Module 10—Introduction to Wave Propagation, Transmission Lines, and Antennas
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2-21 to 2-30
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2-40 to 2-47
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Figure 3-9.—Equivalent circuit of a two-wire transmission line.
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.
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 3-10.—Distributed inductance
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 3-11.—Distributed capacitance.
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.
Figure 3-12.—Distributed resistance.
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
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
Conductance in transmission lines is expressed as the reciprocal of resistance and is usually given in
micromhos per unit length.
Figure 3-13.—Leakage in a transmission line.
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
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
Figure 3-14.—Fields between conductors.
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.
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.
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.
Figure 3-15.—Short section of two-wire transmission line and equivalent circuit.
Q19. Describe the leakage current in a transmission line and in what unit it is expressed.
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?
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).
Figure 3-16.—Characteristic impedance.
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
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.
Figure 3-17.—Termination of a line.
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.
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
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?
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.
Transmission of Energy
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.
Dc Applied to a Transmission
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. 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.
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.
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.
Dc Applied to an
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.
Introduction to Matter, Energy, and Direct Current,
to Alternating Current and Transformers, Introduction to Circuit Protection,
Control, and Measurement
, Introduction to Electrical Conductors, Wiring Techniques,
and Schematic Reading
, Introduction to Generators and Motors
Introduction to Electronic Emission, Tubes, and Power Supplies,
Introduction to Solid-State Devices and Power Supplies
Introduction to Amplifiers, Introduction to
Wave-Generation and Wave-Shaping Circuits
, Introduction to Wave Propagation, Transmission
Lines, and Antennas
, Microwave Principles,
, Introduction to Number Systems and Logic Circuits, Introduction
to Microelectronics, Principles of Synchros, Servos, and Gyros
Introduction to Test Equipment
, Radar Principles,
The Technician's Handbook,
Master Glossary, Test Methods and Practices,
Introduction to Digital Computers,
Magnetic Recording, Introduction to Fiber Optics