# Response to: Comment on “Storage and Dissipation Limits in Resonant Switched-Capacitor Converters”

Authors: Jonathan C. Mayo-Maldonado; Thabiso M. Maupong; Jesus E. Valdez-Resendiz; Julio C. Rosas-Caro
Journal: Mathematical Problems in Engineering
 (2018)

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## 1. Existing Contributions regarding Energy Losses Quoted in Our Paper

This issue of existing contributions on this topic had been already pointed out by a reviewer during the reviewing process. For this reason, we already acknowledged those contributions that developed relevant and similar conclusions in energy quantification such as[13, 15, 21] of the original manuscript. Please note that these references are specifically onresonant switched capacitor power converters. Moreover, we completely acknowledged the fact that there exist several approximations to the topic and alternative mathematical routes to study switching phenomena in resonant converters.We are always happy to quote the work of other authors in similar topics as a way to enrich the literature review. In this case, we were unfortunately unaware of Prof. Goknar’s contribution published in 1972, perhaps because he deals with electrical circuits with jumps, rather than resonant switched capacitor convertersper se.It is also important to point out that we cited one of Prof. Goknar’s more recent contributions, which we believe is much closer to the topic under study:[9] R. Frasca, M. K. Camlibel, I. C. Goknar, L. Iannelli, and F.Vasca, “Linear Passive Networks with Ideal Switches: Consistent Initial Conditions and State Discontinuities,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 57, no. 12, pp. 3138–3151, 2010.Different from the 1972’s contribution,[9] provides a study of power converters, where issues in the current traditional modeling of such devices and alternatives to deal with them by using a linear complementarity framework are proposed. This contribution and ours thus share similar convictions, since we also propose a new mathematical modeling framework.We must say that even now that we are aware of the existence of the aforementioned 1972’s contribution, we still believe that his contribution[9] is much closer to the topic under study and our approach and convictions.


## 2. Contributions in Our Manuscript

Since there are several approaches to energy quantification of switching electrical circuits, as also pointed out by Prof. Göknar, it had already been necessary to point out the contributions in our manuscript, to prove that other important novelties that deserve publication were also proposed.The main purposes of our manuscript, rather than finding an alternative route to develop similar results or drawing equivalent conclusions, consist in(1)
introducing a modeling framework for switched capacitor and resonant converters. This framework is the pivotal figure of a systematic analysis that involves quadratic functionals. In particular, we introduce polynomial algebra tools based on the calculus ofquadratic differential forms (cf. [20]) to obtain information about power and energy, please see Sec. II, pp. 1-2 and Sec. III, pp. 2-3. Please note that such procedure is not possible when resorting to the traditional state space averaging technique as argued in [7, 13];(2)
showing the relationship between the energy transfer mechanism between switched capacitor and resonant switched capacitor structures. Please see Prop. 3, p. 6, and Lemma 1, p. 7. Please note that the circumvention of energy losses associated with redistribution of charge not only is associated with the fact that an inductor is included in the loop or that parasitic resistances are neglected, but also follows from a zero current switching technique: e.g., if the switch is arbitrarily open when the resonant inductor is charged, energy losses will occur and, moreover, if the switch occurs exactly when the inductor reaches its maximum voltage level, the loss will be equal to that of its pure SC counterpart. This paper aims at clarifying these type issues;(3)
introducing a mathematical formula to compute resonant inductances on the basis of maximum peak current specifications, rather than resorting to trial and error designs (see, e.g.,[3, 12]). Such equation is derived from the results of Prop. 3, p. 6, and is provided in (20), p. 7;(4)
providing a relative loss factor to characterize the efficiency of the converters. Please see equation (21), p. 7.


## 3. Points of Disagreement

Given the reasons pointed out above, regarding the relevance of the 1972’s contribution with respect to our particular publication “Storage and Dissipation Limits in Resonant Switched-Capacitor Converters”, we beg to disagree with the suggestion of using a different approach to obtain the same results. The reasons are the following:(i)
Our approach is also general and exemplified using simple (resonant) switched capacitor cells only for illustration purposes, since (resonant) switched capacitor converters are usually constructed in a modular way using such cells.(ii)
Changing our approach (based on switched linear differential systems and quadratic differential forms, as anew modular way to model switching dynamics and energy functions), would leave aside our main contribution which is the introduction of a mathematical framework to study switching phenomena in resonant converters. Please note that this contribution is not trivial, since, currently, the predominant averaging modeling approach to power converters cannot be applied to resonant switched capacitor converters.(iii)
Following the scope of the MPE Journal, we developed a mathematical framework to provide practical solutions to engineering problems, in particular, modeling and design specifications. Consequently, although one can find an alternative mathematical route using the 1972’s contribution, one would need to develop a systematic modeling technique for power converters and a way to associate an efficient parametric exploration with performance and benchmarking specifications, as we illustrated in our paper.


## 4. Conclusion and Recommendation

(i)
We conclude that Prof. Göknar’s contribution can be used as an alternative route to develop some of our technical results that though original, they are not the main contribution in our published paper. (ii)
The 1972’s contribution is not about resonant switched capacitor converters, but electrical circuits with jumps. Although concepts such as energy quantification of capacitors and inductors can be expressed in mathematical terms in both (his and ours) settings, there is still a considerable amount of nontrivial work to do based on such contribution in order to involve the technicalities of power converters. (iii)
We already cited Prof. Göknar’s contribution (see[9] in the published manuscript) that, we strongly believe, is much closer to the approach and convictions argued in our published paper, since it deals with a new mathematical framework to power converters as well. (iv)
Accepting to change our mathematical framework by the one in the 1972 paper would leave aside the main contribution in our manuscript, which is the introduction of a switched linear differential system approach to model and study the overall properties of power converters. (v)
We believe that, though interesting, showing that using an alternative route to develop some of our results would immediately raise the critiques that we faced during the reviewing process, since doing so was not considered as a publishable contribution. Our proposed modeling framework, the developed power converter analyses, and design tools were, on the other hand, considered as such.



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