The coaxial folded monopole lies at the base of the coaxial collinear array and a number of other antennas. The difficulty that designers and analysts have with the antenna lies partly in our inability to effectively model the relatively simple device. However, in principle, the antenna is as easy to understand as a simple folded monopole. Fig. 1 outlines the basic considerations.
The antenna has two "legs." One extends from the ground upward to the top of the structure and bridges to a second conductor running back to the ground. Here, the term "ground" may mean a ground-plane assembly of radials, a perfect ground in models, or a mirror of the upper structure to form a coaxial dipole. In our efforts to model the coaxial monopole, we shall use perfect ground for simplicity. Any changes that we make in length to set the antenna at resonance will not depend upon the radius of a radial set. Hence, we may easily compare one model to the next.
What differentiates the coaxial monopole from a standard folded monopole is the fact that the conductors have a concentric arrangement, with one conductor wholly inside the larger one. Like any folded monopole, the coaxial monopole has two sets of currents. One set is the transmission-line current, where at any given point along the length, the currents on the two conductors are equal in magnitude and opposite in polarity. These currents exist on the outer surface of the inner conductor and the inner surface of the outer conductor. As well, the coaxial monopole should exhibit radiation currents, which in principle have the same polarity on both conductors. However, skin effect places the radiation current on the outer surface of the outer conductor. In principle, the coaxial monopole should have the same gain as a simple single-wire monopole.
Fig. 1 shows the source terminals in series with the bottom edge of the two coaxial conductors. Efforts to model the coaxial monopole cannot proceed with the actual layout of the final product because neither MININEC nor NEC can handle coaxial structures effectively. Rather than using an outer solid-wall tube as the outer conductor, most models employ a number of wires that are parallel to the inner or center wire, each equally spaced from the center conductor and from each other. Fig. 2 shows the general layout used in most models.
The sketch shows the outline of the type of model that has proven most successful and that also has numerous implementations in actual practice. The source connects to each of the outer wires in parallel. The perimeter wire at the base of the monopole only suggests the parallel connection, but does not show the best method for achieving as close to a true parallel connection as possible in a model. At the top, bridge wires connect each outer wire to the center wire that returns to the ground. Theoretically, we may reverse the source connection. We may place it on the base of the center wire, with each of the outer wires returned to ground. Herein lies a problem.
Casual models of the center-fed coaxial dipole appear to produce very disheartening results. If we believe such models, the center-fed coaxial monopole should not be worth the effort of building. Nevertheless, many coaxial collinear vertical arrays are in service and appear to perform to specification. In the past, a number of large arrays of coaxial dipoles were built for narrow-beam radar use. (The patents for the coaxial collinear array go back to Germany in the 1930s.) Either something is wrong with the casual models or something is wrong with the antenna design.
The plan for these notes is to examine the basic modeling requirements that affect models of coaxial monopoles and then to review some data drawn from the version of the antenna that has proven relatively reliable. Only after looking in some detail at the outer-wire-fed coaxial dipole will we be positioned to approach the center-fed version with some idea of what may count as successful modeling.
Some Basic Constraints and Specifications
The models in this exercise will use a test frequency of 299.7025 MHz, where 1 meter = 1 wavelength. All dimensions will be in millimeters to allow for an easy conversion of element lengths to a fraction of a wavelength. All models will use a perfect ground in NEC to allow for ready comparisons among models.
All wires in the assemblies will be 5-mm in diameter, using both perfect or lossless wire and copper wire in tabulated results. (At the test frequency and element diameter selected, we normally find only minuscule differences in the reported performance numbers, so that departures from that condition may prove instructive.) The center-to-center spacing between the fed wire(s) and the return wire(s) will be 20-mm, about 0.8". If the outer wires are numerous enough to form a rough approximation of a circle, the diameter will be about 40-mm. As we shall discover, 6 wires is enough to approach but not quite reach the status of a full circle enclosing the center wire.
Wherever relevant, we shall adjust the length of the center and outer wires together to reach a resonant length with no more than +/-j1 Ohm of reactance at the feedpoint. The values that we obtain in the first series of tests will provide a guide for what sort of feedpoint impedance emerges from a sensible model that uses a center-fed wire.
Besides this condition, we shall be very interested in several model properties. The Average Gain Test (AGT) is critical to testing the models and to adjusting reported values of both gain and feedpoint resistance. We shall use the AGT score to correct the feedpoint resistance value, and we shall use the AGT converted to dB to provide a corrected gain report. Some appearances of both superior and inferior performance will disappear once we make the corrections. However, we must also be aware that the AGT score is also a measure of a model's reliability or adequacy as a model. Some suggested interpretations of AGT scores supply the following meanings to attach to the numbers.
AGT Value Meaning 0.95 to 1.05 Highly reliable model 0.90-0.95 and 1.05-1.10 Good model, adequate for most purposes 0.85-0.90 and 1.10-1.15 Fair model, broadly adequate but may be refined <0.85 or >1.15 Poor model, inadequate for most purposes
We shall also be interested in the current magnitude distribution along the wires of the model. Some erroneous models of coaxial dipoles reveal their deficiency by showing both aberrant feedpoint impedance vales and an inappropriate set of current magnitude curves.
Outer-Wire-Fed Multi-Wire Folded Dipoles
The folded monopole need not consist simply of two vertical wires with a bridge wire at the top. Because the most familiar form of folded monopole performs an upward impedance transformation to a value that many user do not wish, some designers have increased the number of wires to form 3-, 4-, 5-, and 6-feed-wire assemblies. Rarely do we need to go beyond three outer fed wires to obtain a source impedance compatible with common coaxial cables (50 Ohms). Nevertheless, I extended this exercise to 6 outer fed wires to prepare the way for later models.
In fact, the exercise begins with a single-wire monopole as a reference point. Then it proceeds to models with 2 through 6 outer fed wires, all using the same spacing from the center return wire and all forming a regular polygon around the center wire. Each vertical wire uses 25 segments, for a segment length between 9 and 10 mm, depending upon the particular model. Each top bridge wire is the same diameter as the outer wires and the center wire to eliminate errors due to NEC angular junctions of wires having dissimilar diameters.
Fig. M-1 shows the EZNEC wire table for one of the models in the series. It uses 4 outer fed wires with a center return wire. The segmentation of the wires is clear, along with the equal spacing of the outer wires. At the bottom of the list is a very short (5-mm long) 1-mm diameter wire. Its function is to serve as the source wire.
A single perimeter wire with a single feedpoint does not normally produce equal current magnitudes at the bases of all of the conductors. Even with lossless wire, the length of the wire between the source segment and the bottom of each outer wire differs enough to produce unequal current magnitudes at each corresponding segment among the 4 wires. To ensure parallel feeding of all four outer wires, I use a short wire separated physically from the antenna by an amount sufficient to prevent significant interaction between the antenna structure and the new wire. Between the source wire and each vertical leg to be fed, I run a transmission line, using the NEC TL facility. Fig. M-2 shows the lines relevant to the model in Fig. M-1. The significant aspect of the lines is our ability to select a length independent of the actual physical distance between the new wire and the outer wires. Hence, each line has the same length. Moreover, each line is exceptionally short: 0.001-mm in this case. Such a line length is too short to exhibit any impedance transformation relative to the specified source magnitude and phase angle. Hence, the characteristic impedance (Zo) is arbitrary within very broad limits. The 20-Ohm value shows no differences when changed to 50 Ohms, for example. The net effect is to create a virtual short circuit between the source wire and the base segment of each fed leg. Therefore, for a source current magnitude of 1.0, the current magnitude on each base segment is 1/n, where n is the number of fed legs. At the same time, the reported source impedance, within the limits of the AGT value, is the parallel combination of all outer legs.
The short-TL method of deriving parallel feedpoints is not strictly necessary. You can place a source in the lowest segment of each out wire, take the average value of all 6 sources and then divide by 6 for a net feedpoint source. The result will be the same. For example, I converted the 7-wire/6-feed model into separate sources and obtained a reported feedpoint impedance of 41.0 Ohms, the same value that appears in the tabulated values below. The short-TL technique simply saves some post-modeling calculations by letting the program do the work.
The exercise using models of multi-wire folded dipoles fed on the outer wires produced both reliable models and interesting numbers. Table 1 provides the relevant tabular data.
Perhaps the most important columns in the table are the ones devoted to the AGT. The single-wire monopole and the 2-wire folded monopole achieve AGT values of 1.000. As we add more fed wires, the AGT values depart from the ideal value, but the more outer and bridge wires that we add, the closer to ideal the AGT becomes. No AGT value is any worse than good, which gives us confidence in applying the correctives. The gain corrective consists of subtracting the converted AGT score (10*log(10)AGT) from the reported gain. The corrected feedpoint resistance results from multiplying the AGT times the reported source resistance value.
Although the raw NEC reports might suggest that there are small but noticeable differences in the far-field performance of the various models, the correct gain values show a remarkable equality of gain. The absence of differences between the zero-loss wire models and the copper models gives us greater confidence in this result. The corrected feedpoint impedance values show numerical differences but no operationally significant differences relative to the raw values. The impedance ratio between the single-wire monopole and the 2-wire folded monopole is 3.92:1, where a theoretically perfect value would be 4:1. However, the theoretical calculation does not take into account the 20-mm bridge wire.
Fig. 3 shows two different facets of the series of folded monopoles from 2 to 7 wires total (1 to 6 fed outer wires). First, the required length of the monopole does not change more than about 1% over the sampled range. Second, the resonant source resistance values--both raw and adjusted--undergo an interesting progression downward as we add more fed outer wires. Moving from a 2-wire to a 3-wire model results in a 50% reduction in the feedpoint impedance. However, the impedance ratio for additional fed wires does not keep pace with the increase in the number of fed wires that gradually enclose the center return wire. We might innocently expect the 6-fed-wire model (7 total wires) to show 1/6 the impedance of the 2-wire folded monopole. However, the impedance is about 29% of the 2-wire model value at 41 Ohm reported and 41.9 Ohms corrected.
It is likely that the feedpoint impedance of the multi-wire folded dipole could not go below the 36-Ohm value for a single wire monopole. The 6-outer-wire/7-total-wire model does not approach that value largely because it does not provide complete virtual coverage of the center return wire. (If the models had provided complete coverage, the resulting coaxial cable would have a Zo of approximately 115 to 125 Ohms, depending on the exact virtual position of the inner surface of the outer ring of wires.) Nevertheless, the 41-Ohm source impedance of the 7-wire model represents a good benchmark value for use when evaluating other kinds of models of this antenna structure. Part of this status results from its emergence from a model that we may rate as highly reliable by reference to the AGT score.
As we increase the number of outer wires, we may also note a change in the current magnitude distribution along the wires, with special attention to the difference between the curves for the fed wires and for the center return wire. Fig. 4 shows the progression of relative current magnitudes on each of the sampled models. The single wire monopole provides a standard current distribution that runs from 1.0 at the base to about 0.1 at the top. The reported current does not go to zero in a NEC model, since the virtual sampling point is in the middle of the topmost segment, not at the wire tip. In contrast and easily expected are the current magnitude curves for the 2-wire folded monopole. The curves do not approach zero due to the fact that the assembly yields both radiating and transmission-line currents. The latter reach a peak value at the antenna top end, as the radiating current goes toward zero.
As we add more outer fed wires to the assembly, we may note changes in the overall current magnitude distribution of the center wire. The 2-wire model shows maximum current on the return wire at the antenna base. The 3-wire model shows nearly equal overall current magnitudes at both ends. By the time we reach 4 wires, the center return wire shows maximum current at the top and minimum current at the base. Additional coverage with outer fed wires simply increases the ratio of top current magnitude to bottom current magnitude. Table 3 shows some of the differences as numerical values of the current magnitudes for 2-wire, 3-wire, 4-wire, and 7-wire models. The progression skips a few steps since the rate of change shows dramatically after the assembly consists of at least 4 wires. The table shows the top 6 segments and the bottom 6 segments of each sampled wire or wire set. Note that "top" and "bottom" reverse positions in this extraction from the NEC current tables.
The sum of the fed wires results simply from multiplying the current magnitude on 1 of the outer wires by the number of fed wires. The procedure is accurate within the limits of the table. As we increase the number of fed wires, we may note that maximum current does not occur on the lowest segment of the fed-wire set. Rather, it occurs 1 to 2 segments higher. This condition results from the fact that the transmission-line component of the current does not change magnitude or phase angle at the same rate as the radiating current. Therefore, the highest vector sum of these current may not always occur on the source segment.
As we move up the fed-wire set of current values, we find that the composite current on the topmost segment (1) systematically decreases as we add more outer fed wires. The same situation occurs at the top of the return wire at the other end of the bridge wires. As we move down the return wire, the 3-wire system's return-wire currents barely manage parity between the top and bottom segments. However, from the 4-wire system upward, the bottom segment of the turn wire shows less current magnitude than the top, and the reduction increases with the number of outer wires.
The situation suggests that the behavior of the assembly changes with respect at least to transmission-line currents as we more completely surround the return wire, that is, as we approach a coaxial situation. One way to view the difference quantitatively is to sort the radiating and the transmission-line currents from at least two examples. The first sample uses the 2-wire folded monopole, the current patterns for which should be reasonably familiar. Table 3 separates the radiating and transmission-line currents for all 25 segments of the 2-wire folded monopole.
The radiating-current column shows a pattern that exactly parallels the current distribution on the single-wire monopole. The only significant difference is that the folded monopole radiating currents change phase angle more than do the single-wire monopole currents (about 13 degrees for the folded monopole and 9 degrees for the single-wire monopole). The transmission-line current on this model shows its highest value at the top end of the structure and the lowest value at the ground plane. The phase angle is nearly a constant 90 degrees with respect to the phase angle of the source current in the model.
To apply the same sorting process to the 7-wire model requires that we again multiply the current magnitudes (but not phase angles) of a sample fed wire by 6 to obtain a very close approximation of the total current in the fed wires. We may use the return wire without change. The results appear in Table 4.
If we examine the radiating-current column, we find a progression that is almost the same as for the 2-wire folded monopole. Of course, we might wish to increase the maximum value to 1.0, but on a strictly relative scale, we may simply use the values shown and trace the curve from bottom to top. The curve is essentially a typical monopole curve of current magnitude along the wire assembly. The phase angle for the radiating current changes somewhat more radically as we added outer fed wires and reaches about 24 degrees for the 7-wire model.
The greatest difference between a standard 2-wire folded monopole and the 7-wire model appears in the transmission-line columns. The 2-wire model showed large changes in magnitude but almost no change in phase angle. The model that approaches (but does not reach) the status of a coaxial assembly reverses the trends. The transmission-line current magnitude shows only a very small change (about 0.15), while the phase angle undergoes nearly a 75-degree change. It is likely that a perfect coaxial system (assuming a velocity factor of 1.0) would show slightly over 80 degrees of phase shift for a structure that is just under 90% of a quarter wavelength. (The bridge wires, at 20-mm each, are more than trivially long.) Although we must be cautious in evaluating the level of success in the model capturing a coaxial system, the small difference between the theoretical limit of phase change and the level achieved in the model is one preliminary measure of success.
Hypothetically, we might continue the modeling process by adding further outer wires. However, practical systems that I have encountered have not risen to a 6-fed-outer-wire system. Moreover, such systems generally use thin wires around a large-diameter center return structure (such as a tower), which would present a massive difficulty for NEC. Finally, the use of 6 outer wires results in a spacing between each pair of outer wires that is the same as the spacing between the center wire and each outer wire. The sense of symmetry adds an aesthetic reason for closing the book (at least temporarily) on outer-fed quasi-coaxial systems.
Center-Wire-Fed Multi-Wire Folded Monopoles
In theory, we need only shift the position of the source within the 7-wire model to obtain a model of the coaxial monopole with a source on the inner wire so that the outer wires become the returns to the perfect ground. Since all outer wires terminate at the ground, we likely need not retain the TL-based parallel short-circuit among the lowest segments on the outer wires. I shall presume that some such arrangement underlies the models of a coaxial monopole that I have heard about from time to time. Accompanying reports of such models have been commentaries to the effect that the arrangement tends to have a low performance level, almost as if the outer return wires were trapping the energy within the structure. Admittedly, such comments have a casual basis, and the models are not available to me. So at most, we can only see if we can find a reason (not necessarily the reason) why one might reach such a conclusion.
Therefore, I constructed a series of models, of which we shall sample two that I classify as potential casual models of a center-wire-fed coaxial monopole with 6 outer wires. The wires and model environment are identical to the conditions set for the outer-feed structures that we have explored. The first sample appears in the EZNEC wire table shown in Fig. M-3. The model uses no transmission lines. The bridge wires at the top of the structure are 20-mm long with the same diameter wire as the vertical legs of the structure. The segmentation is also the same as in earlier models.
The outline of the structure appears on the left in Fig. 5. The figure also shows an alternative version of the structure. In order to ensure that the return wires electrically terminate at the end of the segment containing the source, I ended the vertical outer wires 5 mm above ground. From each "loose" end, I ran a 2-segment wire to the bottom of the center wire at the ground-plane level (Z=0). The right-hand outline in Fig. 5 should make the plan clear. The wire table in Fig. M-4 will confirm the wire structure.
Each of these models uses the vertical leg-length assigned to the 2-wire folded dipole. I did not proceed further in length adjustment because each of these models mis-performs beyond the limits of reliability of the AGT score. The top entries in Table 5 show the situation. The model designated M-3 or Top Only has an AGT score of 0.538, when a sensible limit to even a poor but usable model is a score of about 0.85. The model reports an exceptionally low far-field gain value and perhaps underlies the sense that the outer wires somehow trap the energy in the center wire. The problem does not lie in the antenna, but rather in its model. If we use the AGT score as a correction on the gain value, we obtain a quite normal value for a monopole over perfect ground. We obtain further evidence of complete model inadequacy from the gain difference between the version using lossless wire and the version using copper wire. Ordinarily we expect a gain change no greater than about 0.01 dB. Any larger difference indicates significant trouble in the model, which the AGT score confirms.
The model called M-4 or Top/Bot suffers an inverse but equal difficulty. The reported AGT score is 2.466, when the limits of a poor but usable model might be about 1.15. The raw gain report seems to promise nearly magical performance in the 9-dBi range. However, if we correct that gain value by reference to the AGT value, the gain drops to a more normal monopole level. Once more, we find an inordinate decrease in the gain of the model version using copper wire relative to the version using lossless wire, another indication of the model's inadequacy.
In both casual models, the reported source data falls well outside of what we might expect, if we use our experience with outer-fed 7-wire models as a reference. Moreover, adding a corrected source resistance column would only exacerbate the difficulty. The low AGT value for M-3 would reduce the already low raw impedance value even further. Likewise, the high AGT value for M-4 would raise an already high value of source impedance. Moreover, the high value of reactance in the reported source impedance does not decrease to zero until the model height shrinks to the 80-85-mm region, and even then, the impedance is unstable. Very small changes in height create very large changes in the reactance value.
The casual models, then, prove to be wholly inadequate to capture the performance of a center-wire-fed 7-wire monopole structure. Table 5 also contains performance reports on three models (out of many that I constructed along the way) that may prove more useful. I modeled them to yield reasonable gain values, good AGT scores, and source impedance values that seem to be relatively typical for 7-wire folded structures. Hence, I call them careful models, but not necessarily ideal models. Each model records an AGT of 1.000, since nothing in the structure comes close to pressing any NEC limit. All of the wires use a simple straight uniform diameter geometry. Fig. 6 shows some suggestive outlines.
Model Top-Bot forms the baseline for the series. At the top, it eliminates the bridge wires and replaces them with TL short circuits of the same type used for the feed system of the outer-fed monopoles. However, each top TL runs between the top segment of the outer wire and the top segment of the center wire. At the base of the model, I also used TL short circuits to connect together the lowest segments of the outer wires. The only difference between these TL entries and those of Fig. M-2 is that the source is not on the external junction wire, but on the lowest segment of the center wire. This model forms perhaps the key comparator with the performance of the 7-wire outer-fed monopole.
The data in Table 5 shows very normal gain performance. The slightly high gain value (by 0.03 dB) results from the fact that the overall height of the model is taller than the models for the corresponding outer-fed monopole. In fact, the length is about as much longer than the single-wire monopole as the outer-fed systems are shorter than the single wire monopole. The extra length stems from replacing the bridge wires--which do affect current distribution--with TL shorts that have no practical length. The correlation cannot be precise because the TL connections extend from the virtual centers of the segments to which they are attached, and each segment is between 9 and 10 mm long.
The reported feedpoint impedance is about 45.2 Ohms, only slightly higher than the value yielded by our final outer-fed model. As a consequence, we may account this model to be a reasonable facsimile of an actual center-fed 7-wire folded monopole. For functional analyses, the model may prove sufficient. However, it is not perfect, since the replacement of bridge wires at the top with TL shorts modifies the current distribution relative to a physical structure. The modification becomes more evident if we sort out the radiating and transmission-line currents, using the same method employed for the corresponding outer-fed 7-wire model. Compare Table 4 with the data in Table 6.
The radiating current column shows a progression of magnitude values not very different from the ones in the earlier table. Indeed, one might account any magnitude differences to the slight height differences between the two models. The change in radiating current phase angle is somewhat greater than in the outer-fed model, but does not reach suspicious or doubtful levels. The key difference lies in the transmission-line current columns, especially in the range of phase angles from bottom to top. The outer-fed system showed a considerable (74-degree) shift in phase angle, while the center-fed model shows slightly less than 45 degrees. It is likely that the bridge-wire replacement with TL shorts is the major source of the difference, but the degree of error attributable to each model remains unknown in the present context.
Both the center-fed and the outer-wire-fed models show impedances in the 40-45-Ohm region. The curves shown by Fig. 3 suggest that a truer coaxial structure would gradually approach the source impedance of the single-wire monopole. Therefore, I performed a 2-step experiment, adding TL shorting structures around the outer wires of the center-fed model. The first step added a set of shorts half way up the structure. The final step added two more encircling shorts at the one-quarter and three-quarter length points. Fig. 6 shows the outlines of these models, while the final lines of Table 5 provide the resulting data. Since the TLs add nothing to the actual geometry, they do not affect the AGT score.
As we add the TLs, the resonant height of the structure decreases somewhat. As a consequence, the reported maximum far-field gain also decreases slightly. Perhaps more significant within this context is the continued decrease in the source impedance value. The final figure in the table is almost identical to the value for a single-wire monopole. However, we should not too quickly equate the two, since the effective diameter of the 70-wire model is many time the diameter of the single-wire monopole.
The final models are an attempt (but not necessarily a fully successful one) at capturing the performance of a true coaxial monopole composed of a center wire and a virtually solid outer shell. The model suggests that the idea of a coaxial monopole is sound. The radiating (or common-mode) currents yield normal operation for a monopole. At the same time, the transmission-line currents remain within the structure, that is, between the outer surface of the inner conductor and the inner surface of the outer conductor. The short circuit at the monopole top ensures that the impedance at the bottom will be a parallel combination of a very high impedance value 1/4 wavelength from the short and the resonant monopole impedance. The result, under ideal conditions, would simply be the monopole impedance.
The simplified impedance analysis presumes a velocity factor of 1.0 and no further geometric disturbances to the structure. However, bridge wires at the top of the structure modify the current distribution. If we add the normal material found within real coaxial cables, the current distribution may show further non-ideal conditions. Therefore, cutting a real coaxial monopole to length and still having a usable impedance value may prove to be a somewhat finicky task. Nevertheless, the existence of successful coaxial collinear monopoles and dipoles attest to that fact that the task is feasible.
Conclusion
Our goal has been to explore the possibility of modeling the coaxial monopole. To that end, we set up a test environment over perfect ground and created a series of models at 299.7925 MHz. The first series used well-established techniques for parallel feeding the outer wires of such systems while adhering to tight NEC guidelines for equality of diameter throughout the model. These models showed that we can model structures using up to 6 relatively fat outer wires and obtain results that both are reasonable and show a progression of feedpoint values and other transitions toward a true coaxial situation.
We next turned to center-fed models and had to modify the antenna geometry and the method of connecting wires to avoid wholly unreasonable AGT values. One result was a model that replicated very closely the performance of the corresponding outer-fed model, while preserving an ideal AGT score. We further pressed the center-fed model toward true coaxial conditions, but remained cognizant of limitations other than the AGT score they may reduce the accuracy of the model relative to a physical implementation of a coaxial monopole.
Since there is a theoretical (tight) and practical (rough) equality between the outer-fed and the center-fed 7-wire models, the choice of which to use within a more complex structure remains a modeler option. In some ways, the 7-wire outer-fed model may be simpler and more direct to use when modeling such arrays as the coaxial monopole or dipole. Notwithstanding, the exercise has demonstrated that there are ways to model both feedpoint positions with multi-wire folded monopoles.
Updated 03-01-2007. © L. B. Cebik, W4RNL. The original item appeared in AntenneX for February, 2007. Data may be used for personal purposes, but may not be reproduced for publication in print or any other medium without permission of the author.
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