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<!DOCTYPE html>
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<title>Converging Toward Excellence</title>
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<body>
<center>
<h1>1. Converging Toward Excellence</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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Whether you are using one of the versions of MININEC or one of the versions of NEC-2, a temptation will overcome you somewhere along the way in your antenna modeling career. You will ignore the convergence test. And in some cases, you may regret it, especially when your copper or aluminum antenna turns out to be well off the mark set by the model.
<p>So let's take a patient look at convergence testing. Our aim will be not just to be able to do it, but as well to understand why we should always do it.</p>
<p>Both NEC and MININEC calculate antenna properties based on the division of the antenna elements into a collection of short wires. Within the program limits, the shorter, the better. However, both NEC and MININEC make use of complex calculation matrices that take time to fill and resolve. The more segments we have, the longer the run time.</p>
<p>Because we are impatient, a bad habit has set in among many program users: to divide the antenna elements into as few segments as possible, again within certain generalizations in the instruction manuals. For MININEC, the recommended minimum number of segments per half wavelength is 10, while for NEC-2 it is 9 or 11. (When center feeding an antenna element, use an even number of segments for MININEC and an odd number of segments for NEC.)</p>
<p>What we often forget is that neither program is absolutely stable, and the best we can hope for is relative stability. What stability refers to is the fact that if we change the number of segments per half wavelength, without changing the overall dimensions, the program will produce different output data. What we strive for is a number of segments per half wavelength that--as we change the segmentation--produces output data differentials that are too small to be operationally significant. Of course, what counts as being operationally significant depends on the modeling goals for a given project.</p>
<p>We can illustrate the instability of the program by making some runs with an ordinary 20-meter dipole, as shown in Figure 1. (In all figures, the heavy black "dot" represents the source or feedpoint. It is shown as a dot rather than as a break in the wire to indicate that the wire is continuous and the source is inserted in series with the wire.) The only factor we shall change from model run to model run is the number of segments into which we divide this half wavelength of wire.</p>
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<img src="amod1-1.gif" width="388" height="264" border="2" vspace="5" hspace="9">
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<p>For NEC-2, we can make a table of the results:</p>
<pre>Segments Gain in dBi Source Impedance (R+/-jX)
11 7.97 67.51 - j0.22
15 7.98 67.50 + j0.34
21 7.98 67.52 + j0.24
25 7.99 67.53 + j0.32
31 7.99 67.54 + j0.42
35 7.99 67.55 + j0.46
41 7.99 67.56 + j0.53
45 7.99 67.56 + j0.56</pre>
<p>This example is the source of the fatal temptation to always use the minimum number of segments. It is operationally stable at the lowest level of segmentation for any purpose I can think of, since I have no instruments capable of trimming an antenna to the variations shown in the numbers.</p>
<p>However, the table does illustrate that the numbers in fact do change with the level of segmentation. As the first 4 lines of the source impedance entries show, the numbers may drift in either direction until some large number of segments are used.</p>
<p>A similar situation exists with MININEC. The segmentation varies from the NEC table to accommodate the source location.</p>
<pre>Segments Gain in dBi Source Impedance (R+/-jX)
10 7.83 67.51 - j6.78
16 7.85 68.19 - j7.71
20 7.85 68.40 - j7.57
26 7.86 68.59 - j7.22
30 7.86 68.67 - j7.01
36 7.86 68.76 - j6.70
40 7.86 68.81 - j6.50
46 7.86 68.87 - j6.26</pre>
<p>The intrinsically higher level of reactance is a function of MININEC's growing error in resonant length as the frequency climbs. Some programs correct for this error; others do not.</p>
<p>Nevertheless, we find a similar drift in values with increasing segmentation. However, for operational purposes, the numbers give us confidence that a dipole constructed from these dimensions would work perfectly well with only such trimming as our yard clutter and local terrain might make necessary.</p>
<p>One common way in which we often fail to observe even the minimum recommended segmentation is when we increase the frequency of operation of an antenna without altering the number of segments. Figure 2 shows the same antenna, but now operated at 28.0 MHz.</p>
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<img src="amod1-2.gif" width="389" height="258" border="2" vspace="5" hspace="9">
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<p>The results for NEC-2 are the following:</p>
<pre>Total Segments Gain in dBi Source Impedance (R+/-jX)
11 9.58 3813 + j1438
23 9.61 4306 - j 376
33 9.62 4120 - j 945
45 9.62 3928 - j1266
55 9.62 3810 - j1415
67 9.62 3705 - j1529</pre>
<p>Although the gain stabilizes as soon as we use at least 11 segments per half wavelength, the source impedance continues to move around considerably. since we would likely feed this antenna via a parallel transmission line and antenna tuner, we might easily overlook the variance.</p>
<p>However, the antenna as presently used can be viewed as two end-fed half- wavelength dipoles physically coupled. End feeding near resonance yields large changes in source impedance with very small changes in physical dimensions. Hence, changes in segmentation yield possibly significant differences in source impedance.</p>
<p>MININEC shows a similar set of figures:</p>
<pre>Total Segments Gain in dBi Source Impedance (R+/-jX)
10 9.55 3401 - j1646
22 9.56 3537 - j1519
32 9.56 3485 - j1568
44 9.56 3431 - j1616
54 9.56 3399 - j1643
66 9.56 3366 - j1669</pre>
<p>Interestingly, the source impedance figures for MININEC appear to be more stable than those for NEC-2. It should be remembered that all MININEC source impedance figures are calculated based on perfect ground, whatever the ground used to determine the far field figures. (Medium earth is used throughout these examples.) NEC-2 calculates source impedance based on the Sommerfeld-Norton ground conditions specified for the antenna.</p>
<p>In determining whether a satisfactory level of convergence has been achieved, the program user must determine how much variance is necessary to make it significant. The following driver-director 10-meter Yagi, shown in Figure 3 at a height of 1/2 wavelength, might be instructive here.</p>
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<img src="amod1-3.gif" width="371" height="204" border="2" vspace="5" hspace="9">
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<p>NEC-2 tabulates the results as follows:</p>
<pre>Segments/Element Gain in dBi Front-to-Back in dB Source Impedance (R+/-jX)
11 10.84 14.11 21.11 - j23.57
15 10.88 14.34 20.66 - j22.64
21 10.91 14.55 20.30 - j21.87
25 10.92 14.64 20.15 - j21.54
31 10.93 14.73 19.99 - j21.19
35 10.94 14.78 19.91 - j21.01
41 10.94 14.84 19.81 - j20.78
45 10.95 14.87 19.75 - j20.66</pre>
<p>It might appear that, although the gain and the source impedance are fairly stable, especially in models using at least 21 segments per element, the front-to-back ratio continues to change at higher rate. However, from an operational standpoint, changes of a dB in this ratio are not especially detectable, and hence, the model might well be said to converge at about 21 segments per element--or less, if the needs are less critical.</p>
<p>Moreover, since we would likely use a matching system, such as the beta match or a series feedline section, between the feedpoint and the main 50- ohm coaxial feedline, the variations in the source impedance are unlikely to fall beyond normal adjustment limits. Hence, the home constructor might well judge the model to be well converged at the lowest level of segmentation. In terms of comparative design analysis, we might hold out for segmentation of at least 20 segments per element.</p>
<p>The corresponding MININEC model was run at the same frequency. Hence, its performance figures will be less impressive, since they reach a peak at a different frequency. However, for our purposes, the table shows a similar level of stability as the NEC-2 table above.</p>
<pre>Segments/Element Gain in dBi Front-to-Back in dB Source Impedance (R+/-jX)
10 10.49 10.71 21.91 - j29.20
14 10.53 10.90 21.72 - j28.79
20 10.56 11.18 21.29 - j27.75
24 10.61 11.34 21.01 - j27.11
30 10.64 11.47 20.81 - j26.40
34 10.65 11.57 20.63 - j25.79
40 10.68 11.71 20.37 - j25.27
44 10.68 11.75 20.31 - j24.94</pre>
<p>Although stable enough for most purposes at the lowest level of segmentation, the figures are highly stable once we use more than 20 segments per elements.</p>
<p>Symmetrical structures, even of some complexity tend to show a higher level of stability than non-symmetrical structures, even of similar types. To illustrate the point, let's compare a pair of triangular designs, vertically oriented. The first is a right-angle delta for 7.15 MHz, shown in Figure 4.</p>
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<img src="amod1-4.gif" width="433" height="432" border="2" vspace="5" hspace="9">
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<p>Despite its angularity, the antenna is quite stable once the segments are shorter than the maximum recommended length. The segmentation column shows the number of segments in each angular leg, and then the horizontal leg.</p>
<pre>Segments Gain in dBi Source Impedance (R+/-jX)
6/6/8 1.73 63.83 + j1.49
12/12/17 1.90 60.95 - j0.29
18/18/25 1.89 60.99 - j0.77
24/24/34 1.89 61.03 - j1.03
30/30/42 1.89 61.06 - j1.18</pre>
<p>The lowest level of segmentation results in segments longer than the maximum recommended length. Beyond that level, the model shows excellent stability over many levels of segmentation, despite the fact that the source is moving slightly each time, since it is specified as a percentage of the left leg.</p>
<p>One might be tempted to also model this antenna via MININEC. However, to do so would be to violate the limits of reliability with respect to the height of the antenna above ground. In general, as an antenna is brought closer to ground than about 0.2 wavelength, the gain increases inaccurately. For example, a fully length-tapered version of this antenna shows a MININEC source impedance of 64.5 - j23.6 ohms, somewhat off the NEC-2 mark, but not drastically. However, the antenna also shows a wholly unrealistic gain of 3.63 dBi.</p>
<p>For comparison, let's look at model of another triangle, one developed by W6RCA and shown in Figure 5.</p>
<center>
<img src="amod1-5.gif" width="427" height="431" border="2" vspace="5" hspace="9">
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<p>Relative to the plane of the antenna facing us, the antenna is not symmetrical on either side of a line one might draw at to bisect the source. Although the angularity of the antenna might lead us to believe that its model will behave similarly to the right-angle delta, the NEC-2 table tells a different story when using a split-current source feed. (This feature is available on programs such as EZNEC. The model shows much less instability when using a split-voltage feed.) The segmentation column refers to the number of segments in the vertical leg, the horizontal leg, and the diagonal leg, respectively.</p>
<pre>Segments Gain in dBi Source Impedance (R+/-jX)
4/8/9 4.33 62.05 - j10.08
7/12/14 2.87 47.99 - j 5.77
9/16/19 2.45 44.10 - j 2.08
11/21/23 2.32 42.65 - j 0.26
13/25/28 2.24 41.83 + j 1.95
16/29/33 2.21 41.37 + j 3.73
18/33/37 2.19 41.19 + j 4.53</pre>
<p>As with the right-angle delta model, the initial segmentation yields segment lengths longer than the maximum recommended values, and this fact is reflected in the unrealistically high gain and source impedance. By the point of maximum segmentation in the exercise, the model has stabilized to an operationally useful point--sufficiently so that the dimensions promise to make a good starting point for building such an antenna. However, the rate of change from one level of segmentation to the next is still significantly higher than it is for symmetrical antennas such as the dipoles, the Yagi, or the right-angle delta.</p>
<p>Inadequate segmentation on the split current-fed model does not just affect the gain and source impedance numbers. In fact, the entire far field pattern changes when comparing minimal segmentation models with adequately segmented models. Figure 6 compares the elevation pattern perpendicular to the face of the triangle for the 21-segment model and the same pattern produced by the 88-segment model. The latter is typical of the patterns of the group of more nearly converged models. The difference is not merely a matter of gain, but as well the correct portrayal of higher-angle radiation from the antenna. (Again, the model does not show the dramatic change in pattern shape when split-voltage-fed.)</p>
<center>
<img src="amod1-6.gif" width="582" height="288" border="2" vspace="5" hspace="9">
</center>
<p>Unlike the right-angle delta, the W6RCA triangle can be modeled via MININEC, since the bottom is about 0.22 wavelengths above ground. Even though MININEC advises against modeling with small numbers of segments when an antenna geometry contains right or acute angles, the exercise may be instructive. Again, for this example, the model is split-current fed.</p>
<pre>Segments Gain in dBi Source Impedance (R+/-jX)
4/8/9 3.43 64.17 - j210.2
7/12/14 3.28 50.00 - j125.6
9/16/19 3.32 45.72 - j 89.5
11/21/23 3.37 42.66 - j 68.4
13/25/28 3.38 42.02 - j 54.2
16/29/33 3.39 41.68 - j 43.3
18/33/37 3.39 41.68 - j 36.8</pre>
<p>Although the model comes to show reasonable stability, its reported gain is well above that of the NEC-2 model, largely because it is near the border of reliability. This fact is significant, since many MININEC users think of the 0.2 wavelength height limit as a kind of breaking point. It is not.</p>
<p>Instead, it is a region where gain reports on different types of antennas become unreliable at different actual heights above ground.</p>
<p>The large swing of reactance in the source impedance is due to the fact that MININEC models in effect cut off sharp interior corners. The longer the length of individual segments meeting at an acute angle, the more wire is effectively trimmed in the computation of impedance and current interactions. As the segments grow shorter, the antenna gradually approaches its physical full size. A fully length-tapered version of the antenna model showed a gain of 3.38 dBi (still unreliably high), and a source impedance of 40.32 + j5.20 ohms, not far off the NEC-2 model.</p>
<p>The lessons taught by the W6RCA triangle are numerous and useful. First, it is unsafe in terms of model reliability to use the minimum number of segments per half wavelength recommended for wires in the model by the program. These program recommendations are based upon linear elements and apply at most to symmetrical antenna geometries.</p>
<p>Second, it is also unsafe to assume that some arbitrary larger number of segments per half wavelength will automatically yield a reliable model. The W6RCA triangle remained less stable with 44 segments per half- wavelength than many of the other models with under half that number.</p>
<p>The designer must make a series of convergence tests and reach a decision concerning the adequacy of stability in the model based on standards brought to the modeling exercise from a knowledge of the goals of the overall project. In some cases, stability adequate to one task may not suffice for another.</p>
<p>Third, inadequate convergence holds the potential for producing a model that misrepresents antenna performance in every way: gain, front-to-back ratio (if applicable), source impedance, and field pattern shape and strength. Although convergence alone cannot guarantee the adequacy of a model, it is one necessary condition to that goal.</p>
<p>Antenna modeling can be a very significant short-cut in the design and building of antennas. However, we can achieve the savings only if we do not take short-cuts with the modeling process itself. Convergence testing is one of those steps we should never omit, lest the minutes we save at the computer testing our models end up costing us hours of frustrating time spent trying to adjust an antenna that is based upon an inadequate model.</p>
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<p><a href="modeling.html"><img src="../images/return.gif" align="bottom" width="40" height="40"> <b>Go to Main Index</b></a></p>
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<!DOCTYPE html>
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<title>Tapering to Perfection</title>
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<center>
<h1>10. Tapering to Perfection</h1>
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<center>
<h3>L. B. Cebik, W4RNL</h3>
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<img src="../images/colorbar.gif" width="591" height="3">
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Everyone who models HF (and low VHF) beams is faced with dealing with elements that do not have a constant diameter. We refer to such elements as tapered-diameter (or tapered radius) elements, in contrast to elements with a uniform diameter, that is, elements that are untapered. The latter have somehow picked up in some literature the self-contradictory name of mono-tapered elements.
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<img src="amod10-1.gif" width="482" height="265" border="2" vspace="5" hspace="9">
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<p><b>Figure 1</b> exaggerates--but only to a degree--the amount of stepping involved in most elements composed of several diameters of aluminum tubing. Normally, the maximum diameter shift from one section to another is about 0.25". However, some designers use a short, fat center wire in the element to simulate the effects of element-to-boom mounting plates and hardware. In these cases, the diameter shift may amount to as much as 2".</p>
<p>Native NEC-2 cannot handle tapered-diameter elements. It simply yields inaccurate results--ordinarily by over-reporting the gain and under-reporting the source impedance of the element. Consequently, in some implementations of NEC-2, correction factors have been installed to yield more accurate results.</p>
<p>Let's begin by comparing some simple cases of uncorrected and corrected NEC-2 reports for some simple tapered-diameter elements set up as dipoles in the 14 MHz range. The following table repeats an exercise we first encountered in installment 3 of this series. The entries in the table beneath each numbered label represent, first, the distance from the elements center at which a diameter shift occurs and, second, the diameter(s) of the element between shift points. The "Cor." entries give the length from center and the diameter of the substitute element used by NEC-2 to provide corrected results.</p>
<pre> Length(s) in inches Free Space Gain Source Impedance
Diameter(s) in inches in dBi (R +/- jX Ohms)
1. Uniform element (no corrective needed)
-201.25...0 2.12 71.8 - j 0.6
1.0
2. One step, far out
-204...-150...0 No Cor. 2.14 73.0 + j 4.4
.75 1.0
-201.45...0 Cor. 2.12 72.0 + j 0.4
0.966
3. One step, near center
-204...-50...0 No Cor. 2.22 72.4 + j 5.2
.75 1.0
-201.661...0 Cor. 2.13 71.8 - j 0.5
0.792
4. Two steps, modest taper
-205.75...-100...-20...0 No Cor. 2.32 72.5 + j10.6
.75 1.0 1.25
-201.49...0 Cor. 2.13 71.9 + j 0.1
0.889
5. Two steps, more extreme taper
-208.5...-100...-20...0 No Cor. 2.82 67.6 + j17.1
.75 1.0 2.5
-201.63...0 Cor. 2.13 72.1 + j 0.9
0.895</pre>
<p>Notice especially the trends. Without correction, the closer the diameter step to the element center (and the source), the higher the reported gain and the lower the reported source impedance. The more numerous the steps and the greater the step sizes, the higher the reported gain and the lower the reported source impedance. By entry number 5, the figures have grown incredible. In contrast, the corrected elements yield figures that are all quite reasonable for a resonant 14 MHz dipole.</p>
<p>The simple lesson from this exercise is always to use the stepped-diameter correction feature when modeling stepped-diameter elements. However, there are numerous limitations to the tapered-diameter element correction scheme, as well as some cautions to be observed in using it. So let's spend a little time understanding the feature and learning to use it wisely. In addition, let's also pick up the habit of making sure that it is in effect when we want it to be.</p>
<h3>A Bit of Background</h3>Wherever the tapered-diameter correction feature is implemented, it creates a substitute element for the tapered one. The substitute element has a uniform diameter and a length that yield essentially the same self-impedance as the tapered diameter element. Thus, the substitute can be expected to behave--either by itself or in an array of elements--in the same way as the tapered element.
<p>The first method of calculating substitute elements was developed (to the best of my knowledge) by Jim Lawson, W2PV, in chapter 7 of his classic <i><b>Yagi Antenna Design</b></i>. Brian Beezley, K6STI, created improve algorithms for use in his NEC-Wires program. A quite different system was developed and explained by Dave Leeson, W6QHS, in chapter 8 of his equally classic <i><b>Physical Design of Yagi Antennas</b></i>. EZNEC, by W7EL, implements the Leeson correctives, as does NECWin Plus.</p>
<p>The Leeson correctives were calibrated against the MN (K6STI) version of MININEC. Indeed, MININEC is considered to be the standard of comparison for all tests of tapered-diameter calculations. It exhibits no problems with tapering schedules of all sorts. (A "tapering schedule" is simply where along an element the diameter changes and by how much.) We shall a bit farther on do some comparing of MININEC with both corrected and uncorrected NEC-2 outputs.</p>
<p>The basic principle of the substitute element is to derive an element with a uniform diameter and the same self-impedance as the original tapered- diameter element. The calculation method is accurate at and near element resonance, where the current distribution along the element is nearly sinusoidal (that is, nearly a sine wave in shape). This condition sets a number of limitations for the use of the corrective.</p>
<h3>Some Practical Boundaries for Substitute Elements</h3>In practical terms, the limitations are straightforward if we remember the what the calculation method requires.
<p>1. The tapered-diameter element must be within about 15% of resonance for the substitute element to be an accurate replacement. The near-resonance requirement applies to the physical length of the antenna without regard to loads, transmission lines, and sources that may be present. EZNEC, for example, cuts off the availability of the correction factor if the element does not meet this standard.</p>
<p>In antennas with many elements, only those meeting this (and other) constraints will have substitute elements calculated. Consequently, a tri-band beam might have the elements for one band (as determined by the frequency selection) substituted, but not those for other bands. This places a requirement for caution in using the results of such a model. For example, on some tri-band beams, the forward 20-meter director can affect performance on 10-meter, especially in limiting the upper frequency performance on that band. Depending upon beam design, the 20 meter director may carry significant current on 10 meters and be somewhat long as a full wavelength element on the upper band. However, it would not ordinarily have an uniform-diameter substitute calculated for it when the antenna is modeled at 10 meters.</p>
<p>2. If the tapered-diameter element is open at both ends (a dipole), it must be symmetrical about a center point along (but not necessarily on) one of the axes of the coordinate system. This requirement demands that the steps in a dipole's taper occur at equal distances from the center point of the element and in equal diameter steps for each corresponding step. EZNEC, for example, flags non-symmetries. In modeling, perhaps the chief reason for the absence of symmetry is transposing numbers when entering them in the wires table.</p>
<p>3. Loads, transmission lines, and sources are permitted only at the center point. Only one of each is permitted if the midpoint of the symmetrical element has no wire junction. Where the center is at a wire junction, split sources and loads may be used on segment immediately adjacent to the center-point junction. A load at a distance from the center--whether a resistance/reactance load, a resistance/inductance/capacitance load, or a shorted or open length of transmission line--would create an abrupt non-sinusoidal change in current magnitude along the element length, thus violating one of the conditions of accuracy. Center loads, transmission lines, and sources do not ordinarily disturb the current magnitude curve, although they may limit its highest value. Once more, EZNEC identifies misplaced loads, sources, and transmission lines.</p>
<p>Note that a serious loading coil used to shorten an element severely would ordinarily be used with an element already too short to pass the length test. Hence, such antennas could not access the corrective process.</p>
<p>For monopoles, connection to ground is essential for the substitution system to work. In these cases, the antenna element must be within length boundaries with respect to 1/4 wl resonance, and all source, load, and transmission line attachments must be to the segment immediately adjacent to the ground.</p>
<p>4. The taper-diameter element must be truly linear--that is, all wires making up the tapered-diameter element must be collinear. Ends must be open or, at most, one end may be grounded. Therefore, tapered diameter sections of elements that make up a complex geometry disable the corrective system. As a result, tapered-diameter elements used in Moxon rectangles, quad loops, capacity-hat dipoles/monopoles, and similar antennas cannot be directly modeled in NEC-2.</p>
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<img src="amod10-2.gif" width="580" height="299" border="2" vspace="5" hspace="9">
</center>
<p><b>Figure 2</b> summaries the rules by calling attention to what goes beyond normal boundaries. The upshot is that the corrective system is quite limited in its applications. However, the design of HF Yagi beams and other arrays with linear tapered-diameter elements is such a widespread application of NEC-2 that the corrective system is considered essential to most users.</p>
<h3>Some Practical Examples and a Subtle Limitation</h3>Let us look at some examples of the tapered-diameter correction factor in operation to further our familiarity with it and to gain some sense of what counts as reasonable expectations of it. We shall look at some wire tables drawn from the EZNEC implementation of NEC-2. Then we shall examine 5 sets of output reports: 1. the results of a MININEC version of the model in AO; 2. the results of uncorrected EZNEC NEC-2; 3. the results of corrected EZNEC NEC-2; 4. the results of creating a single-wire uniform element of the same length, diameter and total number of segments as the substitute group of wires in the corrected NEC-2 report; and 5. the results of (uncorrected) NEC-4. The AO models will have their center sections decreased by one segment to properly place the source at the junction of segments and at the element center. NEC-4 models hypothetically should need no correction. The models are all of the driven elements of large Yagi antennas, so their independent source impedances may not be quite resonant.
<p>1. A 14.175 MHz driven element of W6NGZ design:</p>
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<pre>W6NGZ Frequency = 14.175 MHz.
Wire Loss: Aluminum -- Resistivity = 4E-08 ohm-m, Rel. Perm. = 1
--------------- WIRES ---------------
Wire Conn.---End 1 (x,y,z : in) Conn.---End 2 (x,y,z : in) Dia(in) Segs
1 -205.95, 79.800, 0.000 W2E1 -156.00, 79.800, 0.000 6.25E-01 4
2 W1E2 -156.00, 79.800, 0.000 W3E1 -120.00, 79.800, 0.000 7.50E-01 3
3 W2E2 -120.00, 79.800, 0.000 W4E1 -72.000, 79.800, 0.000 8.75E-01 4
4 W3E2 -72.000, 79.800, 0.000 W5E1 72.000, 79.800, 0.000 1.00E+00 13
5 W4E2 72.000, 79.800, 0.000 W6E1 120.000, 79.800, 0.000 8.75E-01 4
6 W5E2 120.000, 79.800, 0.000 W7E1 156.000, 79.800, 0.000 7.50E-01 3
7 W6E2 156.000, 79.800, 0.000 205.950, 79.800, 0.000 6.25E-01 4
-------------- SOURCES --------------
Source Wire Wire #/Pct From End 1 Ampl.(V, A) Phase(Deg.) Type
Seg. Actual (Specified)
1 7 4 / 50.00 ( 4 / 50.00) 1.000 0.000 I
Ground type is Free Space</pre>
<p>This antenna is symmetrical in the X-axis. Leaving the Y coordinate at its original spacing from the reflector is no problem for implementing the correction factor. The AO model uses 12 segments in the long center element. Note that the center element occupies about 35% of the total element length. Here are the reports.</p>
<pre>Report Source Gain dBi Source Z (R+/-jX)
AO 2.13 77.0 + j 11.6
NEC-2 (No Cor.) 2.21 77.3 + j 23.6
NEC-2 (Cor.) 2.14 74.7 + j 11.9
NEC-2 (1-Wire) 2.14 74.7 + j 11.9
NEC-4 2.17 76.1 + j 12.9</pre>
<p>This 35-segment model shows only a slight displacement from the MININEC standard. The key difference is in the reactance report, which seriously displaces resonance in uncorrected NEC-2, while the corrected NEC-2 report coincides with the AO report. The 1-wire substitute element agrees exactly with the subdivided group that composes the substitute model of the element: both show a source resistance that is about 2.3 Ohms lower than the AO report. Finally, note that the NEC-4 report, while closer by far to the corrected NEC-2 and AO reports than to the uncorrected NEC-2 report, is still a bit high in gain and also in the reported source reactance.</p>
<p>2. A 14.175 MHz driven element of WB0DGF design:</p>
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<pre>WB0DGF Frequency = 14.175 MHz.
Wire Loss: Aluminum -- Resistivity = 4E-08 ohm-m, Rel. Perm. = 1
--------------- WIRES ---------------
Wire Conn.---End 1 (x,y,z : in) Conn.---End 2 (x,y,z : in) Dia(in) Segs
1 -170.00,-209.50, 0.000 W2E1 -170.00,-156.50, 0.000 4.38E-01 7
2 W1E2 -170.00,-156.50, 0.000 W3E1 -170.00,-132.50, 0.000 6.25E-01 3
3 W2E2 -170.00,-132.50, 0.000 W4E1 -170.00,-82.000, 0.000 8.75E-01 6
4 W3E2 -170.00,-82.000, 0.000 W5E1 -170.00,-36.000, 0.000 1.12E+00 6
5 W4E2 -170.00,-36.000, 0.000 W6E1 -170.00,-10.000, 0.000 1.25E+00 3
6 W5E2 -170.00,-10.000, 0.000 W7E1 -170.00, 10.000, 0.000 1.25E+00 3
7 W6E2 -170.00, 10.000, 0.000 W8E1 -170.00, 36.000, 0.000 1.25E+00 3
8 W7E2 -170.00, 36.000, 0.000 W9E1 -170.00, 82.000, 0.000 1.12E+00 6
9 W8E2 -170.00, 82.000, 0.000 W10E1 -170.00,132.500, 0.000 8.75E-01 6
10 W9E2 -170.00,132.500, 0.000 W11E1 -170.00,156.500, 0.000 6.25E-01 3
11W10E2 -170.00,156.500, 0.000 -170.00,209.500, 0.000 4.38E-01 7
-------------- SOURCES --------------
Source Wire Wire #/Pct From End 1 Ampl.(V, A) Phase(Deg.) Type
Seg. Actual (Specified)
1 2 6 / 50.00 ( 6 / 50.00) 0.707 0.000 V
Ground type is Free Space</pre>
<p>Two features of this element design (symmetrical in the Y-axis) are notable for our purposes. First, the center section (composed of three wires of 1.25" diameter) is half the length of the W6NGZ design, so that the first diameter step occurs closer to the source. Second, the diameter steps cover a greater range than the W6NGZ design, now ranging from 1.25" down to 0.4375" (in contrast to the W6NGZ range from 1.0" to 0.625"). Now let's look at the reports.</p>
<pre>Report Source Gain dBi Source Z (R+/-jX)
AO 2.13 76.2 + j 0.8
NEC-2 (No Cor.) 2.33 78.3 + j 30.9
NEC-2 (Cor.) 2.13 72.4 + j 1.7
NEC-2 (1-Wire) 2.13 72.4 + j 1.7
NEC-4 2.21 75.8 + j 6.8</pre>
<p>The gain and source reactance reports of AO and corrected NEC-2 are aligned, while the source resistance differs by about 3.8 Ohms, slightly more than the difference reported in the W6NGZ design. Note the high gain and inductive reactance difference for uncorrected NEC-2. Finally, note that NEC-4--for this more highly tapered-diameter model--shows a gain value that is no longer trustworthy.</p>
<p>3. A 14.175 MHz driven element of K6STI design:</p>
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<pre>K6STI Frequency = 14.175 MHz.
Wire Loss: Aluminum -- Resistivity = 4E-08 ohm-m, Rel. Perm. = 1
--------------- WIRES ---------------
Wire Conn.---End 1 (x,y,z : in) Conn.---End 2 (x,y,z : in) Dia(in) Segs
1 -203.50, 72.000, 0.000 W2E1 -138.00, 72.000, 0.000 5.00E-01 8
2 W1E2 -138.00, 72.000, 0.000 W3E1 -96.000, 72.000, 0.000 6.25E-01 6
3 W2E2 -96.000, 72.000, 0.000 W4E1 -48.000, 72.000, 0.000 7.50E-01 6
4 W3E2 -48.000, 72.000, 0.000 W5E1 -4.000, 72.000, 0.000 8.75E-01 5
5 W4E2 -4.000, 72.000, 0.000 W6E1 4.000, 72.000, 0.000 3.42E+00 1
6 W5E2 4.000, 72.000, 0.000 W7E1 48.000, 72.000, 0.000 8.75E-01 5
7 W6E2 48.000, 72.000, 0.000 W8E1 96.000, 72.000, 0.000 7.50E-01 6
8 W7E2 96.000, 72.000, 0.000 W9E1 138.000, 72.000, 0.000 6.25E-01 6
9 W8E2 138.000, 72.000, 0.000 203.500, 72.000, 0.000 5.00E-01 8
-------------- SOURCES --------------
Source Wire Wire #/Pct From End 1 Ampl.(V, A) Phase(Deg.) Type
Seg. Actual (Specified)
1 1 5 / 50.00 ( 5 / 50.00) 1.000 0.000 V
Ground type is Free Space</pre>
<p>This design uses a short, wide-diameter center section to simulate the effects of element-to-boom hardware. Apart from this short center section, the subdivision of the element is intermediate between the W6NGZ and WB0DGF designs.</p>
<pre>Report Source Gain dBi Source Z (R+/-jX)
AO 2.12 72.7 - j 13.1
NEC-2 (No Cor.) 4.66 42.1 + j 5.1
NEC-2 (Cor.) 2.16 68.9 - j 11.4
NEC-2 (1-Wire) 2.12 69.6 - j 11.5
NEC-4 3.06 59.1 - j 6.2</pre>
<p>This 51-segment element makes every effort to equalize segment lengths from wire to wire within the group, letting the 8" center length determine the segment lengths in the other wires. Even so, the element presses the correction factor limits, showing some difference in gain between the corrected NEC-2 result and the 1-wire substitute element. The effect of a fat short center segment is clearly apparent in the uncorrected NEC-2 result, with its very high gain and very low source impedance report. Note that with this type of segmentation, uncorrected NEC-4 yields quite untrustworthy results.</p>
<p>Despite the fact that even close attention to segmentation density yields corrected NEC-2 results that press the limits of the system, I encounter many carelessly segmented tapered-diameter elements. Perhaps the modelers believe that any old arrangement of segments adding up to more than 10 per half wavelength will do the job. However, the segments adjoining the source segment must be very close in length to the source segment itself. Moreover, it is always good practice to equalize segments lengths in any element. Let's see what happens if we carelessly segment the K6STI element.</p>
<p>3. A 14.175 MHz driven element of K6STI design (carelessly segmented):</p>
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<pre>K6STI Frequency = 14.175 MHz.
Wire Loss: Aluminum -- Resistivity = 4E-08 ohm-m, Rel. Perm. = 1
--------------- WIRES ---------------
Wire Conn.---End 1 (x,y,z : in) Conn.---End 2 (x,y,z : in) Dia(in) Segs
1 -203.50, 72.000, 0.000 W2E1 -138.00, 72.000, 0.000 5.00E-01 4
2 W1E2 -138.00, 72.000, 0.000 W3E1 -96.000, 72.000, 0.000 6.25E-01 3
3 W2E2 -96.000, 72.000, 0.000 W4E1 -48.000, 72.000, 0.000 7.50E-01 3
4 W3E2 -48.000, 72.000, 0.000 W5E1 -4.000, 72.000, 0.000 8.75E-01 2
5 W4E2 -4.000, 72.000, 0.000 W6E1 4.000, 72.000, 0.000 3.42E+00 1
6 W5E2 4.000, 72.000, 0.000 W7E1 48.000, 72.000, 0.000 8.75E-01 2
7 W6E2 48.000, 72.000, 0.000 W8E1 96.000, 72.000, 0.000 7.50E-01 3
8 W7E2 96.000, 72.000, 0.000 W9E1 138.000, 72.000, 0.000 6.25E-01 3
9 W8E2 138.000, 72.000, 0.000 203.500, 72.000, 0.000 5.00E-01 4
-------------- SOURCES --------------
Source Wire Wire #/Pct From End 1 Ampl.(V, A) Phase(Deg.) Type
Seg. Actual (Specified)
1 1 5 / 50.00 ( 5 / 50.00) 1.000 0.000 V
Ground type is Free Space</pre>
<p>The element is obviously the same as the one just run, but the segmentation has been cut to 25 segments total.</p>
<pre>Report Source Gain dBi Source Z (R+/-jX)
AO 2.12 72.7 - j 13.1
NEC-2 (No Cor.) 5.04 38.2 + j 2.5
NEC-2 (Cor.) 2.68 61.1 - j 10.8
NEC-2 (1-Wire) 2.12 69.5 - j 12.0
NEC-4 3.93 48.2 - j 5.8</pre>
<p>The AO numbers have not changed, but the only NEC report that shows sensible values is the 1-wire substitute. Both sets of figures are consistent with those for the carefully segmented version of the element. Both the raw NEC-2 and NEC-4 values are sufficiently off the mark to be completely unusable.</p>
<p>Of great importance is the fact that the corrected NEC-2 figures for this modeled element are also beyond the limits of reliability by a considerable margin. In previous models, we watched the margin of error in corrected NEC-2 reports grow, even in well-segmented elements. In this model, the corrected NEC-2 model and the 1-wire substitute have the same diameter, length, and total number of segment. The only difference between them is the length of segments, especially around the source point: the NEC-2 corrected model has very unequal segment lengths, while the segment length of the 1-wire substitute is uniform throughout the element. This demonstration shows that careful attention to segmentation of tapered-diameter elements is vital to deriving reliable results from the correction system.</p>
<h3>Some Practical Conclusions</h3>This episode has been light on the graphics and heavy on the tables, but for a purpose. The tables show the data that can help us improve our modeling techniques when employing the tapered-diameter correction factor implemented in various versions of NEC-2.
<p>The first step is to carefully model the tapered-diameter element so that it meets the basic criteria for implementing the correction system.</p>
<p>The second step is to check to ensure that the correction has been implemented. It is easy--despite programmers' best efforts--to overlook some error in the model set-up that may have prevented the automated correction system from working.</p>
<p>Third, even models that pass the basic tests require careful attention to segmentation to ensure that segments are as equal as the structure will permit.</p>
<p>Remember that errors tend to be cumulative so that unrealistically high gain reports on individual elements usually result in very significant over-estimates in the gain of an array within which the tapered-diameter elements are used. While such over-estimates may be momentarily pleasing to the designer's ego, they can be most assuredly embarrassing if caught by someone else.</p>
<p>Finally, even if you have access to NEC-4, do not assume that it is accurate in all cases. While NEC-4 improves on the performance of NEC-2 with respect to tapered-diameter elements, it has limitations which suggest that the correction system should also be used with it in all but the simplest tapered-diameter models.</p>
<p>The tapered-diameter element correction system is a valuable feature in any version of NEC-2 that implements it. However, it deserves to be used with both caution and care.<br>
<br></p>
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<h1>100. The Dipole and the Coax</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>The most usual way to feed a resonant dipole uses a simple coaxial cable. Most basic handbooks and texts recommend that the system builder insert a common-mode choke at the feedpoint--between the dipole feedpoint terminals and the coaxial cable. Readily available forms of common-mode choke include a transmission line transformer with a 1:1 impedance ratio and a ferrite-bead collection following the designs proposed initially by Walt Maxwell (W2DU). In either case, the device establishes a compatibility between the balanced feedpoint of the dipole and the seemingly single-ended construction of the coaxial cable.</p>
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<p><b>Fig. 1</b> presents one traditional way to portray the situation at the dipole feedpoint. Its general purpose is to show why the insertion of a balun is important as a precautionary measure in dipole construction. By extension, the situation applies to any split element fed by coaxial cable. The dipole is merely the most fundamental case.</p>
<p>At any given instant, the currents between the coaxial cable center conductor and the inner side of the shield or braid are equal and opposite. However, according to the graphic, once the cable terminates at the dipole feedpoint, the current has multiple paths of travel. The diagram suggests that the center conductor has only a single path: the left leg of the dipole within the figure. However, the current from the braid has 2 paths: the right leg of the dipole in the figure and the outer side of the coaxial cable braid. The function of the choke is then to attenuate so far as possible the current that would appear on the coaxial cable braid outer side. It performs this function largely by introducing a large inductive impedance to such currents. The function is similar to that of an RF choke within an amplifier circuit. However, the arrangement must have a form that does not disturb the currents between the center conductor and the inner side of the cable braid.</p>
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<p>As suggested by <b>Fig. 2</b>, we then have 2 sets of currents to consider. The left portion not only portrays the current as directional, but also indicates the field between the center conductor and the inner side of the braid, so that we have at any point of measurement currents of equal magnitude but opposite polarity or phase angle. These are transmission-line currents in the conventional sense. If we replace the resonant antenna with a resistor of vanishingly small dimension (but still capable of converting the RF energy into heat without self-destruction), we should measure only transmission line currents at any measurement point. If we place a complex load of similarly small lumped components at the cable end, we shall obtain the same results, although the lack of a match between the cable characteristic impedance and the load will alter the pattern of current values along the line.</p>
<p>On the right in <b>Fig. 2</b>, we have the common-mode currents that appear on the surface side of the coaxial cable braid. Common-mode currents in theory may derive from either conductor, but always appear on the coaxial cable outer side due to skin effect. Therefore, in theory, the current on the braid outside side is the sum of currents other than transmission line currents on the entire coaxial cable structure. Since the transmission-line currents are equal in magnitude but opposite in phase angle, they cancel. The common mode currents are the remainder, whatever their source. Because common mode currents appear on the braid outer side, they are capable of radiation, just as the current on the antenna legs proper. Because those currents may appear all along the coaxial cable length, they may also be found at the transmitting equipment, where the cases form an irregular extension of the coaxial cable outer side. (Note: many writers would simply refer to the coaxial cable braid outer surface. However, at any frequency, there is always some depth to the current-conducting portion of the braid. Hence, I have used the term "outer side" instead. The depth of penetration, of course, is a function of frequency.)</p>
<p>The feedpoint of a dipole element represents a small gap in the antenna. Between the terminals of the gap, the feedline provides a series source of energy for the antenna, thus completing the path between those terminals. This very basic fact is important, because it drives the conventional method of trying to model the effects of common mode currents within both NEC and MININEC antenna modeling software. Neither software core is capable of physically modeling conventional coaxial cables. The transmission line function within NEC creates lossless non-radiating mathematical models of lines and hence cannot capture common mode radiation. Therefore, the method used to show common mode radiation is to place a third leg into the dipole. Its feedpoint end connects as closely as possible to one side of the feedpoint segment or pulse, depending upon the software used.</p>
<p>In these notes, we shall not question the appropriateness of the model as a means for effectively modeling common-mode currents on a coaxial cable feedline. That discussion belongs to another context. Within the prescribed modeling effort, there are a number of modeling issues that deserve review.</p>
<p><b>Modeling the "Coax" Wire With a Dipole</b></p>
<p>Therefore, let's begin with a simple dipole that is resonant at 29.97925 MHz, where 1 wavelength = 10 meters. If we use 1-mm diameter wire and make the dipole 0.485 wavelength (4.85 meters), the antenna will show a free-space resonant impedance in both NEC and MININEC. The MININEC model will use 30 segments so that the center feedpoint falls on a pulse. The corresponding NEC model uses 31 segments so that its center feedpoint falls at the center of a segment. Both models show a free-space gain of 2.14 dBi. The reported MININEC source impedance is 71.84 - j0.56 Ohms, while the reported NEC-4 source impedance is 71.99 - j0.43 Ohms. The two values are close enough to qualify the models as the same for the purposes of the exercise to follow. The MININEC software used here is Antenna Model, while the NEC-4 software is EZNEC Pro/4.</p>
<p>We shall try to model the coaxial cable using a 6.35-mm (1/4") diameter wire connected as close to the model source as possible. The conductor size corresponds roughly to the outer diameter of the braid in such cables as RG-58 and RG-8X. For the exercise, we shall use a third-wire length of 0.25 wavelength (2.5 meters). For the purposes of the exercise, the wire will run from its connection point straight downward, relative to a horizontal dipole. In modeling terms, we construct the dipole proper in the X-Y plane, with the third wire representing the coaxial cable modeled along the Z axis. <b>Fig. 3</b> shows--for the MININEC model--the difference between the simple dipole and the dipole plus its "coax" wire. The outline of the NEC model would be similar.</p>
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<p>The diagram does list the ends of each wire. That aspect of the figure will change as we move from one model to another. <b>Fig. 4</b> shows why, at least in part.</p>
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<p>The MININEC model places a source on a pulse, which occurs at the junction of two segments or wires. Since a junction of segments or wires contains an ambiguity relative to which of the segments has the pulse, the convention is to place the pulse on the higher-number segment (using an absolute segment count). Hence, the source pulse appears in the MINNEC model in the middle of <b>Fig. 4</b> on the second or right-leg wire of the dipole. In order to calculate current correctly, we must bring both wire 1 (the left dipole leg) and wire 3 (the coax wire) together so that both wires have end 2 at the junction of wire 2, end 1.</p>
<p>The model that we have just described brings the coax wire as close to the source as is possible. Presumably, this procedure adheres most closely to reality, as earlier described. The alternative modeling procedure in the lower part of <b>Fig. 4</b> creates a 2-segment center wire for the dipole itself. The coax wire connects to one side of the center wire. We shall use this model only briefly to make a comparison near the end of these notes.</p>
<p>In a NEC model, we cannot connect a coax wire directly to the source point. As the upper part of <b>Fig. 4</b> shows, the source occupies a segment, and by convention, we mark this fact by placing the source indicator in the center of the segment. At best, we must connect the coax wire to the segment or the wire junction occurring at one or the other end of the source segment. Since every segment has a definite length, the coax wire junction will be offset from the true center of the dipole. Part of the exercise will be to explore the effects of that offset. In this case, the source appears on the last segment of wire 1, which extends past the dipole center so that the source segment is centered within the overall dipole length.</p>
<p>NEC is sensitive to having the source segment be equal in length to the adjacent segments. The closer we can come to meeting this condition, the more accurate will be the reported source impedance. Hence, for both MININEC and NEC models, the goal was to use segment lengths as close to equal as feasible within the overall 30/31 segment structure of the 4.85-meter dipole.</p>
<p><b>Some MININEC Results</b></p>
<p>The MININEC model, as suggested by <b>Fig. 4</b>, uses 3 wires. The 1-mm diameter dipole wires are equal in length, and each has 15 segments. The length of the individual segments is 161.667 mm. The 6.35-mm-diameter coax wire is 2.5 m long and also uses 15 segments. Hence, the individual segments are 166.667 mm long, about 5 mm longer than those in the dipole.</p>
<p>The following table summarizes the reported characteristics of both the dipole alone (reference to "dpl1" on graphics) and the dipole with its coax wire added (reference to "dpl4" on graphics). AGT refers to the average gain test value.</p>
<pre>
MININEC Models: Standard Dipole vs. Dipole Plus Coax Wire: Free Space
Model Maximum Free-Space Source Impedance AGT
Gain dBi R+/-jX Ohms
Dipole 2.14 71.84 - j 0.56 0.9999
Dipole Plus 2.04 45.68 - j12.85 0.9994
</pre>
<p>The currents on each side of the source segment of the standard dipole are, of course, exactly equal for the symmetrical element. However, for the dipole + coax wire model, we find the following values. W1e2 means wire 1, end 2, etc. W2e1 is the source pulse for the antenna.</p>
<pre>
Relative Current Levels at the Dipole + Coax Wire Feedpoint: MININEC
Current Components Current Magnitude/Phase Angle
Wire/End Ireal Iimag Imag Iphase
W2e1 (SO) 1.00000E+00 0.00000E+00 1.00000E+00 0.000
W1e2 3.01968E-01 -3.47609E-01 4.60453E-01 -49.019
W3e2 6.98032E-01 3.47609E-01 7.79795E-01 26.473
</pre>
<p>The real and imaginary components of the non-source wire ends add up to equal the current value of the source-wire end. The model, then, carries with it the presumption that the current on one side of the source pulse divides between the two existing wires on the other side. The wires are all lossless in the models. The higher current on the coax wire is a function of its greater diameter, its greater length, and the non-linearity of its direction relative to the dipole. MININEC does not have the NEC limitation relative to junction of wires having different diameters, as indicated by the AGT score of the dipole + coax wire model. Hence, the model's reported values require no correctives.</p>
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<p><b>Fig. 5</b> presents the 3-D patterns in free space for both the simple dipole and the dipole + coax wire models. The simple dipole "donut" pattern is useful for reference in gauging the differences that are part of the dipole + coax wire pattern. To what degree the pattern bulges appear in the pattern of the antenna over a real ground requires that we revise the model. So I moved the dipole to place it 1 wavelength (10 meters) above average ground (conductivity 0.005 S/m; relative permittivity 13). The open end of the coax wire is now 7.5 m above ground. The following table summarizes the performance data for the simple dipole vs the dipole + coax wire.</p>
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MININEC Models: Standard Dipole vs. Dipole Plus Coax Wire: 1 WL Above Average Ground
Model Maximum TO Angle Source Impedance
Gain dBi degrees R+/-jX Ohms
Dipole 7.65 14 70.29 - j 5.62
Dipole Plus 6.68 14 45.11 - j14.75
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<p><b>Fig. 6</b> presents azimuth and elevation patterns for both antennas. In order to gather a feel for the maximum gain reduction associated with the dipole + coax wire model, focus on the pattern places marked "Note." In the azimuth pattern, note the shallower nulls off the dipole ends, with the side toward the coax wire being shallower than the opposite side of the pattern. As well, in the elevation pattern, note the shallower null between elevation lobes. Energy that creates these shallower nulls is not available in the main bi-directional lobes that determine the maximum gain of the antenna.</p>
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<p><b>Some NEC Results</b></p>
<p>As shown at the top of <b>Fig. 4</b>, the NEC model must have a slightly different structure relative to the MININEC model. The standard dipole uses 31 segments, each 156.452 mm long. The source is on segment 16 at the exact center of the dipole. When we add a coax wire, it must be at a junction of wires or segments. For simplicity within EZNEC, I cut the dipole wire into two pieces. Wire 1 is 2.503226 m long and has 16 segments. The source is on the last segment of this wire. Wire 2 is 2.346774 m long and has 15 segments. As a result of this division, the segment length remains unchanged and is the same on both wires. The coax wire, wire 3, begins at the junction of wires 2 and 3 and runs downward for 2.5 m. It uses 16 segments, each of which is 156.25 mm long, very close to the length of the segments in the dipole wires.</p>
<p>The dipole + coax wire model contains two problematical features. First, the junction of the coax wire and the dipole wire is displaced from the exact dipole center by half the length of a model segment. Second, the coax wire, with a diameter of 6.35 mm, differs from the 1-mm diameter of the dipole wires, creating an angular junction of wires with dissimilar diameters. NEC has a known limitation in such cases, and the error is greater in NEC-2 than NEC-4. The following table summarizes the free-space performance reports of the 2 cores for both the simple dipole and the dipole + coax wire models. There is no significant difference between NEC-2 and NEC-4 with respect to the simple dipole.</p>
<pre>
NEC Models: Standard Dipole vs. Dipole Plus Coax Wire: Free Space
Model Maximum Free-Space Source Impedance AGT AGT-dB Corrected
Gain dBi R+/-jX Ohms Gain dBi
Dipole 2.14 71.99 - j 0.43 1.000 +/-0.0 2.14
Dipole Plus
NEC-4 1.72 44.95 + j16.99 0.958 -0.19 1.91
NEC-2 1.66 45.70 + j17.33 0.943 -0.25 1.91
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<p>The source resistance of the models is close to the value for the corresponding MININEC value. However, the reactive component of the source impedance is about 30 Ohms distant from the MININEC value. Therefore, I created the alternative MININEC model shown in <b>Fig. 4</b>. In this model, the junction of the coax wire is 1 pulse/segment away from the source. All of the segments in the dipole portion of the model have the same length. Placing the junction another half-segment-length away from the source has interesting consequences. First, the free space gain of the model is 1.92 dBi, virtually the same as the corrected gain values for the NEC-2 and NEC-4 models. Second, the source impedance report is 45.99 + j32.94 Ohms. The resistive component has not changed by much, but the reactive component is considerably more inductive than the NEC reports. The junction position of the coax wire with the dipole wire, relative to the source, appears to make a consistent and systematic difference to the reported reactance at the source.</p>
<p>The current reports for the NEC model also differ from those emerging from MININEC. NEC currents refer to specific segments, with the segment center taken as the virtual read-out position. The source segment current by assignment was 1 A RMS (the convention used within EZNEC software). Since the NEC core uses peak values, the NEC output report shows a corresponding value of 1.4142E0 as the peak value. The comparison of values must use the segment current on each side of the source segment. Hence, the perfect addition that we experienced with the MININEC model is not likely to appear. The question is to what degree we find the current division holding to the MININEC model standard. The following table tells us some of the story. The source is located on wire 1, segment 16. Wire 1, segment 15 is the adjacent segment on the dipole side without the coax wire. Wire 2, segment 1 is the first segment of the remainder of the dipole that meets with wire 3, segment 1, the first segment of the coax wire. The values are for NEC-4</p>
<pre>
Relative Current Levels at the Dipole + Coax Wire Feedpoint: NEC-4
Current Components Current Magnitude/Phase Angle
Wire/Segment Ireal Iimag Imag Iphase
W1s16 (SO) 1.4142E+00 -3.4217E-16 1.4142E+00 0.000
W1s15 1.4047E+00 -7.9009E-03 1.4047E+00 -0.322
W2s1 3.0643E-01 -3.9072E-02 3.0891E-01 -7.226
W3s1 1.1108E+00 3.0984E-02 1.1112E+00 1.598
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<p>The sum of the coax-side real current components is 1.4172E+00, which is only slightly higher than the real current on W1S15. The sum of the imaginary components is -8.088E-03, very close to the imaginary component on W1S15. Within the limits of the model adequacy, as indicated by the less-than-perfect AGT score, the currents add up correctly to indicate a current division on the coax-wire side of the dipole. However, relative to the MININEC model, we find two anomalies. First, the ratio of coax wire current to dipole wire current in the NEC-4 model is about 3.62:1, whereas in the MININEC model, the ratio is about 2.31:1. This difference is likely due to a combination of the position of the wire junction relative to the source position and the error introduced by the junction of wires with differing diameters.</p>
<p>The second anomaly between the cores is less easily explained. The MININEC model showed widely divergent phase angles between the currents on the dipole and coax wires: -49 degrees and +26.5 degrees, respectively. The NEC-4 model shows only a small divergence of the same current phase angles: -7.2 and +1.6 degrees, respectively. It is likely that the lack of coincidence between the wire junction and the source forms the ultimate reason for the small difference in phase angles within the NEC-4 models. The lack of coincidence of the phase angles would require careful measurement of an actual antenna situation in order to decisively tell us which modeling core provides the more realistic report.</p>
<p>Over real ground, using the same conditions as in the MININEC model, we obtain similar results, as shown by the following table. The dipole + coax wire model uses NEC-4 data.</p>
<pre>
NEC Models: Standard Dipole vs. Dipole Plus Coax Wire: 1 WL Above Average Ground
Model Maximum TO Angle Source Impedance
Gain dBi degrees R+/-jX Ohms
Dipole 7.64 14 70.68 - j 5.64
Dipole Plus 6.08 14 45.28 + j14.69
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<p><b>Fig. 7</b> provides comparative simple dipole and dipole + coax wire azimuth and elevation patterns. The simple dipole patterns--as well as the tabular data--are virtually identical to those for the corresponding MININEC model. The gain data for the NEC model of the dipole + coax wire is 0.6-dB lower than in the MININEC model, a value that exceeds the free-space AGT correction factor. However, the pattern elements that reduce maximum gain relative to the simple dipole are evident in the figure. Of course, due to the position of the source relative to the wire junction, the azimuth pattern shows a reversal between the deeper and shallower nulls off the dipole ends. The patterns also show the relative strengths of the vertical and horizontal components of the total far field.</p>
<p><b>Some Tentative Conclusions</b></p>
<p>The test models use an artificial situation to allow some detailed comparisons between MININEC and NEC models of a dipole + coax wire. The MININEC version of the model, using the highly corrected Antenna Model implementation of the core, still requires careful construction in order to effectively model the coax wire as an alternative path for currents right at the feedpoint of the dipole. NEC has several limitations that prevent such exacting models, including the source placement within a segment rather than at its junction and the weakness of its ability to handle angular junctions of wires having dissimilar diameters.</p>
<p>The results show both areas of correspondence and divergence. The resistive component of the source impedance shows a good correlation between models, although the reactive component is apparently sensitive to the position of the wire junction relative to the source. Gain and pattern values are comparable, if we allow for the imperfect AGT scores of the NEC models. Perhaps the greatest divergence appears in the reported current phase angles on the joining coax and dipole wires, along with the reported ratio of currents in each wire.</p>
<p>The exercise has aimed to compare NEC and MININEC models attempting to capture the coax braid as an alternative path for antenna currents. As such, it sets up an artificial and simplified situation. Any complete model of the situation must include all of the factors shown in <b>Fig. 8</b>. Indeed, the list of factors is incomplete, but suggests that modeling the coax wire is not a simple task. Any results that presumes an open-ended wire termination will be suspect with respect to reality. Even a high-impedance choke or balun added to a line at the end of a 1/4 wavelength coax run will not necessarily terminate the common-mode currents, since the antenna element impedance is also very high at that point. Nothing short of a complete model will do for modeling an actual situation.</p>
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<p>Throughout, I have referred to the currents on the outer side of the coax braid as common mode currents. It is not wholly clear that the usage is correct, if we conceive of such currents as an alternative path for the current on the inner side of the braid at the junction with the antenna feedpoint. However, since common-mode currents will appear only on the outer side of coax braid, they are in some respects indistinguishable from what we might otherwise call "alternate-path" currents.</p>
<p>Whether the alternate-route portrayal of these currents is complete in itself may rest on the measured feedpoint impedance of the dipole with the coax attached. (For a related set of experiments measuring dipole element and coax braid currents, see Roy Lewallen, W7EL, <a href="https://archive.org/details/arrlacv3/ARRL_AC_v1/page/n154/mode/1up" target="_blank">"Baluns: What They Do and How They Do It,"</a> <i>The ARRL Antenna Compendium</i>, Vol. 1, pp. 157-164.) In the test situation, we find a very large difference in the resistive components between the simple dipole and the dipole + coax wire models, regardless of the modeling software used. Confirmation of the simple alternate path scenario is as simple as measuring the impedance in the coaxial cable at the termination of the 0.25 wavelength line (without introducing any paths to ground or other disruptive conditions) and then back calculating to the actual feedpoint impedance at the dipole feedpoint.</p>
<p>The situation does demonstrate that modeling is not an end in itself. In many situations, the results of modeling require experimental confirmation, if only to show that a model either is or is not a model of the situation under analysis. Unlike the present model set--with the definite difference between the source resistance of a simple dipole and of the dipole + coax--not all proposed models present clear cut cases for deciding whether or not the model captures a given electrical situation.</p>
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<h1>101. Modeling the Un-Modelable</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>An alternative title for this episode might be <b>"The Intimate Connection Between Modeling and Measuring--Both Before and After Modeling."</b> 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.</p>
<p><b>NEC Limitations</b></p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p><b>Types of Models</b></p>
<p>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.</p>
<p>1. <i>Design Models</i>: 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.</p>
<p>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 <b>Fig. 1</b>. 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.</p>
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<p>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.</p>
<p>2. <i>General Guidance Models</i>: 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.</p>
<p>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.</p>
<p>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 <b>Fig. 2</b> for a sample pattern.</p>
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<p>3. <i>Proof-of-Principle Models</i>: 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 <b>Fig. 3</b> 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.)</p>
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<p>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 <b>Fig. 4</b>. 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.</p>
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<p>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.</p>
<p>4. <i>Suggestive Models</i>: 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.</p>
<p>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 <b>Fig. 5</b> 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.</p>
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<p>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.</p>
<p><b>Physical Post-Modeling Testing</b></p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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 <b>Table 1</b>.</p>
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<p>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.</p>
<p><b>Pre-Modeling Physical Testing</b></p>
<p>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. <b>Fig. 6</b> 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.</p>
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<p>The questions of NEC's ability to model a structure emerge from situations other than the simple insulated wire. <b>Fig. 7</b> shows some typical cases that have often occasioned e-mail questions.</p>
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<p>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.</p>
<p>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.</p>
<p>The situation of the center sketch in <b>Fig. 7</b> 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.</p>
<p>The right-most portion of <b>Fig. 7</b> 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.</p>
<p>There are methods for creating a conversion table between the structures on the left and right in <b>Fig. 7</b> 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.</p>
<p><b>Fig. 8</b> 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.</p>
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<p>Prior to modeling, we may calibrate any of these materials to a selected round-wire diameter by using the dipole-resonance technique. <b>Table 2</b> 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 <b>Table 1</b>.</p>
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<p>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.</p>
<p>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.</p>
<p>Desite 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.</p>
<p><b>Conclusion</b></p>
<p>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.</p>
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<h1>102. True Azimuth Models<br>
NSI Software</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>After a short time in modeling with NEC, the core conventions become almost second nature. You model the antenna geometry using Cartesian conventions for each wire-end coordinate set. The core data produces phi and theta patterns. A phi pattern counts degrees counterclockwise. A theta pattern counts angles from the zenith downward toward the horizon. The pattern conventions are just the opposite of everyday and field engineering conventions. The latter use azimuth angles, generally counted from 0 at North clockwise. Elevation angles count from the horizon upward toward the zenith. The 2 systems appear in <b>Fig. 1</b>. Phi and theta angles are inside the circles, while azimuth and elevation angles are outside.</p>
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<p>Software implementations of NEC use various means of presenting polar plots. Virtually all commercial implementations of NEC do an easy conversion of theta plots into elevation plots. They simply subtract the theta angle from 90 to obtain an elevation angle. Azimuth patterns are a bit more difficult to handle. For example, EZNEC simply uses the phi conventions, but calls its plots "azimuth." There is a compass plot that we shall work with in a subsequent episode.</p>
<p><b>Simple Rotational Models in NEC-Win Plus</b></p>
<p>NSI products (NEC-Win Plus, NEC-Win Pro, and GNEC) handle the polar-plot situation in a different manner. They offer the user the option of pairing phi and theta plots or of selecting azimuth and elevation plots. Elevation plots convert theta plots by the usual method. Phi and azimuth plots use the outer-ring angle markings appropriate to each pattern, as shown in <b>Fig. 2</b>.</p>
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<p>By placing 0 degrees at the top of the plot, NSI polar plots can use the very same data, angle-by-angle, for both plots. Whether the plot on the right is a true azimuth pattern depends in part on the symmetry of the pattern. For antennas that produce symmetrical patterns, we cannot tell a true azimuth model from one that is casually modeled in the Cartesian coordinate system.</p>
<p>There are some modeling tasks that require not just patterns that bear azimuth labels, but as well, true azimuth models. The NSI VOACAP output depends on using compass or azimuth bearings. Broadcast antennas make FCC submissions using azimuth concepts both to describe the antenna and to provide sample patterns. These submissions may include MW BC towers and antennas, or they may involve one or more antennas on a tower--and each such antenna may point in a different direction on the compass. No less complex are some of the fields of antennas used by government, military, and even advanced amateur installations. Hence, it may pay to learn how easy it is to create true azimuth models to make use of the polar plot labels to yield true azimuth patterns. We shall move from that point to creating true azimuth models.</p>
<p>The process begins by attending to the correlation within NEC of the Cartesian coordinate system to the phi angles (and as a direct consequence, to azimuth angles). Consider <b>Fig. 3</b>.</p>
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<p>The circle on the right is identical to its right-side counterpart in <b>Fig. 1</b>. The circle on the left provides the connection between the coordinates used to set up the model and the polar plots. Zero degrees always corresponds to values extended along the +X axis of the coordinate system. If we select a phi pattern, then the pattern angles count counterclockwise toward 90 degrees phi or toward the +Y axis. The values proceed around through 180 and 270 degrees phi before return home to zero degrees.</p>
<p>However, if we wish a true azimuth pattern, then we must proceed clockwise in the polar plot, with zero degrees representing North. Moving in a clockwise position toward East or 90 degrees azimuth, we end up at values extended along the -Y axis. From that point, we proceed to South or the -X axis, and further to West or the +Y axis. Finally, we return home to North and zero degrees once more.</p>
<p>The trick to obtaining a true azimuth model is to set up the geometry adhering to the directions that will eventually yield a true azimuth pattern for the result. For antennas that have symmetrical patterns, such as the pair shown in <b>Fig. 2</b>, the process is simple: extend the boom or direction of radiation from the -X toward the +X direction. That will align the model and pattern toward North, with identical pattern features on both side of the North-South line. <b>Fig. 4</b> shows a NEC-Win Plus version of a model designed precisely in accord with this instruction. In fact, it is the basis for the patterns in <b>Fig. 2</b>.</p>
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<p>The values in the X1 and X2 columns show that the antenna structure proceeds from a reflector at X=0 toward a director at X=186 (inches, in the case of this 14-MHz Yagi). The elements are linear and extend equally toward +Y and toward -Y. The result is a true azimuth model, and so the azimuth pattern is also true.</p>
<p>Suppose that we need the antenna pointed Northeast, that is, at 45 degrees azimuth or compass bearing. The procedure is simple and may use more than one means. Programs like NEC-Win Pro and GNEC give access to the GM command. By specifying a 45-degree rotation around the Z-axis, we can effect the change of heading, so long as we remember to rotate clockwise. Since GM rotation would follow the phi conventions, we would specify the rotation angle as -45 degrees.</p>
<p>NEC-Win Plus (which is also an "insert" within both Pro and GNEC) uses a different method. There is a main screen rotation control labeled with a circular arrow. To use this control, we first block the entire antenna geometry, that is, the entire set of wire entries on the main screen. Then, we click on the rotation button to open a screen that is similar to the help screen in Pro and GNEC for the GM command. One option is to rotate the blocked wires around the Z-axis by a specified amount. For the antenna to point Northeast, we select a rotation angle of +45 degrees. <i>Note: the NEC-Win Plus rotation control around the Z-axis operates in accord with azimuth conventions, not in accord with phi conventions.</i>.</p>
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<p><b>Fig. 5</b> shows the resulting true azimuth pattern for the rotated beam. By comparing the original and new patterns, we can easily see the success of our maneuver. The result is not only a correct or true azimuth pattern for the antenna, one that includes the correct angular labels. As well, we have a true azimuth model. <b>Fig. 6</b> shows the wire coordinates of the resulting beam.</p>
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<p>Although inspection may initially make the numbers seem to be a jumble of coordinate values, you may do a little mental math on the center wires for each complete element (wires 4, 19, and 18) to see that the beam's boom extends along a line that is 45 degrees from either axis and that the elements extend at right angles to the boom. On one side of the boom, the end-1 coordinates extend toward +X and +Y, and on the other side, the end-2 coordinates extend toward -X and -Y. Remember that these are directions and hence relative. End-1 coordinates are simply more positive and less negative than end-2 coordinates.</p>
<p>Suppose that we had 2 identical beams in a stack, one above the other. In such a case, we might wish to see the outcome of rotating one or both beams until there is some desired angular separation in the boom directions. Normally, a stack of Yagis will have each antenna mounted so that the center of mass equalizes the boom-forward and boom-rearward moments. The mounting center will be very close to, but usually not precisely at the center of the boom, as measured from the rear-most element to the forward-most element. For most purposes. we may model the mast position as the boom center.</p>
<p>To reposition each antenna with a boom-center at coordinates 0,0, we can use one of several procedures. The most straightforward would be to subtract one-half of the boom length (0.5*186 or 93 inches) from each X dimension while the boom is still aligned North and South. A second alternative is the use the NEC-Win Plus translation control and effect a block movement of the same amount on all wires in the model. A third way to the same goal is to use the GM command (if available) to effect a translation along the X-axis by the same amount. If we wish to move the antenna along the boom-line and to rotate it 45 degrees by using the GM command, we must use 2 separate GM commands. The GM command rotates before it translates, but our goal is to translate before we rotate. Hence, we cannot combine the two movements into a single command.</p>
<p>To create a stack of 2 Yagis from our original model (<b>Fig. 4</b>), we shall illustrate the process by block copying the first antenna and pasting the result below the first 21 wire lines. To effect a vertical separation (arbitrarily 800" for the example), we set the Z-value for 21 wires to that number--or we may use the translation facility to make that move. Next, as shown by the left side of <b>Fig. 7</b>, we shall block all of the wires and move or translate both antennas -93 inches so that each is centered on its boom along the X-axis. The final step, shown on the right in <b>Fig. 7</b>, is to rotate one of the antennas. We block the wire lines for the antenna or 21-wire set of choice and enter the rotation as 45 degrees, remembering that in NEC-Win Plus, the rotation control system operates clockwise.</p>
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<p>The result of the work appears on the right side of <b>Fig. 8</b>. Note that the boom center is close to but not on the coordinate system center (0,0). Had we gotten the order of operations reversed and rotated before translating, we might have stacked the antennas as shown on the left in <b>Fig. 8</b>. The result might not have produced serious errors in this case, but in other cases it might yield very wrong results.</p>
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<p><b>True Azimuth Models in NEC-Win Plus</b></p>
<p>Not all efforts to create a true azimuth model and plot combination are quite so simple as rotating an antenna having a symmetrical pattern. The are many modeling problems in which we begin the process with one or another form of azimuth data describing the geometry of the antennas. We shall explore a couple of simpler cases in order to maintain clarity on the principles involved. Suppose that we had a set of 3 monopoles that are 1/4 wavelength at 1 MHz. Since a wavelength at 1 MHz is 300 m, the monopoles are each 75-m long. Although a real-world exercise might include a buried radial system for each monopole, we shall use a perfect ground for our exercise. As well, the individual antennas have an arbitrary diameter of 0.1 m and use a conductivity appropriate to steel. Also for simplicity, we shall feed the individual antennas in phase.</p>
<p>In many situations, one monopole may serve as a key against which we determine the positions of the others. We might receive a data list such as the following.</p>
<pre>
Tower Distance Azimuth Bearing
Number Meters Degrees
1 --- ---
2 150 060
3 300 060
</pre>
<p>We shall read this data by using tower 1 as the key, with the subsequent distances and bearings referenced to it. Without much difficulty, we recognize that this system forms a set of 3 monopoles fed in phase, with each monopole spaced by 1/2 wavelength from an adjacent monopole.</p>
<p>NEC, and most commercial NEC input system, do not allow inputs using distances and headings derived from azimuth or compass data. We still need to translate this data into X and Y coordinates that create a true azimuth model. (Of course, we also need to handle the Z-coordinates, but they will each be 0 at end 1 and 75 at end 2 with meters as the unit of measure.) 60 degrees lies in the first azimuth quadrant. Hence, the towers will form a line between North and East, that is, between the +X and the -Y axes.</p>
<p>We define the extension along the +X axis by the cosine of 60 degrees (0.5) and the extension along the -Y axis by the sine of 60 degrees (0.866). These two simple trig operations allow us to translate the original table so as to yield corresponding coordinates for each data entry.</p>
<pre>
Tower Distance Azimuth Bearing +X coordinate -Y coordinate
Number Meters Degrees
1 --- --- 0 0
2 150 060 75 -129.9
3 300 060 150 -259.8
</pre><b>Fig. 9</b> visually portrays the layout in terms of both the original and the derived data. The figure uses the distance between the towers rather than the cumulative distance from the origin. On the right is an azimuth pattern that confirms the accuracy of our set-up work.
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<p>Since the pattern is symmetrical across the line formed by the towers, strict adherence to the conventions of correspondence between azimuth headings and coordinates may seem excessively finicky. However, many FCC and other filings require a pattern that accurately reflects the gain (and often the field strength) in all map directions. Hence. a precision model is more than a desire; it is a necessity. <b>Fig. 10</b> provides a screen view of the resulting model.</p>
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<p>An alternative form of initial data would use each succeeding monopole as a reference for the next. Let's examine such a case, again keeping the background elements simple. We shall retain the steel 0.1-m diameter monopoles and feed them in phase, although they will not form a straight line in this case. In fact, the initial data might take the form of the following table.</p>
<pre>
Tower Reference Distance Azimuth Bearing
Number Tower Meters Degrees
1 --- --- ---
2 1 150 060
3 2 150 030
</pre>
<p>We can proceed in a cumulative manner, as in the first problem. However, that route would require us to calculate the end position in terms of a distance and angle from the origin. For the present case, that task is simple, but many other cases might involve solutions to irregular triangles. For now, it is easier to solve the two positions successively. For the second case, we shall initially assume a start at the origin and then simply add the +X and -Y values to the new +X and -Y values to arrive at the final coordinates. To keep the math simple, I selected the 30-degree azimuth heading for the 3rd monopole since the values of sine and cosine are simply flipped relative to the values for the 60-degree heading. The final data table prior to creating the model itself resembles the following one. Remember that the X values are positive and the Y values are negative.</p>
<pre>
Tower Reference Distance Azimuth Bearing +X coordinate -Y coordinate
Number Tower Meters Degrees
1 --- --- --- 0 0
2 1 150 060 75 -129.9
3 2 150 030 204.9 -204.9
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<p>As <b>Fig. 11</b> shows, we have a bent line of monopoles, each a half wavelength from an adjacent monopole. The line formed from monopole 1 to monopole 3 has an azimuth bearing of 045 degrees. When fed in phase, the array produces the pattern shown to the right (over a perfect ground). Note that, relative to the line of towers if overlaid on the polar plot, the pattern is no longer symmetrical. We might easily contrive any number of non-symmtrical patterns by altering the feedpoint current magnitude and phase angle for each monopole. <b>Fig. 12</b> provides the NEC-Win Plus model that produces this pattern and arrangement of monopoles.</p>
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<p>It is not necessary to use the first monopole as the key. In fact, in any array, we may set any point as the coordinate center and calculate from that point. For example, consider a system of 6 towers with a virtual center in a 2-by-3 arrangement. If we have the field dimensions and the bearing along the rectangle formed by the monopoles, we can easily calculate the azimuth bearing and distance to each monopole from the field center. With that data, simple sine and cosine operations will yield the required coordinates to produce a true azimuth model.</p>
<p>Alternatively, we might have a field of rotatable directional beams. We would use the same techniques to locate the coordinates of the center of each beam antenna in the field relative to the point selected as the coordinate center. Since NEC-Win and NEC rotational commands use one of the coordinate axes as the center of rotation, we might have to use a multi-step process to place each antenna in its correct position pointing along the correct bearing. Create the antenna --centered along its boom--at the coordinate center and rotate it to the correct heading. Then move (translate) the antenna to the final position of the boom center at its field location.</p>
<p>In any complex modeling exercise, it pays to pre-plan each maneuver and set up an order of operations--<i>on paper</i>. Although the exercises just suggested only count as moderately complex, it is still easy to lose track of what move occurs next in the progression. Hence, developing a detailed checklist that includes not just the order of operations, but also the quantities involved in each move, can go a long way toward making the process second nature, smooth, and (most important of all) accurate.</p>
<p>The key to the effort lies in understanding the relationship between the polar plot in its azimuth form and the Cartesian coordinates that result in a true azimuth model. +X is always North or zero degrees azimuth. Clockwise, East or 90 degrees azimuth corresponds to the -Y direction on the coordinate system.</p>
<p>Note also that these directions apply to the NSI implementations of NEC. They only apply to other software if that software follows the same conventions for translating a phi pattern into an azimuth pattern. One limitation in some software is to place zero degrees phi on the far right, allowing 90 degrees phi to occur at the top of the polar plot. This system is at odds with standard azimuth conventions in which zero degrees or North is always at the top of a plot. Such systems do not permit us to use the same set of rules for forming a true azimuth model.</p>
<p>One of the implementations of NEC using the alternative polar-plot set up is EZNEC. Still, the program does have a compass plot facility. In the next episode, we shall explore how to create true azimuth models within the program.</p>
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<h1>103. True Azimuth Models<br>
EZNEC Software</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>In the first of our 2 episodes on creating true azimuth patterns, we explored NSI software to see how to develop both true azimuth patterns and the models that yield those patterns. Because the NEC-Win polar plot places zero degrees at the top of the plot, the apparent difference between a phi plot (the inherent NEC plot from the radiation pattern tables) and an azimuth or compass plot is simple. The phi plot counts degrees in a counterclockwise direct, while a compass plot counts degrees in a clockwise direction.</p>
<p>The creation of true azimuth models, however, requires something more from us as modelers. North equates with the +X-axis in the Cartesian coordinate system. Hence, East or 90 degrees azimuth corresponds to the -Y axis. For irregular sets of positions in the X-Y plane, we need to do some careful planning to obtain a model whose geometry produces a correct azimuth plot. Each step in the process is simple enough, even when we receive initial data in the form of distances and bearings. However, we can easily develop a model of moderate complexity and lose track of what comes next. Hence, I recommend a detailed pencil-and-paper procedure for setting up such models.</p>
<p>Nevertheless, in NSI software, we can in a straightforward way create true azimuth or compass-oriented models that are useful for VOACAP, BC, and large antenna field analyses. In fact, we may do the same using EZNEC software. However, the procedures will not be identical, simply because the polar plot display conventions in EZNEC differ from those used by NSI software. Once we master the conventions applicable to EZNEC, we shall discover that every step possible in one software package is available in the other. They will simply differ in accord with convention differences. One result is that a true azimuth model created for one software set will not be readable as a true azimuth pattern by the other without significant revision of the model geometry.</p>
<p><b>Simple Rotational Models in EZNEC</b></p>
<p>EZNEC (Version 4) comes in 3 sizes: standard, plus, and pro. The last version is available with either the NEC-2 or NEC-4 core. However, all required functions to create true azimuth models are available on even the most basic version of the program.</p>
<p>The process of creating true azimuth models begins with an understanding of the EZNEC azimuth plot. In its raw form, without the supplemental data shown, the plot is somewhat opaque, as suggested by <b>Fig. 1</b>. The plot shows the pattern of the same 3-element Yagi used to start the NSI notes, with the boom aligned along the X-axis. Thus, we discover the first difference between NSI software and corresponding EZNEC software. EZNEC places the standard plot zero-degree position on the far right. This procedure allows a pattern that coincides roughly with the normal way of presenting the Cartesian X and Y axes on a flat surface: The X-axis receives a horizontal line and the Y-axis receives a vertical line. In the plot shown, +Y corresponds to the top position of the plot circle.</p>
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<p>Because the plot does not show the degrees on the outer ring of the plot circle, the nature of the plot may not be immediately apparent. There is a data set attached to each plot to record various headings of interest in plot analsis. However, in this plot transfer, the only way to obtain degree markings is to add them with a paint program. Although EZNEC refers to the plot as an azimuth plot, it is actually a phi plot and counts degrees counterclockwise. (The EZNEC elevation plots do count degrees from the horizon upward.) For a myriad of antennas with symmetrical patterns on each side of the virtual boom line, the difference does not make a difference relative to understanding the antenna's operation.</p>
<p>Among its options, EZNEC does offer a "Compass" plot. The compass plot counts degrees clockwise, with zero degrees positioned at the top of the plot circle. Note that there is a 90-degree difference between the zero-points of the standard plot and the compass plot. If we wish to create a model of an antenna pointed North, we must adopt a new set of convention. <b>Fig. 2</b> shows the relationship of the plot conventions to the Cartesian geometry conventions.</p>
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<p>Since North corresponds to the Cartesian Y-axis, to point an antenna North requires that we form the model with the boom pointed toward +Y. The revised model will then have its elements extended (for a symmetrical model with linear elements) along the -Y to +Y axis, with East corresponding to the +X direction. If we revise our 3-element Yagi model accordingly, we can obtain both a standard and a compass pattern, as shown in <b>Fig. 3</b>.</p>
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<p>Note that there is no difference between the graphical portions of each plot. The differences appear in the data beneath the plot. The standard plot shows the cursor and the maximum gain heading to be 90 degrees, which we would expect of the phi pattern for a plot oriented 90 degrees counterclockwise to the plot in <b>Fig. 1</b>. In contrast, the compass plot shows the cursor and maximum gain at zero degrees, corresponding to the model's boom pointing North along the +Y axis.</p>
<p><b>Fig. 4</b> provides us with the wire table for the model. If you compare it with the wire table from NEC-Win Plus in the preceding episode, you will find that all X-column Values are now in the Y column and vice versa.</p>
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<p>Let's rotate the antenna 45 degrees toward the Northeast (compass bearing 045 degrees) to parallel the same operation in the last episode. To set up the antenna for rotation, we should first move the wires by 93" so that the rotation point (coordinates 0,0) is at the center of the boom. EZNEC has move and rotate functions comparable to those in NSI software. They appear as options within the wire table array of possible modifications. Just like using the GM command, we must move the wires first and then rotate them. <b>Fig. 5</b> shows the relevant maneuvers in terms of the assistance screens that appear. To center the antenna along its boom for a compass pattern, we move the wires -93" along the Y-axis or boom line. Then we rotate the antenna 45 degrees clockwise around the Z-axis to move the boom direction from North to Northeast.</p>
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<p>For reference, <b>Fig. 6</b> shows a composite wire table after each maneuver. The table shows only the end-1 values, with the post-move set on the left and the post-rotation set on the right. If we had wished to place the antenna after rotation at some other position within a field of antennas, we would next perform another move operation, setting the distance along each axis into the appropriate X and Y boxes of the move screen. In EZNEC, it is possible to rotate a wire set around the center of a wire, but this operation would not preserve the centering of the antenna on its overall boomlength from reflector to director. In most cases, making adjustments to a model's position yields an accurate result if taken step-by-step in accord with a plan first noted on paper.</p>
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<p>Having set up the Yagi in true azimuth fashion, we obtain a compass plot that is accurate as a true azimuth plot. <b>Fig. 7</b> shows the plot, its data, and the outline view of the antenna as its is aligned in the X-Y plane. The plot has its maximum gain at a 045-degree azimuth bearing, although to be certain, we must consult the data below the actual plot. The antenna outline confirms that the plot accurately portrays the antenna performance, given its alignment. It also confirms that we have moved and rotated the antenna correctly in accord with the compass-plot conventions that apply to EZNEC. The view also let's us know that the virtual boom center is offset by a small amount from the center or driver element. The feedpoint circle and the axis center circle do not coincide.</p>
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<p>Since we have turned the antenna after moving it, let's create a stack of 2 Yagis. We shall want the 2 antennas in the stack to point in different directions, perhaps one headed Northeast and the other North. EZNEC provides a variety of ways to create the stack. One way to proceed is to copy the existing wires to create a new set. The next step is to use either the height function or the move function to create a spacing between the 2 sets of wires. For this small exercise, I have selected 800" as the arbitrary separation. We may then rotate the new wires 45 degrees counterclockwise to point them North along the boom. If we wish to test other angles of separation between the antennas in the stack or other separation distances, we can use the move and rotate functions appropriately.</p>
<p>Since we already have a beam pointing North, we can also import the file description from that model. For identical antennas in a stack, this procedure might be more cumbersome, since we would need to separate the antennas and ensure that each rotated on its boom center. However, there are many exercises in which we may have different antennas in the modeled stack. In that case, importing the added antenna model may prove to be the most practical maneuver.</p>
<p>The operations to create stacks are available also in NSI software. The spreadsheet main face of NEC-Win Plus allows block copying of the first Yagi into a second version. The translation (move) function then allows us to set the separation. To create a stack of different antennas, we may block copy a set of wires and appended sources and loads from one model file to another. In advanced software that makes the full NEC command set available, we can accomplish the copy and move functions with the GM command.</p>
<p>The results of our EZNEC stacking operations appear in <b>Fig. 8</b>. Although the wire table is too long to reproduce, the plot, the data, and the antenna model outline should suffice to establish that we have a successful true azimuth model and plot of the stack. I set the radiation pattern increment to 0.5 degrees so that the heading for maximum gain would read correctly (rather than showing the nearest integer value). Since the model is in free space, the maximum gain occurs rather exactly between the headings of the identical beams. Placing the stack over ground might amend the heading of maximum gain due to slightly different ground effects on the two antennas in the stack.</p>
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<p>The are many good reasons for studying the patterns of a stack with the individual beams pointing in different directions--assuming that this condition is among the planned modes of operation. Forward gain of the composite pattern is only 1 of several interesting facets of the stack. You may also wish to compare the beamwidth for a single antenna (66 degrees) with the beamwidth for the stack (77 degrees). Note also the change in the rearward lobes. Then mentally overlay two sets of individual-antenna rear lobes at 45 degrees to each other. Although the two antennas have some interaction with each other, the overlay process goes a long way toward showing the revised shape of the rearward radiation pattern in the stack.</p>
<p><b>True Azimuth Models in EZNEC</b></p>
<p>To complete our tour of setting up true azimuth models in EZNEC software, we must also tackle the types of situations that we examined for NSI software. There are many modeling problems in which we begin the process with one or another form of azimuth data describing the geometry of the antennas. We shall explore a couple of simpler cases in order to maintain clarity on the principles involved. Suppose that we had a set of 3 monopoles that are 1/4 wavelength at 1 MHz. Since a wavelength at 1 MHz is 300 m, the monopoles are each 75-m long. Although a real-world exercise might include a buried radial system for each monopole, we shall use a perfect ground for our exercise. As well, the individual antennas have an arbitrary diameter of 0.1 m (100 mm) and use a relatively low conductivity. Also for simplicity, we shall feed the individual antennas in phase.</p>
<p>As we did with the NSI exercises, we may begin with data provides in the form of bearings and distances, perhaps against a map or site plat.</p>
<pre>
Tower Distance Azimuth Bearing
Number Meters Degrees
1 --- ---
2 150 060
3 300 060
</pre>
<p>The system once more consists of the monopole in a line, with 1/2 wavelength spacing between the individual antennas. For the exercise, we shall feed each monopole at its base in phase with the other two monopoles. Because the heading is 060 degrees, the line extends from Southwest to Northeast. If we let the first monopole be the key and place it at coordinates 0,0, then the remaining monopoles will fall somewhere between North and East. Once more, we can make use of the sine and cosine functions of 60 degrees to form multipliers for the distances and derive the proper coordinates. Because the second and third monopoles fall in the first quadrant, both X and Y will be positive.</p>
<p>Since the Y-axis is North or zero degrees and the X-axis is 90 degrees, the X coordinate will be the sine of the angle times the distance or 0.866 * 150 = 129.9 m. The Y coordinate will be the sine of the angle times the distance or 0.5 * 150 = 75 m. The monopoles form a single line, so the second coordinate set can simply use the cumulative distance for X = 259.8 m and Y = 150 m. The completed table is below.</p>
<pre>
Tower Distance Azimuth Bearing X coordinate Y coordinate
Number Meters Degrees
1 --- --- 0 0
2 150 060 129.9 75
3 300 060 259.8 150
</pre>
<p><b>Fig. 9</b> provides a visual presentation of the monopole layout against the Y and X axes that form the South-to-North lines and the West-to-East line. The annotated antenna view has been turned so that the axes assume their proper positions relative to an azimuth map. The compass plot confirms that the main lobes of the radiation pattern are broadside to the line of monopoles, with maximum gain at 135 and 315 degrees azimuth.</p>
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<p><b>Fig. 10</b> shows the wire table for the model to confirm the use of the cordinates in the tables. As noted in the preceding episode, there are many applications in which precision azimuth pattern are required. Hence, the small bit of pre-modeling calculation that it takes to set up the true azimuth model is minuscule compared to having a pattern rejected by a licensing agency.</p>
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<p>The second exercise involves 3 towers that do not form a single line. Instead, the third tower moves off from the second at a different angle. We shall retain the in-phase feeding system over perfect ground to preserve the model's simplicity, since our goal is to get a handle on how to organize the coordinates within EZNEC to obtain a true azimuth model.</p>
<pre>
Tower Reference Distance Azimuth Bearing
Number Tower Meters Degrees
1 --- --- ---
2 1 150 060
3 2 150 030
</pre>
<p>We may easily calculate the coordinates for the second monopole using the techniques employed for the first monopole system. Next, we can assume that the second monopole is at the coordinate center and calculate its coordinates. To the values for X and Y, we simple add the values for each coordinate derived for the position of monopole 2. The result of the small trig exercise appears in the following completed table.</p>
<pre>
Tower Reference Distance Azimuth Bearing X coordinate Y coordinate
Number Tower Meters Degrees
1 --- --- --- 0 0
2 1 150 060 129.9 75
3 2 150 030 204.9 204.9
</pre>
<p>The results appear more graphically in <b>Fig. 11</b>, where the annotated antenna outline shows the final positions along with the original data and the calculated coordinates. The compass plot is identical to the one produced using NEC-Win Plus and the pattern is broadside to the virtual line from the first to the last of the monopoles. The lack of perfect symmetry on each side of that lines reveals the effects of the irregular line formed by the 3 antennas.</p>
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<p>As a confirmation of the model's use of the calculated coordinates, <b>Fig. 12</b> shows the model's wire table.</p>
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<p>Producing true azimuth patterns in EZNEC turns out to be as straightforward as it did using NSI software. Once we located North (the +Y axis) and East (the +X axis), the remain steps became a matter of calculating coordinates from any data that might be supplied in terms of distance and azimuth bearing. Of course, we can always begin with coordinate data, so long as we remember to place the antenna or its parts against a paper version of the X-Y system with North set along the correct axis line. Figuring the antenna coordinates then becomes a matter of arranging the coordinates. If an antenna has an area as defined in the X-Y plane, then it may be easiest to arrange the antenna along one or another azis and then to move and/or rotate the antenna to the desired orientation.</p>
<p>EZNEC and NEC-Win Plus both use proprietary file formats, neither of which is the standard ASCII NEC-input file format. However, NEC-Win Plus will save files in the standard format, and EZNEC Pro will do so as well. Hence, my own work often involves moving from one piece of software to the other by way of an intervening NEC model file. There is a relationship between the systems needed to produce true azimuth models in each type of software. When moving from NEC-Win Plus to EZNEC, rotate the antenna or antenna field 90 degrees counterclockwise in EZNEC. When moving from EZNEC to NEC-Win Plus (or other NSI software), rotate the antenna or antenna field 90 degrees clockwise in the NSI program. The results will yield true a azimuth model if the initial model was truly an azimuth model within its software.</p>
<p>Producing true azimuth models is certainly not necessary for a large part of the modeling enterprise. However, for applications demanding true azimuth models and patterns, the two bodies of software that we have sampled provide guides to almost any other software implementing NEC. The software must have an azimuth pattern that counts degrees clockwise. With that available, the rest of the task has only 2 steps. The first is to determine how the azimuth pattern relates to the software's standard phi pattern. The second is to calculate the necessary coordinates for the antenna to produce a true azimuth model and a true azimuth pattern.</p>
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<h1>104. PS: I Change</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>Suppose that we create a model of a ground-plane monopole with 4 radials. Let's use 7.15 MHz as the test frequency. The monopole will be 10.071 m long and 0.0125 m in radius. The radials will be 11.57 m long and 0.002 m in radius. We shall equip the model with a GM line so that we can readily change the height of the antenna assembly over ground, using the radials as the base height.</p>
<pre>
CM monopole 7.15 MHz 25-mm dia.
CM over ave gnd
CM 90-deg radials
CE
GW 1 11 0 0 0 0 0 10.071 .0125
GW 2 11 11.57 0 0 0 0 0 .002
GM 1 3 0 0 90 0 0 0 2 1 2 11
GM 0 0 0 0 0 0 0 83.858
GE -1 -1 0
GN 2 0 0 0 13.0000 0.0050
EX 0 1 1 0 1 0
FR 0 1 0 0 7.15 1
RP 0 181 1 1000 -90 0 1.00000 1.00000
EN
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<p>The model might begin as high as 2 wavelengths over ground (using the constants for average ground in this case). However, we shall be interested in lower heights, down to and including placing the radials below ground. We shall have to change the model design just slightly to accommodate NEC-4 guidelines that require a wire or segment junction at Z=0. One easy way to achieve this is to run the monopole down to the ground. Then let the innermost segment of each radial slope from Z=0 down to the radial level. The remaining radials segments form a flat plane. The radials are 0.001 wavelength below ground, about 0.042 m or 1.65".</p>
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CM monopole 7.15 MHz 25-mm dia.
CM buried radials .001 wl
CM 90-deg radials
CE
GW 1 11 0 0 0 0 0 10.071 .0125
GW 2 1 1.05 0 -.042 0 0 0 .002
GW 2 10 11.57 0 -.042 1.05 0 -.042 .002
GM 1 3 0 0 90 0 0 0 2 1 2 11
GE -1 -1 0
GN 2 0 0 0 13.0000 0.0050
EX 0 1 1 0 1 0
FR 0 1 0 0 7.15 1
RP 0 361 1 1000 -90 0 1.00000 1.00000
EN
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<p>If we plot the data for models as they approach and then penetrate ground, we obtain an interesting set of discontinuities. All data use the usual units of measure.</p>
<pre>
4-Radial 90-Degree Ground-Plane Monopole From 2-WL Up to Below Ground: Performance with Various Ground Types
Free-Space Reference Performance Monopole: Len 10.071 m, Radius 0.0125 m Freq.: 7.15 MHz
Rad-L m Rad-R m Gain dBi Bmwidth Resist React
11.57 0.002 1.35 102 21.34 0.06
Ground Type Average: C 0.005, P 13
Height wl Height m Gain dBi TO Angle Resist React
0.1 4.193 0.24 70 24.35 -4.84
0.05 2.096 0.10 67 28.64 -3.48
0.025 1.048 -0.04 65 31.75 0.22
0.01 0.419 -0.22 64 34.64 7.85
0.005 0.210 -0.38 64 36.52 16.53
0.001 0.042 -1.37 64 47.26 52.34
-0.001 -0.042 -2.37 64 64.75 7.98
-0.005 -0.21- -2.49 64 66.25 10.30
</pre>
<p>Note that as we enter the ground, the gain drops rapidly. More significantly, the feedpoint impedance changes considerably. If we add more radials, the transition will be less extreme, but the 4-radial model makes a useful tool, if for no other reason than to arouse a bit of curiosity.</p>
<p><b>Fig. 1</b> shows the relative current magnitude along the vertical monopole and along 2 of the 4 radials forming the ground plane. The model showing the current distribution is in free space, but any model having all of its wires above ground would show the same set of characteristics. I have set the maximum monopole current on the 4th segment of the radial as a marker. Note that the maximum radial current is just above the first monopole segment, indicating that we have very close to equal currents on the radials and on the monopole. Of course, each radial carries 1/4 of the total current below the feedpoint.</p>
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<p>Both current curves have very similar shapes. However, current magnitude does not tell us the complete story. The following list shows both the relative current magnitude and the current phase angle based on a source current of 1.0 at 0&deg;. If we count upward on the monopole to the 5th entry, we find a magnitude of 0.81245. The corresponding entry for the radial, counting downward, is 0.21420, very close to the 1/4 wavelength monopole value. As well, note the similarity of current-phase value at the monopole top and the radial end. Remember that the radial has a much smaller radius than the monopole.</p>
<pre>
Monopole Radial
Magnitude (A.) Phase (Deg.) Magnitude (A.) Phase (Deg.)
.09260 -2.93 .25147 -0.08
.24609 -2.71 .24966 -0.27
.38339 -2.49 .24298 -0.43
.50941 -2.26 .23114 -0.56
.62371 -2.01 .21420 -0.67
.72512 -1.75 .19237 -0.78
.81245 -1.47 .16600 -0.89
.88456 -1.17 .13553 -0.99
.94034 -0.82 .10146 -1.09
.97902 -0.42 .06417 -1.19
1.0000 0.00 .02318 -1.29
</pre>
<p>If we lower the antenna so that the radials are below ground, we shall have to modify the model slightly. The monopole will just touch the ground. The radial wires will have 2 sections. The innermost segment will slope from ground level at the monopole base down to 0.001 wavelength below ground (about 0.042 m or 1.65"). The remaining segments will extend to 11.57 m to produce the same total length as the radials in the above-ground model. <b>Fig 2</b> shows the model and the relative current magnitudes, followed by a tabular listing of magnitudes and phase angles.</p>
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Monopole Radial
Magnitude (A.) Phase (Deg.) Magnitude (A.) Phase (Deg.)
.09286 -6.00 .24242 -7.83
.24693 -5.66 .21041 -28.95
.38486 -5.30 .17479 -51.49
.51154 -4.90 .14094 -73.58
.62647 -4.47 .11038 -95.00
.72839 -3.99 .08337 -115.4
.81606 -3.44 .05966 -134.6
.88806 -2.81 .03914 -152.5
.94333 -2.06 .02223 -170.1
.98087 -1.10 .00972 170.08
1.0000 0.00 .00222 139.95
</pre>
<p>Since the models differ very slightly in construction due to the need to develop subsurface radials, the monopole current magnitudes are very close, but not identical, to the values for the preceding model. However, we do find a difference in the current phase angle range. For identical source values (1.0 A. at 0&deg;), the preceding model tip current phase angle was only -2.93&deg;, whereas the model with buried radials has a tip-segment phase angle of -6.0&deg;. The above-ground model is a free-space version of the antenna with a source impedance of 21.35 + j0.07 Ohms. The model with buried radials uses average ground (C 0.005 S/m, P 13) and reports a source impedance of 64.69 - j 7.84 Ohms.</p>
<p>The differences between the radial currents for the above-ground and the buried radial models are far more dramatic. Visually, the radial curve differs by rapidly decreasing in current magnitude as we move from the hub outward. The table confirms the curve. At the 5th entry upward, the monopole shows a value of 0.81606, while the corresponding radial entry shows a value of 0.11038, only half the magnitude for that position on the above-ground radial. The current phase changes along the buried radial are far more radical than those along the above-ground radial. One NEC convention is to maintain all phase reports in the 0&deg;-180&deg; range. The outer-most value is equivalent to a value of -220.05&deg;, about 214&deg; out of phase with the tip of the monopole. Note that these values are not true tip values, but the values at a position roughly comparable to the center of the relevant wire segments in the models.</p>
<p>The comparison makes clear that the common above-ground portions of the two antennas yield essentially the same current distribution. However, the parts that move from above ground to below ground change their current distribution. Most modelers seem to be wholly unaware of this phenomenon. So it bears some exploration. Let's begin by reviewing some fundamentals about NEC's treatment of ground.</p>
<p>For any given ground quality, we measure (or find in some table) values for conductivity (sigma) and relative permittivity (epsilon-r). Relative permittivity rests on the permittivity in free space (epsilon-0). Essentially, the program combines the listed values for conductivity and permittivity into a complex relative permittivity (epsilon-g):</p>
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<p>The term f is the frequency in Hz. As f changes, so too does the value of epsilon-g. Therefore, the effects of ground on buried-radial ground-plane antenna performance vary with conductivity, permittivity, and frequency.</p>
<p>NEC also calculates another value called ks, the wave number in the sinusoidal current expansion in NEC. This value applies to any wire within a medium other than free-space (or a vacuum). Hence, it applies to all insulated wires and to any wires below ground level (assuming that a real ground is operative in the model). The value of ks modifies the length of a wave for the calculation of current along a wire. Hence,</p>
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<p>The current-propagation wave number has the effect of lengthen every applicable segment with respect to current calculations. The exact amount of lengthening depends upon the frequency and ground constants that the modeler selects.</p>
<p>We can easily determine the effect of the wave number on segment length by employing the PS command in NEC-4. The command requires only the command letters, with no following numerical entries. In fact, we can perform segment-length adjustment calculations without further model execution by following the PS command with EN, as in the following sample model. The model contains all of the GW, GN, FR, and EX elements to form a complete model, except that it lacks an output request other than the PS command. Hence, calculations stop after the PS command has done its work.</p>
<pre>
GW 1 11 0 0 0 0 0 10.067 .0125
GW 2 10 10.4823 0 -.04193 .953 0 -.04193 .002
GW 2 1 .953 0 -.04193 0 0 0 .002
GM 1 3 0 0 90 0 0 0 2 1 2 11
GE -1 -1 0
GN 2 0 0 0 13 .005
EX 0 1 1 0 1 0
FR 0 1 0 0 7.15 1
PS
EN
</pre>
<p>Although the report in the NEC output file shows the value of ks, we may confine our attention to the effects on segment lengths. I ran models of 160-m and 40-m ground-plane monopoles with 4 buried radials through the PS command, using 3 diverse ground-quality values, those for very good (C 0.0303 S/m, P 20), for average (C 0.005 S/m, P 13), and for very poor (C 0.001 S/m, P 5) soil. I also ran the same model in free space to illustrate how much wire values change as we bury the radials. Note that in the following lists, all values are normalized to fractions of a wavelength. As well, I converted the entries from engineering to decimal notation. In both cases, the monopole physical segment length is 0.02183 wavelength with a physical radius of 0.000298 wavelength.</p>
<pre>
Radial Segment Length and Radius
Frequency Soil Quality Segment Length Segment Radius
1.85 MHz None 0.02273 WL 0.0000477 WL
Very Good 0.3904 0.0008194
Average 0.1612 0.0003383
Very Poor 0.0713 0.0001577
7.15 MHz None 0.02273 WL 0.0000477 WL
Very Good 0.2017 0.0004233
Average 0.09664 0.0002028
Very Poor 0.05376 0.0001128
</pre>
<p>To extract a simple example from the listing, the 7.15-MHz average-ground segment length is about 4.3 times the physical length, that is, the length in free-space when normalized to a fraction of a wavelength. With respect to current expansion, the radial point corresponding to the 5th monopole entry in the earlier model (<b>Fig. 4-2</b>) actually lies just inside the second segment, where we find a current magnitude of about 0.21. However, notice that the radius increases to the same degree, resulting in what appears to be a more rapid change of current phase angle. Since the current does not go to zero until we reach the radial tip, most of the table entries for the buried radial show very low values compared to the free-space model.</p>
<p>The NEC-4 manual recommends that we use lambda-s as the basis for calculating segment lengths for any wire within a medium other than free space, where free space includes any region above a real ground. Examining the PS command report shows that the calculated segment length for current expansion along a buried radial no longer agrees with the segment length for the monopole that is above ground. The segment-length difference appears at one end of the source segment, suggesting a possible error source in the model. The AGT cannot show this potential error, since the test uses free space as its venue. So the next question is what degree of error we might expect from not adjusting the segment length in accord with the value of lambda-s.</p>
<p>To obtain a sense of what error might be possible, I used the 40-m monopole and ran it in two forms, using very good, average, and very poor soil. The first run used the standard segmentation of the sloping-radial construction with a total of 11 segments per radial. The following lines sample the model over average ground.</p>
<pre>
GW 1 11 0 0 0 0 0 10.067 .0125
GW 2 10 10.4823 0 -.04193 .953 0 -.04193 .002
GW 2 1 .953 0 -.04193 0 0 0 .002
GM 1 3 0 0 90 0 0 0 2 1 2 11
GE -1 -1 0
GN 2 0 0 0 13.0000 0.0050
EX 0 1 1 0 1 0
FR 0 1 0 0 7.15 1
RP 0 181 1 1000 -90 0 1.00000 1.00000
EN
</pre>
<p>Next, I adjusted the number of segments in the radial entries (GW2) so that the calculated segment length for current expansion would more evenly match the monopole segment length. Again, here is a sample over average ground. Note the use of the PS command to allow confirmation of the segmentation.</p>
<pre>
GW 1 11 0 0 0 0 0 10.067 .0125
GW 2 40 10.4823 0 -.04193 .953 0 -.04193 .002
GW 2 4 .953 0 -.04193 0 0 0 .002
GM 1 3 0 0 90 0 0 0 2 1 2 44
GE -1 -1 0
GN 2 0 0 0 13 .005
EX 0 1 1 0 1 0
FR 0 1 0 0 7.15 1
PS
RP 0 181 1 1000 -90 0 1.00000 1.00000
EN
</pre>
<p>For the adjusted models, the outlines in <b>Fig. 3</b> show the distribution of segments between the sloping and the straight portion of the radials. The monopole, of course, remains unchanged. The look of the model outline represents the physical dimensions and not the electrical length of segments as calculated by NEC for use in the current expansion.</p>
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<img src="amod104-3.gif" width="800" height="300" border="2" vspace="5" hspace="9">
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<p>From the total of 6 models, I obtained the following results. The entries showing 11 segments per radial represent the unaltered models. The alternate model segment numbers represent values that yield calculated segment lengths about equal to those in the monopole.</p>
<pre>
7.15-MHz Ground-Plane Monopole with 4 Buried Radials
Soil Quality Segments/Radial Gain dBi Source Impedance
Very Poor 11 -4.24 88.94 + 29.22 O
26 -4.13 86.63 + j26.22
Average 11 -2.36 64.60 + j7.29 O
46 -2.35 64.44 + j4.28
Very Good 11 0.46 50.91 + j8.19
98 0.24 53.53 + j5.75
</pre>
<p>None of the possible error differences are either fatal or unambiguous. For example, the amount of difference is greatest for the antenna over very good soil, but so too is the increase in the radial wire radius. The calculated radius is greater than the monopole radius. In some instances, using the calculated values of segment length and radius as a basis for adjustment may lead to an impossible conflict among NEC guidelines. For precision work, the problem would require considerable thought before finalizing a model. However, for general guidance in determining trends and rough properties, using unaltered models with 11 segments per radial will largely suffice. Remember that NEC ground systems have other limitations. For example, they presume a homogenous ground from horizon to horizon and from the surface downward. In many cases, the actual ground will be stratified, and the exact values of conductivity may not be measurable to the depth of RF penetration.</p>
<p>Within the context of NEC models using buried radials, the exercise does provide a foundation for understanding the different current distributions that we find in those radials, relative to above-ground models.</p>
<p><b>A Note on IS and PS</b></p>
<p>Because the IS or insulated sheath command also places a non-free-space medium around a specified wire, NEC-4 will adjust the segment lengths and radii for implementing the current expansion. Out of curiosity, you may invoke the PS command in trial models to see what happens. Let's begin with a simple dipole for 7.15 MHz using a 0.001-m radius wire.</p>
<pre>
CM 7.15-MHz dipole in free space
CM Radius 0.001 m
CE
GW 1 21 0 -10.2 0 0 10.2 0 .001
GE
FR 0 1 0 0 7.15 1
EX 0 1 11 0 1 0
PS
RP 0 1 361 1000 90 0 1.00000 1.00000
EN
</pre>
<p>With a length of 20.4 m, the dipole is resonant, showing a source impedance of 70.11 - j0.08 Ohms. The free-space gain of this lossless wire is 2.14 dBi.</p>
<p>Next, let's add an IS command to place an insulating sheath around the wire. We shall use a high-quality plastic with a relative permittivity of 3 and a conductivity of 1E-10. The insulation will be 2-mm thick, resulting in a sheath radius of 0.003 m (around the 0.001-m radius wire). An insulation thickness that equals the wire diameter might fall into the relatively heavy insulation category, although thicker insulations certainly exist.</p>
<pre>
CM 7.15-MHz dipole in free space
CM Radius 0.001 m
CM insulated
CE
GW 1 21 0 -9.726 0 0 9.726 0 .001
GE
IS 0 1 1 21 3 1e-10 .003
FR 0 1 0 0 7.15 1
EX 0 1 11 0 1 0
PS
RP 0 1 361 1000 90 0 1.00000 1.00000
EN
</pre>
<p>To obtain resonance we must shorten the antenna to 95.4% of its bare-wire form. The source impedance reports 66.34 - 0.04 Ohms and a free-space gain of 2.11 dBi.</p>
<p>Note that both models implement the PS command, since it takes so little run time and report space. However, only the report for the insulated wire model is relevant to satisfy our curiosity. If we explore the PS portion of the report, we encounter some entries in normalized form, that is, expressed as fractions of a wavelength. These values are for the adjusted segment length and adjusted wire radius. We must calculate the normalized physical dimensions by dividing the physical segment length and radius by 41.92902 m, a wavelength at 7.15 MHz. Now we can compare the results within the limits expressed by the report entries.</p>
<pre>
Insulated 7.15-MHz Dipole
Segment Length Segment Radius
Normalized Physical Dimensions 2.209E-2 2.385E-5
Adjusted Dimensions for Current Expansion 2.209E-2 2.385E-5
</pre>
<p>Within reporting limits, there is no difference in the values, although the effects of the insulation show up in the performance reports. However, the extent of the medium change is so small, that for all practical purposes, the modeler can ignore any segment length changes and use the same segmentation as he or she used for a bare wire model.</p>
<p>If the modeler specifies an upper medium (UM), usable only with the RCA ground system, the situation would be similar to specifying a ground, but apply to the region formerly treated as a vacuum or free-space. Under those conditions, application of the PS command in order to evaluate overall model segmentation is certainly in order.</p>
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<h1>105. Models, Symmetry, and Loads: A Couple of Reminders</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>While diagnosing an ailing model, I encountered a couple of areas in which a bit of clarification might be useful to other modelers. Since the points that we shall cover may be second nature to some modelers and temptations to error for others, let's just call these notes reminders.</p>
<p><b>The Destruction of Symmetry</b></p>
<p>The NEC manuals provide a list of conditions that "automatically reset the symmetry condition." The first item on the list--which is all that we need for this exercise--reads this way: "Addition of a wire or patch (GW, GH, CW, SP, etc.) will destroy all symmetry." The conditions apply equally to the GX and the GR symmetry commands.</p>
<p>The most common interpretation that newer modelers seem to apply to the statement is that the core will fail to produce the symmetric structure specified in either the GX or GR command. This interpretation is incorrect. Rather, the core produces the desired structure, but does not use the shorter form of calculation to arrive at the matrix values. Instead, the model is reset for the entire structure to a non-symmetry mode of calculation.</p>
<p>Consider the following models, all of which have the properties shown in <b>Fig. 1</b>.</p>
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<img src="amod105-1.gif" width="623" height="306" border="2" vspace="5" hspace="9">
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<p>The most direct form for this model would create all 5 wires in separate GW commands, using the following lines.</p>
<pre>
CM Test model: 4 vertical dipoles, inner 2 fed
CM 5th passive vertical dipole at a distance
CM Full version
CE
GW 1 11 0 -.75 -.245 0 -.75 .245 .001
GW 2 11 0 -.25 -.245 0 -.25 .245 .001
GW 3 11 0 .25 -.245 0 .25 .245 .001
GW 4 11 0 .75 -.245 0 .75 .245 .001
GW 5 11 .5 0 -.245 .5 0 .245 .001
GE 0 -1 0
FR 0 1 0 0 299.7925 1
EX 0 2 6 0 1 0
EX 0 3 6 0 1 0
RP 0 1 361 1000 90 0 1.00000 1.00000
EN
</pre>
<p>The relevant performance data for the model is a maximum gain of 6.04 dBi, with a 36-degree beamwidth in the main or maximum-gain lobe. The reported source impedance is 49.383 -j21.986 Ohms. Both of the variations on this model will yield the same results to the last decimal place. The free-space azimuth pattern has the shape shown in <b>Fig. 2</b>. Note that the main lobe (by a tiny fraction of a dB) is away from the 5th vertical dipole.</p>
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<img src="amod105-2.gif" width="426" height="415" border="2" vspace="5" hspace="9">
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<p>An alternative to listing GW 3 and GW 4 is to use the GX commands to replicate the first 2 wires symmetrically across the Y-axis. This alternate form would take this appearance.</p>
<pre>
CM Test model: 4 vertical dipoles, inner 2 fed
CM 5th passive vertical dipole at a distance
CM GX, symmetry defeated
CE
GW 1 11 0 -.75 -.245 0 -.75 .245 .001
GW 2 11 0 -.25 -.245 0 -.25 .245 .001
GX 1 010
GW 5 11 .5 0 -.245 .5 0 .245 .001
GE 0 -1 0
FR 0 1 0 0 299.7925 1
EX 0 2 6 0 1 0
EX 0 3 6 0 1 0
RP 0 1 361 1000 90 0 1.00000 1.00000
EN
</pre>
<p>Since a GW entry (GW 5) follows the GX command, the common misconception is that wires 3 and 4 will not appear. Instead, the correct interpretation is that the GX command produces the required 2 wires, but does not invoke symmetry in its calculations. The following extract from the NEC output report gives us the telltale line.</p>
<pre>
TOTAL SEGMENTS USED= 55 NO. SEG. IN A SYMMETRIC CELL= 55 SYMMETRY FLAG= 0
</pre>
<p>If symmetry had been invoked, the number of segments in the symmetric cell would have been 22, the number of segments in the first 2 wire entries. As well, the symmetry flag would equal 1 rather than zero. The total number of segments in the model is 55, indicating that the core has created the requisite new wires or tags. The performance reports and the segment and current tables also show the existence of the created wires, but without the symmetry calculation shortcuts. In fact, the run times for this test model and for the first one were identical.</p>
<p>On some occasions, it may be necessary to successfully invoke symmetry due to model size or other factors. That route is irrelevant to the present array, but the model may be useful in its simplicity for illustrating one way to go about the process. The technique involves the use of a Numerical Green's File (NGF). We have examined NGFs in past episodes, but always in the context of saving the file for use with multiple new models that call up the results. We need not use separate NGF formation and use files. Instead, we can create and use the NGF results within the same model or .NEC file. We only need to use the NX or next structure command between the saving of the NGF and the call-up of its contents. Note that NEC-2 requires that the first command following NX must be a comment (CM or CE). However, NEC-4 permits a geometry command to directly follow the NX command. We may use the following model as a guide.</p>
<pre>
CM Test model: 4 vertical dipoles, inner 2 fed
CM 5th passive vertical dipole at a distance
CM GX, symmetry via NGF
CE
GW 1 11 0 -.75 -.245 0 -.75 .245 .001
GW 2 11 0 -.25 -.245 0 -.25 .245 .001
GX 1 010
GE 0 -1 0
FR 0 1 0 0 299.7925 1
WG test3.ngf
NX
CE
GF test3.ngf
GW 5 11 .5 0 -.245 .5 0 .245 .001
GE 0 -1 0
EX 0 2 6 0 1 0
EX 0 3 6 0 1 0
RP 0 1 361 1000 90 0 1.00000 1.00000
EN
</pre>
<p>The first half of the model, prior to the NX command, creates the first 2 vertical dipoles and requests their replication via symmetry across the Y-axis. As in all NGF operations, we include ground, loading, and frequency information for the affected wires within this file. The WG command saves the results. The program used for these exercises allows the user to specify any file name and extension for the data storage. However, raw NEC-2 has limitations of the NGF filename, and different implementations of NEC-2 may use different specifications for this entry. Finally, note that we do not use the EN command before the NX command, because EN would terminate calculations.</p>
<p>Following the NX command, I have inserted a CE line so that the model will run on NEC-2 or NEC-4. The GW entry established the 5th dipole, while the EX lines provides the two sources. The results of this model are identical to those of the preceding tests, although we want to examine one special line-set of the NEC output report.</p>
<pre>
TOTAL SEGMENTS USED= 44 NO. SEG. IN A SYMMETRIC CELL= 22 SYMMETRY FLAG= 1
STRUCTURE HAS 1 PLANES OF SYMMETRY
</pre>
<p>In the first set of core calculations, the GX command creates 22 new segments, for a total of 44. The symmetry cell has the initial 22 segments and the flag is set to 1, indicating that symmetry is invoked for 1 plane. Because the model is so small and contains both parts of the process, the run time is no different than for the first 2 tests. However, for this demonstration, run time was not in question. For very large models, it may well become a key reason for using this technique. Our goal was to show how we can invoke symmetry and complete our modeling all within the same .NEC file.</p>
<p>If we wish to consider matters of file size and run times, we can turn to a different sort of model and simultaneously illustrate the same points for the GR or rotational symmetry command.</p>
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<p><b>Fig. 3</b> shows the general properties of the models that we shall construct for this exercise. At the usual test frequency (299.2975 MHz, where 1 wavelength = 1 meter), we shall place 64 1/4 wavelength radials 0.05 m below ground. At the center, we shall construct a single monopole wire that meets the radials below ground and extends 0.25 wavelength above ground. By assigning 11 segments to the monopole wire, the junction of the second and third segments will intersect Z = 0, the ground level. The following model illustrates the simple set-up, using the GM command to replicate the radials 63 times beyond the original GW 1 specifications.</p>
<pre>
CM 64 buried radials with monopole
CM GM
CE
GW 1 11 0 0 -.05 .25 0 -.05 .001
GM 1 63 0 0 5.625 0 0 0
GW 65 11 0 0 -.05 0 0 .25 .001
GE -1 -1 0
FR 0 1 0 0 299.7925 1
EX 0 65 3 0 1 0
GN 2 0 0 0 5 .001
RP 0 181 1 1000 -90 90 1.00000 1.00000
EN
</pre>
<p>The model yields a maximum gain of -0.42 dBi at a TO angle of 59 degrees (theta). <b>Fig. 4</b> shows the elevation/theta pattern for the model. The model made no attempt to achieve resonance, but used instead easily remembered dimension values. Hence, the reported source impedance is 46.125 - j730.025 Ohms. All versions of the model report identical results.</p>
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<img src="amod105-4.gif" width="425" height="255" border="2" vspace="5" hspace="9">
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<p>To shorten run times and further simply model formation, many modelers replace the GM command with a GR command. The GR command allows rotational symmetry and calculates the angle between each radials, so the modeler does not have to remember that a 64-radial system has a 5.625-degree angular separation between radial wires. The replacement model appears in the following lines.</p>
<pre>
CM 64 buried radials with monopole
CM GR
CE
GW 1 11 0 0 -.05 .25 0 -.05 .001
GR 1 64
GW 65 11 0 0 -.05 0 0 .25 .001
GE -1 -1 0
FR 0 1 0 0 299.7925 1
EX 0 65 3 0 1 0
GN 2 0 0 0 5 .001
RP 0 181 1 1000 -90 90 1.00000 1.00000
EN
</pre>
<p>Note that the GR entry is followed by the monopole GW entry. We could not place the monopole ahead of the GR command. That move would have resulted in the creation of 64 monopoles in the same position. By placing the monopole wire after the GR command, we obtain only one monopole, we create only a single monopole, but we defeat the invocation of symmetry. However, we do not defeat the creation of the radials themselves. The critical line from the NEC output report for the model confirms these notes.</p>
<pre>
TOTAL SEGMENTS USED= 715 NO. SEG. IN A SYMMETRIC CELL= 715 SYMMETRY FLAG= 0
</pre>
<p>The entire model--all 715 segments--form one symmetric cell, and the symmetry flag = 0; hence, the radials exist in the model, but it does not use symmetry in the calculations. The run time for this version of the model is identical to the run time required by the version using the GM command.</p>
<p>To invoke symmetry, we may employ the model style used for the final version of the vertical dipole array. We may create and run the radials alone and save the results in an NGF file. Within the same model, we may use the NX command to begin a second run that calls up the stored results and applies them to the enlarged model that now contains the monopole. The final model in this series appears in the following lines.</p>
<pre>
CM 64 buried radials with monopole
CM GR + NGF
CE
GW 1 11 0 0 -.05 .25 0 -.05 .001
GR 1 64
GE -1 -1 0
FR 0 1 0 0 299.7925 1
GN 2 0 0 0 5 .001
WG radials.ngf
NX
GF radials.ngf
GW 65 11 0 0 -.05 0 0 .25 .001
GE -1 -1 0
EX 0 65 3 0 1 0
RP 0 181 1 1000 -90 90 1.00000 1.00000
EN
</pre>
<p>If you run this model on NEC-2, insert at least a CE command prior to the GF entry that follows NX. The key NEC output report lines tell us that the first portion of the multi-run model has invoked symmetry.</p>
<pre>
TOTAL SEGMENTS USED= 704 NO. SEG. IN A SYMMETRIC CELL= 11 SYMMETRY FLAG= -1
STRUCTURE HAS 64 FOLD ROTATIONAL SYMMETRY
</pre>
<p>To create the NGF data, the first run uses only the total number of segments in the radials. The remaining 11 segments in the monopole do not appear until the second run. The symmetric cell shows only the number of segment in the initial radial, and the symmetry flag = -1, indicating rotational symmetry.</p>
<p>The total file begins to show the run-time advantage of using the GR command. The total run time for this latest version of the model is about 1/3 the time for the other two versions. Neither time is truly long for the current generation of PCs. However, the savings are indicative of what may accrue for larger models.</p>
<p>To illustrate that point, I enlarged the radial system to 120 1/4 wavelength radials. To replicate a common BC industry practice, I also inserted a set of 120 shorter radials between the longer radials. The total system-- or at least a portion of it--appears in <b>Fig. 5</b>.</p>
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<img src="amod105-5.gif" width="800" height="411" border="2" vspace="5" hspace="9">
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<p>Only the short radials appear fully, with all segments shown to give an idea of the model construction. However, the model-file appearance does not change much compared to the 64-radial models that we just viewed. Indeed, the only difference is a second GW line (which uses the same tag number as the first wire).</p>
<pre>
CM 240 buried radials with monopole
CM GR + NGF
CE
GW 1 11 0 0 -.05 .25 0 -.05 .0001
GW 1 2 0 0 -.05 .04546 .0012 -.05 .0001
GR 1 120
GE -1 -1 0
FR 0 1 0 0 299.7925 1
GN 2 0 0 0 5 .001
WG radials.ngf
NX
GF radials.ngf
GW 65 11 0 0 -.05 0 0 .25 .0001
GE -1 -1 0
EX 0 65 3 0 1 0
RP 0 181 1 1000 -90 90 1.00000 1.00000
EN
</pre>
<p>The symmetry-cell size and the flag setting both tell us that the model invoked symmetry. Even so, the increased model size required about 21 times longer to run than the 64-radial model using the same technique. As we increase the number of segments, run times increase exponentially. Without symmetry, we would need to add a further multiplier to the run time.</p>
<pre>
TOTAL SEGMENTS USED= 1560 NO. SEG. IN A SYMMETRIC CELL= 13 SYMMETRY FLAG= -1
STRUCTURE HAS 120 FOLD ROTATIONAL SYMMETRY
</pre>
<p>The model style shown has one drawback. We are likely to forget that the file called radials.ngf remains in the directory (or folder) where we store models. Using symmetry, the file is not very large, only 314 KB. But it does take up space. Directory cleaning is an important adjunct function for almost all modeling. (Some implementations of NEC do not save any files except the model file unless the modeler makes a specific request to save something else.) Indeed, if we had created the radial system using the GM command and stored the results for possible future use, the .NGF file might have reached 10 MB of storage space, depending upon the number of segments in each radial. So the GR function has a second benefit besides run time: it produces smaller NGF files.</p>
<p><b>A Loaded Reminder</b></p>
<p>In the NEC manual, there is a chart of control commands that collects them into 3 groups. The following list is from the NEC-4 manual</p>
<blockquote>
<p>Group I: FR, GN, IS, JN, LD, UM, VC</p>
<p>Group II: EX, NT, TL</p>
<p>Group III: CP, EN, GD, LE, LH, NE, NH, NX, PL, PQ, PS, PT, RP, WG, XQ</p>
</blockquote>
<p>The 3 groups correspond roughly to the 3 steps of solution generation. First comes the interaction matrix: calculation and factoring in preparation for solving for currents. The next step is to solve for currents with a given excitation. Finally comes the calculation of near and/or far fields. The first step depends only upon geometry commands and control commands in Group I. The current solutions depend upon commands in both Group I and II. Following a Group II entry by a Group I entry may result in a repetition of the current solution, since the Group I entry would result in a change of the conditions necessary for that solution. The re-calculation would depend upon the placement in the control command sequence of commands that execute, such as XQ and RP.</p>
<p>In the course of examining our first 3 test cases, involving the collection of vertical dipoles, the importance became apparent of arranging to the extent possible all control commands in the sequence of the groups. <b>Fig. 6</b> shows the set up for this modified version of the dipole array in free space. The loads will all by LD4 types with zero reactance and resistance values as shown in the diagram (in Ohms).</p>
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<p>The first of the models uses a straightforward separate specification for each vertical dipole. It follows the simple geometry entries with the Group I entries of FR and LD, with all 3 LDs placed sequentially. Next come the 2 Group II EX commands. A pattern request completes the model.</p>
<pre>
CM Test model: 4 vertical dipoles, inner 2 fed
CM 5th passive vertical dipole at a distance
CM Full version-LD4s all before EX
CE
GW 1 11 0 -.75 -.245 0 -.75 .245 .001
GW 2 11 0 -.25 -.245 0 -.25 .245 .001
GW 3 11 0 .25 -.245 0 .25 .245 .001
GW 4 11 0 .75 -.245 0 .75 .245 .001
GW 5 11 .5 0 -.245 .5 0 .245 .001
GE 0 -1 0
FR 0 1 0 0 299.7925 1
LD 4 1 6 6 1e6 0
LD 4 4 6 6 1e6 0
LD 4 5 6 6 100 0
EX 0 2 6 0 1 0
EX 0 3 6 0 1 0
RP 0 1 361 1000 90 0 1.00000 1.00000
EN
</pre>
<p>The model reports a maximum gain of 4.73 dBi with a 54-degree beamwidth. The reported source impedance is 61.157 - j12.362 Ohms. <b>Fig. 7</b> shows the phi/azimuth pattern for the model.</p>
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<p>Now let's revise the model to reflect a tendency among some modelers. Suppose that we had only specified the 1-MOhm loads for GW 1 and GW4, followed by the 2 EX commands. Then we went back to add a load of 100 Ohms resistance to the 5th dipole. We might inattentively insert the new LD4 command just before the request for pattern (RP) command. The following lines show the basic model file.</p>
<pre>
CM Test model: 4 vertical dipoles, inner 2 fed
CM 5th inert vertical dipole at a distance
CM Full version-LD4 on GW5 separated from other LD4s by EX
CE
GW 1 11 0 -.75 -.245 0 -.75 .245 .001
GW 2 11 0 -.25 -.245 0 -.25 .245 .001
GW 3 11 0 .25 -.245 0 .25 .245 .001
GW 4 11 0 .75 -.245 0 .75 .245 .001
GW 5 11 .5 0 -.245 .5 0 .245 .001
GE 0 -1 0
FR 0 1 0 0 299.7925 1
LD 4 1 6 6 1e6 0
LD 4 4 6 6 1e6 0
EX 0 2 6 0 1 0
EX 0 3 6 0 1 0
LD 4 5 6 6 100 0
RP 0 1 361 1000 90 0 1.00000 1.00000
EN
</pre>
<p>Interestingly, the model shows a maximum gain of 6.74 dBi with a beamwidth of 36 degrees. <b>Fig. 8</b> shows the phi/azimuth pattern. The model reports a source impedance of 58.156 - j14.714 Ohms.</p>
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<p>The revised pattern alone is enough to show that something has gone astray in at least one of the 2 models. Just what went amiss becomes clear from the following revised model. It simply removes the high-resistance loads on GW1 and GW4 from the model.</p>
<pre>
CM Test model: 4 vertical dipoles, inner 2 fed
CM 5th inert vertical dipole at a distance
CM Full version-LD4 on GW5 only
CE
GW 1 11 0 -.75 -.245 0 -.75 .245 .001
GW 2 11 0 -.25 -.245 0 -.25 .245 .001
GW 3 11 0 .25 -.245 0 .25 .245 .001
GW 4 11 0 .75 -.245 0 .75 .245 .001
GW 5 11 .5 0 -.245 .5 0 .245 .001
GE 0 -1 0
FR 0 1 0 0 299.7925 1
EX 0 2 6 0 1 0
EX 0 3 6 0 1 0
LD 4 5 6 6 100 0
RP 0 1 361 1000 90 0 1.00000 1.00000
EN
</pre>
<p>The new model with no loads on the outer passive vertical dipoles returns the identical results that we obtained from the version with the load on the 5th dipole following the EX commands and separated from the other LD4 loads. <b>Fig. 8</b> is the correct phi/azimuth pattern for this model. In the earlier model, separating the load entries by the EX commands resulted in a re-calculation that in effect omitted the earlier loads. If we want to include the loads on the outer dipoles in the model results, we have to use a modeling format similar to the first model in the final sequence of tests.</p>
<p>The lesson of these models is that it pays to group commands first by group level and second by type. The payback involves more important factors than mere run time. It also pays in terms of the accuracy of the output data.</p>
<p><b>Conclusion</b></p>
<p>Occasionally, we encounter interesting models that require diagnosis, and in the process, we either learn something new about the operation of NEC or we are reminded of some fine points about model construction that we might have let slip over months and years of modeling. This month's reminders are examples of that process. In fact, I might not have thought to mention either of the main ideas without the presence of a set of models to diagnose. While not every modeler might need these reminders, I thought I would pass them along while a. they were on my mind and b. I had a reasonable set of simple models by which to illustrate them.</p>
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<h1>106. Refining Our Notions of Azimuth Patterns</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>Many of these columns evolve from e-mail correspondence that I receive relative to antenna modeling. Sometimes, I receive over a short period of time multiple notes on a single subject. At other times, I receive seemingly diverse inquiries that turn out to have a common thread. The latter situation provides the basis for these notes in pursuit of a better understanding of so-called azimuth patterns.</p>
<p>The generic term in NEC for patterns taken along or parallel to the X-Y plane is a phi pattern. Starting with the X-axis, we count counterclockwise in degrees around the circle created by any phi pattern. When we speak of azimuth, we are resorting to the language of seamen, airmen, surveyors, and field engineers. That language presumes that there is a ground level at zero degrees elevation. (We shall, like NEC, presume a flat earth.) We count degrees from the North in a clockwise compass rose. In free-space, NEC may still have its phi pattern, since free space has a Cartesian X-Y plane. However, the idea of azimuth becomes problematical, because we lack a ground. So we often speak in terms of E-plane and H-plane patterns, but these patterns designations depend on how we have oriented the antenna, assuming a relative linear polarization. Software makers--for simplicity--tend to label all X-Y plane patterns as azimuth patterns. However, in many cases, the pattern produced is really a phi pattern that counts counterclockwise.</p>
<p>As daunting as the mere labeling of phi/azimuth patterns can be, the problem is small compared to others that may occur is we are not careful. In these notes, we shall deal with only 2. One is the continuing subterranean discussion about the use of log vs. linear polar plots. The other involves azimuth patterns over ground when the pattern's elevation angle is more than a few degrees above the horizon.</p>
<p><b>Log vs. Linear Polar Plots</b></p>
<p>Although 3-dimensional plotting capabilities come with much modeling software, the most telling data usually derives from 2 dimensional patterns. We have already described the phi/azimuth pattern. Corresponding to it, but with a vertical dimension, is the theta/elevation plot. Most often, we set the phi/azimuth angle for a theta/elevation plot by reference either to the centerline of an antenna array or with respect to the direction of the strongest forward lobe. We can use other angles as the need arises. Theta/elevation patterns seem to present no major problems to newer antenna modelers. Even the translation between theta and elevation conventions is simple. An elevation angle is 90 - a theta angle (degrees) and vice versa. Perhaps the only difficulty involves setting up an elevation pattern's range of angles to be sure that we obtain a full circle in free space.</p>
<p>Over ground, the phi/azimuth pattern causes the most discussion. The discussion tends to extend itself into free space because the plots with and without a ground beneath have such similar shapes. The discussion hinges around whether we ought to be using a logarithmic or a linear scale for the rings in the pattern. Some modelers, especially radio amateurs, may have never seen a linear plot, since most publications in the field use one or another form of log plot. Many developmental and field engineers are most experienced with linear plots, sometimes because the regulatory submissions that they make set requirements of plots--and they are almost always on linear scales.</p>
<p>Over the years, the discussion has acquired some spurious reasoning. For example, a few linear plot proponents claim that we receive unrealistic plots using a log scale, since power gains are already in decibels, a log concept. More interesting is the claim that a log scale makes the main forward lobe of an antenna more dominant, giving a more favorable impression of the antenna performance than it deserves. Since most proponents of a log scale prefer it by habituation, there have been few replies, although log plots tend to dominate the presentations in most publications.</p>
<p>So our question is simple: what difference does it make whether I use a linear or a log scale when making a polar plot? We can settle some of the issues by pursuing a simple exercise. Let's set up a model in free space and plot its phi/azimuth/E-plane pattern. My model is the following simple NEC listing.</p>
<pre>
CM 12-element Yagi, 223.5 MHz
CE
GW 1 19 0. .3225463 0. 0. -.3225463 0. .003175
GW 2 19 .2628857 .3125082 0. .2628857 -.3125082 0. .003175
GW 3 19 .3613731 .2856135 0. .3613731 -.2856135 0. .003175
GW 4 19 .5981218 .2833407 0. .5981218 -.2833407 0. .003175
GW 5 19 .8807049 .2799316 0. .8807049 -.2799316 0. .003175
GW 6 19 1.209123 .2767118 0. 1.209123 -.2767118 0. .003175
GW 7 19 1.577314 .2738708 0. 1.577314 -.2738708 0. .003175
GW 8 19 1.971453 .2712192 0. 1.971453 -.2712192 0. .003175
GW 9 19 2.385479 .2691358 0. 2.385479 -.2691358 0. .003175
GW 10 19 2.819392 .2672419 0. 2.819392 -.2672419 0. .003175
GW 11 19 3.272813 .2655372 0. 3.272813 -.2655372 0. .003175
GW 12 19 3.745931 .2640221 0. 3.745931 -.2640221 0. .003175
GE 0 0 0
LD 5 0 0 0 2.5E+07 1.
FR 0 1 0 0 223.5 1
GN -1
EX 0 2 10 0 1.00000 0.00000
RP 0 1 361 1000 90 0. 1.00000 1.00000 0.
EN
</pre>
<p>The dimensions of the 12-element Yagi are in meters. The LD5 entry casts the antenna in aluminum. The operating frequency is in the middle of the U.S. 220-MHz band. As shown in the outline sketch in <b>Fig. 1</b>, the second element is the driver, fed at its center.</p>
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<p>The antenna design rests on the work of DL6WU, Guenter Hoch. I am less interested at the moment in the overall performance of the antenna than in the fact that DL6WU designs tend to have significant forward sidelobes that are down from the main forward lobe by between 15 and 18 dB. That will give our polar phi/azimuth/E-plane plot a certain look. In fact, the look will not change if we uses a log scale and move from one software package to another, assuming that we normalize the plot, that is, set the plotting software so that the outer ring and the maximum forward lobe just meet. Therefore, GNEC and EZNEC Pro/4 patterns will appear identical. A plot produced by an ARRL publication may initially look different, but will be essentially the same. ARRL likes to place lines in 3 or 6 dB intervals, while most software uses 10 dB as a major circle in the plot. In short, the plot will look like the one at the upper left of <b>Fig. 2</b>.</p>
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<p>Perhaps the most significant reason for using a log-based polar plot of phi/azimuth/E-plane patterns is the fact that for any antenna, if we do not change the antenna placement or environment, the plot will be the same no matter which software we use for the plotting. We cannot make the same claim for linear polar plots.</p>
<p>The plots shown comes from NSI software for a reason. EZNEC allows only one form of linear plot, but NSI software let's the user specify the inner limits and the ring increment for linear plots. As we change the inner limit and the increment, our polar plots may change their shape. Precisely here lies one of the problems with linear plotting of antenna patterns. If we do not have an external standard to direct our plotting, we can make the antenna look as sorry or as distinguished as we please. There are numerous applications in which we find external standards and some in which we find corporate standards that rest on long experience. In either case, we may directly compare plots and conduct fair evaluations.</p>
<p>In the absence of such standards, the field is wide open. For example, the upper right pattern of <b>Fig. 2</b> uses an inner limit of -100 dB with a 20-dB increment in the plotting circles. Using these scale factors, we can tell almost nothing about the antenna except perhaps that its front-to-back ratio is about 20 dB. The middle pair of linear scales use -50 and -40 dB inner limits. At this level, some of the antenna pattern details become much clearer. In fact, the -40-dB version of the plot is essentially the linear scale provided by EZNEC software. Where the scale rings have good labeling, we would have no trouble gleaning essentially the same information from both linear patterns. In fact, newer modelers may wish to track both patterns against the log pattern at the upper left in order to gain a more intuitive grasp of the differences between where on the plot various pattern levels occur.</p>
<p>Just as the upper right pattern seems to obscure important data by virtue of its bulbous shape, the two patterns at the bottom of <b>Fig. 2</b> may obscure important data by using an inner limit and increment level that are too small. In both cases--one somewhat more extreme than the other--the plot tends to enhance the pattern and to give it a purity that simply does not exist. I have seen plots of this order used in older articles, most recently in a review of professional literature on rhombic antennas. The plots could give the impression to the unwary reader that a rhombic antenna has no sidelobes worth mentioning, when in fact, the antenna sprays sidelobes everywhere, with the strongest ones only 8 to 12 dB below the strength of the main lobe.</p>
<p>The linear pattern has plenty of uses and is not inherently simply a means to yield false impressions of an antenna. Field strength readings in volts/meter lend themselves to linear plotting. However, even in this enterprise, one needs a set of plotting standards to facilitate comparisons among plots. The one sure way to make such plots yield misimpressions is to allow scaling decisions to be idiosyncratic. The log plot of power gains obviates the danger by using a relatively constant scale, with only ring values as a plotting option. Having the scale prescribed and in unison with patterns produced by others is a good assurance to less experienced modelers that their results will in fact be comparable with other work. (This positive fact does not eliminate the many other ways in which one can mess up a model or the polar plot that we take of it. One common problem for azimuth plots over ground is the switching of the values for ground conductivity and relative permittivity. I have also seen plots that have intentionally used high values of conductivity and permittivity solely to enhance an antenna pattern. However, the connecting thread of these notes is not the ways in which we can give false impressions of an antenna's performance.) In the end, the use of plots--whether based on log or linear scales--requires good sense and standards if we are to derive from them all of the information available. If a linear plot is necessary for a power gain or other antenna property pattern, then in all cases the plotter needs to annotate the plot so that the user knows exactly how to interpret the data it encases. As well, the text should provide a rationale for the selected inner limit or a reference to an external standard, if it is applicable to the plot.</p>
<p><b>Azimuth Plots Over Ground</b></p>
<p>Phi/azimuth patterns taken over ground hold another potential problem--a lack of reader appreciation of what such patterns really are. Even those who make such plots using standard plotting techniques (for example, the usual log plot) may not fully appreciate what the data from the patterns may be telling us. The most common far-field plots shown on phi/azimuth patterns use relatively shallow elevation angles--somewhere between 1 and 25 degrees. At higher angles, we very often have an interest only in the general pattern shape and the maximum gain, so a lack of full understanding of the terms of an azimuth pattern holds no especial dangers. However, when we begin to look at the fuller data attached to such patterns, we often encounter hurdles for which we are not prepared.</p>
<p>Once more, lets use a specific example. In this case, I shall select a very long-boom Yagi for 432 MHz. The antenna design specifically aims for high performance in as many categories as possible. The gain level is normal for the boom length. The good front-to-back values are overshadowed by the very high level of sidelobe attenuation. In addition, the antenna provides a direct 50-Ohm match and a passband wide enough to cover the entire 70-cm amateur band. <b>Fig. 3</b> provides the free-space patterns and data, as well as a scale outline of the array. In this case, the patterns are log scale and from EZNEC Pro/4.</p>
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<p>One application for an antenna of this size is earth-moon-earth (EME) communications, using the moon as a passive and low-efficiency reflector. EME service requires that we be able to control the azimuth and elevation angles of the antenna. Therefore, one interest we may have in the antenna is its performance at various elevation angles. The antenna requires a minimum boom-center height of about 7.5 wavelengths to keep the reflector at least 1 wavelength above ground when pointed straight upward. I sampled the antenna performance at elevation angles from zero degrees through 90 degrees in 15-degree increments. The results appear in the series of elevation and azimuth plots shown in <b>Fig. 4</b>. The following table (<b>Table 1</b>) shows the values that correlate with the patterns.</p>
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<p>If we first examine only the elevation plots, we can observe an interesting pattern. As we elevate the antenna above an angle where the free-space vertical beamwidth interacts with the surface of the ground, the vertical beamwidth stabilizes at or very near to the free-space vertical beamwidth. We essentially achieve that situation by an elevation angle of 30 degrees.</p>
<p>If we next examine the azimuth patterns, we find an oddity (at least, at first sight). The horizontal beamwidth increases steadily until we pass the 75-degree elevation level. Indeed, the azimuth pattern forward and rearward elements appears to broaden with each increase in elevation angle. However, the patterns for an elevation angle of 90 degrees return essentially to free-space values, with the exception that there are no longer any rearward lobes. Of course, we have changed the procedure at a 90-degree elevation angle and substituted H-plane and E-plane elevation patterns for the expected elevation and azimuth patterns. Below 90 degrees elevation, we do not have this option. Our task is to understand why the horizontal beamwidth and other aspects of the azimuth pattern broaden with an increasing elevation angle.</p>
<p>The answer lies in simple but often overlooked aspects of the conical geometry that is a function of taking phi/azimuth patterns over ground. In every case, we use an elevation angle that is greater than zero. (Over perfect ground, we may take a phi/azimuth pattern at zero degrees elevation, but over real ground, the results are either a negligible set of power gain values or a message informing us that we have requested an illicit phi/azimuth pattern. If we need far-field values very near to ground level, we may specify a very small elevation angle, such as 0.1 degree and also set a finite distance for the pattern's field strength readings.)</p>
<p>The result of our phi/azimuth pattern request is a pattern that involves something other than a flat circle. Instead, the field strength and power gain values occur on the surface of a cone. Many directional antenna patterns do not show the same elevation angle for the strongest forward lobe as for the strongest rearward lobe. When we examine the 180-degree front-to-back ratio calculated from pattern data by NEC software implementations, we discover that the ratio rests on the strength of the rearward lobe at the same elevation angle that we set for the forward lobe. At very low angles, the polar plot pattern does not show any significant distortion that results from the transfer of the conical surface data to a flat circular or polar plot. However, as we increase the angle, foreshortening distortions occur. <b>Fig. 5</b> shows a sample of the situation, although the sketch is itself imperfect.</p>
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<p>The conical section on the left shows an elevation angle of 60 degrees (or a theta angle of 30 degrees). Let us suppose that we have sliced the cone vertically at page or screen level so that we see only half of the cone's surface. The visible section shows us the forward lobe of a hypothetical beam. The angle marked at the top of the sketch shows the beamwidth of the visible lobe. However, even this angle is slightly off the mark, since it transfers a curved surface that slopes away (or toward) the page or screen directly onto the flat page or screen. Nevertheless, the angle is close enough to the actual value to demonstrate what happens when we create a phi/azimuth pattern from this information.</p>
<p>On the right is a polar plot of the pattern. The circles represent rings around the cone. For simplicity, the polar plot uses a linear set of ring scales so that the distance between rings is the same for both the conical section and for the polar plot. When we plot the points of contact between the rings and the pattern width onto the polar plot, we obtain the pattern in the flat polar plot. The plot produced by any graphic addition to NEC uses the data in its radiation pattern report, and this data results in the plot shown, assuming that we have specified the phi/azimuth pattern situation shown on the left.</p>
<p>The conical section assumes that the half-power points of the pattern happen to coincide with the intersection of the pattern line with the next-to-outermost ring. Each point will shows a gain value that is 3 dB lower than the maximum power gain for the antenna. Those points will yield the same gain values when transferred to the flat polar plot. If we create the angle included between those points on the polar plot, we obtain an angle that is a bit over twice as wide as the angle included by the corresponding points on the surface of the cone. In fact, the proper relationship for the 60-degree elevation angle is 2:1, but remember that the representation of the cone surface is distorted by its representation on a flat surface.</p>
<p>For most cases where the antenna is considerably above ground, we may determine (or approximate) the actual horizontal beamwidth from the reported value that results from the foreshortening effect of transferring a conical section to a flat circle.</p>
<blockquote>
<p><b>BWa = BWr * cos(elevation) or BWa = BWr * sin(theta)</b></p>
</blockquote>
<p>BWa is the actual horizontal beamwidth, BWr is the NEC report of the beamwidth, and the indicated angles are the elevation or theta angle at which we take the phi/azimuth pattern. For example, at an elevation angle of 45.4 degrees (the take-off angle that results from pointing the antenna at an angle of 45 degrees upward), we have a reported horizontal beamwidth of 27.8 degrees. The cosine of 45.4 degrees is 0.702. Multiplied times the reported horizontal beamwidth, we obtain 19.5 degrees actual beamwidth, a value that falls between the listed free-space value and the E-plane value that occurs with an elevation angle of 90 degrees.</p>
<p>To confirm that the horizontal beamwidth has not actually undergone any significant change as we elevate the antenna, let's examine a few 3-dimensional patterns. <b>Fig. 6</b> shows the subject antenna's patterns at aiming angles of 60, 75, and 90 degrees relative to ground. The pattern resolution is only 5 degrees, so considerable detail is missing from the graphics. The alternative would be to use a very small pattern increment and end up with patterns that appear only as black blobs.</p>
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<p>Despite te missing detail, we can observe that the main lobe of the beam retains its essential shape throughout the changes in elevation angle.</p>
<p><b>Conclusion</b></p>
<p>These notes represent an attempt to improve our understanding of oft-overlooked aspects of phi/azimuth patterns. The first exercise explored the differences between logarithmic and linear plots. The log plot emerged essentially to lend to phi/azimuth patterns a uniformity that had been lacking in commonly used linear plots. Since we may select the inner limit and the increment for a linear polar plot, these patterns may vary widely for applications that are not subject to external standards. Any user-developed linear polar plot of power gain values should plainly show the inner limit used. As well, the text should explain the rationale for the inner limit selection, even if the explanation is no more than a reference to an external or corporate standard.</p>
<p>Phi/azimuth patterns over ground represent patterns taken on a conical surface. At small elevation angles, polar plots do not severely distort the conical section by transferring the data to a flat circle. However, as we increase the elevation angle for the phi/azimuth plot, the transfer does introduce appreciable foreshortening or broadening of the actual pattern. One result is a misreport of the horizontal beamwidth of an antenna, although we may use a simple calculation to produce a reasonable approximation of the correct value.</p>
<p>These are but two of many aspects of NEC modeling results that we tend to overlook in the process of trying to produce usable data from a carefully constructed model geometry. However, even the best model is subject to misinterpretation of the output data unless we are very careful in how we present it and how we read the presentation.</p>
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<h1>107. Scaling Models</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>Scaing antennas from one frequency to another is common practice. In general, we wish to capture certain performance characteristics that we find at one frequency and transfer them to another. The most straightforward way to achieve this goal is to scale the antenna that has the desired characteristics.</p>
<p>We often need or desire to scale antenna models from one frequency to another. The techniques of model scaling are identical to those of direct antenna scaling, although the way in which we go about the task may differ with the implementation of NEC or MININEC that we are using. Many implementations of modeling cores have a pre-core-run facility for either full or partial model scaling. As well, we may use one of the commands within the full NEC command set to arrive at the same result--a sort of unadvertised special.</p>
<p><b>Basic Scaling</b></p>
<p>To scale a model from one frequency to another requires us to change the element length, the element spacing, and the element diameter (or radius) everywhere in a model or antenna. Let's call the general idea of something to change when scaling the "dimension." We shall have an old or starting set of dimensions and a new or final set of dimensions. The amount of change depends on the old frequency or wavelength and the new frequency or wavelength. The basic formula is simple:</p>
<pre>
new wavelength old frequency new dimension
-------------- = ------------- = -------------
old wavelength new frequency old dimension
</pre>
<p>Note that the dimension changes are directly proportional to the ratio of ratio of wavelengths and inversely proportional to the ratio of frequencies. Since we generally work with frequency in most initial thinking, we simply take the ratio of the old to the new frequency and multiply every old dimension by the value. The easy way to check on whether we are using the correct ratio is to remember that elements get shorter and thinner as we scale up in frequency and longer and fatter as we scale down in frequency.</p>
<p>Some implementations of NEC and MININEC allow you to enter the wire diameter as an AWG value. In many instances, we forget to scale the wire diameter along with the element length and spacing. The cure is to remember to look at the wire gauge or diameter before declaring that the scaling is complete. In some cases, very thin wire antenna elements will show only small performance changes if we forget to change the wire size and if we are only scaling over a limited frequency range--for example, converting a 14-MHz antenna design to 18 MHz. However, the fatter the element or the greater the scaling ratio, the more significant it becomes to make certain that we scale the element diameter or radius along with the other dimensions.</p>
<p>In NEC and MININEC, precise scaling will yield identical performance at both the old and the new frequencies only if certain model conditions exist. First, the antenna material should be "perfect" or "lossless." Even high conductivity materials such as copper show different working values of conductivity as we change frequency. The skin effect changes slowly as we change frequency--enough to show up in results that only make a relatively small frequency change during scaling. Second, the antenna environment should be free space or a perfect ground. Ground effects from any lossy ground change with frequency, again enough to show up in models that we scale over a relatively small range if the antenna is within a few wavelengths of the earth's surface.</p>
<p>In most cases, it is useful to remove material losses and a lossy ground from an antenna model prior to scaling. Then proceed with the scaling calculations and test the model using the new values--remembering to a. change the test frequency for the revised model and b. save the model under a revised file name. Once you are satisfied that the scaling operation is correct, you can add in the material loss factor (LD5) and the lossy or real ground constants (GN).</p>
<p><b>Sample Program Scaling Facilities</b></p>
<p>Many implementations of NEC and MININEC contain facilities to assist in the scaling process. Let's sample a full scaling facility and a partial scaling facility. For a subject antenna, we shall begin in all cases with a 2-element Yagi designed for 28.5 MHz. To further focus on the scaling activity, I have placed it in free space and used lossless elements. So our only project will be to scale the antenna to 14.25 MHz. The ratio of the old frequency to the new frequency is 2:1 to keep the arithmetic obvious.</p>
<p>We also have the option of manually calculating the new dimensions and manually entering them into the wire or GW entries. However, in some programs, such as EZNEC, we need not go through this process. Consider the 2 screens in <b>Fig. 1</b> taken from the initial 10-meter antenna model. The upper screen shows the wire entries with all dimensions in meters. The lower screen shows our design frequency.</p>
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<p>EZNEC implements scaling through the frequency selection screen. Note the box marked "Rescale." Let's revise the frequency to 14.25 MHz and also click on the rescale box. The result will be the set of wire entries shown in <b>Fig. 2</b>. The user made none of the changes in the wire tables. Rather, the program made the necessary changes after the user entered the new frequency and checked the rescale box.</p>
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<p>Element length values appear in the Y-columns, with the spacing values in the X-columns. The program uses wire diameter as the entry dimension, and it has its own column. A quick comparison of <b>Fig. 1</b> and <b>Fig. 2</b> shows that EZNEC's scaling facility has doubled the element length, the element spacing, and the element diameter for the entire model geometry. The version of EZNEC on which I ran these models used NEC-4. For both models--the 10-meter and the 20-meter versions--the free-space gain reported as 6.28 dBi, with a 180-degree front-to-back ratio of 11.25 dB. In both cases, the feedpoint or source impedance was 32.59 - j0.30 Ohms.</p>
<p>Not all programs that provide scaling facilities scale everything. Consider the same model in its NEC-Win Plus form, as shown by the upper portion of <b>Fig. 3</b>. The 10-meter antenna has the same dimensions as the EZNEC model. Because NEC-Win Plus uses NEC-2, we expect very small changes in the output report values. The free-space gain is 6.28 with a front-to-back value of 11.25. The source impedance is 32.58 - j0.55 Ohms. The differences from the NEC-4 report are wholly insignificant.</p>
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<p>The lower portion of the figure shows the scaling screen that the user accesses by clicking on a special button near the top of the NEC-Win Plus screen. The screen contains a wealth of information on the scaling maneuver, showing both initial and final values before the user locks the changes into place. To use this screen effectively, I blocked the full wire lines for the model from the left to the right extremes. The blocking encompassed everything, including segmentation, coordinates, wire diameter, and wire material.</p>
<p>The scaling operation requires the user to enter the scaling factor, in this instance, 2.0. As well, the user must remember to change the model frequency before running the revised model. After making these changes and locking in the re-scaling, I ran the model. It returned a gain of 6.18 dBi, a front-to-back ratio of 11.72, and a source impedance of 33.57 - j70.01 Ohms. Something has gone astray, and in this case, it was carrying over operational expectations from one program to another.</p>
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<p><b>Fig. 4</b> shows us what went wrong by not reading the instructions appropriate to this implementation of NEC-2. The NEC-Win Plus scaling facility changes only the element lengths. However, as the X-column shows, it does not automatically change the spacing. As well, the diameter column shows that the program does not automate the diameter scaling. Hence, the model that gave us the aberrant results is not a true scaling of the original antenna. If we double both the element spacing and wire diameter, the corrected 20-meter scaling of the 10-meter Yagi produces the same output reports.</p>
<p>Different implementations of NEC and MININEC will handle re-scaling a model in ways that range from a totally manual operation to complete automation. Since scaling is such a simple arithmetic operation, it usually makes little difference how we make the required changes. The key is to make all of them.</p>
<p><b>The NEC GS Command</b> The GS command in the NEC set is so easy to use that most instruction sets tend to overlook it. Even the NEC manuals give it only brief treatment. The command structure has only 3 entries, two integers and one floating decimal.</p>
<pre>
CMD I1 I1 F1 (FSCALE)
GS 0 0 0.3048
</pre>
<p>The command's common use is to convert the units of measure in preceding geometry commands (such as GW) into meters, if they are not already in meters. The command must appear after all of the geometry commands that use an alternative unit of measure and before the GE or geometry section end command. NSI software provides a help screen for entering the scaling values, as shown in <b>Fig. 5</b>.</p>
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<p>For manual entry of the conversion unit, the integer entries are zero. Some manuals say they are not used. However, some core versions have automated conversion entries. If I1 = 1, then the conversion is from feet to meters (0.3048). If I1 = 2, then the conversion is from inches to meters. In these cases, I2 and F1 might not appear, or I2 might be zero. However, if both I1 and I1 are zero, then F1 must have a conversion factor value. The user can insert any appropriate value. For example, if the geometry entries are in millimeters, then the value of F1 is 0.001.</p>
<pre>
CM 2el Yagi 7.9/8.67/4.25 fs
CE
GW 1 21 0. 8.67 0. 0. -8.67 0. .04167
GW 2 21 4.25 7.9 0. 4.25 -7.9 0. .04167
GS 0 0 0.304800
GE 0
FR 0 1 0 0 28.5 0
GN -1
EX 0 2 11 0 1.00000 0.00000
RP 0 1 361 1000 90. 0. 1.00000 1.00000 0.
EN
</pre>
<p>The sample model is essentially the same 2-element 10-meter Yagi, but with the 2 elements entered in feet. Hence, the conversion factor in the GS line is 0.3048. Since the dimensions are rounded, we shall find a very slight variance in the output reports, but at levels that are wholly insignificant. For example, the reported source impedance for the Yagi is 32.54 - j0.36 Ohms using the NEC-4 core.</p>
<p>Suppose that we enter our coordinates and wire radius in meters. Since we do not need to convert the units to meters, we might overlook the GS card. The top portion of <b>Fig. 6</b> shows the 10-meter Yagi entered in meters. However, the GS card is still useful to us in scaling the antenna to 14.25 MHz from its original frequency, 28.5 MHz. Note the "custom" conversion factor shown in <b>Fig. 5</b>. The lower portion of <b>Fig. 6</b> shows the revised and scaled Yagi model with the GS card used to do the scaling--along with the required revision of the FR entry.</p>
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<p>To confirm the correct functioning of the GS command, we may examine a 1-line extract from the NEC output report's "Segmentation Data" section that lists the coordinates of all segments in the model. The first line for one end of the GW1 command appears for both the pre-scaled and the post-scaled models. The Y-coordinate, the segments lengths, and the wire radius entries show the scaling accomplished by the GS command. Both models in NEC-4 report a gain of 6.28 dBi, a front-to-back ratio of 11.25 dB, and a source impedance of 32.58 - j0.33 Ohms.</p>
<pre>
28.5-MHz Yagi Model
SEG. COORDINATES OF SEG. CENTER SEG. ORIENTATION ANGLES WIRE CONNECTION DATA TAG
NO. X Y Z LENGTH ALPHA BETA RADIUS I- I I+ NO.
1 0.00000 2.51714 0.00000 0.25171 0.00000 -90.00000 0.01270 0 1 2 1
14.25-MHz Yagi Model
SEG. COORDINATES OF SEG. CENTER SEG. ORIENTATION ANGLES WIRE CONNECTION DATA TAG
NO. X Y Z LENGTH ALPHA BETA RADIUS I- I I+ NO.
1 0.00000 5.03429 0.00000 0.50343 0.00000 -90.00000 0.02540 0 1 2 1
</pre>
<p><b>Other GS Potentials</b></p>
<p>The GS command scaling potential is not limited to simple one-shot frequency conversions. The following practice is one sample of what we may do with the command in a more systematic way. We shall initially enter all geometry commands in meters. We shall also change the subject antenna to a 6-element Yagi. However, our basic design will use a frequency of 299.7925 MHz. At this frequency, 1 meter = 1 wavelength. Our basic model might resemble the following lines.</p>
<pre>
CM 6-el 300 Yagi
CE
GW 1 21 0. .251 0. 0. -.251 0. .001
GW 2 21 .1251 .2484 0. .1251 -.2484 0. .001
GW 3 21 .1705 .2312 0. .1705 -.2312 0. .001
GW 4 21 .321 .225 0. .321 -.225 0. .001
GW 5 21 .4617 .225 0. .4617 -.225 0. .001
GW 6 21 .6713 .2167 0. .6713 -.2167 0. .001
GE 0
FR 0 11 0 0 295 1
GN -1
EX 0 2 11 0 1.00000 0.00000
RP 0 1 361 1000 90. 0. 1.00000 1.00000 0.
EN
</pre>
<p>We may scale this Yagi to any frequency whatever using the GS command and one easy calculator step. The required GS custom conversion entry adheres to a simple equation where CF is the conversion factor.</p>
<pre>
CF = 299.7925 / new frequency
</pre>
<p>Suppose that we wish to scale the Yagi to 15 MHz. The value of CF is 19.986167 (or any usable rounding of that value). The revised model would have the following appearance.</p>
<pre>
CM 6-el 300 Yagi
CM Scaled to 15 MHz via GS
CE
GW 1 21 0. .251 0. 0. -.251 0. .001
GW 2 21 .1251 .2484 0. .1251 -.2484 0. .001
GW 3 21 .1705 .2312 0. .1705 -.2312 0. .001
GW 4 21 .321 .225 0. .321 -.225 0. .001
GW 5 21 .4617 .225 0. .4617 -.225 0. .001
GW 6 21 .6713 .2167 0. .6713 -.2167 0. .001
GS 0 0 19.986167
GE 0
FR 0 1 0 0 15 1
GN -1
EX 0 2 11 0 1.00000 0.00000
RP 0 1 361 1000 90. 0. 1.00000 1.00000 0.
EN
</pre>
<p>Both models return a free-space gain of 10.28 dBi, a 180-degree front-to-back ratio of 31.12 dB, and a source impedance of 58.77 + j10.09 Ohms at 299.7925 MHz and at 15 MHz. The purpose of using the 6-element Yagi lies in the original model's frequency sweep specification. For a center frequency of about 300 MHz, the usable operating passband is about 10 MHz. Within that passband, the design has a free-space gain of at least 10.1 dBi, a 180-degree front-to-back ratio of 19.5 dB or higher, and a 50-Ohm SWR of under 1.3:1. When we apply the scaling conversion factor, we must also apply it to the operating passband. Hence, at 15 MHz, the equivalent passband is only about 0.5 MHz wide.</p>
<p>The scaling technique is perfectly general and may be useful in a variety of modeling activities. However, it also has limitations. Perhaps the most notable limit is the wire radius, which is just under 20 mm at 15 MHz. Unless we are very judicious in constructing our master models, we may still have to optimize them at other frequencies for changes in wire radius, as well as for tapered element diameter schedules in HF antennas. However, any method of scaling an antenna design will result in these same supplementary tasks.</p>
<p>One tendency among modelers is to freeze the potential for the GS command into an unbreakable habit. For example, some modelers believe that the GS command must precede the GE command and follow all other geometry commands. However, the command is more flexible than this over-simplified view. The GS command must simply follow the set of geometry commands that use a unit of measure other than meters. The following artificial sample mixes measures. GW1, the reflector, has its units in feet, while GW2, the driver, uses meters. We may as an exercise retain the mixed measures by inserting the GS card immediately after GW1 to convert its dimensions to meters. The following GW2 entry is unaffected by the action of the GS command.</p>
<pre>
CM 2el Yagi 7.9/8.67/4.25 fs
CE
GW 1 21 0. 8.67 0. 0. -8.67 0. .04167
GS 0 0 0.304800
GW 2 21 1.295 2.408 0. 1.295 -2.408 0. .0127
GE 0
FR 0 1 0 0 28.5 0
GN -1
EX 0 2 11 0 1.00000 0.00000
RP 0 1 361 1000 90. 0. 1.00000 1.00000 0.
EN
</pre>
<p>The technique just described has very limited application. Using a uniform set of coordinate measures throughout the geometry section of a model is always good practice. Nevertheless, the technique does illustrate that the GS command is a bit more flexible than we might have previously thought.</p>
<p>A more common situation might be to develop an initial model in a preferred set of units that are not meters. Then we might wish to scale the antenna. Although we shall use our simple 2-element Yagi as an illustration, the time factor required for manual revision of all geometry entries will increase very sharply for highly complex models. However, we might ease the task by using multiple GS cards. Consider the 28.5-MHz Yagi with its dimensions in feet. Next, we wish to scale the antenna to 14.25 MHz. The resulting model might have the following appearance.</p>
<pre>
CM 2el Yagi 7.9/8.67/4.25 fs
CM Scaled to 14.25 MHz via GS
CE
GW 1 21 0. 8.67 0. 0. -8.67 0. .04167
GW 2 21 4.25 7.9 0. 4.25 -7.9 0. .04167
GS 0 0 0.304800
GS 0 0 2
GE 0
FR 0 1 0 0 14.25 0
GN -1
EX 0 2 11 0 1.00000 0.00000
RP 0 1 361 1000 90. 0. 1.00000 1.00000 0.
EN
</pre>
<p>The example shows that we may use strings of GS entries to accomplish different tasks. We might also have combined the two conversion factors externally. However, Murphy's Law tells us that in 3 months, we will no longer be able to remember what the GS value of 0.6096 means. Sometimes, separate entries (each with an appropriate side notation) are more useful. The small time it takes to enter 2 GS commands can save much head-scratching time later on.</p>
<p><b>Conclusion</b></p>
<p>Frequency scaling of antenna designs has a wide range of uses. They encompass practical design work as well as systematic explorations of antenna properties at various frequencies. As noted early on, scaling best occurs using a free-space or perfect-ground environment, with other model variables introduced after verification of successful scaling.</p>
<p>The methods available to us for scaling designs also cover a considerable territory. We may manually calculate and enter scaled geometry values. We may use the facilities provided by the interface programming of the particular NEC or MININEC implementation that we are using. If we have access to the complete NEC command set, then we may achieve the same ends via the GS or geometry scaling command. The command is useful for more activities than just converting units of measure to meters. To confirm correct command use or to record the resulting coordinate and radius values, we must consult the NEC output report, using the segmentation-data section as our primary resource. I do not anticipate widespread use of the GS command for frequency scaling exercises in NEC. Nevertheless, it is useful to be aware of the command's potential and versatility.</p>
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<h1>108. Dipoles: Variety and Modeling Hazards<br>
Linear, V, and Folded Dipoles in NEC</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>A dipole is in basic texts any antenna that exhibits a single current maximum to minimum transition from the center feedpoint to the wire end, and conversely, transitions from minimum to maximum voltage as we move from the center to the outer end of the wire. Basic antenna theory rests to some degree on the performance or the behavior of very short dipoles. In practical antenna circles, the term "dipole" is generally shorthand for a specific subset of these antenna. A practical dipole is a center-fed, resonant or near resonant 1/2 wavelength antenna usually using linear (wire or tubing) construction. When the antenna grows too long, we no longer have the current and voltage transitions that define the dipole, and so the antenna becomes a doublet--a center-fed wire with otherwise relatively undefined characteristics.</p>
<p>In these notes for relative newcomers to antenna modeling, I shall use the practical-dipole definition. Practical dipoles include the idea of resonance, that is, a center feedpoint impedance that is entirely resistive (or nearly so), with no (or very little) remnant reactance. These antennas are very practical for many applications. At this point, we begin to encounter physical variations on the usual linear dipole construction. The key parameter shifts from the pattern of current an voltage along the wire to resonance, and we use resonance to determine that an antenna is an electrical half wavelength long and hence a dipole. Under this revised view of a practical dipole, we encounter many configurations, all of which count as dipoles if we use resonance and an electrical half wavelength as the key determining factors. <b>Fig. 1</b> shows a number of these configurations, but by no means all of them.</p>
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<p>The standard linear dipole and its impedance-transforming partner, the folded dipole, are most familiar to virtually all antenna users. The fan dipole is actually 2 dipoles for separate frequencies with a common feedpoint. The V dipole occurs most commonly as an inverted V, although we may set the V angle at any value. At 90 degrees, with the legs parallel to the ground, we obtain a quadrant antenna. We may also form Ls and inverted-Ls with corner feedpoints as variations on the Vee. The zigzag dipole is simply any set of relative small departures from a truly linear arrangement, usually done to fit a dipole within a restricted space. The hatted dipole uses symmetrical structures at each end of the main element to obtain resonance with a shorter overall length than a full-length dipole requires. The bent or inverted-U form achieves element shortening with single extensions. However, these extensions radiate with vertical polarization, whereas the hat structures usually have self-canceling fields. The final example of a shortened dipole uses fold-back structures. The sketch shows only one of many possible forms, sometimes called the lazy N.</p>
<p>These notes are not designed to evaluate the relative merits of the various configurations. Instead, we want to examine the modeling challenges that some of them might present. Some of the practical dipoles can easily result in unreliable models if we do not attend to software limitations, especially of NEC. So we shall examine the dipoles using both NEC-2 and NEC-4. One of our tools will be the Average Gain Test (AGT), which assesses the reliability of a model by sampling a lossless model in free space (or over perfect ground, which is not relevant here) over a full sphere of points and comparing the average gain to an isotropic source. Hence, an AGT value of 1.000 is excellent, but values above and below that value indicate potential degrees of unreliability. For source impedances close to resonance, we can derive corrected values by multiplying the AGT value times the reported source resistance. We can also convert the AGT value to a value in decibels. If the derived AGT-dB value is positive, we subtract it from the reported gain to obtain a correct gain value. (This note applies at any vector from the antenna, not just to maximum gain.) If the derived AGT-dB value is negative, we add its absolute value to the reported gain, since it tells us by how much (approximately) the gain report is low. We must always remember that the AGT is a necessary but not a sufficient condition of model adequacy. A good AGT score does not guarantee model adequacy.</p>
<p>This first portion of our journey will deal only with modeling some of the dipoles in NEC (both -2 and -4). In the next episode, we shall compare our results with modeling the same antennas in at least two versions of MININEC. Once we have the basic terms and limits of each program under our belts, we can take up the remaining versions of the dipole using both general types of antenna modeling programs.</p>
<p><i>The Standard Linear Dipole</i></p>
<p>We may begin with the linear dipole. For our tests, we shall use a frequency of 28 MHz, with a 1" diameter lossless element. The only challenge is not skimping on the segments. Although the minimum segmentation per wavelength is 10, the recommended minimum is 20. Of course, in NEC, we use an odd number of segments to obtain a center source position. However, we need not always strive for the minimum recommended number of segments. The test model uses 41 segments. Our only opposing danger would be to use so many segments that we press the segment-length to wire-diameter (or radius) ratio. 41 segments with a 1" diameter comes nowhere close to such pressure. <b>Fig. 2</b> shows the 3-dimensions free-space pattern and overlays the typical E-plane (azimuth) figure-8 pattern with which we are familiar. The listed gain figure sometimes surprises newer modelers who hear that a lossless dipole in free space has a gain of 2.15 dBi. That value applies to dipoles using vanishingly thin elements, which would be significantly longer than our 1" element. Shortening a linear dipole, even if only enough to restore resonance due to using a fat element, results in a gradual gain reduction.</p>
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<p>The impedance figure is from NEC-4. NEC-2 yields 71.95 + j 0.17 Ohms. Both values are from the EZNEC cores through version 4.0.20. Different implementations of NEC-2 and NEC-4 may use different Fortran compilers and result in values that differ even from these. In fact, different CPUs in various computer types can also result in variations, although all will be harmlessly small. Both NEC-2 and NEC-4 agree on the AGT score: 1.000, which requires no corrective on the gain or source resistance report.</p>
<p>The total antenna length for our sample model is 199.4", although it will be more convenient for us to speak in half-lengths: +/-99.7". For many of our "deviant" dipoles, the half-length will be a useful catalog number. Although the antenna is an electrical half wavelength, as indicated by the resonant source impedance, the physical wire is only 0.473 wavelength.</p>
<p><i>The V Dipole</i></p>
<p>V dipoles occur using almost any included angle between the wires. Let's restrict ourselves to a single angle of 90 degrees between wires. That is equivalent to bending each leg 45 degrees from the linear configuration. In terms of modeling, the action will necessarily result in a model with at least 2 wires. MININEC models will place the source on the junction of the two wires at the apex. However, uncorrected MININEC 3.13 will provide two kinds of errors if we are not careful. First, there is a frequency offset that increases with rising frequency. Second, MININEC creates errors in current calculations at sharp angular wire junctions. We can largely overcome the second error condition by using a very large number of segments, since the amount of error varies with the segment lengths. However, many later implementations of MININEC 3.13 provide corrections for both error sources within the code and yield very reliable results. However, unless you possess more than one version of MININEC 3.13, you may not be able to assess the level of correction available within the program you use. I have provided some performance comparisons over a number of MININEC potential trouble spots in a past column.</p>
<p>NEC users face a slightly different challenge. Since the apex of the V will be a junction of 2 wires and since NEC sources lie within segments, we must figure out where to place the source. <b>Fig. 3</b> shows 3 of our options. The black squares are segment junctions, while the red dots are potential source locations. For all of our tests here, we shall stay at 28 MHz and retain the 1" diameter lossless element.</p>
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<p>The top version of the V model uses a simple procedure of placing the source on one of the segments adjacent to the apex of the V. The justification lies in the fact that at high current regions of most antennas, the current changes very slowly as we move away from the precise antenna center. Hence, the potential error is very small, especially since these models use the same total number of segments as the linear dipole (actually 42). The bottom model uses a split or dual source, with a source placed on each side of the apex. EZNEC will internally total the sum of the two source impedance values, but manual addition is simple enough. Otherwise, the model is identical to the top offset-source model.</p>
<p>The middle model uses a different technique. It creates a level wire at the center. The wire uses 3 segments so that the segments adjacent to the center source segment are of equal length with the source segment. That model design maneuver tends to yield maximum accuracy in the current calculations. Each wire extends +/- 7.5" from the center point. The sloping legs each have 19 segments, for a total of 41 segments in the model, and the leg segment lengths are close to the length of the segments in the source wire.</p>
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<p><b>Fig. 4</b> shows two relevant facts about the models. The wire layouts have superimposed current magnitude curves that verify the current distribution as appropriate to a dipole as defined earlier. The E-plane patterns broadside to the plane of the V structures are virtually identical. To a casual viewer, the models might seem indistinguishable. However, the following table shows that there are indeed a few important distinctions among them.</p>
<pre>
Comparison of V-Dipole Models in NEC-2 and NEC-4
All models use 1" diameter lossless elements in free space.
Source Core Wire Length Gain Source Impedance AGT AGT-dB Corrected
Method Inches dBi R +/- j X Ohms Gain dBi
Offset NEC-2 +/-103.3 1.52 44.75 + j0.02 0.949 -0.23 1.75
NEC-4 +/-103.3 1.50 45.00 - j0.22 0.943 -0.25 1.75
Split NEC-2 +/-103.3 1.52 44.75 + j0.22 0.949 -0.23 1.75
NEC-4 +/-103.3 1.50 45.01 + j0.02 0.943 -0.25 1.75
3-Wire NEC-2 +/-102.3 1.72 41.31 + j0.25 1.004 0.02 1.70
NEC-4 +/-102.3 1.73 41.26 + j0.00 1.005 0.02 1.71
Note: Wire Length is the total half-element length from the apex to the tip in offset and split-source versions and from the source point to the wire tips in the 3-wire version.
</pre>
<p>Between the offset-source and the split-source models we find only tiny differences. For each core, the difference falls mostly in a 0.2 consistent differential in the source reactance. We might assign the gain difference between NEC-2 and NEC-4 versions to differences in the core. This assignment would be correct, but not merely due to random compiler or CPU operations. If we move to the AGT and AGT-dB columns, we find values that are far less than perfect. The differences in the AGT values relative to the 2 cores are small but significant. If the initial gain report seemed low, it was. A corrected gain report brings the value more within expectations for a full-size antenna, even if we allow for some broadside gain loss to compensate for the shallower side nulls in the E-plane patterns.</p>
<p>The less than perfect AGT values result from the use of relatively fat elements. Each segment is about 5" long, and the wire diameter is 1". At the apex of the V, the wire segments that join inter-penetrate a considerable, but not fatal distance relative to any use to which one might apply the model. If we had used thin wires, the small wire diameter would have resulted in much less penetration toward the junction segment centers, and the AGT values would have been closer to ideal values. Obviously, I have constructed the sample model to reveal the effect, rather than hiding it by the use of thin wires.</p>
<p>The 3-wire version of the model achieves a set of AGT values much closer to ideal Whether the very small difference in correct gain values is a function of the core or of the slightly different structure is indeterminate from the available data, but we do note that the total amount of element from the source point to the element tip differs by an inch from the 2-wire models. As well, we may note that the 3-wire models result in lower source resistance values. However, you may wish to compare the values after multiply each reported resistance by the basic AGT value.</p>
<p>Even if we discount this sequence of models as unlikely candidates for implementation, they do reveal the importance of making constant reference to the AGT values of a model. Had we used the raw reports from the 2-wire models as a comparator to the linear dipole, we would have drawn very wrong conclusions about the degree of gain deficit. As well, we should not be too hasty in discounting the models, since the design, turned 90 degrees so that all elements parallel the ground surface, forms a common structure of some antennas with forward-sweep elements.</p>
<p><i>Folded Dipoles</i></p>
<p>Folded dipoles are actually hybrid constructions. They are indeed dipoles, but the folded structure adds an extra set of currents. These transmission-line currents are a function of the folded dipole's impedance transformation relative to a linear dipole. The upper left portion of <b>Fig. 5</b> shows the result of combining the 2 sets of currents. The transmission line currents are relatively constant in magnitude. Hence, the radiation currents overlay them. The resulting curve rises above and falls below the average current magnitude, but does not go to zero at the ends of the wires, as we would find for a linear dipole.</p>
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<p>Once we separate the two currents, the radiation currents are virtually identical to those that we find in a linear dipole. Hence, the radiation pattern on the upper right is indistinguishable from the patterns of standard dipole. The resonant folded dipole will be shorter than a line dipole, since the effective diameter of the two parallel wires, each the same diameter as the single wire in a linear dipole, is considerable greater. The 28-MHz model of a folded dipole with 1" diameter elements is 4.5" shorter at resonance than the linear dipole that we reviewed earlier. The modeled folded dipole uses a wires separation of 2" center-to-center, with 41 segments in each long element section.</p>
<p>With both elements having an equal diameter, the resonant impedance is between 281 and 282 Ohms, or about 4 times the impedance of a single element dipole (70 Ohms). However, the exact ratio of impedance transformation is only 4:1 if both elements have the same diameter. The lower half of <b>Fig. 5</b> shows the key elements in calculating the transformation ratio, R. The equation involves the spacing (s) between elements and the relative diameters of the two element sections, where d1 is the section with the feedpoint and d2 is the section parallel to the driven section. Note that s, d1, and d2 must use the same units of measure for the equation to produce usable results. If both elements have the same diameter, then the ratio of logs reduces to 1, and the ratio turns out to be 4. If the driven section is smaller, then the ratio is greater than 4:1. If the undriven section is smaller, then the ratio is less than 4:1 but always greater than 1:1.</p>
<p>We may reduce the undriven section to any diameter. Let's use 0.1". The equation yields 1.89:1 as the transformation ratio. If we assume a linear dipole impedance of 70 Ohms, then the folded dipole with a larger driven section and a smaller undriven section should show an impedance close to 132.5 Ohms. Slight variations will occur because the equation does not take the end connecting wires into account. We shall use the same segmentation level (41 segments per section) that we used on the folded dipole with equal-diameter sections. Since the transformation involves only the transmission-line currents, the gain and pattern of the folded dipole with unequal-diameter elements should be the same (within close tolerances) as the corresponding results from the equal-diameter folded dipole. The following table summarizes the results for both kinds of folded dipole using both NEC-2 and NEC-4.</p>
<pre>
Comparison of Folded-Dipole Models in NEC-2 and NEC-4: Equal and Unequal Diameter Elements
All models use 1" diameter lossless driven elements in free space. Second elements are 1"
or 0.1" diameter lossless wires.
Diameters Core Wire Length Gain Source Impedance AGT AGT-dB Corrected
Driven/Other Inches dBi R +/- j X Ohms Gain dBi
1"/1" NEC-2 +/-97.45 2.13 281.6 - j0.54 1.000 0.00 2.13
NEC-4 +/-97.45 2.13 281.5 - j0.96 1.000 0.00 2.13
1"/0.1" NEC-2 +/-99.2 4.34 230.2 - j10.59 1.662 2.21 2.13
NEC-4 +/-99.2 2.96 164.5 + j0.60 1.209 0.82 2.14
Calculated Impedance 132.5
Note: Wire Length is the total half-element length from the center to the tip.
</pre>
<p>The equal-diameter version of the folded dipole shows good results from both NEC-2 and NEC-4, since both cores produce an AGT of 1.000. The gain values are identical, and the reported impedance values are almost identical. However, the reported values for the folded dipole with unequal diameter sections fall way off the mark. NEC-2 is considerably worse than NEC-4 with respect to anticipated values of gain and impedance. In fact, both NEC-2 and NEC-4 become error prone with angular junctions of wires having dissimilar diameters. NEC-4 improves upon the performance of NEC-2 in this regard, but falls seriously into error as the change of diameter increases. In this model, the ratio is 10:1, so that even NEC-4 yields an unacceptable AGT value. Note that the AGT-dB values bring the gain report close to the anticipated value. However, at a certain level, the AGT value itself becomes useless in correcting the reported source resistance. MININEC does not share the NEC limitation relating to changes in element diameter at wire junctions. Hence, it is generally able to handle models like the folded dipole with unequal-diameter element sections with no problems. If the version of MININEC 3.13 has had other limitations corrected, then it should yield quite accurate results for our test case.</p>
<p>Both models that we have just examined used 41 segments on each long wire. The results are a set of well-aligned segment junctions in the closely spaced parallel wires. Although providing each wire with the same number of segments seems quite natural, we sometimes encounter cases of somewhat careless segmentation. <b>Fig. 6</b> shows the center portions of equal-diameter folded dipoles that differ only in the assignment of segments per wire. (All end connection wires use 1 segment.) In one case, we have 61 segments on the driven wire and 31 on the other wire. The second case reverses the assignment. At the bottom, the figure shows the segmentation of the preferred model.</p>
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<p>The upper two models clearly do not show a pattern of well-aligned segment junction. The following table shows the results of the misalignment, with all other factors being the same for all three models.</p>
<pre>
Comparison of Folded-Dipole Models in NEC-2 and NEC-4: Careless vs. Careful Segmentation
All models use 1" diameter lossless elements in free space.
Segments Core Wire Length Gain Source Impedance AGT AGT-dB Corrected
Driven/Other Inches dBi R +/- j X Ohms Gain dBi
61/31 NEC-2 +/-97.45 1.88 342.5 + j2.31 0.946 -0.24 2.12
NEC-4 +/-97.45 1.71 329.1 - j2.33 0.909 -0.41 2.12
31/61 NEC-2 +/-97.45 1.84 263.6 + j2.23 0.937 -0.28 2.13
NEC-4 +/-97.45 2.00 273.1 + j1.26 0.971 -0.13 2.12
41/41 NEC-2 +/-97.45 2.13 281.6 - j0.54 1.000 0.00 2.13
NEC-4 +/-97.45 2.13 281.5 - j0.96 1.000 0.00 2.13
Note: Wire Length is the total half-element length from the center to the tip.
</pre>
<p><b>Fig. 6</b> provides the NEC-4 reported data for each model. The degree to which the unevenly segmented models depart from correct values depends on both the core used (NEC-2 or NEC-4) and the ratio of segments in the 2 wires, where the ratio shows the misalignment. Admittedly, the sample models provide cases of extreme misalignment. However, even small misalignments can draw the results away from the ideal AGT values attained by the model with well aligned segments. Alignment becomes ever more critical as we close the spacing between wires. Once more, it pays dividends to check the AGT values for any model. Checking those values becomes even more important when we need or wish to compare the reports of one model with another, whether we are using similar or dissimilar geometries.</p>
<p><b>Conclusion to Part 1</b></p>
<p>We have not gone very far in our exploration of resonant half wavelength dipoles, and already we have seen some modeling snares. Some of those traps are modeling practices that we can easily avoid. Others involve limits to the NEC cores that we cannot avoid except by deferring the model construction. Folded dipoles with unequal elements form one of those NEC limitations and require the use of antenna modeling software that lacks the particular limitation involved. MININEC is one of those usable cores.</p>
<p>In addition to running into potential pitfalls, we have also run out of room in this column. Therefore, we shall have resume our journey in the next episode. We shall discover whether or not MININEC can handle those odd folded dipoles--and more.</p>
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