Just Sharing This Calculation of the focal length of an offset satellite dish antenna, Revisited

a33

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When you want to check or calculate the focal point position of your offset dish, the well known equation of John Legon (from his article Calculation of the focal length of an offset satellite dish antenna) might be handy.

His formula for the focal length of a paraboloid offset dish is quite simple, but he uses the depth at the deepest point of the dish as one of the input measures ~ which is not the easiest point of the dish to ascertain.

I managed to derive an equation, using the depth measure at the center of the dish, instead of the depth measure at the deepest point.
The resulting equation is a bit longer than John Legon's, but not really difficult.
Finding the depth at the center of the dish, however, and measuring thát, I find relatively easy.

The equation is:

A33 Focal length equation for offset paraboloid dishes.png

Attached here is also the article, in which I describe the steps of the derivation: "Calculation of the Focal Length of an Offset Satellite Dish Antenna, Revisited". The base of the derivation is a special property of paraboloid dishes, that I discovered not long ago ~ but it might be common knowledge to mathematicians, I don't know.

I wouldn't expect that everyone would be able to follow the math in the derivation, but for all those who are interested, all the steps are there.

The equations described in the article give exactly the same outcomes as the more extensive calculation procedures that I used before in my (offset) satellite dish spreadsheet calculator, so I'm quite satisfied that all the equations are correct.


So I hope you'll enjoy the reading, and the use of the equation!

Greetz,
A33
 

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a33

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To add to my previous post:

In the article I also describe an equation to calculate the offset angle for an offset paraboloid dish, that has a non-flat dish face. Finding this offset angle, is needed to be able to calculate the focal length and other dish specifications.

For the calculation of the offset angle, only four input measures are needed. There is no need anymore to use the 'water method', that is described in the article of John Legon.
So again, I think the measurement phase for finding all the dish specs is much easier.


The equation is:

A33 Equation for the offset angle of a NON-FLAT paraboloid offset dish.png

Again: enjoy the use of this equation!

Greetz,
A33
 

a33

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I managed to derive an equation, using the depth measure at the center of the dish, instead of the depth measure at the deepest point.
The resulting equation is a bit longer than John Legon's, but not really difficult.
Finding the depth at the center of the dish, however, and measuring thát, I find relatively easy.

The equation is:


BTW. For the non-members, that do not see the 'picture' with the equation, here is a text version of the equation:


A33 focal length equation for offset paraboloid dishes, using depth at the center:

f = W^3 * d * [ (1/(4*(d^2))) + (1/H^2) - (1/W^2) ] / (4 * H)

(f=focal length, W=width of dish, d=depth of dish at the center, H=height of dish.)


Greetz,
A33
 

ozumo

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First attempt at measuring Mk2 Raven Sky Zone 2 in centimetres:

Width = 74.8
Width depth = 10.1
Height = 55.3
Height depth = 4.75
 

a33

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Thank you, @ozumo!

Here are the calculated specs (in column V).

The clearance of this dish is about zero: y-null is -0.25. That means that a multifeedholder, when taking space above the LNBs, will take away signal from the dish.
( I don't know if that also is the case with the Sky dish in this recent topic: Multi sat on a sky dish But it might indeed be an issue, as this dish shows! )


Zone2 dish  Parabolic Specs   Dish ozumo  Schermafdruk_2021-11-16.png


NB LNB viewing angle, etc, is calculated for a flat-faced dish with height 55.3, and corresponding width 51.43
Focal point position can be found by top string length, and bottom string length.

In the (small) collection of offset dishes specifications that I now have, LNB viewing angle is usually between 70 and 80 degrees (or, average, about 73 to 74). (But a few exceptions below 70, or above 80. And I'm not an expert on this!)
This dish has 77 degrees. I guess this would mean, that a normal Ku LNBf (as opposed to a special feedhorn Sky LNB ) would 'see' the whole vertical part of the dish (no spill-over), but just a small horizontal part of the dish.

Specification of the indicated points is shown in this picture (I guess I must try to make it a bit smaller, I think...?) :

A33  Offset satellite dish specs calculator  Picture of the relevant points.png


So I'm curious now, whether the LNB holder actually puts the LNB in the right position?

Greetz,
A33

Edit: a few additions, to clarify.
 
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ozumo

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Thanks for this, very interesting! I will do some more measurements with these figures tomorrow.

The Sky LNB which came with the dish (c.2004) only slightly outperforms current 40mm universal LNBs, though I will have to try a newer Sky specific LNB now they are available in 40mm.
 

ozumo

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TF & BF / focal point is within the LNB feed horn, just in front of the 'throat' opening of the LNB (in/out movement of the LNB is fixed). When the dish face is vertical the LNB is at a 23° angle.
 

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TF & BF / focal point is within the LNB feed horn, just in front of the 'throat' opening of the LNB (in/out movement of the LNB is fixed).
So that is absolutely fine. :)

When the dish face is vertical the LNB is at a 23° angle.
You mean that it then aims 23 degrees up, from horizontal?
So it aims higher than the G-spot? So higher than BG=23cm?

As far as I know, aimingpoint between G-spot and deepest point of the dish would be the desired aiming area to use.

Greetz,
A33
 

ozumo

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You mean that it then aims 23 degrees up, from horizontal?
So it aims higher than the G-spot? So higher than BG=23cm?

As far as I know, aimingpoint between G-spot and deepest point of the dish would be the desired aiming area to use.
Yes 23 degrees up from horizontal, I will recheck tomorrow. It is possible the plastic LNB holder has sagged a little over the years. On an identical dish with a multi LNB bracket hanging from the central LNB I have to add extra support under the holder to stop it bending.

I did not properly check the aiming as I ran out of time :-doh
 

a33

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@ozumo :
By the way. Congratulations that you are now booked as the first official user of my non-flat paraboloid dish specs calculator!
It is nice, to have the calculator filled with real input data, instead of the fictitious data I used before....

M:W
Greetz,
A33
 

ozumo

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It's my pleasure :Y

I nearly bought a Channel Master non-flat dish last year, but it was the middle of lockdown so couldn't travel to pick it up :(

Regarding the Zone 2, I don't have an exact way of measuring the aiming point, but using a flimsy cardboard tube (from kitchen paper) in the LNB holder and marking where it meets the dish face, the aiming point centre is around 13mm above the G point. I rechecked the LNB angle and got 22.9° today.
 

a33

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I don't have an exact way of measuring the aiming point

The real fanatics would use a laser pointer for that, I believe. ;-)

However, the exact aiming point is not quintessential, as far as I know; and far less important than having the feedhorn phase center exactly at the focal point.
The LNB central aiming angle would always be a compromise between the least spill-over (at the G-spot), and the highest signal density (further above the G-spot, away from the Vertex position), as I understand it.

With your dish:
BD would be a little over 26 cm, and BG was 23 cm, so with your 13mm above G-spot you are well within the aiming area that I mentioned.
I don't expect real improvement by changing it.

So Sky/Raven did their homework, it seems, for this dish! :-)

Greetz,
A33

Edit: small additions.
 
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ozumo

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It would interesting to know if the later versions of the Zone 2, or those from different manufacturers, are different in any way other than the mounting hardware.
 

a33

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It would interesting to know if the later versions of the Zone 2, or those from different manufacturers, are different in any way other than the mounting hardware.

I would share such interest.

In the Netherlands one series of 60cm dishes that came with CanalDigitaal had a too long LNB arm, so that the LNB feedhorn could not be placed in the focal point. It was a long time, before that was discovered.

We tend to trust the manufacturers, and most of the time things are OK.
But I remember a topic on satelliteguys, with the title (motto): "Dish specs, Trust but verify!".
It certainly inspired me, to find some (new) equations, that can be helpful to measure up (out-of-the-ordinary formed) dishes! :)

Greetz,
A33
 
A

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@a33
Is there any convenient mathematical formula to calculate diameter/radius of mother parabola/paraboloid using values of offset parabola/paraboloid like (f=focal length, W=width of dish, d=depth of dish at the center, H=height of dish) ?
 

a33

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Hello and welcome to the forum, @st2@88e!

Well yes, there is:
Diameter.of.mother.PF.parabola = ( 4 * focal.length * abs(tan(offset.angle)) ) + Width.of.offset.dish.

(where "abs" means the 'absolute' value, and assuming the dish face of the offset dish is flat.
The 'abs'-function is only needed for possible negative offset angles.)

The derivation of the equation is discussed in this blog: Antenas Offset (8)

Of course, the focal length and the offset angle can be calculated with the 'common' equations.
So, if wanted, you could also derive the above equation to an equation, that just needs inputs Height, Width, and Depth at the center. (Not so difficult....)

Is this what you want/mean?

Greetz,
A33
 
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Yes I exactly mean that.:Y
I assumed mother/parent parabola as PF (prime focus) having much lesser F/D ratio than the offset parabola and flat faced offset dish means all four vertex/extreme points of width and height of offset parabola lie on a plane.

One more curious question regarding offset geometry parabola (you can skip it if you think I am asking too much).
How to find d1, d2 and illumination angle (θ) using same set of inputs from offset parabola?
Offset.jpg
 

a33

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Answer to that can be found in the John Legon article:
d2 is called BF by him,
d1 is called AF by him.

Finding angle theta:
consider triangle Focus - TopOfDish - BottomOfDish,
you already know d1 and d2,
other side of the triangle = height of the dish.
Now apply the cosine rule, to find angle theta.

Edit: By the way, there are of course other approaches possible, for the calculation of theta. But this one is rather easy to follow, I guess, so I chose to mention this one.


Is there a little project coming?
Please share that with us (but maybe in a new/separate thread, to keep this thread about equations).
We're always interested...

greetz,
A33
 
A

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Answer to that can be found in the John Legon article:
d2 is called BF by him,
d1 is called AF by him.

Finding angle theta:
consider triangle Focus - TopOfDish - BottomOfDish,
you already know d1 and d2,
other side of the triangle = height of the dish.
Now apply the cosine rule, to find angle theta.

Edit: By the way, there are of course other approaches possible, for the calculation of theta. But this one is rather easy to follow, I guess, so I chose to mention this one.
Thanks for pointing to John Legon article, will read it.
Is there a little project coming?
Please share that with us (but maybe in a new/separate thread, to keep this thread about equations).
We're always interested...

greetz,
A33
No project :)
Only curiosity to know more (more ends at mathematical equation, some complex few simple ) about this peculiar geometrical shaped antenna installed on roof/wall of many households including mine.
 

a33

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No project :)
Only curiosity to know more (more ends at mathematical equation, some complex few simple ) about this peculiar geometrical shaped antenna installed on roof/wall of many households including mine.

Yes, that is a nice curiosity!
With me it started with wanting to know the difference between spherical and paraboloidal dishes. :)
Some discoveries of parabolic forms followed.
Now I'm having multiple ways, to calculate specs for parabolic dishes: flatfaced, non-flatfaced, and multifeed dishes; PFdishes with a hole in the center (so that depth at the center cannot be measured); ....

One more curious question regarding offset geometry parabola (you can skip it if you think I am asking too much).
How to find d1, d2 and illumination angle (θ) using same set of inputs from offset parabola?
As you had found that drawing, I wonder if the formulae for d1 and d2 were there, too?
That is because I've seen that exact picture in the (french) offset_v1_0.xls file by F6AGR, on the base of the articles by F4BAY in about the year 2000....

greetz,
A33
 
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