Highly variable depending on the system in question. There are vehicles with very poorly designed headlamp and beam selector swhitches -- in such cases, the switches are usually the primary bottleneck, so the gain from simple restringing (which is seldom done, BTW) would be small. In some cases, though -- particularly those in which the factory system already includes relays but uses thin wire, and those in which the feed side of the circuit is generally OK but body sheetmetal is relied upon for ground -- simple restringing removes the bottleneck so the improvement is large. More specifics below.
There was a guy some time ago who swore it was not possible for there to be any significant improvement by installing relays, and claiming to be able to prove it with calculations involving total system resistance and suchlike. We parked his vehicle -- a Ford Thunderbird from the early 1980s
-- 13 feet from a wall and I used an illuminance meter placed at the hot spot of each headlamp. The RH headlamp was visibly less intense than the LH headlamp, which was his original complaint and reason for seeking help, and though he'd replaced both headlamps several times with no success, he was certain the wiring could not be to blame because "the DOT wouldn't let a company sell a car with inadequate headlight wiring". Left the existing headlamps in place. Installed relays and 12ga wiring per
That's a 15.2% left-to-right difference, which is just above the 12% threshold of significance in terms of target detection under mesopic vision (the kind we use when driving at night), and is an easily visible difference when looking at the beam patterns against the wall.
LH low beam hot spot, after: 150 fc, =~ 25,350 cd RH low beam hot spot, after: 151 fc, =~ 25,519 cd
That's essentially identical performance left-to-right, and is a 52.5% improvement on the RH side and a 31.6% improvement on the LH side -- WELL above the 12% threshold! I don't have colorimetry equipment, which is kind of too bad, because it would've been fun to quantify the visually dramatic change from brownish to *white* light.
The guy's car obviously had some *very* serious voltage drop in the headlamp circuit, which is not at all uncommon to find in Fords and many other cars similarly set up -- long, thin wires, poor switches, body sheetmetal grounds and no relays.
Well, no, you'll never see truly *melted* wire with puddles of copper -- there'll be an electrical fire or, hopefully, some of the circuit protection built into most cars will kick in before that point. I have seen the wires simply barbecue (insulation seriously burnt, conductor turned into frangible threads of black copper oxide. This amount of damage very seldom occurs from plain old voltage drop, it takes something catastrophic like a dead short and inoperative or too-slow circuit protection to cause it. I have seen ample evidence of wire overheating due to simple voltage drop, in the form of premature insulation hardening and discoloration -- not to mention those instances, usually encountered with overwattage bulbs installed on stock wiring, in which the wires grow too hot to hold comfortably!
The other thing to remember is that while longer wires must be of larger gauge than shorter wires to carry a given load without exceeding a given voltage drop, there is a counterforce at work: longer wires have greater surface area than shorter wires over which to radiate heat! So it's very possible to have voltage drop sufficient to seriously degrade headlamp performance for years without any physical symptoms apparent in the wires.
All that said, yes, the connectors can indeed be a significant bottleneck
-- again especially where overwattage bulbs are installed on stock wiring. This can present the double whammy of many bulb types having contact pins completely adequate for the load of the design wattage filament, but marginally adequate at best for the 100% overload of an overwattage filament. The connector pin itself in such cases is going to heat up, and so will the socket, *and* so will the socket-to-wire junction. And all of this heating will take place in a very small area thoroughly jacketed in plastic.
A good(?!) example of this is the guy who installed 100/90W (high/low)
9004 bulbs in his '87 Mercedes, trying to cure the piss-poor headlamps. He lost on all counts! He couldn't really see any better and got high levels of backglare in bad weather because the headlamp optics as a whole (including the 9004 bulb design) were causing his problem, rather than it just being a case of insufficient source light. He kept meaning to reinstall the original bulbs but didn't get around to it until one of them burned out......at which point he found it was too late. The intense localized heat had welded the plastic bases of his overwattage bulbs to the plastic reflectors of his headlamps, leaving him with a set of very expensive, thoroughly useless "sealed beams".
AGREED!
I don't see where manufacturers who allow such circuits out of their factories deserve fairness. Nevertheless, to be perfectly fair, I seldom see 20ga wire used for other than a run of perhaps a foot from the headlamp's common/ground terminal to an adjacent piece of sheetmetal. That said, body sheetmetal isn't a particularly good ground when new, and does not improve with age. But even taking the much closer comparison of 18ga (6.6 ohms/1000ft) vs. 12ga (1.7 ohms/1000ft), the diference is still quite large.
Your 10' example is a good one -- that's on the order of total wire length found in a typical system without relays. When you factor in losses across switches, grounding losses, losses across imperfect connectors...
Quite nonlinear! The formula, again, is:
rated output in lumens [(new volts /old volts) ^3.4] = lumens @new volts
So for simplicity's sake, let's take a 9006 low beam bulb rated 1000 lumens at 12.8 Volts and plug in different voltages.
10.5V : 510 lumens 11.0V : 597 lumens 11.5V : 695 lumens 12.0V : 803 lumens 12.5V : 923 lumens 12.8V : 1000 lumens