Gamma Match

119. Modeling Odd Structures: the Gamma Match
Part 2. Gamma Assembly Variables

L. B. Cebik, W4RNL (SK)




In the preceding episode, we began an exploration of modeling "odd" structures that we find within the geometry of some antennas, using the gamma match as a good example. The gamma match has both basic and additional properties that are worth modeling in order to discover what effect a number of structural variables may have on the performance of the antenna. Because the gamma match assembly is primarily an impedance transformer, we have been less interested in such antenna performance properties as forward gain and front-to-back ratio than we have in the structure necessary to arrive at a desired feedpoint impedance.

We began with a "pre-gamma" model to arrive at the antenna source impedance prior to adding the gamma-match structure. Our basic subject model used a test frequency of 28 MHz with 0.5" elements in a 2-element driver-reflector Yagi configuration. The driver was 190.7" long, while the reflector was 211.9" long. The spacing was about 0.12-wavelength or 50.58". Without a gamma match, Antenna Model reports the feedpoint impedance as 29.84-j25.73 Ohms. The AGT score for the basic or pre-gamma model was 0.9997, where the ideal value is 1.0000.

Our decision to use Antenna Model software, a highly corrected version of MININEC 3.13, rested on its ability to handle--so far as tests internal to modeling can tell--accurate models of elements within a gamma-match assembly. NEC is subject to systematic errors that make its use difficult and potentially questionable. Less corrected versions of MININEC may yield equally erroneous results. Although Antenna Model may produce the most trustworthy models within the collection of extant wire antenna simulation programs, we must remember that our exercises lack one important step: the calibration of the simulation results with physical antennas. However, our goal in tracking these exercises has not been to yield a building guide. Rather, we have been concerned to explore what counts as good modeling practice for structures as "odd" as the gamma-match assembly.

Our initial gamma-match model employed a basic structure consisting of several new wires and some modifications to the main driver element. As shown in the upper portion of Fig. 1, the main element now has 3 sections. We selected 100 segments as the total of the segments on all three element sections, with 50 assigned to the non-gamma side of the element. The ratio of segments on the gamma side, divided between the gamma assembly section and the remainder of the element, rests on equalizing to the degree possible the lengths of segments along the element. We assigned 2 segments to the source wire and to the shorting bar wire both to equalize segment lengths throughout the model and to place the source at the center of its wire. The latter consideration rests on the need to place MININEC sources (and loads) on a pulse or segment junction. Our initial spacing between the main element and the gamma bar was 4", yielding 2" segments that roughly correspond to the length of segments in the main element.

The goal of the modeling was to produce a model that gave us a 50-Ohm feedpoint impedance, the value that corresponds to general amateur practice for the use of main feedlines. For the exercise, we arbitrarily set a close tolerance limit of 0.1 Ohm in the resistive and the reactive components. For many purposes, this limit is too fine, although it is useful for some systematic modeling exercises. We also began with a gamma-rod diameter of 0.375", although we left the end wires at the same 0.5"-diameter as the main wire. With a gamma rod that is 18.47" long and a series capacitance of 38.32 pF, we obtained a gamma-match feedpoint impedance of 50.04+j0.07 Ohms.

The lower portion of Fig. 1 shows us a number of ways in which an actual gamma match may differ from the idealized model that we have been using. The variations fall into two general categories that we might call simple and complex variations. Simple variations involve selected modifications of single changes to one of the initial dimensions of the ideal gamma-match model. For example, before we closed the preceding episode, we explored the effects of changing the diameter of only the gamma rod, Wire 5 on the basic model wire table. Fig. 2 replicates that table from Antenna Model so that we may identify both the change that we have so far made and changes yet to come.

The starred entry shows the entry with which we were concerned in our first foray into systematic modeling with the gamma-match assembly. As we changed the gamma rod diameter from 0.125" to 0.625" in 0.125" increments, we adjusted the gamma-rod length to arrive at a 50-Ohm impedance. This process involved changes to the wire-table entry wherever the initial wire-table shows 18.47. Because the source and the series capacitance load are on the same pulse, we could independently arrive at the desired source resistance and then modify the load capacitance value to remove any remnant reactance. Table 1 shows the results of our efforts.

During our model modifications, we kept a close eye on the amount by which the gamma rod length changed in order to assure ourselves that the segment lengths on the main element remained stable. In this case, the total length change from the thinnest to the thickest gamma rod was about 0.25", so we could maintain the 10:40 segment division on the gamma side of the element. Of course, the gamma rod and the gamma section of the main element will have the same number of segments, and the two values on the gamma side of the element will add up to 50.

The initial exercise left us with a number of unanswered questions that we might pose to the model, based on normal gamma construction methods and on the lower portion of Fig. 1.. For example, we may wish to know the consequences of varying the spacing between the main element and the gamma rod. We may wish to understand the effects of leaving a gamma rod extension beyond the shorting bar. Do variations in the connecting-wire diameters have significant effects on the gamma-rod length or the series capacitance? Finally, what are the consequences of moving the series capacitor position from its ideal modeling position? Some of these questions involve equally simple modifications to our initial gamma match model, while others may involve more complex modifications.

The Gamma Spacing Question and Its Models

The systematic testing of various gamma-rod diameters is only one of many such tests that we may perform to obtain a general idea of the properties of a gamma-match assembly. We may perform a similar test of gamma-rod-to-main-element spacing. Our basic model used a 4" center-to-center spacing between the rod and the element. We might wish to know something about what happens with narrower and wider spacing values.

As a sample, we might specify alternative spacing values--perhaps 2" and 6"--in order to bracket our initial design. We shall retain the 0.5" main element and the original 0.375" rod diameters. Fig. 3 shows preliminary outlines of the three modeling situations and alerts us to some modeling cautions.

The first caution concerns the change in the segment length for the connecting wires. A 2" spacing value will reduce the segment length to 1", while the 6" spacing value will increase the segment length to 3". Because we wish to keep the source centered in its connecting wire, we have no practical way to obtain exact 2" segment lengths. Therefore, we shall have to pay closer attention to the AGT scores for each model to assure ourselves that the results are reasonably reliable. The initial model with a 4" spacing value achieved a score of 1.0005, very close to ideal. We shall be interested in the scores for the alternative spacing values.

The second caution concerns the segment length within the gamma rod. Casual modelers likely would simply try the new spacing and then adjust the gamma rod length (and the series capacitance), forgetting to be certain that the segment lengths throughout the entire driver element are as equal as feasible. However, we have already noted that unequal segment lengths among the sections of the driver may result in deviant source impedance reports. Hence, we shall reset the relative segmentation between the gamma section and the remainder in order to achieve the most ideal AGT score possible by obtaining the most equal segment lengths feasible.

If we attend to these cautions, then we might obtain results such as those in Table 2.

The table suggests a number of modeled gamma-assembly properties. First, despite the changes that we made in the relative segmentation, the models for the alternative spacing values are farther from the ideal score than the original model. Nevertheless, the probable accuracy of our 50-Ohm source impedance is within 1%. Therefore, the general trends shown by the length and capacitance numbers are reasonably reliable. Second, the gamma-rod length grows shorter as we increase rod spacing. Although not precise, the trend suggests that length changes linearly as the ratio of one spacing value to the next. Third, the required series capacitance decreases with increased spacing, although the amount is small and likely within the adjustment range of any type of capacitor used for this function.

Note that these trends apply so far only to the particular initial feedpoint impedance of the pre-gamma model. For a truly systematic exploration of gamma behavior, we might wish to construct alternative models using a variety of pre-gamma feedpoint impedances. These impedances should not only include several resistance values, but as well a number of reactance values. From such a systematic survey, we might glean a wider view of gamma assembly trends and develop some rational expectations of gamma-assembly behavior at interpolated values for gamma spacing and rod diameter.

Other Model Variations

Let's return to our basic gamma model with the 0.5" and 0.375" diameter assembly elements. Fig. 2 showed us two conditions that might or might not have a significant impact on the assembly final dimensions. One factor is the extension of the gamma rod beyond the shorting bar. The second is the diameter that we assign to the two connecting wires to simulate the plates and bars that we might use to connect the gamma rod to the main element at each end. To ensure that we do not confuse potential effects, we should treat each case separately.

Let's first add extensions to the gamma rod to simulate what might happen if the shorting bar does not come close to the end of the rod that we construct. The rod extensions will use the same 0.375" diameter as the main section of the rod. The extension lengths will consist of a new wire that we add to the model. The wire joins the junction of the shorting bar and the main section of the rod and runs parallel to the main element. Fig. 4 shows the outline of the construct, and points to a caution that we should observe.

We have striven to equalize to the degree possible the lengths of all segments in the model. The average segment length is close to 2". Therefore, we should add gamma rod extensions in 2" increments, using 1 segment per each 2" of extension. Within these restrictions, we might usefully test 2", 4", and 6" extensions.

For this test, we might bypass the earlier procedure of carefully adjusting the rod length and the series capacitance until we see if such a tedious procedure is necessary. Instead, let's retain the 18.47" rod length and the 38.32-pF series capacitance. What we shall observe is the effect of the extensions on the reported feedpoint impedance. The results appear in Table 3.

We had set 0.1-Ohm limits on the initial search for the proper gamma rod and series capacitance for a 50-Ohm match. With no extensions, the source impedance values fall within the limits. As we add successively longer extensions, the reactance does not change in any way that drives it outside the initial limits. Hence, for any reasonable rod extension, the existence of that extension is not likely to require a series capacitor adjustment. The extensions do have a small affect on the source resistance. However, a full 6" extension displaces the match impedance by only about 1.6 Ohms. This change is likely to be well within the field adjustment variables. Hence, it appears not to be a significant factor. For the initial pre-gamma beam, at least, one might cut off the extension or to leave it according to non-electronic reasons.

We should perform an additional test after returning the model to its initial gamma configuration. Let's explore what happens if we vary the 4" connecting wire diameters. We initially set them at 0.5", the same diameter as the main element. We might see what happens with connecting wires ranging from 0.125" to 1.0" in 0.125" increments. To simplify the exercise, we shall change both connecting wires at the same time.

Because we shall be creating some radical differences in the wire diameters at gamma-assembly junctions, we shall pay close attention to the AGT scores to ensure that we do not exceed program limits. Because we might have to adjust both the gamma length and the series capacitance to obtain source impedance values within our limits, we can expect this exercise to be somewhat tedious. However, this feature of systematic modeling is unavoidable. The results of our survey appear in Table 4.

Note that this and all other tables try to include all of the information about the starting points. After several exercises in systematic modeling, scratch pads notes and incomplete table data can obscure the initial model to which the new table's information may apply. Not every modeling caution applies to something that we enter into the software program--some apply to our record keeping procedures.

The table shows that for the initial model, changing the size of the connecting wires by an 8:1 ratio creates only a quarter-inch change in the required gamma length. The required series capacitance changes by only about 3 pF. Some sources suggest that the connecting wire effects are much greater. However, for the present initial gamma model, we do not find significant effects from connecting wires ranging from very thin to very thick, relative to the element and rod diameters. The consistency of the AGT scores suggests that nothing in the modeling itself artificially washes out the anticipated effects. Hence, we now have a real question that we might pose to reality: are the supposed effects as great as some sources assume?

Series Capacitor Placement

The final part of our survey of gamma properties concerns the placement of the gamma capacitor. Fig. 1 shows a concentric capacitor installed on the gamma line. The capacitance is distributed along the section of line in which the two tubes overlap (with a dielectric between them). Hence, we have two difficulties at first sight. One is the changing diameter of the gamma rod. However, as we saw in one of our earlier surveys, the likely increment of diameter change is unlikely to create any significant difference in performance.

The second potential problem concerns the capacitor position. At best, a model can only approximate the distributed capacitance. The version of MININEC in the Antenna Model implementation further limits the potential position of a gamma capacitor to the end of a wire or at its center--assuming that we have an even number of segments on the wire. Fig. 5 shows the relevant positions. The numbered dots represent not only available positions, but also the most popular positions for series capacitors in actual gamma-match construction.

Position 1, which we seldom find in practice, is the ideal position on the same pulse as the source. All of our previous models have used this capacitor position. Builders tend to favor position 2 when using a standard variable capacitor (which they sometimes replace after adjustment with a fixed capacitor having the closest value). Position 3 approximates the position of a concentric tubular capacitor. Although not precise, it is useful for tracking the trends in gamma dimensions as we move the capacitor from one position to the next.

Table 5 tracks the effects of the change of capacitor position, using the ideal position as the starting point. The required capacitance goes down as we move the capacitor away from the source, but not by enough to exceed construction variables in most cases. The major change lies in the required gamma rod length to achieve a 50-Ohm resonant feedpoint. The simple 2" move to the corner (position 2) requires an additional half-inch of gamma rod. Moving the capacitor to midway along the gamma rod calls for a further 1.5" increase in rod length. The 2" total exceeds 10% of the originally modeled gamma rod length--a significant amount.

One further aspect of moving the series capacitor position does not show up in tabular results, but does appear as we work with the model trying to obtain the correct gamma rod length and the correct capacitor value. When the capacitor is on the same pulse as the source, the resistive and reactive components of the source impedance are independent. We adjust the rod length for an acceptable resistive component. Then we adjust the capacitor value for an acceptable reactive component. However, when we move the capacitor away from the ideal position, we discover that the rod length and the capacitor value interact in the model. Changing the value of the capacitor at position 3 alters both the reactance and the resistance at the source. In many cases, the resistance may not change enough to require a change in rod length, but occasionally, we may need to adjust both components until we find the correct values to yield the desired feedpoint impedance. Indeed, under these conditions, model adjustment reflects what we encounter when making field adjustments on a gamma match.

Conclusion

Although we may now conclude this exercise, we are far from concluding the modeling explorations that may go into understanding and designing a gamma match for a given antenna. Our model used a sample antenna that showed a moderate resistance and a considerable reactance at the source relative to the final gamma-match impedance. The trends that we observed apply to this situation. We have not explored other beam feedpoint impedance values, both resonant and non-resonant. We should not presume in advance of modeling that all trends applicable to our subject antenna would hold true of all antenna designs. Even if the trends are generally applicable, the rates of change may vary with the initial feedpoint values prior to adding the gamma assembly. Hence, other initial impedance values may show different levels of sensitivity to small changes than we found within the example.

For each test, we also froze all but one dimension in order to explore systematically a single variable. In many cases, you may need to perform multiple explorations in order to determine for a given test frequency, initial impedance, and main element diameter the most promising gamma rod diameter, capacitor position, and rod length. One additional situation that we have not examined at all is modeling for boom effects. One technique often used to simulate boom effects is to install at the element center a short wire having a large diameter. Such a wire often creates a large difference in diameter within the main element in the gamma area, thus changing the surface-to-surface spacing between the main element and the gamma rod along the rod's length. As well, the short fat wire that compensates for the NEC and MININEC inability to handle transverse boom currents may also make it more difficult to maintain element length equality along the main element. Indeed, the diameter values often used for such exercises may exceed the segment-length limits that we imposed on our models to obtain adequate segmentation of the connecting wires and proper source placement. In all such cases, we need to keep a close eye on the AGT scores to assure ourselves of a reasonably adequate model.

In fact, this episode and the preceding one have not aimed to provide a complete analysis of the gamma match--not even for the beam design used as our focal sample. Rather, the two-part series has tried to show many--but not all--of the modeling challenges that may go into modeling the gamma match as an example of an odd structure relative to normal or simple antenna construction. We varied our efforts along the way as we took into account as many of the physical aspects of the structure that may form variables in the overall element geometry. In the process, we showed a few ways in which we might evaluate whether the structure was relatively sensitive or relatively insensitive to changes in each variable. Even for the single beam example that we selected as a starting point, there is considerably more--and sometimes tedious--work ahead before we complete the portrait of the assembly. Systematic modeling is not a synonym for tedium, but the latter is at least the second cousin of the former.

The fundamental principles involved in modeling odd structures are themselves simple. We need to take into account every physical aspect possible for the structure. We need to devise models that permit systematic variation of each significant aspect of the physical structure while remaining within the limits of the software to yield reliable models. Finally, we need to be as systematic as possible both in obtaining model reports and in recording them so that we produce reliable data--if not to the last numeric place, at least in terms of trends that will likely appear in the physical antenna.


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