Magnetically Coupled Circuits XIII.Frequency Response3. XIV.The Laplace Transform.[r]
(1)Electric Circuit Theory
(2)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
(3)F(s) = 0
(algebraic) I(s), V(s), …
Laplace Transform Inverse Transform
The Laplace Transform
f(t) = 0
(4)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
(5)Definition
t
( ) f t
0
[ ] 0
( ) ( ) ( ) st
F s = L f t = ∫∞ f t e dt−
s = +σ jω
0 ( )
t
f t e σ dt
∞ −
< ∞ ∫
[ ]
1
1 1
( ) ( ) ( )
2
j
st j
f t L F s F s e ds
j σ σ π
+ ∞ −
− ∞
(6)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
(7)Two Important Singularity Functions (1)
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
<
− =
>
(8)Two Important Singularity Functions (2)
t
( ) u t
0 1
Ex 1
Determine the Laplace transform for the waveform?
0
( ) ( ) st
F s = ∫∞u t e dt−
0 1
st
e dt
∞ −
= ∫
0
1 st e s
∞ −
= −
1 s
(9)Two Important Singularity Functions (3) Ex 2
Determine the Laplace transform for the waveform?
0
( ) ( ) st
F 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
(10)Two Important Singularity Functions (4) Ex 3
Determine the Laplace transform for the waveform?
0
( ) [ ( ) ( )] st
F s = ∫∞ u t −u t − a e dt−
0
1 ( ) st
u t e dt s
∞ −
= ∫
0 ( )
st
st e
u t a e dt
s − ∞ − − = ∫ 1 1 ( ) as as e e F s
s s s
− − −
→ = − = t
( )
u t a
(11)Two Important Singularity Functions (5)
t ( )t
δ
0
t
(t a)
δ −
0 a
( ) 0 0
( ) 1 0
t t t dt ε ε δ δ ε − = ≠ = > ∫
( ) 0
( ) 1 0
a a
t a t a
t a dt
ε ε δ δ ε + − − = ≠
− = >
∫
2 ( )
( ) ( )
t f a t a t
f t δ t − a dt = < <
(12)Two Important Singularity Functions (6) Ex 4
Determine the Laplace transform of an impulse function?
0
( ) ( ) st
F s = ∫∞δ t − a e dt−
2
1 2
1 2
( ) ( ) ( )
0 ,
t t
f a t a t
f t t a dt
a t a t
δ − = < <
< >
∫
( ) as
F s e−
(13)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
(14)Transform Pairs (1) Ex 1
Find the Laplace transform of f(t) = t?
0
( ) st
F s = ∫∞te dt−
1
Let u t & dv e stdt du dt & v e stdt e st s
− − −
= = → = = ∫ = −
2
0 0
1
( ) 0
st st
st
t e e
F s e dt
s s s s
∞
∞ − −
∞ −
(15)Transform Pairs (2) Ex 2
Find the Laplace transform of f(t) =cosωt?
0
( ) cos st
F s = ∫∞ ωte dt−
0 2
j t j t
st
e e
e dt
ω − ω
∞ + −
= ∫
( ) ( )
0 2
s j t s j t
e e
dt
ω ω
− − − +
∞ +
= ∫
1 1 1
2 s jω s jω
= +
− +
(16)f(t) F(s)
Transform Pairs (3)
( )t
δ
1
( )
u t
1
s
at
e−
1
s + a
t
2 1
s
at
te−
2
1 (s + a)
sin at
2 a s + a
cos at
(17)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
(18)Properties of the Transform (1)
Property f(t) F(s)
1 Magnitude scaling Addition/subtraction Time scaling
4 Time shifting Frequency shifting Differentiation 7 Multiplication by t 8 Division by t Integration 10 Convolution
( )
Af t AF s( )
1( ) 2( )
f t ± f t F s1( ) ±F s2( ) ( )
f at 1 F s
a a
( ) ( ), 0
f t −a u t −a a ≥ e−asF s( ) ( )
at
e− f t F s( +a)
( ) ( ), 0
f t u t −a a ≥ e−asL f t[ ( +a)] ( ) /
n n
d f t dt s F sn ( )−sn−1f (0)−sn−2 f1(0) −s fo n−1(0) ( )
n
t f t ( 1)n n ( ) / n
d F s ds
− ( ) /
f t t ( )
s F λ λd
∞ ∫
0 ( )
t
f λ λd
∫ F s( ) /s
1 2
( ) * ( ) t ( ) ( )
f t f t = ∫ f λ f t −λ λd
1( ) 2( )
(19)Properties of the Transform (2) Ex 1
Find the Laplace transform of 10
( ) 5 t cos 20 ?
f t = +e− − t
1( ) 2( ) 1( ) 2( )
f t ± f t → F s ± F s
10
( ) [5] [ t] [cos 20 ]
F 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 t L e s − = +
2 2
[cos 20 ]
(20)Properties of the Transform (3) Ex 2
Find the Laplace transform of the waveform?
t
0 5
1 2 3
t
0 5
1 2 3
t 0
5
1 2 3
( ) 5 ( 1) 5 ( 2)
f t = u t − − u t −
2
2
5
( ) 5 5 ( )
s s
s s
e e
F s e e
s s s
− −
− −