According to this video the BBC Micro's power supply provided a 5 Volt supply to the motherboard.
Could a BBC Micro have it's internal power supply replaced with a USB power connector?
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No, or at least not without consideration.
According to the sticker atop the PSU it delivers:
A quick check of the BBC Model B (*2) reveals that
So if you do not intend to read cassette tapes, a single +5 V PSU will do it. The stated 3.75 A is for the whole system, including expansions (tube, etc.). A rough guess would place the motherboard at somewhere between 2 and 2.5 A.
Basic USB is defined as 5 V and 0.5 A. USB 3.0 lifted that to 0.9 A and allows 'charging' ports of up to 1.5 A (see this answer on Superuser). The connector itself is specified for up to 5 A.
As a result almost all USB ports will be unable to power a BBC (*2).
For USB wall warts the situation is a bit different, as there are some that can deliver higher currents. I happened to find several 2 and 2.5 A ones with a quick Google search. When looking for one, it is helpful to check for power (in Watts) as well, as some are described that way. Since Voltage is fixed at 5V, 10 W would mean 2 A, 12.5 W ⇒ 2.5 A, and so on. If the PSU has multiple ports, it must be checked whether that rating is per port or total.
External USB power adaptors may do the trick if rated at or above 2.5 A or 12.5 W. (*3)
In addition it might be a good idea to check any cheap USB PSU for leftover noise before using it, as DmytroL mentions. While most deliver a rather good signal, it's still possible to save on components used for noise reduction. Except the savings are so small that they are impossible to be seen by looking at the (final) sales price, so buying an expensive one will not guarantee a great supply – only good sales profits. :(
Now, if you want to use the cassette port, then generating the -5V from +5V isn't as hard as it might sound. Some 7660 based voltage inverter will do the trick of delivering -5 V (*4). There are plenty ready-to-use boards on Ebay, Alibaba and most part dealer websites. They are as small as 2 by 3 cm and all deliver at least the 100 mA.
Easy to hide inside the BBC, so a single USB plug is possible.
If you're going to that length, it's maybe a good idea to add some noise filtering as well. While it may mean making your own (little) board, it will not add much in size or cost, and guarantee long enjoyable usage.
Time to get pliers and a soldering iron out, isn't it?
*1 - Don't have any other at hand.
*2 - 1.5 A may in fact get it to work, as the base unit is rather low power, but I wouldn't bet on it.
*3 - More is better, especially if there are some changes and/or add-ons.
*4 - More exactly some 4.8 to 4.9 Volt, but that's still within acceptable range for the cassette interface.