Calculating focal length of a PrimeFocus dish that has a hole in the middle

[a33]

Hi guys!

For calculating the focal length and f/D of a prime focus dish, you have to measure width and depth of the dish. That is commonly known.
Focal length = W^2 / (16 * d)

Has anyone ever had the problem of wanting to measure the exact depth of a dish, that has a hole in the middle? Or that has a plate in the middle (holding the arm for the LNB), that doesn't continue the parabolic shape of the reflector?

I don't know if this really could be a problem, but if it is, there is a simple solution to still get exact measurements of the dish, to calculate the focal length from.
For this solution you don't need measurements at the dish center anymore.

You just need the length of a chord, somewhere on the dishface ("A chord of a circle is a straight line segment whose endpoints both lie on the circle").
And at the exact midpoint of that chord, you need to measure the depth (perpendicular to the dish face, of course).
From those inputs you can again calculate the focal length: Focal length = (chordlength)^2 / (16 * depth) .

So, it is basically the same equation as above, but it can be used for any chord on the PF dish; not just the chord at the middle.

The principle behind this method is a special property of paraboloid dishes, that is described in the attachment of this post: Calculation of the focal length of an offset satellite dish antenna, Revisited - SatsUK
Any slice taken out of a parabola, parallel to the symmetry axis of the paraboloid, will give the necessary inputs to calculate the one and only focal length of that paraboloid (see e.g. figure 1 on page 2).

For prime focus dishes this property is even simpler than for offset dishes, as the dish face of a PF dish is always (by definition) perpendicular to the symmetry axis of the paraboloid. So you can use any chord.
(For offset dishes, the choice of chords is limited to chords perpendicular to the long (height) axis of the dish; so: parallel to the dish width axis.)


I hope this method can be of use to someone!
So I thought I'd share it with you.

Greetz,
A33

 

[a33]

BTW.
To add about calculating the focal length of a PF dish with a hole at the center (copy of what I posted elsewhere):


Today I applied a new equation that I derived for calculating the focal length of offset parabolic dishes, to a prime focus dish (with offset angle ZERO).
(The original offset equation is using a single depth measurement at a (randomly) chosen height of the dish, as opposed to measuring depth at the center, or depth at the deepest point.)

Doing that (applying the equation to a PF dish), I found an other equation to easily determine the focal length of a prime focus dish that has hole in the middle.
For that, you have to measure:
1. the depth 'd' (of course, perpendicular to the dish face), somewhere along the vertical center of the dish;
2. the height 'h' where you do the depth measurement (measured along the vertical dish face; from bottom of dish, to the depth measurement location);
3. the total height 'H' of the dish (which is of course equal to the width of the dish).

The equation for the exact focal length of the dish then is:
Focal length = 4 * h * ( H - h ) / (16 * depth)

(offtopic: )
For those that can do a bit of mathematics:
When you apply this formula to a measurement at the center of a PF dish (where h = H/2), you can rearrange the formula to the well known equation for PF dishes: focal length = (diameter)^2 / (16 * depth.at.the center).
(backontopic: )

Maybe this approach is helpfull, too.
Of course the measurements must be done with the necessary care and precision....

greetz,
A33

 

[arlo]

Been there, done that. No matter what you'll still find you have to move the feed components in and out to tweak for max. signal and twist H/V signal peaking.
I guess you could slap a coffee saucer in the middle. Don't spill your coffee.
Good stuff though.

 

[a33]

No matter what you'll still find you have to move the feed components in and out to tweak for max. signal

That, I guess, would be because of measuring inaccuracies, or parabolic surface inaccuracies.

The nice thing of the last equation is: with multiple measurements, you can calculate a 'mean' focal length. So that you are not dependent on just one measurement, with its (parabolic shape-, or measuring-) inaccuracies...

But you're right, always check reception, in practice!

Greetz,
A33