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.
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.
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.
Log vs. Linear Polar Plots
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.
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.
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.
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.
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
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 Fig. 1, the second element is the driver, fed at its center.
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 Fig. 2.
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.
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.
In the absence of such standards, the field is wide open. For example, the upper right pattern of Fig. 2 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.
Just as the upper right pattern seems to obscure important data by virtue of its bulbous shape, the two patterns at the bottom of Fig. 2 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.
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.
Azimuth Plots Over Ground
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.
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. Fig. 3 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.
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 Fig. 4. The following table (Table 1) shows the values that correlate with the patterns.
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.
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.
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.)
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. Fig. 5 shows a sample of the situation, although the sketch is itself imperfect.
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.
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.
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.
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.
BWa = BWr * cos(elevation) or BWa = BWr * sin(theta)
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.
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. Fig. 6 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.
Despite te missing detail, we can observe that the main lobe of the beam retains its essential shape throughout the changes in elevation angle.
Conclusion
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.
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.
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.
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