power triangle

power triangle

1. What Power Triangle Is

Power triangle = right-angled triangle with sides **real power $P$ (W, horizontal), reactive power $Q$ (VAR, vertical), and hypotenuse apparent power $S$ (VA):

      S = √(P²+Q²)  (hypotenuse, VA)
     /|
  Q / |  (VAR, vertical: + for inductive, - for capacitive)
   /  |
  /_φ |
  P   (W, horizontal)

Where

Complex power:

(with $I*$ conjugate).

2. Components

Power	Unit	Meaning	Formula
Real ($P$)	watt (W)	useful work/heat (dissipated in $R$)	$P=VI cosφ$
Reactive ($Q$)	VAR	stored in $L/C$ fields, exchanged	$Q=VI sinφ$ (+ inductive, - capacitive)
Apparent ($S$)	VA	total supplied (vector sum)	$S=VI =√(P²+Q²)$

Scaling: If $Z=R+jX$, then

.

3. Lagging vs Leading

• Lagging (inductive, $φ>0$): $Q>0$, $S$ in first quadrant, current lags.
• Leading (capacitive, $φ<0$): $Q<0$, $S$ in fourth quadrant (negative $Q$).

Power factor $pf=cosφ =P/S$ is adjacent/hypotenuse.

4. Worked Examples

(a) V=230V, I=10A, φ=30° lagging: S=2300VA, P=2300·0.866=1992W, Q=1150 VAR ind., triangle P=1992, Q=1150, S=2300.

(b) Need to correct pf: Qc = -Q to bring Q→0, triangle becomes flat.

(c) R=3Ω, X=4Ω, I=2A: P= I²R=12W, Q=I²X=16 VAR, S=20VA ⇒ pf=0.6 ✓

(d) For pure R, φ=0 ⇒ Q=0, S=P; for pure L, φ=90° ⇒ P=0, Q=S.

5. Exam Points

• Draw power triangle with $P$, $Q$, $S$, $φ$ — 5 marks, very frequent (almost certain to ask to draw and label).
• State $P=VI cosφ$, $Q=VI sinφ$, $S=VI$, $S=√(P²+Q²)$ — 5-7 marks, very frequent.
• Distinguish lagging vs leading $Q$ sign — 3-5 marks, frequent.
• Connect to $pf=cosφ=P/S$ — 3 marks.