Antennas radiate in a spherical pattern and antenna gain is typically given as a function of angle off-boresight. If we want x% spherical coverage, we need to use the lowest antenna gain within that spherical coverage. We need to have a formula that relates the angle off-boresight to the spherical coverage. This spherical coverage is the area of a spherical cap. The circumference of a circle is defined as: $C=2\pi r$ In this case $r$ is the radius of the bottom of the spherical cap. The equation can be rewritten as a function of the total radius $R$ and the polar angle $\phi$: $C=2\pi R\sin{\phi}$ The arc length of a curve is: $s=R\phi$ Where $\phi$ is the polar angle in our case. The differential arc length is thus: $ds=Rd\phi$ ![[4 - Attachments/image 9.png|image 9.png]] This is a sideview of a sphere with a shaded spherical cap. The differential area of a spherical cap is the circumference times the differential arc length: $dA_{sc}=2\pi R \sin{\phi}\cdot Rd\phi$ For an given angle off-boresight $\theta$, the area of the spherical cap is given by: $A_{sc}=\int_0^\theta2\pi R^2\sin{\phi} \ d\phi=\left.2\pi R^2(-\cos{\phi}\right|_0^\theta$ $A_{sc}=2\pi R^2(1-\cos{\theta})$ Finally the spherical coverage is given by a percentage so we need to divide the area of the spherical cap by the total area of the sphere: $A_{sc[\%]}=\frac{2\pi R^2(1-\cos{\theta})}{4\pi R^2}=\frac{1}{2}(1-\cos{\theta})$ ## Application Note that we can put two antennas on either side of a spacecraft and assuming we are sufficiently far away, will each be able to form a half-sphere cap and combine to have fully spherical coverage. For our spacecraft we want a spherical coverage of 91%: $91\%=1-\cos{\theta}$ $\theta=\cos^{-1}(1-0.91)=84.8\%$ For this pattern, the 84.8% works out to have an antenna gain of 71%. ![[4 - Attachments/image 1 3.png|image 1 3.png]] # Now Flipping it on a Sphere The gain of an isotropic antenna is even across the entire sphere, but we can concentrate the power by starting with the isotropic antenna $\theta=2\pi$ We get: $G_T(\theta)=\frac{\text{area over isotropic antenna}}{\text{area of narrower spherical cap}}$ $G_T(\theta)=\frac{4\pi R^2}{2\pi R^2 (1-\cos{\theta})}=\frac{2}{(1-\cos{\theta})}$ We can see that when we have an isotropic antenna, that is, $\theta=2 \pi$, $G_T(\theta=2\pi)=\frac{2}{(1-\cos{2\pi})}=1$ As we narrow the beam, say to $\theta=21.3^o$ (GNSS $\alpha$) $G_T(\theta=21.3^o)=\frac{2}{(1-\cos{21.3^o})}= 29.28$ As $\theta$ gets closer to 0, the gain increases dramatically, but this bumps up against the limits of physics, we cannot have a completely linear beam or we end up in laser, optical physics. More research must be done on this. ![[Connect#Connect]]