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.
3. Higher-Order Functionals
In the study of electrical circuits we are very often not only interested in the analysis of the system variables, such as voltages/currents, but also interested in the study of some functionals of those variables, such as power and energy. Quadratic functionals for higher-order differential systems can be expressed in terms of quadratic differential forms (QDFs) induced by two-variable polynomial matrices (see [20]). Such polynomial characterization brings relevant algorithmic advantages that will be continuously exploited in this paper. In the following, and as a standing assumption in the rest of the paper, we assume that QDFs act only on infinitely differentiable trajectories, whose set is denoted by , avoiding the presence of dirac impulses and their derivatives.
3.1. Algebraic Parametrization
Let ; thenwith inducing the quadratic differential form (QDF) defined byNote that the variables and act as placeholders for the derivatives of and their transpose.
Associated with is a coefficient matrix defined by . Note that since acts on infinitely differential trajectories, the matrix is an infinite matrix with only a finite number of nonzero entries (cf. [20], p. 1708). Moreover, is called symmetric if , or equivalently if ; in the rest of the paper we will focus on the symmetric case.
A relevant advantage in the calculus of QDFs is their differentiation, which is a straightforward matter in terms of two-variable polynomial matrices. For instance, the derivative of is the QDF defined byIn terms of two-variable polynomials, this derivative holds if and only if (see [20], p. 1710)In brief, this approach shows that while sets of linear differential equations are most conveniently associated with one-variable polynomial matrices, quadratic functionals are best characterized in terms of two-variable ones.
3.2. Positivity, Negativity, and Reformulation in Terms of Auxiliary Variables
is nonnegative with respect to (4) if for all that satisfies (4) with and positive with respect to (4) if for all that satisfies (4) with and . The concepts of nonpositive and negative QDF are defined analogously.
QDFs can be reformulated in terms of the auxiliary variable, which simplifies certain computations including positivity and negativity tests. Given , if and are related to (4), definingimplies , and consequently it follows that for satisfying (4) if and only if on . We adopt the notation to refer to a QDF acting on the external variable and , to denote the associated QDFs acting on auxiliary variables.
4. Energy-Based Analysis
We now study energy functions in electrical systems. We afterwards extend this analysis to SC converters.
As previously illustrated, external variables in port circuits can be denoted by , where and are vectors of conjugate variables (input voltages and currents, resp.) with the same number of components. For instance, the input power describing the energy flow into a circuit isBy using (4), we can equivalently computeA fundamental principle in passive circuits states that the rate of change of energy stored inside the circuit, denoted by , is never greater than the power that is being supplied; that is,This means that a portion of the supplied energy has been stored in the circuit, while the rest of it has been dissipated. If we denote such dissipation by , the above inequality yieldswhere is the energy dissipated as heat by resistors.
Example 2. Consider the circuit in Figure 1 where and is an auxiliary variable. Using (7), we define the input power as in (14). After some straightforward computations we obtainSincethen (17) is equivalent to
Since in general energy functions are not necessarily known, nor easy to compute in the time domain for complex circuits, we can use quadratic differential forms to facilitate such computations. In the case of the dissipation, this can be easily done in the frequency domain.
4.1. Losses in the Frequency Domain
The integral version of (16) in terms of the auxiliary variable can be expressed asthen, using Parseval’s theorem, we have that where corresponds to the Fourier transform of the auxiliary variable . From this relationship between the time and frequency domain and the fact that we can conclude that (see proof of Proposition of [20])which is a convenient way to compute dissipation.
Example 3. Given the circuit in Figure 1, we have thatThen we computeThe above equation corresponds to the dissipation
In a special case when , the system is called lossless. This case will play a main role in our analysis since, as we later show, it corresponds to the case where conduction losses on SC converters are neglected.
4.2. Energy Storage in Lossless Circuits
The energy stored in a lossless circuit can be easily computed algebraically; we first compute the two-variable polynomial version of asThen, according to Section 4.1 if the system is lossless (zero dissipation), that is,it follows that . We can easily compute in two-variable polynomial terms; that is,It follows thatThen induces , which corresponds to the stored energy.
The study of lossless systems and the computation of the stored energy is of interest in this paper, since these systems can, almost paradoxically, lose energy if a switching mechanism is involved as in the case of SC converters; we briefly revisit this very well-known issue in the following section using the proposed mathematical framework.
5. Energy Losses in Switched-Capacitors
We review the problem of determining energy losses that are induced by charging/discharging the capacitors in Figure 2. Using a modeling specification as in (4), before closing the switch, the dynamics of the circuit can be described byAnalogously, after closing the switch, the new dynamics can be described byConsider the following proposition.
Proposition 4. Consider the circuit in Figure 2 where the switch is closed at . Define as the energy stored in the circuit before the switch and as the one after the switch. Define ; therefore,
Proof. In order to prove the claim note first that, using , from (30) and (31), respectively, we can verify thatwhich implies that each dynamic regime is individually lossless. In order to compute we use the principle of conservation of charge, which establishes that the total charge within the circuit is the same before and after the switch; that is,Using this equation and taking into account the fact that , due to the parallel connection of and , it follows thatUsing quadratic differential forms and (30)-(31), we can compute the stored energy for both subcircuits by usingThe two-variable polynomials and induce the following QDFs:Finally, using (35) to define the value of , we compute ; that is,The claim follows by straightforward algebraic reductions.
Equation (32) represents the difference between the energy stored before and after the switch and is usually called charging/discharging loss. In real-life such loss occurs due to the presence of a current between capacitors and its mechanism of dissipation is the parasitic resistance of the loop. It is important to emphasize that, even when such parasitic resistance is neglected, the current appears in the form of a Dirac impulse (cf. [8, 11]), that is induced by the voltage difference between capacitors at the switching instant.
Remark 5. The results presented in this section can be compared to those in [9, 11]. In our case we obtain the result using the modeling of external dynamics and quadratic differential forms. Our method can be easily extended to more complex converter topologies as exemplified in [7], for which the algorithmic advantages of the polynomial approach can be exploited.
6. Lossless Resonant Switched-Capacitors
In this section we show that, unlike traditional SC (inductorless) topologies, the transfer of charge between the capacitors that occurs in the time interval in a resonant cell, as in Figure 3, is lossless. Consider the following proposition.
Proposition 6. Consider the circuit in Figure 3 where Mode 1 corresponds to the circuit configuration when the switch is open and Mode 2 to the one when the switch is closed. Define a zero-current switching signal (cf. [16]) (for ease of exposition we use as the time axis, where corresponds to the characteristic frequency of Mode 2) as Define as the energy storage function of the circuit before the switch and as the one after the switch. Defining , it follows that .
Proof. We first obtain the linear differential models of the resonant cell in Figure 3 corresponding to Mode 1 before and Mode 2 after the switch, respectively; that is,In order to compute , we can use the time domain solution of . Note that and are continuous and their initial conditions and are fixed but otherwise arbitrary. Moreover, zero-current switching ensures that is equal to zero at every switching instant. Therefore, we can defineThe time domain solution (cf. [22]) of the dynamic equations of Mode 2 in (40) is given bywhereUsing (42), we obtainwhere . Now, compute the instantaneous power for both dynamic modes; that is,where , , andIn polynomial terms it follows thatIt can be easily verified that both modes are lossless; that is, for . Then we can compute the energy functions:These polynomial matrices induce the following QDFs:In order to compute , we can use (44) to firstly computethen it can be easily corroborated using (41), (44), and (50) that The claim is proved.
By comparing the results of Proposition 6 with respect to those of Proposition 4, we can conclude that the charge transferring mechanism of resonant switched-capacitor converters does not involve charging/discharging losses, since the energy stored before and after the switch is conservative. In the following section we will further explore the relationship between energy storage in resonant SC and pure SC converters.
Remark 7. In Figure 4, we show the traces obtained from a time domain solution (44) for the general case when , for the interval of time at which the switch is closed. Note that according to Proposition 6 the lossless mechanism in resonant SC has the zero-current switching as a necessary condition. For instance, if the switch is open when , then the energy will be lost since it will be instantaneously forced to be zero as well.
7. Energy Transferring Comparison between Resonant SC and Pure SC Converters
In this section, we prove that (32) is a physical fundamental quantity in both resonant SC and pure SC converters that represents the upper bound of the energy losses. By means of this analysis we open the door to the computation of two relevant performance parameters, the maximum peak current value, and a relative loss factor in resonant converters. Such computation is instrumental to achieve benchmarking specifications during a design process. In order to do so, we first consider the following proposition.
Proposition 8. Under the assumptions of Proposition 6, using the inductor current variable and defining , it follows that
Proof. Consider Mode 2 (when the switch is closed); note that the maximum instantaneous value of occurs when , which occurs according to (44) at . It follows that the maximum amount of energy stored by the inductor equals To finish the proof simply substitute from (44) evaluated at . The claim is proved.
From Proposition 8 we conclude that the maximum level of energy storage in inductors of resonant SC topologies accounts for the energy that would be dissipated by its pure SC counterpart. This result enables the computation of the resonant inductor according to specifications. Let us define the maximum current through the inductor as ; then according to Proposition 8where . Consequentlywhere the values of and the capacitors can be easily determined when the topology is given; see, for example, the parameter selection in [24] and the voltage gain computed in [25]. The value of can be thus defined according to desired specifications for the semiconductor devices (see, e.g., [26], where a peak value is approximated experimentally in switched-capacitor converters). Note that our computation of can be regarded as a refinement of the results in [26] applied in our case to resonant SC converters.
Finally, we introduce a result that considers the relationship between losses in the presence of parasitic resistances in resonant SC converters and pure SC converters. We show that the losses in such a realistic circuit are bounded by the fundamental quantities introduced in Propositions 4 and 6. In order to do so, we consider the circuit in Figure 5.
Lemma 9. Consider the circuit in Figure 5 where Mode 1 corresponds to the circuit configuration when the switch is open and Mode 2 to the one when the switch is closed. Assume and define the switching signalDefine as the energy storage function of the circuit before the switch and as the one after the switch. Let be the imaginary part of the complex eigenvalues associated with Mode 2. Define The following statements hold: (1).(2).(3).(4).
Proof. Statements (1) and (2) follow directly from Propositions 4 and 6, respectively. In order to prove statements (3) and (4) let us consider the equations of Mode 2; that is,Then considering its time domain solution, that is,withand evaluating and in an analogous way as in the proof of Proposition 6, we obtainStatements (3) and (4) follow readily.
The results of Lemma 9 not only permits generalizing the results obtained along this paper but also provides two important implications: (1) it shows that is a fundamental physical quantity that bounds the energy losses in both SC and resonant SC converters and (2) it permits developing a relative loss factor that is useful to characterize the efficiency of a resonant converter with parasitic resistances with respect to the worst case scenario. We elaborate the latter point in the following.
Consider the losses in a resonant SC converter with a parasitic series resistance as in the proof of Lemma 9; that is,Define the relative loss factor aswhich represent the percentage of charging/discharging losses with respect to the worst case scenario. Note also that depends on the ratio of the parasitic resistance and the inductive reactance, that is, , which implies that the efficiency of the resonant SC converter also depends on the resonant frequency. We illustrate the relative loss factor with respect to such ratio in Figure 6.
8. Experimental Results
Consider the resonant switched-capacitor cell in Figure 3, where , , . Moreover, we chose a maximum peak value , which according to (55) corresponds to an inductor , which is approximated to a commercial value equal to in the experiment.
In Figure 7 we show the voltage traces as well as the resonant inductor current. It can be noticed that the maximum peak value maintains agreement with the selected value of with a very small error of , that is, , which can be associated to the damping induced by conduction losses.
9. Conclusions
We showed that resonant SC converters are lossless with respect to charging/discharging processes of capacitors. This analytical proof is supported by the use of a linear and quadratic differential framework, which is introduced for SC converters in this paper. We showed that the maximum energy storage in the resonant inductor equals the dissipated energy by its pure SC counterpart. We also generalize the results by involving a realistic scenario where parasitic resistances are involved. We also develop a relative loss factor that permits designing resonant SC cells according to efficiency specifications. Moreover, we showed theoretically and experimentally that the resonant inductors can be computed with respect to their maximum stored energy and/or their maximum current value. The results presented in this paper indicate that migration from pure SC converter topologies to their resonant counterpart is the most compelling solution to well-known disadvantages such as high peak currents among capacitors and charging losses.
Conflicts of Interest
The authors declare that they have no conflicts of interest.