This is why FEM codes exist - the analytical expressions get you most of the
way there, but then as you start to find equations for more of the
interactions, pretty soon you wind up at NEC - which basically models
everything as segments of conductors with a current, and the interaction of
each segment with all the other segments.
NEC does use a simplified model for the "wire" - it takes into account
dielectrics and skin effect (I'm in the middle of figuring out what ZINT does,
which is the core of the "resistance and inductance of a segment", so I can
make a Python version that's not just a copy of the Fortran). It does NOT
model things like proximity effect (that the current along a conductor next to
another conductor is not evenly distributed around the circumference), nor does
it model currents flowing around the conductor.
For that, you need a fancier tool - Maybe HFSS does that.
But it would correctly model the interaction of the hairpin with the element
and the boom.
On Thu, 11 Jan 2024 09:42:19 -0800, Brian Beezley <k6sti@att.net> wrote:
Accurately calculating rod length and antenna reactance for a hairpin
match turned out to be more complicated than I expected. I realized
pretty quickly that my boom model was dubious. I abandoned it. When a
hairpin shorting strap effectively passes through the boom in the style
of Hy-Gain HF Yagis, the boom reduces its inductance, which increases
the required rod length. Even simple contact with a boom will have some
effect. I may address this problem at some point with a finite-element
magnetics model. For now I provide some guidance in the documentation.
In addition to predictability, there is another reason to insulate the
shorting strap: Any antenna imbalance may cause stray current to flow
onto the boom. This can degrade the antenna pattern and bypass a
carefully designed current choke at the feedpoint.
I decided to take advantage of the telegrapher's equations to calculate
loss. The program models several aluminum alloys as well as copper. It
displays hairpin inductance and Q. You can use the figures to design a
coil inductor to replace the hairpin. In the cases I tried, hairpin Q
was almost always substantially greater than that of a practical coil. A
hairpin match is very efficient.
Since it was easy to do, I decided to calculate the mutual inductance
between the leads and hairpin rods. The level of interaction surprised
me. I knew that lead self-inductance could affect a hairpin match, but I
didn't realize that magnetic interaction with the rods could be so
substantial.
Much of the complexity I encountered came from trying to model hairpins
with the feedpoint offset from the rods. I've seen photos of such
designs, which effectively put part of the driven element in the hairpin
circuit. I wanted to allow for them. This geometry spawns several
additional self- and mutual inductance calculations. The program handles
most of them, but I found mutual inductance between the leads and rods
too complicated to calculate for this geometry. It can be done, but I
ran out of steam. Keeping the leads and rods coincident provides the
most predictable design.
A hairpin match is easy to adjust. The purpose of my program is to make
adjustment unnecessary.
http://ham-radio.com/k6sti/hairpin.htm
Brian
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