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Your starter doesn't need boosted 13.5 or 14.0 volts. I think you can connect it directly to a lead acid battery, and also connect your boost converter to the same battery and run the rest of the car off that. Higher voltage going to the starter might not be good for it.
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The starter motor will be drawing 100+Amps. Any DC/DC step-up converter I might use to get the battery voltage up to the alternator output voltage of 14.5v is only going to output 30 Amps at the very most, more likely between 10 and 20 Amps, so I don't think the starter motor will 'see' a significantly increased voltage. If you start a car with the headlights on, they often dim as the starter turns, right? That's because the starter is drawing so much current that the battery voltage is pulled right down, even though the starter motor has its own dedicated cable.
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The headlights didn't do as well, running on a 12V lead acid deep cycle battery. For these, I now want to use some of those Chinese 10A DC-DC boost converters you mentioned.
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As far as the headlight dimming goes, have you measured the actual voltage drop between the battery and the headlights? Both positive and negative? It's common to find a significant drop there which can be removed by feeding the headlights direct from the battery via thick cable and a relay triggered by the existing wiring to them. (The stock wiring is usually too thin and runs a long way to the headlight switches on the dash and back again, hence the voltage drop.) I'm going to try that first. My headlights get at least a volt less than the battery puts out, so when the battery is at 12v, the headlights are getting less than 11v. If they were truly getting at least 12v at all times my hunch is that they would be fine. The same MAY be true of the wiper motor, although I haven't measured that yet.
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Question 1:
How many of those 10A DC-DC booster converters would I need, supporting only the headlights? The car uses 2-20A fuses for headlights, one each left and right. So theoretically 4x10A converters should provide 40A. However I think maybe it's better to provide 50-60A of capacity so as not to overtax the boosters. What do you think? The car uses separate high beam and low beam (dip) bulbs. At approx 10A each, when all four are lit that's 40A. This car lights all four bulbs when using the high beams. I don't think it's practical to reconfigure the circuit to use only one bulb set at a time - even on cars built to do that, there's always an instant when all bulbs are lit, when switching between high/low beams.
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I would forget the fuse ratings. Headlight bulbs use about 55w, or about 5Amps. One 10Amp converter should be adequate for each bulb.
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Question 2:
What do those boosters do when they get less than the specified 10V required input? Damage to the boosters? Or do they just not provide your specified output voltage briefly, and then go back to normal when the input voltage goes over the 10V minimum? Do you know?
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No, sorry, I don't know. I have never actually used a DC/DC converter, and I expect they vary in design and performance.
One thing to watch out for is that they usually don't have a common ground or common negative. That means that you have to isolate both the positive AND the negative outputs from the vehicle's chassis and from the rest of the car's wiring. Car wiring is not usually designed that way and almost always uses the car body as the negative return path, so unless you can find a converter that uses a common negative ground (input negative terminal connected directly to the output negative terminal) then that does potentially make it tricky to use them in a car for individual components. (I bought a 30Amp PWM for my blower motor that turned out to use a common positive rather than common negative, so that was a waste of money! My fault: I should have asked before buying.)
Using DC/DC converters to feed power into the WHOLE of the car's electrical circuit however is not so complicated as the only place where you have to carefully isolate a negative from the chassis is between the donor battery's negative terminal and the DC/DC converter. The output negative from the converter can be connected direct to the chassis as long as the input negative is not. I hope that makes sense. As I say I have not yet used one of these converters so take anything I say for what it's worth!
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I'm considering also using the boosted battery to power the wipers. The wipers never were very happy on the 12.x volts provided by the lithium pack. Just a matter of routing power to the wiper fuse patching it in.
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Again, that MAY be all that is required, provided you can find a converter with a common negative ground.
I have found just ONE such converter so far...
E914 12A 12v-12v Battery Charger (12V Turn On)
It is designed as a charger for an auxilliary battery to charge it from the vehicle's main cranking battery, so I'm not sure it will really be suitable as far as shut-off/switch-on voltage levels go, and the output seems to be a fixed 14.0v, which may be half a volt too low for my purposes, but at least it has a common negative!