ltm laplace transform 2011a mk
TRANSCRIPT
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N u ễn Côn Phươn
The Laplace Transform
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Contents
1. Basic Elements Of Electrical Circuits
2. Basic Laws
3. Electrical Circuit Analysis
4. Circuit Theorems
5. Active Circuits
6. Capacitor And Inductor
7. First Order Circuits. econ r er rcu s
9. Sinusoidal Steady State Analysis
10. AC Power Analysis
. -12. Magnetically Coupled Circuits
13. Frequency Response
15. Two-port Networks
The Laplace Transform 2
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The Laplace Transform
f (t ) = 0
(integrodifferential)i (t ), v (t ), …Circuit
F (s ) = 0
(algebraic)I (s ), V (s ), …
The Laplace Transform 3
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The Laplace Transform
• Definition
• Two Important Singularity Functions• Transform Pairs
• Properties of the Transform
• Inverse Transform• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer FunctionThe Laplace Transform 4
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Definition ( ) f t
t 0 0( ) ( ) ( )st
s L f t f t e dt
s j
0( ) t f t e dt
1
1
1 1( ) ( ) ( )
2
j st
j f t L F s F s e ds
j
The Laplace Transform 5
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The Laplace Transform
• Definition
• Two Important Singularity Functions• Transform Pairs
• Properties of the Transform
• Inverse Transform• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer FunctionThe Laplace Transform 6
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Two Important Singularity Functions (1)
( )u t 10 0
( )t
u t
t 0
( )u t a
t 0 a
( )1
u t at a
The Laplace Transform 7
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Two Important Singularity Functions (2)Ex. 1
Determine the Laplace transform for the waveform?
( )u t 1
( ) ( ) st F s u t e dt
t 0
0 1
st
e dt
0
1 st e s
1
s
The Laplace Transform 8
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Two Important Singularity Functions (3)Ex. 2
Determine the Laplace transform for the waveform?
( ) ( ) st F s u t a e dt
( )u t a
1
0 0 1
a st
adt e dt
t 0 a
1 st
a
e s
ase
s
The Laplace Transform 9
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Two Important Singularity Functions (4)Ex. 3
Determine the Laplace transform for the waveform?
( ) [ ( ) ( )] st F s u t u t a e dt
t 0 a
0
1
( )
st
u t e dt s
( )u t
1
0( )
st st e
u t a e dt s
t 0
1 1as ase e F s
t
( )u t a
0
The Laplace Transform 10
s s s 1a
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Two Important Singularity Functions (5)( )t
0 0t t
t 0
( ) 1 0t dt
( )t a
0t a t a
t 0 a( ) 1 0
a
at a dt
2 1 2( )t f a t a t t t a dt
The Laplace Transform 11
1 1 20 ,t a t a t
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Two Important Singularity Functions (6)Ex. 4
Determine the Laplace transform of an impulse function?
0( ) ( )
st
F s t a e dt
2 1 2( )
( ) ( )t
t
f a t a t f t t a dt
( ) as F s e
The Laplace Transform 12
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The Laplace Transform
• Definition
• Two Important Singularity Functions• Transform Pairs
• Properties of the Transform
• Inverse Transform• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 13
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Transform Pairs (1)Ex. 1
Find the Laplace transform of f (t ) = t ?
0( ) st F s te dt
1Let & & st st st u t dv e dt du dt v e dt e s
200 0
1( ) 0t st st t e e F s e dt
s s s s
The Laplace Transform 14
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Transform Pairs (2)Ex. 2
Find the Laplace transform of f (t ) =cosωt ?
0( ) cos st F s te dt
0 2
j t j t st e e
e dt
0 2
e edt
2 s j s j
s
The Laplace Transform 15
2 2 s
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Transform Pairs (3)
f (t ) ( )t ( )u t at e t at te sin at cosat
F ( s) 1 s s a 2 s 2( ) s a 2 2
a
s a 2 2
s
s a
The Laplace Transform 16
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The Laplace Transform
• Definition
• Two Important Singularity Functions• Transform Pairs
• Properties of the Transform
• Inverse Transform• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 17
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Properties of the Transform (1) Property f (t ) F ( s)
1. Magnitude scaling
2. Addition/subtraction
( )t ( ) F s
1 2( ) ( ) f t f t 1 2( ) ( ) F s F s
1 s .
4. Time shifting
a a ( ) ( ), 0 f t a u t a a ( )
ase F s
( ) ( ), 0 f t u t a a [ ( )]
as
e L f t a
. requency s t ng
6. Differentiation
7. Multiplication by t
( )e t s a
( ) / n nd f t dt
1 2 1 1( ) (0) (0) ... (0)
n n n o n s F s s f s f s f
( )nt f t ( 1) ( ) / n n nd F s ds
8. Division by t
9. Integration
( ) / f t t ( ) s
F d
0( )
t
f d ( ) / F s s
The Laplace Transform 18
10. Convolution1 2 1 2
0( ) * ( ) ( ) ( )
t
f t f t f f t d 1 2( ) ( ) F s F s
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Properties of the Transform (2)Ex. 1
Find the Laplace transform of 10( ) 5 cos20 ?
t f t e t
1 2 1 2( ) ( ) ( ) ( ) f t f t F s F s
10( ) [5] [ ] [cos20 ]t F s L L e L t
( ) ( ) f t AF s
[5] 5 [1] L L 5
[5] L s [1] L
s
10 1[ ]t L e
2 2 2[cos20 ]
20 400
s s L t
s s
The Laplace Transform 19
2 2
5 1 5 2400 4000( )
10 400 ( 10)( 400)
s s s F s
s s s s s s
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Properties of the Transform (3)Ex. 2
Find the Laplace transform of the waveform? 5
t 0 1 2 3
5
( ) 5 ( 1) 5 ( 2) f t u t u t
t 0 1 2 32
25 s s se e
5 s s s
The Laplace Transform 20
t 0 1 2 3
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Properties of the Transform (4)Ex. 3
Find the Laplace transform of the waveform? 5
t 0 1 2 3( ) ( 5 10)[ ( 1) ( 2)] f t t u t u t
5 ( 1) 10 ( 1)tu t u t
5 ( 2) 10 ( 2)tu t u t
5 ( 1) 5( 1 1) ( 1)tu t t u t
t 0 1 2 3
5 ( 2) 5( 2 2) ( 2)tu t t u t
5( 2) ( 2) 10 ( 2)t u t u t
1( ) 5( 1) ( 1) 5 ( 1)
10 ( 1)
f t t u t u t
u t
t 0 1 2 3
The Laplace Transform 2110 ( 2)
t u t u t
u t
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Properties of the Transform (5)Ex. 3
Find the Laplace transform of the waveform? 5
t 0 1 2 3( ) ( 5 10)[ ( 1) ( 2)] f t t u t u t
5( 1) ( 1) 5 ( 1)t u t u t
10 ( 1)
5( 2) ( 2) 10 ( 2)
u t
t u t u t
t 0 1 2 3
u t
5( 1) ( 1) 5 ( 1)t u t u t
12
2 2
5( ) 5 5
s s se e
F s e s s s
The Laplace Transform 22
t 0 1 2 3
2
5
(1 )
s se
s e s
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Properties of the Transform (6)Ex. 4
Find the Laplace transform of the waveform? 5
t 0 1 2 3
The Laplace Transform 23
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The Laplace Transform
• Definition
• Two Important Singularity Functions• Transform Pairs
• Properties of the Transform
• Inverse Transform• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 24
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The Laplace Transform
f (t ) = 0
(integrodifferential)i (t ), v (t ), …Circuit
F (s ) = 0
(algebraic)I (s ), V (s ), …
The Laplace Transform 25
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Inverse Transform (1)1
1 1 0( ) ...m m
m m P s a s a s a s a
1
1 1 0( ) ...n n
n nQ s b s b s b s b
1 2 n K K K
1 2
...n
s p s p s p
1( )Complex- conjugate poles : ( ) P s F s 1
*
1 1 ...
s s s
K K
s j s j
1
1 1
( )Multiple poles : ( )
( )( )n
P s F s
Q s s p
The Laplace Transform 26
11 12 1
2
1 1
... ...( ) ( ) ( )
n
n s p s p s p
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Inverse Transform (2)
1 2
1 2
Simple poles : ( ) ...
( )
n
n
F s
Q s s p s p s p
( )( ) 0 ... 0 0 ... 0
( )i
i i
s p
P s s p K
Q s
1 i p t ii
K L K e
i
1 2 n p t p t p t
The Laplace Transform 27
n
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Inverse Transform (3)Ex. 1
Find the inverse La lace transform of 225 300 640 s s
F s
( 4)( 8) s s s
1 2 3 ( )( ) ; ( ) 0 ... 0 0 ... 0i i
K K K P s F s s p K
i s p
2 2
25 300 640 25 300 640 640 s s s s 1 0
0 0( 4)( 8) ( 4)( 8) 4 8 s
s s s s s s s
2 2
2 4
4 4
2
( 4) ( ) ( 4) ( 4)( 8) ( 8)
25( 4) 300( 4) 640
s
s s K s F s s s s s s s
The Laplace Transform 28
( 4)( 4 8)
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Inverse Transform (4)Ex. 1
Find the inverse La lace transform of 225 300 640 s s
F s
( 4)( 8) s s s
1 2 3 ( )( ) ; ( ) 0 ... 0 0 ... 0i i
K K K P s F s s p K
1 220; 10 K K
2 225 300 640 25 300 640 s s s s
i s p
3 8
8 8
2
( 4)( 8) ( 4)
25( 8) 300( 8) 6405
s
s s s s s s s
( 8)( 8 4)
20 10 5 4 8t t
The Laplace Transform 29
4 8 s s s
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Inverse Transform (5)Ex. 2
Find the inverse La lace transform of 100( 6) s
F s
( 1)( 3) s s
The Laplace Transform 30
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Inverse Transform (6)*
1 1 1( )Com lex- con u ate oles : ...
P s K K F s
1( )( )( )Q s s j s j s j s j
1 1
( )( )
P s s j K K
s j
*
1 1 K K
1( ) F s 1
s j
1 1... ...
s j s j s j
( ) ( ) ( ) ( )... ... j j t j j t t j t j t t K e e K e e K e e e
cos sin je j
cos sin cos sin ...t t K e t t t t
The Laplace Transform 31
12 cos( ) ...t K e t
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Inverse Transform (7)Ex. 3
Find the inverse La lace transform of 24 76 s s
F s
( 2)( 6 25) s s s
1 2 3( )K K K
F s
1 1
( )( ) ; ( ) 2 cos( ) ...
( )
t P s K s j f t K e t
Q s
2 2
3 2 2
4 76 4 76( 2) 8
( 2)( 6 25) 6 25
s s s s K s
s s s s s
2
1 2
3 4
4 76( 3 4) 6 8 10
( 2)( 6 25) s
s s K s j j
s s s
o53.1
The Laplace Transform 32
3 o 2 3 o 2( ) 2 10 cos(4 53.1 ) 8 20 cos(4 53.1 ) 8t t t t f t e t e e t e
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Inverse Transform (8)Ex. 4
Find the inverse La lace transform of 5( 2) s
F s
( 4 5) s s s
The Laplace Transform 33
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Inverse Transform (9)1 11 12 1
2
( )Multiple poles : ( ) ... ...n
n n
P s K K K F s
1 1 1 1
11 1( ) ( )n
n s p s p F s K
1
1 1 1[( ) ( )]n
n
s p
d s p F s K
ds
2
1 1 22[( ) ( )] (2!)n
n
d s p F s K
ds
1
1 1
1[( ) ( )]
n jn
j n
d K s p F s
The Laplace Transform 34
1 s p
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Inverse Transform (10)Ex. 5
Find the inverse La lace transform of
210 34 27( )
s s F s
s s
11 12 21 12
1( ) ; [( ) ( )]
n jn
j n j
K K K d F s K s p F s
2 22 2
12 23
10 34 27 10 34 27( 3) ( ) ( 3) 5
3 s
s s s s K s F s s
s s s
1 s p
s s
22
11
3 3
10 34 27[( 3) ( )]
s s
d d s s K s F s
ds ds s
2
2
3
(20 34) (10 34 27)7
s
s s s s
s
The Laplace Transform 35
2 20
0
( ) 3( 3) s
s
s s K sF s s
s s
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Inverse Transform (11)Ex. 5
Find the inverse La lace transform of
210 34 27( )
s s F s
s s
11 12 21 12
1( ) ; [( ) ( )]
n jn
j n j
K K K d F s K s p F s
7 5 3 K K K
1 s p
7 5 3 F s
3 ( 3) s s s
3 3t t
The Laplace Transform 36
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Inverse Transform (12)Ex. 6
Find the inverse La lace transform of 5( 3) s
F s
( 1)( 2) s s
The Laplace Transform 37
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The Laplace Transform
• Definition
• Two Important Singularity Functions• Transform Pairs
• Properties of the Transform
• Inverse Transform• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 38
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Initial-Value & Final-Value Theorems (1)
0t s s s
0Final value theorem : lim ( ) lim ( )
t s f t sF s
The Laplace Transform 39
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Initial-Value & Final-Value Theorems (2)Ex.
Find the initial and final values of 5( 1) s
F s
( 2 2) s s s
2
5( 1)
(0) lim ( ) lim 0
s
f sF s
20 0
5( 1)( ) lim ( ) lim 2.52 2 s s
s f sF s s s
The Laplace Transform 40
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The Laplace Transform
• Definition
• Two Important Singularity Functions• Transform Pairs
• Properties of the Transform
• Inverse Transform
• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 41
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Ex.
Find the current i(t )?
t = 0
+
200
Method 1
Laplace Circuit Solutions (1)
–
1V ( )i t 1 L R
div v e L Ridt
di n
dt
n e e e
3
2002000
100 10
R
2000t
ni Ke
10.005A
200 f
ei
R
2000
0.005
t
f ni i i Ke
2000 0(0) 0.005 0.005 0 0.005i Ke K K
The Laplace Transform 42
2000( ) 0.005(1 ) At i t e
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Laplace Circuit Solutions (2)
f (t ) = 0
(integrodifferential)i (t ), v (t ), …Circuit
F (s ) = 0(algebraic)
I (s ), V (s ), …
The Laplace Transform 43
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Laplace Circuit Solutions (3)Ex.
Find the current i(t )?
t = 0
+
200
Method 2 –
1V( )i t
0.1 200 1 L R
div v e idt
0.1 200 [1] 0.1 [200 ] L i L L L idt dt
1
[1] L s
[200 ] 200 ( ) L i I s
1 2 1 1( )nn n n o nd f t ...
n
dt
0.1 0.1[ ( ) (0)] 0.1 ( )di
L sI s i sI s
The Laplace Transform 44
10.1 ( ) 200 ( ) sI s I s
s
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f (t ) = 0
(integrodifferential)i (t ), v (t ), …Circuit
F (s ) = 0(algebraic)
I (s ), V (s ), …Circuitin s -domain
The Laplace Transform 46
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The Laplace Transform
• Definition
• Two Important Singularity Functions• Transform Pairs
• Properties of the Transform
• Inverse Transform
• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 47
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Circuit Element Models (1)
Rv t
RV s
v Ri
( ) ( ) Af t AF s ( ) ( )V s RI s
The Laplace Transform 48
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Circuit Element Models (2)( )i t ( ) I s
L
sL
(0)i
+ –
(0) Li
v L dt
( )df t
( ) [ ( ) (0)]V s L sI s i
The Laplace Transform 49
dt
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Circuit Element Models (3)( )i t ( ) I s
C
1
sC
–+ (0)v
s
0
1( ) (0)
t
v i x dx vC
0( )( )t s f d s
1 (0)( ) ( ) vV s I s sC s
The Laplace Transform 50
(0)v s
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Circuit Element Models (4)
++ + sM +––
1 1 2
1(0)i2(0)i
1 2 –2
–1 1 s 2 s
–2
–1
1 2( ) ( )di t di t
1 2 1 2
1 1
dt dt 1 1 1 1 1 2 2
2 1( ) ( )di t di t
The Laplace Transform 51
2 2dt dt
2 2 2 2 2 1 1
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Circuit Element Models (5)
i t I s
R( )v t R( )V s
L
sL
( )v t (0)i
( )V s + –
(0) Li
The Laplace Transform 52
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Circuit Element Models (6)( )i t ( ) I s
C ( )v t
1
sC ( )V s
–+ (0)v
s
++ + sM +––
1 1 2(0) (0) L i Mi 2 2 1(0) (0) L i Mi
1 L 2 L
+
–2( )v t
+
–1( )v t
1 sL 2 sL
+
–2( )V s
+
–1( )V s
The Laplace Transform 53
1( )i t 2( )i t 1( ) I s 2( ) I s
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The Laplace Transform
• Definition
• Two Important Singularity Functions
• Transform Pairs
• Properties of the Transform
• Inverse Transform
• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 54
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Analysis Techniques (1)
KVL/KCL : ( ) ( ) ... ( ) 0 x t x t x t
1 2KVL/KCL : ( ) ( ) ... ( ) 0n X s X s X s
i (t ), v (t ), …Circuit
Inverse TransformCircuit Element Models DC circuit analysis techniques
, , ,
mesh analysis, sourcetransformation, superposition,
Thevenin/Norton equivalent, …)
The Laplace Transform 55
I (s ), V (s ), …in s -domain
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Analysis Techniques (2)Ex. 1
Find the current i(t )?
t = 0
+
200
–
1V( )i t (0) 0i
1200 ( ) 0.1 ( ) 0.1 (0) 200 ( ) 0.1 ( ) I s sI s i I s sI s
+ –
0.1 s
1 21 10( )
(0.1 200) ( 2000) 2000
K K I s
s s s s s s
1
s( ) s
+–0.1 (0)i
1
0
100.005
2000 s
K s
2
2000
100.005
s K s
The Laplace Transform 56
. .( )
2000 I s
s s
( ) 0.005(1 ) At i t e
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Analysis Techniques (3)Ex. 1
Find the current i(t )?
=
+
200
100mH2000( ) 0.005(1 ) At i t e
1. Solve for initial capacitor
voltages & inductor currents2. Draw an s-domain circuit
–
1V ( )i t 3. Use one of DC circuit
analysis techniques to solve
for voltages or/and currents
-(0) 0i Circuit Element Models
200
4. Find the inverse Laplace
transform to convert them
back to the time domain
+ – 0.1 s
1
1200 ( ) 0.1 ( ) I s sI s
10( ) I s
The Laplace Transform 57
s( ) s
+–
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Analysis Techniques (4)Ex. 2
Find the voltage v(t )?
t = 0
10k
+
4
6
4
110
1 525 10( ) // ( )1 2
sV s R J s sC s
(0) 0v 25 At e 25 F –
v
625 10 s 10k
5A
2 s
–
( )V s4
1 24 10 K K
1. Solve for initial capacitor voltages &
6
1
25 10 s
4
4
1
2
4 102 10
4 s
K s
n uc or curren s
2. Draw an s-domain circuit3. Use one of DC circuit analysis
techniques to solve for voltages or/and
-
4
4
2
4
4 10 2 102
s
K s
The Laplace Transform 58
-
4. Find the inverse Laplace transform to
convert them back to the time domain4 2 4( ) 2 10 ( ) Vt t v t e e
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Analysis Techniques (5)Ex. 3
Find the current i(t )? ( )i t ( ) s + 4
t
8 +8
(0) 1A
8
i ( )v t
(0)i
sL
( )V s + –
–
2 H
8V
( )i t –
12V
2( )
2 4
s I s s
(0) Li
+– 42 + –
1. Solve for initial capacitor voltages &1 2
6
( 2) 2
s K K
s s s s
s s
n uc or curren s
2. Draw an s-domain circuit3. Use one of DC circuit analysis
techniques to solve for voltages or/and
-
1
0
632 s
s K s
2
( ) 3 2 At
i t e
The Laplace Transform 59
-
4. Find the inverse Laplace transform to
convert them back to the time domain2
2
62
s
s K
s
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Analysis Techniques (6)Ex. 4
Find the voltage v(t )? + 4
t
8 +( )i t 1
( ) I s
12 81 s s
(0) 8 Vv –
2 F8V
( )v t –
12V –
C ( )v t
sC
( )V s
+
12 42
s
s
0.5 K K
–+ 4
1
8
s + –+
–
(0)v
s
1. Solve for initial capacitor voltages &( 0.125) 0.125 s s s s
0.54 K
2 s s
s –
n uc or curren s
2. Draw an s-domain circuit3. Use one of DC circuit analysis
techniques to solve for voltages or/and
-
00.125 s s
2
0.54 K
The Laplace Transform 60
-
4. Find the inverse Laplace transform to
convert them back to the time domain. s
0.125( ) 4(1 ) Vt v t e
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Analysis Techniques (7)Ex. 5
Write the mesh equations in the s-domain? 1C 1
R 22 L
+ –
+ –
1( )e t 2 ( )e t 3
L3(0)i2 (0)i
– +
+
1(0)v( )i t
sL
( ) I s
3 –3
1 R 2 2
sL1(0)v
s +
(0)i
+ –
+ –
+ –
1
1
sC 3 sL
2 2(0) L i
+ – +–
( )i t ( ) I s
1( ) E s2( ) E s
3 3
(0) L i
3(0)v+ –
+–
C ( )v t
1
sC ( )V s
(0)v
The Laplace Transform 61
3
1
sC
s
–
s
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Analysis Techniques (8)Ex. 5
Write the mesh equations in the s-domain? 1C 1
R 22
L
+ –
+ –
1( )e t 2 ( )e t 3
L3(0)i2(0)i
– +
+
1(0)v1
1
1: ( ) A A R I s
sC
3 –3
3
3
1 3
[ ( ) ( )]
(0) (0)
A B sL I s I s sC
v v
1 R 2 2
sL1(0)v
s +1 3 3 s s
+ –
+ –
1
1
sC 3 sL
2 2 (0) L i
+ – +–
2 2: ( ) B B R sL I s
( ) A I s ( ) B I s1( ) E s
2( ) E s
3 3
(0) L i
3(0)v+ –
+–
3
3
1 [ ( ) ( )] B A sL I s I s sC
The Laplace Transform 62
3
1
sC
s3
3 3 2 2 2(0) (0) ( )v
L i L i E s s
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Analysis Techniques (9)Ex. 6
Write the node equations in the s-domain? +–
2C
2(0)v
–1
L
1(0)i2 (0)i
2 L
3(0)v
3C
+1( ) j t 1 R 3 R
3( ) j t
The Laplace Transform 63
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Analysis Techniques (10)Ex. 7
Solve for v(t ) ?8
(0) 0; (0) 0; L C i v
+
1H
2
+
–
( )v t
15( )
5 ( ): 012
a
a
V s
V s sa s s
–
5 ( ) Au t 15 ( ) Vu t s
2
3 2
10 35 15
( )a
s s
V s
8
s
1
s
a
2
3 2
( ) 10 35 15( ) 2 2
1aV s s s s
V s
+ –5 15
s
2
+
–
( )V s s
11 1210( 3) s K K
The Laplace Transform 64
2 2( 1) 1 ( 1) s s s
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Analysis Techniques (11)Solve for v(t ) ?
8 1
a
Method 1
Ex. 7
+5 15
s
2
s +( )V s
11 12
2 2
10( 3)( )
( 1) 1 ( 1)
s K K V s
s s s
2
12 2 1
1
10( 3)( 1) 10( 3) 20
( 1) s
s
s K s s
s
–
s s
2
11 2
11
10( 3)( 1) 10 3 10
( 1) s s
d s d K s s
ds s ds
10 20( ) ( ) 10(2 1) Vt V s v t t e
The Laplace Transform 65
s s
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Analysis Techniques (12)Solve for v(t ) ?
8 1
a
Method 2
Ex. 7
+5 15
s
2
s +( )V s
( ) A I s
Suppose the current source flows via the inductor.
5 1 15( ) 2 ( ) A A s I s I s
s s s
–
s s
2
3( ) 5
( 1) A
s I s
s
2
3( ) 10
sV s
The Laplace Transform 66
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Analysis Techniques (13)Solve for v(t ) ?
8 1
a
Method 3
Ex. 7
+5 15
s
2
s +( )V s
52
25 10( 0.5)
( )1 ( 1)
2ab
s
s s s
V s s s s
–
s s
10( 0.5) 2 10 s s8
s
1
s
a
52 21
( 1) ( 1)2 s s s s
52
+
–
5( ) s
V s
The Laplace Transform 67
b
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Analysis Techniques (14)Solve for v(t ) ?
8 1
a
Method 3
Ex. 7
+5 15
s
2
s +( )V s
152
15( )
1 ( 1)2
C s
s I s s s
s
–
s s
8
s
1
s
a
15
2
( ) 2( 1) s
V s s
+ –
15
s
2
+
–
15( ) s
V s
The Laplace Transform 68
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Analysis Techniques (15)Solve for v(t ) ?
8 1
a
Method 3
Ex. 7
+5 15
s
2
s +( )V s8 1
s
a
–
s s5
2
10( )
( 1) s
sV s
s
5
2
+
–
5( ) s
V s
s
b
a 15 15( ) ( ) ( ) s s
V s V s V s
1515
( ) 2V s
8
s
1
s + 2 2
10 30
( 1) ( 1)
10 3
s
s s
s
The Laplace Transform 69
s s –
15
s
– s 2( 1) s
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Analysis Techniques (16)Solve for v(t ) ?
8 1
a
Method 4
Ex. 7
+5 15
s
2
s +( )V s( ) Z s s
–
s s
2
5 15 3( ) 5
s E s s
s s s
Z s
1
s2
35
3( ) 5
1 ( 1)2
s s s I s s s
+ –( ) E s
2
+
–
( )V s
s
2 2
3 3( ) 2 5 10
s sV s
The Laplace Transform 70
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Analysis Techniques (17)Solve for v(t ) ?
8 1
a
Method 5
Ex. 7
+5 15
s
2
s +( )V s8
s1 s
a
–
s s( )eq Z s ( )eq Z s s s
+( )e E s
eq s
2
+
–
( )V sa
–
+ s s
( )eq E s
5 15
( )eq E s s s s
3 s
( )( ) 2
eq E s
V s
The Laplace Transform 71
+ –5
s
15
s –
eq s s
2
310
( 1)
eq
s
s
0.5v
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0.5v
Analysis Techniques (18)Ex. 8
Solve for i(t ) ?
x
v
+
–
2F
6
i
(0) 0; (0) 0C Lv i
6( ) 2 ( ) ( ) ( 2) ( ) 0 x A c A
V s I s I s s I s 1H
6
( ) 2 ( ) 0.5 ( ) ( 2) ( ) 0 x A x AV s I s V s s I s
s
.c x
2 4( ) 2 ( ) 2 ( ) x A AV s I s I s
s s
( ) xV s
+2
2 2
6
s
( ) I s I s
2 ( ) 2 ( ) 0.5 2 ( )
( 2) ( ) 0
A A A
A
I s I s I s s s s
s I s
The Laplace Transform 72
s
–
s8 12( ) ( )
( 2)( 6) A
s I s I s
s s s
0.5v
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Analysis Techniques (19)Ex. 8
Solve for i(t ) ?
x
v
+
–
2F
6
i
1 2 38 12( )
( 2)( 6) 2 6
s K K K I s
s s s s s s
1H
1
0
8 121
( 2)( 6) s
s K
s s
.c x
2
2
8 120.5
( 6) s
s K
s s
( ) xV s
+2
2 2
6
s
( ) I s
36
1.5( 2) s
s K
s s
The Laplace Transform 73
s
–
s( ) 1 0. 1. Ai t e e
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Analysis Techniques (20)Ex. 9
Find the current i(t )?0t
2
152
8(0) 1A
8
i + –
4
2 H8V ( )i t
8 + –
2
.9( )2 4 ( 2)( 9)
s s I s s s s
*
1 2 2 K K K
2 3 3 s s j s j 2
1 2
16.51.58
s K
+– 42
( ) I s
+ –
152 s
2
2
16.50.35
( 2)( 3)
s K
s s j
o146.3
2 9 s
The Laplace Transform 74
2 o( ) 1.58 0.70cos(3 146.3 ) At i t e t
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Analysis Techniques (21)
i (t ), v (t ), …Circuit
Inverse TransformCircuit Element Models DC circuit analysis techniques
(KVL, KCL, nodal analysis,
Circuit
mesh analysis, source
transformation, superposition,
Thevenin/Norton equivalent, …)
, , …in s -domain
The Laplace Transform 75
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The Laplace Transform
• Definition
• Two Important Singularity Functions
• Transform Pairs
• Properties of the Transform
• Inverse Transform
• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 76
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Convolution Integral (1)1 2 1 2 1 2
0 0( ) ( )* ( ) ( ) ( ) ( ) ( )
t t
f t f t f t f t f d f f t d
2( ) f t
t 0
t
t 0
1( ) f t 2( ) f t
The Laplace Transform 77
0 0
2
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Convolution Integral (2)Ex. 1
Find the convolution of the two signal?1
1( ) f t
t 0 1 2 3 4
2
1 2 1 2 1 2
0 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t t
f t f t f t f t f d f f t d
t 0
2
1 2 3 4
22( ) f
1
1
( ) f t
22( ) f
1
0 1 2 3 4
0 1 2 3 4
1( ) f t 1 2
0 1: 1; 0t f f
The Laplace Transform 78
1 2( )* ( ) 0 f t f t
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Convolution Integral (3)Ex. 1
Find the convolution of the two signal?1
1( ) f t
t 0 1 2 3 4
2
1 2 1 2 1 2
0 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t t
f t f t f t f t f d f f t d
t 0
2
1 2 3 4
22( ) f
1
1
( ) f t
0 1 2 3 41 2
0 1: ( )* ( ) 0t f t f t
1 21 2 : 1; 2t f f
The Laplace Transform 79
1 2 1 2 11 1( )* ( ) ( ) ( ) 1 2 2 2( 1)
t t t f t f t f t f d d t
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Convolution Integral (4)Ex. 1
Find the convolution of the two signal?1
1( ) f t
t 0 1 2 3 4
2
1 2 1 2 1 2
0 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t t
f t f t f t f t f d f f t d
t 0
2
1 2 3 4
22( ) f
1
1
( ) f t
0 1 2 3 41 2
0 1: ( )* ( ) 0t f t f t
1 2 : ( )* ( ) 2( 1)t f t f t t
1 22 3: 1; 2t f f
The Laplace Transform 80
1 2 1 2 11 1( )* ( ) ( ) ( ) 1 2 2 2
t t t
t t t f t f t f t f d d
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Convolution Integral (5)Ex. 1
Find the convolution of the two signal?1
1( ) f t
t 0 1 2 3 4
2
1 2 1 2 1 2
0 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t t
f t f t f t f t f d f f t d
t 0
2
1 2 3 4
22( ) f 1
1
( ) f t
0 1 2 3 41 2
0 1: ( )* ( ) 0t f t f t
1 2 : ( )* ( ) 2( 1)t f t f t t
1 23 4 : 1; 2t f f 1 22 3: ( )* ( ) 2t f t f t
The Laplace Transform 81
3 3 3
1 2 1 2 11 1( )* ( ) ( ) ( ) 1 2 2 8 2
t t t f t f t f t f d d t
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Convolution Integral (6)Ex. 1
Find the convolution of the two signal?1
1( ) f t
t 0 1 2 3 4
2
1 2 1 2 1 2
0 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t t
f t f t f t f t f d f f t d
t 0
2
1 2 3 4
22( ) f
1
1
( ) f t
0 1 2 3 41 2
0 1: ( )* ( ) 0t f t f t
1 2 : ( )* ( ) 2( 1)t f t f t t
1 24 : 1; 0t f f 1 22 3: ( )* ( ) 2t f t f t
1 23 4 : ( )* ( ) 8 2t f t f t t
The Laplace Transform 82
1 2( )* ( ) 0 f t f t
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Convolution Integral (7)Ex. 1
Find the convolution of the two signal?1
1( ) f t
t 0 1 2 3 4
2
1 2 1 2 1 2
0 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t t
f t f t f t f t f d f f t d
t 0
2
1 2 3 41 2
0 1: ( )* ( ) 0t f t f t
1 21 2 : ( )* ( ) 2( 1)t f t f t t
1 22 3: ( )* ( ) 2t f t f t
2
1 2( )* ( ) f t f t
1 24 : ( )* ( ) 0t f t f t
1 2:t t t t
The Laplace Transform 83
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Convolution Integral (8)Ex. 2
Find the convolution of the two signal?1
( )w t
t 0 1 2 3 41 2 1 2 1 2
0 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t t
f t f t f t f t f d f f t d 2 e
2
2e
( )w t
t 0 1 2 3 4
2( )
000 2 : ( ) 2 1 2 2(1 )t t t t f t e d e e
2 t 0 1 2 3 4
The Laplace Transform 84
002 : ( ) 2 1 2 2( 1)t t t t f t e d e e e
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Convolution Integral (9)Ex. 2
Find the convolution of the two signal?1
( )w t
t 0 1 2 3 41 2 1 2 1 2
0 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t t
f t f t f t f t f d f f t d 2 e
22 t e
( )w
t 0 1 2 3 4
2
000 2 : ( ) 1 2 2 2(1 )t t f t e d e e
t t 0 1 2 3 4
The Laplace Transform 85
222 : ( ) 1 2 2 2( 1) t
t t t f t e d e e e
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Convolution Integral (10) Property f (t ) F ( s)
1. Magnitude scaling
2. Addition/subtraction
( )t ( ) F s
1 2( ) ( ) f t f t 1 2( ) ( ) F s F s
1 s .
4. Time shifting
a a ( ) ( ), 0 f t a u t a a ( )
ase F s
( ) ( ), 0 f t u t a a [ ( )]ase L f t a
. requency s t ng
6. Differentiation
7. Multiplication by t
( )e t s a
( ) / n nd f t dt
1 2 1 1( ) (0) (0) ... (0)
n n n o n s F s s f s f s f
( )nt f t ( 1) ( ) / n n nd F s ds
8. Division by t
9. Integration
( ) / f t t ( ) s F d
0
( )t
f d ( ) / F s s
The Laplace Transform 86
10. Convolution1 2 1 2
0( ) * ( ) ( ) ( )
t
f t f t f f t d 1 2( ) ( ) F s F s
C l i I l (11)E 3
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Convolution Integral (11)Ex. 3
Find vo(t )?+
+ v
–
( )iv t
0.2F
( )ov t e
( ) 5 5 50.2( ) ( )1( ) 1 5 11
C o i
C
Z s sV s V s Z s s s s
Method 1: 5( ) ( ) 6.25( ) Vt t
o oV s v t e e
Method 2: ( ) ( ) ( ) ( ) ( )* ( )o i o iV s H s V s v t h t v t
55
( ) ( ) 55
t
H s h t e s
5 5 4 5 4
t t t t
t t t
The Laplace Transform 87
00 0 0
5
.
6.25( ) V
o i
t t e e
Th L l T f
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The Laplace Transform• Definition
• Two Important Singularity Functions
• Transform Pairs
• Properties of the Transform
• Inverse Transform
• Initial-Value & Final-Value Theorems
• Laplace Circuit Solutions
• Analysis Techniques
•
• Transfer Function
The Laplace Transform 88
T f F ti (1)
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Transfer Function (1)( )in I s ( )out I s
+
( )inV s
+
( )out V s( ) s – –
( )Out s
( ) In s
If ( ) ( ) ( ) 1 ( ) ( )in t t In s H s Out s
The Laplace Transform 89
T f F ti (2)E 1
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Transfer Function (2)Linear
++Ex. 1
Find the transfer function h(t ) of the filter?an pass
filter – –
iv t ov
( ) 10 ( )iv t u t
10( ) ( ) ( ) ( )o iV s H s V s H s
1( ) ( )o s sV s
1 ( )odv t
The Laplace Transform 90
10 dt
T f F ti (3)Ex 2
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Transfer Function (3)Ex. 2
Find the transfer function H ( s)? +
–
( )iv t 1
0.2F
( )ov t
1( ) 50.2C Z s s
1( ) 51
0.2
o i i i iC Z s s
s
( ) 5( )( ) 5
o
i
V s H sV s s
The Laplace Transform 91
Transfer F nction (4)
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Transfer Function (4)
A circuit is stable if : lim ( ) finite x
h t
j
1 2
( )( )
( )( )...( )n
N s H s
s p s p s p
1 2 ... n p t p t p t h t k e k e k e u t
n
The Laplace Transform 92
lie in the left half of the s-plane
Transfer Function (5)Ex 3
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Transfer Function (5)Ex. 3
An active filter has the transfer functionk
2 (4 ) 1 s k s
For what values of k is the filter stable?
j A circuit is stable when all the poles of its transfer
function H ( s) lie in the left half of the s-plane
2
1,2
(4 ) (4 ) 4
2
k k p
4 0k
The Laplace Transform 93
4k
Transfer Function (6)Ex 4
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Transfer Function (6)Ex. 4
+Given the transfer function5
–
( )iv t 1
C
( )ov t
5 s
Find C ?
( ) 1( ) ( ) ( ) ( )
1( ) 11
C o i i i
C
Z s sC V s V s V s V s R Z s Cs
++
( ) 1 5( ) oV s
H s –
i s 1
sC
( )oV s
i
5 1C
The Laplace Transform 94
0.2FC