--- title: "RMS and average of half-wave rectified sine" subject: "Fundamental of Electrical and Electronics Engineering" chapter: 2 key: "semester_1/fundamental_of_electrical_and_electronics_engineering:rms-average-half-wave" source: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/rms-average-half-wave site: BCT Engineering --- # RMS and average of half-wave rectified sine Fundamental of Electrical and Electronics Engineering · Chapter 2: Average and RMS Values - Asked in: 2081 Baishakh (new course), 2+4 marks - Video: [How to calculate the RMS and Average value of Half wave Rectifier and Full wave rectifier](https://www.youtube.com/watch?v=A2SMI31EgMA) — ALL ABOUT ELECTRONICS - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/rms-average-half-wave - Whole chapter: https://www.bctengineer.com/md/semester_1/fundamental_of_electrical_and_electronics_engineering/chapter-2 ## Question Define RMS and average value of ac signal. Find the RMS and average value of half wave rectified sinusoidal waveform V = Vm sin ωt. ## Short answer The RMS (effective) value is the equivalent DC value that produces the same heating effect, given by the square root of the mean of the squared waveform. The average value is the arithmetic mean over one period. For a half-wave rectified sine wave v = Vm sin ωt, V_avg = Vm/π ≈ 0.318 Vm, and V_rms = Vm/2 = 0.5 Vm. ## Answer #### Definitions - **RMS (effective) value:** the value of a steady DC current that produces the same heat in a resistor, in the same time, as the AC current. Mathematically, the square root of the mean of the squared values over one period: I_rms = √[(1/T) ∫₀ᵀ i² dt]. - **Average value:** the arithmetic mean of all instantaneous values over one period (or half period for a symmetrical wave): V_avg = (1/T) ∫₀ᵀ v dt. It equals the DC component of the wave. #### Half-wave rectified sine wave ```text v = Vm sin ωt for 0 ≤ ωt ≤ π v = 0 for π ≤ ωt ≤ 2π Period = 2π (in terms of θ = ωt) Vm | __ __ | / \ / \ | / \ / \ 0 +/------\_____/------\____ ωt 0 π 2π 3π 4π ``` **Average value** ```text V_avg = (1/2π) ∫₀^π Vm sin θ dθ = (Vm/2π) [−cos θ]₀^π = (Vm/2π) [1 + 1] = Vm/π ≈ 0.318 Vm ``` **RMS value** ```text V_rms² = (1/2π) ∫₀^π Vm² sin²θ dθ = (Vm²/2π) ∫₀^π (1 − cos 2θ)/2 dθ = (Vm²/4π) [θ − (sin 2θ)/2]₀^π = (Vm²/4π) × π = Vm²/4 V_rms = Vm/2 = 0.5 Vm ``` **Answer: V_avg = Vm/π ≈ 0.318 Vm, V_rms = Vm/2 = 0.5 Vm.** Related factors: form factor = V_rms/V_avg = (Vm/2)/(Vm/π) = π/2 ≈ 1.57; peak factor = Vm/V_rms = 2. For comparison, a full sine wave has V_rms = 0.707 Vm. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the RMS value of an AC current represent? A. The maximum value of the wave B. The mean of instantaneous values over a period C. Half the peak-to-peak value D. The DC value producing the same heating in a resistor Answer: D. RMS is the effective value: equal DC gives the same heat in the same time. (Key point: Meaning of RMS) 2. What is the average value of a half-wave rectified sine of peak Vm over a full period? A. Vm/pi B. 2Vm/pi C. Vm/2 D. Vm/sqrt(2) Answer: A. (1/2pi) x integral of Vm sin(theta) from 0 to pi = (Vm/2pi) x 2 = Vm/pi = 0.318 Vm. (Key point: Average of half-wave) 3. What is the RMS value of a half-wave rectified sine wave? A. Vm/sqrt(2) B. Vm/2 C. Vm/pi D. 2Vm/pi Answer: B. V_rms^2 = Vm^2/4, so V_rms = Vm/2; the zero half-cycle halves the mean square compared with a full sine. 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