Part 2: 10- and 15-Turn Helical Antennas Over Perfect Ground and Modeling Helices in NEC-2

In our introduction to axial-mode helical antennas for radio amateurs, we examined helix basics, setting up NEC-4 models over a perfect ground, and the main characteristics of 5-turn helix performance. However, as we move up the amateur bands, we find longer helical antennas. Therefore, we need to look at 10-turn and 15-turn helices in the same environment. We shall see if there are any significant differences in performance beyond the anticipated gain increase and beamwidth decrease. Since a significant part of our investigation involves how to model these antennas, we shall also explore what we need to do if we only have NEC-2 with which to work.

**A 10-Turn Helix Using 2-mm Diameter Wire**

A 5-turn helix is just barely larger than the 4-turn limit for the application of axial-mode theory to antenna performance. Doubling that length to 10 turns yields a middle-size antenna, perhaps long for the amateur 2-meter band, but still shorter than those in use at 1296 MHz. Using 2-mm diameter wire and restricting ourselves to pitches of 14° and 12°, we obtain helical antennas that vary from 1.6WL to almost 3.4WL in total length. Our primary question will be whether there are any changes in the characteristics of these antennas other than the anticipated increase in maximum gain.

**Table 3** provides the modeling data for the series of 10-turn helices. The overall increase in gain is accompanied by an equally expected reduction in beamwidth as we compare data for each level of circumference. However, for the smaller circumferences, the source impedance does not change by any very significant amount, a fact that is consistent with established helix theory and measurement. As well, for this modeling investigation, the AGT values for both helix lengths are consistent with each other.

**Fig. 12** presents the gain data for the range of circumference covered by this study. Note that, compared with the 5-turn helices, both pitch versions reach their peak gain values at circumferences about 0.1WL smaller. The 12°-pitch version reaches peak gain at a circumference of about 1.2WL. Small-circumference gain values approach the peak gains of large-circumference versions. However, we suspect in advance that the usable region of the curves is smaller than the total span. As well, the minimum gain values occur at about 0.95WL circumference levels. The tilt of the curves in this area suggests that the actual minimum value for the 12° curve occurs with a slightly smaller circumference than for the 14° model.

The source resistance and reactance data also show some degree of stability shift toward the smaller circumference values. The 12°-pitch version of the helix begins to show wider variations of both resistance and reactance at a circumference of about 1.15WL. The lower gain 14°-pitch model appears stable with respect to reactance to about 1.2WL, but the resistance begins to show instabilities just shy of that circumference. These trends appear in **Fig. 13** and **Fig. 14**. At the low end of the scale, instabilities begin to appear only for the smallest circumference within the survey.

The end result is that the 10-turn helix appears to be stable to almost the same circumference as the 5-turn helix when we restrict ourselves to 14° and 12° pitch levels. At most, we shrink the stability limit by one step, down to a circumference of 1.15WL.

We are as interested in pattern shapes as we are in other performance factors. Here, **Fig. 15** can be useful, as it presents the Y-axis elevation patterns for the 12° pitch 10-turn helix at the same circumferences used in **Fig. 7**. The patterns make several things clear. First, past the gain peak, a 10-turn helix is quite unsatisfactory as an axial mode antenna. In fact, the single dome pattern reappears at various larger circumferences, but it has lost the consistency that marks axial mode operation.

Second, just as the gain peaked at a smaller circumference than for the 5-turn antenna, the sidelobes begin their appearance at a smaller circumference in 10-turn antennas. The sidelobes of the 10-turn helix with a 0.95WL circumference are nearly as distinct as the ones that we encounter in the 5-turn helix only at a circumference of about 1.15WL.

Third, the sidelobe structure becomes more complex, even if we restrict our attention to patterns for circumferences up to 1.15WL. Rather than having single or double side lobes, we have a wider sidelobe that suggests the inclusion of several overlapping sidelobes. Although the peak values are not as high as in the case of some 5-turn patterns, the total energy within the sidelobes may be equal to single stronger lobes. In terms of the pick-up of unwanted noise and signals in directions other than the focus of the main lobe, the effects may be similar.

Whether the trends continue or whether the helices have stabilized requires one more set or models.

**A 15-Turn Helix Using 2-mm Diameter Wire**

At the design frequency, 15-turn axial-mode helices with circumferences between 0.75WL and 1.35WL yield antennas between 2.4WL and 5.0WL long. We may generally classify these antennas as long helices. Longer helical antennas are possible, but these would be quite rare in amateur service. The long antennas yield their anticipated further increase in gain along with the accompanying decrease in beamwidth. Despite the increased length, the long helices do not yield perfectly aligned patterns, as evidenced by the continuing divergence between X-axis and Y-axis beamwidth data in **Table 4**.

The gain data captured in **Fig. 16** shows a further compression of the gain curve toward the smaller circumferences. The gain peaks are at least one step smaller in circumference, with more precipitous declines in gain above the circumference of peak gain. Even if we press the circumference for maximum gain, the limit of utility is well below the maximum circumference (about 1.3WL) for which axial-mode helical antenna theory is most often rated.

The 15-turn source impedance data also suggest that circumferences below 1.15WL are best suited for stable operation, as shown in **Fig. 17** and **Fig. 18**. With respect to source resistance, the 14° version is more stable. The variations in the 12°-pitch version are not an artifact of modeling, since the AGT values of the 15-turn helices are comparable to those of the 5-turn and 10-turn antennas for each pitch shown. However, it is notable that the region of relatively stable resistance values continues to move toward small circumferences.

Although the resistance of the 14° model shows the greater stability, reactance is the opposite. With respect to reactance, the 12° version shows greater stability, but not by much. At the upper end of the circumference scale, both pitches show radically divergent values with respect to the stable region, suggesting that the longer the axial-mode helix, the more that circumference limits represent a sudden threshold rather than a gradual transition into unreliable operation.

The net result of the data is the suggestion that the home builder of a long helix should use great care if he wishes to press the limits of gain and source impedance at the 15-turn level. Construction variables may easily push the antenna over the limits of stable performance. Conservative design using smaller circumference values may make the resulting antenna easier to tame than versions using large values for the circumference.

The trends in pattern formation that we have observed through the 5-turn and 10-turn helices continue in the longer version. Sidelobes development occurs at smaller circumferences, with a definitive and an incipient sidelobe already apparent in the pattern in **Fig. 19** for a circumference of 0.95WL. The pattern is relatively free of side lobes only at the smallest circumferences surveyed in these notes. Only the patterns through a circumference of 1.15WL are suitable for axial-mode operation. (Note from the pattern for a circumference of 1.25WL that a narrow half-power beamwidth does not clearly indicate a desirable axial-mode pattern.) Even with single central forward lobes, the sidelobes of the long helices should be a matter of concern, since they are down from the main lobe by less than 10 dB in these models using a perfect ground as the modeling ground plane.

**Some Comparisons**

To summarize part of what the data show, we may compare the gain curves for both the 14° and the 12° versions of the three levels of helices studied here. **Fig. 20** compares the gain curves for the 14° antennas, while **Fig. 21** does the same for the 12° models. For both pitches, note that there is strong parallel among the three curves. However, the longer the helix, the greater is the displacement of the curve toward the smaller values of circumference.

Equally apparent is the parallelism among curves with respect to the gain decrease above the peak gain point. As the pattern graphics have shown, operation above the peak gain level yields generally unsatisfactory axial-mode radiation patterns, not to mention operation in regions of unstable source resistance and reactance values. How far one may push the circumference smaller than the limit shown on the graph to capitalize on a clean pattern and higher gain on this side of the main minimum would require further study. However, in terms of real helices using a self-contained ground plane and positioned some distance above the actual ground, there will be limitations that the perfect-ground models do not show.

We may also compare the gain values derived from this exploration with at least a couple of the shirt-pocket estimation schemes proposed in basic literature about axial-mode helical antennas. Similar calculation systems for gain and the half-power beamwidth appear in Kraus (p. 310) and in Stutzman and Thiele (p. 237). The terms for these calculations have the following meanings. CWL = Circumference of helix in WL. SWL = Turn spacing of helix in WL. n = number of turns in helix. WL = wavelength(s)

1. Gain (Directivity): Kraus Stutzman & Thiele D = 12 CWL^2 n SWL D = 6.2 CWL^2 n SWL Note: Gain (dBi) = 10 log10(D) 2. -3dB (half-power) Beamwidth: Kraus Stutzman & Thiele HPBW = 52°/(CWL SQRT(n SWL) HPBW = 65°/(CWL SQRT(n SWL) 3. Terminal Resistance: R = 140 CWL R = 140 CWL

The original coefficient for directivity in Kraus was 15, but he had reduced this number to 12 by the release of the 2nd edition of *Antennas*. Although the equation for terminal resistance equation is common to virtually all systems, we shall not try to evaluate it, since the source values for a NEC model do not occur at the true terminal point of a helix.

As a test case, let's examine the data for helices with a 1.15WL circumference (using 2-mm wire) and compare the modeled values for 5-, 10-, and 15-turn versions. Gain is in dBi and BW (half-power beamwidth) is in degrees. Modeled data also appear for the 5-turn, 5-mm wire helix.

Turns LWL = n SWL Modeled Kraus Stutzman & Thiele Gain BW D Gain BW D Gain BW 5 1.222WL 9.93 53 16.9 12.3 41 8.7 9.4 51 5 (5-mm) 1.222WL 10.62 53 10 2.444WL 12.23 39 33.7 15.3 29 17.4 12.4 36 15 3.666WL 13.76 28 50.6 17.0 24 26.1 14.2 26

As a second test case, let's evaluate helices with 0.85WL circumferences in the same manner.

Turns LWL = n SWL Modeled Kraus Stutzman & Thiele Gain BW D Gain BW D Gain BW 5 0.903WL 9.49 71 9.2 9.6 58 4.8 6.8 76 5 (5-mm) 0.903WL 9.75 69 10 1.807WL 10.69 58 18.4 12.7 46 9.1 9.8 57 15 2.710WL 11.42 50 37.2 14.4 27 14.2 11.6 47

The calculating schemes can grow to into fairly complex affairs, as is evident in the more elaborate equations found in the Emerson and the King and Wong references. However, none take the wire radius (or diameter) into account. Hence, all remain shirt-pocket estimators and not precise calculators of the properties of axial-mode helical antennas, at least as modeled in this study. For practical purposes, that is, initial planning and the like, the Stutzman and Thiele simple formulas are as good as any. However, they remain seriously off the mark for shorter helices with smaller circumferences that fall on the curve below the gain minimum. The two simpler schemes presume a steadily rising gain across the span of allowed axial-mode circumferences, and that presumption is not correct with respect to the models surveyed here..

This exercise has used NEC-4 to produce the modeled results. NEC-2 also has a helix command, although its structure is quite different from the one used in NEC-4. The GH command does not appear in the original NEC-2 manual (NOSC TD 116, Vol. 2), but does appear in numerous implementations of NEC-2. The only significant difference between the model that we showed for NEC-4 and the NEC-2 counterpart appears in the GH entry in the following sample. (The GE entry also has a slight difference between the NEC-2 and NEC-4 versions.)

CM NEC-2 GH helical antenna over perfect ground CE GH 1 100 .249328 1.24664 .159155 .159155 .159155 .159155 .001 GE 1 GN 1 EX 0 1 1 00 1 0 FR 0 1 0 0 299.7925 1 RP 0 181 1 1000 -90 90 1.00000 1.00000 RP 0 181 1 1000 -90 0 1.00000 1.00000 EN

**Fig. 22** shows GNEC and NEC-Win Pro help screens for the GH entry to aid in explaining the differences. Whereas the NEC-4 entry uses the number of turns and the total helix length to internally calculate the turn spacing, the NEC-2 version uses the turn spacing and total length to calculate the number of turns. NEC-4 uses a single helix radius (at both top and bottom), but NEC-2 requires entries of radius for both X- and Y-axes to allow for oval spirals. Restricting the helix to a circular form in NEC-4 opens a floating decimal position that permits one to choose between log spirals and Archimedes spirals, which yield a difference in the turn positions only if the radii differ at the top and bottom of the spiral.

The GH command in both systems yields a wire segment structure that runs from Z=0 to a Z-value that equals the total length of the helix. The circumference of the helix is centered at X=0 and Y=0. In both systems, the modeler must use the GM command to change the position or orientation of the helix, a maneuver that we have not yet needed for this exercise.

The key question is whether we can expect any significant differences in the reported output between NEC-2 and NEC-4. **Table 5** provides a negative answer with respect to 5-turn helices using 2-mm diameter wire for pitches of 14° and 12°. (The table repeats the data from **Table 1** to facilitate the comparison.) I ran the series using the sample model shown and with the additional EK (extended thin-wire kernel) command that is useful in NEC-2 when the segment length begins to approach the wire diameter. The EK command made no difference to the data output. This test also confirms that the chief source of low AGT values is the position of the source and the orientation of the wire segment on which it is placed. The bottom line is simply that NEC-2 is fully adequate for modeling axial-mode helices so long as the modeler uses the AGT value to correct the gain reports and recognizes the limitation inherent in the source impedance report.

**Next Time. . .**

In this episode, we explored longer helixes to identify how their characteristics vary from those of the initial set of 5-turn helices. Besides seeing the gain curves rise in peak value and generally slide to smaller circumferences, we also saw the increase in sidelobe development at smaller circumference levels. Finally, we made some comparisons, including a brief primer on how the NEC-2 and NEC-4 GH (helix) commands differ.

However, the entire study has a major limitation. Few amateur antennas have the opportunity to operate over indefinitely extended perfect grounds. Therefore, we need to explore, if only partially, how axial-mode helical antennas perform over self-contained ground planes that are elevated from the actual ground. That exercise will put us a position to do a rudimentary comparison between helices and other antennas amateurs might use the obtain high-gain circular polarization.

*Updated 10-20-2005. © L. B. Cebik, W4RNL. Data may be used for personal purposes, but may not be reproduced for publication in print or any other medium without permission of the author.*