The Band Numbers: Wavelength, Antenna Scale, and Frequency Loss
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.
See the relationship before changing it
The figure reads from left to right. The blue card is carrier frequency. The middle card applies this page's rule. The green card is quarter-wave antenna length. Walk the arrows once: set the input, apply the rule, then read the result with its unit.
The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only carrier frequency, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 2400 MHz.
- 2
Name the relationship. quarter wave = 75,000 / frequency in MHz
- 3
Substitute with units. 75,000 / 2,400 = 31.25 mm
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change carrier frequency
Try Predict the direction of quarter wave = 75,000 / frequency in MHz. Test another carrier frequency, then compare quarter-wave antenna length.
Observe Higher carrier frequency shortens the free-space quarter-wave antenna. Reset carrier frequency to 2400 and compare quarter-wave antenna length.
Explain Higher carrier frequency shortens the free-space quarter-wave antenna.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Compute quarter-wave lengths at 900 MHz, 2.4 GHz, and 5 GHz, then compare the same-distance frequency loss for 2.4 GHz versus 900 MHz.
Antenna scale falls from about 8.3 cm to 3.1 cm to 1.5 cm, while moving 900 MHz → 2.4 GHz costs roughly 8.5 dB of free-space margin.
Wavelength is inversely proportional to carrier frequency, making resonant elements shorter at higher bands, but the 20 log10(f) term simultaneously increases free-space loss.
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.
Technical boundaries: This free-space quarter-wave comparison excludes enclosure detuning, ground-plane size, antenna efficiency and matching, body absorption, walls and diffraction, polarization, legal power limits, and available bandwidth.