## Parabolic Reflection

The gain of a parabolic reflector can be defined using the equation

$G = \frac{\pi^2 * d^2}{\lambda^2}*e_a$

where

- $ G $ = the gain of the dish
- $ d $ = the diameter of the dish
- $ \lambda $ = the wavelength requested
- $ e_a $ = the efficiency of the aperture (.55 to .70) is common for dishes

with the beamwidth of the antenna being

$\theta = \frac{k*\lambda}{d} $

where

- $\theta$ = the half-power (3 dbI down) beamwidth
- $k$ = a factor varying on the shape and reflective pattern of the dish (typical ~70 when $\theta$ is in degrees
- $\lambda$ = the wavelength of the requested frequency
- $d$ = diameter of dish

So with some basic math, the gain of the 6ft dish at GPS frequencies 1575.42 MHz,

- $d$ = 1.829m
- $\lambda$ = 0.19029m

$G = \frac{\pi^2 * 3.345241m^2}{0.0362102841m^2}*.55 = 501.4849$ times mag

$G = \frac{\pi^2 * 3.345241m^2}{0.0362102841m^2}*.70 = 638.2535$ times mag

To convert that into dB isotropic, $ dbI = 10*log(power)$

$G_{dbI} = 10 * log_{10}(501.4849) = 27.002 dbI$ gain

$G_{dbI} = 10 * log_{10}(638.2535) = 28.049 dbI$ gain

with a beamwidth of

$ \theta = \frac{70 * 0.19029m}{1.829m} = 7.282°$

So with some basic math, the gain of the 6ft dish at GPS frequencies 10.455 GHz,

- $d$ = 1.829m
- $\lambda$ = 0.02855m

$G = \frac{\pi^2 * 3.345241m^2}{0.0008151025m^2}*.55 = 22278.0729$ times mag

$G = \frac{\pi^2 * 3.345241m^2}{0.0008151025m^2}*.70 = 28353.9109$ times mag

$G_{dbI} = 10 * log_{10}(22278.0729) = 42.478 dbI$ gain

$G_{dbI} = 10 * log_{10}(28353.9109) = 44.526 dbI$ gain

with a beamwidth of

$ \theta = \frac{70 * 0.02855}{1.829m} = 1.0926°$