## Modeling Folded Monopoles

### L. B. Cebik, W4RNL

he folded monopole is an interesting variation on the standard linear monopole. Essentially, the folded monopole is one-half of a folded dipole. As such it retains two important properties. First, the act of folding results in an increase in the feedpoint resistance relative to the linear or open-ended monopole. The exact ratio of impedance transformation depends on the relative diameters of the fed and the "other" wire. The transformation ratio answers to the same equation that we have often seen for the folded dipole. Although the ratio of wire diameters provides the key variable in the equation, the spacing between the wires plays a significant role in two ways. The terms of the ratio itself are each ratios of diameter to spacing. For reference, the following equation appears in many texts, where R is the ratio of impedance compared to the open-ended linear antenna, s is the center-to-center spacing of the wires, and d1 and d2 are the 2 diameters, with d1 representing the fed wire.

As well, the wires must be close enough to each other to ensure that the pair forms a transmission line and not a simple wide-spaced half loop. The fact that the folded monopole is itself a transmission line comprises the second major property of the folded monopole.

If the folded monopole is shorter than a self-resonant length, that is, is shorter than an electrical quarter wavelength, then the transmission-line aspect of folded monopole behavior is a shorted transmission line. Alternatively expressed, the line is an inductive reactance up to about 50 degrees electrical length, depending upon the diameters and spacing of the wires in the folded monopole. The characteristic impedance of the line is mostly a function of the folded monopole physical properties. For a two-wire monopole, the key elements that set the characteristic impedance are the center-to-center element spacing and the diameter of the elements. For a given construction (that is, for a given element diameter, spacing, and total height), the characteristic impedance does not change with frequency (if we ignore the effects of a real or lossy ground for the moment). The inductive reactance becomes a function of the folded monopole height measured in terms of a wavelength. The inductive reactance is then a tangent function of the electrical length translated into degrees or radians. We then may calculate the equivalent inductance by using the standard relationship among reactance, inductance, and frequency. However, the source impedance of the folded monopole, being a function of both radiation and transmission-line characteristics, will not be the same as the inductive reactance of a shorted transmission having the same length.

Many antenna designers for the MF and lower HF ranges prefer the folded monopole to the linear monopole, especially when the overall height must be shorter than a resonant length. Since the transmission-line aspect of the antenna's behavior always yields an inductive reactance for such lengths, tuning networks may use inherently high-Q capacitors exclusively and avoid low-Q inductors. The effort to design such antennas leads to modeling attempts in NEC or MININEC. The seemingly simple antenna should yield equally simple models. Unfortunately, all too often the simple models prove to be quite inadequate. In these notes, I want to review a number of places in which the modeling may go astray.

For uniformity through the exercise, I shall use some constants. The modeling frequency will be 3.5 MHz. The lossless wire will be 0.1" in diameter, although we shall also use a 1" wire under certain specified conditions. To simplify ground radial aspects of the model--which might create some ungainly models--and to avoid the variable of the ground losses, I shall place the model on perfect ground. This ground type will satisfactorily reveal most of the modeling dangers that we may encounter. I shall also use two different programs and cores. One core is NEC-4, which performs better than NEC-2, but still not perfectly. I shall contrast the NEC-4 results with MININEC outputs from Antenna Model, perhaps the most corrected version of MININEC 3.13 available.

Fig. 1 shows the initial evolution of the models that we shall examine in detail. The linear resonant monopole will become the standard against which we may compare the other models. Next, we shall turn to the two-wire or "hairpin" folded monopole. Finally, we shall examine a simple 5-wire cage folded monopole. In all cases, we shall designate a prime or fed wire. In the case of the cage, we shall use the center wire to simplify feeding. We shall later briefly note how to feed the outer wires in parallel. For reasons that we shall explain as we move along, the linear monopole uses more segments for essentially the same height as the outlines of the folded monopoles. The brief reason is that folded monopole models using the higher segment density would not show the space between wires very well.

Both linear and folded monopoles show about the same gain and the same pattern shape, as revealed in Fig. 2. There is one very minor exception to this statement. A two-wire folded monopole will show a very slight difference between the gain broadside to the pair of wires than in line with the wires. Gain on the fed side will always be numerically but not operationally higher than gain in the direction of the unfed wire. The exact differential varies with the wire spacing, but generally is less than 0.1 dB for all practical spacing and wire-diameter values.

Let's model the linear monopole over perfect ground in both NEC-4 and MININEC as a baseline data set for future reference. Table 1 shows the results of supplying the models with 69 segments, with the feedpoint or source on the lowest segment.

The table shows that both cores provide the same output reports, just as we might expect for this modest frequency and well-segmented model. Each segment is about 1' long. Both models show an ideal or nearly ideal Average Gain Test (AGT) score. Antenna Model actually provides the raw calculated number, which over perfect ground is twice the value shown. I have adjusted the number to coincide with the EZNEC or free-space value. My reason is simple. 10 times the common log of the free-space AGT score provides the adjustment factor necessary to correct the gain report when the AGT score is not ideal. The AGT dBi entry provides the calculated correction factor, which is 0.0 for this simple model.

We are now ready to create a model of a two-wire folded monopole. Fig. 3 will provide some guidance.

Good modeling practice dictates that we adhere to certain guidelines when constructing the wires of a model. In NEC, keeping the segment length more than twice the diameter or 4 times the wire radius ensures the most accurate current calculations. We can reduce that ratio in NEC-2 by invoking the EK command, and NEC-4 generally is accurate with a 2:1 segment-length-to-diameter ratio. Antenna Model recommends that the segment length be at least 1.25 times the wire diameter for best accuracy with its modified MININEC core. The use of 1' segment lengths with 0.1" diameter wire ensures that we shall not have problems in this department.

Second, adjacent segments should have similar lengths, especially in NEC and especially in high-current regions of the antenna assembly. The two-wire portion of the figure shows a spacing (and 1-segment end wire) that is 1' long, matching the segment length of the long wires. The resulting folded monopole will have 139 segments, not very high for today's fast computers. Some programs, like EZNEC, provide for a length-tapering feature to reduce the segment count while keeping the segment length at one or both ends of the wire at a selected minimum length (in this case 1'). The NEC GC command achieves the same goal but uses a different length-tapering algorithm. We may bypass these steps by creating a 139-segment model both in EZNEC and in Antenna Model. The results appear in Table 2.

Once more, the output reports from the two programs are almost identical. Both programs can handle the folded monopole composed of a single wire diameter throughout with all of the other modeling guidelines in order. The height of the resonant folded monopole is shorter than the height of the linear monopole because, with respect to its radiation behavior, the double wire acts like a single fatter wire. The gain entry shows a sample of the in-line gain differential, while the main part of that entry shows the broadside value.

The resistive component of the source impedance is 143.5 Ohms. The fact that the wires have the same diameter with a constant spacing between them yields the familiar 4:1 impedance transformation ratio. 4 times 36 Ohms (for the linear monopole) is 144 Ohms. So far, all is simple and well.

Let's now create a cage-type folded monopole with the center wire fed, using the right side of Fig. 3 as a guide. The outer wires that return to ground use the same segmentation as the single return wire of the model that we have just reviewed. The end wires are each 1' long to preserve the identity of segment lengths throughout the model. Since we shall be reducing the overall antenna height, the total segment count for the model is 319. Table 3 shows the results.

The models show several interesting facts about modeling cage-type folded monopoles. Starting with the impedance, we notice a very great increase in the resistive component. (The reactive component did not go to zero using height increments of 0.1', so the table shows the lowest value attained.) The cage monopole is essentially a form of coaxial transmission line structure. Ideally, the wire diameter values used to calculate the impedance transformation would use the center wire value and the effective diameter of the ring of cage wires. For these cases, the transformation equation will not work. If we feed the center wire, then 2S/d2 is 1 and its log is zero, leading to a calculational error or to an indefinitely large value for the ratio, depending on whether you are using a computer or a scratch pad. Likewise, if we feed the outer wires in parallel, then 2S/d1 becomes 1 and its log is zero. This partial result leads to a ratio of 1.0 for all cases. However, we cannot be certain that the value of d2 is in fact 2' for this example, since the structure has more open area than closed area, and the cage wires are thin. The reported values from the model are sufficiently high to establish that we have a thin fed wire and a very fat "other" or return wire, just the conditions that yield a very large transformation value by standard calculations.

The other significant fact to note is the AGT of the NEC-4 output report. 0.988 is not an ideal value and requires a correction of 0.05 dB to the gain report. As the table suggests, when the calculated adjustment factor is negative, we increase the raw report by the absolute value of the adjustment factor. Had the calculated adjustment factor been positive, we would have subtracted it from the raw gain report. The MININEC AGT score shows no need for adjustment, and the raw gain report is very close to the adjusted NEC-4 number.

NEC-4 appears to drift somewhat off an ideal AGT value due to the proximity of 4 wires to the fed wire, as well as the current division at the top of the model. A single return-wire did not produce this consequence. NEC-2 is even further distant from an ideal AGT score, yielding a value of 0.928 for the same model. Hence, its gain report would be 0.32-dB low. Some scales of model adequacy as measured by the AGT score use limits of 1.05 and 0.95 as marking the ends of truly reliable models. I tend in most of my work to use even tighter limits. The NEC-2 AGT value lies well outside even fairly loose limiting values.

Although the Average Gain Test is a necessary, but not a sufficient, condition of model adequacy, it is an important test for all models. Modelers should routinely apply the test, since it may minimally require correction of the gain reports. If one is conducting systematic modeling exercises designed to show performance trends, then the AGT is essential lest one misread the trends.

If we are searching for general properties of a particular type of antenna, we may alter the model to overcome some of the limitations that we might encounter initially. Fig. 1 showed two-wire and five-wire folded monopoles that used fewer segments per unit of antenna height. However, each return wire used a wider spacing from the center fed wire. Let's explore this avenue of modeling to see what emerges. First, we may increase the spacing from the center wire to 3' and still have an effective folded monopole at 3.5 MHz. The end wires will use a single segment. To keep all segments roughly the same length, we shall reduce the total number of segments in the 2-wire monopole to 22. The overall height will be 66.3', in keeping with the increase in the "fat wire" effect of placing the wires 3' apart. Table 4 shows the results from the new model using each program.

Increasing the spacing yields a two-wire NEC-4 model that is almost identical in performance to the more narrowly spaced two-wire model. The AGT is ideal and the impedance a virtually the same at 4 times the linear monopole value (within less than 1 Ohm). Interestingly, the MININEC results in Antenna Model show a nearly ideal AGT value and a very close impedance coincidence to the NEC-4 model. The gain remain on target (using the value broadside to the plane of the 2 wires).

We have enough data to let us try a five-wire model in each system using the wider spacing and the reduced overall segmentation. The five-wire model will be shorter than the two-wire model, so the vertical wires will use 21 segments each. Table 5 provides the output reports.

Both models provide excellent AGT values. As well, the difference between the gain values has decreased to merely 0.03 dB. However, we see a considerable difference in the reported impedance values. It is likely that some, if not all, of the difference stems from the fact that with a high impedance, very small differences between models will result in outsized changes of some calculated results. We normally think of such changes as modifications that we might make to the geometric structure. However, in this case, the difference is most likely a product of the difference in the calculation methods.

Most folded monopole systems consist of one fat element (often a tower structure) and one or more thin wires. Without changing anything else, we may explore what happens when we increase the diameter of the fed wire to 1", that is, increase the diameter by a factor of 10. 1" is well below the effective diameter of most towers, but the differential with the return wire should be enough to reveal any calculation difficulties that might be a core function. If we apply the fed-wire diameter increase to the two-wire folded monopole, we obtain the results in Table 6. Note that we did not change the overall antenna height relative to the previous two-wire model.

The MININEC version of the model shows one advantage of its system: it is less sensitive than NEC to junctions of wires having dissimilar diameters, whether those junctions are linear or angular. The AGT value is very close to the ideal and requires no gain report change. Both the gain and the impedance reports are closer to the calculated value than is the NEC report, although the difference is small.

NEC-4 shows an AGT score that is off the mark by a noticeable amount. Hence, the raw gain report is about 0.12-dB too high, but after correction, it returns to the expected value. (NEC-2 under the same conditions produced an AGT score of 1.137, a wholly unreliable score for a model. The correction factor requires a raw gain reduction of 0.56 dB. The result might be in the ballpark for expectations, but we likely could not trust the impedance reports.) NEC-4's raw impedance report is fairly close to the MININEC report. However, trying to adjust it with the AGT multiplier carries it further from the MININEC value. In effect, we are now modeling in a region where NEC (-2 and -4) does not produce the most accurate results due to the junctions of wires with different diameters.

The impedance reports are less than 4 times the impedance of the (36-Ohm) linear monopole because the fed wire is considerable larger in diameter than the return wire. As the fed wire increases in diameter for a constant-diameter return wire, the transformation ratio will continue to decrease, but it can never descend below a 1:1 ratio. A folded monopole (or a folded dipole) cannot be an impedance down-converter.

When we apply the same fed-wire diameter increase to the five-wire model, we obtain interesting results. Once more, we retain the same overall antenna height that we used in the previous five-wire model with its uniform wire diameter. Table 7 shows the model reports.

The NEC-4 model shows near resonance almost accidentally. However, the most important aspect of the report is the further departure from an ideal AGT score and the requirement for a sizable correction factor to the raw gain report. (NEC-2 produced an AGT score of 1.177 for the same model, indicating a required correction of 0.71 dB to the raw gain report.) In contrast, the MININEC model--as provided by Antenna Model--remains close to ideal in its AGT score. The raw gain report is also very close to the NEC-4 corrected value and well-suited to the shorter overall height of the model, relative to the 68.4' linear monopole. (Hence, the slightly higher two-wire gain reports remain an anomaly in the overall model progression.)

Simple models of folded monopoles tend to ignore the remaining structure of a tower or mast that has been converted to folded monopole use. We may easily test whether or not we are well or poorly advised to ignore such upper-end lengths of structure beyond the limits of the folded-monopole proper. For these initial tests, we may add an extension to the 1" fed or center wire of both folded monopoles that we have just tested without an extension. Let's add 10' of 1" diameter wire to the top of the structure and use about 5 segments. Because the extension is a low-current region of the antenna, the exact segmentation will make only small differences to the source impedance report and almost none to the gain and AGT reports. The height of the antenna up to the extension is the same as in the previous two-wire and five-wire folded monopoles. See Fig. 4. Our goal is to see if the mast extension makes a difference, and if there is a difference, we expect it to appear most prominently in the source impedance report.

If we run both models using EZNEC/4 and Antenna Model, we obtain the results in Table 8. In this case, we may pass over the AGT issues and focus on the source impedance. Compared to the values shown in Table 6, the new values clearly show that the extension raises the resistive component of the impedance and adds a very significant inductive reactance to the overall impedance.

The five-wire version of the folded monopole shows the same general pattern of impedance. Compare Table 9 to Table 7. The inductive reactance has grown significantly--faster than in the two-wire model.

Although feeding the center wire/fat wire of the various folded monopole models has served well to reveal some features of modeling, it does not reflect what AM BC and amateur applications may encounter in practice. Normally, we would feed the thinner wire, while leaving the central tower or fat wire grounded. The two-wire monopole requires only a small adjustment to correct this situation, but the five-wire models will need a different strategy. Fig. 5 shows our options.

The two-wire model only requires that we move the source from the fat wire to the thin wire. We shall use the same model whose results appear in Table 6: a 66.3' folded monopole. This time we shall place the source on the lowest point of the 0.1" thin wire and leave the 1" fat wire as a return to perfect ground. The results of this model revision appear in Table 10.

We do not lose the deficiencies of NEC by moving the source point. The AGT value shows that we need to adjust the gain report by 0.19 dB to obtain the expected value of about 5.15 dBi. However, note that the AGT value is now less than an ideal 1.0 rather than being greater than 1.0. The Antenna Model version of the revised model shows an admirable AGT value and gain value. Finally, note the increase in the source resistance that results from using a thin fed wire and a fat return wire (relative to the reverse situation shown in Table 6). Once more, the MININEC core reports an impedance that is closer to the calculated value.

The five-wire monopole presents us with a different challenge. We might be tempted to end all 4 outer thin wires above ground by a foot and then connect all of them around the center wire. A short wire to ground from one corner would become the source wire. This procedure might seem to reflect the actual physical structure of a folded cage-type monopole, but it creates a number of modeling error sources. The segment lengths in the model are 3' long, so the series of connecting wires and the source wire introduces aberrant segment lengths. As well, the source segment and the adjacent segments would not have equal lengths, a desirable situation especially in NEC for the most accurate calculations. Moreover, the segment junctions of the outer wires would no longer parallel the segment junctions of the center wire, an undesirable condition in NEC for highest accuracy. Finally, the current division among the outer wires would not be equal, resulting in possible pattern and impedance errors

A more secure method to retain whatever accuracy the model has would be to use 4 sources, one on each outer thin wire. This technique is equally applicable to NEC and to MININEC. To arrive at a net single-source impedance value, we need only divide one of the source impedance reports by 4, taking the resistive and the reactive components separately. If we use the model whose result appear in Table 7 as our starting point, we need only replace the single source on the center 1" wire with 4 sources, one on each of the outer thin wires. We shall leave the modeled antenna height at 61.3'. The results of our efforts appear in the top portion of Table 11.

Because we are now feeding the outer conductor, the effective diameter of which is greater than 1", the net impedance is considerably lower than the value found in Table 7, the model the feeds the inner and effectively thinner conductor of this concentric model. However, do not be fooled by the deceptively attractive resistive value. An actual tower situation will place a much larger diameter center conductor into the model. As a useful but imperfect guide, BC engineers conventionally use the following diameters as 1-wire substitutes for antenna towers. For a triangular tower, use a wire with a radius of 0.37 times the face dimension. For a rectangular tower, use a wire with a radius that is 0.56 times the face dimension. You can adjust the spacing of the wires from the central tower and their diameter (and even their number) to arrive at a desired impedance, since the value (over perfect ground) will not go below 36 Ohms in an adequate model.

The parallel-source method of feeding the cage folded monopole is convenient for initial modeling, but it will not suit models to which we wish to add matching components or loads. We may use in NEC an alternative technique that arrives at the same source impedance value, but uses only a single source. The right portion of Fig. 5 shows the essential elements of the technique. We terminate a transmission line at each of the former source segments. The other set of terminations appear on a single segment wire that is a considerable geometric distance from the antenna. The distance is sufficient to prevent the wire from interacting significantly with the main antenna wires. In the present case, the wire happens to be about 140' from the antenna. As well the wire is very short (0.3' in this case) and may be very thin, although I retained the 0.1" diameter used with other wires. The 4 transmission lines that terminate on this new wire are in parallel with each other and in parallel with a source that we place on the wire. The physical position of the new wire only prevents wire interactions, but does not itself determine the length of the transmission lines. We may set these lines to the shortest length feasible. I used 1E-10 feet in EZNEC, but you may use simply the shortest length allowed by your particular core. Because the line is not a physical line and plays no role in the matrix calculations, line routing is unimportant. As well, because the line is so short, its characteristic impedance is unimportant. I used 200 Ohms, which roughly corresponds to the impedance of the individual former source segments. The transmission-line technique places all 4 outer-wire segments in parallel. The single source on the remote wire records the parallel source value. The lower portion of Table 11 shows that we obtain the same source impedance that we derived from the parallel source technique.

The alternative transmission-line feeding system has an important advantage. We may extend the remote wire structure and incorporate loading/matching impedances. Essentially, we use a matrix of very short and thin wires to replicate the structure of a network, adding components to the series and/or parallel legs as needed to make up the actual network. (NEC2GO has a built-in method for creating source point networks.)

However, the method also carries a caution. NEC-4 shows a usable but less than ideal AGT score. As we have seen, NEC-2 AGT values are much worse when we have junctions of wires with different diameters. The networks that we add can only be as accurate as the initial source impedance values prior to adding the new components, and NEC-2 source impedance values may be inaccurate.

A folded monopole also tends to imply a buried radial system in the MF and lower HF regions of the spectrum. We have not looked at the effects of adding such a system or how best to model such a system. However, experience has taught that none of the alternative modeling systems that we might use will adequately substitute for a NEC-4 set of below-ground radials. (There are copious notes on this situation with respect to standard monopoles in Ground-Plane Notes, available from antenneX.) The folded monopole requires careful treatment as we approach ground level to ensure that we do not violate good modeling procedures while developing a common ground point for all necessary wires.

Those who build and install wire cages for AM BC antennas tend to call them "skirts," although the general informal name is often associated with an alternative use for the cage. Between the cage base and ground, installers may place tuning elements and thus detune the tower relative to a give frequency. The technique has application to cellular and other UHF antenna towers that fall within the quite large near-field radius of an existing AM BC antenna, where unwanted interactions might distort the certified pattern for the AM antenna system. It may also apply to the antennas of different stations whose antennas lie within the near-field of each other. In most cases, skirt assemblies have standard sizes (at least with respect to outside diameter). Installers have developed a considerable number of techniques for bringing the overall tuned filtering frequency to the desired point. If you model a commercial assembly, note the presence of [periodic spacers and shorting rings. These assemblies serve both mechanical and electrical purposes. Hence, the shorting ring belongs in the model.

Some engineers who employ cages on towers to convert the monopole into the alternative transmitting assembly also prefer the term "skirt," since the assembly (as noted above) does not answer to standard equations for folded dipoles/monopoles. See, for example, the NAB 1997 paper by Rackley, Cox, Moser, and King ("An Efficiency Comparison: AM/Medium-Wave Series-Fed vs. Skirt-Fed Radiators"). Other engineers retain the term "folded monopole" or use the expression "folded isopole." Whatever the preferred label, the cage-style folded monopole retains its incomplete shielding by the cage and hence leaves an impedance that one may best approximate by appropriate models, subject to field testing and adjustment.

For those interested in ferreting out the effects of a relatively open but surrounding skirt, modeling may provide more data than just the anticipated feedpoint impedance. With respect to 2-wire folded or hairpin monopoles, Kuecken's method of separating transmission line from radiation currents (see pp. 224 ff of his Antennas and Transmission Lines) has proven effective for the analysis of both folded monopole and folded dipole models that use 2 wires. NEC and MININEC both provide a record of relative current levels along the wires of a caged or skirted antenna, and investigators might well use the data to develop the relative roles of transmission-line and radiation currents with these antennas.

As incomplete as this treatment may be, it still provides some guidance on the initial modeling of both two-wire and cage-type folded monopoles. With due attention to AGT values and correctives, as well as to the reasonableness of reported output values from the calculating core in use, we may successfully model folded monopoles using either NEC or MININEC. However, as always, hasty or careless modeling leads to relatively useless results. The rule of GIGO strictly applies to antenna modeling.

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~ antenneX August  2007 Online Issue #124 ~
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