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I am interested to know what are the smallest antenna sizes that are efficient transmitters for skywave propagation/ham radio applications. Has anyone been able to design a transmit antenna to work in the entire range of 2-30 MHz with the antenna size of 5-10 m or even lower?

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You first need to define "efficient" :)

For narrow band antennas the Chu Harrington limit gives the maximum possible efficiency and bandwidth of an antenna. See the wikipedia article and its references for some details and graphs. There are lots of compact amateur antennas, say for HF on a vehicle, which have low efficiency because of this.

If you want to cover the entire HF band simultaneously without adjusting the antenna or tuning network, there are several travelling wave antenna designs that do this. None of them are compact. All of these have some loss but give fairly good efficiency over the band. Look at

A large conical monopole could work over the whole band, but it needs to be a full quarter-wave tall which is impractical for most hams at 2 MHz.

Moving up in price, there are various log-periodic structures that can give quite wide band coverage and good efficiency. The LPDA is about half a wave wide at the back and about 1 wavelength long, it needs to be half a wavelength off the ground. The monopole version stands nearly a quarter wave tall.

To answer your question:
For a full-band no-tuning transmitting antenna the best you could do would probably be a monopole or dipole, as fat as possible, with a 4:1 transformer and 400 ohm resistor load at the feed point. The efficiency of this would be ~50% from 15-30 MHz, dropping to maybe 0.1% at 2 MHz. Whether this is good enough is up to you.

Here's a real product example:
The DIPL-A0035 is 6 m long, resistor loaded (but no transformer) and has a gain of -10 dBi at 20 MHz (10% efficiency), dropping at about 10 dB per octave. From an antenna twice as big, at 2 MHz, you can expect about -40 dBi or 0.01% efficiency. A transformer will help a bit.

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Short answer: no.

Long answer: there are compact HF antennas, but they're not efficient. However they are practical enough for some applications. Any ground-plane antenna smaller than a quarter-wavelength (λ/4) or non-ground-plane antenna smaller than a half-wavelength (λ/2) will be less efficient than a "full-size" antenna. The smaller the antenna gets, the worse the efficiency gets. Lots of small HF antenna designs exist; they are all compromises (OK, any product design is a compromise). For some situations, the compromises are worthwhile.

For those who would like to get on HF but are severely space-constrained, I can think of a couple antenna designs that might be worthy of consideration. There are several multi-band vertical antennas on the market that are about 8.2 m (27 feet) high. Typically the bandwidth of such an antenna, meaning the range of frequencies where the SWR is less than 2:1, doesn't cover the entire 40m or 80m bands. For those who have transmatches (antenna tuners), a transmatch will allow more bandwidth. The bandwidth of my HF6V on 80m is quite narrow, 50 kHz at most IIRC, but it works OK inside that narrow range.

Vertical antennas mounted on typical soil, rocks, or pavement require radials; the rule of thumb is at least 16 radials at least λ/4 long. More radials are better, and longer radials are better. If the radials are shorter, the performance of the antenna is compromised. If the radials are longer or shorter in one direction than others, then the performance of the antenna radiating in that direction will be better or worse.

The other type of antenna I'll mention is a magnetic loop antenna. As the size of an antenna decreases, its radiation resistance decreases also; if the resistance of the magnetic loop antenna is carefully managed to make the resistance as low as possible, then the transmit efficiency of a magnetic loop antenna can be tolerable. Small magnetic loop antennas must be tuned with a capacitor, and the smaller the loop gets, the more finicky the antenna is to tune. I've heard that magnetic loop antennas can be difficult to work with, and that they like to cause RFI problems in nearby equipment, but for some situations, they are worthwhile. Beware of marketing claims though.

I suppose I should mention "screwdriver" antennas also. These are compact HF vertical antennas that are meant to be mounted on vehicles. They have large loading coils; the larger the coil the better the efficiency, generally speaking. But since the ground plane of a vehicle-mounted vertical is the size of the vehicle, which is often much smaller than a quarter-wavelength, the transmit efficiency of such an antenna is often quite poor, but any antenna can be better than no antenna, so people buy them. The loading coil is usually tuned remotely using an electric motor; one of the first models allegedly used the same motor as a motorized screwdriver, which is how they got their nickname. Other compact vertical antennas are on the market for people who like backpacking with QRP radios and who can't or won't erect temporary wire antennas hung on trees.

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