Shane Colton blogs about an alternative, faster algorithm to compute an arctangent in phase measurements.

Normally, to get the phase angle of a set of (assumed balanced) three-phase signals, I’d do a Clarke Transform followed by a atan2(β,α). This could be atan2f(), for single-precision floating-point in C, or some other approximation that trades off accuracy for speed. The crudest (and fastest) of these is a first-order approximation atan(x) ≈ (π/4)·x which has maximum error of ±4.073º over the range {-1 ≤ x ≤ 1}.

It’s possible to extend this method to three inputs, a set of three-phase signals assumed to be balanced. Instead of quadrants, the input domain is split based on the six possible sorted orders of the three-phase signals. Within each sextant, the middle input (the one crossing zero) is divided by the difference of the other two to form a normalized input, analogous to selecting x = β/α or x = α/β in the atan2() implementation.

For this three-phase approximation the maximum error is ±1.117º, significantly lower than the four-quadrant approximation. If starting from three-phase signals anyway, this method may also be faster, or at least nearly the same speed.

See the details in the post here.