Everyone who models conscientiously using either NEC or MININEC worries over the results. Essentially, we find two general categories of worry. First is the basic adequacy of a model. Within NEC--and assuming adherence to the guidelines--the chief measure of model adequacy is the average gain test (AGT), which we have discussed extensively in past columns. Ideally, a free-space lossless model should show an AGT of 1.000 (2.00 for models using a perfect ground). There are no absolute rules for when an AGT score other than 1.000 renders a model inadequate. However, the more comparative and systematic a modeling exercise is, the closer to 1.000 that we require the AGT values to validate comparisons among models.

The AGT rests upon performing a full spherical far-field scan for a model in free space (a hemisphere over perfect ground). Using sufficient and equally spaced (angularly, of course) increments for the sample, the average far-field gain should equal the reference gain so that the ratio is 1.000. As we examine more complex geometries, we should reduce the increment between each sampled point on the sphere to obtain the most accurate average value. (Automated AGT scans offered by some implementations of NEC often use 5-degree increments, which is normally adequate for linear elements in various arrays. However, changing the increment to a lower value is a good check to ascertain the best value to use in a given exercise.)

We may often obtain corrected values for raw NEC gain reports by converting the reported AGT value into dB. The conversion consists of taking the log of the reported AGT value and multiplying by 10. A reporter AGT score of 1.010 converts to 0.04 dB. Since the AGT score is greater than 1, we subtract the converted value from the reported gain. Hence a raw gain report of 5.00 dBi becomes 4.96 dBi. Reported AGT values below 1.000 result in increases to the raw far-field gain report. An AGT report of 0.960 becomes -0.18 dB. Had the gain report been 5.00 dBi, the corrected gain value would be 5.18 dBi.

We may also correct the resistive component of the reported source impedance using the AGT directly. The correction works best when the impedance is close to resonant, that is, has a relatively low reactive component. (Results with high reactive components appear to be mixed, with some results appearing to coincide with range test results and some appearing to diverge considerably.) To correct the impedance value, simply multiply the AGT score times the resistive component. Assume a reported value of 150 Ohms. An AGT score of 1.01 would convert this value to 151.5 Ohms. An AGT value of 0.960 would correct the reported resistance to 144 Ohms. How significant these corrects are depends upon the terms of the modeling exercise. If I were building an antenna with a target feedpoint impedance of 150 Ohms, I might expect construction variables to outweigh the range of variation within the example. However, for a sequence of modeled antenna geometry variations, I might wish to use corrected source impedance values in order to obtain a reasonable sequence of values associated with the variations, especially if each variation produces a different AGT value.

The notion of construction variables leads us to the second major category of worry. The basic concern is how well a model conforms to a reasonable physical implementation of a modeled antenna. As we move ever higher in frequency, the bumps and the short leads that we usually do not model take on increased importance, since they grow as a function of a wavelength at higher frequencies. We might usefully make a catalog of compensatory measures that we sometimes take to overcome anticipated construction variations. However, we shall save that level of concern for some future column. Assuming that we would use general care in constructing a physical implementation of a modeled antenna, we shall restrict our concerns to general expectations for relative gain among relevantly comparable antennas and for their feedpoint impedances.

Modeling yields numbers derived from calculations within the particular modeling core that we might use. Many implementations of NEC include graphical representations of those numbers, but the basic NEC output is a large collection of numbers. When we find differences between two sets of numbers, we tend to worry. However, only in some cases is the worry justified. To distinguish between justified worries and unjustified worries, let's look at a case study.

**Closed 1-Wavelength Loop Elements**

A closed 1-wavel;ength loop element with a single feedpoint provides major bi-directional far-field lobes broadside to the plane of the loop. There are two significant properties for us to note initially. First, the 1-wavelength loop provides directional gain over a simple half-wavelength resonant dipole. In fact, we may usefully analyze the 1-wavelength loop as constituting two dipoles fed in phase with an average distance apart of about 1/4-wavelength. Second, whereas a physical dipole will by shorter than 1/2-wavelength at resonance, a full-wavelength loop circumference will be greater than 1-wavelength at resonance. Part of the lengthening results from the mutual coupling between the two dipole elements presumed to be in the loop: resonance requires that we lengthen each of them by a small amount.

Amateur antenna builders continually look for so-called "cutting formulas," those seemingly magical simplifications that supposedly give the right answer for element length using only a single constant and the design frequency. Unfortunately, the world of antennas is more complex than cutting formulas imagine. Even in free space, the required length for a dipole or a closed loop--at resonance--will vary with the wire diameter and to a much lesser degree on the material loss of the antenna wire or tubing. For our exercise, all antennas will be in free space and use 0.002-m (2-mm) lossless or perfect wire. Horizontally polarized antennas are also prone to variations in the required length for a resonant impedance as a function of the antenna height above ground, especially below 1-wavelength.

The exercise that led to these notes
emerged from a variety of claims that one hears about various closed
loop configurations. One claim is that a perfectly circular loop has a
significantly higher gain than a conventions square-sided loop
(arranged to form either a square or a diamond). Presumably, the circle
is a figure with an indefinitely high number of sides, compared to the
4 sides of the conventional HF quad loop. However, that account is at
odds with claims made for a triangular or delta closed loop with equal
side lengths, namely, that it is just as good as the 4-sided loop,
again with equal side lengths. So I decided to example some models
(free-space using 0.002-m diameter lossless wire) involving the
configurations shown in **Fig. 1**. The test frequency is 300 MHz.

The dipole, of course, provides a reference against which to measure the increased gain offered by each closed-loop configuration. The sequence of models included both square and diamond-shaped 4-sided loops. To place the feedpoint at conventional positions required a mid-side position for the square loop and a position at the lower point for the diamond-shaped version. The two delta or triangular loops are both equilateral triangles and differ only in the feedpoint position. Since the model is in free space, moving the mid-side position to the top wire makes no difference in performance relative to inverting the entire structure. The loops have from 33 to 44 segments total, so the circular loop uses 45 wires to complete the simulation of a true circle. In some implementations of NEC, one may use the GA (arc) command to create the circle. The software used here--EZNEC-Pro/4--creates circles using separate wires. However, the results are identical to circles created with the GA command.

The sequence of seemingly comparable models yields the data shown in **Table 1**.
With respect to the AGT values, we may note that the dipole and the
circular loop achieve virtually perfect values, while the two angular
loops fed at mid-side positions have identical very good values. Both
require only a 0.02-dB decrease in the reported gain value.

The point-fed angular loops, however, tell a
different story. The AGT values diverge more widely from the ideal
value as the angle between the sides decreases. As indicated in **Fig. 1**,
both point-fed models use a "split-source" feed, that is, a pair of
sources on segments adjacent to the point. The reported source
impedance value is the sum of these series sources. Removing one of the
sources does not change the result (or the AGT value) significantly.

The table provides corrected maximum far-field gain values based on the raw report and the AGT score. We may note in more than a passing way that the corrected gain value for each point-fed model coincides very closely with the corresponding value for a mid-side source position. (In fact, the gain values form a progression from low to high as we move from the most angular structure--the delta or triangle--to the least angular--the circle.)

The table does not list corrected source resistance values. Only the values for the point-fed positions would change noticeably. The corrected source resistance for the point-fed square-diamond becomes 131.6 Ohms, only 1 Ohm different from the mid-side fed square. Likewise, the adjusted point-fed delta source resistance becomes 126.0 Ohms, about 5 Ohms away from the mid-side source value.

**A MININEC Trial**

We may note two facts about the NEC models. First, NEC models place a source within a segment. Hence, we cannot obtain a perfect point-fed source location. For many purposes, the split-source work-around to this small limitation proves to be very satisfactory. However, in this particular exercise, the split-source technique does not provide models that are sufficiently adequate to let us make the comparisons required by the exercise goals, at least, not without correcting the reported gain and source resistance values. Second, the AGT departs more widely from the ideal value as we sharpen the angle between wires at the point feedpoint. The wire angles themselves--in these NEC-4 models--do not create a problem, since the mid-side feedpoints yield very good AGT values. Only when we place the feedpoint at the angular junction of two wires do we encounter the difficulty.

MININEC (3.13) does not suffer one of
the problems that we have just noted. We may place a source directly on
a diamond or delta point, since MININEC places sources (as well as
loads) on pulses, that is, on the junction between segments. However,
uncorrected MININEC 3.13 suffers other difficulties, especially with
angular junctions. Antenna Model is a highly corrected version of
MININEC that has to a very large measure overcome the limitations of
raw MININEC. It has several other features as well, including the
ability to provide an AGT value and the ability to import NEC models.
The latter feature is useful in ensuring that the models used in
MININEC trials are as precise to their NEC-4 originals as is feasible.
The result is a set of models with the outlines shown in **Fig. 2**.

Since MININEC requires a pulse or segment junction for the mid-side fed models, Antenna Model automatically increases the segment count on the affected wire by 1 to obtain an even number of segments. To assure equal segments on equal-length wires, I increased the count on the other wires of the square and the mid-side fed delta to coincide with the count on the source wire. Otherwise, the models are identical to their NEC originals.

**Table 2** lists the results of the trials.
The reference dipole and the circular loop provided excellent AGT
values and showed resonance within the project limit of +/-j1.0 Ohm
reactance. However, all other models required some dimensional
alteration to bring them to resonance within the same limits. The table
shows the initial impedance reports as well as the circumference of the
corrected models. (Of course, the side lengths are the circumference
divided by 4 for the square models and divided by 3 for the deltas.)
All of the models display excellent AGT values. The gain values for
each type of loop coincide relative to mid-side and point feeding, and
the progression is virtually the same as we found in the NEC models,
with increasing gain as we move from the triangle toward the circle. In
a general way, our worries over the deviant AGT values provided by the
NEC models were well founded.

However, one worry was not as significant as it might seem at first sight. The angular models showed a departure from resonance of at least j10 Ohms, occasioning the work of revising the dimensions to bring the model back to resonance. If we bring a mere numerical sense of resonance to the project, the departure may seem large. However, we should also note that the feedpoint resistance ranges from about 120 to 150 Ohms. Therefore j10 Ohms departure is far less troublesome in any respect than the same reactance with a feedpoint resistance of 50 Ohms.

To see how much of a worry the
departure from resonance might be, I calculated the change of dimension
between the imported NEC models and the corrected versions. The results
appear in **Table 3**. The table lists the ratio of corrected to
uncorrected circumference, the amount of change as a percentage, and
the SWR of the original model relative to the reported feedpoint
resistance.

All but one of the models show less than a 1% required dimensional change. For these models, it would be inappropriate to worry about the differences relative to an eventual physical implementation of the loop structures. Construction variables would likely wash out the difference between the original and the adjusted dimensions. In practice, we likely could not distinguish between construction and modeling variations. As the title of this column suggests, some modeling differences are worth the worry; others are not.

The one case in which we needed to
change dimensions by more than 1% involves the point-fed delta loop.
The 2.4% increase in the circumference to restore resonance in the
MININEC model leaves us with a small worry that is more prominent in
this program than in NEC. From **Table 2**, we find a corrected
feedpoint impedance that is very much higher than expected. Indeed, the
value is higher than the source impedance of the circular loop. The
model from which we derived the feedpoint impedance uses only 16
segments per side. To MININEC models we may apply the convergence test,
raising the number of segments per wire until the values for gain and
impedance stabilize. For the point-fed delta, gain is not a problem,
but the source impedance may be. Therefore, I increased the
segmentation to 44 segments per side in small increments. At this
level, the Antenna Model AGT value became 0.999, with a gain that was
still 3.05 dBi. However, the resonant source impedance dropped to 132.6
- j 0.3 Ohms. Arriving at resonance at the converged value required a
reduction in the delta circumference to 1.122 wavelengths. This
circumference is very close to the original NEC-model value. At the
same level of segmentation, the delta using a mid-side feedpoint showed
no change in either the far-field gain of the source impedance.
However, the circumference at resonance decreased to 1.109 wavelengths,
virtually the same as required by the original NEC model.

**A Final Test**

The MININEC divergence between source impedance values using a point-fed source position and a mid-side source position remains somewhat worrisome when we compare those values to values derived from NEC models and corrected with reference to the AGT values. The worries may be minuscule in the context of developing a model that prepares us to construct a physical implementation of the antenna. However, from the perspective of systematic modeling for comparative purposes, the worries grow in both size and number.

Instead of a single concern, we now have two, one general to modeling, the other specific to the present exercise. First, the MININEC AGT scores are universally exceptionally good, despite shortcomings that we found in some of the models. Whereas NEC AGT values are valuable up to an indefinable limit in providing corrected values to the reported gain and impedance, they are less valuable in MININEC. Instead, the convergence test provides perhaps the most critical test for MININEC model adequacy.

Second, the point-fed model source impedance values do not show reasonable agreement between the NEC and the MININEC models, although all other models show good coincidence. Moreover, the MININEC models suggest that the impedance of a point-fed model (both diamond and delta) will be higher than the value exhibited by mid-side fed models. In contrast, NEC models suggest that the source impedance for mid-side and point feeding will be quite similar.

In order to provide some resolution
to this question, I compared three NEC models of the delta using a
point source location. The first is our original models using split
source, but with the segmentation density increase to 33 segments per
wire. The second model blunted the feedpoint by inserting a 1-segment
"bridge" wire to replace the sharp point. The segment length is
identical to the segment length used in the long wires. The third model
increased the length of the bridge wire to accommodate 3 segments so
that the segments adjacent to the source segment are in the same plane
as the source segment and have the same length. **Fig. 3** shows the point end of the deltas under all three conditions.

The results from all three models appear in **Table 4**.
The data show a progressive increase in the AGT value toward the ideal,
with the final model only slightly imperfect. The nature of the AGT
values strongly suggests that the angle of the wires that approach the
source segment plays a strong role in yielding less than adequate
models. Only when we isolate the source segment from the angled wires
do we arrive at a model that passes NEC AGT muster.

The bridge wire technique has two interesting consequences relative to the reported and corrected output data. First, the gain report is higher than for the original mid-side fed delta. It is likely that the length of the 3-segment bridge wire that is parallel to the far side of the delta may account for the small increase in reported gain--an amount that is numerically signicant in this comparative exercise but not in operational terms. The second result is the corrected source impedance value, which is identical to the value yielded by the mid-side fed model of the delta. This result suggests that ideal models would show very comparable source values for resonant 1-wavelength closed loops of the same shape, regardless of whether the loop uses a mid-side or point source location.

**Conclusion**

Systematic modeling exercises raise numerous concerns along the way. The present exercise sought to compare 1-wavelength closed loops to examine a number of conflicting claims about their performance. Our interest in the exercise has less to do with the conclusions that we might reach about such loop structures than about the worries raised by the modeling effort. Some worries proved to be relatively insignificant. Other gave us pause and required us to employ all relevant model adequacy tests and even work-arounds to resolve. In the process, we also considered different contexts, some of which made the worries important and other of which reduce them to mere footnotes on the modeling effort.

Knowing when to worry, how much to worry, and what to do about the worry are all parts of mastering the art and craft of antenna modeling.