One of the examples in my text, Basic Antenna Modeling: A Hands-On Tutorial, involves applying a beta match to a 3-element Yagi. The challenge is to place a reactive load in parallel with the source. Since several of the techniques require a rather high level of segmentation, we shall use a model already set up for the job. Fig. 1 shows the evolution through which we shall go before departing from it.
The reflector and the director of the Yagi, set up for 14.175 MHz, follow the segmentation pattern in the top driver option. However, the actual driver uses a 3-wire set-up, with the 1-segment wire in the center having the same length as the remaining segments in the driver. In this way, we assure a correct source impedance calculation regardless of what we do in order to place the load in parallel with the source.
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Fig. 2 shows the basic Yagi model in its present form. If we place the source on wire 3 with no matching system, then the impedance will be about 23.4 - j 24.6 Ohms. The impedance is ripe for matching to a 50-Ohm coax line with a beta match. A beta match is made up of an L-network with a reactance in series with the load and a shunt reactance across the source--the coax in this case. The series reactance is already present in the antenna driver source impedance. Hence, the beta match physically consists of the deceptively simple placement of a reactance across the terminals to which the coax connects with the driver.
Standard L-network calculations, summarized in the equation set (1), provide the level of reactance necessary to effect a good 50-Ohm match. For the case at hand we need just about 50 Ohms (46.9 Ohms, to be more and perhaps spuriously precise) of inductive reactance for the parallel component to go with the nearly 25 Ohms of series capacitance. Indeed, the transformation of 23.4 Ohms to 50 Ohms calls for a series capacitive reactance of 24.95 Ohms, and we have 24.6 Ohms in place--a very close setting of the driver length.
Essentially, we have two ways to achieve the inductive reactance: a standard inductor or a shorted length of transmission line. Before we finish, we shall look at both methods, but let's start with the coil. It will involve the more complex modeling.
We cannot simply place a reactive load on the source segment. Every such load will wind up in series with the source, when our goal is to place it in parallel with the source. If we wish to place a reactive load in parallel with the source, one technique is to model the antenna so that there is a physical place for the load to be. To make a place, we must model a set of wires such that they are in parallel with the source segment.
The lowest portion of Fig. 1 shows the physical arrangement. We simply add three wires to form a square with the 1-segment source wire. The high level of segmentation is designed--well within the limits of NEC's segment length to wire radius limitations--to keep the assembly as small as feasible so that it does not contribute significantly to the radiation pattern of the antenna, thereby distorting the performance reports. Fig. 3 shows the resulting model with its new square.
To Wire 5, the one that is physically parallel to the source wire, we shall add a load. The next questions are "what kind?" and "how much?" Spot-frequency modeling can use a simple R-X load in which we specify the resistance and reactance of the load. Let's assume a coil Q of 200. With a 46.9-Ohm reactance, the resistance will be 0.2345 Ohms at the specified Q. Fig. 4 shows the entry of the 2 values on the load set-up screen, along with added data on the load magnitude and phase angle.
With the load specified, the new source impedance is 57.6 + j 1.5 Ohms. This value is correct for 14.175 MHz. However, R-X loads have a limitation. A coil will change reactance for every change in frequency. An R-X load will hold the reactance at the same value for every frequency we might check. Hence, an R-X load will not give a true picture of the source impedance across the 20 meter band.
We have an easy alternative. On wire 5, instead of using an R-X load, we can employ an R-L-C load. Since there is no capacitance in the load we are applying, we shall leave its value at zero. The resistance remains 0.2345 Ohms. From fundamental equations that appear in every handbook and that are repeated in equation set (2), we can calculate the inductance that provides a reactance of 46.9 Ohms at 14.175 MHz: 0.5266 micro-H.
Fig. 5 shows the entry screen for R-L-C loads, with the resistance and inductance properly entered. When we run the model, it shows a feedpoint impedance of 57.6 + j 1.4 Ohms, the same as with the R-X load. Where differences will appear is in source impedance reports at a distance from the design frequency. In general, an R-X load will provide too optimistic a portrait of the source impedance and resulting 50-Ohm SWR. A resistance-inductance load is necessary to arrive at a more conservative but more correct set of curves.
Incidentally, throughout this sequence of models, both the ones covered so far and those yet to be examined, the free-space gain of the Yagi varied by only 0.06 dB and the front-to-back ratio varied by less than 0.1 dB. Since the feedpoint impedance reported so far coincides with L-network calculations within a few Ohms, the techniques are accurate and harmless to the model.
If we are dissatisfied with the 7-Ohm deviation of the matched model from the calculated nearly perfect 50-Ohm value--perhaps due to some task-driven requirement--then we can check the original model with the addition of the square to see if the square in fact changed the source impedance significantly. To do this, we need only return to version with the R-X load. We can change the values for R and X to 1E10 to simulate an open circuit. With the present driver dimensions (+/-4.947 m), the source impedance in our check model reports as 23.3 - j 24.6 Ohms. The added structure does not significantly affect the source impedance (originally 23.4 - j 24.6 Ohms).
Nonetheless, the structure is composed of "fat" wire (25 mm or about 1") in order to avoid NEC difficulties with angular junctions of wires having dissimilar diameters. As well, the beta shunt component is placed at a slight distance from the actual source. In order to arrive at a value of source impedance within about 1 Ohm of calculations, it is necessary to juggle two aspects of the model: the driver length and the reactance (or inductance) of the shunted load. For the model that we have been using, I arrived at a source impedance of 50.6 + j 0.3 Ohms with a reactance of 48.5 Ohms and a driver length of +/-4.963 m. The actual values do not necessarily reflect what the physical antenna would require, but the juggling is typical of the adjustment procedure used with beta matches to arrive at an acceptable, if not perfect, match. Modeling does have more than numerical analogs to antenna construction, if we are alert to identify them.
One limitation of the "added-square" system of putting a load in parallel with the source is that it will disable the use of Leeson corrections, if our model uses a tapered-diameter schedule of tubing. However, the uniform diameter model can be used for analytical purposes, with the tapered diameter version later developed to determine the exact element lengths required for the physical antenna.
If we wish to avoid the use of a square of wires at the feedpoint, we can still arrive at a model of a beta match. The techniques will employ either a transmission line or a network. Either modeling option requires that we create an "arbitrary" wire at a distance from the antenna. The wire should be far enough away to create no detectable effects on the model. As well, the wire can be perfect or lossless, be a single segment, and be so short and thin that it is virtually invisible to radiated RF energy. Fig. 6 shows the revised model with the square missing but with the arbitrary wire added into the wire set.
The required shunt or parallel beta inductive reactance can be obtained not only from an actual inductor, but as well from a shorted transmission line stub, the proverbial beta "hairpin." Transmission lines connected to the same segment as a source appear in parallel with the source. Therefore, we can simply create a transmission line using the "TL" facility built into NEC.
Fig. 7 shows the required transmission line screen. The transmission line goes from the source wire (3) to the new distant wire (6). To make sure that the transmission line appears as a short, we enter very high values of admittance, the equivalent of entering exceptionally low values of impedance. The real and imaginary components are the conductance and the susceptance, the inverse of resistance and reactance. We enter values such as 1E10 in both places. Some programs automate this feature so that the user only needs to enter the request for a shorted (or open) stub.
Every beta hairpin or stub will have a characteristic impedance based upon the diameter of the wires making up the line and the distance between them. For this exercise, we have arbitrarily set the characteristic impedance at 600 Ohms, indicating a fairly wide spacing between wires. The length is determined from the fact that the inductive reactance of a shorted stub is the product of the line characteristic impedance and the tangent of the lines length in electrical degrees. The ratio of the inductive reactance to the line impedance gives us the tangent of the line length in degrees, which we can then convert into a fraction of a wavelength and from there into a physical length (assuming a velocity factor of 1.0). For this case, the required line length is 0.262584 meters (since the entire dimension set for the model has been in meters). With this length, we obtain a source impedance of 49.2 - j 0.0 Ohms. This value is a few Ohms different from the values using the load applied to the parallel wire square and is likely more accurate, since it involves the addition of nothing physical to the antenna.
Like the R-L-C type load, the transmission line stub will yield correct results on a frequency sweep. Since the stub is specified as a set of physical dimensions (including the characteristic impedance, which is derived from physical properties of the line), it will correctly modify the source impedance over a wide frequency range. In addition, even if the ultimate antenna will employ a beta shunt inductor rather than a hairpin stub, the transmission line stub can be used with a tapered diameter element set as a substitute for the coil for design purposes--even if the SWR curves will not exactly coincide with those for the coil. Use of the coil to determine the operating SWR bandwidth can be done with the uniform-diameter model. The two significant items that a TL line will not reveal are a. any losses in the line (not usually a problem with short stubs) and b. the effects of the physical line on the radiation pattern.
We have not yet exhausted the ways in which we can add a parallel load to a source segment. For example, we have not yet employed a "network," the NT function of NEC. We can convert the required values of series resistance and reactance to their equivalents for use in a short-circuit admittance matrix. An example of the network screen appears in Fig. 8.
The source wire (3) represents the first port, while the "arbitrary" wire (6) represents the second port. Into the Y11 boxes we insert the real and imaginary values required for a parallel or shunt admittance. We leave Y12's values as zeroes. We create a short circuit at the Y22 entry place by using very high values (1E10) for both the real and imaginary components. All we need to do now is determine the real and imaginary components for Y11.
The equations in set (3) provide us with the conversion formulas to apply to the series resistance (0.2345 Ohms) and reactive (46.9 Ohms) in order to arrive at usable numbers. The resulting values appear in Fig. 8. The result, when the model is run, is a source impedance of 49.0 - j 0.0 Ohms, which is in very close agreement with the transmission stub result we just finished examining. However, the network result is as limited as the R-X load technique: it is correct for only the single frequency for which it is specified.
Some programs do not provide the user with ready access to the network (NT) facility, but they do permit use of the transmission line (TL) facility. In such as case, we can simulate the network load across the source. Let's examine Fig. 9, the same transmission line screen that we used for the shorted stub. However, we shall approach it from a different angle.
The line runs from wire (3) to wire (6). Let's set the characteristic impedance at the value of the line to which we wish to effect a match, that is, to 50 Ohms, the value of the coax that we want matched to the antenna. The length of a transmission is independent of the distance between wires (3) and (6) and can be set into the TL call. We do not want the line to create any significant impedance transformation, so we shall make the line very short. The arbitrary length used in this exercise is 0.01 meters.
Next, let's place the (overly precise) shunt admittance values that we calculated into the real and imaginary (conductance and susceptance) boxes at the "far" end of the very short line. We now have the load across the source. The source impedance that NEC reports from this model is 49.0 - j 0.3 Ohms, almost precisely the value obtained by using the network. However, like the network technique, this application of the transmission line facility is limited to a single test frequency.
Of the 5 techniques that we have shown for modeling a beta match, the beta inductor on the extra wire square and the transmission line stub are certainly the most useful. Both are responsive to changes in frequency and therefore produce relative accurate impedance and SWR curves for estimating the operating bandwidth of the loaded source antenna. One can change the value of the inductor's Q and develop sets of curves for the operating bandwidth using different values of Q. However, the "added-square" technique is not applicable if tapered-diameter elements are used with the Leeson corrections enabled.
In spite of this judgment, this exercise has had something of an ulterior motive. The network facility available within NEC (-2 or -4) is rarely used by modelers, especially modelers with less experience in handling shunt admittance networks. Indeed, most amateurs are far more familiar with the concepts of resistance, reactance, and impedance than with their very useful inverse concepts of conductance, susceptance, and admittance. For such reasons as these, some implementations of NEC omit the network facility altogether (along with many other lesser-used facilities) from basic software packages. Other implementations include the facility, but pass over it in silence in manuals. Indeed, the NEC user manual itself is somewhat opaque on the subject for anyone not having significant previous experience with networks. I once asked a mail-list group of NEC users if any would share a few non-proprietary applications of networks to antenna modeling problems. I received one request to share whatever I might receive. Unfortunately, the request brought no sample applications at all. I would still like to receive some samples.
The examples that I have contrived for this exercise are in many ways unnecessary for effectively modeling a load across a source. However, they do call attention to two facts: 1. the network facility is available and can be put to service, and 2. the transmission line facility is a special case of the network facility.
Networks can place loads of various orders across any segment. For very large models, an added advantage of the network is that the load can be changed without causing a recalculation of the structure matrix, as is required when using standard loads (the LD facility). As well, networks are in parallel with sources on the same segment, unlike loads (LD), which are in series with sources on the same segment. To offset this advantage, we may note once more that the network value does not automatically scale with frequency changes as does an R-L-C load.
It may be the case that the increasing speed of desk-top computers and the ease with which one may form a work-around for paralleling a source and a load have largely obviated the advantages of using networks for small to fairly large size models. In any event, we have at least made a passing acquaintance with networks, and that may be enough for one exercise.
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