standing wave

Chapter 13 Transmission Lines
13-1: Transmission-Line Basics
13-2: Standing Waves
13-3: Transmission Lines as Circuit Elements
13-4: The Smith Chart
13-1: Transmission-Line Basics
13-2: Standing Waves
13-3: Transmission Lines as Circuit Elements
13-4: The Smith Chart
 Transmission lines in communication carry
telephone signals, computer data in LANs, TV
signals in cable TV systems, and signals from a
transmitter to an antenna or from an antenna to a
 Transmission lines are also circuits.
 Their electrical characteristics are critical and
must be matched to the equipment for successful
communication to take place.
The two primary requirements of a transmission
line are:
1. The line should introduce minimum attenuation
to the signal.
2. The line should not radiate any of the signal as
radio energy.
Types of Transmission Lines
 Parallel-wire line is made of two parallel
conductors separated by a space of ½ inch to
several inches.
 The most widely used type of transmission line is
the coaxial cable. It consists of a solid center
conductor surrounded by a dielectric material,
usually a plastic insulator such as Teflon.
 Twisted-pair cable uses two insulated solid copper
wires covered with insulation and loosely twisted
Figure 13-1: Common types of transmission lines. (a) Open-wire line. (b) Open-wire
line called twin lead. (c) Coaxial cable (d) Twisted-pair cable.
Balanced Versus Unbalanced Lines
Transmission lines can be balanced or
A balanced line is one in which neither wire is
connected to ground.
The signal on each wire is referenced to ground.
In an unbalanced line, one conductor is
connected to ground.
Open-wire line has a balanced configuration.
Figure 13-2: (a) Balanced line. (b) Unbalanced line
Characteristic Impedance
When the length of transmission line is longer
than several wavelengths at the signal frequency,
the two parallel conductors of the transmission
line appear as a complex impedance.
An RF generator connected to a considerable
length of transmission line sees an impedance
that is a function of the inductance, resistance,
and capacitance in the circuit—the
characteristic or surge impedance (Z0).
l = 300,000,000 (m/s) / f(Hz)
Figure 13-9
A transmission line appears as a distributed
low-pass filter to any driving generator.
(a) A distributed line with lumped components
(b) Simplified equivalent circuit
Figure 13-10 A transmission line whose load is resistive
and equal to the surge impedance appears as an equal
resistance to the generator.
Zo =
Velocity Factor
The speed of the signal in the transmission line is
slower than the speed of a signal in free space.
The velocity of propagation of a signal in a cable
is less than the velocity of propagation of light in
free space by a fraction called the velocity
factor (VF).
VF = Vp/Vc
where Vp is the velocity in the transmission line and
Vc the velocity in free space
Time Delay
 Because the velocity of propagation of a
transmission line is less than the velocity of
propagation in free space, any line will slow down
or delay any signal applied to it.
 A signal applied at one end of a line appears some
time later at the other end of the line.
 This is called the time delay or transit time.
 A transmission line used specifically for the
purpose of achieving delay is called a delay line.
Figure 13-11: The effect of the time delay of a transmission line on signals.
(a) Sine wave delay causes a lagging phase shift. (b) Pulse delay
Transmission-Line Specifications
 Attenuation is directly proportional to cable length
and increases with frequency.
 A transmission line is a low-pass filter whose cutoff
frequency depends on distributed inductance and
capacitance along the line and on length.
 It is important to use larger, low-loss cables for
longer runs despite cost and handling
 A gain antenna can be used to offset cable loss.
Example 13-3
A l65-ft section of RG-58A/U at 100 MHz is being used to
connect a transmitter to an antenna. Its attenuation for 100 ft
at 100 MHz is 5.3 dB. Its input power from a transmitter is
100 W. What are the total attenuation and the output power
to the antenna?
Table 13-12 Table of common transmission line characteristics
Figure 13-14: Attenuation versus length for RG-58A/U coaxial
cable. Note that both scales on the graph are logarithmic
Example 13-4
A 150-ft length of RG-62AIU coaxial cable is used as a
transmission line. Find
(a) the load impedance that must be used to terminate the
line to avoid reflections,
(b) the equivalent inductance per foot,
(c) the total attenuation in decibels.
13-1: Transmission-Line Basics
13-2: Standing Waves
13-3: Transmission Lines as Circuit Elements
13-4: The Smith Chart
If the load on the line is an antenna, the
signal is converted into electromagnetic
energy and radiated into space.
If a resistive load equal to the characteristic
impedance of a line is connected at the end
of the line, the signal is absorbed by the load
and power is dissipated as heat.
If the load at the end of the line is an open or
a short circuit or has an impedance other
than the characteristic impedance of the line,
the signal is not fully absorbed by the load.
Matched Lines
A matched transmission line is one terminated
in a load that has a resistive impedance equal to
the characteristic impedance of the line.
Alternating voltage (or current) at any point on a
matched line is a constant value. A correctly
terminated transmission line is said to be flat.
The power sent down the line toward the load is
called forward or incident power.
Power not absorbed by the load is called
reflected power.
Figure 13-16: A transmission line must be terminated in its
characteristic impedance for proper operation
 When a line is not terminated properly,
some of the energy is reflected and
moves back up the line, toward the
 This reflected voltage adds to the forward
or incident generator voltage and forms a
composite voltage that is distributed along
the line.
 The pattern of voltage and its related
current forms a standing wave.
 Standing waves are not desirable.
Figure 13-8 Standing waves on a shorted transmission line
Figure 13-9 Standing waves on a open-circuit transmission line
Figure 13-20 Transmission line with mismatched load and
the resulting standing waves
Calculating the Standing Wave Ratio
The magnitude of the standing waves on a
transmission line is determined by the ratio of the
maximum current to the minimum current, or the
ratio of the maximum voltage to the minimum
voltage, along the line.
These ratios are referred to as the standing
wave ratio (SWR).
Imax Vmax
Imin Vmin
V V
SWR  1
 = max min =
Vmax  Vmin SWR  1
1 
1 
Example 13-5
An RG-11/U foam coaxial cable has a maximum voltage
standing wave of 52 V and a minimum voltage of 17 V. Find
(a) the SWR, (b) the reflection coefficient, and (c) the value of
a resistive load.
a. SWR = Vmax / Vmin = 52 / 17 = 3.05
b. Γ = (Vmax – Vmin) / (Vmax + Vmin) = (52 -17) / (52 + 17) = 0.51
c. SWR = 3.05
Z0 = 75Ω
SWR = Z0 /Zl or Zl /Z0
Zl = Z0 (SWR) = 75(3.05) = 228.75 Ω
Zl = Z0 / (SWR) = 75 / 3.05 = 24.59 Ω
13-1: Transmission-Line Basics
13-2: Standing Waves
13-3: Transmission Lines as Circuit Elements
13-4: The Smith Chart
 The standing wave conditions resulting from open- and
short-circuited loads must usually be avoided in working
with transmission lines.
 However, with one-quarter and one-half wavelength
transmissions, these open- and short-circuited loads can
be used as resonant or reactive circuits.
Figure 13-25: Impedance and reactance variations of
shorted and open lines for lengths up to one wavelength
13-1: Transmission-Line Basics
13-2: Standing Waves
13-3: Transmission Lines as Circuit Elements
13-4: The Smith Chart
The Smith Chart is a sophisticated graph
that permits visual solutions to transmission
line calculations.
Despite the availability of the computing
options today, this format provides a more or
less standardized way of viewing and solving
transmission-line and related problems.
Figure 13-30: The Smith chart
The horizontal axis is the pure resistance or
zero-reactance line.
The point at the far left end of the line
represents zero resistance, and the point at
the far right represents infinite resistance.
The resistance circles are centered on and
pass through this pure resistance line.
The circles are all tangent to one another at
the infinite resistance point, and the centers
of all the circles fall on the resistance line.
Any point on the outer circle represents a
resistance of 0 Ω.
The R = 1 circle passes through the exact
center of the resistance line and is known as
the prime center.
Values of pure resistance and the
characteristic impedance of transmission line
are plotted on this line.
The linear scales printed at the bottom of
Smith charts are used to find the SWR, dB
loss, and reflection coefficient.
 The remainder of the Smith chart is
completed by adding reactance circles
 All circles meet at the infinite resistance
 Each circle represents a constant
reactance point, with the inductivereactance circles at the top and the
capacitive-reactance circles at the
Example 1 for Fig. 13-34. The operating frequency for a 24-ft
piece of RG-58A/U coaxial cable is 140 MHz. The load is
resistive, with a resistance of 93 Ω. What is the impedance
seen by a transmitter?
Example 2 for Fig. 13-36. An antenna is connected to the 24ft 53.5-Ω RG-58A/U coaxial cable. The load is 40 + j30 Ω.
What impedance does the transmitter see?

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