Chrono Boost Math

As you know, every CB puts the building into an "accelerated state" in which it'll build/upgrade 50% faster for 20 seconds (those 20 seconds are "game seconds", and can be counted by the sc2 time counter).

In normal circumstances, a building builds/researches at a speed of 1 (30 seconds to complete a 30 "production units" unit/research).
The acceleration factor (50%) tells us that, during a CB's duration, the speed at which it builds/reseaches is 1.5 (20 seconds to complete a 30 "production units" unit/research).

That of "production units" is confusing. This is what I mean:
If you leave your mouse cursor over a construction bar, you'll see as your chrono that the speed at which the numbers go up increases. So the building is doing more "production units" by game second.

I didn't mean to make this a "math postulate". I'm just expaining how I get the numbers in case:
a) You're curious and you really care.
b) Something MIGHT be wrong because nobody is perfect. In that case, more talented people can review it and I'll fix it.

In you don't care about the math, just skip all that and go to the juicy parts: conclusions and numbers.

You can deduce, or do enough experiments, to get this formula.

Cr= Cd*Cf-Cd= Cd(Cf-1)

Where:
Cr= Chrono Reduction. Save of time in seconds caused by one CB.
Cd= Chrono duration (20 secs).
Cf= Chrono factor (1.5).

Computing the numbers you get:

CR= 10 seconds.

So: one Chrono Boost == you save up 10 seconds.

If you apply more than one CB, then of course you get linear results:
Two chronos= 20 seconds.
Three chronos = 30 seconds.
And so on.

So the general formula to get a Bt modified by CB is:

Bt'=Bt-N*Cr

where N is the number of chronos applied.

If you apply continuous CB (one after another) is like you have accelerated time by 1.5. So the fastest BT' you can get is BT/1.5.
Which is 50% faster. This is the limit.

To get the maximum number of CB, you just solve the equation for N using BTmin.

BTmin=Bt/1.5=Bt-N*Cr

We'd like get a formula that can tell us how many continuous CB (one Chrono every 20 secs) can we get from a Nexus.
This can also be used to, for example, know much energy is needed to CB Warpgate research.
That formula is (contribution of Jaeger, thanks):
Nc*Ec=Nn[Ei+ Rg* Dc*( Nc-1)]

where:
Nc= Amount of continuous CB we can get.
Nn= Amount of Nexi.
Ei= Initial Energy of Nexi.

And we solve for Nc or Ei, whichever we want to get.
We can get interesting results from this. More on conclusions.

I accidently found out this thing. Don't know if it's wide known but I haven't heard of this anywhere.

The fact: CB accelerates the speed at which shields recharge by 50% too!

I've tested this on a Nexus, testing how much it takes to replenish its shields without and with continuous CB on the Nexus.
Normally, when the building is no longer being attacked, shields recharge a constant rate of 2 SHP per second.
With CCB, I've found that they recharge faster, at a rate of 3SHP/sec.

You can see the results inside the Excel chart.

*First conclusion: One Chrono Boost == you save up 10 game seconds.

*Second conclusion: CB is linear. The more you chrono, the more you save up and always with the same proportion.

*Third conclusion: You can get something up to 50% faster using CB. No more. This is the limit.
Note that 50% faster DOESN'T mean that it takes half of the time. It takes 66% of the original time to build (or 33% less time).

*Fourth conclusion: Buildings shields recharge 50% faster when CB! Test it yourself


Assuming continuous CB:

Colossus
*BT'= 50 (you save 25 seconds).
*CB needed= 3.

Inmortal
*BT'= 37 (you save 18 seconds).
*CB needed= 2.

Void Ray
*BT'= 40 (you save 20 seconds).
*CB needed= 2.

Warp Gate research
*BT'= 93 (you save 47 seconds).
*CB needed= 5.

+1 Ground Weapons Research
*BT'=107 (you save 53 seconds).
*CB needed= 5.

CB and Nexus
-Empty Nexi (Ei=0).
You can get 1 CB every 45 seconds.
If you have more than you Nexus, then time is split. Example: 2 Nexi, then 1 CB every 22.5 secs, which is almost enough to get continuous CB.

-Full Nexi (Ei=100).
You can continuously CB 6 times (waiting a little you'll have 7). For instance, that's more than enough to fully CB Warpgate research (only need 5 CB).

More data on the Excel spreadsheet.