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Microwave Transmission Lines
Engr. Ghulam Shabbir
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Transmission Lines:
Fundamental Concepts
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Transmission Line Theory
Introduction:
In an electronic system, the delivery of powerrequires the connection of two wires between the
source and the load. At low frequencies, power isconsidered to be delivered to the load through the wire.
In the microwave frequency region, power isconsidered to be in electric and magnetic fields that are
guided from place to place by some physical structure.Any physical structure that will guide an electromagneticwave, place to place is called a Transmission Line.
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• A lumped circuit is one where all the terminalvoltages and currents are functions of timeonly. Lumped circuit elements includeresistors, capacitors, inductors, independentand dependent sources.
• An distributed circuit is one where the
terminal voltages and currents are functionsof position as well as time. Transmission linesare distributed circuit elements.
Circuit Theory:
Lumped vs. Distributed Systems
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Lumped vs. Distributed Systems
• A lumped system is one in which the dependent variables of
interest are a function of time alone. In general, this will
mean solving a set of ordinary differential equations (ODEs )
• A distributed system is one in which all dependent variables
are functions of time and one or more spatial variables. In
this case, we will be solving partial differentialequations (PDEs )
• For example, consider the following two systems:
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• The first system is a distributed system, consisting of an
infinitely thin string, supported at both ends; the dependentvariable, the vertical position of the string is indexed
continuously in both space and time.
• The second system, a series of ``beads'' connected by
massless string segments, constrained to move vertically,can be thought of as a lumped system, perhaps an
approximation to the continuous string.
• For electrical systems, consider the difference between a
lumped RLC network and a transmission line.
• The importance of lumped approximations to distributed
systems will become obvious later, especially
for waveguide -based physical modeling , because it enables
one to cut computational costs by solving ODEs at a few
points, rather than a full PDE (generally much more costly)
Lumped vs. Distributed Systems
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Lumped vs. Distributed Systems
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Circuit Theory: Capacitors Circuit
i(t) +
-
v(t)
The
rest
of
the
circuit
dt
t dvC t i
)()(
t
dx xiC
t v )(1
)(
t
t
dx xiC
t vt v
0
)(1)()( 0
)(2
1
)(
2
t Cvt wC Energy stored in the
capacitors
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Circuit Theory: Inductors Circuits
dt
t di Lt v
)()( i(t)
+
-
v(t)
The
rest
of
the
circuit
L
t
dx xv L
t i )(1
)(
t
t
dx xv L
t it i
0
)(1
)()( 0
)(2
1
)(2
t Lit w L Energy stored in the
inductor
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Propagation Velocity
• Physical example:
Wave propagates in z direction
• Circuit:L = [nH/cm]
C = [pF/cm]
t
I Ldz dz
z
V
• Total voltage change across Ldz
(use ):dt
dI LV
• Total current change across Cdz (use ):
dt dV C I
t V Cdz dz
z I
[1]
[2]
• Simplify [1] & [2] to get the
Telegraphist’s Equations
[3a]
t
I L z
V
t
V C
z
I
[3b]
I
V
Ldz
Cdz
dz
V+ dzdV
dz
I+ dzdI
dz
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Propagation Velocity (2)
• Phase velocity definition:
LC v
1 [7]
• Equation in terms of current:
2
2
22
2
2
2 1
t
I
t
I LC
z
I
[8]
•
Equate [4] & [5]: [6]2
2
22
2
2
2
1t V
t V LC
z V
• Differentiate [3b] by z: [5] z t
I L
z
V
2
2
2
• Differentiate [3a] by t : [4]2
22
t
V C
t z
I
• Equation [6] is a form of the wave equation.
• The solution to [6] contains forward and backward traveling
wave components, which travel with a phase velocity.
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Equivalent Circuit of a Real Transmission Line
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Characteristic Impedance (Lossless)
Ldz
Cdx
Z1 Z2 Z3
Ldz
Cdz
Ldz
Cdz
dz dz
V1
V3
V2 to
a
f ed
cb
dz
dz= segment length
C = capacitance per segment
L = inductance per segment
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Characteristic Impedance (Lossless)
•
The input impedance (Z1) is the impedance of the firstinductor (Ldz) in series with the parallel combination of the
impedance of the capacitor (Cdz) and Z2.
[.9] Cdz j Z Cdz j Z Ldz j Z
/1
/1
2
21
0/1/1/1 2221 Cdz j Z Cdz j Z Ldz jCdz j Z Z
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Characteristic Impedance (Lossless)
• Assuming a uniform line, the input impedance should be
the same when looking into node pairs a-d , b-e, c - f , and so
forth. So, Z2 = Z1 = Z0.
0/1/1/1 0000 Cdz j Z Cdz j Z lLdz jCdz j Z Z [10]
Cdz j
Ldz jdz LZ j Z
Cdz j
Z
Cdz
Ldz dz LZ j
Cdz j
Z Z
0
2
00
002
0 0
• Allow dz to become very small, causing the frequency
dependent term to drop out:
00
2
0 C
Ldz LZ j Z [11]
02
0 C
L Z [12]
• Solve for Z 0 called as Characteristic Impedance:
C L Z 0
[13]
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Visualizing Transmission Line Behavior
• Water flow
– Potential = Wave height [m]
– Flow = Flow rate [liter/sec]
I
I
V
+++++++
- - - - - - -
• Transmission Line
• Potential = Voltage [V]
• Flow = Current [A] = [C/sec]
• Just as the wave front of the water flows in the pipe,
the voltage propagates in the transmission line.
The same holds true for current.
•
Voltage and current propagate as waves in thetransmission line.
h flow
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Visualizing Transmission Line Behavior
•
Extending the analogy – The diameter of the pipe relates the flow rate and height of the
water. This is analogous to electrical impedance.
– Ohm’s law and the characteristic impedance define the relationship
between current and potential in the transmission line.
• Effects of impedance discontinuities
– What happens when the water encounters a ledge or a barrier?
– What happens to the current and voltage waves when the
impedance of the transmission line changes?
– The answer to this question is a key to understanding transmissionline behavior.
– It is useful to try to visualize current/voltage wave propagation on a
transmission line system in the same way that we can, for water
flow in a pipe.
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General Transmission Line Model (No Coupling)
•
Transmission line parameters are distributed (e.g.capacitance per unit length).
• A transmission line can be modeled using a network of
resistances, inductances, and capacitances, where the
distributed parameters are broken into small discrete
elements.
R L
G C
R L
G C
R L
G C
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General Transmission Line Model #2
Parameter Symbol Units
Conductor Resistance R W•cm-1
Self Inductance L nH•cm-1
Total Capacitance C pF•cm-1
Dielectric Conductance G W-1•cm -1
Parameters
Characteristic ImpedanceC jG
L j R Z
0 [14]
Propagation Constant jC jG L j R [15]
= attenuation constant = rate of exponential attenuation = phase constant = amount of phase shift per unit length
p
Phase Velocity [16]
In general, and are frequency dependent.
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Frequency DependenceFrom [14] and [15] note that:
• Z 0 and depend on the frequency content of the signal.• Frequency dependence causes attenuation and edge rate
degradation.
Attenuation
Edge rate degradation
Output signal from lossy transmission line
Signal at driven end of transmission line
Output signal from
lossless transmission line
• R and G are sometimes
negligible, particularly
at low frequencies• Simplifies to the
lossless case:
• no attenuation
&
• no dispersion
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Lossless Transmission Lines
Quasi-TEM Assumption
• The electric and magnetic fields are perpendicular to the
propagation velocity in the transverse planes.
x
z y
H E
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Lossless Line Parameters
• Lossless line characteristics are frequency independent.
• As noted before, Z 0 defines the relationship between
voltage and current for the traveling waves. The units are
ohms [].• defines the propagation velocity of the waves. The units
are cm/ns.
– Sometimes, we use the propagation delay, (units are
ns/cm).
C
L Z
0
LC v
1
Characteristic Impedance
Propagation Velocity
[17]
[18]
Lossless transmission lines are characterized by the
following two parameters:
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Lossless Line Equivalent Circuit
• The transmission line equivalent circuit shown on the left
is often represented by the coaxial cable symbol.
L
C
L
C
L
C
Z , v , length Z 0 , , length
L= Self Inductance per segment
C = Total Capacitance per segment
C
L Z
0
LC v
1
Characteristic Impedance
Propagation Velocity
Length of segment = dz
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Homogeneous Media
• A homogeneous dielectric medium is uniform in all
directions.
– All field lines are contained within the dielectric.
• For a transmission line in a homogeneous medium, the
propagation velocity depends only on material
properties:
r r r
nscmc LC
v
/3011 0 [19]
0 r Dielectric Permittivity
cm F x 140 10854.8 Permittivity of free space
cm H x
8
0 10257.1 Magnetic Permeability
0 Permeability of free space
er is the relative permittivity or dielectric constant.
Note: only r is
required to
calculate .
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Non-Homogeneous Media
• A non-homogenous medium contains multiple materials
with different dielectric constants.
• For a non-homogeneous medium, field lines cut across
the boundaries between dielectric materials.
• In this case the propagation velocity depends on the
dielectric constants and the proportions of the materials.Equation [19] does not hold:
11
LC
v
• In practice, an effective dielectric constant, er,eff is often
used, which represents an average dielectric constant.
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Phase Constant (Lossless Case)• Recall the basic voltage divider circuit:
R1
R2 V 1
+
V 2
-
I
• We want to find the ratio of the input voltage, V 1, to the
output voltage, V 2, to the output voltage, V 2..
• Now, we apply it to our transmission line equivalent
circuit.
0211
IR IRV 21
1
R R
V I
21
2122
R R
RV IRV
2
1
2
21
2
1 1 R
R
R
R R
V
V
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Phase Constant (Lossless Case)• The analogous transmission line circuit looks like this:
• The phase shift is the ratio of V 1 to V 2:
• Substitute the expressions for Z C , Z L, and Z 0 :
00
0
00
0
2
1 11111
Z Z Z
Z Z
Z Z Z
Z Z
Z
Z Z
Z Z Z
V
V
C
L
C
C L
C
L
C
C L
L Z R 1
0
002
Z Z Z Z Z Z R
C
C C
Cdz j Z C
1
Ldz j Z L
C L Z 0
LC dz j LC dz j Ldz j
Cdz jCdz j Ldz j
Z Z Z
V
V
C
L
222222
02
1 1111
1
LC dz j LC dz V
V
22
2
1
1
Ldz
CdzV1
+
V2
-
Z0
I
CharacteristicImpedance
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Phase Constant (Lossless Case) #3
• The amplitude of the phase constant is:
• The phase angle, denoted as tanl , is:
• Now, we make the assumption that dz is small enough that
the applied frequency, , is much smaller than the
resonant frequency, , of each subsection, so that:
LC dz LC dz V
V 22222
2
1
1
LC dz
LC dz l 22
1
tan
LC dz 1
122 LC dz
• The phase angle becomes: LC dz l tan
• Since , tanl is, very small. Therefore:
LC dz l l tan
122
LC dz
Ph C t t (L l C ) #4
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Phase Constant (Lossless Case) #4
• The phase shift per unit length is:
p
• l represents the amount by which the input voltage, V 1,
leads the output voltage, V 2.
• We can simplify the amplitude ratio by using the condition
of small l :
• So, there is no decrease in the amplitude of the voltage
along the line, for the lossless case. Only a shift in phase.• From our definition of phase velocity in equation [16] we
get
LC dz
l
1122222
2
1 LC dz LC dz V
V
C
L p
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Transmission-Lines Equations
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Transmission-Lines Equations
A typical engineering problem involves the transmission of a
signal from a generator to a load. A transmission line is thepart of the circuit that provides the direct link between
generator and load.
Transmission lines can be realized in a number of ways.
Common examples are the parallel-wire line and the coaxialcable.
For simplicity, we use in most diagrams the parallel-wire
line to represent circuit connections, but the theory applies
to all types of transmission lines.
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Examples of Transmission-Lines
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Microwave Transmission Lines and Solutions
Conventional two-conductor transmission lines are
commonly used for transmitting microwave energy. If a line is properly matched to its characteristic impedance
at each terminal, its efficiency can reach a maximum.
In ordinary circuit theory, it is assumed that all impedance
elements are lumped constants. This is not true for long transmission line over a wide range
of frequencies. Frequencies of operation are so high that
inductances of short lengths of conductors and capacitances
between short conductors and their surroundings cannot be
neglected.
These inductances and capacitances are distributed along
the length of a conductor, and their effects combine at each
point of the conductor.
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Microwave Transmission Lines and Solutions
Since the wavelength is short in comparison to the physical
length of the line, distributed parameters cannot berepresented accurately by means of lumped-parameter
equivalent circuit.
Thus microwave transmission lines can be analyzed in terms
of voltage, current, and impedance only by distributedcircuit theory.
If the spacing between the lines is smaller than the
wavelength of the transmitted signal, the transmission line
must be analyzed as a waveguide.
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Transmission-Lines Equations and Solutions
If you are only familiar with low frequency circuits, you are
used to treat all lines connecting the various circuitelements as perfect wires, with no voltage drop and no
impedance associated to them (lumped impedance
circuits).
This is a reasonable procedure as long as the length of thewires is much smaller than the wavelength of the signal. At
any given time, the measured voltage and current are the
same for each location on the same wire.
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Transmission-Lines Equations and Solutions
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Transmission-Lines Equations and Solutions
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Transmission-Lines Equations and Solutions
d l
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Transmission-Lines Equations and Solutions
i i i i d l i
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Transmission-Lines Equations and Solutions
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Why is the study of transmission lines (TL) and
waveguide important in understanding
nanotechnology?
One of the practical applications of nanotechnology is the
generation and transmission of light.
The short wavelength of light makes almost every small
conducting tree appear as a TL or waveguide that can divert
light from its intended path.
Laser diodes, for example, are dependent on the creation of waveguides in the nm range for operation.
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Transmission-Line Theory:
Two Basic Approaches to Analysis
Lumped – Element Modelo Uses discrete components and circuit theory
o Assumes dimensions smaller than the wavelength
Field Analysiso Uses distributed components and Maxwell’s Equations
o Considers dimensions vs. wavelength
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Transmission-Line Theory:
Two Basic Approaches to Analysis
Lumped circuits: resistors, capacitors, inductors
neglect time delays (phase)
account for propagation and
time delays (phase change)
Distributed circuit elements: transmission lines
We need transmission-line theory whenever the length of a
line is significant compared with a wavelength.
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Lumped – Element Circuit Model
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Lumped – Element Circuit Model
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Lumped – Element Circuit Model
z
,i z t
+ + + + + + +
- - - - - - - - - - ,v z t x x xB
R z L z
G z C z
z
v( z + z , t )
+
-
v( z , t )
+
-
i( z , t ) i( z + z , t )
z
z
T i i Li
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Transmission Line
2 conductors
4 per-unit-length parameters:
C = Shunt capacitance/unit length [F/m
]
L = Series inductance/unit length (both conductors) [H/m]
R = Series resistance/unit length (both conductors [/m]
G = Shunt conductance/unit length [ /m or S/m]
z
L l T i i Li
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Loss-less Transmission Line
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Loss-less Transmission Line
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Loss-less Transmission Line
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Loss-less Transmission Line
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Loss-less Transmission Line
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Loss-less Transmission Line
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Loss-less Transmission Line
Loss less Transmission Line
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Loss-less Transmission Line
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Lossy Transmission Line
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Lossy Transmission Line
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Lossy Transmission Line
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Lossy Transmission Line
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Lossy Transmission Line
Lossy Transmission Line
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Lossy Transmission Line
Lossy Transmission Line
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Lossy Transmission Line
Lossy Transmission Line
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Lossy Transmission Line
Lossy Transmission Line
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Lossy Transmission Line
Lossy Transmission Line
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Lossy Transmission Line
T i i Li E ti d S l ti
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A transmission line can be analyzed either by the solution of
Maxwell’s field equations or by the methods of distributed
circuit theory.
The solution of Maxwell’s equations involves three space
variables in addition to the time variable. The distributed –circuit method, however, involves only one
space variable in addition to the time variable.
Transmission-Line Equations and Solutions:Transmission-Line Equations
Based on uniformly distributed-circuit theory, the schematiccircuit of a conventional two-conductor transmission line
with constant parameters R, L, G and C is shown as under:
T i i Li E ti d S l ti
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Elementary section of a transmission line
The parameters are expressed in their respective names per
unit length, and the wave propagation is assumed in the
positive direction.
Transmission-Line Equations and Solutions:Transmission-Line Equations
Transmission-Line Equations and Solutions:
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By Kirchhoff’s voltage law, the summation of the voltage
drops around the central loop is given by
Rearranging this equation, dividing it by ∆z, and then
omitting the argument (z, t), which is understood, we obtain
Using Kirchhoff’s current law, the summation of the currents
at point B in above Figure can be expressed as
Transmission-Line Equations
Transmission-Line Equations and Solutions:
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By rearranging the preceding equation, dividing it by ∆z,omitting ((z, t), and assuming ∆z equal to zero, we have
Transmission Line Equations and Solutions:Transmission-Line Equations
Then by differentiating Eq. (3-1-2) with respect to z and Eq.
(3-1-4) with respect to t and combining the results, the final
transmission-line equation in voltage form is found to be
Transmission-Line Equations and Solutions:
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Also, by differentiating Eq.(3-1-2) with respect to t and Eq.(3-
1-4) with respect to z and combining the results, the final
transmission-line equation in current form is
Transmission Line Equations and Solutions:Transmission-Line Equations
All these transmission-line equations are applicable to the
general transient solution.
Transmission-Line Equations and Solutions:
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The voltage and current on the line are the functions of bothposition z and time t.
The instantaneous line voltage and current can be expressed as
Transmission Line Equations and Solutions:Transmission-Line Equations
where Re stands for “real part of ” .
The factors V(z) and I(z) are complex quantities of thesinusoidal functions of position z on the line and are known as
phasors.
Transmission-Line Equations and Solutions:
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Transmission Line Equations and Solutions:Transmission-Line Equations
The phasors give the magnitudes and phases of the sinusoidal function at each position of z, and they can be expressed as
Where V+ and I+ indicate complex amplitudes in the positive zdirection, V- and I- signify complex amplitudes in the negative z
direction, α is the attenuation constant in nepers per unit
length, and β is the phase constant in radians per unit length.
Transmission-Line Equations and Solutions:
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Transmission-Line Equations
If we substitute j ω for ∂/∂t in Eqs. (3-1-2), (3-1-4), (3-1-5), and(3-1-6) and divide each equation by e jωt, the transmission-line
equations in phasor form of the frequency domain become
In which the following substitutions have been made:
using the phasor representation
Transmission-Line Equations and Solutions:
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Transmission-Line Equations
For a lossless line, R = G = 0, and the transmission-line
equations are expressed as
Transmission-Line Equations and Solutions:
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Transmission-Line Equations
It is interested to note that Eqs. (3-1-14) and (3-1-15) for atransmission line are similar to Eqs (3-1-21) and (3-1-22) of
electric and magnetic waves, respectively.
The only difference is that the transmission-line equations are
one-dimensional.
Transmission-Line Equations and Solutions:
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Solutions of Transmission-Line Equations
The one possible solution for Eq. (3-1-14) is
The factors V+ and V- represent complex quantities.
The term involving e-jβz shows a wave travelling in the positive
z direction, and the term with the factor e jβz is a wave going in
the negative z direction.
The quantity β z is called the electrical length of the line and is
measured in radians.
Similarly, the one possible solution for Eq. (3-1-15) is
Transmission-Line Equations and Solutions:
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Solutions of Transmission-Line Equations
In Eq. (3-1-24) the characteristic impedance of the line is
defined as
The magnitude of both voltage and current waves on the line is
shown in Fig.
Fig. Magnitude of voltage and current travelling waves
At microwave frequencies it can be seen that
Transmission-Line Equations and Solutions:
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Solutions of Transmission-Line Equations
By using the binomial expansion, the propagation constant can
be expressed as
Therefore, the attenuation and phase constants are,
respectively, given by
Transmission-Line Equations and Solutions:
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Solutions of Transmission-Line Equations
Therefore, the attenuation and phase constants are,
respectively, given by
Similarly, the characteristic impedance is found to be
Transmission-Line Equations and Solutions:
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Solutions of Transmission-Line Equations
From eq. (3-1-29) the phase velocity is
The product of LC is independent of the size and separation of
the conductors and depends only on the permeability µ and
permittivity of ϵ of the insulating medium.
If a lossless transmission line used for microwave frequencies
has an air dielectric and contains no ferromagnetic materials,
free-space parameters can be assumed.
Thus the numerical value of 1/ for air-insulatedconductor is approximately equal to the velocity of light in
vacuum. That is,
Transmission-Line Equations and Solutions:
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Solutions of Transmission-Line Equations
When the dielectric of a lossy microwave transmission line is
not air, the phase velocity is smaller than velocity of light invacuum and is given by
In general, the relative phase velocity factor can be defined as
A low-loss transmission line filled only with dielectric medium,
such as a coaxial line with solid dielectric between conductors,
has a velocity factor on the order of about 0.65.
Example 3-1-1:Li Ch i i I d d P i C
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Line Characteristic Impedance and Propagation Constant
Reflection Coefficient and
T i i C ffi i
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Transmission CoefficientReflection Coefficient
In the analysis of the solutions of transmission-line equation,the traveling wave along the line contains two components:
o One traveling wave in the positive z direction
o other traveling the negative z direction.
+
V +( z )
-
I + ( z )
z
+
V -( z )
-
I - ( z )
z
0
( )
( )
V z Z
I z
0
0
( )
( )
z
z
V z V e
I z I e
00
0
V Z
I
( Z 0
is a number, not a
function of z .)
so
0
( )
( )
V z Z
I z
0
( )
( )
V z
Z I z
soNote: The reference directions for voltage and current are the same as for the forward wave.
General Case (Waves in Both Directions)
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00
0 0
j z j j z
z z
z j z V e e
V z V e V
V e e e
e
e
0
0 cos
c
, R
os
ej t
z
z
V e t
v z t V z
z
V z
e
e t
Note:
wave in + z
direction wave in - z direction
General Case
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A general superposition of forward and backward
traveling waves:
Most general case:
Note: The reference
directions for voltage
and current are the
same for forward and
backward waves.
+
V ( z )
-
I ( z )
z
1
2
12
0
0 0
0 0
0 0
z z
z z
V z V e V e
V V I z e e
Z
j R j L G j C
R j L Z
G j
Z
C
phase velocity v p
[m/s] pv
The Terminated Lossless Transmission Line
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The Terminated Lossless Transmission Line
• In the analysis of the solutions of transmission-line equations,
the traveling wave along the line contains two components:• One traveling in the positive z direction and the other
traveling the negative z direction.
The Terminated Lossless Transmission Line
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The Terminated Lossless Transmission Line
•
Figure shows a lossless transmission line terminated in anarbitrary load impedance Z ℓ .
• This problem will illustrate wave reflection on transmission
lines, a fundamental property of distributed systems.
• Assume that an incident wave of the form V +e ─ is
generated from a source at z < 0.
• The ratio of voltage to current for such a traveling wave is
Z 0 , the characteristic impedance of the line.
• However, when the line is terminated in an arbitrary load
Z ℓ ≠ Z 0 , the ratio of voltage to current at the load must beZℓ .
The Terminated Lossless Transmission Line
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The Terminated Lossless Transmission Line
• Thus, a reflected wave must be excited with the appropriate
amplitude to satisfy this condition.• The total voltage and current on the line can then be written as
a sum of incident and reflected waves:
• Similarly, the total current on the line is described by
Reflection Coefficient andTransmission Coefficient
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Transmission CoefficientReflection Coefficient
It is usually more convenient to start solving the
transmission-line problem from the receiving rather than the
sending end, since the voltage-to-current relationship at the
load point is fixed by the load impedance.
Fig: Transmission line terminated in a load impedance
If the load impedance is equal to the line characteristicimpedance, the reflected traveling wave does not exist.
Figure below shows a transmission line terminated in an
impedance Zℓ.
Reflection Coefficient and
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Transmission CoefficientReflection Coefficient
The incident voltage and current waves traveling along the
transmission line are given by
in which the current wave can be expressed in terms of the
voltage by
If the line has a length of ℓ, the voltage and current at thereceiving end become
Reflection Coefficient and
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Transmission CoefficientReflection Coefficient
The ratio of the voltage to the current at the receiving end is
the load impedance. That is,
• The amplitude of the reflected voltage wave normalized to the
amplitude of the incident voltage wave is defined as the voltage
reflection coefficient, which is designated by (gamma), as
Reflection Coefficient and
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Transmission CoefficientReflection Coefficient
If Eq. (3-2-6) is solved for the ratio of the reflected voltage at
the receiving end , which is V-eℓ ,
to the incident voltage at
the receiving end, which is V+e ─ℓ, the result is the reflection
coefficient at the receiving end:
From Fig., let = ℓ − . Then the reflection coefficient atsome point located a distance from the receiving end is
Reflection Coefficient and
T i i C ffi i t
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Transmission CoefficientReflection Coefficient
The reflection coefficient at that point can be expressed in
term of the reflection coefficient at the receiving end as
Here the phase of Γℓ is assumed to be θℓ .
, the reflection coefficient Γ at a distance of d from
the load end is .
Its magnitude will be with a phase
Reflection Coefficient and
T i i C ffi i t
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Transmission CoefficientReflection Coefficient
This is a very useful
equation for determining
the reflection coefficient
at any point along the
line.
For a lossy line, both the
magnitude and phase of the reflection coefficient
are changing in an
inward –spiral way as
shown:
Reflection coefficient for a lossy line
Reflection Coefficient and
Transmission Coefficient
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Transmission CoefficientReflection Coefficient
For a lossless line, = , the magnitude of the reflectioncoefficient remains constant, and only the phase of is
changing circularly towards the generator with an angle of
− as shown
Reflection coefficient for a lossless line
Reflection Coefficient and
Transmission Coefficient
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Transmission CoefficientReflection Coefficient
It is evident that ℓ will be zero and there will be no
reflection from the receiving end when the terminating
impedance is equal to the characteristic impedance of the
line.
Thus a terminating impedance that differs from thecharacteristic impedance will create a reflected wave
traveling towards the source from the termination.
The reflection, upon reaching the sending end, will itself be
reflected if the source impedance is different from the linecharacteristic impedance at the sending end.
Reflection Coefficient andTransmission Coefficient
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Transmission CoefficientTransmission Coefficient A transmission line terminated in its characteristic
impedance Z0 is called a properly terminated line.
Otherwise it is called an improperly terminated line.
There is a reflection coefficient at any point along an
improperly terminated line.
According to the principle of conservation of energy, theincident power minus the reflected power must be equal to
the power transmitted to the load. This can be expressed as
The letter T represents the transmission coefficient, which is
defined as
Reflection Coefficient andTransmission Coefficient
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Transmission CoefficientTransmission Coefficient The following figure shows the transmission of power along a
transmission line where P inc is the incident power, P ref the
reflected power, and P tr the transmitted power.
Let the travelling waves at the receiving end be
Reflection Coefficient andTransmission Coefficient
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Transmission CoefficientTransmission Coefficient
Multiplication of Equation (3-2-16) by Zℓ and submission of the result in Equation (3-2-15) yields
which, in turn, on submission back into Equation (3-2-15),
results in
• The power carried by the two waves in the side of the
incident and reflected waves is
Reflection Coefficient andTransmission Coefficient
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Transmission CoefficientTransmission Coefficient
The power carried to the load by the transmitted wave is
• By setting Pinc
= Ptr
and using Equation (3-2-17) and (3-2-
18), we have
•
This relation verifies the previous statement that thetransmitted power is equal to the difference of the
incident power and reflected power.
Example 3-2-1
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Example 3 2 1
Solution
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