STATE PLANE ANALYSIS, AVERAGING,

Transcription

1 CHAPER 3 SAE PLAE AALYSIS, AVERAGIG, AD OHER AALYICAL OOLS he sinusoidal approximaions used in he previous chaper break down when he effecs of harmonics are significan. his is a paricular problem in he case of disconinuous conducion modes, where harmonics canno be ignored. o obain a complee undersanding of he behavior of resonan converers, anoher approach is needed. In his chaper, he fundamenal principles necessary for an exac ime domain analysis of resonan converers are explained. hese principles are used in laer chapers o examine no only he series and parallel resonan converers, bu also quasiresonan converers. he sae plane can be used o reduce he complicaed ank waveforms of resonan converers o geomery. When properly normalized, he ank waveforms are described by segmens of circles, lines, and/or oher simple figures in he sae plane. Deerminaion of converer seadysae characerisics is ofen a maer of piecing ogeher hese segmens, hen solving a few riangles or oher figures. Equally imporan is he use of averaging, in which he dc and lowfrequency ac componens of he converer erminal waveforms are found while neglecing high frequency swiching harmonics. he average oupu curren of he series resonan converer is relaed o he charge ha flows hrough is oupu erminals per swiching period. his charge also flows hrough he ank capacior, where i excies an ac volage. he load curren and ank capacior volage magniude are herefore closely relaed, and considerable insigh can be gained by use of some simple argumens regarding he flow of charge during a swiching period. Similarly, he average oupu volage of he parallel resonan converer is he volseconds (flux linkages) applied o he ank inducor per swiching period, and herefore i is also relaed o he peak ank curren. hus, some simple charge and fluxlinkage argumens are discussed in his chaper, and are used in laer chapers o easily relae he ank waveforms o he dc erminal volages and currens. he various fundamenal principles which describe he flow of charge and flux linkages in a resonan circui, and heir relaions o he average erminal waveforms, are colleced in secion 3., and are illusraed using he series and parallel resonan converers as examples. Sysems of

2 Principles of Resonan Power Conversion noaion and normalizaion, a perennial source of confusion in any discussion of resonan converers, are described in secion 3.. In secions 3.3 and 3.4, he ringing responses of series and of parallel resonan ank circuis are derived, and hey are ploed in he sae plane. I is apparen ha an exac ime domain analysis of resonan converers is more complex han he use of he sinusoidal approximaions and frequency domain mehods of chaper. oneheless, simple and exac closedform soluions can be obained for he many coninuous and disconinuous conducion modes of he series resonan converer, as well as for he parallel resonan and many quasiresonan converers. hese ideas are also useful in modeling he dynamics of hese converers, and he basic ideas developed in his chaper are used hroughou he remainder of his monograph. 3.. Averaging and Relaed Conceps he signals in a power elecronics sysem generally conain subsanial swiching harmonics. By specificaion and design, he magniude of hese harmonics mus be negligible a he converer oupu. Hence, when analyzing he behavior of a converer, we usually neglec he swiching harmonic componens of he converer erminal waveforms, and model only heir dominan dc and lowfrequency ac componens. his simplifies he analysis considerably, and allows a much beer undersanding of he converer properies. he basic argumens used o average he converer waveforms were described by Weser and Middlebrook []. Alhough hese argumens were originally developed for modeling PWM converers, hey also assis he analysis of resonan converers. Averaging he sysem signals over a period does no significanly aler he waveforms, so long as he period is shor compared o he sysem s naural response imes. his is similar o passing he waveforms hrough a lowpass filer; if he filer corner frequency is sufficienly high, hen he imporan dc and lowfrequency ac componens are no affeced. In paricular, i is useful o average he erminal waveforms over one swiching period. his effecively removes he swiching and ringing harmonics wihou modifying he desired dc and low frequency ac response, and significanly simplifies he analysis. his approximaion is jusified because i is normally required ha swiching harmonics be negligibly small a he load, and herefore sufficien lowpass filering is incorporaed ino any welldesigned converer. he implicaion is ha he converer oupu curren can be adequaely represened if we simply find he oal charge which flows ou of he oupu por during one swiching period. Dividing his charge by he swiching period yields he average oupu curren. Dual argumens allow represenaion of he oupu volage knowing he oal fluxlinkages, or volseconds, which he converer applies o he oupu during one swiching period. In his secion, some basic

3 Chaper 3. Sae Plane Analysis principles are discussed which allow he average erminal volages and currens o be direcly relaed o he ank ringing waveforms. Averaging: charge argumens Le us consider how o average a dependen erminal curren i of a swich nework, as illusraed in Fig. 3.. If i is periodic wih period, hen he average value can be wrien < i > = i () d = q (3) where q = i () d is he ne charge ransferred a he por over period. he form <i > = q / is useful because, as shown laer, q can be relaed o oher salien feaures of he resonan nework waveforms. In paricular, q is a funcion of he change in ank capacior charge (and hence also he change in ank capacior volage) over a porion of he swiching period. COVERER v i i Por.. Por v Load i () area = q q < i > = Fig. 3.. Arbirary oupu of swich nework; average oupu curren compued from charge ransfer.

4 Principles of Resonan Power Conversion Averaging: fluxlinkage argumens Dependen erminal volages can be averaged using dual argumens. Consider a dependen erminal volage v of a swich nework, as illusraed in Fig. 3.. If v is periodic wih period, hen he average value can be wrien < v > = v () d = λ (3) where λ = v () d is he ne volseconds applied a he por over period. he form < v > = λ / is useful because, as shown laer, λ can be relaed o oher salien feaures of he resonan nework waveforms. In paricular, λ is a funcion of he change in ank inducor flux linkages (and hence also he change in ank inducor curren) over a porion of he swiching period. COVERER v i i Por.. Por v Load v () area = λ < v > = λ Fig. 3.. Compuaion of average erminal volage <v > using flux linkages λ.

5 Chaper 3. Sae Plane Analysis ank capacior charge variaion Over one swiching cycle, charge is ransferred from he swich power inpu, hrough he ank capacior, o he oupu. he amoun of charge ransfer can be direcly relaed o he capacior volage waveform. In paricular, over a given inerval ( α, β ), if a given amoun of charge q is deposied on he ank capacior, hen we know ha he capacior volage changes from v C ( α ) o v C ( β ), where q = C (v C ( b ) v C ( a )) (33) Hence, he capacior volage iniial and final values v C ( α ) and v C ( β ) are relaed o he charge ransfer, and herefore also o he swich average erminal curren (by Eq. 3). i C v C C For example, consider he circui of Fig I is desired o compue he average, or dc componen, of he bridge recifier oupu curren i <i >, and o relae i o he capacior volage waveform v C (). ypical waveforms are skeched in Fig he average value of i () is given by: R <i > = i () d Fig Demonsraion of direc relaion beween dc componen of load curren = q and peakopeak capacior volage. (34) where q = i () d During he inerval /, he capacior curren i C () is idenical o he bridge recifier oupu curren i (), and hence he same ne charge q is deposied on he capacior. he maxima and minima of he capacior volage waveform v C () coincide wih he zero crossings of he curren i C (), and hence he capacior volage changes from is minimum value V CP o is maximum value V CP during his inerval. he capacior charge relaion is herefore q = C (V CP ( V CP )) = CV CP (35) Eliminaion of q from Eqs. (34) and (35) yields <i > = 4CV CP (36)

6 Principles of Resonan Power Conversion i c area = q area = q Hence, he average value, or dc componen, of he resisor curren <i > and he peak capacior volage V CP are direcly relaed. hese argumens are used in chaper 4 o derive a nearly idenical relaion beween he peak ank capacior volage and he load curren of he series resonan converer. area = q area = q ank inducor flux linkage variaion i = i C < i > v C V CP V CP he dual of he ank capacior charge relaion follows from he definiion λ = L i. Inducor fluxlinkage λ has he dimensions of volseconds, and is he inegral of he applied volage as defined in Eq. (3). During one swiching cycle, volseconds are ransferred from he swich power inpu, hrough he ank inducor, o he oupu. So over a given inerval ( α, β ), if a given amoun of flux linkages λ are sored in he ank inducor, hen he inducor curren changes from i L ( α ) o i L ( β ), where λ = L (i L ( b ) i L ( a )) Fig Waveforms for he circui of Fig Hence, he inducor curren boundary values i L ( α ) and i L ( β ) are relaed o he volsecond ransfer, and hence also o he converer average erminal volage (by Eq. 3).

7 Chaper 3. Sae Plane Analysis i L v L v Fig Sinusoidal curren source driving an inducor in parallel wih bridge recifier and resisor. <v > = v () d R For example, consider he circui of Fig We wish o compue he average, or dc componen, of he bridge recifier oupu volage <v >, and o relae i o he inducor curren waveform i L (). ypical waveforms are skeched in Fig he average value of v () is given by where λ = v () d = λ (38) During he inerval /, he inducor volage v L () is idenical o he bridge recifier oupu volage v (), and hence he same ne flux linkages λ are sored in he inducor. he maxima and minima of he inducor curren waveform i L () coincide wih he zero crossings of he volage v L (), and hence he inducor curren changes from is minimum value I LP o is maximum value I LP during his inerval. he inducor flux linkage relaion is herefore λ = L (I LP ( I LP )) = LI LP (39) Eliminaion of λ from Eqs. (38) and (39) yields <v > = 4LI LP (3)

8 Principles of Resonan Power Conversion v L area = λ Hence, he average value, or dc componen, of he resisor volage <v > and he peak inducor curren I LP are direcly relaed. Similar argumens are used in chaper 5 o derive a relaion beween he peak ank inducor curren and he load volage of he parallel resonan converer. v = v i L I LP I LP L area = λ Kirchoff s laws in inegral form We know from Kirchoff s Curren Law (KCL) ha he oal currens flowing ino a given node mus be zero: i k = k (3) he ne charge which eners he node over a given inerval ( α, β ) mus also be zero: q k = k (3) where q k = a b i k () d Fig Waveforms for he circui of Fig he inegral form of Kirchoff s Curren Law is useful for relaing erminal charge quaniies o he change in ank capacior charge. For example, consider he circui of Fig his circui is similar o he ank circuis of boh he parallel resonan converer and he zerocurren resonan swich, during one ringing subinerval. In conjuncion wih he deerminaion of he average inpu curren of his nework, we wish o compue he charge conained in i during he given ringing subinerval: q β = α α β i () d (33) By KCL, we know ha i = i C i. Hence, q β = q Cβ q β (34) where q Cβ = i C () d α α β < v >

9 Chaper 3. Sae Plane Analysis i L i V C i c I q B = q CB q B i area = q β α β i c area = q Cβ α i area = q β α α β Fig Illusraion of use of inegral form of KCL for a ypical ank nework. and q β = i () d α α β herefore, he ringing inerval inpu charge q β is relaed o q Cβ, he change in ank capacior charge over he ringing inerval, and o q β, he charge ransferred o he oupu during he ringing inerval. Some of he inpu charge is sored on he ank capacior, while he remainder flows o he oupu.

10 Principles of Resonan Power Conversion An inegral form of Kirchoff s Volage Law (KVL) is also useful. he oal volage around a nework loop is zero: v k = k (35) he oal volseconds applied over a given inerval ( α, β ) across he elemens of his loop mus also be v L v L C v I zero: λ k k = (36) v λ B = λ LB λ B area λ β where λ k = b v k () d a When elemen k is an inducor or ransformer, λ k has he physical inerpreaion of winding flux linkages. he inegral form of KVL relaes erminal volsecond quaniies o he change in ank inducor flux linkages. For example, in conjuncion wih he deerminaion of he average oupu volages of he parallel resonan converer and he zero curren resonan swich, we wish o compue he volseconds conained in v during he ringing inerval (see Fig. 3.8): v L v α α β area λ Lβ area λ β α β α β λ β = α β v () d (37) α By KVL, we know ha v = v v L. herefore, λ β = λ β λ Lβ (38) where λ β = α α β v () d α Fig llusraion of he use of he inegral form of KVL. λ Lβ = α β v L () d α herefore, he ringing inerval oupu volseconds λ β is relaed o λ Lβ, he change in ank inducor flux linkages over he ringing inerval, and o λ β, he volseconds applied o he inpu during he ringing inerval. Some of he inpu volseconds are sored in he ank inducor, while he remainder are applied o he swich oupu.

11 Chaper 3. Sae Plane Analysis Seadysae capacior charge balance When a capacior operaes wih periodic seadysae waveforms, hen he iniial and final values of he capacior volage waveform are idenical. In consequence, no ne charge is deposied in he capacior, and he inegral of he capacior curren waveform over one complee cycle is zero. he dc componen of capacior curren is zero. Formally, his follows from he definiion i C () = C dv C() (39) d Inegraion over one complee period yields v C () v C () = C i C () d In periodic seady sae, v C () = v C (). Hence, = i C () d (3) (3) Division by he period hen shows ha he average value, <i C >, or dc componen, mus also be zero: i C () d = <i C > = (3) his is he wellknown principle of seadysae capacior ampsecond, or charge, balance. I is rue for any capacior which operaes wih periodic seadysae waveforms. i () For example, consider he capaciive oupu filer i () Fig. 3.9 i CF C F v () Capaciive filer circui. R circui of Fig In seady sae, no ne charge is deposied on capacior C F during a swiching period, and hence he average recifier oupu curren i is equal o he dc componen I of he load curren i(). By Ohm s law, he dc componen V of he load volage v() is V = IR. Hence, we have V = <i > R (33) For his example, he inpu curren, i (), is a recified sinusoid i () = I P sin(ω S ) (34) whose average is S <i > = i () d = S π I P (35) Subsiuion of his expression ino Eq. (33) yields V = π I P R (36)

12 Principles of Resonan Power Conversion i () I P < i > Oupu volage ripple can also be esimaed using charge argumens. In Fig. 3., he capacior curren waveform is skeched for he case in which C F is large i CF () area = q C area = q C 8 36 ω S ω S enough ha is volage ripple (induced by he swiching harmonics in i ()) is small compared o he dc componen V. In his case, he volage harmonics applied o resisor R are also small, and hence by Ohm s law he curren hrough R is essenially dc. herefore, he dc componen of i () flows exclusively hrough R, while he swiching harmonic of i () flows overwhelmingly hrough C F. he capacior curren waveform is hen given by v () V Eq. (37) over his inerval: q C = v C Fig. 3.. ypical waveforms for he circui of Fig ω S I P ( sin(ω S ) π ) d(ω S) ω S i CF () = i () i() I P sin(ω S ) π I P (37) he capacior curren is posiive over he inerval < ω S < During his inerval, he capacior volage increases by an amoun v C, from is minimum value o is maximum value. his corresponds o an increase in charge q C given by he inegral of (38) Evaluaion of he inegral yields q C =.67 I P S (39) he peakoaverage capacior volage ripple is herefore v C =.67 I P S (33) C Or, in erms of he load curren, v C =.56 I S (33) C his gives a simple esimae which is useful for choosing he oupu filer capaciance. However, i does no include he effecs of capacior esr (equivalen series resisance), which can cause he volage ripple o be significanly larger han ha prediced by Eq. (33).

13 Chaper 3. Sae Plane Analysis Seadysae inducor fluxlinkage balance When an inducor operaes wih periodic seadysae waveforms, hen he iniial and final values of he inducor curren waveform are idenical. In consequence, no ne flux linkage is induced in he inducor, and he inegral of he inducor volage waveform over one complee cycle is zero. he dc componen of inducor volage is zero. Formally, his follows from he definiion v L () = L di L() (33) d Inegraion over one complee period yields i L () i L () = L v L () d In periodic seady sae, i L () = i L (). Hence, = i C () d (333) (334) Division by he period hen shows ha he average value, <v L >, or dc componen, mus also be zero: v L () d = <v L > = (335) his is he wellknown principle of seadysae inducor volsecond, or fluxlinkage, balance. I is rue for any inducor which operaes wih periodic seadysae waveforms. 3.. ormalizaion and oaion he geomeries of he sae plane plos of he nex secions are simplified considerably when he waveforms are normalized using a base impedance R base equal o he ank characerisic impedance R. he normalizing base volage V base can be chosen arbirarily, and is usually chosen o be equal o he power inpu volage V g. Oher normalizing base quaniies can hen be derived: base impedance R base = R = L / C base volage V base = V g base curren I base = V base / R base = V g / R base power P base = V base I base = V g / R (336) In he sysem of noaion developed a he Universiy of Colorado (CU), he symbol for a normalized volage conains he same subscrips and case as he original unnormalized volage, bu he characer V is replaced by M. For example, M = V / V base normalized load volage m C () = v C () / V base normalized ank capacior volage (337)

14 Principles of Resonan Power Conversion For currens, I is replaced by J : J = I / I base normalized load curren j L () = i L () / I base normalized ank inducor curren (338) When a converer conains a ransformer, he base quaniies should be referred o he proper side of he ransformer by muliplicaion by he appropriae funcion of he ransformer urns raio. Some oher auhors use he same normalizing base quaniies, bu denoe normalized variables using he subscrip n. he symbol q is also someimes used o denoe he normalized oupu volage. eiher of hese convenions is used here. I is convenien o normalize frequency using he ank resonan frequency f, and o conver ime o angular form: f base = f = / π LC base frequency ω = / LC ank resonan angular frequency F = f S / f normalized swiching frequency α = ω X angular lengh of inerval X (339) where f S is he swiching frequency, and S = /f S is he swiching period. he following noaion is also radiional in he series resonan converer lieraure: γ = ω S / = π / F angular lengh of one half swiching period α = ω α diode conducion angle β = ω β ransisor conducion angle (34) When performing exac imedomain or saeplane analysis, Q is defined using he acual load resisance R (as opposed o he effecive resisance R e of chaper ): Q = R / R for he series resonan converer Q = R / R for he parallel resonan converer (34) ξ = 3 ec. k = 3 f f k = ξ = f k = f s Fig. 3.. Swiching frequency ranges over which various mode indices k and subharmonic numbers ξ occur. Final definiions for he series resonan converer are he mode index k and subharmonic number ξ, as follows. he coninuous conducion mode k occurs over he frequency range f k < f s < f k or k < F < k (34) for ineger k. he subharmonic number is hen

15 Chaper 3. Sae Plane Analysis ξ = k ()k (343) For example, coninuous conducion mode operaion a swiching frequency f S =.4 f would imply ha k = and ξ = Sae Plane rajecory of a Series ank Circui Le us nex examine he imedomain response of a series i resonan circui. A series ank circui, excied by a consan L volage V, is shown in Fig. 3.. As shown in he nex chaper, V he series resonan converer can be reduced o a circui of his v form during each subinerval. he sae equaions of his circui C are: L di L() = V d v C () Fig. 3.. Series ank circui, C dv (344) excied by consan C() = i d L () volage V. Le us normalize he sae equaions according o he convenions of secion 3.. oe ha L = R ω and C = ω R (345) where ω is he ank angular resonan frequency and R is he ank characerisic impedance, as defined in Eqs. (336) and (339). Division of Eqs. (345) by V g and use of he ideniies (346) and (336) (339) yields dj L () = M ω d m C ( dm C () (346) = j ω d L () where M = V / V g. he soluion of his secondorder sysem of linear differenial equaions is m C () = A cos(ω ϕ) M j L () = A sin(ω ϕ) (347) where he consans A and ϕ depend on boundary condiions. I can be seen ha he soluion conains a dc erm m C = M (or, v C = V ) which represens he dc soluion of he circui, plus a sinusoidal erm which represens he ac ringing response of he resonan ank.

16 Principles of Resonan Power Conversion y j L A Y A θ M ϕ m C X x x = X A cos(θ) y = Y A sin(θ) Fig Parameric represenaion of a circle. Fig ormalized sae plane rajecory for he circui of Fig. 3., corresponding o Eq. (347). he normalized sae plane he normalized sae plane is a plo of m C () vs. j L (), wih as an implici parameer. As shown in Fig. 3.4, he soluion (347) above describes a circle in he normalized sae plane, of radius A. If we le he radius go o zero, we can see ha he circle is cenered a m C = M, j L =, which coincides wih he dc soluion of he circui. As ime increases, he soluion moves in he clockwise direcion around he cener; his mus be rue because he capacior is in series wih he inducor, and if he normalized inducor curren is posiive, hen he capacior charges and m C increases. In general, he normalized sae plane rajecory of an undamped woelemen resonan circui is circular and is cenered a he dc soluion of he circui. he radius depends on he iniial values of j L and m C, and remains consan. I can also be seen from Eq. (347) and Fig. 3.4 ha ime is relaed o he angle hrough which he rajecory moves. During an inerval of ime, he rajecory moves hrough an arc of angle ω. So he lengh of ringing inervals and heir angles in he normalized sae plane can be easily relaed. oe, his is no necessarily rue for sysems oher han he resonan ank circui considered here. he ank circui of he parallel resonan converer he name of he parallel resonan converer can presen some confusion, because alhough is load is conneced in parallel wih he ank capacior, he ank capacior and inducor are effecively in series. In consequence, he ime domain response and normalized sae plane rajecory are quie similar o ha of he series resonan converer ank circui.

17 Chaper 3. Sae Plane Analysis L As shown in chaper 5, during each i L subinerval of he operaion of he parallel resonan converer, is ank circui can be reduced o a V C v C I configuraion of he form shown in Fig his differs from he circui of Fig. 3. only by he addiion of consan curren source I. he Fig ank circui, driven by effec of his exra source is o shif he dc soluion consan volage source, V, and consan curren source, I of he circui, and hence also he cener of he. circular rajecory. he sae equaions of he circui are L di L() = V d v C () C dv C() = i d L () I (348) In normalized form, he sae equaions become dj L () ω d = M m C ( dm C () ω d = j L () J (349) where J = I R / V g. he soluion is m C () = M (m C () M ) cos(ω ϕ) (j L () J ) sin(ω ϕ) j L () = J (j L () J ) cos(ω ϕ) (m C () M ) sin(ω ϕ) (35) j L j () L J m () C r. M m C Fig. 3.6 ormalized sae plane rajecory for he circui of Fig As shown in Fig. 3.6, his represens a circular arc cenered a he dc soluion m C = M, j L = J, whose radius r depends on he iniial condiions and is given by r = (m C () M ) (j L () J ) (35) As in he case of he circui of Fig. 3., he lengh of ringing inervals and heir angles in he normalized sae plane can be easily relaed. During an inerval of ime, he rajecory moves hrough an arc of angle ω. All of he fundamenal conceps necessary for an exac analysis of he series, parallel, and oher resonan converers have now been discussed. Various argumens involving he flow of charge and fluxlinkages can be used o relae he ank waveforms o he average erminal volages and currens of he converer. he waveforms can be normalized, which causes he sae plane rajecories o assume circular pahs. As seen in he nex wo chapers, hese conceps allow

18 Principles of Resonan Power Conversion closedform analyical soluion of he characerisics of he series and parallel resonan converers in a direc manner. hey also aid in he undersanding of resonan swich converers. REFERECES [] G.W. Weser and R.D. Middlebrook, Low Frequency Characerizaion of Swiched DcDc Converers, IEEE ransacions on Aerospace and Elecronic Sysems, vol. AES9, May 973, pp [] Seven G. raber and Rober W. Erickson, "SeadySae Analysis of he Duy Cycle Conrolled Series Resonan Converer", IEEE Power Elecronics Specialiss Conference, 987 Record, pp [3] R. Orugani and F.C. Lee, Resonan Power Processors, Par : Sae Plane Analysis, IEEE ransacions on Indusry Applicaions, vol. IA, ov/dec 985, pp [4] R. Orugani and F.C. Lee, Sae Plane Analysis of he Parallel Resonan Converer, IEEE Power Elecronics Specialiss Conference, 985 Record, pp. 5673, June 985. [5] C.Q. Lee and K. Siri, Analysis and Design of Series Resonan Converer by Sae Plane Diagram, IEEE ransacions on Aerospace and Elecronic Sysems, vol. AES, no. 6, pp , ovember 986.