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Laplace Circuit Analysis
Electrical and Computer Engineering DepartmentUniversity of Tennessee
Knoxville, Tennessee
wlg
Laplace Circuit Analysis
Circuit Element Modeling
i(t) I(s)
+_ +
_v(t) V (s )
Laplace Circuit Analysis
Circuit Element Modeling
Resistance
R
R
G
+
_
+
_
+
_
v(t) = R i(t)
V (s) = R I(s)
V (s) =I(s) G
+
_
+
_
+
_
v(t)
V (s)
V (s)
Time Domain
Complex Frequency Domain
Laplace Circuit Analysis
Circuit Element Modeling
Inductor
+
_
i(t)
I (s )
I (s )
V L (s )
+
_
sL
_
+L i(0 )
V L (s ) = sL I(s) - L i(0 )
V L (s )i(0 ) s
+
_
sL
B est fo r m esh
B est fo r n o d a l
di(t)v t = LL dt
Laplace Circuit Analysis
Capacitor +
_
+
_
V C (s )
1s C
+
_
i(t)
V C (0 )
s
I(s )
V C (s ) =I(s )s C
v c(0 )s
+
1s Cv c(0 )C
+
_
v C (s)
I (s )
M esh
N od al
1( ) ( ) (0)0
tv t i t dt vc cC
Laplace Circuit Analysis
+
_
+
_
i1 (t) i2 (t)L 1 L 2
M
+
_
+
_
i1 (t) i2 (t)L 1 L 2
M
+
_
+
_
I 1 (s ) I 2 (s )s L 1 s L 2
s M
v 1 (t) v 2 (t)
L 1 i(0 ) + M i2 (0 ) L 2 i2 ( 0 ) +M i1 (0 )_ + + _
V 1 (s ) V 2 (s )
Linear Transformer
1 21 1
1 1 1 1 1 2 2
2 12 2
2 2 2 2 2 1 1
( ) ( )( )
( ) ( ) (0) ( ) (0)
( ) ( )( )
( ) ( ) (0) ( ) (0)
di t di tv t L M
dt dtV s sL I s L i sMI s Mi
di t di tv t L M
dt dtV s sL I s L i sMI s Mi
Laplace Circuit Analysis
Time domain to complex frequency domain
R ! R 2
+
_
_
+
C 2
L 2
C 1
V A (t) V B (t)L 1i1 (t)
i2 (t)
+ __
+
v 1 (0 )
v 2 (0 )
R ! R 2s L 2
s L 1
+
_
+
_
+ _+ _
_
+
1
s C 1
1
s C 2
V A (s ) V B (s )
v 1 (0 )
s
v 2 (0 )
s
+
_
L 2i2(0 )
L 1i1(0 )
Oh man!
What a mess.
Laplace Circuit Analysis
Circuit Application:
Given the circuit below. Assume zero IC’s. Use Laplace to find vc(t).
The time domain circuit:
+
_2 u(t) V
1 0 0
0 .0 0 1 F v c (t)
+
_
t = 0
+
_
1 0 0
+
t = 0
V c(s )
_
2
s s1 0 0 0
I(s )
Laplace circuit
)10(
20)(
1000100
10002
)(
sssV
s
sssV
c
c
Laplace Circuit Analysis
Circuit Application:
+
_
1 0 0
+
t = 0
V c(s )
_
2
s s1 0 0 0
I(s )
)(22)(
10
22
)10(
20)(
10 tuetv
sssssV
tc
c
Laplace Circuit Analysis
Circuit Application:
Given the circuit below. Assume vc(0) = - 4 V. Use Laplace to find vc(t).
The time domain circuit:
+
_2 u(t) V
1 0 0
0 .0 0 1 F v c (t)
+
_
+
_
1 0 0
+
t = 0
t = 0
V c(s )
_
_
+
2
s4s
s1 0 0 0
I(s )
Laplace circuit:
10
6)(100
1000100)(
42
ssI
ssI
ss
Laplace Circuit Analysis
Circuit Application:
+
_
1 0 0
+
t = 0
V c(s )
_
_
+
2
s4s
s1 0 0 0
I(s )
)(10
62
0)()(1002
sVss
sVsIs
c
c
)(62)(
10
62)(
10)10(
204)(
10 tuetv
sssV
s
B
s
A
ss
ssV
t
c
c
Check the boundary conditions
vc(0) = - 4 V
vc(oo) = 2 V
1
2
3
Laplace Circuit Analysis
Circuit Application:
+
_e -tu (t)4 u (t)
1 2
1
2 H
1 F
+
_
1 2
1 1s
1
s + 1
2 s
4s I 1 (s ) I 2 (s ) I 3 (s )
i0 (t)
T im e D o m a in
L ap lace
Find i0(t) using Laplace
Circuit Application: Find i0(t) using Laplace
+
_
1 2
1 1s
1
s + 1
2 s
4s I 1 (s ) I 2 (s ) I 3 (s )
Mesh 1
4)()()1(
4)()(
)1(
21
21
sIsIs
ss
sIsI
s
s
Laplace Circuit Analysis
+
_
1 2
1 1s
1
s + 1
4s I 1 (s ) I 2 (s ) I 3 (s )
Circuit Application: Find i0(t) using Laplace
Laplace Circuit Analysis
Mesh 2
ssIsssIs
s
ssIssI
ssI
s
ssI
s
sIsIs
ssI
s
)()13)(1()()1(
1)()13()(
01
1)(
13)(
1
0)()(13
)(1
21
21
21
321
Circuit Application: Find i0(t) using Laplace
Laplace Circuit Analysis
+
_1 1
s1
s + 1
4s I 1 (s ) I 2 (s ) I 3 (s )
4)()()1(21
sIsIs
ssIsssIs )()13)(1()()1(21
Add these 2 equations
4)()43(2
ssIss
Circuit Application: Find i0(t) using Laplace
Laplace Circuit Analysis
+
_1 1
s1
s + 1
4s I 1 (s ) I 2 (s ) I 3 (s )
4)()43(2
ssIss
)(]3
21[)(
343
21
)34(
)4(31
)(
34
2
2
tueti
ssss
ssI
t
Is final value of i2(t) reasonable?
That’s all Folks !
