﻿ Modeling, Steady-State Analysis of a SEPIC dc-dc Converter Based on Switching Function and Harmonic Balance Technique Journal of Power and Energy Engineering, 2014, 2, 704-711 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jpee http://dx.doi.org/10.4236/jpee.2014.24094 How to cite this paper: Taiwo, A.S. and Oricha, J.Y. (2014) Modeling, Steady-State Analysis of a SEPIC dc-dc Converter Based on Switching Function and Harmonic Balance Technique. Journal of Power and Energy Engineering, 2, 704-711. http://dx.doi.org/10.4236/jpee.2014.24094 Modeling, Steady-State Analysis of a SEPIC dc-dc Converter Based on Switching Function and Harmonic Balance Technique Ajayi Samuel Taiwo, Joseph Yakubu Oricha Electrical and Computer Engineering Department, Ahmadu Bello University, Zaria, Nigeria Email: tz4dabest@gmail.com, okaitojyo@yahoo.co.uk Received November 2013 Abstract The paper presents modeling approach of a Single Ended Primary Inductance Converter (SEPIC) system. The complete model derivation of the SEPIC converter system has been presented in dif-ferent modes of operation. Steady state and small signal analysis was carried out on the converter dynamic equations using the method of Harmonic balance Technique. The steady state variables and their respective ripple quantities obtained were plotted against duty ratio D. The results ob-tained for a supply input voltage of volts60 to the converter at a duty ratio of 0.8D=, compares well with simulation results. Keywords SEPIC Conve rte r; Switching Function; Harmonic Balance Technique 1. Introduction Fourth order converter had made applications of power possible where the demand for such requires less input voltage and high output voltage. Zeta, Cuk and SEPIC converter are examples of these. These converters have the ability to either buck or boost the voltage applied to their inputs depending on their applications. Analysis of these converters using averaging technique and waveform approach for transient and steady state study is very tedious and takes much computational time because of two storage element inclusion to the circuit. Solar based systems find interesting application when used with these converters to furnish the load with power. The SEPIC converter system as an example could be integrated with rural lightening systems, solar water pumping, Tele-communication industries and electric vehicle charging systems due to it buck-boost abilities. Attempts have been made by authors in , to use SEPIC converter to determine the I-V characteristics of solar generators. The performance of a dc water pump set driven by a PV source through a boost converter is reported in . Studies and analysis of a buck converter large signal average model around an operating point with constant power load was reported in . In  a novel technique for selecting passive components for the power stage of fourth-order dc-dc converter for an optimize system frequency response is reported. The authors’ in  reported that dc-dc response characteristics and stability determination using state space averaging tech- A. S. Taiwo, J. Y. Oricha 705 nique are one key point for accurate performance of a dc-dc converter feeding a load with power. In   a model and steady state analysis of their respective converters for different power source were set forth for dif-ferent operating condition of the load. In this paper a switching function based technique is presented. The derivatives are used to study the large signal and small signal dynamic behavior of the SEPIC converter on a large scale computer simulation in conti-nuous current mode (CCM). In view of these, the resulting non-linear equation obtained due to this switching function approach coupled with applying harmonic balance technique help to produce dynamic equations which predict the average and ripple quantities respectively. This paper is organized as follows: section II is the studied SEPIC converter it’s model derivations in CCM together with the harmonic balance technique, section III present Fourier series of the switching functions. Section IV is the simulation and steady state results. Section V is the conclusion. 2. Studied Converter Figure 1, Shows the circuit diagram of the SEPIC converter used for this paper, it consist of an input capacitor0C, which serves as a voltage bank to the input of the SEPIC converter. Two inductors 1L and 2L with each parasitic resistance 1Lr and 2Lr. Two capacitors 1C and 2C with each associate equivalent series parasitic resistance 1Cr and 2Cr. One active switch and a passive switch. For continuous conduction modes (CCM), there are two states of the switch within the switching period T. 1S could be a Mosfet, power bipolar junction transistor (BJT), Insulated gate bipolar transistor (IGBT), or any high power switching device which is switched on by a pulse width modulation signal (PWM) generated from a control module. During the first switching in-terval 1S is on for a period defined by [ ]0,t DTε, while 2S is off. During the next interval 1S is off and 2S is on for a period within the interval [ ],tDT Tε. 2.1. Model Derivation Equation (1) Gives the equation that describes two modes of operations of the SEPIC converter. Note; dpdt= 22CLCLrRrRϕ=+, 2LCLRrRβ=+; throughout this paper. ( )( )( )( )11112 1122122112 212 1212221121212221 2220221 2 1 LLLLCLC CLC LLLCL LCCL LC LLCCLC LLL pivirSirivvLpiS virSirSiiSvC pviSSiC pvSiivrRvvSi iϕ ϕβϕβββϕ−−×++×++−−−+×−×−++ ×−=====++ +× (1) Current and voltages of the transistor and diode are; Figure 1. SEPIC converter. C0rL1L1S1 ITrC1C1rL2L2rC2VC2RLV0IdC2VC1IL2-++-IC2IIL1IC1V+-+ -S2 A. S. Taiwo, J. Y. Oricha 706 ( )( )[ ]11 22 1122121 221 2121T LLTCLLC Cd LLdLCC Ci SiivSriiv viSii Sv Sirvvϕ ϕββ++−−+= ==+−=− (2) 2.2. Harmonic Balance Technique The technique unlike transient analysis assumes that the circuit steady state consist of a sum of sinusoids that best describe the differential equations. The method will involve imposing on Equation s (1) and (2) values, with average values together with their respective a.c variations, (ripple quantities). From these analysis, the steady state response solution of the system can be obtained and their respective ripple quantities. These equations are used to predict the performance of the system at steady state. ( )( )( )( )()( )( )( )( )1 10112 20211 10112 20210101010101 sssssssssjLL LjLL LjCC CjCCCjTT Tjdd djTT Tjdd djii iiiRe ieiiRe ievvRevevvRe veiiReieiiReievvRe vevVRe veSdRe d evvθθθθθθθθθ= += += += +== += += += += + (3) whe r e ; v, 10Cv, 20Cv, 0Tv, 0dv, 10Li, 20Li, 0di, 0Ti, 0id are the average (dc) values of supply voltage, ca-pacitor voltage 1C, capacitor voltage 2C, transistor voltage T, diode voltage D, inductor current 1L, in-ductor current 2L, transistor current, diode current and switching function d. Moreso, 11Cv, 21Cv, 1Tv, 1dv, 11Li, 21Li, 1Ti, 1di, 1id are complex ripple harmonic peaks of capacitor voltage 1C, capacitor voltage 2C, inductor current 1L, inductor current 2L, transistor Current, diode current and switching function d. Where I = 1, 2, 3···, ,?sstθω= whe r e sω angular switching frequency in radians/seconds and t is time in seconds. For continuous conduc-tion mode (CCM), then the switching of the converter is conforms to this constraint 211SS= −. The Equatio ns (1) and (2) have been used for simulation of the model. Putting Equation (3) into Equations (1) and (2) and con-sequently applying harmonic balance technique separating the steady state and ripple quantities, the dynamic equations for the average quantities and their ripple quantities counterpart are derived. 3. Fourier Series of the Switching Function The switching function applied to Eq uations (1) and (2) presents some difficulties in getting analytical close- form solutions. In solving this problem Fourier analysis is applied to get rid of these discontinuities. The expres-sion for the average (dc) components of the switching function can be found for this analysis. The function so derived is used for the derivation of the dynamic equation for the average and ripple quantities using harmonic balance Technique. The technique adopted is an effective method for studying the steady state of the system; which encompasses average, ripple quantities and could be use to study the stability of the converter which is not reported in this paper. A. S. Taiwo, J. Y. Oricha 707 S = 1, for [ ]0,t DTε and S = 0, for [ ],tDT Tε The Fourier expression of the periodic sequence of the pulse can be represented by a trigonometric series represented of the form . ( )( )01sinn snnftaDn tω∞== ++∅∑ (4) whe r e: ( )()( )( )( )00221001 tan2cos ?2sin ?Tn nnnnTnsTnsaftdtTD ababaftnt dtTbftnt dtTωω−==+∅= ==∫∫∫ (5) T is the period, represented as 1sfT= where sf denote the switching frequency. sω is the radian frequency define by 22ssfTπωπ= = For 1n= The average value and the ripple quantities can be quickly found as a decided advantage for the value of 1. For this case, ( )100t DTft DTt T≤≤=≤≤ (6) ( )( )( )( )( )( )( )( )( )( )( )00 101011012211102cos ?1sin 22sin ?11cos 21sin 21cos 2DTsDTsa DTDTad Daftnt dtTaDbftnt dtTbDdD Dωππωπππππ= −== =====−= +−∫∫ (7) The results of Equation (7) put in to Equation (3) and substituted into Equation s (1) and (2), are used for steady state average quantity calculations and ripple quantity calculations which are as presented in the next sec-tion. 4. Simulation and Steady-State Results Discussions Table 1 shows the parameters of the SEPIC converter used for the computer simulation. Steady state computer A. S. Taiwo, J. Y. Oricha 708 simulation results for of the inductor currents, capacitor voltages show in Figure s 2 and 3. Voltages and currents impressed on the transistor and diode shown in Figures 4 and 5. The simulation and steady state results are ob-tained at duty ratio of 0.8. The steady state results plots against the duty ratio D, is shown in Figure s 6 and 7. Also results for ripple quantities are shown in Figures 8 and 9. By comparing the simulation and calculated re-sults there is a good correlation between them Figures 2-5. Shows simulation results of the converter variables operating at duty cycle 0.8 the graph shows their simulation steady state results. Figure s 6-9 shows the result obtained after the application of harmonic balance technique to the converter model equations. A close observa-tion to the graph of Figure 6(f) shows that as the duty ratio varies between 0 and 0.9 the efficiency η falls slightly. But as the duty ratio increases the efficiency falls significantly. In that note It can also be noted that av-erage value of the simulation graph of Figure 3(d) at a duty ratio of 0.8 are closed to that obtained from har-monic balance model at steady state at that particular point as shown in Figure 6. The graph of Figure 9 at a particular duty ratio give one the choice of selecting the voltage withstand capabilities of switching devices used. Figure 2. Simulation: state variables at steady state for 0.8D=. (a) 1Li; (b) 2Li. Figure 3. Simulation: state variables at steady state for 0.8D= (c) 1Cv; (d) 2Cv. 0.499 0.4991 0.4992 0.4993 0.4994 0.4995 0.4996 0.4997 0.4998 0.4999 0.5859095100105110(a)IL1 (A)0.499 0.4991 0.4992 0.4993 0.4994 0.4995 0.4996 0.4997 0.4998 0.4999 0.51520253035(b)Time (s)IL2 (A)0.4990.4991 0.4992 0.4993 0.4994 0.49950.4996 0.4997 0.4998 0.49990.552545658606264(c)VC1 (V)0.4990.4991 0.4992 0.4993 0.4994 0.49950.4996 0.4997 0.4998 0.49990.5235236237238239240241(d) Time (s)VC2 (V) A. S. Taiwo, J. Y. Oricha 709 Table 1. SEPIC converter parameters. voltage supply 60 V Inductance 250 µH uctanceInd 250 µH Capacitance 300 µF Capacitance 470 µF Rated load 10 Ω Duty ratio 0.8 switching Frequency 10 KHZ Figure 4 . Simulation: transistor current and voltage at steady state for 0.8D=. (e) Ti; (f) Tv. Figure 5 . Simulation: diode current and voltage at steady state for 0.8D=. (g) di; (h) dv. 0.4990.4991 0.4992 0.4993 0.4994 0.49950.4996 0.4997 0.4998 0.4999 0.5-50050100150(e)IT (A)0.499 0.4991 0.4992 0.4993 0.4994 0.4995 0.4996 0.4997 0.4998 0.4999 0.5-50050100150200250300350(f)Time (s)VT (V)0.499 0.4991 0.4992 0.4993 0.4994 0.4995 0.4996 0.4997 0.4998 0.4999 0.5-50050100150(g)Id (A)0.499 0.4991 0.4992 0.4993 0.4994 0.4995 0.4996 0.4997 0.4998 0.4999 0.5-300-200-1000100(h)Time (s)Vd (V) A. S. Taiwo, J. Y. Oricha 710 Figure 6. Steady state average quantities (state variables), (a) 10Li; (b) 20Li; (c) 10Cv; (d) 20Cv; (e) 00v; (f) η. Figure 7. Steady state average quantities (devices current and voltage), (a) 0Ti; (b) 0Tv; (c) 0di; (d) 0dv. Figure 8 . Steady state average ripple quantities (state variables), (a) 11Li; (b) 21Li; (c) 11Cv; (d) 21Cv; (e) 01v. 00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.910200400600 Input inductor current vs duty ratioiL10 (A)D(a)00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.91050100150 Output inductor current vs duty ratioiL20 (A)D(b) 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91050100 Intermediate Capacitor voltage vs duty ratiovC10 (V)D(c)00.1 0.2 0.30.4 0.5 0.6 0.7 0.8 0.9 1050010001500 Output Capacitor voltage vs duty ratiovC20 (V)D(d) 00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.91050010001500 Output voltage vs duty ratiov00 (V)D(e)00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.91050100Efficiency vs duty ratioηD(f)00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10100200300400500600Transistor T1 Current vs duty ratioiT0 (A)D(a)00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1020406080100120140Diode d1 Current vs duty ratioid0 (A)D(c )00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1010203040506070Transistor T1 Voltage vs duty ratiovT0 (V)D(b)00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1400-1200-1000-800-600-400-2000Diode d1 voltage vs duty ratiovd0 (V)D(d)00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.910510 Input inductor current ripple vs duty ratioiL11 (A)D(a)00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.910510 Output inductor current ripple vs duty ratioiL21 (A)D(b)00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.91051015 Intermediate Capacitor ripple voltage vs duty ratiovC11 (V)D(c )00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.910510 Output Capacitor ripple voltage vs duty ratiovC21 (V)D(d)00.1 0.2 0.30.4 0.5 0.60.7 0.8 0.910510 Output ripple voltage vs duty ratiov01 (V)D(e) A. S. Taiwo, J. Y. Oricha 711 Figure 9 . Steady state ripple quantities (devices current and voltage), (a) 1Ti; (b) 1Tv; (c) 1di; (d) 1dv. 5. Conclusion In this paper, the steady state and the dynamic behavior of the SEPIC converter are studied. Based on the graphs obtained, it can be seen that the results from the simulation and steady state results are in agreement. The har-monic balance technique simplifies the analysis for the non-linear equations of the state variables and devices. The steady state average and ripple quantity calculation was obtained by varying the duty ratio from 0 to 1. The advantage of the method is that switching stress withstand capability on the devices due to switching action can be predicted. References  Aranda, E.D., Galan, J. A.G., de Cardona, M.S. an d Marquez, J.M.A. (2009) Measuring the I-V Curve of PV Genera-tors. Industrial Electronic Magazine, 3, 4-14. http://dx.doi.org/10.1109/MIE.2009.933882  Akbaba, M. and Akbaba, M.C. 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