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While early tubes used the directly heated filament as the cathode, most ( but not all ) more modern tubes employed indirect heating.
A separate element was used for the cathode.
Inside the cathode, and electrically insulated from it, was the filament or heater.
Thus the heater did not function as an electrode, but simply served to heat the cathode sufficiently for it to emit electrons by thermionic emission.
This allowed all the tubes to be heated through a common circuit ( which can as well be AC ) while allowing each cathode to arrive at a voltage independently of the others, removing an unwelcome constraint on circuit design.

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