + k e 2 (1.46)
n
1
2
s
s
Case II: If D 2 P = Z 2 0 , the roots and are real, negative, and equal. This results in the critically
1
2
damped natural response and has the form
– D t
P
v t = e k + k t (1.47)
2
1
n
2 2
s
s
Case III: If Z ! D P , the roots and are complex conjugates. This results in the underdamped
2
0
1
or oscillatory natural response and has the form
– D t – D t
P
P
v t = e k cos Z nP t + k sin Z nP t = k e cos Z nP t + M (1.48)

3
2
1
n
1.6 Response of Parallel GLC Circuits with DC Excitation
C
G
R
L
Depending on the circuit constants (or ), , and , the natural response of a parallel GLC cir-
cuit may be overdamped, critically damped or underdamped. In this section we will derive the total
response of a parallel GLC circuit which is excited by a DC source using the following example.
Example 1.3
For the circuit of Figure 1.11, i 0 = 2A and v 0 = 5V . Compute and sketch vt for t ! 0 .
L
C
1-16 Circuit Analysis II with MATLAB Applications
Orchard Publications