Might an x-ray tube have the capability produce more power than it consumes?

Question

In transformers, when the voltage is powered up, the current is proportionally powered down.

In the x-ray tube’s transformer, this also happens.

However, the current of an x-ray tube is not the powered down current from its transformer. It is the current that results from electrons produced due to thermionic emission at the cathode (the tube current).

Supposing there is an amount of power (let us call it x) that if fed into the thermionic emission circuit of the x-ray tube would enable the tube current to rise to a value just above the powered down current.

Let us label the power fed into the x-ray tube’s transformer as y.

Let us refer to the electrical equivalent of heat produced at the anode per second as a.

We shall refer to the electrical equivalent of x-rays produced per second as b.

x and y are the only power inputs. a and b the only outputs.

If x added to y is less than a added to b, then the x-ray tube would be producing more power than it was given.

I have considered that the maximum temperature we can allow the Tungsten cathode to reach is that just below its melting point.

However, I was wondering whether this entire assertion could be pitched against the goal of producing cleaner more affordable energy in the world.

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Answer 1

The power in the heater is wasted power, it just boils off some electrons to permit current to flow. The power in the anode motor (if any) is also wasted.

All the power dissipated in the (typically rotating) anode is due to the high voltage current (a short pulse of maybe 300-800mA at 50-150kV). A bit of that power (maybe 1%) comes off as X-rays, the rest is absorbed as heat in a few rotations of the anode.

So mostly an X-ray machine turns electrical power into waste heat.

There is no free lunch, conservation of energy and all that.

Answer 2

"the current [...] is the current that results from electrons produced due to thermionic emission at the cathode (the tube current)."

Congratulations! You reinvented the thermionic generator. Just don't use the X-ray tube stuff around it and don't forget that you have to supply heat (at one end and cooling at the other!) from outside.

https://www.britannica.com/technology/thermionic-power-converter says

They can provide efficiency in the range of 12 to 15 percent at temperatures of 900 to 1,500 K (about 600 to 1,200 °C, or 1,200 to 2,200 °F). Since these converters function best at high temperatures, they may eventually be developed for use as topping devices in conventional fossil fuel power plants. Their currently available efficiencies make them suitable power sources for terrestrial application in certain remote or hostile environments.

Answer 3

Although, yes, the thermionic emission driven current is not coupled directly to the input transformer/heater current, the entire power in the system is constant. If anything, using the x-ray tube, the power would be significantly decreased due to the large heat dissipation inherent in thermionic emission. The system may trick an output system to have a spike in power for a short period while the emission is taking place, but the overall average power has to be conserved.