Page 49 - Circuit Analysis II with MATLAB Applications

Summary

2 2
If Z ! D S , the roots s 1 and s 2 are complex conjugates. This is known as the underdamped or
0
oscillatory natural response and has the form
– D t – D t
i t = e S k cos Z t + k sin Z t = k e S cos Z t + M

n
1
2
3
nS
nS
nS
x For a parallel GLC circuit, the roots and are found from
s
s
2
1
2
2
2
s s = – D P r D – Z = – D P r E P if D ! Z 2 0

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

1
0
0
2
where
G
1
2
D P = ------- Z = ----------- E P = D P 2 – Z 2 0 Z nP = Z – D P 2
0
0
2C
LC
2
s
s
If D ! Z 2 0 , the roots and are real, negative, and unequal. This results in the overdamped nat-
P
1
2
ural response and has the form
s t s t
2
1
v t = k e + k e
n
1
2
2 2
s
s
If D P = Z 0 , the roots and are real, negative, and equal. This results in the critically damped
2
1
natural response and has the form
– D t
P
v t = e k + k t
2
1
n
2 2
s
s
If Z ! D P , the roots and are complex conjugates. This results in the underdamped or oscil-
2
1
0
latory 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

n
2
1
3
x If a second order circuit is neither series nor parallel, the natural response if found from
s t s t
y = k e 1 + k e 2
1
n
2
or
s t
1
y = k + k t e

n
2
1
or
y = e – Dt k cos Et + k sin Et = e – Dt k cos Et + M

n
4
5
3
depending on the roots of the characteristic equation being real and unequal, real and equal, or
complex conjugates respectively.
1-37 Circuit Analysis II with MATLAB Applications
Orchard Publications