Beam Radius Calculations

The half-power point occurs when the power drops to half of its maximum value. Mathematically, this is when

\[G\left( \theta_{HPBW} \right) = \ 4 \times \ \left| \frac{J_{1}\left( kasin(\theta_{HPBW}) \right)}{ka \cdot sin\left( \theta_{HPBW} \right)} \right|^{2} = \frac{1}{2}\]

Taking square root

\[\ \left| \frac{2 \cdot J_{1}\left( kasin(\theta_{HPBW}) \right)}{ka \cdot sin\left( \theta_{HPBW} \right)} \right| = \frac{1}{\sqrt{2}}\]

Solving numerically for \(J_{1}(x) = \frac{x}{2\sqrt{2}}\), we get

\[kasin\left( \theta_{HPBW} \right) \approx 1.616\]

Substituting \(k = 2\pi/\lambda\) and \(a = \frac{D}{2}\)

\[\frac{2\pi}{\lambda} \cdot \frac{D}{2}\sin\left( \theta_{HPBW} \right) \approx 1.616\]

Using the fact that for small angles \(\sin(\theta) \approx \theta\), and on converting to degrees, we get

\[\theta_{HPBW}\lbrack degrees\rbrack \approx \frac{70\lambda}{D}\]

For aperture, \(a\) and wavelength \(\lambda\), since \(2a = d\) and substituting \(\lambda = \frac{c}{f}\) we get

\[\theta_{HPBW\ }\lbrack degrees\rbrack = \frac{35 \times c}{a \times f}\]

Parameters	Values
Configuration	LEO 1200	MEO 10000	GEO35786
Altitude (km)	1200	10000	35786
Band	S band	S band	S band
Antenna Aperture Radius (m)	1	5	11
Channel Frequency (GHz)	2	2	2
\(\theta_{3dB}\) (or \(\theta_{HPBW})\) (Calculation provided below)	5.25	1.05	0.476
Beam Radius (km)	110.26	183.28	297.31

Table-1: Antenna and Beam Parameters for LEO, MEO, and GEO Satellite Configurations

\[\theta_{HPBW} = \frac{35 \times \lambda}{a} = \ \frac{35 \times c}{a \times f}\]

\[LEO\ 1200\ \ \theta_{HPBW} = \frac{35 \times 3\ \times 10^{8}}{1 \times 2 \times 10^{9}} = 5.25{^\circ}\]

\[MEO\ 10000\ \ \theta_{HPBW} = \frac{35 \times 3\ \times 10^{8}}{5 \times 2 \times 10^{9}} = 1.05{^\circ}\]

\[GEO\ 35786\ \ \theta_{HPBW} = \frac{35 \times 3\ \times 10^{8}}{11 \times 2 \times 10^{9}} = 0.476{^\circ}\]

Parameters	Values
Configuration	LEO 1200	MEO 10000	GEO35786
Altitude (km)	1200	10000	35786
Band	K band	K band	K band
Antenna Aperture Radius (m)	0.25	1	2.5
Channel Frequency (GHz)	20	20	20
\(\theta_{3dB}\) (or \(\theta_{HPBW})\) (Calculation provided below)	2.1	0.525	0.21
Beam Radius (km)	44	91.63	131.16

Table-2: Antenna and Beam Parameters for LEO, MEO, and GEO Satellite Configurations

\[MEO\ 10000\ \ \theta_{HPBW} = \frac{35 \times 3\ \times 10^{8}}{1 \times 20 \times 10^{9}} = 0.525{^\circ}\]

\[GEO\ 35786\ \ \theta_{HPBW} = \frac{35 \times 3\ \times 10^{8}}{2.5 \times 20 \times 10^{9}} = 0.21{^\circ}\]

Hexagonal tessellation based on beam radius

In a hexagonal tessellation:

Each beam is represented by a hexagon.
The beams are arranged in a honeycomb pattern to maximize coverage and minimize overlap.
The beam radius, \(R\) is the distance from the centre of the beam to its edge, where the power drops to a specified level (e.g., half-power or ‑3 dB).
In a hexagonal grid, the distance between beam centers (beam spacing) is \(\sqrt{3} \cdot R\)