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What challenges are you facing? It’s a small mercy that antenna fundamentals are basically the same across all frequencies. The resonant element must be sufficiently long as to present an impedance match (or be brought down through via a transformer; 9:1, 49:1, or whichever flavor you need).
On 40m (~7Mhz) a dipole would need to be ~67’ long. You can get away with shorter, bearing in mind the compromises which come with that.
What are the compact options short of a small wire and palm tree? I could probably etch that length on a couple of a4 size sheets of copper clad from ABC at around my minimum resolvable pitch size.
I've never really wrapped my head around impedance in this kind of context to the point of a functional fundamental understanding. I know the basic explanations and definitions well. It is like AC resistance; high impedance means low current potential, aka needs buffering or gain; low impedance is deadly microwave transformers and welding type stuff. I have wound my own I/E core transformers and built a dozen switching supplies, but I do not understand what you mean here by using impedance matching with a transformer. I would like to.
Great questions, one which highlights my own knowledge gap beyond knowing that for a given feedline and antenna combination, you’ll have some measure of impedance. At the most basic level, your radio will “see” some impedance value. In the amateur radio world this is generally 50Ω. If our antenna system (feedline + radiator) presents 450Ω (quite common), we use a 9:1 transformer to get it to match. This allows us to use our radio on that system without (1) stray current returning to the radio and damage our transmission circuits, and (2) at full power but with inherent loss of signal owing to antenna inefficiency.
Case in point, I have a commercially-purchased multi-band EFHW antenna which presents varying amounts of impedance to the radio. This system includes a transformer (I think it’s 9:1) so that on the bands of interest, there’s a resistance match and as a result an SWR that’s suitable to make decent transmissions on.
As a tangential example, J-pole antennas have a built-in matching system which uses no special parts. It’s composed of a matching section and radiator. The combination of matching section, radiator length, and physical feedpoint allow this type of antenna to sort of self-manage impedance.
The difference here is that a j-pole is a monoband antenna, and a long wire with transformer can often be functional on many bands, depending on length, where the lowest useable frequency is the inverse of its length.
So by using a transformer, I think you are essentially lowering the voltage to exchange for current. What I do not understand, assuming that simple assumed relationship is correct, is why it is still at thing people do. Like if we lived in the old days of vacuum tubes or bipolar junction transistors, sure it makes sense that a little more current might help. Now, in the era of rail to rail op amps with JFET inputs, I don't understand why anyone needs to create the Eddy current losses of a transformer. Maybe it is safety from transients? But then why attenuate... and why not resistively for a more simple RLC element... very curious now...
There is certainly a lot to learn, and you would benefit greatly from joining the hobby officially. If you are US-based, you can take the amateur radio exam after memorizing the answers for the exam (a legal and encouraged practice), the exam itself can be administered remotely via Zoom.
I am beyond my technical knowledge if I tried to explain why we use transformers to get an impedance match; I only know what we do.
transformers are not the only way to do this, and some other circuits can be used instead. if you take a transformer with 1:2 winding ratio, then if on one side current is 1 and voltage is 1, then on the other current will be 0.5 and voltage 2, which means that impedance increases 4x. in EFHW, it's 1:7 winding ratio and impedance ratio is 49x, which works for end-feeding a half-wave dipole, just as expected (from 50 ohm to ~2500 ohm). that transformer is a limitation on power usable in this antenna and main reason to use this type of antenna is mechanical
most importantly, transformers work nicely only if you have real impedances, so your antenna has to be resonant anyway. l- or pi-network tuner will also handle complex impedances so doublet or random wire will work nicely with it, as long as you can accept weight and losses in tuner
if you have an antenna with impedance of, say, 200 ohms, then you need to match it to transmitter impedance of 50 ohms, or else most of power output from transmitter would be bounced back and will damage output stage of amplifier. because on receive currents are tiny, you can wing it by just using it without any match and connecting it directly
how compact and what do you want to do with it? if it's for receive only, the most compact you can get is ferrite rod antenna, but it's very different from usual wire antennas used for transmission. if using wire antennas, random wire would be fine