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RMS and average of half-wave rectified sine

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.

Define RMS and average value of ac signal. Find the RMS and average value of half wave rectified sinusoidal waveform V = Vm sin ωt.

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

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

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

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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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