Magnetically Coupled Circuits XIII.Frequency Response3. XIV.The Laplace Transform.[r]
Trang 1Electric Circuit Theory
The Laplace Transform
Trang 2I 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
Trang 3F(s) = 0
Laplace Transform Inverse Transform
The Laplace Transform
f(t) = 0
(integrodifferential) i(t), v(t), … Circuit
Trang 4The 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
Trang 5t
( )
f t
0
F s = L f t = ∫∞ f t e dt−
s = + σ j ω
0∞ f t e ( ) −σtdt < ∞
∫
1
2
j
st j
j
σ σ π
+ ∞
−
− ∞
Trang 6The 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
Trang 7Two Important Singularity Functions (1)
t
( )
u t
0 1
t
u t − a
0
1
a
( )
t
u t
t
<
=
>
0
1
t a
u t a
t a
<
>
Trang 8Two Important Singularity Functions (2)
t
( )
u t
0 1
Ex 1
Determine the Laplace transform for the waveform?
0
F s = ∫∞u t e dt−
0∞1 e dt−st
= ∫
0
1 st
e s
∞
−
= −
1
s
=
Trang 9Two Important Singularity Functions (3)
Ex 2
Determine the Laplace transform for the waveform?
0
F s = ∫∞u t − a e dt−
0a0 1 st
a
dt ∞ e dt−
1 st
a
e s
∞
−
= −
as
e s
−
=
t
u t − a
0 1
a
Trang 10Two Important Singularity Functions (4)
Ex 3
Determine the Laplace transform for the waveform?
0
F s = ∫∞ u t − u t − a e dt−
0
1 ( ) st
u t e dt
s
∞ −
=
∫
st
st e
u t a e dt
s
−
∫
( )
F s
u t a
1
−
0
a
t
( )
u t
0 1
t
0 1
a
Trang 11Two Important Singularity Functions (5)
t
( ) t
δ
0
t
( t a )
δ −
t dt
ε
ε
δ
−
∫
a
a
t a dt
ε
ε
δ
+
−
∫
( )
∫
Trang 12Two Important Singularity Functions (6)
Ex 4
Determine the Laplace transform of an impulse function?
0
F s = ∫∞δ t − a e dt−
2
1
( ) ( ) ( )
t
t
∫
( ) as
F s e−
Trang 13The 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
Trang 14Transform Pairs (1)
Ex 1
Find the Laplace transform of f(t) = t?
0
F s = ∫∞te dt−
1 Let u t & dv e st dt du dt & v e st dt e st
s
2 0
1
st
∞
∞
−
Trang 15Transform Pairs (2)
Ex 2
Find the Laplace transform of f(t) =cosωt?
0
F s = ∫∞ ω te dt−
j t j t
st
e dt
ω − ω
= ∫
s j t s j t
dt
= ∫
2 s j ω s j ω
Trang 16F(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
a
s + a
cos at
s
s + a
Trang 17The 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
Trang 18Properties of the Transform (1)
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
( )
1( ) 2( )
( )
F
( )
at
( ) ( ), 0
( ) /
d f t dt s F sn ( ) − sn−1f (0) − sn−2 f1(0) − s fo n−1(0)
( )
n
− ( ) /
s∞F λ λ d
∫
0t f ( ) λ λ d
0
( ) * ( ) t ( ) ( )
Trang 19Properties of the Transform (2)
Ex 1
( ) 5 t cos 20 ?
f t = + e− − t
1( ) 2( ) 1( ) 2( )
f t ± f t → F s ± F s
10
Af t → AF s
1 [1]
L
s
=
5 [5]
L
s
[ ]
10
t
L e
s
+
[cos 20 ]
Trang 20Properties of the Transform (3)
Ex 2
Find the Laplace transform of the waveform?
t
0
5
t
0
5
t
0
5
2
2
5