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(^332) l E E E TRANSACTIONS ON POWER ELECTRONICS, VOL. 7. NO. 1. A P R I L 1992
Rectifier Design for Minimum Line-Current
Harmonics and Maximum Power Factor
Arthur W. Kelley, Member, IEEE, and William F. Yadusky, Member, IEEE
Abstract-Rectifier line-current harmonics interfere with proper power system operation, reduce rectifier power factor, and limit the power available from a given service. The recti- fier's output filter inductance determines the rectifier line-cur- rent waveform, the line-current harmonics, and the power fac- tor. Classical rectifier analysis usually assumes a near-infinite output filter inductance, which introduces significant error in the estimation of line-current harmonics and power factor. This paper presents a quantitative analysis of single- and three-phase rectifier line-current harmonics and power factor as a^ function of the output filter inductance. For the single-phase rectifier, one value of finite output filter inductance produces maximum power factor and a different value of finite output filter induc- tance produces minimum line-current harmonics. For the three- phase rectifier, a near-infinite output filter inductance produces minimum line-current harmonics and maximum power factor, and the smallest inductance that approximates a near-infinite inductance is determined.
I. INTRODUCTION
IGH line-current harmonics and low power factor
Hhave recently received increased scrutiny from both power-systems and power-electronics perspectives due to increased installation of rectifiers in applications such as machine drives, electronic lighting ballasts, and uninter- ruptible power supplies. Line-current harmonics prevent full utilization of the installed service by increasing the rms 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 with
the operation of sensitive electronic equipment sharing the
rectifier supply. Full-wave rectifiers, as shown in Fig. 1 convert a sin-
gle-phase ac source us or a three-phase ac source usA,usB,
and usc to a high-ripple dc voltage ox. The output filter,
consisting of Lo and CO, 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 North Carolina State University and supported by the National Science Founda- tion under grant ECS-8806171. The 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 Carolina State University when this work was performed and is now with Exide Electronics, 3201 Spring 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 Vo to the
load. 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 elements modeled as ideal switches with zero-forward voltage drop when on, zero reverse leakage current when off, and in-
stantaneous switching. The output filter inductor Lo and
capacitor CO are linear and lossless. A near-infinite output
capacitor CO and a near-zero-ripple capacitor voltage zic
are also assumed. Using these assumptions, the output fil- ter inductance Lo determines the rectifier line-current har- monics and power factor, but this important relationship is frequently subverted by the further assumption of a
near-infinite Lo resulting in a near-zero-ripple inductor
current ix. Overly simplified waveshapes for single-phase
rectifier line current is and three-phase rectifier line cur-
rents isA, isB, and isc result, and grossly inaccurate esti-
mates of rectifier line-current harmonics 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 0 1992 IEEE
KELLEY A N D 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 output filter inductance Lo. For the single-phase rectifier, the classical near-infinite output filter inductor is shown to produce maximum power factor, but not to produce min- imum rectifier line-current harmonics, and the inductance that produces minimum rectifier line-current harmonics is determined. 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 to approximate an infinite inductor is determined.
11. NORMALIZATION
Before the rectifier analysis and design relationships are developed, the rectifiers of Fig. 1 are normalized with re- spect to the set of references shown in Table I to produce the normalized rectifiers shown in Fig. 2. Note that an
- N appended to a subscript indicates a normalized quan- tity. 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 numerical
value in both the normalized and circuit domains, and the
- N is omitted from the subscript for angles. Normalization is based on three principal references: a voltage reference, a power reference, and a frequency (or time) reference. For the single-phase rectifier, the nomi- nal rms value VS(nom) of the source voltage v s is the volt- age reference VREF. For the three-phase rectifier, the nom-
balanced line-to-neutral source voltage is the voltage ref-
erence. The nominal rectifier output power Po(,o,)^ is the
power reference PREF. The nominal source frequency fS(nom) is the frequency reference fREF, where the nominal
period of the source TS(,,,) = 1 /fS(nom) is the companion
time reference TREF = l/fREF. The voltage and power references are used to derive a current reference ZREF = P R E F / VREF, which is the rms cur- rent drawn from a voltage source VREF^ by a linear load of apparent power equal in magnitude to PREF. Similarly, the impedance reference ZREF = VREF/ZREF = V/2REF/PREF is the impedance of a linear load connected to VREF drawing apparent 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 circuit
quantities into dimensionless normalized quantities for comparison with the normalized design curves found in this paper. For example, as shown by (N.4) the normal- ized single-phase source voltage is the dimensionless
ratio of the actual source voltage v s to the voltage refer-
ence VREF; as shown by (N.5) the normalized single-phase
rectifier 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. For example, the normalized impedance Z,,., of the output filter inductor
inal rms value VS.4(nom) - -^ VSB(nom) - - vSC(nom) of the
TABLE I NORMALIZATIONREFERENCES
Quantity Symbol Value
Voltage VREF =^ V,,,,,,-(single-phase rectifier) nominal rms source voltage I.',,(,,,,-(three-phase rectifier) 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 frequency of source
Time TKEF = Ts(nom) = 1 /f,(nom)-nominal period of source
Current IKEF = PREF/ VR,,-rms current
(derived) drawn from voltage reference V,,, by linear load of apparent power P R E P
Irn p eda n ce (derived)
ZKEF =^ V~,,/P,,,--irnpedance^ of linear load drawing apparent power P,,, from source V,,F
'0-N
(b) Fig. 2. Rectifiers of Pig. 1 normalized with respect to voltage, power, and frequency references of Table I. (a) Normalized single-phase rectifier. (b) Normalized three-phase rectifier.
Lo is the ratio of the inductor impedance Z,, at the ref-
erence frequency to the reference impedance ZREF:
ZLO ( ~ ~ ~ R E F ) L o
ZREF REF I h F
z,,, = - =
KELLEY AND YADUSKY: RECTIFIER DESIGN FOR MINIMUM LINE-CURRENT HARMONICS 335
itor voltage uCO+ The state equations for both rectifiers
are:
The values of LO-N and CO-N are constant. In addition, a
constant output power load is assumed and PO.N is held
constant. Therefore, (2) is nonlinear due to the reciprocal
relationship of output current io.N and capacitor voltage
Z I ~ for constant output power ~. ~ PO.N.
For the single-phase rectifier, the nonlinearity of diodes DI-D4 is embedded in (3) because the filter input voltage zjX.,,, is a function of the diodes' state as determined by the voltage source and the state variables iX-N and u C 0. N. If iX.N > O or 1 > v ~ ~. ~ , then one pair of diodes is on with z^ /^ ~^ =.^ ~^ I^ 1 ,^ and^ iX.,,,^ circulates through the voltage source. If = 0 and 1 L J ~. ~ J< uCO.N, then all diodes are off, z / ~. ~= and the current through the voltage source is zero. Similarly for the three-phase rectifier, the nonlinearity
of diodes DI-D6 is embedded in (3) because the filter in-
put voltage u ~ is a function of the diodes' state as de- - ~
termined by the three-phase voltage source Z I ~ ~. ~ , vSB.N, z i S C - N , and the state variables iX-N and vCO.,,,. If iX.N > 0 or the largest positive line-to-line voltage is greater than z / ~ ~. ~ ,then the pair of diodes associated with the largest positive line-to-line voltage is on with Z I ~. ~equal to this
line-to-line voltage, and iX-N circulates through the two
line-to-neutral voltage sources associated 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 ~ ~ ~ and 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 program [4] numerically inte- grates the state equations with respect to time to find the circuit waveforms in the periodic steady-state condition
(PSSC). A Newton-Raphson-based iterative method [ 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 normalized
period TS.N with m = 1024 points per period and integra-
tion time step T,.,/m. The m discrete points of normal-
ized time are
j
t N ( j ) = - m TS.N ( j^ =^ 0,^ 1,^ 2,^ *^ *^ *^ ,^ m^ -^ 1).^ (4)
For the single-phase rectifier, the discrete-time represen-
tation of source voltage Z I ~. ~ ( ~ ) ,line current is i S. N ( j ) ,
filter input voltage ~ ~. inductor current ~ ( j ) iX.N( j ) , and
capacitor voltage uCO.N ( j ) are saved in a file for further
Qnalvcic Fnr the three-nhane rectifier. the discrete-time
A line current i S A W N ( j ) , filter input voltage ~ ~. ~ ( j ) , in-
ductor current iX.N( j ) , and capacitor voltage u ~ ( j ) ~ are - ~
saved in a file for further analysis. The analysis of these waveforms is described in the next section.
V. DEFINITIONSAND ANALYSIS
This section defines rectifier line-current harmonics and
power factor, where IEEE Standard 5 19 definitions are
used, if possible [6]. In addition, analysis of discrete-time
representations of rectifier waveforms for harmonic con- tent and power factor is described. The definitions and analysis are illustrated with respect to the single-phase rectifier shown in Fig. 2(a), but when taken on a per-phase basis 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 illustrated in Fig. 4. The single-phase rectifier is driven by a sinusoidal volt- age source:
The rectifier current i S N ( j ) is nonsinusoidal and periodic
and is represented by the Fourier series:
and is composed of a fundamental i S ( l ). N ( j ) and higher order harmonics i S ( h ) - N ( j ) where h > 1. The positive-
going zero crossing of the source voltage at t N ( j ) = 0 is
the 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). A discrete Fourier transform (DFT) is used to find the rms value IS(l)-N and phase angle +s(l) of the fundamental as
illustrated in Fig. 4, and the rms values ZS(h).N and phase
angles +S(h) of the harmonics of iS.N(j).
of iS-,,,(j) is the square root of the
sum of the squares of the fundamental and the harmonics:
The rms value
Therefore, the rectifier current harmonics increase the rms value of the rectifier current above that of the fundamental alone. As a check, the rms value is found by taking
the square root of the mean value of the squares of the m
discrete values of iS.N(j) over one period:
The real power P.F.N supplied by the source is obtained
336
-3 - \ ’ I (^) I I (^) I I (^) I I (^) I 0.000 0.250 0.500 0.750 1. tN
Fig. 4. Example of nonsinusoidal rectifier line current ,, i drawn from a sinusoidal source us., for finite output filter inductor. The line-current fun- damental is,,,., lags the source voltage by angle $J~,~,.
~ ~. ~. (iS.N( j )j ) over one period:
The circuit is lossless, and PS.N, as calculated from sim-
ulation waveforms, is checked by comparison with the
constant output power PO.N. Since the source voltage
j ) is a fundamental-frequency sinusoid, only the
rectifier current fundamental iS(I).N ( j ) contributes to real
power PS.N, and (9) reduces to
1 m - l PS-N = - m (^) j = O u S - N ( J ’ ) i S - N ( j ).
Ps-N = VS-N~S(I)-N COS & ( I ). (10) The rectifier power factor PF, is the ratio of the real power P,, to the apparent power VS.,,, * Is-N delivered by the source:
As a check, the rectifier power factor is calculated using both (11) and (12). The expression for rectifier power factor (12) contains
the familiar displacement power factor term cos in
which 4s(1)is the angle between the sinusoidal source
voltage ( j ) and the sinusoidal rectifier current fun-
damental iS(I).N( j ). The displacement power factor is
made unity by reduction of 4s(I)to zero. In addition, the expression for rectifier power factor contains the term
ZS(l).N/ZS.N, which embodies the effects of rectifier line
current harmonics on the overall power factor. The au- thors have been unable to find a generally accepted name for this term and have called it the “purity factor” (de- spite the moral overtones) because ZS(I).N/ZS.N = l implies
that is.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-pure sinusoid with greater harmonic content. The purity factor is easily related to the familiar total harmonic distortion (THD) expressed in percent by
IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 7. NO. 2. APRIL 1991
Examination of (12) shows that a near-unity power fac- tor rectifier must simultaneously have both a near-unity
displacement power factor and a near-unity purity factor
(or near-zero THD). In the remainder of the paper, the explicit dependence of the discrete-time circuit wave-
forms on ( j ) is omitted, for example, ~ ~. ~ is written ( j )
as etc.
VI. DERIVATIONOF DESIGNRELATIONSHIPS
The analysis by simulation determines the rectifier line- current fundamental and harmonics, displacement power factor, purity factor, and overall power factor for one combination of Vs.N or VsA.N = VsB.N = V S C - N , fs.N, LO-N, CO.,,,, and Po-N. Design relationships for line-current fun- damental and harmonics, displacement power factor, pu- rity factor, and overall power factor as a function of Lo.,,, are derived by holding Vs-N or 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 at
regular intervals. The rectifier is simulated using nominal source voltage
with VS-N = 1 or VSA., = VsB.N = VSc., = I , nominal
source frequency withf,-, = 1, and with nominal output
power so that PO.N = 1. A conveniently large value of
CO-N = 1000 is found to be sufficient to keep the peak-to-
peak output voltage ripple less than 0.2% for all values
of L0.N. However, smaller Co.N values do not dramati-
cally affect the results especially for larger L0.N.
VII. VERIFICATION The simulation’s accuracy is verified by comparison to waveforms and partial power factor data available in the
literature [7]-[ 121. In addition, rectifiers were constructed
in the laboratory, and waveforms acquired by a Tektronix 11401 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. The laboratory 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 DFT. The laboratory measurements compare extremely well with the simulated data, both of which are shown in the following two sec- tions that provide design relationships for line-current harmonics and power factor as a function of LO-N for sin- gle- and three-phase rectifiers.
VIII. SINGLE-PHASERECTIFIERDESIGNRELATIONSHIPS
This section presents design relationships for the rms value and phase angle of rectifier line-current fundamen- tal and harmonics, and for rectifier displacement power factor, purity factor, and overall power factor as a func-
tion of normalized output filter inductance LO.N for the
single-phase rectifier shown in Fig. 2(a). Portions of this problem have been examined by previous investigators
(^338) IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 7. NO. 2. APRIL 1997
DCM I DCM II CCM
4 - 4 5 1.5 (^) i t , t , t 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 values Zs(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 the harmonics with respect to the source voltage U,., as a function of nornial- ized output filter inductance lo^#. Simulation data are shown as continuous curves, and laboratory data are shown as discrete points.
DCM I DCM II CCM
Key: ..... A ...... Is ( ~ ) - N/IS-,, --0-cOS@q1), -PFs Fig. 7. Single-phase rectifier displacement power factor cos &.,,,, purity factor Z s ( l ). N / Z s. N , and overall power factor PF,yas a function of output filter inductance 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 the previously cited literature and match those shown in Fig.
As shown in Fig. 6(b), the line-current fundamental
phase angle @scl,is near zero for small Lo.,, rises to the
largest-magnitude phase angle I # I ~ ~ ~ ) J= 38" for Lo-, = 0.033, and returns to near zero for large LO.,,,. Therefore,
as shown in Fig. 7, the displacement power factor cos
q 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 value 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 a
minimum for Lo., slightly less than that required to enter
DCM 11, slightly larger for Lo., in the DCM I1 range, and
approach their near-infinite-inductance values in the CCM
range. Therefore, in Fig. 7, the purity factor is low for small Lo.,, at a maximum of 0.94 for Lo-, = 0.030, and slightly lower at 0.90 for large Lo.,. As shown by (12), the overall power factor PF, is the product of the purity factor Zs,,,.,v/Zs_, and the displace-
ment power factor cos qbscl), and, as shown in Fig. 7, these
two influences are in conflict. For small Lo.,, the overall power factor is at a global minimum, despite the near- unity displacement power factor, 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. for LO., = 0.016 because of the reduction in waveform distortion and the increase of purity factor. However, the overall power factor is at a local minimum PF, = 0.
for LO.N = 0.039 because of the worsening displacement
power 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-,. However, operation in this condition requires
an uneconomically large and impractical output filter in-
ductor. As 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 this operating condition. The minimum overall line-current
harmonics 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 = 1200 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 factor
design. Substitution of the normalization references and
LO., = 0.030 into design equation (D.12) gives Lo =
6 mH a for minimum line-current harmonics design.
4SCI).
IX. THREE-PHASERECTIFIERDESIGNRELATIONSHIPS
This section presents design relationships for the rms value and phase angle of rectifier line-current fundamen- tal and harmonics, and for rectifier displacement power factor, purity factor, and overall power factor as a func-
tion of normalized output filter inductance LO.N for the
KELLEY A N D YADUSKY: RECTIFIER DESIGN FOR MINIMUM LINE-CURRENT HARMONICS
2 0
I 8 s
339
1-q A fl A A A I I (^) 1 ' 1 ' 1 ' 1
-2 -
l - n A-^ IX-N -
I I l l I 1 1 ' A
7 'SA-N 1 - 0 -1 - -2 -
1 ,'X-N 0 1 I I I I I , I 1 1 , I D. ON D, ON D, ON
D. ON De ON D" ON
(C)
t N
Fig. 8. Normalized three-phase rectifier waveforms: phase-A source volt- age phase-A rectifier line current rectifier output voltage vX.,,,, and output filter inductor current iX.N for (a) discontinuous conduction mode I (DCM I ) for Lo.,,, = 0.0024, (b) discontinuous conduction mode I1 (DCM 11 ) for Lo.,,, = 0.0064, and (c) continuous conduction mode (CCM)for Lo.,,, = 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 this problem have been treated by previous investigators [ 101- [ 121, and the three-phase design relationships are pre- sented for completeness, unification, and verification of
prior work and for contrast with the single-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.N that are not reproduced
in this paper. Representative normalized three-phase rectifier time waveforms of phase-A source voltage vSA-N, phase-A rec- tifier current filter input voltage v ~ - ~ and output- , filter-inductor current i X - N are shown in Fig. 8 for three values of normalized output filter inductance. The con- duction time interval for each diode is also indicated. The time waveforms measured in the laboratory are essentially
identical to the waveforms shown in Fig. 8. The wave-
forms for phases B and C are produced by shifting the
waveforms for phase A by one-third period and two-thirds period, respectively. The three-phase rectifier also ex- hibits DCM I, DCM 11, and CCM operation, as illustrated
in Fig. 8. As with the single-phase rectifier, a comparison
of Fig. 8 with Fig. 3(b) reveals the significant error that results from assuming a finite L O - N^ to be near-infinite. The description of the three modes of operation in the previous section for the single-phase rectifier is applicable to the three-phase rectifier, with the exception that three-phase rectifier has six diodes and six conduction intervals as op- posed to the four diodes and two conduction intervals for the single-phase rectifier. The boundary between DCM I
and I1 occurs for L O - N = 0.0050, and the boundary be-
tween DCM I1 and CCM occurs for L O - N = 0.0083.
Figs. 9 and 10 show design relationships for three-phase
rectifier 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 , IsA(I I ) - N , and ISA(13)-K-andFig. 9(b) shows the phase angles-
fundamental, and the fifth, seventh, eleventh, and thir-
teenth current harmonics, respectively, of the phase-A
rectifier line current iSA.N as a function of L 0. N. As ex- pected, all even-order harmonics and all harmonics that
are 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 in Figs. 9 and 10. The laboratory measurements are nearly identical to the simulation confirming the accuracy of both. In contrast to the single-phase rectifier, the line-current fundamental phase angle +SA(I) as shown in Fig. 9(b) is
near zero regardless of the value of L 0. N. The largest-
magnitude phase angle = 12" and the minimum displacement 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 = 0.33 because one third of
is delivered by phase A and the remaining two thirds
of 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
KELLEY AND YADUSKY: RECTIFIER DESIGN FOR MINIMUM LINE-CURRENT HARMONICS 34 1
ACKNOWLEDGMENT [I21 M. Sakui, H. Fujita, and M. Shioya, “A method for calculating har- manic currents of a three-phase bridge uncontrolled rectifier with dc filter,” 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 measurements.
111 I
[
[
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 the two-coil model of the ferroresonant transformer with a rectified output in the low-line heavy-load minimum-frequency condition,” 1983 In- 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 bridge converter with input filter,” 1985 Power Electronics Specialists ’ Con Rec. (PESC’85), Toulouse, France, June 1985, pp. 502-508. G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Merh- ods f o r Mathematical Computations. Englewood Cliffs, NJ: Pren- tice-Hall, 1977, ch. 6. F. R. Colon and T. N. Trick, “Fast periodic steady-state analysis for large-signal electronic circuits,” IEEE^ J.^ Solid-State Circuits,^ vol. SC-8, no. 4, pp. 260-269, Aug. 1973. “IEEE guide for harmonic control and reactive compensation of static power 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 subcritical inductance,” IEEE Trans. ind. Electron. Contr. Instrum., vol. IECI-20, no. 2, pp. 61-68, May
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 Center, pp. 1-49, May
M. Grotzbach, B. Draxler, and J. Schorner, “Line harmonics of con- trolled six-pulse bridge converters with dc ripple,’’ Rec. I987 IEEE industry Applications Society Annu. Meet., part I, Atlanta, GA:, Oct. S. W. H. De Haan, “Analysis of the effect of source voltage fluctua- tions of the power factor in three-phase controlled rectifiers,” IEEE Trans. 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 1957 in Norfolk, VA. He received the B.S.E. degree from Duke University, Durham, NC, in 1979. He continued at Duke as a James B. Duke Fellow and received the M.S., and Ph.D. degrees in 1981, and 1984, respectively. From 1985 to 1987, Dr. Kelley was employed as a Senior Research Engineer at Sundstrand Cor- poration, Rockford, IL, (^) where he worked on power electronics applications to aerospace power systems. He ioined the facultv of the Department of Electrical and Computer Engineering at North Carolina State University in 1987 where he currently holds the rank of Assistant Professor. His in- terests in power electronics include PWM dc-to-dc converters, line-inter- faced ac-to-dc converters and power quality, magnetic devices, magnetic materials, and computer-aided analysis and design of nonlinear circuits. Dr. Kelley is a member of Sigma Xi, Phi Beta Kappa, Tau Beta Pi, and Eta Kappa Nu.
William F. Yadusky (S’87, M’90) received the B.A. degree in English from the University of North Carolina, Chapel Hill, in 1982, and the B.S.E.E. from North Carolina State University, Raleigh, in 1987. He was certified as an Engineer in Training in 1987. Mr. Yadusky worked as a Graduate Research Assistant with the Electric Power Research Center at North Carolina State University under Dr. Ar- thur W. Kelley in 1988 and 1989. 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 printed cirucits. As an electrical design engineer on the Advanced Technology De- velopment team, Mr. Yadusky is presently responsible for designing and developing high-voltage PWM rectifiers and high-frequency magnetic structures for unintermptible power systems, frequency converters, and power conditioners.
