Formula for sat dish offset
 

On Fri, 7 Oct 2011 08:54:36 +0100, John Legon
wrote:

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.

Except, surely, the position of the LNB *affects* the offset angle, in
the sense that where the position of the LNB is slightly out of true
but is not corrected, which it won't be by most installers, the
elevation of the dish will have to be slightly different than it would
be if the LNB were in the correct position?

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.

Ok, I worded that loosely, partly because for some reason or other I
was thinking you have a rotor like myself, thus making the Clarke belt
'horizontal' wrt to the face of the dish no matter at which bit of the
belt the dish is pointing. But what I meant is still true, the LNB's
will lie on a line lying in the focal surface of the dish.

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.

Good. If possible, I'd like to see at least an outline of how to
prove this formula, to save me working it out for myself.

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!

But, as mentioned above, if the LNB arm is positioned wrongly, that
would change the effective offset of the dish anyway.

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.

Yes, it's definitely looking to me as though the 'boresight'
calculation is the least reliable of the three so far discussed. The
other two show good agreement though, which is encouraging for both. I
will have to give some thought, and maybe some maths, as to how an LNB
holder being slightly out of true will effect the accuracy of each
method.

But again, thanks for the detailed trouble you're going to, to help.
It really is much appreciated!
--
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Formula for sat dish offset
 

On 06/10/2011 17:15, John Legon wrote:
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.



Agreed. The acceptance angle of a dish improves as the dish size gets
larger. That is one of the problems with the mini-dish and why it is
wider than its height. AFAIR a 60cm dish has an acceptance angle of a
bit under 3 degrees which is about the spacing of N. Europe Sats around
the Clarke Belt.

Naturally getting the focus spot-on must improve things. The parabola
has two benefits - it accepts a signal from an infinite distance
(parallel beam) and it has the same path length from everywhere on the
source to the focus, thus eliminating destructive phasing.

I have an 80cm with 3 LNBs and a 120 cm for feed hunting. Great hobby
but almost too many channels to waste time watching them :-)

Formula for sat dish offset
 

At 13:24:17 Fri, 7 Oct 2011, Java Jive wrote:
On Fri, 7 Oct 2011 08:54:36 +0100, John Legon
wrote:

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.

Except, surely, the position of the LNB *affects* the offset angle, in
the sense that where the position of the LNB is slightly out of true
but is not corrected, which it won't be by most installers, the
elevation of the dish will have to be slightly different than it would
be if the LNB were in the correct position?

Of course, the position of the LNB will determine whether the tilt of
the dish when mounted corresponds to the theoretical ideal, and most
people won't care if it isn't, as long as the system works. It's still
the case, however, that the "correct" offset angle is defined by the
geometry of the dish, and not by the position of the LNB.

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.

Ok, I worded that loosely, partly because for some reason or other I
was thinking you have a rotor like myself, thus making the Clarke belt
'horizontal' wrt to the face of the dish no matter at which bit of the
belt the dish is pointing. But what I meant is still true, the LNB's
will lie on a line lying in the focal surface of the dish.

I do have a dish with a rotor, and also a dish with three LNBs...

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.

Good. If possible, I'd like to see at least an outline of how to
prove this formula, to save me working it out for myself.

Don't know about proof - this is just something I sketched out the other
day on a scrap of paper. But since you ask...

Take the vertical section through an offset dish with origin (0,0) of
the parabolic curve x = y^2/4a located on the lower rim. Take a chord,
length d, from the origin to a point (xt,yt) located on the upper rim.
The offset angle of the dish will be atan(xt/yt).

From the mid-point of the chord (xt/2, yt/2), drop a perpendicular
(parallel to the x-axis) on to the y-axis. It will intersect the
parabola at the point (xt/4, yt/2). Hence the distance along this line
from the mid-point of the chord to the parabola is also xt/4.

This distance, which I call k, could be measured from the dish itself if
the offset angle theta was known, but the angle isn't known and we can
only measure the distance p perpendicular to the chord.

Now to a close approximation, p = k*cos(theta). This is because the
gradient of the parabolic curve at the chosen half-way point is parallel
to the chord. For the same reason, the distance p is effectively the
maximum distance perpendicularly from the chord to the curve.

Hence the distance of 2k from the mid-point of the chord to the y-axis,
measured parallel to x-axis, is 2p/cos(theta). The mid-point is also at
a distance of d/2 along the chord from the origin, and the offset angle
can be calculated as

sin(theta) = 2p/(cos(theta)) / (d/2)

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

As regards the approximation to k in the above procedure, I've worked
out that it amounts to about 0.2 mm in the actual measurement, or about
0.1 degree in the final result.

--
John Legon

Formula for sat dish offset
 

On Tue, 04 Oct 2011 21:35:06 +0100
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. 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.

Did you ask for feedback?
--
Davey.

Formula for sat dish offset
 

On Fri, 7 Oct 2011 19:18:09 +0100, Davey
wrote:

Did you ask for feedback?

On Thu, 22 Sep 2011 13:19:44 +0100, Java Jive
wrote:

I'd be interested to hear from anyone else willing to try and verify
this new formula by measurement, particularly against a dish where the
offset is known from the manufacturer's specifications.

I'd also be interested in people's views on where the focus of the
dish is in relation to the LNB, where it's present, or just the holder
where it is not. Although LNBs must surely vary at least a little,
they must sit at the correct focal point, and as the only given is the
LNB holder, it must be possible to say that the focal point must be x
mm directly in front of the holder, but what is x?
--
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Formula for sat dish offset
 

On Fri, 07 Oct 2011 19:28:59 +0100
Java Jive wrote:

On Fri, 7 Oct 2011 19:18:09 +0100, Davey
wrote:

Did you ask for feedback?

On Thu, 22 Sep 2011 13:19:44 +0100, Java Jive
wrote:

I'd be interested to hear from anyone else willing to try and verify
this new formula by measurement, particularly against a dish where
the offset is known from the manufacturer's specifications.

I'd also be interested in people's views on where the focus of the
dish is in relation to the LNB, where it's present, or just the
holder where it is not. Although LNBs must surely vary at least a
little, they must sit at the correct focal point, and as the only
given is the LNB holder, it must be possible to say that the focal
point must be x mm directly in front of the holder, but what is x?

Ok. I was looking in the Original Post, not further down the thread.
--
Davey.

Formula for sat dish offset
 

On Fri, 7 Oct 2011 19:09:59 +0100, John Legon
wrote:

Of course, the position of the LNB will determine whether the tilt of
the dish when mounted corresponds to the theoretical ideal, and most
people won't care if it isn't, as long as the system works. It's still
the case, however, that the "correct" offset angle is defined by the
geometry of the dish, and not by the position of the LNB.

Well, I guess that depends on how you actually define the offset. As
we are interested in knowing it because of how it effects what
elevation we set on the dish, I would suggest that the useful
definition is the difference in elevation between the type of dish
under discussion and an axi-symmetric equivalent. Under that
definition, the position of the LNB does indeed affect the offset.

Don't know about proof - this is just something I sketched out the other
day on a scrap of paper. But since you ask...

Take the vertical section through an offset dish with origin (0,0) of
the parabolic curve x = y^2/4a located on the lower rim.

I don't like this assumption. Although, like the boresight
assumption, it makes sense, I don't think we can absolutely rely on
all manufacturers to produce dishes that we think makes sense.

However, going with the flow for the mo ...

For future reference let O be the origin (0,0)

Take a chord,
length d, from the origin to a point (xt,yt) located on the upper rim.
The offset angle of the dish will be atan(xt/yt).

From the mid-point of the chord (xt/2, yt/2)

Let M be midpoint of the chord (xt/2, yt/2)

drop a perpendicular
(parallel to the x-axis) on to the y-axis. It will intersect the
parabola at the point (xt/4, yt/2).

Let P this point of intersection with the parabola (xt/4, yt/2)

Hence the distance along this line
from the mid-point of the chord to the parabola is also xt/4.

This distance, which I call k,

So MP = k

could be measured from the dish itself if
the offset angle theta was known, but the angle isn't known and we can
only measure the distance p perpendicular to the chord.

Now to a close approximation, p = k*cos(theta).

Surely it's actually EXACTLY that?

The gradient of the parabola at any point is given by ...
d sqrt(4ax) / dx
.... which evaluates to ...
sqrt(a/x)

Therefore, to find x where the tangent is parallel to the chord, we
equate the gradients:

sqrt(a/x) = (yT/xT)

.... which after squaring and substituting yT^2 = 4a.xT gives ...

x = xT/4

So the tangent is parallel exactly at P. If we drop a perpendicular
from P onto the chord and call the point where it meets it Q, then PQ
is p. The angle QMP = 0MP = 90 - theta, so MPQ = theta, and ...
p = k.cos(theta).

Hence the distance of 2k from the mid-point of the chord to the y-axis,
measured parallel to x-axis, is 2p/cos(theta). The mid-point is also at
a distance of d/2 along the chord from the origin, and the offset angle
can be calculated as

sin(theta) = 2p/(cos(theta)) / (d/2)

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

Yes, agreed.

As regards the approximation to k in the above procedure, I've worked
out that it amounts to about 0.2 mm in the actual measurement, or about
0.1 degree in the final result.

I don't think any approximation is actually involved. I'd be
interested to hear your thoughts after reading mine.

I think the next step with this is to see if it can be generalised it
to remove the assumption that the bottom of the dish B is at (0,0). If
we could do that, and produce equations for the offset and perhaps the
focal point that only rely on the two dimensions measured, that would
have great potential use. However, if we can't remove that
assumption, I would favour my new formula over this method.
--
================================================== =======
Please always reply to ng as the email in this post's
header does not exist. Or use a contact address at:
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Formula for sat dish offset
 

At 23:20:02 Fri, 7 Oct 2011, Java Jive wrote:
On Fri, 7 Oct 2011 19:09:59 +0100, John Legon
wrote:

Take the vertical section through an offset dish with origin (0,0) of
the parabolic curve x = y^2/4a located on the lower rim.

I don't like this assumption. Although, like the boresight
assumption, it makes sense, I don't think we can absolutely rely on
all manufacturers to produce dishes that we think makes sense.

I don't much care for the assumption myself, but haven't as yet found
any reason to doubt its validity.

However, going with the flow for the mo ...

For future reference let O be the origin (0,0)

Take a chord,
length d, from the origin to a point (xt,yt) located on the upper rim.
The offset angle of the dish will be atan(xt/yt).

From the mid-point of the chord (xt/2, yt/2)

Let M be midpoint of the chord (xt/2, yt/2)

drop a perpendicular
(parallel to the x-axis) on to the y-axis. It will intersect the
parabola at the point (xt/4, yt/2).

Let P this point of intersection with the parabola (xt/4, yt/2)

Hence the distance along this line
from the mid-point of the chord to the parabola is also xt/4.

This distance, which I call k,

So MP = k

could be measured from the dish itself if
the offset angle theta was known, but the angle isn't known and we can
only measure the distance p perpendicular to the chord.

Now to a close approximation, p = k*cos(theta).

Surely it's actually EXACTLY that?

Not quite. See below.




The gradient of the parabola at any point is given by ...
d sqrt(4ax) / dx
... which evaluates to ...
sqrt(a/x)

Therefore, to find x where the tangent is parallel to the chord, we
equate the gradients:

sqrt(a/x) = (yT/xT)

... which after squaring and substituting yT^2 = 4a.xT gives ...

x = xT/4

So the tangent is parallel exactly at P. If we drop a perpendicular
from P onto the chord and call the point where it meets it Q, then PQ
is p. The angle QMP = 0MP = 90 - theta, so MPQ = theta, and ...
p = k.cos(theta).

The tangent is exactly parallel at P, but the position of P depends on
the offset angle which we're trying to find. For this reason, my
distance p isn't measured from P to the chord at point Q, but is instead
the distance along the perpendicular bisector to the chord through the
midpoint M. This line intersects the curve at a point where the tangent
isn't exactly parallel to the chord, and there is a slight approximation
in taking k to be p/cos(theta).

However, the error can be eliminated by taking p to be the maximum
distance from the chord to the curve. This being the case, it is isn't
necessary to specify exactly where the measurement of p is made, but it
should in theory correspond to PQ.

--
John Legon

Formula for sat dish offset
 

On Fri, 07 Oct 2011 18:31:17 +0100, Demonic wrote:

I have an 80cm with 3 LNBs and a 120 cm for feed hunting. Great hobby
but almost too many channels to waste time watching them :-)

So many channels, so little time!
http://www.nonags.com/funimg/sat.jpg
--
Peter.
The gods will stay away
whilst religions hold sway

Formula for sat dish offset
 

On Sat, 8 Oct 2011 08:12:05 +0100, John Legon
wrote:

At 23:20:02 Fri, 7 Oct 2011, Java Jive wrote:

I don't like this assumption. Although, like the boresight
assumption, it makes sense, I don't think we can absolutely rely on
all manufacturers to produce dishes that we think makes sense.

I don't much care for the assumption myself, but haven't as yet found
any reason to doubt its validity.

Let me give you a possible one ...

I've managed to find two pictures of my last dish, which I no longer
have. I've temporarily put them up on my site so that anyone
interested can check my working:
http://www.macfh.co.uk/Temp/OldDish1.png
http://www.macfh.co.uk/Temp/OldDish2.png

From these, by knowing the rest of the dimensions which I measured
previously to getting rid of it, and counting pixels in the pictures
(in PSP, this is easily done by choosing selection areas exactly
enclosing the item of interest, reading off the dimensions of the
selection as it is being made, and doing a Pythagoras calculation), I
have been able to estimate the perpendicular depth p.

In the first the dish is mounted in use, the picture being taken from
the ground vertically underneath the mounting. The scale of the
picture is determined by the known width of the dish as below. The
min figure is from the number of pixels between the parallel lines,
the max from the full length of the perpendicular line.

In the second, the dish is taken from exactly edge on, but not from
exactly in the middle of the side. The scale was therefore assumed to
be that the line across the rim was the average of the height and the
width, 825.

Dimensions (mm):
Width: 800
Height: 850
B2LNB: 535
T2LNB: 860
First pic ...
Min estimated depth: 65
Max estimated depth: 90
Ave estimated depth: 75 (approx)
Second pic ...
Estimated depth: 65
(all rounded to nearest 5mm)

Offset Calculations (deg):
Yours min: 18.86
Yours max: 28.95
Yours ave: 22.45
Yours #2: 18.86
Boresight: 19.75
Universal: 22.48

You can see that for your method, while the average from the first pic
agrees closely and encouragingly with mine, the min and max are each
further out even than the boresight method, itself a long way out.
However, I think the second pic is more reliable, and this is very
close to the min estimate from the first pic. All this suggests that:
:-( In this case, the underlying assumption, that the bottom of
the dish B is coincident with the origin O, is likely to be wrong;
:-( Your method is sensitive to errors in measuring the depth;
:-( Therefore it is probably also sensitive to the correctness of
the assumption that B is coincident with O.

I think the sensitivity problem lies in the 8 times factor in the asin
argument.

However, the error can be eliminated by taking p to be the maximum
distance from the chord to the curve. This being the case, it is isn't
necessary to specify exactly where the measurement of p is made, but it
should in theory correspond to PQ.

I must have misunderstood what you originally wrote, I thought the
above was what you were actually doing.
--
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Please always reply to ng as the email in this post's
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