Over the years, I have developed a number of highly accurate design aids for some very basic antennas, such as the Moxon rectangle, quads from 1 to 4 elements, 3-element Yagis, and a dual-element wideband dipole. Although the aids are available in a variety of forms, I have decided to combine all 11 calculation programs into a single spreadsheet. You may download the spreadsheet using the following links:

Almost all of the calculating programs are also available as NEC-2 antenna models in NEC-Win Plus (.nwp) format. This format is perhaps the most preferred because it allows you to see antenna patterns and the calculated feedpoint impedance as you enter the design frequency and the element or wire diameter. Some are available as programs (AC6LA's Moxgen), scripts (Moxon), and may appear elsewhere on-line. However, the Quattro-Pro and Excel spreadsheets may make them available as a group to some antenna builders who lack modeling software.

All of the programs have some of the same limitations. They all presume that the antenna uses a uniform diameter element material throughout. This provision is normally not a problem for quad antennas, but many HF and lower VHF Moxons and Yagis will use stepped diameter elements. The programs have no provisions for stepped-diameter elements and so will be useful only as a starting point for significant additional design work. As well, all of the programs presume that the wire is bare; insulated wire will required significant adjustments. Finally, all of the programs presume that all elements are insulated and electrically isolated from any conductive support structure. The calculations do not consider boom effects for elements directly connected to a conductive support structure.

The spreadsheet pages also have limitations, the foremost of which is that nothing has been protected from accidental or intentional change. You may wish to store an archival copy of the spreadsheet you download and use only a work copy.

All of the programs rest on a common set of procedures for each type of antenna. For each basic design, I developed a large number of optimized NEC models translating each design for measurements in wavelengths. I then subjected each dimension collection to regression analysis to obtain working equations that allowed me to "connect the dots." The equations have no electrical significance in themselves, but provide continuity between the steps in the optimized models. In virtually all cases, the difference between a calculated model and an independently optimized model at the same intermediate place resulted in less than 1% variation. The calculations have been calibrated for use between 3 and 300 MHz. However, I have successfully designed antenna from these equations for frequencies as low as 1 MHz and as high as 900 MHz.

Unlike a large number of oversimplified antenna design aids (including those notoriously inaccurate "cutting formulas"), the programs take seriously the fact that the dimensions of an antenna design of any complexity will change as we change the element or wire diameter. In fact, the user must enter only two pieces of numerical data. One is the design frequency in MHz. The other is the diameter of the wire or element. The programs have been calibrated for wire diameters as small as 1E-5 wavelength up to about 1E-2 wavelength. However, you will enter the wire diameter in a selected unit of measure. Each program provides options for AWG wires sizes as well as diameters given in inches or in millimeters. The program calculates the wire diameter as a fraction of a wavelength. With 3 entries (frequency, unit of wire size, and the wire size itself), each program produces a set of dimensions given in wavelengths, meters, feet, and inches. You may not be able to take them to the bank, but you will be able to take them to any NEC modeling software and confirm the performance. Then, you may build the antenna.

Every program page contains a reference diagram to identify each dimension against the sketch. However, since these floating diagrams may not translate to all versions of either Quattro Pro or Excel, we shall run down the programs briefly in these notes. Page A of the spreadsheet contains two programs for Moxon rectangles, the antenna that started me down the path of using regression analysis on optimized models. Page B has programs for 1 and 2 element quads. Page C provides calculations for 3-element quad beams, with a wide-bandwidth and a high-gain version. Page C supplies calculations for a high-performance 4-element quad. Page (E) provides calculations for three different 3-element Yagi designs: one for maximum practical gain, one for maximum front-to-back ratio, and one for very-wide-band operation. The final page (F) contains the equatrions for the dual-element wideband dipole. It is unlikely that I shall be able to add further programs to this set of 11, because every additional increase in design complexity multiplies the number of variables to be tracked when optimizing designs for various increments of wire diameter.

**The Moxon Rectangles**

The Moxon rectangle is a compact two-element driver-reflector parasitic array that delivers almost the gain of a 2-element Yagi with significant improvement of the front-to-back performance. Because the element ends fold toward each other, we must create some calculation dimensions in addition to the usual element-length and element-spacing pair. **Fig. 1** diagrams the dimensions.

The tails are dimensions B and D, and the fairly critical gap is dimension C. E is simply the sum of B through D as an arithmetic check. A, of course, is the maximum side-to-side dimension of the antenna. In its normal form, it is about 70% as wide as a comparable 2-element Yagi.

By normal form, I mean simply a version that has a resonant feedpoint of about 50 Ohms. This form appears in the first of the two programs. A 50-Ohm feedpoint impedance allows direct connection to a 50-Ohm main feedline, although the use of a common-mode current suppressor is advisable.

An alternative form uses a somewhat squarer structure to yield a feedpoint impedance between 90 and 100 Ohms. This form is useful when creating circularly polarized turnstile pairs of Moxons for fixed satellite antenna use. RG-62 makes a good phase line, and the net impedance allows use with a 50-Ohm main feedline. The gain is somewhat lower than for the 50-Ohm version, but when placed over ground and pointed straight up, ground reflections wash out the differences. The second program on Page A calculates the required dimension. Note that each program is separate and requires new user inputs.

For further information on Moxon rectangles, see the index of articles on Moxon Rectangles and Online Calculator. Alternatively, see my volume on *Moxon Rectangle Notes* on the Books Page.

**The Quads**

Volume 2 of *Quad Notes* or the index of "New Quad Studies" provides references to the articles relating to the calculation aids for the entries in this category. The process began with single-element quad loops and progressed into beams. **Fig. 2** shows the critical dimensions of the smallest quad components. One- and two-element calculations appear on Page B of the spreadsheet.

With a square quad loop, we are usually concerned with the length of each side to see how much room the antenna will occupy and with the loop circumference to see how much wire we need to make the antenna. I included the single loop since cutting formulas for single quad loops are so very far off the mark. If you make a diamond loop, you may use the same side and circumference calculations and simply rotate the array 45 degrees. The 125-Ohm feedpoint impedance is ripe for a quarter wavelength section of 70-75-Ohm cable as a matching section for our normal 50-Ohm main cable.

When we turn to 2-element quad beams, we encounter a major quad limitation, the relatively narrow operating bandwidth for its feedpoint impedance and front-to-back ratio. Therefore, I optimized the 2-element quad for maximum front-to-back ratio and for maximum operating bandwidth. As a result, the calculated spacing between the elements will be somewhat wider than we find in most articles and handbooks. Narrower spacing may raise the gain at the design frequency by a very small amount, but the SWR and front-to-back bandwidth will drop very noticeably. The wide-band 2-element quad does have a fairly high feedpoint impedance--about 135 Ohms. A quarter wavelength section of 93-Ohm RG-62 will usually provide a good match to the main feedline.

**Fig. 3** shows the outline of critical dimensions for a 3-element quad beam. When we add a third element (a director), we are faced with options. We may design for maximum operating bandwidth or we may design for maximum gain--but we can not do both within the same design. Page C of the spreadsheet, however, does give you a chance to design quads of each type and to compare their dimensions.

The wide-bandwidth version at the top of the page shows reduced gain (by about 0.7 dB) relative to the high-gain version at the bottom of the page. However, for any operating parameter, such as the SWR curve or the front-to-back ratio, the bandwidth of the high-gain version is only 60-70% of the value for the wide-band version. The wide-band version of the beam has a 75-80-Ohm feedpoint impedance, while the high-gain version is closer to 50-Ohms.

On page D of the spreadsheet, we find calculations for a high-performance 4-element quad. Obtaining over 10-dBi free-space gain with the widest possible operating bandwidth requires fairly wide spacing among the elements. There have been spot-frequency designs of equal performance using shorter booms, but they do not lend themselves to smooth curves that allow a very wide range of wire diameters and frequencies. The feedpoint impedance values of the 4-element quad design hover in the 60-Ohm region. **Fig. 4** shows the outline of the beam for parts identification.

Like the Moxon rectangle, the quad and its calculations presume a monoband beam. Since the dimensions all rest on fractions of wavelengths, combining them into a concentric multi-band quad is no easy task. Elements interact so that the monoband dimensions may not be right as a foundation for the more complex array.

**3-Element Yagis**

All 3-element Yagi designs share some common features, most notably, the ones shown in **Fig. 5**. We have 3 elements, with 2 spacing values to consider. However, spreadsheet page E has three separate calculation programs, indicating that there is more to 3-element Yagi design than having 3 elements.

Over the years, I have identified 3 fairly distinct species within the genus of 3-element Yagi: a high-gain type, a maximum front-to-back ratio type, and a very-wide-band type. The top of the spreadsheet calculates the dimensions of the high-gain Yagi. With a free-space gain of about 8.1 dBi, it still has a design-frequency front-to-back ratio of about 25 dB. However, it is somewhat narrow-banded, straining to cover the first MHz of 10 meters. The resonant feedpoint impedance is 25 Ohms, a convenient value for use with a quarter wavelength matching section, a beta match, or a gamma match.

Immediately below the graphic on the spreadsheet are the calculations for the maximum front-to-back version of the antenna. Although the gain drops to about 7.8 dBi, the rear lobe has a deep null on the design frequency. The null may approach 60 dB below the forward lobe. Hence, the antenna is useful for single-frequency direction finding. The typical resonant feedpoint impedance is close to 30 Ohms.

At the bottom of the page is the wide-bandwidth version of the antenna. The gain is about 7 dBi on the design frequency, although the front-to-back ratio remains above 20 dB. The antenna's key advantage is its ability to maintain most of its design-frequency performance across a wide bandwidth, such as all of 10 meters or all of 2 meters. A second advantage is the 50-Ohm resonant feedpoint impedance, a direct match for the typical 50-Ohm main feedline.

Each of these 3-element Yagi designs has applications for which it is most suited. The key limitation lies in the programs. They calculate for uniform-diameter elements, restricting their use to VHF and UHF beams, and possibly some low-HF wire Yagi installations. To create a stepped-diameter version of any of the Yagis requires the use of NEC-type software. Ordinarily, stepped-diameter elements do not require any changes of element spacing. However, having the Leeson corrections available can ease the problem of arriving at equivalent-length stepped-diameter elements relative to the uniform versions used in the calculations.

For additional information on these antenna designs, see "Modeling Yagis by Equation"

**The Dual-Element Wideband Dipole (DEWD)**

UR0GT developed an interesting wide-band dipole antenna that employs 2 elements in an interesting close-spaced relationship. See **Fig. 6**. The center cross-wire, marked "spacing," contains the feedpoint, normally a 50-Ohm coaxial cable. Each end of the antenna consists of 2 wires, one longer and one shorter. By judicious selection of the 3 wire lengths, one can cover all of the 80-75-meter band using common wire sizes. Fatter wire sizes (or tubing) yield even broader bandwidth values. For example, one might make a single vertical antenna to cover both the 10- and 11-meter bands. Likewise, one can easily broaden the bandwidth of a UHF planar reflector array with a DEWD driver.

The design program is based on using the center resonant frequency (with other resonant points at the low and high ends of the passband) set to about 70 Ohms for a 50-Ohm SWR at that point of less than 1.5:1. By entering the element diameter and the center or design frequency, the spreadsheet calculates the length of the spacing wire and the two pairs of end wires to produce a wideband dipole with under 1.5:1 50-Ohm SWR for the calculated bandwidth shown as the last entry. The bandwidth is simply the upper frequency of a 1.5:1 SWR minus the lower frequency with the same SWR divided by the center frequency and multiplied by 100 to arrive at a percentage.

The calculation program is not a final design product unless you want it to be that. It will also provide starter values for modeling wider-band DEWDs that use a higher center SWR (perhaps 2:1) or special purpose very low SWR DEWDs, such as an element designed just to cover the entire 10-meter band with less than 1.1:1 50-Ohm SWR.

**Conclusion**

The 11 calculation programs included in these spreadsheet pages represent an advance beyond the over-simplified formulas all too often used to design antennas with marginal performance. Each system rests on careful optimized modeling and regression analysis and has been spot checked with physical prototypes. When used within their limitations, they can speed the design of utility antennas of the types included in the collection. However, like any system of calculation, they do not account for the set of construction variables that accompany the building of any antenna. Those variables are the user's responsibility.

For an additional set of spreadsheets that may be useful when matching antenna feedpoints to main feedlines, see "Antenna Matching"

*Updated 06-26-2006, 04-04-2008. © L. B. Cebik, W4RNL. Data may be used for personal purposes, but may not be reproduced for publication in print or any other medium without permission of the author.*