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I am in the early stages of reading about converting 120V AC to 240V AC to power an electric dryer.

One way to create 240V AC is to "find" two out of phase 120V outlets.

My question is HOW does an electrician establish which outlets will have what phase relationship? Can I measure these phase relationships at the level of the circuit breaker box, rather than searching the house for pairs that create 0v versus 240v?

How it's done by electricians is that they are familiar with the busing inside the service panel in question, and they know what they're punching down into. Also, anytime phase matters, they are using a multipole breaker (or at least handle-ties) which enforce phase choice for them, via keying.

Generally, there are only four different regimes: the standard alternating stabs (CH, BR, QO, HOM, Siemens), standard with space-straddling (GE and FPE), side-by-side busing (Pushmatic), or both-at-once busing (Zinsco). All the named panels except GE are obsolete.

Don't even think about it

What you're trying to do there is extremely bad at a variety of levels.

And it isn't going to work anyway, because the outlets you're trying to tap can't possibly deliver enough power. Not by a long stretch.

The best you can hope for is to rearrange the dryer's wiring so its heating unit runs on 120V instead of 240V (whilst the tumble motor and controls continue to get 120V). This will give 1/4 the heat, so drying will take much longer, but will draw within the delivery range of a 120V outlet.

You could also do something outlandish with AC-DC-AC double conversion using fully isolated DC supplies, but we're over the moon at this point, and it would be cheaper just to pay the landlord to fit a 240V circuit. (And no, you can't run the heating element on DC because the switchgear can't handle that because DC is a very nasty customer above about 40V.)

Buying an electric dryer was just a bad idea. Sell it on Craigslist and get a gas dryer and hook it up the normal way. If you weren't paying attention when you bought the dryer, welcome to the world of people who bought 3-phase machine tools without thinking that mattered.

Why suicide cords don't work

The first problem is one they share with any cord that has 2 male connections (like a generator cord built by someone who has apparently not heard of inlets): When one is plugged in, the other is lethal. Even if they are on opposite poles, you can't rely on unplugging the load first, because someone might knock out a cord by mistake.

Then you have the various sides of the circuit not running together in the same cable - making a big loop. (This blindsides DC electronics people everytime because it doesn't matter in DC). AC circuits throw considerable EMF. That's why we run conductors together, so the EMF cancels out. And this is a lot of current - a reed switch can operate on 10 ampere-turns, and you'll have 23 ampere-turns anywhere along the wire! Everything in the middle becomes the core of a transformer. This causes vibration, wire chafing and eddy current heating on anything nearby that's metallic. Like the receptacle itself! This vibration and heating takes energy, which reduces the effective capacity of the circuit.

Length is also an issue. Unequal lengths means phase and echo differences, that can themselves cause wire heating. It's complicated, but another energy loss.

There's just not nearly enough power on the circuits. A typical 120V outlet is good for 1800W intermittent, and your dryer is pulling 5600W or so continuous, which calls for a 125% derate, so 5600W becomes 7000W.

Even paralleling two 1800W circuits to make 240V, you're still only at 3600W - not even close.

I gather this isn't for a one-time use, so now you have the problem of routing all these cables. I gather the core issue is that a landlord won't give permission for a 240V connection, so I gather they won't give permission to fish or run Legrand Wiremold all over the walls, so no competent wiring method will be possible. So this becomes an ad-hoc connection with extension cords draping across doorways, which is a wretched fire hazard. This problem will also haunt the AC-DC-AC conversion method. The only way that can work is if a huge storage battery is involved, but 5600W for an hour is a whole lot of storage.

A typical electric clothes dryer in the US runs on 120V/240V. 120V for controls, motor (could be 120V or 240V) & light, 240V for the heat. The key is the 240V for the heat - that is the most practical way to get enough heat with an electric dryer. In almost all cases, the incoming power will provide 120V/240V without much effort.

However, you need to have the proper equipment installed - a double-pole breaker (not two separate breakers, and most definitely not "use a couple of existing circuits") to provide power in a safe manner (double-pole means it will trip if there is an overload in either 1/2 of the circuit) and a proper 4-wire receptacle installed (there are older 3-wire receptacles but they are not for new installations, and any dryer built in the last 25 years (probably longer) can be easily switched from 3-wire to 4-wire connection). The wire between the breaker and the receptacle needs to be sized correctly as well - typical 10 AWG for a 30 Amp circuit.

Don't give up on the idea just yet. Post pictures or additional information about the breaker panel and it is likely (but not guaranteed) that there is a reasonable solution.