## 2 PFC Converter Total Harmonic Distortion (THD) and Power Factor (PF)  
In a power system, the THD value should be kept as low as possible; a lower THD produces a higher power factor, higher efficiency, and lower current stress in power system components due to fewer peak currents. Usually, the THD value should follow the standard specifications, such as IEC61000-3-2 or IEEE 519-20142, for different classes of equipment. Ignoring the effect of current distortion, the input PF of the PFC converter can be expressed as  
\[\text{PF}=\cos(\theta_{\text{vs}}-\theta_{\text{is}}), \tag{1}\]  
where \(\theta_{\text{vs}}\) is the voltage angle and \(\theta_{\text{is}}\) is current angle for the input supply. Equation (1) is not the full definition of the converter PF and usually, this mathematical expression is called the displacement factor (dF). The power factor calculation using Equation (1) can be applied only if the supply voltage and current waveforms are completely sinusoidal. In most power electronic circuits, including the PFC boost converters, operation techniques and rapid changes between on and off states of these converters act as nonlinear loads, which can change the nature of the current so it is no longer a sinusoidal waveform. Figure 2 shows the supply current and voltages of the PFC boost converter, demonstrating the current distortion when the boost converter switch changes rapidly from on to off states. This point is usually called the zero-crossing point. The conventional control techniques using PI controllers cannot fully remove the current distortion of the input current in the zero-crossing point due to the slow dynamic response of the current controller at this point [15], and the inductor current cannot track the reference current when the supply current changes rapidly from positive to negative values. The root mean square value of the input current (I\({}_{\text{s\_rms}}\)) for the PFC converter can be expressed as  
\[\text{I}_{\text{s\_rms}}=\sqrt{\text{I}_{\text{dc}}^{2}+\sum_{n=1}^{\infty} \text{I}_{\text{n\_rms}}}\, \tag{2}\]  
where I\({}_{\text{dc}}\) is the average current component and n is the harmonic order. When n = 1, I\({}_{\text{1\_rms}}\) is the fundamental component of the supply input current. With an ideal voltage source that offers voltage only at the fundamental frequency (Vs\({}_{\text{s\_rms}}\) = V\({}_{\text{1\_rms}}\)), the converter input average power (P\({}_{\text{avg}}\)) can be calculated as  
\[\text{P}_{\text{avg}}=\text{V}_{\text{1\_rms}}\times\text{I}_{\text{1\_rms}} \times\text{dF}. \tag{3}\]  
Apparent power (S), which represents the supply active and reactive power components, still includes all of the current harmonics and can be expressed as  
\[\text{S}=\text{V}_{\text{s\_rms}}\times\text{I}_{\text{s\_rms}}=\text{V}_{ \text{1\_rms}}\times\sqrt{\text{I}_{\text{dc}}^{2}+\sum_{n=1}^{\infty}\text{I }_{\text{n\_rms}}}. \tag{4}\]  
From Equations (3) and (4), the full definition of PFC converter power factor that considers the current distortion can be expressed as  
\[\text{PF}=\frac{\text{I}_{\text{1\_rms}}}{\sqrt{\text{I}_{\text{dc}}^{2}+ \sum_{n=1}^{\infty}\text{I}_{\text{n\_rms}}}}\times\text{dF}=\text{DF}\ \times\text{dF} \tag{5}\]  
where DF represents the current distortion factor and is equal to \(\frac{\text{I}_{\text{1\_rms}}}{\text{I}_{\text{s\_rms}}}\). The THD value related to the current distortion factor can be expressed as  
\[\text{DF}=\sqrt{\frac{1}{1+\text{THD}^{2}}}. \tag{6}\]  
So, the power factor (PF) is the product of two factors called displacement factor (DF) and the distortion factor (DF), as given in Equation (5). The displacement factor (DF) depends on the phase shift between the voltage and current of the input supply, which is usually very low for AC/DC power converters (less than five degrees). The distortion factor (DF) depends on the total harmonic distortion of the current waveform. Figure 2: Supply voltage-current waveforms for PFC-boost converter with conventional PI current controller.