Transmission Line Equations & Their Solutions in Phasor Form 5.. Transmission Lines of Finite Length 8.. Transmission Line Equations & Their Solutions in Phasor Form 5.. Transmission Lin
Trang 1Engineering Electromagnetics
Transmission Lines
Nguy ễ n Công Ph ươ ng
Trang 2I Introduction
II Vector Analysis
III Coulomb’s Law & Electric Field Intensity
IV Electric Flux Density, Gauss’ Law & Divergence
V Energy & Potential
VI Current & Conductors
VII Dielectrics & Capacitance
VIII Poisson’s & Laplace’s Equations
IX The Steady Magnetic Field
X Magnetic Forces & Inductance
XI Time – Varying Fields & Maxwell’s Equations
XII Transmission Lines
XIII The Uniform Plane Wave
XIV Plane Wave Reflection & Dispersion
XV Guided Waves & Radiation
Trang 3Transmission Lines
1 Introduction
2 The Transmission Line Equations
3 Lossless Propagation
4 Transmission Line Equations & Their Solutions in Phasor Form
5 Wave Reflection at Discontinuities
6 Voltage Standing Wave Ratio
7 Transmission Lines of Finite Length
8 Some Transmission Line Examples
9 Graphical Method
10 Transients Analysis
Trang 5The Transmission Line Equations (1)
Trang 6The Transmission Line Equations (2)
Trang 7The Transmission Line Equations (3)
Trang 8Transmission Lines
1 Introduction
2 The Transmission Line Equations
4 Transmission Line Equations & Their Solutions in Phasor Form
5 Wave Reflection at Discontinuities
6 Voltage Standing Wave Ratio
7 Transmission Lines of Finite Length
8 Some Transmission Line Examples
9 Graphical Method
10 Transients Analysis
Trang 145 Wave Reflection at Discontinuities
6 Voltage Standing Wave Ratio
7 Transmission Lines of Finite Length
8 Some Transmission Line Examples
9 Graphical Method
10 Transients Analysis
Trang 15Transmission Line Equations
& Their Solutions in Phasor Form (1)
Trang 16Transmission Line Equations
& Their Solutions in Phasor Form (2)
dI
dz
ω ω
Trang 17Transmission Line Equations
& Their Solutions in Phasor Form (3)
Ex.
A lossless transmission line is 100 cm long & operates at a frequency of 600 MHz The line
parameters are L = 0.25 μH/m & C = 100 pF/m Find the characteristic impedance, the phase
constant, & the phase velocity?
Trang 18Transmission Line Equations
& Their Solutions in Phasor Form (4)
1 1 8
Trang 19Transmission Line Equations
& Their Solutions in Phasor Form (5)
1 1 8
Trang 20Transmission Line Equations
& Their Solutions in Phasor Form (6)
0
1 Re 2
Trang 21Transmission Lines
1 Introduction
2 The Transmission Line Equations
3 Lossless Propagation
4 Transmission Line Equations & Their Solutions in Phasor Form
5 Wave Reflection at Discontinuities
6 Voltage Standing Wave Ratio
7 Transmission Lines of Finite Length
8 Some Transmission Line Examples
9 Graphical Method
10 Transients Analysis
Trang 22Wave Reflection at Discontinuities (1)
V V
V V
Trang 23Wave Reflection at Discontinuities (2)
* 2
0 0 0 2
z j z
P P
→ = ΓΓ = Γ
Trang 24Wave Reflection at Discontinuities (3)
Trang 25Wave Reflection at Discontinuities (4)
sT
V z = V e− β + Γ e β φ+ ( ) (2 1)
Trang 26Wave Reflection at Discontinuities (5)
sT
V z = V e− β + Γ e β φ+ ( ) 2
Trang 27Wave Reflection at Discontinuities (6)
−
2 2
φ π β
− −
4 2
φ π β
− −
6 2
φ π β
− −
2
φ π β
− −
3 2
φ π β
− −
5 2
φ π β
Trang 28Transmission Lines
1 Introduction
2 The Transmission Line Equations
3 Lossless Propagation
4 Transmission Line Equations & Their Solutions in Phasor Form
5 Wave Reflection at Discontinuities
6 Voltage Standing Wave Ratio
7 Transmission Lines of Finite Length
8 Some Transmission Line Examples
9 Graphical Method
10 Transients Analysis
Trang 29Transmission Lines of Finite Length (1)
LZ
0
z =
L
Vin
j z j z sT
( )
sT w
0
cos( ) sin( ) ( )
cos( ) sin( )
L w
Trang 30Transmission Lines of Finite Length (2)
in
Zin
0
cos( ) sin( ) ( )
cos( ) sin( )
L w
Trang 31Transmission Lines of Finite Length (3)
0 0
0
cos( ) sin( ) cos( ) sin( )
L in
λ
=
( 0,1, 2, ) 2
0
z =
L
Vin
Trang 32Transmission Lines of Finite Length (4)
03
in
Z Z
Trang 33Transmission Lines
1 Introduction
2 The Transmission Line Equations
3 Lossless Propagation
4 Transmission Line Equations & Their Solutions in Phasor Form
5 Wave Reflection at Discontinuities
6 Voltage Standing Wave Ratio
7 Transmission Lines of Finite Length
9 Graphical Method
10 Transients Analysis
Trang 34Some Transmission Line Examples (1)
Ex 1
0 0
300 300
0
300 300
L L
0.8 rad/m 2.5 10
V
–
60V 100MHz
V
–
60V100MHz
Trang 35Some Transmission Line Examples (2)
Ex 2
0 0
V
–
60V 100MHz
300 Ω +
−
0 0
V
–
60V100MHz
Trang 36Some Transmission Line Examples (3)
o
120 60 300
120 60 300 0.447 153.4
V
–
60V 100MHz
V
–
60V100MHz
Trang 37Some Transmission Line Examples (4)
Ex 4
o 0
V
–
60V 100MHz
V
–
60V100MHz
Trang 38Transmission Lines
1 Introduction
2 The Transmission Line Equations
3 Lossless Propagation
4 Transmission Line Equations & Their Solutions in Phasor Form
5 Wave Reflection at Discontinuities
6 Voltage Standing Wave Ratio
7 Transmission Lines of Finite Length
8 Some Transmission Line Examples
10 Transients Analysis
Trang 39Graphical Method (1)
0
0
Z Z
Z Z
normalized (
Trang 40Graphical Method (2)
22
2
} Re{
1
}
Re{
} {
Im }
= Γ +
→
L
L L
L
z
z z
z
} Re{
1
1 }
Trang 41Graphical Method (3)
2 2
2
} Re{
1
} Re{
} {
Im }
Re{
1
} Re{
= Γ +
L
z
z z
z
} { Im
1 }
Im{
1 }
Im{
1 }
2 2
− Γ
Trang 42Graphical Method (4)
2 2
2
} Re{
1
} Re{
} {
Im }
Re{
1
} Re{
L
z
z z
z
} Re{ zL
} Re{ Γ
Trang 43Graphical Method (5)
} Im{ zL
s =
} { Im
1 }
Im{
1 }
Im{
1 } Re{
2
2 2
− Γ
1 ,
Equation of a circle, centered at & a radius of
} Im{ Γ
} Re{ Γ
Trang 44Graphical Method (6)
1 Find the normalized load impedance
2 Find the circle corresponding to Re{z L }
3 Find the arc corresponding to Im{z L }
4 The intersection of the circle & the arc is Γ.
} Im{
}
Re{
0
L L
L
Z Z
Trang 454 Γ is the intersection of the
circle & the arc.
Γ = 0.52 + j0.64
Trang 47Transmission Lines
1 Introduction
2 The Transmission Line Equations
3 Lossless Propagation
4 Transmission Line Equations & Their Solutions in Phasor Form
5 Wave Reflection at Discontinuities
6 Voltage Standing Wave Ratio
7 Transmission Lines of Finite Length
8 Some Transmission Line Examples
9 Graphical Method
10 Transients Analysis
Trang 48Transient Analysis (1)
0
V
LV
L
g g
Trang 49g
V Z V
+ =
+
LR
Trang 505 4
L v
11 4
L v
13 4
L v
Trang 525 4
L v
11 4
L v
13 4
L v
Trang 53Transient Analysis (6)
Ex 1
LR
R
Rg = Z0 = 50 Ω, RL = 25 Ω, V0 = 10 V The
switch is closed at time t = 0 Determine
the voltage at the load resistor and the
current in the battery as functions of time.
25 50
0.33
25 50
L L
50 50
0
50 50
g g
V
I
Z
+ + = = =
1 1
( 1.67)
0.033A 50
V I
Z
−
− = − = − − =