Modeling Radiating Surfaces

L. B. Cebik, W4RNL (SK)

The notes in this exercise derive from my attempts to determine if it is possible to model with reasonable accuracy the results obtain by an experimental exercise conducted in 1952 by two RCA researchers, George H. Brown and O. M. Woodward, Jr. Among their numerous contributions to the development of VHF and UHF antennas, including the emergent television antenna industry, was an experimental characterization of conical and triangular antennas. (See "Experimentally Determined Radiation Characteristics of Conical and Triangular Antennas," RCA Review, Dec., 1952, pp, 425-452.) The work eventuated in the widespread use of solid-surface fan dipoles in TV antennas, especially for the new UHF channels from about 480 to 920 MHz. It even resulted in the bent bow-tie dipole used in corner-reflector TV antennas. I had some limited success in capturing in NEC models some, but by no means all, of the capabilities of the corner reflector with a bent bow-tie in Planar and Corner Reflectors.

Brown and Woodward wanted to experimentally characterize the properties of bi-conical dipoles and fan dipoles, antennas that had undergone extensive theoretic analysis, but with what Brown and Woodward saw as "simplifying assumptions and approximations in order to satisfy the required boundary conditions and to reduce the mathematical difficulties." (p.425) As shown in simplified form in Fig. 1, they reduced the dipoles to UHF solid-surface monopoles with a very large highly conductive ground-plane surface to simulate a perfect ground (PEC).

The practical interest in these antenna types does not involve their 1/2-wavelength impedance. Instead, it involves the impedance and radiation properties of these antenna shapes as they approach and surpass a 1-wavelength electrical length at a given operating frequency. A linear dipole shows a very high impedance at 1-wavelength, with regular repetitions of the impedance peak at integral multiples of that length. However, the biconical shape shows a regular decrease in the peak impedance as the angle formed by opposite sides of the cone gradually increases. At very large angles--about 60 degrees apex angle--the reactance swing associated with a linear dipole decreases to a level that is manageable for wide-band antenna service. In addition, the difference between the impedance at odd multiples of 1/2-wavelength and integral multiples of 1 wavelength also decreases. As a result, the biconical shape--with a sufficiently large apex angle--results in the potential for an antenna with a 300-Ohm SWR of less than 2:1 over a very broad frequency range. For reasons of pattern shape that we shall see along the way, VHF and UHF use of the phenomenon is limited to about a 2:1 frequency range. This characteristic is eminently convenient for TV antennas that typically use a 300-Ohm feedpoint impedance.

Since the biconical dipole is somewhat complex as a manufactured item, Brown and Woodward also explored the easier-to-make flat triangular dipole, again with a solid surface. For equivalent apex angles, it showed similar characteristics, but not quite as flat a bandwidth impedance as the biconical antenna. However, the result was good enough to allow Brown and Woodward to report on a successful flat, solid, fan dipole using a 60-degree apex angle to cover 480 to 920 MHz with a 300-Ohm SWR level of less than 2:1.

The Modeling Interest

A number of questions arise from the Brown and Woodward work for the antenna modeler. First, one may ask whether it is possible to replicate in models the core of the Brown and Woodward experimental effort. Given the use of solid surface cones and triangles in the original experiments, this question does not have as easy an answer as we might like to give. NEC employs round wires to model the geometry of radiating elements. Its surface-patch facilities were intended to simplify the construction of surround bodies and objects--such as the hulls of ships--that might affect the antenna's radiation characteristics. Their function was not intended to serve as radiating elements with direct voltage sources. Hence, the NEC surface patches use considerably simplified equations to speed core runs. If we are to simulate the Brown and Woodward antennas, we must use round wires to form the surfaces.

The second question that we shall explore is whether we may extrapolate the VHF/UHF results using solid-surface antenna elements down to the HF range. Wide-band center-fed elements have proven to be highly desirable over the decades since the Brown and Woodward paper, for example in curtain arrays for both SW broadcasting and over-the-horizon HF radar systems. However, the ability to extrapolate the experimental results will depend upon our ability to translate solid surface structures into structures using individual wire elements components. Hence, from the perspective of modeling, the first question is the key to the second. Although we may easily model wire-based extended-range elements at HF, relating their properties via models to the Brown and Woodward experiments requires that we be able to successfully (within reasonable limits) model the solid surfaces with round wires.

The Biconical Dipole

One advantage of modeling in NEC is that we may proceed directly to the biconical dipole and not pass through the monopole stage. Modelers have long known that we may model a solid-surface biconical dipole with reasonable success by using a collection of longitudinal wires, so long as we use enough of them. Fig. 2 shows the outline of a bi-conical dipole with a 40-degree apex angle. Each cone uses 12 wires. Experience has shown that there is very little difference in the output of models using 12, 25, and 45 wires per cone. To achieve a usable average gain test score (AGT), the more wires that we use, the thinner must be each wire, since all cone wires meet at a junction. As we increase the number of wires, the angle between wires at the junction becomes narrower, increasing the length along the first segment of each wire in which we have surface interpenetration for a given wire size. As we add more wires, the interpenetration increases unless with use thinner wires. In addition, the length and segmentation of the center or source wire connecting the two cones may require custom treatment to achieve the AGT score nearest to the ideal.

The simulated biconical dipole in the figure is 16.8" long, with an end diameter of 5.2". The wire diameter is 0.002". The intended useful frequency range for the antenna is from 480 MHz to at least 920 MHz, to coincide with the Brown and Woodward TV fan dipole. To achieve this range, the 1/2-wavelength self-resonant frequency of the antenna is about 231 MHz in this free-space model. However, we are not interested in the first self-resonant point. Rather, we are interested in the antenna's behavior as it approaches and passes 1 wavelength.

The biconical dipole meets the Brown and Woodward requirements for a UHF TV dipole. Those requirements include having a bi-directional pattern and a 300-Ohm SWR that is less than 2:1 across the passband. Table 1 shows the key antenna characteristics and the AGT value (with adjusted gain values) at the passband ends and at the approximate geometric mean frequency.

Table 1.  Biconical dipole performance

Frequency Source Impedance 300-Ohm Raw Gain AGT AGT-dB Adj. Gain
MHz R +/- jX Ohms SWR dBi dBi
480 346 + j187 1.80 2.59 1.012 0.05 2.54
665 345 - j 58 1.26 3.32 1.012 0.05 3.27
920 167 + j 33 1.83 3.30 1.012 0.05 3.25

With respect to the desired SWR values, the tabular entries seem to describe a curve. The bottom half of Fig. 3 confirms the impression. The top half of the figure extends the SWR curve to 1500 MHz to establish the general pattern of biconical behavior. As we increase the electrical length of the antenna by increasing the operating frequency, the resistive and reactive components--as indicated indirectly by the SWR values--continue to decrease the difference between the values at odd multiples of a half-wavelength and integral multiples of a full wavelength. In short, the higher the frequency of operation or the longer the antenna length for the 40-degree apex angle biconical antenna, the flatter the SWR curve grows.

Our interest in this behavior is not to establish it. That has long been done via theoretical analysis and physical experimentation. Our interest lies in seeing whether the 12-wire biconical model can effectively and reasonably capture that behavior. The model does that job. There are variations on the model that will affect its dimensions, but with no great change in the results. For example, we may connect the outer ends of the cone wires to form a circle. Essentially, the end wires add length to each individual wire, a length that is roughly but not exactly half the distance between the wire tips. Hence, for the same performance curve, we would have to reduce the physical length of the cone. As well, adding or subtracting wires from the assembly will slightly change the required length for the same performance curve, although each such change should be accompanied by a change in the wire diameter within the limits of NEC's ability to handle angled junctions of wires at the center.

Applications for the biconical antenna in TV antennas that might make use of a planar or a corner reflector rarely make use of the extended impedance stability of the antenna shape. All such arrays depend upon using a fed element with a bi-directional pattern. Fig. 4 shows sample free-space E-plane patterns for the biconical dipole within the passband of intended use with one extra pattern a bit beyond the upper limit.

A linear dipole would show multiple lobes with stronger angular lobes than broadside lobes by the point at which the antenna is about 1.5 wavelengths long. The biconical dipole extends the range of bi-directional patterns to nearly the 2-wavelength point, although the 920-MHz pattern shows significant but non-fatal sidelobe development. By 1150 MHz, the pattern has become completely useless for a directional beam with a planar or corner reflector. (We shall be interested in comparing these patterns with a corresponding set for a flat-face fan dipole.)

For some applications, the change in pattern is less important. For example, scaled by a factor of 100, the antenna would provide a very wide-band antenna for general purposes. The lowest frequency would by about 4.8 MHz, with an undetermined upper limit for a 300-Ohm feedpoint impedance that would not require a tunable matching network. A single transformation of 300 Ohms to 50 Ohms (by way of a balun) would satisfy the impedance requirements for most common transceiving equipment.

The limitation for this antenna is that it would require a length of 140' with end diameters of 43.3'. As well, it likely would require considerable mounting height to overcome the effects of ground proximity on the feedpoint impedance across the operating span. The scaled wire size would be 0.2", equivalent to AWG #4 wire. However, one might easily use thinner wire by increasing the number of wires in each cone. With an additional scaling factor of 2, the antenna would cover 80 through 10 meters. If we increase the added scaling factor to 2.7, we might add 160 meters, but the chances of having a support system that would handle 115' diameter ends at a sufficient height to avoid deleterious ground effects on the source impedance would dwindle to the day-dream level.

Our notes on modeling a biconical dipole have not sought to establish a number-by-number correlation to the Brown and Woodward experiments using a mono-conical element with a perfectly conducting ground-plane surface. Rather, our goal has been only to establish that we can simulate a biconical dipole with NEC's round wires. The successful result is not surprising, since there are many examples of physical antennas that employ the same technique. In fact, there are HF discones, a first cousin to the biconical dipole and a more immediate kin to the Brown and Woodward test antennas, that employ wire structures for successful operation.

A Model of a Solid-Surface Fan (Triangular) Dipole

Brown and Woodward also reported on their experiments with a triangular monopole using the same large highly conductive ground plane surface. The antenna was equivalent to one-half of a fan dipole. However, as suggested by Fig. 5, their element used a solid sheet rather than a simple outline of a fan. To see something of the performance difference, Table 2 provides performance figures for an outline fan dipole that is 14" from one end to the other and 7.8" tall at the ends. The apex angle is 60 degrees so that the element half, exclusive of the short center source wire, forms an equilateral triangle. This shape is very close to the UHF fan dipole created by Brown and Woodward for use from 480 to 920 MHz.

Table 2.  Fan outline dipole performance

Frequency Source Impedance 300-Ohm Raw Gain AGT AGT-dB Adj. Gain
MHz R +/- jX Ohms SWR dBi dBi
480 718 + j190 2.59 3.32 1.010 0.04 3.28
665 93 - j276 6.13 6.42 1.018 0.08 6.34
920 337 - j212 1.95 0.37 1.012 0.05 0.32 (multiple lobes)

In effect, the fan outline operates more like a linear dipole than a biconical dipole. With a half-wavelength resonance at 250 MHz, the 4-lobe structure at 920 MHz is an approach to an electrical 2-wavelength equivalent. To obtain performance that more closely approaches the Brown and Woodward results, we must fill the outline to simulate a solid surface. The right side of Fig. 5 shows the pattern used in the test model. One triangle results from an exercise using NEC-Win Synth (program may not be compatible with OS newer than Win2k) and saved as a .NEC file. The 60-degree triangle was then re-opened in EZNEC Pro/4 v.5 as an incomplete model. EZNEC prefers to create models by using a single wire between junctions, but does accept without an error report a set of wires where wire crossings occur at segment junctions. I moved the triangle to one correct position for a dipole half and then replicated the structure and rotated it by 180 degrees to form the other dipole half. A short connecting wire between triangle apex points for the source completed the model.

To simulate a solid surface requires that we use a sufficient wire diameter. Selecting the diameter is a compromise between true electrical solidity at all operating frequencies and an acceptable AGT score. A 0.1" diameter wire yielded AGT values that averaged about 1.06, not quite ideal but usable on the premise that we are seeking the operating trends and not construction guidance from the model. Table 3 provides the reported data from the model at the three sample frequencies. The half-wavelength self-resonant frequency for the model is 252 MHz.

Table 3.  Wire-grid fan dipole performance

Frequency Source Impedance 300-Ohm Raw Gain AGT AGT-dB Adj. Gain
MHz R +/- jX Ohms SWR dBi dBi
480 278 + j194 1.94 2.91 1.057 0.24 2.67
665 392 - j 56 1.37 3.89 1.059 0.25 3.64
920 171 - j 61 1.86 5.42 1.067 0.28 5.14

The SWR values are much closer to the biconical of Table 1 than they are to the fan outline values of Table 2. In fact, the SWR curve at the bottom of Fig. 6 is very close to the curve obtain by the Brown and Woodward fan dipole (Fig. 42 on p. 452 of the referenced article). The upper portion of the SWR charts shows the extended SWR curve to 1500 MHz. It almost replicates the smoothness of the corresponding biconical curve but shows a slight compression of values relative to frequency, suggesting that the wire-grid simulation of the flat fan dipole does not achieve the broad-banding effect to the same degree as the biconical element.

The gain data for the wire-grid simulation of the fan does not quite match the performance reported by the Brown and Woodward experimental solid-surface fan. If we subtract the gain of a linear dipole (about 2 dB) from the values on the Brown and Woodward curve, the gain curves correspond. However, the pattern shapes reported by Brown and Woodward do not coincide with the curves in Fig. 7. The Brown and Woodward pattern for 920 MHz shows no sidelobe development, although the wire-grid fan pattern for the same frequency shows moderate sidelobe strength. By increasing the wire diameter of the wire-grid simulation of the solid surface, it is possible to reduce the sidelobe development in the 920-MHz pattern, but the model becomes wholly unreliable before the sidelobes diminish completely. Hence, the model fails to capture the pattern shapes reported by Brown and Woodward for the upper end of the operating spectrum.

Although outside the operating passband of the antenna, the additional pattern for 1150 MHz allows some initial comparison with the performance of the biconical dipole at the same frequency. With respect to gain and pattern formation, the flat 60-degree fan shows a broader bandwidth than the 40-degree biconical antenna. The 1150-MHz pattern still shows the broadside lobes that are characteristic of a linear dipole just slightly longer than 1.5-wavelengths. In contrast, the biconical pattern at the same frequency displayed a pattern closer to a 2-wavelength linear dipole. The difference is largely due to the difference in the self-resonant half-wavelength frequencies needed to obtain the desired SWR curves in the defined operating passband.

The wire-grid fan dipole does manage to capture most of the data reported by Brown and Woodward for their UHD TV dipole, even if imperfectly. It also shows its relationship to the biconical shaped element and to the fan outline element. However, there are alternative structures used historically to simulate a solid flat fan shape. We now have enough modeling data that we are positioned to evaluate them as potential methods of capturing the Brown and Woodward solid-surface fan dipole.

The Multi-Wire Fan Dipole Model

One popular way sometimes used in the HF range to simulate a solid surface in a fan dipole is the use of a series of wires in each fan triangle, with each wire extending from the apex to the opposite side. Fig. 8 shows such a structure using 5 wires. In all dimensions, the fan dipole is identical to the wire-grid simulation of a solid surface. It is 14" from end to end and 7.8" high at the outer ends, with a 60-degree apex angle. The wire diameter for this model is also 0.1". The question for modeling is whether this model, much simpler to form, is an adequate simulation of the solid surface antenna upon which Brown and Woodward developed their data.

If we were concerned about the half-wavelength self-resonant performance of the antenna, the simplified model might serve. The self-resonant frequency is 260 MHz, which is not far from the 252-MHz self-resonant frequency of the wire-grid version. In both versions, the highest current occurs relatively close to the apex of each triangle, where the wire density in the multi-wire version is highest. However, the stable impedance performance of the solid-surface fan antenna depends upon using the antenna at a length that approaches and passes 1 wavelength. Current maximums occur at positions well away from the center source wire. In the multi-wire version of the fan dipole, the wire density diminishes steadily as we move away from the center source wire. Table 4 shows some of the consequences of the decreasing wire density toward the fan ends.

Table 4.  Multi-wire fan dipole performance

Frequency Source Impedance 300-Ohm Raw Gain AGT AGT-dB Adj. Gain
MHz R +/- jX Ohms SWR dBi dBi
480 285 + j188 1.87 2.79 1.027 0.12 2.67
665 382 - j 76 1.39 3.86 1.048 0.20 3.66
920 149 - j 58 2.10 5.68 1.077 0.32 5.36

The results are both promising and disappointing. In some respects, the impedance data appears to be comparable to the data in Table 3 for the wire-grid model. However, the resistive component of the multi-wire model's impedance has a wider range of variation across the operating passband than we find in the wire-grid model's data. The consequences for the 300-Ohm SWR curve, in the lower half of Fig. 9, are a reduction in the 2:1 SWR bandwidth. The extended SWR curve at the top of the same figure shows essentially the same general pattern that we found in the wire-grid curve (Fig. 6), but with a much wider variation in value.

With respect to gain and pattern shape, the multi-wire model using relatively fat (0.1" diameter) wires shows less variation from the wire-grid model than the SWR curves. The adjusted gain values are similar for the two models. As revealed by Fig. 10 (when compared to Fig. 7), the pattern shapes are very nearly twins, even at 1150 MHz. For some purposes, the simpler multi-wire model may be a suitable substitute for the more complex wire-grid model. However, the multi-wire version alone does not disclose its shortcomings with respect to capturing the solid-surface SWR curve.

The 60-degree fan dipole might form a wide-band antenna in the HF range with suitable scaling. Roughly speaking, scaling the dimensions by a factor of 100 would yield an antenna theoretically covering 4.8 thought 9.2 MHz--and beyond, if the pattern shape is not a matter of concern. The resulting antenna would be a bit under 117' long with an end spread of 65'. However, the limiting factor in the scaling is the requirement to multiply the wire diameter by 100 to maintain the wire density. 10" diameter conductors generally fall outside the realm of feasibility for most (but by no means all) installations.

The temptation is to use common wire sizes. The original UHF model wire is 0.1", corresponding closely to AWG #10 wire. If we retain this practical wire size, the multi-wire model loses its ability to capture the properties of a solid surface. In the UHF model, the wire size would scale to 0.001". Table 5 provides the kind of data that we obtain for such a model (and for an HF antenna using 0.1" wire).

Table 5.  Multi-thin-wire fan dipole performance

Frequency Source Impedance 300-Ohm Raw Gain AGT AGT-dB Adj. Gain
MHz R +/- jX Ohms SWR dBi dBi
480 289 + j391 3.48 2.50 0.965 -0.15 2.65
665 864 - j 29 2.89 3.71 0.967 -0.15 3.86
920 155 - j102 2.23 6.01 0.975 -0.11 6.12

The impedance information from the thin-wire version of the 60-degree fan dipole shows very large excursions of both resistance and reactance. The 300-Ohm SWR curve for the operating range in Fig. 11 shows only one sudden dip below the 2:1 level, behavior that we might normally associate with a linear dipole/doublet at the self-resonant 3/2-wavelength mark. The frequency of the SWR minimum is 710 MHz in the model.

The association of the thin-wire model with the behavior of a linear doublet also shows up in the sample free-space E-plane patterns, shown in Fig. 12. The 920-MHz pattern is especially interesting for the crisp sidelobes, similar to those we might find in the pattern for a 1.25-wavelength center-fed wire. All of the broadband patterns for both the biconical element and the fat-wire fans show far less of a null between the sidelobes and the main bi-directional lobes. The crispness of the lobe structure in the thin-wire model carries over into the pattern for 1150 MHz. Compare the null depth values to those in Fig. 10 for the fat-multi-wire version of the same antenna.

In models and in physical antennas using multiple wires to simulate the performance of a solid surface, wire diameter and wire density both make a difference to the performance. As a fan element decreases the wire diameter without increasing the number of wires, the antenna gradually creases to perform like a solid surface. It becomes a version of a linear wire antenna with a somewhat wider bandwidth at the 1/2-wavelength resonant frequency region. However, using the necessary wire diameter and increasing the wire density are strategies that have undesirable consequences in physical antennas.

Conclusion

This exercise has explored the modeling of solid surface antennas, so easily fabricated for UHF frequencies, through the use of various round-wire modeling techniques. The Brown and Woodward experimental data from 1952 provided a standard against which we could measure to some degree the success of the modeling techniques. Biconical properties, with a few reservations, prove amenable to using a multiple-wire simulation, a method reflected in the construction of practical biconical antennas.

The flat solid-surface fan elements explored by Brown and Woodward required that we use some form of wire-grid structure to simulate the surface adequately. Even fat-wire models composed of multiple linear elements showed some departure from the performance curves in the original experiments. As we discovered in the final exercise, the use of wires that are too thin degraded the performance from its desired levels completely with respect to broadband coverage.

Even the wire-grid model failed to capture every nuance of solid-surface performance. Nevertheless, it proved productive to capturing most of the data experimentally derived by Brown and Woodward. Within the range of what the model successfully simulated and in what ways it fell short lie some lessons for effective modeling. 

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