This thesis carries out two basic studies for the flux function and the global observability of IPMSMs, leading to, respectively, two lightweight sensorless algorithms. First, we show that the flux function almost always gives a one-to-one correspondence between the rotor angle and flux, implying that the familiar IPMSM flux equations admit a unique position solution. This result eliminates the need for the arctan/atan2 function, leading to a simple flux estimation-based algorithm. Second, we show that the pair (speed, position) of IPMSMs is not globally observable, and the number of all indistinguishable pairs is at most four, an invariant independent of motor parameters and coincides with the number of the solutions of the two fundamental IPMSM equations. This result, which caps the worst-case scenario sensorless IPMSMs can behave, follows as a corollary by characterizing the observability condition involving infinitely many equations as a special limited form of the injectivity nature of finitely many polynomials. This bigger problem is approached through algebraic methodologies. The FOC scheme causes the “collapse” of at most two indistinguishable pairs and joins forces with the knowledge of motor/generator mode to turn the sensorless task “observable”, leading to another simple algorithm. / Thesis / Doctor of Philosophy (PhD)