AC power and three-phase

Từ vựng

English	Ý nghĩa
complex power	Công suất phức
power factor	hệ số công suất
power-factor correction	Parallel capacitance supplying the load reactive power locally.
three-phase	ba pha

English

Power: real, reactive, apparent — and the correction

Complex power $S = VI^{*}$ splits into $P = VI\cos θ$ (billed, W) and $Q = VI\sin θ$ (borrowed, var); the power factor $\cos θ$ is the share of apparent power $|S|$ that does real work. The loss formulas $P = |I|^{2}R$, $Q = |I|^{2}X$ never touch the angle — they are the check.
Power-factor correction supplies Q locally with $C = Q/(ωV^{2})$; a balanced three-phase load is analysed per phase, with $V_{L} = \sqrt{3}\,V_{phase}$ in star and total power $\sqrt{3}\,V_{L} I_{L}\cos φ$.

Example

The RL load: $P = 100 \times 2 \times 0.6 = 120$ W, $Q = 160$ var — and $|I|^{2}R = 4 \times 30 = 120$, $|I|^{2}X = 4 \times 40 = 160$ agree. A 50.9 µF capacitor at 100 V, 50 Hz cancels the 160 var. Escalate to a star motor on 415 V with $10 + j10$ Ω per phase: phase current $239.6/14.14 = 16.94$ A, and $P = 3 \times 239.6 \times 16.94 \times 0.7071 = 8611$ W by the per-phase route and $\sqrt{3} \times 415 \times 16.94 \times 0.7071 = 8611$ W by the line route.