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Contents
20.1 Transmission Line Basics
20.1.1 Fundamentals
20.1.2 Matched and Mismatched Lines
20.1.3 Reflection Coefficient and SWR
20.1.4 Losses in Transmission Lines
20.2 Choosing a Transmission Line
20.3 The Transmission Line as Impedance
Transformer
20.3.1 Transmission Line Stubs
20.3.2 Transmission Line Stubs as Filters
20.3.3 Project: A Field Day Stub Assembly
20.4 Matching Impedances in the Antenna System
20.4.4 Matching the Line to the Transmitter
20.4.5 Adjusting Antenna Tuners
20.4.6 Myths About SWR
20.5 Loads and Balancing Devices
20.5.1 Quarter-wave Baluns
20.5.2 Broadband Baluns
20.6 Using Transmission Lines in Digital Circuits
20.7 Waveguides
20.7.1 Evolution of a Waveguide
20.7.2 Modes of Waveguide Propagation
20.7.3 Waveguide Dimensions
20.7.4 Coupling to a Waveguide
20.8 Glossary of Transmission Line Terms
20.9 Further Reading
20.4.1 Conjugate Matching
20.4.2 Impedance Matching Networks
20.4.3 Matching Antenna Impedance at the
Antenna
Chapter
20
Transmission Lines
20.1 Transmission Line Basics
There are three main types of transmission lines used by radio amateurs: coaxial, open-wire
and waveguide. The most common type is the
coaxial
line, usually called
coax
, shown in vari-
ous forms in
Fig 20.1
. Coax is made up of a center conductor, which may be either stranded
or solid wire, surrounded by a concentric outer conductor with a
dielectric
center insulator
between the conductors. The outer conductor may be braided shield wire or a metallic sheath.
A flexible aluminum foil or a second braided shield is employed in some coax to improve
shielding over that obtainable from a standard woven shield braid. If the outer conductor is
made of solid aluminum or copper, the coax is referred to as
hardline
.
The second type of transmission line uses parallel conductors, side by side, rather than the
concentric ones used in coax. Typical examples of such
open-wire
lines are 300-Ω TV ribbon
line or
twin-lead
and 450-Ω ladder line (sometimes called
window line
), also shown in Fig
20.1. Although open-wire lines are enjoying a sort of renaissance in recent years because
of their inherently lower losses in simple multiband antenna systems, coaxial cables are far
more prevalent because they are much more convenient to use.
The third major type of transmission line is the
waveguide
. While open-wire and coaxial
lines are used from power-line frequencies to well into the microwave region, waveguides are
used at microwave frequencies only. Waveguides will be covered at the end of this chapter.
RF power is rarely generated right
where it will be used. A transmitter
and the antenna it feeds are a good
example. The most effective antenna
installation is outdoors and clear of
ground and energy-absorbing struc-
tures. The transmitter, however, is
most conveniently installed indoors,
where it is out of the weather and is
readily accessible. A
transmission line
is used to convey RF energy from the
transmitter to the antenna. A trans-
mission line should transport the RF
from the source to its destination with
as little loss as possible. This chapter,
written by Dean Straw, N6BV, and
updated by George Cutsogeorge,
W2VJN, explores transmission line
theory and applications.
20.1.1 Fundamentals
In either coaxial or open-wire line, currents flowing in the two conductors travel in opposite
directions as shown in Figs 20.1E and 20.1I. If the physical spacing between the two parallel
conductors in an open-wire line, d, is small in terms of wavelength, the phase difference be-
tween the currents will be very close to 180°. If the two currents also have equal amplitudes,
the field generated by each conductor will cancel that generated by the other, and the line
will not radiate energy, even if it is many wavelengths long.
The equality of amplitude and 180° phase difference of the currents in each conductor
in an open-wire line determine the degree of radiation cancellation. If the currents are for
some reason unequal, or if the phase difference is not 180°, the line will radiate energy. How
such imbalances occur and to what degree they can cause problems will be covered in more
detail later.
In contrast to an open-wire line, the outer conductor in a coaxial line acts as a shield,
confining RF energy within the line as shown in Fig 20.1E. Because of
skin effect
(see the
RF Design Techniques
chapter), current flowing in the outer conductor of a coax does so on
the inner surface of the outer conductor. The fields generated by the currents flowing on the
outer surface of the inner conductor and on the inner surface of the outer conductor cancel
each other out, just as they do in open-wire line.
VELOCITY FACTOR
In free space, electrical waves travel at the speed of light, or 299,792,458
meters per sec-
ond. Converting to feet per second yields 983,569,082. The length of a wave in space may
be related to frequency as wavelength = λ = velocity/frequency. Thus, the wavelength of a
Transmission Lines
20.1
Fig 20.1 — Common types of transmission lines used by amateurs. Coaxial cable, or “coax,” has a center conductor surrounded by
insulation. The second conductor, called the shield, cover the insulation and is, in turn, covered by the plastic outer jacket. Various
types are shown at A, B, C and D. The currents in coaxial cable low on the outside of the center conductor and the inside of the
outer shield (E). Open-wire line (F, G and H) has two parallel conductors separated by insulation. In open-wire line, the current lows
in opposite directions on each wire (I).
1 Hz signal is 983,569,082 ft. Changing to a
more useful expression gives:
λ=
983.6
f
lator which slows the wave travel down. The
actual velocity of the wave is a function of
the dielectric characteristic of that insulator.
We can express the variation of velocity as
the
velocity factor
for that particular type of
dielectric — the fraction of the wave’s ve-
locity of propagation in the transmission line
compared to that in free space. The velocity
factor is related to the dielectric constant of
the material in use.
So the wavelength in a real transmission
line becomes:
λ=
983.6
VF
(3)
f
(1)
where
λ = wavelength, in ft
f = frequency in MHz.
As an example, many coax cables use poly-
ethylene dielectric over the center conductor
as the insulation. The dielectric constant for
polyethylene is 2.3, so the VF is 0.66. Thus,
wavelength in the cable is about two-thirds as
long as a free-space wavelength.
The VF and other characteristics of many
types of lines, both coax and twin lead, are
shown in the table “Nominal Characteristics
of Commonly used Transmission Lines” in the
Component Data and References
chapter.
Thus, at 14 MHz the wavelength is 70.25 ft.
Wavelength (λ) may also be expressed in
electrical degrees. A full wavelength is 360°,
1
⁄
2
λ is 180°,
1
⁄
4
λ is 90°, and so forth.
Waves travel slower than the speed of light
in any medium denser than a vacuum or free
space. A transmission line may have an insu-
1
VF
=
(2)
ε
where
VF = velocity factor
ε = dielectric constant.
20.2
Chapter 20
There are differences in VF from batch
to batch of transmission line because there
are some variations in dielectric constant
during the manufacturing processes. When
high accuracy is required, it is best to actually
measure VF by using an antenna analyzer to
measure the resonant frequency of a length
of cable. (The antenna analyzer’s user manual
will describe the procedure.)
Fig 20.2 — Equivalent of an ininitely long lossless transmission line using lumped
circuit constants.
CHARACTERISTIC IMPEDANCE
A perfectly lossless transmission line may
be represented by a whole series of small
inductors and capacitors connected in an in-
finitely long line, as shown in
Fig 20.2
. (We
first consider this special case because we
need not consider how the line is terminated
at its end, since there is no end.)
Each inductor in Fig 20.2 represents the
inductance of a very short section of one wire
and each capacitor represents the capacitance
between two such short sections. The induc-
tance and capacitance values per unit of line
depend on the size of the conductors and the
spacing between them. Each series inductor
acts to limit the rate at which current can
charge the following shunt capacitor, and in so
doing establishes a very important property of
a transmission line: its
surge impedance
, more
commonly known as its
characteristic imped-
ance
. This is usually abbreviated as Z
0
,
ance. Practical open-wire lines exhibit char-
acteristic impedances ranging from about 200
to 800 Ω, while coax cables have Z
0
values
between 25 to 100 Ω. Except in special in-
stances, coax used in amateur radio has an
impedance of 50 or 75 Ω.
All practical transmission lines exhibit
some power loss. These losses occur in the re-
sistance that is inherent in the conductors that
make up the line, and from leakage currents
flowing in the dielectric material between the
conductors. We’ll next consider what happens
when a real transmission line, which is not
infinitely long, is terminated in an actual load
impedance, such as an antenna.
20.1.2 Matched and
Mismatched Lines
Real transmission lines do not extend to
infinity, but have a definite length. In use they
are connected to, or
terminate
in, a load (such
as an antenna), as illustrated in
Fig 20.3A
.
If the load is a pure resistance whose value
equals the characteristic impedance of the
line, the line is said to be
matched
. To current
traveling along the line, such a load at the end
of the line acts as though it were still more
transmission line of the same characteristic
impedance. In a matched transmission line,
energy travels outward along the line from
the source until it reaches the load, where it
is completely absorbed (or radiated if the load
is an antenna).
Fig 20.3 — At A the coaxial transmission
line is terminated with resistance equal to
its Z
0
. All power is absorbed in the load.
At B, coaxial line is shown terminated in
an impedance consisting of a resistance
and a capacitive reactance. This is a
mismatched line, and a relected wave will
be returned back down the line toward
the generator. The relected wave adds to
the forward wave, producing a standing
wave on the line. The amount of relection
depends on the difference between the
load impedance and the characteristic
impedance of the transmission line.
L
Z
C
where L and C are the inductance and
capacitance per unit length of line.
The characteristic impedance of an air-
insulated parallel-conductor line, neglecting
the effect of the insulating spacers, is given
by
≈
0
2S
Z
=
276 log
(4)
0
10
d
where
Z
0
= characteristic impedance
S = center-to-center distance between
conductors
d = diameter of conductor (in same units
as S).
The characteristic impedance of an air-
insulated coaxial line is given by
will go to zero, while the current will rise to a
maximum. Since the current can’t develop any
power in a dead short, the energy will all be
reflected back toward the source generator.
If the short at the end of the line is replaced
with an open circuit, the opposite will hap-
pen. Here the voltage will rise to maximum,
and the current will by definition go to zero.
The phase will reverse, and all energy will be
reflected back towards the source. By the way,
if this sounds to you like what happens at the
end of a half-wavelength dipole antenna, you
are quite correct. However, in the case of an
antenna, energy traveling along the antenna is
lost by radiation on purpose, whereas a good
transmission line will lose little energy to ra-
diation because of field cancellation between
the two conductors.
For load impedances falling between
the extremes of short- and open-circuit, the
phase and amplitude of the reflected wave
MISMATCHED LINES
Assume now that the line in Fig 20.3B is
terminated in an impedance Z
a
which is not
equal to Z
0
of the transmission line. The line
is now a
mismatched
line. RF energy reaching
the end of a mismatched line will not be fully
absorbed by the load impedance. Instead, part
of the energy will be reflected back toward
the source. The amount of reflected versus
absorbed energy depends on the degree of
mismatch between the characteristic imped-
ance of the line and the load impedance con-
nected to its end.
The reason why energy is reflected at a
discontinuity of impedance on a transmis-
sion line can best be understood by examining
some limiting cases. First, consider the rather
extreme case where the line is shorted at its
end. Energy flowing to the load will encounter
the short at the end, and the voltage at that point
b
Z
=
138 log
(5)
0
10
a
where
Z
0
= characteristic impedance
b = inside diameter of outer conductors
a = outside diameter of inner conductor
(in same units as b).
It does not matter what units are used for S,
d, a or b, as long as they are the
same
units. A
line with closely spaced, large conductors will
have a low characteristic impedance, while
one with widely spaced, small conductors will
have a relatively high characteristic imped-
Transmission Lines
20.3
will vary. The amount of energy reflected
and the amount of energy absorbed in the
load will depend on the difference between
the characteristic impedance of the line and
the impedance of the load at its end.
What actually happens to the energy re-
flected back down the line? This energy will
encounter another impedance discontinuity,
this time at the generator. Reflected energy
flows back and forth between the mismatches
at the source and load. After a few such jour-
neys, the reflected wave diminishes to noth-
ing, partly as a result of finite losses in the
line, but mainly because of absorption at the
load. In fact, if the load is an antenna, such
absorption at the load is desirable, since the
energy is actually radiated by the antenna.
If a continuous RF voltage is applied to the
terminals of a transmission line, the voltage at
any point along the line will consist of a vec-
tor sum of voltages, the composite of waves
traveling toward the load and waves traveling
back toward the source generator. The sum of
the waves traveling toward the load is called
the
forward
or
incident
wave, while the sum
of the waves traveling toward the generator
is called the
reflected wave
.
load impedance is not exactly known, since
an antenna usually terminates a transmission
line, and the antenna impedance may be influ-
enced by a host of factors, including its height
above ground, end effects from insulators, and
the effects of nearby conductors. We may also
express the reflection coefficient in terms of
forward and reflected power, quantities which
can be easily measured using a directional
RF wattmeter. The reflection coefficient and
SWR may be computed as
2
2
(120
−
50)
+−
( 90)
ρ=
=
0.593
2
2
(120
+
50)
+−
( 90)
Note that if R
l
in equation 6 is equal to R
0
and X
l
is 0, the reflection coefficient, ρ, is 0.
This represents a matched condition, where
all the energy in the incident wave is trans-
ferred to the load. On the other hand, if R
l
is
0, meaning that the load is a short circuit and
has no real resistive part, the reflection coeffi-
cient is 1.0, regardless of the value of R
0
. This
means that all the forward power is reflected
since the load is completely reactive.
The concept of reflection is often shown in
terms of the
return loss
(RL), which is given
in dB and is equal to 20 times the log of the
reciprocal of the reflection coefficient.
P
P
r
f
ρ=
(10A)
and
P
r
f
1
+
P
(10B)
P
SWR
=
r
f
RL (dB)
=−
10 log
=−
20 log( )
ρ
(8)
P
P
r
f
1
−
P
In the example above, the return loss is 20 log
(1/0.593) = 4.5 dB.
If there are no reflections from the load, the
voltage distribution along the line is constant
or
flat
. A line operating under these conditions
is called either a
matched
or a
flat
line. If
reflections do exist, a voltage
standing-wave
pattern will result from the interaction of the
forward and reflected waves along the line.
For a lossless transmission line at least
1
⁄
4
-
λ
long, the ratio of the maximum peak voltage
anywhere on the line to the minimum value
anywhere along the line is defined as the
volt-
age standing-wave ratio
, or VSWR. (The line
must be
1
⁄
4
λ or longer for the true maximum
and minimum to be created.) Reflections from
the load also produce a standing-wave pattern
of currents flowing in the line. The ratio of
maximum to minimum current, or ISWR, is
identical to the VSWR in a given line.
In amateur literature, the abbreviation
SWR
is commonly used for standing-wave
ratio, as the results are identical when taken
from proper measurements of either current
or voltage. Since SWR is a ratio of maximum
to minimum, it can never be less than one-to-
one. In other words, a perfectly flat line has
an SWR of 1:1. The SWR is related to the
magnitude of the complex reflection coef-
ficient and vice versa by
where
P
r
= power in the reflected wave
P
f
= power in the forward wave.
If a line is not matched (SWR > 1:1) the
difference between the forward and reflected
powers measured at any point on the line is
the net power going toward the load from that
point. The forward power measured with a
directional wattmeter (often referred to as a
reflected power meter or
reflectometer
) on a
mismatched line will thus always appear
greater than the forward power measured on
a flat line with a 1:1 SWR.
The software program
TLW
,
written by
Dean Straw, N6BV, and included on the
ARRL Antenna Book
CD solves these com-
plex equations. This should come as a big
relief for most radio amateurs. The character-
istics of many common types of transmission
lines are included in the software so that real
antenna matching problems may be easily
solved. Detailed instructions on using the pro-
gram are included with it. The various ex-
amples in this chapter have been solved with
TLW
. An introductory version of
TLW
, called
TLA
, is included on the Handbook CD.
Equation 6 is also solved in a normalized
form with a graphical method called the Smith
Chart. Matching problems can be handled
much easier than solving complex arithme-
tic equations by using this polar chart. More
detailed information about using the Smith
Chart is included in
The
ARRL Antenna Book
.
Many references to Smith Charts and their use
may be found on the Web.
20.1.3 Relection Coeficient
and SWR
In a mismatched transmission line, the ratio
of the voltage in the reflected wave at any one
point on the line to the voltage in the forward
wave at that same point is defined as the
volt-
age reflection coefficient
. This has the same
value as the current reflection coefficient. The
reflection coefficient is a complex quantity
(that is, having both amplitude and phase)
and is generally designated by the Greek let-
ter ρ (rho), or sometimes in the professional
literature as Γ (Gamma). The relationship
between R
l
(the load resistance), X
l
(the load
reactance), Z
0
(the line characteristic imped-
ance, whose real part is R
0
and whose reac-
tive part is X
0
) and the complex reflection
coefficient ρ is
Z
−
Z
(R
± −±
jX )
(R
jX )
(6)
l
0
l
l
0
0
ρ=
+
± +±
Z
Z
(R
jX )
(R
jX )
l
0
l
l
0
0
For most transmission lines the charac-
teristic impedance Z
0
is almost completely
resistive, meaning that Z
0
= R
0
and X
0
≅ 0.
The magnitude of the complex reflection co-
efficient in equation 6 then simplifies to:
1
+ρ
(9A)
SWR
=
1
−ρ
and
2
2
(R
−
R )
+
X
l
0
l
ρ=
(7)
2
2
SWR
−
1
(R
+
R )
+
X
(9B)
l
0
l
ρ=
SWR
+
1
For example, if the characteristic imped-
ance of a coaxial line is 50 Ω and the load
impedance is 120 Ω in series with a capaci-
tive reactance of
–
90 Ω, the magnitude of the
reflection coefficient is
20.1.4 Losses in
Transmission Lines
A real transmission line exhibits a certain
amount of loss, caused by the resistance of the
The definitions in equations 8 and 9 are
valid for any line length and for lines which
are lossy, not just lossless lines longer than
1
⁄
4
λ at the frequency in use. Very often the
20.4
Chapter 20
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