Find Vrms/Vavg with Half Wave Rectifier Integration

[kimmy510]

while finding Vrms or Vavg we use integration and we use d(wt) and integrate between the limits 0 to 2pie. When 'w' is a constant how can we take it as d(wt).and if we take as d(wt) only then why are the limits not changed to 0 to 2(w)(pie)?

 

[uart]

Yes you're kind of right. You would change the limits of integration as appropriate. But remember that the original limits of integration would normally be t=0 to t=T (the period). But T = 2 Pi / w therefore wT = 2Pi is new limit after the change of variable.

Mathematically when we make the substitution $\theta = \omega t$ we get:

$$\int_{t=0}^T f(\omega t) \, dt = \int_{\theta=0}^{\omega T} f(\theta)\, d\left(\frac{\theta}{\omega}\right) = \frac{1}{\omega} \int_0^{2 \pi} f(\theta) \, d\theta$$