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Six-level single-leg flying capacitor converter voltage balancing dynamics analysis
Abstract
Natural voltage balancing property is an attractive feature of a flying capacitor converter. In this paper, time domain approach based on stitching piece-wise analytical solutions for consecutive switching intervals is applied to a six-level single-leg flying capacitor converter. The small parameter approximation analysis yields physically meaningful, simple, and accurate expressions for average voltage balancing dynamics giving an in-depth insight into parameters and carrier frequency impact for carrier-based phase-shifted DC PWM.
... The alternative time domain approach based on "stitching" of analytical transient solutions for consecutive PWM period switching subintervals has been demonstrated in [12][13][14][15][16]. It delivers simple final expressions for balancing time constants, frequencies, and overall capacitor voltage dynamics. ...
... Periodic (oscillating) voltage natural balancing dynamics analysis is not covered by this approach. Such oscillating dynamics happen for FC converters with two and more capacitors and inductance dominated RL-load [9,10,[13][14][15][16]. Aperiodic balancing dynamics take place for multilevel FC converters with pure resistive loads [17] or balance boosters used to speed up balancing dynamics rate for highly inductive loads [18,19]. ...
... For pure resistive load and zero source voltage, the average dissipated power on a switching period is (16) and the decay factor is ...
Reported capacitor average natural voltage balancing dynamics analysis methods are mostly based on frequencydomain Fourier transformations that involve infinite series. Therefore, these methods require high mathematical skills, are not truly analytical and difficult to use in engineering practise. This study suggests a time-domain power averaging-based approach to the analysis of a multilevel DC-DC flying capacitor converter (or, more generally, switched capacitive converter) aperiodic (non-oscillating) average voltage balancing dynamics. Simple analytical solutions are illustrated by single and dual capacitor converter examples that include pure resistive load and balance booster circuit. Theoretical results are supported by simulations and experimental verification.
... Six-level DC-DC FCC natural balancing degradation for lead (lag) PS-PWM in the vicinity of D=0.5 was predicted in [11], [7]. This paper investigates into natural balancing for PS-PWM strategies different from the lead one by carrier order that make use of converter topologies with three and four capacitors in hope that this may improve natural balancing rate. ...
... Though oscillating step response of all required LCRcircuits is assumed (relatively small capacitance), the results also hold for relatively large capacitances (11) are found in the same form of power series expansion in . ...
... The series expansion coefficients in (12) are found by substitution of (12) in (11) and successively equating to zero the coefficients of the resulting series. ...
... For a four capacitor FCC, the known target ratios are 1/5, 2/5, 3/5, 4/5 [6,10] 1 0 0 1 0 2 0 1 0 -1 3 0 1 -1 1 4 1 -1 0 0 Table 6: FCC unbalanced implementation of TR=2/5: capacitor balanced average voltages V1=1/5; V2=2/5; V3=3/5. (Table 10) and the previous analysis by other authors assumed the use of 4 topologies out of 5. ...
... The charges (14) are significantly reduced compared with (10). Moreover, by replicating states 1, 2 and 4 it is possible to obtain the balanced switching with a period of 8 phases. ...
This paper deals with Switched Capacitor Converters (SCC) with multiphase switching and addresses two major subjects. The first one is additional Target Ratios (TR) for the known SCC - Fibonacci Converter and Flying Capacitor Converter. The second subject is Minimal Norm Principle for analysis of SCC phase charge flow that is described by underdetermined systems of linear algebraic equations and its application to single- and dual-output SCC.
... Such an analysis was first applied to describe an elementary three-level FCC with a single flying capacitor in (Margaliot et al., 2011Ruderman et al., 2008). This was followed by calculations for multi-level FCC up to seven-level ( Ruderman and Reznikov, 2011). All these articles provide natural balancing dynamics time constant (and oscillating frequency where applicable) expressions, but the calculation methodology is described very briefly (by literally several phrases) due to the conference paper size limitations. ...
The paper considers a discrete state-space model for transients in a three-level flying capacitor DC–DC converter. A transition matrix is obtained for a pulse width modulation (PWM) period. The matrix elements are expanded into a power series using a selected small parameter. The matrix eigenvalues that determine the natural balancing dynamics transients are presented in the form of power series as well. Four separate transients are constructed based on four possible PWM period initial states (topologies). Inductor current and capacitor voltage transients are found for the voltage source power-up as the arithmetic average of the four separate transients. The discrete solutions are replaced by continuous ones. The resulting transients that are elementary functions of the circuit parameters, PWM period, and voltage reference demonstrate good agreement with the simulation results.
This article investigates behavior of flying capacitor multilevel (FCML) converters in single-phase buck-type power factor correction (PFC) applications. Recent developments in FCML converters using gallium nitride (GaN) transistors are leveraged to improve power density of a single-phase 240
$\text{V}_{\text{RMS}}$
ac to direct 48 V dc conversion stage in data center power delivery applications. Here, we experimentally demonstrate this concept in a digitally controlled six-level FCML converter hardware prototype. The experimental prototype can deliver 4.5 A average output current at 48 V, resulting in 216 W output power. A key contribution of this article is experimental demonstration of an FCML buck converter in a single-phase PFC application where the flying capacitor voltages follow fractions of the rectified input voltage by swinging at twice-line frequency.
Purpose
The flying capacitor converter (FFC) balances the clamping capacitor voltages naturally when phase shifted carrier modulation is used. Several models that describe this mechanism are discussed in the literature. However, due to the model complexity, the stability of the mechanism is inferred from the circuit operation. This paper aims to show that the expressions describing the balancing mechanism can be simplified and used to prove Lyapunov stability.
Design/methodology/approach
The FCC is analysed in the frequency domain. An equivalent circuit that describes the converter operation in terms of total and difference parameters is used. A concerted effort is made to simplify the resulting convolution expressions to their most basic forms by using the characteristics of the phase shifted switching functions' Fourier series coefficients.
Findings
It is shown that the system matrix decomposes naturally into the sum of a symmetric and a skew‐symmetric matrix. Through use of Lyapunov's theorem it is shown that the system is stable if the symmetric part of the decomposition is positive definite. A proof is provided that this matrix is positive semidefinite and the system is therefore Lyapunov stable.
Research limitations/implications
The simplified expressions describing the convolution and the decomposition of the system matrix can be used in future studies to provide maximum bounds on the rebalancing time constant.
Originality/value
This study provides a proof that the natural voltage balancing mechanism is stable. This stability had to be inferred from circuit operation in previous studies. Secondly, the decomposition of the system matrix provides an avenue for future research.
Purpose
– This study aims to propose a mathematical model for stacked multicell converters (SMCs), to be exploited in the analytic determination of natural voltage balancing dynamics of the flying-capacitor (FC) stacked multicell multilevel converters, i.e. investigations of the start-up behavior, dynamic response, and natural voltage balancing phenomenon.
Design/methodology/approach
– The crux of the proposed strategy is based on the closed-form analytic solution derivation for the switching functions used in the switching of the SMCs operated under phase disposition (PD) and phase shifted carrier (PSC) pulse width modulation (PD-PSC-PWM) technique. Hence, the suggested approach develops an analytic solution for the Fourier series and associated Fourier coefficients pertinent to the switching functions of the SMCs by obtaining the switching instants of the PD-PSC-PWM modulator in terms of Kapteyn series when the frequency of the triangular carrier waveform (fc) and that of the sinusoidal reference waveform (fr) have an integer ratio, i.e. f c · f r −1= k , k ∈N.
Findings
– This approach results into a model, first order differential equation based model, which can be readily developed for the SMCs with any number of levels expediting the investigation of their performance. Furthermore, by an experimental scrutiny conducted on a 4×2-cell-nine-level topology of an SMC, it is inferred that under PD-PSC-PWM modulation technique, FC voltages balance naturally for higher number of stacks and cells, therefore the natural balancing exist for high-level SMCs.
Research limitations/implications
– Despite the sophistication of the proposed methodology and mathematical model, this study presents an alternative approach with high potential of applicability for derivation of the multilevel converter mathematical model exploiting the Kapteyn (Bessel-Fourier) series.
Practical implications
– Numeric computation results of the proposed analytic model for the SMCs and the simulation results as well as investigational measurements taken from 2×2-cell-five-level and 4×2-cell-nine-level experimental set-ups are presented in order to substantiate the suggested approach, derived model, and verification of natural balancing.
Originality/value
– This article and its innovations are original.
Reported are the results of simple analytical comparison of natural voltage balancing mechanisms in three-level H-bridge flying capacitor and diode-clamped converters with carrier-based PWM strategies. While the optimal voltage quality of the nearest level switching is obtained for both of them, natural voltage balancing dynamic properties are essentially different. It is shown that a flying capacitor H-bridge converter with appropriate phase-shifted carrier-based PWM delivers much faster natural voltage balancing dynamics, especially, for small DC voltage commands and AC modulation indices. Along with reduced parts (diodes) count, this gives a three-level H-bridge flying capacitor converter a competitive advantage over other three-level H-bridge converter topologies with natural balancing.
Two active capacitor voltage balancing schemes are proposed for single-phase (H-bridge) flying-capacitor multilevel converters. They are based on the circuit equations of flying-capacitor converters. Consequently, they can be implemented using straightforward control rules. In particular, the first technique is based on an algorithm which follows the standard multilevel modulation. Then, it utilizes a redundant state selection table for capacitor voltage balancing. In the second method, multiple duty cycles are defined and modulated in direct response to the capacitor voltages. The most important advantage of these two proposed methods is that they can be utilized to converters with any desired number of levels in their output voltage. Moreover, the analysis and implementation of both methods are straightforward. Through simulation and experimental implementation, these methods are shown to be effective on capacitor voltage regulation in flying-capacitor multilevel converters.
Multilevel converters are considered today as the state-of-the-art power-conversion systems for high-power and power-quality demanding applications. This paper presents a tutorial on this technology, covering the operating principle and the different power circuit topologies, modulation methods, technical issues and industry applications. Special attention is given to established technology already found in industry with more in-depth and self-contained information, while recent advances and state-of-the-art contributions are addressed with useful references. This paper serves as an introduction to the subject for the not-familiarized reader, as well as an update or reference for academics and practicing engineers working in the field of industrial and power electronics.
Flying capacitor multilevel PWM converter with a natural voltage balance is an attractive multilevel converter choice because it requires no voltage balance control effort. Flying capacitor converter practically does not suffer from voltage balance imposed performance limitations as opposed to multiple point clamped converter. Voltage balance dynamics analytical research methods reported to date deal mostly with an AC modulation case and are essentially based on a frequency domain analysis using double Fourier transform. Therefore, these methods require high mathematical skills, are not truly analytical and rather difficult to use in an everyday practice by electrical engineer. In this paper, we consider a DC modulation case to demonstrate that a straightforward time domain approach based on switching intervals piece-wise analytical solutions makes it easy to obtain time-averaged discrete and continuous models for voltage balance dynamics simulation. A primitive single-phase single-leg three-level converter analytical investigation yields a surprisingly simple accurate expression for capacitor charge / discharge related time constant revealing its dependence on inductive load parameters, carrier frequency, and duty ratio.
The attractive feature of a flying capacitor converter is the natural voltage balance property. The reported voltage balance dynamics analytical research methods are based on heavy frequency domain transformations (Fourier transform, Bessel functions etc) and are rather algorithmic and difficult to use in an everyday engineering practice. Suggested time domain approach uses stitching of piece-wise analytical solutions for consecutive switching intervals. The small parameter analysis of a five-level single-leg converter yields physically meaningful, simple, and accurate expressions for average voltage balance dynamics giving an in-depth insight into parameters, carrier frequency, and modulation strategy impact for both DC and AC PWM.
A novel topology of multilevel modular capacitor clamped dc-dc converter (MMCCC) will be presented in this paper. In contrast to the conventional flying capacitor multilevel dc-dc converter (FCMDC), this new topology is completely modular and requires a simpler gate drive circuit. Moreover, the new topology has many advantageous features such as high frequency operation capability, low input/output current ripple, lower on-state voltage drop, and bi-directional power flow management. This paper discusses the construction and operation of the new converter along with a comparison with a conventional converter. Finally, the simulation and experimental results validate the concept of this new topology.
This paper presents a strategy for the analytic determination of the natural voltage balancing dynamics of flying capacitor converters. The approach substitutes Double Fourier series representations of the PWM switching signals into a nonlinear dynamic circuit model of the converter. The result reduces to a linearised state space model that can be readily solved, with the Fourier solution coefficients defining the state space matrix terms. The solution can be readily developed for converters of any level, and allows rapid analytical investigation of the dynamic (and static) balancing behaviour over a wide range of conditions. Furthermore, the approach allows powerful strategies such as root locus to be used to investigate the converter's performance as a function of changes in parameters such as modulation index and load. The analysis approach has been fully verified by comparing it against experimental results on a low voltage prototype converter.
An integrated and comprehensive theory of PWM The selection of the best algorithm for optimum pulse width modulation is an important process that can result in improved converter efficiency, better load (motor) efficiency, and reduced electromagnetic interference. However, the identification of the best approach is a complex process requiring extensive mathematical manipulation. Pulse Width Modulation for Power Converters: Principles and Practice is the first single-volume resource written to help researchers in the field attain a working knowledge of the subject. The authors bring together todays seemingly diverse approaches into a single integrated and comprehensive theory of modulation. The book provides a generalized approach to the fundamentals of PWM, looking at:
Active switch pulse width determination
Active switch pulse placement within a switching period
Active switch pulse sequence between phase legs and across switching periods
The main benefit of this generalized concept is that once the common threads are identified, the selection of a modulation strategy for any converter topology becomes immediately clear, leaving only secondary factors, such as practical performance, cost, and difficulty of implementation to consider. Additionally, it allows the performance of any particular converter topology and PWM strategy to be quickly and easily identified without complex and time-consuming analysis. Pulse Width Modulation for Power Converters: Principles and Practice enables the reader to achieve optimum PWM results for any application.
This paper presents a strategy for the analytic determination of the natural voltage balancing dynamics of flying capacitor converters. The approach substitutes double Fourier series representations of the pulsewidth modulation (PWM) switching signals into a nonlinear dynamic circuit model of the converter. The result reduces to a linearized state-space model that can be readily solved, with the Fourier solution coefficients defining the state-space matrix terms. The solution can be readily developed for converters of any level, and allows rapid analytical investigation of the dynamic (and static) balancing behavior over a wide range of conditions. Furthermore, the approach allows powerful strategies such as root locus to be used to investigate the converter's performance as a function of changes in parameters such as modulation index and load. The analysis approach has been fully verified by comparing it against experimental results on a low voltage prototype converter.