The Band Numbers: Wavelength, Antenna Scale, and Frequency Loss
Ada re-derives the chapter’s band wavelengths, quarter-wave antenna sizes, and same-distance free-space loss
ADA · CALCULATION AUDIT
The Band Numbers: Wavelength, Antenna Scale, and Frequency Loss
Band choice is physics bookkeeping: calculate the wavelength, check whether the antenna can fit, then remember that higher carrier frequency also spends more path-loss budget at the same distance.
A wearable device has only a 2 cm antenna budget, which the chapter says makes 2.4 GHz far more feasible than a sub-gigahertz band, while a battery sensor sending a few bytes per hour across a warehouse can still justify going sub-gigahertz for reach. Both choices trace back to the same physics: wavelength equals wave speed divided by frequency, so the example bands near 900 MHz, 2.4 GHz, and 5 GHz each imply a different quarter-wave antenna length. This audit asks the question that trade-off invites: exactly how much physical antenna, and how much extra free-space path loss, does moving from sub-gigahertz to 2.4 GHz or 5 GHz actually cost?
Companion to the chapter Radio Waves and Frequency Bands — every number here comes from that chapter.
Use the chapter's own example bands
1. Use the chapter's own example bands. The table above uses sub-GHz near 900 MHz, 2.4 GHz, and 5 GHz. With wave speed c = 3 x 10^8 m/s, the wavelength calculation is direct.
| Band check | Arithmetic | Audited result |
| Sub-GHz near 900 MHz | (3 x 10^8) / (900 x 10^6) = 0.333 m; 0.333 / 4 = 0.083 m | Wavelength about 0.33 m; quarter wave about 8.3 cm. |
| 2.4 GHz | (3 x 10^8) / (2.4 x 10^9) = 0.125 m; 0.125 / 4 = 0.031 m | Wavelength about 0.125 m; quarter wave about 3.1 cm. |
| 5 GHz | (3 x 10^8) / (5 x 10^9) = 0.060 m; 0.060 / 4 = 0.015 m | Wavelength about 0.06 m; quarter wave about 1.5 cm. |
| Antenna scale change | 8.3 / 3.1 = 2.68; 3.1 / 1.5 = 2.07 | A 900 MHz quarter-wave element is about 2.7x the 2.4 GHz element, and a 2.4 GHz element is about 2.1x the 5 GHz element. |
Frequency also spends path-loss budget
2. Frequency also spends path-loss budget. At the same distance, the frequency term in free-space path loss is 20·log10(f). Comparing two bands cancels the shared distance term, so the extra loss is 20·log10(f2 / f1).
| Same-distance comparison | Arithmetic | Audited result |
| 2.4 GHz versus 900 MHz | 20·log10(2400 / 900) = 8.52 dB | About 8.5 dB more free-space loss before walls, fading, or antenna mistakes. |
| 5 GHz versus 2.4 GHz | 20·log10(5000 / 2400) = 6.38 dB | About 6.4 dB more free-space loss at the same distance. |
| 5 GHz versus 900 MHz | 20·log10(5000 / 900) = 14.89 dB | About 14.9 dB more free-space loss, so the shorter antenna is not free. |
What the mathematics buys you: the lower band helps range because it has a longer wavelength and lower same-distance free-space loss. The higher band helps compact antennas and capacity, but the link budget must pay for the extra loss and the field test must prove enough margin remains.
Every number above is taken from the chapter’s own example bands and re-derived step by step.