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Yagi-Uda antenna driven element impedance question

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E Kafeman:
>Even for a plain old half wavelength dipole, the often quoted "73 " is only valid if the antenna is around one wavelength or more above ground

It is not correct if it is antenna impedance. If we ditch reactance part and only focus on resistance is it somewhat more correct. At one wavelength above ground have the antenna resistance crossed 73 Ohm four times from lower position and upwards.This height can be expressed as [0.23 lambda + (n x 0.25 x lambda)] as first height above ground in air is around 0.23 wavelengths when first crossing of antenna radiation resistance of 73 Ohm occur for a simple center feed dipole antenna.
This resistive value is same resistive value as for antenna radiation resistance in free space why it can be a bit interesting when trying to understand dipole antenna nature.
Reactive part do not share this behavior as can be seen in my above dipole measurements and shown in Smith chart so I assume it is resistance you is referring to, not impedance.

Antenna impedance and resistance are different animals not living in same dimension.
Impedance is a value calculated for alternating voltages.
It is an expression of voltage/current wave ratio and phase difference.

Antenna radiation resistance is neither same as measured antenna resistance, as the later includes antenna resistivity losses.
1-10 % of total resistance are normal losses for a typical well designed horizontal wire-dipole in range 10-100 MHz at decent height and common type of ground but can be a lot more if ground below antenna is a less ideal electrical reflector.
In controlled environment can losses be lover.

>Other heights offer different impedances, & some common heights are a good match to 50 .

Maybe are you thinking about 0.18 lambda which is the only height above an ideal ground which an ideal horizontal antenna dipole resistivity radiation is reduced to around 50 Ohm due to ground height.
Even lower height -> lower resistance. It is however a steep resistance curve not really good for stable antenna design.
All other higher heights for an ideal dipole will all have an resistance higher then 50 Ohm and will stabilize at an impedance of 83∠300 Ohm for heights closer to free space.

Impedance variation dependence with height above ground is a bit more complex then the resistive variation.
Both your statements are covered by above by me previous shown Smith chart. It is an dipole antenna which impedance is measured at different heights above ground.
Check these measurements and compare how it agrees with your statements.
Smith chart is mayor method to show antenna impedance.

Basic knowledge about "plain old half wavelength dipole" impedance behavior needs some theory to be understand.
I know that the word "theory" is scaring for many lacking antenna knowledge but with minor knowledge can big mistakes be avoided.Below 7 minutes can be well invested dipole knowledge intended for beginners in EM-theory.
Dipole antenna impedance math formulas can be a bit scaring when first seen but it is very simple explained in below video.
Knowledge about AC circuits and reactive component math in general is assumed prior knowledge.
Linked site below is well known because it explain many aspects of antenna theory in simplest possible way without making it wrongly oversimplified and very little prior knowledge is needed:

EZNEC 7.0 is now free, and 4NEC2 also works reasonably well.  These are method-of-moments simulators that can be used for calculating feedpoint impedance the radiation pattern, and should produce a result that is fairly close to reality, though inevitably a design may require minor tweaking depending on deviations of the environment and antenna materials from the idealized method-of-moments calculation.  These programs can take a little of the sting out of writing NEC2 models.

A conventional current balun for HF would be something like wrapping a coaxial cable several times around a ferrite toroid.  At VHF/UHF, the capacitance between adjacent turns of the winding, even though it is on the order of a picofarad, greatly decreases the choking impedance of the balun.  A better choice might be to pass a thin coaxial cable, for example RG-174 or RG-316, through several binocular core apertures.  Such binocular core apertures are used to make the common 75/300 ohm 1:4 balun used between 75 ohm RG6 cable video transmission coax and 300 ohm antennas to receive terrestrial broadcasting.

It would be fairly easy to measure the feedpoint impedance of an antenna using a VNA, even a common one such as the NanoVNA, but quite a bit harder to measure the gain and radiation pattern of the antenna.  Shield rooms, goniometers, and biconical antennas are often used for such measurements, as they are often used to measure other possibly unintentional radiators for EMI.  It may be very difficult to eke out the last dB out of a homebrew design without a good measurement setup.  Even still, if the ground is nearby, or there are other objects nearby, this could well frustrate achieving the best gain anyways. 

I will note that the idealized calculation of dipole impedance with antenna length (which I have included a graph of below from "Antenna Theory and Design" Figure 8.16 by Balanis) a point at which feedpoint reactance is zero is actually for an antenna slightly shorter than 0.5 waves, actually around 0.47 to 0.48 waves, at which the feedpoint resistance is actually fairly close to 50 ohms.  It's this reason why generally one gets a pretty good match trimming an antenna, because the resistance is going to be closer to 50 ohms when you trim it for minimum SWR.


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