ELC-24

Three-phase power

P = √3 VL IL cosφ. Balanced three-wire, line quantities.

Reading speed

PowerIEC 60034

Governing equation

P=\sqrt{3}\,V_L I_L\cos\varphi

$\cos\varphi \in \left[0.1, 1\right]$

where

V_L: Line voltage (V)
I_L: Line current (A)
\cos\varphi: Power factor (—)
P: Active power (kW)
S: Apparent power (kVA)

The figure, part by part

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①

Governing law — red

P=\sqrt{3}\,V_L I_L\cos\varphi

$\cos\varphi \in \left[0.1, 1\right]$

$\cos\varphi = 0.1$ $P = 3.464$

Along the curve

②

Domain of the variable — blue

\cos\varphi_{\min} = 0.1,\quad \cos\varphi_{\max} = 1

$\cos\varphi \in \left[0.1, 1\right]$

$\cos\varphi = 0.1$ $\cos\varphi = 0.1$

Along the curve

③

Response curve — green

P=\sqrt{3}\,V_L I_L\cos\varphi

$\cos\varphi \in \left[0.1, 1\right]$

$\cos\varphi = 0.1$ $P = 3.464$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}P}{\mathrm{d}\cos\varphi}

$\cos\varphi \in \left[0.1, 1\right]$

$\cos\varphi = 0.106$ $P^{\prime} = 34.64$

Along the curve

Lecture brief

Historical brief

Ohm (1827), Kirchhoff (1845) and Maxwell’s circuit reduction still run every board: RLC transients, transformers, skin effect and three-phase power. The sheets are those network laws, not a SPICE deck. This sheet (ELC-24 — Three-phase power) is the form associated with IEC 60034. Working symbols: $V_L$ , $I_L$ , $\cos\varphi$ $\rightarrow$ $P$ , $S$ . Three 120° phasors of line-to-line voltage and line current; active power is √3 VL IL cosφ.

Purpose

Purpose: compute

P

S

from

V_L

I_L

\cos\varphi

in Electrical via

P=\sqrt{3}\,V_L I_L\cos\varphi

P = √3 VL IL cosφ. Balanced three-wire, line quantities. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

V_L = 400.000\,\mathrm{V}

I_L = 50.000\,\mathrm{A}

\cos\varphi = 0.850\,\mathrm{—}

, the governing relation

P=\sqrt{3}\,V_L I_L\cos\varphi

yields

P = 29.44\,\mathrm{kW}

S = 34.64\,\mathrm{kVA}

. Three lines, a load triangle, a power arrow. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Line voltage V_L400 VLine current I_L50 APower factor \cos\varphi0.85 —

Outputs

Active power P29.44 kW
Apparent power S34.64 kVA

Reading speed

Watch on YouTube

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Narration of this film

Three lines, a load triangle, a power arrow.

Three 120° phasors of line-to-line voltage and line current; active power is √3 VL IL cosφ.

Reading speed

Watch on YouTube