power factor

power factor

1. What Power Factor Is

Power factor ($pf$ or $cosφ$) = ratio of real power to apparent power, also

where $φ$ = phase angle between $V$ and $I$ ($φ = θ_V - θ_I = tan^{-1}(X/R)$).

• $φ>0$: $v$ leads $i$ ⇒ inductive, $i$ lags $v$ (lagging pf, e.g., motor).
• $φ<0$: $v$ lags $i$ ⇒ capacitive, $i$ leads $v$ (leading pf, e.g., capacitor).
• $φ=0$: resistive or resonance, $pf=1$ (unity) ⇒ all power real.

Range

(leading/lagging indicates sign of $φ$). Lagging pf = inductive, leading pf = capacitive.

2. Why Power Factor Matters

• Real power (useful, W):

• Reactive power (stored, VAR):

• Apparent power (supplied, VA):

Utilities must supply $I = P/(V pf)$; low $pf$ ⇒ larger $I$ for same $P$ ⇒ more losses $I²R$ in lines and larger equipment. So high $pf$ (≈1) desired.

3. Correction

Inductive load ($pf$ lagging) can be corrected by adding capacitor in parallel (leading $Q$) to bring $φ$ toward 0, reducing $I$.

4. Worked Examples

(a) Series RL: R=3Ω, X_L=4Ω ⇒ |Z|=5Ω, pf= R/|Z|=0.6 lagging (φ=53.1°) ⇒ P= V·I·0.6

(b) Load 1kW at 230V, pf=0.6 lagging ⇒ I=1000/(230·0.6)=7.25A vs pf=1 ⇒ 4.35A ⇒ 66% more current, more loss.

(c) Need to correct pf 0.6 to 0.9: add capacitor Qc = P(tanφ_old - tanφ_new)

(d) Pure resistor pf=1, pure L/C pf=0 (no real power).

5. Exam Points

• Define $pf=cosφ=P/(V I)=R/|Z|$ — 5 marks, very frequent (almost certain to ask definition).
• Show $P=VI cosφ$, $Q=VI sinφ$, $S=VI$ — 5-7 marks, very frequent.
• Distinguish lagging (inductive) vs leading (capacitive) — 5 marks, frequent.
• Need for power factor correction with capacitor — 5 marks, frequent.