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Chapter 6 : Metallic Transmission Lines
Chapter contents


6.1 Introduction
6.2 Metallic Transmission Lines (pg 512 – 525 of textbook)



6.3 Metallic Transmission Line Equivalent Circuit

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
conductor losses, dielectric heating losses, radiation losses coupling losses, corona
6.6 Incident and Reflected Waves


6.4.1 Velocity factor and dielectric constant
6.5 Transmission Line Losses


6.3.1 Uniformly distributed transmission line
6.3.2 Transmission characteristics
6.4 Wave Propagation on a Metallic Transmission Line


6.2.1 Parallel-Conductor Transmission Lines
6.2.2 Coaxial (Concentric) Transmission Line
Resonant/nonresonant transmission line, reflection coefficient
6.7 Standing waves

SWR, open line/shorted line SW
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6.3 Metallic Transmission Line Equivalent Circuit
6.3.1 Uniformly Distributed Transmission Lines

characteristics of a transmission line are determine by :



the above properties determine the primary electrical constant of the
transmission line :






Electrical properties – wire conductivity, insulator dielectric constant
Physical properties – wire diameter, conductor spacing
series DC resistance (R) : occurs along the line
series inductance (L) : occurs along the line
shunt capacitance (C) : occurs between the conductors
shunt conductance (G) : occurs between the conductors
these primary constants are uniformly distributed throughout the length of the
line and commonly called as distributed parameters.
to simplify analysis, distributed parameters are normally given in unit length of
cable to form an artificial model of a line – lump parameters (e.g. ohm per
meter).
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6.3.1 Uniformly Distributed Transmission Lines

electrical equivalent circuit for a metallic two-wire parallel transmission line
showing the relative placement of the various lumped parameters :
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6.3.2 Transmission Characteristics


transmission characteristics of a transmission line are called secondary constants
and are determined from the previous four primary constants.
the secondary constants are :


characteristic impedance
propagation constant
6.3.2.1 Characteristic impedance

for a maximum power transfer from the source to the load (i.e. no reflected
power), a transmission line must be terminated in a purely resistive load equal to
the characteristic impedance of the transmission line.

characteristic impedance (Z0) is defined as the impedance seen looking into an
infinitely long line or the impedance seen looking into a finite length of line that
is terminated in a purely resistive load with a resistance equal to the
characteristic impedance of the line.
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6.3.2.1 Characteristic Impedance

an infinitely long line can be simulated if a finite line is terminated in a purely
resistive load equal to Z0, where all the energy that enters the line from the
source is dissipated in the load (totally lossless line).

Below figure shows a single section of a transmission line terminated in load ZL
equal to Z0. The impedance seen looking into a line of n such section is
determined from the following :
2
Z
L
(7.3-1)
Z 0 2  Z 1Z 2 
n
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6.3.2.1 Characteristic Impedance
where n is the number of sections. For infinite number of sections, ZL2/n
approaches 0 if
ZL2
lim
n
Z0 
then
n  
0
Z 1Z 2
(7.3-2)
where
Z 1  R  jL
1
1
1
Y2 


 G  jC
Z2
Rs 1 / jC
1
Z2 
G  jC
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6.3.2.1 Characteristic Impedance
Therefore,
Z0 

( R  jL)
1

G  jC
(7.3-3)
for extremely low frequency signals, the resistance dominate and (7.3-3)
simplifies to
Z0 

R  jL
G  jC
R
G
(7.3-4)
for extremely high frequency signals, the inductance and capacitance dominate
and (7.3-3) simplifies to
Z0 
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j L
jC
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(7.3-5)
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6.3.2.1 Characteristic Impedance


From (7.3-5), it can be seen that for high frequencies, the characteristic
impedance of a transmission line approaches a constant, is independent of both
frequency and length and is determined solely by the distributed inductance and
capacitance.
Therefore, Z0 looks purely resistive and all the incident energy is absorbed by
the line.
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6.3.2.1 Characteristic Impedance

From a purely resistive approach, the impedance seen looking into a
transmission line made up of an infinite number of sections approaches the
characteristic impedance as shown below.

mathematically, Z1 is
Z1  R  Rs  10 100  110

adding a second section, Z2 gives
RsZ 1
100 110
Z2  R 
 10 
 10  52.38  62.38
Rs  Z 1
100  110
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6.3.2.1 Characteristic Impedance

and the third section, Z3 is
Z3  R 

RsZ 2
100  62.38
 10 
 10  38.42  48.32
Rs  Z 2
100  62.38
if the above process is continued, the impedance seen looking into the line will
decrease asymptotically toward 37 Ω, which is the characteristic impedance of the
line.
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6.3.2.1 Characteristic Impedance

if the previous transmission line were terminated in a load resistance ZL = 37 Ω,
the impedance seen looking into any number of sections would equal 37 Ω, the
characteristic impedance. For a single section of line, Z0 is
Rs  ZL
100  37
3700
Z 0  Z1  R 
 10 
 10 
 37
Rs  ZL
100  37
137

adding a second section, Z2 is
Z0  Z2  R 

Rs  Z 1
100  37
3700
 10 
 10 
 37
Rs  Z 1
100  37
137
therefore, if this line were terminated into a load resistance ZL = 37 Ω, Z0 = 37 Ω, no
matter how many sections are included.
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6.3.2.1 Characteristic Impedance

mathematically, the characteristic impedance, Z0 is
E0
Z0 
I0
where Z0 = characteristic impedance (ohms)
E0 = source voltage (volts)
I0 = transmission line current (amps)

the characteristic impedance of a two-wire parallel transmission line with an air
dielectric can be determined from its physical dimensions and the formula
D
Z 0  276 log 10
r
where Z0 = characteristic impedance (ohms)
D = distance between the centers of the two conductors (inches)
r = radius of the conductor (inches)
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6.3.2.1 Characteristic Impedance
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6.3.2.1 Characteristic Impedance
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6.3.2.2 Propagation constant





Propagation constant (propagation coefficient) is used to express the attenuation
(signal loss) and the phase shift per unit length of a transmission line.
as signal propagates down a transmission line, its amplitude decrease with
distance travel. Propagation constant is used to determine the reduction in
voltage and current as the TEM wave propagates down a transmission line.
for an infinitely long line, all the incident power is dissipated in the resistance of
the wire as the wave propagates down the line.
therefore, with an infinitely long line or a line that looks infinitely long, such as
a finite line terminated in a matched load (Z0 = ZL), no energy is returned or
reflected back toward the source.
mathematically, the propagation constant is
    j
where γ = propagation constant
α = attenuation coefficient (nepers per unit length)
β = phase shift coefficient (radians per unit length)
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(7.3-6)
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6.3.2.2 Propagation constant

the propagation constant is a complex quantity defined by
 

( R jL )( G  jC )
because a phase shift of 2πrad occurs over a distance of one wavelength
 

(7.3-7)
2
(7.3-8)

at intermediate and radio frequencies, ωL > R and ωC > G ; thus
GZ 0
R
 

and  
2Z0
2
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(7.3-9)
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6.3.2.2 Propagation constant

The current and volatage distribution along a transmission line that is terminate
in a load equal to its characteristic impedance ( matched line ) are determined
from the formulas
I  Ise
 l
V  Vs e
(7.3-10)
 l
(7.3-11)
where Is = current at the source end of the line (amps)
Vs = voltage at the source end of the line (volts)
γ= propagation constant
l = distance from the source at which the current or voltage is determined

For a matched load ZL = Z0 and for a given length of cable l, the loss in signal
voltage or current is the real part of γl and the phase shift is the imaginary part.
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6.4 Wave Propagation on a Metallic Transmission Line


Electromagnetic waves travel at the speed of light when propagating through a
vacuum and nearly at the speed of light when propagating through air.
However, in metallic transmission line, where the conductor is generally
copper and the dielectric materials vary considerably with the cable type, an
electromagnetic wave travels slower.
6.4.1 Velocity factor and dielectric constant


velocity factor (velocity constant) is defined as the ratio of the actual velocity
of propagation of an electromagnetic wave through a given medium to the
velocity of propagation through a vacuum (free space).
mathematically expressed,
Vp
(7.4-1)
Vf 
c
where Vf = velocity factor (unitless)
Vp = actual velocity of propagation (meter per second)
c = velocity of propagation through a vacuum (3 x 108 m/s)
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6.4.1 Velocity factor and dielectric constant



rearrange (7.4-1),
Vf  c  Vp
the velocity at which an electromagnetic wave travels through a transmission
line depends on the dielectric constant of the insulating material separating the
2 conductors.
the velocity factor is approximated with the formula
Vf 

(7.4-2)
1
r
(7.4-3)
where εr = dielectric constant of a given material
εr can be defined as the permittivity of the material relative to the permittivity
of a vacuum,

r 
(7.4-4)
0
where ε: permittivity of the dielectric, ε0 : permittivity of air
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6.4.1 Velocity factor and dielectric constant

Table of velocity factor and dielectric constant :

dielectric constant depends on the type of insulating material used.
inductor stores magnetic energy and capacitor stores electric energy. It takes a
finite amount of time for an inductor or a capacitor to take on or give u energy.
therefore, the velocity at which an electromagnetic wave propagates along a
transmission line varies with the inductance and capacitance of the cable.


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6.4.1 Velocity factor and dielectric constant

let the time T =
LC. Therefore,
dis tan ce D
Vp 
 
time
T

D
LC
(7.4-5)
if the distance is normalized to 1 meter, the velocity of propagation for a lossless
transmission line is
Vp 
1
meter / sec ond
LC
(7.4-6)
where Vp = velocity of propagation (meter per second)
LC = second
L = inductance per unit length (H/m)
C = capacitance per unit length (F/m)
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6.5 Transmission Line Losses
in analysis, metallic transmission lines are often considered to be totally
lossless.
In reality however, there are several ways in which signal power is lost in a
transmission line.







Conductor loss
dielectric heating losses
Radiation losses
Coupling losses
corona
6.5.1 Conductor losses



because electrical current flows through a metallic transmission line that has a
finite resistance, there is an inherent and unavoidable power loss.
equal to I2R power loss.
to reduce it, shorten the transmission line or using a larger diameter wire.
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6.5.2 Dielectric Heating Losses




potential difference between 2 conductors of a metallic transmission line
causing dielectric heating.
heat is a form of energy and must be taken from the energy propagating down
the line.
for air dielectric transmission line, the heating loss is negligible.
however, for solid-core transmission line, dielectric heating loss increases with
frequency.
6.5.3 Radiation losses





the distance between conductors that is equal to appreciable fraction of length
can cause the transmission line to act as an antenna and transfer the energy to
nearby conductive material.
the energy radiated is called radiation loss and depends on dielectric material,
conductor spacing and length of the transmission line.
shielding the cable can reduce the radiation losses.
shielded cables (e.g. STP, coaxial) have less radiation loss compared to
unshielded cables (e.g. twin lead, open wire, UTP).
radiation loss also directly proportional to frequency.
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6.5.4 Coupling Losses



occurs whenever a connection is made to or from a transmission line or when
2 sections of transmission line are connected together.
mechanical connections are discontinuities which are locations where
dissimilar materials meet.
discontinuities tend to heat up, radiate energy and dissipate power.
6.5.5 Corona


corona is a luminous discharge that occurs between the 2 conductors of a
transmission line when the difference of potential between them exceeds the
breakdown voltage of the dielectric insulator.
generally, when corona occurs, the transmission line is destroyed.
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6.6 Incident and Reflected Waves




an ordinary transmission is basically bidirectional : power can propagate
equally well in both directions.
voltage that propagates from source toward the load is called incident voltage,
and voltage that propagates from load toward the source is called reflected
voltage.
for an infinitely log line, all the incident power is stored by the line and there is
no reflected power.
also if the line is terminated in a purely resistive load equal to the characteristic
impedance of the line, the load absorbs all the incident power (lossless line).
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6.6 Incident and Reflected Waves

reflected power is portion of the incident power that was not absorbed by the
load. Therefore, the reflected power can never exceed the incident power.
6.6.1 Resonant and Nonresonant Transmission Line





transmission line without a reflected power is called a flat or nonresonant line.
a transmission line is nonresonant if it is of infinite length or if it is terminated
with a resistive load equal to the characteristic impedance of the transmission
line.
when the load is not equal to the characteristic impedance of the line, some of
the incident power is reflected back toward the source.
if the load is either shorted or an open circuit, all the incident power is
reflected back toward the source.
if the source were replaced with an open or shorted and the line were lossless,
energy present on the line would reflect back and forth (oscillate) between the
source and load ends – resonant transmission line.
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6.6.2 Reflection coefficient


Reflection Coefficient is a vector quantity that represents the ratio of reflected
voltage to incident voltage or reflected current to incident current.
Mathematically, the reflection coefficient Γis defined as

Er Ir

Ei Ii
(7.6-1)
where Γ= reflection coefficient (unitless)
Ei = incident voltage (volts)
Er = reflected voltage (volts)
Ii = incident current (amps)
Ir = reflected current (amps)
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6.7 Standing Waves




when Z0 = ZL, all the incident power is absorbed by the load. This is called a
matched line.
when Z0 ≠ ZL, some of the incident power is absorbed by the load and some is
returned (reflected) to the source. This is called an unmatched or mismatched
line.
with a mismatched line, there are 2 electromagnetic waves, traveling in
opposite direction, present on the line at the same time (traveling waves).
these 2 waves set up an interference pattern known as a standing wave.
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6.7 Standing Waves



as the incident waves pass
each other, stationary
patterns of voltage and
current are produced on the
line.
these stationary waves are
called standing waves
because they appear to
remain in a fixed position on
the line, varying only in
amplitude.
the standing wave has
minima (nodes) separated
by a half wavelength of the
traveling waves and maxima
(antinodes) also separated
by a half wavelength.
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6.7.1 Standing Waves Ratio (SWR)



Standing wave ratio is defined as the ratio of the maximum voltage to the
minimum voltage or the maximum current to the minimum current of a
standing wave on a transmission line.
often called as voltage standing wave ratio (VSWR), a measure of the
mismatch between the load impedance and the characteristic impedance of the
transmission line.
mathematically expressed as
SWR 


V max
(unitless )
V min
(7.7-1)
voltage maxima (Vmax) occurs when the incident and reflected waves are in phase.
voltage minima (Vmin) occurs when the incident and reflected waves are 180º out of
phase.
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V max  Ei  Er
(7.7-2)
V min  Ei  Er
(7.7-3)
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6.7.1 Standing Waves Ratio (SWR)

therefore, (7.7-1) can be written as
V max Ei  Er
SWR 

V min Ei  Er



voltage maxima (Vmax) occurs when the incident and reflected waves are in phase.
voltage minima (Vmin) occurs when the incident and reflected waves are 180º out of
phase.
from (7.6-1), the SWR can be written in terms of Γas follow.
Ei  Er Ei  Ei Ei (1  ) 1  
SWR 



Ei  Er Ei  Ei Ei (1  ) 1  

(7.7-4)
(7.7-5)
cross multiplying of (7.7-5) gives reflection coefficient Γ
SWR  1

SWR  1
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(7.7-6)
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6.7.1 Standing Waves Ratio (SWR)

when the load is purely resistive, SWR can also be expressed as a ratio of the
characteristic impedance to the load impedance or vice versa.
Z 0 ZL
SWR  or
ZL Z 0

(whichever gives an SWR greater than 1)
(7.7-7)
disadvantages of not having a matched (flat) transmission line :





100% of the source incident power is not absorbed by the load.
the dielectric separating the 2 conductors can breakdown and cause corona as a
result of the high-voltage standing wave ratio.
reflections and re-reflections cause more power loss.
reflections cause ghost images
mismatches cause noise interference.
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6.7.2 Standing Waves on an Open Line


when incident waves of voltage and current reach an open termination, none of
the power is absorbed, it is all reflected toward the source.
the characteristic impedance of a transmission line terminated in an open
condition can be summarized as follow :




the voltage incident wave is reflected back just as if it was to continue.
the current incident wave is reflected back 180º from how it would have continued.
the sum of the incident and reflected current waveforms is minimum at the open.
the sum of the incident and reflected voltage waveforms is maximum at the open.
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6.7.2 Standing Waves on an Open Line



the voltage and current standing waves repeat every one-half wavelength.
the impedance at the open end Z = Vmax / Imin and is maximum.
the impedance one-quarter wavelength from the open Z = Vmin / Imax and is
minimum.
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6.7.3 Standing Waves on a shorted line



none of the incident power is absorbed by the load when a transmission line is
terminated in a short circuit.
however with the shorted line, the incident voltage and current waves are
reflected back in the opposite manner.
the characteristics of a transmission line canbe summarized as :




the voltage incident wave is reflected back 180º reversed from how it would have
continued.
the current incident wave is reflected back the same as if it would continued.
the sum of the incident and reflected current waveforms is max at the shorted point.
the sum of the incident and reflected voltage waveforms is zero at the shorted point.
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6.7.3 Standing Waves on a shorted line



the voltage and current standing waves repeat every one-halfwavelength.
the impedance at the short Z = Vmin / Imax = minimum.
the impedance one-quarter wavelength from the short Z = Vmax / Imin =
maximum.
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