Average and RMS Values

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.

Equation of the waveform

In one period (0 ≤ t < T) the wave rises linearly from 10 to 20:

y(t) = 10 + (20 − 10) t/T = 10 + 10t/T
T = 10 ms

Average value

Y_avg = (1/T) ∫₀ᵀ (10 + 10t/T) dt
      = (1/T) [10t + 5t²/T]₀ᵀ
      = (1/T) (10T + 5T)
      = 15

This is also clear from the shape: a straight ramp from 10 to 20 has mean (10 + 20)/2 = 15.

RMS value

Put x = t/T (x goes from 0 to 1, dt = T dx):

Y_rms² = (1/T) ∫₀ᵀ (10 + 10t/T)² dt
       = 100 ∫₀¹ (1 + x)² dx
       = 100 [(1 + x)³/3]₀¹
       = 100 × (8 − 1)/3
       = 233.33
Y_rms  = √233.33 = 15.275

Answer: Average value = 15, RMS value ≈ 15.28 (in the units of y, e.g. volts).

The value T = 10 ms does not change the result: the average and RMS of a periodic wave depend only on its shape over one period, not on how long the period is. RMS (15.28) is slightly larger than the average (15) because squaring gives more weight to the larger values near 20.