Transmission Lines Short notes | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

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Tr ansmission Lines Notes
Introduction
Tr ansmission lines are structures designed to tr ansmit electromagnetic signals
with minimal loss and distortion. They are critical in applications such as telecom-
munications, RF systems, and power distribution, guiding waves between two
points.
K ey Concepts
• Tr ansmission Line : A pair of conductors (e.g., coaxial cable, microstrip)
that guide electro magnetic waves.
• Types : Coaxial cables, twisted pairs, waveguides, and striplines.
• Applications : Antennas, RF circuits, high-speed digital systems, and power
delivery .
Tr ansmission Line Par ameters
• Resistance (R ) : Resistance per unit length (ohms/m), due to conductor
losses.
• Inductance (L ) : Inductance per unit length (henries/m), due to magnetic
fields.
• Capacitance (C ) : Capacitance per unit length (far ads/m), due to electric
fields between conductors.
• Conductance (G ) : Conductance per unit length (siemens/m), due to dielec-
tric losses .
T elegr apher ’ s Equations
The voltageV(z,t) and currentI(z,t) along a tr ansmission line are governed b y:
?V(z,t)
?z
=-RI(z,t)-L
?I(z,t)
?t
?I(z,t)
?z
=-GV(z,t)-C
?V(z,t)
?t
In the frequency domain (phasor form):
dV(z)
dz
=-(R+j?L)I(z)
dI(z)
dz
=-(G+j?C)V(z)
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Page 2


Tr ansmission Lines Notes
Introduction
Tr ansmission lines are structures designed to tr ansmit electromagnetic signals
with minimal loss and distortion. They are critical in applications such as telecom-
munications, RF systems, and power distribution, guiding waves between two
points.
K ey Concepts
• Tr ansmission Line : A pair of conductors (e.g., coaxial cable, microstrip)
that guide electro magnetic waves.
• Types : Coaxial cables, twisted pairs, waveguides, and striplines.
• Applications : Antennas, RF circuits, high-speed digital systems, and power
delivery .
Tr ansmission Line Par ameters
• Resistance (R ) : Resistance per unit length (ohms/m), due to conductor
losses.
• Inductance (L ) : Inductance per unit length (henries/m), due to magnetic
fields.
• Capacitance (C ) : Capacitance per unit length (far ads/m), due to electric
fields between conductors.
• Conductance (G ) : Conductance per unit length (siemens/m), due to dielec-
tric losses .
T elegr apher ’ s Equations
The voltageV(z,t) and currentI(z,t) along a tr ansmission line are governed b y:
?V(z,t)
?z
=-RI(z,t)-L
?I(z,t)
?t
?I(z,t)
?z
=-GV(z,t)-C
?V(z,t)
?t
In the frequency domain (phasor form):
dV(z)
dz
=-(R+j?L)I(z)
dI(z)
dz
=-(G+j?C)V(z)
1
Char acteristic Impedance and Propagation Constant
• Char acteristic Impedance (Z
0
) : Ratio of voltage to current for a tr aveling
wave:
Z
0
=
v
R+j?L
G+j?C
F or a lossles s line (R = G = 0 ):
Z
0
=
v
L
C
• Propagation Constant (? ) : Describes wave propagation:
? =
v
(R+j?L)(G+j?C) = a+jß
wherea is the attenuation constant (nepers/m) andß =
?
v
is the phase con-
stant (r adians/m).
• Phase V elocity : Speed of wave propagation:
v =
?
ß
=
1
v
LC
(lossless case)
W a ve Propagation
• V oltage a nd Current W aves : F or a line alongz -axis:
V(z) = V
+
e
-?z
+V
-
e
?z
I(z) =
V
+
Z
0
e
-?z
-
V
-
Z
0
e
?z
whereV
+
,V
-
are forward and reflected wave amplitudes.
• Reflection Coefficient ( G ) : At load impedanceZ
L
:
G =
Z
L
-Z
0
Z
L
+Z
0
• V oltage S tanding W ave Ratio (VS WR) :
VS WR =
1+|G|
1-|G|
Smith Chart
• A gr aphical tool to analyze tr ansmission lines, plotting normalized impedance
z =
Z
Z
0
or reflection coefficient G .
• Used for impedance matching, calculatingG , VS WR, and tr ansmission line
lengths.
2
Page 3


Tr ansmission Lines Notes
Introduction
Tr ansmission lines are structures designed to tr ansmit electromagnetic signals
with minimal loss and distortion. They are critical in applications such as telecom-
munications, RF systems, and power distribution, guiding waves between two
points.
K ey Concepts
• Tr ansmission Line : A pair of conductors (e.g., coaxial cable, microstrip)
that guide electro magnetic waves.
• Types : Coaxial cables, twisted pairs, waveguides, and striplines.
• Applications : Antennas, RF circuits, high-speed digital systems, and power
delivery .
Tr ansmission Line Par ameters
• Resistance (R ) : Resistance per unit length (ohms/m), due to conductor
losses.
• Inductance (L ) : Inductance per unit length (henries/m), due to magnetic
fields.
• Capacitance (C ) : Capacitance per unit length (far ads/m), due to electric
fields between conductors.
• Conductance (G ) : Conductance per unit length (siemens/m), due to dielec-
tric losses .
T elegr apher ’ s Equations
The voltageV(z,t) and currentI(z,t) along a tr ansmission line are governed b y:
?V(z,t)
?z
=-RI(z,t)-L
?I(z,t)
?t
?I(z,t)
?z
=-GV(z,t)-C
?V(z,t)
?t
In the frequency domain (phasor form):
dV(z)
dz
=-(R+j?L)I(z)
dI(z)
dz
=-(G+j?C)V(z)
1
Char acteristic Impedance and Propagation Constant
• Char acteristic Impedance (Z
0
) : Ratio of voltage to current for a tr aveling
wave:
Z
0
=
v
R+j?L
G+j?C
F or a lossles s line (R = G = 0 ):
Z
0
=
v
L
C
• Propagation Constant (? ) : Describes wave propagation:
? =
v
(R+j?L)(G+j?C) = a+jß
wherea is the attenuation constant (nepers/m) andß =
?
v
is the phase con-
stant (r adians/m).
• Phase V elocity : Speed of wave propagation:
v =
?
ß
=
1
v
LC
(lossless case)
W a ve Propagation
• V oltage a nd Current W aves : F or a line alongz -axis:
V(z) = V
+
e
-?z
+V
-
e
?z
I(z) =
V
+
Z
0
e
-?z
-
V
-
Z
0
e
?z
whereV
+
,V
-
are forward and reflected wave amplitudes.
• Reflection Coefficient ( G ) : At load impedanceZ
L
:
G =
Z
L
-Z
0
Z
L
+Z
0
• V oltage S tanding W ave Ratio (VS WR) :
VS WR =
1+|G|
1-|G|
Smith Chart
• A gr aphical tool to analyze tr ansmission lines, plotting normalized impedance
z =
Z
Z
0
or reflection coefficient G .
• Used for impedance matching, calculatingG , VS WR, and tr ansmission line
lengths.
2
Pr a ctical Consider ations
• Impedance Matching : Maximize power tr ansfer b y matching Z
L
= Z
0
,
minimizing r eflections ( G = 0 ).
• Losses : Conductor losses (R ) and dielectric losses (G ) increase attenuation.
• Frequency Dependence : At high frequencies, skin effect increases R , and
dispersion a ffects v .
• Applications : Design of RF circuits, antennas, and high-speed PCB tr aces
requires c areful line par ameter selection.
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