Bài giảng Electric circuit theory - Chapter XIV: The Laplace Transform - Nguyễn Công Phương
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Functio
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Electric Circuit Theory
The Laplace Transform
Nguyễn Công Phương
Contents
I. Basic Elements Of Electrical Circuits
II. Basic Laws
III. Electrical Circuit Analysis
IV. Circuit Theorems
V. Active Circuits
VI. Capacitor And Inductor
VII. First Order Circuits
VIII.Second Order Circuits
IX. Sinusoidal Steady State Analysis
X. AC Power Analysis
XI. Three-phase Circuits
XII. Magnetically Coupled Circuits
XIII.Frequency Response
XIV.The Laplace Transform
XV. Two-port Networks
The Laplace Transform - sites.google.com/site/ncpdhbkhn 2
F(s) = 0
(algebraic) I(s), V(s),
Laplace Transform Inverse Transform
The Laplace Transform
The Laplace Transform - sites.google.com/site/ncpdhbkhn 3
f(t) = 0
(integrodifferential) i(t), v(t), Circuit
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 4
Definition
The Laplace Transform - sites.google.com/site/ncpdhbkhn 5
t
( )f t
0[ ] 0( ) ( ) ( ) stF s L f t f t e dt
∞
−
= = ∫
s jσ ω= +
0
( ) tf t e dtσ∞ − < ∞∫
[ ] 1
1
1 1( ) ( ) ( )
2
j
st
j
f t L F s F s e dsj
σ
σpi
+ ∞
−
− ∞
= = ∫
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 6
Two Important Singularity Functions (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 7
t
( )u t
0
1
t
( )u t a−
0
1
a
0 0( )
1 0
t
u t
t
<
=
>
0( )
1
t a
u t a
t a
<
− =
>
Two Important Singularity Functions (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 8
t
( )u t
0
1
Ex. 1
Determine the Laplace transform for the waveform?
0
( ) ( ) stF s u t e dt∞ −= ∫
0
1 ste dt
∞
−
= ∫
0
1 st
e
s
∞
−
= −
1
s
=
Two Important Singularity Functions (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 9
Ex. 2
Determine the Laplace transform for the waveform?
0
( ) ( ) stF s u t a e dt∞ −= −∫
0
0 1
a
st
a
dt e dt
∞
−
= +∫ ∫
1 st
a
e
s
∞
−
= −
as
e
s
−
=
t
( )u t a−
0
1
a
Two Important Singularity Functions (4)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 10
Ex. 3
Determine the Laplace transform for the waveform?
0
( ) [ ( ) ( )] stF s u t u t a e dt∞ −= − −∫
0
1( ) stu t e dt
s
∞
−
=∫
0
( )
st
st e
u t a e dt
s
−
∞
−
− =∫
1 1( )
as ase eF s
s s s
− −
−
→ = − = t
( )u t a− −
1−
0
a
t
( )u t
0
1
t0
1
a
Two Important Singularity Functions (5)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 11
t
( )tδ
0
t
( )t aδ −
0 a
( ) 0 0
( ) 1 0
t t
t dt
ε
ε
δ
δ ε
−
= ≠
= >∫
( ) 0
( ) 1 0a
a
t a t a
t a dt
ε
ε
δ
δ ε+
−
− = ≠
− = >∫
2
1
1 2
1 2
( )( ) ( )
0 ,
t
t
f a t a tf t t a dt
a t a t
δ < <− =
∫
Two Important Singularity Functions (6)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 12
Ex. 4
Determine the Laplace transform of an impulse function?
0
( ) ( ) stF s t a e dtδ∞ −= −∫
2
1
1 2
1 2
( )( ) ( )
0 ,
t
t
f a t a tf t t a dt
a t a t
δ < <− =
∫
( ) asF s e−→ =
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 13
Transform Pairs (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 14
Ex. 1
Find the Laplace transform of f(t) = t?
0
( ) stF s te dt∞ −= ∫
1Let & &st st stu t dv e dt du dt v e dt e
s
− − −
= = → = = = −∫
20
0 0
1( ) 0
st st
stt e eF s e dt
s s s s
∞∞
− −
∞
−→ = − + = − =∫
Transform Pairs (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 15
Ex. 2
Find the Laplace transform of f(t) =cosωt?
0
( ) cos stF s te dtω∞ −= ∫
0 2
j t j t
ste e e dt
ω ω−
∞
−
+
= ∫
( ) ( )
0 2
s j t s j te e dt
ω ω− − − +
∞ +
= ∫
1 1 1
2 s j s jω ω
= +
− +
2 2
s
s ω
=
+
f(t)
F(s)
Transform Pairs (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 16
( )tδ
1
( )u t
1
s
at
e
−
1
s a+
t
2
1
s
atte−
2
1
( )s a+
sin at
2 2
a
s a+
cos at
2 2
s
s a+
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 17
Properties of the Transform (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 18
Property f(t) F(s)
1. Magnitude scaling
2. Addition/subtraction
3. Time scaling
4. Time shifting
5. Frequency shifting
6. Differentiation
7. Multiplication by t
8. Division by t
9. Integration
10. Convolution
( )Af t ( )AF s
1 2( ) ( )f t f t± 1 2( ) ( )F s F s±
( )f at 1 sF
a a
( ) ( ), 0f t a u t a a− − ≥ ( )ase F s−
( )ate f t− ( )F s a+
( ) ( ), 0f t u t a a− ≥ [ ( )]ase L f t a− +
( ) /n nd f t dt 1 2 1 1( ) (0) (0) ... (0)n n n o ns F s s f s f s f− − −− − −
( )nt f t ( 1) ( ) /n n nd F s ds−
( ) /f t t ( )
s
F dλ λ∞∫
0
( )t f dλ λ∫ ( ) /F s s
1 2 1 20
( ) * ( ) ( ) ( )tf t f t f f t dλ λ λ= −∫ 1 2( ) ( )F s F s
Properties of the Transform (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 19
Ex. 1
Find the Laplace transform of 10( ) 5 cos20 ?tf t e t−= + −
1 2 1 2( ) ( ) ( ) ( )f t f t F s F s± → ±
10( ) [5] [ ] [cos20 ]tF s L L e L t−→ = + −
( ) ( )Af t AF s→
[5] 5 [1]L L→ =
1[1]L
s
=
5[5]L
s
→ =
10 1[ ]
10
tL e
s
−
=
+
2 2 2[cos20 ] 20 400
s sL t
s s
= =
+ +
3
2 2
5 1 5 2400 4000( )
10 400 ( 10)( 400)
s s sF s
s s s s s s
+ +
→ = + − =
+ + + +
Properties of the Transform (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 20
Ex. 2
Find the Laplace transform of the waveform?
t0
5
1 2 3
t0
5
1 2 3
t0
5
1 2 3
( ) 5 ( 1) 5 ( 2)f t u t u t= − − −
2
25( ) 5 5 ( )
s s
s se eF s e e
s s s
− −
− −→ = − = −
t0
1
1 2 3
Properties of the Transform (4)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 21
Ex. 3
Find the Laplace transform of the waveform?
t0
5
1 2 3
t0
5
1 2 3
( ) ( 5 10)[ ( 1) ( 2)]f t t u t u t= − + − − −
5 ( 1) 10 ( 1)
5 ( 2) 10 ( 2)
tu t u t
tu t u t
= − − + − +
+ − − −
5 ( 1) 5( 1 1) ( 1)tu t t u t− − = − − + −
5( 1) ( 1) 5 ( 1)t u t u t= − − − − −
5 ( 2) 5( 2 2) ( 2)tu t t u t− = − + −
5( 2) ( 2) 10 ( 2)t u t u t= − − + −
( ) 5( 1) ( 1) 5 ( 1)
10 ( 1)
5( 2) ( 2) 10 ( 2)
10 ( 2)
f t t u t u t
u t
t u t u t
u t
→ = − − − − − +
+ −
+ − − + −
− −
Properties of the Transform (5)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 22
Ex. 3
Find the Laplace transform of the waveform?
t0
1
1 2 3
t0
5
1 2 3
t0
5
1 2 3
( ) ( 5 10)[ ( 1) ( 2)]f t t u t u t= − + − − −
5( 1) ( 1) 5 ( 1)
10 ( 1)
5( 2) ( 2) 10 ( 2)
10 ( 2)
t u t u t
u t
t u t u t
u t
= − − − − − +
+ −
+ − − + −
− −
5( 1) ( 1) 5 ( 1)
5( 2) ( 2)
t u t u t
t u t
= − − − + − +
+ − −
2
2 2
2
5( ) 5 5
5 (1 )
s s
s
s
s
e eF s e
s s s
e
s e
s
− −
−
−
−
→ = − + +
= − − −
Properties of the Transform (6)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 23
Ex. 4
Find the Laplace transform of the waveform?
t0
5
1 2 3
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 24
The Laplace Transform
The Laplace Transform - sites.google.com/site/ncpdhbkhn 25
F(s) = 0
(algebraic) I(s), V(s),
Laplace Transform Inverse Transform
f(t) = 0
(integrodifferential) i(t), v(t), Circuit
Inverse Transform (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 26
1
1 1 0
1
1 1 0
( ) ...( ) ( ) ...
m m
m m
n n
n n
P s a s a s a s aF s Q s b s b s b s b
−
−
−
−
+ + + +
= =
+ + + +
1 2
1 2
Simple poles : ( ) ... n
n
K K KF s
s p s p s p
= + + +
+ + +
1
1 1
11 12 1
2
1 1
( )Multiple poles : ( ) ( )( )
... ...( ) ( ) ( )
n
n
n
P sF s Q s s p
K K K
s p s p s p
=
+
= + + + +
+ + +
1
1
*
1 1
( )Complex- conjugate poles : ( ) ( )( )( )
...
P sF s Q s s j s j
K K
s j s j
α β α β
α β α β
=
+ − + +
= + +
+ − + +
Inverse Transform (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 27
1 2
1 2
( )Simple poles : ( ) ...( )
n
n
P s K K KF s Q s s p s p s p= = + + ++ + +
( )( ) 0 ... 0 0 ... 0( )
i
i i
s p
P s
s p KQ s
=−
+ = + + + + + +
1 ip ti
i
i
KL K e
s p
−−
= +
1 2
1 2( ) ... np tp t p t nf t K e K e K e−− −= + + +
Inverse Transform (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 28
Ex. 1
Find the inverse Laplace transform of
225 300 640( ) ( 4)( 8)
s sF s
s s s
+ +
=
+ +
1 2 3 ( )( ) ; ( ) 0 ... 0 0 ... 0
4 8 ( )
i
i i
s p
K K K P sF s s p K
s s s Q s
=
= + + + = + + + + + +
+ +
2 2
1 0
0 0
25 300 640 25 300 640 640( ) 20( 4)( 8) ( 4)( 8) 4 8s
s s
s s s sK sF s s
s s s s s=
= =
+ + + +
= = = = =
+ + + + ×
2 2
2 4
4 4
2
25 300 640 25 300 640( 4) ( ) ( 4) ( 4)( 8) ( 8)
25( 4) 300( 4) 640 10( 4)( 4 8)
s
s s
s s s sK s F s s
s s s s s=−
=− =−
+ + + +
= + = + = =
+ + +
− + − +
= =
− − +
Inverse Transform (4)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 29
Ex. 1
Find the inverse Laplace transform of
225 300 640( ) ( 4)( 8)
s sF s
s s s
+ +
=
+ +
1 220; 10K K= =
2 2
3 8
8 8
2
25 300 640 25 300 640( 8) ( ) ( 8) ( 4)( 8) ( 4)
25( 8) 300( 8) 640 5( 8)( 8 4)
s
s s
s s s sK s F s s
s s s s s=−
=− =−
+ + + +
= + = + = =
+ + +
− + − +
= = −
− − +
20 10 5( )
4 8
F s
s s s
→ = + −
+ +
4 8( ) 20 10 5t tf t e e− −→ = + −
1 2 3 ( )( ) ; ( ) 0 ... 0 0 ... 0
4 8 ( )
i
i i
s p
K K K P sF s s p K
s s s Q s
=
= + + + = + + + + + +
+ +
Inverse Transform (5)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 30
Ex. 1
Find the inverse Laplace transform of
225 300 640( ) ( 4)( 8)
s sF s
s s s
+ +
=
+ +
1 2 3( )
4 8
K K KF s
s s s
= + +
+ +
2
1
25 300 640s sK
s
+ +
=
2
0 0
25 300 640 640 20( 4)( 8) 4 8( 4)( 8)
s s
s s
s ss s
= =
+ +
= = =
+ + ×+ +
2
2
25 300 640
( 4)
s sK
s s
+ +
=
+
2
44
25 300 640 10( 8)( 8)
ss
s s
s ss
=−
=−
+ +
= =
++
2
3
25 300 640
( 4) ( 8)
s sK
s s s
+ +
=
+ +
2
88
25 300 640 5( 4)
ss
s s
s s
=−
=−
+ +
= = −
+
20 10 5( )
4 8
F s
s s s
→ = + −
+ +
4 8( ) 20 10 5t tf t e e− −→ = + −
Inverse Transform (6)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 31
Ex. 2
Find the inverse Laplace transform of 100( 6)( ) ( 1)( 3)
sF s
s s
+
=
+ +
Inverse Transform (7)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 32
*
1 1 1
1
( )Complex- conjugate poles : ( ) ...( )( )( )
P s K KF s Q s s j s j s j s jα β α β α β α β= = + ++ − + + + − + +
1 1
( )( ) ( )
s j
P s
s j K KQ s
α β
α β
=− +
+ − = = θ
*
1 1K K= θ−
1( ) KF s→ = 1K
s j
θ
α β ++ −
1 1
... ...
j jK e K e
s j s j s j
θ θθ
α β α β α β
−
−
+ = + +
+ + + − + +
( ) ( ) ( ) ( )
1 1 1( ) ... ...j j t j j t t j t j tf t K e e K e e K e e eθ α β θ α β α β θ β θ− − − − + − + − + → = + + = + +
cos sinje jφ φ φ= +
[ ]1( ) cos( ) sin( ) cos( ) sin( ) ...tf t K e t j t t j tα β θ β θ β θ β θ−→ = + + + + − − + − − +
12 cos( ) ...tK e tα β θ−= + +
Inverse Transform (8)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 33
Ex. 3
Find the inverse Laplace transform of
2
2
4 76( ) ( 2)( 6 25)
s sF s
s s s
+
=
+ + +
1 1
( )( ) ; ( ) 2 cos( ) ...( )
t
s j
P sK s j f t K e tQ s
α
α β
α β β θ−
=− +
= + − = + +
2 2
3 2 2
22
4 76 4 76( 2) 8( 2)( 6 25) 6 25
ss
s s s sK s
s s s s s
=−=−
+ +
= + = = − + + + + +
2
1 2
3 4
4 76( 3 4) 6 8 10( 2)( 6 25)
s j
s sK s j j
s s s
=− +
+
= + − = − =
+ + +
o53.1−
3 o 2 3 o 2( ) 2 10 cos(4 53.1 ) 8 20 cos(4 53.1 ) 8t t t tf t e t e e t e− − − −→ = × − − = − −
1 2 3( )
3 4 3 4 2
K K KF s
s j s j s= + ++ − + + +
Inverse Transform (9)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 34
Ex. 3
Find the inverse Laplace transform of
2
2
4 76( ) ( 2)( 6 25)
s sF s
s s s
+
=
+ + +
2
3
4 76
( 2)
s sK
s
+
=
+
2
22
22
4 76 8
6 25( 6 25)
ss
s s
s ss s
=−
=−
+
= = −
+ ++ +
2
1
4 76
( 2) ( 3 4)
s sK
s s j
+
=
+ + −
3 4
6 8 10( 3 4)
s j
j
s j
=− +
= − =
+ +
o53.1−
3 o 2 3 o 2( ) 2 10 cos(4 53.1 ) 8 20 cos(4 53.1 ) 8t t t tf t e t e e t e− − − −→ = × − − = − −
1 2 3( )
3 4 3 4 2
K K KF s
s j s j s= + ++ − + + +
Inverse Transform (10)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 35
Ex. 4
Find the inverse Laplace transform of 2
5( 2)( ) ( 4 5)
sF s
s s s
+
=
+ +
Inverse Transform (11)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 36
1 11 12 1
2
1 1 1 1
( )Multiple poles : ( ) ... ...( )( ) ( ) ( ) ( )
n
n n
P s K K KF s Q s s p s p s p s p= = + + + ++ + + +
1
1 1( ) ( )n ns ps p F s K=−+ =
1
2
1 1 22 [( ) ( )] (2!)n n
s p
d
s p F s K
ds −
=−
+ =
1
1 1 1[( ) ( )]n n
s p
d
s p F s K
ds −
=−
+ =
1
1 1
1 [( ) ( )]( )!
n j
n
j n j
s p
dK s p F s
n j ds
−
−
=−
= +
−
Inverse Transform (12)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 37
Ex. 5
Find the inverse Laplace transform of
2
2
10 34 27( ) ( 3)
s sF s
s s
+ +
=
+
2 2
2 2
12 23
3 3
10 34 27 10 34 27( 3) ( ) ( 3) 5( 3)s
s s
s s s sK s F s s
s s s=−
=− =−
+ + + +
= + = + = = −
+
2
2
11
3 3
2
2
3
10 34 27[( 3) ( )]
(20 34) (10 34 27) 7
s s
s
d d s sK s F s
ds ds s
s s s s
s
=−
=−
=−
+ +
= + = =
+ − + +
= =
2
2 20
0
10 34 27( ) 3( 3)s
s
s sK sF s s
s s=
=
+ +
= = =
+
1
11 12 2
1 12
1( ) ; [( ) ( )]
3 ( 3) ( )!
n j
n
j n j
s p
K K K dF s K s p F s
s s s n j ds
−
−
=−
= + + = +
+ + −
Inverse Transform (13)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 38
Ex. 5
Find the inverse Laplace transform of
2
2
10 34 27( ) ( 3)
s sF s
s s
+ +
=
+
2
7 5 3( )
3 ( 3)F s s s s→ = − ++ +
3 3( ) 7 5 3t tf t e te− −→ = − +
11 12 27; 5; 3K K K= = − =
1
11 12 2
1 12
1( ) ; [( ) ( )]
3 ( 3) ( )!
n j
n
j n j
s p
K K K dF s K s p F s
s s s n j ds
−
−
=−
= + + = +
+ + −
Inverse Transform (14)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 39
Ex. 5
Find the inverse Laplace transform of
2
2
10 34 27( ) ( 3)
s sF s
s s
+ +
=
+
11 12 2
2( ) 3 ( 3)
K K KF s
s s s
= + +
+ +
2
2
10 34 27s sK
s
+ +
=
2
22
0 0
10 34 27 3( 3)( 3)
s s
s s
ss
= =
+ +
= =
++
2
12 2
10 34 27
( 3)
s sK
s s
+ +
=
+
2
33
10 34 27 5
s
s
s s
s
=−
=−
+ +
= = −
2
11 2
10 34 27
( 3)
d s sK
ds s
+ +
=
+
2
33
2
2
3
10 34 27
(20 34) (10 34 27) 7
s
s
s
d s s
ds ss
s s s s
s
=−
=−
=−
+ +
=
+ − + +
= =
2
7 5 3
3 ( 3)s s s= − ++ +
3 3( ) 7 5 3t tf t e te− −→ = − +
Inverse Transform (15)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 40
Ex. 6
Find the inverse Laplace transform of 2
5( 3)( ) ( 1)( 2)
sF s
s s
+
=
+ +
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 41
Initial-Value & Final-Value Theorems (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 42
0
Initial value theorem : lim ( ) lim ( )
t s
f t sF s
→ →∞
− =
0
Final value theorem : lim ( ) lim ( )
t s
f t sF s
→∞ →
− =
Initial-Value & Final-Value Theorems (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 43
2
5( 1)(0) lim ( ) lim 0
2 2s s
sf sF s
s s→∞ →∞
+
= = =
+ +
20 0
5( 1)( ) lim ( ) lim 2.5
2 2s s
sf sF s
s s→ →
+
∞ = = =
+ +
Ex.
Find the initial and final values of 2
5( 1)( ) ( 2 2)
sF s
s s s
+
=
+ +
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 44
The Laplace Transform - sites.google.com/site/ncpdhbkhn 45
Ex.
Find the current i(t)?
t = 0
+
–
200Ω
100 mH
1V ( )i t1L R
di
v v e L Ri
dt
+ = → + =
0n n
diL Ri
dt
+ = tni Ke
α−→ = 0t tLK e RKeα αα − −→ − + = 0L Rα→ − + =
3
200 2000
100 10
R
L
α
−
→ = = =
×
2000t
ni Ke
−→ =
1 0.005A
200f
ei
R
= = =
20000.005 tf ni i i Ke
−
= + = +
2000 0(0) 0.005 0.005 0 0.005i Ke K K− ×= + = + = → = −
2000( ) 0.005(1 )Ati t e−→ = −
Method 1
Laplace Circuit Solutions (1)
Laplace Circuit Solutions (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 46
F(s) = 0
(algebraic) I(s), V(s),
Laplace Transform Inverse Transform
f(t) = 0
(integrodifferential) i(t), v(t), Circuit
Laplace Circuit Solutions (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 47
Ex.
Find the current i(t)?
t = 0
+
–
200Ω
100 mH
1V ( )i t
Method 2
0.1 200 1L R
di
v v e i
dt
+ = → + =
0.1 200 [1] 0.1 [200 ]di diL i L L L i
dt dt
+ = = +
1[1]L
s
=
[200 ] 200 ( )L i I s=
1 2 1 1( ) ( ) (0) (0) ... (0)
n
n n n o n
n
d f t
s F s s f s f s f
dt
− − −→ − − −
0.1 0.1[ ( ) (0)] 0.1 ( )diL sI s i sI s
dt
→ = − =
10.1 ( ) 200 ( )sI s I s
s
→ + =
Laplace Circuit Solutions (4)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 48
Ex.
Find the current i(t)?
t = 0
+
–
200Ω
100 mH
1V ( )i t
Method 2
0.1 200 1L R
di
v v e i
dt
+ = → + =
10.1 ( ) 200 ( )sI s I s
s
→ + =
1 21 10( ) (0.1 200) ( 2000) 2000
K KI s
s s s s s s
→ = = = +
+ + +
1
0
10 0.005
2000 s
K
s
=
= =
+
0.005 0.005( )
2000
I s
s s
→ = −
+
2
2000
10 0.005
s
K
s
=−
= = −
2000( ) 0.005(1 ) Ati t e−→ = −
The Laplace Transform - sites.google.com/site/ncpdhbkhn 49
F(s) = 0
(algebraic) I(s), V(s),
Laplace Transform Inverse Transform
f(t) = 0
(integrodifferential) i(t), v(t), Circuit
Circuit
in s-domain
Circuit Element Models
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 50
Circuit Element Models (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 51
v Ri=
( ) ( )Af t AF s→ ( ) ( )V s RI s→ =
R( )v t
( )i t
+
−
R( )V s
( )I s
+
−
Circuit Element Models (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 52
di
v L
dt
=
( ) [ ( ) (0)]df tA A sF s f
dt
→ −
( ) [ ( ) (0)]
( ) (0)
V s L sI s i
sLI s Li
→ = −
= −
L
( )v t
( )i t
+
−
(0)i
sL
( )V s
( )I s
+
−
+– (0)Li
Circuit Element Models (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 53
0
1 ( ) (0)tv i x dx v
C
= +∫
0
( )( )t F sf d
s
λ λ →∫
1 (0)( ) ( ) vV s I s
sC s
→ = +
(0)(0) vv
s
→
C
( )v t
( )i t
+
−
1
sC( )V s
( )I s
+
−
–
+ (0)v
s
Circuit Element Models (4)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 54
1 2
1 1
( ) ( )( ) di t di tv t L M
dt dt
= + 1 1 1 1 1 2 2( ) ( ) (0) ( ) (0)V s sL I s L i sMI s Mi→ = − + −
1( )i t
1L 2L
+
–
M
2 ( )i t
2 ( )v t
+
–
1( )v t
2 1
2 2
( ) ( )( ) di t di tv t L M
dt dt
= + 2 2 2 2 2 1 1( ) ( ) (0) ( ) (0)V s sL I s L i sMI s Mi→ = − + −
1( )I s
1sL 2sL
+
–
sM
2 ( )I s
2 ( )V s
+
–
1( )V s
+ –+–
1 1 2(0) (0)L i Mi+ 2 2 1(0) (0)L i Mi+
1(0)i 2 (0)i
Circuit Element Models (5)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 55
R( )v t
( )i t
+
−
R( )V s
( )I s
+
−
L
( )v t
( )i t
+
−
(0)i
sL
( )V s
( )I s
+
−
+– (0)Li
Circuit Element Models (6)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 56
C
( )v t
( )i t
+
−
1
sC( )V s
( )I s
+
−
–
+ (0)v
s
1( )i t
1L 2L
+
–
M
2 ( )i t
2 ( )v t
+
–
1( )v t
1( )I s
1sL 2sL
+
–
sM
2 ( )I s
2 ( )V s
+
–
1( )V s
+ –+–
1 1 2(0) (0)L i Mi+ 2 2 1(0) (0)L i Mi+
Circuit Element Models (7)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 57
Ex. 1
Transfer the circuit into Laplace domain?
t = 0
+
–
200Ω
100 mH
1V ( )i t
+
–
200
0.1s
1
s ( )I s +
–
0.1 (0)i
+
–
1
s
+
–
1V
200 200
0.1s
+
–
0.1 (0)i
0.1H
(0)i
Circuit Element Models (8)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 58
Ex. 2
Transfer the circuit into Laplace domain?
t = 0
+
–
4A
0.01H 1mF
24V
6Ω
4Ω
i
+
–
v
2Ω
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 59
Analysis Techniques (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 60
I(s), V(s),
Inverse Transform
i(t), v(t), Circuit
Circuit
in s-domain
Circuit Element Models DC circuit analysis techniques
(KVL, KCL, nodal analysis,
mesh analysis, source
transformation, superposition,
Thevenin/Norton equivalent, )
1 2KVL/KCL : ( ) ( ) ... ( ) 0nx t x t x t+ + + =
1 2KVL/KCL : ( ) ( ) ... ( ) 0nX s X s X s+ + + =
Analysis Techniques (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 61
Ex. 1
Find the current i(t)?
t = 0
+
–
200Ω
100 mH
1V ( )i t
+
–
200
0.1s
1
s ( )I s +
–
0.1 (0)i
+
–
1
s
+
–
1V
200 200
0.1s
+
–
0.1 (0)i
0.1H
(0)i
Analysis Techniques (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 62
Ex. 1
Find the current i(t)?
t = 0
+
–
200Ω
100 mH
1V ( )i t
+
–
200
0.1s
1
s ( )I s +
–
0.1 (0)i
(0) 0i =
1200 ( ) 0.1 ( ) 0.1 (0) 200 ( ) 0.1 ( )I s sI s i I s sI s
s
+ − = = +
1 21 10( ) (0.1 200) ( 2000) 2000
K KI s
s s s s s s
→ = = = +
+ + +
1
0
10 0.005
2000 s
K
s
=
= =
+
0.005 0.005( )
2000
I s
s s
→ = −
+
2
2000
10 0.005
s
K
s
=−
= = −
2000( ) 0.005(1 ) Ati t e−→ = −
The Laplace Transform - sites.google.com/site/ncpdhbkhn 63
Analysis Techniques (4)
Inverse Transform
Circuit Element Models
t = 0
+
–
200Ω
100 mH
1V ( )i t
+
–
200
0.1s
1
s ( )I s +
–
0.1 (0) 0i =
1200 ( ) 0.1 ( )I s sI s
s
+ =
10( ) ( 2000)I s s s= +
2000( ) 0.005(1 )Ati t e−= −
1. Solve for initial capacitor
voltages & inductor currents
2. Draw an s-domain circuit
3. Use one of DC circuit
analysis techniques to solve
for voltages or/and currents
in s-domain
4. Find the inverse Laplace
transform to convert them
back to the time domain
(0) 0i =
Ex. 1
Find the current i(t)?
The Laplace Transform - sites.google.com/site/ncpdhbkhn 64
Analysis Techniques (5)
4
6
4
6
1101 525 10( ) // ( ) 1 210
25 10
sV s R J s
sC s
s
−
−
×
= = × + +
×
1. Solve for initial capacitor voltages &
inductor currents
2. Draw an s-domain circuit
3. Use one of DC circuit analysis
techniques to solve for voltages or/and
currents in s-domain
4. Find the inverse Laplace transform to
convert them back to the time domain
(0) 0v =
Ex. 2
Find the voltage v(t)?
t = 0
10kΩ
25 Ate− 25 Fµ –
+
v
10k
5 A
2s +
6
1
25 10 s−×
–
+
( )V s
4
1 24 10
( 2)( 4) 2 4
K K
s s s s
×
= = +
+ + + +
4
4
1
2
4 10 2 10
4
s
K
s
=−
×
= = ×
+
4
4
2
4
4 10 2 10
2
s
K
s
=−
×
= = − ×
+
4 2 4( ) 2 10 ( ) Vt tv t e e− −→ = × −
The Laplace Transform - sites.google.com/site/ncpdhbkhn 65
Analysis Techniques (6)Ex. 2
Find the voltage v(t)?
t = 0
10kΩ
25 Ate− 25 Fµ –
+
v
4 2 4( ) 2 10 ( ) Vt tv t e e− −= × −
-500 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
The Laplace Transform - sites.google.com/site/ncpdhbkhn 66
Analysis Techniques (7)
122
( )
2 4
sI s
s
+
=
+
1. Solve for initial capacitor voltages &
inductor currents
2. Draw an s-domain circuit
3. Use one of DC circuit analysis
techniques to solve for voltages or/and
currents in s-domain
4. Find the inverse Laplace transform to
convert them back to the time domain
8(0) 1A
8
i = =
Ex. 3
Find the current i(t)?
1 26
( 2) 2
s K K
s s s s
+
= = +
+ +
1
0
6 3
2 s
sK
s
=
+
= =
+
2
2
6 2
s
sK
s
=−
+
= = −
2( ) 3 2 Ati t e−→ = −
L
( )v t
( )i t
+
−
(0)i
sL
( )V s
( )I s
+
−
+–
(0)Li
+
–
4Ω
2 H8V ( )i t
0t =
8Ω +
–
12 V
+
– 4
2s
2
( )I s
+
–
12
s
The Laplace Transform - sites.google.com/site/ncpdhbkhn 67
Analysis Techniques (8)Ex. 3
Find the current i(t)? 2( ) 3 2 Ati t e−= − +
–
4Ω
2 H8V ( )i t
0t =
8Ω +
–
12 V
-2000 -1000 0 1000 2000 3000 4000 5000
0
0.5
1
1.5
2
2.5
3
+
–
8V 0i
8Ω
4Ω
2 H ( )ni t
4Ω
( )fi t
+
–
12 V
The Laplace Transform - sites.google.com/site/ncpdhbkhn 68
Analysis Techniques (9)
12 8
1 8( ) 12 4
2
s sV s
s s
s
−
= × +
+
1. Solve for initial capacitor voltages &
inductor currents
2. Draw an s-domain circuit
3. Use one of DC circuit analysis
techniques to solve for voltages or/and
currents in s-domain
4. Find the inverse Laplace transform to
convert them back to the time domain
(0) 8 Vv =
Ex. 4
Find the voltage v(t)?
1 28 1.5
( 0.125) 0.125
s K K
s s s s
+
= = +
+ +
1
0
8 1.5 12
0.125 s
sK
s
=
+
= =
+
2
0.125
0.5 4
s
K
s
=−
= = −
0.125( ) 12 4 Vtv t e−→ = −
+
–
4Ω
2 F8V
( )v t
0t =
8Ω +
–
12 V
+
–
–
+ 4
1
2s
8
s
( )V s
+
–
12
s
+
–
C
( )v t
( )i t
+
−
1
sC
( )V s
( )I s
+
−
–+
(0)v
s
The Laplace Transform - sites.google.com/site/ncpdhbkhn 69
Analysis Techniques (10)Ex. 4
Find the voltage v(t)? 0.125( ) 12 4 Vtv t e−= − +
–
4Ω
2 F8V
( )v t
0t =
8Ω +
–
12 V
+
–
-5 0 5 10
x 104
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
+
–
8V 0v
8Ω +
–
4Ω
( )nv t
+
–
4Ω
( )fv t
+
–
12 V
+
–
The Laplace Transform - sites.google.com/site/ncpdhbkhn 70
Analysis Techniques (11)Ex. 5
Write the mesh equations in the s-domain?
+
–
+
–
1( )e t
2 ( )e t
1C 1R 2R
2L
3L
3C
3(0)i
2 (0)i
–+
–
+
1(0)v
3(0)v
+
–
+
–
1( )E s 2( )E s
1
1
sC
1R 2R 2sL
3sL
3
1
sC
3 3(0)L i
2 2 (0)L i
1(0)v
s
3(0)v
s
+ – + –
+
–
+
–
L
( )v t
( )i t
+
−
(0)i
sL
( )V s
( )I s
+
−
+–
(0)Li
C
( )v t
( )i t
+
−
1
sC( )V s
( )I s
+
−
–+ (0)v
s
The Laplace Transform - sites.google.com/site/ncpdhbkhn 71
Analysis Techniques (12)Ex. 5
Write the mesh equations in the s-domain?
+
–
+
–
1( )e t
2 ( )e t
1C 1R 2R
2L
3L
3C
3(0)i
2 (0)i
–+
–
+
1(0)v
3(0)v
( )AI s ( )BI s
1
1
3
3
1 3
1 3 3
1
: ( )
1 [ ( ) ( )]
(0) (0)( ) (0)
A
A B
A R I s
sC
sL I s I s
sC
v vE s L i
s s
+ +
+ + − =
= − + −
+
–
+
–
1( )E s 2( )E s
1
1
sC
1R 2R 2sL
3sL
3
1
sC
3 3(0)L i
2 2 (0)L i
1(0)v
s
3(0)v
s
+ – + –
+
–
+
–
( )2 2
3
3
3
3 3 2 2 2
: ( )
1 [ ( ) ( )]
(0) (0) (0) ( )
B
B A
B R sL I s
sL I s I s
sC
v L i L i E s
s
+ +
+ + − =
= − − −
The Laplace Transform - sites.google.com/site/ncpdhbkhn 72
Analysis Techniques (13)Ex. 6
Write the node equations in the s-domain? +–
+
–
1( )j t
1L
1(0)i
2 (0)i
1R 3R
2L
2C
2(0)v
3(0)v
3C
3( )j t
The Laplace Transform - sites.google.com/site/ncpdhbkhn 73
Analysis Techniques (14)Ex. 7
Solve for v(t) ?
+
–
5 ( ) Au t 15 ( ) Vu t
8Ω
1H
2Ω
1F
+
–
( )v t
15 ( )5 ( )
: 012
a
a
V s V ssa
s s
s
−
+ − =
+
+
–
5
s
15
s
8
s
2
1
s
+
–
( )V s
a
2
3 2
10 35 15( )
2a
s sV s
s s s
+ +
→ =
+ +
2
3 2
( ) 10 35 15( ) 2 21 2 2 12
aV s s s sV s
s s s s
s
+ +
→ = = ×
+ + ++
11 12
2 2
10( 3)
( 1) 1 ( 1)
s K K
s s s
+
= = +
+ + +
(0) 0; (0) 0;L Ci v= =
The Laplace Transform - sites.google.com/site/ncpdhbkhn 74
Analysis Techniques (15)
Solve for v(t) ?
2
12 2 1
1
10( 3)( 1) 10( 3) 20( 1) s
s
sK s s
s =−
=−
+
= + = + =
+
( )211 2
11
10( 3)( 1) 10 3 10( 1) ss
d s dK s s
ds s ds
=−
=−
+
= + = + = +
+
–
5
s
15
s
8
s
2
1
s
+
–
( )V s
a
11 12
2 2
10( 3)( ) ( 1) 1 ( 1)
s K KV s
s s s
+
= = +
+ + +
2
10 20( ) ( ) 10(2 1) V
1 ( 1)
tV s v t t e
s s
−→ = + → = +
+ +
Method 1
Ex. 7
The Laplace Transform - sites.google.com/site/ncpdhbkhn 75
Analysis Techniques (16)
Solve for v(t) ?
5 1 15( ) 2 ( )A As I s I s
s s s
− + + =
+
–
5
s
15
s
8
s
2
1
s
+
–
( )V s
a
Method 2
( )AI s
2
3( ) 5 ( 1)A
sI s
s
+
→ =
+
2
3( ) 10 ( 1)
sV s
s
+
→ =
+
Ex. 7
5
s
The Laplace Transform - sites.google.com/site/ncpdhbkhn 76
Analysis Techniques (17)
Solve for v(t) ?
5 2
12 5 10( 0.5)( ) 1 ( 1)2
ab
s
s
ssV s
s ss
s
+ +
= × =
++ +
+
–
5
s
15
s
8
s
2
1
s
+
–
( )V s
a
Method 3
5 2 2
10( 0.5) 2 10( ) 1( 1) ( 1)2s
s sV s
s s
s
+
→ = × =
+ ++
5
s
8
s
2
1
s
+
–
5( )
s
V s
a
b
Ex. 7
The Laplace Transform - sites.google.com/site/ncpdhbkhn 77
Analysis Techniques (18)
Solve for v(t) ?
15 2
15
15( ) 1 ( 1)2
C
s
sI s
ss
s
= =
++ +
+
–
5
s
15
s
8
s
2
1
s
+
–
( )V s
a
Method 3
15 2
15( ) 2 ( 1)sV s s→ = + +
–
15
s
8
s
2
1
s
+
–
15( )
s
V s
a
Ex. 7
The Laplace Transform - sites.google.com/site/ncpdhbkhn 78
Analysis Techniques (19)
Solve for v(t) ?
+
–
5
s
15
s
8
s
2
1
s
+
–
( )V s
a
Method 3
5 2
10( ) ( 1)s
sV s
s
→ =
+
15 2
15( ) 2 ( 1)sV s s→ = +
5
s
8
s
2
1
s
+
–
5( )
s
V s
a
b
+
–
15
s
8
s
2
1
s
+
–
15( )
s
V s
a 15 15
2 2
2
( ) ( ) ( )
10 30
( 1) ( 1)
10( 3)
( 1)
s s
V s V s V s
s
s s
s
s
→ = +
= +
+ +
+
=
+
Ex. 7
The Laplace Transform - sites.google.com/site/ncpdhbkhn 79
Analysis Techniques (20)
Solve for v(t) ?
+
–
5
s
15
s
8
s
2
1
s
+
–
( )V s
a
Method 4
+
–( )E s
( )Z s
2
1
s
+
–
( )V s
( )Z s s=
2
5 15 3( ) 5 sE s s
s s s
+
= + =
2
35 3( ) 51 ( 1)2
s
ssI s
ss
s
+
+
= =
++ +
2 2
3 3( ) 2 5 10( 1) ( 1)
s sV s
s s
+ +
= × =
+ +
Ex. 7
The Laplace Transform - sites.google.com/site/ncpdhbkhn 80
Analysis Techniques (21)
Solve for v(t) ?
+
–
5
s
15
s
8
s
2
1
s
+
–
( )V s
a
Method 5
+
–( )eqE s
( )eqZ s
2
+
–
( )V s
8
s
1
s
( )
eqZ s
a
+
–
5
s
15
s
8
s
1
s
+
–
( )
eqE s
a
1( )eqZ s s
s
= +
5 15( )eqE s s
s s
− =
3( ) 5eq
sE s
s
+
→ =
2
( )( ) 2
2 ( )
310 ( 1)
eq
eq
E s
V s
Z s
s
s
→ =
+
+
=
+
Ex. 7
The Laplace Transform - sites.google.com/site/ncpdhbkhn 81
Analysis Techniques (22)Ex. 8
Solve for i(t) ?
2 ( ) Au t
xv
+
–
0.5 xv
2Ω
2Ω 2Ω
1H
1 F
6
i
(0) 0; (0) 0C Lv i= =
2
s
( )xV s
+
–
( ) 0.5 ( )c xI s V s=
2
2 2
s
6
s
( )I s( )AI s
[ ]6( ) 2 ( ) ( ) ( 2) ( ) 0x A c AV s I s I s s I s
s
− + + − + + =
[ ]6( ) 2 ( ) 0.5 ( ) ( 2) ( ) 0x A x AV s I s V s s I s
s
→ − + + − + + =
2 4( ) 2 ( ) 2 ( )
x A AV s I s I s
s s
= − = −
4 6 42 ( ) 2 ( ) 0.5 2 ( )
( 2) ( ) 0
A A A
A
I s I s I s
s s s
s I s
→ − − + + − − +
+ + =
8 12( ) ( )( 2)( 6)A
sI s I s
s s s
+
→ = =
+ +
The Laplace Transform - sites.google.com/site/ncpdhbkhn 82
Analysis Techniques (23)Ex. 8
Solve for i(t) ?
2 ( ) Au t
xv
+
–
0.5 xv
2Ω
2Ω 2Ω
1H
1 F
6
i
2
s
( )xV s
+
–
( ) 0.5 ( )c xI s V s=
2
2 2
s
6
s
( )I s
1 2 38 12( ) ( 2)( 6) 2 6
s K K KI s
s s s s s s
+
= = + +
+ + + +
1
0
8 12 1( 2)( 6)
s
sK
s s
=
+
= =
+ +
2
2
8 12 0.5( 6)
s
sK
s s
=−
+
= =
+
3
6
8 12 1.5( 2)
s
sK
s s
=−
+
= = −
+
2 6( ) 1 0.5 1.5 At ti t e e− −→ = + −
The Laplace Transform - sites.google.com/site/ncpdhbkhn 83
Analysis Techniques (24)
22
2
152 16.59( )
2 4 ( 2)( 9)
ssI s
s s s
+ ++
= =
+ + +
8(0) 1A
8
i = =
Ex. 9
Find the current i(t)?
*
1 2 2
2 3 3
K K K
s s j s j= + ++ − +
2
1 2
2
16.5 1.58
9
s
sK
s
=−
+
= =
+
2
2
3
16.5 0.35( 2)( 3)
s j
sK
s s j
=
+
= =
+ +
o146.3−
2 o( ) 1.58 0.70cos(3 146.3 ) Ati t e t−→ = + −
+
– 4
2s
2
( )I s
+
–
2
15
9s +
+
–
4Ω
2 H8V ( )i t
0t =
8Ω +
–
5sin 3 Vt
The Laplace Transform - sites.google.com/site/ncpdhbkhn 84
Analysis Techniques (25)Ex. 9
Find the current i(t)?
2 o( ) 1.58 0.70cos(3 146.3 ) Ati t e t−= + −
+
–
4Ω
2 H8V ( )i t
0t =
8Ω +
–
5sin 3 Vt
-2000 -1000 0 1000 2000 3000 4000 5000-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
The Laplace Transform - sites.google.com/site/ncpdhbkhn 85
Analysis Techniques (26)Ex. 10
Find the current i(t)?
+
–
4Ω
2 H
8V
( )i t
0t =
8Ω +
–
20sin 3 Vt
+
–
6j
0I
8Ω
20 V
0
20 2
8 6j= =+I
o36,9 A−
o
0( ) 2sin(3 36,9 ) Ai t t→ = −
o(0) 2sin( 36,9 ) 1.20Ai→ = − = −
+
– 4
2s
2.40−
( )I s
+
–
8
s
82.40 1.2 4( ) A
2 4 ( 2)
ssI s
s s s
− +
− +
= =
+ +
1 2
2
K K
s s
= +
+
1
0
1.2 4 2;
2 s
sK
s
=
− +
= =
+ 2 2
1.2 4 3.2
s
sK
s
=−
− +
= = −
2( ) 2 3.2 Ati t e−→ = −
The Laplace Transform - sites.google.com/site/ncpdhbkhn 86
Analysis Techniques (27)Ex. 10
Find the current i(t)?
+
–
4Ω
2 H
8V
( )i t
0t =
8Ω +
–
20sin 3 Vt
2( ) 2 3.2 Ati t e−= −
t
(A)i
2
1.2−
Analysis Techniques (28)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 87
I(s), V(s),
Inverse Transform
i(t), v(t), Circuit
Circuit
in s-domain
Circuit Element Models DC circuit analysis techniques
(KVL, KCL, nodal analysis,
mesh analysis, source
transformation, superposition,
Thevenin/Norton equivalent, )
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 88
Convolution Integral (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 89
t0
t0
λ0 λ0
2 ( )f t
1( )f t
1( )f λ
1( )f t λ− 2 ( )f t λ−
2( )f λ
1 2 1 2 1 20 0
( ) ( ) * ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
Convolution Integral (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 90
Ex. 1
t0
1
1
1( )f t
2 3 4
t0
2
2( )f t
1 2 3 4
1 2 1 2 1 20 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
λ0
2
2( )f λ
1 2 3 4
1
1( )f t λ−
λ0
2
2 ( )f λ
1 2 3 4
1
1( )f t λ−
1 20 1: 1; 0t f f< < = =
1 2( )* ( ) 0f t f t =
Find the convolution of the two signals?
Convolution Integral (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 91
Ex. 1
t0
1
1
1( )f t
2 3 4
t0
2
2( )f t
1 2 3 4
λ0
2
2 ( )f λ
1 2 3 4
1
1( )f t λ−
1 21 2 : 1; 2t f f< < = =
1 2 1 2 11 1
( ) * ( ) ( ) ( ) 1 2 2 2( 1)t t tf t f t f t f d d tλλ λ λ λ λ == − = × = = −∫ ∫
1 2 1 2 1 20 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
1 20 1: ( )* ( ) 0t f t f t< < =
Find the convolution of the two signals?
Convolution Integral (4)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 92
Ex. 1
t0
1
1
1( )f t
2 3 4
t0
2
2( )f t
1 2 3 4
λ0
2
2 ( )f λ
1 2 3 4
1
1( )f t λ−
1 22 3: 1; 2t f f< < = =
1 2 1 2 11 1
( ) * ( ) ( ) ( ) 1 2 2 2t t t
tt t
f t f t f t f d d λλ λ λ λ λ = −
− −
= − = × = =∫ ∫
1 2 1 2 1 20 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
1 20 1: ( )* ( ) 0t f t f t< < =
1 21 2 : ( ) * ( ) 2( 1)t f t f t t< < = −
Find the convolution of the two signals?
Convolution Integral (5)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 93
Ex. 1
t0
1
1
1( )f t
2 3 4
t0
2
2( )f t
1 2 3 4
λ0
2
2 ( )f λ
1 2 3 4
1
1( )f t λ−
1 23 4 : 1; 2t f f< < = =
3 3 3
1 2 1 2 11 1
( ) * ( ) ( ) ( ) 1 2 2 8 2
tt t
f t f t f t f d d tλλ λ λ λ λ = −
− −
= − = × = = −∫ ∫
1 2 1 2 1 20 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
1 20 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< < =
Find the convolution of the two signals?
Convolution Integral (6)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 94
Ex. 1
t0
1
1
1( )f t
2 3 4
t0
2
2( )f t
1 2 3 4
λ0
2
2 ( )f λ
1 2 3 4
1
1( )f t λ−
1 24 : 1; 0t f f> = =
1 2( )* ( ) 0f t f t =
1 2 1 2 1 20 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
1 20 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< < =
1 23 4 : ( )* ( ) 8 2t f t f t t< < = −
Find the convolution of the two signals?
Convolution Integral (7)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 95
Ex. 1
t0
1
1
1( )f t
2 3 4
t0
2
2( )f t
1 2 3 4
1 2 1 2 1 20 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
1 20 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< < =
1 24 : ( )* ( ) 0t f t f t> =
1 23 4 : ( )* ( ) 8 2t f t f t t< < = −
t0
2
1 2( )* ( )f t f t
1 2 3 4
Find the convolution of the two signals?
Convolution Integral (8)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 96
Ex. 2
t0
1
1
( )w t
2 3 4
1 2 1 2 1 20 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
t0
2 2 te−
1 2 3 4
( )
00
0 2 : ( ) 2 1 2 2(1 )t tt t tt f t e d e eλλ λ
λ
λ =− − − −
=
< < = × = = −∫
2 2( ) 2
00
2 : ( ) 2 1 2 2( 1)t t tt f t e d e e eλλ λ
λ
λ =− − − −
=
> = × = = −∫
t0
2
2e λ−
1 2 3 4
( )w t
t0
2
1 2 3 4
Method 1
Find the convolution of the two signals?
Convolution Integral (9)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 97
Ex. 2
Find the convolution of the two signals?
t0
1
1
( )w t
2 3 4
1 2 1 2 1 20 0
( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫
t0
2 2 te−
1 2 3 4
00
0 2 : ( ) 1 2 2 2(1 )t t tt f t e d e eλλ λ
λ
λ =− − −
=
< < = × = − = −∫
2
22
2 : ( ) 1 2 2 2( 1)t t t
tt
t f t e d e e eλλ λ
λ
λ =− − −
= −
−
> = × = − = −∫
t0
2 2 te−
1 2 3 4
( )w λ−
t0
2
1 2 3 4
Method 2
Convolution Integral (10)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 98
Property f(t) F(s)
1. Magnitude scaling
2. Addition/subtraction
3. Time scaling
4. Time shifting
5. Frequency shifting
6. Differentiation
7. Multiplication by t
8. Division by t
9. Integration
10. Convolution
( )Af t ( )AF s
1 2( ) ( )f t f t± 1 2( ) ( )F s F s±
( )f at 1 sF
a a
( ) ( ), 0f t a u t a a− − ≥ ( )ase F s−
( )ate f t− ( )F s a+
( ) ( ), 0f t u t a a− ≥ [ ( )]ase L f t a− +
( ) /n nd f t dt 1 2 1 1( ) (0) (0) ... (0)n n n o ns F s s f s f s f− − −− − −
( )nt f t ( 1) ( ) /n n nd F s ds−
( ) /f t t ( )
s
F dλ λ∞∫
0
( )t f dλ λ∫ ( ) /F s s
1 2 1 20
( ) * ( ) ( ) ( )tf t f t f f t dλ λ λ= −∫ 1 2( ) ( )F s F s
Convolution Integral (11)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 99
Ex. 3
Find vo(t)? +– +
–
( )iv t 1Ω
0.2 F
( )ov t
t0
iv 5 te−
1 2 3 41
( ) 5 5 50.2( ) ( ) 1( ) 1 5 11
0.2
C
o i
C
Z s sV s V s
R Z s s s s
s
= = × = ×
+ + + ++
Method 1: 5( ) ( ) 6.25( ) Vt t
o o
V s v t e e− −→ = −
Method 2: ( ) ( ) ( ) ( ) ( ) * ( )
o i o iV s H s V s v t h t v t= → =
55( ) ( ) 5
5
tH s h t e
s
−
= → =
+
5( ) 5 4 5 4
0
0 0 0
5
( ) ( ) ( ) 5 5 25 6.25
6.25( ) V
t t t
tt t t
o i
t t
v t h t v d e e d e d e
e e
λλ λ λ λ
λ
λ λ λ λ λ =− − − − + − +
=
− −
= − = = =
= −
∫ ∫ ∫
The Laplace Transform
1. Definition
2. Two Important Singularity Functions
3. Transform Pairs
4. Properties of the Transform
5. Inverse Transform
6. Initial-Value & Final-Value Theorems
7. Laplace Circuit Solutions
8. Circuit Element Models
9. Analysis Techniques
10. Convolution Integral
11. Transfer Function
The Laplace Transform - sites.google.com/site/ncpdhbkhn 100
Transfer Function (1)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 101
( )inI s
+
–
( )inV s
( )outI s
+
–
( )outV s( )H s
( )( ) ( )
Out sH s
In s
=
If ( ) ( ) ( ) 1 ( ) ( )in t t In s H s Out sδ= → = → =
Transfer Function (2)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 102
Linear
bandpass
filter
+
–
+
–
( )iv t ( )ov t
Ex. 1
Find the transfer function h(t) of the filter?
( ) 10 ( )iv t u t=
10( ) ( ) ( ) ( )o iV s H s V s H s
s
= =
1( ) ( )
10 o
H s sV s→ =
1 ( )( )
10
odv th t
dt
→ =
Transfer Function (3)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 103
Ex. 2
Find the transfer function H(s)? +–
+
–
( )iv t 1Ω
0.2 F
( )ov t
1
( ) 50.2( ) ( ) ( ) ( ) ( ) ( )1( ) 51
0.2
C
o i i i i
C
Z s sV s V s V s V s H s V s
R Z s s
s
= = = =
+ ++
( ) 5( ) ( ) 5
o
i
V sH s
V s s
→ = =
+
Transfer Function (4)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 104
A circuit is stable if : lim ( ) finite
x
h t
→∞
=
1 2
( )( ) ( )( )...( )n
N sH s
s p s p s p
=
+ + +
1 2
1 2( ) ( ... ) ( )np tp t p t nh t k e k e k e u t−− −→ = + + +
σ
jω
A circuit is stable when all the poles of its transfer function H(s)
lie in the left half of the s-plane
Transfer Function (5)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 105
Ex. 3
2
1,2
(4 ) (4 ) 4
2
k k
p
− − ± − −
=
An active filter has the transfer function
2( ) (4 ) 1
kH s
s k s
=
+ − +
For what values of k is the filter stable?
σ
jω
A circuit is stable when all the poles of its transfer
function H(s) lie in the left half of the s-plane
4 0k→ − >
4k→ <
Transfer Function (6)
The Laplace Transform - sites.google.com/site/ncpdhbkhn 106
Ex. 4
+–
+
–
( )iv t 1Ω
C
( )ov t
1
( ) 1( ) ( ) ( ) ( )1( ) 11
C
o i i i
C
Z s sCV s V s V s V s
R Z s Cs
sC
= = =
+ ++
( ) 1 5( ) ( ) 1 5 5
o
i
V sH s
V s Cs Cs
→ = = =
+ +
Given the transfer function 5( )
5
H s
s
=
+
Find C?
+–
+
–
( )iV s 1 1
sC
( )oV s
5 1C→ =
0.2 FC→ =
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