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.

how parabolic dish work?

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 mn.

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 = x2 / 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

mt = a / 2f

The slope of the tangent m
t and the slope of the normal mn at point M are related by

mt*mn = -1. Hence

mn = -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 mr = tan(n - i).

also
i + n = 90 degrees.

Combine the last two to obtain

mr = tan(2*n - 90)

tan(2*n - 90) can be written as

tan(2*n - 90) = -cot(2*n)

= -1 / tan(2*n) = - (1-tan2 n)/(2*tan n)
.

Hence

mr = - (1-tan2 n)/(2*tan n)

Substitute
tan(n) = -2f / a into the above and simplify to obtain

mr = (a2 - 4f2) / 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 = mr (x - a)

The y intercept of the above line is given by

y-intercept = b - a*mr

Since point M is on the parabola,
b = a2 / 4f

Substitute b and m
r in the expression that gives the y-intercept to obtain

y-intercept = b - a*mr

= a2 / 4f - a*(a2 - 4f2) / 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.