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.

  • 3
    $\begingroup$ So what were the paragraphs before and after each of these formulae saying? $\endgroup$
    – Solar Mike
    Commented Nov 30, 2021 at 21:25
  • $\begingroup$ 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)". $\endgroup$
    – RMS
    Commented Nov 30, 2021 at 21:30
  • 1
    $\begingroup$ 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, $\endgroup$ Commented Nov 30, 2021 at 22:20
  • $\begingroup$ The first representation is the vector representation (in the complex plane). The second is the magnitude. ∠30° means cos(30°) + i sin(30°). $\endgroup$
    – user10249
    Commented Dec 2, 2021 at 15:30

1 Answer 1


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drawing by SSR

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} $).

The phase angles are only relevant when looking at phasor diagrams, so they are usually left off for clarity.


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