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I forgot to charge my time machine and now I'm sitting in medieval Europe and need to recharge the batteries. Luckily, those are only car batteries and not the old plutonium powered ones.
What's the easiest way to generate a sufficiently stable voltage/current output to charge my car battery?
Assume I can talk to other people and buy what's widely accessible, but if I tell them too much they'll probably burn me as a witch.
Bonus question, can I get some kV AC?
The first piece of good news is that witch burning is much more an Early Modern phenomenon than a Medieval phenomenon. The second piece of good news is that you can pass for a very respectable alchemist, especially if you have a white beard and know how to make simple explosives such as nitrocellulose, also known as gun cotton (and as fulmicoton in French, a splendid name). The third piece of good news is that copper wire was readily available, as were various acids and zinc (available for export in India since the 12th century). Make a number of zinc-carbon cells, connect them in a suitable way (enough parallel rows of 9 or 10 cells each) and charge your lead-acid car battery.
Depending on when exactly you are stranded, you may have to build the ships to go to India to buy the zinc, but that's a minor consideration. (If you want to get zinc from European sources or if you must wait for Vasco da Gama to open the trade with India then that's the 16th century; as some users here tell me, the tendency nowadays is to consider the Renaissance a major historical period on its own instead of simply the last part of the Middle Ages, so that would be outside the indicated time frame.)
For those kilovolts AC you need alternators. Iron is available, copper wire is available, shellac you can buy from India: you are all set.
By the way, you have advanced the knowledge of explosives, of electrochemistry and of electricty in general by several centuries. You may have also brought the Age of Discovery a few centuries before its time. The world you will return to won't be the world from where you started your journey.
Fortunately you already have an AC generator built into your car, the alternator. All you need to do is turn it fast enough to generate electricity. Usually that's done by the car's engine, but you have to get the engine started first.
If you were REALLY smart you'd have brought a jump-starter with a built in battery and just jumped the car with that.
Put it right next to the Chilton's Guide, textbooks, spare gasoline and fluids, spark plugs, fuses, and all the other spare parts you also thought to bring with you before time traveling (or just have in your trunk because you're Ash Williams).
If you're smart, your time traveling car is a manual transmission. It's easier to repair and can be push started. A modest 5 to 10 mph is all that's required, rolling down a shallow hill or getting a few folks from the local inn to push. This provides enough mechanical energy to turn the alternator and generate electricity for the spark plugs to start the gasoline engine. Once the engine is turning, the alternator will charge the battery.
(If you're smart you'd also use a diesel set up to run bio-diesel. If you run out of gas you can use vegetable oil. It also needs no spark plugs, one less thing to go wrong.)
Push starting an automatic is much harder and might be possible around 30-40 mph, but I wouldn't risk irreparable damage to the transmission.
Instead, you'll have to turn the alternator manually. Most alternators need 500 to 1500 RPMs to charge and need less than 1 horse power. This can be done with leather belts (or adapt the ones already in the engine), simple gearing, and some way to get 1 horse power in medieval Europe... perhaps a draft animal, a wind mill, water wheel, some friendly peasants. I guess a horse will work, too.
Getting the RPMs just right isn't very important, any car alternator in the last 50 years will have a voltage regulator.
For payment there are any number of gadgets and tools in a car which would be worth a fortune in medieval Europe. A knife, lighter, calculator, clocks, flashlights, foam padding... even something as simple as a watertight plastic bag, or tiny springs and screws are amazing technology in medieval Europe.
The good news is that the Middle Ages in Europe were the first big expansion of non-animal generated power in history. Europe is pretty uniquely advantaged (along with Eastern North America...hmmm where did those Industrial Revolutions take place...) with having lots of perennial streams that have reasonably high speed and are constant year-round. This reliable energy source is going to give you what you need to juice up your time machine.
The technology of water wheels were pretty well developed by the 'Middle Ages', and certainly more developed in the later parts. By the 13th century, dozens of smithies had relocated to the hills around Milan to use hydro-powered bellows to crank out steel for the Milanese armorers, merchants in Arezzo in Tuscany pioneered using hydro-power to mechanize throwing silk, etc. Check out further applications in Cathedral, Forge, and Waterwheel by the Gies'.
In any case, that is by far your best bet for a power source. A watermill can be expected to keep pretty constant flow on the timescale of day and will provide a steady rotational speed, which is of course the key to providing a reliable source of DC.
Now you just have to build a dynamo to generate some electricity, and potentially some transformers or whatnot. This would be very challenging for someone who didn't have experience building one from scratch. But again, the upside is that the necessary skills are there. Bell-casters, particularly in the 14th century onwards when they started casting cannon, can make solid iron cores. Mail-makers have extensive experience with drawing the necessary wires. Now, your wires aren't going to be thin, and you will have to find some sort of insulating coating to put on the wires before you coil them into your generator's armature, but no one said this would be easy.
The most challenging part is actually going to be the axle for your generator's rotor. There are no ball-bearings available, so you will have to use a greased wooden axle and hope for the best. You probably should monitor this apparatus constantly to be able to disconnect the power source from your machine in case of an (inevitable) accident.
As for AC, this method would be easier for AC since you wouldn't need to make the brushes. This would make the copper-smith's job easier, and reduce the potential for mishaps (of the fire creating sort...high voltage plus grease and wooden parts...). Good luck! And definitely read up on DC generators (or bring a book) before you travel!
Only improvements to the other answers:
What AlexP means are voltaic piles. The problem is that only a small current and low voltage is delivered by one element. So you need + to -, + to - (serial circuit) to increase the voltage n times and + to +, - to - (parallel circuit) to decrease the inner resistance, increasing the current with the same resistance.
At least you will be busy for a time because if your time machine is battery powered, I expect that you need to rent a barn for the voltaic pile. This is the next problem: I hope your car has a comfortable and safe charger because evil things could happen if you increase the voltage/current too high or if you forget to stop charging. Lead acid batteries don't like such things, they are getting warm and can even begin literally to cook.
The kingledion approach is also valid, but will also make you busy because you need to wind coils (and naturally a good, powerful watermill or windmill) to get kV AC. The only other way is a treadmill, so if you are able to get some people with endurance.....
What does not work:
Lightning: Will only destroy your battery. Too much power in a too short timeframe, you have no way to get a consistent load and you cannot store the energy.
Solar: No photovoltaic cells available. You need silicon/germanium which is not available, it needs to be doped which needs technology you do not have, etc. etc.
For the kV AC you could build a Tesla coil, which is an RF step-up transformer. The AC would be generated from the DC supply by a spark gap in the primary side. You would need some hundreds of volts or more to keep the spark alive, depending on what material you had for the gap- preferably some graphite sharpened into points. That's just a matter of making enough cells in series (and they have to be fat enough to generate enough power that the arc heats the graphite sufficiently to allow a reasonably wide gap).
RF will be much easier than trying to build a conventional alternator plus low frequency step-up transformer as you would not require a ferromagnetic core for either, fewer turns would be required, allowing cruder wire and insulation, and you would not have to find a water wheel or windmill for motive power.
If you can manage to get some metalworking tools and a permanent magnet, you can build a homopolar generator, which generates a DC potential difference between the disc's axis of rotation and its edge (link one electrode to the disc's axle, the other will be on a brush touching the edge of the disc). Then you can hand-crank it or link it to a waterwheel or windmill. Just make sure that you figure out which way to turn it to get positive voltage to the positive electrode! It'll depend on which orientation you put the magnet in...
Regarding magnets, naturally-magnetic magnetite ore was known in Mediaeval times (under the name lodestone in England), and there was a modest demand for them as aphrodisiacs (!), so a large city might have some way of procuring them. Of course, if you can repurpose an existing magnet from your time machine (from the door of a mini-fridge, for example) it will be much easier on you.
As a last-ditch alternative, it might be possible to link the generator's electrodes in parallel with a solenoid coil and make that into an electromagnet. I haven't calculated the circuit, but it might be possible that the Earth's magnetic field could provide enough magnetic field strength to bootstrap the electromagnet...
You could make a massive lemon battery using an entire orchard of lemons if need be. I'm sure you could calculate just how many lemons you would need.
