Author name used in this publication: Chi K. TseAuthor name used in this publication: Kam C. Tam2005-2006 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Tam, KC

Tse, CKM

Wong, SC

IEEE Transactions on Circuits & Systems II Express Briefs

Author name used in this publication: Chi K. TseAuthor name used in this publication: Kam C. Tam2005-2006 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishedVoR allowe

PolyU Institutional Repository

690 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 10, OCTOBER 2005An Improved Wavelet Approach for FindingSteady-State Waveforms of Power ElectronicsCircuits Using Discrete ConvolutionKam C. Tam, Student Member, IEEE, Siu-Chung Wong, Member, IEEE, and Chi K. Tse, Senior Member, IEEEAbstract—Due to the switching action and the presence ofparasitics, waveforms arising from power electronics circuitsoften contain high-frequency ringings embedded in slowly varyingsegments. Such a feature is consistent with the localization prop-erty of wavelets which has previously been exploited for fastapproximations of steady-state waveforms. This paper proposesan improved and more robust approach for calculating the waveletcoefficients, exploiting the orthogonal property of the Chebyshevpolynomials. Simulation results demonstrate the effectiveness ofthe new algorithm.Index Terms—Chebyshev polynomials, power electronics cir-cuits, steady-state solutions, wavelet transforms.I. INTRODUCTIONSTEADY-STATE waveforms of power electronics circuitsare important practical consideration for engineers, as theyprovide complete information on the operation of the circuitsas well as the extent of current and voltage stresses on specificdevices [1]–[3]. Obtaining the steady-state waveforms are,however, usually time consuming and may suffer from numer-ical instabilities, especially for power electronics circuits whereslow and fast variations are present in different parts of thesame waveform [3]. When describing waveforms arising frompower electronics circuits, a frequently encountered problemis the fast variations near the switching instants. This usuallyappears as waveform ringings during the switching transitionsand is caused by the presence of high-frequency resonantloops formed by small parasitic capacitances and inductances.Thus, the resulting waveforms contain largely low-frequencycomponents with high-frequency details concentrated nearthe switching instants. Such a feature is consistent with thelocalization property of wavelets which are known in the signalprocessing literature to be highly suitable for describing fastchanging edges embedded in slowly varying backgrounds [4].It therefore appears that power electronics waveforms can beefficiently approximated by an appropriate set of wavelets.Some preliminary work on the analysis of power electronicswaveforms using wavelets was reported by Wernekinck, Valen-zuela and Anfossi [5]. Recently, a systematic algorithm for ap-Manuscript received April 27, 2004; revised October 29, 2004. This work wassupported by the Hong Kong Polytechnic University under Grant A-PD68 andby the Hong Kong Research Grants Council under the CERG Grant. This paperwas recommended by Associate Editor T. Ueta.The authors are with the Department of Electronic and InformationEngineering, Hong Kong Polytechnic University, Hong Kong (e-mail:encktse@polyu.edu.hk).Digital Object Identifier 10.1109/TCSII.2005.852167proximating steady-state waveforms arising from power elec-tronics circuits using wavelets has been reported by Liu, Tseand Wu [6]. The advantages of using wavelets for the anal-ysis of power electronics circuits have also been demonstrated.The original algorithm has employed a standard curve-fittingmethod, which uses the boundary conditions together with extrainterpolation points for solving the wavelet coefficients. The ac-curacy of the approximated solution is therefore dependent uponthe positions of the interpolation points relative to the actualwaveform to be found and limited by the wavelet levels used. Ithas been found that the solution is very sensitive to the choice ofinterpolation points for obtaining wavelet coefficients. A smallmismatch between the set of interpolation points with the wave-form to be approximated may lead to significant errors in theapproximated solution, reducing the robustness of the method.The aim of this paper is to tackle the limitations in thestandard curve-fitting method. We will provide a new techniquefor obtaining better approximated waveforms with the samenumber of wavelet levels.II. REVIEW OF ORTHOGONALITY IN CHEBYSHEV POLYNOMIALSOne important property of Chebyshev polynomials is thatthey are orthogonal in the sense that their inner product [7]–[9],defined by(1)has the following property:ififif .(2)It is always possible to convert a (continuous) orthogonalityrelationship, as defined in (1), into a discrete orthogonality rela-tionship simply by replacing the integral with a summation. Ingeneral, of course, the result is only approximately true. How-ever, where trigonometric functions or Chebyshev polynomialsare involved, there are many cases in which the discrete orthog-onality can be shown to hold exactly, e.g.,for (3)where are the zeros of . Here, we have(4)1057-7130/$20.00 © 2005 IEEEAuthorized licensed use limited to: IEEE Xplore. Downloaded on December 17, 2008 at 22:30 from IEEE Xplore.  Restrictions apply.TAM et al.: IMPROVED WAVELET APPROACH FOR FINDING STEADY-STATE WAVEFORMS 691satisfying the orthogonality condition, i.e.,ififif .(5)III. WAVELET APPROXIMATION OF POWER ELECTRONICSWAVEFORMS USING DISCRETE CONVOLUTIONIt has been shown [6] that wavelet approximation can beuseful for approximating steady-state solutions of nonlinearcircuits without solving systems of nonlinear equations. Inmost cases, power electronics circuits can be represented by atime-varying state-space equation(6)where is the -dim state vector, is an time-varyingmatrix, and is the input function. Specifically, we write............(7)...(8)In the steady state, the solution satisfiesfor (9)where is the period. For an appropriate translation and scaling,the boundary condition can be mapped to the closed interval[ 1, 1], i.e.,(10)Assume that the basic time-invariant approximation equationisfor (11)where is any wavelet basis of size ( beingthe wavelet level) constructed from Chebyshev polynomials,1is a coefficient vector of dimension, which is to be found.The wavelet transformed equation of (6) is(12)1The construction of wavelet basis has been discussed in detail in Liu et al.[6].where (see [6])............(13)Thus, (12) can be written generally as(14)where is a matrix, as given by the equa-tion shown at bottom of page, and is a -dimvector given by(15)Note that since the unknown is of dimension ,we need equations. Now, the boundary condition(10) provides equations, i.e.,for (16)In the original curve-fitting approach, equations areobtained by interpolating at distinct points in the closedinterval [ 1, 1], and the interpolation points are chosen fromthe zeros of Chebyshev polynomials of the second kind, i.e.,(17)It is obvious from (16) that one of the end points of eitheror must be excluded from the interpolationpoints, as will be discussed in Section IV. In contrast, our newapproach here takes advantage of the orthogonality in Cheby-shev polynomials. The more equations are obtained bytaking discrete convolution over the closed interval [ 1, 1] withfor . The number of points for discreteconvolution is therefore no longer limited by the wavelet levelin the calculation. Also, as discrete convolution is performedover individual wavelets, the relative weightings of the waveletbases required become irrelevant and the exact locations of thedata points are unimportant. Thus, this discrete convolution ap-proach is expected to give a more robust and more accurate so-lution. In discrete convolution, the approximate solution is(18)where(19)..............................Authorized licensed use limited to: IEEE Xplore. Downloaded on December 17, 2008 at 22:30 from IEEE Xplore.  Restrictions apply.692 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 10, OCTOBER 2005and , , , and are given by the equation at bottom ofpage, and.........with representing a new matrix whose elementsare the corresponding convolutions with . By solving (18),we obtain all the coefficients necessary for generating an ap-proximate solution for the steady-state system.IV. SIMULATIONS AND EVALUATIONSThe results are evaluated using the mean relative error (MRE)and mean absolute error (MAE), which are defined as(20)(21)where is the wavelet-approximated value and is theSPICE simulated result. The SPICE result, being generatedfrom exact time-domain simulation of the actual circuit atdevice level, can be used for comparison and evaluation.In discrete forms, MAE and MRE are simply given by(22)(23)where is the total number of points sampled along the in-terval [ 1, 1] for error calculation. In the following, we use uni-form sampling (i.e., equal spacing) with , includingboundary points.Fig. 1. Simple boost converter circuit.TABLE ICOMPONENT AND PARAMETER VALUES FOR THE BOOST CONVERTERExample 1: Simple Boost Converter: We consider a simpleboost converter, as shown in Fig. 1. Suppose the switch has a re-sistance of when it is turned on, and is practically open-cir-cuited when it is turned off. The diode has a forward voltagedrop of and an on-resistance of . The basic system equa-tion can be readily found as(24)where , and and are given by(25)(26)with defined byforforfor all .(27)The circuit parameters and component values used in the calcu-lations and simulations are shown in Table I.We compare the standard curve-fitting method and the pro-posed discrete convolution method. The approximated wave-forms for the inductor current is shown in Fig. 2. By inspection,the discrete convolution method almost always gives more accu-rate waveforms than the standard curve-fitting method. In orderto compare the results quantitatively, the values of the MREs areprovided in Table II. Finally, we note that the inductor currentwaveform has been very well approximated by using only 33............Authorized licensed use limited to: IEEE Xplore. Downloaded on December 17, 2008 at 22:30 from IEEE Xplore.  Restrictions apply.TAM et al.: IMPROVED WAVELET APPROACH FOR FINDING STEADY-STATE WAVEFORMS 693TABLE IICOMPARISON OF MREs FOR APPROXIMATING WAVEFORMS IN BOOST CONVERTER. DATA WITH DENOTES POOR REFERENCE PROVIDED BY SPICEFig. 2. Inductor current waveforms of boost converter. Solid line iswavelet-approximated waveform using discrete convolution, dotted line iswaveform from SPICE simulation and dot-dashed line is waveform usingcurve-fitting method, (a) using wavelets of level  1 to 4; (b) using wavelets oflevel  1 to 5; (c) using wavelets of level  1 to 6; and (d) using wavelets oflevel  1 to 7.wavelets of level up to 4 in the proposed method with extremelysmall MREs.It should be noted that the MRE generally decreases as morewavelet levels are used. Moreover, as evident from Table II, thetrend appears to revert when wavelets up to a certain level areused, apparently implying that there may be an optimal waveletlevel beyond which no further reduction in error can be ex-pected. However, upon close inspection, the apparent increasein MRE when wavelet levels up to 8 (for the proposed method)and 10 (for the curve fitting method) are used is actually due tothe limited accuracy of the SPICE reference waveforms [10]. Inother words, SPICE results have become less accurate and hencetheir use as reference have deceptively given a larger MRE.Thus, data corresponding to any higher wavelet levels are notmeaningful.Example 2: Flyback Converter With Ringings: We nowconsider the flyback converter shown in Fig. 3(a). This is amore realistic model as the parasitic capacitance across theswitch and leakage inductance of the transformer are deliber-ately included. The operation is as follows. When the switchis turned on, current flows through the magnetizing inductanceFig. 3. (a) Flyback converter model with transformer leakage inductance andparasitic capacitance. (b) On-time circuit models. (c) Off-time circuit models.and the leakage inductance , with the transformer sec-ondary opened and the diode not conducting. When the switchis turned off, the transformer secondary conducts through thediode, clamping the primary voltage (i.e., voltage across )to the output network (assuming a 1:1 turns ratio). Thus,discharges through the transformer primary, while the leakageand the parasitic capacitance form a damped resonantloop around the input voltage source. Figs. 3(b) and (c) show thedetailed circuit models for the on-time and off-time durations.The state equation of this converter is given by(28)where , and and are given by(29)(30)with defined earlier in (27) and the ’s and ’s being deriv-able from the circuit topologies.The parameters for simulation are listed in Table III. We havecompared the approximated waveforms of the leakage inductorcurrent for both discrete convolution and curve-fitting method.Fig. 4(a) and (b) shows the approximated waveforms using dis-crete convolution at two different wavelet levels. Fig. 4(c) and(d) shows the approximated waveforms using the curve-fittingapproach, excluding and , respectively.Authorized licensed use limited to: IEEE Xplore. Downloaded on December 17, 2008 at 22:30 from IEEE Xplore.  Restrictions apply.694 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 10, OCTOBER 2005TABLE IIICOMPONENT AND PARAMETER VALUES FOR FLYBACK CONVERTERFig. 4. Inductor waveforms of flyback converter. Dashed line is waveformfrom SPICE simulation. Solid line is approximated waveform. (a) Using discreteconvolution wavelets of level  1 to 5. (b) Using discrete convolution waveletsof level 1 to 7. (c) Using curve-fitting wavelets of level 1 to 5 excluding theinterpolation point  (=  1). (d) Using curve-fitting wavelets of level 1to 5 excluding the interpolation point  (= 1).As there is no systematic method to predict which exclusionwould produce better result, the original curve-fitting methodis less robust.TABLE IVCOMPARISON OF MAEs FOR APPROXIMATING LEAKAGE INDUCTOR CURRENTIN FLYBACK CONVERTER. DATA WITH DENOTES POOR REFERENCEPROVIDED BY SPICESince the waveform contains a substantial portion where thevalue is near zero, we use the MAE as the evaluation metric.Better results from the original method are selected for compar-ison in Table IV which clearly verifies the advantage of usingdiscrete convolution for wavelet approximation. The problemof SPICE’s failing to provide an accurate reference is also ap-parent here.As a final remark on the computational speed, we generallyobserve a 10 to 20 times reduction in the computation time of-fered by the proposed method over the curve fitting method. Forexample, to achieve an MAE of less than 0.18 for , the pro-posed method takes 14.9 s on a Pentium III machine, whereasthe curve fitting method takes 265.2 s. To achieve an MAE ofless than 0.37, the proposed method and the curve fitting methodtake 2.516 and 31.98 s, respectively.V. CONCLUSIONIn this paper, we propose a robust algorithm for approxi-mating waveforms of power electronics using wavelets. The al-gorithm exploits the orthogonality property to provide the nec-essary alternative equations for solving the wavelet coefficients.Furthermore, it has been shown that the discrete convolutionmethod offers good robustness as well as numerical accuracyover the original curve-fitting approach.REFERENCES[1] D. Li and R. Tymerski, “A comparison of steady-state methods for powerelectronic circuits,” in IEEE Power Electronics Specialists Conf. Rec.,vol. 2, 1998, pp. 1084–1090.[2] P. T. Krein, Elements of Power Electronics. New York: Oxford Uni-versity Press, 1998.[3] Y. S. Lee, Computer-Aided Analysis and Design of Switch Mode PowerSupplies. New York: Marcel Dekker, 1993.[4] I. Daubechies, “Time-frequency localization operators: a geometricphase space approach,” IEEE Trans. Inf. Theory, vol. 34, pp. 605–612,1988.[5] E. Wernekinck, H. Valenzuela, and I. Anfossi, “On the analysis of powerelectronics circuits waveforms with wavelets,” in Proc. Power Conver-sion Conf., 1993, pp. 544–549.[6] M. Liu, C. K. Tse, and J. Wu, “A wavelet approach to fast approximationof steady-state waveforms of power electronics circuits,” Int. J. CircuitTheory Appl., vol. 31, no. 6, pp. 591–610, Nov. 2003.[7] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials. BocaRaton, FL: Chapman and Hall/CRC, 2003.[8] T. Kilgore and J. Prestin, “Polynomial wavelets on the interval,” Constr.Approx., vol. 12, pp. 95–110, 1995.[9] B. Fisher and J. Prestin, “Wavelets based on orthogonal polynomials,”Math. Comput., vol. 66, no. 220, pp. 1593–1618, 1997.[10] L. O. Chua and P. M. Lin, Computer-Aided Analysis of Electronic Cir-cuits. Englewood Cliffs, NJ: Prentice-Hall, 1975.Authorized licensed use limited to: IEEE Xplore. Downloaded on December 17, 2008 at 22:30 from IEEE Xplore.  Restrictions apply.

An improved wavelet approach for finding steady-state waveforms of power electronics circuits using discrete convolution

Abstract

Similar works

Full text

Available Versions

PolyU Institutional Repository