Rectifier Design

8/12/2019 Rectifier Design

http://slidepdf.com/reader/full/rectifier-design 1/10

332 l E E E TRANSACTIONS ON POWER ELECTRONICS, VOL. 7. NO. 1. A P R IL 1992

Rectifier Design for Minimum Line-Current

Harmonics and Maximum Power FactorArthur W . Kelley, Member, IEEE, and Will iam F. Yadusky, Member, IEEE

Abstract-Rectifier line-cu rrent harm onics interfe re withproper power system operat ion, reduce rect i fier power factor,and limit the power available from a given service. The recti-fier's output fil ter inductance determines the rectifier line-cur-rent waveform, the l ine-current harmonics, a nd the power fac-tor. Classical rectifier analysis usually assumes a near-infiniteoutput fil ter inductance, which introduces significant error inthe estimation of line-current harm onics and power facto r. Thispaper presents a quan titative analysis of single- and three-phaserecti fier l ine-current harmon ics and power factor as a functionof the output fil ter inductance. For the single-phase rectifier,one value of finite output fi l ter inductance produces maximumpower factor and a different value of finite outpu t fil ter induc-tance produces minimum line-current harmo nics. Fo r the three -phase rectifier, a near-infinite output fil ter induc tance producesminimum l ine-current harmonics and maximum power factor,

and the smal lest inductance that approxim ates a near-infiniteinductance is determined.

I . INTRODUCTION

IGH line-current harmonics and low power factorH ave recently received increased scrutiny from both

power-systems and power-electronics perspectives due toincreased installation of rectifiers in applications such asmachine drives, electronic lighting ballasts, and uninter-ruptible power supplies. Line-current harmonics prevent

full utilization of the installed service by increasing therms line current without delivering power and reduce rec-tifier power factor. Line-current harmonics cause over-heating of power system components, and trigger protec-

tive devices prematurely. In addition, propagation of line-current harmonics into the power system interferes withthe operation of sensitive electronic equipment sharing therectifier supply.

Full-wave rectifiers, as shown in Fig. 1 convert a sin-gle-phase ac source usor a three-phase ac source usA,usB,and usc to a high-ripple dc voltage ox. The output filter,

consisting of Lo and C O , attenuates the ripple in ox and

Manuscript received November 28, 1989; revised August 1, 1991. This

research was sponsored by the Electric Power Research Center of NorthCarolina State University and supported by the National Science Founda-tion under grant ECS-8806171. T he original version of this paper was pre-sented at the 1989 Applied Power Electronics Conference (APEC'89 ), Bal-

timore, MD, March 13-17, 1989.

A. W. Kelley is with the Department of Electrical and Computer Engi-neering, North Carolina State University, Raleigh, NC 27695-791 1.

W. F . Yadusky was with North Carolin a State University w hen this workwas performed and is now with Exide Electronics, 3201 Sp ring Forest Road,

Raleigh, NC 27604.IEEE Log Number 9106960.

L O

-

'sc

(b)

Fig. 1 . Rectifiers commonly used for conversion of power system ac volt-age to unregulated dc. (a) Single-phase rectifier. (b) Three-phase rectifier.

supplies a low-ripple unregulated dc voltage V o to theload. In a simplified analysis, os is a zero-impedance

source, and osA vsE,and oSc are balanced zero-imped-ance sources. The diodes are piecewise-linear elementsmodeled as ideal switches with zero-forward voltage dropwhen on, zero reverse leakage current when off, and in-

stantaneous switching. The output filter inductor Lo andcapacitor COare linear and lossless. A near-infinite outputcapacitor COand a near-zero-ripple capacitor voltage z ic0

are also assumed. Using these assumptions, the output fil-ter inductance L o determines the rectifier line-current har-monics and power factor, but this important relationshipis frequently subverted by the further assumption of anear-infinite Lo resulting in a near-zero-ripple inductor

current ix. Overly simplified waveshapes for single-phaserectifier line current is and three-phase rectifier line cur-

rents isA, i sB,and isc result, and grossly inaccurate esti-mates of rectifier line-current harmon ics and power factor

are produced.This paper describes a computer-simulation-based

analysis of rectifier line-current harmonics and power fac-

0885-8993/92$03.00 992 IEEE



KELLEY AND YADUSKY: RECTIFIER DESIGN FOR MINIMUM LINE-CURRENT HARMONICS 333

tor that is verified by comparison with laboratory mea-

surement and presents design curves for rectifier line-cur-

rent harmonics and power factor as a function of a outputfilter inductance Lo. For the single-phase rectifier, theclassical near-infinite output filter inductor is shown to

produce maximum power factor, but not to produce min-

imum rectifier line-current harmonics, and the inductancethat produces minimum rectifier line-current harmonics isdetermined. For the three-phase rectifier, a near-infinite

output filter inductor is shown to produce minimum rec-tifier line-current harmonics and maximum power factor,

and the minimum value of output inductance needed toapproximate an infinite inductor is determined.

11. NORMALIZATION

Before the rectifier analysis and design relationships a redeveloped, the rectifiers of Fig. 1 are normalized with re-spect to the set of references shown in Table I to producethe normalized rectifiers shown in Fig. 2. Note that an- appended to a subscript indicates a normalized quan-t i ty. However, angles in either degrees or radians are al-

ready normalized with respect to the period of the fun-damental of source voltage, have the same numericalvalue in both the normalized and circuit domains, and the-N is omitted from the subscript for angles.

Normalization is based on three principal references: avoltage reference, a power reference, and a frequency (ortime) reference. For the single-phase rectifier, the nomi-

nal rms value VS nom)f the source voltage vs s the volt-age reference VREF. For the thre e-pha se rectifie r, the nom -

balanced line-to-neutral source voltage is the voltage ref-erence. The nominal rectifier output power Po(,o,) is thepower reference PREF. The nominal source frequencyfS nom) is the frequency reference fREF, where the nominal

period of the source TS(,,,) = 1 fS(nom) is the companiontime reference TREF = l/fREF.

The voltage and power references are used to derive acurrent referen ce ZREF = P R E F / VREF, wh ich is the rms cu r-rent drawn from a voltage source VREF by a linear load ofapparent power equal in magnitude to PREF. Similarly, theimpedance reference ZREF = VR E F/Z R E F = V/2REF/PREF isthe impedance of a linear load connected to VREF drawingapparent power P R E F .

Table I1 shows the normalization and design relation-ships based on the references of Table I. The normaliza-tion relationships, (N. 1)-(N. 13), translate rectifier circuitquantities into dimensionless normalized quantities forcomparison with the normalized design curves found inthis paper. For example, as shown by (N.4) the normal-ized single-phase sourc e voltage is the dimensionlessratio of the actual source voltage vs o the voltage refer-ence VREF; as shown by (N.5) the normalized single-phaserectifier current is is the ratio of the actual rectifier cur-rent is to the current reference ZRE.. The normalized value

of a circuit element is found from the normalized imped-ance of that element at frequency f&F. Fo r example, thenormalized impedance Z, of the output fi lter inductor

- -inal rms value VS.4(nom) - VSB(nom) - vSC(nom) of the

TABLE INOR M ALI ZATI ONEF ER ENC ES

Quantity Symbol Value

= V,,,,,,-(single-phaseoltage VREFrectifier) nominal rms source

voltage

I.',,(,,,,-(three-phaserectifier) nominal rms line-to-neutral source voltage

= vS,4(nom) = VSB(nom) =

Power PREF = P,(,,,,-nominal rectifier

output power

Frequency f R E F = j,(,,,,-nominal freq uenc y of

source

Time TKEF = Ts(nom) = 1 f,(nom)-nominalperiod of source

Current IKEF = PREFVR,,-rms current

(derived) drawn from voltage referenceV by linear load of

apparent power P R E P

I np e d anc e(derived)

ZKEF = V~,,/P,,,--irnpedance oflinear load drawing apparent

power P,,, from source VF

(4

0-N

(b)

Fig. 2. Rectifiers of Pig. 1 normalized with respect to voltage, power, andfrequency references of Tabl e I. (a) Normalized single-phase rectifier. (b)Normalized three-phase rectifier.

Lo is the ratio of the indu ctor impe dan ce Z,, at the ref-erence frequency to the reference impedance ZREF:

ZLO ( ~ ~ ~ R E F ) L o

ZREF REF I h=



334 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 7. NO. 2. APRlL 1991

1

TABLE I1NOR M ALI ZATI ONN D D E S I G NELATIONSHIPS

Normalization Eqn . Design Eqn.Relationships No. Relationships No.

o < I I I I

and the normalized output filter inductance Lo.N is defined

by (N.12).The design relationships, (D . 1)-(D. 13) of Table 11, are

simply the inverse of the no rmalization relationships and

translate dimensionless normalized quantities into actualcircuit quantities. Rectifier design for minim um line-cur-

rent harmonics and maximum power factor uses the de-sign relationships to choose actual circuit quantities basedon desired normalized quantities. Examples of single- and

three-phase rectifier design are presented subsequently.

111. CLASSICAL NALYS IS

Traditionally, the principal output-filter design crite-rion is attenuation of output vo ltage ripple, and the rela-tionship between output filter design and rectifier line-cur-rent harmonics and power factor is ignored. Th e classical

line-current analysis assumes a near-infinite output filtercapacitance Co.N causing a near-zero-ripple output filter

capac itor voltag e and also assum es a near-infiniteoutput filter inductance LO.N causing a near-zero-ripple

output filter inductor current iX.N [l]. The single- andthree-phase rectifier waveforms that result from these as-

sumptions are shown in Fig. 3. The rectifier line-currentwaveforms are easily analyzed for line-current harmonicsand power factor, which explains the popularity of theclassical analysis.

1 ,XN

o l , l l l l l l , l l l l

0 000 0.167 0.333 0.500 0.667 0.833 1.000

tN

(b)

Fig. 3. Normalized rectifier waveforms for near-infinite output filter in-

ductance and near-zero-ripple output filter inductor current. (a) Single-phaserectifier source voltag e u ~ ~ , ~ .ectifier line current is-,,,, rectifier output volt-age v ~ . ~ ,nd output filter inductor current ix N. (b) Three-phase rectifierphase-A sou rce volt age phase-A rectifier line curre nt iSA-,,,. ectifieroutput voltage u ~ ~ ~ ,nd output filter inductor current ix-,v.

In practice, the output filter capacitor Co.N is usuaZ/y

large enough to produce sufficiently near-zero-ripple o ut-put filter capacitor voltage uo.N with respect to rectifierline-current harmonics and power factor, and this as-sumption is often justified. H oweve r, the assumption of

near-infinite output filter inductance and near-zero-ripple

output filter inductor current iX.Nis almost never met inpractice and produces gross errors in the classical analysisof rectifier line-current harmon ics and power factor. Prac-

tical rectifier line-current waveshapes are much morecomplex than those shown in Fig. 3, and the followingsection describes a computer simulation for determining

these waveshapes.

IV. C O M P U T E RIMULATION

The normalized rectifier waveforms in the periodic

steady-state condition are determined by time-domaincomputer simulation. Subsequent analysis of the wave-

forms determines the line-current harmonics and powerfactor. The time-domain simulation of circuits is a well-

developed analysis technique, and has been often de-scribed in the literature [2], [ 3 ] . The simulation tech-niques used to produce the results reported in this paper

are summarized in this section.The nonlinear differential equations that describe both

single- and three-phase rectifiers are cast in the same state-

variable formulation. The two state variables are the out-

put filter inductor current iX .N and the output filter capac-



KELLEY AND YADUSKY: RECTIFIER DESIGN FOR M I NI M UM L I NE - CURRE NT HARM ONI CS 335

itor voltage uCO+ The state equations fo r both rectifiers

are:

(3)

The values of L O - N and CO-N re constant. In addition, aconstant output power load is assumed and PO.N is heldconstant. Therefore, (2) is nonlinear due to the reciprocalrelationship of output current io.N and capacitor voltageZ I ~ ~ . ~or constant output power PO.N.

For the single-phase rectifier, the nonlinearity of d iodesD I - D 4 s embedded in (3) because the filter input voltagezjX. is a function of the diodes' state as determined by the

voltage source and the state variables i X - N andu C 0 . N . If iX.N> O or 1 > v ~ ~ . ~hen one pair ofdiodes is on with z / ~ . ~I 1 and iX.,,, circulates through

the voltage source. If = 0 and L J ~ . ~ J uCO.N,henall diodes are off, z / ~ . ~ and the current throughthe voltage source is zero.

Similarly for the three-phase rectifier, the nonlinearity

of diodes DI-D 6 is embedded in (3) because the filter in-put voltage u ~ - ~s a function of the diodes' state as de-termined by the three-phase voltage source Z I ~ ~ . ~ ,SB.N,

z i S C - N , and the state variables i X - N and vCO.,,,. If iX .N > 0

or the largest positive line-to-line voltage is greater thanz / ~ ~ . ~ ,hen the pair of diodes associated with the largestpositive line-to-line voltage is on with Z I ~ . ~qual to thisline-to-line voltage, and iX-N circulates through the two

line-to-neutral voltag e sources associa ted with this line-to-line voltage. The current through the third line-to-neu-tral voltage source is zero. If iX .N = 0 and the absolute

value of the largest line-to-line voltage is less than V ~ O - N ,

then all diodes are off, v ~ . ~Y ~ ~ - ~ ,nd the currents

through all voltage sources are zero.The two state equations (2) and (3) are coded as a pair

of Fortran subroutines, and a publicly available variable-time-step Runge-Kutta progra m [4] num erically inte-grates the state equations with respect to time to find thecircuit waveforms in the periodic steady-state condition(PSSC). A Newton-Raphson-based iterative meth od [5]is used to aid convergence to the PSSC. After the PSSC

is reached, the simulation generates fixed-time-step dis-crete-time waveforms in the PSSC over one normalizedperiod TS.Nwith m = 1024 points per period and integra-

tion time step T,.,/m. The m discrete points of normal-

ized time are

jt N j ) = SN j = 0, 1, 2, * * * , m 1). (4)

m

For the single-phase rectifier, the discrete-time represen-tation of source voltage Z I ~ . ~ ~ ) ,ine current is i S . N ( j ) ,filter input voltage ~ ~ . ~ j )nductor current iX.N j ) , and

capacitor voltage uCO.N ) re saved in a file for furtherQnalvcic Fnr the three-nhane rectifier. the discrete-time

A line current i S A W N j ) ,ilter input voltage ~ ~ . ~ j ) ,n-

ductor current iX.N j ) , nd capacitor voltage u ~ ~ - ~) resaved in a file for further analysis. The an alysis of these

waveforms is described in the next section.

V . DEFINITIONSN D ANALYSIS

This section d efines rectifier line-current harmo nics andpower factor, where IEEE Standard 5 19 definitions areused, if possible [6]. In addition, analysis of discrete-time

representations of rectifier waveforms for harmonic con-tent and power factor is described. The definitions andanalysis are illustrated with respect to the single-phase

rectifier shown in Fig. 2(a), but when taken on a per-phasebasis are identical for the three-phase rectifier shown in

Fig. 2(b). An example of time waveforms for the single-phase rectifier with finite filter inductance are illustratedin Fig. 4.

The single-phase rectifier is driven by a sinusoidal volt-

age source:

The rectifier current i S N ( j ) s nonsinusoidal and periodicand is represented by the Fourier series:

(6)

and is composed of a fundamental i S l ) . N j )nd higherorder harmonics i S ( h ) - N ( j )here h > 1 . The positive-

going zero crossing of the source voltage at t N j )= 0 isthe phase reference for the Fourier representation in (6) ,and the phase angles + S h) of harmonic h in (6) are refer-enced to the fundamental (rather than the harmonic). Adiscrete Fourier transform (DFT) is used to find the rmsvalue IS l)-N nd phase angle +s l ) of the fundamental as

illustrated in Fig. 4, and the rms values Z S h ) . N and phaseangles +S(h) of the harmonics of i S . N ( j ) .

of iS-,,,(j) is the square root of the

sum of the squares of the fundamental and the harmonics:

The rms value

(7)

Therefore, the rectifier current harmonics increase the rms

value of the rectifier current above that of the fun damentalalone. As a check, the rms value is found by taking

the square root of the mean value of the squ ares of the mdiscrete values of i S . N ( j )over one period:

The real power P.F.Nupplied by the source is obtained



336

-3 - \ ’ I I I I I I I I

0 000 0.250 0.500 0 750 1.000

tN

Fig. 4. Example of nonsinusoidal rectifier line current i drawn from asinusoidal source us., for finite output filter inductor. Th e line-current fun-damental is,,,., lags the source voltage by angle J~,~,.

~ ~ . ~ ( j )iS.N( j ) over one period:

(9)

The circuit is lossless, and PS.N, as calculated from sim-ulation waveforms, is checked by comparison with theconstant output power PO.N. Since the source voltage

j ) is a fundamental-frequency sinusoid, only the

rectifier current fundamental iS I).N ( j ) ontributes to realpower PS.N, and (9) reduces to

1 m - l

PS-N = u S - N ( J ’ ) i S - N ( j ) .m j O

Ps-N = VS-N~S I)-NOS ( I ) . (10)

The rectifier power factor PF, is the ratio of the realpower P,, to the apparent power VS.,,, * Is -N delivered by

the source:

As a check, the rectifier power factor is calculated usingboth (11) and (12).

The expression for rectifier power factor (12) containsthe familiar displacement power factor term cos in

which 4s(1)s the angle between the sinusoidal sourcevoltage ( ) and the sinusoidal rectifier current fun-damental iS (I) .N( j ) . The displacement power factor ismade unity by reduction of 4s(I)o zero. In addition, theexpression for rectifier power factor contains the termZ S l ) . N / Z S . N , which embodies the effects of rectifier linecurrent harmonics on the overall power factor. The au-thors h ave been unable to find a generally accepted namefor this term and have called it the “purity factor” (de-spite the moral overtones) because ZS(I).N/ZS.N = l impliesthat i s . N( j ) is a pure sinusoid with no harmonic content,whereas Zs(l).N/Zs.N < 1 implies that is.N(j) is a less-puresinusoid with greater harmonic content. The purity factor

is easily related to the familiar total harmonic distortion

(THD) expressed in p ercent by

IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 7. NO. 2. APRIL 1991

Exam ination of (12 ) show s that a near-unity powe r fac-tor rectifier must simultaneously have both a near-unitydisplacement power factor and a near-unity purity factor(or near-zero THD). In the remainder of the paper, theexplicit dependence of the discrete-time circuit wave-

forms on ( j ) s omitted, for example, ~ ~ . ~ ( j )s writtenas etc.

VI. DERIVATIONF DESIGN ELATIONSHIPS

The analysis by simulation determines the rectifier line-current fundamental and harmonics, displacement powerfactor, purity factor, and overall power factor for one

combination of Vs.N o r VsA. N = VsB.N = V S C - N , f s .N , L O - N ,

CO.,,,, and Po-N.Design relationships for line-current fun-damental and harmonics, displacement pow er factor, pu-

rity factor, an d overall power fac tor as a function of Lo.,,,are derived by holding Vs-Nor VSA., = VsB.N = VSC.,v,

f S . N , C O - N , and PO.N, constant and analyzing rectifier op-eration in the PSSC for discrete values of LO.,,, spaced atregular intervals.

The rectifier is simulated using n ominal source voltagewith VS-N = 1 or VSA., = VsB.N = VSc., = I , nominalsource frequen cy withf,-, = 1, and with nominal output

power so that PO.N = 1. A conveniently large value ofCO-N= 1000 is found to be sufficient to keep the peak-to-peak output voltage ripple less than 0.2 for all valuesof L0.N. However, smaller Co.N values do not dramati-cally affect the results especially for larger L0.N.

VII. VERIFICATION

The sim ulation’s accuracy is verified by com parison towaveforms and partial power factor data available in the

literature [7]-[ 121. In addition , rectifiers were construc tedin the laboratory, an d waveforms acquired by a Tektronix11401 digitizing oscilloscope were transferred to a Ma-intosh IIx computer over the IEEE-488 instrumentation

bus using a National Instruments IEEE-488 card and Lab-view software. The fidelity of the simulated time wave-

forms to the laboratory time waveforms is excellent. Th elaboratory time waveforms were also analyzed for line-current fundamental and harmonics, displacement power

factor, purity factor, and overall power factor as a func-tion of L0.N using (6)-(12) and the DF T. Th e laboratorymeasurements compare extremely well with the simulateddata, both of which are shown in the following two sec-tions that provide design relationships for line-currentharmonics and power factor as a function of L O - N for sin-gle- and three-phase rectifiers.

-3

VIII. SINGLE-PHASEECTIFIER ESIGN ELATIONSHIPS

This section presents design relationships for the rmsvalue and phase angle of rectifier line-current fundamen-

tal and harmonics, and for rectifier displacement powerfactor, purity factor, and overall power factor as a func-tion of normalized output filter inductance LO.N for the

single-phase rectifier shown in F ig. 2 (a). Portions of thisproblem have been examined by previous investigators



KELLEY AND YADUSKY: RECTIFIER DESIGN FOR MINIMUM LINE-CURRENT HARMONICS

I I I I

337

[7]-[9], but minimum rectifier line-current harmonic s and

their effect on power factor have not been previously di-rectly examined. The previous investigators also providedesign relationships for Vo-Nas a function of LO-N that arenot reproduced in this paper.

Representative normalized single-phase rectifier simu-lation time waveforms of source voltage rectifiercurrent is-N, filter input vo ltage v ~ - ~nd output-filter-in-ductor current iXWNre shown in Fig. 5 for three values ofnormalized output filter inductance. The conduction time

intervals for each diode are also indicated. The timewaveforms measured in the laboratory are essentiallyidentical to the waveforms shown in Fig. 5. The wave-

forms of iX-N nd are dramatically different for differ-ent output filter inductances, and Fig. 5 illustrates threedistinct modes of operation which, adopting the nomen-clature of Dewan [8], are the discontinuous conductionmode I (DCM I), discontinuous conduction mode I1(DCM 11), and continuous conduction mode (CCM ), re-spectively.

The DCM I occurs for LO-N < 0.027 nd is illustratedin Fig. 5(a) for Lo-N = 0.010. The is-,,,and iX-N aveforms

are characterized by two short-duration high-peak-valuecurrent pulses during which either D1 and D4 conduct orD2 and D3 conduct. Two zero-current intervals separatethe single D nd D4 conduction interval and the singleD2 and D3 conduction interval. The DCM I1 occurs for0.027 < LO-N < 0.043 and is illustrated in Fig. 5(b) for

LO-N = 0.037.The DCM I1 is characterized by a zero-current interval separating two D1 nd D4 onduction in-tervals and com mutation from D1 and D4 to D2 and D3 attN = 0.5, and a zero-current interval separating two D2

and D3 conduction intervals with commutation from D2and D3 to D, and D4 at t N = 1.0. The CCM occurs for

Lo-N > 0.043 and is illustrated in Fig. 5(c) for LO-N =

0.090. Diode conduction alternates at tN = 0.5 and at tN= 1.Obetween a single D1 and D4 conduction interval and

a single D2 and D3 conduction interval. As the name im-plies, no zero-current interval exis ts in the CCM. Com-parison of Fig. 5(a) , (b), and (c) shows the wide variation

in rectifier-current waveforms that result from differingvalues of Lo-N and comparison to Fig. 3(a) shows thesevere error that is incurred by using a near-infinite-in-ductance approximation to represent finite L0.N.

Figs. 6 and 7 how design relationships for single-ph aserectifier line-current harmonics and power factor as ob-tained from both simulation and laboratory measurement.The simulation data are shown as continuous curves andthe laboratory data are shown as discrete points. Fig. 6(a)shows the rms values-Zs(l)-N, I S 3 ) - N , ZS ~)-~,S(7)-N, andZs(9+-and F ig . 6(b) shows the phase angles-4s(l, ,

4~(3), s ( s ) , ~ (7 ), nd 4s(9)-0f the current fundam ental,and the third, fifth, seventh, and ninth current harmonics,respectively, of the rectifier current is-,,, as a function of

L0-N.As expected , all even-order harmonics are zero. Fig.

7 hows the relationship between the displacement powerfactor cos cl , , the purity factor Z S I ) - N / Z S - N , and the over-all power factor PFs as a function of L0-N. The range of

-1

-2-3

'X-N-

1 -

0 1

- X-N,

I I 1D, ON D ON

D4ON

r I

0.000 0.250 0 500 0 750 1 oootN

(a)

3 1 ..

-3 I 1 I 1

D, ON D, ON

D4 ON

I I

0.000 0.250 0.500 0 750 1 @O@

tN

(b)

3 1

-2I

-3 II I I I I I

D. ON D,

D ON D ON

I I I

0 000 0.250 0.500 0 750 I oootN

(C)

Fig. 5 . Normalized single-phase re ctifier waveforms: sourc e voltage

rectifier line current rectifier output voltage and output filter in-ductor current iX.N for (a) discontinuous conduction mode I (DCM I) forLO.N= 0.010 (b) discontinuous conduction mode I1 (DCM 11) for LO.N=

0.037 and (c) continuous conduction mode (CCM) for LO.N= 0.090. Theconduction intervals for diode s D,- D, are indicated by the shaded areas

below the waveforms.

Lo-N over which each condu ction mode occurs is also in-dicated in Figs. 6 and 7. The laboratory measurements arenearly identical to the simulation confirming the accuracyof both. In add ition, the design relationships for displace-



338 IEEE TRANSACTIONS ON POWER ELECTRONICS. V O L . 7. NO. 2. APRIL 1997

DCMI DCMII CCM5 1.5 t t t-4

i i

b - N

Key: -- s(lp -O-@s(3)? '&45p 4 4 q 7 ) , - 4s(9)

(b)

Fig. 6 . Harmonics of single-phase rectifier line current i y .N . (a) The nor-malized rms value Z,,,,., of the fundamental, and the normalized rms valuesZs 3 ) .N , Zs 5 ) .N , I . and Zs 9)., of the harmonics. (b ) The phase angle c~~ I ,of the fundamental, and the phase angles 9s(z,.[ S J . and ,yls, of theharmonics with respect to the source voltage U . s a function of nornial-ized output filter inductance lo^ . Simulation data are shown as continuouscurves, and laboratory data are shown as discrete points.

DCM I DCM CCM

Key: .....A .....I ( ~ ) - N/IS-,, --0-cOS@q1), -PFs

Fig. 7. Single-phase rectifier displacement power factor cos &.,,,,purityfactor Z s l ) . N / Z s . N , and overall power factor PF,y s a function of ou tput filterinductance Lo.,. Simulation data are shown as continuous curves, and lab-

oratory data are shown as discrete points.

ment power factor and overall power factor appear in thepreviously cited literature and match those shown in Fig.7.

As shown in Fig. 6(b), the line-current fundamentalphase angle @ scl ,s near zero for small Lo.,, rises to thelargest-magnitude phase angle I ~~~) J38 for Lo-, =

0.033, and returns to near zero for large LO.,,,. Therefore,

as shown in Fig. 7, the displacement power factor cosq b S , ) is near unity for small Lo.,. falls to 0 .79 for Lo.*, =

0.033 and rises again to near unity for large Lo., . Ex-

amination of I O ) shows that since V,., and Po., = P,.,

are constant, the rms v alue of Z,,,)., of current fundamen-tal as shown in Fig. 6(a) is inversely proprotional to cos

As seen in Fig. 6(a), the normalized rms values Z,,,,.,

of the current harmonics are large for small LO- are at aminimum for Lo., slightly less than that required to enterDCM 11, slightly larger for Lo., in the DCM I1 range, andapproach their near-infinite-inductance values in the CCMrange. Therefore, in Fig. 7, the purity factor is low forsmall Lo.,, at a maximum of 0.94 for Lo-, = 0.030, andslightly lower at 0.90 for large Lo.,.

A s shown by (12 ), the overall pow er factor PF, is the

product of the purity factor Zs,,,.,v/Zs_, and the displace-ment power factor cos qbscl), and, a s shown in Fig. 7, thesetwo influences are in conflict. For small Lo.,, the overall

power factor is at a global minimum, despite the near-unity displacement power facto r, because the rectifier cur-rent is very distorted and the punty factor is low. The

overall power factor is at a local maximum PFs = 0.76for LO., = 0.016 because of the reduction in waveformdistortion and the increase of purity factor. However, the

overall power factor is at a local minimum PF, = 0.73for LO.N = 0.039 because of the worsening displacementpower factor. The overall power factor is at a global max-imum PF, = 0.90 for a near-infinite Lo., due to the im-provement of displacement power factor and the rela-tively good purity factor.

The maximum overall PF, = 0.90 occurs for near-in-finite LO-,. Howe ver, operation in this condition requiresan uneconomically large and impractical outpu t filter in-ductor. A s noted by Dewan [SI, maximum practical PFs

= 0.76 for a reasonably-sized output filter inductor oc-curs in DCM I with Lo.,v = 0.016. The waveforms used

in Fig. 4 to illustrate power factor are obtained in thisoperating condition. The minimum overall line-currentharmonics occur not for near-infinite Lo., but for =

0.030 where Zs(l,.,v/Zs-N = 0.94.

As a design example, a Pocnom) 1200 W , Vs/S nom)

120 V , fS(,, ,,,)= 60 Hz rectifier has normalization refer-ences VRE, = 120 V , P R E F = 120 0 W,fRE, = 60 Hz, ZREF

= 10 A , and ZRE, = 12 Q. Substitution of the normal-ization references and LO.,v = 0.016 into design equation(D.12) gives LO = 3.2 mH for a maximum power factordesign. Substitution of the normalization references and

LO., = 0.030 into design equation (D.12) gives L o =

6 mH a for minimum line-current harmonics design.

4SCI).

IX . THREE-PHASEECTIFIER ESIGNRELATIONSHIPS

This section presents design relationships for the rmsvalue and phase angle of rectifier line-current fundamen-tal and harmonics, and for rectifier displacement powerfactor, purity factor, and overall power factor as a func-

tion of normalized output filter inductance LO.N for the



KELLEY AND YAD USKY: RECTIFIER DESIGN FOR MINIMUM LINE-CURRENT HARMONICS

2

0

8

339

- - . - - -IX-N

--1 q l

1 1 1 1

101 --2-

0

IX-Nl n A

-A

I I l l I 1 1

7 SA-N

101 --2 -

1 ,X-N

0 1 I I I I I 1 1 , I

D. ON D, ON D, ON

D. ON DeON D ON

0.000 0.167 0.333 0 500 0.666 0.833 1.000

(C)

t N

Fig. 8. Normalized three-phase rectifier waveform s: phase-A source volt-age phase-A rectifier line current rectifier output voltage vX.,,,,

and output filter inducto r current iX.Nfor (a) discontinuous conduction modeI (DCM I for Lo.,,, = 0.0024 (b) discontinuous conduction mode I1 (DCM

11) for Lo.,,, = 0.0064 and (c) continuous conduction mode CCM) orLo.,,, = 0.10. The conduction intervals for diodes DI-D, are indicated by

the shaded areas below the waveforms.

three-phase rectifier shown in Fig. 2(b). Aspects of thisproblem have been treated by previous investigato rs [101-

[121, and the three-phase design relationships are pre-sented for completeness, unification, and verification of

prior work and for contrast with the sin gle-phase rectifier.

As with the single-phase rectifier, the previous investi-gators also provide three-phase rectifier design relation-ships for Vo-N as a function of LO.Nhat are not reproducedin this paper.

Representative normalized three-phase rectifier timewaveforms of phase-A source voltage vSA-N, phase-A rec-

tifier current filter input voltage v ~ - ~ ,nd output-filter-inductor current i X - N are shown in Fig. 8 for threevalues of normalized output filter inductance. The con-

duction time interval for each diode is also indicated. Thetime w aveforms measured in the laboratory are essentiallyidentical to the waveforms shown in Fig. 8. The wave-

forms for phases B and C are produced by shifting thewaveforms for phase A by one-third period and two-thirdsperiod, respectively. The three-phase rectifier also ex-hibits DCM I, DCM 11, and CCM operation, as illustratedin Fig. 8. As with the single-p hase rectifier, a comparisonof Fig. 8 with Fig. 3(b) reveals the significant error thatresults from assuming a finite L O - N to be near-infinite. Thedescription of the three modes of operation in the previoussection for the single-phase rectifier is applicable to the

three-phase rectifier, with the exception that three-phaserectifier has six diodes and six conduction in tervals as op-posed to the four diodes and two conduction intervals forthe single-phase rectifier. The boundary between DCM Iand I1 occurs for L O - N = 0.0050, and the boundary be-tween DCM I1 and CC M occurs for L O - N = 0.0083.

Figs. 9 and 10show design relationships for three-phaserectifier line-current harmonics and power factor as ob-tained from both simulation and laboratory measurement.The simulation data are shown as continuous curves and

the laboratory data are shown as discrete points. Fig . 9(a)

shows the rms values-hA(l)-N, I S A 5 ) - N , I S , 4 7 ) - N , I s A I I ) - N ,

and ISA(1 3)-K-a nd ig. 9(b) shows the phase ang les-

fundamental, and the fifth, seventh, eleventh, and thir-

teenth current harmonics, respectively, of the phase-Arectifier line current iSA.N s a function of L 0 . N . As ex-pected, all even-order harmonics and all harmonics thatare a multiple of three are zero. Fig. 10 shows the rela-tionship between the displacement power factor cos + S A I ) ,

the purity factor Z S A l ) - N / Z S A . N , and the overall power fac-tor PFsA as a function of L 0 . N . The range of L O - N over

which each conduction mode occurs is also indicated inFigs. 9 and 10. The laboratory measurements are nearlyidentical to the simulation confirming the accuracy ofboth.

In contrast to the single-phase rectifier, the line-currentfundamental phase angle + SA (I ) as shown in Fig. 9(b) isnear zero regardless of the value of L 0 . N . The largest-

magnitude phase angle = 12 and the minimumdisplacement power factor cos + SA (I ) = 0.98 occur for

L O - N = 0.0055 as shown in Figs. 9(b) and 10, respec-tively. Since the displacement power factor cos =1 regardless of L O - N , Z S A I ) - N = 0.3 3 because one third of

is delivered by phase A and the remaining two thirdsof P O - N is delivered equally by phases B and C. Therefore,

+SA(I), +SA 5)9 +SA 7) , +SA I I ) , and +SA(13)-Of the current



340 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 7, NO. 2. APRIL 1992

D C M I D C M I I C CMt , , t I

v-0.4 -, 1

I / I

D C M I D C M I I C CMt3 360 1 1

Fig. 9. Harmonics of three-phase rectifier phase-A line current isa.N. (a)Normalized rms value IsA l .Nof the fundamental, and the normalized rms

values I,,,,, N , I,,,,,.,, I s c lJ.N, and ISacl3, f the harmonics. (b) The phaseangle of the fundamental, and phase angles QsacI ,, and

asa function of normalized output f ilter inductance Lo.N. Simulation data areshown as continuous curves, and laboratory data are shown as discretepoints.

of the harmonics with respect to the line-to-neutral voltage us,,

D C M I D C M I I C CM

Fig. 10. Three-ph ase rectifier displacem ent pow er factor cos purityfactor Isacl, / I s A - N and overall power factor PF, of phase-A as a function

of output filter inductance Lo-,. Simulation data are shown as continuous

curves, and laboratory data are shown as discrete points.

LO.Nhas almost no influence on displacement power fac-tor, and the rectifier line-current harmonics and the purity

factor are the dominant influences on the overall powerfactor.

As shown by Fig. 9(a), the line-current harmonics are

high for small L 0 . N values and, except for a small portion

of Z s A 1 3 ) - N , decrease uniformly for larger L 0 . N values,reaching a minimum at L 0 . N = 0.10. Therefore, the pu-rity factor Z s A I ) . N / Z s A - N and the overall power factor PFsAare low for small L 0 . N values and incrase to a maximumZ s A I ) . N / Z s A . N = PFsA = 0.96 for L 0 . N = 0.10. A larger

value of L 0 . N does not significantly improve displacementpower factor or overall power factor. Therefore, L O - N =

0. 10 is a reasonable approximation to a near-infinite o ut-put-filter inductan ce.

Consider a modification of the previous design examp le

in which P o n o m ) = 1200 w, VsA(nom) = VSB(nom) - VSC(nom)

= 120 V line-to-neutral, andfs(nom)= 60 Hz. The nor-malization references are V R E F = 120 V , P R E F = 1200 W ,

f RE F = 60 Hz, ZREF = 10 A , and Z R E F = 12 a. Substitutionof the normalization references and LO.N= 0.10 into de-sign equation ( D.1 2) gives Lo = 20 mH.

A designer might be tempted to conclude that the sin-gle-phase rectifier requires a physically smaller inductorby comparing Lo = 20 mH for a maximum-power-factor

three-phase rectifier to Lo = 3.2 mH for a maximum-power-factor single-phase rectifier. How ever, comparison

of Figs. 5 and 8 shows a substantially higher v ~ . ~nd asubstantially lower iX.N for the three-phase rectifier as

compared to the single-phase rectifier for the same nor-malized output power P O - N = 1. Since inductor physicalsize depends both on inductance value and inductor cur-rent, conclusions based on inductance value alone aremisleading.

-

X. SUMMARY

Classical rectifier analysis based on near-infinite outputfilter inductance and near-constant filter inductor currentbecomes less satisfactory as line-current harmonics and

power factor issues increasingly concern power-systemsand power-electronics engineers. This paper providesquantitative design data for line-current harmonics and

power factor for single and three-phase rectifiers for re-alistic design situations w ith finite output-filter inductanceand appreciable current ripple. These data provide a ref-erence for designers of new equipment and for the eval-

uation of harmonic and power factor problems caused byexisting equipment.

The maximum power factor for a single-phase rectifierusing an infinite output filter inductor is 0.90, but themaximum power factor for a reasonably-sized finite-in-

ductance rectifier is 0.76. However, this operating con-dition does not result in minimum rectifier line-currentharmonics. Design of the rectifier for minimum line-cur-

rent harmonics produces a purity factor of 0.94. The max-imum power factor for a three-phase rectifier is 0.96 andoccurs for an infinite output inductance. This operating

condition also results in minimum line-curren t harmonics.Pow er factor is not significantly improved , nor are line-current harmo nics significantly reduced for outpu t filter

inductances larger than a crucial value.



KE L L EY AND YADUS KY: RE CT I F I E R DE SI GN F OR M I NI M UM L I NE - CURRE NT HARM ONI CS 34

ACKNOWLEDGMENT [I21 M. Sakui, H . Fuji ta, and M. Shioya, “A method for calculat ing har-manic currents of a three-phase bridge uncontrolled rectifier with dcfilter,” IEEE Trans. Ind. Electron., vol. IE-36, no. 3, pp. 434-440,Aug. 1989.

The authors thank L. Hall and E. Reese for their assis-tance in assembling the instrumentation and conducting

line-current harmonics and power factor m easurements.

111

I21

[31

[41

REFERENCES

J. Schaefer, Rectifier Circuits, Theory and Design. New York:Wiley, 1965.A. W. Kelley, T. G. Wilson, and H. A. Owen, Jr . , “Analysis of thetwo-coil model of the ferroresonant transformer with a rectified outputin the low-line heavy-load minimum-frequency condition,” 1983In-ternational Telecommunications Energy Con$ Rec. ( INTELEC ’83),

Tokyo, Japan, October 1983, pp. 374-381.E. B. Sharodi and S . B. Dewan, “Simulation of the six-pulse bridgeconverter with input filter,” 1985Power Electronics Specialists ’ Con5Rec. (PESC’85), Toulouse, France, June 1 985, pp. 502-508.G . E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer M erh-ods for Mathematical Computations. Englewood Cliffs, NJ: Pren-

t ice-Hall , 1977, ch. 6.F. R. Colon and T. N. Trick, “Fast periodic steady-state analysis forlarge-signal electronic circuits,” IEEE J . Solid-State Circuits, vol.SC-8, no. 4, pp. 260-269, Aug. 1973.“IEE E guide for harmonic con trol and reactive compensation of staticpower converters,” IEEE/ANSI Standard 519, 1981.F. C . Schwarz, “Time-domain analysis of the power factor for a rec-tifier-filter system with over- and sub critical inductance,” IEEE Trans.ind. Electron. Contr. Instrum., vol. IECI-20, no. 2, pp. 61-68, May1973.S . B . Dewan, “Optimum input and output filters for single-phase rec-tifier power supply,” IEEE Trans. Industry Applications, vol. IA-17,no. 3, pp. 282-288, May/June 1981.California Institute of Technology, Power Electronics Group, “In-put-current shaped ac-to-dc converters, final report,” NASA-CR-176787, prepared for NASA Lewis Research Cente r, pp. 1-49, May1986.M . Grotzbach, B. Draxler, and J . Schorner, “Line harmonics of con-trolled six-pulse bridge converters with dc ripple,’’ Rec. I987 IEEEindustry Applications Society Annu. M eet., part I, Atlanta, G A:, Oct.

S . W. H. De Haan, “Analysis of the effect of source voltage fluctua-tions of the power factor in three-phase controlled rectifiers,” IEEETrans. Industry Applications, vol. IA-22, no. 2, pp. 259-266, March/April 1986.

1987, pp. 941-945.

Arthur W. Kelley (S’78, M’85) was born in 1957in Norfolk, VA. He received the B.S.E. degree

from Duke University, Durham , NC, in 1979. Hecontinued at Duke as a James B. Du ke Fellow andreceived the M.S., and Ph.D. degrees in 1981,and 1984, respectively.

From 1985 to 1987 , Dr. Kelley was employedas a Senior Research Engineer at Sundstrand Cor-poration, Rockford, IL, where he worked onpower electronics applications to aerospace powersystems. He ioined the facultv of the Department

of Electrical and Compu ter Engineering a t North Carolina State Universityin 1987 where he currently holds the rank of Assistant Professor. H is in-terests in power electronics include PWM dc-to-dc converters, line-inter-faced ac-to-dc converters and power quality, magnetic devices, magneticmaterials, and computer-aided analysis and design of nonlinear circuits.

Dr. Kelley is a member of Sigma Xi, Phi Beta Kappa, Tau Beta Pi, andEta Kappa Nu.

William F. Yadusky (S’87, M’90) received theB.A. degree in English from the University ofNorth Carolina, Chapel Hill, in 1982, and theB.S.E.E. from North Carolina State University,Raleigh, in 1987. He was certified as an Engineerin Training in 1987.

Mr. Yadusky worked as a Graduate ResearchAssistant with the Electric Power Research Centerat North Carolina State University under Dr. Ar-thur W. Kelley in 1988 and 198 9. In 1990, Mr.Yaduskv ioined the Exide Electronics Technology“ _ _

Center, Raleigh, NC, where he helped develop selectable-input/selectable-

output on-line unintermptible power systems, PWM inverters, and printedcirucits. As an electrical design engineer on the Advanced Technology De-velopment team, Mr. Yadusky is presently responsible for designing anddeveloping high-voltage PWM rectifiers and high-frequency magneticstructures for unintermptible power systems, frequency converters, andpower conditioners.

Rectifier Design

Documents