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<title>Impedance-Transformation Properties of Common 4:1 Balun Types Index</title>
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<h1>The Impedance-Transformation Properties<br>
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of Common 4:1 Balun Types</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>Baluns, as we commonly construe them in amateur radio, have three main functions. First, they convert an antenna feedpoint or a parallel transmission line from a balanced circuit to a single-ended (sometimes called an unbalanced) configuration, where the single-ended configuration is necessary for cables and equipment using coaxial connectors in which the outer conductor is connected to ground somewhere in the system. Second, we often employ baluns to attenuate common-mode currents to keep them out of equipment and off the surface of coaxial cable braids or sheaths.</p>
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<p>Third, we employ some baluns to transform load impedance values to an alternative value. Although there are designs for baluns capable of many impedance ratios (from 1:1 upward), the most usual balun impedance ratio is 4:1. Indeed, the amateur radio marketplace offers dozens of 4:1 units from different makers, with many different styles of construction for many different applications.</p>
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<p>Although many measurements exist for the power handling capabilities of existing 4:1 balun, with additional measurements for their ability to attenuate common-mode currents, I have seen no measurements of the impedance transformation properties of baluns across the full HF spectrum from 3 to 30 MHz. The AIM 4170 antenna analyzer unit and associated software provides a convenient method of filling this vacuum, at least partially. The notes in the following series of preliminary measurement reports make a start toward characterizing different types of balun designs with respect to impedance transformation.</p>
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<blockquote>
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<p><b><a href="41balun-1.pdf">Part 1: Essential Background</a></b></p>
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<p><b><a href="41balun-2.pdf">Part 2: The Dual-Ferrite-Bead HF Balun: Some Preliminary Measurements</a></b></p>
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<p><b><a href="41balun-3.pdf">Part 3: Voltage Baluns: Some Preliminary Measurements</a></b></p>
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<p><b><a href="41balun-4.pdf">Part 4: Toroidal Current Baluns: Some Preliminary Measurements</a></b></p>
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</blockquote>
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<p>Since each item in the sequence is in PDF format, you will have to use the "Back" button to return to this page to access the following item.</p>
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<br>
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<p><i>Updated 04-01-2008.© 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.</i></p>
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<p><a href="../radio.html"><img src="../images/return.gif" align="bottom" width="40" height="40"> Return to Amateur Radio Page</a></p>
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<p class="topics-footer">Topics: <a href='../topics.html#balun'>Balun</a></p><br>
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<h1><img src="../images/yagicad.gif" align="left" width="125" height="62" border="2" vspace="10" hspace="10"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10"> Antenna Design</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>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:</p>
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<blockquote>
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<p><b><a href="../download/ant-design.xls">Antenna-Design in .XLS format (Excel)</a></b></p>
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<p><b><a href="../download/ant-design.qpw" download="ant-design.qpw">Antenna-Design in .QPW format (Quattro Pro)</a></b></p>
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</blockquote>
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<p>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.</p>
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<p>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.</p>
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<p>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.</p>
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<p>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.</p>
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<p>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.</p>
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<p>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.</p>
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<p><b>The Moxon Rectangles</b></p>
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<p>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. <b>Fig. 1</b> diagrams the dimensions.</p>
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<p>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.</p>
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<p>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.</p>
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<p>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.</p>
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<p>For further information on Moxon rectangles, see the index of articles on <a href="../moxon/moxpage.html">Moxon Rectangles and Online Calculator</a>. Alternatively, see my volume on <i>Moxon Rectangle Notes</i> on the <a href="../bookant.html">Books Page</a>.</p>
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<p><b>The Quads</b></p>
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<p>Volume 2 of <i>Quad Notes</i> or the index of <a href="../quad/quadlist.html">"New Quad Studies"</a> 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. <b>Fig. 2</b> shows the critical dimensions of the smallest quad components. One- and two-element calculations appear on Page B of the spreadsheet.</p>
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<p>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.</p>
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<p>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.</p>
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<p><b>Fig. 3</b> 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.</p>
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<p>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.</p>
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<p>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. <b>Fig. 4</b> shows the outline of the beam for parts identification.</p>
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<p>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.</p>
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<p><b>3-Element Yagis</b></p>
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<p>All 3-element Yagi designs share some common features, most notably, the ones shown in <b>Fig. 5</b>. 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.</p>
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<p>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.</p>
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<p>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.</p>
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<p>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.</p>
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<p>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.</p>
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<p>For additional information on these antenna designs, see <a href="../yagi/eq-yag-1.html">"Modeling Yagis by Equation"</a></p>
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<p><b>The Dual-Element Wideband Dipole (DEWD)</b></p>
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<p>UR0GT developed an interesting wide-band dipole antenna that employs 2 elements in an interesting close-spaced relationship. See <b>Fig. 6</b>. 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.</p>
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<p>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.</p>
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<p>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.</p>
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<p><b>Conclusion</b></p>
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<p>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.</p>
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<p>For an additional set of spreadsheets that may be useful when matching antenna feedpoints to main feedlines, see <a href="ant-match.html">"Antenna Matching"</a></p>
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<p><i>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.</i></p>
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<p><a href="../radio.html"><img src="../images/return.gif" align="bottom" width="40" height="40"> Return to Amateur Radio Page</a></p>
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<p class="topics-footer">Topics: <a href='../topics.html#design'>Design</a>, <a href='../topics.html#dipole'>Dipole</a>, <a href='../topics.html#matching'>Matching</a>, <a href='../topics.html#moxon'>Moxon</a>, <a href='../topics.html#quad'>Quad</a>, <a href='../topics.html#yagi'>Yagi</a></p><br>
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<h1><img src="../images/feedbut.gif" align="left" width="145" height="75" border="2" vspace="5" hspace="10"><img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10"> Antenna Matching</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>The following notes have a very limited purpose: to orient you to the calculation aids available in spreadsheet form for determining the dimensions of a few common ways of matching an antenna--especially a beam--to your feedline. You may download the spreadsheet using the following links:</p>
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<blockquote>
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<p><a href="../download/ant-match.xls"><b>Antenna-Match in .XLS format (Excel)</b></a></p>
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<p><a href="../download/ant-match.qpw" download="ant-match.qpw"><b>Antenna-Match in .QPW format (Quattro Pro)</b></a></p>
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<p>The spreadsheet pages include series matches (including the 1/4 wavelength transformer, the Bramham system, and the Regier system), the beta or hairpin match, the gamma match (using separate sheets for the Healey-Wheeler system and the Tolles-Nelson-Leeson system), and finally the match-line and stub method. The pages themselves are simple and direct, with only numerical content, plus a few notes here and there. There are no hidden entries, so you may alter them as you wish--although accidental ruination is also possible. Be sure to have entries for each input data slot, or the results will not be usable. The sheets have very few protections against non-calculable situations, such as division by zero. All pages have undergone reasonably extensive testing via antenna models.</p>
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<p>The following notes orient you to the pages, but do not provide extensive explanations of how each matching system work.</p>
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<p><b>Series Matching</b></p>
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<p>Series matching includes 3 system, ranging from the most specific to the most general.</p>
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<p><i>1. The 1/4 Wavelength Transmission-Line Transformer</i></p>
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<p>The 1/4 wavelength transmission-line transformer is perhaps the best known of the series matching systems. <b>Fig. 1</b> outlines the basic application of the system.</p>
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<p>We may insert a 1/4 wavelength section of transmission line between a resonant antenna impedance and a feedline if the transformer section Zo is the geometric mean between the antenna and the feedline impedance. For example, if a beam has an impedance of 25 Ohms and we have a 50-Ohm feedline, then a transformer section of 35-37 Ohms will effect the required impedance transformation. We may use RG-83 or parallel sections of RG-59 to create the transformer. We may also step up or step down: the only requirement is that the transformer Zo be roughly the geometric mean of the two end values.</p>
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<p>For further information on the 1/4 wavelength transmission-line transformer, as well as ways to use variations of it, see <a href="quarter.html">"When is a Quarter Wave Not a Quarter Wave?"</a>.</p>
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<p><i>2. The Bramham System</i></p>
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<p>The Bramham system of series matching tackles a special problem: matching a resonant antenna impedance to a different feedline Zo. The basic problem and solution appear in outline form in <b>Fig. 2</b>.</p>
|
||||
<center>
|
||||
<img src="ant-match-2.gif" width="450" height="220" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>By using two inserted line sections (that never total more than 1/4 wavelength), we can arrive at a perfect match. However, we require that the initial antenna impedance be resonant. For further information on the Bramham system, see <a href="ser.html">"Series Matching: A Review"</a>.</p>
|
||||
<p><i>3. The Regier General Series-Matching Solution</i></p>
|
||||
<p>3 decades ago, Regier developed a general solution to series matching any antenna impedance to a given line with a single line insertion. The general outline appears in <b>Fig. 3</b>.</p>
|
||||
<center>
|
||||
<img src="ant-match-3.gif" width="350" height="220" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>There are limits to what combinations of Zo and Z1 we may use and still obtain a desired match. In general, the closer the values of Zo and Z1, the smaller the range of antena impedance values that we can match. Note that we can combine parallel and coaxial lines in creating the match. For further information on the Regier system, see <a href="ser.html">"Series Matching: A Review"</a>."</p>
|
||||
<p>The spreadsheet page for series matching has undergone many modeling tests using the NEC TL facility. The results are exact. All series matching systems presume a driven element that is insulated and isolated from any conductive support boom.</p>
|
||||
<p><b>The Beta or Hairpin Match</b></p>
|
||||
<p>On a separate spreadsheet page, you will find calculations for the beta or hairpin match. Essentially, the beta match is a form of L-network specifically arranged to transform a higher line Zo to a lower antenna impedance. In the process, the network uses a shortened element that has capacitively reactance in the feedpoint impedance as one of the reactive components in the L-network. <b>Fig. 4</b> shows the general evolution of the beta or hairpin match.</p>
|
||||
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|
||||
<img src="ant-match-4.gif" width="550" height="320" border="2" vspace="5" hspace="9">
|
||||
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|
||||
<p>For the highest level of effectiveness for a given resistive component of feedpoint impedance, the beta match requires a certain capacitive reactance. We obtain this value by adjusting the element length. The only component that we need to add to the system is the parallel or shunt element. If the element has a capacitive reactance, the shunt element must be inductive (and vice versa).</p>
|
||||
<p>Some folks distrust the beta match because one form of shunt inductance seems to be a short circuit across the feedpoint. <b>Fig. 5</b> shows 3 typical forms of adding inductive reactance across the feedpoint terminals, which are insulated and isolated from any conductive support boom.</p>
|
||||
<center>
|
||||
<img src="ant-match-5.gif" width="398" height="202" border="2" vspace="5" hspace="9">
|
||||
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|
||||
<p>A solenoid inductor is feasible and generally has little loss, since its reactance will normally be quite low. However, shorted transmission-line stubs may generally provide the same inductive reactance with even lower loss. The hairpin or shorted parallel transmission line section is the version that most worries new users. However, the beta match in any form is as effective as virtually any other system in effecting a low-loss match between the element and the feedline--when the element resistive component is less than the feedline Zo. In addition, one may also lengthen an element to make it inductively reactive. Then the shunt component becomes a capacitance. Both versions of the beta match have undergone extensive modeling confirmation.</p>
|
||||
<p>For further information on the beta or hairpin match, see <a href="beta.html">"The Beta Match: 2 Views"</a>."</p>
|
||||
<p><b>The Gamma Match</b></p>
|
||||
<p>Two pages devote themselves to gamma match calculations. The gamma match differs from the previous matching systems in that the calculations are not precise. Rather, they produce starter values that will require careful field adjustment (the gentler sounding term for trial and error). <b>Fig. 6</b> shows some of the reasons why the calculations are less than fully precise.</p>
|
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|
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<img src="ant-match-6.gif" width="490" height="250" border="2" vspace="5" hspace="9">
|
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</center>
|
||||
<p>The gamma system begins with a larger number of variables, some of which are the physical dimensions of the assembly components. We need to know or decide upon the main element diameter, the gamma rod diameter, and the center-to-center spacing between these two parts. Calculations usually proceed (although there have been variations) by treating the gamma assembly as a section of parallel transmission line, shorted at the far end. The end result is a change in the position of the antenna feedpoint relative to the element without the gamma assembly. Most calculation systems do not take into account the far-end shorting bar structure or the structure that supports the feedline connector.</p>
|
||||
<p>Practical gamma matches also include a number of variations on the ideal situation used in calculations. The rod may extend beyond the shorting bar. The required series capacitor may not be at the feedpoint, but be somewhere along the gamma rod. The gamma system is the only matching system in this group that permits a direct connection of the main element center to the boom. However, we cannot easily obtain the initial feedpoint impedance when we connect the element to the boom, and the boom will have an affect upon the feedpoint impedance.</p>
|
||||
<p>The two calculation systems have different sources but similar starting points. The Healey-Wheeler system derives from some of the earliest work on the gamma and requires the user to insert trial values of the gamma rod length until the resulting resistive component at the new feedpoint matches the target line Zo. The Tolles-Nelson-Leeson system (distributed in Basic format by ARRL with <i>The ARRL Antenna Book</i>) calculates the gamma rod length.</p>
|
||||
<p>The two systems do not produce identical results. As well, the results differ from the results of antenna modeling. Because NEC cannot effectively handle the gamma match, only a highly corrected version of MININEC (such as Antenna Model) is adequate to the modeling taks. However, even MININEC cannot show the required variations that emerge from connecting the element to a central boom. Since gamma matches receive only spot checks rather than systematic comparison of calculations and/or models with physical antennas, all three methods are tentative guides, useful for beginning the process of designing a gamma match, but always needing extensive field adjustment.</p>
|
||||
<p>For further information on the beta or hairpin match, see <a href="gamma.html">"Some Preliminary Notes on the Gamma Match"</a>." Despite the lesser precision of the calculations relative to building a gamma matching assembly, the gamma match itself is capable of matching a wide range of antenna impedance vales--both resonant and non-resonant--to standard feedlines.</p>
|
||||
<p><b>The Match-Line and Stub Matching System</b></p>
|
||||
<p>Of all of the matching systems included in the spreadsheet pages, the match-line and stub system may be the oldest. We use it to match odd antenna impedance values--such as values we might obtain from an extended double zepp antenna--to a standard feedline. <b>Fig. 7</b> outlines what we require.</p>
|
||||
<center>
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<img src="ant-match-7.gif" width="277" height="248" border="2" vspace="5" hspace="9">
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</center>
|
||||
<p>If we select a match-line Zo that is high enough relative to the antenna resistive component, at some line length, the resistive component will be the Zo of the feedline. In almost all cases, there will be a remnant reactance for which we may compensate with a shorted or open stub of the same type of line that we use for the match-line section. In most cases, there will be a solution in which the combined length of the match-line and the stub is less than 1/4 wavelength. In fact, the match-line and stub system is a variation of the Regier series matching system, but it is sufficiently special to deserve separate treatment.</p>
|
||||
<p>There are values of match-line impedance that will not effect the desired match. The spreadsheet shows how to detect them. Wherever we may create the desired match, we shall find two match-line lengths that will do the job. Each line length will have both open and shorted stub reactance compensation values. Of the four options, we normally select the one that yields the shortest combined match-line and stub length. (In fact, it is possible by adding 1/2 wavelength match-line sections to bring the main feedline connection point close to ground level--so long as we do not let the stub drag on the ground.)</p>
|
||||
<p>For further information on the match-line and stub method, see <a href="stub.pdf">"Stub Matching: A Review"</a>."</p>
|
||||
<p><b>Conclusion</b></p>
|
||||
<p>I am making these spreadsheet pages available solely as an aid to amateurs and others who may have occasion to calculate one or more common matching systems. Despite subjecting them to numerous tests, I cannot certify that they will always produce precise results, since I have not tested all possible cases. (Gamma match results, of course, will not be precise, although the two versions shown yield results that coincide with their sources.) Even where the results are good, every antenna builder must remember that construction variables may call for field adjustments in even the most precise systems.</p>
|
||||
<p>There are many more ways to match antennas than I have included in these few sheets. General L-, T-, and PI-network solutions appear in many programs, and so I have omitted them. Instead, I have chosen methods that, despite their wide use, have not usually been included in matching programs. Applying them to situations more significant than satisfying curiosity about the calculations is a user responsibility.</p>
|
||||
<p>For an additional set of spreadsheets that may be usful when designing some kinds of antennas, see <a href="ant-design.html">"Antenna Design"</a></p>
|
||||
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<p><i>Updated 06-26-2006. © 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.</i></p>
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<p><a href="../radio.html"><img src="../images/return.gif" align="bottom" width="40" height="40"> Return to Amateur Radio Page</a></p>
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<p class="topics-footer">Topics: <a href='../topics.html#matching'>Matching</a></p><br>
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<h1><img src="../images/feedbut.gif" align="left" width="145" height="75" border="2" vspace="5" hspace="10"> The Beta Match: 2 Views <img src="../images/w4rnlwwr.gif" align="right" width="99" height="83" border="2" vspace="5" hspace="10"></h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>The beta or hairpin match is so deceptively simple in appearance that many radio amateurs distrust it immediately. Surely adding a coil across the feedpoint terminals--and nothing else except the coax--must foul up the performance and produce dreadful results. The coil or the hairpin is so small that it must act like a short circuit.</p>
|
||||
<p>Unlike a gamma match, which permits the builder to make a direct connection between the driver and the boom, the beta-matched driver requires insulation and isolation from a conductive boom. But, beyond that, the driver connections are simpler. In fact, we can use the same terminals to connect both the beta match and the coax. Every reduction in something that can degrade with time and weather is a blessing in home-built antennas.</p>
|
||||
<p>The two common forms of the beta match appear in <b>Fig. 1</b>. The antenna terminals connect to a coaxial cable, usually 50 Ohms. The beta component is anything that produces a required inductive reactance. How much inductive reactance we need will appear shortly. The two most common ways to produce an inductive reactance are with a solenoid inductor or coil and with a shorted transmission-line stub. The stub form is responsible for the name "hairpin" match. Apparently, the term "beta match" was originally copyrighted by Hy-Gain, a company that has used the system extensively over the years. Sensitivity to that situation has led some folks to avoid the simple term "beta match" and to use only the expression "hairpin match." The result has been some odd locutions. For example, the solenoid or coil method of producing the required inductive reactance has been called a "hairpin inductor." Since I have acknowledged the potential copyright holder and have no commercial interests, I shall call the system by its simplest name.</p>
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<img src="beta-1.gif" width="399" height="232" border="2" vspace="5" hspace="9">
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<p>The ARRL <i>Antenna Book</i> has long had a correct explanation of how a beta match works, along with the conditions for setting up the system. Part of what we shall see here parallels that account. However, there is also a second way to look at the beta match, one that takes us back to even more basic terms. By looking at the beta match in two ways, perhaps we can give you a way to avoid confusion about the system.</p>
|
||||
<p>First things first: we call the system a matching system. This label implies that we have an antenna whose feedpoint impedance is not a good match for the ubiquitous 50-Ohm coaxial cable that we so often use as our feedline. We only use the beta match when the antenna feedpoint impedance is less than the cable impedance. For this case, we are talking of impedances well under 50 Ohms. Although we can use a beta match with impedances as high as 35 Ohms, we normally reserve the system for impedances of 25 Ohms or less.</p>
|
||||
<p>If our driver is resonant at about 25 Ohms, then we do not need a beta, gamma, or Tee match. Instead, we can use a 1/4 wavelength section of 37-Ohm cable to effect an impedance match. Although there is a 35-Ohm coaxial cable available, it is not easy to find and runs about $3.00/foot. A simpler scheme is to connect 2 lengths of 70-Ohm cable in parallel, with the center conductors connected and the braids connected at both ends. A pair of RG-59 cables in parallel will just fit inside of the common male UHF connector shell without significant deformation. Hence, we can create and waterproof the required cable with fair ease. Remember to account for the velocity factor of the line that you actually use when constructing such a cable.</p>
|
||||
<p>Of course, the line length will vary not only according to the line's velocity factor, but also as a function of the frequency that you use as your design frequency. The design frequency is normally the center frequency of the span of frequencies for the antenna (assuming a monoband antenna). If the passband (another name for the spread of frequencies used) is wide and the antenna has any odd properties, you may adjust the design frequency so that you achieve satisfactory performance at the edges of the span. For wider amateur bands, I often recommend that an antenna like the Moxon rectangle be designed for a point about 1/3 the way up from the low end of the passband. Other antennas may require different treatment.</p>
|
||||
<p>I used to calculate the length of a wave at the design frequency from scratch. 299.7925/F in MHz gives a wavelength in meters. Divide that by 0.3048 for the length in feet. Multiply by 12 for the length in inches. Of course, if you need the length in cm or mm, then simply make an adjustment by 100 or 1000 to the length in meters. Nowadays, I simply keep my antenna modeling program running and let it do the calculating for me.</p>
|
||||
<p>If we opt for a 1/4 wavelength matching section, then we have no need to use a beta match. That situation would mean that this little article would end right here. So let's assume that we do not have any 70-Ohm cable or do not want to cut a section out of our cable TV system. What we do have is some leftover house wiring material, specifically, some AWG #12 copper wire with a 0.0808" diameter. Ultimately, we shall apply it to a 3-element 10-meter Yagi. However, we first need a general procedure for calculating everything that we need for the beta match.</p>
|
||||
<h4>A Beta Match: the Network Way</h4>
|
||||
<p>One of the conventional representations of a beta matched antenna element appears on the left in <b>Fig. 2</b>. Let's examine it more closely, because it tells us part of what we have to do to set up the antenna's driven element to receive a beta match.</p>
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<center>
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<img src="beta-2.gif" width="650" height="275" border="2" vspace="5" hspace="9">
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</center>
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<p>Although the figure shows stubs to indicate that we are dealing with an antenna element, the load for the energy arriving via the coaxial cable is essentially between the terminals. Note that we must have a complex load. First, there is a series resistance (Rs). Next, there is a series capacitive reactance (XCs). That much alone tells us that the antenna element itself must not be resonant. If we lengthen the element beyond resonance, it will present a complex impedance composed of a series resistance and an inductive reactance. That condition is a possible one for a beta match, but by habit, amateurs want lighter elements, not heavier ones. So they traditionally use an element that is shorter than resonant. That physical situation results in a complex impedance consisting of a series resistance and a series capacitive reactance, as shown in the sketch. If we had started with a 1/2-inch resonant 10-meter element about 196.97" long, then we must reduce its length to about 192.2" to obtain a usable combination of resistance and capacitive reactance.</p>
|
||||
<p>Before we look at why my measurements are so fussy, let's clear up another misconception, this one bred by the appearance of the left-hand sketch in <b>Fig. 2</b>. Because we have only 1 capacitor symbol, numerous newcomers tend to think of the beta match as an unbalanced system. Now the gamma match is inherently an unbalanced system, but the beta match is a balanced system. The center part of <b>Fig. 2</b> shows a way of drawing the situation that gives us a balance of capacitor symbols. We have 1/2 of the capacitive reactance on each side of the resistor. (If we were using real capacitors as lumped components, we would make each one be twice the value of the virtual capacitor in the left-hand sketch, so that each one has 1/2 the total series reactance. But remember that we are not using a lumped component. Instead, we are using the reactance of an element that is shorter than its resonant length.)</p>
|
||||
<p>Now we can return to the fussy element length measurements. How long or short we make the element is a function of the series resistance that it presents. If we know that value, then we can calculate everything else using some very simple equations that apply to an L-network when the energy source has a higher resistive impedance than the load (antenna element). To see how this works, look at the right-hand part of <b>Fig. 2</b>. There, I have redrawn the center sketch in the form of a balanced L-network. You calculate a balanced network the same way as an unbalanced one, except that you later divide the series capacitive reactance into 2 equal parts, one part for each side of the line.</p>
|
||||
<p>The L-network consists of a series capacitive reactance (XCs) and a parallel inductive reactance (XLp) across the line on the source or cable side of the network. XLp is the beta-match component, either a solenoid inductor or a shorted transmission-line stub. (Note: XLp is also called a shunt component, but I shall stick to the term parallel. I do not want to confuse matters by using too many terms beginning with "s.")</p>
|
||||
<p>There are several ways of calculating the required values of XCs and XLp, but at root, they depend on knowing the value of Rs, the resistive part of the antenna or load impedance. Let's create and define a term having the name "delta." Nowadays, we call this term the loaded Q, the working Q, or the network Q, but its original name was delta. In equations, we represent the name with the lower case Greek letter. We can show the definition of the delta of a down-converting L-network by a simple equation (and down-converting simply means that the source-end impedance is higher than the load-end impedance).</p>
|
||||
<center>
|
||||
<img src="betae1.gif" width="107" height="62">
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||||
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||||
<p>How do we find out the antenna feedpoint resistance. If we have our antenna set up for a field test, we can always measure the impedance with one of the several antenna analyzers on the market. Or we can go to an antenna modeling program and let it calculate the impedance. Let's suppose that we come up with 25 Ohms. The ratio of the 50-Ohm source end impedance to 25 Ohms is 2:1. Subtract 1 from that and you get the square root of 1, which is 1. That is the value of delta.</p>
|
||||
<p>Next, let's figure out how much capacitive reactance we need in series with the 25-Ohm resistive value to satify the network requirements. This step is also an easy calculation.</p>
|
||||
<center>
|
||||
<img src="betae2.gif" width="121" height="32">
|
||||
</center>
|
||||
<p>We simply multiply the value of delta by the value of the series resistance that is the load, and the answer appears. 1 times 25 gives us 25 Ohm reactance. Notice that in this network calculation, we do not obtain a + or - sign. The absence of a sign simply means that the reactance can be either inductive or capacitive under a special condition. The condition is that the parallel reactance that we will next calculate must be (for ordinary cases) the opposite type from the series reactance. Since we want a shorter element with capacitive reactances as the series reactance, we must use an inductive reactance for the parallel reactance.</p>
|
||||
<center>
|
||||
<img src="betae3.gif" width="113" height="45">
|
||||
</center>
|
||||
<p>The source-side or parallel reactance is simply the value of the source resistance (50 Ohms) divided by the value of delta. Since delta = 1, then the parallel reactance must be 50 Ohms. Remember that these steps apply to the situation in which we are matching a higher impedance (the 50-Ohm cable that is the source of emergy) to a lower value of resistive impedance (the value of Rs that is the resistive component of the antenna element impedance). When receiving, the situation is reversed. the antenna element is the source of energy at a lower resistive impedance that we up-convert to the 50-Ohm cable impedance. The same component values apply. However, to help keep things straight, we usually only think in terms of transmitting. So the cable end becomes our energy source and the antenna becomes the load.</p>
|
||||
<p>That is all there is to doing the basic calculations for the beta-match as a down-converting L-network. We have not yet translated these reactance values into physical components. Of course, we have only one component needing the translation: the parallel inductive reactance across the terminals. The capacitive reactance is contained in the antenna feedpoint impedance. However, before we effect the translation, let's step back and take another look at what we just did.</p>
|
||||
<h4>A Beta Match: the Parallel-Circuit Way</h4>One of the shortcomings of most handbooks designed for radio amateurs is that they do not give enough space to a fundamental fact about impedances. We receive the information about the antenna feedpoint impedance and do not stop to think about the orientation of the information. In our ideal case, the shorter antenna elements has an impedance of 25 - j25 Ohms. We normally forget that the values for each component of the impedance are series values.
|
||||
<p>Why is the qualification necessary? The answer lies in the fact that for every series impedance, there is an equivalent parallel combination of impedance values. Although the handbooks give the conversion equations in each direction, they do not give enough practical examples of using them. Our beta match presents us with a very significant use, since it may teach us something about impedances at the same time that it allows us to flex those long-dormant equations. What we shall do is convert the left side of <b>Fig. 3</b> into a form that reflects the right side.</p>
|
||||
<center>
|
||||
<img src="beta-3.gif" width="600" height="250" border="2" vspace="5" hspace="9">
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||||
</center>
|
||||
<p>Note that the L-network consists of source and load resistances, with a combination of series and parallel reactances. Suppose that we converted the series antenna impedance components into their parallel equivalent values. For reference, here are the series-to-parallel equations. Note the similarities that let you simplify the actual key presses needed on a hand calculator.</p>
|
||||
<center>
|
||||
<img src="betae4.gif" width="266" height="52">
|
||||
</center>
|
||||
<p>Since both the series resistance and series reactance values had numerical values of 25, we can quickly arrive at the parallel equivalent resistance (50 Ohms) and the parallel equivalent reactance (-j50 Ohms). Now note that something amazing has occurred. The parallel resistive component of the load--with its associated capacitive reactance at work--yields a 50-Ohm resistance.</p>
|
||||
<p>Now be amazed by a fact that is well known to people to match things up for a living: The series and parallel values of the resistive component of an impedance are the same when the is no series reactance or there is an indefinitely high parallel reactance. We cannot simply re-resonate the antenna element, because that would yield a 25-Ohm resistive impedance. We need a way to keep the capacitive reactance working while getting rid of the series reactance net value-- or driving the parallel reactance upward and off scale for all practical purposes.</p>
|
||||
<p>One very practical way to accomplish this is to place a parallel reactance of the opposite type scross the initial parallel reactance value. Parallel reactances that are equal make up a resonant circuit. Parallel reactances operate in the same manner as parallel resistances with respect to arriving at a net value for 2 values.</p>
|
||||
<center>
|
||||
<img src="betae11.gif" width="232" height="50">
|
||||
</center>
|
||||
<p>Since the reactances are equal within this idealized case, the denominators add up to zero, because the two reactance values have opposite signs. Division by zero is not possible, of course. But we know that the value of the result becomes higher without limit as the denominator approaches zero. One common computer program technique of handling division by zero, when zero is within a range of possible values, is to set the result at an arbitrary but very high number. Typically, programs use 1E10. We may use that number here in order to see what has happened to the series values for R and X that we might encounter as the impedance at the coax terminals.</p>
|
||||
<center>
|
||||
<img src="betae5.gif" width="269" height="50">
|
||||
</center>
|
||||
<p>The equations tell us how to back calculate into the more familiar series forms of resistance and reactance at the coaxial cable terminals. Remember that Rp is still 50 Ohms. Adding the square of 50 to the square of 1E10 has no effect. Any hand calculator will return 1E20 as the result because the square of 50 (2500) only appears at the far end of the result, outside the range of what the readout will show. The resistance numerator is 50 times 1E20. When we divide this by 1E20, we end up with 50 Ohms as the series resistance. The reactance conversion back into a series value has a quite different result. We end up with 2500 times 1E10 divided by 1E20. The result is 2.5E-7 or about 250 nano-Ohms. Effectively, the reactance is zero, just as we predicted.</p>
|
||||
<p>The calculations that we have just performed are commonplace in matching situations. In another item at the site are ways to calculate the match-line-and-stub system of matching. We find the place along a transmission line at which the parallel resistive impedance is 50 Ohms. Then we add a parallel transmission-line stub to match the parallel reactance value at that point, but use a stub that produces a reactance of the opposite type. The result is a parallel resistive impedance in paralel with a very high value of parallel reactance. When we reconvert to series values, the series resistance equals its parallel counterpart, and the reactance goes to zero or very close to it. Then we can run coax from that point back to the transmitter. Of course, there is no magic in using 50 Ohms as the desired impedance. It just happens to be the most common line impedance to which we normal gear our matching efforts.</p>
|
||||
<p>Although the series-parallel-series conversion equations seem more complex than the simplified L-network equations, they are in fact more fundamental. Indeed, you now have a more complete idea of what those network equations are actually calculating.</p>
|
||||
<p>One of the main questions facing the would-be beta matcher is knowing what length to make the antenna element. One of the advantages of the L-network equations is that they provide guidance here, where the series-parallel conversions require that we almost hunt out the correct series reactance value by trial and error. Since we most often calculate for a 50-Ohm source resistance, let's make a small chart of beta-match reactance values using small increments of change in the series load resistance.</p>
|
||||
<pre>
|
||||
Some Beta-Match Resistance and Reactance Values
|
||||
|
||||
Rso = 50 Ohms for all cases
|
||||
|
||||
Rs (Ohms) 35 25 17 12.5 10 8.3
|
||||
Delta 0.65 1.0 1.4 1.7 2.0 2.2
|
||||
Xs (Ohms) 22.9 25.0 23.6 21.7 20.0 18.6
|
||||
Xp (Ohms) 76.4 50.0 35.4 28.9 25.0 22.4
|
||||
</pre>
|
||||
<p>One way to use this chart is as a guide to setting the length of the driven element in an array that might use a beta match. If you have antenna modeling software, you can try shorter elements relative to resonance to arrive at vaue very close to the optimum for the match. It is also likely that you might be able to interpolate between listings for the entire set of reactance values for your own installation.</p>
|
||||
<h4>Finishing the Job</h4>Let's set up a worked problem so that we can track the techniques and finish the job of creating all of the beta-match components. We may use the 3-element Yagi shown in <b>Fig. 4</b> as an example that is not far from the idealized numbers that we have used. The antenna uses 1/2" aluminum elements.
|
||||
<center>
|
||||
<img src="beta-4.gif" width="550" height="330" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<pre>
|
||||
3-Element 10-Meter Yagi Dimensions in Inches
|
||||
Design Frequency: 28.5 MHz Elements: 0.5" aluminum
|
||||
Element Length Spacing from Reflector
|
||||
Reflector 206.28 ----
|
||||
[Driver (resonant) 196.97 62.40]
|
||||
Driver (shortened) 192.20 62.40
|
||||
Director 185.33 134.54
|
||||
</pre>
|
||||
<p>The dimension table shows the original resonant driver. With this element, the feedpoint impedance was 25.7 Ohms. In the process of shortening the element for the beta match, the resistive component dropped a bit to 23.76 Ohms, with a capacitive reactance of -j24.27 Ohms. The numbers that I am citing are too precise for practical purposes, but they certainly cannot hurt our progression of calculations. Both the resistive and reactive components of the modeled feedpoint impedance fit the table nicely, being slightly below those in the column for a 25-Ohm resistive component.</p>
|
||||
<p>To calculate delta, we would take the ratio of 50 Ohms, the cable or source resistive impedance, to the value of Rs at the antenna, 23.76 Ohms. The value for delta is (again, in overly precise terms) 1.051. Delta times Rs gives us the series reactance: -j24.97 Ohms or very close to the modeled value, and certainly too close to require any more trial changes of the antenna length. The source resistance divided by delta gives us the reactance of the parallel inductive component or j47.58 Ohms.</p>
|
||||
<p>To check out work, lets convert the series impedance terms at the antenna terminals to their parallel equivalents. With the series-to-parallel value equations, we can use the modeled feedpoint series values to obtain equivalents. Rp = 48.55 Ohms. Xp = -j47.53 Ohms. To compensate for the capacitive reactance, we need an inductive reactance of j47.53 Ohms in parallel with the equivalent parallel capacitive reactance. This value is only 0.05 Ohm different from the value we calculated with the network equations. Quite frankly, I know of no one who could build the difference. However, let's use that very slight difference to calculate the net parallel reactance. We end up with a net reactance of j45,229.55 Ohms</p>
|
||||
<p>We can now apply the parallel-to-series conversion equations to the new values: Rp = 48.55 Ohms and Xp = j45229.55 Ohms. The value for the new Rs is 48.549, that is, 48.55 Ohms. The final value for Xs is j0.052 Ohms, that is, so close to zero that it does not matter.</p>
|
||||
<p>Our work can now go in two directions. We may construct a solenoid inductor or a shorted transmission-line stub to give us the required parallel inductive reactance. Let's begin with a coil. The relationship between a value of inductive reactance and the inductance that produces it at a given frequency is given in common equations.</p>
|
||||
<center>
|
||||
<img src="betae6.gif" width="202" height="47">
|
||||
</center>
|
||||
<p>Since we know the frequency (f = 28.5 MHz) and the inductive reactance (XL = j47.58 Ohms), we can use the right-side version to obtain an inductance: L = 0.2657 uH. I have purposely used too many decimal places for practical purposes, but they never hurt an example on paper.</p>
|
||||
<p>The next step is to calculate a coil that has this inductance. Remember at the beginning of these notes, I specified that we had some AWG #12 copper house wire. We can build a coil from this material. However, when installing it, we must be certain that the copper does not touch the aluminum elements. Otherwise, we shall see corrsion from electrolysis. However, we can use stainless steel hardware (including washers) both to make a secure connection and to separate the two reactive materials.</p>
|
||||
<p>There is an age-old equation for calculating the inductance of a coil from its construction, so long as the coil is a single-layer solenoid. Of course, we already know the inductance, and so we can turn the equation around to obtain the number of turns. We shall assume a coil diameter of 1" and a total length of 1". These number assure us of a reasonable Q for the coil. As well, the 1" coil length will coincide with the spacing between the connecting bolts and still leave room for a small gap between the element halves.</p>
|
||||
<center>
|
||||
<img src="betae7.gif" width="316" height="51">
|
||||
</center>
|
||||
<p>In the equations, L is the inductance in uH and n is the number of turns. The length of the coil is l, and the diameter is d, both in the same unit of measure. We are using inches. Since the values of l and d are both 1, the value of n is the square root of the inductance times 58 or the square root of 15.41. So n = 3.93 turns. Of course, wwe shall raise that to 4 turns so that the leads both head for the element terminals.</p>
|
||||
<p>Note that the coil-winding equation does not take the wire size into consideration and does not account for the inductance in the coil leads. We shall, of course, keep the leads as short as feasible. Then we may fine tune the coil inductance by spreading or squeezing the turns as needed.</p>
|
||||
<p>Next, let's replace the coil with a shorted transmission-line stub or hairpin. We shall use our AWG #12 wire for the hairpin, using the same precautions at the element connections that we applied to the coil. First, we need to determine the characteristic impedance (Zo) of the parallel line that we shall create. Let's suppose that we retain the 1" center-to-center spacing. The diameter of AWG #12 wire is 0.0808". We can calculate the Zo from another standard equation, where S is the center-to-center spacing and d is the diameter when both are in the same unit of measure.</p>
|
||||
<center>
|
||||
<img src="betae8.gif" width="121" height="40">
|
||||
</center>
|
||||
<p>The equation gives us 276 times the log of 2/.0808 or 384.64 Ohms. Again, the sense of precision in the 2 decimal places is spurious and only reminds us that we are working a paper example. We shall assume a velocity factor of 1.0 for our short line that will be self-supporting. However, we need to know how short--or long--the line must be. Since we want an inductive reactance, we shall use a shorted transmission line. Again, we have some standard equations that relate the inductive reactance (XL) to the line length (l).</p>
|
||||
<center>
|
||||
<img src="betae9.gif" width="299" height="47">
|
||||
</center>
|
||||
<p>Note that the equation requires that we use a trig function on our hand calculator. "Arctan" is simply the inverse of the "tan" function. So we divide 47.58 (XL) by 384.64 (Zo) to get 0.1237 and then take the inverse of the tangent of that number. The result is a line that is 7.052 degrees long. We are now only 2 short steps from the final length.</p>
|
||||
<p>The first step is to divide the electrical length in degrees by 360 to find the fraction of wavelength that the line happens to be: 0.0196 wavelength. That is the equation on the left.</p>
|
||||
<center>
|
||||
<img src="betae10.gif" width="183" height="40">
|
||||
</center>
|
||||
<p>The length in inches will be this fractional length times the length of a wave in inches. At 28.5 MHz, a wavelength is 414.135". So our AWG #12 1"-wide stub will be 8.112" long. One reason that the excessive number of decimal places is spurious lies in the various ways in which we construct beta stubs. <b>Fig. 1</b> showed a rounded end, which is commercially common, since it is easy to create a smooth curve by bending the wire around a cylinder. However, the equation for the electrical length assumes that the shorting wire has effectively zero length and plays no role in the functioning of the stub. Even a flat end to the stub will have a small effect on the inductive reactance, and so too will the connection eyes at the element end. Like the coil, we can also spread or squeeze the stub line into submission.</p>
|
||||
<p>Nevertheless, we may test our calculation of the hairpin by adding a shorted transmission line across the terminals of the Yagi with the shortened driver. We shall specify in NEC's transmission line facility a Zo of 384.64 Ohms and a length of 8.112". (The actual TL command requires the measurement in meters. Many commercial implementations of NEC allow user entry in the same dimensions as the wire entries for the elements. The program then converts this value to 0.2060 meters before turning the core loose to perform its calculations.)</p>
|
||||
<p>Remember that when we performed our series-to-parallel-to series calculations, we ended up with an impedance of 48.55 + j0.05 Ohms. With the hairpin in place within the model, we obtain a NEC-calculated impedance of 48.55 - j0.05 Ohms. Those values are too close (j0.1 Ohm apart) to be coincidental. Indeed, you may wish to run this exercise through a number of trial value combinations just to become familiar with the process and to get a firm grasp on the relationship of the network calculations to those we performed to convert the feedpoint situation into an equivalent pure parallel circuit.</p>
|
||||
<p>Does a beta matched Yagi have an advantages or disadvantages relative to one with a resonant driver? <b>Fig. 5</b> overlays the 27.5-Ohm SWR curve of the version with a resonant driver on the 50-Ohm SWR curve of the final beta-match version using a shorted transmission-line stub. As you can see, the SWR curves are almost indistinguishable. The Yagi will cover between 28.0 and almost 28.9 MHz with under 2:1 SWR at the feedpoint. The curve will be slightly broader at the transmitter end of the coax line.</p>
|
||||
<center>
|
||||
<img src="beta-5.gif" width="572" height="330" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>Both versions of the antenna show a free-space gain of 8.11 dBi. There is a divergence in the 180-degree front-to-back ratio: 27.12 dB vs. 27.13 dB. The pattern in <b>Fig. 4</b> applies equally to both versions of the antenna.</p>
|
||||
<p>This small exercise has tried to accomplish 2 things at the same time. First, it has tried to outline the procedure for creating a beta match, including the initial determination of whether or not such a matching system is in order. We calculated the required element capacitive reactance and the required parallel inductive reactance. Then we converted the inductive reactance into a practical beta inductor and also into a beta shorted transmission-line stub.</p>
|
||||
<p>Our second goal was to understand better how a beta match does its work. We began with the standard L-network treatment, which is very handy for calculating the required reactances once we know the resistive component of the driver feedpoint. However, we supplemented that perspective by converting the circuit into an equivalent purely parallel circuit so that we could see some basic facts about matching situations of all sorts. We seek a series combination of resistance and reactance that--when converted to their parallel equivalents--yields a parallel resistive component of 50 Ohms. Then we parallel the reactance with the opposite type having the same absolute value. When reconverted back into series values, the reactance goes to zero or very close to zero. Under those conditions, the series and parallel resistive components are, for all practical purposes, the same. In the case of the beta match, we obtained a resistive impedance too close to 50 Ohms to notice any fussy numerical difference.</p>
|
||||
<p>One final caution: the Yagi that we used for our example is a very good design. However, the uniform 0.5" diameter elements are impractical In effect, we would need a combination of 5/8 and 1/2 inch elements or, for lighter duty, a combination of 1/2 and 3/8 inch elements. However, even those values, which seem so close to 1/2", will require a complete refiguring the the element lengths. As well, since the beta match results in a balanced set of terminals, use a common-mode supressor at the feedpoint. Something like a W2DU-type bead choke will work fine in the transition from the balanced terminals to the single-ended coaxial line.</p>
|
||||
<center>
|
||||
<img src="../images/colorbar.gif" align="middle" width="540" height="4">
|
||||
</center>
|
||||
<p><i>Updated 02-02-2005. © 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.</i></p>
|
||||
<center>
|
||||
<img src="../images/colorbar.gif" align="middle" width="540" height="4">
|
||||
</center>
|
||||
<p><a href="../radio.html"><img src="../images/return.gif" align="bottom" width="40" height="40"> Return to Amateur Radio Page</a></p>
|
||||
<p class="topics-footer">Topics: <a href='../topics.html#matching'>Matching</a></p><br>
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<meta name="keywords" content="Beamwidth, Quad, Yagi">
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<title>How Accurately Must We Aim a Beam?</title>
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<h1><img src="../images/yagicad.gif" align="middle" width="125" height="62" border="2" vspace="5" hspace="10">How Accurately Must We Aim a Beam?<img src="../images/w4rnlwwr.gif" align="middle" width="99" height="83" border="2" vspace="5" hspace="10"></h1>
|
||||
</center>
|
||||
<center>
|
||||
<h3>L. B. Cebik, W4RNL</h3>
|
||||
</center>
|
||||
<center>
|
||||
<img src="../images/colorbar.gif" align="middle" width="540" height="4">
|
||||
</center>
|
||||
<p>We may divide the question of beam aiming accuracy into two parts: how accurate we can be and how accurate we must be.</p>
|
||||
<h3>How Accurate Can We Be?</h3>The accuracy with which we can aim a directional beam array depends largely on two factors.
|
||||
<p><b>1. The accuracy of the beam assemblies initial alignment</b>: A beam's initial direction depends upon the heading on which we lock down the boom and mast within the rotator and upon the accuracy of the rotator's registry of that direction. Consequently, some installers go to great pains to establish the direction of true North and the departure of the beam from that bearing at the point of installation. Once these values are known, the rotator indicator is also set to this heading. Presumably, when everything is set, we lock down the system. The heading will at least be accurate when the beam is pointed in this initial direction.</p>
|
||||
<p><b>2. The accuracy of the beam heading indicator</b>, normally as registered on the rotator control box meter or translated into a computer numerical value. Regardless of the readout system, the accuracy of the heading indication depends upon the accuracy with which the indications system follows the changes of heading relative to the initial alignment.</p>
|
||||
<p>Many rotator systems use a potentiometer in the rotator motor assembly housing, along with a control voltage from the control box. A perfectly linear potentiometer with no dead region at either end of the turning circumference would provide a very accurate indication of the departure of the beam heading from the setting at initial alignment. However accurate the circuitry for translating the voltage returned by the potentiometer to the control box, the potentiometer itself has an accuracy limit, normally expressed as a potential range of error. A +/-1% error range would translate into a +/-3.6-degree range of heading error. Given the environmental stress under which such potentiometers operate--in terms both of temperature and humidity extremes and of the motor and gear operation--it is dubious that even a 1% potentiometer would long uphold that level of accuracy.</p>
|
||||
<p>We have presumed that the circuitry that translates the incoming voltage from the potentiometer has no error, an unsafe assumption to be sure. Taking all possible sources together, it is unlikely that a rotating system using a potentiometer in the rotator motor housing and standard sorts of indicator circuitry will be better than +/-3% accurate (and possibly less accurate). This accuracy percentage translates into a possible error of up to +/-10.8 degrees. The error is likely to be variable across the horizon in several ways. First, the error level may shift with the bearing. Second, the error level may shift with the direction of rotation, as the wiper contact changes direction. Third, the error level may shift with changes in temperature and humidity. Fourth, the error level may shift with time and the changing conditions of operation of the potentiometer--not to mention the rotator control box indicator circuitry.</p>
|
||||
<p>It is not difficult to resolve the voltage returned by the indicator potentiometer down to several decimal places. Translating this voltage into beam headings in tenths of a degree--as read out on a numeric display--is also routine. However, since the basic accuracy of the return voltage is subject to so many potential error-inducing factors, such a display would be more a satisfying illusion than a true registry of the beam's actual heading.</p>
|
||||
<p>Obtaining a more accurate readout of the beam heading would require an independent measurement system. One such system would include a self-correcting sensing and indicator system of true North or some other preset bearing. It would then measure the difference between the known or reference bearing and the beam heading, possibly taken by real or virtual measurements of the boom vs. the preset heading. Laser interception of the boom and a fixed reference bar could easily achieve 1-degree accuracy for the beam system. However, for almost all installations, the cost would be prohibitive. Likely, it would also require alignment before each use.</p>
|
||||
<p>In the end, the bulk of amateur operations will have to be satisfied with an accuracy of beam heading read-out somewhere in the +/-1 to +/-5 percent region.</p>
|
||||
<h3>How Accurate Must We Be?</h3>The answer to our second question depends upon several factors regarding the beamwidth of directional antennas. Most amateurs are familiar with the registration of antenna beamwidth in terms of half-power or -3 dB power points. Relative to the bearing of maximum forward gain, we may obtain a beamwidth in degrees by determining the bearings on either side of the maximum gain heading at which the gain decreases by 3 dB.
|
||||
<p>However, the -3dB beamwidth is too broad for those types of operation most concerned with maximum precision of beam heading readouts. No standard beamwidth currently exists to evaluate the question at hand. Therefore, solely for the sake of discussion and with absolutely no pretense of adoption, I shall create a standard. For standard voice and CW operation, a drop in gain of 0.5 dB is too small to detect. However, for some of the newer modes of operation, such a drop may mean the difference between a signal being received and not being received. A 0.5-dB change of gain represents a change of less than 0.1 S-unit as currently set into meters at 5 to 6 dB per S-unit. Although not detectable in many types of communications, let's take this unit as the hypothetical least detectable change of gain for at least some types of operation.</p>
|
||||
<p>Using such a standard will allow us to re-evaluate antenna beamwidths by yielding new values. If someone wishes to adopt a more liberal standard, the resulting beamwidths will always be greater than the +/-0.5 dB beamwidth. Hence, any conclusions reached in the following notes can be readily adjusted to the revised standard.</p>
|
||||
<p>With a standard at hand, we have only to apply it to typical antennas used by the amateur community. As a sampling, I scanned several dozen beam antennas from my collection of models to determine both the standard -3 dB and the suggested -0.5 dB beamwidths. The standard figure was produced by the modeling software--NEC-4 as commercially implemented. However, the -0.5 dB beamwidth was determined by exploring the azimuth pattern until the gain decreased by half a dB. Since pattern exploration occurs in increments, I chose as the demarcating value the heading on which the gain was at or just above -0.5 dB relative to the maximum forward gain so as not to be too liberal with the recorded beamwidth value.</p>
|
||||
<p>Using NEC models of beam antennas is convenient, since measurement of the -0.5 dB heading would be difficult, at best. Models presume certain conditions that may or may not be true of real antennas. For instance, the models of standard beam designs all produce symmetrical patterns. The symmetry of the pattern of a real antenna, especially as it increases in frequency and within the tight limits of -0.5 dB points, would be open to question. The models all use free- space gain as the reference in order to fairly compare antennas having vastly different gain values. Over real ground, the take-off angle--or elevation angle of maximum radiation--will vary slightly as we increase the gain of a directional beam, which would make comparisons a bit more difficult.</p>
|
||||
<p>The table below provides model figures for antennas of 2 types: monoband Yagis and monoband quads. Yagis are far more numerous in the listing--numerous enough not only to show progressions, but as well to show occasional slight departures from progression main lines. The collection of quads is barely large enough to form a progression, but it is sufficiently extensive to allow some comparisons with the group of Yagis.</p>
|
||||
<pre> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
||||
Representative Beams and Their Beamwidths
|
||||
|
||||
Free-Space Gain +/-3 dB B/W +/-0.5 dB B/W No. Elements
|
||||
dBi degrees degrees
|
||||
Yagis
|
||||
6.13 70.0 28 2
|
||||
6.27 68.8 20 2
|
||||
6.48 68.8 28 2
|
||||
7.12 66.6 26 3
|
||||
7.80 63.6 26 3
|
||||
8.11 62.8 24 3
|
||||
8.48 61.2 24 4
|
||||
8.90 59.8 24 4
|
||||
9.25 58.8 24 4
|
||||
9.71 55.6 22 4
|
||||
10.05 54.6 22 5
|
||||
10.21 52.6 22 5
|
||||
10.22 53.6 22 5
|
||||
10.23 53.6 22 6
|
||||
10.23 52.6 20 6
|
||||
10.28 52.4 20 5
|
||||
10.53 51.8 20 5
|
||||
11.07 49.2 20 7
|
||||
11.33 49.8 20 6
|
||||
11.89 44.6 18 8
|
||||
12.25 46.2 18 6
|
||||
12.54 42.5 18 8
|
||||
13.27 36.7 16 9
|
||||
13.33 40.2 16 8
|
||||
13.89 37.4 16 10
|
||||
14.21 35.8 14 11
|
||||
15.22 32.6 14 13
|
||||
15.68 32.0 14 14
|
||||
15.68 31.2 12 14
|
||||
16.27 29.0 12 16
|
||||
17.62 25.0 10 21
|
||||
18.59 22.6 10 26
|
||||
18.66 22.4 8 25
|
||||
19.64 20.4 8 31
|
||||
Quads
|
||||
7.06 75.0 30 2
|
||||
7.49 74.0 30 2
|
||||
8.88 65.4 26 3
|
||||
9.56 61.6 24 4
|
||||
9.74 60.2 24 3
|
||||
9.96 57.8 24 4
|
||||
10.00 57.0 24 6
|
||||
11.16 49.6 20 5
|
||||
11.34 49.2 20 5
|
||||
11.89 45.2 18 6
|
||||
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .</pre>
|
||||
<p>The table reveals several notable items about gain vs. beamwidth.</p>
|
||||
<center>
|
||||
<img src="bw-1.gif" width="425" height="340" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p><b>1. The general trend</b>: As shown in <b>Fig. 1</b>, the general trend in each group of directional beams is a narrower beamwidth for a higher gain. This fact has been known since beam antennas first appeared in the dim recesses of radio history. However, few amateurs have a due appreciation of the quantification of this phenomenon. Therefore, the table is worth reviewing. Note also that, within limits, the gain-beamwidth relationship is independent of the number of elements within an array. Rather, the gain derived from the beam is the chief determinant of both -3 dB and -0.5 dB beamwidth--or anything in between. Indeed, the exact boom length is also secondary to the beamwidth, since some nearly equal-gain arrays have differences in operating bandwidth: narrower bandwidth arrays can often achieve a peak gain on a shorter boom. Of course, excessively widening the beamwidth standard can easily let the check points encounter secondary lobes-- although not for the reasonably well-behaved patterns in <b>Fig. 1</b>--and thus empty the figure of meaning.</p>
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<p>For the most part, HF operators work with antennas with less than 12 dBi free- space gain, in other words, less than 8-9 elements on a long boom. The majority work with antennas yielding between 7 and 9 dBi free-space gain. Consequently, the available -0.5 dB beamwidth will be at least 20 degrees (+/- 10 degrees of the desired heading for most operators. Even those working with beams capable of 12+ dBi free-space gain will have a -0.5 dB beamwidth of 18 degrees.</p>
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<p>The situation differs as we increase gain, as is common in VHF and UHF operation. Beamwidths at the -0.5 dB level shrink to single digit levels, increasing the difficulty of accurately aiming an array. Indeed, independent references are often needed to precisely position a very high gain array on its target.</p>
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<p><b>2. Quad vs. Yagi beamwidth</b>: At every level sampled where the gain of a quad array could be roughly equated with the gain of a Yagi array, the quad exhibited wider -3 dB and -0.5 dB beamwidths. As shown in <b>Fig. 2</b>, the difference is most dramatic at lower gain levels: the plots show antennas with 7+ dBi free-space gain levels. The linear plot is necessary to display the two patterns clearly.</p>
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<p>As the gain of the array increases, the differential in beamwidth between Yagis and quads decreases. However, it never disappears entirely within the range of arrays sampled here for the quads. However, since no quad with a free-space gain above 12 dBi was used in the listing (since I do not have any in my collection of models), there may well be a point at which the differential disappears from view.</p>
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<p>What applies to the differential between Yagis and quads also applies to other types of arrays. It is always unwise to assume from a glance at an azimuth pattern that a directional beam has a certain order of beamwidth at any level used as the standard. Careful checking of each design is necessary to know for certain.</p>
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<p><b>3. Exceptions to the trend</b>: Within the list of directional arrays, there are a few exceptions to the general progression of gain vs. beamwidth correlations. Moreover, the progression would not graph out as a smooth curve for either the - 3dB or -0.5 dB levels. <b>Fig. 3</b> demonstrates why the curve would be irregular. Not all patterns are the smooth ovals that we associate with "perfectly designed" parasitic arrays. Although the curve on the left resembles in its main forward lobe outline the middle pattern of <b>Fig. 1</b>, the forward secondary lobes make it difficult to know for sure whether the main lobe has been distorted from the most "well-behaved" condition. Hence, ripples in the general progression are bound to occur wherever secondary forward lobes are present.</p>
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<p>As the right side of <b>Fig. 3</b> demonstrates, even when the secondary lobe structure seems invisible, it may exert an effect on the beamwidth. For equal-gain antennas, the "bullet" pattern to the right has a narrower beamwidth by noticeable amounts at both levels of interest in these notes. The widening of the pattern as the bearings approach the side nulls is an indication of secondary lobe formation.</p>
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<p>In fact, changes in the azimuth pattern that can affect the beamwidth may occur across the operating bandwidth of an antenna, even though gain changes are small to negligible. For well-designed arrays with limited operating bandwidths, such differences are likely to be only of academic interest. However, they do exist and may play a role in performance wherever designs are pressed beyond their limits.</p>
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<h3>Some Tentative Conclusions</h3>
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<p>The second portion of these notes was designed to graphically and tabularly demonstrate the relationship between gain and beamwidth for a span of array gains that cover most amateur operations. However, the basic question for the second part of these notes was how accurate our beam aiming must be to achieve the strongest signals between locations. If we combine the notes for the two portions of this preliminary investigation, some tentative answers emerge.</p>
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<p>1. For lower gain, wide beamwidth arrays--typical of most (but not all) HF operations--excessive concern with the precise heading of a beam seems to be beyond the level of necessity. The potential errors of heading indicators are offset by the very wide -0.5 dB beamwidth--in most cases, better than 20 degrees. Unless on the cusp of detectability, most changes of 5 degrees in beam heading will not yield any change in signal strength that will show up on a meter. For this class of arrays, simply point at "Europe" or "Southeast Asia" (for the U.S. operator) will be accurate enough, although a final tweaking to compensate for unquantified indicator system errors may be useful. For voice and CW operations, where the -0.5 dB standard for detectable signal strength change is too stringent, the available beamwidth before a change of beam direction would make a difference can be considerably wider.</p>
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<p>2. For very high gain system with inherently narrow beamwidths, such as those above 12 dBi free-space gain, the indicator system error and the -0.5 dB beamwidth begin to merge. For such systems, adjunct aiming accessories are certainly warranted as a matter of course for each change of beam direction. Such auxiliary systems may involve manual or automated fine aiming relative to signal strength, or they may make use of adjunct directional aids derived from precise navigation work.</p>
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<p>3. For the intermediate level system with free-space gain levels between 8 and 13 dBi, we have a situation that is difficult to define in terms of operational actions. When used for casual or slow-rate operations, manual system tweaking based on signal strength of the target station (and possibly QRM from other competing stations) often suffices. However, for high-speed and competitive operations with definite target stations, even an operating assistant may not be able to keep pace with the need for adjusting headings to be as precise as necessary for the highest efficiency. It is in these kinds of cases that we find appropriate concern with precision installation settings, indicator system error levels, and (automated) adjunct heading adjustments.</p>
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<p>However, system error potential and antenna beamwidth alone--even when all normal variables are taken into account--do not tell the full story. To these factors we must add the propagation conditions and paths that exist at any given moment of operation. This final collection of variables can sometimes (but certainly not always or even most of the time) void the best results of the most precise system that does not rely on careful listening and re-aiming for the strongest signal.</p>
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<p>In the end, each operator must make his or her own decisions on how much effort to place into each phase of the problem of aiming a directional array. These notes--especially the beam data--are designed to help with, but not substitute for, the process through which each station operator must go in directing his or her array.</p>
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<p><i>Updated 01-22-01. © 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.</i></p>
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<p><a href="../radio.html"><img src="../images/return.gif" align="bottom" width="40" height="40"> Return to Amateur Radio Page</a></p>
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<p class="topics-footer">Topics: <a href='../topics.html#beamwidth'>Beamwidth</a>, <a href='../topics.html#quad'>Quad</a>, <a href='../topics.html#yagi'>Yagi</a></p><br>
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<h1><img src="../images/coax.gif" align="middle" width="155" height="68" border="2" vspace="5" hspace="9"> A Common-Mode Current Picture Show <img src="../images/w4rnlwwr.gif" align="middle" width="99" height="83" border="2" vspace="5" hspace="10"></h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>Common-mode currents on transmission lines come in two varieties: good ones and bad ones. Of course, if we do not know what a common mode current is, then we would have a difficult time telling the two apart.</p>
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<p>On a normal transmission-line, free of anything other than the energy that we wish to transfer from one end to the other, we find on the 2 conductors (of a 2-conductor transmission line) currents of equal magnitude and opposite phase angle. The field between the conductors ideally prevents the line from radiating any energy. However, if we introduce energy onto the lines that has a common magnitude and phase angle on both lines, then the transmission lines becomes a simple antenna wire that happens to be composed of two wires. Many amateurs have pressed old lengths of 300-Ohm TV ribbon cable into antenna use by joining the wires at each end and using the cables as a single wire. Under these conditions, the wires have only antenna currents. We might also call them radiation currents. Finally, we can call them common-mode currents, since the two wires have a common current magnitude and phase angle.</p>
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<p>Now suppose that we could combine transmission-line currents and antenna/radiation/common-mode currents on the same pair of wires. We do so every time we construct or use a folded dipole or a folded unipole (monopole). In <a href="../wire/fdpl.html">"Unfolding the Story of the Folded Dipole"</a>, I noted that a folded dipole is also a transmission line, and I showed a method for sorting the radiation from the transmission-line currents. The transmission-line currents turned out to have a phase angle of just about 90 degrees, indicating a non-power-consuming current. Hence, the energy that we supply to the folded dipole becomes radiation just as efficiently as similar energy supplied to a standard dipole. (An identical relationship exists between the two types of current in a folded monopole and a standard or 1-wire monopole.)</p>
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<p>Note that we want to call the non-transmission-line currents radiation currents when we have a devise like a folded dipole, where our main interest is in the radiation. We tend to call those very same currents common-mode currents when we have a transmission line that we do not wish to radiate. There is another reason for the name change. Common-mode currents exist all along the transmission line (until we attenuate them sufficiently). The end of the line that enters the house and the operating room radiates these currents into other nearby wires and devices, including the transmitting equipment. These currents can cause some metering circuits, such as the one measuring SWR, to produce erroneous readings. It does not take much current to create problems in solid-state circuitry. We may experience the RF energy on sharp corners of equipment cabinets in the form of small surprising shocks. This "bite" is only a small version of what we might experience if we were to touch the antenna wire itself while transmitting. For all of these reasons, common-mode currents tend to represent the bad side of radiation currents. Nevertheless, the two are in principle the very same when contrasted to transmission-line currents.</p>
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<p>Every transmission line has the potential for carrying common mode currents. Common-mode currents occur whenever we have an imbalance between either the magnitude or the phase of the currents along a transmission lines, where we measure such an imbalance at any facing points along the line. The two most common sources of common-mode currents in lines on which we do not want them are an imbalance at the antenna feedpoint and induced currents from the antenna field. Parallel transmission lines show imbalances whenever we feed a wire antenna away from a point of maximum current. This point is normally the center point on a half wavelength antenna. If we move the feedpoint on this wire off-center or to the end, we shall find a current imbalance and resulting common-mode currents (and what we sometimes call feedline radiation). On a parallel transmission line, we can induce common-mode currents by improper routing (that is, routing the line at other than a 90-degree angle to the antenna element). We can sometimes create them by running the line too close to a metallic object, like a down spout. If we have different coupling levels between each wire and the metallic object, an imbalance may result, and so too may common-mode currents.</p>
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<p>In a parallel transmission line, we rarely can easily sort the transmission-line from the common-mode currents. (In fact, few amateurs are equipped to even measure the currents on a parallel transmission line.) Even when we use NEC or MININEC to model a transmission line as a set of wires, the program reports a single set of currents, one for each wire at any facing segment or pulse pair. However, in a future episode of my continuing series on antenna modeling (#123), I shall describe the necessary steps to set up a simple spreadsheet or similar calculating aid to do the sorting.</p>
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<p>Coaxial cables are another matter when it comes to sorting transmission-line from common mode currents. Since introduced by Walt Maxwell, W2DU, in the early 1980s, we have seen pictures like <b>Fig. 1</b> in various handbooks. The sketch portrays the junction of a common dipole or other center-fed wire antenna with a coaxial cable feedline.</p>
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<p>The sketch shows a cutaway view of the coax so that we have a braid line on each side of the center conductor. Between facing surfaces of the coaxial cable, we have transmission-line currents, as suggested by the arrows that have opposite directions. Since the currents are RF (a form of AC), they change direction periodically, so any arrows indicate the directions at one instant of time. At that instant, the currents in the antenna wire have a direction in common with the conductors of the cable, since the cable and its source are in series with the antenna feedpoint.</p>
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<p>The sketch shows, on the right side for convenience (but actually extending all around the braid) an extra arrow having the same direction as the arrow in the antenna wire connected to the braid. The idea that the sketch portrays is that the coax braid forms a second path for antenna currents, otherwise called common-mode currents. Those new to antennas and feedlines may ask how this can be so. Others may ask whether such currents are always a problem.</p>
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<p><b>Fig. 2</b> may help us answer the first of the two questions. Within the coaxial cable, we have two surfaces of concern: the outer surface of the center conductor and the inner surface of the braid or outer conductor. We are concerned about surfaces due to skin effect, which holds the current activity very close to the conductor surface. The indicated field allows the cable to act as a transmission line, since the current magnitudes are the same on any facing points along the way, but the phase angles are 180 degrees apart.</p>
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<p>Any common-mode or radiation currents have the same phase angle, regardless of how they originate. The same phenomenon--skin effect--holds common-mode current activity on the outermost surface of the conductor, in this case, the outer surface of the braid. (The intervening dielectric or insulation in the cable does not disrupt this process. In fact, the fields that force RF to the near-surface area make any interior region a poor conductor for the RF of the common-mode current. Hence, we use tubes for beam elements rather than solid rods. The same effect also allows stranded wire to serve with equal effectiveness as solid wire in antennas.) Essentially, the near-surface RF currents on the coax braid are separated from the transmission-line currents on the inner braid surface (and the center conductor) by the material between those surfaces. In effect, when we use a coaxial cable feedline, common-mode and transmission-line currents are self-separating.</p>
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<p><b>Getting Ready for a Picture Show</b></p>
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<p>The second question wondered whether common-mode currents are always a problem. Some amateurs have never used an attenuator for them and still not experienced any problems. Others have had major problems. We need to see if we can find some of the probable causes, at least in principle. To provide some preliminary glimpses into the world of common-mode currents, we can either measure the currents on many dozens of antennas or we might try to construct a model of the common-mode situation.</p>
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<p>In episode 100 of the antenna modeling series (<a href="../amod/amod100.html">"The Dipole and the Coax"</a>), I pointed out some of the difficulties in constructing a completely adequate model of a simple center-fed dipole antenna connected to a coaxial cable. The problems are especially acute using NEC, since we must construct the coaxial cable wire a a single wire having the properties of the coaxial cable's outer-surface. Even then we shall not have captured the coax completely, since coax normally has an insulating outer jacket. In addition, the coaxial-cable diameter is normally different from the diameter of the regular antenna element wire, and NEC (both -2 and -4) has accuracy difficulties when we have angular junctions of wires having dissimilar diameters. Finally, the source will not be at the exact point where the antenna wire and the simulated coax wire form a junction. The source is on the source segment, but the junction must occur at the end of the source segment. Moreover, NEC is most accurate when the length and diameter of the source segment is the same as the length and diameter of the segments adjacent to the source segment. If we connect the coax wire to the end of the source segment, we violate this condition, but if we add a segment on each side of the source segment before creating the junction of antenna wire and coax wire, we increase the problem of displacement between the source and the junction of interest.</p>
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<p>By comparing models from both NEC and MININEC systems, I reached the conclusion that models may be good enough to show the common-mode phenomenon, but we cannot fully trust their detailed data reports in the absence of a large (and largely unconducted) set of laboratory and/or field measurements. Since our goal is not to evaluate a specific situation involving an antenna element with a certain diameter and a coaxial cable with a certain diameter, we can minimize most of the problems to an acceptable level.</p>
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<p><b>Fig. 3</b> outlines the conditions of the models that we shall use in our picture show. We shall create a resonant dipole (within +/- j1 Ohm) at 29.97925 MHz, that is, exactly at 10 meters. The wire will be lossless (or perfect) and will have a 5-mm (0.1968") diameter. To avoid angular junctions of wires with dissimilar diameters, the simulated coax wire will also be 5-mm in diameter. (The selected diameter falls between the outer-braid diameter of RG-174 and RG-58. Hence, the set-up may not be normal in the everyday sense of amateur installations, but it falls within the range of possible installations.)</p>
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<p>Constructing the antenna will require 3 wires. The first wire has 51 segments and extends beyond the exact center point by half the length of a single segment. (A segment turns out to be 0.047624 m long, about 10 times longer than the wire diameter to ensure good calculating accuracy in NEC.) The source goes on Segment 51 of Wire 1. Hence, it is at the exact element center. Wire 2 extends from the end of wire 1 to the outer end of the element and uses 50 segments. Hence, all segments in the antenna wire have the same length. The coax wire (wire 3) extends downward from the junction of wires 1 and 2. It length will be variable, depending on our demonstration needs, but it will be segmented so that the length of a segment is as close as possible to the length of a segment in the element proper.</p>
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<p>One possible way to model a dipole and its coaxial cable is to also model a transmission line using the TL facility within NEC. We then place the source on the wire that terminates the feedline. However, we do not know in advance what characteristic impedance to use for the line. Moreover, we have a choice of velocity factor values. We shall return to the affects of a transmission line on the situation before we close. For now, we may simply place the source at the position indicated in <b>Fig. 3</b>.</p>
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<p>Unlike our normal concerns for feedlines, which ordinarily require us to pay close attention to the velocity factor, we need only concern ourselves with the physical length of the simulated coax wire in the model. With respect to common-mode or radiation currents, the electrical length of the coax wire is a function of 3 factors: the physical length, the diameter, and the small adjustment for the outer insulating weather jacket. The antenna velocity factor of insulated wire tends to run between 0.95 and 0.98. Since we need not be ultra-precise in selecting coax wire lengths for these demonstrations, we can ignore both the antenna velocity factor and the effects of diameter without losing track of the major shifts in performance.</p>
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<p>In fact, using EZNEC software and NEC-4, we can directly vary the lengths of the coax simulation wire in 1/8 wavelength increments. At the test frequency, exactly 10 m per wavelength, we can always determine the length of any line in meters by multiplying the length in wavelengths by 10. As well, EZNEC's graphic facilities allow us to produce some simply-to-read illustrations suited to these exercises.</p>
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<p>The exercises will consists of paired tables and graphics. Hence, it will amount to a picture show of sorts. Indeed, the pictures may be more significant than some of the precise data, although we shall pause now and again to comment on where the data is adequate and where it may be more questionable.</p>
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<p><b>Exercise 1: Free Space</b></p>
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<p>Let's begin our examination of potential common-mode currents by placing the dipole in free space. Although this environment is not realistic for the average amateur installation, it will provide us with one advantage. We may record an average gain test (AGT) score for each model in the series. We shall begin with a zero-length coax wire, that is, a simple dipole with no appendage. Then we may add the coax wire dangling "below" the dipole to approximate an ideal right-angle position. (The term "below" is relative only to the Z-axis of the coordinate system. Free-space, of course, has no inherent above or below.)</p>
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<p>For each length of coax wire, we shall record several items of data. The feedpoint impedance will be useful to know, as well the maximum gain and the AGT score. In addition, we shall record the relative current magnitude and phase angle on selected segments. The source segment will always have a value of 1.0 at 0 degrees. Our interest will lie in the adjacent segments. On one side, we have a single segment (wire 1). On the other side, we have two segments that join (wires 2 and 3). (The pure dipole, of course, lacks a wire 3.) These current values may seem tedious to gather, but they will turn out to be informative.</p>
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<p>The results of our expedition appear in <b>Table 1</b>.</p>
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<p>Although the text omits quotation marks, all tables use them around the expression "coax wire" to ensure that we do not forget that the wire is a stand-in for a length of coax that serves as both the feedline and the source of added radiation currents. Within the free-space set-up, the lengths of coax used are convenient measures, even if they do not represent lengths we might encounter in an amateur installation. In fact, the free end of the coax wire might be an actual wire end or it might be the point at which a longer coaxial cable is subjected to one or another type of attenuator so that the remaining coaxial cable length is effectively isolated from the length under test.</p>
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<p>Within the data, two coax-wire lengths should attract our attention: 1/4 wavelength and 3/4 wavelength. Of the sampled lengths, only these two show oddly aberrant feedpoint impedance values, lower current magnitude on wire 2, elevated current magnitude on wire 3, less than ideal AGT values, and lower maximum gain levels. The AGT scores are of special note, since they represent a modeling limitation. When wire 3 shows relatively insignificant current levels (less than 0.1 of the maximum value), the AGT score indicates a reliable model requiring no corrective for either the gain value or the reported feedpoint impedance. However, when wire-3 current is significant, the AGT value departs from the ideal by a significant amount, suggesting that the model's numerical data are not reliable. Even if we use the AGT value to correct the gain figure, we find lower than normal maximum gain. Under the prescribed conditions, common-mode currents are sufficient to significantly alter the dipole radiation pattern relative to our expectations.</p>
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<p>One area in which the AGT does not disturb the results is in the calculation of the net current magnitude and phase angle on the right side (wires 2 and 3) of the dipole. The vector sum of the currents on that side should equal the reported current on the left side of the source. We may expect a small and systematic difference, since the sum of the surface areas of wires 2 and 3 is double that of wire 1. However, we anticipate a general coincidence between the net current values. <b>Table 2</b> will inform us of whether or not the models achieve this end.</p>
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<p>The columns marked "wE" provide the required calculated net current values, while those marked "w1" record the current for the single-wire side of the dipole. The pure dipole, of course, shows no difference between the two. For every entry in which we have a coax wire, we find a consistent 0.07 to 0.08 degree difference in the phase angle for all entries. For all entries with an AGT value of 1.000, the wire-1 value for the phase angle is slightly higher than for the simple dipole. I have bold-faced the values for coax wire lengths of 1/4 and 3/4 wavelength, because the phase angle values are out of line with the other values in the list.</p>
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<p>Despite the questions of data reliability, the general phenomenon of common-mode currents shows well in the table. We may also represent the data in more graphical form. For each tabular entry, we may generate E-plane and H-plane patterns for the free-space model. In addition, we may also represent the pattern of current magnitude along the antenna elements and the coax wire. <b>Fig. 4</b>, <b>Fig. 5</b>, and <b>Fig. 6</b> provide a catalog of the current and radiation patterns applicable to the entries in <b>Table 1</b>.</p>
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<p>Let's first look at the current magnitude portions of each figure. For all coax-wire lengths except 1/4 and 3/4 wavelength, the current all along the coax wire is small. I have magnified the levels so that a definitive curve shows for each coax wire length. However, the peak current is never above about 0.2 of the maximum value. Therefore, the current distribution on the antenna element wire remains virtually undisturbed by the presence of the coax wire. In contrast, with 1/4 and 3/4 wavelength coax wires, the current peak for the coax wire occurs at the junction with the antenna element. Hence, we find a significant current division between wire 2 and wire 3. The current in wire 2 (to the left in the graphical representations) is low enough that we might expect to find considerable alteration of the normal dipole free-space pattern.</p>
|
||||
<p>The E-plane and H-plane patterns (azimuth and elevation in conventionalized modeling terms) provide us with two types of data. First, the patterns show both the vertical and the horizontal components of the total field patterns. In virtually all cases, most of the horizontal component disappears beneath the line that records the total field. However, the vertical component is instructive. For example, the pure dipole with no coax wire shows no vertical component in free space. When we add the coax wire, we obtain a distinct vertical component to both patterns. For all but two of the pattern sets, the vertical component is about 20 dB or more down from the maximum gain. In addition, the indicated patterns show side nulls that are at least 20 dB or more lower in gain than the main lobes. The two aberrations are the patterns for 1/4 wavelength and 3/4 wavelength coax wires. The peak vertical components in these cases may be less than 10-dB down from maximum gain values, with strong consequences for the total field dipole radiation pattern. The patterns for lengths of 1/4 and 3/4 wavelength differ, but both equally distort what we think of as the normal dipole pattern.</p>
|
||||
<p><b>Exercise 2: 1 Wavelength above Average Ground</b></p>
|
||||
<p>Before we draw any conclusions about common-mode currents, let's repeat the exercise with a small difference. For the second run, we shall set the dipole exactly 1.01 wavelength above average ground. (The use of 1.01 wavelength as the antenna height ensures that the 1 wavelength coax wire does not touch the ground. We shall look at that option in a separate exercise.) Since we are now dealing with a lossy ground, we cannot make use of AGT score values, since those values require the removal of all resistive losses, using either a free-space or perfect-ground environment. However, we may record the reported gain value and the TO angle (take-off or elevation angle of maximum field strength). In all other ways the models remain intact. <b>Table 3</b> records the results of this exercise. Remember that the "zero-length" model is a simple dipole with no wire 3.</p>
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<p>In this exercise, coax-wire lengths of 1/4 and 3/4 wavelength again show anomalous results compared to the results for other lengths. At all other coax-wire lengths, we obtain about the same maximum gain and very similar feedpoint impedance values. Note that the impedance for the simple dipole entry shows a small reactive component because I did not change the dipole length when moving it from free space to a position above real ground. The impedances for the two sensitive coax-wire lengths are between 20 and 30 Ohms lower than for the other lengths. As well, the maximum gain is more than a dB lower. In the current-value columns, we find the same general pattern of current values on the first segments of wires 2 and 3 that we found in the free-space exercise</p>
|
||||
<p>We may translate the tabular results into the same sorts of patterns that we developed for the first exercise. We omit the current portions of the graphics, since they are virtually identical whether in free space or over real ground. However, <b>Fig. 7</b>, <b>Fig. 8</b>, and <b>Fig. 9</b> provide the elevation and azimuth patterns for each of the entries in <b>Table 3</b>.</p>
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<p>The first difference to note in the patterns relative to free-space models is that even the simple dipole shows a small vertical component over real ground. When we compare this component with the corresponding components for most of the models with coax wires, we find that the strength of the component does not change. It remains about 20 dB below the maximum gain, although the exact shape of the component changes somewhat with the length of the coax wire. In contrast, the maximum gain for the vertical components for 1/4 and 3/4 wavelength coax wires is only about 10 dB down from the maximum gain value. The consequences for the total field elevation patterns are not dramatic. However, we find distinct distortions in the azimuth patterns for coax wire lengths of 1/4 and 3/4 wavelength. These distortions do not make the antenna unusable, but when combined with the gain reduction, they do make a difference. Note that even with the pattern distortions, the TO angle for all of the patterns remains at 14 degrees elevation.</p>
|
||||
<p>At this point, we might be tempted to draw some hasty conclusions. One such conclusion is that we can avoid coaxial cable common mode currents by avoiding line lengths that are odd multiples of 1/4 wavelength. However, this conclusion would be warranted only for cases in which the coax end was either free or was isolated by some attenuation device to the degree that it acted like a free end. In most amateur installations, we do not find these conditions at work. We need to set up a model that might more adequately represent installation conditions.</p>
|
||||
<p><b>Exercise 3: Coax Wire to Ground, with and without a Ground Rod</b></p>
|
||||
<p>In most cases, the coax wire will come to the level of the ground. The route may be direct or indirect. One such indirect route is through the equipment cases to a station grounding system. I cannot represent each possible case. However we might explore some stand-in cases. For example, we might set the dipole at various heights from 3/4 wavelength up to 1-1/4 wavelength, with coax wires that just reach the ground. For this set of cases, we shall expect elevation patterns that show different TO angles, since each entry places the dipole at a different height.</p>
|
||||
<p>We also have two ways to handle the fact that the coax wire just reaches the ground. One way is to simply let the wire end at Z=0. This type of model may not be wholly reliable, since NEC typically shows an inaccurate source impedance under these conditions. So we might (in NEC-4) create a 1/4 wavelength ground rod, using the same 5-mm diameter wire that we have used in the remainder of the model.</p>
|
||||
<p><b>Table 4</b> records the NEC reports for both type of models. In addition, it also records reference values of simple dipoles (with no coax wire) at each of the heights for the other cases in this exercise.</p>
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<img src="cmp-t4.gif" width="705" height="353" border="2" vspace="5" hspace="9">
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<p>The table immediately shows some very interesting results. When the coax wire touches the ground, with or without a good ground rod, the sensitive lengths are no longer odd multiples of 1/4 wavelength. Instead, the sensitive lengths become multiples of 1/2 wavelength--in this case, a length of 1 wavelength. Our previous hasty conclusion turns out to have been far from universal.</p>
|
||||
<p>The second notable feature is the difference in the dipole performance with a 1 wavelength coax wire between having a ground rod and only touching the ground. The no-rod situation shows the greater reduction in the feedpoint impedance and the maximum gain. In comparison, the other coax-wire lengths show only modest differences between a rod and a no-rod condition. Additionally, the gain with a ground rod and a 1 wavelength wire shows the greatest reduction in maximum gain of all the cases when we compare the values with the gain values for a modeled dipole with no coax wire. If we were to carry out the exercise in even smaller increments, we might expect to see the gain differential curve show a non-linear curve to a peak difference at 1 wavelength and then a non-linear downward trend, with repetitions every half wavelength.</p>
|
||||
<p>We may sample the current magnitude and distribution curves, as in <b>Fig. 10</b>. The curves for coax-wire lengths of 0.75 and 1.25 wavelengths begin at the feedpoint junction with very low current values, and those values remain very low all along the coax wire. The samples use ground rods, which are a constant 0.25 wavelength for each case. The curves for the 1 wavelength coax wire begin at the feedpoint with high current on the coax wire. Indeed, the current magnitude is high enough so that we find unequal current curves on the two halves of the antenna element itself. The current pattern presents two significant peaks at heights of 1 wavelength and 1/2 wavelength: these maximums are strong enough to affect the overall radiation pattern.</p>
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<p>Before we turn to the patterns, let's take note of the current distribution on the ground rod. Although the ground rod is physically 1/4 wavelength, its electrical length within the selected ground medium (conductivity 0.005 S/m, permittivity 13) is longer. Hence, the current undergoes 2 peak values. (An underground dipole would not have to be very long physically to be a resonant half wavelength electrically, although the precise physical length will vary with the ground medium.)</p>
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<p><b>Fig. 11</b> translates the tabular data for the models with ground rods into elevation and azimuth patterns. Despite the half-dB gain loss in the model with a 1 wavelength coax wire, we find nothing unusual in either the elevation or the azimuth pattern. The vertical component of the patterns is only slightly greater than for the other patterns. Indeed, with the well-grounded termination of the coax wire, we find that the feedpoint impedance is not very far off the mark for a dipole at a 1 wavelength height with no coax wire at all.</p>
|
||||
<p>We may fairly ask at this point what all of the pictures might be telling us with respect to common-mode currents on the outer side of a coaxial cable' braid. We have examined two sorts of cases, one in which the coax wire had a free end and another in which the coax wire terminated at ground. Both cases have used straight coax wires projecting at a 90-degree angle from the junction with one side of the feedpoint of the dipole antenna element. Within these conditions, we discovered two different situations in which common-mode currents appeared to be significant. With a free end to the coax wire, sensitive coax lengths appeared to be odd multiples of 1/4 wavelength. With the coax terminated at ground, sensitive lengths appeared to be multiples of 1/2 wavelength.</p>
|
||||
<p>One initial impression might be that under one or the other condition, we have a good chance of not incurring common-mode currents to any significant degree. However, as we noted near the beginning of our picture show, we do not require common-mode current levels that distort the radiation pattern to experience unwanted coupling into equipment or household devices making use of solid-state circuitry. Since we do find some common-mode currents on coax braids in nearly all circumstances, the potential for problems exists under nearly all coax-wire conditions.</p>
|
||||
<p>The picture show also suffers severe restrictions. It does not cover a myriad of cable routings that are typical of almost any antenna installation. Cables may travel horizontally for considerable distances at a variety of heights ranging from underground to a considerable distance above ground. Cable routing and switching may make the calculation of the total cable length to a ground connection nearly impossible. As well, we may not be certain of the quality of the cable ground, since a good ground for lightning protection may or may not also be a good ground for RF at the operating frequency.</p>
|
||||
<p>On top of all this, we have not investigated whether non-ideal cable routing may have an affect on a cable's susceptibility to common-mode currents.</p>
|
||||
<p><b>Exercise 4: Coax Wire to Ground with a Ground Rod But Sloping Under the Antenna Element</b></p>
|
||||
<p>We have long believed that improper or non-ideal routing of a coaxial cable may increasing the potential for incurring common-mode currents due to coupling between the antenna element fields and the outer surface of the cable. (A similar situation would also induce common-mode or radiating currents on a parallel transmission line.) We can gather some idea of the level of interaction by a simple demonstration. We must note from the start, however, that the variations on the situation that we shall set up are nearly endless. So the exercise will show only that some influence does exist, but it will not go any further.</p>
|
||||
<p>Let's use a model in which the coax wire reaches the ground and terminates in a 1/4 wavelength 5-mm diameter ground rod with average ground. The coax wire will slope at a 30-degree angle directly under one of the dipole legs. When the angled portion of the coax is at the end of the antenna element, the coax will drop straight down to the ground. The total length of the coax wire is 1 wavelength. Since the sloping leg is 1/4 wavelength, the sloping end is 1/8 wavelength below the end of the dipole. With the remaining 3/4 wavelength of coax wire, the antenna height will have to be about 0.875 wavelength to satisfy the overall geometry.</p>
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<img src="cmp-12.gif" width="410" height="600" border="2" vspace="5" hspace="9">
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<p><b>The top portion of Fig. 2</b> shows the antenna model geometry and also shows the current magnitude and distribution. The lower portion of the figure shows a straight coax run with an equivalent total length of coax-wire, that is, 1 wavelength. Since the antenna heights differ, the total field elevation pattern differences do not reflect any effects of the different coax-wire routings. Any dipole at a height of 7/8 wavelength will show the large upward lobe structure.</p>
|
||||
<p>More significant is the vertical component within the pattern. The peak value for that component appears to be somewhat higher with the sloping coax wire than with the straight one. A similar condition appears on the current distribution section of the graphic. Indeed, the two current portraits seem somehow anomalous. Both raise the peak current value to as close to the same level as feasible. The sloping coax wire appears to show higher current peaks than the straight 1 wavelength coax wire, although as we discovered in the preceding exercise, a coax-wire length of 1 wavelength showed the highest coax-wire current peaks of any of the sampled lengths. In contrast to this appearance, the straight-wire model shows a reduced current magnitude on the dipole leg sharing a connection with the coax braid. However, the sloping-coax model shows a seemingly smooth curve of current magnitude across the feedpoint junction.</p>
|
||||
<p>To resolve the visual situation, we may turn to the data in <b>Table 5</b>. The table lists the modeled data for the sloping case and for two straight-coax cases, the 0.875 and the 1 wavelength coax runs. All three cases use a ground rod.</p>
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<img src="cmp-t5.gif" width="705" height="145" border="2" vspace="5" hspace="9">
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<p>The table entries for the sloping-coax model indeed show that the current magnitude of wire 2 is very close to the magnitude on the single-wire side. What the graphic did not show was the very significant difference in phase angles between the two current magnitudes. As a consequence, the current magnitude on wire 3 is about 30% higher than we find on the same segment in the case of a 1 wavelength straight coax wire. Because the graphic omitted the phase-angle data (a necessity for me to provide 2-D representations), it only appears anomalous. With the phase-angle data added, we can see that the sloping-coax situation increases the common-mode current magnitude on the coax braid. In addition, the situation in the model reduces the maximum gain of the dipole at a 7/8 wavelength height by about 0.6 dB relative to the straight coax dipole at the same height.</p>
|
||||
<p>The single demonstration does not provide a comprehensive picture of how non-ideal cable routing may affect both antenna performance and the level of common-mode currents. However, it does tell us that non-ideal routing can alter antenna performance and provide increased common-mode current levels. We have already seen that we may have difficulty in determining whether we have a sensitive coax length due to problems in measuring the length of the coax between the antenna and ground and uncertainties about the quality of the ground. The safest procedure is to install some form of attenuation for common-mode currents as a standard operating procedure with all coax-fed antennas.</p>
|
||||
<p><b>Exercise 5: Coax Wires with Attenuation</b></p>
|
||||
<p>Common-mode current attenuation generally comes in two forms: the 1:1 balun or transmission-line transformer and the ferrite-bead choke. Simulating a 1:1 balun is beyond the capabilities of the EZNEC implementation of NEC-4. However, the W2DU-type ferrite bead choke is quite simple. It provides an inductive reactance on the outside of the coax braid without disturbing what occurs inside the coaxial cable. Hence, it does not affect transmission-line currents, but it attenuates common-mode currents.</p>
|
||||
<p>With a properly constructed bead balun for the frequency used, it is possible to obtain at least j750 Ohms of inductive reactance. (See the current <i>ARRL Antenna Book</i>, p. 26-25. As well, see Bill Sabin, W0IYH, "Exploring the 1:1 Current (Choke) Balun," <i>QEX</i>, July, 1997, as well as Chapter 21 of Walt Maxwell, W2DU, <i>Reflections II</i>, for background on the ferrite bead choke.) Therefore, we shall use a simple load on the coax wire consisting of j750 Ohms.</p>
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<p>In the first part of our new exercise, we shall use the model of a coax wire with a free end. We found that when the coax wire was 3/4 wavelength long, it showed a high sensitivity to common mode currents. We may install the load (that is, the bead choke simulation) anywhere along the wire. Let's place it at the feedpoint and then at 0.25 wavelength intervals, including the far or open end of the coax wire. <b>Table 6</b> shows the numerical results of the exercise.</p>
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<p>The table includes values for the case without the bead choke for comparison. When we place the choke 0.25 and 0.75 wavelengths away from the feedpoint, it has virtually no effect on the common mode current values. However, when we place it either at the feedpoint or at a point 1/2 wavelength distant from the feedpoint, it reduces the current on the coax wire to nearly equally low levels. To see this situation graphically, we may examine <b>Fig. 13</b>.</p>
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<p>The current distribution graphics for no-load and for loads at 1/4 and 3/4 wavelength positions are virtually indistinguishable from each other, except for the square marking the load location. When we place the load at the feedpoint, common mode current drops to near-zero. When the load is 1/2 wavelength from the feedpoint, the current level between the feedpoint and the load is not quite zero. But, it is very much attenuated compared to the no-load or the misplaced load conditions.</p>
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<p>We may repeat the experiment using the model with a ground rod and a straight 1 wavelength coax wire. This model showed the greatest sensitivity to common mode currents in its series of samples. Once more, we shall use a j750-Ohm load and move it from a position at the feedpoint along the line in 0.25 wavelength increments. Hence, the last position will be just about at ground level. <b>Table 7</b> gives us the outcome of the modeling exercise.</p>
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<p>With the load positioned at the 0.25 and 0.75 wavelength points along the coax wire, we obtain little or no benefit. The impedance reflects the value for a no-load (that is, a no-attenuator) condition. However, when we place the load at the feedpoint or at multiples of 1/2 wavelength from the feedpoint, the antenna impedance and gain values return to the levels that we expect of a dipole at 1 wavelength above average ground with no coax wire in the model. <b>Fig. 14</b> provides a set of current distribution portraits that reflect these conditions.</p>
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<p>The portraits reveal the same general patterns as the portraits in <b>Fig. 13</b>. When the choke is at the feedpoint or at a distance that is a multiple of 1/2 wavelength from the feedpoint, we obtain maximum attenuation of common-mode currents. When the choke is at a distance, the common-mode currents do not go to zero between the feedpoint and the load, but they are very much attenuated relative to their value with a no-load or a misplaced-load condition.</p>
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<p>The feedpoint is the most common location for the installation of a common-mode current attenuator. In fact, commercially made devices often combine the attenuator and a dipole center insulator assembly in one. In terms of reducing common-mode currents on coax braids to a minimum, the feedpoint position is universally the best. However, the demonstration has only covered cases of straight-line coax runs. It has not covered the case of a sloping or otherwise erratic coax run. For this reason, it is not unwise to install a second bead choke at the point where the coax enters the house or equipment room. A good earth ground running from the braid on the equipment side of the attenuator is an additional method to maximize the supplemental attenuation.</p>
|
||||
<p><b>Exercise 6: Coax Wires and Transmission Lines</b></p>
|
||||
<p>One of the markers that we have used to initially spot a coax-wire length that was sensitive to common-mode currents is the feedpoint impedance. In virtually all cases where common-mode currents have been high, the feedpoint impedance has dropped significantly from a more normal dipole value. We have been using the feedpoint impedance as modeled at the center segment of the dipole element. The potentially misleading idea that this situation may suggest is that a normal dipole impedance at the equipment end of the coaxial cable indicates relative freedom from common mode currents, while a lower-than-normal impedance may indicate their presence.</p>
|
||||
<p>Unfortunately, this idea does not necessarily play out in real situations. The electrical length of the coax outer braid is a function of its physical length and its diameter. The outer jacket may contribute an antenna-level velocity factor that might range from 0.95-0.98, depending on the jacket material and its thickness.</p>
|
||||
<p>In contrast, the transmission line currents and the impedance transformation that may occur between the center conductor and the inside of the coax braid have very different velocity factors. For 50- and 70-Ohm cables, we find two common velocity factor values: about 0.66 for solid dielectrics and about 0.78 for foam dielectrics. As a result, a given physical length of a coaxial conductor will have an electrical length in practice that is considerable loger than the electrical length of the outer braid for common-mode currents.</p>
|
||||
<p><b>Table 8</b> provides a simple example using the dipole and a 3/4 wavelength coax wire with a free end. To the model we can add a 3/4 wavelength 70-Ohm transmission line, with a short terminating wire on which we place the source. The table shows a reference line with the values for the same model but with the source located on the center of the dipole. If we arbitrarily assign the transmission line a velocity factor of 1.0, we discover that the line transforms the 52-Ohm impedance up to about 94 Ohms.</p>
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<p>With realistic values for the velocity factor, we obtain different values for the source impedance, values that are not likely in isolation to give us clues to the common-mode current situation. In an actual situation, we are faced with perhaps two alternatives. One direction in which we can go is to devise means by which to measure the level of common-mode currents. If we lack proper measuring equipment, we might rely upon other signs, such as the presence of interference with communications equipment functions of other household devices.</p>
|
||||
<p>The other alternative is to install common-mode current attenuation devices, such as bead chokes, during the installation of the antenna system. The antenna system includes both the antenna and the transmission line. As we noted earlier, the most use position for a common-mode attenuator is the junction of the coaxial cable and the antenna element. However, a secondary attenuator at the house or equipment-room entry point can provide additional protection against potentially harmful effects from even low levels of common-mode currents.</p>
|
||||
<p><b>Conclusion</b></p>
|
||||
<p>Our common-mode-current picture show has reached its end, even though it has surveyed only a few of the possible combinations of antennas and feedlines that might make up an amateur installation. Along the way, we discovered that the level of common-mode currents depends upon several factors, including the length of the coaxial cable and the nature of its termination. Cables with free ends have sensitive lengths that differ from cables that terminate at the ground. Cables that do not leave the antenna feedpoint at right angles tend to be more susceptible to common-mode currents.</p>
|
||||
<p>Although the tabular data is indicative of actual conditions, we should remember that the models used are subject to some restrictions. We had to use a single wire diameter for both the antenna element and the coax wire to minimize model inadequacies. Even with this departure from the reality of most antenna installations, we still encountered non-ideal AGT values when common-mode currents were relatively high. The chief cause for the deviation from an ideal AGT value is the position of the junction between the coax wire and the element wire immediately adjacent to the source segment. Such models fail to implement the conditions under which NEC calculates most accurately the current on the source segment. The problem does not show up when common-mode currents are very low and most of the source current appears on the antenna element with almost none on the coax wire.</p>
|
||||
<p>Notwithstanding the limitations that the situation imposes on the numerical data, the general portrait of common-mode currents shows generally accurate patterns of currents and the conditions that tend to either promote or suppress their development. However, in most amateur installations, we have no accurate method for determining the exact electrical length of our coaxial cable as it proceeds from the antenna to the equipment and then via an earth ground line to the actual ground. For the average amateur antenna installation, the safest strategy relative to common-mode currents is prevention. Adding a common-mode current attenuator to the system should be a standard part of any installation involving a coax-fed antenna.</p>
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<p><i>Updated 08-21-2006. © 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.</i></p>
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<h1><img src="../images/elect.gif" align="middle" width="145" height="75" border="2" vspace="5" hspace="9">ATUs, Delta, and Losses<img src="../images/w4rnlwwr.gif" align="middle" width="99" height="83" border="2" vspace="5" hspace="10"></h1>
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<h3>L. B. Cebik, W4RNL</h3>
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Losses from the insertion of antenna tuning units or transmatches between transmitters and transmission lines have bothered numerous folks for some time. There are two questions: how much loss can I expect? and how do I minimize the loss? Both are good questions with good answers. Unfortunately, they are not super simple answers. But some old material combined with some new material can make the answers easier to come by. Those who like the math involved can focus on the equations, while those interested in operational matters can concentrate on the tables and the resulting rules of thumb to minimize losses.
|
||||
<p>Most ATUs used today are L-C networks. The most common configuration is the C-L-C Tee network. However, L-C-L Tees and C- L-C PIs are in use, as are SPC and Ultimate Transmatch designs. Loss figures can be calculated for all such networks.</p>
|
||||
<p><b>Delta</b>: The old material that contributes to determining ATU losses appears in Terman's <i>Radio Engineers' Handbook</i> (McGraw-Hill, 1943), pages 210-215. Based predominantly on work done in the 1930s by W.I. Everitt, Terman's analysis of classic impedance-matching networks is still referenced by current handbooks. The relevant part for considering ATU losses is the term, delta, which is a justifiably simplified measure of power dissipation in networks. For any impedance matching network, the primary power dissipation culprit is the inductance, and inductor losses can be calculated.</p>
|
||||
<p>The efficiency of L-C networks is dependent upon the impedance transformation ratio and the phase shift. PI networks and L-C-L Tees generally exhibit a large angle of phase retardation, while the C-L-C Tee circuit shows a large angle of phase advance. In contrast, simple L-networks show small angles of phase advance or retardation. Losses increase with increasing transformation ratios and tend to be larger when the phase shift is either very large or vary small. The delta-figure takes both into account. (Note, in some explanations of networks, but not in all, Terman's delta goes under the name of "working Q," "circuit Q," or "network Q.")</p>
|
||||
<p>The concept of delta has certain limitations. First, it ignores capacitor losses. This is justified for well-designed capacitors at HF and below, but capacitor losses may become significant at the boundary between HF and VHF (about 30 MHz). Second, the loss figure ignores the effects of strays, both capacitive and inductive, within a real ATU, especially one cramped inside a metal case. At 10 meters, the principal L and C in a matching circuit may be composed more of strays than of component values. Third, the technique will assume we know the coil Q in figuring ultimate efficiency and losses. On this score, we mostly guess, but a figure of 100 for good air wound coils is a conservative guess. Good powdered iron core toroids may yield more. Of course, the "suck out" effect of shorted coil turns is also ignored. Moreover, only networks, and not inductively coupled ATUs, are covered.</p>
|
||||
<p>Within these limits, the Terman-Everitt delta yields a fair estimate of loss and efficiency. As Terman notes,</p>
|
||||
<center>
|
||||
<img src="ddeltf.gif" width="368" height="74" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>In other, equally approximate terms, network efficiency = 1 - delta/Q. (Multiply this figure times 100 for a percentage value of efficiency).</p>
|
||||
<p>Now if we only knew how to calculate delta, the rest would be simple. Terman provides graphs for estimating delta. Each type of network requires its own calculation, and it can be a bit tedious to calculate delta for several combinations of component settings. However, Brian Egan, ZL1LE, has derived equations for delta- calculations and added them to his program TUNER.BAS. This versatile program is now included in the collection of programs called <a href="../radio.html#HAMCALC">HAMCALC</a>.</p>
|
||||
<p>Among the networks for which ZL1LE includes calculations are PIs, C-L-C Tees, SPCs, Ultimates, and L-C-L Tees. By running the program through a selection of values, one can learn the trends of tuning that will minimize losses. Operationally, that is the best we can hope for with a fixed set of impedance conditions presented by the transmission line from the antenna and with ATU components that we cannot modify, that is, with a coil whose Q we cannot improve. From the general relationship of power lost to power delivered, it is clear that we always want to tune for minimum delta.</p>
|
||||
<p>For CLC T networks, a more complete analysis of losses (or efficiency) appeared in July, 1997, <i>QEX</i>, by Kevin Schmidt, W9CF. In fact, ZL1LE has included Schmidt calculations for this network in his suite of programs, so that you can not only calculate network values for a given match, but also calculate the efficiency of the coupling. Delta turns out to be a good approximation of Schmidt losses for networks with coil Qs above 50.</p>
|
||||
<p>As a preliminary sample, consider a CLC T network of the kind so common in commercial tuners. With an input impedance of 50 ohms resistive and a sample load impedance of 400 + j100 ohms (SWR=8.53:1) at 3.5 MHz, calculations give a maximum transmitter end C1 of 343.7 pF. Using various values of C1 (depending on the components in the tuner we have), we can find the following table of values. Efficiencies are calculated by Schmidt means and delta values by Terman means. A more detailed analysis of CLC networks appears below, along with Figure 2 to make clearer the components C1, C2, and L.</p>
|
||||
<pre>C1 343 pF 250 150
|
||||
L (Q=250) 6.58 microH 7.16 10.77
|
||||
C2 305pF 99 51
|
||||
Efficiency 98.8% 98.1 96.8
|
||||
|
||||
C1 343 250 150
|
||||
L (Q=100) 6.46 7.15 10.82
|
||||
C2 263 96 50
|
||||
Efficiency 97.1 95.4 92.0
|
||||
|
||||
Delta 2.7 4.5 8.0</pre>
|
||||
<p>Note that efficiency is always higher for the coil with the higher Q and also for the lowest value of delta for any given coil Q. Using delta alone to calculate the efficiencies would have given us 98.9 for the Q = 250 case and 97.3 for the Q = 100 case in the first column.</p>
|
||||
<p>If we work the same problem for 1.8 MHz, we find that the maximum capacitance for C1 is 668 pF. Let's arbitrarily choose 645 pF for this capacitor, but also work the cases where C1 = 250 pf and 150 pF.</p>
|
||||
<pre>C1 645 pF 250 150
|
||||
L (Q=250) 12.48 microH 24.11 39.35
|
||||
C2 418 pF 84 50
|
||||
Efficiency 98.7% 96.2 93.7
|
||||
|
||||
C1 645 250 150
|
||||
L (Q=100) 12.36 24.26 39.83
|
||||
C2 393 82 47
|
||||
Efficiency 96.9 90.6 84.4
|
||||
|
||||
Delta 3.0 9.4 15.9</pre>
|
||||
<p>Both of the earlier trends are repeated: efficiency is highest with the highest coil Q and the lowest value of delta. Delta remains a very close approximation of losses (or, in Schmidt terms, efficiency). Note, however, the range of values needed to achieve the highest levels of efficiency.</p>
|
||||
<p>I ran TUNER.BAS through a number of exercises using different types of networks, and here are some results for some configurations. I chose 10.1 MHz, because I was working on a network problem for that band. To standardize the tests, I chose the following load conditions for each network: 150 ± j0 ohms, 150 + 100 ohms, and 150 - j100 ohms. Looking at a straight 3:1 SWR purely resistive load is a good beginning point, while exploring the network with both capacitive and inductive reactances indicates limitations of the network with certain types of loads. This test is limited, since a higher delta at a 3:1 impedance transformation ratio is not indicative of the network's overall capability. Some networks are more efficient with lower transformation ratios, while others grow more efficient as the transformation ratio increases.</p>
|
||||
<p><b>PIs:</b> This familiar low-pass network configuration appears in Figure 1.</p>
|
||||
<center>
|
||||
<img src="dpis.gif" width="445" height="222" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>For those interested, the Egan equation for delta for a PI network (where omega = 2piF wherever it occurs) is the following:</p>
|
||||
<center>
|
||||
<img src="dpif.gif" width="300" height="78" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>Here is a small table of results for 10.1 MHz:<br></p>
|
||||
<pre>Input = 50 ohms Output = 150 ± j0 ohms Maximum permitted L = 1.365 µH
|
||||
L 1.364 C1 172 C2 179 delta 2.2
|
||||
or 1.364 192 185 2.5
|
||||
L 1.2 C1 36 C2 150 delta 1.5
|
||||
L 1.15 C1 15 C2 149 delta 1.5
|
||||
|
||||
Input = 50 ohms Output = 150 + j100 ohms Maximum permitted L = 1.640 µH
|
||||
L 1.639 C1 140 C2 197 delta 2.5
|
||||
L 1.6 C1 84 C2 187 delta 2.2
|
||||
L 1.5 C1 26 C2 182 delta 1.9
|
||||
|
||||
Input = 50 ohms Output = 150 - j100 ohms Maximum permitted L = 1.640 µH
|
||||
L 1.639 C1 140 C2 100 delta 2.5
|
||||
L 1.6 C1 84 C2 90 delta 2.2
|
||||
L 1.5 C1 26 C2 85 delta 1.9</pre>
|
||||
<p>In all three cases, the lists stop where one of the values approaches the minimum usually obtainable with ordinary components. In this case, a value for C1 lower than about 20 pF is not normal for most ATUs. In many instances, as illustrated by the very first entry, two solutions to the matching problem are possible. After the first entry, only the more promising set of values is shown. PI networks are, within limits, equally capable of handling capacitive and inductive reactances in the load, with the limits of both being related to the range of C2. Notice that efficiencies of 98% are possible within the limits of the method. As Terman notes, increasing the ratio of the impedances to be matched increases the value of delta and decreases efficiency. For example, when the resistive component of the output impedance is 300 ohms, with or without a reactive load, the lowest value of delta is about 2.3. To achieve the highest efficiency possible for any given matching situation, we must use the following rule of thumb:</p>
|
||||
<center>
|
||||
<i>PI networks: Use the lowest value of L that still permits C1 to be effective.</i>
|
||||
</center>
|
||||
<p><b>C-L-C Tees:</b> This most common ATU network configuration appears in Figure 2.</p>
|
||||
<center>
|
||||
<img src="dclcs.gif" width="446" height="323" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>To determine delta, use the Egan equation</p>
|
||||
<center>
|
||||
<img src="dclcf.gif" width="246" height="83" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<pre>Input = 50 ohms Output = 150 ± j0 ohms Maximum permitted C1 = 222.9 pF
|
||||
C1 222.8 C2 6027 L 1.65 delta 1.4
|
||||
C1 222 C2 1468 L 1.59 delta 1.5
|
||||
C1 220 C2 992 L 1.56 delta 1.5
|
||||
C1 210 C2 362 L 1.43 delta 1.8
|
||||
C1 200 C2 262 L 1.39 delta 2.0</pre>
|
||||
<p>Note that although a very low value of delta is theoretically possible, a realistic value of 2.0 or higher is to be expected with common components used in ATUs. As in the case of the PI network, the C-L-C Tee yields higher values of delta as the output impedance is increased. At 6:1 ratio of output-to-input impedance, delta does not go below 2.1 with normal ATU components.</p>
|
||||
<pre>Input = 50 ohms Output = 150 + j100 ohms Maximum permitted C1 = 222.9 pF
|
||||
C1 222.8 C2 154 L 1.65 delta 1.4
|
||||
C1 220 C2 132 L 1.54 delta 1.6
|
||||
or 220 196 1.85 1.3
|
||||
C1 210 C2 110 L 1.43 delta 1.8
|
||||
or 210 279 2.12 1.2
|
||||
C1 200 C2 98 L 1.39 delta 2.0
|
||||
or 200 396 2.34 1.2</pre>
|
||||
<p>Note in this table that two sets of matching values are possible, even within the range of normal ATU components. One set drives toward maximal efficiency as C1 is decreased, while the other set moves away from the highest efficiency under the same condition. The primary setting to watch, therefore, is that of C2.</p>
|
||||
<pre>Input = 50 ohms Output = 150 - j100 ohms Maximum permitted C1 = 172.6 pF
|
||||
C1 172.5 C2 91080 L 1.37 delta 2.5
|
||||
C1 150 C2 456 L 1.42 delta 3.0
|
||||
C1 130 C2 223 L 1.52 delta 3.6</pre>
|
||||
<p>With respect to efficiency, the C-L-C Tee is less effective with capacitively reactive loads than with inductively reactive loads. With some load conditions, here a capacitive reactance of 100 ohms combined with our standard 150-ohms resistive component, a very low delta will not be possible with normal components. The best we can do is to keep delta as low as possible by following the common advice for C-L-C Tee tuners:<br></p>
|
||||
<center>
|
||||
<i>C-L-C Tees: Use the maximum capacitance possible, especially for C2, for a given matching situation.</i>
|
||||
</center>
|
||||
<p>However, even with a delta of 3.6 to 4, efficiency with a coil Q of 100 will be in the neighborhood of 96%.</p>
|
||||
<p><b>SPC:</b> The SPC (series-parallel capacitance) network is the more general case of which the C-L-C Tee is a specific instance. Note that the C-L-C results from the SPC if we lower the value of the capacitor in parallel with L to zero. Ordinarily, the parallel capacitor is ganged as a dual unit with either C1 or C2. The requirement for a high-voltage dual-section capacitor has limited commercial production of SPC tuners. Figure 3 shows the more common configuration, with Cp and C2 ganged as equal-value units.</p>
|
||||
<center>
|
||||
<img src="dspcs.gif" width="446" height="321" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>The SPC departs from the C-L-C Tee in two significant operational characteristics. First, the actual values required for a given impedance transformation will differ. Second, the SPC tends to tune much more sharply. I recommend the use of verniers on the capacitors for ease of tuning. However, the equation for determining delta for the SPC is the same as that given for the C- L-C Tee.</p>
|
||||
<pre>Input = 50 ohms Output = 150 ± j0 ohms Maximum permitted C1 = 222.9 pF
|
||||
C1 222.8 C2 6027 L 0.04 delta 58.8
|
||||
C1 210 C2 362 L 0.64 delta 4.7
|
||||
C1 180 C2 176 L 0.69 delta 4.6
|
||||
C1 170 C2 152 L 0.75 delta 4.7
|
||||
|
||||
Input = 50 ohms Output = 150 + j100 ohms Maximum permitted C1 = 222.9 pF
|
||||
C1 222.8 C2 154 L 0.82 delta 2.9
|
||||
C1 220 C2 137 L 0.85 delta 2.8
|
||||
C1 215 C2 118 L 0.87 delta 2.9
|
||||
|
||||
Input = 5 0ohms Output = 150 - j100 ohms Maximum permitted C1 = 172.6 pF
|
||||
C1 172.5 C2 91080 L 0.01 delta 1256
|
||||
C1 140 C2 305 L 0.52 delta 9.1
|
||||
C1 120 C2 172 L 0.76 delta 8.2
|
||||
C1 110 C2 136 L 0.88 delta 8.3</pre>
|
||||
<p>The SPC tuner improves the harmonic rejection characteristics of the C-L-C Tee (which is inherently a high-pass filter), but apparently at the cost of lower efficiency. All three tables show the values surrounding the lowest value for delta obtainable under the given load conditions. The SPC appears to do better with inductive loads than with purely resistive transformations or capacitive loads. The value of delta increases for resistive and inductive loads as the ratio of output-to-input impedance increases, but shows a slight decrease for capacitive loads at higher ratios. Because tuning points are quite specific, no general rule of thumb is possible for the SPC.</p>
|
||||
<p><b>L-C-L Tees:</b> The L-C-L Tee, shown in Figure 4, has suffered from neglect until recent years. It is a natural low-pass filter, like the C-L-C PI shown earlier. However, whereas the intermediate impedance of the PI is lower than either the input or output impedances, the intermediate impedance of the L-C-L Tee is higher than either the input or output impedances. This results in higher voltages and lower currents at the junction of the three components.</p>
|
||||
<center>
|
||||
<img src="dlcls.gif" width="446" height="222" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>Determining delta for the L-C-L Tee requires attention to both inductors, which for simplicity are assumed to have the same Q in the following Egan equation:</p>
|
||||
<center>
|
||||
<img src="dlclf.gif" width="443" height="113" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<pre>Input = 50 ohms Output = 150 ± j0 ohms Minimum permitted L1 = 1.114 µH
|
||||
L1 1.115 L2 0.07 C 152 delta 1.4
|
||||
L1 1.12 L2 0.20 C 157 delta 1.5
|
||||
L1 1.20 L2 0.77 C 176 delta 1.8
|
||||
|
||||
Input = 50 ohms Output = 150 + j100 ohms Minimum permitted L1 = 1.438 µH
|
||||
L1 1.439 L2 0.01 C 181 delta 1.8
|
||||
L1 1.50 L2 0.16 C 180 delta 2.0
|
||||
L1 2.00 L2 1.30 C 159 delta 3.1
|
||||
|
||||
Input = 50 ohms Output = 150 - j100 ohms Minimum permitted L1 = 1.114 µH
|
||||
L1 1.115 L2 1.51 C 145 delta 2.1
|
||||
L1 1.12 L2 1.38 C 140 delta 2.0
|
||||
L1 1.14 L2 1.16 C 129 delta 1.9
|
||||
L1 1.43 L2 0.02 C 85 delta 1.8</pre>
|
||||
<p>Despite the use of two theoretically lossy inductances in the L-C-L Tee, it maintains low values of delta over a wide range of load conditions and transformation settings. Delta climbs more rapidly with inductive loads than with capacitive loads away from optimal settings. The operative rule of thumb for maximum efficiency is this:</p>
|
||||
<center>
|
||||
<i>L-C-L Tees: Choose the lowest value of L2 that permits a match.</i>
|
||||
</center>
|
||||
<p>This operative rule holds true at higher ratios of output-to-input impedance, even though values of delta increase at those ratios. L-C-L Tees are subject to losses from circulating currents in the shorted turns of either switched or rotary solenoid or toroidal coils, losses that the value of delta cannot take into account. Therefore, construction of this network requires special care.</p>
|
||||
<p><b>Ultimate Transmatch:</b> Lew McCoy's Ultimate Transmatch (UT) is an interesting variation on the C-L-C Tee. It places a dual capacitor at the input of the network, one section in series with the signal path, the other section to ground. Figure 4 shows the general outline of the network.</p>
|
||||
<center>
|
||||
<img src="dults.gif" width="443" height="321" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<p>Calculating delta for the UT design requires the following Egan equation:</p>
|
||||
<center>
|
||||
<img src="dultf.gif" width="280" height="107" border="2" vspace="5" hspace="9">
|
||||
</center>
|
||||
<pre>Input = 50 ohms Output = 150 ± j0 ohms Maximum permitted C1 = 750 pF per section
|
||||
C1 749 C2 168 L 0.57 delta 5.8
|
||||
C1 600 C2 161 L 0.68 delta 5.0
|
||||
C1 400 C2 143 L 0.90 delta 4.1
|
||||
C1 250 C2 113 L 1.17 delta 3.8
|
||||
|
||||
Input = 50 ohms Output = 150 + j100 ohms Maximum permitted C1 = 750 pF per section
|
||||
C1 749 C2 81 L 0.57 delta 5.8
|
||||
C1 400 C2 75 L 0.90 delta 4.1
|
||||
C1 250 C2 66 L 1.17 delta 3.8
|
||||
|
||||
Input = 50 ohms Output = 150 - j100 ohms Maximum permitted C1 = 546 pF per section
|
||||
C1 545 C2 >350K L 0.73 delta 4.7
|
||||
C1 300 C2 615 L 1.06 delta 3.8
|
||||
C1 250 C2 400 L 1.17 delta 3.8
|
||||
C1 200 C2 256 L 1.30 delta 3.9</pre>
|
||||
<p>Like other series-capacitance designs, the UT appears most effective with purely resistive and inductive loads: Required values for a match fall easily within the ranges of normal components just as the network approaches maximum efficiency. (If C1 is lowered further in these cases, delta increases.) However, with capacitively reactive loads, maximum efficiency has passed by the time C2 enters the range most normal for ATUs.</p>
|
||||
<p>Although the values for delta are higher than with some other designs, efficiency remains in the 96% ballpark. In fact, the efficiency of the Ultimate Transmatch appears to increase (at least to some limit) with increases in the ratio of output-to-input impedance. Using a 300-ohms output resistance with either no reactance or 100 ohms of inductive reactance, the UT achieved values of delta in the neighborhood of 2.8, while with 100 ohms of capacitive reactance, delta was about 3.5. The rule of thumb for the UT design is not as simple as with some other designs:</p>
|
||||
<center>
|
||||
<i>UT: With normal ATU components (Cmax less than 400 pF), choose the highest value of C2 that permits a match within the range of C1.</i>
|
||||
</center>
|
||||
<p>Remember that, for any of the network designs, as the frequency decreases, some of the efficiencies achieved in these 10.1 MHz tables may not be reached, since proportionately greater values of C and L may be needed for the same load conditions. Moreover, switched inductors may not provide the values necessary for highest efficiencies. This limitation is rarely a problem, since the value of delta in most cases "bottoms out" over a fairly wide range. Nonetheless, the rules of thumb continue to apply.</p>
|
||||
<p>Moreover, remember that these figures are based solely on losses in the inductor, one assumed to have a Q in the range of 100 or more. Additional losses due to poor components, bad wiring and switching, stray inductances and capacitances, and coil "suck-out" are not accounted for by the calculations. If components get warm or arc under matched conditions at 100 watts, even though the calculations indicate less than 5% losses, believe your senses. Then analyze the components and construction of your ATU. Something other than network choice is very wrong.<br>
|
||||
<br></p>
|
||||
<center>
|
||||
<img src="../images/colorbar.gif" width="591" height="3">
|
||||
</center><br>
|
||||
<br>
|
||||
<p><i>First printed in</i> QRP Quarterly<i>, January, 1996. Updated 3-25-98. © 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.</i><br>
|
||||
<br></p>
|
||||
<center>
|
||||
<img src="../images/colorbar.gif" width="591" height="3">
|
||||
</center><br>
|
||||
<br>
|
||||
<p><a href="../radio.html"><img src="../images/return.gif" align="bottom" width="40" height="40"> Return to Amateur Radio Page</a></p>
|
||||
<p class="topics-footer">Topics: <a href='../topics.html#matching'>Matching</a></p><br>
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||||
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<!DOCTYPE html>
|
||||
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|
||||
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||||
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|
||||
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<title>Notes on the Gamma Match</title>
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<h1>Some Preliminary Notes on the Gamma Match</h1>
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<h3>L. B. Cebik, W4RNL</h3>
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<p>My e-mail regularly contains a question that appears in various forms, but has one theme: Why have I not written anything about the gamma match? There are notes at my personal website on various forms of series matching systems and notes on the beta match. These two types of matching systems represent alternatives to the gamma match, especially when the task is to transform the generally low impedance of a Yagi array up to the value of 50 Ohms, as required by the most common feedlines used in amateur and other services.</p>
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<p>Compared to a gamma match, series and beta matching systems are both simple and precise. Both systems make no alteration to the driven element, but add networks composed of transmission lines (usually) to the element feedpoint. The beta match does require that we initially set the driven element length to arrive at an optimal value of reactance relative to the feedpoint resistance, but the beta component is or is equivalent to adding a simple reactance across the feedpoint. The matching systems do not affect the radiation properties of the element.</p>
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<p>As a consequence of these conditions, we may calculate the required values for series-matching or beta-matching components very precisely. In practice, the key factor affecting field adjustment of the systems is the accuracy of the velocity factor that we use in the calculations relative to the value that actually applies to the line used. If we know the velocity factor with measured accuracy (in contrast to the values we find in lists and specification sheets), we can often obtain the desired result with no need for further adjustment.</p>
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<p>These fundamental matching systems are therefore amenable to automated formulation, that is, a utility program or a spreadsheet page that will tell us the required values if we input the feedpoint conditions and other values related to the components. In fact, I keep a spreadsheet on hand in my computer for just such calculations. It contains calculations for 1/4 wavelength matching sections, Bramham transformations, and Regier series matching calculations. In addition, it allows beta-match calculations and returns results for using either a shorted transmission-line stub or a solenoid inductor as the most common beta components. However, it also yields open lines and capacitance values should one choose to lengthen the driven element rather than shortening it. I have also added a page for the match-line and stub system of matching, used with higher-impedance antenna elements. I wrote the pages in Quattro Pro (.qpw), but have also saved them in Excel (.xls). You may download the sheet of your choice from the following addresses.</p>
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<blockquote>
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<p><a href="../download/ant-match.xls"><b>Antenna-Match in .XLS format (Excel)</b></a></p>
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<p><a href="../download/ant-match.qpw" download="ant-match.qpw"><b>Antenna-Match in .QPW format (Quattro Pro)</b></a></p>
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<p><b>The Gamma Difference</b></p>
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<p>The gamma match differs fundamentally from the other matching systems because it alters the physical properties of the driven element in ways indicated by <b>Fig. 1</b>. First, it adds new wires to the element, giving the element a more complex shape. Second, it changes the element feedpoint relative to the original element. The simple element uses a feedpoint position that normally is at the center of the element. The gamma-matched element places the feedpoint on a wire that joins the gamma rod to the main element.</p>
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<p>One gamma-match advantage to many builders is the fact that the element may now connect directly to the boom. Both series and beta matching systems require driven elements that are insulated and isolated from any conductive support boom. In the era in which so-called "plumber's delight" construction methods ruled amateur Yagi construction, the gamma match equally ruled impedance transformation for coaxial feedlines. However, connecting the element to the boom changes its electrical length and therefore the feedpoint impedance prior to creating a gamma match. Therefore, most gamma match users began their calculations or experiments with only an estimated feedpoint impedance for the pre-matched element. NEC and MININEC antenna-modeling software offer no assistance here, since these programs only model axial currents (that is, along the element) and thus could not account directly for boom effects. Some builders have come to believe that a gamma match requires a direct connection between the boom and the element center. However, the connection is only an option, not a mandatory condition for the matching system.</p>
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<p>One criticism of the gamma match accuses it of displacing the pattern in the direction of the match due to the size of the assembly. Effectively, as some believe, the assembly increases the diameter of the gamma side of the element, and this asymmetry of the driven element results in the main lobe's re-aiming. To test this notion, I constructed models of 28-MHz beams with identical reflectors and element spacing. One beam uses a simple driven element. The other uses a driver with the same overall length, but with a gamma match assembly (of course, with no boom). The gamma assembly is in the plane of the two elements and projects forward of the driver. The results of the test appear in <b>Fig. 2</b>.</p>
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<p>The slightly lower forward gain of the gamma-matched version of the beam is an artifact of certain limitations of NEC (which we shall review shortly). The key factor in the overlaid patterns is the degree of lobe displacement, indicated by the lines that I added to the sides of the lobe. Displacement does indeed occur, but at a level too small for any user ever to notice in operation. The two patterns to the right show that the gamma match also has an affect on the free-space side nulls for the array. The simple beam has side nulls that show no limit. However, the gamma-matched beam has limited side nulls that are a mere 40-dB down from the level of maximum gain. I am unaware of any operational use of a beam in which one might be able to detect the difference.</p>
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<p>The modeled test case assumes by its mathematical basis relatively perfect construction of the gamma-match driven element. I have in past years seen range-generated patterns for gamma-matched beams with a significant displacement of the main forward lobe. It would not be possible to perform a full analysis of such patterns without being able to model currents within the driven element and along the boom--if the elements makes a direct connection to the boom. However, in principle and assuming careful construction, pattern displacement is not a hindrance to the use of a gamma match.</p>
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<p><b>Calculating the Gamma Match</b></p>
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<p>H. H. Washburn, W3MTE, introduced the amateur community to the gamma match in his September, 1949, <i>QST</i> article, "The Gamma Match" (pp. 20-21, 102). D. J. Healey, W3PG, provided the first mathematical analysis of the match in "An Examination of the Gamma Match," <i>QST</i>, April, 1969 (pp.11-15, 57). Healy's treatment, however, required the use of nomographs and a Smith chart.</p>
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<p>Since these seminal articles, several alternative analyses have appeared in amateur journals. H. F. Tolles, W7ITB, presented a purely mathematical analysis in "How to Design Gamma Matching Networks" in <i>Ham Radio</i> for May, 1973 (pp. 46-55). Because the Tolles equations proved tedious to many gamma designers, R. A. Nelson, WB0IKN, set them into a Basic program in "Basic Gamma Matching," <i>Ham Radio</i>, January, 1985 (pp. 29-33). ARRL converted Nelson's Apple-Basic program into a version suitable for IBM computers, and a listing appears in <i>The ARRL Antenna Book</i>, 16th Ed. (p. 26-20). In 2000, Dave Leeson, W6NL, corrected portions of the program so that it more accurately serves as a design vehicle to calculate gamma matches. This program is also available within the HamCalc collection of Basic utilities edited by George Murphy, VE3ERP. <b>Fig. 3</b> shows the GW Basic listing for the version of the program distributed by ARRL.</p>
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<p>Since the work of Tolles and Nelson, two alternative mathematical analyses have appeared. Ron Barker, G4JNH, presented "A New Look at the Gamma Match" in QEX, May/June, 1999 (pp. 23-31). Barker changes some of the fundamental assumptions about the key factors in a gamma match to arrive at his results. Unfortunately, his work is less amenable to easy placement in a Basic utility or a spreadsheet, since the calculations require the solution to simultaneous equations. In contrast, Roger Wheeler, G3MGW, returned to the Healey analysis and converted the graphical techniques back into mathematical methods that allow a straightforward spreadsheet set of calculations. Both of these later analyses rely on something that was unavailable to earlier gamma calculations. In most cases, the determination of the initial or pre-match driver feedpoint impedance rested on assumption, guesswork, or rudimentary measurement. Measurement became difficult if the builder connected the driver to the boom and did not allow for a feedpoint gap, even if it would later be closed. Both Barker and Wheeler require the use of antenna modeling software to determine the pre-match driver impedance. Other methods exist, for example, the Brian Beezley, K6STI, module in the overall program YO. However, Beazley has never published his procedures.</p>
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<p>The downloadable spreadsheet contains pages for the Healey-Wheeler (HW) and the Tolles-Nelson-Leeson (TNL) methods of calculating gamma match rods and series capacitors. Every gamma-match calculating system tries to yield a physical value for the length of the gamma rod and a series capacitance value at the feedpoint to leave a pure resistive impedance. The required inputs appear in <b>Fig. 4</b>. We need to know the diameters of the main element in the region of the gamma assembly and of the proposed gamma rod, tube, or wire. As well, we must input the center-to-center spacing between the main element and the gamma rod. Ordinarily, the physical dimensions for the inputs and the outputs are in the same units of measure. The spreadsheets use inches.</p>
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<p>In addition, we require two impedance values. One value is for the simple driven element, designated Ra and jXa in <b>Fig. 4</b>. We also need to specify a desired feedpoint impedance, Rf, which is the target resistive impedance that matches the main feedline. For our samples, we shall use 50 Ohms, since it is the most common value that we encounter in amateur radio applications. However, we may apply the gamma match for virtually any reasonable line impedance.</p>
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<p><b>Fig. 4</b> also shows some of the points in the gamma assembly that exhibit derived impedances calculated somewhere within the system. Z1 is the transformed impedance based on the fact that the gamma assembly forms a shorted transmission line stub. (One misguided criticism of the Healey system was that it treats the gamma assembly as a folded dipole. Every folded dipole exhibits both radiation and transmission-line currents. In the calculation of the gamma line, we are concerned with the transmission-line performance of the assembly.) We can derive the characteristic impedance (Zo) of the stub using conventional equations that involve only the physical dimensions of the line. S is the center-to-center line spacing, and d1 and d2 are the diameters of the gamma rod and the main element, respectively.</p>
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<p>The TNL system uses more fundamental equations involving havercosines. However, in the typical range of Zo (perhaps 200 to 600 Ohms), the differential in results between equations is less than 1% and normally only about 0.1%.</p>
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<p>While we are using only the physical dimensions that we input to the calculation system, we can also calculate a step-up ratio between the original simple-driver impedance and the value shown as Z1 in <b>Fig. 4</b>. The most usually form of the equation again employs the three input physical values, S, d1, and d2.</p>
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<p>Once more, the TNL system uses more fundamental equations, but the differential in result, compared to the more usual engineering formulation is well under 1%. Both the HW and TNL systems use this equation to calculate the value of Z1 (R1 +/- jX1) simply by multiplying Ra and jXa by the value of r. If the diameters of the main element and the gamma rod are the same, then r = 4. If the gamma rod is thinner than the main element, then r > 4. If the gamma rod is thicker than the main element (an unusual but possible situation), then r < 4 but r > 1.</p>
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<p>We may add two side notes here. First, the Barker calculation system does not use the step-up ratio derived from the usual equation. Barker uses the impedance ratio between a simple driver element in isolation (essentially a dipole) to the impedance of the simple driver in service within the beam antenna. Second, at least the HW system does not account for the fact that the impedance undergoes not only a step-up in value, but also a shift in phase angle when we move from the driver without the gamma assembly to the driver with the assembly. If we model a gamma system and place the feedpoint at the position it would occupy on the pre-gamma driver element, we can observe the phase-angle shift in the impedance.</p>
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<p>The HW system calculates the value of Z2, the impedance presumed to exist at the far end of the gamma assembly. If we assume that the current distribution is sinusoidal--which is close to correct but not precise--then we may use a standard equation to determine the values of R2 and X2.</p>
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<p>Theta is the electrical length of a gamma assembly either in degrees or in radians at the design frequency. Since we cannot determine the value of theta without a physical gamma rod, the HW system calls for a trial length. In concert with the remaining calculations, we simply adjust the trial length until the value of Rf becomes 50 Ohms, if that is the target feedpoint impedance.</p>
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<p>The TNL system operates differently. By addressing the required impedance and phase angle at the feedpoint, it calculates the required rod length using the factors already derived plus some ratios that appear in the listing in <b>Fig. 3</b>. The original Tolles article in <i>Ham Radio</i> provides the source of these ratios as they are applicable to the calculations. The Barker system uses neither of these methods, but creates an assumption of what must be the relative impedance at the gamma junction with the main element. He then calculates actual values from the initial driver impedance in a set of simultaneous equations.</p>
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<p>The HW system derives the feedpoint impedance from two values in parallel. One value is the impedance of the gamma assembly as a shorted transmission-line stub having the length, theta, and the characteristic impedance Zo.</p>
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<p>The other values are the values of Z2 as transformed by the same length of transmission line back to the new feedpoint. Wheeler follows Healey in using the following equation for this part of the impedance combination.</p>
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<p>To resolve the equation, of course, one must break it into real and imaginary parts and then recombine them to arrive at the final value of Z3. (The TNL system essentially reverses the procedures and calculates the rod length from the required transformation.) We combine the parallel combination to arrive at an impedance that is the desired Rf in series with a value of Xf that is inductive. From this inductive reactance and the design frequency, we may determine the required series capacitance to leave us with a purely resistive feedpoint impedance.</p>
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<p>I have tracked some of the rudimentary elements of gamma calculating systems to show what sort of thinking goes into them. However, the treatment is in no way complete, nor does it touch a number of the dimensions of the TNL and Barker systems. Instead, it is simply complete enough to allow one to track through the attached spreadsheet formulations of the HW and the TNL systems, in case one wishes to calculate a few typical gamma assemblies.</p>
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<p><b>Testing the Gamma-Calculating Systems</b></p>
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<p>Virtually all published gamma-calculating systems use one or two examples of the system's application and then declare the system adequate. Wheeler applied his formulation to gamma assemblies on quad loops. Tolles preferred VHF Yagis using gamma rods considerably thinner than the main element. Barker uses a single 20-meter beam as his test case. Of course, trying to develop a systematic set of test cases would be nearly impossible if we were restricted to constructing physical antennas having an interesting range of feedpoint impedance values for transformation.</p>
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<p>It is possible to construct a series of antenna models to serve as a surrogate for the physical antennas. However, we cannot do the job in NEC-2 or even NEC-4. As suggested in <b>Fig. 5</b>, the gamma assembly presents NEC with two problems. First, unless the gamma and main elements are the same diameter, we encounter angular junctions of wires with dissimilar diameters. Although NEC-4 improves on the performance of NEC-2 under these circumstances, the results are insufficiently accurate for use as a comparator to the calculated values. In addition, gamma spacing is rather narrow for most beams that use relatively fat element diameters. Under these conditions, NEC tends to yield less than precise results. The relative unreliability appears in the average gain test (AGT) scores, which generally are no better than 0.92 when a perfect score would be 1.00. Since arriving at a feedpoint impedance of 50 Ohms is critical to the comparisons, AGT values in the range of 0.92 are too far from ideal to be useful. Values of 0.98 to about 1.02 are more valuable to the task of comparison.</p>
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<p>Fortunately, MININEC is not sensitive to angular junctions of wires with differing diameters. However, in its raw form, it is subject to limitations related to close wire spacing and to angular junctions in general. One version of MININEC, Antenna Model, has introduced correctives that make it suitable for some first-order comparisons with the calculation systems. Note that I do not call the models "standards" against which we test the calculation systems. At best, the models are comparators so that we may observe some general trends as well as similarities and differences in outcomes.</p>
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<p><b>Fig. 6</b> shows the general set-up for the modeling process. Although unnecessary for the pre-match model, I have assigned to each element the same number of segments used in the gamma model. The number of segments for each element derives from the test values of gamma spacing. All test models will use 28 MHz as the design frequency. The gamma spacing will use 4" as a center value and require 2 segments in the feedpoint wire to place the feedpoint at that wire's center. Hence, a 2" segment length becomes the standard. For some tests, we shall use gamma spacing values of 2" and 6", but the segment length differential will not prove too detrimental to the AGT scores. In fact, in the accumulated data, I shall show not only the modeled gamma rod length and the indicated series capacitor, but as well the AGT score to permit you to reach your own conclusion about the model's reliability.</p>
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<p>From the very start, we recognize some important differences between the models and the calculation systems. Foremost among the differences is the fact that the models contain actual physical structures for the gamma far end shorting bar and for the feedpoint connecting assembly. In all models, I shall use end wires that have the same diameter as the main element, since in gamma assemblies applied to Yagis, the shorting bar and the coax connector plate tend to be substantial. Neither the HW not the TNL calculation system includes any allowance for such structures. Barker does add some fudge factors to his system, and the HW spreadsheet page includes an optional fudge-factor section at the end. The reason for the fudge factors is that calculations tend to call for capacitance values that are too high, and the models will reflect this fact by requiring lower capacitance values than the calculations indicate.</p>
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<p><b>Test 1: A Simple Scaling Project</b></p>
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<p>As an initial test, let's compare 3 perfectly scaled antennas with gamma matches. The test frequencies are 7, 14, and 28 MHz. For each frequency, I created a basic 2-element driver-reflector Yagi with a feedpoint impedance of 29.84 - j25.73 Ohms (typical of 0.12 wavelength element spacing). To ensure perfect scaling, the elements are lossless. (The calculation systems do not take material losses into account, and given the small size and generally large element surface areas used in beams, the losses would indeed be small.) The 10-meter antenna uses a 0.5" main element diameter, with a 0.375" gamma rod. The element-to-rod spacing is 4". All of these dimensions also scale upward as we lower the test frequency. Therefore, the 7-MHz version of the antenna uses a 2" main element, a 1.5" gamma rod, and 16" spacing. Even though the 40-meter dimensions may be somewhat larger than realistic, they will serve well in this test. The top section of <b>Table 1</b> provides a more detailed run-down of the beam dimensions.</p>
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<p>The lower portion of the table shows the calculated length of the gamma rod and the calculated series capacitance using the Healey-Wheeler and the Tolles-Nelson-Leeson system. It also shows the modeled values in the AM MININEC program. Because all dimensions are perfectly proportional, the calculated values of the gamma-line characteristic impedance, Zo, and the step-up-ratio, r, are the same for all three antennas. A review of the basic equations shown earlier will confirm this result.</p>
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<p>As a consequence, we find that the two calculation systems also produce scaled values for both the gamma rod length and the required series capacitance. Likewise, the modeled gamma rod and series capacitance also internally scale within the limits that I set for modeling precision. I adjusted the rod length and the series capacitance until the feedpoint impedance reached 50 Ohms resistive, +/-0.1-Ohm, and j0 Ohms reactive, +/-j0.1 Ohm.</p>
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<p>For this test case, the two calculating systems yield quite similar values for the gamma rod length. However, both values are 16-17% shy of the modeled value. One reason that I selected this initial test case was the fact that the calculation systems yield values smaller than the model. Theoretically, if we simply assume that the calculation system does not take the shorting bar into account, we would expect the calculated lengths to be slightly long. Hence, the systems and the model must have other differences.</p>
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<p>As we expected from general gamma-match experience, both calculation systems over-estimate the required amount of series capacitance needed to bring the feedpoint impedance to a resistive 50-Ohm value. In this early test, we may also note that the two systems yield very different series capacitance values. Therefore, a single method of adding fudge factors into the determination of the final series capacitance value will not work for both systems.</p>
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<p>However, I have to insert a reminder here. The modeled gamma match is only a comparator in this test (and in those yet to appear). It is not a standard against which to measure the adequacy of the two calculating systems. With the exception of the gamma-rod length, the 3 systems of determining the required gamma-rod length and the series capacitance merely yield different results.</p>
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<p><b>Test 2: Changing the Ratios among Element Diameter, Rod Diameter, and Spacing</b></p>
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<p>For a second test, let's modify the initial beam so that it is very close to resonant, with a feedpoint impedance of 32.07 - j0.05 Ohms. For this test, we may use a 28-MHz beam. Like the beam in the first test, the element spacing is 0.12 wavelength, so the only change is to the driver length. The main element diameters are 0.5". Once more, we shall explore the HW and TNL calculating system results and compare them with AM MININEC modeling results.</p>
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<p>This test will be somewhat more complex. We shall explore 3 element-to-rod spacing values: 2", 4", and 6". As well, we shall look at gamma-rod diameters from 0.125" to 0.625" in 1/8" increments. A gamma rod that is fatter than the main element is unusual in reality, but certainly possible. Because we are changing both the diameter ratio between linear parts of the gamma assembly and the spacing between those parts, the values of both Zo and r will change with each sample case. The value of Zo will range from about 233 Ohms (for the 0.625" rod at a 2" spacing) to nearly 465 Ohms (for the 0.125" rod with a 6" spacing). The versions using 2" spacing will show the widest range of step-up ratio (r) values, running from about 3.6 for the fattest rod to 7.1 for the thinnest.</p>
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<p><b>Table 2</b> catalogs the data for the series of tests. The top part of the table provides the initial antenna dimensions, along with the near-resonant pre-match feedpoint impedance. The next part of the table provides results for the three methods of determining the required gamma parameters. The AM section provides an additional column that lists the AGT score for each model in the set. The models using a 2" spacing are the farthest from ideal. It is not wholly clear that MININEC follows the same general AGT rules as does NEC. Hence, we cannot claim with assurance that the 50-Ohm impedance derived from the models (within the limits used in the first test) is off by no more than about 1.25 Ohms. However, the parallels among values for all three spacing values suggest that the models are generally reliable within the limits of the average gain test.</p>
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<p>The three methods of finding gamma parameters differ in almost every category. The HW and TNL systems show various degrees of gamma-rod length increase as we increase the rod diameter, regardless of the spacing. The models show virtually no change in length within each of the spacing groups. The HW system shows a considerable increase in length as the rod diameter increases, while the increase is fairly modest for the TNL system. In all cases, the calculating methods show longer rods than the models. However, we cannot draw conclusions until we review the second part of this test series, using a different initial or pre-match feedpoint impedance.</p>
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<p>The series capacitance values produced by each method present an equally befuddling array of differences within each method and between any two methods. The HW system produces only small changes in value across the span of rod diameters for wide-spacing values, but larger changes for narrow spacing values. The HW changes within spacing groups are in all cases smaller than for the TNL intra-group changes. The modeled values partially parallel the HW values in terms of the amount of change within each spacing group. However, the trends are not consistent between the calculating systems and the modeling system.</p>
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<p>Before we rush to conclusions, we should repeat the very same tests using a different pre-match impedance. We shall retain every other beam detail, except that we shall use the version of the beam that shows a pre-match impedance of 29.84 - j25.73 Ohms. The results of this second survey appear in <b>Table 3</b>.</p>
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<p>With respect to gamma rod length, within each group, the HW system shows an increasing length as we increase the rod diameter. Both the TNL system and the AM models show a decreasing rod length as we increase its diameter. Despite the different trends for the initial impedance of the antenna, the HW and TNL lengths are not very different from each other for any given element-to-rod spacing. However, the rod lengths required by the models are systematically longer. (This fact is exactly the reverse of what we saw when the beam's pre-match impedance was nearly resonant.)</p>
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<p>With the high capacitive reactance of the pre-match impedance, the gamma-rod lengths remain relatively stable for all 3 methods within each increment of element-to-rod spacing. However, the series capacitance is another matter. The HW system shows the greatest rate of increase with increasing rod diameter, while the AM models show the smallest rate of increase. Both calculating systems produce much higher series capacitance values than the models, with the HW system showing values that are 100-150% too high. The TNL and AM series capacitance values are more closely--but not too closely--aligned.</p>
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<p>The trends shown within each of the two test situations generally failed to parallel each other, despite the fact that the only difference between system inputs is the initial or pre-match impedance values. It would appear that we need a further type of test situation.</p>
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<p><b>Test 3: Varying the Input Impedance</b></p>
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<p>The goal of this test sequence is to determine--at least in a preliminary way--the effects of varying the pre-match feedpoint impedance, with special reference to the reactance. We already have sample of resonant and highly capacitively reactive impedances. We may use the same basic model of a 28-MHz 2-element Yagi and vary the driver length to create a reasonably fair sequence of reactance values. We need a positive limiting reactance value that is close to the negative limiting value. As well we need reactance values close to +/-13 Ohms as intermediate values between resonance and the limits. We can arrive at these values just by varying the length of the initial beam driver, as shown in the top section of <b>Table 4</b>. The resistive component of the pre-match feedpoint impedance will increase as the driver grows longer. However, the amount of increase should not be enough to invalidate this highly preliminary test sequence. We shall survey two gamma-rod diameters in order to assure ourselves that any trends are not mere quirks.</p>
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<p>The lower portion of <b>Table 4</b> shows the results of calculating the gamma parameters and of modeling them. Let's examine the results, separating the gamma length from the series capacitance, and also separating the two different gamma-rod diameters. <b>Fig. 7</b> graphs the gamma-rod lengths for the 0.125" diameter gamma rod.</p>
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<img src="gamma-7.gif" width="611" height="407" border="2" vspace="5" hspace="9">
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<p>Both the HW and the TNL calculating systems show roughly equal rod lengths at the extremes of the test series, with the inductively reactive initial driver requiring a considerably longer length than the capacitively reactive driver by a factor of about 2:1. However, between these extremes, the two systems show curves with almost exactly opposite tendencies. Moreover, when we examine the modeled gamma-rod lengths, we find a quite different curve. The rod is shortest when the pre-match impedance is closest to resonance, with increases in length as the impedance becomes more reactive, regardless of the type of reactance.</p>
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<center>
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<img src="gamma-8.gif" width="611" height="407" border="2" vspace="5" hspace="9">
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</center>
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<p>When we increase rod diameter to 0.375", as shown in <b>Fig. 8</b>, the same tendencies repeat themselves, although the calculated lengths increase the ratio between the most inductive and the most capacitive reactance values. However, the modeled gamma-rod lengths show virtually the same values as shown in the curve for the 0.125" rod.</p>
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<p>If we graph the series capacitance values for the 0.125" rod across the spread of pre-match impedances, the curves become quite interesting, as suggested by <b>Fig. 9</b>. Each method of reaching a capacitance value shows a peak, and the peak occurs at a different impedance for each method. The HW system arrives at a peak capacitance value at the intermediate capacitive reactance value, while the TNL system peaks at (or close to) resonance. The modeling method shows its peak near the intermediate inductive reactance value. We should remember that the model contains gamma end wires that are not a part of the calculating systems.</p>
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<center>
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<img src="gamma-9.gif" width="611" height="407" border="2" vspace="5" hspace="9">
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</center>
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<p>The capacitance curves for the 0.375" rod diameter show similar traits to those for the 0.125" rod. See <b>Fig. 10</b>. The calculating systems appear to show peaks with higher levels of reactance than we found to be the case for the thinner gamma rod. Once more, the TNL gamma matches show peak series capacitance values close to an initial resonant impedance, with the HW peak in the capacitive reactance region and the modeled match's peak in the inductively reactive region, relative to the initial or pre-match driver impedance.</p>
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<center>
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<img src="gamma-10.gif" width="611" height="407" border="2" vspace="5" hspace="9">
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</center>
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<p>Perhaps more vividly than any other test, the final series of tests shows one of the chief sources of differences among the three systems. The two calculating systems respond to differences in the feedpoint reactance in similar ways, although the length curves show opposite tendencies as the pre-match impedance approaches and passes resonance. If the test is representative, then we have established that the two systems are the same in principle, although they differ in detail. However, for changes in the gamma-rod length, neither system correlates well with the modeling method of designing a gamma assembly. Otherwise expressed, the modeling system of design fails to correlate well with the methods of calculation.</p>
|
||||
<p>Note that I have emphasized the conditional nature of the test series. We have examined only the impedances that tend to apply to Yagi beams, that is, impedances below the feedpoint impedance. Establishing that the results are in fact representative would require a very large series of test sequences involving many possible impedance combinations relative to the feedpoint target and the pre-match values. At most, this test has established the importance of the pre-match impedance as a factor governing the results from each method.</p>
|
||||
<p><b>Some Tentative Conclusions</b></p>
|
||||
<p>I approached the gamma match out of curiosity. My inquisitiveness arose from the difference between series matching and beta matching calculation systems and the calculation of gamma matching systems. The first two methods produce precise results so that the most significant limitation when implementing one of them surrounds the physical properties of the components involved. For transmission line lengths, the accuracy of the velocity factor (or our ability to make linear measurements of the line) becomes the chief source of error.</p>
|
||||
<p>The gamma matching system calculation methods show far less precise results. Some writers have ascribed most of the error to the lack of end wires in the calculated values. For many implementations, this convenient explanation seemed too weak to account for the differences. Therefore, I took two of the systems that are amenable to straightforward calculation progressions and compared their results to MININEC models using the most reliable version available of that software. The results of our preliminary series of tests--restricted to a 50-Ohm target feedpoint and to pre-match impedance values typical of Yagi arrays--show something else entirely.</p>
|
||||
<p>The two calculation systems--HW and TNL--produce seemingly divergent results, especially with respect to variations among the main element diameter, the gamma rod diameter, and the spacing. However, if we employ a range of pre-match impedance values that vary mostly with respect to reactance, we begin to see an emerging pattern in which the calculated gamma lengths converge at high pre-match reactance values and diverge when the pre-match impedance approaches resonance. (We may bypass capacitance calculations, since they depend on the calculated gamma rod length.)</p>
|
||||
<p>Perhaps the most disturbing aspect of the test series is the fact that neither calculation method approaches--either in values or in trends--what we find when we model a gamma match using MININEC models with nearly ideal AGT scores. However, we cannot in this case give automatic priority to the modeled results because they have not undergone confirming field tests. They simply serve here as a third method that differs in principle from the basic presumptions underlying the two calculation systems that we examined. Nevertheless, the differences in results among the methods strongly suggests that the present methods of calculating gamma match components fall seriously short of being precise.</p>
|
||||
<p>We have had occasion to note that one point at which both the HW and TNL systems make a questionable assumption lies in the use of standard equations for calculating the transformed pre-match impedance to arrive at a value that we have called Z1. Remodeling the gamma to move the feedpoint to the main element at the junction of the gamma-half-element with the non-gamma-half-element suggests at least a phase shift and also significant variation from the calculated value of r that depends only on the physical properties of the assembly. The transformation also appears to relate to the pre-match impedance--especially the reactive component--although our test series is too small to reach two important conclusions. One conclusion would be the derivation of a revised step-up function, either as a correction factor on the usual calculation or as a substitute formulation. The other conclusion that we cannot draw is the adequacy of the modeled gammas to serve as source for such correctives.</p>
|
||||
<p>Even if we could revise the available calculation systems, they would still fall short of the precision that we obtain from the calculations associated with series and beta matching systems. The methods by which we implement a gamma match include significant variables relative to even a precise calculation system. <b>Fig. 11</b> shows some of the factors involved.</p>
|
||||
<center>
|
||||
<img src="gamma-11.gif" width="450" height="230" border="2" vspace="5" hspace="9">
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</center>
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<p>The upper portion of the sketch shows the general situation presumed by all of the methods that we have examined. The elements have a uniform diameter and thus most closely approximate sinusoidal current distribution. The end wires at the gamma far end and at the feedpoint have no weight in the calculation--except as post-facto fudge factors. Third, the series capacitance is precisely at the feedpoint of the gamma system.</p>
|
||||
<p>The lower portion of the sketch shows some typical variations in the theoretical arrangement used in calculations. The elements in the HF range may use stepped-diameter structures that may vary the usual expectations for current distribution. The gamma rod normally extends beyond the shorting bar, leaving a small but definite radiating structure. The shorting bar at the gamma far end and the plate holding the coaxial cable connector have significant proportions that will vary from one installation to another. The sketch also shows the use of a tubular capacitor, a common HF technique to provide the series capacitance without concern for the voltage and current levels on the component and without concern for the effects of weathering. Once set, we may effectively seal the capacitor so that it requires only long-term maintenance. However, equally important to the type of capacitor used is its position. Typically, without regard to the type of capacitor used, gamma-match builders install the component on the gamma rod rather than at the actual feedpoint. <b>Fig. 12</b> illustrates a sample of commercial gamma-match construction. By virtue of its position along the gamma line, the capacitor modifies both its influence on the feedpoint impedance and on the structure of the gamma line. As well, the physical implementation of the gamma must take into account any affects of the boom, if one chooses to connect the driven main element to the boom.</p>
|
||||
<center>
|
||||
<img src="gamma-12.jpg" width="600" height="436">
|
||||
</center>
|
||||
<p>Therefore, even if we were to perfect gamma-matching system calculations, they would not yield the precision that we associate with other matching system calculations. Rather, they would serve only as a general guide to begin a process that only field adjustment can perfect. Indeed, for any selection of main element diameter, gamma rod diameter, and spacing between the two, there is a gamma length and a series capacitance that will effect a usable match to a desired main feedline over a wide range of pre-match impedance values--although not a completely unlimited range. As well, the gamma match will work either with or without a direct connection of the driven element to the boom.</p>
|
||||
<p>Whatever the complexity of the calculation system, its output is simply a starting point to the process. We might as easily replace it with a table encompassing all of the successful implementations of gamma-match systems arranged by frequency, main element diameter, gamma rod diameter, and element-to-rod spacing (with annotations on the method of implementing the series capacitance). Such an archive--if it existed--would likely provide as much guidance to gamma-match dimensions as the current methods of designing them.</p>
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<p><i>Updated 10-01-2006. © L. B. Cebik, W4RNL. This item first appeared in</i> AntenneX<i>, Sep., 2006. 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.</i></p>
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<img src="../images/colorbar.gif" width="591" height="3">
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<p><a href="../radio.html"><img src="../images/return.gif" align="bottom" width="40" height="40"> Return to Amateur Radio Page</a></p>
|
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<p class="topics-footer">Topics: <a href='../topics.html#matching'>Matching</a></p><br>
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