Steady-State Analysis of Switching Converters via Frequency-Domain Circuit Equivalents

Transcription

1 Steady-State Analyss of Swtchng onerters a Frequency-Doman rcut Equalents ccardo Trnchero, Student Member, IEEE, Paolo Manfred, Member, IEEE, Igor S. Steano, Senor Member, IEEE, Flao G. anaero, Fellow, IEEE Abstract Ths bref presents a frequency-doman approach for the steady-state analyss of pulse-wdth modulated conerters and swtched crcuts wth non-deal swtchng behaor. The proposed strategy generalzes recent methodologes based on the Fourer expanson of the steady-state responses of a perodcally swtchng crcut and on the smulaton of an augmented lnear tme-narant system. Ths system s now also gen an nterpretaton n terms of an equalent crcut, whch s smulated at a sngle frequency pont to sole for all the harmoncs. The method offers a modular topologcal approach that s combned wth standard tools for crcut analyss and enables the smulaton of networks wth an arbtrary number of swtches and drng mechansms. Sngle, multple, and possbly non-deal commutaton eents wthn the swtchng perod are handled n the same framework, wthout addtonal complexty. The technque allows for the full frequency-doman characterzaton of both the functonal and the nosy behaor of the crcut responses. The feasblty and strength are demonstrated a comparsons wth smulatons and measurements on two applcaton examples,.e. a full-brdge sngle-phase nerter and a dc-dc boost conerter. Index Terms rcut smulaton, harmonc analyss, perodcally swtched lnear crcuts, pulse wdth modulaton nerters, SPIE, swtchng conerters. I. INTODUTION Perodcally swtched lnear (PSL) crcuts represent a wde and mportant class of tme-aryng electrcal networks that are used to descrbe the behaor of a number of modern deces. A releant example s proded by swtchng power conerters, whch are massely used to supply energy to electrcal and electronc equpment and applances []. PSL crcuts exhbt a complex dynamcal behaor arsng from the perodc actty of the swtches and requre sutable methods to accurately model and predct, n the desgn phase, the oltage and current waeforms at ther steady state. In partcular, the currents absorbed by swtchng crcuts are the man sources of conducted emssons (Es), whch need to be fully characterzed to comply wth the current electromagnetc compatblty (EM) regulatons specfed n terms of maxmum leels oer a wdeband frequency spectrum [2]. Ths work was partally supported by the esearch Foundaton Flanders (FWO-Vlaanderen).. Trnchero, I. S. Steano and F. G. anaero are wth the EM Group, Department of Electroncs and Telecommuncatons, Poltecnco d Torno, 29 Torno, Italy (e-mal: rccardo.trnchero@polto.t, gor.steano@polto.t, flao.canaero@polto.t). P. Manfred s wth the Electromagnetcs Group, Department of Informaton Technology, Ghent Unersty, 9 Gent, Belgum (e-mal: paolo.manfred@ugent.be). opyrght c 25 IEEE. Personal use of ths materal s permtted. Howeer, permsson to use ths materal for any other purposes must be obtaned from the IEEE by sendng an emal to pubs-permssons@eee.org. Some modelng approaches prode a smplfed, aeraged behaor of the steady-state response [3]. These methods, howeer, generate low-frequency models that cannot reproduce the wde-band behaor of swtchng crcuts. Alternate smulaton technques are aalable that prode the frequencydoman representaton of the steady-state response [4] [9]. Operatng drectly n the frequency doman has the releant adantage of aodng the calculaton of possbly long transents and any ssue related to the proper choce of ntegraton tme step and waeform wndowng, whch may mpact the accuracy and the effcency of the smulaton. Nonetheless, the complex mathematcal formulaton and the cumbersome techncal solutons burden the analyss of lnear swtchng crcuts wth arbtrary network confguratons and drng mechansms. To oercome the aforementoned lmtatons, a more straghtforward framework has been recently proposed for the class of PSL crcuts [], [], whch are nterpreted n terms of augmented lnear tme-narant (LTI) networks whose frequency-doman soluton drectly prodes the harmoncs of the steady-state responses. Such a LTI network s readly bult from the topologcal structure of the orgnal swtchng crcut. Unfortunately, the technque s lmted to deal swtches wth sngle and nstantaneous transtons wthn the perod and cannot be appled, as s, to a wde class of more practcal crcuts that nole multple commutatons and/or non-deal (e.g., nosy) swtchng dynamcs. Therefore, the am of ths bref s to extend the exstng methodology by ntroducng an mproed modelng framework that can deal wth an arbtrary behaor of the swtches, ncludng the releant cases of pulse-wdth modulaton (PWM) and of real-lfe MOS and dode elements n swtched-mode power conerters. Furthermore, an equalent crcut nterpretaton of the pertnent augmented LTI network s also proded, thus addtonally renderng the approach compatble wth adanced crcut solers such as SPIE. The equalent models are automatcally generated a a modular and topologcal approach. A general soluton method s thus establshed that oercomes the nherent lmtatons of an ad-hoc deraton and soluton of generc swtchng crcuts (e.g., a statespace crcut descrptons). The aboe feature also enables the analyss of practcal crcuts wth a suffcently large number of swtches and dfferent modulaton schemes. II. POPOSED SIMULATION FAMEWOK Ths secton summarzes the proposed smulaton framework for the soluton of a generc PSL crcut lke the one

2 2 shown n Fg., whch s a representate example wth the key elements of ths class of crcuts (.e. resstors, dynamcal elements, and swtches) and allows for a comprehense llustraton of the method. The swtch S s a tme-aryng element characterzed by a perodc behaor wth ether a sngle commutaton (S) or a multple commutaton (M) wthn the perod T. The swtch behaor s represented, for the moment, by logcal bnary sgnals, wth the leels one and zero correspondng to the closed and open swtch, respectely. The unfed handlng of both S and M operaton modes wthn the same smulaton framework allows for the applcaton of the method to a wde class of swtchng crcuts, rangng from dc-dc conerters to PWM dc-ac nerters, thus extendng the state-of-the-art technque aalable n the lterature. Later on, the swtchng behaor s further generalzed to arbtrary dynamcs. Table I ELATIONSHIPS BETWEEN THE VOLTAGE AND UENT HAMONIS FO THE MAIN LTI ELEMENTS AND A SWITH (n, k = N,..., +N ). element consttute relaton harmonc relaton S L = = L d dt = d dt { = / on = S closed S open V n = I n V n = jnω cli n I n = jnω cv n I n = m,k Y mv k for m + k = n e(t) B S A (t) A L S M Fgure. Example of a dynamcal PSL crcut. The swtch s dren by sngle or multple commutaton eents wthn the swtchng perod T, referred to as S and M operaton modes, respectely. A. Frequency-Doman epresentaton and onsttute elatons Analogously to harmonc balance smulatons, for the frequency-doman analyss of a PSL crcut the ndependent oltage source s defned as a csodal exctaton e(t) = (E /2π) exp(jω t) wth angular frequency ω, and the perodc response at the steady state s expressed as a truncated sum of delta functons. E.g., for the oltage A (t) n Fg. : V A (ω) +N n= N V A,n δ(ω nω c ω ), () where V A,n are the 2N + Fourer coeffcents (harmoncs) of the oltage A (t), and ω c = 2π/T. Analogous approxmatons hold for all the other oltages and currents. Based on the aboe nterpretaton of the crcut arables, approprate consttute relatons are dered for the oltage and current harmoncs of both the LTI elements and the swtches. The substtuton of () nto the - consttute relatons of the standard LTI elements such as resstors, capactors and nductors, yelds the augmented relatons between the oltage and current harmoncs collected n the frst three rows of Table I []. It s worth notng that n ths case no couplng between the harmoncs s noled, n agreement wth the wellknown concepts underlyng the phasor analyss of lnear and tme-narant crcuts. The frequency-doman current-oltage characterstc of a generc swtchng element s gen n the last row of the table nstead. ontrary to classcal LTI crcuts, the presence of a PSL element couples the oltage and current harmoncs, whch can no longer be computed ndependently. T T t t The coeffcents Y m are complex numbers computed as the Fourer transform of the tme-doman perodc behaor of the swtch admttance,.e. T Y m = Π(t) exp( jmω c t)dt, (2) T on m = 2N,...,,..., 2N, where on s the seres resstance of the swtch, whlst Π(t) s the functon defnng the swtch operaton as shown n Fg.. B. Modelng of the S and M Swtchng Modes The Fourer transform (2) s computed accordng to the behaor of the functon Π(t). If the swtch S operates n S mode, t s closed once durng the perod T, thus leadng to the followng perodc behaor: { for t [t, t + DT ] Π(t) = Π S (t) = (3) otherwse For ths smple case, the coeffcents Y m are computed analytcally as [] Y S,m = exp( jmω c t ) exp( jmω c (t + DT )). on T jmω c (4) The preous scenaro s now generalzed to handle the more complex M swtchng behaor. In prncple, ths can be nterpreted as a sum of S square wae functons,.e., Π M (t) = M = Π S,(t), wth M the number of commutatons occurrng n a sngle perod. Gen the lnearty of the Fourer operator, the coeffcents Y M,m are n turn computed as a superposton of the analytcal result (4). Nonetheless, an alternate and more practcal approach s to numercally compute these entres by replacng the contnuous-tme Fourer transform (2) wth a dscrete Fourer transform (DFT) a the followng relaton: N s Π[k] exp on N s k= Y M,m = N s Π[k] exp on N s k= ( ) jk2πm N s f m ( ) jk2π(ns+m) N s f m < (5)

3 3 where N s = 4N + and Π[k] = Π(k t ) s the sampled erson of the contnuous sgnal Π(t) on a sngle perod T at equspaced nterals t = T/N s. The number of samples N s has to be greater than 4N + to obtan all the entres n Table I (from Y 2N to Y 2N ). It s worth to remark that, snce the spectrum of a generc swtchng sgnal Π(t) s of nfnte bandwdth, the samplng wll netably lead to alasng. In order to reduce the alasng effect on the frst 2N + harmoncs of the spectrum of the dscrete sgnal Π[k], a number of samples N s > 8N s suggested. As a further generalzaton, the functon Π(t) can be replaced by any normalzed perodc functon that descrbes the possble non-deal behaor of the swtch. The typcal example s proded by a component that swtches between the on and the off state wth a fnte transton tme and arbtrary (e.g., nosy) commutaton. Ths alternate nterpretaton s partcularly useful when the nformaton on the actual swtchng behaor of real components s aalable (e.g., from measurements) and can be used to mproe the predcted results, as wll be shown n Secton IV.. Equalent rcut Interpretaton Assumng that the node oltages and the branch currents are expanded accordng to (), a correspondng augmented network s created by frst assocatng a node to each oltage harmonc coeffcent. The resultng network has thus a total number of nodes that s 2N + tmes larger. These nodes are then connected wth sutable elements as descrbed n the followng, n accordance wth the orgnal crcut topology. For the case of a resstor, the same current-oltage relatonshp s presered for all the harmoncs (frst row of Table I), and the resstor s smply replcated between the pertnent nodes. For each harmonc of a capactor nstead, the characterstc equaton n the second row of Table I s mapped nto a modfed capactance n, computed a the followng admttance equaton: j(ω + nω c ) = jω n, (6) where Ω > s the smulaton frequency. The modfed capactances are thus defned as n = ω+nωc Ω. It should be noted that the choce of the smulaton frequency Ω s arbtrary and the modfed capactance s computed accordngly. An nductor s smlarly mapped nto modfed nductances. So far, the crcut nterpretaton for LTI elements preseres the component type, as there s no couplng between ther oltage and current harmoncs. On the other hand, the coupled equaton of a PSL element s mplemented by ntroducng, for each term Y m = G m + jb m, a dependent current source that exhbt an admttance transfer functon n the form of k,m + jωk,m, wth k,m = G m and k,m = B m /Ω. Hence, each PSL component s replaced by dependent sources wth a transadmttance equal to the Fourer coeffcents of the swtch tme-doman behaor. Fnally, ndependent sources are also expressed a (), and the correspondng Fourer coeffcents are sutably connected to the nodes of the augmented network. For the sake of llustraton, Fg. 2 shows the augmented LTI counterpart of the example crcut of Fg.. To lmt the crcut sze, only three harmoncs, correspondng to n =,, +, are consdered. The dfferent models for each element are connected based on the same topologcal structure as the orgnal network. The sources E -, E and E are the Fourer coeffents of the tme-doman source e(t). For the frequent case of dc sources (ω = ), only the generator E s non-null. The resultng network s a coupled LTI crcut that s soled once at a sngle smulaton frequency Ω to smultaneously retree all the 2N + harmoncs descrbng the steady-state oltage and current responses of the orgnal PSL crcut and resultng from the exctaton at the angular frequency ω. The equalent crcut s compatble wth adanced commercal SPIE-type smulators and s easly mplemented a aalable components. Alternately, the modfed nodal analyss (MNA) representaton [2] of the network s readly generated from crcut nspecton and t s soled a a sngle lnear nerson, e.g., n MATLAB. The llustraton of Fg. 2 contans all the key buldng blocks that allow to produce the equalent models of more complex crcuts as those consdered n the followng applcaton examples. Furthermore, the outlned topologcal procedure allows for the automatc and transparent generaton of the augmented LTI equalent startng from the orgnal netlst of any arbtrary PSL crcut. It s releant to pont out that no specfc assumpton has been made on the swtch operaton, and the equalent model apples to arbtrary commutng behaors. III. APPLIATION #: NUMEIAL SIMULATION E E E - B B B - Y2VBA - YVBA YVBA Y-VBA YVBA YVBA - YVBA - Y-VBA Y-2VBA A - ω Ω L ω ωc Ω L ω ωc Ω A ω+ωc Ω L ω Ω A ω+ωc Ω S S 3 L n F n F V n n L F S 2 S p 4 (t) Fgure 3. Full-brdge sngle-phase nerter. The crcut elements take the followng alues: V n = 5 V, n = Ω, n = 47 µf, L F = 2 mh, F = 3 µf, p = 3 pf and = 2 Ω. p Fgure 2. Augmented equalent LTI model of the swtchng crcut of Fg. for N =. The notaton V BAn = V Bn V An s used. The effecteness of the proposed method s demonstrated on the PWM full-brdge sngle-phase nerter [] shown n

4 4 Fg. 3. The swtches S k, k =,..., 4 are dren by four PWM sgnals obtaned by comparng a reference snusodal sgnal r(t) of frequency f r = 6 Hz wth a companon perodc trangular carrer c(t) of frequency f c = 96 Hz. The behaor of the swtches S and S 3 of the H-brdge s defned by means of the normalzed functons Π (t) and Π 3 (t) shown n Fg. 4. The remanng swtches S 2 and S 4 behae accordng to the functons Π 2 (t) = Π (t) and Π 4 (t) = Π 3 (t), so that a short crcut across the oltage source V n s aoded. Π(t) Π3(t).5 Swtch control sgnals Tme [ms] Fgure 4. Operaton of the swtches S and S 3 n the scheme of Fg. 3. The methodology ntroduced n Secton II allows to obtan a LTI equalent of the conerter, where the tme-aryng PWM swtches are replaced by ther augmented frequencydoman representatons. Ths equalent representaton of the conerter has been mplemented both n a MATLAB scrpt, by means of the MNA formalsm, and n SPIE, by consderng the crcutal nterpretaton dscussed n Secton II-. Fg. 5 shows the spectrum (top panel) and the correspondng steadystate tme-doman response (bottom panel) of the output oltage, predcted wth the proposed frequency-doman approach n MATLAB (red lne) and SPIE (nner black lne). For comparson, the reference result calculated wth a transent Smulnk smulaton wth t step =. µs s also proded (outer gray lne). The cures n the fgure hghlght the excellent accuracy of the proposed soluton n reproducng the steadystate response wthout the ntal transent. To account for the hgh-frequency components of the steady-state response, the expanson order s set to a suffcently large alue of N = 8. In MATLAB, the proposed method requres a computatonal tme of.4 s only, and achees a speed-up of 3 wth respect to the transent smulaton. IV. APPLIATION #2: EXPEIMENTAL VEIFIATION The proposed methodology s now used to reproduce the measured spectrum of the dfferental mode (DM) E of the boost conerter of Fg. 6. Accordng to the EM standards, the E of the conerter s obtaned by measurng the oltages LG and NG at the output ports of a LISN (Lne Impedance Stablzaton Network) and then the DM emssons are computed offlne as V DM = (V LG V NG )/2. The boost conerter operatng at the steady state s consdered as a PSL crcut, wth the MOS and the dode sutably replaced by two perodc swtches of frequency f c. Specfcally, two dfferent swtchng behaors are consdered, as shown n Spectrum [dbµv] Tme doman [V] Output oltage Frequency [Hz] reference model(matlab) model(spie) Tme [ms] Fgure 5. Steady-state behaor of the nerter output oltage n both frequency (top panel) and tme (bottom panel) doman. Gray lne: reference transent smulaton; red and black lnes: predctons of the proposed technque mplemented n MATLAB and SPIE, respectely. E LISN Ω Ω 5µH µf µf LG NG 5µH.µF 5Ω 5Ω.µF lp L boost f c,d lp Boost onerter Fgure 6. Expermental setup for the measurement of the DM E of a boost conerter. The conerter operates n contnuous mode wth swtchng frequency f c = 5 khz, duty cycle D = 5% and constant nput oltage E = 5 V. The component alues are as follows: L boost = 47 µh, lp =.3 Ω, lp = 27 pf, = 47 µf and = 5 Ω. Fg. 7: ) an deal and complementary commutaton (dashed lnes), n whch the MOS s off when the dode s on, and ce ersa, and ) the actual measured behaor of the oltages across the dode and the MOS elements (sold lnes). In both cases, the correspondng augmented network s generated from the schematc of Fg. 6 usng the rules proded n Secton II and t s smulated n the frequency doman, leadng to the predcton of the E spectrum. The number of harmoncs s set to N = f max /f c = 6 to coer the entre frequency band specfed by EM standards (f max = 3 MHz). Fg. 8 compares the measured (sold gray lne) and predcted (black and red lnes) Es for the two dfferent operatons of the swtches. It s worth notng that the result obtaned by assumng an deal behaor of the swtchng elements (black lne) fals to predct the spectral harmoncs placed at the een multples of the swtchng frequency f c. Howeer, when the

5 5 Π(t) Swtch control sgnals.5 actual MOS deal MOS actual dode deal dode Tme [µs] Fgure 7. Normalzed measured oltages across the dode and the MOS components. Spectrum [dbµv] Dfferental mode oltage measurement smulaton (actual swtch) smulaton (deal swtch) Frequency [khz] Fgure 8. Spectrum of the boost DM E. The measured behaor (gray lne) s compared aganst the predcton of the proposed frequency-doman method wth both the deal (black lne) and the measured (red lne) swtch operaton. actual behaor of the swtches s used n the smulaton, a far better accuracy s ndeed acheed (see the red lne), wth mproements n the predcton of the een harmoncs and of the hgh-frequency response. Such a correcton s not straghtforward n a tradtonal tme-doman smulaton and could not hae been mplemented n the framework n [], []. To further llustrate the nherent, yet crtcal ssues n the transent smulaton of swtched networks, Fg. 9 shows a tme-doman smulaton of the dfferental oltage DM (t) n SPIE. A long transent s requred for the oltage to reach the steady state, whch s readly and accurately captured wth the proposed frequency-doman approach nstead (see the nsert). Ths makes the presented method 8 faster (7.5 s s 35.8 s) despte the augmented sze of the equalent LTI network. Tme doman [V].5.5 Dfferental mode oltage Tme [ms] V. ONLUSIONS In ths bref, a frequency-doman approach for the steadystate smulaton of swtchng conerters s proposed. The method apples to a broad class of lnear dynamcal crcuts consstng of classcal LTI components and swtches dren by PWM sgnals and/or wth possbly non-deal commutaton eents. A flexble and comprehense framework s dered, where the orgnal tme-aryng network s nterpreted n terms of an equalent augmented LTI crcut. The new network arables correspond to the harmoncs of the steady-state crcut responses and are calculated a a sngle crcut smulaton. The equalent crcut s compatble wth standard SPIEtype solers. The method has been appled to the predcton of the steady-state behaor of a PWM nerter and a dcdc boost conerter wth measured behaor of ts swtchng components, and excellent agreement and smulaton speedup were obsered wth respect to the reference results. Other possble and ensaged applcatons nclude the robust desgn and optmzaton of ths class of crcuts. EFEENES [] M. H. ashd, Power Electroncs: rcuts, Deces and Applcatons. 3rd edn., Prentce Hall, 24. [2] E. ondon-pnlla, F. Morel,. Vollare, and J.-L. Schanen, Modelng of a buck conerter wth a S JFET to predct EM conducted emssons, IEEE Trans. Power Electron., ol. 29, no. 5, pp , May 24. [3] A. Daoud, J. Jatskech, and T. De ybel, Numercal state-space aerage-alue modelng of PWM D-D conerters operatng n DM and M, IEEE Trans. Power Electron., ol. 2, no. 4, pp. 3 2, Jul. 26. [4] M.-L. Lou, Exact analyss of lnear crcuts contanng perodcally operated swtches wth applcatons, IEEE Trans. rcut Theory, ol. 9, no. 2, pp , Mar [5] S.. Sanders, J. M. Noworolsk, X. Z. Lu, and G.. Verghese, Generalzed aeragng method for power conerson crcuts, IEEE Trans. Power Electron., ol. 6, no. 2, pp , Apr. 99. [6] F. Yuan and A. Opal, Nose and senstty analyss of perodcally swtched lnear crcuts n frequency doman, IEEE Trans. rcuts Syst. I, eg. Papers, ol. 47, no. 7, pp , Jul. 2. [7] F. Wang, H. Zhang, and X. Ma, Analyss of slow-scale nstablty n boost PF conerter usng the method of harmonc balance and floquet theory, IEEE Trans. rcuts Syst. I, eg. Papers, ol. 57, no. 2, pp , Feb. 2. [8] J. Lang and W.-H. Lao, Steady-state smulaton and optmzaton of class-e power amplfers wth extended mpedance method, IEEE Trans. rcuts Syst. I, eg. Papers, ol. 58, no. 6, pp , Jun. 2. [9] H. Behjat, L. Nu, A. Daoud, and P. L. hapman, Alternate tmenarant mult-frequency modelng of PWM D-D conerters, IEEE Trans. rcuts Syst. I, eg. Papers, ol. 6, no., pp , No. 23. []. Trnchero, I. S. Steano, and F. G. anaero, Steady-state response of perodcally swtched lnear crcuts a augmented tme-narant nodal analyss, J. Elect. omput. Eng., ol. 24, artcle ID 98273, 24. []. Trnchero, I. S. Steano, and F. G. anaero, Steady-state analyss of swtchng power conerters a augmented tme-narant equalents, IEEE Trans. Power Electron., ol. 29, no., pp , No. 24. [2].-W. Ho, A. uehl, and P. Brennan, The modfed nodal approach to network analyss, IEEE Trans. rcuts Syst., ol. 22, no. 6, pp , Jun Fgure 9. Tme-doman response of the boost DM oltage. The reference SPIE smulaton (blue lne) exhbts a long transent. The behaor at the steady state s well predcted by the proposed method (red cure n the nsert).