Shaft, if I understand you correctly you want to know how the SQRT[3] is derived? For simplicity I will define phase-to-neutral voltages as "phase" voltages, and phase-to-phase voltages as "line" voltages.
Reverting to basics, the instantaneous phase voltage of a single-phase system is expressed as V(t) = Vm x Sin(wt), where the wave's amplitude is Vm, and its frequency, in radians, is (wt)
Expressing the wave as a vector, having both magnitude and direction, the phase voltage in RMS, Vrms equals Vm/SQRT[2] at a reference angle, usually taken as zero.
The three-phase system has three phase voltages, Van, Vbn, and Vcn. Their corresponding vector notations are:
Van = Vrms at 0 deg (ref); Vbn = Vrms at -120 deg; Vcn = Vrms at +120 deg.
Because their magnitudes are equal, i.e., Van = Vbn = Vcn = Vrms, and given an A-B-C phase rotation, their corresponding complex phasor notations are:
Van = Vrms(1.0+j0.0) as reference.
Vbn = Vrms(-0.5-j0.866), lags Van by 120 deg
Vcn = Vrms(-0.5+j0.866), lags Van by 240 deg
The phase-phase voltages are related to the phase-neutral voltages as by vectoral addition:
Phil, Thank you for this explanation. This was very useful. For this same example (and using your assumptions) how would you calculate the phase angle for Vab?