# Line to neutral voltage in three-phase power systems

I have the book "Power system analysis and design" by Sarma. In his intro to balanced LN voltages, in chapter 2 p62 he states:

$$V_{LL} = \sqrt{3} V_{LN} \angle30º$$

but later in the chapter, p70, he states for balanced LN voltages:

$$V_{LL} = \sqrt{3} V_{LN}$$

This is a very different statement, why is he saying that after literally saying a few pages before there is a 30º phase shift.

• So what were the paragraphs before and after each of these formulae saying? Nov 30, 2021 at 21:25
• For the first equation: " In a balanced 3-phase Y-connected system with positive sequence sources, the LL voltages are $\sqrt{3}$ times the LN voltages, and lead by 30º. For the second one he says "For 3-phase generators under balanced conditions, (formula 2)".
– RMS
Nov 30, 2021 at 21:30
• If they have to spell it out every time it is used, what happens the first time they forget it. Both are true. First is vector represention and second is magbitude, Nov 30, 2021 at 22:20
• The first representation is the vector representation (in the complex plane). The second is the magnitude. ∠30° means cos(30°) + i sin(30°).
– user10249
Dec 2, 2021 at 15:30

Because $$V_{AB} = V_A - V_B$$ (Vector Subtraction). Dashed line is $$-V_{B} = -(120 \angle -120° V)$$ $$= 120 \angle 60° V$$. Vector addition of $$V_{A}$$ and $$-V_{B}$$ gives line voltage $$V_{AB} = 208 \angle 30° V$$
In a three-phase, wye connected system the line-to-line voltage (line voltage) $$V_{LL}$$ ($$V_{AB}$$, $$V_{BC}$$, $$V_{CA}$$) is $$\sqrt {3}$$ larger than phase to neutral voltages (phase voltage) $$V_{LN}$$ ($$V_{AN}$$, $$V_{BN}$$, $$V_{CN}$$) and leads the phase voltage by 30°.
In the first case the author is making emphasis that the line voltage is a vector (magnitude $$\angle$$ angle or $$V_{LL} = \sqrt{3} V_{LN} \angle30º$$) and the second case talks about magnitude ($$V_{LL} = \sqrt{3} V_{LN}$$).