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Why PWM saves power than DC control? Find a mathematical deduction.
Who said that? I specialize in motor drive circuit design.

Generally speaking, it is similar, but different fields are suitable for different motors.

If you must say so, please see below. ...

DC control is also turned into AC drive motor, but it should be noted that it is turned into square wave here.

The motor driven by PWM inverter can be equivalent to sine wave,

A square wave and a sine wave, which is clear here. ....