Value chain · CLM-1464
In plain terms
Adding antennas to a row that all send together does two things. The beam gets narrower, and it gets stronger straight ahead: twice as many antennas give four times the strength in that direction. A row 16 antennas long, half a wave apart, makes a beam about 6 degrees wide.
The claim as recorded
For N identical elements in a straight row, sending in step, the power straight ahead is N^2 times that of one element, because N equal waves add to N times the field. The full width of the main beam at half its peak power narrows as the row lengthens: at half-wavelength spacing it is 60.0 degrees for 2 elements, 12.80 degrees for 8 and 6.36 degrees for 16. For a long row it approaches 0.886 / (N d) radians, with d the spacing in wavelengths, that is about 0.886 wavelengths divided by the row length.
Evidence behind this: T0: closed form of the sum of N equal waves, solved numerically, and checked against a second implementation that shares no code (CALC-142). A held peer-reviewed paper (SRC-005, PDF 5 [printed 236], equations 4 and 5) uses the same array factor, with a peak of 20 log10 N in decibels and a main-lobe width that falls as 1 / (N d cos theta0). No textbook is held.
Scope and limits
The figures are for the beam straight ahead; a swung beam is wider by about 1 / cos of the swing angle. N^2 is the power in the direction of the peak relative to one element, not the total power fed in, which is N times one element's. Tapered amplitudes widen the beam and lower the side lobes. The 0.886 figure is a long-row limit, and short rows are wider than it gives.
What it assumes
Far field; uniform amplitudes; elements sending in step (beam straight ahead); identical elements whose own pattern is the same in every direction in the plane drawn.
Where it comes from
This is the project's own working, shown in full:
CALC-142 step 2. The sum of N equal unit waves with phase step psi has power sin^2(N psi / 2) / sin^2(psi / 2), equal to N^2 at psi = 0. Half power is where sin(N x) / (N sin x) = 1 / sqrt 2; solved by bisection for N = 2 to 16, and width times N d tends to 0.886 (1.047 at N = 2, 0.894 at 8, 0.888 at 16, 0.886 at 64). The page model's widths agree to within 0.011 degrees and its peaks exactly.
Basis: spacing in wavelengths; beam width in degrees, full width at half the peak power; power relative to one element
Related records
- Answers question
- QST-013 not published in this release
Cite this
energyaperture.com/c/clm-1464
CLM-1464, Energy Aperture, as of 2026-10-17.