How Parabolic Dish Antennas work?
It is shown, mathematically, how parabolic dish antennas work.
An electromagnetic ray parallel to the axis of the parabola (in this case the y axis) is
incident on the inner surface of the parabola; see figure below.
Point M is the point at which the ray hits the parabolic dish. i is the angle made by the incident ray and the normal (in red) which is perpendicular to the tangent (in blue) to the parabola at point M(a,b). r is the angle made by the reflected ray and the normal. According to the laws of reflection, angles i and r are equal.
We will show that all reflected rays due to incident rays, at different positions, intercept the axis of the parabola y axis) at the same.
We first calculate the slope of the tangent m_{t} in order to obtain the slope of the normal m_{n}.
The slope of the tangent is given by the value of the first derivative at point M(a,b).
For a given parabolic reflector (or dish), the equation of the parabola can be written as
y = x^{2} / 4f
where f is a constant.
see Find The Focus of Parabolic Dish Antennas for more details.
The first derivative is given by
y' = x / 2f
The slope of the tangent at point M(a , b) is given by
m_{t} = a / 2f
The slope of the tangent m_{t} and the slope of the normal m_{n} at point M are related by
m_{t}*m_{n} = 1. Hence
m_{n} = 2f / a
Let n be the angle made by the normal and the x axis.
tan(n) = 2f / a (relationship between angles and slopes).
The angle made by the reflected ray and the x axis is equal to
n  r = n  i, since r = i
Let m_{r} be the slope of the reflected ray. Hence m_{r} = tan(n  i).
also i + n = 90 degrees.
Combine the last two to obtain
m_{r} = tan(2*n  90)
tan(2*n  90) can be written as
tan(2*n  90) = cot(2*n)
= 1 / tan(2*n) =  (1tan^{2} n)/(2*tan n).
Hence
m_{r} =  (1tan^{2} n)/(2*tan n)
Substitute tan(n) = 2f / a into the above and simplify to obtain
m_{r} = (a^{2}  4f^{2}) / 4(f*a)
Now that we have the slope of the reflected ray, let us find an equation for the line through point M(a , b) and with the same slope as the incident ray. In point slope form the equation is given by
y  b = m_{r} (x  a)
The y intercept of the above line is given by
yintercept = b  a*m_{r}
Since point M is on the parabola, b = a^{2} / 4f
Substitute b and m_{r} in the expression that gives the yintercept to obtain
yintercept = b  a*m_{r}
= a^{2} / 4f  a*(a^{2}  4f^{2}) / 4(f*a)
= f.
The y intercept is equal to f which is constant for a given parabolic reflector (or dish). It does not depend on the position of the incident ray. Hence all the reflected rays passe by the same point (0 , f) called the focus and f is the focal distance. In practical situations a device is placed at the focus (0 , f) to collect all the incoming electromagnetic energy.
Note: We have used algebra, trigonometry, calculus, physics laws of reflection to explain how a parabolic refelctor work. This is a good example of how are mathematics applied in engineering and physics to solve real life problems.
Exercises:
Find the focus of each of the parabolas whose equations are given below
1) y = 4*x^{2}.
2) y = x^{2} / 8.
3) y = 0.12*x^{2}.
More links and references parabolic reflectors
Tutorial on how to Find The Focus of Parabolic Dish Antennas.
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.

