December 1942 QST
Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
from ARRL's
QST, published December 1915  present. All copyrights hereby acknowledged.

Author T.A. Gadwa employs a standing wave mechanism analogy that
I don't recall having read before  that of a dam on a river. The
river is the transmission line with a lake as the source (presumably)
and then he imagines a dam load. The dam standing waves, per his
description, have phase and amplitude characteristics that depend
on how tall the dam wall is relative to the surface height of the
dammed river. An extensive array of graphs is provided showing how
the current of the dam standing waves react to the dam transmission
line termination impedance. I always wonder when seeing electricalmechanical
parity examples whether, as with this case, there are any dam magazine
articles out there that use an electrical transmission line to help
fellow civil engineers understand their dam designs?
Standing Waves on Transmission Lines
A Method of Line Matching Based on Graphical Comparison
By T. A. Gadwa, SC.D., W2KHM
Standing waves are often a problem to amateurs who attempt to
use untuned transmission lines for their antennas. The elimination
of these waves is often difficult because of a lack of understanding
of the principles involved. Using the analogy of water waves in
a canal is often helpful in visualizing the factors that influence
the operation of transmission lines. Suppose the canal has a dam
at one end and a wave is created at the opposite end. This wave
traveling toward the dam is reflected back to the starting point.
Now if the height of the dam is lowered sufficiently to allow the
initial wave to splash over, then no return wave or reflection is
produced. In the radiofrequency application, the canal corresponds
to the transmission line and the dam to the load or antenna.
Fig. 1  Section of transmission line with its
terminating load.
Terman^{1} has analyzed the position and magnitude of
standing waves on lines for several different types of loads. Everitt^{2}
has derived equations that make it possible to establish the character
of these waves. If one neglects the line losses, which are usually
small for relatively short lengths of line, the calculation is simplified
considerably. At radio frequencies, such lines may be assumed to
behave as pure resistances. The current at any point on the line
for any type of load is given by the equation:
(1)
The voltage at any point is given by the equation:
(2)
where I_{s} = current at any point in the line
I_{r} = load current at output or receiving end of line
E_{s} = voltage at any point on line
E_{r} = voltage at output or receiving end of line
Z_{r} = load impedance
R_{o} = characteristic impedance of line
θ = distance from point to output or receiving end of line
2π radians = 360° = 1 wavelength
indicating 90° phase shift
+ j for inductive reactance
 j for capacitive reactance
The load may be any of the combinations shown in Fig. 2. The
character of the standing waves that are associated with each case
will be discussed.
If a voltage is applied or a current induced at the sending end
of a transmission line and the receiving end is an open circuit,
a wave traveling toward the open circuit is reflected wholly since
no power is absorbed. This reflected wave combines with the incident
wave to form standing waves. Waves that can be measured as average
values of current or voltage are called standing waves. The readings
are all positive since no account is taken of phase between the
current and voltage. The results are represented as positive values
plotted as ordinates above the horizontal axis. At the open circuit,
the voltage is reflected in phase since the incident and reflected
voltages are equal and their sum is not zero, while the current
is reflected out of phase since the incident and reflected currents
are equal and their sum is zero. If the average current or voltage
along the line is measured, maxima and minima are found at regular
intervals from the receiving end of the line. The current distribution
for an open circuit is shown in Fig. 31. Only a half wavelength
is shown as the cycle is repeated for additional lengths of line.
To avoid confusion that might result if voltage were superimposed,
only the current waveforms are plotted. There is always a current
minimum or node at the receiving end and at every halfwavelength
point back along the line, and a current maximum or loop at every
quarter wavelength. Furthermore, the voltage is maximum or a loop
at each current minimum or node and there is a voltage minimum or
node at each current maximum or loop. It is evident that maximum
and minimum values of either voltage or current occur exactly 1/4
wavelength apart.
Fig. 2  Possible combinations of resistance
and reactance which may make up the load impedance, Z_{r}.
If the receiving end is shortcircuited, a wave starting down
the line is reflected and again standing waves are found. Here the
positions of the maximum and minimum have been shifted and appear
as in Fig. 32. At the short circuit, the incident and reflected
voltages are out of phase and their sum is zero, while the current
is reflected in phase and the sum of the two components is not zero.
If an appropriate pure resistance equal to a constant known as the
characteristic impedance of the line is connected as a receiving
load, a wave starting down the line is absorbed completely and no
reflection is possible. The current and voltage are constant at
all points, with no maximum or minimum, as shown in Fig. 35. Since
all actual lines have losses, the current and voltage diminish slowly
toward the load, as indicated in Fig. 36. Such lines are known
as flat lines. This load impedance is dependent only on the physical
properties of the line: the conductor diameter, conductor spacing
and type of insulation or dielectric. Its value for an openair
twowire parallel line is calculated by the formula:
R_{o} = 276 log (2S)/D
(3)
where R_{o} = characteristic impedance of line in ohms
S = spacing between conductor centers in any units.
D = diameter of one conductor in same units
If the load resistance is less than the line impedance but not
a short circuit, the standing waves are similar to the short circuit
load except that the minimum current is greater than zero, as shown
in Fig. 34. If the load is made greater than the line impedance
but not infinite (open circuited), the standing waves are as shown
in Fig. 33. It is evident that the maximum and minimum currents
occur in the same positions as in the case of the opencircuit load,
but the maximumtominimum ratio is less. The ratio approaches the
value of unity as the load approaches the characteristic impedance.
If the load is an inductive reactance equal to the characteristic
impedance, Fig. 2B, no power is absorbed and standing waves are
as shown in Fig. 39. The line behaves similarly to a shortcircuit
load except that the waves are shifted toward the receiving or load
end. The current is zero at 1/8 wavelength from the load and maximum
at 1/4 wavelength farther along and then every 1/2 wavelength to
the sending end. As the inductive reactance is increased from values
less than to greater than the characteristic impedance, the standing
waves are shifted toward the receiver or load end as can be seen
by comparing Figs. 37, 39 and 311.
When the load is a capacitive reactance, Fig. 2C, equal to the
characteristic impedance, no power is absorbed and standing waves
are present in the form given in Fig. 310. The line behaves similarly
to an open circuit except that the standing waves are all shifted
toward the receiving or load end. A current maximum occurs at 1/8
wavelength from the load and every 1/2 wavelength toward the sending
end of the line. As the capacitive reactance is reduced from greater
than to less than the characteristic impedance, the standing waves
are shifted toward the receiver as can be seen by comparing Figs.
312, 310 and 38.
There are many other possible combinations of resistance in series
or parallel with either or both inductive and capacitive reactances,
Figs. 2A to 2F. This discussion is confined to series circuits,
since any parallel circuit can be transformed into an equivalent
series circuit. Theoretically there are 4 X 4 X 4 + 2 or 66 combinations
where the individual components are less than, equal to or greater
than the characteristic impedance. A series circuit of resistance,
capacitance and inductance behaves like a resistance in series with
either inductance or capacitance, depending upon the frequency,
except at resonance where it is resistive only. This limits the
actual number of cases to 3 X 3 X 3 + 2 or 29. For series loads
of resistance and inductance, the wave forms are given in Figs.
313 to 321. For series loads of resistance and capacitance, the
wave forms are given in Figs. 322 to 330.
If the load is composed of resistance and reactance whose total
impedance is equal to the line impedance, the maximum or minimum
current or voltage always occurs at exactly 1/8 wavelength from
the receiver load, regardless of the resistance to reactance ratio.
This can be seen by comparing Figs. 331, 332 and 333 for inductive
and resistive loads and Figs. 334, 335 and 336 for capacitive
and resistive loads. The maximumtominimum ratio of current or
voltage approaches unity as the resistancetoreactance ratio increases.
When the total load impedance and the resistance component are
each greater than the line impedance, an increase in inductive reactance
shifts the waves back from the load end, as can be seen by comparing
Figs. 319, 320 and 321. Similarly, a decrease in capacitive reactance
shifts the waves back from the load end, as can be seen by comparing
Figs. 330, 329 and 328. This effect of reactance change upon
wave shift is in the opposite direction to the shift obtained with
reactance loads only.
With a series inductive reactance and resistance load whose total
impedance is greater than the characteristic impedance, the minimum
current is always less than 1/8 wavelength from the receiver.
With a series capacitive reactance and resistance load whose
total impedance is less than the characteristic impedance, the maximum
current is always less than 1/8 wavelength from the receiver.
Fig. 3  Positions and relative magnitudes of
standing waves for various load impedances. The curves show the
ratio of current at the point on the line considered to the current
in the load.
Some of the curves for current may be applied for the distribution
of voltage as well. The line current for the open circuit load is
also the line voltage for a short circuit load. The line current
for the short circuit load is also the line voltage for an open
circuit load. Similarly, Fig. 322 represents the voltage for load
conditions in Fig. 317 and vice versa. Also Fig. 326 represents
the voltage for load conditions in Fig. 313 and vice versa. The
voltage and current at some points on the line may rise above the
sending and receiving values because of the resonant effect of the
transmission line.
Fig. 4  Reactance variation in a seriesresonant
circuit.
Matching the Antenna
An antenna is a seriesresonant circuit and may act as a load
for the transmission line. In most cases it is inconvenient or impossible
to secure a direct match between the line and antenna. It is then
necessary to insert some sort of impedance transformer between the
antenna and receiver terminals of the line in order to present a
load equal to the line impedance. An antenna acts like a pure resistance
at resonance, is capacitive and resistive at lower frequencies,
and is inductive and resistive at higher frequencies; or, for a
given frequency, the antenna is capacitive and resistive if too
short and inductive and resistive if too long. The reactance of
a series resonant circuit is shown in Fig. 4. The resistance of
an antenna changes with frequency or length and is maximum at resonance.
The resistance and reactance of an antenna may also be represented
as shown in Fig. 5.
Before it is possible to obtain an impedance match and a flat
line, the antenna must be tuned to resonance either by adjustment
of its length or by inserting a series inductance if too short or
a series capacitance if too long. The recognized method is to excite
the antenna parasitically and obtain maximum antenna current by
tuning. This is laborious and requires accurate measuring equipment.
Neither can the exact length of the antenna be calculated for resonance.
Many avoid this step by erecting the complete antenna system and
attempting to obtain a flat line by trial and error in antenna tuning
and impedancetransformer adjustments. This procedure may result,
in rare cases, in obtaining a flat line. It is evident that the
number of variables is too numerous to achieve the desired results
with a minimum of experiment. At this point a working knowledge
of standing waves will enable one to establish the condition of
the antenna or the transmissionline load. A scheme is proposed
whereby, after determining the relative magnitude and position of
the maximum and minimum or loop and node of voltage or current,
the antenna condition is indicated by comparison with various curves
for different types of loads.
The complete set of curves shown in Fig. 3 covers all possible
combinations of loads that may be encountered. By the recognized
mathematical methods of differential calculus, the maximum and minimum
positions have been calculated by differentiating the line current
I_{s} of equations (1) and (2) with respect to the distance θ,
equating to zero and solving for θ. Some of these equations are
of second degree and require solution by the quadratic equation
method.
The idea in adjustment is to shift the minimum or maximum current
to the 1/4 wavelength position (to eliminate the reactive component)
and then to reduce the maximumtominimum ratio to unity by impedance
transformer adjustment. The procedure is to supply sufficient power
to the transmission line, with the antenna system in position, to
permit measurement of the line current or voltage, starting at the
load if possible, and then by measurement at equal small intervals
to establish the location of the maximum and minimum voltage or
current. In most cases it is preferable to locate the current nodes
or minima rather than the voltage, because the change in current
per unit length of line is more rapid and easier to detect. Onesixteenth
wavelength intervals are sufficiently close to enable one to plot
a curve of current or voltage vs. distance to the load. If it is
impracticable to start at the load it is permissible to begin at
any multiple of a half wavelength from the load, since the standing
waves are repeated along the line to the sending end. Radiofrequency
waves travel more slowly on transmission lines than in air, so that
the length of a wavelength for an openwire line is usually about
97.5 per cent of that in air.
If maximum or minimum current or voltage occur at points other
than at multiples of 1/4 wavelength from the load, the antenna is
nonresonant and must first be tuned to resonance by whatever method
is desirable. If the maximum current occurs between the load and
1/4 wavelength, the antenna is capacitive or too short and must
be lengthened or series inductance added.
Fig. 5  Resistance and reactance variation in
an antenna, looking into a current loop.
If minimum current occurs between the load and 1/4 wavelength,
the antenna is inductive or too long and must be shortened or series
capacitance added. If maximum or minimum current or voltage occur
at 1/4 wavelength from the load, the antenna is resonant but the
impedance match is incorrect. The impedance transformer then should
be adjusted until the maximum and minimum values are equal or the
standing wave ratio is unity.
All the wave forms shown can be encountered during the process
of tuning the antenna and matching the line impedance. As the antenna
approaches resonance and the impedance match becomes nearly correct,
the waves may look like Figs. 316 or 325. Poor adjustments may
yield waves like Figs. 321 and 330.
It must be emphasized that no adjustment at the sending or transmitter
end of the line will change the position of the standing waves.
This adjustment will only control the degree of coupling and the
amount of power delivered to the line and antenna. A reactive component
always appears at the sending end if standing waves are present.
It is evidenced by the necessity for resetting the plate tank tuning
capacity to obtain minimum plate current when the line is coupled.
All adjustments must be made first to the antenna and then to the
impedance transformer for the elimination of standing waves. For
efficient reception, a proper impedance match must be made at the
receiving end. In this case the antenna is at the sending end and
the receiver at the output of the line.
With this approach to the standing wave problem, that elusive
flat line should be easily realized by all amateurs using untuned
transmission lines for their antennas.
1 Terman, Radio Engineering.
2 Everitt, Communication Engineering.
Posted February 24, 2014
