ELC-06

Three-phase active power

P = √3 VL IL cos φ for a balanced system.

Reading speed

Power systemsIEC 60038

Governing equation

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

$\varphi \in \left[-60, 60\right]$

where

V_L: Line voltage (V)
I_L: Line current (A)
\varphi: Phase angle (°)
P: Active power (kW)
Q: Reactive power (kvar)

The figure, part by part

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①

Governing law — red

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

$\varphi \in \left[-60, 60\right]$

$\varphi = -60$ $P = 6.928$

Along the curve

②

Domain of the variable — blue

\varphi_{\min} = -60,\quad \varphi_{\max} = 60

$\varphi \in \left[-60, 60\right]$

$\varphi = -60$ $\varphi = -60$

Along the curve

③

Response curve — green

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

$\varphi \in \left[-60, 60\right]$

$\varphi = -60$ $P = 6.928$

Along the curve

④

Rate of change — amber

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

$\varphi \in \left[-60, 60\right]$

$\varphi = -59.25$ $P^{\prime} = 0.208$

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-06 — Three-phase active power) is the form associated with IEC 60038. Working symbols: $V_L$ , $I_L$ , $\varphi$ $\rightarrow$ $P$ , $Q$ . Three 120°-shifted phases each contribute Vφ Iφ cos φ; with VL = √3 Vφ and IL = Iφ this collapses to √3 VL IL cos φ.

Purpose

Purpose: compute

P

Q

from

V_L

I_L

\varphi

in Electrical via

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

P = √3 VL IL cos φ for a balanced system. 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 = 20.000\,\mathrm{A}

\varphi = 25.000\,\mathrm{^{\circ}}

, the governing relation

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

yields

P = 12.558\,\mathrm{kW}

Q = 5.856\,\mathrm{kvar}

. Balanced wye or delta, sinusoidal. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Line voltage V_L400 VLine current I_L20 APhase angle \varphi25 °

Outputs

Active power P12.558 kW
Reactive power Q5.856 kvar

Reading speed

Watch on YouTube

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Narration of this film

Balanced wye or delta, sinusoidal.

Three 120°-shifted phases each contribute Vφ Iφ cos φ; with VL = √3 Vφ and IL = Iφ this collapses to √3 VL IL cos φ.

Reading speed

Watch on YouTube