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.

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.

If V=V0sin(x)+V0sin(x+120), the max value of V is V0.

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] aol.com)

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] 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 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)

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.

>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.

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