AC motors in general are not inefficient, and the most efficient motors are AC, but shaded pole motors used in many fans are quite inefficient, but they’re popular because it’s a very cheap design to manufacture. The “shaded pole” is how the motor spins in a specific direction, despite only having 1 phase power input.
If you see “brushless DC” It’s actually an AC motor(3 phase), which needs extra control circuitry to drive. Basically it’s converting the AC from the wall into DC, then back into AC at whatever frequency gets the fan to spin at the desired speed, and whatever voltage makes manufacturing it cheapest.
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