Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 1054179, 10 pages

https://doi.org/10.1155/2018/1054179

## Storage and Dissipation Limits in Resonant Switched-Capacitor Converters

^{1}School of Engineering and Sciences, Tecnologico de Monterrey, Monterrey, NL, Mexico^{2}College of Sciences, Botswana International University of Science and Technology, Palapye, Botswana^{3}Engineering Faculty, Universidad Panamericana, Guadalajara, JAL, Mexico

Correspondence should be addressed to Jonathan C. Mayo-Maldonado

Received 7 July 2017; Revised 10 November 2017; Accepted 26 November 2017; Published 24 January 2018

Academic Editor: J.-C. Cortés

Copyright © 2018 Jonathan C. Mayo-Maldonado et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The purpose of this manuscript is twofold, first we introduce an energy-based modeling framework for the analysis of resonant switched-capacitor (SC) converters and second we demonstrate that energy storage and dissipation in resonant SC with ideal switches are bounded by a fundamental physical limit that, up until now, has been only associated with the special case of pure SC topologies. For instance, we show that the maximum energy stored in the small size inductors in resonant SC converters is equal to the energy that would be dissipated by their purely SC counterpart. The presented analysis permits the computation of resonant inductances in terms of maximum current peak values, which is experimentally validated. Furthermore, we introduce a relative loss factor that permits determining the efficiency of a design for a general case in the presence of parasitic resistances. These results corroborate that migrating to resonant SC technologies is one of the most compelling alternatives to overcome well-known disadvantages in pure SC topologies.

#### 1. Introduction

DC-DC power converters with switched-capacitors exhibit highly desirable features in energy conversion systems such as high-voltage gains, high-efficiency, and transformerless profiles (see, e.g., [1–5]). An important issue in SC converters is its dynamic analysis, since while state-space modeling techniques such as* state averaging* are effective for some traditional classes of power converters, SC converters cannot be modeled in this way. Although the modeling of standard DC-DC converters is conventionally carried out considering ideal switches, parallel connections between capacitors in SC converters induce voltage discontinuities (see [1, 6]), demanding a more refined analysis (see [7]).

Another important challenge in SC converters is their* efficiency*, for which many approaches and studies have been proposed; see, for instance, [8–11], where losses due to the charging/discharging process of capacitors have been identified besides the standard conduction losses. Moreover, these topologies exhibit also high current peaks due to sudden parallel interconnection of capacitances, demanding high stress on semiconductor devices. For this reason, the use of small inductances that limit such currents is a common alternative (see [12–16]). The justification for the use of such limiting current inductors is mainly based on their size and cost, since resonant inductors are not required to store a significant amount of energy as in other traditional topologies.

In this paper, we introduce a modeling framework for resonant SC and pure SC converters based on switched linear differential systems (see [17–19]). Moreover, a natural, modular way to describe power and energy as quadratic quantities is introduced using the calculus of quadratic differential forms [20]. This setting is the pivotal figure in our ensuing results that encompass the study of efficiency and performance issues in resonant SC topologies, with respect to pure SC converters. Previous contributions that elaborate on these issues include [13], where a study of conduction losses computation is presented. In [15], the authors argue that charging/discharging losses are mitigated in resonant SC topologies and consequently conduction losses are predominant. This contribution is well-supported by interesting discussions and results. In [21], the authors present an extension of the SC converter energy losses analysis, based on the so-called slow- and fast-switching limits, to the case of resonant SC converters. Motivated by the current trends in the study of losses in SC resonant converters, we show a rigorous analytical proof that corroborates that any loss in resonant SC converters must be regarded as conduction losses, due to parasitic (or ESR) resistors. We also generalize the results in resonant SC and pure SC converters involving parasitic resistances by proving that losses are upper bounded by a fundamental physical limit.

The results presented in this paper are not straightforwardly evident nor reached by pure intuition since, for instance, it is well-known that SC converters with ideal switches dissipate energy at switching instants (even when parasitic resistances are neglected), which means that resonant SC converters with negligible inductances are also lossy. Consequently the effect of adding a tiny inductor originally considered to damp peak currents that is by observation, as reported in [15], mitigating energy transfer (discharging) losses, deserves a clear explanation. The study of such energy conservation mechanism is adopted as the main conviction in this paper, while the advantages of the results are illustrated with the design of resonant SC cells with respect to peak value specifications, which is experimentally validated and a relative loss factor that permits computing losses in a general case.

In this paper, we use the following notation. The space of -dimensional real vectors is denoted by and that of real matrices is denoted by ; when one of the dimensions is not specified we use denotes the matrix obtained by stacking the matrix over . The ring of polynomials with real coefficients in the indeterminate is denoted by ; the ring of two-variable polynomials with real coefficients in the indeterminates and is denoted by . denotes the set of all matrices with entries in and that of polynomial matrices in and . denotes an identity matrix of dimensions denotes the transpose the matrix Let and ; then we define and , provided that such limits exist.

#### 2. Higher-Order Modeling

When modeling from first principles, we usually focus on a set of* variables of interest* (e.g., input/output voltages and currents) that are important for analysis, control, simulation, and so on. Occasionally, we need to introduce some* auxiliary variables* (e.g., nodal voltages, mesh currents, and state variables) that permit writing first principle equations in a convenient way. In this paper, we denote a vector of variables of interest by and that of auxiliary variables by .

In general, during the modelling stage we obtain sets of linear differential equations (e.g., by applying first principles, by algebraic elimination of variables, and by using the calculus of impedances) of the formwith , and . This set of equations can be written in a compact form aswhere and , that is, and are* polynomial matrices,* and thus are defined asA special case of (2) arises when equals the identity; that is,Equation (4) is called* image representation* and it can be always achieved when the system under analysis is* controllable* (see [22], Section , pp. 234–236).

An image representation can be also conveniently associated with the concept of* transfer function*, that is, a representation of the form , where and are polynomial matrices of suitable rank and dimensions. Consider an* input-output partition* of the external variables of (4), that is, , possibly permuting the components of . Consequently the matrix is partitioned accordingly as (here we assume that has no pole/zero cancellations; see [22], Section and Theorem )where the number of components in is the same as those in (see [23], Section VI-A). Moreover, in the case of* imittances*, the* inputs * and* outputs * are* conjugate variables*, for example, pairs of input/output port voltages and currents; consequently they have the same number of components.

*Example 1. *Consider the circuit in Figure 1 where the variables of interest are and that correspond to the input voltage and the current through inductor , respectively.

This circuit can be modelled as an impedance , by standard series/parallel reductions; that is,The circuit admits an image representationwhere, as previously mentioned, is the vector of variables of interest and is an auxiliary variable.