I. INTRODUCTION
The establishment of an accurate model for a DC–DC converter is the key step in power electronics analysis and design. A few suitable methods have been proposed in open literature [1]– [3]. For example, to investigate nonlinear phenomena such as chaos, period-doubling bifurcation, border collisions, and Hopf bifurcation in a pulse-width modulated-controlled DC–DC converter, a discrete map of the DC–DC converter is established [1]. The main idea of this discrete map is to solve the differential equations in each operation state of the DC–DC converter and then sample and collect the values of the circuit variables at a specific instant. According to the characteristics of DC–DC converters (i.e., the switching frequency is higher than the inherent frequency), the averaged method has been proposed to model DC–DC converters to investigate dynamical behavior in low frequency regions. The averaged method includes the state space averaged method [2] and circuit-averaging technique [3]. The averaged method is popular in the field of DC–DC converters because this method can be used in the complex frequency domain [4], [5]. However, all of the above results are obtained under the condition that the real inductor and capacitor in the DC–DC converter are considered integer order and are described by integer calculus.
Given the recent developments in integer calculus and thelarge number of discoveries on the fractal dimensions of nature and science/technology fields, fractional calculus and its corresponding applications have attracted considerable research interest and many excellent results have been obtained [6]– [16]. Open literature shows that the fractional order model, which is derived by using fractional calculus, is more accurate than the integer order model, which is derived by using integer calculus, in describing the real dynamical behavior of systems. For example, Reyes-Melo et al. [6] indicated that the dielectric relaxation phenomena in polymeric materials can be clearly described by using a fractional order model. Meral et al. [7] confirmed that the essential properties of viscoelastic materials can be precisely reflected by using a fractional order model. Recent research on the modeling of real inductors and capacitors shows that the nature of these two circuit elements are fractional orders and that these elements should be modeled by using fractional calculus [8]– [16]. For example, for the real inductor, Westerlund [8] indicated that the real inductor is fractional order in nature, that is, the magnetic process should be described by using a fractional order model to satisfy its causality. Westerlund [8] also measured the fractional order inductor of the air core coil, which is equal to 0.97. Tenreiro Machado et al. [9] indicated that a fractional order inductor with different orders can be developed based on the skin effect. For the real capacitor, Jonscher [10] and Bohannan [11] showed that an integer order capacitor cannot exist in nature because the form of its impedance will violate causality. Westerlund et al. [12] experimentally measured the order of the fractional order capacitor under different dielectrics and finding that the fractional order of the capacitor is 0.9776 for polyvinylidenefluoride, 0.9821 for metalized paper, and 0.9978 for polycarbonate. Jesus et al. [13] developed a fractional order capacitor with different orders by choosing different fractal structures, such as the curve of Koch, the carpet of Sierpinski, and the curves of Hilbert. Petráš [14] applied the fractional order inductor and fractional order capacitor to design a circuit that realizes the fractional Chua’s circuit successfully; the experimental results are in good agreement with the simulations. Thus, the real inductor and capacitor should be modeled by using fractional calculus to describe their real electrical characteristics.
A question naturally occurs to researchers and engineers who use integer calculus to derive the integer order model and describe the dynamical behavior of a Buck converter: why are the results of the derived integer order model of the Buck converter in good agreement with the circuit simulations or experiments? The main reason is that the inductor and capacitor orders in the market are near to one; thus, the integer order model can be used to describe its dynamical behavior approximately. These approximations will provide the wrong results if the inductor and capacitor orders are slightly far away from one, for example, the different inductor orders in [9] and the capacitor orders at 0.59 and 0.42 in [13]. Thus, the Buck converter should be modeled by using fractional calculus. In other words, the real Buck converter should be called as the fractional order Buck converter. By using fractional calculus and the circuit-averaging technique, this study investigates the modeling and analysis of the fractional order Buck converter in discontinuous conduction mode (DCM) operations.
The rest of the paper is organized as follows. Section II discusses the circuit operations and averaged circuit model of the fractional order Buck converter in DCM operations. Sections III and IV presents the DC and small signal analysis, respectively. Section V shows the design of the realization forms for the fractional order inductor and fractional order capacitor, as well as the validation of the effectiveness of the theoretical analysis by PSIM circuit simulations. Finally, Section VI concludes.
II. CIRCUIT OPERATION AND AVERAGED CIRCUIT MODEL
The circuit schematic of the fractional order Buck converter and its typical time domain waveforms in DCM operations are shown in Figs. 1(a) and 1(b), respectively. The fractional order inductor is denoted by Lα, where α is its fractional order. Hence, two parameters describe the fractional order inductor, namely, inductance and inductor order α. The fractional order capacitor is denoted by Cβ, where β is its fractional order, and is described by two parameters, namely, capacitance and capacitor order β. The fractional order Buck converter contains the switch S and diode Di. vin is the input voltage, and R is the load.
Fig. 1.Fractional order Buck converter and its typical timedomain waveforms in DCM operation: (a) circuit schematic and (b) inductor current.
The expression of the fractional order inductor of the relationship between its voltage across (vL) and its current through (iL) in the time and complex frequency domains are described in [8, 17].
The expression of the fractional order capacitor of the relationship between its voltage across (vC) and its current through (iC) in the time and complex frequency domains are described in [8, 17].
Fig. 1(b) shows that the inductor current iL equals zero within d3T. So, three operation states exist in DCM operations: d1T, d2T, and d3T, which are different from the operation states of CCM operations. Thus, the above two points should be considered when using the circuit-averaging technique to model the fractional order Buck converter in DCM operations.
Assume that
Figs. 1(a) and 1(b) show that the switch S is ON and the diode Di is OFF within d1T. The voltage vL is equal to the input voltage vin minus the output voltage v0. Note that the input voltage vin and output voltage v0 can be replaced by their averaged value because the amplitude of their ripple is significantly small [3]. Thus, we obtain the following:
According to the fractional calculus [17], the peak value of the current iL can be calculated as follows:
Therefore, the averaged value of the inductor current iL can be derived as follows:
The current through switch S is equal to the current iL within d1T and is zero within d2T and d3T. Thus, the averaged value of the current through switch S is expressed as follows:
From Eqs. (5), (6), and (7), the inductor order has a significant effect on currents iLp,
The voltage across diode Di is equal to the input voltage vin within d1T, is zero within d2T, and is v0 within d3T. Thus the averaged voltage across diode Di is expressed as follows:
The voltage across the inductor equals vin − v0 within d1T, is −v0 within d2T, and is zero within d3T. Thus, the averaged voltage across the inductor is expressed as follows:
However, according to the voltage second balance, the averaged value of the voltage across the inductor equals zero, that is,
By integrating Eq. (10) into Eqs. (6) and (8), the expressions concerning
Thus, the averaged circuit model for the fractional order Buck converter in DCM operations can be described (Fig. 2).
Fig. 2.Averaged circuit model for the fractional order Buck converter in DCM operation.
By using Eq. (3) into Eqs. (6) and (7) and omitting high-order small signal terms, Eqs. (6) and (7) can be changed into the following formulas, respectively:
The DC and small signal analysis of the fractional order Buck converter in DCM operations are provided in the following two sections. The circuit parameters used in this study are as follows: vin = 6 V, Cβ = 470 μF, Lα = 100 μH, D1 = 0.2, R = 20 Ω, T = 10 μs, 0 < α < 1, and 0 < β < 1.
III. DC ANALYSIS
According to the averaged circuit model for the fractional order Buck converter in DCM operations (Section II), the corresponding DC circuit model can be obtained by substituting the averaged values in Fig. 2 into their corresponding DC values. According to Caputo’s fractional derivate [17], the branch circuit of the fractional order capacitor is open because its current is zero. Thus, the DC equivalent circuit model for the fractional order Buck converter in DCM operations can be obtained (Fig. 3).
Fig. 3.DC equivalent circuit model for the fractional order Buck converter in DCM operations.
The DC value of iL can be easily derived by removing the small AC variations from Eq. (13), that is:
The following inequality can be derived from the condition for the Buck converter in DCM operations [18], that is, IL ≥ V0/R:
Fig. 3 shows that IL equals V0/R. Thus, the following equation is obtained:
Therefore, the voltage ratio M can be derived as follows:
Accordingly, the respective expressions for V0 and IL can be rewritten as follows:
Thus, V0, IL, M, and the DCM condition are all affected by the inductor order because they all include this item. Fig. 4 shows the critical resistors about the boundary condition under different inductances, with α = 0.8 and α = 0.7. The DCM region increases with decreasing inductor order. Thus, the fractional order Buck converter easily operates in DCM operations with decreasing inductor order.
Fig. 4.Calculated results for the critical resistors under different inductances with α = 0.8 and α = 0.7.
Fig. 5 shows the calculated voltage ratio M under different inductances with α = 0.8 and α = 0.7. Under the same inductor order α, the voltage ratio M decreases with increasing inductance of the fractional order inductor. However, under the same inductance, the voltage ratio M increases with decreasing inductor order. If α = 1 (the inductor is integer order), the expressions for IL, V0, and M and the boundary condition for DCM operations are the same as in [3].
Fig. 5.Calculated results for the voltage ratio M under different inductances with α = 0.8 and α = 0.7.
IV. SMALL SIGNAL ANALYSIS
The small signal equivalent circuit model for the fractional order Buck converter can be obtained by substituting the average values in Fig. 2 into the corresponding small signal AC items in the complex frequency domain (Fig. 6).
According to circuit theory and Fig. 6, the following equation can be derived:
Fig. 6.Small signal equivalent circuit model for the fractional order Buck converter in DCM operation.
The small signal of can be easily derived by removing the DC values from Eqs. (13) and (14), respectively:
By using Eqs. (22) into (21) and simplifying the equation, the expression for can be derived as follows:
Thus, the transfer function from the input voltage to the output voltage can be derived by setting the duty cycle variation to zero.
The transfer function from the duty cycle to the output voltage can be derived by setting the input voltage variation to zero.
The input impedance of the fractional order Buck converter in DCM operations can be derived by calculating the input voltage variations over the current through the switch S variations at a duty cycle variation of zero.
The output impedance of the fractional order Buck converter in DCM operations can be derived by setting both the input voltage variation and duty cycle variation to zero. Thereafter, the definition of the output impedance is used to derive the formulas.
The above four transfer functions contain the inductor and capacitor orders. Thus, all functions are significantly affected by these two parameters. If α, β = 1 (the inductor and capacitor are integer order), the above four transfer functions are the same as in [3]. Only the Bode diagrams of Gvv(s) and Gvd(s) under different inductor orders and capacitor orders are discussed in this study.
By using α, β = 0.8 and α, β = 0.7 and the other parameter values shown in Section II, the corresponding Bode diagrams of Gvv(s) and Gvd(s) are plotted in Fig. 7 according to the definition of the Bode diagram for the fractional order system [19]. The magnitudes and phases in these two cases (α, β = 0.8 and α, β = 0.7) are very different.
Fig. 7.Bode diagrams of Gvv(s) and Gvd(s) under different inductor orders and capacitor’s orders. (a) Gvv(s) and (b) Gvd(s).
V. PSIM SIMULATION FOR CONFIRMATION
As mentioned in the Introduction, all real inductors and capacitors are considered integer orders. Thus, no established circuit models exist for the fractional order inductor and fractional order capacitor in current popular circuit simulation software, such as PSIM. Thus, to confirm the correctness of the above theoretical analysis, building a circuit model for the fractional order inductor and fractional order capacitor is necessary to construct the fractional order Buck converter. The chain fractance [17] and Oustaloup’s approximation [20] are used to establish the approximate circuit models for these two fractional order circuit elements. PSIM software, which is widely used in simulating power electronics and motor drives [21– 22], is used to simulate the fractional order Buck converter.
The dynamical behavior of the fractional order inductor can be described by using the approximate circuit model in Fig. 8. When we need Lα = 100 μH with α = 0.8, the resistors and inductors in Fig. 8 can be calculated by making the formulas from the input impedance of the approximate circuit model for the fractional order inductor equal to the formulas from Oustaloup’s approximation. The calculated results are as follows: R11 = 238.5 Ω, R12 = 11.4 Ω, R13 = 1.1 Ω, R14 = 118 mΩ, R15 = 12 mΩ, R16 = 1.27 mΩ, R17 = 130 μΩ, R18 = 13.6 μΩ, R19 = 1.4 μΩ, R110 = 0.17 μΩ, L11 = 3.2 μH, L12 = 2.56 μH, L13 = 4.4 μH, L14 = 7.72 μH, L15 = 13.6 μH, L16 = 24 μH, L17 = 42.3 μH, L18 = 74.5 μH, and L19 = 131 μH.
Fig. 8.Fractional order inductor and its approximate circuit model.
When Lα = 100 μH with α = 0.7, the calculated results for the resistors and inductors in Fig. 8 are as follows: R11 = 36. 2Ω, R12 = 3.03 Ω, R13 = 0.4 Ω, R14 = 56 mΩ, R15 = 7.7 mW, R16 = 1.06 mΩ, R17 = 146 μΩ, R18 = 20 μΩ, R19 = 2.76 μΩ, R110 = 0.44 μΩ, L11 = 0.555 μH, L12 = 0.8 μH, L13 = 1.8 μH, L14 = 4.2 μH, L15 = 9.9 μH, L16 = 23.1 μH, L17 = 54.2 μH, L18 = 127 μH, and L19 = 296 μH.
Only the Bode diagrams of Lαsα for Lα = 100 μH with α = 0.8 from the theoretical analysis and PSIM simulations are plotted in Fig. 9. The theoretical analysis is in good agreement with the PSIM simulations. Therefore, substituting the corresponding fractional order inductor with this approximate circuit model is reasonable.
Fig. 9.Bode diagrams of Lαsα for Lα = 100 μH with α = 0.8 based on PSIM simulations and theoretical analysis.
The dynamical behavior of the fractional order capacitor can be described by using the approximate circuit model in Fig. 10. When we need Cβ = 470 μF with β = 0.8, the resistors and capacitors in Fig. 10 can be calculated by making the formulas from the input impedance of the approximate circuit model for the fractional order capacitor equal to the formulas from Oustaloup’s approximation. The calculated results are as follows: R21 = 4.16 mΩ, R22 = 33 mΩ, R23 = 0.32 Ω, R24 = 3 Ω, R25 = 30 Ω, R26 = 289 Ω, R27 = 2794 Ω, R28 = 27 kΩ, R29 = 260 kΩ, R210 = 22 MΩ, C11 = 31 μF, C12 = 65.7 μF, C13 = 115 μF, C14 = 203 μF, C15 = 358 μF, C16 = 630 μF, C17 = 1 mF, C18 = 1.96 mF, C19 = 3.46 mF, and C110 = 2.63 mF.
Fig. 10.Fractional order capacitor and its approximate circuit model.
When we need Cβ = 470 μF with β = 0.7, the calculated results for the resistors and capacitors in Fig. 10 are as follows: R21 = 25.6 mΩ, R22 = 152 mΩ, R23 = 1.11 Ω, R24 = 8 Ω, R25 = 59 Ω, R26 = 427.6 Ω, R27 = 3109 Ω, R28 = 22.6 kΩ, R29 = 164.1 kΩ, R210 = 7.86 MΩ, C11 = 4.3 μF, C12 = 12.4 μF, C13 = 29 μF, C14 = 67.6 μF, C15 = 158 μF, C16 = 370 μF, C17 = 867 μF, C18 = 2 mF, C19 = 4.755 mF, and C110 = 6.35 mF.
Only the Bode diagrams of 1/(Cβsβ) for Cβ = 470 μF with β = 0.8 from the theoretical analysis and PSIM simulations are plotted in Fig. 11. The theoretical analysis is in good agreement with the PSIM simulations. Therefore, substituting the corresponding fractional order capacitor with this approximate circuit model is reasonable.
Fig. 11.Bode diagrams of 1/(Cβsβ) for Cβ = 470 μF with β = 0.8 from the PSIM simulations and theoretical analysis.
The circuit model for the fractional order Buck converter can be constructed by using the above approximate circuit models. The corresponding Bode diagrams of Gvv(s) and Gvd(s) from the theoretical analysis and PSIM simulations under α, β = 0.8 and α, β = 0.7 can be obtained (Fig. 12). Note that the Bode diagrams of Gvv(s) and Gvd(s) from the PSIM simulations are calculated from the original switch model of the fractional order Buck converter, and no average model is required. The results of the PSIM simulations are in basic agreement with the theoretical analysis, except for some discrepancy caused by the approximate circuit models for the fractional order inductor (Fig. 8) and fractional order capacitor (Fig. 10).
Fig. 12.Bode diagrams of Gvv(s) and Gvd(s) from the PSIM simulations and theoretical analysis: (a) Gvv(s), (b) Gvd(s).
VI. CONCLUSIONS
By using DC and small signal analyses for the fractional order Buck converter in DCM operations, the DC values IL and V0, voltage ratio M, and boundary condition for DCM operations are found to contain the inductor order, and all derived transfer functions include the inductor and capacitor orders. The theoretical calculation results for the boundary condition for DCM operations and voltage ratio M show that the inductor order has a major effect on both of them; that is, the DCM operation region increases with decreasing inductor order, the voltage ratio M increases with decreasing inductor order under the same inductance. The theoretical analysis and PSIM simulations for the Bode diagrams of Gvv(s) and Gvd(s) show that both inductor order and capacitor order play an important role in the dynamical behavior of the fractional Buck converter in DCM operations. Therefore, if we directly establish the integer order transfer functions instead of the fractional order transfer functions in describing the fractional order Buck converter in DCM operations, the obtained results will be incorrect. In other words, the fractional order transfer functions of the fractional Buck converter in DCM operations must be established to describe its real dynamical behavior.
References
- C. K. Tse, Complex behavior of switching power converters, CRC Press LLC, 2004.
- R. W. Erickson and D. Maksimovic, Fundamentals of power electronics, 2nd Edition, Kluwer Academic Publishers, 2001.
- W. P. Zhang, Modeling and control of switching power converter, China Electric Power Press, 2005.
- R. Li, T. O'Brien, J. Lee, J. Beecroft, and K. Hwang, "Analysis of parameter effects on the small-signal dynamics of Buck converters with average current mode control," Journal of Power Electronics, Vol. 12, No. 3, pp. 399-409, May 2012. https://doi.org/10.6113/JPE.2012.12.3.399
- Q. M. Luo, S. B. Zhi, W. G. Lu, and L. W. Zhou, "Direct current control method based on one cycle controller for double-frequency Buck converters," Journal of Power Electronics, Vol. 12, No. 3, pp. 410-417, May 2012. https://doi.org/10.6113/JPE.2012.12.3.410
- E. Reyes-Melo, J. Martinez-Vega, C. Guerrero-Salazar, and U. Ortiz-Mendz, "Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials," J. Appl. Polym. Sci., Vol. 98, No. 2, pp. 923-935, Jul. 2005. https://doi.org/10.1002/app.22057
- F. C. Meral, T. J. Royston, and R. Magin, "Fractional calculus in viscoelasticity: an experimental study," Commun. Nonlinear Sci. Numer. Simulat., Vol. 15, No. 4, pp. 939-945, Apr. 2010. https://doi.org/10.1016/j.cnsns.2009.05.004
- S. Westerlund, Dead matter has memory!, Kalmar, Sweden: Causal Consulting, 2002.
- J. A. Tenreiro Machado and A. M. S. F. Galhano, "Fractional order inductive phenomena based on the skin effect," Nonlinear Dyn., Vol. 68, No. 1-2, pp. 107-115, Apr. 2012. https://doi.org/10.1007/s11071-011-0207-z
- A. K. Jonscher, Dielectric relaxation in solids, London: Chelsea Dielectric Press, 1983.
- G. W. Bohannan, "Analog realization of a fractional control element-revisited," in Proc. of the 41st IEEE CDC, Tutorial Workshop 2: Fractional Calculus Applications in Automatic Control and Robotics, 2002.
- S. Westerlund and L. Ekstam, "Capacitor theory," IEEE Trans. Dielectr. Electr. Insul., Vol. 1, No. 5, pp.826-839, Oct. 1994. https://doi.org/10.1109/94.326654
- I. S. Jesus and J. A. Tenreiro Machado, "Development of fractional order capacitors based on electrolyte processes," Nonlinear Dyn., Vol. 56, No. 1-2, pp. 45-55, Apr. 2009. https://doi.org/10.1007/s11071-008-9377-8
- I. Petras, "A note on the fractional-order Chua's system," Chaos, Soliton. Fract., Vol. 38, No. 1, pp. 140-147, Oct. 2008. https://doi.org/10.1016/j.chaos.2006.10.054
- R. Martin, J. J. Quintana, A. Ramos, and I. D. L. Nuez, "Modeling of electrochemical double layer capacitors by means of fractional impedance," J. Comput. Nonlinear Dyn., Vol. 3, No. 021303, pp. 1-6, Feb. 2008.
- N. Bertrand, J. Sabatier, O. Briat, and J. Vinassa, "Fractional non-linear modelling of ultracapacitors," Commun. Nonlinear Sci. Numer. Simulat., Vol. 15, No. 5, pp. 1327-1337, May 2010. https://doi.org/10.1016/j.cnsns.2009.05.066
- I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
- X. Yang, Y.Q. Pei, and Z. A. Wang, Switching power supply technology, China Machine Press, 2004.
- D. Y. Xue and Y. Q. Chen, MATLAB Solutions to Mathematical Problems in Control, Tsinghua University Press, 2007.
- A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, "Frequency-band complex noninteger differentiator: characterization and synthesis," IEEE Trans. Circuits Syst.-I, Vol. 47, No. 1, pp. 25-39, Jan. 2000. https://doi.org/10.1109/81.817385
- B. Singh and G. D. Chaturvedi, "Analysis, design and development of a single switch flyback Buck-Boost AC-DC converter for low power battery charging applications," Journal of Power Electronics, Vol. 7, No. 4, pp. 318-327, Oct. 2007.
- M. Veerachary, "PSIM circuit-oriented simulator model for the nonlinear photovoltaic sources," IEEE Trans. Aerospace Electron. Syst., Vol. 42, No. 2, pp. 735-740, Apr. 2006. https://doi.org/10.1109/TAES.2006.1642586
Cited by
- Controllability of fractional-order Chua’s circuit vol.24, pp.3, 2015, https://doi.org/10.1088/1674-1056/24/3/030203
- Modeling and analysis of fractional order DC-DC converter 2018, https://doi.org/10.1016/j.isatra.2017.06.024
- The fractional-order state-space averaging modeling of the Buck–Boost DC/DC converter in discontinuous conduction mode and the performance analysis vol.79, pp.1, 2015, https://doi.org/10.1007/s11071-014-1695-4
- Fractional-Order 2 × n RLCCircuit Network vol.24, pp.09, 2015, https://doi.org/10.1142/S021812661550142X
- Systematic Dynamic Modeling of an Integrated Single-stage Power Converter vol.10, pp.6, 2015, https://doi.org/10.5370/JEET.2015.10.6.2288
- A fractional-order multifunctional n-step honeycomb RLC circuit network vol.18, pp.8, 2017, https://doi.org/10.1631/FITEE.1601560
- Modeling and Characteristics Analysis for a Buck-Boost Converter in Pseudo-Continuous Conduction Mode Based on Fractional Calculus vol.2016, 2016, https://doi.org/10.1155/2016/6835910
- Attempt to generalize fractional-order electric elements to complex-order ones vol.26, pp.6, 2017, https://doi.org/10.1088/1674-1056/26/6/060503
- Indirect Output Voltage Control in Negative Output Elementary Super Lift Luo Converter Using PIC plus FLC in Discontinuous Conduction Mode vol.07, pp.11, 2016, https://doi.org/10.4236/cs.2016.711310
- Fractional-Order Modeling and Simulation of Magnetic Coupled Boost Converter in Continuous Conduction Mode vol.28, pp.05, 2018, https://doi.org/10.1142/S021812741850061X