Question 5.17: Find the average and rms values of the sinusoid and sawtooth......
Find the average and rms values of the sinusoid and sawtooth waveforms in Figure 5–42.
As noted previously, the sinusoid has an average value of zero. The sawtooth clearly has a positive average value. By geometry, the net area under one cycle of the sawtooth waveform is VA T0/2, so its average value is (1/T0)(VA T0/2) = VA/2. To obtain the rms value of the sinusoid we apply Eq. (5–34) as
V_{rms} = \sqrt{\frac{1}{T} \int_{t}^{t + T}{[υ(t)]^{2} dt} } (5–34)
V_{rms} = \sqrt{\frac{(V_{A})^{2}}{T_{0}} \int_{0}^{T_{0}}{\sin^{2} (2πt/T_{0}) dt} }
= \sqrt{\frac{(V_{A})^{2}}{T_{0}} \left[\frac{t}{2} – \frac{sin(4πt/T_{0})}{8π/T_{0}}\right]^{T_{0}}_{0}} = \frac{V_{A}}{\sqrt{2}}
For the sawtooth waveform the rms value is found as:
V_{rms} = \sqrt{\frac{1}{T_{0}} \int_{0}^{T_{0}}{(V_{A}t/T_{0})^{2} dt}} = \sqrt{\frac{(V_{A})^{2}}{T_{0}^{3}} \left[\frac{t^{3}}{3}\right]^{T_{0}}_{0}} = \frac{V_{A}}{\sqrt{3}}