Formula for sat dish offset

Although I'd really have liked a lot more feedback than the single
example I did get (thanks for the good work, John), the single
underlying assumption is so reasonable and the proof is so simple that
I felt confident enough to release this live. Accordingly the
following pages on my site have been updated:
http://www.macfh.co.uk/JavaJive/Audi...teGeneral.html
There is now an option to use the new formula in:
http://www.macfh.co.uk/JavaJive/Audi...Calculator.php
Gives a diagram and proof:
http://www.macfh.co.uk/JavaJive/Audi...sSettings.html

On Thu, 22 Sep 2011 00:00:08 +0100, Java Jive
wrote:

One of the few advantages of having to sit around while vinyls are
recording in real time is that you can do some mathematical doodling,
as a result of which I now have a formula for the offset of any sat
dish, even a minidish, as long as it is parabolic is section, which
surely they must all be?

It's:
Offset = asin[ (dT - dB) / d ]

Whe
d = chordal distance across dish from top to bottom
dB = distance of bottom of dish to focal point
dT = distance of top of dish to focal point

It really should be as simple as that. What I like about this formula
is that, unlike the 'boresight' one that's currently on my site and
another I derived a year or two ago, the ONLY assumption it relies
upon is that the dish is parabolic in section.
--
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Java Jive wrote:

Although I'd really have liked a lot more feedback than the single
example I did get (thanks for the good work, John), the single
underlying assumption is so reasonable and the proof is so simple that
I felt confident enough to release this live.

I couldn't see that it was any easier than your old formula, in fact it
seems to involve measurement to a vague point, rather than the definite
width and height ...

Yes, the actual previous measurements and calculation were as simple
as the new calculation, but that calculation has the drawback of an
underlying assumption, which may not always be true - the assumption
is the so-called 'boresight' assumption, that, seen from the satellite
and/or the LNB, the dish appears to be circular.

Certainly, with a conventional taller than wide offset dish, this
assumption makes sense, because that is the way to use the dish to
maximum efficiency, but nevertheless the manufacturer cannot be
absolutely relied upon to have made it so, while with minidishes the
resulting formula cannot be used at all.

The ONLY assumption underlying the new system of calculation is that
the dish is parabolic - surely a cast-iron assumption - AND, best
of all, the formula will work for minidishes too.

I agree that the precise position of the focal point a little behind
the front face of the LNB is debatable, but I still think that even
after errors in measurement to this point, the result is likely to be
more accurate than the boresight calculation.

I'm also hoping to do further research to determine exactly where the
focal point of a 'standard' LNB is. Ultimately, whatever their
manufacturing differences, they all, when mounted at a known pont of
reference in the LNB holder, must all focus the beam, or they simply
wouldn't work, so, acknowledging a possible difference between
conventional LNBs and minidish LNBs, those in each group must each
share a common focal point to an acceptable level of accuracy.

Therefore, if we can work out how to calculate where the focal point
is for each of the two types, it should be possible to adapt the
method of calculation so the user can just measure to a definitely
known point such as the front face of the LNB or, perhaps more likely,
the middle of the holder, and the calculator page will correct the
measurements automatically.

On Wed, 05 Oct 2011 21:02:03 +0100, Andy Burns
wrote:

I couldn't see that it was any easier than your old formula, in fact it
seems to involve measurement to a vague point, rather than the definite
width and height ...
--
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At 23:07:20 Wed, 5 Oct 2011, Java Jive wrote:
The ONLY assumption underlying the new system of calculation is that
the dish is parabolic - surely a cast-iron assumption - AND, best
of all, the formula will work for minidishes too.

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...


I agree that the precise position of the focal point a little behind
the front face of the LNB is debatable, but I still think that even
after errors in measurement to this point, the result is likely to be
more accurate than the boresight calculation.

I'm also hoping to do further research to determine exactly where the
focal point of a 'standard' LNB is. Ultimately, whatever their
manufacturing differences, they all, when mounted at a known pont of
reference in the LNB holder, must all focus the beam, or they simply
wouldn't work, so, acknowledging a possible difference between
conventional LNBs and minidish LNBs, those in each group must each
share a common focal point to an acceptable level of accuracy.

But in fact, the LNB can be moved some distance away from the true focal
point and it will still work, if the dish is aligned accordingly. After
all, multiple LNBs can be mounted on a dish for different satellites.
One of my dishes has three LNBs, but only one could possibly give the
correct result for the offset angle using your formula.

Therefore, if we can work out how to calculate where the focal point
is for each of the two types, it should be possible to adapt the
method of calculation so the user can just measure to a definitely
known point such as the front face of the LNB or, perhaps more likely,
the middle of the holder, and the calculator page will correct the
measurements automatically.

A couple of year ago, I analysed curvature of a dish by taking offsets
to the surface from a straight edge laid across from top to bottom, and
worked out that the best fit to a parabolic curve was obtained when the
axis of rotation of the paraboloid was located on the lower rim.

Consequently, it turns out that the offset angle theta can be calculated
using the formula

sin(theta) = 4p/d(cos(theta))

where d is the length of the chord across the dish from top to bottom,
and p is the maximum perpendicular distance from the chord to the dish.

This can be solved by assuming an initial value for theta in the RHS,
say zero, to obtain a more accurate value in the LHS, and then repeating
the calculation with the new value....

The exact solution can be obtained by developing a quadratic equation in
sin^2(theta) and finding the roots in the usual manner, but as the maths
is a bit messy I'll leave that as an exercise for the reader.

Once the offset angle has been found, the "correct" position for the LNB
can be calculated.

--
John Legon

On Thu, 06 Oct 2011 05:47:37 +0100, John Legon wrote:

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

This should be a primary criterion in buying a satellite dish.

If the LNB arm is cheap and flimsy, do not buy the dish and look for
brands and models which have a solidly built and attached LNB support arm.

J G Miller wrote:
On Thu, 06 Oct 2011 05:47:37 +0100, John Legon wrote:

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

This should be a primary criterion in buying a satellite dish.

If the LNB arm is cheap and flimsy, do not buy the dish and look for
brands and models which have a solidly built and attached LNB support arm.

The dish in question may have been cheap, but it certainly isn't flimsy!
In fact it took me as much strength as I could muster to bend the LNB
arm - by an inch or so - and I'm not a weakling!

The issue with this dish was one of manufacturing accuracy, although it
worked well enough as supplied for the major satellites. My interest
was partly theoretical, but I also wanted to get the best performance
possible with a view to bringing in some of the weaker signals.

I would also suggest that there is no guarantee that an expensive dish
will necessarily have accurate geometry - it's something worth checking
from both theoretical and practical viewpoints.

On 06/10/2011 16:24, John Legon wrote:
J G Miller wrote:
On Thu, 06 Oct 2011 05:47:37 +0100, John Legon wrote:

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

This should be a primary criterion in buying a satellite dish.

If the LNB arm is cheap and flimsy, do not buy the dish and look for
brands and models which have a solidly built and attached LNB support
arm.

The dish in question may have been cheap, but it certainly isn't flimsy!
In fact it took me as much strength as I could muster to bend the LNB
arm - by an inch or so - and I'm not a weakling!

The issue with this dish was one of manufacturing accuracy, although it
worked well enough as supplied for the major satellites. My interest was
partly theoretical, but I also wanted to get the best performance
possible with a view to bringing in some of the weaker signals.

I would also suggest that there is no guarantee that an expensive dish
will necessarily have accurate geometry - it's something worth checking
from both theoretical and practical viewpoints.

The deviation from a true parabolic shape is quite small if the LNB is a
bit high or low or even off to the left or right - that is why it is
possible to use a standard dish with 3 or 4 LNBs to get, for example 13,
19.2E and 28.2E as many of us do. Yes, there is a slight loss of gain
for the LNBs that are farthest from the intended focus but not enough to
negate the technique.

Demonic wrote:
On 06/10/2011 16:24, John Legon wrote:
The issue with this dish was one of manufacturing accuracy, although it
worked well enough as supplied for the major satellites. My interest was
partly theoretical, but I also wanted to get the best performance
possible with a view to bringing in some of the weaker signals.

I would also suggest that there is no guarantee that an expensive dish
will necessarily have accurate geometry - it's something worth checking
from both theoretical and practical viewpoints.

The deviation from a true parabolic shape is quite small if the LNB is a
bit high or low or even off to the left or right - that is why it is
possible to use a standard dish with 3 or 4 LNBs to get, for example 13,
19.2E and 28.2E as many of us do. Yes, there is a slight loss of gain
for the LNBs that are farthest from the intended focus but not enough to
negate the technique.

As I mentioned earlier, I have one dish with three LNBs for just those
three satellites, so I appreciate that the technique works. But I also
have a dish on a motor, and found that tweaking the position of the LNB
on that dish gave a distinct improvement. I don't think it's just a
question of signal strength - having the LNB at the true focal point
will almost certainly improve the focusing power and resolution of the
dish, and hence give a increase in signal quality for satellites that
are close to others in the arc.

On Thu, 6 Oct 2011 05:47:37 +0100, John Legon
wrote:

At 23:07:20 Wed, 5 Oct 2011, Java Jive wrote:
The ONLY assumption underlying the new system of calculation is that
the dish is parabolic - surely a cast-iron assumption - AND, best
of all, the formula will work for minidishes too.

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

In that case, depending on amount of the manufacturer's error:

:-) If only slight, it would be even more essential that we use
the most accurate formula possible, in order to keep the total error
to a minimum.

:-( If not, we're f***ed whatever the formula we use - we'd be
reduced to trial and error.

But in fact, the LNB can be moved some distance away from the true focal
point and it will still work, if the dish is aligned accordingly. After
all, multiple LNBs can be mounted on a dish for different satellites.
One of my dishes has three LNBs, but only one could possibly give the
correct result for the offset angle using your formula.

Yes, but that's simply because talk of focal point is a convenient but
inexact simplification, it would be more correct to talk in terms of
focal surface. With a multi-LNB set up, the LNBs must all lie in the
focal surface, and because the Clarke Belt effectively forms a line in
front of the dish, the LNBs will further be constrained to lie in one
horizontal 'line' of the focal surface.

A couple of year ago, I analysed curvature of a dish by taking offsets
to the surface from a straight edge laid across from top to bottom, and
worked out that the best fit to a parabolic curve was obtained when the
axis of rotation of the paraboloid was located on the lower rim.

That's good, I'd be interested to see your workings, if you still have
them, even if only a scan of handwritten notes.

Consequently, it turns out that the offset angle theta can be calculated
using the formula

sin(theta) = 4p/d(cos(theta))

where d is the length of the chord across the dish from top to bottom,
and p is the maximum perpendicular distance from the chord to the dish.

This can be solved by assuming an initial value for theta in the RHS,
say zero, to obtain a more accurate value in the LHS, and then repeating
the calculation with the new value....

Yes, I've seen a similar style of calculation somewhere, ISTR there
was an example somewhere in he
http://www.qsl.net/n1bwt/chap1.pdf

The exact solution can be obtained by developing a quadratic equation in
sin^2(theta) and finding the roots in the usual manner, but as the maths
is a bit messy I'll leave that as an exercise for the reader.

Errm? Multiplying through by cos(theta) suggests use of the double
angle formula:

sind(theta)*cos(theta) = (1/2)*sin(2*theta) = 4p/d

Therefore sin(2*theta) = 8p/d
Therefore 2*theta = asin(8p/d)
Therefore theta = (1/2)*asin(8p/d)

So how does this compare with my formula which you tested, and your
own settings? Oh, and just for the record, is yours a conventional
taller than wide offset dish, in which case it would be also
interesting to know what the 'boresight' method gives, or a wider than
tall minidish?
--
================================================== =======
Please always reply to ng as the email in this post's
header does not exist. Or use a contact address at:
http://www.macfh.co.uk/JavaJive/JavaJive.html
http://www.macfh.co.uk/Macfarlane/Macfarlane.html

At 01:19:10 Fri, 7 Oct 2011, Java Jive wrote:
On Thu, 6 Oct 2011 05:47:37 +0100, John Legon
wrote:

At 23:07:20 Wed, 5 Oct 2011, Java Jive wrote:
The ONLY assumption underlying the new system of calculation is that
the dish is parabolic - surely a cast-iron assumption - AND, best
of all, the formula will work for minidishes too.

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

In that case, depending on amount of the manufacturer's error:

:-) If only slight, it would be even more essential that we use
the most accurate formula possible, in order to keep the total error
to a minimum.

:-( If not, we're f***ed whatever the formula we use - we'd be
reduced to trial and error.

The fundamental starting point is the dish itself. If the curvature
isn't accurate then nothing can be done, but at least the position of
the LNB can be excluded from the calculation of the offset angle.


But in fact, the LNB can be moved some distance away from the true focal
point and it will still work, if the dish is aligned accordingly. After
all, multiple LNBs can be mounted on a dish for different satellites.
One of my dishes has three LNBs, but only one could possibly give the
correct result for the offset angle using your formula.

Yes, but that's simply because talk of focal point is a convenient but
inexact simplification, it would be more correct to talk in terms of
focal surface. With a multi-LNB set up, the LNBs must all lie in the
focal surface, and because the Clarke Belt effectively forms a line in
front of the dish, the LNBs will further be constrained to lie in one
horizontal 'line' of the focal surface.

I don't follow. The belt is inclined upwards from (say) Astra 28E in
the east to (say) Hotbird at 13 E, and the reflection of the arc off the
dish is inclined downwards accordingly. The LNBs are not placed in a
horizontal line but rather above and below that line with (in my set up)
only the LNB for Astra 19 E on the (presumed) focal surface.


A couple of year ago, I analysed curvature of a dish by taking offsets
to the surface from a straight edge laid across from top to bottom, and
worked out that the best fit to a parabolic curve was obtained when the
axis of rotation of the paraboloid was located on the lower rim.

That's good, I'd be interested to see your workings, if you still have
them, even if only a scan of handwritten notes.

It would be good if I could find them! I have, however, found this plot
based on my actual measurements, with a reconstruction of the focus:
http://www.john-legon.co.uk/temp/parab.jpg

Consequently, it turns out that the offset angle theta can be calculated
using the formula

sin(theta) = 4p/d(cos(theta))

where d is the length of the chord across the dish from top to bottom,
and p is the maximum perpendicular distance from the chord to the dish.

This can be solved by assuming an initial value for theta in the RHS,
say zero, to obtain a more accurate value in the LHS, and then repeating
the calculation with the new value....

Yes, I've seen a similar style of calculation somewhere, ISTR there
was an example somewhere in he
http://www.qsl.net/n1bwt/chap1.pdf

It's actually in the appendix at the end of:

http://www.qsl.net/n1bwt/chap5.pdf

Significantly, the method proposed there assumes that the offset angle
of the dish is known, and then proceeds to find the focal length and
location of the origin.


The exact solution can be obtained by developing a quadratic equation in
sin^2(theta) and finding the roots in the usual manner, but as the maths
is a bit messy I'll leave that as an exercise for the reader.

Errm? Multiplying through by cos(theta) suggests use of the double
angle formula:

sind(theta)*cos(theta) = (1/2)*sin(2*theta) = 4p/d

Therefore sin(2*theta) = 8p/d
Therefore 2*theta = asin(8p/d)
Therefore theta = (1/2)*asin(8p/d)

Excellent! It really is that simple :-)

Now using this formula with the data supplied in the above pdf file,
namely for a dish with d = 500 and p = 43, we get an offset angle of
21.7 degrees and hence a complement of 68.3 degrees.

The pdf assumes 66.9 degrees initially, but then - realizing that the
axis of the parabola intersects the lower rim of the dish - obtains the
better result of 68.3 degrees by trial and error, in perfect agreement
with my calculation.


So how does this compare with my formula which you tested, and your
own settings?

Measuring inside the raised edge on the rim of the dish I get d = 644 mm
with a maximum depth of 54 mm. Hence the offset angle will be

(1/2)*asin(8*54/644) = 21.1 degrees

As mentioned earlier in this thread, using your method gives an offset
angle of 20.7 degrees, so that's quite close. However, I only get this
agreement because I had already bent the LNB boom arm to place the LNB
where I thought it ought to be!

Oh, and just for the record, is yours a conventional
taller than wide offset dish, in which case it would be also
interesting to know what the 'boresight' method gives, or a wider than
tall minidish?

The outer dimensions are 605 x 655 mm, giving 22.5 degrees.

--
John Legon