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

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.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







Copyright: 1996  2024 Webmaster:
Kirt Blattenberger, BSEE  KB3UON 
RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling
2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas
and reference material while performing my work as an RF system and circuit design engineer.
The Internet was still largely an unknown entity at the time and not much was available
in the form of WYSIWYG
...
All trademarks, copyrights, patents, and other rights of ownership to images and text
used on the RF Cafe website are hereby acknowledged.
My Hobby Website: AirplanesAndRockets.com

