Find The Focus of Parabolic Dish Antennas

The position of the focus (of a parabolic dish antenna or parabolic reflector) is found in term of the diameter of the dish and its depth.

Focus of a Parabola

We first write the equations of the parabola so that the focal distance (distance from vertex to focus) appears in the equation.
The figure below shows a parabola, its
focus F at (0,f) and its directrix at y = -f.
parabolic dish
We now use the
definition of the parabola. Any point M(x,y) on the parabola is equidistant from the focus and the directrix. Hence the equation:
sqrt[(x-0)
2 + (y - f)2] = sqrt[(x - x)2 + (y - (-f))2]
We now square both sides and expand the squares.
x
2 + y2 + f2 - 2y*f = y2 + f2 + 2y*f
Simplify to obtain the equation of the parabola involving the focal distance f.
y = x2 / 4f

We now look at a more practical situation where we know the dimensions of the dish and we want to find the focal distance which gives the position of the focus relative to the position of the dish as swown in the figure below.
parabolic dish with dimensions: diameter and depth
D is the diameter of the dish, d is the depth of the dish and f is the focal distance.
The points (D/2,d) and (-D/2,d) are on the parabola, hence
d = (D/2)
2 / 4f
Which gives a relationship between the diameter D, the depth d and the focal distance f of the dish.
f = D2 / 16d

The above formula helps in positioning the feed of the parabolic antennas as it gives the focal distance f. Of course in practice the shape of the dish is not a perfect parabola and therefore small adjustments are needed when positioning the feed.

Exercises

1 - Find the focal distance f for a dish that has a diameter D = 80 cm and a depth d = 25 cm.
2 - Find a relationship between the diameter D and the depth d so that the focal distance f is equal to twice the depth d of the parabolic dish.
3 - Find a relationship between the diameter D and the depth d so that the focal distance f is greater than d.

More References and Links to Parabolic Reflector and Antennas

Tutorial on How Parabolic Dish Antennas work?
Interactive tutorial on
Parabolic Reflectors and Antennas work.
Dipole Antennas
Antenna Polarization
Antenna Arrays
Interactive tutorial on the
Equation of a Parabola.
Interactive tutorial on how to
find the equation of a parabola.
Define and
Construct a Parabola.