Modeling the Un-Modelable

L. B. Cebik, W4RNL


wpe6.jpg (1219 bytes)n alternative title for this episode might be "The Intimate Connection Between Modeling and Measuring--Both Before and After Modeling." With appropriate measurements before modeling, we can sometimes model structures that are technically outside the range of what NEC models best: bare round wires. We cannot model everything successfully, but by making some pre-modeling calibration measurements, we can model a good bit more than we might initially think. Let's see if we can approach the subject in a roughly systematic manner.

NEC Limitations

NEC (both -2 and -4) employs algorithms that presume thin round wires. When we model antenna structures that make use only of round wires, we tend to assume that the program is accurate. Often, we assume too much, forgetting that NEC has limitations. Tapered diameter elements plague NEC-2, and extreme tapers can lead to some errors even in NEC-4. Angular junctions of dissimilar diameter wires also lead to errors, and the Leeson corrections that apply to linear tapered diameter elements will not work. As well, the correctives will not work with mid-element loads or transmission lines that disrupt the current stepping from one segment to the next. Some angular junctions of wires with dissimilar diameters can also create a few problems. For example, a fat monopole with a set of thin radials at right angles to the main element tends to model accurately. However, sloping the radials tends to create errors.

For these common cases, we have tests internal to NEC for evaluating and sometimes correcting errors created by at least mild cases of surpassing the round-wire limitations. The Average Gain Test, described in at least 2 past episodes, provides a model-adequacy figure of merit. For a lossless version of the model in free space, a value of 1.00 is ideal (2.00 if tested using a perfect ground). Anything less than 1.00 or greater than 1.00 indicates a level of inadequacy. The greater the departure from the ideal value, the less adequate the model. For some purposes, we can convert the Average Gain Test value into a correction for the reported gain value and for the reported feedpoint resistance value. For structures that include anything more than linear elements, the Average Gain Test is required. However, the Average Gain Test is a necessary and not a sufficient condition of model adequacy.

NEC also comes with considerable advise on obtaining accurate results. For a given linear element, all segments should be the same length. This good-modeling practice is especially important in the region of the source. The source segment should be the same length as the adjacent segments. If wires are closely spaced, the segment junctions should align as closely as feasible for highest accuracy. The segment length should be several times the wire radius. As we create ever-narrower angles, we run risks of adjacent wire surface penetrations that may adversely affect accuracy. This quick scan of some of the normal round-wire modeling guidelines is just a reminder of the total list of good modeling practices.

Even following all of the guidelines, we can still run into situations that defy the precision we tend to assume for models of dipoles and simple Yagis. Consider a right triangle with three sides having different lengths. Now feed the antenna at the most acute angle. Even with perfect proportions--that is, with identical segment lengths throughout--the model may not converge. The chief indicator of the fact that something is amiss is the fact that if we alternatively provide the model with a standard voltage source and then an indirect current source, we may obtain different feedpoint impedance values. This model prevents the source segment--or even the source-segment pair--from obtaining equal current levels in the segments immediately adjacent to each end of the source segment or segments.

This brief review of some--but by no means all--NEC limitations is not designed to cast aspersions on either of the most-used NEC cores. Rather, the catalog does no more than record that NEC has something in common with all software that makes highly complex calculations: the software has limits and ways to determine in large measure how close to those limits a given model might be.

Types of Models

Over the years, I have developed some general categories of modeling efforts to flag what models may be good for. The borders between categories are judgment calls that perhaps only experience can certify. Nevertheless, they may be useful to illustrate the levels at which modeling may be useful.

1. Design Models: the "design-model" category is reserved for antenna models with an existing track record of construction and testing to the model specification. The correlation between model and physical reality is sufficient to build directly from the dimensions specified in the model. Such models, of course, have passed all internal adequacy tests. In addition, they carry with them a set of physical correlation instructions or limitation notations. For example, a model may specify that it is for a non-conductive or well insulated/isolated support boom.

The design category has very high standards, but is not at all unusual. From some Moxon rectangle and monoband quad designs that have been modeled by equation, many implementations have successfully emerged without the need for more than routine initial set-up procedures. The model of a 50-Ohm Moxon rectangle with uniform-diameter elements can be set up for design by entering only the element diameter and frequency. See Fig. 1. Numerous Yagi designs guide commercial production in several countries. Going beyond the limits of NEC, hybrid programs are yielding wireless antenna designs of all sorts.

The design category also includes the other side of the modeling coin--analysis. Recently, I examined data on an interesting small antenna and was able to replicate the design and the test results. Some matters, difficult to measure in the test set-up, move from presumption to confirmed analysis status by virtue of the reliable model produced. One need not begin with a model: instead, modeling often serves as a supplementary analytical tool, not to mention as an educational tool when rightly used.

2. General Guidance Models: models provide general guidance when they are reliable as models, but not necessarily models of an antenna that either exists or will be built exactly as modeled. The models meet all rigorous standards of adequacy, but do not have a direct correlation to any particular existing or anticipated set of construction processes.

General guidance models require further design effort to move to the design-model category. The model would have to take into account significant appurtenances that may affect RF performance in the physical implementation. These added factors may range anywhere from a simple change of materials to lumpy brackets and other hardware within the mutual coupling range of the elements.

Nonetheless, since the models are known to be reliable, they often serve as the basis for systematic modeling studies. Since one may create a sequence of models more efficiently than a sequence of test antennas, the relationship between the model and a test situation is normally not a 1:1 affair. Instead, models may identify key test points to confirm or disconfirm modeling trends that emerge from the study. As well--and as more than a mere incidental--general guidance models often save prototype and test efforts from any unproductive byways. As well, they often turn up unexplored directions in antenna work and provide a first-order quantification of information that has hitherto been only anecdotal. Such was the case, judging by the feedback, from some notes I produced on the loss "knee" frequency and the patterns of a typical terminated wide-band "folded dipole." See Fig. 2 for a sample pattern.

3. Proof-of-Principle Models: This special category of model may be difficult to place properly with an example or two. Consider the gamma match (or the Tee match) used extensively in Yagi design and elsewhere. Ordinary procedures for building such a match employ gamma match tubing or rod that is much smaller in diameter than the elements to which it is attached. NEC does not handle well very closely spaced wires of different diameters and lengths, even when one carefully aligns the segment junctions. As well, there will be angular junctions of wires having dissimilar diameters. To model a gamma match and remain within the boundaries of what NEC does well, one must use gamma rod, connecting rod, and element diameters that are equal. The proportions required for this model do not correspond to normal Yagi construction, but do fall well within gamma match calculations. See Fig. 3 for a rough outline of the differences. Hence, one may not be able to model a given gamma match in NEC, but one can model a gamma match to prove the principle of the matching system as a physical construct and to examine certain properties, such as the currents along the gamma rod. (For a better correlation between models and physical implementations of gamma-matched antenna elements, a well-calibrated version of MININEC is often superior.)

Consider next a solid-sheet fan dipole that might be used at UHF frequencies. The antenna may be too small for effective wire-grid construction, and even a wire-grid may prove problematical in terms of reflecting accurately the current distribution on a fan. One may approximate such a fan as a wire outline, as suggested in Fig. 4. The outline fan may not have the full frequency range of the solid-surface fan, but it will exhibit a good part of the broadband effects. Hence, in comparisons with other types of elements that might be used in an array, it can provide a proof of principle of those effects in a complex array, but is not quite satisfactory for full general guidance or design-level work.

Proof-of-principle models do not have relaxed standards as models. Indeed, to serve as a proof of a principle, they need to be fully adequate as models within the software system being used. As well, the modeler must have enough knowledge of the system being modeled in principle to be able to specify both the correlations to and departures from reality, plus enough understanding of the principles themselves to be able to show that the model falls within the limits of those principles. Proof-of-principle modeling is not simply a matter of approximating an antenna system or getting into a ballpark estimate of what is happening. There must be a reasonably well-understood relationship between the models and physical antennas to be able to confirm that both fall under the same principles of operation. Theoretically, every proof-of- principle model should be susceptible to physical replication and testing, even if no one actually conducts the test.

4. Suggestive Models: Sometimes it is not possible to construct a model that rigorously meets all internal standards of adequacy, but it may come close. In such borderline cases, the model may give every indication that over some part of the of the reported output data, there are reliable trends, even if the specific numerical data for any single item fail to meet reliability standards. Such models--when accompanied by a carefully wrought justification and set of limitations--may be used as suggestive of directions for further study.

I recently approximated a Brown-Woodward bent fan dipole within a corner reflector array. The AGT value for the construct was too far from perfect for high confidence in it as a model of the actual driver. However, the relationship between the planes of the modeled version and of the corner reflector surface yielded some interesting impedance curves, especially when compared to standard fan and linear dipoles. See Fig. 5 for a sample of these curves. At most, these curves are suggestive of how the driver manages to widen the bandwidth of the corner array, but they are not adequate yet as proof-of-principle models.

Categorizing modeling results requires a level of judgment that comes from long experience and solid familiarity with the foundations, procedures, and limitations of antenna modeling within a given software system. The process also requires solid familiarity with antenna theory and practice, as well as construction and testing techniques and practices.

Physical Post-Modeling Testing

I have briefly and incompletely reviewed some of NEC's limitations for two reasons. First, for a large number of modeling enterprises that fall well within the well-known limits of the core, we approach the exercise without thought to the limits. As a result, we tend to assume that the results are accurate to the realities of a physical implementation of an antenna design. That assumption is, of course, a dangerous temptation if we carry it outside the region of well-verified results. Second, even when we do not make assumptions about the correctness of NEC reports relative to corresponding physical antenna structures, most modelers reserve testing and measurement activities to post-modeling exercises. That is, they create a modeled design and then build a prototype to match the model and test its performance.

Post-modeling physical testing is extremely important, and I have no intent to reduce that importance in these notes. However, we do tend to encounter two distinct groups of individuals whenever the measured results for a physical antenna do not agree closely with the reported results for the model. One group tends to almost automatically presume that there is something wrong with either the model or the software. The other group tends to presume that there is something wrong with the physical prototype.

In principle, either possibility may be correct, but never as a presumption. In most cases, those who conduct the physical construction and testing of an antenna are not the same individuals who do the modeling. In sundry consulting activities, I have discovered that one of the chief causes of disparity between test measurements and modeling results is a failure of communication. When communications are clear, concise, and complete, virtually all dissimilarities between test measurements and model reports tend to dissolve.

Dissolution of disparities tends to come in two forms. The first is a refinement of both the testing and the modeling structures and environments so that one can say that the modeled antenna is a very close analog of the physical antenna--and vice versa. The second form of removing conflicts between a model and a physical antenna is a comprehensive understanding of differences between the two. A physical Yagi may connect the elements to a conductive boom, with resultant changes in the required element lengths relative to a model. A model may involve geometric structures that require reference to and correction by the Average Gain Test value for gain and source resistance values. In many cases, apparent differences between a model and a physical antenna may disappear under appropriate calculation to adjust for those differences.

The logic of this situation makes measurement a post-modeling activity, regardless of the temporal order of the modeling and the field testing. Equally, we might call the situation one of post-construction testing against a model. Consider Table 1.

The table provides parallel columns of values for the length of a dipole resonant at 146 MHz. One column lists the measured value of the length of round elements trimmed to resonance. The other column lists the length of modeled dipoles trimmed to resonance within the software. It makes no difference which activity comes first in time. We simply cannot make a comparison until we have both columns filled in.

Pre-Modeling Physical Testing

There are a number of situations that NEC cannot directly model, many of which involve the proximity of the conductor with a non-conductive material. Of course, NEC-4 is able to model wires having an insulated sheath. Fig. 6 illustrates the dimensions involved and shows the parameters involved, as listed on a GNEC assistance screen. NEC-2 lacks this facility, but there is a work-around that is applicable in many situations. See episodes 50 and 83 for further details of modeling insulated wires with NEC.

The questions of NEC's ability to model a structure emerge from situations other than the simple insulated wire. Fig. 7 shows some typical cases that have often occasioned e-mail questions.

The situation that invites the most inquiries involves the left-most sketch, where a relatively thin and flat conductor is bonded to a non-conductive substrate. In fact, NEC has no direct means of dealing with this "PC-board" style of structure, although the use of such materials is common in the UHF region. First, the conductor is not round, and there is no standard list correlating flat strip surface areas to the surface areas of round conductors, at least not in any reliable way. Second, the substrate represents an insulator having a thickness, a relative permittivity, and a conductivity, but bonded to only one side of the conductor. Hence, we can assume that some sort of electrical lengthening occurs, but we cannot say in advance what the velocity factor will be. When we combine the two problems, we usually end up in a quandary about effectively modeling subject antennas without investing in very expensive hybrid software.

The problem set may increase for many antenna designs adapted to printed circuit board materials. For example, a Yagi or a log-periodic dipole array may etch the elements on a single plane so that the adjacent elements expose their thin edges to each other. In addition, the substrate may fill part of the region between the elements, at least on one side of the plane of the element strips. Consequently, the substrate material plays a role not only in determining the electrical length of the elements, but as well in their mutual coupling.

The situation of the center sketch in Fig. 7 often occasions questions, but is in fact quite simple. A number of antenna builders--mostly to use local materials--apply conductive tapes around non-conductive round structures and thereby form antenna elements. The sketch shows one typical case in which the tape completely circles the support and forms a closed cylinder. In this case, we may treat the element as a wire having the outer diameter of the tape surface. Of course, this assumes that the tape forms a fully complete and closed circle around the central support. The result is not dissimilar to the copper-clad steel wire known as copperweld. That steel has some conductivity and the central support for the tape has little or none matters not at all, since the RF currents will be near the surface in both cases.

The right-most portion of Fig. 7 changes matters considerably. We still have a central support and an overlay of conductive tape. However, the conductive surface does not form a complete circle around the center material. If the circle is almost closed, then the element may act as if it were closed. However, if the gap is wide enough, then the semi-circle of tape may act more like the left-most figure. To the best of my knowledge, there are no handy guidelines for converting various forms of the right-most figure into equivalent round-wire values for modeling.

There are methods for creating a conversion table between the structures on the left and right in Fig. 7 and round-wires values that we may model. They involve preliminary test antennas to determine the round wire-equivalent values for a given physical structure. One technique is simplicity in itself: create a dipole at the frequency of interest for the material and then find its corresponding round-wire equivalent diameter for resonance at the same element length.

Fig. 8 shows a variety of shapes of alternative materials used for antennas in the amateur 2-meter band. Most of the materials come from hardware outlets. Builders use some of them because they are locally available. In some cases, builders find stock with flat surfaces easier to work. A few materials, such as measuring tape or rabbit ears, may have special features useful in transporting the antenna or using it in rough terrain.

Prior to modeling, we may calibrate any of these materials to a selected round-wire diameter by using the dipole-resonance technique. Table 2 lists some results of measurements made locally at 146 MHz with some typical materials. Each material lists the resonant length and the round wire diameter with the most similar length from Table 1.

The sample tables are for 146 MHz. It is not known how far from the listed frequency that the equivalencies would apply. For example, stock that is 1/16" thick is about 7.7e-4 wavelength at 146 MHz but only about 7.4e-5 wavelength at 14 MHz. Whether that change in relative thickness brings the 1/16" stock down to measuring tape thickness requires a re-run of the tests for the frequency of interest.

Although the pre-modeling test runs are useful for independent elements, the tests have additional limitations besides their potential frequency restrictions. For arrays in which mutual coupling between elements is critical, very flat and wide stock may show some differences depending upon whether the elements are edge-to-edge or flat-to-flat. As well, without specific pre-modeling tests, one cannot know the effects of a continuous substrate on the mutual coupling between elements. However, with proper equipment, such tests are possible and may lead to round-wire, open-space equivalents.

Despite the limitations, the exercise does demonstrate that all is not lost with respect to modeling if we use materials other than the round wires upon which NEC's algorithms are based. One question is whether all of that work is worth the effort. In most cases, of course, pre-testing or calibration of materials will be confined to only a few selected candidates. As well, once a material has found its round-wire equivalent diameter, then we may construct relatively diverse and complex arrays within the models and reserve prototype testing until we are satisfied with the modeling results. Hence, when we view the design of an antenna as a full-scale activity set, the modeling saves enough time to make the pre-testing phase well worth the effort involved.

Conclusion

We began with the idea that there is an intimate connection between modeling and measurement that extends from pre-modeling calibrations through post-modeling prototype testing. Modeling does not exist in a vacuum, since its results are either a physical antenna or an understanding of the performance of a physical antenna. These columns have tended to focus on modeling's internal working. However, we should never lose sight of the fact that modeling has an integral place within a larger set of activities that may involve measurement both before and after the modeling itself. As well, with appropriate pre-modeling calibration measurements, we may effectively model many (but not all) materials that would otherwise violate the NEC round-wire premises. Without those pre-modeling measurements, trying to model such materials would amount to mere speculation.

-30-

~ antenneX July 2006 Online Issue #111 ~
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