Phase and Phasors: Phase difference

Phase and Phasors: Phase difference

1. What Phase Difference Is

Phase $φ$ = argument of sine at $t=0$:

where $φ$ is initial phase (radians or degrees). Phase difference $Δφ = φ_1 - φ_2$ between two sinusoids of same frequency $ω$ tells relative timing.

• $Δφ =0°$: in-phase (peaks align).
• $0<Δφ<180°$: leading/lagging (one peaks earlier).
• $Δφ=180°$: opposite phase (inverse).
• $Δφ=90°$: quadrature.

2. Why Phase Matters

• In AC circuits, $v$ and $i$ may not be in phase (e.g., inductor $i$ lags $v$ by $90°$). Phase difference determines power factor $cos(Δφ)$.
• For combining signals, phase decides constructive/destructive interference.

3. Worked Examples

(a) v1=10 sin(314t) (φ1=0), v2=10 sin(314t+60°) (φ2=60°): Δφ=60°, v2 leads v1 by 60° (peaks 60° earlier) ✓

(b) v(t)=325 sin(314t+30°) at t=0: v=162.5V ✓

(c) Two signals 90° apart: when one at peak, other at zero.

(d) Need to compare v and i: v=10 sin(314t), i=5 sin(314t-30°) ⇒ i lags v by 30° (or v leads i by 30°) ⇒ power factor cos30°=0.866.

4. Representation

• Time domain: $v(t)$ curves shifted.
• Phasor: vector of length $V_m$ (or RMS) at angle $φ$ rotating at $ω$, projection gives instantaneous.

5. Exam Points

• Define phase $φ$ and phase difference $Δφ$ — 3-5 marks, very frequent.
• Distinguish leading/lagging/in-phase with diagram — 5 marks, very frequent (common to ask "what is phase difference?").
• Give $Δφ=0, 90°,180°$ cases — 3 marks, frequent.
• Show $φ$ in $V_m sin(ωt+φ)$ — 3 marks.