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Can L6-20 charge at 32A?

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See my reply, yeah, that should bother you more than a bad adapter.
As best I can tell from his picture, he has a 20A GE type THQL breaker. So I went and found the time current curve for that breaker:

http://apps.geindustrial.com/publibrary/checkout/GES-9884?TNR=Time Current Curves|GES-9884|PDF&filename=GES-9884.pdf

Based on the trip curve, the breaker should trip between roughly 30 seconds and 4 minutes at 32A. So if OP stopped charging in under 30 seconds after charging current reached 32A, the breaker should not have tripped. If OP stopped charging between ~30 seconds and 4 minutes, the breaker may or may not have tripped, with tripping becoming more likely the longer it was charging.

However, if OP let it run for greater than 4 minutes at 32A and the breaker did not trip, then he has a breaker problem. But from his posts it doesn't sound like it was at 32A for more than a few seconds, so there is no issue with his circuit breaker.
 
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Most breakers are designed to trip instantaneously only at fault level currents, otherwise you would have a lot of nuisance trips every time you start your vacuum cleaner with the coffee maker on. It sounds like you reacted quickly to the ramp, so I doubt it reached its designed trip time for that amount of current, and I definitely wouldn't expect it to trip as soon as it went above 20A.

See if you can find the model number and pull the time current curve for your breaker off the internet, then you can determine the amount of time it should take for your breaker to trip at 32A. I would expect somewhere around 30 seconds to a minute since 32A is less than twice the frame rating of the breaker.

Made-up trip characteristic example:
Breaker XYZ-30 will trip between 15-20 seconds at twice the frame rated current, but at three times the rated current, it will trip in 4-6 seconds. It will trip instantaneously above ten times the rated current.

Well of course you have a time factor to this, but if he's watching it ramp up to 32A, that isn't happening instantaneously. It would've crossed into the TCC for a while. But I wasn't there and the OP hasn't said for how long he had to wait before he shut it off.