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:

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.

The following notes orient you to the pages, but do not provide extensive explanations of how each matching system work.

**Series Matching**

Series matching includes 3 system, ranging from the most specific to the most general.

*1. The 1/4 Wavelength Transmission-Line Transformer*

The 1/4 wavelength transmission-line transformer is perhaps the best known of the series matching systems. **Fig. 1** outlines the basic application of the system.

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.

For further information on the 1/4 wavelength transmission-line transformer, as well as ways to use variations of it, see "When is a Quarter Wave Not a Quarter Wave?".

*2. The Bramham System*

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 **Fig. 2**.

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 "Series Matching: A Review".

*3. The Regier General Series-Matching Solution*

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 **Fig. 3**.

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 "Series Matching: A Review"."

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.

**The Beta or Hairpin Match**

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. **Fig. 4** shows the general evolution of the beta or hairpin match.

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).

Some folks distrust the beta match because one form of shunt inductance seems to be a short circuit across the feedpoint. **Fig. 5** shows 3 typical forms of adding inductive reactance across the feedpoint terminals, which are insulated and isolated from any conductive support boom.

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.

For further information on the beta or hairpin match, see "The Beta Match: 2 Views"."

**The Gamma Match**

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). **Fig. 6** shows some of the reasons why the calculations are less than fully precise.

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.

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.

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 *The ARRL Antenna Book*) calculates the gamma rod length.

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.

For further information on the beta or hairpin match, see "Some Preliminary Notes on the Gamma Match"." 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.

**The Match-Line and Stub Matching System**

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. **Fig. 7** outlines what we require.

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.

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.)

For further information on the match-line and stub method, see "Stub Matching: A Review"."

**Conclusion**

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.

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.

For an additional set of spreadsheets that may be usful when designing some kinds of antennas, see "Antenna Design"

*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.*