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This article describes by example, a method of design of a Gamma match solution using the Smith chart. The algorithm is based on (Balanis 1997), which is consistent with (Healey 1969), probably the seminal work on the subject.
This article does not explain principles of the Smith chart, nor principles of the Gamma match.
The Gamma match is a means of transforming the impedance of a plain dipole to some new value better suited to the application, usually to suit a feed line.
The Gamma arm is assumed to be uniform diameter, and uniformly spaced from the driven element (DE) which is also uniform diameter. This means that the coaxial tuning capacitive stub is inside the gamma arm, and projects only enough to make a connection to it. A lot of Gamma match constructions would not comply with these assumptions, and so the design technique here will not apply so well.
First step is to estimate the impedance at the centre of a plain split dipole DE.
For this example, let us choose 20+j5Ω at 144MHz.
Fig 1 shows that impedance plotted on a Smith chart normalised to 50Ω, Z'=0.4+j0.1Ω.
The diameter of the driven element, the Gamma arm and the spacing of the two determine the first key element of the impedance transformation, and that is the step up ratio due to the apportionment of current between the DE and Gamma arm.
The step up ratio is a function of these three elements, but in the case where the DE and Gamma tubes are the same diameter, the impedance step up ratio is simply 4:1. The 4:1 step up ratio should be usable for plain dipole impedances from about 15+j0Ω to about 40+j0Ω, outside this range unequal tubes will work better, and a more complicated design procedure is needed.
Fig 2 shows the step up applied. Z'=1.6+j0.4Ω.
Now an arc is drawn from Z' to intersect the R'=1Ω circle.
Fig 3 shows the inductive stub added to intersect the R'=1Ω circle. The susceptance of this stub is the difference between B values at both ends. B'=0.490  0.145=0.345S, B=0.345/50=0.0069S.
The reactance X' at the point this arc intersects the R'=1Ω circle is 0.836Ω, X=0.836*50=41.8Ω, this will be used later.
Determine Zo of the transmission line formed by the DE and Gamma arm.
Again, if the DE and Gamma tubes are the same diameter, and assuming the line is air spaced, Zo is simply 120acosh(D/d). Lets assume the tube diameters d are 10mm, and the centre to centre distance D is 40mm, then Zo=247Ω. Alternatively, you could calculate this using TWLLC .
Design the Gamma arm stub of Zo calculated in Step 4 to offset the B value calculated in Step 3. In this example Zo is 247Ω and target B is 0.0069S, and normalised to 247Ω, B'=0.0069*247=1.704S.
Fig 4 shows the arc from Y=∞ to Y=0j1.704S. The length of the arc read from the outside circle is 0.3340.25=0.084λ=30.4°
The the series reactance looking into the Gamma arm, found in Step 3 must be 'tuned out' with a series o/c stub.
In this example, the stub will be made from 5mm tube centred inside the 10mm tube using some insulating bushes. Note that the insulating bushes alters average permittivity and Zo, but if the bushes are small, the effect is small and the solution for an air spaced line is close enough and can be fine tune by adjustment of the length inserted within the outer tube.
Let us calculate Zo for the coaxial tubes.
If we take the wall thickness of the 10mm tube to be 1.0mm, the ID is 8.0mm and Zo for the coaxial line can be calculated as 138log(8.0/5.0)=28.2Ω.
Let us now calculate the length of an o/c stub of Zo found in Step 6 to have cancel the residual reactance to cancel that found in Step 3.
We now design an o/c stub of 28.2Ω line to have a reactance X=41.8Ω, X'=41.8/28.2=1.48Ω.
Fig 5 shows the arc from Z=∞ to Z=0j1.48Ω. The length of the arc read from the outside circle is 0.3440.25=0.0944λ=34°.
Fig 6 shows the complete solution from point X, the impedance of a plain dipole DE, to O, the desired 50+j0Ω load for the feed line:
It is possible to design a Gamma match using a Smith chart.
The procedure given is for pencil and paper, electronic Smith chart software usually allows building the components into a network which allow rapid determination of a solution.
Version  Date  Description 
1.01  09/10/2013  Initial. 
1.02  
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1.05 
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