This will be the last word I have on this topic.

I have placed two files he

www.k6mhe.com/n7ws/N7WS_Yagi_Resonant.EZ

and he

www.k6mhe.com/n7ws/N7WS_Yagi_Shortened.EZ

The files are models of my 20-meter three-element Yagi that are as
nearly representative of the physical antenna and its location as I
can make them. The only difference between them is the half-length of
the DE. The actual antenna uses the shortened version with a stub
(Beta) matching/balun arrangement.

Two photos of the details of this are he

www.k6mhe.com/n7ws/YagiFeed-1a.jpg

and he

www.k6mhe.com/n7ws/YagiFeed-2a.jpg

The photos were taken with the antenna mounted on the tower and the
tower folded over in case you're wondering about the orientation.

For the purposes of the discussion I have removed the stub matching
system from the model.

The following transformation and matching exercise can be performed
using a Smith Chart, your favorite computer program or with pencil on
the back of an envelope. I happen to prefer, and highly recommend,
AC6LA's XLZIZL.xls Excel workbook for this stuff.

First let's analyze the full-length, resonant DE version. After
running the analysis we (should) have a feedpoint Z of 26.76 +j0 and a
gain at the selected elevation of 12.91 dBi. The SWR is 1.87:1.

Instead of the integral stub, which Yuri believes is part of the
antenna that is "folded back" along the boom, I will move the matching
system away from this location using an ideal ½ wavelength (34.7 foot)
transmission line with an ideal current balun at the antenna end. I
don't believe anyone would argue that the feedpoint impedance is not
replicated exactly at the input end of this line.

At the input end of the lossless line, the Z is of course, 26.76 +j0.
Because, as will be shown, the stub matching system is nothing more
than an L-network; I will use the same at the input of the half-
wavelength line.

I begin by inserting a series capacitor, C = 448 pF and Q = 1000. At
the input side of this capacitor the Z is now: 26.785 -j25.062. If
using XLZIZL, the loss in this capacitor is shown as 0.004 dB, because
Q is not infinite.

Continuing, I place a shunt inductor, L = 0.6, Q = 200 across the
input of the series capacitor. The resulting input Z = 50.00 +j0.2.
The total network loss is 0.02 dB. This is the baseline.

Returning to the shortened driven element version, after analysis, we
find that the input Z = 24.55 -j25.2 and the gain is unchanged at
12.91 dBi. At the input end of our magical ½ wavelength line, the Z
remains 24.55 -j25.2.

Once again using the L-network system, I find that the series
capacitor is unnecessary and I can proceed by adding a shunt
inductance. Rather than using Yuri's "preferred" discrete inductor,
let's use a "lossy" stub. Instead of using the large diameter,
parallel tube stub of the actual antenna, I'll use a standard
transmission line for the stub. XLZIZL has a number if transmission
lines and their parameters "built in," including the Wireman ladder
lines. The parameters for these are those I derived in my ladder line
paper.

http://www.k6mhe.com/n7ws/Ladder_Line.pdf

Selecting Wireman 553, shorting one end and placing the other in
parallel with the input to the lossless line and doing a little
manipulation and I find that a 14.85" length makes the Z = 50.12
+j0.18. The network loss remains 0.02 dB. So much for this less than
ideal stub vs. Yuri's preferred discrete inductor.

There you have it. The stub matching method is equal to a discrete L-
network in efficiency, it does not detract from the antenna efficiency
one bit, it can incorporate the balun function without additional
components, it grounds the feedpoint, with a little sealant on the
cable, it is weatherproof and unlike Gammas and Tee-matches, it will
handle full power without being prone to capacitor breakdown. It is
not part of the radiator; it is part of the matching network. Period.