Circular R-X Graphs

L. B. Cebik, W4RNL (SK)

The NEC calculation cores produce only a tabular output. Any graphical outputs that we may wish represent post-core-run manipulations of the output data. Most of the work involves finding and parsing the data into a form that a graphing module may use to present the information in a more useful visual manner. Various implementations of NEC provide some of these graphs, namely, the ones most often required or desired by antenna modelers.

Graphs that accompany an implementation of NEC generally come in two forms: polar and rectangular. Polar graphs generally apply to the radiation pattern outputs, since the data values appear in terms of angles and magnitudes. In past episodes, we have examined some of the considerations that go into the forms and plot ring arrangements for such graphs. It is also possible to present these graphs in rectangular form, using the X-axis of the graph for the angular information and the Y-axis for the magnitude. Rectangular graphs are also very useful to present other information, such as the current magnitude and/or phase angle along one or more wires in the model or the resistance, reactance, and SWR information over a specified frequency range.

Some graphical outputs from NEC implementations involve post-core-run calculations. The most common calculation is the SWR relative to a user-selected (or a default) resistive impedance. In addition, some implementations have created polar plots of the left-hand and the right-hand circular components of radiations patterns using calculations based on the radiation pattern data. We have also examined some of these calculations in past episodes.

There is one type of graph that is both very useful and very absent from implementations of NEC. In fact, the only NEC-related program that makes the graph available--to the best of my knowledge at the time of writing--is AC6LA's Multi-NEC. This Excel application does not use a core, but taps into the cores of a number of popular programs for the core run itself. However, Multi-NEC does provide a large collection of facilities unavailable in most NEC implementations.

The graph to which I am referring is the X-Y graph. It is available in many graphing and in most spreadsheet programs. Instead of plotting the magnitude of Y against a progression of set values for the X-axis, the graph plots both X and Y as points on a field. The graphing facility normally calculates the field area needed to contain the points and then creates X- and Y-axes to accommodate the values. Some graphing programs allow the user to modify the axes limits and subdivisions. In addition, most X-Y graphing facilities add a line connecting the successive data points in the series.

X-Y graphs have numerous uses. With respect to antennas, one of the most useful versions is a plot of resistance and reactance across a large frequency span. There is much that we can glean from a close examination of X-Y graphs of R +/-jX. So let's probe a bit further. Once we catch on to how we can create our own X-Y graphs of NEC output data and look at a few comparative situations, we may transform such graphs from mere interesting oddities into genuinely useful data presentations that we are likely to use often in the future.

The Exercise

To see what we might learn from X-Y graphs, we shall need a few antennas. Fig. 1 shows the four that we shall use. The first three are linear dipoles, but with very different length-to-diameter ratios (1000:1, 100:1, and 20:1). The last of the sequence is a biconical dipole composed of 4 wires simulating the element cones. To simplify graphing, I have resonated all four antennas at 300 MHz as 1/2-wavelength elements. We shall be interested in the impedance behavior of each antenna over an 8:1 frequency range (3 octaves).

To obtain the requisite data, we need to conduct frequency sweeps across the prescribed range. I have selected a 30-MHz increment to yield 71 sweep steps as defined for the FR command, beginning at 300 MHz. We have enough steps to produce some interesting graphs, but not so many as to tax our patience while the core generates the necessary data. Some programs have tabular facilities to collect the impedance data from each run for each frequency into a single table. The partial table in Fig. 2 samples the information as presented by software. The entire table would be unnecessarily long, since we may view it more compactly for each of our subject antennas in graphical form.

We shall be interested in the Z(real) and Z(imag) columns, since we wish to plot R and X. However, we might have as easily selected Z(mag) and Z(phase)--or any other pair of data items--for our work. R and X simply give us some focus to develop a sense of what we might eventually learn from the graphing exercise.

Unless we are using Multi-NEC, we shall have to create graphs for ourselves externally to the NEC implementation. The first step is to perform whatever re-shaping we might need to do to enable us to import the data from the table into a spreadsheet. Many spreadsheets create separate columns only when the separator between data values in a table is of a certain sort. TAB is perhaps the most universal separator for easy spreadsheet entry. If we save the data table and open it in a word processing program, then we can find and replace a uniform series of spaces with a TAB code. From that point, we can copy and paste the revised table into a spreadsheet.

The rest of the job is simply creating a graphic using the X-Y format, along with axis labels, titles, and any marker notations that we might find useful. For example, the X- and Y-axes of our graph will note only the range of values for R and X. They will not locate specific frequencies. Therefore, we may wish to add a few marker notations to facilitate comparing graphs.

The 1000:1 L/d Dipole

The very thin-wire dipole provides a useful starting point. At 300 MHz, its length is 0.4810 wavelength (or meters), with a diameter of 0.000481 wavelength (or meters). Both dimensions translate directly into meters at this frequency. Let's begin with a very conventional graph of the feedpoint or source resistance, reactance, and 72-Ohm SWR values. Fig. 3 provides the data as developed via EZPlots, another AC6LA program for analyzing frequency sweeps taken with EZNEC.

The chart seems clear enough, despite the three data curves. Between the points of minimum SWR, which also generally mark the low values of R, we can see the peaking of the values of X as we increase frequency. As we increase the frequency, we can also see that the peak values of SWR systematically decrease, along with the peak values of R and X. The SWR peaks seem to correspond roughly to frequencies at which the antenna element is an integral multiple of 1 wavelength--but not exactly.

If we take an X-Y graph of the resistance and the reactance, we obtain a chart with the appearance of Fig. 4. I have purposely shrunk the width of the chart so that the X- and the Y-divisions are about equal in space, even if not in numerical values. Since most such charts that appear in texts have a relatively square form, producing nearly circular patterns of data values, this shape gives the sample an air of familiarity. Unfortunately, my spreadsheet does not have a spline function to round the curves, and we do not have enough data points to yield a good round outer curve to the spiral.

The spiral itself captures some of the essential features of the linear graph and sets them into fairly bold relief. As we increase the operating frequency, the resistance at the low-impedance resonances increases with each passage. In addition, with each increase in frequency, the peak values of resistance and reactance decline. The data points for resistance and reactance are the same ones that appeared in Fig. 3. However, the presentation in Fig. 4 allows us to see some of the interesting relationships more clearly.

Note that I have added a few frequency markers. One points to the beginning of the curve at 300 MHz. The others mark the frequencies at which we might have expected the antenna to show a resonance as the reactance makes its sudden transition from a very high inductive value to a very high capacitive value. However, due to end effect and other factors, these frequencies to not mark resonant points on the curve.

Rather than probe this single graph for various further details, let's turn to a second dipole. Some of the utility of X-Y graphs lies in comparing one with another--so long as the antennas involved are indeed comparable.

The 100:1 L/d Dipole

A fit comparator for the dipole with a length-to-diameter ratio of 1000:1 is another dipole with a 100:1 ratio. At 300 MHz, the length is 0.4676 wavelength (or meters) with a diameter of 0.004676 wavelength (or meters). Like the first antenna, it will be resonant at 300 MHz as a 1/2-wavelength dipole. The question that we may pose to our resistance vs. reactance X-Y graphs is how the two antennas behave similarly and differently between 300 and 2400 MHz.

Fig. 5 shows the conventional graph of resistance, reactance, and 72-Ohm SWR for the fatter dipole, using a frequency-based X-axis. In many respects, the curves for the two dipoles are very similar in shape. However, if we compare the values on the left and the right Y-axes, we shall see that the peak values are far lower in every category.

The left part of Fig. 6 shows the X-Y graph of resistance vs. reactance for the 100:1 dipole. The spiral resembles the one in Fig. 4, but with a few exceptions. For example, the peak values are rather vividly lower. The version of the graph on the right uses the same axis range as Fig. 4, and the smaller range of values in the new antenna's spiral becomes very clear.

In addition, note the positions of the frequency markers on the graphs in Fig. 4 and Fig. 6. The fatter version of the antenna places the markers further along the spiral than does the thinner dipole. In addition, we may note that in both of the resistance vs. reactance graphs, we see a more extreme value of capacitive reactance than of inductive reactance.

These are by no means new discoveries about dipoles of varying length-to-diameter ratios. Basic college texts will contain a number of equations by which to calculate the impedance behavior. The function of the X-Y graph is to present the data in a manner that naturalizes it so that it becomes part of our expectations of dipole behavior.

The 20:1 L/d Dipole

Two instances do not themselves establish a trend. Therefore, let's add one more dipole to our collection, this time with a length-to-diameter ratio of 20:1. The length at 300 MHz is 0.45 wavelength (or meters), with a diameter of 0.0225 wavelength (or meters). This dipole is about as fat as we dare let the model go while still expecting reliable data. Fig. 7 provides the resistance, reactance, and SWR data in the conventional format.

Once more, the curves have their by-now familiar shapes, but the peak values have declined even further. Above about 1200 MHz, the resistance begins to flatten so that a value of about 100 Ohms becomes the median value. Indeed, a center-fed element with a 20:1 length-to-diameter ratio becomes a candidate for being a broadband antenna, were it not for the fluctuations in the reactance. For example, the antenna exhibits a 400-Ohm SWR of under 2:1 from about 400 through 575 MHz.

The X-Y graph in Fig. 8 provides the spiral perspective on the fattest of our dipoles. The smaller range of both resistance and reactance values removes much of the distortion from the actual smooth curves of the transition between values. In fact, the 20:1 dipole shows total ranges of both resistance and reactance that are about 1/3 of the range shown by the 100:1 element and well under 20% of the ranges displayed in the spiral for the 1000:1 center-fed antenna. Nonetheless, all three spirals share the common trait of shrinking ranges of both resistance and reactance with rising frequencies. If we keep the 3:1 range difference in mind between the fatter two dipoles, we can also see that the frequency markers are farther along the spirals for the thicker of the two, continuing the potential trend that we saw when comparing the first two antennas in our exercise.

We also raised the question as to whether the apparent domination of the spirals by capacitive rather than inductive reactance was a real phenomenon or an artifact of the increment selected for creating the curves. In fact, the phenomenon is real. From 400 to 600 MHz, the capacitive reactance peaks at about -j250 Ohms. However, the inductive reactance never quite reaches j100 Ohms. One might leave the explanation for this condition--reflected to lesser degrees in the thinner dipoles--as "an exercise for the reader," but we should not forget the capacitance between the element halves at the feedpoint gap created in the model and in real antennas. Most cage elements (assuming periodic rings around the wire collection to ensure even current distribution) bring the wires forming the fat dipole together in a sloping point, a structure that reduces the capacitance. Some wide-band elements may create a biconical structure for up to half the length of each side of the center feedpoint.

And that last note brings us to the final element of our collection.

The Biconical Dipole

As a contrast to the uniform-diameter dipoles with which we have been working in order to develop an appreciation of resistance vs. reactance X-Y graphs, we may examine a sample biconical dipole having the structure shown in Fig. 1. We shall use 4 wires to simulate the cone, brining the ends of each wire together at the center of each end. The nominal slope of reach cone is 10 degrees relative to the dipole's centerline. The value is nominal, since the feedpoint region consists of a short 3-segment wire, with the middle segment serving as the source segment. Hence, there is about a 0.2-degree difference between the angle of each wire relative to where it joins the source wire and the virtual angle taken from the exact center to the outer tip of the cone wires. The differential is not sufficient to invalidate the very general outline of impedance behavior for the biconical antenna between 300 and 2400 MHz.

Each of wires has a diameter at 300 MHz of 0.002 wavelength (meters). The overall length is 0.3522 wavelength (meters). The maximum distance across the extreme end of the element is 0.061 wavelength (meters). Whether we can call this dimension the diameter of the cone at its widest opening depends upon the degree to which 4 wires simulates a solid-surface cone, a consideration requiring a different context and discussion from the present topic. As well, because the biconical element changes its diameter along its length, we cannot readily assign to it a length-to-diameter ratio. However, see Kraus, Antennas, 2nd Ed., Section 9-11 for a discussion of and equations for calculating the impedance of thin cylinder and biconical elements. Imperfect as the simulated biconical structure may be, it does provide a good indication of biconical properties. Note, for example, the overall length, resonant at 300 MHz (with an impedance of 52 Ohms), in comparison to the lengths of the uniform-diameter dipoles, the shortest of which is 0.45-wavelength (meters)

The conventional graph of resistance, reactance, and (52-Ohm) SWR in Fig. 9 displays much of what we expect in any dipole. The undulations of all three properties recorded in the graph resemble those of the 20:1 L/d dipole, although the biconical element shows slightly higher peak values in the first SWR cycle. For example, the fat dipole shows a peak resistance of about 400 Ohms, while the biconical antenna has a peak resistance of about 600 Ohms.

Perhaps the most notable difference between the uniform-diameter dipoles and the biconical element becomes evident when we count SWR cycles. All of the cylindrical dipoles show an average of about 3-1/4 SWR cycles between 300 and 2400 MHz. In the same span, the biconical antenna exhibits about half a cycle less. Although we see a progressive broadening of the bandwidth as we increase the diameter of the dipoles, the biconical simulation outstrips the dipole progression by a significant margin.

The X-Y graph of resistance and reactance in Fig. 10 reveals some additional properties that may, under certain circumstances, be useful to know. Unlike the uniform-diameter dipoles, the biconical antenna shows nearly equal inductive and capacitive reactance peaks in the first cycle of the spiral. However, as we raise the operating frequency, capacitive reactance begins to dominate each cycle. An average reactance line drawn across the face of the graph would fall in the vicinity of the -j50-Ohm marker. We must moderate this average by noting that the dominance of the capacitive reactance in the source impedance appears to become stronger with each successive cycle. In contrast, the uniform-diameter models seem to present a near symmetry of reactance in each cycle once we establish an average value line on the graph. To what degree modeling limitations may enter the values for the higher frequencies would become a necessary consideration for an actual antenna that might be under analysis.

One of the important external additions to the graph is annotating the curves with frequency markers at critical points. In the case of the biconical dipole, these notes allow us to see clearly to what degree the antenna geometry has spread the undulations of resistance and reactance across a wider range than we found for the cylindrical dipoles.

Conclusion

Our excursion into the sample dipoles has not tried to establish anything new about these fundamental antennas. The exercise examples have simply served as a convenient way to illustrate the benefits of adding to what modeling programs provide by creating an external graphing functions. In this case, we have extracted the source information from NEC core runs over a wide frequency sweep to produce X-Y graphs of resistance and reactance. The result is a spiral graph of the values that served to reveal some properties more clearly than standard linear graphs. Although we have used an external spreadsheet to produce the sample graphs, Multi-NEC provides this facility as part of its spreadsheet shell for using a variety of NEC cores. Fig. 11 shows a resistance-reactance plot for a sample antenna in the Multi-NEC application.

Although the resistance-reactance X-Y graph is the most common form in antenna work, other pairs of values may prove relevant for X-Y graphing in many contexts. Even at a fundamental level, the R-X plot has significant use. For example, amateurs often install single-wire antennas design to serve all frequencies from about 3.5 MHz through 30 MHz. A series of X-Y plots of resistance and reactance can assist the average ham in finding a wire length at the proposed height and ground conditions that best avoids radically high and radically low antenna impedance values in each amateur band so as to minimize losses in the selected parallel transmission-line to the station's antenna tuner. Since some current programs allow the entry of transmission lines with their listed velocity factors and loss factors, one might model the entire system using various trial element lengths to arrive at the best combinations that is usable within the antenna construction site.

In short, additional post-core and post-program manipulation of data can serve useful purposes, and the X-Y graph is only one of them. -

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