Went to aviation sites.
From what I gather after a half-dozen sites intended for aviator and aviation fans, black boxes have three power sources.
Both black boxes are powered by one of two power generators that draw their power from the plane's engines. One generator is a 28-volt DC power source, and the other is a 115-volt, 400-hertz (Hz) AC power source.
https://science.howstuffworks.com/transport/flight/modern/black-box.htm
This is the same specs found on multiple sites, so probably good (or at least not challenged, that I can find). But notice, the generators draw power from the engines, they are *not* batteries nor do they have separate fuel supplies.
Some sites explicitly refer to a battery power supply. This was confusing because others refer to the two generators. Turns out both are likely correct:
The beacon sends out pulses at 37.5 kilohertz (kHz) and can transmit sound as deep as 14,000 feet (4,267 meters). Once the beacon begins pinging, it pings once per second for 30 days. This beacon is powered by a battery that has a shelf life of six years. In rare instances, the beacon may get snapped off during a high-impact collision.
So the recorders, that take more juice, are connected to an external powers supply. That fails, when the magnetic field in the generator collapses there's no more current to run the recorders. But the beacon's power supply will keep on powering the beacon.
So if the power from the engines stops for whatever reason, no more recorded telemetry. And, yes, since the black boxes aren't located anywhere near the cockpit, if something interrupts the source of the telemetry--like the transmission lines or wifi signal is stopped--telemetry is lost and if they have power they record nothing.
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