## 4 Dynamic Response  
Reformulating (6) as  
\[C_{DC}v_{DC}(t)\frac{dv_{DC}(t)}{dt}=0.5v_{M}(t)i_{M}(t)-p_{L}(t)-\underbrace{ 0.5v_{M}(t)i_{M}(t)\cos\left(2\int\omega(t)dt\right)}_{p_{2}(t)} \tag{30}\]  
and linearizing around the operating point by substituting  
\[ \begin{array}{c}v_{DC}(t)=V_{DC}^{*}+\widetilde{v}_{DC}(t),\quad i_{M}(t)= \frac{2p_{L}}{V_{M}}+\widetilde{i}_{M}(t),\quad v_{M}(t)=V_{M}+\widetilde{v}_ {M}(t),\\ p_{L}(t)=P_{L}+\widetilde{p}_{L}(t),\quad p_{2}(t)=P_{L}\cos(2\omega_{G}t)+ \widetilde{p}_{2}(t),\end{array}  \tag{31}\]  
Figure 4: Plots of \(f(\alpha)\) for different base frequencies and values of \(\xi_{f}\). into (31) and neglecting high-order small-signal terms yields  
\[C_{DC}V_{DC}^{s}\frac{d\widetilde{v}_{DC}(t)}{dt}=0.5V_{M}\widetilde{v}_{M}(t)+ \frac{P_{L}}{V_{M}}\widetilde{v}_{M}(t)-\widetilde{p}_{L}(t)-\widetilde{p}_{2}( t). \tag{32}\]  
The resultant control structure is depicted in Figure 5[22]. The corresponding loop gain is obtained as  
\[ \begin{split} L(s)&=\frac{0.5V_{M}}{C_{DC}V_{DC}^{ s}}C_{V}(s)=\\ &\frac{0.5V_{M}K}{C_{DC}V_{DC}^{s}}\frac{Ts+1}{s^{2}+2(200\pi)^{2 }}\frac{s^{2}+(200\pi)^{2}}{s^{2}+2\xi_{f}(200\pi)s+(200\pi)^{2}}\frac{s^{2}+ (240\pi)^{2}}{s^{2}+2\xi_{f}(240\pi)s+(240\pi)^{2}}=\\ &\omega_{n}^{2}\frac{\frac{Ts}{2\omega_{n}}s+1}{s^{2}}NF_{1}(s) NF_{2}(s),\ \ \ \ \omega_{n}=\sqrt{\frac{0.5 KV_{M}}{C_{DC}V_{DC}^{s}}},\ \ \ \ \xi_{n}=\omega_{n}\frac{\tau}{2}.\end{split}  \tag{33}\]  
Denoting the crossover frequency of the voltage loop as \(\omega_{CV}\), the corresponding phase and gain contributions of the notch terms are given by  
\[ \begin{split}\arg NF_{1}(\omega_{CV})&=-\tan^{-1} \left(\frac{2\xi_{f}^{s}}{\frac{200\pi}{\omega_{CV}}-\frac{\omega_{CV}}{200\pi }}\right),\ \ \arg NF_{2}(\omega_{CV})=-\tan^{-1}\left(\frac{2\xi_{f}^{s}}{\frac{200\pi}{ \omega_{CV}}-\frac{\omega_{CV}}{200\pi}}\right),\\ |NF_{1}(\omega_{CV})|&=\sqrt{\frac{1}{1+\frac{4 \xi_{f}^{2}}{\left(\frac{200\pi}{\omega_{CV}}-\frac{\omega_{CV}}{200\pi} \right)^{2}}}},\ \ \ \ |NF_{2}(\omega_{CV})|=\sqrt{\frac{1}{1+\frac{4\xi_{f}^{2}}{\left(\frac{200\pi}{ \omega_{CV}}-\frac{\omega_{CV}}{200\pi}\right)^{2}}}},\end{split}  \tag{34}\]  
respectively. The Bode diagram of dual-notch term \(NF_{1}(s)\cdot NF_{2}(s)\) is depicted in Figure 6 for different values of \(\xi_{f}\) and \(\omega<200\pi\) (since \(\omega_{CV}<200\pi\) in practice). It is well-evident that increasing the value of \(\xi_{f}\) escalates both the phase and gain contributions of the dual-notch term. It is well-evident that the gain contribution may be neglected within the whole frequency range considered, thus  
\[|L(\omega_{CV})|\cong\ \frac{\omega_{n}^{2}}{\omega_{CV}^{2}}\sqrt{\left(2\xi_{n} \frac{\omega_{CV}}{\omega_{n}}\right)+1}=1 \tag{35}\]  
should be satisfied at the crossover frequency \(\omega_{CV}\), yielding the crossover frequency  
\[\omega_{CV}=\underbrace{\xi_{n}\sqrt{2+2\sqrt{1+\frac{1}{4\xi_{n}^{4}}}}}_{ \theta_{n}}\omega_{n}=\theta_{n}\omega_{n}. \tag{36}\]  
Figure 5: Linearized simplified control structure. On the other hand, the phase contribution in (34) must be taken into account, imposing  
\[\arg\left(L(\omega_{CV})\right)= \tag{37}\] \[tg^{-1}\left(2\xi_{n}\frac{\omega_{CV}}{\omega_{n}}\right)-\tan^{-1 }\left(\frac{2\xi_{f}}{\frac{200\pi}{\omega_{CV}}-\frac{\omega_{CV}}{20\pi N}} \right)-\tan^{-1}\left(\frac{2\xi_{f}}{\frac{200\pi}{\omega_{CV}}-\frac{\omega _{CV}}{20\pi N}}\right)-\pi=-\pi+PM_{V}^{*}\]  
at the crossover frequency with \(PM_{V}^{*}\) indicating the desired voltage loop phase margin.