## 3 Total Harmonic Distortion  
According to (10), (13) and Figure 3, there is  
\[I_{M}^{*}=\frac{2P_{L}}{V_{M}}-\Delta V_{DC}|C_{V}(2\omega_{G})|\sin(2\omega_{ G}t+\arg C_{V}(2\omega_{G})) \tag{18}\]  
in steady state. Consequently (cf. (15)), there is  
\[|i_{G}(t)|=I_{M}^{*}|\sin\omega_{G}t|=\frac{2P_{L}}{V_{M}}\bigg{(}1-\frac{V_{ M}\Delta V_{DC}}{2P_{L}}|C_{V}(2\omega_{G})|\sin(2\omega_{G}t+\arg C_{V}(2 \omega_{G}))\bigg{)}|\sin\omega_{G}t| \tag{19}\]  
in case of a diode bridge-based rectifier. If  
\[\frac{V_{M}\Delta V_{DC}}{2P_{L}}|C_{V}(2\omega_{G})|<<1 \tag{20}\]  
Figure 3: Simplified control structures for (**a**) a bridge-rectifier-based AC/DC converter; and (**b**) a bridgeless rectifier/inverter.  
then (19) implies grid current [51]  
\[\begin{array}{l}i_{\rm{G}}(t)=\frac{2p_{\rm{I}}}{V_{\rm{M}}}\Big{(}1-\frac{V_{ \rm{M}}\Delta V_{\rm{DC}}}{2P_{\rm{L}}}|C_{V}(2\omega_{\rm{G}})|\sin(2\omega_{ \rm{G}}t+\arg C_{V}(2\omega_{\rm{G}}))\Big{)}\sin\omega_{\rm{G}}t\\ =\frac{2p_{\rm{I}}}{V_{\rm{M}}}\Bigg{(}\begin{array}{c}\sin\omega_{\rm{G}}t +\frac{V_{\rm{M}}\Delta V_{\rm{DC}}}{4P_{\rm{L}}}|C_{V}(2\omega_{\rm{G}})|\cos (\omega_{\rm{G}}t+\arg C_{V}(2\omega_{\rm{G}}))-\\ -\frac{V_{\rm{M}}\Delta V_{\rm{DC}}}{4P_{\rm{L}}}|C_{V}(2\omega_{\rm{G}})|\sin (3\omega_{\rm{G}}t+\arg C_{V}(2\omega_{\rm{G}}))\end{array}\Bigg{)},\end{array} \tag{21}\]  
formed by the first and the third harmonic components only. Note that (21) also holds in the case of bridgeless grid-interfacing converter implementation. Consequently, the grid-side current THD is given by (cf. (20))  
\[THD\cong\frac{V_{M}\Delta V_{\rm{DC}}}{4P_{L}}|C_{V}(2\omega_{\rm{G}})|. \tag{22}\]  
Furthermore, (20) may be reformulated as  
\[2\cdot THD<<1, \tag{23}\]  
which is typically true since THD values of 5% and below are typically considered. Hence, (20) is justified and (21) is a valid approximation.  
It may be concluded from (21) that the pulsating ripple component of the DC link voltage (13) imposes a non-zero grid-side current THD. Denoting the maximum tolerable THD value as \(THD^{*}\), the combination of (13) and (21) yields the following constraint  
\[\frac{V_{M}}{8\omega_{\rm{G}}V_{\rm{DC}}^{*}C_{\rm{DC}}}|C_{V}(2\omega_{\rm{G }})|=THD^{*}\Rightarrow|C_{V}(2\omega_{\rm{G}})|=\frac{4(2\omega_{\rm{G}})V_ {\rm{DC}}^{*}C_{\  
\[K\frac{\sqrt{\left(2\omega_{G}\tau\right)^{2}+1}}{2\omega_{G}}f(\alpha)=\frac{4(2 \omega_{G})V_{DC}^{*}C_{DC}}{V_{M}}THD^{*} \tag{28}\]  
with  
\[f(\alpha)=\frac{1}{\sqrt{1+4\xi_{f}^{2}\frac{\alpha^{2}}{\left(1-\alpha^{2} \right)^{2}}}}\cdot\frac{\left\{\frac{1}{\sqrt{\frac{1+4\xi_{f}^{2}\frac{ \left(\hat{\xi}\right)^{2}}{\left(1-\left(\hat{\xi}\right)^{2}\right)^{2}}}}}} \cdot\frac{\left\{\frac{1}{\sqrt{\frac{1+4\xi_{f}^{2}\frac{\left(\hat{\xi} \right)^{2}}{\left(1-\left(\hat{\xi}\right)^{2}\right)^{2}}}}}},\quad\omega_{G }=\alpha 100\pi\right.} \tag{29}\]  
denoting the multiplication of tuned and un-tuned notch term gains, respectively. Plots of \(f(\alpha)\) versus \(0.99\leq\alpha\leq 1.01\) (signifying 1% mains frequency uncertainty) are depicted in Figure 4 for different values of \(\xi_{f}\). As expected, the gain of the notch term is zero at nominal frequency (theoretically implying \(THD=0\)), increasing with detuning. Moreover, the highest total \(f(\alpha)\) is attained at \(\alpha_{\min}\) for 50 Hz mains and at \(\alpha_{\max}\) for 60 Hz mains. Nevertheless, since the magnitude response of the PI term in (17) monotonically decreases until the effect zero begins to kick in, the gain at \(\omega_{G}=\alpha_{\min}\cdot 100\pi\) may be expected as the worst case one. In addition, it is well-evident that increasing \(\xi_{f}\) improves robustness to frequency variations. It should be emphasized that if \(\xi_{f}=0\) is selected, the control structure in (17) reduces to the typical PI regulator.