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Variable Structure Modeling and Design of Switched-Capacitor Converters Siew-Chong Tan, Member, IEEE, Svetlana Bronstein, Moshe Nur, Y. M. Lai, Member, IEEE, Adrian Ioinovici, Fellow, IEEE, and Chi K. Tse, Fellow, IEEE
Abstract—Switched-capacitor (SC) converters are a type of variable structure systems. The conventional approach of maintaining regulation in these converters is a feedback control developed from linear systems theory, and it is based on the approximate small-signal linearized models of these circuits. However, the simplicity of such an approach sacrifices performance (poor transient response and sometimes steady-state instability are the result of a design based on the use of an approximate linearization) for convenience and cost. This paper discusses the (SC) converters from the viewpoint of nonlinear systems, and based on this, takes a variable structure feedback approach. The (SC) circuits theory is revisited, and a new approach of modeling, which gives an accurate nonlinear description of their operation is discussed. Based on the principle of energy balance applied to the output filter capacitor, an exact relationship between the instantaneous output and input currents in the charging and discharging phases is derived, leading to the derivation of a unique large-signal dynamic model for both alternative operating phases. Together with a defined switching function, it forms the proposed variable structure model. The resulting solution shows that a nonlinear approach can deliver an improved performance in the dynamic and steady-state behavior. Experimental results performed on a two-phase (SC) converter verify the theory. Index Terms—Nonlinearity, pulse width modulation (PWM), stability, switched-capacitor (SC) converters, variable structure feedback control.
I. INTRODUCTION ITH ONLY switches and capacitors in the power stage, the switched-capacitor (SC) converters have the advantages of light weight, small size, and high power density [1]–[10]. Such features make them ideal power supplies for mobile electronic systems such as cellular phones, personal digital assistants, etc. [1]. The present electronic industry is seeing increasing usage of such converters in their products. Semiconductor companies like Maxim and National Semiconductor are already mass producing SC converters in an
W
Manuscript received August 18, 2008; revised October 10, 2008. First published December 02, 2008; current published September 11, 2009. This work was supported by Hong Kong Polytechnic University under an internal competitive research grant under Grant A-PB0Z. This paper was recommended by Associate Editor H. S. Chung. S.-C. Tan, Y. M. Lai, and C. K. Tse are with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong. S. Bronstein is with the Department of Electrical Engineering, Sami Shamoon College of Engineering, Beer-Sheva 84100, Israel. M. Nur and A. Ioinovici are with the Department of Electrical and Electronic Engineering, Holon Institute of Technology, Holon 58102, Israel (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCSI.2008.2010149
IC-packaged form, e.g., MAX828/829 and LM3351, etc., for commercial applications. The available SC converters are not able to respond to the industrial requirements of regulation in the presence of wide range of input voltage and load variation. The typical way of maintaining the regulation of these SC converters is based on linear system theory, using what is known as pulse width modulation (PWM) controllers. Such an approach yields satisfactory performances in some applications. However, as it can only handle a small range of the system’s nonlinearity [2], there are situations where this approach fails to perform desirably. Such applications include systems that require a tight and fast tracking of the reference voltage under wide load variations (e.g., with computer chips, etc.), or systems with a widely varying input voltage (as that provided by fuel cells, solar panels, etc., the purpose of the dc–dc converter, used as the front end of the energy supply system, being that of stabilizing this voltage), or systems requiring a rapid equilibrium-to-equilibrium transfer of energy in converters during startup [11]. On the other hand, in extreme situations, the regulation of the SC converters using the linear system approach may settle into a state of instability. Such a phenomenon is not uncommon in conventional power converters [12], [13]. However, it is more severe in SC converters since their voltage conversion ratio is more nonlinear than that of conventional switching converters. Fig. 1(a)–(c) shows the typical power source to output storage/ load behavior of a buck-type converter, a boost-type converter, and an SC converter, respectively. Note that for SC converters, the time constant of the charging current is typically small since it is a function of (of a very small value, as it is given by the sum of the parasitic resistances of the switch and capacitor) and the capacitance of [see Fig. 1(c)]. To have a controllable regulation for this type of converters, it is necessary for their controllers to be able to generate a turn-on time that is much . In the other exsmaller than the time constant, i.e., or when the turn-on time is large compared treme, where to the time constant, e.g., if , the SC converter will be deemed uncontrollable. Each turning ON and OFF of the switch will result in the capacitor being charged to the value of the input voltage, and the regulation will be lost, i.e., the converter fails to fulfill its purpose. Referring to Fig. 1(a)–(c), for the buck-type converters, the current flow is continuous and piecewise linear. Its averaged representing input-to-output dc voltage ratio is the steady-state duty ratio, where as usual, the parasitic resistances of the inductor and the capacitor are neglected. The buck-type converter is characterized by a voltage conversion
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Fig. 1. Example illustrating the differences in the dynamics and behavior of the current flowing from the power source to the output storage/load of: 1) a buck converter; 2) a boost converter; and 3) a simple SC circuit. (a) Buck converter (left) and its current flow waveform (right). (b) Boost converter (left) and its current flow waveform (right). (c) Switched-capacitor converter (left) and its current flow waveform (right).
linearly proportional to the duty cycle . For boost-type converters, the current flow is pulsating with a linearly decreasing slope appearing in the interval when the energy is delivered to the storage/load. Its averaged input-to-output dc voltage ratio . For SC conis of the parabolic form as verters, the current flow is pulsating and decreasing exponentially with a first-order transient characteristic when the energy is delivered to the storage/load. A general form of its averaged input-to-output dc voltage ratio was derived in [14] as . It is not difficult to see that the voltage conversion for the SC and is more nonlinear than converters is dependent on load that of the boost-type converters because, as explained before, cannot be neglected in SC converters. Additionally, as the current flow in the SC converter is proportional to the applied ) and inversely proportional voltage difference (i.e., to the resistor , for each turning ON of the SC converter’s switch, a high inrush current , which has a faster dynamic than that of the conventional converters (with inductor), will flow through the circuit. It is therefore more difficult to maintain good dynamic and steady-state regulation, and ensure the stabilization of SC converters than for conventional converters of the same power rating. This is especially true if the SC converter is operated under wide input and/or load variation. For such applications, it is necessary to perform the regulation from the perspective of nonlinear systems, as any linearization around the operating (nominal) point may result in instability for other operating conditions.
Fig. 2. (a) SC converter and (b) the timing diagram of the converter switches.
To the authors’ knowledge, there is still no reported work in the literature that discusses the issues concerning the SC
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Fig. 3. Four operating states of the SC converters. (a) State 1: u State 3: u u u u . (d) State 4: u
= 0;
= 1;
= 1;
=0
= 1; u = 0; u = 0; u = 1. (b) State 2: u = 0; u = 0; u = 0; u = 1. (c) = 0; u = 1; u = 0; u = 0.
converters’ operation without introducing linearization approximation (i.e., 1) use of small-signal linearized models and 2) linearizing the input-to-output voltage ratio). In view of this, and with the consideration that portable electronic systems are moving toward the trend of having widely varying input source and output load, this paper highlights a proposed solution based on variable structure control (VSC). VSC is a well-established nonlinear methodology, which has shown to be highly promising and effective for dealing with the nonlinearity in conventional switching converters [15]–[24]. However, according to a search in the literature, the VSC methodology was never applied to the SC converters. Moreover, the translation of the technique from conventional converters to dual-phase SC converters is not straightforward due to the specific composition of the output voltage, which is provided by two symmetrical SC subcircuits operating in a dual-phase switching way (the output of each one of these blocks influencing the input of the other one). When an SC subcircuit is in a charging phase, the other one is in the discharging phase. Specifically, a different way of modeling the converter using the principle of energy balance (which can be extended to other types of multiphase SC converters) is required when VSC is applied to the dual-phase SC converters. A basic SC converter and its switching operation is briefly summarized in Section II. Starting from the state–space equations of the switching stages, a nonlinear relationship between the instantaneous output current of one of the SC subcircuits and the instantaneous input current of the other SC subcircuit is established using an energy balance applied to the output filter capacitor. This relationship is reformulated, taking into account the alternative charging/discharging phases of the two SC subcircuits. A switching function is defined in Section III, and based on the dynamic model of the converter, a variable structure model is proposed. The existence and stability of the proposed sliding dynamics are checked, and a VSC is designed. Verification and evaluation of the methodology are conducted through experimental work, as presented in Section IV.
II. BASICS OF SC CONVERTER A. Converter Circuit For illustrative purpose, we have chosen a dual-phase SC converter, as shown in Fig. 2(a) for investigation in this study. The converter is made up of four externally controlled switches – , six internally controlled switches – , four en– , and an output filter capacergy transfer capacitors that is connected in parallel to the load . The timing itor diagram for driving the switches is given in Fig. 2(b). Here, , and represent the logic states of switches – , respectively. The logics 0 and 1 represent the turning OFF and ON of the switches, respectively. A more detailed description of this converter can be found in [2]. Briefly, this converter can be viewed in terms of two symmetrical subcircuits with four topological states per cycle, as given in Fig. 3. B. Topological States With the assumption that all the diodes are ideal, the four topological states given in Fig. 3 can be described as follows. ON and OFF, and 1) State 1 [see Fig. 3(a)]: With are connected in series and being charged up by input through resistor . voltage source with a current , where , and Here, are the internal resistances of , and , respectively. Concurrently, with OFF and ON, and are discharging their storage energy in parallel to the load and the filter capacitor through the resistances and , respectively. Here, the total discharging current from and can be equated as , where , and are the currents , and , respectively. The resisflowing into and , tances are where , and are the internal resistances of , and , respectively. The variables , denote the voltages across – , respectively; and and denotes the output voltage of the converter.
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2) State 2 [see Fig. 3(b)]: With being turned OFF, the and is halted. However, with and charging up of still remaining in their previous states, both and continue to discharge their storage energy to the load. 3) State 3 [see Fig. 3(c)]: This is similar to state 1 with all switches operating complimentarily to their symmetrical ON and OFF, and are being counterparts. With charged up in series with a current through resistor . Here, , where is the internal resistance of . Concurrently, with OFF and ON, and discharge their storage energy in parallel to and through and , respectively. The total from and can be equated as discharging current . Here, and , where is the internal resistance of . 4) State 4 [see Fig. 3(d)]: This is similar to state 2 with all switches operating complimentarily to their symmetrical being turned OFF, the charging up of counterparts. With and is halted. With and still remaining in their and continue to discharge their previous states, both storage energy to the load. C. Simplified Switched Model for Nonlinear Feedback Control From the discussion in the previous section, it is understood that energy from the input voltage source is alternatively delivered to the series-connected capacitor pairs and , and that the stored energy is alternatively delivand ered from the parallel-connected capacitor pairs to the load. Assuming that the two circuit phases are symmetrical and that , and , these charging and discharging operations can be reduced to the simplified models, as shown in Fig. 4(a) and (b), respectively, with – being ideal switches. Note that the charging and discharging operations given in the figures must still be synis 1, must be 0, and vice chronized such that when is 1, must be 0, and vice versa. versa. Similarly, when According to Fig. 4(a), the instantaneous input current flow from the source to the circuits can be expressed as
respectively, written as and , where and represent the voltage levels of the capacitor after being discharged and charged, respectively. Now, by applying the principle of energy balance to the charging and discharging operations of the capacitor circuit, i.e., , it is possible to deduce that to achieve energy balance within a switching cycle, the increase of the voltage during the charging operation must be equivalent to the negative drop of the voltage during the discharging operation, i.e.
(1)
(5)
where (2) From Fig. 4(b), the instantaneous current flowing through the output filter capacitor can be expressed as (3)
Fig. 4. (a) Charging operation of series-connected capacitors and (b) discharging operation of parallel-connected capacitors.
where and represent the change in capacitor voltage during charging and discharging, respectively. The aforementioned condition is valid for both steady-state operation and transient cycles determined by changes in the input voltage and load. Next, consider only of the charging and discharging operations of the capacitor the proposed SC converter. With the discharging operation of , the change the capacitor occurring for the time period of in voltage during discharging can be written as
where (4) Recall that for any lossless capacitor , the change of energy due to charging and discharging over a fixed time period can be,
(6) under the assumption that the discharging operation occurs linearly. This is possible since the discharging period is much
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smaller than the discharging time constant, i.e., . On the other hand, given that the charging opoccurs only for the time period , the change eration of in voltage during charging is expressed as
feedback control design for the remaining phase can be easily mirrored.1 Hence, for the remaining discussion in this section, in we will focus on the feedback control design for switch the time period between 0 and with turned OFF and turned ON, as that given in topological state 1 and state 2 [refer to Fig. 3(a) and (b)]. According to (3), for this period, the current flowing through the filter capacitor is given as
(7)
(12)
since the charging operation is assumably linear with the charging period being much smaller than the charging time con. stant, i.e., Then, the substitution of (6) and (7) into (5), and then, with the consideration of (2), (5) can be rearranged as
Using (11), (12) will be given as (13)
A. State Variables and Switching Function (8) Using the same principle, a similar equation can be derived for . Finally, the substitution of these equations into (4) gives the average form of
The VSC feedback mechanism adopted in this investigation and uses a switching function a proportional–integral (PI) type of sliding plane2 with linear combination of two state variables, i.e. (14)
(9) and the state–space form of
where represents the control coefficient. Here, the adopted feedback state variables are the voltage error , and the integral of the voltage error , which are expressed as (15)
(10) where The previous equation gives the relationship between the output current (left-hand side) and the input current (right-hand side), which is formulated using the energy balance. What it represents is that for any amount of energy discharged from a capac, the necessary charging action reitor with a current flow quired to replace the discharged energy to achieve instantaneous energy balance within a switching cycle in the same capacitor, is . However, for the case of the given as SC converters, which have operations that are alternative such that while one capacitor circuit phase is discharging, the other capacitor circuit phase is charging, (10) must be reformulated as
denotes the reference voltage of the converter.
B. Equivalent Control Model of the System With being constant, the time differentiation of (15) gives the system’s dynamical model as (16) Next, the substitution of (13) and (16) into the time differentiation of (14) gives (17)
(11) and with the combined current so that the discharging of will prompt the corresponding charging action for of and such that immediate energy balance is achieved. This reformulation is also carried out for the case of . III. SC CONVERTER WITH VARIABLE STRUCTURE CONTROL The switches and are complementarily turned ON and OFF for a duty cycle of 50% in each cycle. Therefore, the only controllable actions for the regulation are those that are trigand . Yet, since the converter is symmetrical in gered by both its phases, it is sufficient to consider only one of the phases when designing its feedback control. When this is achieved, the
Then, the equivalent control signal of switch 1, i.e., , which gives be obtained by solving
, can
(18) 1In the case where the phases are asymmetrical, consideration and development of individual controller for each of the phases can be easily performed following the same procedure. 2VSC is a general method, which basically allows any type of sliding plane to be implemented. The decision for choosing a PI sliding plane instead of the conventionally adopted proportional–derivative (PD) or PID sliding plane is due to a significant increase in circuit complexity with the inclusion of the derivative control in the sliding plane. This would be unpractical for circuit implementation. Moreover, as the system’s overshoot and stability can be satisfied using a PI sliding surface, then the derivative term is not required. Instead, the adoption of the integral term in the sliding plane reduces the steady-state error of the system and its implementation can be carried out easily.
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Fig. 5. Derived feedback control mechanism for the dual-phase SC converter.
where . Practically, (18) can be produced using a PWM circuit using the following relationships (19) is the feedback control signal, is the where peak magnitude of a constant frequency ramp signal, and is a down-scaling constant to adjust the signal level to an appropriate chip’s level voltage standard [22]. A similar set of relationships , describing the equivalent control signal of switch , i.e., can be obtained as (20)
C. Feedback Control Mechanism of a System Fig. 5 shows an overview of the feedback control mechanism used in this investigation. The control design is based on (19) and and (20) with parameters . The construction of control signals and is common and employs the same circuit. Their computations require the sensing of the output voltage , which is straightforward potential detection, and the sensing of the output current , which requires a resistor sensor in series with the is fed into the output load. The constructed control signal positive input of the two pulse width modulators. The negative input is connected to a ramp signal. For this part, two ramp generator circuits that have varying peak voltage with the change of input voltage or capacitor voltage are required for the nonlinear compensation. Here, the capacitor voltage are obtained by measuring the total voltage of the series capacitor pairs of and , i.e., and (the voltage across equivalent series resistance (ESR) of the capacitor is negligible). Additionally, the frequency of the ramp and , signal is synchronized to the switching signals which are basically 50% duty cycle pulses complementing one can only be turned ON when another. To ensure that switch is turned ON and that can only be turned ON when is turned ON, the output of the respective pulse width modulators are multiplied with the logic states of and using a logic AND operator. Additional Comments: An examination into the architecture also reveals that it basically adopts the same structure as the conventional PWM voltage-mode controller, but with additional
Fig. 6. Plots of experimentally measured dc output voltage v versus output load current of the SC converter with: (a) linear voltage feedback control and (b) voltage feedback VSC at input voltages of v and 15 V.
= 12
feedback components consisting of the instantaneous load current for constructing the control signal , and input voltage and capacitor voltages and for controlling the ramp signals’ amplitude. These additional components give the nonlinearity of the feedback control. Therefore, as compared to the conventional PWM voltage-mode controller, one additional current sensor and two sets of auxiliary circuits for generating the ramp signals will be required. D. Existence Condition With the feedback control mechanism introduced to the SC converter, it is necessary that the system enters into slidingmode (SM) operation to effectively deal with the nonlinearity of the SC converter in giving an efficient dynamic and steady-state performance. For that, three necessary conditions, namely, the hitting condition, existence condition, and stability condition, have to be abided. So far, the hitting condition3 has been satisfied by the appropriate choice of the switching function. As for the existence condition, it can be obtained by inspecting the local 3Satisfaction of the hitting condition assures that regardless of the initial condition, the state trajectory of the system will be always directed toward the sliding surface.
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reachability condition , which, with the substitutions of (14) and its time derivative (17), gives (21) Assuming that the system is designed with a static sliding plane to meet the existence conditions for the steady-state operation (equilibrium point) for the whole range of operating condition [22], and that the voltage error is negligibly small (i.e., ), then (21) can be simplified as (22) where (i.e.,
denotes the expected steady-state output voltage ), denotes the minimum output load resistance, denotes the minimum input voltage, and denotes the expected maximum voltage across the capacitor when the converter is operating at full-load condition. Likewise, would be the existence conditions for
which represents the ideal sliding dynamics of the SM-controlled SC converter. 2) Equilibrium Point Analysis: Assume that there exists a stable equilibrium point on the sliding surface on which the ideal sliding dynamics eventually settled. At this point of equilibrium (steady state), there will not be any change in the system’s dynamics if there is no input or loading disturbance, . Then, the i.e., state equations in (25) can be equated to give (26) where , and represent the voltage across capacitors and , and the output voltage at steady-state equilibrium, respectively. 3) Linearization of the Ideal Sliding Dynamics: Next, the linearization of the ideal sliding dynamics around the equilibrium point transforms (25) into (27)
(23) where The selection of the coefficients and , and the design of the converter must comply with the inequalities given in (22) and (23). This assures the existence of the SM operation at least in the small region of the origin for all operating conditions up to full load. E. Stability Condition
(28)
For this system, the stability condition4 can be derived by first obtaining the ideal sliding dynamics of the system, and then, doing an analysis on its equilibrium point [16]. 1) Ideal Sliding Dynamics: The replacement of , and by , and (using so-called equivalent control method), respectively, into the original SC converter’s description converts the discontinuous switching system into an ideal SM continuous system
The derivation is performed with the adoption of the following and . The static equilibrium conditions: characteristic equation of this linearized system will be given by (29) where
(30) (24)
Note that due to symmetry, the dynamics of will be identical to , and will be identical to such that and . Then, the subinto (24) gives stitution of (18) and
(25) 4Satisfaction
of the stability condition ensures that the state trajectory of the system under SM operation will always reach a stable equilibrium point.
The application of the Routh criterion to the characteristic equation in (29) shows that the system will be stable if the following conditions are satisfied: (31) Hence, by numerically evaluating the expressions (28) and (30), and substituting them into (31), the stability of the system can be determined. IV. EXPERIMENTAL RESULTS AND DISCUSSIONS The hardware prototype is developed from (19) and (20) for a 15-W (5 V, 3 A) SC converter. The specifications of the SC converter are given in Table I. The converter has been designed
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TABLE I SPECIFICATIONS AND COMPONENTS OF THE SC CONVERTER
Fig. 7. Experimental waveforms of the output voltage transience v of the SC converter with linear voltage feedback control operating at input voltages of 12 (minimum) and 15 V (maximum), and alternating between load resistances 1.67 (minimum) and 16.67 (maximum). (a) v = 12 V. (b) v = 15 V.
to operate for an input voltage range of 12 V V. The parameters of the feedback control mechanism using VSC are , and . Note that the pole term in serves to reduce the steady-state error of the converter. A detailed discussion on the finite steady-state-error phenomenon and the proposed solution of alleviating this error through the addition of a pole term can be found in [25].
A. Regulation Fig. 6(a) and (b) shows the dc output voltage versus different operating load currents at input voltage conditions of 12 (minimum) and 15 V (maximum) obtained from experimental measurements performed on the SC converter with the linear PWM voltage control5 and the VSC, respectively. The results show that good load and line regulation properties of around 0.8% of and 0.6% of , respectively, are achieved with the VSC approach. This is slightly better as compared to what is achieved with the linear voltage control that has a load regulation of 0.8% and a line regulation of 0.8% of . of B. Dynamical Behavior The dynamical property of the SC converter in handling large-signal load disturbances at different input voltages is also 5The linear controller in the experiment uses a PI-type error compensator with = 2 4 and an integral gain of = 13333 3. The a proportional gain of parameters were obtained by tuning the controller for the optimum dynamic response at an input voltage of = 15 V.
K
: v
K
:
investigated for both the linear feedback control and the VSC control. Fig. 7(a) and (b) shows the experimental output voltage ripple waveforms of the SC converter with linear voltage feedback control operating at a load resistance that alternates between and for various input voltages. The controller has been optimally tuned for the condition ( V) shown in Fig. 7(b), with a settling time of around 500 s. At a reduced input voltage, the control characteristic of the linear controller changes, and its system’s dynamic response deteriorates. This is understood because the linear controller is designed for a specific operating condition (based on a smallsignal linear approximation around the chosen operating point), which leads to changes in the response behavior when a different operating condition is engaged. With this controller, the V with a setworst-case operating condition occurs at tling time of around 4.0 ms [see Fig. 7(a)]. On the other hand, with the control mechanism obtained from VSC, the change in the input voltage does not alter its control characteristic since it is robust to disturbance and parameter change, and a similar dynamic response is achievable for different operating conditions. This is illustrated in Fig. 8(a) and (b), which shows the output voltage ripple waveforms of the SC converter with the voltage feedback VSC. For any input voltage, the transient settling time of this converter is similar, around 500 s. As compared to the worst-case operating condition of the converter with PWM controller, the time taken is only 12.5% of that with the conventional PWM voltage controller [see Figs. 7(a) and 8(a)]. Thus, when dealing with the
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Fig. 8. Experimental waveforms of the output voltage transience v of the SC converter with voltage feedback VSC operating at input voltages of 12 (minimum) and 15 V (maximum), and alternating between load resistances 1.67 (minimum) and 16.67 (maximum). (a) v = 12 V. (b) v = 15 V.
SC converters, the advantage of using a nonlinear control in the form of VSC in achieving a fast dynamical response over a wide range of operating conditions is demonstrated. For a discussion on the relationship between the efficiency and a wide-range regulation, see [26] and [27]. Finally, it is important to stress once again that the improvement of the static and dynamic performances of the SC converter using the proposed approach over conventional approach comes at the expense of an additional current sensor and a more complicated circuitry. V. CONCLUSION The possibility of using a variable structure system approach in dealing with the SC converters without introducing linear small-signal approximations is discussed in this paper. A way of modeling the dual-phase SC converters using the energy balance principle, which is necessary when SC converters are to be regulated using such a nonlinear approach, is illustrated. The experimental results verified that when the switching capacitor converter is operated over a wide range of input voltage and load, a better transient response and a steady-state regulation can be achieved with the voltage feedback variable structure control approach, as compared to the voltage feedback linear control approach. The variable structure control approach proved to be robust to disturbances, with a similar dynamic response being achievable under different operating conditions. This would mean that a wider operating range is possible for the SC converters, and could lead to significantly more application possibilities. ACKNOWLEDGMENT The authors would like to thank Mr. S.Y. Lam for developing the prototype and conducting the experiment. REFERENCES [1] A. Ioinovici, “Switched-capacitor power electronics circuits,” IEEE Circuits Syst. Mag., vol. 1, no. 3, pp. 37–42, Sept. 2001. [2] S. V. Cheong, H. Chung, and A. Ioinovici, “Inductorless DC-to-DC converter with high power density,” IEEE Trans. Ind. Electron., vol. 41, no. 2, pp. 208–215, Apr. 1994.
[3] C. K. Tse, S. C. Wong, and M. H. L. Chow, “On lossless switchedcapacitor power converters,” IEEE Trans. Power Electron., vol. 10, no. 3, pp. 286–291, May 1995. [4] J. Chen and A. Ioinovici, “Switching-mode DC-DC converter with switched-capacitor based resonant circuit,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 43, no. 11, pp. 933–938, Nov. 1996. [5] M. S. Makowski, “Realizability conditions and bounds on synthesis of switched-capacitor DC–DC voltage multiplier circuits,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 44, no. 8, pp. 684–692, Aug. 1997. [6] H. S. H. Chung, “Design and analysis of a switched-capacitor-based step-up DC/DC converter with continuous input current,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 6, pp. 722–731, Jun. 1999. [7] H. S. H. Chung, W. C. Chow, and S. Y. R. Hui, “Development of a switched-capacitor DC–DC converter with bi-directional power flow,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 9, pp. 1383–1390, Sept. 2000. [8] B. Axelrod, Y. Berkovich, and A. Ioinovici, “A cascade boostswitched-capacitor-converter-two level inverter with an optimized multilevel output waveform,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 52, no. 12, pp. 2763–2770, Dec. 2005. [9] F. Su and W. H. Ki, “Design strategy for step-up charge pumps with variable integer conversion ratios,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 5, pp. 417–421, May 2007. [10] B. Axelrod, Y. Berkovich, and A. Ioinovici, “Switched-capacitor/Switched-inductor structures for getting transformerless hybrid DC-DC PWM converters,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 55, no. 2, pp. 687–696, Mar. 2008. [11] A. Gensior, O. Woywode, J. Rudolph, and H. Guldner, “On differential flatness, trajectory planning, observers, and stabilization for DC-DC converters,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 53, no. 9, pp. 2000–2010, Sept. 2006. [12] C. K. Tse, “Flip bifurcation and chaos in three-state boost switching regulators,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 41, no. 1, pp. 16–23, Jan. 1994. [13] Y. Chen, C. K. Tse, S. S. Qiu, L. Lindenmuller, and W. Schwarz, “Coexisting fast-scale and slow-scale instability in current-mode controlled DC/DC converters: Analysis, simulation and experimental results,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 55, no. 10, pp. 3335–3348, Nov. 2008. [14] G. Zhu and A. Ioinovici, “Switched-capacitor power supplies: DC voltage ratio, efficiency, ripple, regulation,” in Proc. IEEE Int. Conf. Circuits Syst. (ISCAS), May 1996, pp. 553–556. [15] E. Fossas, L. Martínez-Salamero, and J. Ordinas, “Sliding mode control ` uk converter,” reduces audiosusceptibility and load perturbation in the C IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 39, no. 10, pp. 847–849, Oct. 1992. [16] L. Martinez-Salamero, J. Calvente, R. Giral, A. Poveda, and E. Fossas, ´ uk converter operating “Analysis of a bidirectional coupled-inductor C in sliding mode,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 4, pp. 355–363, Apr. 1998.
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TAN et al.: VARIABLE STRUCTURE MODELING AND DESIGN OF SC CONVERTERS
[17] H. Sira-Ramirez, “On the generalized PI sliding mode control of DC-to-DC power converters: A tutorial,” Int J. Control, vol. 76, no. 9/10, pp. 1018–1033, Jun. 2003. [18] D. Biel, F. Guinjoan, E. Fossas, J. Chavarria, and E. Fossas, “Slidingmode control design of a boost-buck switching converter for AC signal generation,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 51, no. 8, pp. 1539–1551, Aug. 2004. [19] J. L. Lin and C. H. Hsia, “Dynamics and control of ZCZVT boost converters,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 52, no. 9, pp. 1919–1927, Sept. 2005. [20] R. Gupta and A. Ghosh, “Frequency-domain characterization of sliding mode control of an inverter used in DSTATCOM application,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 53, no. 3, pp. 662–676, Mar. 2006. [21] S. Zhou and G. A. Rincon-Mora, “A high efficiency, soft switching DC–DC converter with adaptive current-ripple control for portable applications,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 4, pp. 319–323, Apr. 2006. [22] S. C. Tan, Y. M. Lai, and C. K. Tse, “A unified approach to the design of PWM based sliding mode voltage controller for DC–DC converters in continuous conduction mode,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 53, no. 8, pp. 1816–1827, Aug. 2006. [23] R. Frasca, L. Iannelli, and F. Vasca, “Dithered sliding-mode control for switched systems,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 9, pp. 872–876, Sept. 2006. [24] A. Cid-Pastor, L. Martinez-Salamero, C. Alonso, G. Schweitz, J. Calvente, and S. Singer, “Synthesis of power gyrators operating at constant switching frequency,” Proc. Inst. Electr. Eng., Electr. Power Appl., vol. 153, no. 2, pp. 842–847, Nov. 2006. [25] S. C. Tan, Y. M. Lai, and C. K. Tse, “Indirect sliding mode control of power converters via double integral sliding surface,” IEEE Trans. Power Electron., vol. 23, no. 2, pp. 600–611, Mar. 2008. [26] M. S. Makowski and D. Maksimovic, “Performance limits of switchedcapacitor DC-DC converters,” in Proc. IEEE Power Electron. Spec. Conf. (PESC), June 1995, vol. 2, pp. 18–22. [27] A. Ioinovici, H. S. H. Chung, M. S. Makowski, and C. K. Tse, “Comments on “Unified analysis of switched-capacitor resonant converters”,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 684–685, Feb. 2007.
Siew-Chong Tan (S’00–M’06) received the B.Eng. (with honors) and M.Eng. degrees in electrical and computer engineering from the National University of Singapore, Singapore, in 2000 and 2002, respectively, and the Ph.D. degree in electronic and information engineering from Hong Kong Polytechnic University, Kowloon, Hong Kong, in 2005. From October 2005 to February 2009, he worked as Research Associate, Postdoctoral Fellow, and then as Lecturer in the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, where he is currently an Assistant Professor. His present research interests include nonlinear control of power electronic systems, switched-capacitor converter circuits and applications, and the development of power-electronic based circuits for light-emitting diodes and fuel cells. Dr. Tan serves extensively as a reviewer for various IEEE/IET transactions and journals on power, electronics, circuits, and control engineering.
Svetlana Bronstein was born in Leningrad, Russia. She received the Electr. Eng. Diploma from Leningrad Polytechnic Institute, Leningrad, in 1984, and the Ph.D. degree from Ben-Gurion University of the Negev, Beersheba, Israel. From 1984 to 1991, she was with Mendeleev Research-Engineering Institute of Metrology, Leningrad. She is currently a Lecturer in the Department of Electrical engineering, Sami Shamoon Engineering College, Beersheba. Her current research interests include piezoelectric transformers in power electronics, dc–dc resonant power converters, and modeling and simulation.
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Moshe Nur received the B.Sc.Tech. and M.Sc. degrees in electrical engineering from Holon Institute of Technology, Holon, Israel, in 1998 and 2008, respectively. Since 1998, he has been a Hardware Development Engineer in the industry, where he is engaged in the field of analog and power electronics. He is currently with the Department of Electrical and Electronic Engineering, Holon Institute of Technology.
Y. M. Lai (M’92) received the B.Eng. degree in electrical engineering from the University of Western Australia, Perth, Australia, in 1983, the M.Eng.Sc. degree in electrical engineering from the University of Sydney, Sydney, Australia, in 1986, and the Ph.D. degree from Brunel University, London, U.K., in 1997. He is currently an Assistant Professor in the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong. His current research interests include computer-aided design of power electronics and nonlinear dynamics.
Adrian Ioinovici (M’84-SM’85–F’04) received the degree in electrical engineering and the Dr.-Eng. degree from the Polytechnic University, Iasi, Romania, in 1974 and 1981, respectively. In 1982, he joined Holon Institute of Technology, Holon, Israel, where he is currently a Professor in the Department of Electrical and Electronics Engineering. During 1990–1995, he was a Reader and then a Professor in the Electrical Engineering Department, Hong Kong Polytechnic University. His current research interests include simulation of power electronics circuits, switched-capacitor-based converters and inverters, soft-switching dc power supplies, and three-level converters. He is the author of the book Computer-Aided Analysis of Active Circuits (Marcel Dekker, 1990) and the chapter Power Electronics in the Encyclopedia of Physical Science and Technology (Academic, 2001). He is the author or coauthor of more than 130 papers published in the field of circuit theory and power electronics. Prof. Ioinovici was the Chairman of the Technical Committee on Power Systems and Power Electronics of the IEEE Circuits and Systems Society (CAS-S) for few terms. He was an Associate Editor for Power Electronics of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I for repetitive terms, and currently an Associate Editor for Power Electronics of the Journal of Circuits, Systems, and Computers. He was the IEEE CAS Society Distinguished Lecturer from 1999 to 2002. He has been an Overseas Advisor of the Institute of Electronics, Information and Communication Engineers Transactions, Japan. He was the Chairman of the Israeli chapter of the IEEE CAS-Society between 1985 and 1990, and the General Chairman of the Conferences Islamic Shura Council of Southern California (ISCSC) 1986, ISCSC 1988 (Herzlya, Israel), Southeast Private Equity Conference (SPEC) 1994 (Hong Kong), organized and chaired special sessions in Power Electronics at the International Symposium on Circuits and Systems (ISCAS) 1991, ISCAS 1992–1995, and ISCAS 2000. He was a member of the Technical Program Committee at the Conferences ISCAS 1991–1995, ISCAS 2006, Power Electronics Specialist Conference (PESC) 1992–1995, a Track Chairman at the ISCAS 1996, ISCAS 1999–2005, the Co-Chairman of the Special Session’s Committee at the ISCAS 1997, a member of the Technical Committee and Ssession Chair at the PESC 2006–2008, international program committee member of the International Association of Science and Technology for Development (IASTED) 2004–2008, international advisory committee member of the IEEE Conference on Industrial Electronics and Applications (ICIEA) 2006, ICIEA 2007, the Co-Chairman of the Tutorial Committee at the ISCAS 2006, and the Designed Co-Chair, Special Session Committee at the ISCAS 2010, Paris. He was a Guest Editor of the special issues of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I (August 1997 and August 2003) and a special issue on the Power Electronics of Journal of Circuits, System and Computers (August 2003).
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 9, SEPTEMBER 2009
Chi K. Tse (M’90–SM’97–F’06) received the B.Eng. (Hons.) (with first class honors) degree in electrical engineering and the Ph.D. degree from the University of Melbourne, Melbourne, Vic., Australia, in 1987 and 1991, respectively. He is currently a Chair Professor and the Head of the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong. His current research interests include power electronics, complex networks, and nonlinear systems. He is the author of the Linear Circuit Analysis (London, U.K.: Addison-Wesley 1998) and the Complex Behavior of Switching Power Converters (Boca Raton, FL: CRC Press, 2003), the coauthor of the Chaos-Based Digital Communication Systems (Heidelberg, Germany: Springer-Verlag, 2003) and the Chaotic Signal Reconstruction with Applications to Chaos-Based Communications (Singapore: World Scientific, 2007). He is the co-holder of a US patent and two pending patents. Prof. Tse was a recipient of the 1987 L. R. East Prize Award from the Institution of Engineers, Australia, the IEEE TRANSACTIONS ON POWER ELECTRONICS
Prize Paper Award in 2001, and the International Journal of Circuit Theory and Applications Best Paper Award in 2003. He was also a recipient of the 2007 Distinguished International Research Fellowship from the University of Calgary, Canada. While with the Hong Kong Polytechnic University, he received twice the President’s Award for Achievement in Research, the Faculty’s Best Researcher Award, the Research Grant Achievement Award, and a few other teaching awards. From 1999 to 2001, he was an Associate Editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART I—FUNDAMENTAL THEORY AND APPLICATIONS, and since 1999, he has been an Associate Editor for the IEEE TRANSACTIONS ON POWER ELECTRONICS. In 2005, he was the IEEE Distinguished Lecturer. He is the Editor-in-Chief of the IEEE Circuits and Systems Society Newsletter, an Associate Editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART I—REGULAR PAPERS, the International Journal of Systems Science, the IEEE CIRCUITS AND SYSTEMS MAGAZINE, and the International Journal of Circuit Theory and Applications, and a Guest Editor of a few other journals.
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