How is the minimum conductor size (of some type) for a certain run determined?
The key is that a 100A line isn't 100A all the time

... Or, ever.  

Go read the brochure website on any power plant, particularly a nuke or solar installation, it will say something like "this 1.5 gigawatt plant serves 1.5 million homes".   Um.   So they are saying the average home takes 1kw (about 4.1 amps at 240V).  Seriously? 

Yeah, seriously.  They are saying your house spends most of its time asleep, drawing even less than 1 amp because everything's off.  The dryer pulls 23A but only 90 minutes a week.  Averaged through the year the A/C draws only 2 hours a day.  

Your electric bill proves it. 1KW is 1KWH per hour or 720 KWH per month. At 12 cents a KWH, with fees, that amounts to about a $100 power bill.  If your house drew 100A all the time, your monthly power bill would be $10,000.  It's clearly not.  

So you have to pause and think about your realistic load - the level of load you care about.   It really helps to have experience with a whole-house power monitor.  Then, in those odd moments, those excessive peaks when the dryer is running while someone is cooking and the A/C is kicking -- how much voltage drop can you live with? 

There isn't anything wrong with heating a little wire in those 1-in-1000 moments.  As long as you follow the specs in those charts (NEC 310.16) it guarantees the wire cannot get too warm, no matter how much length or voltage drop you have. 

Those voltage drop calcs are a menace...

... because they don't think about that.  One guy had a calc tell him to run #6 for a 20A circuit of only 135'. Datz kray.  I showed him for realistic loads, #12 would be fine and #10 would be overkill. 

It's pretty much my "hat" here to have someone coming in complaining that the calc told to use 600kcmil for his 2000' run to a 100W pole light... And then I show him how to do it with 14AWG wire, the smallest size you can use on mains wiring.

So to start with, set your amp values to the realistic common number.   Put "acceptable percentage" to 5% just so it's not bumping you up a size just because the smaller size would be 3.06% (that happened in the 20A case).  

Try a couple other numbers and up the permissible percentage.  

Then do punch in the nameplate amp number, but put 99% for permissible voltage drop.  And see just how bad the worst case is.  If it gives you a gory number like 19%, then back the number off to 18% to force it to go one size larger at a time. 

That said, why not provision for 200A?

I seriously doubt you plan to use 50-100A in the garage.  So why not go ahead and plan for 150-200A wire for the "main" circuit to the house?  That will certainly remove any concern of voltage drop at 100A usage levels, and give you the headroom to use your entire 200A panel capacity.  That would be 250KCmil aluminum for 2.64% drop at 200A and 1.32% drop at 100A and you never have to give it another thought. 

By the way, a person wouldn't even think of using copper at these large sizes, unless they were tragically misinformed.  Aluminum has 12 times the conductivity of copper, when measuring by dollar of raw metal.  (and twice the conductivity when measuring by weight.)  

I take it the generator will be in the garage?  Normally you're not allowed to run two parallel services between buildings, but a special usage like a generator is permitted.   On that, #4 aluminum will give you 3.06% voltage drop at 50A. **



** remember what I was saying about how those calculators getcha?  When you did it, you got #3 aluminum because  you told the calculator to limit drop to 3%, which 3.06% failed.  Passes in my book.  That's why I started by telling the calc 99% was acceptable, to see what it would say. 

Oh and it's coincidence that it was 3.06% in my earlier example.  This last was an edit.  I just did the calc now.  This happens a lot.