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How to calculate phase to phase voltage
Calculation of phase to phase voltage

For 3 phase 4 wire network, how to calculate Phase to phase voltage from Line to neutral volage.

Square root of 3 times line to neutral voltage for Y config.

By Trevor Ousey \(list\) on 7 October, 2007 - 3:20 pm

For standard 120 degree between phases it is the L-N voltage multiply by the square root of 3.

Why is it the square root of 3?
If V=V0sin(x)+V0sin(x+120), the max value of V is V0.

1 out of 1 members thought this post was helpful...

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:

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[3] times phase (ph-neutral) voltage. (QED)

Regards, Phil Corso (cepsicon [at]

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?

1 out of 1 members thought this post was helpful...

Yusuf C... the calculation follows:

Referring to the vectoral addition Vab = Van-Vbn = Vrms(1.5+j0.866) yields SQRT[3]Vrms@+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)

>Why is it the square root of 3?
>If V=V0sin(x)+V0sin(x+120), the max value of V is V0.

Vab = Van-Vbn = V0sin(x)-V0sin(x+120).

Try to derive the rms value of V0sin(x)-V0sin(x+120) and you will get sqrt(3). Vector analysis is much easier to analyse.