126. "Ideal" Polar Plots

L. B. Cebik, W4RNL (SK)

NEC outputs are strictly tabular. Most NEC users obtain a commercially available package that includes--as a matter of course--a polar plot facility for graphing the radiation pattern of an antenna model. Very often, we lose track of the fact that the software writer as a service to users adds these modules in order to make the results of a simulation more (visually) accessible to the modeler. Especially if we have used only one package, we face the temptation of thinking that the graphing facility is an integral part of the NEC calculation core.

As post-calculation facilities, polar plot facilities come in many flavors. Commercial packages tend to develop modules to serve the widest range of users equally. However, numerous firms have developed their own modules for special purposes, for example, satisfying FCC requirements for antenna patterns. Many polar plot facilities offer a wide range of plotting alternatives. Some offer a choice between linear or log scaling of the plot. Many also offer a selection of what data to graph, including the total gain, component gain (either major-minor axis or vertical-horizontal), and possibly the phi and theta voltage components of the far field. A few offer left-hand and right-hand circularly polarized patterns. One package even offers azimuth patterns for ground-wave calculations.

Polar plots are generally available for either azimuth/phi or elevation/theta patterns, as dictated by the selections available for calculation within NEC. These pattern choices use conventions inherent to the NEC coordinate system. An azimuth or phi pattern within NEC software generally is a pattern in the X-Y plane, with a user-selected angle relative to that plane (elevation) or to the zenith (theta). An elevation or theta pattern calculates values with the Z-axis as the basis and on a plane that the user selects by specifying a phi or azimuth angle, where zero degrees coincides with the X-axis in the model geometry. Free-space elevation/theta patterns may use a full 360-degrees plot, but models employing a ground are restricted to the hemisphere above ground.

Within the polar plots, software writers try to provide a variety of critical information needed by most modelers. Therefore, patterns usually show a line from the pattern center along the heading of maximum gain. Where possible, the plot will also indicate with lines the limits of the half-power or 3-dB beamwidth. Some plotting facilities also show by a line the strongest lobe other than the maximum gain lobe. Finally, the plot may show one or another form of front-to-back ratio, usually in an inset to the plot and listed in tabular format. Indeed, in most cases, the key data registered by the bearing lines appears in the inset table.

As excellent as are the polar plot facilities offered by various implementations of NEC, they are not quite ideal from the perspective of a heavy user. These notes enumerate some of the shortcomings of polar plot facilities and some of the features that I would like to have available. These features would be part of what I would like to think of as an ideal polar plot facility. I shall list them without regard to their programming feasibility. Inevitably, we shall encounter situations that will tell us why I am unlikely ever to see a polar plot facility that provides them. Even if I specify that they do not all need to appear at the same time, there are limitations arising from the nature of the antennas that we model that make many of the features impractical, if not impossible, to supply.

Some "Back-and-Forth" Desires

Many patterns (phi or theta) display multiple lobes in a context in which we can easily identify a forward and a rearward direction. Modules that identify both the strongest lobe and the next strongest lobe suffer a limitation. In some patterns, the second strongest lobe may be a secondary forward lobe, while in others, the second strongest lobe may be the strongest rearward lobe. Fig. 1 provides a sample of each type of pattern.

Several difficulties arise from the fact that Sample A and Sample B identify different lobes as the second strongest, that is, as the major sidelobe. The accompanying data inset may identify the sidelobe, its bearing, and its strength (and relative strength to the main forward lobe). However, only Sample B gives us information on the main forward sidelobe; Sample A gives data for the main rearward lobe. Now suppose that we perform a frequency sweep for the subject antenna. In the course of the sweep, both patterns may appear, although at different frequencies. The sweep (for example, in EZNEC) will often record the data for the sidelobe strength. Unfortunately, such data is not usable as a record of the main forward sidelobe strength, since some of the values belong to a different lobe.

If we focus only upon directional antennas of the type whose patterns appear in Fig. 1, we may arrive at a nearly ideal request for addition information on the plot. Fig. 2 summarizes in a series of color-coded lines a more satisfactory user situation. In viewing the figure, also imagine an enlarged inset with tabular data for all of the lines.

First the plot should identify forward and rearward quadrants. This task is relatively easy for antennas with deep nulls that are 90 degrees off the main forward gain direction. As usual, the plot identified the main forward heading with a line, and also shows the 3-dB beamwidth limits. In addition to this information, the plot also identifies each of the remaining forward lobes. Immediately, we encounter a challenge, since there is no single standard for distinguishing a true lobe from a bulge in the pattern. We might apply a 3-dB rule, that is, to identify a lobe as such when there is a 3 dB difference between the lobe peak gain and both adjacent null regions (regions with lower gain values). This rule would give each lobe its own half-power points. Anything less than 3-dB would be a bulge, a common feature of many long Yagis used above their design frequencies.

The more usual technique in identifying a lobe is simply to register when a pattern, following the progression of sampled plotting angles, shows an increase in gain relative to the values at both the preceding angle and the following angle. This method relieves the plotting facility of the need to determine if the lobe meets a 3-dB measure. The technique does not eliminate mere bulges, as illustrated in Fig. 3. The patterns are for the same Yagi, but with an operating frequency change of 100 kHz.

Since the patterns are both symmetrical, the locating dot does not obscure the fact that each pattern shows an apparent projection at the dot location. In the left pattern, the dot at a phi angle of 63 degrees shows a higher gain than we obtain at 62 and 64 degrees. The progression, in dBi/degrees, is -9.68/62, -9.67/63, -9.70/64. A change of 0.01 dB suffices to mark a lobe, as the plot line indicates. 100 kHz higher, we find the following progression of values: -8.35/60, -8.40/61, -8.48/62, -8.58/63, -8.72/64. The progression shows a steady decrease in gain, but the rate of decrease changes through the sampled range. The changing rate of gain decrease is enough to yield a pattern bulge. (In some cases, but not this one, we may convert a bulge into a lobe by using a smaller increment between samples, such as 0.1 degrees.) Although we may consider bulges to be incipient lobes, their location will likely always be a viewer task.

To the rear, we find in Fig. 2 a 180-degrees line that simply continues the line from the point of maximum gain. The line is marginally useful in distinguishing arrays with symmetrical patterns from those with non-symmetrical patterns (such as parasitic beams based on a corner-fed half square). More significant is the identification of each rearward lobe--with associated data shown on the tabular inset.

Bringing Up the Rear

Once we move to the rear quadrants of a directional antenna, the data becomes more complex, since we are very often interested in some kind of front-to-back ratio. Since it is the easiest to calculate, most polar plot facilities provide the 180-degree front-to-back ratio value. The 180 front-to-back ratio is the main lobe forward gain (or the maximum antenna gain) minus the gain of the lobe (however big or small) that is 180 away from the heading of the maximum forward gain. If the main forward lobe is split or does not align with the graph heading, the 180 front-to-back ratio is 180 away from the direction of maximum pattern strength. Hence, the value may not be for a direction directly to the rear of the antenna structure. Since a Yagi is usually symmetrical, the maximum gain will normally be directly forward, and the 180 front-to-back ratio will indicate the relative strength to the direct rear. Note that if we use a normalized scale, we can read the front-to-back ratio directly from the plot--between 25 and 30 dB relative to the maximum gain of the antenna in the leftmost pattern. The left portion of Fig. 4 shows a part of a Yagi pattern in which the 180-degree lobe is the strongest rearward lobe.

In Fig. 4, the center pattern shows a 180-degree gain of very tiny proportions. Hence, the 180 front-to-back ratio is very large (over 40 dB compared to a "mere" 27 dB for the leftmost pattern). Yet, we find rearward lobes that have considerable strength. The line through one of those lobes indicates the direction of maximum strength. It is only about 22 dB weaker than the maximum gain. Some sources call this the worst-case front-to-back ratio, and its value is the maximum forward gain minus the highest value of gain in either rearward quadrant. For this antenna, the 180 front-to-back ratio does not give a true picture of the QRM levels from the rear, so some folks prefer to use this figure as a better indicator. The worst-case front-to-back ratio provides the most conservative value for rearward suppression of QRM. The rightmost graphic in Fig. 4 shows that the 180 and the worst-case front-to-back values do not require separate lobes, even thought the values differ.

We are not done with front-to-back ratios. Each sketch in Fig. 4 contains an arc going from 90 on one side of the line of maximum gain around the rear to the other point that is 90 from the maximum gain line. Suppose that we add up all of the gain values at the headings that pass through the arc. Next take their average value. Subtract the average gain value to the rear from the maximum forward gain and you arrive at what some call the front-to-rear ratio. Others call this the averaged front-to-back ratio. A 5-interval between rearward readings is often sufficient for this sampling. The rationale behind using the front-to-rear ratio is that it provides an averaged total picture of the rearward QRM suppression.

Although I am not aware of any software that provides an average front-to-rear ratio, at least one maker (NSI) provides a user selection between the 180-degree and the worst-case values. The worst-case value is detected by using the main forward lobe bearing and creating sampling limits 90 degrees to the rear of that bearing. Within the rearward quadrants, the program then identifies the strongest lobe and uses its gain in the front-to-back calculation. The process sounds simple enough (at least arithmetically) until we encounter cases like the ones illustrated in Fig. 5.

The azimuth pattern shown applies to a Moxon rectangle, which does not place its side nulls at the 90-degree mark relative to the heading of maximum gain. Rather, its nulls occur closer to 110 degrees off the main heading. Hence, the preset limit line for determining the worst-case front-to-back ratio uses a portion of the forward lobe as a legitimate direction for the worst-case front-to-back ratio calculation.

The same consideration--that is, the location of the deep side nulls--applies equally to the calculation of the average front-to-rear gain value. If one were to implement this additional front-to-back calculation, one might use the preset limits that gave the odd heading for the worst-case rearward lobe or one might use some sort of comparative scheme to detect the deep side nulls and then to use the reduced scanning region to determine the average front-to-rear ratio.

Although the challenges of determining "proper" values of the worst-case and the average front-to-back ratios are significant, they are not insurmountable. However, not all azimuth patterns have detectable side nulls that occur--either at all or anywhere close to the side of the beam pattern. Fig. 6 shows two cases--very typical for vertically oriented arrays--in which perhaps only the 180-degree front-to-back ratio makes good sense. On the left, we have a common cardioidal pattern with only one null directly opposite the main forward lobe direction. A worst-case direction would have to choose between the 180-degree direction or involve the main lobe. An averaged front-to-back ratio would have little, if any, meaning in this case.

The right-side pattern has a rearward lobe, but it is problematical. Once more, the idea of a worst-case front-to-back ratio becomes co-terminal with the 180-degree front-to-back ratio. In theory, we can take an averaged front-to-rear reading using only the rearward lobe. However, the angles included in the calculation are so few as to make the calculation less than useful.

Which Way Did He Go?

Our hope for an ideal polar plot facility that includes on screen all of the information that we might wish to see--whether all at once or serially--has significant limitations in the rearward mode for directional antennas. Very likely, programming various scans and detection systems is less of a problem than having the system know when certain criteria are relevant and when they are not. Antenna models do not advertise themselves as having certain characteristics calling for the application of certain variations on standard measures. Rather, the patterns emerge from the data. At present, post-pattern interpretation remains a user operation.

Of course, many antenna patterns are not directional in the sense of having strict forward and rearward headings and a single forward lobe. A primary example is the simple dipole, which has an E-plane bi-directional pattern. The left pattern in Fig. 7 shows the difficulty facing a polar plot facility.

The plot identifies one of the dipole lobes as the main lobe and relegates the other to the status of a rear, side, or secondary lobe. Ordinarily, the software decides on the main lobe by selecting the first lobe exhibiting maximum gain that it encounters in sampling the gain values starting at zero degrees. Zero degrees conventionally aligns with the X-axis in the coordinate system of the model geometry. Hence, for a dipole with the element ends aligned parallel to the -Y/+Y axis, the main lobe will be at zero degrees. Had the element been aligned parallel to the -X/+X axis, the main lobe would be at 90 degrees, counting in the phi or counterclockwise manner. (Special note: many polar plot systems also sample adjacent angles for a span that extends until the gain value changes; the system then centers the indicator line within the range of angles showing the maximum gain value. The NEC output report records gain values in dB using 2 decimal places. Hence, for broad lobes like those of the dipole or for small angular increments such as 0.1 degree, there may well be a considerable set of angles having the same gain value. Similar techniques may be applied to any of the headings identified in terms of gain, such as rear and side lobes.)

In fact, the identification of the first-encountered lobe as the main lobe and the other dipole lobe as non-main is arbitrary and a function of the polar plot system design. Electrically, both lobes are equal. However, it remains a user task to examine the plot and the data included with it to establish that the pattern is equally bi-directional. Certain cases, such as very closely spaced wires of which only one has a source, may exhibit bi-directional patterns broadside to the wire pair but show a very tiny (and normally operationally insignificant) differential in gain in the plane of the wires. For cases of true bi-directionality, the availability of a clear graphic representation of the pattern and confirming data tends to make the arbitrary designation of one lobe as the main lobe harmless.

The right side of Fig. 7 present the same situation in a different context. Although the pattern is directional when looking at the plot from left to right, we find a symmetry when viewing the pattern vertically. (The pattern is for a single end-fed unterminated long-wire antenna, with the source at a position corresponding to the left side of the pattern.) Since this particular facility counts counterclockwise, the main lobe is at the first maximum-gain angle greater than zero. The sampling technique does not encounter the second and equal lobe until it is approaching 360 degrees in its sequential scan. Therefore, the lobe with the lower angular value receives the main lobe designation, with its equal mirror lobe receiving secondary status. Once, more, the user must use a careful review of the plot and associated data to confirm that the two lobes are equal.

We may complicate the task facing a polar plot facility even further by increasing the number of main-lobe equivalents. The pattern shown in Fig. 8 is for a center-fed doublet that is many wavelengths long. The model extends the wire parallel to the -X/+X axis, so that the broadside direction is up and down relative to the page or screen. In this case, the program identifies the main lobe as the first lobe of maximum gain that it encounters counting counterclockwise from zero. The secondary lobe is the next lobe of maximum gain that it encounters as the angle of sampling increases. This lobe becomes the side lobe. However, two other lobes in the pattern have maximum gain values but receive no markers. In this case, the user who restricts himself to the graphics and data within the polar plot facility can only presume that the remaining main-lobe equivalents are in fact the symmetrical matches of the lobes bearing designations. Confirming true symmetry in the model requires attention to the detailed radiation pattern data in the NEC output report or to any convenient truncated version provided by the software. For simple models, such as those used to illustrate these notes, a presumption of symmetry may be justified. However, for more complex geometries, the presumption may hide subtle differences.

By now, it should be clear that the desired identifications of all relevant lobes on a plot is likely impractical. This conclusion applies not only to the differentiation of all forward and rearward lobes for a directional antenna, but as well to antennas such as shown in Fig. 7 and Fig. 8. Indeed, a plot with lines indicating each lobe would clutter the graphical presentation and obscure the pattern shape. At the same time, scanning the tabular data is often cumbersome, since a full azimuth/phi scan with 1-degree increments will have 360 entries. If we use an increment of 0.1-degree, the number of entries increases tenfold.

However, there is a technique that might be used to create a compact table that shows only the headings of lobes and nulls within the overall pattern. No system is likely to be perfect in the sense of recording where lobes should be but are mere bulges. For example, a vertical antenna place well above ground will show a depression of both gain and null-depth values in elevation/theta patterns at the pseudo-Brewster angle. Nevertheless, a suitably programmed sampling and comparison routine could identify all of the lobes and nulls in a pattern using a simple comparison with values at adjacent sampled angles. When placed in tabular form, the table might look something like Table 1, which records the lobes and nulls for the pattern in Fig. 8.

One might embellish the table with whatever other data one might wish to have. Pattern component information is readily available from the NEC output report. The table itself calculates the ratio between the pattern's maximum gain and the lobe or null value. Note that NEC uses a limiting value of -99.99 dB for values lower than that number. Hence, the calculations for the ratio also use 99.99 as a placeholder. One version of MININEC provides lobe data within its polar plot screen, so the technique is not beyond possibility. However, for extremely complex plots, like those we might obtain from an unterminated rhombic with extremely long legs, the table may require considerable size. Hence, an independent table might be a better program facility.

The existence of such a table can be quite useful. By examining the table and the polar plot together, the user may decide which lobes count as forward and rearward lobes for a non-standard directional antenna plot. As well, the short table allows quick identification of all equivalent main lobes and their headings. Moreover, the comparative null depth values in a many-lobed pattern can be very instructive. (Unfortunately, the information in Table 1 does not result from software, but only from an eyeball scan of the full radiation pattern data in the NEC output file. However, the exercise may have converted my eyes into a different kind of software.)


In my exploration of the "perfect" polar radiation pattern plot, I have purposely exceeded the boundaries of what we should expect such plots to show us, especially in the face of the great variety of possible antenna patterns that might emerge from models. The pretext has provided an opportunity for us to refine that ways in which we look at such plots to derive from them--and the accompanying tabular data--the most data. At the same time, the exercise has alerted us to limitations inherent in converting tabular data into a graphic form.

We may overcome some of the limitations by creating external tables. For example, we might export the radiation pattern portion of the NEC output data to a spreadsheet which then automatically calculates the lobe and null information contained in Table 1. As well, we might use that data to create a rectangular graph of the lobes and nulls for easier identification. (NSI implementations of NEC provide such a facility. Fig. 9 provides a sample rectangular version of the polar plot in Fig. 8.) Nevertheless, tabular data remains the most precise method for identifying and quantifying the many lobes and nulls in a radiation pattern, as well as obtaining an accurate measure of their relative values and the rates of change.

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