Source text. Published from the recorded source PDF for NEETS Module 10: Wave Propagation, Transmission Lines, and Antennas.
NEETS MODULE 10-Wave Propagation, Transmission Lines,
and Antennas
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3 PRINCIPLES OF TRANSMISSION LINES
LEARNING OBJECTIVES
After you finish this chapter, you should be able to do the following:
1. State what a transmission line is and how transmission lines are used.
2. Explain the operating principles of transmission lines.
3. Describe the five types of transmission lines.
4. State the length of a transmission line.
5. Explain the theory of the transmission line.
6. Define the term LUMPED CONSTANTS in relation to a transmission line.
7. Define the term DISTRIBUTED CONSTANTS in relation to a transmission line.
8. Define LEAKAGE CURRENT.
9. Describe how the electromagnetic lines of force around a transmission line are
affected by the distributed constants.
10. Define the term CHARACTERISTIC IMPEDANCE and explain how it affects
the transfer of energy along a transmission line.
11. State how the energy transfer along a transmission line is affected by
characteristic impedance and the infinite line.
12. Identify the cause of and describe the characteristics of reflections on a
transmission line.
13. Define the term STANDING WAVES as applied to a transmission line.
14. Describe how standing waves are produced on a transmission line and identify the
types of terminations.
15. Describe the types of standing-wave ratios.
3.1 INTRODUCTION
A TRANSMISSION LINE is a device designed to guide electrical energy from one point
to another. It is used, for example, to transfer the output rf energy of a transmitter to an
antenna. This energy will not travel through normal electrical wire without great losses.
Although the antenna can be connected directly to the transmitter, the antenna is usually
located some distance away from the transmitter. On board ship, the transmitter is located
inside a radio room and its associated antenna is mounted on a mast. A transmission line
is used to connect the transmitter and the antenna.
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The transmission line has a single purpose for both the transmitter and the antenna. This
purpose is to transfer the energy output of the transmitter to the antenna with the least
possible power loss. How well this is done depends on the special physical and electrical
characteristics (impedance and resistance) of the transmission line.
3.2 TERMINOLOGY
All transmission lines have two ends (see figure 3-1). The end of a two-wire transmission
line connected to a source is ordinarily called the INPUT END or the GENERATOR
END. Other names given to this end are TRANSMITTER END, SENDING END, and
SOURCE. The other end of the line is called the OUTPUT END or RECEIVING END.
Other names given to the output end are LOAD END and SINK.
You can describe a transmission line in terms of its impedance. The ratio of voltage to
current (Ein/Iin) at the input end is known 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 transmission line
and its source. If an infinitely long transmission line could be used, the ratio of voltage to
current at any point on that transmission line would be some particular value of
impedance. This impedance is known as the CHARACTERISTIC IMPEDANCE.
Figure 3-1 Basic transmission line
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Q1. What connecting link is used to transfer energy from a radio transmitter to its
antenna located on the mast of a ship?
Q2. What term is used for the end of the transmission line that is connected to a
transmitter?
Q3. What term is used for the end of the transmission line that is connected to an
antenna?
3.3 TYPES OF TRANSMISSION MEDIUMS
The Navy uses many different types of TRANSMISSION MEDIUMS in its electronic
applications. Each medium (line or wave guide) 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 chapter include
PARALLEL-LINE, TWISTED PAIR, SHIELDED PAIR, COAXIAL LINE, and
WAVEGUIDES. The use of a particular line depends, among other things, on the applied
frequency, the power-handling capabilities, and the type of installation.
NOTE: In the following paragraphs, we will mention LOSSES several times. We will
discuss these losses more thoroughly under "LOSSES IN TRANSMISSION LINES."
3.3.1 Two-Wire Open Line
One type of parallel line is the TWO-WIRE OPEN LINE illustrated in figure 3-2. 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 for power lines, rural telephone lines, and
telegraph lines. It is sometimes used as a transmission line between a transmitter and an
antenna or between 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.
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Another type of parallel line is the TWO-WIRE RIBBON (TWIN LEAD) illustrated in
figure 3-3. This type of transmission 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 embedding the two wires in a
low-loss dielectric, usually polyethylene. Since the wires are embedded in the thin ribbon
of polyethylene, the dielectric space is partly air and partly polyethylene.
Figure 3-2 Parallel two-wire line
Figure 3-3 Two-wire ribbon type line
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3.3.2 Twisted Pair
The TWISTED PAIR transmission line is illustrated in figure 3-4. 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.
3.3.3 Shielded Pair
The SHIELDED PAIR, shown in figure 3-5, consists of parallel conductors separated
from 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.
Figure 3-4 Twisted pair
Figure 3-5 Shielded pair
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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 magnetic fields.
3.3.4 Coaxial Lines
There are two types of COAXIAL LINES, RIGID (AIR) COAXIAL LINE and
FLEXIBLE (SOLID) COAXIAL LINE. The physical construction of both types is
basically the same; that is, each contains two concentric conductors.
The rigid coaxial line consists of a central, insulated wire (inner conductor) mounted
inside a tubular outer conductor. This line is shown in figure 3-6. In some applications,
the inner conductor is also tubular. The inner conductor is insulated from the outer
conductor by insulating spacers or beads at regular intervals. The spacers are made of
Pyrex, polystyrene, or some other material that has good insulating characteristics and
low dielectric losses at high frequencies.
The chief advantage of the rigid line is its ability to minimize radiation losses. The
electric and magnetic fields in a two-wire parallel line extend into space for relatively
great 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.
Figure 3-6 Air coaxial line
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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; and (3) although
high-frequency losses are somewhat less than in previously mentioned lines, they are still
excessive enough to limit 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 pumped
into the dielectric space of the line at a pressure that can vary from 3 to 35 pounds per
square inch. The inert gas is used to dry the line when it is first installed and pressure is
maintained to ensure that no moisture enters the line.
Flexible coaxial lines (figure 3-7) 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.
Because of the high-frequency losses associated with rubber insulators, polyethylene
plastic was developed to replace rubber and eliminate these losses. Polyethylene plastic is
a solid substance that remains flexible over a wide range of temperatures. It is unaffected
by seawater, gasoline, oil, and most other liquids that may be found aboard ship. The use
of polyethylene as an insulator results in greater high-frequency 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.
Figure 3-7 Flexible coaxial line
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3.3.5 Waveguides
The WAVEGUIDE is classified as a transmission line. However, the method by which it
transmits energy down its length differs from the conventional methods. Waveguides are
cylindrical, elliptical, or rectangular (cylindrical and rectangular shapes are shown in
figure 3-8). The rectangular waveguide is used more frequently than the cylindrical
waveguide.
The term waveguide can be applied to all types of transmission lines in the sense that they
are all used to guide energy from one point to another. However, usage has generally
limited the term to mean a hollow metal tube or a dielectric transmission line. In this
chapter, we use the term waveguide only to mean "hollow metal tube." It is interesting to
note that the transmission of electromagnetic energy along a waveguide travels at a
velocity somewhat slower than electromagnetic energy traveling through free space.
A waveguide may be classified according to its cross section (rectangular, elliptical, or
circular), or according to the material used in its construction (metallic or dielectric).
Dielectric waveguides are seldom used because the dielectric losses for all known
dielectric materials are too great to transfer the electric and magnetic fields efficiently.
Figure 3-8 Waveguides
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The installation of a complete waveguide transmission system is somewhat more difficult
than the installation of other types of transmission lines. The radius of bends in the
waveguide must measure greater than two wavelengths at the operating frequency of the
equipment to avoid excessive attenuation. The cross section must remain uniform around
the bend. These requirements hamper installation in confined spaces. If the waveguide is
dented, or if solder is permitted to run inside the joints, the attenuation of the line is
greatly increased. Dents and obstructions in the waveguide also reduce its breakdown
voltage, thus limiting the waveguide’s power-handling capability because of possible arc
over. Great care must be exercised during installation; one or two carelessly made joints
can seriously inhibit the advantage of using the waveguide.
Q4. List the five types of transmission lines in use today.
Q5. Name two of the three described uses of a two-wire open line.
Q6. What are the two primary disadvantages of a two-wire open line?
Q7. What type of transmission line is often used to connect a television set to its antenna?
Q8. What is the primary advantage of the shielded pair?
Q9. What are the two types of coaxial lines in use today?
Q10. What is the chief advantage of the air coaxial line?
Q11. List the three disadvantages of the air coaxial line.
Q12. List the two common types of waveguides in use today.
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3.4 LOSSES IN TRANSMISSION LINES
The discussion of transmission lines so far has not directly addressed LINE LOSSES;
actually some line losses occur in all lines. Line losses may be any of three types—
COPPER, DIELECTRIC, and RADIATION or INDUCTION LOSSES.
NOTE: Transmission lines are sometimes referred to as rf lines. In this text the terms are
used interchangeably.
3.4.1 Copper Losses
One type of copper loss is I2R LOSS. In rf lines the resistance of the conductors is never
equal to zero. Whenever current flows through one of these conductors, some energy is
dissipated in the form of heat. This heat loss is a POWER LOSS. With 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. When dc flows through a
conductor, the movement of electrons through the conductor's cross section is uniform.
The situation is somewhat different when ac is applied. The expanding and collapsing
fields 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 opposition to the flow 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. When the frequency applied is 100 megahertz or higher, the
electron movement in the center is so small that the center of the wire could be removed
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 minimized 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
flow through the silver layer. The tubing then serves primarily as a mechanical support.
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3.4.2 Dielectric Losses
DIELECTRIC LOSSES result from the heating effect on the dielectric material between
the conductors. Power from the source is used in heating the dielectric. The heat
produced is dissipated into the surrounding medium. When there is no potential
difference between two conductors, the atoms in the dielectric material between them are
normal and the orbits of the electrons are circular. When 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 change in the path of electrons requires
more energy, introducing a power loss.
The atomic structure of rubber is more difficult to distort than the structure of some other
dielectric materials. The atoms of materials, such as polyethylene, distort easily.
Therefore, polyethylene is often used as a dielectric because less power is consumed
when its electron orbits are distorted.
3.4.3 Radiation and Induction Losses
RADIATION and INDUCTION LOSSES are similar in that both are caused by the fields
surrounding the conductors. Induction losses occur when the electromagnetic field about
a conductor cuts through any nearby metallic 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 magnetic lines of force about a conductor do not
return to the conductor when the cycle alternates. These lines of force are projected into
space as radiation and this results in power losses. That is, power is supplied by the
source, but is not available to the load.
Q13. What are the three types of line losses associated with transmission lines?
Q14. Losses caused by skin effect and the I 2R (power) loss are classified as what type of
loss?
Q15. What types of losses cause the dielectric material between the conductors to be
heated?
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3.5 LENGTH OF A TRANSMISSION LINE
A transmission line is considered to be electrically short when its physical length is short
compared to a quarter-wavelength (1/4 ) of the energy it is to carry.
NOTE: In this module, for ease of reading, the value of the wavelength will be spelled
out in some cases, and in other cases, the numerical value will be used.
A transmission line is electrically long when its physical length is long compared to a
quarter-wavelength of the energy it is to carry. You must understand that the terms
"short" and "long" are relative ones. For example, a line that has a physical length of 3
meters (approximately 10 feet) is considered quite short electrically if it transmits a radio
frequency of 30 kilohertz. On the other hand, the same transmission line is considered
electrically long if it transmits a frequency of 30,000 megahertz.
To show the difference in physical and electrical lengths of the lines mentioned above,
compute the wavelength of the two frequencies, taking the 30-kilohertz example first:
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Now, computing the wavelength for the line carrying 30,000 megahertz:
Thus, you can see that a 3-meter line is electrically very short for a frequency of 30
kilohertz. Also, the 3-meter line is electrically very long for a frequency of 30,000
megahertz.
When power is applied to a very short transmission line, practically all of it reaches the
load at the output end of the line. This very short transmission line is usually considered
to have practically no electrical properties of its own, except for a small amount of
resistance.
However, the picture changes considerably when a long line is used. Since most
transmission lines are electrically long (because of the distance from transmitter to
antenna), the properties of such lines must be considered. Frequently, the voltage
necessary to drive a current through a long line is considerably greater than the amount
that can be accounted for by the impedance of the load in series with the resistance of the
line.
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3.6 TRANSMISSION LINE THEORY
The electrical characteristics of a two-wire transmission line depend primarily 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 magnetic field about them when electrical 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 example, 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 reactances of the line
depend on the frequency applied. Since no dielectric is perfect, electrons manage to move
from one conductor to the other through the dielectric. Each type of two-wire
transmission line also has a conductance value. This conductance value represents the
value of the current flow that may be expected through the insulation. If the line is
uniform (all values equal at each unit length), then one small section of the line may
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 transmission lines. We will explain the theories using LUMPED
CONSTANTS and DISTRIBUTED CONSTANTS to further simplify these principles.
3.6.1 Lumped Constants
A transmission line has the properties of inductance, capacitance, and resistance just as
the more conventional circuits have. Usually, however, the constants in conventional
circuits are lumped into a single device or component. For example, a coil of wire has the
property of inductance. When a certain amount of inductance is needed in a circuit, a coil
of the proper dimensions is inserted. The inductance of the circuit is lumped into the one
component. Two metal plates separated by a small space, can be used to supply the
required capacitance for a circuit. In such a case, most of the capacitance of the circuit is
lumped into this one component. Similarly, a fixed resistor can be used to supply a
certain value of circuit resistance as a lumped sum. Ideally, a transmission line would
also have its constants of inductance, capacitance, and resistance lumped together, as
shown in figure 3-9. Unfortunately, this is not the case. Transmission line constants are
distributed, as described below.
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3.6.2 Distributed Constants
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.
3.6.2.1 Inductance of a Transmission Line
When current flows through a wire, magnetic lines of force are set up around the wire. As
the current increases and decreases in amplitude, the field around the wire expands and
collapses accordingly. The energy produced by the magnetic lines of force collapsing
back into the wire tends to keep the current flowing in the same direction. This represents
a certain amount of inductance, which is expressed in microhenrys per unit length. Figure
3-10 illustrates the inductance and magnetic fields of a transmission line.
Figure 3-9 Equivalent circuit of a two-wire transmission line
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3.6.2.2 Capacitance of a Transmission Line
Capacitance also exists between the transmission line wires, as illustrated in figure 3-11.
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 capacitor.
Figure 3-10 Distributed inductance
Figure 3-11 Distributed capacitance
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3.6.2.3 Resistance of a Transmission Line
The transmission line shown in figure 3-12 has electrical resistance along its length. This
resistance is usually expressed in ohms per unit length and is shown as existing
continuously from one end of the line to the other.
Q16. What must the physical length of a transmission line be if it will be operated at
15,000,000 Hz? Use the formula:
Q17. What are two of the three physical factors that determine the values of capacitance
and inductance of a transmission line?
Q18. A transmission line is said to have distributed constants of inductance, capacitance,
and resistance along the line. What units of measurement are used to express these
constants?
Figure 3-12 Distributed resistance
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3.6.2.4 Leakage Current
Since any dielectric, even air, is not a perfect insulator, a small current known as
LEAKAGE CURRENT flows between the two wires. In effect, the insulator acts as a
resistor, permitting current to pass between the two wires. Figure 3-13 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-ohms per unit length.
3.7 ELECTROMAGNETIC FIELDS ABOUT A TRANSMISSION LINE
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 flow and voltage exist in a transmission 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 important to your understanding of energy transmission.
There are two kinds of fields; one is associated with voltage and the other with current.
The field associated with voltage is called 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 exert a force on any magnetic pole placed in it. Figure 3-
14 illustrates the way in which the E fields and H fields tend to orient themselves
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 direction of the lines of force. Both fields
normally exist together and are spoken of collectively as the electromagnetic field.
Figure 3-13 Leakage in a transmission line
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3.8 CHARACTERISTIC IMPEDANCE OF A TRANSMISSION LINE
You learned earlier that the maximum (and most efficient) transfer of electrical energy
takes place when the source impedance is matched to the load impedance. This fact is
very important in the study of transmission lines and antennas. If the characteristic
impedance of the transmission line and the load impedance are equal, energy from the
transmitter will travel down the transmission line to the antenna with no power loss
caused by reflection.
3.8.1 Definition and Symbols
Every transmission line possesses a certain CHARACTERISTIC IMPEDANCE, usually
designated as Z0. Z0 is the ratio of E to I at every point along the line. If a load equal to
the characteristic impedance is placed at the output end of any length of line, the same
impedance will appear at the input terminals of the line. The characteristic impedance is
the only value of impedance for any given type and size of line that acts in this way. The
characteristic impedance determines the amount of current that can flow when a given
voltage is applied to an infinitely long line. Characteristic impedance is comparable to the
resistance that determines the amount of current that flows in a dc circuit.
Figure 3-14 Fields between conductors
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In a previous discussion, lumped and distributed constants were explained. Figure 3-15,
view A, shows the properties of resistance, inductance, capacitance, and conductance
combined in a short section of two-wire transmission line. The illustration shows the
evenly distributed capacitance as a single lumped capacitor and the distributed
conductance as a lumped leakage path. Lumped values may be used for transmission line
calculations if the physical length of the line is very short compared to the wavelength of
energy being transmitted. Figure 3-15, view B, shows all four properties lumped together
and represented by their conventional symbols.
Q19. Describe the leakage current in a transmission line and in what unit it is expressed.
Figure 3-15 Short section of two-wire transmission line and equivalent circuit
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Q20. All the power sent down a transmission line from a transmitter can be transferred to
an antenna under what optimum conditions?
Q21. What symbol is used to designate the characteristic impedance of a line, and what
two variables does it compare?
3.8.2 Characteristic Impedance and the Infinite Line
Several short sections, as shown in figure 3-15, can be combined to form a large
transmission line, as shown in figure 3-16. Current will flow if voltage is applied across
points K and L. In fact, any circuit, such as that represented in figure 3-16, view A, has a
certain current flow for each value of applied voltage. The ratio of the voltage to the
current is the impedance (Z).
Recall that:
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Figure 3-16 Characteristic impedance
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The impedance presented to the input terminals of the transmission line is not merely the
resistance of the wire in series with the impedance of the load. The effects of series
inductance and shunt capacitance of the line itself may overshadow the resistance, and
even the load, as far as the input terminals are concerned.
To find the input impedance of a transmission line, determine the impedance of a single
section of line. The impedance between points K and L, in view B of figure 3-16, can be
calculated by the use of series-parallel impedance formulas, provided the impedance
across points M and N is known. But since this section is merely one small part of a
longer line, another similar section is connected to points M and N. Again, the impedance
across points K and L of the two sections can be calculated, provided the impedance of
the third section is known. This process of adding one section to another can be repeated
endlessly. The addition of each section produces an impedance across points K and L of a
new and lower value. However, after many sections have been added, each successive
added section has less and less effect on the impedance across points K and L. If sections
are added to the line endlessly, the line is infinitely long, and a certain finite value of
impedance across points K and L is finally reached.
In this discussion of transmission lines, the effect of conductance (G) is minor compared
to that of inductance (L) and capacitance (C), and is frequently neglected. In figure 3-16,
view C, G is omitted and the inductance and resistance of each line can be considered as
one line.
Let us assume that the sections of view C continue to the right with an infinite number of
sections. When an infinite number of sections extends to the right, the impedance
appearing across K and L is Z0. If the line is cut at R and S, an infinite number of sections
still extends to the right since the line is endless in that direction. Therefore, the
impedance now appearing across points R and S is also Z0, as illustrated in view D. You
can see that if only the first three sections are taken and a load impedance of Z0 is
connected across points R and S, the impedance across the input terminals K and L is still
Z0. The line continues to act as an infinite line. This is illustrated in view E.
Figure 3-17, view A, illustrates how the characteristic impedance of an infinite line can
be calculated. Resistors are added in series parallel across terminals K and L in eight
steps, and the resultant impedances are noted. In step 1 the impedance is infinite; in step 2
the impedance is 110 ohms. In step 3 the impedance becomes 62.1 ohms, a change of
47.9 ohms. In step 4 the impedance is 48.5 ohms, a change of only 13.6 ohms. The
resultant changes in impedance from each additional increment become progressively
smaller. Eventually, practically no change in impedance results from further additions to
the line. The total impedance of the line at this point is said to be at its characteristic
impedance; which, in this case, is 37 ohms. This means that an infinite line constructed as
indicated in step 8 could be effectively replaced by a 37-ohm resistor. View B shows a
37-ohm resistor placed in the line at various points to replace the infinite line of step 8 in
view A. There is no change in total impedance.
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In figure 3-17, resistors were used to show impedance characteristics for the sake of
simplicity. Figuring the actual impedance of a line having reactance is very similar, with
inductance taking the place of the series resistors and capacitance taking the place of the
shunt resistors. The characteristic impedance of lines in actual use normally lies between
50 and 600 ohms.
Figure 3-17 Termination of a line
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When a transmission line is "short” compared to the length of the radio-frequency waves
it carries, the opposition presented to the input terminals is determined primarily by the
load impedance. A small amount of power is dissipated in overcoming the resistance of
the line. However, when the line is "long” and the load is an incorrect impedance, the
voltages necessary to drive a given amount of current through the line cannot be
accounted for by considering just the impedance of the load in series with the impedance
of the line. The line has properties other than resistance that affect input impedance.
These properties are inductance in series with the line, capacitance across the line,
resistance leakage paths across the line, and certain radiation losses.
Q22. What is the range of the characteristic impedance of lines used in actual practice?
3.9 VOLTAGE CHANGE ALONG A TRANSMISSION LINE
Let us summarize what we have just discussed. In an electric circuit, energy is stored in
electric and magnetic fields. These fields must be brought to the load to transmit that
energy. At the load, energy contained in the fields is converted to the desired form of
energy.
3.9.1 Transmission of Energy
When the load is connected directly to the source of energy, or when the transmission
line is short, problems concerning current and voltage can be solved by applying Ohm’s
law. When the transmission line becomes long enough so the time difference between a
change occurring at the generator and the change appearing at the load becomes
appreciable, analysis of the transmission line becomes important.
3.9.2 Dc Applied to a Transmission Line
In figure 3-18, 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 reached 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, respectively. 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 force represent the electric
field that exists between the opposite kinds of charge on the wire sections from A to A’
and B to B’. Crosses (tails of arrows) indicate the magnetic field created by the electric
field moving down the line. The moving electric field and the accompanying magnetic
field constitute an electromagnetic wave that is moving from the generator (battery)
toward the load. This wave travels at approximately the speed of light in free space. The
energy reaching the load is equal to that developed at the battery (assuming 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.
Figure 3-18 Dc voltage applied to a line
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3.9.3 Ac Applied to a Transmission Line
When the battery of figure 3-18 is replaced by an ac generator (fig. 3-19), each
successive instantaneous value of the generator voltage is propagated down the line at the
speed of light. The action is similar to the wave created by the battery except that the
applied voltage is sinusoidal instead of constant. Assume that the switch is closed at the
moment the generator voltage is passing through zero and that the next half cycle makes
point A positive. At the end of one cycle of generator voltage, the current and voltage
distribution will be as shown in figure 3-19.
Figure 3-19 Ac voltage applied to a line
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In this illustration the conventional lines of force represent the electric fields. For
simplicity, the magnetic fields are not shown. Points of charge are indicated by plus (+)
and minus (−) signs, the larger signs indicating points of higher amplitude of both voltage
and current. Short arrows indicate direction of current (electron flow). The waveform
drawn below the transmission line represents the voltage (E) and current (I) waves. The
line is assumed 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. Waves traveling from the generator to the load are called
INCIDENT WAVES. Waves traveling from the load back to the generator are called
REFLECTED WAVES and will be explained in later paragraphs.
3.9.4 Dc Applied to an Infinite Line
Figure 3-20 shows a battery connected to a circuit that is the equivalent of a transmission
line. In this line the series resistance and shunt conductance are not shown. In the
following discussion the line will be considered to have no losses.
As the switch is closed, the battery voltage is applied to the input terminals of the line.
Now, C1 has no charge and appears, effectively, as a short circuit across points A and B.
The full battery voltage appears across inductor L1. Inductor L1 opposes the change of
current (0 now) and limits the rate of charge of C1.
Figure 3-20 Dc applied to an equivalent transmission line
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Capacitor C2 cannot begin to charge until after C1 has charged. No current can flow
beyond points A and B until C1 has acquired some charge. As the voltage across C1
increases, current through L2 and C2 charges C2. This action continues down the line and
charges each capacitor, in turn, to the battery voltage. Thus a voltage wave is traveling
along the line. Beyond the wavefront, the line is uncharged. Since the line is infinitely
long, there will always be more capacitors to be charged, and current will not stop
flowing. Thus current will flow indefinitely in the line.
Notice that current flows to charge the capacitors along the line. The flow of current is
not advanced along the line until a voltage is developed across each preceding capacitor.
In this manner voltage and current move down the line together in phase.
3.9.5 Ac Applied to an Infinite Line
An rf line displays similar characteristics when an ac voltage is applied to its sending end
or input terminals. In figure 3-21, view A, an ac voltage is applied to the line represented
by the circuit shown.
Figure 3-21 Ac applied to an equivalent transmission line
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In view B the generator voltage starts from zero (T1) and produces the voltage shown. As
soon as a small voltage change is produced, it starts its journey down the line while the
generator continues to produce new voltages along a sine curve. At T2 the generator
voltage is 70 volts. The voltages still move along the line until, at T3, the first small
change arrives at point W, and the voltage at that point starts increasing. At T5, the same
voltage arrives at point X on the line. Finally, at T7, the first small change arrives at the
receiving end of the line. Meanwhile, all the changes in the sine wave produced by the
generator pass each point in turn. The amount of time required for the changes to travel
the length of the line is the same as that required for a dc voltage to travel the same
distance.
At T7, the voltage at the various points on the line is as follows:
If these voltages are plotted along the length of the line, the resulting curve is like the one
shown in figure 3-22, view A. Note that such a curve of instantaneous voltages resembles
a sine wave. The changes in voltage that occur between T7 and T8 are as follows:
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Figure 3-22 Instantaneous voltages along a transmission line
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A plot of these new voltages produces the solid curve shown in figure 3-22, view B. For
reference, the curve from T7 is drawn as a dotted line. The solid curve has exactly the
same shape as the dotted curve, but has moved to the right by the distance X. Another
plot at T9 would show a new curve similar to the one at T8, but moved to the right by the
distance Y.
By analyzing the points along the graph just discussed, you should be able to see that the
actions associated with voltage changes along an rf line are as follows:
1. All instantaneous voltages of the sine wave produced by the generator travel down the
line in the order they are produced.
2. At any point, a sine wave can be obtained if all the instantaneous voltages passing the
point are plotted. An oscilloscope can be used to plot these values of instantaneous
voltages against time.
3. The instantaneous voltages (oscilloscope displays) are the same in all cases except that
a phase difference exists in the displays seen at different points along the line. The phase
changes continually with respect to the generator until the change is 360 degrees over a
certain length of line.
4. All parts of a sine wave pass every point along the line. A plot of the readings of an ac
meter (which reads the effective value of the voltage over a given time) taken at different
points along the line shows that the voltage is constant at all points. This is shown in view
C of figure 3-22.
5. Since the line is terminated with a resistance equal to Z 0, the energy arriving at the end
of the line is absorbed by the resistance.
3.10 VELOCITY OF WAVE PROPAGATION
If a voltage is initially applied to the sending end of a line, that same voltage will appear
later some distance from the sending end. This is true regardless of any change in
voltage, whether the change is a jump from zero to some value or a drop from some value
to zero. The voltage change will be conducted down the line at a constant rate.
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Recall that the inductance of a line delays the charging of the line capacitance. The
velocity of propagation is therefore related to the values of L and C. If the inductance and
capacitance of the rf line are known, the time required for any waveform to travel the
length of the line can be determined. To see how this works, observe the following
relationship:
This formula shows that the total charge or quantity is equal to the current multiplied by
the time the current flows. Also:
This formula shows that the total charge on a capacitor is equal to the capacitance
multiplied by the voltage across the capacitor.
If the switch in figure 3-23 is closed for a given time, the quantity (Q) of electricity
leaving the battery can be computed by using the equation Q = IT. The electricity leaves
the battery and goes into the line, where a charge is built up on the capacitors. The
amount of this charge is computed by using the equation Q = CE.
Figure 3-23 Dc applied to an equivalent transmission line
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Since none of the charge is lost, the total charge leaving the battery during T is equal to
the total charge on the line. Therefore:
As each capacitor accumulates a charge equal to CE, the voltage across each inductor
must change. As C1 in figure 3-23 charges to a voltage of E, point A rises to a potential
of E volts while point B is still at zero volts. This makes E appear across L2. As C2
charges, point B rises to a potential of E volts as did point A. At this time, point B is at E
volts and point C rises. Thus, we have a continuing action of voltage moving down the
infinite line.
In an inductor, these circuit components are related, as shown in the formula
This shows that the voltage across the inductor is directly proportional to inductance and
the change in current, but inversely proportional to a change in time. Since current and
time start from zero, the change in time (∆T) and the change in current (∆I) are equal to
the final time (T) and final current (I). For this case the equation becomes:
If voltage E is applied for time (T) across the inductor (L), the final current (I) will flow.
The following equations show how the three terms (T, L, and C) are related:
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For convenience, you can find T in terms of L and C in the following manner. Multiply
the left and right member of each equation as follows:
This final equation is used for finding the time required for a voltage change to travel a
unit length, since L and C are given in terms of unit length. The velocity of the waves
may be found by:
Where: D is the physical length of a unit.
This is the rate at which the wave travels over a unit length. The units of L and C are
henrys and farads, respectively. T is in seconds per unit length and V is in unit lengths
per second.
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3.11 DETERMINING CHARACTERISTIC IMPEDANCE
As previously discussed, an infinite transmission line exhibits a definite input impedance.
This impedance is the CHARACTERISTIC IMPEDANCE and is independent of line
length. The exact value of this impedance is the ratio of the input voltage to the input
current. If the line is infinite or is terminated in a resistance equal to the characteristic
impedance, voltage and current waves traveling the line are in phase. To determine the
characteristic impedance or voltage-to-current ratio, use the following procedure:
Take the square root:
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Example:
A problem using this equation will illustrate how to determine the characteristics of a
transmission line. Assume that the line shown in figure 3-23 is 1000 feet long. A 100-foot
(approximately 30.5 meter) section is measured to determine L and C. The section is
found to have an inductance of 0.25 millihenries and a capacitance of 1000 picofarads.
Find the characteristic impedance of the line and the velocity of the wave on the line.
If any other unit length had been considered, the values of L and C would be different,
but their ratio would remain the same as would the characteristic impedance.
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3.12 REFLECTIONS ON A TRANSMISSION LINE
Transmission line characteristics are based on an infinite line. A line cannot always be
terminated in its characteristic impedance since it is sometimes operated as an OPEN-
ENDED line and other times as a SHORT-CIRCUIT at the receiving end. If the line is
open-ended, it has a terminating impedance that is infinitely large. If a line is not
terminated in characteristic impedance, it is said to be finite.
When a line is not terminated in Z0, the incident energy is not absorbed but is returned
along the only path available—the transmission line. Thus, the behavior of a finite line
may be quite different from that of the infinite line.
3.12.1 Reflection of Dc Voltage from an Open Circuit
The equivalent circuit of an open-ended transmission line is shown in figure 3-24, view
A. Again, losses are to be considered as negligible, and L is lumped in one branch.
Assume that (1) the battery in this circuit has an internal impedance equal to the
characteristic impedance of the transmission line (Zi = Z0); (2) the capacitors in the line
are not charged before the battery is connected; and (3) since the line is open-ended, the
terminating impedance is infinitely large.
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Figure 3-24 Reflection from an open-ended line
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When the battery is connected to the sending end as shown, a negative voltage moves
down the line. This voltage charges each capacitor, in turn, through the preceding
inductor. Since Zi equals Z 0, one-half the applied voltage will appear across the internal
battery impedance, Zi, and one-half across the impedance of the line, Z0. Each capacitor is
then charged to E/2 (view B). When the last capacitor in the line is charged, there is no
voltage across the last inductor and current flow through the last inductor stops. With no
current flow to maintain it, the magnetic field in the last inductor collapses and forces
current to continue to flow in the same direction into the last capacitor. Because the
direction of current has not changed, the capacitor charges in the same direction, thereby
increasing the charge in the capacitor. Since the energy in the magnetic field equals the
energy in the capacitor, the energy transfer to the capacitor doubles the voltage across the
capacitor. The last capacitor is now charged to E volts and the current in the last inductor
drops to zero.
At this point, the same process takes place with the next to the last inductor and capacitor.
When the magnetic field about the inductor collapses, current continues to flow into the
next to the last capacitor, charging it to E volts. This action continues backward down the
line until the first capacitor has been fully charged to the applied voltage. This change of
voltage, moving backward down the line, can be thought of in the following manner. The
voltage, arriving at the end of the line, finds no place to go and returns to the sending end
with the same polarity (view C). Such action is called REFLECTION.
When a reflection of voltage occurs on an open-ended line, the polarity is unchanged.
The voltage change moves back to the source, charging each capacitor in turn until the
first capacitor is charged to the source voltage and the action stops (view D). As each
capacitor is charged, current in each inductor drops to zero, effectively reflecting the
current with the opposite polarity (view C). Reflected current of opposite polarity cancels
the original current at each point, and the current drops to zero at that point. When the
last capacitor is charged, the current from the source stops flowing (view D).
Important facts to remember in the reflection of dc voltages in open-ended lines are:
• Voltage is reflected from an open end without change in polarity, amplitude, or
shape.
• Current is reflected from an open end with opposite polarity and without change
in amplitude or shape.
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3.12.2 Reflection of Dc Voltage from a Short Circuit
A SHORT-CIRCUITED line affects voltage change differently from the way an open-
circuited line affects it. The voltage across a perfect short circuit must be zero; therefore,
no power can be absorbed in the short, and the energy is reflected toward the generator.
The initial circuit is shown in figure 3-25, view A. The initial voltage and current waves
(view B) are the same as those given for an infinite line. In a short-circuited line the
voltage change arrives at the last inductor in the same manner as the waves on an open-
ended line. In this case, however, there is no capacitor to charge. The current through the
final inductor produces a voltage with the polarity shown in view C. When the field
collapses, the inductor acts as a battery and forces current through the capacitor in the
opposite direction, causing it to discharge (view D). Since the amount of energy stored in
the magnetic field is the same as that in the capacitor, the capacitor discharges to zero.
Figure 3-25 Reflection from a short-
circuited line
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Now there is no voltage to maintain the current through the next to the last inductor.
Therefore, this inductor discharges the next to the last capacitor.
As each capacitor is discharged to zero, the next inductor effectively becomes a new
source of voltage. The amplitude of each of these voltages is equal to E/2, but the polarity
is the opposite of the battery at the input end of the line. The collapsing field around each
inductor, in turn, produces a voltage that forces the current to continue flowing in the
same direction, adding to the current from the source to make it 2I. This action continues
until all the capacitors are discharged (view E).
Reflected waves from a short-circuited transmission line are characterized as follows:
• The reflected voltage has the opposite polarity but the same amplitude as the
incident wave.
• The reflected current has the same polarity and the same amplitude as the incident
current.
3.12.3 Reflection of Ac Voltage from an Open Circuit
In most cases where rf lines are used, the voltages applied to the sending end are ac
voltages. The action at the receiving end of the line is exactly the same for ac as for dc. In
the open-ended line, shown in figure 3-26, view A, the generated ac voltage is distributed
along the line, shown in view B. This voltage is distributed in such a way that as each
instantaneous voltage arrives at the end, it is reflected with the same polarity and
amplitude. When ac is used, this reflection is in phase. Each of the reflected voltages
travels back along the line until it reaches the generator. If the generator impedance is the
same as the line impedance, energy arriving at the generator is absorbed and not reflected
again. Now two voltages are on the line.
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View B shows how two waves of the same frequency and amplitude moving in opposite
directions on the same conductor will combine to form a resultant wave. The small solid
line is moving steadily from left to right and is the INCIDENT WAVE (from the source).
The broken-line waveform is moving from right to left and is the REFLECTED WAVE.
The resultant waveform, the heavy line, is found by algebraically adding instantaneous
values of the two waveforms. The resultant waveform has an instantaneous peak
amplitude that is equal to the sum of the peak amplitudes of the incident and reflected
waves. Since most indicating instruments are unable to separate these voltages, they show
the vector sum. An oscilloscope is usually used to study the instantaneous voltages on rf
lines.
Figure 3-26 Formation of standing waves
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Since two waves of voltage are moving on the line, you need to know how to distinguish
between the two. The voltages moving toward the receiving end are called INCIDENT
VOLTAGES, and the whole waveshape is called the INCIDENT WAVE. The wave
moving back to the sending end after reflection is called the REFLECTED WAVE. The
resultant voltage curve (view B of figure 3-26) shows that the voltage is maximum at the
end of the line, a condition that occurs across an open circuit.
Another step in investigating the open-circuited rf line is to see how the current waves
act. The incident current wave is the solid line in figure 3-26, view C. The voltage is
represented by the dotted line. The current is in phase with the voltage while traveling
toward the receiving end. At the end of the line, the current is reflected in the opposite
polarity; that is, it is shifted 180 degrees in phase, but its amplitude remains the same.
The reflected wave of current is shown by dashed lines in view C. The heavy-line curve
represents the sum of the two instantaneous currents and is the resultant wave. Notice that
current is zero at the end of the line. This is reasonable, since there can be no current flow
through an open circuit.
Views B and C of figure 3-26 show the voltage and current distribution along a
transmission line at a point about 1/8 after a maximum voltage or current reaches the end
of the line. Since the instantaneous values are continuously changing during the
generation of a complete cycle, a large number of these pictures are required to show the
many different relationships.
Figure 3-27 shows the incident and reflected waveshapes at several different times. The
diagrams in the left column of figure 3-27 (representing voltage) show the incident wave
and its reflection without change in polarity. In figure 3-27, waveform (1), the incident
wave and the reflected wave are added algebraically to produce the resultant wave
indicated by the heavy line. In waveform (2), a zero point preceding the negative-going
cycle of the incident wave is at the end of the line. The reflected wave and incident wave
are 180 degrees out of phase at all points. (The reflected wave is the positive cycle that
just preceded the negative cycle now approaching the end of the line.) The resultant of
the incident and reflected waves is zero at all points along the line. In waveform (3), the
waves have moved 1/8 along the line; the incident wave has moved 45 degrees to the
right, and the reflected wave has moved 45 degrees to the left. The resultant voltage,
shown by the heavy line, has a maximum negative at the end of the line and a maximum
positive 1/2 from the end of the line.
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Figure 3-27 Instantaneous values of incident and reflected waves on an open-
ended line
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In waveform (4), the incident wave is at a maximum negative value at the end of the line.
The wave has moved another 45 degrees to the right from the wave in the preceding
illustration. The reflected wave has also moved 45 degrees, but to the left. The reflected
wave is in phase with the incident wave. The resultant of these two waves, shown by the
dark line, again has a negative maximum at the end of the line and a positive maximum
1/2 from the end of the line. Notice that these maxima have a greater amplitude than
those in waveform (3).
In waveform (5), the incident wave has moved another 45 degrees to the right and the
reflected wave 45 degrees to the left. The resultant again is maximum negative at the end
and positive maximum 1/2 from the end. The maxima are lower than those in waveform
(4). In waveform (6), the incident and reflected wave have moved another 1/8 . The two
waves again are 180 degrees out of phase giving a resultant wave with no amplitude. The
incident and reflected waves continue moving in opposite directions, adding to produce
the resultant waveshapes shown in waveforms (7) and (8). Notice that the maximum
voltage in each resultant wave is at the end and 1/2 from the end.
Study each part of figure 3-27 carefully and you will get a clear picture of how the
resultant waveforms of voltage are produced. You will also see that the resultant voltage
wave on an open-ended line is always zero at 1/4 and 3/4 from the end of the
transmission line. Since the zero and maximum points are always in the same place, the
resultant of the incident and the reflected wave is called a STANDING WAVE of
voltage.
The right-hand column in figure 3-27 shows the current waveshapes on the open-ended
line. Since the current is reflected out of phase at an open end, the resultant waveshapes
differ from those for voltage. The two out-of-phase components always cancel at the end
of the transmission line, so the resultant is always zero at that point. If you check all the
resultant waveshapes shown in the right-hand column of figure 3-27, you will see that
zero point always occurs at the end and at a point 1/2 from the end. Maximum voltages
occur 1/4 and 3/4 from the end.
When an ac meter is used to measure the voltages and currents along a line, the polarity is
not indicated. If you plot all the current and voltage readings along the length of the line,
you will get curves like the ones shown in figure 3-28. Notice that all are positive. These
curves are the conventional method of showing current and voltage standing waves on rf
lines.
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When an rf line is terminated in a short circuit, reflection is complete, but the effect on
voltage and current differs from that in an open-ended line. Voltage is reflected in
opposite phase, while current is reflected in phase. Again refer to the series of pictures
shown in figure 3-27. However, this time the left column represents current, since it
shows reflection in phase; and the right column of pictures now represents the voltage
changes on the shorted line, since it shows reflection out of phase.
The composite diagram in figure 3-29 shows all resultant curves on a full-wavelength
section of line over a complete cycle. Notice that the amplitude of the voltage varies
between zero and maximum in both directions at the center and at both ends as well but,
one-fourth of the distance from each end the voltage is always zero. The resultant
waveshape is referred to as a standing wave of voltage. Standing waves, then, are caused
by reflections, which occur only when the line is not terminated in its characteristic
impedance.
Figure 3-28 Conventional picture of standing waves
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Figure 3-29 Composite results of instantaneous
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The voltage at the center and the ends varies at a sinusoidal rate between the limits
shown. At the one-fourth the three-fourths points, the voltage is always zero. A
continuous series of diagrams such as these is difficult to see with conventional test
equipment, which reads the effective or average voltage over several cycles. The curve of
amplitude over the length of line for several cycles is shown in figure 3-29, view B. A
meter will read zero at the points shown and will show a maximum voltage at the center,
no matter how many cycles pass.
As shown in view D, the amplitude varies along the length of the line. In this case it is
zero at the end and center but maximum at the one-fourth and three-fourths points. The
entire diagram of the open ended line conditions is shown in view E. The standing waves
of voltage and current appear together. Observe that one is maximum when the other is
minimum. The current and voltage standing waves are one-quarter cycle, or 90 degrees,
out of phase with one another.
3.12.4 REFLECTION OF AC VOLTAGE FROM A SHORT CIRCUIT
Reflection is complete when an rf line is terminated in a short circuit, but the effect on
voltage and current differs from the effect obtained in an open-ended line. Voltage is
reflected in opposite phase, while current is reflected in phase. Again look at the series of
diagrams in figure 3-27. The left column represents current, and the right column shows
voltage changes on the shorted line. The standard representation of standing waves on a
shorted line is shown in figure 3-30; the voltage is a solid line, and the current is a dashed
line. The voltage is zero at the end and center (1/2 ) and maximum at the 1/4 and 3/4
points, while the current maximum at the end and center and minimum at the 1/4 and
3/4 points.
Figure 3-30 Standing waves on a shorted line
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As we discussed voltage and current waves on transmission lines, we pointed out several
differences between open and shorted lines. Basic differences also appear in the standing-
wave patterns for open and shorted lines. You can see these differences by comparing
figure 3-29, view E, and figure 3-30. Notice that the current and voltage standing waves
are shifted 90 degrees with respect to the termination. At the open end of a line, voltage is
maximum (zero if there are no losses in the line). At a short circuit, current is maximum
and voltage is minimum.
Q23. Two types of waves are formed on a transmission line. What names are given to
these waves?
Q24. In figure 3-27, which waveforms on the left have a resultant wave of zero, and what
is indicated by these waves?
Q25. On an open-ended transmission line, the voltage is always zero at what distance
from each end of the line?
3.13 TERMINATING A TRANSMISSION LINE
A transmission line is either NONRESONANT or RESONANT. First, let us define the
terms nonresonant lines and resonant lines. A nonresonant line is a line that has no
standing waves of current and voltage. A resonant line is a line that has standing waves of
current and voltage.
3.13.1 Nonresonant Lines
A nonresonant line is either infinitely long or terminated in its characteristic impedance.
Since no reflections occur, all the energy traveling down the line is absorbed by the load
which terminates the line. Since no standing waves are present, this type of line is
sometimes spoken of as a FLAT line. In addition, because the load impedance of such a
line is equal to Z0, no special tuning devices are required to effect a maximum power
transfer; hence, the line is also called an UNTUNED line.
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3.13.2 Resonant Lines
A resonant line has a finite length and is not terminated in its characteristic impedance.
Therefore reflections of energy do occur. The load impedance is different from the Z0 of
the line; therefore, the input impedance may not be purely resistive but may have reactive
components. Tuning devices are used to eliminate the reactance and to bring about
maximum power transfer from the source to the line. Therefore, a resonant line is
sometimes called a TUNED line. The line also may be used for a resonant or tuned
circuit.
A resonant line is sometimes said to be resonant at an applied frequency. This means that
at one frequency the line acts as a resonant circuit. It may act either as a high-resistive
circuit (parallel resonant) or as a low-resistive circuit (series resonant). The line may be
made to act in this manner by either open or short-circuiting it at the output end and
cutting it to some multiple of a quarter-wavelength.
At the points of voltage maxima and minima on a short-circuited or open-circuited line,
the line impedance is resistive. On a short-circuited line, each point at an odd number of
quarter-wavelengths from the receiving end has a high impedance (figure 3-31, view A).
If the frequency of the applied voltage to the line is varied, this impedance decreases as
the effective length of the line changes. This variation is exactly the same as the change
in the impedance of a parallel-resonant circuit when the applied frequency is varied.
Figure 3-31 Sending-end impedance of
various lengths and terminations
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At all even numbered quarter-wavelength points from the short circuit, the impedance is
extremely low. When the frequency of the voltage applied to the line is varied, the
impedance at these points increases just as the impedance of a series-resonant circuit
varies when the frequency applied to it is changed. The same is true for an open-ended
line (figure 3-31, view B) except that the points of high and low impedance are reversed.
At this point let us review some of the characteristics of resonant circuits so we can see
how resonant line sections may be used in place of LC circuits.
A PARALLEL-RESONANT circuit has the following characteristics:
• At resonance the impedance appears as a very high resistance. A loss-free circuit
has infinite impedance (an open circuit). Other than at resonance, the impedance
decreases rapidly.
• If the circuit is resonant at a point above the generator frequency (the generator
frequency is too low), more current flows through the coil than through the
capacitor. This happens because XL decreases with a decrease in frequency but XC
increases.
A SERIES-RESONANT circuit has these characteristics:
• At resonance the impedance appears as a very low resistance. A loss-free circuit
has zero impedance (a short circuit). Other than at resonance the impedance
increases rapidly.
• If the circuit is resonant at a point above the generator frequency (the generator
frequency is too low), then XC is larger than XL and the circuit acts capacitively.
• If the circuit is resonant at a point below the generator frequency (the generator
frequency is too high), then XL is larger than XC and the circuit acts inductively.
Since the impedance a generator sees at the quarter-wave point in a shorted line is that of
a parallel resonant circuit, a shorted quarter-wave- length of line may be used as a
parallel-resonant circuit (figure 3-31, view C). An open quarter-wavelength of line may
be used as a series-resonant circuit (view D). The Q of such a resonant line is much
greater than can be obtained with lumped capacitance and inductance.
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3.13.3 Impedance for Various Lengths of Open Lines
In figure 3-32, the impedance (Z) the generator sees for various lengths of line is shown
at the top. The curves above the letters of various heights show the relative value of the
impedances presented to the generator for the various line lengths. The circuit symbols
indicate the equivalent electrical circuits for the transmission lines at each particular
length. The standing waves of voltage and current are shown on each length of line.
Figure 3-32 Voltage, current, and impedance on open line
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At all odd quarter-wave points (1/4 , 3/4 , etc.), the voltage is minimum, the current is
maximum, and the impedance is minimum. Thus, at all odd quarter-wave points, the
open-ended transmission line acts as a series resonant circuit. The impedance is
equivalent to a very low resistance, prevented from being zero only by small circuit
losses.
At all even quarter-wave points (1/2 , 1 , 3/2 , etc.), the voltage is maximum, the
current is minimum, and the impedance is maximum. Comparison of the line with an LC
resonant circuit shows that at an even number of quarter-wavelengths, an open line acts
as a parallel-resonant circuit. The impedance is therefore and extremely high resistance.
In addition, resonant open lines may also act as nearly pure capacitances or inductances.
The illustration shows that an open line less than a quarter-wavelength long acts as a
capacitance. Also, it acts as an inductance from 1/4 to 1/2 wavelength, as a capacitance
from 1/2 to 3/4 wavelength, and as an inductance from 3/4 to 1 wavelength, etc. A
number of open transmission lines, with their equivalent circuits, are shown in the
illustration.
3.13.4 Impedance of Various Lengths of Shorted Lines
Follow figure 3-33 as we study the shorted line. At the odd quarter-wavelength points,
the voltage is high, the current is low, and the impedance is high. Since these conditions
are similar to those found in a parallel-resonant circuit, the shorted transmission line acts
as a parallel-resonant circuit at these lengths.
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At the even quarter-wave points voltage is minimum, current is maximum, and
impedance is minimum. Since these characteristics are similar to those of a series-
resonant LC circuit, a shorted transmission line whose length is an even number of
quarter-wavelengths acts as a series-resonant circuit.
Figure 3-33 Voltage, current, and impedance on shorted line
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Resonant shorted lines, like open-end lines, also may act as pure capacitances or
inductances. The illustration shows that a shorted line less than 1/4 wavelength long acts
as an inductance. A shorted line with a length of from 1/4 to 1/2 wavelength acts as a
capacitance. From 1/2 to 3/4 wavelength, the line acts as an inductance; and from 3/4 to 1
wavelength, it acts as a capacitance, and so on. The equivalent circuits of shorted lines of
various lengths are shown in the illustration. Thus, properly chosen line segments may be
used as parallel-resonant, series-resonant, inductive, or capacitive circuits.
3.14 STANDING WAVES ON A TRANSMISSION LINE
There is a large variety of terminations for rf lines. Each type of termination has a
characteristic effect on the standing waves on the line. From the nature of the standing
waves, you can determine the type of termination that produces the waves.
3.14.1 Termination in Z0
Termination in Z0 (characteristic impedance) will cause a constant reading on an ac meter
when it is moved along the length of the line. As illustrated in figure 3-34, view A, the
curve, provided there are no losses in the line, will be a straight line. If there are losses in
the line, the amplitude of the voltage and current will diminish as they move down the
line (view B). The losses are due to dc resistance in the line itself.
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Figure 3-34 Effects of various terminations on
standing waves
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3.14.2 Termination in an Open Circuit
In an open-circuited rf line (figure 3-34, view C), the voltage is maximum at the end, but
the current is minimum. The distance between two adjacent zero current points is 1/2 ,
and the distance between alternate zero current points is 1 . The voltage is zero at a
distance of 1/4 from the end of the line. This is true at any frequency. A voltage peak
occurs at the end of the line, at 1/2 from the end, and at each 1/2 thereafter.
3.14.3 Termination in a Short Circuit
On the line terminated in a short circuit, shown in figure 3-34, view D, the voltage is zero
at the end and maximum at 1/4 from the end. The current is maximum at the end, zero at
1/4 from the end, and alternately maximum and zero every 1/4 thereafter.
3.14.4 Termination in Capacitance
When a line is terminated in capacitance, the capacitor does not absorb energy, but
returns all of the energy to the circuit. This means there is 100 percent reflection. The
current and voltage relationships are somewhat more involved than in previous types of
termination. For this explanation, assume that the capacitive reactance is equal to the Z0
of the line. Current and voltage are in phase when they arrive at the end of the line, but in
flowing through the capacitor and the characteristic impedance (Z0) connected in series,
they shift in phase relationship. Current and voltage arrive in phase and leave out of
phase. This results in the standing-wave configuration shown in figure 3-34, view E. The
standing wave of voltage is minimum at a distance of exactly 1/8 from the end. If the
capacitive reactance is greater than Z0 (smaller capacitance), the termination looks more
like an open circuit; the voltage minimum moves away from the end. If the capacitive
reactance is smaller than Z0, the minimum moves toward the end.
3.14.5 Termination in Inductance
When the line is terminated in an inductance, both the current and voltage shift in phase
as they arrive at the end of the line. When XL is equal to Z0, the resulting standing waves
are as shown in figure 3-34, view F. The current minimum is located 1/8 from the end
of the line. When the inductive reactance is increased, the standing waves appear closer
to the end. When the inductive reactance is decreased, the standing waves move away
from the end of the line.
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3.14.6 Termination in a Resistance Not Equal to the Characteristic Impedance (Z0)
Whenever the termination is not equal to Z0, reflections occur on the line. For example, if
the terminating element contains resistance, it absorbs some energy, but if the resistive
element does not equal the Z0 of the line, some of the energy is reflected. The amount of
voltage reflected may be found by using the equation:
If you try different values of RL in the preceding equation, you will find that the reflected
voltage is equal to the incident voltage only when RL equals 0 or is infinitely large. When
RL equals Z0, no reflected voltage occurs. When RL is greater than Z0, ER is positive, but
less than Ei. As RL increases and approaches an infinite value, ER increases and
approaches Ei in value. When RL is smaller than Z0, ER has a negative value. This means
that the reflected voltage is of opposite polarity to the incident wave at the termination of
the line. As RL approaches zero, ER approaches Ei in value. The smaller the value of ER,
the smaller is the peak amplitude of the standing waves and the higher are the minimum
values.
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3.14.7 Termination in a Resistance Greater Than Z0
When RL is greater than Z0, the end of the line is somewhat like an open circuit; that is,
standing waves appear on the line. The voltage maximum appears at the end of the line
and also at half-wave intervals back from the end. The current is minimum (not zero) at
the end of the line and maximum at the odd quarter-wave points. Since part of the power
in the incident wave is consumed by the load resistance, the minimum voltage and
current are less than for the standing waves on an open-ended line. Figure 3-34, view G,
illustrates the standing waves for this condition.
3.14.8 Termination in a Resistance Less Than Z0
When RL is less than Z0, the termination appears as a short circuit. The standing waves are
shown in figure 3-34, view H. Notice that the line terminates in a current LOOP (peak)
and a voltage NODE (minimum). The values of the maximum and minimum voltage and
current approach those for a shorted line as the value of RL approaches zero.
A line does not have to be any particular length to produce standing waves; however, it
cannot be an infinite line. Voltage and current must be reflected to produce standing
waves. For reflection to occur, a line must not be terminated in its characteristic
impedance. Reflection occurs on lines terminated in opens, shorts, capacitances, and
inductances, because no energy is absorbed by the load. If the line is terminated in a
resistance not equal to the characteristic impedance of the line, some energy will be
absorbed and the rest will be reflected.
The voltage and current relationships for open-ended and shorted lines are opposite to
each other, as shown in figure 3-34, views C and D. The points of maximum and
minimum voltage and current are determined from the output end of the line, because
reflection always begins at that end.
Q26. A nonresonant line is a line that has no standing waves of current and voltage on it
and is considered to be flat. Why is this true?
Q27. On an open line, the voltage and impedance are maximum at what points on the
line?
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3.15 STANDING-WAVE RATIO
The measurement of standing waves on a transmission line yields information 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 takes place.
You have probably noticed that the variation of standing waves shows how near the rf
line is to being terminated in Z0. A wide variation in voltage along the length means a
termination far from Z0. A small variation means termination near Z 0. Therefore, the
ratio of the maximum to the minimum is a measure of the perfection of the termination of
a line. This ratio is called the STANDING-WAVE RATIO (swr) and is always expressed
in whole numbers. For example, a ratio of 1:1 describes a line terminated in its
characteristic impedance (Z 0).
3.15.1 Voltage Standing-Wave Ratio
The ratio of maximum voltage to minimum voltage on a line is called the VOLTAGE
STANDINGWAVE RATIO (vswr). Therefore:
The vertical lines in the formula indicate that the enclosed quantities are absolute and that
the two values are taken without regard to polarity. Depending on the nature of the
standing waves, the numerical 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 vswr is to be a useful quantity, the power losses along
the line must be small in comparison to the transmitted power.
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3.15.2 Power Standing-Wave Ratio
The square of the voltage standing-wave ratio is called the POWER STANDING-WAVE
RATIO (pswr). Therefore:
This ratio is useful because the instruments used to detect standing waves react to the
square of the voltage. Since power is proportional to the square of the voltage, the ratio of
the square of the maximum 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.
3.15.3 Current Standing-Wave Ratio
The ratio of maximum to minimum current along a transmission 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 magnetic field along a line. It gives the same results as vswr
measurements.
Q28. At what point on an open-circuited rf line do voltage peaks occur?
Q29. What is the square of the voltage standing-wave ratio called?
Q30. What does vswr measure?
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3.16 SUMMARY
This chapter has presented information on the characteristics of transmission lines. The
information that follows summarizes the important points of this chapter.
TRANSMISSION LINES are devices for guiding electrical energy from one point to
another.
INPUT IMPEDANCE is the ratio of voltage to current at the input end of a transmission
line.
OUTPUT IMPEDANCE is the ratio of voltage to current at the output end of the line.
TWO-WIRE OPEN LINES are parallel lines and have uses such as power lines, rural
telephone lines, and telegraph lines. This type of line has high radiation losses and is
subject to noise pickup.
TWIN LEAD has parallel lines and is most often used to connect televisions to their
antennas.
A TWISTED PAIR consists of two insulated wires twisted together. This line has high
insulation loss.
A SHIELDED PAIR has parallel conductors separated by a solid dielectric and
surrounded by copper braided tubing. The conductors are balanced to ground.
RIGID COAXIAL LINE contains two concentric conductors insulated from each other
by spacers. Some rigid coaxial lines are pressurized with an inert gas to prevent moisture
from entering. High-frequency losses are less than with other lines.
FLEXIBLE COAXIAL LINES consist of a flexible inner conductor and a concentric
outer conductor of metal braid. The two are separated by a continuous insulating
material.
WAVEGUIDES are hollow metal tubes used to transfer energy from one point to
another. The energy travels slower in a waveguide than in free space.
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COPPER LOSSES can result from power (I 2R) loss, in the form of heat, or skin effect.
These losses decrease the conductivity of a line.
DIELECTRIC LOSSES are caused by the heating of the dielectric material between
conductors, taking power from the source.
RADIATION and INDUCTION LOSSES are caused by part of the electromagnetic
fields of a conductor being dissipated into space or nearby objects.
LUMPED CONSTANTS are theoretical properties (inductance, resistance, and
capacitance) of a transmission line that are lumped into a single component.
DISTRIBUTED CONSTANTS are constants of inductance, capacitance and resistance
that are distributed along the transmission line.
LEAKAGE CURRENT flows between the wires of a transmission line through the
dielectric. The dielectric acts as a resistor.
An ELECTROMAGNETIC FIELD exists along transmission line when current flows
through it.
CHARACTERISTIC IMPEDANCE, Z0, is the ratio of E to I at every point along the
line. For maximum transfer of electrical power, the characteristic impedance and load
impedance must be matched.
The VELOCITY at which a wave travels over a given length of transmission line.
A transmission line that is not terminated in its characteristic impedance is said to be
FINITE.
When dc is applied to an OPEN-ENDED line, the voltage is reflected back from the
open end without any change in polarity, amplitude, or shape. Current is reflected back
with the same amplitude and shape but with opposite polarity.
When dc is applied to a SHORT-CIRCUITED line, the current is reflected back with
the same amplitude, and polarity. The voltage is reflected back with the same amplitude
but with opposite polarity.
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When ac is applied to a SHORT-CIRCUITED line, voltage is reflected in opposite
phase, while current is reflected in phase.
A NONRESONANT line has NO STANDING WAVES of current and voltage and is
either infinitely long or terminated in its characteristic impedance.
A RESONANT line has STANDING WAVES of current and voltage and is of finite
length and is NOT terminated in its characteristic impedance.
There are a variety of TERMINATIONS for rf lines. Each termination has an effect on
the standing waves on the line.
A transmission line can be terminated in its characteristic impedance as an open- or short-
circuit, or in capacitance or inductance.
Whenever the termination on a transmission line is NOT EQUAL TO Z 0, there are
reflections on the line.
When the termination on a transmission line EQUALS Z0, there is NO reflected voltage.
The measurement of standing waves on a transmission line yields information about
operating conditions. If there are NO standing waves, the termination for that line is
correct and maximum power transfer takes place.
The STANDING WAVE RATIO is the measurement of maximum voltage (current) to
minimum voltage (current) on a transmission line and measures the perfection of the
termination of the line. A ratio of 1:1 describes a line terminated in its characteristic
impedance.
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ANSWERS TO QUESTIONS Q1. THROUGH Q30.
A1. Transmission line.
A2. Input end, generator end, transmitter end, sending end, and source.
A3. Output end, receiving end, load end and sink.
A4. Parallel two-wire, twisted pair, shielded pair, coaxial line and waveguide.
A5. Power lines, rural telephone lines, and telegraph lines.
A6. High radiation losses and noise pickup.
A7. Twin lead.
A8. The conductors are balanced to ground.
A9. Air coaxial (rigid) and solid coaxial (flexible).
A10. The ability to minimize radiation losses.
A11. Expensive to construct, must be kept dry, and high frequency losses limit the
practical length of the line.
A12. Cylindrical and rectangular.
A13. Copper, dielectric, and radiation.
A14. Copper losses.
A15. Dielectric losses.
A16. = 20 meters.
A17. (1) Type of line used, (2) dielectric in the line, and (3) length of line.
A18. Inductance is expressed in microhenrys per unit length, capacitance is expressed
in picofarads per unit length, and resistance is expressed in ohms per unit length.
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A19. The small amount of current that flows through the dielectric between two wires
of a transmission line and is expressed in micromhos per unit length.
A20. When the characteristic impedance of the transmission line and the load
impedance are equal.
A21. Z0 and it is the ratio of E to I at every point along the line.
A22. Between 50 and 600 ohms.
A23. Incident waves from generator to load. Reflected waves from load back to
generator.
A24. 2 and 6 have zero resultant wave and they indicate that the incident and reflected
waves are 180 degrees out of phase at all parts.
A25. One-fourth the distance from each end of the line.
A26. The load impedance of such a line is equal to Z0.
A27. Even quarter-wave points (1/2
A28. At 1/2 wavelength from the end and at every 1/2 wavelength along the line.
A29. Power standing-wave ratio (pswr).
A30. The existence of voltage variations on a line.