in-phase quantities

in-phase quantities

1. What In-Phase Means

In-phase = phase difference $Δφ =0°$ — two sinusoids of same frequency peak and zero-cross at same instants, their waveforms overlap (scaled) and add arithmetically.

If $v(t)=V_m sin(ωt)$ and $i(t)=I_m sin(ωt)$ (both $φ=0$), they are in-phase: $v$ and $i$ rise/fall together.

2. When Does In-Phase Occur?

• Pure resistor ($R$): $v = i·R$ ⇒ $v$ and $i$ are in-phase ($φ=0°$) — Ohm's law with no $L/C$ phase shift.
• Resonance in series/parallel RLC: net $X_L = X_C$ ⇒ net reactance zero ⇒ circuit behaves resistive ⇒ $v$ and $i$ in-phase (power factor $1$).
• Two voltages in phase means they are coherent and add: $V_{total}=V_1+V_2$ (not vector).

3. Worked Illustration

(a) Resistor 10Ω with v=10 sin(314t) ⇒ i=1 sin(314t) ⇒ both peak at same time, zero together ⇒ in-phase ✓

(b) At resonance series RLC: X=0 ⇒ Z=R ⇒ i = V/R in-phase with V ⇒ P=V·I maximal (power factor 1).

(c) Two in-phase sources 10 sinωt and 5 sinωt ⇒ sum =15 sinωt (magnitudes add).

(d) Two opposite phase (180°): 10 sinωt and 10 sin(ωt+180°)= -10 sinωt ⇒ cancel to zero.

4. Exam Points

• Define in-phase as $Δφ=0°$, peaks align — 3-5 marks, very frequent.
• State that resistor $v$ and $i$ are in-phase — 5 marks, very frequent (common example).
• Show that at resonance, $v$ and $i$ become in-phase — 3-5 marks, frequent.
• Distinguish from lagging/leading — 3 marks.