This helical antenna worked amazingly well for a poor FM radio. I was receiving only 13 channels according to auto-preset function. This adjustment resulted in 27 presets, more than double the number of channels received in clear stereo.
Quick update: I am getting better reception by directing it sideways instead of down.
Executive summary of instructions:
For FM radio, just cut 1.5 or 3 meters meter (59" or 118") of electrical wire, coated or exposed, and wrap it around something like the thick part of a baseball bat. Take it off the "bat" and put a small "hook" on one end of exposed wire to hang off your existing antenna that is retracted (if the telescoping type) as shown in my 2nd picture. Set the radio to a noisy channel. Slowly stretch this "spring" out until you hear the signal get better. If you pull it too far it will start getting worse again, so you should easily see it get better and worse with the stretching. Diameter did not seem to make a huge difference, but it might have had a preference for larger diameters. So a baseball bat is better than a broom handle, but a paint can diameter might be best. Total wire length is determined by speed of light divided by target wavelength. For FM radio: 3E8 / 100 MHz = 3 meters. 1/2 wavelength (1.5 meters) might be better than 3 meters because "ground reflection" might be acting as the other 1/2. With a ground plane and reading the rest of these instructions, you can make it directional. Wire diameter does not matter. I like thick electrical wire because it can get bumped around more without losing its shape. You want at least 7 loops, and more than 15 is not needed.
Easy fine tuning for your radio and specific channels
Pull or squeeze this "slinky" to make the spacing between turns more and less in order to fine tune for any particular channel and radio. Low FM around 88 MHz was improved by slightly closer spacing. Longer wavelengths around 107 MHz needed it to be stretched a little for fine tuning. Longer spacing between turns is less inductance per turn. 4 cm seemed to work sufficiently good excellent for any wavelength.
I was not able to get it to be better than the stock antenna of a good portable radio, but I did not use the ground plane which requires opening the radio up and finding the correct solder points for a coaxial cable that terminates correctly at the helical.
1) Cut wire length equal to a 1/2 multiple of your wavelength (wavelength=3E8 meters / frequency). "1/2 multiple" means 0.5, 1, 1.5, 2, 2.5, 3 ...etc. I don't know if 1.5 is better than 1, but 1 was better than 0.5, and 0.8 was not as good as 0.5.
2) Larger diameter loops are better for a given number of loops, but try any diameter you want. Keep diameter less than wavelength. 5 to 15 loops are normally used. 15 loops is twice the signal as 5.
3) Stretch the "spring" while listening to the signal to find best spacing between loops.
4) Wire diameter does not matter.
5) Ideal circumference is supposed to be = to wavelength (diameter*pi) and "pitch" distance between loops (s) is supposed to be 0.21 to 0.26 of the wavelength. I found 0.17 to be best, maybe because my circumference was a lot less than the wavelength, so step 2 is more important that theory. I found I could get away with much smaller circumferences, like 1/10 the wavelength and still do good. I also found the wire length being a 1/2 multiple of the wavelength to be crucial, which I was not expecting, but then realized "well of course it is" because it needs to resonate in the wire.
Example of directional type of helical: Radio-controlled quadcopter at 2.4 GHz (same as WiFi) is not receiving very far. Wavelength W=3E8/2.4E9 = 12.5 cm. A video shows N=7 loops requires good aiming, so use a little less, N=6. Assume pitch S=0.17*W (stretching will in testing with likely increase). Interestingly, but not affecting reception, 6 loops * S is the coil's constructed length which happens to be about 1 W. A paper tells me circumference C should be equal to W, so total wire length L is about 6*W. I must have L fall on an integer multiple of 1/2 of W. (0.5, 1.0, 1.5, 2.0), so 7 is good. Forcing L=6, I I wonder what effect this has on N, so here's the exact calculation:
L=N*SQRT ( S^2 + C^2)
6*W=N*SQRT((0.17*W)^2 + W^2)
N=5.92 which is close enough to 6, and I can't change L away from 6.0*W.
The diameter will be 1.56". The length, 5". If a ground plane is added correctly and the coaxial cable used correctly, and if the right copter antenna (cloverleaf) is use, it will be hard to beat. I might work great vertically, especially if the copter carries an identical one, which is possible by just hanging a thin wire coil from it's bottom.
Example: 433 Mhz. Wavelength = 3E8/433M = 0.69 meters. Choose wire length 1 times wavelength: 0.69*1 = 0.69 meters. Choose 12 loops and solve for desired diameter: pi*D*12 = 0.69 => D=0.019 meters (1.9 cm). Theoretical ideal spacing between loops, to be changed during testing: D*3/4 = 1.4 cm. Total length = 1.4*12 =17 cm. If this is too long, start over with 0.5 times wavelength, or hope the antenna works better at 1/2 the 1.4 cm loop spacing during your testing, which is very possible.
This is a "normal mode" helical antenna
This is a helical antenna in "normal mode". That means the circumference of a loop is small compared to the wavelength. This makes it non-directional, meaning pointing it in different directions will not help. A ground plane is supposed to help, but my brief testing of that did not. There is also "impedance matching" of the wires leading into the circuit that I can't do.
5 to 15 loops loops are considered a useful number of loops. 15 loops is 2 times better reception or better, but the 5 loops is a much smaller antenna and space is usually a consideration.
"Axial mode" helical antennas are much better
"Axial mode" is the other type of helical where the circumference of a single loop is equal to the wavelength and the spacing between loops is 0.24 times the wavelength (this gives a pitch angle of 13 degrees for maximum gain). I got these numbers from reading the following article that describes how to make axial mode helical antennas smaller with stubs: scholar.lib.vt.edu/theses/available/etd-11242003.../Barts_etd_CH4.pdf
Using a spacing of 0.17 times the wavelength might be better if space is limited. This increases the number of loops at a cost of not having the perfect pitch angle. For FM radio (3 meter wavelength) the 0.24 factor would be 3/pi= 0.96 meter diameter loops and 3*0.24 = 0.74 meters between loops....a whopping 3.7 meters long for 5 loops using the perfect pitch angle. These large things are actually used for TV reception (UHF/VHF)which is very close to FM radio.
In axial mode you point it towards the target, so it is much better. A ground plate is supposed to be used which may double reception. You may have seen DIY versions used for wireless-G reception. I did not find a small ground plate to be helpful in this simple radio case.
How to get world-class reception
I once had a Pontiac 6000 car with the stock Delco radio and the stock antenna, a pair of L-shaped antennas embedded in the front window. Sitting still I could get a nearly-clear channel in every other click stop which is 100 channels. This was in Montgomery AL back in the 1990's. Montgomery had about 7 stations. Mobile, Birmingham, and Atlanta were 100 to 200 miles away. I've never seen anything like it and "experts" would probably say this is impossible. It probably was doing more than 'line of sight", although that entire region is very flat. So if you want great reception, I would look for an old Delco car radio (probably made in U.S. back then) and simulate the antenna connection (looks like an RCA connector but it is coaxial) and the antenna pair. Maybe the entire car acting like a ground plane also helped.