What are the basic principles?
• Recall that when you multiply a pair of complex numbers, their phases (angles) add and their magnitudes multiply. Similarly, when you multiply one complex number by the conjugate of the other, the phase of the conjugated one is subtracted (though the magnitudes still multiply). Therefore: To add R’s phase to C: C’ = C·R Ic’ = Ic·Ir – Qc·Qr Qc’ = Qc·Ir + Ic·Qr To subtract R’s phase from C: C’ = C·R* Ic’ = Ic·Ir + Qc·Qr Qc’ = Qc·Ir – Ic·Qr • To rotate by +90 degrees, multiply by R = 0 + j1. Similarly, to rotate by -90 degrees, multiply by R = 0 – j1. If you go through the Algebra above, the net effect is: To add 90 degrees, multiply by R = 0 + j1: Ic’ = -Qc Qc’ = Ic (negate Q, then swap) To subtract 90 degrees, multiply by R = 0 – j1: Ic’ = Qc Qc’ = -Ic (negate I, then swap) • To rotate by phases of less than 90 degrees, we will be multiplying by numbers of the form “R = 1 +/- jK”. K will be decreasing powers of two, starting with 2^0 = 1.0. Therefore, K = 1.0, 0.5, 0.25, etc. (We use
