How is quarter wave antenna length calculated?
In free space the formula L = 300 / f_MHz / 4 is used: it gives 17.3 cm for 433.92 MHz, 8.6 cm for 868 MHz and 3.1 cm for 2.45 GHz. On a real conductor the velocity factor is applied; L(m) = (75 × VF) / f_MHz. On thin wire antennas VF is taken as about 0.95, that is L(m) = 71.25 / f_MHz; the lengths then become 16.4 cm, 8.2 cm and 2.9 cm respectively.
The calculation has two steps. First find the wavelength: λ(m) = 300 / f_MHz. Then divide by four. At 433.92 MHz λ = 69.1 cm, and a quarter of that is 17.3 cm.
The second step is usually skipped but it matters. An electromagnetic wave travels slightly slower than the speed of light in copper wire; this ratio is the velocity factor and is taken as about 0.95 on thin wire antennas. The corrected formula is L(m) = (75 × VF) / f_MHz; for VF = 0.95 it becomes L(m) ≈ 71.25 / f_MHz and gives 16.4 cm for 433.92 MHz and 8.2 cm for 868 MHz. This is why 82 mm of wire is recommended in practice on 868 MHz modules.
A λ/4 monopole does not work on its own; it uses the conductive ground plane beneath it as a mirror and behaves electrically like a dipole. If the ground plane is missing or very small, the radiation pattern is distorted, impedance shifts and the outer braid of the coaxial cable starts to radiate.
In practice, keeping the antenna length "approximately" right is not enough. If the length is wrong, impedance moves away from 50 Ω, SWR rises and part of the transmitter output comes back. Do not shorten or lengthen the antenna supplied with the unit; do not use it in a different band.
Full article: 433 MHz or 2.4 GHz?