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# antenna stacking formula in 1.2

## Question

What was the final result for the formula for stacking antennas?

I know the initial was the square root of the product of the sum of the ranges. I know that a modifier was added but I can't remember where it was placed and what the value was.

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This is the most recent official-ish pronouncement I've seen on the topic:

On 9/21/2016 at 2:20 AM, NathanKell said:

Make combinability of antennas more complex: they now support a combinability exponent. Default is 0.75. Final strength of a combined set is (the single best strength) * (total strength / single best strength)^weighted exponent where the exponent used is the weighted (by antenna strength) average of all combinable antennas' exponent values.

The above pronouncement was made during pre-release, so I'm not sure whether it may have changed since then, but I'm guessing this is it.

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2 hours ago, steuben said:

What was the final result for the formula for stacking antennas?

I know the initial was the square root of the product of the sum of the ranges. I know that a modifier was added but I can't remember where it was placed and what the value was.

Anyways, it's the square root of the range rating (5k in command pods) * other range rating.

ie.

command pod has a 5k (5,000) rating and DSN level 1 has a range of 2G (2,000,000,000)

SQRT (5000*2000000000) = 3162277 meters = 3.16 Mm

Edited by ToukieToucan
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I believe the question was about stacked antennas.

Haven't heard anything regarding a modifier, last I checked it was a simple matter of adding the antennas together before running the formula. So a probe with four 1k antennas connecting to a probe with one 1k antenna would give a range of SQUARE ((1000x4)x1000)=2k.

If a modifier was added I too would love to know about it.

Edited by JohnnyPanzer
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@JohnnyPanzer - there are diminishing returns, so there is definitely more to it.

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The whip antennas stack with no penalty -- all the others have diminishing returns.

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15 hours ago, Streetwind said:

@JohnnyPanzer - there are diminishing returns, so there is definitely more to it.

The square root portion of the formula already creates a diminishing return so you don't really need anything else.  It's possible there still is another component, but for the purpose of getting a diminishing return the formula already has that.  If you want double the range you need 4x the number of antennas, if you want 5x the range then you need 25 antennas, etc.  Each additional antenna gives you less additional range than the previous one did - that's your diminishing return.

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@Kelderek - then how do you explain what bewing said, above? o_O

It seems like this topic really needs some empirical testing, since everyone seems to have a different idea.

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52 minutes ago, Streetwind said:

@Kelderek - then how do you explain what bewing said, above? o_O

It seems like this topic really needs some empirical testing, since everyone seems to have a different idea.

I would assume the devs felt that returns should diminish even more?

I know it worked that way in early pre, because I ran the numbers. I honestly don't see the need to nerf it further as the squared formula (just like Kelderek says) combined with the large difference in strength between antenna tiers allready means you'd need stupid stacks to even come close to the next tier up. Let's use a made up antenna with a strength of 1Mm as an example:

Say you have a sat with such an antenna in LKO and now you wanna place a sat around the Mun capable of reaching it using stacks of the same 1Mm antenna. To get a healthy 12Mm range your second sat would need 150 antennas!

SQUARE ((150x1000km)x1000km)=12 247km

So is the modifier needed for game balance?

Edited by JohnnyPanzer
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3 hours ago, Streetwind said:

@Kelderek - then how do you explain what bewing said, above? o_O

It seems like this topic really needs some empirical testing, since everyone seems to have a different idea.

Quite agree for the testing.
However, @bewing is correct, while @Kelderek was just giving his opinion that it is unnecessary, while also being correct.
I think part of the confusion is because there are two sides to the equation - the two ships (or ground network+ship) in question.
So to double the range by combining antennae on only one ship, you need 4 antennae (+1 on the other ship : total 5). But If you combine antennae on each side, you only need to double them up (2 on each ship : total 4).

Just for info:

The spreadsheet uses the basic formula given during 1.2 pre. Put simply, for identical antennae, it is:

•    total power = antenna power*((number of antennae)^(combining exponent))

So two antennae with an exponent of 0.75 will have 1.68x rather than 2x the power (and 1.3x rather than 1.4x the range) of one. To double your range, you need 6.5 antennae rather than 4...

If you mix antennae, it should be:

• (highest individual power)*((number of multiples of that highest power)^(weighted combining exponent))

However, last time I looked, the spreadsheet didn't try to work out how to produce a weighted exponent, so it just gave:

• (highest individual power)*((number of multiples of that highest power)^(average combining exponent))

This necessarily causes problems if you add a Communotron 16 to the spreadhseet, since its exponent is 1, but it's a minor limitation (it should be obvious that adding a single '16 to a bunch of relays would not magically extend your range by several Mm...).

Personally, I don't have a problem with the exponent - its effect is relatively minor for small numbers of combined antennae.

But it would be nice to know.

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1 hour ago, Plusck said:

Quite agree for the testing.
However, @bewing is correct, while @Kelderek was just giving his opinion that it is unnecessary, while also being correct.
I think part of the confusion is because there are two sides to the equation - the two ships (or ground network+ship) in question.
So to double the range by combining antennae on only one ship, you need 4 antennae (+1 on the other ship : total 5). But If you combine antennae on each side, you only need to double them up (2 on each ship : total 4).

Just for info:

The spreadsheet uses the basic formula given during 1.2 pre. Put simply, for identical antennae, it is:

•    total power = antenna power*((number of antennae)^(combining exponent))

So two antennae with an exponent of 0.75 will have 1.68x rather than 2x the power (and 1.3x rather than 1.4x the range) of one. To double your range, you need 6.5 antennae rather than 4...

If you mix antennae, it should be:

• (highest individual power)*((number of multiples of that highest power)^(weighted combining exponent))

However, last time I looked, the spreadsheet didn't try to work out how to produce a weighted exponent, so it just gave:

• (highest individual power)*((number of multiples of that highest power)^(average combining exponent))

This necessarily causes problems if you add a Communotron 16 to the spreadhseet, since its exponent is 1, but it's a minor limitation (it should be obvious that adding a single '16 to a bunch of relays would not magically extend your range by several Mm...).

Personally, I don't have a problem with the exponent - its effect is relatively minor for small numbers of combined antennae.

But it would be nice to know.

Superb information, thank you!

While the exponent isn't a major issue, I still fail to see the point of it. As stated earlier, even without it stacking was far from OP, so all it does is add confusion and complexity for complexity's own sake. Actual antenna range for 100% signal strength was allready something that required spreadsheets, formulas and calculators to even get a reasonable guesstimate. It's as if in order to get the actual ISP of an engine you had to use a modifier on the number shown in-game, that was different for each engine and had never been documented...

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Look, the deal is that if there was no modifier, and you stacked a stupid number of 100G antennas on a relay ship, you could easily create something with more range than a Tier 3 Tracking Station. Which sounded dumb to everyone, so NathanKell put a limit on it. The point was to make it so nobody would bother trying to stack stupid numbers of antennas in a real game, OK? Generally, you don't need to know the precise ranges. If you're talking to the Tracking Station, and you want double the range, then stack 3 antennas. If you're talking to another relay, and you want to double the range, use 6. Beyond that, you need to figure out some other clever solution.

Edited by bewing
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8 hours ago, Plusck said:

However, last time I looked, the spreadsheet didn't try to work out how to produce a weighted exponent, so it just gave:

• (highest individual power)*((number of multiples of that highest power)^(average combining exponent))

This necessarily causes problems if you add a Communotron 16 to the spreadhseet, since its exponent is 1, but it's a minor limitation (it should be obvious that adding a single '16 to a bunch of relays would not magically extend your range by several Mm...).

Personally, I don't have a problem with the exponent - its effect is relatively minor for small numbers of combined antennae.

But it would be nice to know.

This is correct and was unfortunately something I could not easily rectify on Google Sheets without doing some major rework to the spreadsheet. I (ignorantly) presume that users will not combine the C16 antenna with much more powerful ones.

I presume a Weighted Exponent takes the relative power between antennas and uses the combinability exponent relative to the other antenna power levels on that vessel?

Anywho, for a quick and dirty explanation, I did try to update the CommNet Wiki page somewhat to provide some more accurate information as well as that Google Sheets page: http://wiki.kerbalspaceprogram.com/wiki/CommNet

EDIT:

@Plusck I have edited my Spreadsheet so that the Combinability Exponent value used in the Signal Strength calculation is done by the following method:

Sum((Antenna 'n' Power * Antenna 'n' Exponent):(Antenna 'n+1' Power * Antenna 'n+1' Exponent)) / Sum(Antenna 'n' Power):(Antenna 'n+1' Power)

i.e. A vessel with a Comm 88-88 (100e9 @ 0.75) and also a Comm 16 (500e3 @ 1.00) would have the following:

((100e9 * 0.75) + (500e3 * 1.00)) / (100e9 + 500e3) = 0.75000125 ... being the Weighted Average Combinability Exponent for the Vessel.

Basically what I am doing is weighting the Combinability Exponent value used for any antenna proportionally against its total power level when working out the Combined Combinability Exponent for the vessel.

I've tested this in-game and it seems to work well. Weirdly in the Vessel Info tab where it shows the link details the value for the signal strength (0 < x < 1) is almost always 0.01 lower than the Percentage Signal strength shown at the top of the screen in the CommNet toolbar.

Edited by Poodmund
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8 hours ago, Plusck said:

Quite agree for the testing.
However, @bewing is correct, while @Kelderek was just giving his opinion that it is unnecessary, while also being correct.
I think part of the confusion is because there are two sides to the equation - the two ships (or ground network+ship) in question.
So to double the range by combining antennae on only one ship, you need 4 antennae (+1 on the other ship : total 5). But If you combine antennae on each side, you only need to double them up (2 on each ship : total 4).

Wow, I was only looking at it from one side which is a narrower view than I should have used.  I think perhaps that I was thinking of the range to a ship with the tracking station as one side and the ship as the other.  Also, I also think of it in terms of moving away from Kerbin and needing to add more antennas as you move further away and the distances get longer and longer -- it's not very practical to send up 100+ relay satellites, so it helps to have more and more antenna range per satellite the farther you get from the sun.  Thanks for pointing this out though, I was only looking at one side of the equation.

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2 hours ago, Poodmund said:

I presume a Weighted Exponent takes the relative power between antennas and uses the combinability exponent relative to the other antenna power levels on that vessel?

Anywho, for a quick and dirty explanation, I did try to update the CommNet Wiki page somewhat to provide some more accurate information as well as that Google Sheets page: http://wiki.kerbalspaceprogram.com/wiki/CommNet

To be honest, I have no clue how you would produce a weighted exponent.

My first instinct was to treat it like a weighted average. Multiply the power times the exponent for each antenna, and divide by the sum of all powers combined.
It gives a sort of acceptable answer - for 2x 5Mm antenna you have a power of 8.41Mm. If you add a Communotron 16 to that, it increases to 8.80Mm. That's an increase of 0.39Mm, or 78% efficiency for the added antenna (whereas if it had had the same combinability exponent as the others, its addition would only have added 62% of its power).

But as I say, I have no idea if that is how it works.

edit: ok I just read your tutorial thread and I think you came to the same conclusion... about 2 hours before I got there

edit 2: and you edited your post here a while ago too... while I was slowly faffing about with numbers

Edited by Plusck
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1 minute ago, Plusck said:

To be honest, I have no clue how you would produce a weighted exponent.

My first instinct was to treat it like a weighted average. Multiply the power times the exponent for each antenna, and divide by the sum of all powers combined.
It gives a sort of acceptable answer - for 2x 5Mm antenna you have a power of 8.41Mm. If you add a Communotron 16 to that, it increases to 8.80Mm. That's an increase of 0.39Mm, or 78% efficiency for the added antenna (whereas if it had had the same combinability exponent as the others, its addition would only have added 62% of its power).

But as I say, I have no idea if that is how it works.

Yeah thats pretty much what I have deduced, as per my 'Edit' to my previous post. It makes sense logically and seems to reflect the in-game readouts when calculating Signal Strength,

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5 minutes ago, Poodmund said:

Yeah thats pretty much what I have deduced, as per my 'Edit' to my previous post. It makes sense logically and seems to reflect the in-game readouts when calculating Signal Strength,

Excellent.

Yes, I was far too slow to write my response. I started thinking it should be weighted average, then decided it might require some other operation like the average of the result of the exponent, before deciding it was silly and seeing that the weighted average looked quite reasonable. By that time you'd already sorted your spreadsheet...

Hence my own pair of 'edits'

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Okay. Think I have it.

I'm leaving aside signal strength since my probes are usually science return, I'm also assuming uniform antennas, and a level 3 station. But, I need a sanity check peer review of my math, and processes.

R1= AP*((n*AP)/AP)(3/4)
= AP*n(3/4)

R2=250e9

R=(R1*R2)(1/2)
=(250e9 * Ap*n(3/4))(1/2)
=500e3 (Ap*n(3/4))(1/2)

Now for my use case. A Jool-Kerbin relay.

Design constraints:
Mass: None
Cost: None
Part Count: None
Launch Count: None
Tech level: level 5 or lower. This means I will have to use HG-5.

Design case, High power straight line J-K relay

Required range:
R= 73e9+14e9, worst case range just before and just after Jool is eclipsed by Kerbol as viewed from Kerbin
=  87e9 -> 90e9, for ease of carry.

90e9=500e3(5e6*n(3/4))(1/2) turning it inside out to find n.
8100e18=250e9(5e6*n(3/4))
810e9/25=5e6*n(3/4)
810e3/125=n(3/4)
n=(810e3/125)(4/3)
n= 120,809 -> 121,000 for ease of carry

note: formula is
n= (R
2/(R1*R2))(4/3)

TLDR; It looks like I'm in the DPF range for part counts. But, let the slide show begin.

Edited by steuben
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4 hours ago, steuben said:

Okay. Think I have it.

I'm leaving aside signal strength since my probes are usually science return, I'm also assuming uniform antennas, and a level 3 station. But, I need a sanity check peer review of my math, and processes.

R1= AP*((n*AP)/AP)(3/4)
= AP*n(3/4)

R2=250e9

R=(R1*R2)(1/2)
=(250e9 * Ap*n(3/4))(1/2)
=500e3 (Ap*n(3/4))(1/2)

Now for my use case. A Jool-Kerbin relay.

Good thing you crossed out "sanity check" because you lost it right there

4 hours ago, steuben said:

90e9=500e3(5e6*n(3/4))(1/2) turning it inside out to find n.
8100e18=250e9(5e6*n(3/4))
810e9/25=5e6*n(3/4)
810e3/125=n(3/4)
n=(810e3/125)(4/3)
n= 120,809 -> 121,000 for ease of carry

note: formula is
n= (R
2/(R1*R2))(4/3)

TLDR; It looks like I'm in the DPF range for part counts. But, let the slide show begin.

Um, I'm confused by the first line.

Rather than “ 90e9=500e3(5e6*n(3/4))(1/2)"

Then:
8100e18=500e9(5e6*n(3/4))
810e9/50=5e6*n(3/4)
810e3/250=n(3/4)
n=(810e3/250)(4/3)
n= 47,930

No?

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Tha's one of those skipped steps. Probably shouldn't have when iI was writing it on the blackboard. The 500e3 is the square root of the range the third level tracking station.

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19 minutes ago, steuben said:

Tha's one of those skipped steps. Probably shouldn't have when iI was writing it on the blackboard. The 500e3 is the square root of the range the third level tracking station.

Ah, ok then, I can't find any error in what you did then. But I am by no means a maths professor.

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