Skip to main content

Average and RMS of trapezoidal wave

Short answer

For the trapezoidal wave, the half-cycle average is found by area: Vav = (5/6)Vm = 0.833 Vm. The RMS value, computed from squared integral, is Vrms = (√7/3)Vm = 0.882 Vm.

Calculate the average (half period) value and rms value of the waveform shown below. [Figure: trapezoidal wave of peak Vm; rises linearly from 0 at θ = 0 to Vm at π/6, stays at Vm until 5π/6, falls linearly to 0 at π; negative half cycle is the mirror image up to 2π.]

The waveform rises linearly from 0 to Vm over 0 to π/6, stays at Vm from π/6 to 5π/6, and falls linearly to 0 at π. The negative half is a mirror image, so we calculate over the half period 0 to π.

Average (half-period) value

Vav = (1/π) ∫₀^π v dθ

The area has three parts:

  • Rising ramp: ½ × (π/6) × Vm = πVm/12
  • Flat top: Vm × (5π/6 − π/6) = 2πVm/3
  • Falling ramp: πVm/12

Total area = πVm (1/12 + 8/12 + 1/12) = 10πVm/12 = 5πVm/6

Vav = (1/π)(5πVm/6) = 5Vm/6 = 0.833 Vm

RMS value

Vrms² = (1/π) ∫₀^π v² dθ

  • Rising ramp: v = Vm θ/(π/6) = 6Vmθ/π, so ∫₀^{π/6} (6Vmθ/π)² dθ = (36Vm²/π²) × (π/6)³/3 = πVm²/18
  • Falling ramp: the same, πVm²/18
  • Flat top: Vm² × 2π/3 = 12πVm²/18

Sum = πVm² (1 + 1 + 12)/18 = 14πVm²/18 = 7πVm²/9

Vrms² = (1/π)(7πVm²/9) = 7Vm²/9

Vrms = (√7 / 3) Vm = 0.882 Vm

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Basic Electrical Engineering chapter 5 questions →