While standard FM antennas are designed for picking up stations operating at a particular frequency range, this isn't always sufficient for amateur "ham" radio broadcasting and reception that operates at different frequencies. Calculating the optimum antenna length is possible with a simple formula, though you must take into account the way the antenna is designed.
Confirm the frequency range within which you want to operate. This will depend on your location, the allocated frequencies in your area (see Resources), and the terms of any license under which you are allowed to broadcast.
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Calculate the middle of the frequency range within which you will operate. That is, take the upper frequency, subtract the lower frequency, and divide by two. The result is the megahertz figure you'll use to calculate antenna length.
Divide 468 by the megahertz frequency figure. The result is the total optimum antenna length in feet. (0.083 feet = one inch, 0.166 feet = two inches, and so on.)
Divide the optimum antenna length by two if you are using a dipole antenna, which simply means "two poles." The result is the optimum length for each of the two poles. (Alternatively, just divide 234 by the frequency.)
In practice, you may need to tweak the antenna length from the calculated figure to get the best signal strength for your specific location and setup. Because of this, if you are making and adjusting your own antenna, it makes sense to overestimate the length rather than underestimate. For example, you could start calculating using a figure towards the higher end of the frequency range you are targeting, rather than the lower end. The benefit of doing this is that if you need to tweak the antenna once you've tried it out, it's usually much easier to trim down your antenna as needed, than it is to increase the antenna length.
Rather than using the middle of the frequency range for the calculations, you can use a specific frequency you want to target. This could be useful for trying to receive or broadcast on a particular frequency where the signal is relatively weak, for example because of the broadcast equipment involved at either end.
There isn't a specific ratio for calculating the length of the coaxial cable that you run between the antenna and your radio equipment. Instead, keep it as short as is practical to minimize signal degradation.