Understanding Converter Power Factor

The input power factor (PF) for a converter with non-sinusoidal current is determined by the combined effect of the phase difference between voltage and current fundamentals and the presence of harmonic components.

The power factor can be calculated as:

$ PF = \frac{\text{Real Power (P)}}{\text{Apparent Power (S)}} $

For waveforms containing harmonics, a practical approach is using the product of the fundamental power factor ($PF_{fundamental}$) and the distortion factor (DF):

$ PF = PF_{fundamental} \times DF $

Calculating Fundamental Power Factor ($PF_{fundamental}$)

The input voltage is given as:

$ v(t) = 300 \sin(\omega t) \text{ V} $

This is a pure sinusoidal voltage at the fundamental frequency ($\omega$). The phase of the fundamental voltage component is $\phi_v = 0$.

The input current is:

$ i(t) = 10\sin\left(\omega t - \frac{\pi}{6}\right) + 2\sin\left(3\omega t + \frac{\pi}{6}\right) + \sin\left(5\omega t + \frac{\pi}{2}\right) \text{ A} $

The fundamental component of the current is:

$ i_1(t) = 10\sin\left(\omega t - \frac{\pi}{6}\right) \text{ A} $

The phase of the fundamental current component is $\phi_{i1} = -\frac{\pi}{6}$.

The phase difference ($\phi_1$) between the fundamental voltage and current is:

$ \phi_1 = \phi_v - \phi_{i1} = 0 - \left(-\frac{\pi}{6}\right) = \frac{\pi}{6} $

Therefore, the fundamental power factor is:

$ PF_{fundamental} = \cos(\phi_1) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \approx 0.866 $

Calculating Distortion Factor (DF)

First, find the RMS (Root Mean Square) values of the current components.

• Fundamental RMS current: $I_{1,rms} = \frac{10}{\sqrt{2}}$ A
• 3rd Harmonic RMS current: $I_{3,rms} = \frac{2}{\sqrt{2}}$ A
• 5th Harmonic RMS current: $I_{5,rms} = \frac{1}{\sqrt{2}}$ A

The total RMS current ($I_{rms}$) is calculated as:

$ I_{rms} = \sqrt{I_{1,rms}^2 + I_{3,rms}^2 + I_{5,rms}^2} $

$ I_{rms}^2 = \left(\frac{10}{\sqrt{2}}\right)^2 + \left(\frac{2}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 $

$ I_{rms}^2 = \frac{100}{2} + \frac{4}{2} + \frac{1}{2} = 50 + 2 + 0.5 = 52.5 $

$ I_{rms} = \sqrt{52.5} \text{ A} $

The distortion factor (DF) is the ratio of the fundamental RMS current to the total RMS current:

$ DF = \frac{I_{1,rms}}{I_{rms}} = \frac{10/\sqrt{2}}{\sqrt{52.5}} = \frac{10}{\sqrt{2} \times \sqrt{52.5}} = \frac{10}{\sqrt{105}} $

$ DF \approx \frac{10}{10.247} \approx 0.9759 $

Calculating Total Power Factor (PF)

Multiply the fundamental power factor by the distortion factor:

$ PF = PF_{fundamental} \times DF $

$ PF \approx 0.866 \times 0.9759 $

$ PF \approx 0.8451 $

The input power factor of the converter is closest to 0.845.