Welcome to the Raspberry Pi SE site! Your question may be judged as off-topic for this site (not saying it will, but it may be), but personally, I'm happy that you've asked it. I say that only because radio applications for the RPi don't seem to get a lot of coverage here - which suggests to me that folks are missing out on one of the best application areas for this amazing little computer: Software Defined Radio!
That said, it seems that your question is mostly concerned with the antenna design, and calculations presented in the tutorial. I'll make that assumption, and formulate my answer accordingly. However, if I've missed the mark in my assumption, please let me know, and we'll adjust fire.
For the most part, the equations used in the tutorial you've referenced are based on general guidelines for antenna design. Briefly, here are a few items that you already know, but others might not:
- Wavelength is defined as the velocity of wave (near the speed of light for rf, or ~ 3x108 m/sec)) propagation divided by the frequency of the wave; L = v/f.
- Dipole antennas operate most efficiently when they are at or near resonance, and resonance naturally occurs at specific multiples of the wavelength, the shortest of which is 1/2 wavelength.
Armed with this information, we can calculate the length of a half-wave dipole antenna for resonance at 107.9 MHz:
L = v/(2*f) = 3x108/(2 * 107.9x106)
= 1.39 meters
And so if you built a dipole antenna to this length, connected it to the Pin 4 GPIO (PWM), and drove that pin at a center/carrier frequency of 107.9 MHz, you'd have a reasonably efficient (and probably illegal) transmitter. The software in the tutorial (
Sox) applies the frequency modulation to the combined baseband + RF carrier, pumps it out over the PWM GPIO pin, and now you've got a reasonably efficient (and probably illegal) FM radio transmitter! Which is very cool! (but probably illegal!).
Finally to your question re. why the
100 factor is apparently due to the author's desire to express his result in
cm rather than
m! This conclusion due to the diff between the exponents for the speed of light (v = 3x108) and frequency (f = 107.9x106). But as he's already accounted for this (using 300 instead of 3), and he expresses his answer in
cm rather than
m, then it must be a conversion to
cm. And note that if you ignore the
100 factor, you will get the same answer as the tutorial's author did, but in meters.
Second, I guess the
16 factor is to deliberately reduce the antenna to a size that makes it very inefficient, thereby reducing the radiated power, and perhaps keeping this radiated power below the legal limit.
Note however that the tutorial author muddies things again by dividing by
2! This seems unnecessary as the antenna is no longer a half-wave dipole. In my opinion, this is either the result of a "copy-and-paste" operation gone wrong, or perhaps the author's (or my) failure to comprehend.
How to calculate the range of transmission?
You would need to calculate the Effective Isotropic Radiated Power (EIRP). And since you're building an "electrically small", inefficient antenna, you would need to construct a model of the antenna (or find someone who's already done that). There are some free antenna models available that are reasonably good - EZNEC is one. Be warned that antenna modeling can be rather arcane and very time-consuming!
Once you've got EIRP calculated, you can use that value in the Friis Equation to calculate power vs. range, or more simply calculate the free space path loss, and subtract that value from EIRP.
But understand this before you attempt these calculations: You're going to get a very small number, and the number you get won't be generally accurate at all. If I were doing this, I'd simply use the antenna size provided in the tutorial. I'd then use a small hand-held FM radio, tuned to the transmit frequency, and experimentally determine the range (Hint: It won't be very far from the transmitter using the antenna lengths provided in the tutorial).
Does the voltage and antenna diameter improve the range transmission?
Increasing the voltage will increase EIRP, and thereby increase range.
Increasing antenna diameter will, in general, increase bandwidth very slightly, but will in general not increase EIRP.
May the signal be transmitted without a wire antenna; i.e. only by the pin (PIN 4 GPIO on a 5V)?
Yes. This will (probably) further reduce antenna efficiency, and reduce range. Again, experiment as above using a hand-held FM radio.