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Power-transmission Capacity of Open-wire Transmission Lines
Author: Edmund A. Laport
The following discussion will apply to transmission lines that are guiding energy in the form of a traveling wave. The line is assumed to be terminated exactly in its characteristic impedance. When standing waves exist, the rating of the system is limited by conditions at some one place along the line where a critical factor, such as potential, determines the limitation. The losses in a line are minimum for the traveling-wave condition and increase with increasing standing-wave ratio as shown in Fig. 4.1. The power-transmission capacity is as much an economic matter as one of physical limitations due to heating or flashover. For this reason the rating of a system is generally indeterminate within the limits set by flashover, and the latter in turn is so dependent upon the empirical conditions of the construction of the system and its location that flashover potential may also be indeterminate.
As the frequency becomes higher, ionization of the air, which involves a time factor, does not occur as visible corona. The first sign of voltage overstress is manifest by the formation of a standing arc, or plume. The intense heat of a plume can be very destructive, though in some cases it serves to smooth off, by fusion, any small projection in the metallic conductors that produces an excessive gradient. When a system is operated at potentials close to critical pluming or flashover, care must be exercised in all the fine details of splicing wires, removing projecting ends, nicks on the wires, sharp points and corners on the insulator and assembly hardware, binding wires, etc. The potential rating of a system decreases with increased altitude, also. Installations at high altitude require special precautions, and the design of the system is made such that it would be suitable for much higher power if installed at sea level.
It is well known that corona and flashover, especially self-propagating arcs or streamers extending to ground or to other wires of the system, depend upon the energy of the system as well as the voltage. This is familiar qualitatively to radio engineers, who use different space factors for the same potentials in small transmitters, for instance, as compared with high-power transmitters. As soon as an incipient arc starts, current flows into the capacitance of the arc, which lowers its resistance. This changes the electric field configuration in such a way as to reduce the gradients near its base, which in turn moves the regions of critical gradient outward, increasing the capacitance and the entering current at its base, with great heat developing in the plume. This continues until a condition of electric and thermal stability is reached, which determines the size of the plume. The current required to sustain a plume of a given size increases with the frequency because the size of the plume determines its capacitance. In high-power systems, a plume can eventually extend itself to the point where it will bridge the line and cause flashover. Actual flashover is seldom experienced except in rather high power systems, where the size of a plume is not limited by the energy of the system to distances small with respect to the wire spacings.
The potential limit of a feeder system at any given altitude is set by those gradients, distributed or localized, which can cause pluming. The potential gradients are reduced by the use of relatively large conductors or by the use of two or more conductors in parallel to divide the current. At bends and turns, entrances, and branching points and at insulating points, localized gradients can be very high, therefore creating points of weakness, unless proper precautions are taken in design and construction. Potential-grading rings are sometimes permissible if they do not introduce a disturbance in the line parameters as an irregularity. It is possible to strengthen such points with a coating of insulating material, ceramic, plastic, or varnish, to provide potential grading by the use of intermediate values of dielectric constant between that of unity for the surrounding air and infinity for the metallic conductor. The use of such an insulated control of flux density can in some instances raise the pluming potential of a system at a particular weak point without introducing an intolerable irregularity in the line parameters. It functions in two ways - to prevent ionization near the conductor, where the gradient is maximum, and to limit the amount of current that can flow into an incipient arc. When used, the insulated control is most effective when it is homogeneous. Thickness can be increased to increase its effectiveness.
The maximum potential gradients that can be sustained in air before corona, pluming, or flashover occurs has been the subject of a great deal of research over many years. Thornhill and Beasley59 have made a synthesis of this available information and have reduced a complex subject to relatively simple form. The values of critical gradients and the equations for maximum working potentials on two elementary types of electrodes have been adapted from their work.
For two clean parallel cylinders in still air at an atmospheric pressure of 28 inches of mercury and a temperature of 45 degrees centigrade, with direct potentials applied between them, the critical potential gradient at which visible corona starts is about 66.7 kilovolts per inch. Under the same conditions concentric cylinders have a critical gradient of about 69.3 kilovolts per inch. However, between the conductor and the zone of visible corona there is formed an ionized conducting layer which has the effect of increasing the radius of the conductor. The actual potential gradient at the surface of the conductor is higher than the critical gradients for visible corona. The amount that it is higher is a function of the geometry of the system. The potential gradient E0 at the conductor surface that produces visible corona at direct current under these same atmospheric conditions is given by the following relations: For parallel cylinders,
For concentric cylinders,
For very low frequencies, these would represent crest values. As the frequency increases, up to about 2 megacycles, the critical voltage falls to a value of about 80 percent of direct-current value. For frequencies above 2 megacycles it rises again, eventually approaching the direct-current value, or even more, at around 200 megacycles. As a general principle one may use the 80 percent value for practical engineering purposes for radio frequencies up to 30 megacycles. Then for ordinary system design the above equations can be rewritten to include this reduction and transformed to root-mean-square values, giving for parallel cylinders in air at 28 inches of mercury and a temperature of 45 degrees centigrade (worst sea-level conditions)
and for concentric cylinders under the same conditions
These values will be further reduced by corrosion of conductor surfaces, moisture films, drip water, sharp points, irregularities due to roughness, bends, and other physical conditions. These are all taken into account qualitatively by employing design safety factors. The factor selected will depend upon the consequences of a flashover or the system losses. The safety factor for an open wire line may be less than for a concentric line for these reasons.
The critical gradients will vary with atmospheric pressure and with ambient temperature, which determine the relative molecular density of the air D, which can be found from the relation in which P is the barometric pressure in terms of inches of mercury and T is the ambient temperature in degrees centigrade.
In the preceding equations for critical gradients, the value used was D = 0.877.
Table 4.3 gives values of D as functions of barometer reading P in inches (corrected for temperature) and the temperature, together with approximate equivalent altitudes based on a corrected sea-level pressure of 30 inches of mercury.
These relations can be combined in a single equation involving the geometry of the line to give directly the maximum voltage that can be safely applied when the line is terminated in its characteristic impedance. When this is done for a two-wire balanced open-wire line,
and for an air-dielectric concentric line
In these equations the symbols have the following meanings:
Vm = maximum root-mean-square applied voltage, kilovolts S = design safety factor, usually between 1 and 3
a = center-to-center wire spacing, inches
ρ = wire radius, inches ρ1 = radius, inches, of the center conductor ρ2 = radius, inches, of the inside of the outer conductor D = relative molecular air density
The influence of the spacing of the wires is not immediately obvious from these formulas. Breakdown occurs due to ionization in the gap between the wires, which forms ions and free electrons.
At the low frequencies, there is time for both ions and free electrons to be swept into the wires during an alternation, so that there is no accumulated space charge unless the peak voltage applied exceeds the ionization voltage by a considerable amount. As the frequency is made higher, transit time for the heavy ions becomes too great for the space charge to be cleared during an alternation. Thus the free electrons move into the wires, and a space charge of ions accumulates. This space charge distorts the field and reduces the breakdown voltage. At about 2 megacycles the heavy ions are unable to traverse the gap dimensions, but the electrons, being lighter, are absorbed during the negative alternation. During the positive alternation, they move at high velocity and cause more ionization by impact. Under these circumstances the breakdown potential is at its lowest. This occurs in the high-frequency range, in the vicinity of 2 megacycles.
As the frequency is further increased, the transit time for electrons becomes great with respect to one period. Free electrons are then left in space with the ions, and the space charge is reduced. Finally, at around 25 megacycles, the inability of electrons to be cleared away during the negative alternation leaves the field of the gap in substantial equilibrium, with consequent increase in the breakdown voltage to nearly that for direct current. Experiment shows that, at the worst frequency, the breakdown voltage falls to about 80 percent of the direct-current value. A design value of 80 percent of direct-current breakdown voltage is chosen as a safe working limit.
Spurious flashovers may occur at random owing to a number of unusual conditions not necessarily associated with underdesign. Strong ultraviolet bombardment of the air can produce abnormal ionization in still air and cause flashover. Air turbulence tends to minimize this effect on open-wire systems. Ionization due to cosmic rays may be another spurious effect. Insects or birds on the wires may set off a plume or a flashover. High transient potentials may momentarily exceed critical values and strike an arc. Circuits without static drains may accumulate high direct potentials superimposed upon the existing radio-frequency potentials and cause arcing. The same effect can occur when transient potentials are induced into the system by lightning in the vicinity. Drip water on a feeder operated at near-maximum potential can introduce an increase in local gradient and cause flashover. Abnormally high peaks of amplitude modulation are another cause at times. To these must be added momentary mismatches due to arc-overs in the antenna system which cause standing waves.
During on-off keying, and also amplitude modulation, a plume is automatically extinguished with the first interruption of radio-frequency power. Spurious arcs and flashovers due to most of the above causes may not reoccur. Types of emission that maintain constant radio-frequency power would not extinguish an arc, and unless it is blown out by the wind it is necessary manually to interrupt power to extinguish an arc. For this reason, higher safety factors are usually required for such systems.
The power rating of feeder design also depends upon the total losses resulting from a given feeder design. The power-transmission rating of a feeder thus requires one to weigh the cost of line for a given loss against the annual cost of the power lost in the system. The price of programed radio-frequency power, for instance, reaches some rather high values per kilowatt-hour, and some organizations are accustomed to reckon that a certain reduction in feeder loss would justify an increase in capital outlay for a lower loss system. Long feeders are typical for high-frequency telegraph and broadcasting stations where several antennas are used with several transmitters. Long feeders are also required for many medium-frequency broadcast stations with extensive directive antennas. In such applications, the feeder losses are far from negligible, and considerable engineering may be required to design feeders for higher efficiency.
If we take as an example a two-wire balanced feeder 3,000 feet long, designed for transmitting 100 kilowatts of amplitude-modulated power, its attenuation at 20 megacycles may be about 2 decibels, corresponding to an efficiency of 63 percent. If a four-wire balanced line is used instead, with about twice as much copper, its efficiency can be increased to something of the order of 73 percent. A further increase in efficiency can be obtained with more copper and other materials at correspondingly greater cost. This case emphasizes the importance of planning the station and its antenna layout to use feeders of the shortest possible length as the best approach to economy of construction and to over-all efficiency.
Experience has shown that the losses increase in a feeder as the wires become corroded. When feeders are to be located in places subject to salt-spray atmosphere, sulphur smoke or fumes, or other corrosive conditions, the conductors may require protection in the form of electrical enamel, lacquer, or plastic coatings. With such coatings, the initial loss may be higher than for bright, new, bare copper, but it will remain constant over a much longer time than unprotected copper. Thick plastic coatings can provide a substantial increase in maximum operating potential, but at the sacrifice of increased dielectric losses.
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