I bought a Sinclair ZX81 on a garage sale some days ago for around 12 USD as a non-tested item as the guy said he had no wires to even try and run it. I have a couple of RF cable kicking about from my Atari 2600 and ZX Spectrum, so I have this basis covered.
According to the manual LINK it takes 9V 700mA unregulated (I'm not sure what it means) 3.7mm jack power supply. I have a 9V DC 1A jack power supply I bought for the Atari (it's not an original one - a unbranded one) and I tried to use it, but to no success. Manual tuning my TV same as with Spectrum and Atari, and no signal.
I was wondering if maybe the power supply is not up to spec? I was thinking that maybe 1mA is too much to power it up? But I went online to look for an unbranded ZX81 power supply and they are running 2A so I'm not sure what's going on here.
Is it busted? Or should I get one of those unbranded power supplies that say they are made for ZX81 to test it?
Short answer: your power supply should work - you just need the right voltage, current rating, polarity, and connector.
An unregulated power supply is a kind of power supply whose output voltage varies with current draw. In other words, it does not put out a constant voltage - it is only rated to supply a minimum voltage at a certain current draw. In this case, you will need a power supply that can provide 9V DC at 700mA. Because most devices require a constant voltage, you can always use a higher rated supply.
If you test an unregulated power supply with a multimeter, you will likely see the voltage is higher than what is on the label, say 9.5V DC. That is because the multimeter is not putting a heavy load on the power supply. As the current draw increases, the supply voltage will drop. That is why you can use a higher rated device, but not a lower one.
In your post you said it uses a 3.7mm plug, but the manual link says 3.5mm with a positive tip. I assume that it means it uses a 3.5mm phone connector for power, like what you would see on a pair of wired headphones. The important part is that the positive 9V is connected to the tip, and ground on the sleeve. Sometimes, reversing these connections can damage devices, but there is usually a protection circuit inside that prevents this, and I would wager that this is no exception.
To simplify the current-aspect: When in comes to powersupplies, it helps to think about mA or Amperes as "capacity". Under normal circumstances, too much won't hurt. In many cases, too little might actually work too. So when searching for a suitable powersupply, the Voltage should be the same, amperes should be "enough", and the polarity must match.
If your hardware lists Watts instead of Amperes, you can use Ohms law to calculate between the two:
P = U x I
Or in other words: Watts (W) = Voltage (V) x Current (A)