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I must disagree. Can you describe the simplest situation that the more complicated lane balancer solves, that the simpler (single splitter) will not?
For OP's use-case, the simpler single-spliter design solves that case perfectly:
An input balancer balances the ingoing belts or lanes, assuring they are consumed equally.
An output balancer balances the outgoing belts or lanes, assuring they are filled up equally.
OP's target is to "balance the two lanes on the one belt, so that one lane doesn't sit there while the other lane gets used." I.e. their goal is to ensure equal draw, not equal distribution.
They aren't looking to divert one lane to totally different production line than the other lane, and ensuring both lines get 50% of whatever flows in to work with. They want whatever is being consumed to draw 50% evenly on both lanes of the input belt.
What they want is input balancing; not output balancing.
As I said before: the 'simple' lane balancer is a newb-trap. I don't mean that to be offensive to anyone's skill level. I simply mean to say: it is very deceiving in what it actually achieves vs what it doesn't; and unless you're already intimately familiar with how balancers work, it will fool you. At least once or twice.
If you use one input belt and one output belt, half the contents won't be lane-balanced and will pass through on the straight bypass before being merged back in. In effect you get one belt with half the contents lane-balanced and the other half as it were, i.e. not balanced.
The single incoming belt will be split 50-50 by the first splitter.
The internal 'simple' 1:1 lane balancer will output balance the contents to evenly distribute over the two lanes 50-50. Then the final splitter will recombine it with the non-balanced other half into one output belt.
Because a 50-50 distributed belt ends up being recombined with a non 50-50 distributed remainder, there will be instances where either lane will temporarily block and in pessimistic cases this can compound to having one of the input lanes temporarily block.
And therefore it is not throughput unlimited.
It's also not universal - because it doesn't actually fully properly balance unless all the inputs and outputs are used and moving. Universality is something practically no balancer has though. Even the basic and well known 4:4 'hourglass' -- which actually is throughput unlimited -- is in fact not universal.
If you would connect up one input belt and three output belts to it, you wouldn't get a 1 -> 0.33 : 0.33 : 0.33 ratio. you'd get a 1 -> 0.25 : 0.25 : 0.5 ratio.
You can make a 4:4 universal balancer as well; but it's at least an approximate 5~6x larger in footprint.
See also: https://alt-f4.blog/ALTF4-27/
Sure, let's call it "input" balancing -- I don't want to get hung on up terminology.
No offense taken. I'm generally interested in learning more. What situation(s) does the more complex lane balancers solve (or solves better) that the simpler one does not handle (or does not handle as well as)? I only use the simpler one and I have quite literally never ran into a case where I think to myself "this isn't working, let me find a more complicated lane balancer design".
All these different lane balancer designs have the same desired effect: filling gaps in an output lane, using excess materials in the other lane.
This is also what I've experienced myself, I've experienced no problems with it, and I don't actually see any mathematical reason why it wouldn't work either.
Simply put, I quite believe you misunderstand how it works. Because the point is simply to switch (there's no actual balancing by just switching, you need more than that) on one part and not switch on the other, then merge those.
"The internal 'simple' 1:1 lane balancer will output balance the contents to evenly distribute over the two lanes 50-50." No, it won't. It'll switch the sides, with possible overflow. The "'simple' 1:1 lane balancer" isn't a lane balancer AFAIK, it's a lane switcher with overflow. That's the reason some other designs use undergrounds, to switch sides without any possible overflow. I have found no good reason to actually use undergrounds for this in practice. The only time overflow will matter is if the downstream is managing to pull more than the input is giving, since the other situation where something backs up will lead to one lane per side in the "'simple' 1:1 lane balancer" being backed up always, essentially working like the version with undergrounds that only switch the sides.
I've also used it for 2:2 belt lane balancing too, and the input belts and lanes have also always been pulled from in equal amounts.
It's essentially just a basic splitter (2:2 belt balancer) with some extra within it. And the extra within it does not affect the throughput.
EDIT: Essentially what I'm saying is that if the one I prefer to use isn't TU, then the spliter itself isn't TU either. And I'm pretty sure everyone is going to agree that the splitter itself is TU.
If you want a "proper" balancer you need something a bit more complex but that was also provided in the fourth answer.
Without items on the belt the answer is not obvious for me
I followed your explanation till "sideways underground belts" ... there I asked myself why underground is necessary and lost you.
I'm not seeing what's "quick and dirty" about it. It does the same as what you're calling the "proper" solution, but in a much simpler way, without needless additional underground belts and splitters.
If the items are only ever going to be on one side of the input belt, you're going to end up with the same amount of items on each side of the belt once it gets rejoined either way. All that making it complicated does is introduce a bigger possibility of it going wrong later.
I am still beginner but assumed that this underground belts are superfluous.
If a problem in Factorio has a simple solution, never underestimate the community's ability to recommend an overly complicated one.
is basically a lane switcher with overflow (unless only having one lane input, since then the overflow on one side becomes an important part to making the output belt have 2 lanes with items).
The one shared by Fel uses the fact that undergrounds block one lane to control which lane goes where, then merge those two back together. The top lane goes to the top belt, but on the bottom lane, the bottom lane (of the input) goes to the bottom belt and also on the bottom lane. Then the splitter puts half of the input (which is 1 full belt worth of items only on the bottom lane) straight to the right on the bottom lane, and half of the input to the top of the splitter output, whereupon it is placed on the top lane of the output.
Thus no matter the input ratio the output of the one Fel showed will give equally distributed output. If the input is less than max and the output has one lane backed up it will also draw equally from both input lanes onto the free output lane.
It is definitely good enough if all you want is to balance the output, but it isn't balancing the input.
What that means is that if the right lane is the only one really moving after the balancer, it will mean that only the left lane will move before the balancer.
And in many cases, that's perfectly fine, which is why I called it "quick and dirty but close enough", most people including me use this all the time too because it's small in terms of space, very easy to make by hand, and it fits the purpose well enough to not really need something more complec.
But there are cases where you want both lanes before the balancer to be "consumed" at the same speed (assuming both are full or coming at the same rate of course) regardless of what happens after the balancer (even if only one lane ends up moving for example).
In those cases, you will want to use the more complex one.
But since it is a bit more complex, you are probably going to want to use a blueprint (or make it part of a book) for it to make sure you don't make mistakes when building it, which is the same as when making belt balancers really.
The complex lane balancer pulls equally from both lanes and outputs equally to both lanes.
And, if you consume items on the output, but take from one side here, the input still pulls both sides equally.