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I tried to understand the physics of Fuchs antenna and there is one thing that I don't quite understand. This is probably a very silly question, but I spent a lot of time looking for an answer (books, online) and didn't manage to find it.

Modern Fuchs antenna looks like this:

enter image description here

The L1,L2 is a transformer, typically 64:1 or 49:1. It is required because the end fed half wave wire has an impedance about 2500..3200 Ohm, assuming a counterpoise wire is about 0.05 lambda. OK, this is clear.

Also the L2,Cv circuit has to resonate on the frequency of interest (e.g. 7.1 Mhz for 40m band). This is the part I don't quite understand. Why using only a transformer is not enough to convert 2500..3200 Ohm to 50 Ohm of the feed line and feed the antenna as a regular dipole? Why we also need a capacitor to form a resonant circuit? Or, in other words, what happens if we remove Cv?

I'm not an expert in antennas or physics. I would appreciate very much if you could explain it like I'm 10 years old.

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L2 and Cv builds an resonant circuit (or tuned circuit) to the desired frequency.

If you remove the C or change it to a small fixed value and hold on this transformer ratio, you change the antenna "type" to an EFHW.

Generally a transformer has an high bandwidth than an resonant circuit. It has a very small bandwidth.

This means a Fuchs antenna does some filtering on the used wire and a EFHW not so much. So the EFHW antenna should give more noise from differed frequencys to the output than a Fuchs antenna. But it needs not to tuned across the resonant ham bands.

In any case you will not feed the antenna like a dipole. The radiation diagram looks close like it but dipole means always an center-fed antenna. This antenna types are all end-fed.

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It may be simpler to think of the matching network this way:

enter image description here

The high-impedance antenna (in the example, R1 = $3500\Omega$) is connected to a parallel-resonant network consisting of an inductor (L1+L2: note the coupling indicated by the "k1" statement) in parallel with a capacitor (C1). L1 is, in reality, a "tap" on a single inductor comprising (L1+L2). As the "tap" approaches ground - that is, as L1 becomes a smaller part of the total L1+L2 - the impedance presented to the generator (V1) becomes smaller.

This is the impedance presented to the generator vs. frequency:

enter image description here

Judicious selection of components produces the desired match at the operating frequency. Air-spaced and vacuum-variable capacitors are often used to accommodate: the difficulty of knowing the antenna impedance at a particular frequency; the narrow operating bandwidth; the high voltages that come with high-impedance levels. This method, commonly referred to as "voltage feed," is often used with end-fed antennas such as half-squares, etc.

The coupling link, L1, shown in your question is another way to "tap" down on the impedance across L2.

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