124. Modeling (with) Parabolic Reflectors

L. B. Cebik, W4RNL (SK)




Modeling with non-uniformly shaped wire-grid structures carries a considerable collection of cautions and warnings. Newer modelers are likely to overlook many of them while focusing on the complexities of the structure itself. Therefore, it may be useful to review some of the potentials and the limitations of models that include them. We shall focus on the parabolic reflector as one of the most popular wire-grid structures, paralleling the popularity of physical implementations in many aspect of UHF communications. However, virtually all of the notes along the way will apply equally to other not-uniform shapes.

We employ uniform wire-grid structures when we create the rectangular shapes that form planar and corner reflectors. In these structures, the individual wires or wire segments are almost equal in length in both directions. Therefore, we may size the wire easily to form a good simulation of a solid surface. As we shall see, parabolic reflectors do not admit of such easy calculations.

A parabola, of course, is a graphical solution to a certain type of quadratic equation. The antenna structures that we commonly call parabolas are paraboloids with uniform dimensions, as suggested by the 2 views in Fig. 1. The center of the dish is the vertex. The distance from the vertex to a point that is in line with the lip of the dish is the depth (d). The distance across the widest pair of points on the lip of the dish is the diameter (D). The parabolic reflector also has a focal point, and we can calculate the distance from this point to the vertex by a common equation.

The focal length will equal the depth of the parabolic reflector under the condition that the depth is 1/4 the diameter. Most antenna handbooks will provide further information on parabolic reflectors. Our main concern is modeling the device.

Most applications allow us to create the dish model by using a shape-synthesizing program of some sort. For this exercise, we shall use NSI's NEC-Win Synth. To create a parabola, we need only specify the depth, the radius, the wire diameter, and the desired segmentation. The program allows us to specify the coordinates for the 2 radii (that is, X-Y, Y-Z, etc.) and also allows us to use separate values for the pair. We shall use a circular outline for simplicity.

The native output of NEC-Win Synth is a special file format that is directly compatible with NSI's NEC-Win Plus program that uses a spreadsheet format for its files. However, we can also save the file that we generated in .NEC format for direct importation to any NEC implementation. However, like other synthesis programs, we must use caution. The product is not a complete model file, but only the geometry of a model, along with the frequency specified.

CM Generated by NEC-Win Synth 1.0
CE
GW 1 1 0.31623 0.00000 0.05000 0.30075 0.09772 0.05000 0.00500
GW 2 1 0.30075 0.09772 0.05000 0.25583 0.18587 0.05000 0.00500
GW 3 1 0.25583 0.18587 0.05000 0.18587 0.25583 0.05000 0.00500
GW 4 1 0.18587 0.25583 0.05000 0.09772 0.30075 0.05000 0.00500
GW 5 1 0.09772 0.30075 0.05000 0.00000 0.31623 0.05000 0.00500
-----
GW 396 1 0.00000 -0.94868 0.45000 0.00000 -1.00000 0.50000 0.00500
GW 397 1 0.29316 -0.90225 0.45000 0.30902 -0.95106 0.50000 0.00500
GW 398 1 0.55762 -0.76750 0.45000 0.58779 -0.80902 0.50000 0.00500
GW 399 1 0.76750 -0.55762 0.45000 0.80902 -0.58779 0.50000 0.00500
GW 400 1 0.90225 -0.29316 0.45000 0.95106 -0.30902 0.50000 0.00500
GS 0 0 1.000000
FR 0 1 0 0 299.800000 1

To complete the model, we must add a driver--if we are not exciting one of the wires in the assembly. We must also specify the excitation, add any material or other loading that we need, and request some for of output. Therefore, the lower portion of the incomplete sample shown requires some variation of the following set of lines. The request for far-field patterns is only one of numerous outputs that we might request.

GW 396 1 0.00000 -0.94868 0.45000 0.00000 -1.00000 0.50000 0.00500
GW 397 1 0.29316 -0.90225 0.45000 0.30902 -0.95106 0.50000 0.00500
GW 398 1 0.55762 -0.76750 0.45000 0.58779 -0.80902 0.50000 0.00500
GW 399 1 0.76750 -0.55762 0.45000 0.80902 -0.58779 0.50000 0.00500
GW 400 1 0.90225 -0.29316 0.45000 0.95106 -0.30902 0.50000 0.00500
GW 501 11 0 -.24 .5 0 .24 .5 .005 !driver
GS 0 0 1.000000
GE
FR 0 1 0 0 299.800000 1
EX 0 501 6 0 1 0
RP 0 361 1 1000 -90 90 1.00000 1.00000
RP 0 361 1 1000 -90 0 1.00000 1.00000
EN

If we were to run the incomplete model, all that we would obtain for our trouble is an output report that provides a set of segments and their connections. However, even this much information can be useful to us, since we may use the information to evaluation why the parabolic reflector is a non-uniform wire-grid structure. In lieu of the data lines, we may examine more closely the graphic portrait of the reflector, as shown in Fig. 2.

The wire-grid assembly uses straight wires to approximate curved surfaces. It consists of several radials connected at semi-regular points by circles of wires. Normally, in a synthetic structure, each wire has 1 segment. The dish shown is 2 wavelengths in diameter and 0.5 wavelengths deep. Note that virtually no junction of wires at a right angle involves wires of equal length. The wires forming the circle increase their length systematically as we move from the vertex to the lip of the dish. The wires forming radials use equal length segments except at the innermost section. An ideal radial would have one more circle to achieve equal length segments throughout. However, the wires for the missing innermost circle would become exceptionally short.

Missing from the line graphic is any indication of the wire diameter used in the parabolic structure. The selection of wire diameter interacts with the selection of the number of radials at the vertex, where all radials join. The model shown uses 20 radials as a minimum value to form a close approximation of a circle. Adjacent wires at the vertex form a small angle. As we increase the number of radials or as we increase the diameter of the wire, press the NEC limits for the inter-penetration of wires at the vertex. However, if we make the wires too thin, we run the risk of creating a leaky reflector, that is, one that does not approximate a solid surface.

To examine the consequences, let's provide our 2-wavelength-diameter dish with a dipole placed at the focal point. Since the depth is 1/4 of the diameter, the focal point is 0.5-wavelength from the vertex, that is, even with the lip of the dish. We shall begin with a reflector-wire diameter of 0.01-wavelength and increase it up to 0.08-wavelength. Table 1 shows the results of these trials.

This simple experiment has several dimensions. First, as we examine the maximum gain column, we notice a steady rise in gain solely by virtue of the increase in wire diameter. The gain appears to peak at the last entry. However, we should also note the average gain test (AGT) column, which shows a value that departs slowly but surely away from the ideal value of 1.0 for the free-space lossless model. We may convert the AGT value to decibels and correct the reported gain. The corrected gain shows a maximum value with 0.07-wavelength diameter wire. In addition, the 0.08-wavelength version of the model contains numerous warnings about wire inter-penetration.

Given the curve of peak gain values, wire diameters between 0.05-wavelength and 0.07-wavelength are likely equally usable in most applications as the closest approximations of a solid surface parabolic dish. The technique shown here or a reasonable variation is the only way to discover how close to a solid surface that we may approximate with the non-uniform structure of the assembly.

Second, we may also test the assembly by examining the far-field patterns produced by the parabola and its driver. Fig. 3 shows patterns for our basic dipole driver and the dish using thin wires and thick wires. The thin-wire version of the dish yields results that are far from those of a solid-surface dish. The fat-wire version is closer to the mark.

Third, numerous modelers are surprised by the fact that even a solid surface--or our closest approximation of it--yields a set of rearward lobes, however, small that may be. Most texts focus on how rays intercept the dish surface and reflect in the forward direction. However, a parabolic dish shares some significant properties with rectangular reflectors. There are both semi-shadow and shadowed areas. As well, we encounter diffraction at the dish lip. Hence, every parabolic dish will have rearward far-field lobes. Advanced techniques of feeding the dish concentrate on minimizing these lobes while illuminating the dish to maximum advantage.

Finally, we should note the gain values for the dipole driver. A number of texts provide an equation to calculate the gain of a dish relative to its diameter at the frequency of use.

The equation has a fudge factor, k, the efficiency factor, given as 0.00 through 1.00. Most sample calculation use values between 0.5 and 0.55. However, the highest gain value in our table only emerges if we reduce the value of k to about 0.4. Most initial gain calculations presume an isotopic source of excitation. If we exchange the single dipole for a pair of turnstiled dipoles, the gain values do not change, but the source comes closer to be isotropic in free space. So we are left with a quandary: is the efficiency presumption behind the gain calculations off the mark, or is the model deficient enough to account for the difference between reported gain and pre-calculated gain? One of the limitations of modeling parabolic dishes in the absence of appropriate range or chamber tests is that we lack any means of forming an answer to the question.

One interesting facet of modeling parabolic reflectors is the optimal placement of the driver assembly relative to the focal point of the parabola. Our initial test showed very little variation in position relative to reflector wire thickness. However, we did not survey the gain behavior of the total antenna as we changed the position of the driver. A small survey may be useful in terms of showing what to expect from modeled parabolic assemblies. Therefore, let's use our 2-wavelength diameter dish with a depth of 0.5-wavelength and a focal length of 0.5-wavelength. For comparison, we may create a 3-wavelength diameter dish that also uses a depth of 0.5-wavelength. The equation with which we began or simplified view of parabolas tells us that the focal length is 1.125-wavelengths. In both cases, we shall use a turnstiled-dipole driver for simplicity. (Remember that these notes do not focus on parabolic reflector technology, but upon modeling the reflector. Therefore. we may justifiably use simplified driver assemblies.)

The smaller dish uses 0.07-wavelength wire in the reflector assembly. Table 2 provides us with some basic data on the modeled performance as we move the driver from 0.3-wavelengths to 1.1-wavelengths away from the vertex. The peak gain occurs at a distance (within the limits of the sampling) of 0.5-wavelengths. Maximum gain does not occur at the same distance as maximum front-to-back ratio. The reported beamwidth at the maximum gain distance coincides closely with the estimating equation, which sets the beamwidth as equal to 70 times a wavelength divided by the dish diameter (or 35 degrees, in this case). However, note the fluctuation in beamwidth as we move the driver position.

Fig. 5 overlays 3 patterns for the smaller dish and its turnstiled dipole driver at the driver position for maximum gain. The three traces are almost indistinguishable. The smaller dish appears to show a coincidence with basic calculations in every way, except perhaps for the gain deficit. (We may note in passing that we find a secondary gain peak with the driver at 1.0-wavelength away from the vertex, but this peak is considerably lower than the main peak.)

Let's turn to the larger dish that is 1.5 times the diameter of the smaller one. The diameter is 6 times the largest dimension of the driver. The depth remains at 0.5-wavelength. The focal point by calculation is 1.125-wavelengths from the vertex. For this dish, a wire of 0.06-wavelength diameter proved to be the largest usable value. If we move the turnstiled dipole driver from a closer point to a further point from the vertex, we obtain the data in Table 3.

The optimal distance for maximum gain is not the focal point, but a different point considerably closer to the vertex. At a distance of 0.75-wavelength, we obtain maximum gain with a beamwidth about 3 degrees wider than the calculated value. A secondary gain peak occurs at a distance of 1.375-wavelengths and is only about 0.3-dB lower than the near-position gain. The beamwidth at the farther position is closer to the calculated value.

The modeling result produces gain peaks that depart from standard rudimentary theory (but not perhaps from more advanced calculations). As well, the secondary peak is now a contender for use. Fig. 6 compares the patterns for the two position--again using an overlay of 3 patterns for each driver position. The two driver positions yield very distinct low-angle lobe structures. What we cannot specify solely on the basis of the models is whether the pictures are accurate to the behavior of a solid-surface dish on a test range.

The gain values produced by the larger dish are also deficient compared to standardized calculations. Once more, a value of about 0.4 for k, the efficiency factor, would align the model report with the standard calculation of peak gain. One limitation of the system is that fully half of the radiation of the driver is away from the reflector. Therefore, we might wish to explore in perhaps the most crude manner what happens if we focus more energy on the reflector.

Fig. 6 shows our final experiment. It replaces the simply driver with crossed or turnstiled Yagi elements forming a circularly polarized 3-element beam. The performance is modest, with about 7.5-dBi free-space forward gain and 11-dB front-to-back ratio. The array reflector elements are just over 0.5-wavelength long, and the boom length is exactly 0.5-wavelength. We shall direct the Yagi toward the reflector and try to find a mounting position that maximizes the gain. Table 4 summarizes the results for the 3-wavelength diameter dish. The listed distances are to the reflector, with the director 0.5-wavelength closer to the vertex.

Once more we find a pair of distances at which the array gain peaks: 1.0-wavelength and 1.4-wavelength. (Subtract 0.25-wavelength for the distance to the center of the Yagi and 0.5-wavelength for the distance to the director.) Unlike the dipole driver, the Yagi gain values are very comparable. (Corrected for the AGT values, the are almost identical.) As well, we find nothing to choose in the beamwidth and front-to-back values. The comparative patterns for the two positions appear in Fig. 8 and show may major reasons in the sidelobe structure for selecting one position over the other.

The gain values recorded by the Yagi driver with the 3-wavelength dish are above the pre-calculated gain values for dishes, largely as a result of the improved focus of energy from the driver onto the parabolic surface. Indeed, for some purposes, a modeler may wish to examine the current distribution on the modeled dish wires under various circumstance. Solely as an example of how we might find differences, I set the range of currents from 1.5e-3 down to 3.0e-4 to provide a range of color variation on the dish wires. Using the 2 positions of maximum gain, I obtained the graphical representations in Fig. 9.

The two current magnitude plots show very different patterns of current distribution. (The closer position shows some red lines at the center; these are from the Yagi elements.) The farther driver position appears to illuminate the radials to a considerable degree, in contrast to the closer position situation. Whether this factor has a bearing on the reliability of the wire-grid reflector as a model for a solid-surface reflector remains unknown if we remain completely at the level of modeling.

Conclusion

These notes have not attempted to answer any questions about parabolic reflectors as physical antenna structures. Instead, we have been focusing on the parabola as an example of a non-uniform wire-grid modeling structure. We initially concentrated on the ways in which such structures press the limits of NEC guidelines. We found limits beyond which we could not go, especially in terms of the number of radials that we might use vs. the wire diameter that we might assign to the reflector. However, since the reflector is wholly passive--or at most parasitic--pushing the limits does not yield unusable AGT values.

We also looked at some rudimentary models of parabolic arrays. Our goal was to develop some preliminary expectations of such models with a wire-grid parabolic reflector. We found ways in which the models fail to coincide with fundamental calculations associated with reflectors of this order. However, modeling alone does not provide corrections either to itself or to the basic equations. (I have for the most part resisted the temptation to suggest first-order possible explanations for some phenomena. For example, the driver assemblies are all within coupling range of the reflector, and it is uncertain whether the results deviate from a solid surface due to coupling to individual wires within the wire grid.)

The net result--applicable to any non-uniform wire-grid structure--is that we can only use such structures with caution and with attention to all of the tests that we might apply to the model. The models are eminently useful so long as our expectations are suitably modest.


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