Shaft, if I understand you correctly you want to know how the SQRT 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 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:
Vab = Van-Vbn = Vrms [(1.0+j0.0)-(-0.5-j0.866)] = Vrms [1.5+j0.866)] = Vrms [1.5+j0.866] = SQRT(3)Vrms
Thus, the line voltage ( ph-to-ph)) is SQRT times phase (ph-neutral) voltage. (QED)
Regards, Phil Corso (cepsicon [at] aol.com)
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?
Yusuf C... the calculation follows:
Referring to the vectoral addition Vab = Van-Vbn = Vrms(1.5+j0.866) yields SQRTVrms@+30Deg.
In other words, for the sequence ABC, Vab leads Van by 30Deg, and Vrms is phase-to-neutral voltage.
FYI, phase-to-phase voltage is often referred to as Line-Voltage, while phase-neutral voltage is referred to as Phase-Voltage.
Regards, Phil Corso (cepsicon [at] aol [dot] com)