
A Review of Transmission Lines as Circuit Elements
November 1966 QST Article


November 1966 QST
[Table
of Contents]These articles are scanned and OCRed from old editions of the
ARRL's QST magazine. Here is a list of the
QST articles I have already posted. All copyrights (if any) are hereby acknowledged. 
Mr.
Wilfred Jensby wrote an incredibly detailed article for the November
1966 edition of QST that delves deeply into the subject of using transmission
lines as distributed circuit elements. I did a search on his name, figuring
that he likely had other publications of like sort, but nothing was
found. Information contained herein is similar to what you would expect
to find in a Master's level engineering course textbook or in a $100+
technical book from Artech House, Cambridge University Press, John Wiley &
Sons, etc. The brainzapping equations are omitted with only a great,
laymanlevel discussion of the concepts and some really nice illustrations
and graphs. This is definitely an article you will want to check out
and pass on to colleagues.
See all available
vintage QST articles.
A Review of Transmission Lines as Circuit Elements
By Wilfred Jensby, WA6BQO
Fig. 1  Table of equivalent circuits using resonant lines. Voltage
and current relationships are illustrated for open and shorted line.
Fig. 2  Chart showing reactance of lines, expressed in terms of
Z _{0}, is illustrated at A. At B, a chart showing the characteristic
impedance of lines from 0220 ohms.
Fig. 3  Approximate voltage and current distribution in onequarter
wavelength (A), onehalf wavelength (B), and threequarter wavelength
(C) resonant coaxial lines. The field strength, E, is also shown.
At B, an illustration of magnetic and electrical coupling to coaxial
cavity circuits.
Fig. 4  Graphic representations of coaxial line characteristics
are shown at A. At B, a chart showing Q in connection with element
diameters and frequency, for concentric lines based on b/a = 3.6,
using copper lines and air dielectric.
Fig. 5  Illustrations of various applications for parallel line
sections as discussed in the text, at A. Phaseshift characteristics
for line sections are shown at B.
Many amateurs active on the v.h.f. bands enjoy building their own equipment.
The r.f. circuits often consist of hardware or plumbing which involves
considerable metal work. Cutandtry methods involve much more time
and expense than at the lower frequencies.
I will review some
of the design details involved in highfrequency circuit construction,
so that most of the cutandtry work can be done on paper.
Transmissionline
sections are used as circuit elements at v.h.f. because of their desirable
impedance properties. Lines that are used for such purposes are usually
opencircuited or shortcircuited at the receiving end, and do not serve
to actually transmit energy. The term" transmission line" is used for
purposes of clarity.
Equivalent Circuits
If we consider only what appears at the input terminals, a shortcircuited
quarterwavelength line and a parallelresonant circuit, of coil and
capacitor, have these characteristics in common; both present extremely
high impedance at one particular frequency; with both, the impedance
at resonance is resistive and the impedance drops rapidly if the frequency
varies slightly from resonance. Both will carry direct current freely
while effectively blocking the frequency to which they are resonant.
An inherent difference is that the transmission line displays
similar resonance at all odd multiples of its lowest resonant frequency;
and has the inverse resonance characteristics of a shorted halfwavelength
line at the even multiples.
An opencircuited quarterwavelength
line is similar to a seriesresonant circuit of coil and capacitor.
It has extremely low impedance at the resonant frequency, is resistive
at resonance while being inductive above and capacitive below this frequency.
It blocks direct current while freely passing the resonantfrequency
r.f. energy. Like a shortcircuited line (but unlike a circuit of lumped
constants), its characteristics tend to repeat at odd multiples of the
lowest resonant frequency, whereas at even multiples the inverse characteristics
appear.
An opencircuited halfwavelength line is similar to
a shortcircuited quarterwavelength line in that both have the same
Q and are thus equally selective in a resonant circuit. However, at
radio frequencies other than the desired resonant frequency (such as
half and double the fundamental resonant frequency), the open and shortcircuited
lines have quite different characteristics. This may be important in
connection with harmonics.
'With a quarterwave line, the closest
resonant frequencies to the fundamental occur at odd multiples such
as 3, 5 and 7 times the fundamental frequency. With a halfwave line,
they occur at multiples of 2, 3 and 4 times the fundamental. A quarterwave
resonant line, therefore, gives greater separation of the higherresonant
frequencies from the fundamental.
Parallel Lines
Parallel lines are most often used with pushpull circuits,
in either quarterwave or halfwave configuration. With halfwavelength
lines, the B plus is connected at the electrical center of the lines,
and often a coil, resonant at a lower frequency, is placed here to give
multiband operation.
Parallel lines are relatively easy to construct.
Their electrical length may be readily changed with shortcircuiting
bars, and when they are used with appropriate types of tubes, the connections
between lines and tube terminals can be short and direct. Furthermore,
these connections and the portions of the tube leads inside the envelope
become parts of the resonantline system. For very high frequencies,
the tube leads may constitute the principal part of this system but
are largely inaccessible for purposes of poweroutput coupling. In some
cases, the portion of the circuit from which power is to be coupled
may be operated at a multiple length of the shortest possible line;
e.g., threequarter rather than onequarter wavelength.
Since
open parallel lines radiate electromagnetic energy when excited, it
is necessary to shield these lines for optimum performance. The parts,
such as sides and covers, of the metal boxes used as the shield should
be well bonded together, either with screws or by contact fingers. This
is because electromagnetic shielding depends on the flow of induced
currents in the metal of the shield. For the same reason, the shield
should be constructed from material of high conductivity. For ultrahigh
frequencies, silver plating is desirable.
Several methods of
tuning are available. An adjustable shortcircuiting strap can be used,
which must make good electrical contact. If the line is also shortcircuited
at the end by a large disk of copper or other good conducting material,
it will be more effective. A butterfly capacitor, or a parallelplate
capacitor, may be placed anywhere along the line the tuning effect becoming
less pronounced as the capacitor is located nearer the shorted end of
the line.
The characteristic impedance of parallel conductors
may be calculated as follows:
where b is the centertocenter spacing of the conductor and a is
the radius of the conductors. This relationship is shown in Fig. 2.
For twowire lines, minimum attenuation theoretically will occur
when b/a = 2.7. However, when proximity effect is included, the optimum
b/a ratio is about 4. The b/a ratio to give maximum impedance to a shortcircuited
quarterwavelength 2wire line i about 8.0.
Coaxial
Lines
When the various characteristics (Fig. 4) of
a coaxial transmission line are considered, such as attenuation, resonant
impedance, breakdown voltage, and powercarrying capacity, an optimum
ratio of b/a = 3.6 is found to exist, where b is the inner radius of
the outer conductor, and a is the outer radius of the inner conductor.
Minimum attenuation occurs at this value, which also corresponds to
a characteristic impedance of 77 ohms for a line with air dielectric.
This is an important reason for the widespread practical use of lines
with approximately this impedance.
Physically, if the inner
conductor is smaller than the optimum size, its resistance is higher
and loss is increased. If the inner conductor is larger than optimum,
the increased capacitance lowers the value of Z and hence more current
is required to transmit a certain amount of power, with the result that
loss is again increased.
However, a line designed for minimum
attenuation is not best for all purposes. A line may be designed to
transmit maximum power. The limiting factor is electric field strength
at the surface of the inner conductor; if a critical value of field
strength (about 30,000 volts per centimeter) is exceeded, corona or
sparking results. The optimum value of b/a for maximum power transmission
is 1.65, and the corresponding characteristic impedance is 30 ohms.
When a line is designed to act as a resonant circuit, other
values of b/a may be preferred. For a shortcircuited resonant coaxial
line to have maximum impedance, b/a should be 9.2, corresponding to
Z_{0} equals 133 ohms for an airinsulated line. For an opencircuited
resonant line to have minimum impedance, the inner conductor of the
coaxial line should be as large as possible, requiring Z_{0}
to approach zero.
CoaxialLine Oscillators and Amplifiers
The adoption of conventional oscillator and amplifier circuits to
u.h.f. use is facilitated by the use of coaxial lines as circuit elements.
The high inherent Q of concentric lines as resonant circuits, the very
low radiation, and the possibility of isolation of the circuits, contribute
to successful design. The lighthouse tube is designed especially for
such circuits. The cylindrical, or dish construction, is carried through
from the external terminal of the tube to the active part of the tube
elements. A high degree of circuit isolation is thus possible, and coupling
between circuits is reduced to a minimum.
The groundedgrid
circuit is often used for oscillators and amplifiers at u.h.f. and is
particularly advantageous in amplifier operation. The feedback or coupling
capacitance between output and input circuits is the platecathode capacitance,
which is reduced to a minimum in most tubes suitable for coaxial circuit
use. Thus, regeneration through interelectrode feedback is materially
reduced by grid shielding.
The similarity between the gridseparationtype
oscillator and amplifier circuits is considerable. The conversion of
an oscillator to an amplifier consists primarily of removing the external
feedback system, the addition of a source of driving energy, and retuning.
The platecircuit loaded Q will influence both the frequency stability
and modulated bandwidth of an oscillator and, for a given loaded resonant
impedance, will depend on line dimensions, tube capacitance, and the
operating mode.
LoadedQ Consideration
Whereas in the ideal case, the expression for the input impedance
of the coaxial line is frequently treated as a pure reactance, it should
not be forgotten that the line is actually a circuit element with distributed
constants, both inductive and capacitive. While the inductive reactance
of a shortcircuited line less than 90 degrees in length may be used
to tune out a terminating capacitive reactance, the total capacitance
in the resonant circuit is materially increased by that which is distributed
in the line.
The distributed capacitance of a coaxial line is
a function of the characteristic impedance. This is of importance where
high operating Q must be considered for its limitation on the modulated
bandwidth or, in the case of an oscillator, for its influence on frequency
stability. A given input reactance might be obtained with a short highcharacteristicimpedance
line or a long lowcharacteristicimpedance line. The resonant circuit
Q of the short line when shuntloaded with a given resistance will be
lower than that of the longer line if the electrical length of the lines
is less than 90 degrees. The extra storage of energy in the lowimpedance
line will increase its operating Q over that of the highimpedance line.
Where physical dimensions are concerned, low and high might be considered
to be about 20 and 90 ohms, respectively.
Fig. 6  Nomograph for determining
physical lengths of lines at various frequencies with relation to
terminating capacitance.
Limitations on Tuning Range
A practical
limitation on the lowfrequency range of a coaxial oscillator or amplifier
is the actual physical length of the line elements, which rapidly increases
as the frequency is lowered. This can be appreciated when the actual
physical quarterwavelength is considered at low frequencies, for the
resonant lines approach this length quite closely as the reactance of
a fixed terminating capacitance increases with the decrease in frequency.
When overall physical length is an important consideration,
it is helpful to remember that a given terminating capacitance may be
resonated, with a fixedmaximum length of line, to a lower frequency
with a line of higher characteristic impedance.
Fig.7A photo of a typical 432Mc. amplifier coaxial cavity (upper),
and a 432Mc. coaxial filter, with crystal diode detector added
(lower).
Fig. 8  A block diagram illustrating three typical applications
for coaxial filters.
Physical dimensions also influence the practicable upperfrequency limit
of coaxial lines as resonant circuit elements. This results from the
ability of cavities of large radial electrical dimensions to support
interfering waveguide and spurious coaxialresonance modes. The principal
interfering higherorder coaxialresonance mode is the TE mode, which
can exist only at wavelengths less than the cutoff value given by:
where a is the radius of the inner conductor, and b the radius of
the outer conductor. In any event, this TE mode should not interfere
if the resonantcircuit line lengths are less than 90 degrees.
Coaxial Filters
Preselectors, or bandpass
filters, are often made using quarterwave or threequarterwave coaxial
resonators. These can be nearly identical to coaxial v.h.f. amplifiers
except that they are passive circuits. A preselector is a device used
to pass discrete bands of frequencies within a limited operating range,
while rejecting signals at frequencies outside its passband. It can
be very useful in suppressing transmitter harmonics and in reducing
receiver overloading due to strong signals outside the amateur v.h.f.
bands.
When designing a filter, it is necessary to know the
minimum passband attenuation and bandwidth desired. If it is made tunable,
then the filter can be adjusted for minimum loss at any particular frequency.
Nearly all the characteristics of a coaxial filter can be related to
Q_{u} and Q_{L} where Q_{u} is the unloaded
Q of the filter, and Q_{L} is the loaded Q of the filter. The
unloaded Q of a cavity depends on the frequency and the impedance and
size of the cavity. The theoretical Q_{u} of a coaxial cavity
can be obtained from the equation
where b is in centimeters, ƒ is in c.p.s. and H a factor related
to b/a as shown in Fig. 4, at A. The Q of resonant coaxial lines of
optimum proportions (b/a = 3.6) is shown in Fig. 4, at B. Usually, these
values must be derated from 10 to 50 percent because of lower conductivity
than predicted, contact resistance between movable and fixed parts of
a cavity, capacitive loading effects of coupling elements and end plates,
and other unavoidable imperfections.
Losses in coaxial filters
are of two kinds  mismatch and dissipation. If the filter is simply
inserted in a 50 or 70ohm line, a good match can be obtained if the
input and output loops have the same size and shape and are located
at points of equal intensity. Usually, the effect of selfinductance
of the coupling loops is merely to shift the resonant frequency slightly.
Dissipation (or resistive) loss is an important factor in narrowband
filters because of the relatively high values of Q_{L} required
for narrow passbands.
The passband insertion loss, due to dissipation
alone, for a single resonant circuit is given by
where A is the dissipative loss in db. To have an insertion loss
of less than 1 db., Q_{u} must equal 10 Q_{L}.
The Q of a resonant circuit may also be defined as the ratio of
the mean passband frequency to the 3db. bandwidth F/ƒ or
A v.h.f. coaxial filter showing input and output coupling lines.
The tuning capacitor is tapped down on the resonant element.
Since selectivity and insertion loss are directly related to Q_{L},
both functions can be adjusted for any particular need by making the
coupling variable (such as rotatable loops).
If two or more
cavities are used in series to increase the selectivity, they should
be spaced an electrical onequarter wavelength from center to center.
The position of the loops, with respect to the center conductor
of the cavity, also has an effect on Q_{L}. The closer the coupling
the lower the Q_{L} and the greater the bandwidth.
In
practice, a certain amount of electrical coupling will be combined with
the magnetic coupling of the loop, depending on the size of the loop.
As an example, a coaxial filter for two meters might be designed
to cover the entire band of 4 megacycles. Thus,
To keep the insertion loss A below 1 db., Q_{u} should be
365. From Fig. 4B, a coaxial cavity of 1/2inch outer diameter has a
theoretical Q of about 600. Usually, more selectivity than this is desired,
and a previous article listed typical cavity dimensions for the various
v.h.f. bands.
A filter such as this can be made tunable either
by changing the length of the inner conductor or by capacitive loading.
The latter is generally less difficult to accomplish.
"The
World Above 50 Mc.," QST. February, 1961.
Additional
Considerations
The best method in constructing transmitters,
converters or filters using resonant line elements is to follow the
ideas in articles in the handbooks and magazines. A typical circuit
for parallelline construction is the 2meter transmitter described
in QST.^{2} A coaxialline amplifier for 2 meters is described
in an earlier issue of QST.^{3}
An important consideration,
when constructing similar equipment, is to determine the length of the
quarterwave section of transmission line. The equation used to solve
this problem is
where d = quarterwave resonant length in inches. c = velocity
of propagation in a vacuum (1.18 X 10^{10} inches/sec.).
n = index of refraction of the dielectric medium = 1 for air. ƒ
= operating frequency in cycles/second. CT = Terminating capacity
in farads. Z_{0} = Characteristic impedance in ohms and
tan^{1} is in degrees.
This equation is illustrated
graphically in Fig. 6, relating line length to terminating capacity
for various frequencies. For these curves, Z = 71 ohms and n = 1.
These curves may be used for resonant lines having a characteristic
impedance other than 71 ohms by using the conversion
where C_{0} is the terminating capacitance normalized with
respect to the 71ohm impedance. To use this chart, determine the
total minimum capacitance across the end of the line, including tube
or tubes and tuning capacitor. Find the length of the line at the highest
frequency used. Remember, the line can be lengthened electrically, or
lowered in frequency by adding capacitance, but it can only be shorted
electrically by cutting it off.
Construction Notes
The ideal way to build a coaxline amplifier or coaxial filter
would be to use copper or brass tubing, silver plated on the conducting
surfaces, and with all joints soldered. However, satisfactory results
can be obtained with less effort. As an example, a coaxial filter for
use on 6 and 2 meters was constructed, using a 3 X 4 X 17inch aluminum
chassis box and a 13 1/2inch length of 5/8inch copper tubing. If 1inch
diameter tubing is used, a length of 14.12 inches should be about right.
A 2 3/4 X 3 3/4inch plate was soldered to one end of the tubing and
mounted in the box. Input and output connectors were mounted on opposite
sides and about 4 inches up from the base. Wire loops, the shape of
an L, were spaced about 1/8 inch from the center conductor. A 330pf.
capacitor was connected halfway up the line. This provided enough capacitance
to tune the line to resonance at 6 meters. The filter was tried on each
band, with a power output of about 40 watts, into a wattmeter and 50ohm
load. The insertion loss was approximately 1 db. at center frequency.
Spurious emissions and harmonics outside the bands should be suppressed
by 40 to 50 db. Birdies and interference from TV and f.m. stations should
also be similarly suppressed. When using a multiband antenna on 6 and
2, a filter such as this should help to prevent 6meter thirdharmonic
energy from being radiated by the 2meter section.
References
VHF Techniques. Vols. 1 and 2. Radio Engineering
Handbook, Terman. Radar Circuit Analysis, USAF. Lighthouse Tubes,"
General Electric ETX110. Moreno, Microwave Transmission Design
Data, Dover Publications, Inc., New York, N. Y. "Narrow Band Preselectors,"
Microlab Catalog No. 11A. Penfield, "Design of QuarterWave Resonant
Lines," Electrical Design News, June. 1959.
Posted 2/25/2013



