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Whats with the Gamma Match Equations

By Jacob Rose,2014-11-25 18:19
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Whats with the Gamma Match Equations

What’s with the Gamma Match Equations?

Background

    Recently I needed to attempt a gamma matching solution to an antenna, with a parasitic element, with coax feed and I wanted to find the sensitivity to the various parameters to make a selection of gamma dimensions. The current ARRL Antenna Book comes with a PC calculator application (“Gamma”) that allows you to input the gamma wire sizes and

    spacing to then calculate the length and capacitance needed for those wires to provide a match to your specified feed line and unmatched antenna impedance. It turns out that there are gamma calculators supplied on CD with another ARRL book plus others have placed on the web for online use by helpful hams. (Sadly, the various calculators do not always give the same answers.) However, if you want to answer the question as to what the output impedance will be from a specified gamma match of known dimensions acting on a given input antenna impedance, that cannot be done directly with the calculators. Such a capability may be of interest for looking at the sensitivities to selection of gamma parameters and adjusting the length of the antenna driven element. Beyond that, inquiring minds just want to know.

    Yet more sadly, the information on gamma matches in The ARRL Antenna Book (21st) and the ARRL ON4UN’s Low-Band DXing book (4th) have some unclear drawings and

    inconsistent editing for designations of the parameters, which make the matter a bit confusing. There is also a question about the application for dipole-type antennas (including yagis) versus vertical-type antennas (including gamma-fed towers).

The Equations

    So here is what you need to know to calculate gamma transform and matching on your own using the conventional model. First the diagram of the set up:

Figure 1. Gamma match schematic.

    The basic quantities needed (consistent with common notation where possible) are:

    Za - the complex impedance of the unmatched antenna (Za = Ra + j Xa, normally measured with dipole halves split)

    S - center-to-center spacing of the circular antenna element to the circular gamma rod

D or d2 diameter of the circular antenna element

    d or d1 diameter of the circular gamma rod

    L length of the gamma rod

    C the added series capacitance used to null any resulting inductive reactance

    Not all authors are super careful in their drawings to indicate that the gamma rod spacing definition is center-to-center with the driven element, but this usage in the math seems to be universal.

    The gamma rod, along with the driven element to which it runs parallel, can be viewed as a two wire transmission line with (potentially) different sizes of wire. This transmission line (as is well known among the EE types) has a characteristic impedance, almost always called Zo, and its value in ohms is

     -1222Zo = (376.73/2) cosh( (4S-D-d)/(2Dd) ) (1)

Here the 376.73/2 is sometimes set to 60 (true value is 59.96…). The 376.73 ohms is

    the well known, nature-given, impedance of free space. This expression is fine for any -1consistent length units. The cosh function is the inverse hyperbolic cosine function (aka,

    arc-cosh or acosh) that is not always available for a calculator or programming language. However, it can be evaluated exactly by

     -121/2cosh(x) = log(x+(x-1)) . (2) e

    Now the conventional story is that a short length L of gamma match transmission line acts like a shunting inductance with reactive impedance of

     j X = j Zo tan (2 L / ) (3) ;

where is the wavelength. Sometimes the quantity 2L/ is expressed in degrees of

    phase to describe the length. Of course, L and must be in the same units.

    It turns out that tapping the driven element (no longer split) of the antenna off center with a gamma section of spacing S, driven element diameter D and gamma rod diameter d provides a “step-up” in impedance by a factor we will call “SU” here. It is far short of

    obvious that

     -1222-12222 SU = [1 + cosh( (4S-D+d)/(4Sd) )/ cosh( (4S+D-d)/(4SD) ) ] (4)

This SU is the famous factor “Z Ratio,” associated with a folded dipole, that is often

    plotted for various S/D (or S/d2) and D/d (or d2/d1) values such as below. Note that it is not dependent on the length of the gamma rod.

    Figure 2. The Impedance Step-Up for folded dipole and other configurations.

     This figure is taken from W3PG’s article on gamma matching (QST, April 1969) - it is

    said to be originally taken from an unspecified edition of the ARRL Antenna Book. The results are consistent with the above equation for SU.

     thIn ON4UN’s Low-Band DXing book (4 ed., p13-36) there is a reproduction of this plot,

    rather than a copy, BUT the vertical axis is incorrectly labeled d1/d2 where it should be d2/d1.

    Now we have all the components needed to calculate the effects of a gamma match impedance transformation. The standard equivalent circuit used consists of a stepped up

antenna impedance, Za*SU, shunted by the inductance impedance, j X , of the short ;

    transmission line formed by the added gamma rod as seen at the input end of the gamma rod. In later discussion there comes up the question as to whether the stepped up impedance should be Za*SU or Za*SU/2. This seems to be a point not fully resolved among gamma match publications. For the moment we will carry on without the ? but it will be addressed later.

Figure 3. Conventional gamma match equivalent circuit.

    So calling the complex output impedance Zout, and knowing how to combine parallel impedances, we arrive at the complex equation

1/ Zout = 1/(Rout+jXout) = 1/(j X ) + 1/(SU*(Ra+jXa)) . (5) ;

Thus given a gamma rod length to calculate X from (3) one can solve for the Zout with a ;

    simple bit of complex number arithmetic. So the matter of the transformation equations is resolved (with the possible exception of the ?.).

    However, the frequent question is just how to select the gamma parameters to provide a good match to a feed line of known characteristic impedance (call it Ro). It is possible to do this by iteration (guess and correct) on the above equation but that is not necessary if you start with S, D, and d as specified quantities (which may still require some iteration if you don’t like the answer you get). Generally there is not a unique solution for S, D, d, L

    and series capacitance C that you can find by just requiring Rout to be to equal to the feed line impedance and also requiring that Xout be nulled with that additional capacitor.

Solve for Length/Capacitance

    If you specify desired S, D, and d (which then gives SU and Zo) and you know the unmatched antenna impedance, it is possible to solve directly for L and C (if a solution exists) by a direct algebraic approach with Eq (5). This can be done by inverting the expression to read

Rout+jXout = 1 / [ 1/(j X ) + 1/(SU*(Ra+jXa)) ] . (6) ;

    Then using just the REAL part, set the desired Rout to be the desired feed line impedance Ro (say 50 ohms), solve for X and finally convert that to L. This solving is ;

    straightforward but a tedious (as they say in math class) and it leads to a quadratic

    equation for which the solution with the correct sign must be taken, the one that provides a positive X. Then using this result you can return to the IMAG part of the equation and ;

    calculate the resulting Xout, which will be positive (inductive) if all is well. Finally using the frequency, f, and setting Xout = 1/(2fC), you find the C that must be placed in

    series with Zout to cancel Xout and make it all look like a pure resistance to match the feed line with an SWR of 1:1. The equivalent circuit is below, where X is found from ;

    the calculated L needed to make Rout=Ro as indicated. The ARRL code Gamma (the Basic source code for it, GAMMA.BAS, is included) has the solution coded up in real number format. Beware that solutions are not always possible especially if Ra is small compared to Ro and/or if Xa is too inductive or even too close to being inductive. For a 2solution SU* |Za|/Ra>Ro is required, so smaller Za requires a bigger step up.

Figure 4. Fully gamma matched equivalent circuit.

The Calculators and Tables

    There are two gamma calculators that have some degree of distribution plus another of interest that have been found. All three find L and C using S, D, d, feed line impedance and frequency as inputs. There is the ARRL Antenna Book CD with “Gamma,” the

    ON4UN’s ARRL Low-Band DXing book CD with “ON4UN's Low-Band DXing

    software” which has a GAMMA /OMEGA/HAIRPIN MATCHING feature (but no

    source code) and finally there is (or was) a MATLAB code found on the website of a McMaster University EE Professor. (After I sent an email inquiry about that code, which was not answered, public access to the MATLAB codes appears to have gone away. Hmmm.) Notes with the code indicates it is based on the book by Balanis.

    Apart from two matters, the three codes appear to carry out essentially the same operations, although the details differ and there maybe some approximations in the ON4UN code that provide a modest deviation relative to the others. One matter is the factor of 1/2.

    This factor is highlighted up front for the ON4UN code for which dipole-type antennas are the default use. For vertical antennas, the code gives the explicit instruction that this case, you should take the unmatched vertical antenna impedance, Za, and DOUBLE it when asked to type in the antenna impedance.

    It turns out that if the ON4UN code is run in its vertical mode (instituted by the user doubling Za before input) it provides results that are quite similar to the (not doubled) ARRL “Gamma” code. Without that doubling they are rather different. The MATLAB code, which appears to carry out the solution steps described above, explicitly says it is working with a dipole. When this code is run, it generates results that are quite similar to the ON4UN code run in standard (dipole-like) mode. Finally if you go in and modify the MATLAB code (which transforms the unmatched antenna impedance to the tap point impedance by (Ra+jXa) -> SU*(Ra+jXa)/2 (note the factor of 1/2), by removal of the 1/2 in the transformation (or multiply the input Za by 2 before typing it in), the altered results are virtually identical to those from the ARRL “Gamma.”

    The second issue of interest is the discovery that the ARRL “Gamma” code will produce wildly incorrect results for some parameters. A brief investigation of the math suggests that the Gamma code is fine so long as SU*Ra > Ro is satisfied. When 2SU* |Za|/Ra> Ro > SU*Ra the full complex equations indicate that there are legitimate solutions but if an Ro in this range is input to Gamma, the results bear no relation to the actual solution and a negative capacitance is recommended (which may be a tipoff that there is a problem). This bad outcome is the result of the selection of the wrong quadratic equation root. Whatever the basis for the erroneous output, there are solutions that cannot be found with Gamma. Interestingly enough, if the ON4UN code is run in the double the Za mode, it can find the solutions for those Ro > SU*Ra. All this is discussed in grim detail in another note.

    There are additional materials in the two ARRL books that need to be noted.

     st In the Antenna Book (21ed) page 26-10, two examples of the “Gamma” code use are

    provided. The first example is for a yagi and the second for a shunt fed (and we assume grounded) tower which is a vertical. All the calculated values for the two examples quoted are completely consistent with the “Gamma” code from the CD going with that

    book. However, in the text for the yagi, it says “enter the choice for a dipole” but (the current?) “Gamma” does not offer any such option.

     thFinally, the ON4UN book (4 ed) contains Table 13-10 with yagi gamma match

    examples. Generally the results quoted are close to (but sometimes not identical to) the results from the ON4UN code that came with the same book. However, for the cases for Xa= -20 and 0, the quoted results for C are clearly wrong (and different from the ON4UN code calculated results). This same table was copied in the Silver article published in QST in Dec 2002. The ARRL has been informed of the issue of the table columns.

Dipole versus Vertical

    There seems to be some undertone of technical disagreement on the matter of treatment of dipole-types and monopole-type verticals for gamma matching. The ON4UN code is quite explicit and in his book (see Fig 13-56, p13-36, lower right) the splitting of Za into equal parts is very clear.

    Yet in the BASIC source code for the ARRL Gamma calculator there is a notation:

     “12 REM Removed corrections RA/2 and XA/2 per W6NL, Apr 1, 2000”

    which suggests that the different treatment was once considered and is now rejected.

    I have exchanged some emails with Dave Leeson, W6NL, who has contributed a variety of antenna expertise in ARRL and other publications. He provided a preliminary copy of his in-progress technical article addressing some of the issues about gamma (and tee) matches. In the context of the current discussion, the upshot of W6NL’s conclusions is

    that there should not be a factor of ? appearing in the gamma transformation and the existence of it at all resulted from a misinterpretation of early work using monopoles that used a factor of 2 to convert to the dipole equivalent impedance. From W6NL’s

    discussion, this seems to have resulted in a common but improper inclusion of the factor of ?, apparently taken by some from the Balanis book, for the transformed antenna tap point impedance ( i.e., the apparently incorrect SU*Za/2) and this was then propagated into the gamma literature. Finally W6NL points out that for a monopole the factor ? should also not be present when the Za used is the impedance of that monopole. (Again note that if the monopole impedance is taken as that for an equivalent dipole (i.e., twice the impedance of the monopole), then and only then should the ? be used.)

Other’s Experimental Data

    I have not seen any reports of serious experimental data taken to verify the gamma transform theory, but then again this is a niche market. Examples seen in the ham domain generally have the property that a calculator is used to estimate L and C and then (sometimes much) adjustment ensues. Healey’s article is a mix of theory and experiment

    but it appears that the theoretical approach has aspects different from conventional understanding, plus it is not really explained how the step-up factor is used, and then there are the dreaded Smith charts.

    My motivation for looking at the gamma match arose from the matching needs of a two element parasitic vertical array constructed for 30 meters. This was encouraged by an article by Hulick (QST) who used a gamma match for his 80m array with shortened loaded wire elements. Unfortunately the author does not reveal what the starting unmatched antenna impedance actually is so it is hard to draw any firm conclusions there.

    One pretty clean example of data was provided in W6NL’s unpublished article where a half-folded dipole (i.e., one with only one of its elements was folded, and using the same tube size for the gamma rod as the driven element, d=D) was a part of a larger yagi system. Here is was found that the original (not folded) impedance (25 ohms) was transformed into a 4X impedance (100, not 50) by actual measurement. For this approximately quarter wave length gamma, the transmission line contribution drops out (X is very large) so this indicates that the factor of ? should not be present. ;

“Calculated” Data

    By way of qualification, the phrase “calculated data” is, of course, a oxymoron but they

    may provide insights. Generally, the models used are certainly not full solutions to Maxwell’s equations and there are a number of approximations, omissions and fits based on the original modeler’s judgment. With that in mind:

W6NL’s unpublished article also mentions some instances where some antenna modeling

    colleagues have tested the much discussed factor of ? by direct construction of gamma matches with wires as part of the models. The reports were always that the ? factor was not appropriate.

    EZNEC provides a ready tool to get a handle on some of the issues raised. Using it, a set of models were generated for a dipole, and variants, and a vertical, and variants.

Dipole

    Test of gamma matching transformation using EZNEC with the following sequence of dipole configurations:

Figure 5. Dipole models and variants.

    In EZNEC, first take a free space dipole, and for the sake of round numbers, make it 60’ each leg with 1” diameter wires. Result is resonance at fo=3.95 MHz and Z(fo) = 72.1-

    j0.2 within the frequency resolution used. In all cases, the segments on the longer wires were approximately 2’.

    Next, make it into a folded dipole with second wire also with 1” diameter and with separation of 2’. Now fo=3.85 MHz with Z(fo)= 286.4-j.07 (Note that 4 X 72 = 288 so this is very close to the expected result when SU=4 for this D=d case.) At 3.95 MHz, Z(3.95) = 309.4+j 87.

    Now into less charted territory, we make the folded dipole into a HALF-FOLDED dipole so it is fed like a gamma match where the gamma rod extends all the way to the end of the full dipole. Now fo = 3.895 MHz and Z(fo) = 287-j0.3. And Z(3.95) = 297+j56. Therefore a half folded dipole (at least with SU=4 for D=d) will have the essentially the same impedance as a full folded dipole with that same 4X step up.

    So this says that the gamma match transform over a full half dipole should provide a 4X impedance change. This goes directly to the old factor of 1/2 and it says that the 1/2 should not appear for this (dipole) case. (Note that for a dipole with quarter wave legs, the shunt inductance reactance X in the gamma transform becomes infinite and so it ;

    does not contribute.)

    Next we can compare the way the impedance changes from a gamma transform versus EZNEC for some more general cases. Take the prior simple dipole and shorten it from 60’ to 50’ per leg. Now the Z(3.95) = 42.5 – j204.3 . This will now be an example of

    unmatched antenna impedance, Za, that can be changed with a gamma match.

Now we add a gamma feed to our shortened dipole in the standard way with a 2’

    separation and D=d=1”. The gamma rod length, Lg, will be varied and the effects determined on the impedance at the output input of the center gamma rod feed. At the same time we can calculate (see Eq (6)) the expected resultant impedance using the same numerical gamma transform discussed before which is the same as appears in the gamma match calculators. This will be done using two versions of the transformation, one with the factor 1/2 included (such as the ON4UN dipole version) and then without that factor (such as the ARRL Gamma code). The tabulated results for the resistance and reactance at the gamma input for five gamma rod lengths are provided below.

DipoleEZNECXform / 2 Xform NO /2

    Lg (ft)RgXgRgXgRgXg

    301969492007-272190860

    256758881199975522

    2027.438718458532.6364

    1511.325550.531713.5235

    104.2515813.41654.7139

    Table 1. Shortened dipole gamma transform effects.

    To ease the analysis, these results are now plotted with the two transformed versions (with and without the ?) normalized with the EZNEC values for both Rg and Xg. Of course, a result that is close to EZNEC will have a value near 1 (the green line). Red is for factor 1 and blue is for factor ?.

    14

    Rg(1)12

    Xg(1)

    10Rg(/2)

    Xg(/2)

    8

    6

    4

    2

    0R and X normalized to EZNEC

    010203040

    -2

    Lg (ft)

    Figure 6. Shortened dipole model results for gamma transforms with EZNEC versus gamma length variations.

It seems clear that the transform without the ? agrees very well with EZNEC while the

    case with ? included is very poor. Note that EZNEC does not have any implicit model for a gamma match so this simulation is done purely by the addition of wires to the model. (If wire loops are too small, EZNEC will refuse to do the computation but the loops used here are acceptable to the code how accurate the results may be is subject to discussion.)

Vertical

    Next we test the gamma matching transformation using EZNEC with the following sequence of perfect ground vertical configurations:

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