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ECE 201 Lecture 25
2
nd
order circuits: RLC with constant inputs
(continue….)
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View Full Document REVIEW  Source Free 2
nd
Order Circuit
and
given
with
characteristic equation
Problem is reduced to solving the
linear
differential equation:
Case 1 (
two distinct real roots s
1
, s
2
):
Case 2 (
two identical real roots s
1
=s
2
):
Case 3 (
two conjugate complex roots s
1
=
σ
+j
ϖ
d
, s
2
=
σ
j
ϖ
d
):
2
2
0
L
L
L
d i
di
LC
RC
i
dt
dt
+ + =
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
0.5
0
0.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
0.5
0
0.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20
10
0
10
20
Typical Solutions
Series LCR: Case 3
Series LCR: Case 3 (when R=0)
2
R
L
σ=
Unstable system (Not in ECE201)
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View Full Document Plot of v
C
(t)
0
0.05
0.1
0.15
0.2
1
0.5
0
0.5
1
1
1

x t 0
,
(
)
x t 2
,
(
)
x t 5
,
(
)
x t 10
,
(
)
0.2
0
t
0 0.05 0.1 0.15 0.2
1
0.5
0
0.5
0.459
1

x t 10
,
( )
x t 20
,
( )
x t 30
,
( )
x t 40
,
( )
x t 80
,
( )
0.2
0
t
ϖ
0
( )
2
π
⋅
31.831
=
ϖ
2
( )
2
π
⋅
31.791
=
ϖ
5
( )
2
π
⋅
31.581
=
ϖ
10
(
)
2
π
⋅
30.82
=
0
Ω
2
Ω
5
Ω
10
Ω
R
Series
LCR
10
Ω
20
Ω
30
Ω
40
Ω
ϖ
0
( )
2
π
⋅
31.831
=
ϖ
10
(
)
2
π
⋅
30.82
=
ϖ
20
(
)
2
π
⋅
27.566
=
ϖ
30
(
)
2
π
⋅
21.054
=
ϖ
40
(
)
2
π
⋅
0
=
Case 3
Underdamped
Critically damped
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
5
4
3
2
1
0
1
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This note was uploaded on 10/26/2010 for the course ECON 002 taught by Professor Eudey during the Spring '08 term at UPenn.
 Spring '08
 EUDEY
 Macroeconomics

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