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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.
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