and RMS values of alternating signals

and RMS values of alternating signals

1. What RMS Is

RMS (Root Mean Square) $V_{rms}, I_{rms}$ = effective value — the DC equivalent that produces same heating power in a resistor:

For resistor $R$, power

(if $v,i$ in phase). For sine, RMS relates to peak.

2. RMS for Sine and Other Waves

• Sine:

;

• Half-wave rectified sine:

(since half cycle zero)

• Full-wave rectified sine: same as sine $V_m/√2$ (since $|v|$ squared same).
• Square wave ($±V_m$, 50%):

(since $v²=V_m²$ always)

• Triangular (peak $V_m$):

3. Why RMS Matters (vs peak/average)

• Power, heating, meter reading (true RMS): AC voltmeters (true RMS) read $V_{rms}$ directly; 230V mains is RMS (peak 325V).
• Insulation uses peak, average for rectified, RMS for power. Do not mix.

4. Worked Relations

230V RMS mains ⇒ V_m=325V, V_avg (half)=207V (2·325/π), not used.

If V_m=10V sine: V_rms=7.07V, V_avg(half)=6.37V ⇒ Form factor=1.11, Peak factor=1.414

Square wave V_m=5V: V_rms=5V, V_avg (full rectified)=5V ⇒ Form=1, peak=1.

Need to find power in 10Ω resistor with v=14.14 sinωt (V_m=14.14, V_rms=10): P=10²/10=10W.

5. Exam Points

• Define RMS as $√(1/T∫v² dt)$ and $V_{rms}=V_m/√2$ for sine — 5-7 marks, very frequent (almost certain to ask $V_{rms}$ vs $V_m$).
• Compute $V_{rms}$ for sine, half-wave, square, triangular — 5 marks, frequent.
• Distinguish RMS for power vs peak for insulation vs average for charge — 5 marks, frequent.
• Mention 230V RMS ⇒ 325 peak — 3 marks.