We cannot model every possible antenna structure in NEC. Some structures are best suited for other types of software, for example, strip elements with a substrate on one side that has a definite dielectric constant. In fact, strip elements in free space may alone prove problematical unless we first perform some external equivalency tests to determine what size round wire best approximates the performance of a strip element. Very often, creating a series of simple dipoles at a design frequency will suffice, although there may be more critical situations in which we find a difference in the mutual coupling between strips relative to round wires.

In some cases, the limitations of NEC-2 and NEC-4 may limit our modeling abilities. NEC-2 provides various degrees of error in the output report for linear elements with stepped diameters. NEC-4 largely, but not completely, corrects this error. For small diameter changes between steps, NEC-4 is highly accurate, but becomes less accurate as the diameter steps grow larger, especially in high-current regions of a structure. Both cores tend to show errors with changes in wire diameter at angular junctions. Many software packages have modules to create substitute elements using the Leeson corrections to form uniform-diameter elements out of stepped-diameter elements. However, an angular junction will normally prevent the functioning of these modules or facilities. In addition, NEC cannot directly model coaxial wire structures. Hence, we cannot capture the physical aspects of an antenna element composed of coaxially arranged wires.

We have not listed all of the limitations of the NEC cores, but we have enough to give us sufficient reason to develop substitute models where NEC may not directly go. In a broad way, we may divide the types of substitute models into three groups.

Type 1. Substituting one geometry for another

Type 2. Substituting the electrical equivalent for the original structure where we may check our work with the original structure

Type 3. Substituting the electrical equivalent for the original structure where we may not check our work by modeling the full structure

Each type of substitute has different consequences for our trust in the models and different cautions for creating the substitute model. Therefore, let's examine a sample of each type to see what we might learn.

**Substituting on Geometry for Another**

Perhaps the most common form of geometry substitution consists of replacing a highly complex multi-legged tower with a uniform-diameter round wire with the same height. As earlier episodes in the series established, the BC industry has developed some very reliable guidelines for the substitution. Extensive cross checks between the substitutes and full models of multi-legged towers have yielded the following equivalencies.

Recommended Substitute Single-Wire Dimensions for Multi-Face Towers Tower Type Diameter Radius Triangular D = 0.74 * Face Width R = 0.37 * Face Width Square D = 1.12 * Face Width R = 0.56 * Face Width Note: D and R are in the same units as the Face Width

Modelers also face another problem--especially when using NEC-2--even for round tower sections when the tower steps the diameters of the sections. Let's consider a 60' tall tower over perfect ground (for simplicity of modeling) at 3.5 MHz. The tower consists of 6 10' sections, each 1/2" smaller in diameter than the next lower one. Let's use 3.5" for the base section diameter and taper it to 1" for the top section. The total tower is still short to achieve resonance at 3.5 MHz, so we shall add two short section of 1" diameter material to form a T at the tower top. We shall select lengths of tubing that just bring the tower to resonance over the prescribed ground. We shall place the source on the lowest segment of the lowest tower section.

We cannot simply invoke the Leeson corrections for the tower in this case, since the presence of the T-top will normally block the calculations, since the corrections only apply to linear section under certain conditions. Therefore, we shall have to proceed in steps, as indicated in **Fig. 1**.

First, we remove the T-top tubes from the tower. Now we may perform the Leeson correction calculations on the tower sections alone. Once we have derived the correct length and diameter of a uniform-diameter element that is equivalent to the original tower, we can replace the T-top and proceed to the final output reports that we might need. **Fig. 2** shows the process as it might proceed using EZNEC's version of the correctives.

Some less-practiced modelers might object to the lower section of the wire table, since it includes angular junctions between very different wire diameters. Therefore, let's tabulate the results and see what we obtain. Raw Gain indicates the direct NEC output report. AGT is the average gain test score, and converts to the gain adjustment value in the AGT-dB column. The adjusted gain appears in Adj Gain. The Feed Z column reports the source impedance.

Results of substituting one geometry for another: sample 60' tower with a T-top Model Raw Gain dBi AGT AGT-dB Adj Gain dBi Feed Z (R +/- jX Ohms) NEC-2 Original 5.41 1.062 0.26 5.15 35.8 + 14.4 NEC-2 Substitute 5.10 1.000 0.00 5.10 32.7 - j9.1 Substitute with 7.7' T elements 5.11 1.000 0.00 5.11 34.2 + j0.9 NEC-4 Original 5.22 1.018 0.08 5.14 35.4 - j0.3

The sample problem shows several things, not the least of which is that nothing critical is at stake except perhaps for the correct array gain value. We easily bring this into line by adjusting it with the AGT score. As well, the model shows that the symmetrically placed T-top elements at 90 degrees to the tower do not create serious errors. Unlike a single bend, as we might find in an inverted-L configuration, symmetrical elements (from two to many) result in virtually complete field cancellations from these low-current additions and do not adversely affect the AGT score or the general reliability of the results.

The original model, as shown by the last line of the table, emerged from a NEC-4 exercise using no correctives. This model is the origin of the 6.8' T elements. However, even NEC-4 shows a small but not insignificant departure from the ideal AGT score, enough to require an adjustment to its gain report. Therefore, it also has a degree of unreliability that--while smaller than for the uncorrected NEC-2 model--casts some doubt on the accuracy of the 6.8' length for the T-top elements. Increasing their length to 7.7' each brings the corrected model to resonance. In fact, the only difference between the 2 cores with respect to the substitute model is a minuscule difference in the report source impedance. NEC-4 reports 34.0 - j0.3 Ohms.

The sample also informs us that geometric substitutions are not perfect solutions if we plan to build the modeled structure. Assuming that we could simulate perfect ground and construct the tower as originally specified, the exercise would alert us to allow for considerable adjustment range in the lengths of the T-top elements if our goal happened to be to bring the antenna to resonance at 3.5 MHz. Of course, we may increase the level of modeling complexity by adding an appropriate real ground and bury some radials (in NEC-4) according to the number we plan to place at the tower base. Nevertheless, in all of its simplicity and final indefiniteness, the sample illustrates one of the typical processes of using a substitute geometry to arrive at a more adequate, if not quite perfect, model of the tapered tower and T-top situation.

**Substituting the Electrical Equivalent for the Original Structure Where We May Check Our Work with the Original Structure**

We may sometimes simplify the modeling process by replacing complex wire structures with simplified electronic equivalents. The process is especially applicable if we can first establish the equivalence between the substitution technique and an all-wire structure. Once confirmed, we may apply the technique with confidence in situations where we might not be able to accurately produce an all-wire model.

One such situation is the placements of 1/4 wavelength phasing stubs composed of parallel transmission line between successive 1/2 wavelength sections in a collinear array. The use of stubs keeps all sections of the array in phase. Because the stubs occur at high-impedance points along the wire, where voltage and current are changing very rapidly, the use of the NEC TL facility is not recommended. Therefore, modelers normally create all-wire models of the collinear array, as suggested by the top sketch in **Fig. 3**. We shall explain the lower half shortly.

The accuracy of such models often is restricted to cases in which the phase-line wires and the main element wires have the same diameter. For example, suppose that we wished the 2 wavelength array to use AWG #12 wire. We might construct the phase lines from the same wire, perhaps using a spacing of 6". Such an arrangements for a 15-meter (21.225-MHz) antenna might be quite practical. However, if we were to apply the same principles to a vertical array for the VHF or UHF range, it is more likely that the main vertical element and the phasing line would have very different diameters.

Vadim Demidov recently sent me a note outlining an alternative procedure that does not require the very high segmentation often required of all-wire (sometimes called "brute-force") models. As Vadim explained his reasoning, "After splitting TEM and common-mode phenomena in the stub I suggested considering it as an ideal auto-transformer with its midpoint "grounded" by means of a quarter wavelength wire. In this type of model, a stub is represented by its common-mode equivalent, which is a single wire (without need for too fine segmentation), while the phasing transformer is made by a short transmission line linking two segments joining it." The result is the model in the lower sketch in **Fig. 4**, where the EZNEC designator "T" marks the location of the ideal transformer, and the vertical wire is the common-mode element.

To confirm the exercise, I converted the collinear 21.225-MHz array into a Demidov model. **Fig. 4** shows what is involved. However, understand that this is a proof-of-principle exercise. Therefore, both models use the same level of segmentation on all wires in order to minimize modeling differences. My goal was to discover to what degree we can trust the Demidov electrical substitute as an accurate representation of the all-wire model of presumed accuracy.

The upper section of the model shows the original all wire structure, with the 6" spacing between the wires of the phasing lines. The lines are each 132" long. The middle section shows the simplified wire table of the Demidov substitute. The junctions between the main element wire and the common-mode wires appear at points exactly half way between the feedpoint and the wire outer ends. The bottom section of the figure shows the two ideal transformers. Each uses as close to an infinitesimal length as one's modeling program will permit. Anything from 1e-5 wavelength and shorter will do. The idea is to use a length of transmission line that is so short that no significant impedance transformation can occur along its length. The 300-Ohm characteristic impedance is largely arbitrary, as values between 50 and 600 Ohms work as well.

Comparative results using NEC-4, single precision, on all-wire and a Demidov models of a 2-wavlength 21,225-MHz collinear array Model Gain dBi Beamwidth Feed X (R +/- jX Ohms) AGT All-wire (AWG #12) 11.97 26.2 deg 2189 + j29 1.001 Demidov substitute 12.05 26.2 deg 2133 - j57 1.001

The differences between the results are insignificant, especially in view of the fact that the critical junctions and the source position occur at very high impedance positions on the model. In fact, the models shown in **Fig. 4** contain an illusion. The length of the common-mode stub in the sample is just about 3" longer than the parallel line stubs in the original model. The illusion is that the common-mode stub accounts for the original stub length plus 1/2 the spacing between stubs. In fact, if we change cores and run the same substitute model in each, variously using single and double precision versions of each core, we obtain different values for the source impedance. They are all very high resistively and fall on the very steep curve that marks the ordinary reversal of reactance. Hence, the differences do not make a difference. In a real construction situation, a builder would need to adjust the length of the stub for best performance, taking into account the velocity factor of the actual phase line used. A quarter wavelength at 21.225 MHz is 139", the length of the Demidov common-mode line shown. However, with real wires having a small copper loss plus any dielectric shortening required, the physical length in most cases will be shorter, as it is in the original model.

The exercise does show an example of a substitute modeling technique that can be verified against an all-wire model. Once confirmed, we may use the technique in other comparable situations, even those where a direct comparison may not be feasible due to the large size of the all-wire model or the inability to handle velocity factors easily.

**Substituting the Electrical Equivalent for the Original Structure Where We May Not Check Our Work by Modeling the Full Structure**

There are some types of antennas that we cannot directly model within NEC (or MININEC). By directly model, I mean to replicate the physical structure within the confines of the NEC wire facilities. One type of structure that we cannot effectively model physically is a coaxial element, where the physical antenna may use a coaxial cable as part or all of the structure. In some cases, especially with fairly simple structures, we may be able to construct reasonable replicas using a series of wires surrounding a central wire. However, in most cases, we must resort to various techniques to ensure that the surrounding wires form a single relevantly continuous conductor around the central wire.

One antenna type that has recently seen renewed popularity is the coaxial collinear array, especially in vertical form for VHF and UHF use. The antenna has a very long history, but came into prominence in the 1950s as a potential VHF mobile array and also in radar uses due to the potential for developing a very narrow bi-directional beamwidth in a horizontal orientation. In the 21st century, the amateur search for an ideal omni-directional vertical array with very high gain for line-of-sight paths has brought on a surge of interest. With the interest has come an urge to model the antenna.

**Fig. 5**, on the left, shows the outline of the form most amateur envision using. A shorted stub 1/4 wavelength top section completes the array. At the base, the lowest section consists of a 1/4 wavelength section with 4 radials to form the feed portion. Between the top and bottom section, we may place any number of 1/2 wavelength sections, 1 through n. Although the sketch shows only 2, the number is limited only by the physical space available to hang the somewhat floppy coaxial array.

The sections of the array consist of length of coaxial-cable transmission line. Hence, each section, whether 1/2 wavelength or 1/4 wavelength is electrically only that long. The physical length is shorter, since we multiply each electrical length by the velocity factor of the line used. At each junction, we reverse the connections of the lines so that we end up with the equivalent of a 1/4 wavelength phasing stub without the need to install one. The required phase reversal (that actually produces a phase continuation) results from the line connection reversals at each junction.

On the right of **Fig. 5** we find a modeling work-around that has been proposed to capture the antenna's performance. We separate the TEM or transmission line currents from the radiating currents by using two separate sets of connections between sections. The physically modeled wire that is solid in the sketch does the radiating. The dotted line represents transmission-line section connected from one end to the other end of each section wire. Note that in this idealized model, the top and bottom sections are bare wire without transmission lines. As well, the feedpoint comes between sections rather than at the base of the antenna. Hence, the model will not simulate directly the conception of a coaxial collinear antenna sketched on the left. But it may give us some idea of what happens if we successfully manage to phase successive 1/2 wavelength sections of wire (with the length adjusted for the line velocity factor of the proposed cable).

**Fig. 6** shows the wire and transmission-line tables from an EZNEC version of the antenna. For convenience, the original modeler has used a separate 1-segment wire between sections on which to make the connections for both the source and the transmission lines. However, the total length of each section consisting of a longer wire and the connecting section is 0.41-wavelement, the result of multiply 1/2 by the line velocity factor of 0.82. The antenna begins 0.5 wavelength above a perfect ground in the ideal model.

The model provides us with two important outputs. As shown in **Fig. 7**, the current distribution is in phase and quite even along the length of the antenna. Hence, the array attenuates high angle radiation very well, as shown in the elevation pattern to the right. (Over average ground, the gain drops to 10.35 dBi, nearly 4-dB lower than over perfect ground. The elevation angle is about 4 degrees, equivalent to a single dipole at a height of over 3 wavelengths, but with a gain advantage to the coaxial collinear array.)

Unlike the horizontal phased array that we previously discussed, we cannot compare the idealized coaxial collinear model with a physical version of the same antenna. Therefore, we must approach the substitute model with all due caution. For example, the reported impedance at the source is 269 + j54 Ohms. However, the model is exceptionally sensitive to changes in the velocity factor. Decreasing the value by only 0.01 drops the reported impedance to just above 100 Ohms, with a sizable remnant reactance. Variations in the velocity factor of cables between lots may vary by several percent. What the model cannot tell us is whether the physical implementation of the antenna will be equally sensitive to variations in the line velocity factor. Moreover, the model does not reveal what effect a revised lower section might have, should one wish to replicate the more use practical of using a base section that is 1/4 wavelength long with a set of radials. We may model such an arrangement, but casual modeling in this direction shows reduced gain, stronger high-angle lobes, and a departure from the smooth current magnitude curves of the ideal model. I shall not show any models taken in this direction because they leave us with the same difficulties in correlating the model with a real antenna.

It is possible in the abstract to create models that seemingly are the electrical equivalents of physical structures that fall outside the boundaries of direct capture in a wire structure. Many of these models may prove useful in seeing some basic properties of antenna types. However, they remain limited in their reliability as models--despite nearly perfect AGT scores--due to the fact that we have no way to compare the models with physically accurate versions. In most cases, we also lack detailed information on performance from rated test ranges. In the present case, just such information would be necessary to determine if the sensitivity to small changes in the cable velocity factor is a physical feature of the coaxial collinear antenna or an artifact of the idealized model.

**Conclusion**

We have examined several different types of substitute models ranging from simple geometry substitutions to replacing physical structures with their electrical equivalents. The goal is not to discourage the use of substitute models. Rather, the aim has been to alert modelers to the level of caution necessary to bring to the models. Especially in cases where we cannot model the antenna as a physical set of wires, we should exert the highest levels of caution.