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TRANSMISSION LINES AND WAVEGUIDES

UNIT-I
TRANSMISSION LINE THEORY

INTRODUCTION TO TRANSMISSION LINE THEORY

Transmission Lines and Waveguides

A TRANSMISSION LINE is a device designed to guide electrical energy from one point
to another. It is used, for exam ple, to transf er the output rf energy of a transm itter to an
antenna. This energy will not travel through norm al electric al wire without great losses.
Although the antenna can be connected directly to the transm itter, the antenna is usually
located some distance away from the transmitter.
On board ship, the transm itter is located inside a radio room, and its associated
antenna is m ounted on a mas t. A transm ission line is used to connect the transm itter and the
antenna. The transm ission line has a single purpose f or both the transmitter and the antenna.
This purpose is to transfer the energy output of the transm itter to the antenna with the least
possible power loss. How well this is done depends on the s pecial physical and electrical
characteristics (impedance and resistance) of the transmission line.

TRANSMISSION LINE THEORY

The electrical characteristics of a two-wire transm ission line depend prim arily on the
construction of the line. The two-wire line acts like a long capacitor. The change of its capacitive
reactance is noticeable as the frequency applied to it is changed.
Since the long conductors have a m agnetic field about them when elec trical energy is
being passed through them, they also exhibit the properties of inductance. The values of
inductance and capacitance presented depend on the various physical factors that we
discussed earlier.
For exam ple, the type of line used, the dielectric in the line, and the length of the line
must be considered. The effects of the inductive and capacitive reactance of the line depend on
the frequency applied. Since no dielectric is perfect, electrons m anage to m ove from one
conductor to the other through the dielectric.
Each type of two-wire transm ission line also has a conductance value. This
conductance value represents the value of the current f low that m ay be expec ted through the
insulation, If the line is uniform (all values equal at each unit length), then one small section of

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the line m ay represent several feet. This illustration of a two-wire transmission line will be used
throughout the discussion of transmission lines; but, keep in mind that the principles presented
apply to all transm ission lines.W e will explain the theories using LUMPED CONSTANTS and
DISTRIBUTED CONSTANTS to further simplify these principle.

LUMPED CONSTANTS

A transmission line has the properties of inductance, capacitance, and resistance just as
the m ore conventional circuits have. Usually, however, the constants in conventional c irc uits
are lum ped into a single device or com ponent. For exam ple, a coil of wire has the property of
inductance. W hen a certain am ount of inductance is needed in a circuit, a coil of the proper
dimensions is inserted.
The inductance of the circuit is lum ped into the one com ponent. Two m etal plates
separated by a small s pace, can be used to supply the required capacitance for a circuit. In
such a case, most of the capacitance of the circuit is lum ped into this one component. Similarly,
a fixed resistor can be used to supply a certain value of circuit resistance as a lum ped sum.
Ideally, a transm ission line would also have its constants of inductance, capacitance, and
resistance lumped together, as shown in figure 3-1. Unfortunately, this is not the
case.Transmission line constants are as described in the following paragraphs.
D I S T R I B U TE D C O N S T A N T S
Transmission line constants, called distributed constants, are spread along the entire
length of the transmission line and cannot be distinguished separately. The amount of
inductance, capacitance, and resistance depends on the length of the line, the size of the
conducting wires, the spacing between the wires, and the dielectric (air or insulating medium)
between the wires. The following paragraphs will be useful to you as you study distributed
constants on a transmission line.

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Two-wire transmission Iine.

Inductance of a Transmission Line

W hen current flows through a wire, m agnetic lines of force are set up around the wire.
As the current increases and dec reases in am plitude, the f ield around the wire expands and
collapses accordingly. The energy produced by the magnetic lines of force
collapsing back into the wire tends to k eep the current flowing in the same direction. This
represents a certain amount of inductance, which is expressed in m icrohenrys per unitlength.
Figure illustrates the inductance and magnetic fields of a transm ission
line.

Capacitance of a Transmission Line

Capacitance also exists between the transmission line wires, as illustrated in figure 3-3. Notice
that the two parallel wires act as plates of a capacitor and that the air between them acts as a
dielectric. The capacitance between the wires is usually expressed in picofarads per unit length.
This electric field between the wires is similar to the field that exists between the two plates of a
c a p a c it o r .
Resistance of a Transmission Line

The transm ission line shown in figure 3-4 has electrical resistance along its length. This
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resistance is usually
expressed in ohms per unit length and is shown as existing continuously from one end of the line to
the other..
Leakage Current
Since any dielectric, even air, is not a perfect insulator, a small current known as LEAKAGE
CURRENT f lows between the two wires. In effect, the insulator acts as a resis tor, perm itting
current to pass between the two wires. Figure 3-5 shows this leakage path as resistors in
parallel connected between the two lines. This property is called CONDUCTANCE (G) and is
the opposite of resistance. Conductance in transmission lines is expressed as the reciprocal of
resistance and is usually given in micro mhos per unit length.

ELECTROMAGNETIC FIELDS CHARACTERISTIC IMPEDANCE

The distributed constants of resistance, inductance, and capacitance are basic

properties common to all transmission lines and exist whether or not any current flow exists. As
soon as current f low and voltage exist in a transm ission line, another property becomes quite

evident. This is the presence of an electromagnetic field, or lines of force, about the wires of the
transmission line.
The lines of force themselves are not visible; however, understanding the force that an
electron experiences while in the field of these lines is very im portant to your understanding of
energy transmission. There are two kinds of fields; one is associated with voltage and the other
with current. The field assoc iated with voltage is c alled the ELECTRIC (E) FIELD. It exerts a
force on any electric charge placed in it. The field associated with current is called a
MAGNETIC (H) FIELD, because it tends to extra force on any m agnetic pole placed in it. Figure
3-6 illustrates the way in which the E fields and H f ields tend to orient them selves between
conductors of a typical two-wire transmission line. The illustration shows a cross section of the
transmission lines. The E field is represented by solid lines and the H field by dotted lines. The
arrows indicate the direc tiowww.Vidyarthiplus.com
n of the lines of force. Both f ields norm ally exist together and are
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spoken of collectively as the electromagnetic Field.

FieIds betw een conductors.

You can describe a transm ission line in terms of its im pedance. The ratio of voltage to
current (Ein/Iin) at the input end is k nown as the INPUT IMPEDANCE (Zin). This is the
impedance presented to the transmitter by the transmission line and its load, the antenna.
The ratio of voltage to current at the output (EOUT/IOUT) end is known as the OUTPUT
IMPEDANCE (ZOUT). This is the impedance presented to the load by the transm ission line and
its source. If an infinitely long transmission line could be used, the ratio of voltage to current at
any point on that transm ission line would be some particular value of im pedance. This
im pedance is known as the CHARACTERISTIC IMPEDANCE. The m aximum (and most
efficient) transfer of electrical energy takes place when the source impedance is matched to the
load impedance. This fact is very im portant in the study of transm iss ion lines and antennas. If
the characteristic impedance of the transm ission line and the load im pedance are equal, energy
from the transmitter will travel down the transmission line to the antenna with no power loss
caused by reflection.

LI N E L O S S E S

The discussion of transmission lines so far has not directly addressed LINE LOSSES;
actually some losses occur in all lines. Line losses may be any of three types
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1. COPPER, DIELECTRIC,
2. RADIATION or INDUCTION LOSSES.

NOTE: Transmission lines are sometimes referred to as RF lines. In this text the terms are
used interchangeably.
Copper Losses
One type of copper loss is I2R LOSS. In RF lines the resistance of the conductors is
never equal to zero. W henever current flows through one of these conductors, some energy is
dissipated in the form of heat. This heat loss is a POW ER LOSS. W ith copper braid, which has
a resistance higher than solid tubing, this power loss is higher.
Another type of copper loss is due to SKIN EFFECT. W hen dc flows through a
conductor, the m ovem ent of electrons through the conduc tor’s cross section is uniform, The
situation is som ewhat different when ac is applied. The expanding and collapsing f ields about
each electron encircle other electrons. This phenomenon, called SELF INDUCTION, retards the
movement of the encircled electrons.
The flux density at the center is so great that electron movement at this point is reduced.
As frequency is increased, the oppos ition to the f low of current in the center of the wire
increases. Current in the center of the wire becomes smaller and most of the electron flow is on
the wire surface. W hen the frequency applied is 100 m egahertz or higher, the electron
m ovem ent in the center is so sm all that the center of the wire could be rem oved without any
noticeable effect on current. You should be able to see that the effective cross-sectional area
decreases as the frequency increases.
Since resistance is inversely proportional to the cross-sectional area, the resistance will
increase as the frequency is increased. Also, since power loss increases as resistance
increases, power losses increase with an increase in frequency because of skin effect.
Copper losses can be m inim ized and conductivity increased in an RF line by plating the
line with silver. Since silver is a better conductor than copper, most of the current will f low
through the silver layer. The tubing then serves primarily as a mechanical support.
DieIectric Losses
DIELECTRIC LOSSES result from the heating effect on the dielectric material between
the conductors. Power f rom the s ource is used in heating the dielectric. The heat produc ed is
dissipated into the surrounding m edium. W hen there is no potential difference between two
conductors, the atoms in the dielectric m aterial between them are norm al and the orbits of the
electrons are circular.
W hen there is a potential difference between two conductors, the orbits of the electrons
change. The excessive negative charge on one conductor repels electrons on the dielectric
toward the positive conductor and thus distorts the orbits of the electrons.
A c hange in the path of electrons requires m ore energy, introduc ing a power los s . The
atom ic structure of rubber is m ore difficult to distort than the s truc ture of some other dielectric
materials. The atoms of materials, such as polyethylene, distort easily. Therefore, polyethylene
is often used as a dielectwww.Vidyarthiplus.com
ric because less power is consum ed when its electron orbits are
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distorted.
RADIATION AND INDUCTION LOSSES
RADIAION and INDUCTION LOSSES are sim ilar in that both are caused by the f ields
surrounding the c onductors. Induction losses occur when the electrom agnetic field about a
conductor cuts through any nearby m etallic object and a current is induced in that object. As a
result, power is dissipated in the
Object and is lost. Radiation losses occur because some m agnetic lines of force about a
conductor do not return to the conduc tor when the cyc le alternates. These lines of force are
projected into space as radiation, and this result in power losses. That is, power is supplied by
the source, but is not available to the load.
VOLTAGE CHANGE
In an elec tric circuit, energy is s tored in elec tric and m agnetic f ields. These fields must
be brought to the load to transm it that energ y. At the load, energy contained in the f ields is
converted to the desired form of energy
Transmission of Energy
W hen the load is connected directly to the source of energy, or when the transm iss ion
line is s hort, problems concerning current and voltage can be solved by applying Ohm’s law.
W hen the transm ission line becom es long enough so the time difference between a c hange
occurring at the generator and a change appearing at the load becom es appreciable, analysis
of the transmission line becomes important.

Dc AppIied to a Transmission Line

In figure 3-7, a battery is connected through a relatively long two-wire transmission line
to a load at the far end of the line. At the instant the switch
is closed, neither current nor voltage exists on the line.
When the switch is closed, point A becomes a positive potential, and point B becomes
negative. These points of difference in potential move down the line. However, as the initial
points of potential leave points A and B, they are followed by new points of difference in
potential, which the battery adds at A and B.
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This is merely saying that the battery maintains a constant potential difference between
points A and B. A short time after the switch is closed, the initial points of difference in potential
have reac hed points A’ and B’; the wire sections from points A to A’ and points B to B’ are at
the same potential as A and B, respec tively. The points of charge are represented by plus (+)
and minus (-) signs along the wires, The directions of the currents in the wires are represented
by the arrowheads on the line, and the direction of travel is indicated by an arrow below the
line.
Conventional lines of f orce represent the electric f ield that exists between the oppos ite
kinds of charge on the wire sections from A to A’ and B to B’. Crosses (tails of arrows) indicate
the m agnetic field created by the electric f ield moving down the line. The m oving electric field
and the accom panying magnetic f ield constitute an electrom agnetic wave that is m oving from
the generator (battery) toward the load.
This wave travels at approxim ately the speed of light in free space. The energy reaching
the load is equal to that developed at the battery (assum ing there are no losses in the
transmission line). If the load absorbs all of the energy, the current and voltage will be evenly
Distributed along the line.
Ac AppIied to a Transmission Line
W hen the battery of figure 3-7 is replaced by an ac generator (fig. 3-8), each successive
instantaneous value of the generator voltage is propagated down the line at the speed of light.
The action is sim ilar to the wave created by the battery, except the applied voltage is sinusoidal
instead of c onstant. Assume that the s witch is c losed at the mom ent the generator voltage is
passing through zero and that the next half cyc le m akes point A pos itive. At the end of one
cycle of generator voltage, the current and voltage distribution will be as shown in figu
In this illustration the conventional lines of force represent the electric fields. For
simplicity, the m agnetic f ields are not shown. Points of charge are indicated by plus (+) and
minus (-) signs, the larger signs indicating points of higher am plitude of both voltage and
current. Short arrows indic ate direc tion of current (electron f low). The wavef orm drawn below
the transmission line represents the voltage (E) and current (I) waves.
The line is assum ed to be infinite in length so there is no reflection. Thus, traveling
sinusoidal voltage and current waves continually travel in phase from the generator toward the
load, or far end of the line. W aves traveling from the generator to the load are called INCIDENT
WAVES. W aves traveling from the load back to the generator are called REFLECTED W AVES
and will be explained in later paragraphs.

TRANSMISSION MEDIUMS
The Navy uses many different types of TRANSMISSION MEDIUMS in its electronic
applications. Each medium (line or waveguide) has a certain characteristic impedance value,
current-carrying capacity, and physical shape and is designed to meet a particular requirement.
The five types of transmission mediums that we will discuss in this topic.

1. PARALLEL-LINE,
2. TW ISTED PAIR,
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3. SHIELDED PAIR,
4. COAXIAL LINE, and
5. W AVEGUIDES.

The use of a particular line depends, among other things, on the applied
frequency, the power-handling capabilities, and the type of installation.

Power Standing-Wave Ratio ParaIIeI Line

The square of the vswr is called the POW ER One type of parallel line is the TWO-W IRE
OPENSTANDING-WAVE RATIO (pswr). Therefore: LINE, illustrated in figure

TRANSMI
TTER

I N PU T EN D TRANSMISSION LINE OUTPUT END

AN T EN N A

This line consists of two wires that are generally spaced from 2 to 6 inches apart by
insulating spacers. This type of line is most often used f or power lines, rural telephone lines,
and telegraph lines. It is som etimes us ed as a transm ission This ratio is useful bec ause the
instrum ents used to line between a transm itter and an antenna or between detect standing
waves react to the square of the an antenna and a receiver.
An advantage of this type of line is its simple construction. The principal disadvantages
of this type of line are the high radiation losses and electrical noise pickup because of the lack
of shielding.

Radiation losses are produced by the changing fields created by the changing current in
each conductor. Another type of parallel line is the TWOW IRE RIBBON (TW IN LEAD) LINE,
illustrated in figure 3-10. This type of transm ission line is commonly used to connect a television
receiving antenna to a home television set.
This line is essentially the same as the two-wire open line except that uniform spacing is
assured by em bedding the two wires in a low-loss dielectric, usually polyethylene. Since the
wires are em bedded in the thin ribbon of polyethylene, the dielectric space is partly air and
partly polyethylene.

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Twisted Pair

The TW ISTED PAIR transmission line is illustrated in figure 3-11. As the name implies,
the line consists of two insulated wires twisted together to form a flexible line without the use of
spacers. It is not used for transmitting high frequency because of the high dielectric losses that
occur in the rubber insulation. When the line is wet, the losses increase greatly.

Two-wire ribbon Iine.

ShieIded pair.

The SHIELDED PAIR, shown in figure, consists of parallel conduc tors separated f rom
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each other and surrounded by a solid dielectric. The conductors are contained within a braided
copper tubing that acts as an electrical shield. The assembly is covered with a rubber or flexible
composition coating that protects the line from moisture and mechanical damage. Outwardly, it
looks much like the power cord of a washing machine or refrigerator.
ShieIded pair.
The principal advantage of the shielded pair is that the conductors are balanced to
ground; that is, the capacitance between the wires is uniform throughout the length of the line.
This balance is due to the uniform spacing of the grounded shield that surrounds the wires
along their entire length. The braided copper shield isolates the conductors from stray m agnetic
f ields.
C o a x i a I Li n e s

There are two types of COAXIAL LINES,

1. RIGID (AIR) COAXIAL LINE
2. FLEXIBLE (SOLID)COAXIAL LINE.

The physical construction of both types is basically the same; that is, each contains two
concentric conductors.

R I G I D C O A X I A L LI N E
The rigid coaxial line consists of a central, insulated wire (inner conductor) m ounted
inside a tubular outer conductor. This line is shown in f igure 3-13. In some applications, the
inner conductor is also tubular. The inner conductor is insulated from the outer c onductor by
insulating spacers or beads at regular intervals. The spacers are made of Pyrex, polystyrene, or
some other m aterial that has good insulating charac teristics and low dielectric losses at high
frequencies.
ADVANTAGES OF RIGID CO AXIAL LINE
The chief advantage of the rigid line is its ability to minimize radiation losses. The
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electric and magnetic fields in a two-wire parallel line extend into space for relatively great
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distances and radiation losses
occur. However, in a coaxial line no electric or magnetic fields
extend outside of the outer conductor. The fields are confined to the space between the two
conductors, resulting in a perfectly shielded coaxial line. Another advantage is that interference
from other lines is reduced.
DIS ADVANTAGES OF RIGID CO AXIAL LINE
The rigid line has the following disadvantages:

(1) It is expensive to construct;

(2) It must be kept dry to prevent excessive leakage between the two
Conductors;
(3) Although high-f requency losses are som ewhat less than in previously m entioned
lines, they are still excessive enough to lim it the practical length of the line. Leakage caused by
the condensation of moisture is prevented in some rigid line applications by the use of an inert
gas, such as nitrogen, helium, or argon. It is pum ped into the dielectric space of the line at a
pressure that c an vary from 3 to 35 pounds per s quare inch. The inert gas is us ed to dry the
line when it is first installed and pressure is m aintained to ensure that no m ois ture enters the
line.
F L E X I B L E C O A X I A L LI N E
Flexible coaxial lines (fig. 3-14) are made with an inner conductor that consists of
flexible wire insulated from the outer conductor by a solid, continuous insulating material. The
outer conductor is made of metal braid, which gives the line flexibility. Early attempts at gaining
flexibility involved using rubber insulators between the two conductors. However, the rubber
insulators caused excessive losses at high frequencies.

FIexibIe coaxiaI Iine.

Because of the high-frequency losses associated with rubber insulators, polyethylene

plastic was developed to replace rubber and eliminate these losses.
Polyeth ylene plastic is a solid substance that rem ains f lexible over a wide range of
temperatures. It is unaffected by seawater, gasoline, oil, and most other liquids that m ay be
found aboard s hip. The use of polyeth ylene as an insulator res ults in greater high-f requency
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losses than the use of air as
an insulator. However, these losses are still lower than the losses
associated with most other solid dielectric Materials.
This concludes our study of transmission lines. The rest of this chapter will be an introduction
into the study of waveguides.

THE TRANSMISSION LINE EQUATION – GENERAL SOLUTION

A circuit with distributed parameter requires a method of analysis somewhat different
from that employed in circuits of lumped constants. Since a voltage drop occurs across each
series increment of a line, the voltage applied to each increment of shunt admittance is a
variable and thus the shunted current is a variable along the line.

Hence the line current around the loop is not a constant, as is assumed in lumped
constant circuits, but varies from point to point along the line. Differential circuit equations that
describes that action will be written for the steady state, from which general circuit equation will
be defined as follows.
R= series resistance, ohms per unit length of line( includes both wires)
L= series inductance, henrys per unit length of line
C= capacitance between conductors, faradays per unit length of line
G= shunt leakage conductance between conductors, mhos per unit length
Of line
ωL = series reactance, ohms per unit length of line
Z = R+jωL
ωL = series susceptance, mhos per unit length of line
Y = G + jω C
S = distance to the point of observation, measured from the receiving end of the line
I = Current in the line at any point
E= voltage between conductors at any point
l = length of line
The below figure illustrates a line that in the limit may be considered as made up of cascaded
infinitesimal T sections, one of which is shown.

This incremental section is of length of ds and carries a current I. The series line impedance
being Z ohms and the voltage drop in the length ds is
dE = IZds (1)
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dE = IZ (2)
ds

The shunt admittance per unit length of line is Y mhos, so that

The admittance of thr line is Yds mhos. The current dI that follows across the line or from one
conductor to the other is
dI = EYds (3)
dI
= EY (4)
ds
The equation 2 and 4 may be differentiat ed with respwect to s

d2E dI
2
=Z ,
ds ds
d 2I dE
2
=Y
ds ds
d2E
= ZYE (5)
ds 2

d 2I
= ZYI (6)
ds 2
These are the ifferential equations of the transmission line, fundamental to circuits of distributed
constants.
This results indicates two solutions, one for the plus sign and the other for the minus sign
before the radical. The solution of the differential; equations are

ZY s
E = Ae AYs
+ Be - (7)
ZY s
I = Ce ZY s
+ De - (8)
Where A,B,C,D are arbitrary constants of integration.
Since the distance is measured from the receiving end of the line, it is possible to assign
conditions such that at

s = 0, I = I R, E = E R

The n equation 7 & 8 becomes

ER = A + B
(9)
IR = C + D
A second set of boundary condition is not available, but the same set may be used over again if
a new set of equations are formed by differentiation of equation 7 and 8. Thus

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(10)
dE ZY s ZY s
= A ZY e - B ZY e -

ds
ZY s ZY s
(11)
IZ = A ZY e - B ZY e -
Y Y -
I = A e ZY s -
B e ZY s

Z Z
dI (12)
ZY s ZY s
=C ZY e - ZY e -
ds
Z ZY s - Y - (13)
E = C e D e ZY

Y Z
Y Y
IR = A - B
Z Z
Z Z
E R =C - D Simultaneous solution of equation 9 ,12 and 13, along
Y Y
with the fact that E R = I R Z R and that Z Y has

been identified as the Z 0 of the line,leads to solution for the constants of the above equations

as

ER I R Z ER � Z �
A= + = ��1 + 0 ��
2 2 Y 2 � ZR �
ER I R Z ER � Z 0 �
B= - = �1 - �
2 2 Y 2 �� Z R ��
I R ER Y I � ZR �
C= + = R ��1 + �
2 2 Z 2 � Z0 �
�

I R ER Y I R �� �
D= - = 1 -
ZZR ��
2 2 Z 2 �� �

The solution of the differential equations of the transmission line may be written

ER � Z � ZY s E �Z �
E= ��1 + 0 �e
� + ��1 R- 0 �e
� - ZY s

2 � ZR � 2 � ZR �
( 14)
IR � ZR � �I R
ZY s �Z R
I= �1 + � + ��1 - �
-2 Z�Y s Z 0 �e 2 Z 0 �e
� � � �
The above equations are very useful form for the voltage and current at any point on a
transmission line.

fter simplifying the above equations we get the final and very useful form of equations
for voltage and current at any point on a k=line, and are solutions to the wave equation

E = E R cosh ZY s + I R Z 0 sinh ZY s
ER
I = I R cosh ZY s + sinh ZY s
Z0
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WaveIength, VeIocity of propagation

WaveIength
The distance the wave travels along the line while the phase angle is changed through
2∏ radians is called wavelength.

λ =2 п / ß
The change of 2п in phase angle represents one cycle in time and occurs in a distance of one
wavelength,
λ= v/f
V e I oc i t y
V= f λ
V=ω/ ß
This is the velocity of propagation along the line based on the observation of the change in the
phase angle along the line.It is measured in miles/second if ß is in radians per meter.
We know that
Z = R + j ωL
Y= G + j ω C
Then

γ= α+j ß = ZY

= RG -� 2 LC + j� ( LG + CR)
Squaring on both sides

α 2 + 2 jαβ - β 2 = RG -� 2 LC + j� ( LG + RC)
Equating real parts and imaginary parts we get

RG -� 2 LC + ( RG -� 2 LC) 2 +� 2 ( LG + CR)
α =
2
And the equation for ß is

� 2 LC - RG + ( RG -� 2 LC) +� 2 ( LG + CR)
β =
2

In a perfect line R=0 and G = 0 , Then the above equation would be

β = � LC
And the velocity of propagation for such an ideal line is given by
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�
v=
β
Thus the above equation showing that the line parameter values fix the velocity of propagation.

DISTORTION
Wave-form distortion
The value of the attenuation constant α has been determined that

RG -� 2 LC + ( RG -� 2 LC) 2 +� 2 ( LG + CR)
α =
2
In general α is a function of frequency. All the frequencies transmitted on a line will then not be
attenuated equally. A com plex applied voltage, such as voice voltage containing m any
frequencies, will not have all frequencies transm itted with equal attenuation, and the received
for will be identical with the input waveform at the s ending end. This variation ic=s k nown as
frequency distortion.
Phase Distortion
The of propagation has been stated that

� 2 LC - RG + ( RG -� 2 LC) +� 2 ( LG + CR)
β =
2

It is apparent that ωand β do not both involve f requency in same m anner and that the ve locity
of propagation will in general be some function of frequency.
All the frequencies applied to a transm ission line will not have the same time of
transm ission , some frequencies delayed m ore than the others. For an applied voice voltage
waves the rec eived waves will not be identic al with the input wave f orm at the receiving end,
since some com ponents will be delayed m ore than those of the other frequencies. This
phenomenon is known as deIay or phase distortion.
Frequenc y distortion is reduced in the transmission of high quality radio broadcast
programs over wire line by use of equalizers at line terminals

These circuits are networks whose frequency and phase characteristics are adjusted to
be inverse to those of the lines, resulting in an over all unif orm frequency response over the
desired frequency band.
Delay distortion is relatively minor importance to voice and music transm ission because
of the characteris tics of ear. It can be very series in circuits intended f or picture transm ission,
and applications of the co axial cable have been made to over come the difficulty.
In such cables the internal inductance is low at high frequencies because of skinn effect,
the resistance small because of the large conductors, and capacitance and leakance are small
because of the use of air dielec tric with a m inimum spacers. The velocity of propagation is
raised and made more nearly equal for all frequencies.
The distortion Iess Iine
If a line is to have neither fwww.Vidyarthiplus.com
requency nor delay distortion,then attenuation constant and velocity
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of propagation cannot be function of frequency.

We know that
�
v=
β
Then the phase constant be a direct fuction of frequency.

� 2 LC - RG + ( RG -� 2 LC) +� 2 ( LG + CR)
β =
2
The above equation shows that if the the term under the second radical be reduced to equal

( RG +� 2 LC) 2
Then the required condition for ß is obtained. Expanding the term under the internal radical and
forcing the equality gives

R 2 G 2 - 2� 2 LCRG +� 4 L2 C2 +� 2 L2 G 2 + 2�2 LCRG +�2 C2 R 2 = ( RG +�2 LC) 2

This reduces to

- 2� 2 LCRG +� 2 L2 G 2 +� 2 C2 R 2 = 0
( LG - CR) 2 = 0
Therefore the condition that will make phase constant a direct form od=f frequency is
LG = CR

A hypothetic al line m ight be built to f ulf ill this c ondition. The line would then have a value of ß
obtained by use of the above equation.
Already we know that the formula for the phase constant

β = � LC
Then the velocity of propagation will be

v = 1/ LC
This is the same for the all frequencies, thus eliminating the delay distortion.
We know that the equation for attenuation constant

RG -� 2 LC + ( RG -� 2 LC) 2 +� 2 ( LG + CR)
α =
2
May be made independent of frequency if the term under the internal radical is forced to reduce
to

( RG +� 2 LC) 2

Analysis shows that the condition for the distortion less line LG = CR , will produce the desired
result, so that it is possible to make attenuation constant and velocity independent of frequency
simultaneously . Applying the condition LG= RC to the expression for the attenuation gives

α = RG
This is the independent of fwww.Vidyarthiplus.com
requency, thus elim inating frequency distortion on a line. To achieve
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this condition
LG = CR
L R
=
C G
Require a very large value of L, since G is sm all. If G is intentionally increas ed, α and
attenuation are increased, resulting in poor line efficiency.
To reduce R raises the size and cost of the conductors above econom ic lim its, so that the
hypothetical results cannot be achieved.
THE TELEPHONE CABLE
In the ordinary telephone cable the wired are insulated with paper and twisted in pairs.
This construction results in negligible values of inductance and conductance so that reasonable
assumptions in the audio range of frequencies are that
Z=R
Y = j�C

We know that γ RG -� 2 LC + j� ( LG + CR)

=

γ = j�CR =

� CR
α =
2
� CR
With L=0, this equation becomes β =
2

� 2�
v= =
β CR
It should be observed that both α and the velocity are functions of frequency, such that
the higher frequencies are attenuated more and travel faster than the lower frequencies. Very
considerable frequency and delay distortion is the result on the telephone cable.
INDUCTANCE LOADING OF TELEPHONE CABLE
A distortion less line with distributed parameters sugest a rem edy for the severe
frequency and delay distortion experienced on long cables. It was indicated that it was
necessary the L/C ratio to achieve distortion less conditions. Heaviside suggested that the
inductance be increased,
And Pupin developed the theory that m ade possible this increase in the induc tance by
LUMPED INDUCTORS spaced at regular intervals along the line. This use of inductance is
c a l le d
In some submarine cables, distributed or uniform loading is obtained by winding the
cable with a high perm eability steel tape such as permalloy. This method is employed because
of the practical difficulties ofwww.Vidyarthiplus.com
designing lumped loading coils for such underwater circuits.
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For simplicity, consider first a
uniformly loaded cable circuit for which it may be assumed
that G= 0 and for which L has been increased so that �L is large with respect to R. Then
Z = R + j�L
Y = j�C
S in c e ,
π R
Z = R 2 +� 2 L2 � - tan -1
2 �
L
Then

γ =
Y
= π R π
R 2 +� 2 L2 � - tan -1 x�C�
2 �L 2
=� R2 π 1 R
LC 4 1 + 2
� - tan -1
� 2 2 �L
L2
R2
In view of the fact that R is small with respect to �L , the term may be dropped, and m ay
�2
L2
propagation constant becomes

π 1 R
γ = � LC � tan -1
- 2 2 �L
π 1 -1 R
If θ = - tan
2 2 �L
π 1 R 1 R �
cosθ = cos( tan -1 ) = sin �� tan -1 �
- 2 2 �L �2 �
�L

For a small angle

sin θ = tan θ = θ
R
cosθ =
2�
L
Finally the propagation constant may be written as,

� R �
γ = � LC (cosθ + j sin θ ) = � LC� + j�
� 2� �
L
Therefore, for the uniformly loaded cable,

R L
α =
2 C
β = � LC
� 1
v= =
β LC
It is readily obs erved that, www.Vidyarthiplus.com
under the assum ptions of G=0and �L large with respect to R,
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the
attenuation and velocity are both independent of frequency and the loaded cable will be
distortion less. The expression f or attenuation constant shows that the attenuation m ay be
reduced by increasing L, provided that R is not also increased too greatly.
Continuous or uniform loading is expensive and ac hieves only a sm all increase in L per unit
le n g t h .

Lum ped loading is ordinarily preferred as a means of transm ission improvem ent f or
cables. The im provem ent obtainable on open wire line is usually not sufficient to justify the
extra cost of the loading inductors.
C A M P B E L L ’ S E Q U A TI O N
An analysis for the performance of a line loaded at uniform intervals can be obtained by
considering a symmetrical section of line from the center of one loading coil to the center of the
next, where the loading coil of the inductance is Zc.

The section line may be replaced with an equivalent T section having symmetrical series
arms. Adopting the notation of filter circuits one of these series arms is called Z1/2 and is
Z1 Nγ
= Z 0 tanh
2 2

W here N is the num ber m iles between loading coils and γ is the propagation constant
per m ile. Upon including half a loading coil, the equivalent series arm of the loaded section
b e co m e s

Z 1' Z c Nγ
= + tanh
2 2 2
The shunt z2 arm of the equivalent T section is
Z0
Z2 =
sinh Nγ
An equation relating that γ and the circuit elem ent of a T section was already derived, which

may be applied to the loaded T section as

Z 1'
cosh Nγ ' = 1 +
2Z 2
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Z c / 2 + Z 0 tanh(Nγ / 2)

=1+
Z 0 / sinh Nγ

By use of exponential it can be shows that

Nγ cosh Nγ - 1
tanh =
2 sinh Nγ
So that the above equation reduces to
Zc
cosh Nγ ' = sinh Nγ + cosh Nγ
2Z 0
This expression is known as CampbeII’s Equation and perm its the determination of a value for
'
γ of a line section consisting partially of lumped land partially of distributed elements.
Campbell’s equation m akes possible the calculation of the effects of loading coils in
reducing attenuation and distortion on lines.

For a cable Z2 of the above f igure is essentially c apacitive and the cable c apacitance
plus lumped inductances appear similar to the circuit of the Iow pass fiIter
It is found that for frequencies below thw cutoff, given by
1
f0 =
π LC
The attenuation is reduced as expected, but above cutoff the attenuation rises as a
result of f ilter action. This cutoff frequency forms a definite upper lim it to successful
transmission over cables.
It can be raised by reducing L but this expedient alloes the attenuation to rise.
The c utoff frequency also be reduced by spacing the clos er together, thus reduc ing C
and more closely approximating the distributed constant line, but the cost increases rapidly.
In practice, a truIy distortion Iess Iine is not obtained by Ioading, because R and L
are to some extent functions of frequency. Eddy current Iosses in the Ioading inductors
aggravate this condition. However, a major improvement is obtained in the Ioaded cabIe
for a reasonabIe frequency range.

INPUT IMPEDANCE AND TRANSFER IMPEDANCE

The input impedance of a transmission line has already been obtained as

� Z R cosh γl + Z 0 sinh γl �
Z s = Z 0 = �� ��
� Z 0 cosh γl + Z R sinh γl �
In terms of exponentials, this is

� e γl + Ke γl �
Z S = Z 0 �� γl γl
�
� e - Ke �
�
If the voltage at the sending –end terminals is known, it is convenient to have the transfer
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impedance so that the
received current can be computed directly. The sending end voltage Es
is

E R (Z R + Z 0 ) γl
Es = (e + Ke - γl )
2Z R
I R (Z R + Z 0 ) γl
Es = (e + Ke - γl )
2
For which the transfer impedance is

E s (Z R + Z 0 ) γl
ZT = = (e + Ke - γl )
IR 2
By substituting for K, The above equation becomes

� e γl + e -γl � � e γl - e -γl �
Z T = Z R �� �� + �� ��
� 2 � � 2 �
This is recognizable as
Z T = Z R cosh γl + Z 0 sinh γl
If the expression is desired in terms of the hyperbolic functions.

Open and short circuited lines

As limited cases it is convenient to consider lines terminated in open circuit or short
circuit, that is with Z R = ∞or Z R = 0. The input impedance of a line of length l is

� Z R cosh γl + Z 0 sinh γl �
Z s = Z 0 = �� ��
� Z 0 cosh γl + Z R sinh γl �
And for the short circuit case Z R =0., so that

Z s = Z 0 tanh γl
Before the open circuit case is considered, the input impedance should be written

� cosh γl + (Z 0 / Z R ) sinh γl �
Z s = Z 0 �� ��
� (Z 0 / Z R ) cosh γl + sinh γl �
The input impedance of the open circuited line of length l, with ZR = ∞ , is

Z oc = Z 0 coth γl
By multiplying the above two equations it can be seen that

Z 0 = Z oc Z sc

This is the same result as was obtained for a lumped network. The above equation supplies a
very valuable means of experimentally determining the value of z0 of a line.
Also fro the same two equations

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Z sc
tanh γl =
Z oc
Z sc
γl = tanh -1
Z oc

Use of this equation in experimental work requires the determination of the hyperbolic tangent
of a complex angle. If
tanh γl = tah(α + jβ )l = U + jV
Then it can be shown that
2U
tanh 2αl
= 1+U 2 +V
and 2U
tanh 2βl 1-U 2 -V
=

he value of β is unc ertain as to quadrant. Its proper value may be selected if the approximate

velocity of propagation is known.

REFLECTION FACTOR AND REFLECTION LOSS
REFLECTION FACTOR

| 2 Z1 Z 2 |
K=
| Z1 + Z 2 |
The term K denotes the reflection factor. This ratio indicates the change in current in the load
due to reflection at the mismatched junction and is called the reflection factor.
REFLECTION LOSS
Reflection loss is defined as the number of nepers or decibles by which the current in
the load under image matched conditions would exceed the current actually flowing in the load.
This reflection loss involves the reciprocal of the reflection factor K.

Z1 + Z 2
Reflection loss, nepers= ln | |
2 Z1 Z 2

Z1 + Z 2
Reflection loss, db = 20 log | |
2 Z1 Z 2
INSERTION LOSS
Insertion loss of the line or network is defined as the number of nepers or decibels by
which the current in the load is changed by the insertion.
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T A N D п E Q U I V A LE N T T O LI N E S
The design of an equivalent T section from measurement on a network. These relations
were

Z 1 = Z 1oc - Z 2 oc (Z 1oc - Z 2 oc )
Z 2 = Z 2 oc - Z 2 oc (Z 1oc - Z 2 oc )
Z 3 = Z 2 oc (Z 1oc - Z 2 oc )

The input impedance of open circuited and short circuited lines were already developed.

Z0 � e γ l + e -γ l �
Z 1 oc = = Z 0 �� γ l �
tanh � e - e -γl

γl � �
� e γ l - e -γ l �
Z 1 sc = Z 0 tanh γ l = Z 0 �� γ l �
� e + e� �
- γl

Since a line is symmetrical network,

Z1oc = Z 2 oc

The Z 3 or shunt element of a T section that will be equivalent, in so far as external voltages

and currents are connected, to the long line can then be readily obtained as

Z0 � Z0 �
Z3 = �� - Z 0 tanh γl ��
tanh � tanh γl �
γl
Z0
=
sinh γl

The series elements for the equivalent section then are

� e γl + e - γ l 2 �
Z 1 = Z 2 = Z 1oc = Z 3 = �� γl - γl ��
�e -e
-γ l
e - e -γl �
� (e γl / 2 + e -γl / 2 ) 2 �
= Z 0 �� γl / 2 -γl / 2 γl / 2 -γ l /
��
� (e - e 2 )(e + e ) �
Z 1 = Z 2 = Z 0 tanh γ l / 2
The T – section equivalent for the long line, made up of these elements, is shown in the below
figure. It is useful in certain types of line calculations.

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A π - section equivalent for the line may likewise be determined from the terminal
measurements.
Because of symmetry,

Z 2oc Z 1sc
Z A = ZC =
Z 2 oc Z 2 oc (Z 1oc - Z 1sc )
Z 02
=
� e γl + e -γl � 2Z 0
Z 0 �� γl -γl
� - γl -γl
� e - e� � e - e
Therefore
Z0
ZA = ZC =
tanh(γl / 2)

The ZB arm simply obtained as

Z 2oc Z 1sc
ZB =
Z 2 oc (Z 1oc - Z 1sc )

Z 02
ZB = = Z 0 sinh
Z 0 / sinh γl γl

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U N IT – II

HIGH FREQUENCY TRANSMISSION LINES

INTRODUCTION
W hen a line, either open wire or coaxial, is used at frequencies of a megacycle or mor, it
is f ound that certain approxim ations m ay be em ployed leading to simplified analysis of line
performance . The assumptions usually made are:
1. Very considerable skin effect, so that currents m ay be assum ed as f lowing on a
conductor surfaces, internal inductance then being zero.
2. That � L>>R whe com puting Z. This assum ption is justifiable bec ause it is found

that the resistance increases because of skin effect with f while the line resistance
increases directly with f.
3. The lines are well enough constructed that G may be considered zero
The analysis is m ade in either of two ways, depending on whether R is merely sm all with
respect to � L or R is small, the line is considered completely negligible compared with � L.
If R is sm all, the line is considered one of sm all dissipation, and this concept is useful when
lines are em ployed as circ uit elem ents or where resonance properties are involved. If losses
were neglected then infinte current or voltages would appear in calculations, and and physical
reality would not be achieved.
In applications where losses may be neglegted, as in transmission of power at high efficiency,
R may be considered as negligible, and the line as one of zereo dissipation. These methods will
be studied separately.
STANDING WAVES
When the transmission line is not matched with its load i.e., load impedance is not equal
to the characteristic impedance ( Z R = Z 0 ) , the energy delivered to the load is reflected back to

the source.
The combination of incident and reflected waves give rise to the standing waves.
STANDING-WAVE RATIO
The m easurement of standing waves on a transm ission line yields inf orm ation about
equipment operating conditions. Maximum power is absorbed by the load when ZL = Z0. If a
line has no standing waves, the termination for that line is correct and maximum power transfer
t a k e s p la c e .
| VMAX|
VSW R =
|VMIN |
You have probably notic ed that the variation of standing waves s hows how near the rf
line is to being terminated in Z0. A wide variation in voltage along

31

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the length m eans a term ination far from Z0. A sm all variation m eans term ination near Z0.
Therefore, the ratio of the m aximum to the m inim um is a m easure of the perfection of the
term ination of a line. This ratio is called the STANDING-W AVE RATIO (SW R) and is always
expressed in whole num bers. For example, a ratio of 1:1 describes a line term inated in its
characteristic impedance (Z0).
V oI t a g e S t a n d i n g - W a v e R a t i o
The ratio of m aximum voltage to m inimum voltage on a line is called the V OLTAGE
STANDING-WAVE RATIO (VSWR). Therefore: The vertical lines in the formula indicate that the
enclosed quantities are absolute and that the two values are tak en with out regard to polarit y,
Depending on the nature of the standing waves, the num eric al value of VSWR ranges from a
value of 1 (ZL = Z0, no standing waves) to an infinite value for theoretically complete reflection.
Since there is always a small loss on a line, the minimum voltage is never zero and the
VSWR is always some finite value. However, if the VSW R is to be a useful quantity. the power
losses along the line must be sm all in com parison to the transm itted power voltage. Since
power is proportional to the s quare of the voltage, the ratio of the s quare of the m aximum and
minimum voltages is called the power standing- wave ratio. In a sense, the name is misleading
because the power along a transmission line does not vary.
Current Standing-Wave Ratio
The ratio of maximum to minimum current along a transm ission line is called CURRENT
STANDING- WAVE RATIO (ISWR). Therefore: This ratio is the same as that for voltages. It can
be used where measurements are made with loops that sample the m agnetic field along a line.
It gives the same results as VSWR measurements.
ST AN D I N G W AVE R AT I O
The ratio of the maximum to minimum magnitudes of voltage or current on a line having
standing waves is called the standing wave ratio or voltage standing wave ratio (VSW R)

V max I max
S= =
V min I min
Voltage equation is
V R (Z R + Z 0 ) - jβx
V = [e +jβ
x Ke ]
2Z R
Maxima of voltage occurs at which the incident and reflected waves are in phase

V R (Z R + Z 0 )
V m ax = [1 + K ]
2Z
R

Minima of voltage occurs at which the incident and reflected waves are out of phase

32
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V R (Z R + Z 0 )
V min = [1- K ]
2Z R

Vmax 1+ K
=
V min 1- K
V max
-1
V min
K =
V max
+1
V min

V max - V min
K =
| V min | +V min

|K|

SWR

This figure shows the relation between standing wave ratio S and reflection coefficient
O N E E I G T H W A V E LI N E
For the transmission line the voltage and current at any point x from the receiving end of
the transmission line is
V R (Z R + Z 0 )γx
V = [e + Ke -γx ]
2Z R

I R (Z R + Z 0 )γx
I= [e - Ke -γx ]
2Z 0

The term with γx is identified as the incident wave progressing forward from the source
-γx
to the load, where as the term involving e is the reflected wave traveling from load back
towards the source.

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33

For the line of zero dissipation, the attenuation constant is zero.

γ = jβ anZd 0 = R0

V R (Z R + Z 0 ) - jβx
V = [e jβ
x + Ke ]
2Z R
After simplification of the above equation for standing wave |K| = 1

V =V coR s β x + jI R R si0n β x.
Similarly, for the current on the transmission line

I = I R cos βx + jV R / R0 sin βx.

The input impedance of a dissipation line is
V
Zs =
I
V R cos βx + jI R0Rsin βx
=
I R cos βx + jV R / R0 sin βx

� Z R cos βx + jR 0 sin βx �
= R0 � �
� R0 cos βx + jZ R sin �
βx
Z � + jR t an β x
Z = � R 0
Or
R�0 + jZ �tan β �
R �
x
For an eighth wave line
2π λ
x = λ / 8, βx = .
λ 8
Z � + jR0 tan(π / 4) �
Z s = R0 R�
R0�+ jZ R tan(π / 4) � �
Z � + jR0 �
Z s = R0 R�
R0�+ jZ R � �

If such a line is terminated with pure resistance

Z R = RR
� R R + jR0 �
ZS = � �
� R0 + jR� R
Since, both the numerator and denominator have identical magnitudes, then

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34

| Z s |= R0
Thus an eighth – wave line m ay be used to transfer any resistance to im pedance with a
magnitude equal to R0 of the line, or obtain a m agnitude m atch between a resistance of any

value and R0 , the internal resistance of the source

Q U A R T E R W A V E LI N E A N D I M P E D A N C E M A T C H I N G
The input impedance of a dissipation transmission line is

Z R� + jR0 tan βx �
Z s = R0 � �
R0�+ jZ R tan βx �
Z R�/ tan βx + jR0 �
Z s = R0 �
R0�/ tan βx + jZ R � �

For a quarter wave line

x = λ / 4, βx = 2π / λ * λ / 4 = λ / 2

Substituting the param eter value in the above equation the sending end im pedance of the
quarter wave transformer is

2
R0
Zs =
ZR

A quarter wave section of line m ay be c onsidered as a transform er to m atch a load of

'
Z R to a source of Z s . Such a m atch can be obtained if the characteristic im pedance R0 of the
matching quarter wave section of the line is properly chosen.

R0 =| ' Z s Z R |

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UNIT III

IMPEDANCE MATCHING IN HIGH FREQUENCY

APLLICATIONS OF QUARTER WAVE TRANSFORMAER

 A quarter wave transformer may also be used if the load is not a pure resistance
.
 The quarter wave transformer is a single frequency or narrow band
device. The bandwidth may be increased by using two or more
Quarter wave section in series.
 A quarter wave transformer may be considered as an impedance
Inverter in that it can transform a low impedance into a high
Impedance and vice versa.

35

Zs ZR

λ 4

H AL F W AVE L I N E
Already we know that The input impedance of a dissipation less line is

Z � + jR0 tan βx �
Z s = R0 R� �
R0�+ jZ R tan βx �
For a quarter wave line
x = λ / 2, βx = 2π / λ * λ / 2 = π

� Z+R jR0 tanπ �

Z s = R0 � �
R0�+ jZ R tanπ �

ZR
R0
R0
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Zs = ZR

This line may be considered as one to one transformer.

APPL I C AT I O N :
It is used in connecting a load to a source in cases when the load and source cannot be
made adjacent.
ST U B M AT C H I N G
N e e d f or s t ub m a t c h i ng
To match the load impedance to be equal to the input impedance.
T YPES
1. SINGLE STUB MATCHING
2. DOUBLE STUB MATCHING

36

SINGLE STUB MATCHING ON A LINE

For greatest eff iciency and delivered power, a high frequency transm ission line s hould
be operated as a sm ooth line or with an R0 term ination. However, the usual loads, s uch as
antennas, do not in general have resistance of value equal to R0, so that m any cases it is
necessary to introduce some form of impedance – transform ing action between line and load to
make the load appear to the line as a resistance value R0.
The quarter wave line or transformer and the tapered line are such impedance matching
devices. Another means of accomplishing the desired result is the use of an open or closed
stub line of suitable length as a reactance shunted across the transmission line at the load to
resonance with an antiresonant resistance equal to R0.

Voltage minimum before insertion of the stub

S1
d S1
A

ZR
Ys Yd
A

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Since the input conductance of the of a line is S/R0 at a voltage maximum and S/R0 at
a voltage m inimum, then at some interm ediate [point A the real part of the input adm ittance
may be an intermediate value of 1/R0 or the input admittance at A has a value
1
Ys = ± jβ
R0
The susceptance B is the shunt value at the point in question. Af ter the point having a
conductance equal to 1/R0 is loc ated, a s hort stub line having input s usc eptance of � β m ay

be connected across the transmission line. The input admittance at this point then is
1 1
Ys = ± jβ � jβ =
R0 R0
Or the input impedance of the line at point A looking towards the load is
Z s = R0

37

Since both the location and length of the stub must be determ ined, two independent
measurements must be made on the original line and load to secure sufficient data.
The most easily obtained measurements are the standing wave ratio S and the position
of a voltage m inimum, usually the m inimum nearest to the load. A voltage m inimum is chosen
rather than a maximum, since its position usually can be determined more accurately.
If the location of the stub is fixed with respect to an original voltage m inimum, no
knowledge of the load impedance is needed.
Because of the paralleling of elem ents, it is most convenient to work with adm ittances.
The input im pedance equation is looking towards the load f rom any point on the line, m ay
w r it t e n a s

1 �|1-K�φ - 2βs �
Ys = �� 1+ ��
R0 �| K�φ - 2βs �
Writing G0=1/R0 and changing to rectangular coordinates gives

1- �| K | cos(φ - 2βs) - j | K | sin(φ - 2βs) �

Ys = G0 �1- �
�| K | cos(φ - 2βs) + j | K | sin(φ - 2βs) �
And upon rationalizing,

1-�| K | 2 -2 j | K | sin(φ - 2βs) �

Ys = G0 �1- 2 �
� | K | +2 j | K | cos(φ - 2βs)�
Expressing the shunt conductance as a dimensionless ratio Gs/Go, or on a per unit basis,

Gs � 1- | K | 2 �
�1=- 2 | K | cos(φ - 2βs)+ | K |2 �
G0 � �
And the shunt susceptance on a per unit basis is

Bs
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� �
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1- | K | 2 -2 j | K | sin(φ - 2βs)
� | K | 2 +2 j | K | cos(φ - 2βs)
�
�

After simplifying the above equations we get the location and distance of the stub
The distance d from the voltage minimum to the point of stub connection is
d = s 2 - s1
cos -1 | K |
d=
2β
s -1 λ�
cos -1 �� �
s +1� �4
=
π

Before connection of the stubthe equation is

λ S -1
L = t an
2π S -1

38

The stub length should be

λ
L' = -L
2

The CircIe diagram for dissipation Iess Iine

A somewhat similar circle diagram may be obtained, however, that solves the
impedance equation and simplifies the design of dissipationless lines considerably. The input
impedance equation for a dissipation less line may be written as
Z s 1+ | K | �φ - 2βs
=
R0 1- | K | �φ - 2βs
Zs
= ra + jx a
R0
An actual circle will have the radius
1
2 S-
S -1 S
r= =
2S 2
And the center of the circle on the positive is
1
S+
S + 12 S
c= =
2S 2
A family of circles may be drawn for successive values of S as in fig. In drawing particular
circles it is interesting to note that for any circle the intercept near origin is at 1/S, and that far
removed from the origin is at S units on the ra axis

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1.5 S=3 s =2 . 5
1 222
0.5 2222 s =2
11111 s =1 .5

- 0 .5
-1
- 1 .5

The minimum value for S is unity.The above figure shows that all S circles must
surround the 1,0 point. In fact, the circle for S = 1 is represented by the 1,0 point.

39

The maximum value of S is infinity, for the case of open circuit or short

circuit line
termination. As S increase, the radius of the S increases, and the center moves to the right; for
the limiting case of S = infinity, the circle becomes the xa axis.
The line impedance is maximum. And

Zs 1+ | K |
=S=
R0 1- | K |

Zs
W hen terminates at the circle intercept 1/S, the line impedance has a minimum
R0
v a lu e , a n d
Z s 1 1- | K |
==
R0 S 1+ | K |
After some simplification we get the final equation, that
2
� 1 � 1
ra2
+ � x a �+ �� = 1 + 2
� t an � t an
1 2βs 2βs
= 2
sin 2βs
Lines of equal βs are the seen to be circles of radius

1
= 2
sin 2βs
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With the shift of center downward on the xa axis (ordinate)

1
=-
tan 2 2βs
S- CIRCLE
2 2
� � S 12 + �� S � + 122 �
�g a -
� 2S �
� � + b a = � 2S ��
�
� � � �� � �

AppIication of the circIe diagram

 Used to find the input impedance of a line of any chosen length
An open circuited line has S∞= , the corresponding S circ le appearing as the
vertical axis. The input im pedance is then pure reactanc e, with the value for
various electrical lengths determ ined by the intersections of the corresponding
βs with the vertical axis.
 The input admittance of the line may be found by this method.

A short circuited lin may be solved by determining its admittance. The S circle is
again the vertical axis, and susceptance values may be read off at appropriate
intersections of the βs circles with the vertical axiS

SM I T H C H AR T

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An impedance Smith chart (with no data plotted)

. The Sm ith Chart can be used to repres ent many param eters including impedances,
admittances, refIection coefficients, scattering parameters, noise figure circles, constant
gain contours and regions for unconditional stability. The Smith Chart is most frequently used at
or within the unity radius region. However, the remainder is still mathematically relevant, being
used, for example, in oscillator design and stability analysis
The Smith Chart is plotted on the complex reflection coefficient plane in two dim ensions
and is scaled in norm alized im pedance (the most comm on), norm alized adm ittance or both,
using different colors to distinguish between them. These are often k nown as the Z, Y and YZ
[7]
Smith Charts respectively. Normalized scaling allows the Smith Chart to be used for problems
involving any characteristic impedance or system im pedance, although by far the most
commonly used is 50 ohms. W ith relatively simple graphical construction it is straightforward to
convert between normalized im pedance (or norm alized adm ittance) and the corresponding
complex voltage reflection coefficient.

The Sm ith Chart has circumferential scaling in wavelengths and degrees. The
wavelengths scale is used in distributed com ponent problems and represents the distance
measured along the transm ission line connected between the generator or source and the load
to the point under consideration. The degrees scale represents the angle of the voltage
reflection coefficient at that point. The Sm ith Chart m ay also be used for lum ped elem ent
matching and analysis problems.

Use of the Sm ith Chart and the interpretation of the res ults obtained us ing it requires a
good understanding of AC c ircuit theory and transm ission line theor y, both of which are pre-
requisites for RF engineers.

As impedances and admittances change with frequenc y, problems using the Sm ith
Chart can only be solved m anually using one frequency at a tim e, the result being represented
by a point. This is often adequate for narrow band applications (typically up to about 5% to 10%
bandwidth) but for wider bandwidths it is usually necessary to apply Smith Chart techniques at
m ore than one frequenc y ac ross the operating frequency band. Provided the frequenc ies are
sufficiently close, the resulting Sm ith Chart points m ay be joined by straight lines to create a
locus.

A locus of points on a Smith Chart covering a range of frequencies can be used to visually
represent:

• How capacitive or how inductive a load is across the frequency range

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• How difficult matching is likely to be at various frequencies

• How well matched a particular component is.

The accuracy of the Smith Chart is reduced for problems involving a large spread of
im pedances or adm ittances, although the scaling can be m agnif ied for individual areas to
accommodate these.
Regions of the Z Smith Chart

If a polar diagram is m apped on to a cartesian coordinate system it is conventional to

measure angles relative to the positive x-axis using a counter-clock wise direc tion f or positive
angles. The m agnitude of a com plex num ber is the length of a straight line drawn from the
origin to the point representing it.

The Sm ith Chart uses the same c onvention, noting that, in the norm alized im pedance
plane, the positive x-axis extends from the center of the Smith Chart at to the point . The region
above the x-axis represents induc tive im pedances and the region below the x-axis represents
capacitive impedances. Inductive im pedances have positive im aginary parts and capacitive
impedances have negative imaginary parts.

If the termination is perfectly matched, the reflection coefficient will be zero, represented
effectively by a circle of zero radius or in fact a point at the centre of the Smith Chart. If the
termination was a perfect open circuit or short circuit the magnitude of the reflection coefficient
would be unity, all power would be reflected and the point would lie at some point on the unity
circumference circle.

CircIes of Constant NormaIized Resistance and Constant NormaIized Reactance

The normalized impedance Sm ith Chart is composed of two families of circles: circles of
constant norm alized resistance and circ les of constant norm alized reac tance. In the com plex
ref lection coefficient plane the Sm ith Chart occupies a circle of unity radius centered at the
origin. In Cartesian coordinates theref ore the circ le would pass through the points (1,0) and (-
1,0) on the x-axis and the points (0,1) and (0,-1) on the y-axis.

Working with both the Z Smith Chart and the Y Smith Charts

In RF circuit and m atching problems som etimes it is m ore convenient to work with
adm ittances (representing conductance’s and susceptances) and sometim es it is more
convenient to work with im pedances (representing resistances and reactance’s). Solving a
typical matching problem will often require several changes between both types of Sm ith Chart,
using norm alized im pedance for series elements and norm alized adm ittances for parallel
elements. For these a dual (normalized) im pedance and admittance Smith Chart may be used.
Alternatively, one type may be used and the scaling converted to the other when required.

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In order to change from normalized im pedance to normalized adm ittance or vice versa,
the point representing the value of ref lection c oefficient under consideration is m oved through

43

exactly 180 degrees at the same radius. For example the point P1 in the example representing
a reflection coefficient of has a norm alized im pedance of. To graphic ally change this to the
equivalent norm alized admittance point, say Q1, a line is drawn with a ruler from P1 through the
Smith Chart centre to Q1, an equal radius in the opposite direction. This is equivalent to m oving
the point through a circular path of exactly 180 degrees. Reading the value from the Sm ith
Chart for Q1, remembering that the scaling is now in normalized admittance, gives .

Once a transformation from impedance to admittance has been performed the scaling changes
to normalized admittance until such time that a later transformation back to normalized
impedance is performed.

PROBLEMS ON SINGLE STUB MATCHING

1.Determine the Iength and the distance of the stub from the Ioad. Given that a
compIex Ioad ZL= 50-j100 is to be matched to a 75 ohm transmission Iine using a short
circuited stub.

G iv e n
Characteristic impedance of the transmission line Z0= 75ohm
Load impedance to be matched to the transmission line ZL= 50-j100
To find
1D. istance of the stub from the load
2L.ength of the stub from the load
S oI u t i o n
1T.he norm alized im pedance is determ ined by dividing the load im pedance by the
characteristic impedance of the transmission line.
Z L 50 - j100
ZL = = = 0.667 - j1.33
Z0 75
2T. he normalized im pedance, ZL is plotted on the smith chart by determ ining the point
of intersection between the constant R circle with R = 0.667 and constant X circle
w it h X = 1 . 3 3
The impedance circle is drawn.
Because the stubs are connected in parallel with the load, adm ittances can be much
easily used rather than impedances to simplify the calculations.
3T. he normalized adm ittance is determ ined from the smith chart by sim ply rotating the
im pedance plot, by 180degree. This is sim ply done by drawing a line from point A
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through the center of the chart
to the opposite side of the circle, point B.

4. the admittance point is rotated clockwise to a point on the impedance circle where it
intersects the characteristic im pedance Z 0 . At the point C. The real com ponent of

the input im pedance at this point is equal to the characteristic impedance Z0 . At this

point C, the admittance is y=1+j1.7.

5. The distance from point B to point C, in terms of the wavelength is how far from load
the stub must be placed,
The stub m ust have a zero resistive com ponent im pedance and susceptance that
has the opposite polarity.
6. To determine the length of the shorted stub that has an opposite reactive component
to the input adm ittance, the outside of the Sm ith chart (R=0) is m oved around with
the starting point at D {since at point D t= 0 and hence γ = ∞ }, until an adm ittance y

= 1.7isfound
7. The distance between point D and E is the length of the stub. For this quantity the
from the smith chart,

DOUBLE STUB MATCHING

2. Using DoubIe stub matching, match a compIex Ioad of ZL = 18.75+j56.25 to a
Iine with characteristic impedance Z0 = 75ohm.
Determine the stub Iengths, assuming a quarter w aveIength spacing are
maintained between the two short circuited stubs.

A spacing of λ / 4 is m aintained between the stubs, stub2 and s tub1. For sm ooth line
operation of the transm ission line the input im pedance looking into the term inals 2,2 of
the line should be,

Y2,2 = 1/ Z 0

The stub at 1,1 must be capable to transform the admittance at the terminating
impedance end to the circle B which is displaced from the circle A; R=1 by ‘ λ / 4 ’.
The quarter wavelength line will further transform the admittance into a value at 2,2
which plot on the circle A. Thus the line to load distance between position 2,2 is not
required to be determined.

45
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ZT
Terminating impedance

Ls2
Ls1

The normalized load impedance

Zi 18.75 + j 56.25
ZL = =
Z0 75
Z L = 0.25 + j 0.75
Plotting the normalized impedance on the Smith chart, the impedance circle is drawn
with distance between the point (1,0) and the point of the normalized impedance as the
radius {distance, OA}

1. Moving by 180 degree (0.25 λ ) on the impedance circle , that is at a diam etrically
opposite point to the point A, i.e., point B will give the normalized admittance.
From the smith chart YL= 0.4-j1.2
2. Circle A is the constant R circle for R = 1. Circle B is the locus of all the points on the
circle A is displaced by λ /4, quarter wavelength. The stub 1 adds a suscepatance of all
the points on the circle B.
Since stub 1 cannot alter the conductance , to a point on the circle B, point C,

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46

Y(at point C)= 0.4-j0.5

3. Transferring the point C to the point D on the circle A, since the line between 1,1 and
2,2 is a quarter wave line that transforms the admittance at 1,1 to 2,2 such that the
conductance equals the characteristic conductance, 1/ Z0.
Y ( A t p o in t D ) = 1 . 0 + j1 . 2
4. The stub length at 2,2 should cancel the imaginary part of the above admittance of the
stub at 2,2 must be -1.2.
5. To find the length of the stub with an admittance ,
(a) +j0.7 and (b) –j1.2
The outside circle of the smith chart (the circle, R=0), is moved around having a
reference at a point P, until

An admittance y= -1.2 is found at point E and

An admittance y= +0.7 is found at point F.

6. From the smith chart,

Length of the stub 1= distance between P and F Ls1=0.348 λ
Length of the stub 2= distance between P and F Ls2=0.11 λ

3. Determine the foIIwing:

(a) Standing wave ratio(VSWR)
(b) Load Admittance
(c) Impedance of the transmission Iine at the maximum and minimum of the
stationary waves aIong the Iine
(d) Distance between Ioad and first voItage maximum. For a transmission Iine with
characteristic impedance of 50 ohm with a receiving end of 100+j121. The
waveIength of the eIectricaI signaI aIong the Iine is 2.5m.
G i v e n:
Characteristic impedance Z0=50 ohm
Load impedance ZL= 100+j121ohm
Wavelength of the electrical signal λ =2.5

S o lu t io n
100 + j121
1. Normalized impedance = =2+j2.42
50
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Ploting the point p on the smith chart . The impedance circle is drawn with O(1+j0) centre and
radius as (OP), the distance between centre and the normalized voltage standing Wave Ratio =
5
2. The point Q diametrically opposite to the normalized impedance point on the impedance
circle is the normalized admittance of the load.
Y
= 0.22 - j 0.25
G0
YZ 0 = 0.22 - j 0.25
Load impedance
1
Y = (0.22 - j 0.25)
50
= 0.0044 - j0.005 mho

3. Impedance at the first voltage maximum from load =

5× Z0
= 25 0 o h m

Impedance at the first voltage minimum=

0.2 × Z 0
= 10ohm

Distance between load and first voltage maximum=

0.042λ
= 0,042 × 2.5m
= 0.105m

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UNIT -IV PASSIVE FILTERS

1. Neper

A neper (Symbol: Np) is a logarithmic unit of ratio. It is not an SI unit but is

accepted for use alongside the SI. It is used to express ratios, such as gain and loss,
and relative values. The name is derived from John Napier, the inventor of
logarith ms.

Like the decibel, it is a unit in a logarithmic scale, the difference being that where
the decibel uses base-10 logarithms to compute ratios, the neper uses base e ≈
2.71828. The value of a ratio in nepers, Np, is given by

where x1 and x2 are the values of interest, and ln is the natural logarithm.

The neper is often used to express ratios of voltage and current amplitudes in
electrical circuits (or pressure in acoustics), whereas the decibel is used to express
power ratios. One kind of ratio may be converted into the other. Considering that
wave power is proportional to the square of the amplitude, we have

and

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The decibel and the neper have a fixed ratio to each other. The (voltage) level is

Like the decibel, the neper is a dimensionless unit. The ITU recognizes both units.

2. Decibel

The decibel (dB) is a logarithmic unit of measurement that expresses the

magnitude of a physical quantity (usually power or intensity) relative to a specified
or implied reference level. Since it expresses a ratio of two quantities with the
same unit, it is a dimensionless unit. A decibel is one tenth of a bel, a seldom-used
unit.

The decibel is widely known as a measure of sound pressure level, but is also used
for a wide variety of other measurements in science and engineering (particularly
acoustics, electronics, and control theory) and other disciplines. It confers a
number of advantages, such as the ability to conveniently represent very large or
small numbers, a logarithmic scaling that roughly corresponds to the human
perception of sound and light, and the ability to carry out multiplication of ratios
by simple addition and subtraction.

The decibel symbol is often qualified with a suffix, which indicates which
reference quantity or frequency weighting function has been used. For example,

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"dBm" indicates that
the reference quantity is one milliwatt, while "dBu" is referenced to 0.775
volts RMS.[1]
The definitions of the decibel and bel use base-10 logarithms. For a similar unit
using natural logarithms to base e, see neper.

Definitions

A decibel is one-tenth of a bel, i.e. 1 B=10 dB. The bel (B) is the logarithm of the
ratio of two power quantities of 10:1, and for two field quantities in the ratio
[8]
. A field quantity is a quantity such as voltage, current, sound pressure,
electric field strength, velocity and charge density, the square of which in linear
systems is proportional to power. A power quantity is a power or a quantity
directly proportional to power, e.g. energy density, acoustic intensity and luminous
intensity.

The calculation of the ratio in decibels varies depending on whether the quantity
being measured is a power quantity or a field quantity.

Power quantities

When referring to measurements of power or intensity, a ratio can be expressed in

decibels by evaluating ten times the base-10 logarithm of the ratio of the measured
quantity to the reference level. Thus, if L represents the ratio of a power value P1 to
another power value P0, then LdB represents that ratio expressed in decibels and is
calculated using the formula:

P1 and P0 must have the same dimension, i.e. they must measure the same type of
quantity, and the same units before calculating the ratio: however, the choice of
scale for this common unit is irrelevant, as it changes both quantities by the same
factor, and thus cancels in the ratio—the ratio of two quantities is scale-invariant.
Note that if P1 = P0 in the above equation, then LdB = 0. If P1 is greater than P0 then
LdB is positive; if P1 is less than P0 then LdB is negative.

Rearranging the above equation gives the following formu la for P1 in terms of P0
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and LdB:

Since a bel is equal to ten decibels, the corresponding formulae for measurement in
bels (LB) are

Field quantities

When referring to measurements of field amplitude it is usual to consider the ratio

of the squares of A1 (measured amplitude) and A0 (reference amplitude). This is
because in most applications power is proportional to the square of amplitude, and
it is desirable for the two decibel formulations to give the same result in such
typical cases. Thus the following definition is used:

This formula is sometimes called the 20 log rule, and similarly the formula for
ratios of powers is the 10 log rule, and similarly for other factors.[citation needed] The

equivalence of and is of the standard properties of

logarithms.

The formula may be rearranged to give

Similarly, in electrical circuits, dissipated power is typically proportional to the

square of voltage or current when the impedance is held constant. Taking voltage
as an example, this leads to the equation:

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where V1 is the voltage being measured, V0 is a specified reference voltage, and

GdB is the power gain expressed in decibels. A similar formula holds for current.

An example scale showing x and 10 log x. It is easier to grasp and compare 2 or 3

digit numbers than to compare up to 10 digits.

Note that all of these examples yield dimensionless answers in dB because they are
relative ratios expressed in decibels.

• To calculate the ratio of 1 kW (one kilowatt, or 1000 watts) to 1 W in

decibels, use the for mula

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• To calculate the ratio of to in decibels, use the

formula

Notice that , illustrating the consequence from the

definitions above that GdB has the same value, , regardless of whether it is
obtained with the 10-log or 20-log rules; provided that in the specific system being
considered power ratios are equal to amplitude ratios squared.

• To calculate the ratio of 1 mW (one milliwatt) to 10 W in decibels, use the

f o r mu l a

• To find the power ratio corresponding to a 3 dB change in level, use the

formula

A change in power ratio by a factor of 10 is a 10 dB change. A change in power

ratio by a factor of two is approximately a 3 dB change. More precisely, the factor
is 103/10, or 1.9953, about 0.24% different from exactly 2. Similarly, an increase of
3 dB implies an increase in voltage by a factor of approximately , or about 1.41,
an increase of 6 dB corresponds to approximately four times the power and twice
the voltage, and so on. In exact terms the power ratio is 10 6/10, or about 3.9811, a
relative error of about 0.5%.

Merits

The use of the decibel has a number of merits:

• The decibel's logarithmic nature means that a very large range of ratios can
be represented by a convenient number, in a similar manner to scientific
no tation. This allows one to clearly visualize huge changes of some quantity.
(See Bode Plot and half logarithm graph.)

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• The mathematical properties of logarithms mean that the overall decibel gain
of a multi-component system (such as consecutive amplifiers) can be
calculated s imp ly by summing the decibel ga ins of the individual
co mpone nts, rather than needing to multiply amp lif ication factors.
Essentially this is because log(A × B × C × ...) = log(A) + log(B) + log(C) +
...
• The human perception of, for example, sound or light, is, roughly speaking,
such that a doubling of actual intensity causes perceived intensity to always
increase by the same amount, irrespective of the original level. The decibel's
logarithmic scale, in which a doubling of power or intensity always causes
an increase of approximately 3 dB, corresponds to this perception.

Absolute and relative decibel measurements

Although decibel measurements are always relative to a reference level, if the

numerical value of that reference is explicitly and exactly stated, then the decibel
measurement is called an "absolute" meas urement, in the sense that the exact value
of the measured quantity can be recovered using the formula given earlier. For
example, since dBm indicates power measurement relative to 1 milliwatt,

• 0 dBm means no change from 1 mW. Thus, 0 dBm is the power level
corresponding to a power of exactly 1 mW.
• 3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power level
corresponding to 103/10 × 1 mW, or approximately 2 mW.
• -6 dBm means 6 dB less than 0 dBm. Thus, -6 dBm is the power level
corresponding to 10-6/10 × 1 mW, or approximately 250 μW (0.25 mW).

If the numerical value of the reference is not explicitly stated, as in the dB gain of
an amplifier, then the decibel measurement is purely relative. The practice of
attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu,
dBA, etc, is not permitted by SI. [10] However, outside of documents adhering to SI
units, the practice is very common as illustrated by the following examples.

Absolute measurements

Electric power

dBm or dBmW

dB(1 mW) — power measurement relative to 1 milliwatt. X dBm = XdBW + 30.

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d BW

dB(1 W) — similar to dBm, except the reference level is 1 watt. 0 dBW =

+30 dBm; -30 dBW = 0 dBm; XdBW = XdBm - 30.

Vo l t a g e

Since the decibel is defined with respect to power, not amplitude, conversions of
voltage ratios to decibels must square the amplitude, as discussed above.

A schematic showing the relationship between dBu (the voltage source) and dBm
(the power dissipated as heat by the 600 Ω resistor)

d BV

dB(1 VRMS) — voltage relative to 1 volt, regardless of impedance. [1]

dBu or dBv

dB(0.775 VRMS) — voltage relative to 0.775 volts.[1] Originally dBv, it was

changed to dBu to avoid confusion with dBV. [11] The "v" comes from "volt",
while "u" comes from "unloaded". dBu can be used regardless of impedance,
but is derived from a 600 Ω load dissipating 0 dBm (1 mW). Reference
vo lta ge

dBmV

dB(1 mVRMS) — voltage relative to 1 millivolt across 75 Ω[12]. Widely used

in cable television networks, where the no minal strength of a single TV
signal at the receiver ter minals is about 0 dBmV. Cable TV uses 75 Ω
coaxial cable, so 0 dBmV corresponds to -78.75 dBW (-48.75 dBm) or ~13
nW.

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dBμV or dBuV

dB(1 μVRMS) — voltage relative to 1 microvolt. Widely used in television

and aerial amplifier specifications. 60 dBμV = 0 dBmV.

3. Properties of Symmetrical Networks and Characteristic impedance of

Symmetrical Networks

A two-port network (a kind of four-ter minal network or quadripole) is an electrical

circuit or device with two pairs of ter minals connected together internally by an
electrical network. Two ter minals constitute a port if they satisfy the essential
requirement known as the port condition: the same current must enter and leave a
port. Examples include small-signal models for transistors (such as the hybrid-pi
model), filters and matching networks. The analysis of passive two-port networks
is an outgrowth of reciprocity theorems first derived by Lorentz[3].

A two-port network makes possible the isolation of either a complete circuit or part
of it and replacing it by its characteristic parameters. Once this is done, the isolated
part of the circuit becomes a "black box" with a set of distinctive properties,
enabling us to abstract away its specific physical buildup, thus simplifying
analysis. Any linear circuit with four terminals can be transformed into a two-port
network provided that it does not contain an independent source and satisfies the
port conditions.

There are a number of alternative sets of parameters that can be used to describe a
linear two-port network, the usual sets are respectively called z, y, h, g, and ABCD
parameters, each described individually below. These are all limited to linear
networks since an underlying assumption of their derivation is that any given
circuit condition is a linear superposition of various short-circuit and open circuit
conditions. They are usually expressed in matrix notation, and they establish
relations between the variables

Input voltage
Output voltage
Input current
Output current

These current and vo ltage variables are most use ful at low-to- moderate
frequencies. At high frequencies (e.g., microwave frequencies), the use of power
and energy variables iwww.Vidyarthiplus.com
s more appropriate, and the two-port current–vo lta ge
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approach is replaced by
an approach based upon scattering parameters.
The terms four-terminal network and quadripole (not to be confused with
quadrupole) are also used, the latter particularly in more mathematical treatments
although the term is becoming archaic. However, a pair of terminals can be called
a port only if the current entering one terminal is equal to the current leaving the
other; this definition is called the port condition. A four-ter minal network can only
be properly called a two-port when the terminals are connected to the external
circuitry in two pairs both meeting the port condition.

4. voltage and current ratios

In order to simplify calculations, sinusoidal voltage and current waves are

commonly represented as complex-valued functions of time denoted as and .[7][8]

Impedance is defined as the ratio of these quantities.

Substituting these into Ohm's law we have

Noting that this must hold for all t, we may equate the magnitudes and phases to
obtain

The magnitude equation is the fa miliar Ohm's law applied to the voltage and
current amplitudes, while the second equation defines the phase relationship.

Validity of complex representation

This representation using complex exponentials may be justified by noting that (by
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Euler's formula):

i.e. a real-valued sinusoidal function (which may represent our voltage or current
waveform) may be broken into two complex-valued functions. By the principle of
superposition, we may analyse the behaviour of the sinusoid on the left-hand side
by analysing the behaviour of the two complex terms on the right-hand side. Given
the symmetry, we only need to perform the analysis for one right-hand term; the
results will be identical for the other. At the end of any calculation, we may return
to real-valued sinusoids by further noting that

In other words, we simply take the real part of the result.

Phasors

A phasor is a constant complex number, usually expressed in exponential form,

representing the complex amplitude (ma gnitude and phase) of a sinusoidal function
of time. Phasors are used by electrical engineers to simplify computations
involving sinusoids, where they can often reduce a differential equation problem to
an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor
voltage across the element to the phasor current through the element, as determined
by the relative amplitudes and phases of the voltage and current. This is identical to
the definition from Ohm's law given above, recognising that the factors of
cancel

5. Propagation constant

The propagation constant of an electromagnetic wave is a measure of the change

undergone by the amplitude of the wave as it propagates in a given direction. The

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quantity being measured can be the voltage or current in a circuit or a field vector
such as electric field strength or flux density. The propagation constant itself
measures change per metre but is otherwise dimensionless.

The propagation constant is expressed logarithmically, almost universally to the

base e, rather than the more usual base 10 used in teleco mmunications in other
situations. The quantity measured, such as voltage, is expressed as a sinusiodal
phasor. The phase of the sinusoid varies with distance which results in the
propagation constant being a complex number, the imaginary part being caused by
the phase change.

Alternative names

The term propagation constant is somewhat of a misnomer as it usually varies

strongly with ω. It is probably the most widely used term but there are a large
variety of alternative names used by various authors for this quantity. These
include, transmission parameter, transmission function, propagation parameter,
propagation coefficient and transmission constant. In plural, it is usually implied
that α and β are being referenced separately but collectively as in transmission
parameters, propagation parameters, propagation coe fficie nts, transmission
constants and secondary coefficients. This last occurs in transmission line theory,
the term secondary being used to contrast to the primary line coefficients. The
primary coefficients being the physical properties of the line; R,C,L and G, from
which the secondary coefficients may be derived using the telegrapher's equation.
Note that, at least in the field of transmission lines, the term transmission
coefficient has a different meaning despite the similarity of name. Here it is the
corollary of reflection coefficient.

Definition

The propagation consta nt, symbol γ, for a given system is defined by the ratio of
the amplitude at the source of the wave to the amplitude at some distance x, such
that,

Since the propagation constant is a complex quantity we can write;

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wh e r e

α, the real part, is called the attenuation constant

β, the imaginary part, is called the phase constant

That β does indeed represent phase can be seen from Euler's formula;

which is a sinusoid which varies in phase as θ varies but does not vary in amplitude
because;

The reason for the use of base e is also now made clear. The imaginary phase
constant, iβ, can be added directly to the attenuation constant, α, to form a single
complex number that can be handled in one mathematical operation provided they
are to the same base. Angles measured in radians require base e, so the attenuation
is likewise in base e.

For a copper transmission line, the propagation constant can be calculated from the
primary line coefficients by means of the relationship;

where;

, the series impedance of the line per metre and,

, the shunt admittance of the line per metre.

Attenuation constant

In telecommunications, the term attenuation constant, also called attenuation

parameter or coefficient, is the attenuation of an electromagnetic wave propagating
through a medium per unit distance from the source. It is the real part of the
propagation consta nt a nd is measured in nepers per metre. A ne per is
approximately 8.7dB. Attenuation constant can be defined by the amplitude ratio;

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The propagation constant per unit length is defined as the natural logarithm ic of
ratio of the sending end current or voltage to the receiving end current or voltage.

Copper lines

The attenuation constant for copper (or any other conductor) lines can be
calculated from the primary line coefficients as shown above. For a line meeting
the distortionless condition, with a conductance G in the insulator, the attenuation
constant is given by;

however, a real line is unlikely to meet this condition without the addition of
loading coils and, furthermore, there are some decidedly non-linear effects
operating on the primary "constants" which cause a frequency dependence of the
loss. There are two main components to these losses, the metal loss and the
dielectric loss.

The loss of most transmission lines are do minated by the metal loss, which causes
a frequency dependency due to finite conductivity of metals, and the skin effect
inside a conductor. The skin effect causes R along the conductor to be
approximately dependent on frequency according to;

Losses in the dielectric depend on the loss tangent (tanδ) of the material, which
depends inversely on the wavelength of the signal and is directly proportional to
the frequency.

Optical fibre

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