What’s with the Gamma Match Equations?
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).
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 2？L/； 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
-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.