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

In this article from a 1942 issue of
QST magazine, 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
Fig. 1  Section of transmission line with its terminating
load.
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.
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.
Fig. 2  Possible combinations of resistance and reactance
which may make up the load impedance, Z_{r}
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.
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.
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.
Fig. 4  Reactance variation in a seriesresonant circuit.
Fig. 5  Resistance and reactance variation in an antenna,
looking into a current loop.
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.
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.
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.
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 November 13, 2019 (updated from original post on 2/24/2014)
