Cigarette lighter using rechargeable AA batteries …

NiCads have a lower internal resistance than NiMH. SLA even lower.

Having said that, I'd approach this as an academic exercise rather than as a practical project opportunity

This page http://www.buchmann.ca/article4-page1.asp (and I think they mean milliohms rather than milliwatts in the table) gives you some indication of internal resistances for cell types. (remember to divide these figures by the number of cells to get the per-cell internal resistance)

I think you'd have a better chance with D cells (certainly SLA cells are available in that size)

A major issue would be the connection to the cells. You would have to use cells that are terminated with solder tags rather than bare cells -- the connection resistance would be far too high.

You would need to look at the datasheets on individual cells to determine if the discharge rates are possible for the cell.

Another issue would be that you would almost certainly need to generate another higher voltage source to power your regulator. It is difficult to imagine any of the more efficient designs starting up on their own from 1.5V. You might need a more specialised "joule thief" type of inverter to generate an initial 12V rail to power the main inverter before using the generated 12V rail for continuous operation (or not -- you could use 2 regulators, it's not like a bit of inefficiency here would be a real issue).

More practically, you may be better off creating your own specialised "cigarette lighter" from a coil of nichrome wire of sufficient length to glow red hot from just a 1.2V supply. Since this device would be small enough to turn on when brought to the cigarette, it need not require the relatively large thermal mass of a conventional car cigarette lighter.

I would estimate that you could probably create a device that used perhaps only 10W (i.e. 8A) and would only need to be operated for a couple of seconds. The major issue here would be the contact resistance and the switch. It may be sensible to use a small inverter to provide gate voltage for a high current mosfet that has a very low RDSon.

At a minimum, I think you'd still be looking at a sub-C sized cell.

OK, here's the specs I found on a sub-C cell. It is rated for up to 30A discharge. http://www.gpbatteries.com/pic/330SCH-ZRS1161rev2.pdf Note the voltage at 30A, also note that the effective capacity is much lower, still it looks like you'd get 6 minutes use at 30A which is pretty good.

This page has more battery types listed: http://www.rechargeable-battery-review.com/data-sheets/battery-specs/gp-battery-data-sheets.html

From a quick look, it appears that the AA sized cells top out at a recommended max of 6.6A. (D cells go to 50A)

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Cigarette lighter using rechargeable AA batteries ...

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