Rectifier Design for Minimum Line Current Harmonics and Maximum Power Factor

Rectifier Design for Minimum Line Current Harmonics and Maximum Power Factor Arthur W. Kelley and William F. Yadusky Department of Electrical and Computer Engineering, Box 791 1 North Carolina State University Raleigh, North Carolina 27695-791 1, U.S.A. (919) 737-7357 Absrracf - Rectifier line-current harmonics interfere with proper power system operation, reduce rectifier power factor, and limit the power available from a given service. The rectifier s output filter inductance determines the rectifier-line-current waveform, the line-current harmonics, and the power factor. Classical rectifier analysis usually assumes a near-infinite output filter inductance which introduces significant error in the estimation of linecurrent harmonics and power factor. This paper presents a quantitative computer-simulation-based analysis of singleand three-phase rectifier line-current harmonics and power factor as a function of the output filter inductor. A finite output filter inductor is shown to produce minimum singlephase rectifier line-current harmonics, and if used with an input displacement power factor correction capacitor, produce maximum overall power factor. A near infinite output filter inductor is shown to produce minimum threephase rectifier line-current harmonics and maximum overall power factor, and the smallest inductor that approximates a near-infinite inductor is determined. INTRODUCTION High rectifier line-current harmonics and low rectifier power factor have recently come under increased scrutiny by both power system and power electronics engineers. The number of rectifiers installed in the ac power system has increased mainly through such applications as machine drives and unintermptible power supplies, and the line-current harmonics created by these circuits can cause undesirable interactions with the power system. Rectifier line-current harmonics cause overheating of transformers, lines, and neutral conductors at power levels below the service rating. The current harmonics can also propagate into the system and interfere with the operation of sensitive equipment sharing the rectifier supply [ 11. The most common solution to a power system harmonic problem is to add one or more harmonic traps, each of which is tuned to the frequency of an undesirable current harmonic. However, the rectifier should be purposely designed to inject minimum current harmonics thereby keeping the number of additional power system components to a minimum. Rectifier line-current harmonics do not directly affect the quality of power delivered to the rectifier load. However, the ac source is usually protected by an rms-sensing circuit breaker. Rectifier line-current harmonics do not deliver any power to the rectifier, unnecessarily increase the rms value of the line current, and trip the breaker before the full amount of available power is delivered to the load. This problem acutely affects personal computers and workstations designed to operate on a standard single-phase 120-V 15-A supply 121. This research was sponsored by the Electric Power Research Center of North Carolina State University. Figure 1. Common rectifiers used for conversion of power system ac voltage to dc: (a) Single-phase rectifier, and three-phase rectifier. The single- and three-phase full-wave rectifiers shown in Figs. l(a) and, respectively, are the most common circuits used to convert the power system ac voltage vs to an unregulated dc voltage. The ripple in the rectifier output voltage vx is attenuated by the output filter inductor and the output filter capacitor CO, and a low-ripple dc voltage Vo is supplied to the load. Historically, the principal criterion for the inductive-capacitive output filter design is to prevent the rectifier output voltage ripple from reaching the load and to prevent undesirable interactions between the filter and certain types of constant-power loads, such as dc-to-dc converters. The output filter design also determines the rectifier line current waveform ir, and often little attention is paid to the rectifier line current when designing the filter. Classical rectifier analysis is based on the simplifying assumption of near-infinite output filter inductance which permits the assumption of near-zero ripple in the output filter inductor current ix[3]. This assumption is rarely met in practice, and produces an incorrect estimate of rectifier line-current harmonics and power factor. Conversely, the voltage ripple across the output filter capacitor is usually negligible with respect to its effect on the rectifier line-current waveform. 13 CH2719-3/89/0000-0013 $1.00 0 1989 IEEE

Therefore, for the purpose of determining the rectifier linecurrent waveform, a near-infinite output capacitor assumption is reasonable and near-zero-ripple output voltage is obtained in practice. This paper provides a quantitative analysis of current harmonics and power factor for rectifiers in which the ripple current through is appreciable and the ripple voltage across CO is negligible. Using this analysis, design curves are presented which allow rectifier design for minimum linecurrent harmonics and maximum power factor. In addition, the input capacitor CI is shown to be useful for correcting the displacement power factor of the single-phase rectifier of Fig. l(a). Ideal zero-loss rectifier components are assumed. However, if desired, the inclusion of most nonideal component behavior for a particular rectifier circuit is straightforward. The circuits shown in Fig. 1 are normalized with respect to a set of references, and a computer simulation determines the normalized rectifier line-current waveform in the periodic steady-state condition. The rectifier current harmonics are determined with a Fast Fourier Transform (FFT). The power factor is calculated from,the waveforms of rectifier current and source voltage. The analysis is repeated at regular intervals over a range of filter inductor values to produce design curves of current harmonics and power factor as a function of output filter inductance. For the single-phase rectifier, a near-infinite output filter inductor is shown not to be the best choice for reducing rectifier line-current harmonics, and the value of inductance that produces minimum rectifier current harmonics is determined. However, the input capacitor is necessary to achieve an acceptable value of power factor for this value of inductance. For the three-phase rectifier, a near-infinite output filter inductor is shown to produce minimum rectifier current harmonics and maximum power factor. The minimum value of output inductance needed to approximate an infinite inductor is deteimined. NORMALIZATION Before the rectifier analysis and design relationships are developed, the circuits of Fig. 1 are normalized with respect to a set of reference quantities to produce the normalized circuits shown in Fig. 2. The normalization references are shown in Table I. The normalization procedure requires three references: a voltage reference, a current reference, and a time reference. The rectifier analysis and design relationships based on the normalized circuit are applicable to any rectifier once values for the normalization references are chosen. The voltage reference for the single-phase rectifier is the nominal rms value VS(nom) of the ac line voltage vs. The voltage reference for the three-phase rectifier is the nominal rms value VS(nom) of the ac line-to-neutral voltage. The current supplied by a source of rms voltage VS(nom) to a linear resistive load of power PO(nom) is the current reference Is(nom) for both the single- and three-phase rectifiers. The time reference is the nominal period TS(nom) of the sinusoidal ac source. For example, a 1200-W recnfier connected to a 120-V 60-Hz ac source would have voltage, current, and time references of VS(nom) = 120 V, IS(nom) = 1200 W/120 v = 10 A, and TS(nom) = 16.67 ms, respectively. The normalization and design relationships are shown in Table 11. Note that an "-N" appended to a subscript indicates TABLE I NORMALIZATION REFERENCES - References Voltage - nominal rms line (single-phase rectifier) or lineto-neutral (three-phase rectifier) voltage: Vs(mm) Current - rms current drawn from ac source by linear resistive load of nominal power Po(,): IS(nom) = PO(nom)/VS(nom) Time - nominal period of sinusoidal source: Ts(nom) S-N Figure 2. Rectifiers of Fig. 1 normalized with respect to the nominal source voltage vs(nom), nominal output power Po(nom), and nominal source period IS(nom): (a) normalized single-phase rectifier, and normalized three-phase rectifier. a normalized quantity. The normalization relationships, equations N.l to N.16, translate rectifier circuit quantities into normalized quantities for comparison with the normalized design curves found in this paper. The design relationships, equations D.l to D. 16, are simply the normalization relationships solved for the circuit quantity in terms of the normalized quantity and the references, and translate normalized component values found by designing in the normalized domain into actual circuit values. An example of the design process is presented for both the single- and threephase rectifiers. The normalized time t~ is simply the ratio of time t to the time reference TS(nom). Similarly, a normalized voltage VN is the ratio of the circuit voltage v and the voltage reference Vs(nm), and a normalized current in is the ratio of the circuit current i and the current reference IS(nom). The normalization relationship for a circuit element is found from the circuit law describing the element. For example, the circuit law that describes the output filter inductor is: dil VL=LOx 14

TABLE II NORMALIZATION AND DESIGN RELATIONSHIPS FOR RECTIFIER CIR Normalization Relationships JIT QUANTITIES I Design Relationshius 3 2 I 1 0-1 -2-3 0.00 0.33 tn 0.67 1.00 where is the voltage across the inductor and il is the current through the inductor. Substitution of the appropriate design relationships - VL = VL-N-VS nom) for voltage, il = il-wis(nom) for current, and t = h.$s(nom) for time - gives: Rearranging gives: The circuit law for the inductor must also hold in the normalized domain, as shown in (3), and therefore the normalized value LO-N of LO is defined: (4) Finally, note that angles, whether in degrees or radians, are the same in both the circuit and normalized domains. DEFINITIONS This section provides definitions that are the foundation for characterization of the rectifiers. The definitions are illustrated with respect to the single-phase rectifier, and are identical for the three-phase rectifier when taken on a per-phase basis. Where applicable, IEEE Standard 519 definitions are used [4]. Initially, consider a sinusoidal voltage source VS-N connected to an linear inductive-resistive load as shown in Fig. 3(a). The voltage source is sinusoidal with rms value VS-N, peak value fi*vs.~ and period Ts-N: 0.00 0.33 tn 0.67 1.oo Figure 3. Examples of current drawn from a sinusoidal source VS-N: (a) sinusoidal current is-n for linear inductive-resistive load, and nonsinusoidal rectifier current ir-n and its fundamental ir(l)-n for nonlinear rectifier load. The positive-going zero crossing of VS-N is the reference point from which all other phase angles are. measured. The current is-n drawn from the source is also sinusoidal, with rms value Is-N, peak value ~ Is-N, period Ts-N, and a phase angle @S-N < 0: is.n =.iz IS-N sin [(2n/ TS-N )tn+@s-n] (6) The overall source power factor PF is the ratio of the real power Ps-N to the apparent power VS.N*IS-N delivered by the ac source: TS-N I VS-N is-n dtn ps-n = 0 PF = VS-N IS-N VS-N IS-N Note that PS-N = PO-N because the circuit is lossless. Both VS-N and is-n are sinusoids of the same frequency, and Ps-N evaluates to the product of the rms value of the source voltage, the rms value of the source current, and the cosine of the angle between them: The power factor is simply the cosine of the angle between the source current and the source voltage. The rectifier, shown in Fig. 3, is a nonlinear load, and when connected to a sinusoidal source draws a nonsinusoidal rectifier current ir-n. However, ir-n is periodic with period Ts-N and is represented by Fourier series as the sum of its current fundamental ir(l)-n and higher-order current (7) 15

harmonics ir(h)-n: ir-n = fi IR(l)-N sin [ ('"/TS-N)tN $R( 1)-N] (9) and the power factor as seen by the ac source. The timedomain simulation of circuits is a well-developed analysis technique, and has been described in the literature by [5][61 and many others. The simulation techniaues used to Droduce the resuits reported in this paper are summarizedsin this section. where the current fundamental is sinusoidal with rms value IR(I)-N, peak value &-IR(~~-N, period Ts-N, and phase angle $R(~)-N < 0. CWent harmonic ir h)-n has rms Value IR@)-N, peak value &-IR(h)+& period TS-Njh, and phase angle $R(h)-N. The rms value IR-N of ir-n is the square root of the sum of the squares of the rms values of the current fundamental and the current harmonics: It is assumed that all components in the rectifier are ideal. The ac input voltage is assumed to be a sinusoidal zero-impedance source. For the single-phase rectifier, the source voltage is defined by (5). For the three-phase rectifier, the voltage source is balanced and defined by: VS1-N = fi VS-N sin[(2x/ts-n) bi] IR-N VS2-N = fi VS-N sin[(2a/ts-n) tn - (2"/3)] (10) (14) h > l VS3-N = fi VS-N sin[(2"/ts-n) tn - (4"/3)] (15) Therefore, the rectifier current harmonics increase the rms value of the rectifier current above that of the current The output filter inductor and capacitor are assumed to be fundamental alone. linear and lossless. The diodes are nonlinear elements modeled as ideal switches with zero forward voltage drop The power factor definition in (7) applies to the nonlinear load when ON, zero reverse leakage current when OFF, and as well as the linear load, and the rectifier power factor PF is: instantaneous switching. The rectifier load is assumed to draw constant power and is therefore also nonlinear because the PF TS-N output current ~O-N varies inversely with output voltage VCO-N. I VS-N i ~-N d t ~ PS-N 0 (11) The differential equations that describe the rectifiers are = VS-N IR-N VS-N IR-N nonlinear due to the constant-power load and the diode switching. The nonlinear differential equations that describe Since the voltage source is a fundamental-frequency sinusoid, the circuit are cast in a state-variable formulation. The two only the rectifier current fundamental contributes to real power state variables are the output filter inductor current ix-n and the Ps-N, and (11) becomes: output filter capacitor voltage VCO-N. The state equations for the circuit are: (13) The rectifier power factor definition retains the familiar COS$R(~)-N term which is called the displacement power factor. The displacement power factor is improved by reduction of the phase angle $~(1 N between the source voltage VS-N and the rectifier current Lndamental ir(i)-n. In addition, the rectifier power factor definition contains the term 'R(i)-N/IR.N e 1 which embodies the effects of rectifier current harmonics on the overall power factor. This term, which to the authors' knowledge has no standard name, is called the purity factor for the purposes of this paper because it relates the shape of the rectifier current to that of a pure sine wave. The purity factor is improved by reduction of the rectifier current harmonics IR(h)-N so that the rms value IR-N of the rectifier current is nearly the same as the rms value IR(~)-N of the current fundamental, and the waveform of ir-n is more purely sinusoidal. Therefore, a high-power-factor rectifier must simultaneously have both a near-unity displacement power factor and a near-unity purity factor. Reduction of rectifier current harmonics, therefore, is essential to power factor improvement. COMPUTER SIMULATION AND ANALYSIS The normalized circuit waveforms of the rectifiers shown in Figs. 2(a) and are determined by time-domain computer simulation. Subsequent analysis of the waveforms determines the current fundamental and current harmonics of the rectifier The normalized values of I+N, G-N, and PO-N are constant. Note that (16) is nonlinear because the rectifier output voltage VX-N at any instant is determined by the state of the diodes, and (17) is nonlinear due to the constant output power. At any point in time during the operation of the rectifier, the rectifier output voltage VX-N is a function of the source voltage or voltages, and the output filter capacitor voltage. The rectifier output voltage also depends on whether the output filter inductor current is zero or nonzero. All diodes can be OFF with the source or sources disconnected from the output filter, no output filter inductor current ix-n flowing, and no rectifier current ir-n flowing. Alternately, two diodes can be ON with the source or sources connected to the output filter and a nonzero rectifier current ir-n = fix-n flowing from the source or sources into the filter. If two diodes are ON, one odd-numbered diode connected to the +VX-N node is.on, and one even-numbered diode connected to the -VX-N node is ON. All odd-even two-diode combinations are examined to determine the rectifier output voltage that would result if that combination were to conduct. The largest possible value of rectifier output voltage that could result from any of the several odd-even two-diode combinations is VX(max)-N. 16

The single-phase rectifier has two odd-even two-diode combinations: a D1-D4 combination and a D2-D3 combination. If the Dl-Dq combination is ON, the source voltage connected to VX-N is +VS-N, and if the D2-D3 combination is ON, the source voltage connected to VX-N is -VS-N. Therefore, if +VS-N > -VS-N (i.e., VS-N > 0), then VX(max)-N = +VS-N. Alternately, if -VS-N > +VS-N (i.e., vs-n < O), VX(max)-N = -VS-N. This relationship is expressed more compactly by: +VS-N (Di-Dq ON) vx(max)-n = MAX [-VS.N (D2-D3 ON) I Where the MAX [ 1 operator chooses the largest positive value of the two arguments +VS-N and -VS-N. The three-phase rectifier has six odd-even two-diode combinations: a DI-D~ combination, a D2-D3 combination, a D3-Dg combination, a D4-D5 combination, a D5-D2 combination, and a Dfj-Dl combination. If the DI-D~ combination is ON, the source voltage connected to VX-N is the line-to-line voltage +(vsi-n - VS2-N). If the D2-D3 combination is ON, the source voltage connected to VX-N is the line-to-line voltage -(VSI-N- VS~-N). The same relationship holds true for +(VS2-N - VS3-N) with D3-Dg ON, -(VS2-N - VS3-N) with D4-D5 ON, +(vs~-n - vs1-n) with D5-D2 ON, and -(vs~-n- VSI-N) with Dg-Di ON. The largest possible rectifier output voltage VX(max)-N is the largest positive value of the six line-to-line voltages. This relationship is expressed more compactly by: +(VSl-N - VS2-N) (Dl-D4 ON) 1 The odd-even two-diode combination that produces the largest possible rectifier output voltage VX(max)-N will conduct if VX(max)-N exceeds the output filter capacitor voltage VCO-N thereby forward biasing the two diodes, or if the output filter inductor current 1X-N is greater than zero. VX-N = VX(~)-N if ix-n > 0 or VX(max)-N > VCO-N (20) As shown by (16), if VX-N > VCO-N then dix-n/dtn > 0 and ix-n is increasing. If VX-N < VCO-N, dix-n/dtn < O and ix-n is decreasing. The output filter inductor current ix-n decreases to zero and all diodes turn OFF in certain modes of periodic steady-state operation. If the output filter inductor current ix-n is zero and the maximum rectifier output voltage VX(max)-N is less than the output filter capacitor voltage VCO-N, all diodes are OFF, the voltage across LO-N is zero, and the rectifier output voltage is equal to the voltage across the output filter capacitor: VX-N = VCO-N if ix-n = 0 and VX(~=)-N < VCO-N (21) In this condition, dix-n/dtn = 0 and the current remains zero until VX(ma)-N again exceeds VCO-N. Equations (3,(16), (17), (IS), (20) and (21) describe the single-phase rectifier, and (13), (14), (13, (16), (17), (19), (20), and (21) describe the three-phase rectifier. The two sets of equations are coded as a pair of function subroutines, and a publicly-available software package [7] is employed to numerically integrate the nonlinear differential equations with respect to time. Beginning with arbitrary initial conditions, ix-n[ 0) and VCO-N[ 0), for the state variables, the simulation runs until the periodic steady-state condition (PSSC) is reached. The simulation is deemed to have reached the PSSC after k cycles at time k-ts-n when the values of the state variables, ix-~[k*ts-~) and VCO-N[~ TS.N), at the end of a period are the same, within a certain tolerance E, as the values of the state variables, ix - N [ (k- 1 ).T s - N ) and VCO-N[ (k-l) Ts-N), at the end of the prior period: Iix-N[ k*ts-~) - ix-n[ (k-l).ts-~) I < E A typical tolerance is E = (22) The simulation is continued for one cycle after the PSSC is reached, with p = 1024 points per period and integration time step Ts-N/~. The p discrete points of source voltage VS-N( t ~], rectifier current ir-n[ tn), and output voltage VCO-N[ tn) versus discrete time tn = J/p-TS.N (j = 0, 1, 2,..., p-i) are saved in a file for further analysis. The current fundamental and current harmonics of 1R.N are subsequently determined by a publicly available Fast Fourier Transform (FFT) pro ram. The rms value IR-N of ir-n, the purity factor IR(I)-N~~.~, the displacement power factor COS$R(~)-N, and the overall power factor PF are calculated from the rms values of the current fundamental IR(I N and the current harmonics IR(h)-N (h>l) of ir-n, and tke current fundamental phase angle $R(])-N using (IO) and (12). As a check, the rms value IR-N of ir-n is found by taking the square root of the sum of the squares of the p discrete values of ~R-N [ t~ I over one period Ts-N: The real power Ps-N supplied by the source is obtained by averaging the instantaneous power vs-~[ tn)-ir-n[ tn) over one period: The circuit is lossless and PS-N is checked by comparison with the constant output power Po-N. In addition, since VS-N is known, a check on PF is obtained by direct evaluation of (1 1). Both methods of calculating IR-N, Po-N, and PF produced nearly identical results giving increased confidence in the simulation results. In addition, the simulation s accuracy is verified by comparison to published waveforms, and published current harmonic and power factor data as referenced in the sections on single- and three-phase rectifiers. The simulation described above determines the rectifier current fundamental and current harmonics, purity factor, displacement power factor, and overall power factor for one set of VS-N, b-~, Q-N. and Po-N. Given values of VS-N, CO-N, and Po-N, the simulation is repeated at regular intervals 17

over a wide range of L(-JN values to produce design curves of current fundamental, current harmonics, and power factor versus b-~. During the runs, the other three circuit values are held constant with the following values: VS-N = 1, PO-N = 1, and Q-N = 1OOO. If desired, the value of VS-N could be set lower or higher than unity to investigate high- and low-line voltage conditions. The value of PO-N = 1 is for full output power and is assumed to be the worst-case condition for the generation of rectifier current harmonics. The value of CO-N is sufficiently large so that the ripple in the output voltage VCO-N is small regardless of the value of b-~. This condition is almost always met in practice. The FFT analysis is also run on VCO-N to ensure that this assumption is me regardless of the value of b-~. The peak-to-peak output voltage ripple is on the order of 0.2%for all values of LON. 3 2 1 0-1 -2-3 D,,.... DM3,....... (a) SINGLE-PHASE RECTLFIER This section presents curves of current fundamental and current harmonics, displacement power factor, purity factor, and overall power factor as a function of normalized output filter inductance Q-N for the single-phase-rectifier shown in Fig. 2(a). Many aspects of this problem have been treated by previous investigators [8][9][ 101, but minimum rectifier current harmonics and their effect on power factor have not been previously directly examined. Consider initially single-phase rectifier operation without the input capacitor CI-N. Figures 4(a),, (c), and (d) show time waveforms of source voltage VS-N, rectifier current ir-n, and illustrate the rectifier current fundamental ir 1) N for four values of output filter inductor LO-N = 0.010, b.037, 0.090, and 1.0, respectively. The rectifier current waveform is dramatically different for different output filter inductor values, and Figs. 4(a),, and (c) illustrate three distinct modes of operation which, adopting the nomenclature of Dewan [9], are the discontinuous conduction mode I (DCM I), discontinuous conduction mode I1 (DCM 11), and continuous conduction mode (CCM), respectively. Figure 4(d) illustrates the classical CCM waveforms for near-infinite output filter inductance and near-zero output-filter-inductor ripple current. The DCM I shown in Fig. 4(a) occurs for b-~ < 0.027, and the rectifier current is characterized by short-duration highpeak-value current pulses. The positive pulse occurs during the D1-D4 conduction interval during which ir-n = +ix-n, and the negative pulse occurs during the D2-D3 conduction interval during which ir-n = -ix-n. The rectifier and output filter inductor currents are zero for two time intervals during which all four diodes are OFF. The zero-current intervals occur between the single D1-D4 conduction interval and the single D2-D3 conduction interval. For 0.027 < LO-N < 0.043, the rectifier is in DCM I1 as illustrated in Fig. 4 for LO-N = 0.037. The larger LO-N value extends the declining tail of the positive D1-D4 current pulse beyond t~ = 0.5, and results in a brief negative D2-D3 current pulse before the current falls to zero. The current remains zero for an interval before a second D2-D3 conduction interval. The tail of the negative pulse extends beyond tn = 1.0, and results in a brief positive D1-D4 current pulse before the current falls to zero. As with DCM I, the rectifier current and the output filter current are zero for two time intervals during which all four diodes are OFF. In contrast to DCM I, one zero-current interval separates the two D1-D4 conduction intervals and the other zero-current interval separates the two D2-D3 conduction intervals. -24 I D1&4 - D2&3 1 I I c (c) 0.00 0.25 0.50 0.75 1.oo t N Figure 4. Single-phase rectifier waveforms: source voltage VS-N, rectifier current ir-n, and rectifier current fundamental ir(l)-n for: (a) discontinuous conduction mode I (DCM I) for h-~ = 0.010, discontinuous conduction mode 11 (DCM 11) for LO-N = 0.037, (c) continuous conduction mode (CCM) for h-~ = 0.090, and (d) CCM for near-infinite LO-N = 1.0. Diode conduction intervals are indicated by - for ON and for OFF. 18

For LO-N > 0.043 the rectifier is in CCM as illustrated in Fig. 4(c) for LO-N = 0.090. Diode conduction alternates between a D1-D4 conduction interval and Dz-D3 interval. As the name implies, no zero-current interval exists in the CCM. Figure 4(d) shows the near-square-wave rectifier current that results for the near-infinite LO-N = 1.0. This operating condition forms the basis for the classical analysis of rectifier operation. However, comparison of Figs. 4(a) to (d) shows the wide variation in rectifier-current waveforms that result from differing values of b-~, and the severe error that is incurred by using a near-infinite-inductance approximation to represent finite b~. Figure 5(a) shows the rms values - IR I)-N. IR(~)-N, IR(~)-N, IR(~)-N, and IR(~)-N - and Figure 5[b) shows the phase angles - $R 1) N, @R(3)-N3@R 5) N @R(7 N,. and $R(9)-N - of the current lundamental, and ihd third; fifth, seventh, and ninth current harmonics, respectively, of the rectifier current ir-n as a function of b-~. Figure 6 shows the relationship between the displacement power factor COS$R 1) N, the purity factor IR(I)-N/IR.N, and the overall power (faitor PF as a function of b-~. The curves of displacement power factor and overall power factor appear in the literature [8][9][lO] and match those shown in Fig. 6. As shown by (12), the overall power factor PF is the product of the purity factor and the displacement power factor, and is subject to two conflicting influences. For small LO-N, the overall power factor is at an overall minimum, despite the near unity displacement power factor, because the rectifier current is very distorted and the purity factor is low. The overall power factor is at a local maximum PF = 0.76 for LO-N = 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 = 0.73 for LO-N = 0.037 because of the worsening displacement power factor. The overall power factor is at a overall maximum PF = 0.900 for a near infinite b-~ due to the improvement of displacement power factor and the relatively good purity factor. The maximum overall PF = 0.900 occurs for near-infinite Lo-N. However, operation in this condition requires an uneconomically large and impractical output filter inductor. As noted by Dewan [9], maximum practical PF=0.76 for a reasonably-sized output filter inductor occurs in DCM I with LO-N = 0.016. The waveforms used in Fig. 3 to illustrate power factor are obtained in this operating condition. Y m g z 1.5 U 1.0 B 2 0.5 i? z" 0.0 0 1 0.027 0.030 0.033 0.043 '0-N (a) 180,2 120 e! $ 60 a, 9 0 The current fundamental phase angle $R(~)-N is near zero for -60 small b-~, rises to -38' for LO-N = 0.033, and returns to near zero for large b-~. Therefore, the displacement power 3-120 factor COS@R 1) N is near unity for small b-~, falls to 0.79 for LO-N = 0.035 hd rises again to near unity for large b-~. -180 The rms value IR(~)-N of current fundamental is inversely 1 proportional to the displacement power factor since 0.027 0.030 0.033 0.043 PO-N = Ps-N = 1 = COS$R(~)-N-IR 1) N. The rms value of the '0-N current fundamental is near unity io; small b-~, rises to 1.27 for b-~ = 0.033 and falls to near unity for large b-~. As seen in Fig. 5(a), the normalized rms values IR(h)-N of the Figure 5. Harmonics of single-phase rectifier Current ir-n: current harmonics are large for small h-~, are at a minimum (a) normalized rms values - IR(~)-N, IR(~)-N, for LO-N slightly less than that required to enter DCM I1 IR(s)-N, IR(~)-N, and IR(~) - and phase angles range, slightly larger for LO-N in the DCM I1 range, and - @R(l)-N, @R(3)-N, @R(5)-N, qr(7i-n. and @R(9) approach their near-infinite-inductance values in the CCM - as a function of output filter inductance LO-N. range. Therefore, in Fig. 6, the purity factor is low for small Lo-N, at a maximum of 0.94 for LO-N = 0.030, and slightly lower at 0.90 for large b-~. LO-N Figure 6. Displacement power factor COS@R(I)-N, punty factor IR(I)-N/I~.~, and overall power factor PF as a function of output filter inductance LO-N for the single-phase rectifier. 19

The conflict between displacement power factor and punty factor produces a low overall power factor for all practical LO-N values. However, addition of a properly-selected input capacitor CI-N corrects the displacement power factor to unity for any value of LO-N and increases the overall power factor subject only to the rectifier current harmonics represented by the purity factor. The input capacitor is connected in parallel with the stiff sinusoidal voltage source VS-N, and the capacitor current ic1-n is a cosinusoidal current in quadrature with VS-N. The rectifier current ir-n is unaffected by the presence of the input capacitor because VS-N is a stiff source. Therefore, the source current is-n is the sum of the rectifier current fundamental, the rectifier current harmonics, and the capacitor current. is-n = ir( 1)-N -k c ir(h)-n + ici-n h > l = fi IR(I)-N cos@r(l)-n sin[(21r/ts-n)tn ] + fi IR(I)-N Sin$R(l)-N cos[ (21rfiS.N>tN] -k fi xir(h)-n sin [(2xh/T~-~)tN-k @R(h)-N] h > l Note in (26) that the rectifier current fundamental ir(i)-n is separated into a sinusoidal component which is in phase with VS-N and a cosinusoidal component which is in quadrature with VS-N. The rms value of the in-phase component is IR(~)-N*cos$R(~)-N = Ps-N = PO-N = 1, and the rms value of the quadrature component is IR(i)-N*sin$R(i)-N = 0.75. The capacitor is selected so that the in-quadrature capacitor current ICI-N is equal to and of opposite sign to the inquadrature component of rectifier current fundamental, so that only the in-phase component of rectifier current fundamental remains. The required capacitor is found from: When examining (27) recall that $R(I)-N < 0. Therefore, when CI-N is chosen for unity displacement power factor according to (27), the source current in (26) reduces to: h > l which is the sum of a unity-rms-value in-phase sinusoidal current fundamental and the rectifier current harmonics. Using (10) and (12), the overall power factor in the unitydisplacement-power-factor condition is found from: If CI-N is chosen for unity displacement power factor in accordance with (27), only the nonzero rectifier current harmonics prevent unity power factor. 0.00 0.33 tn 0.67 1.00 (a) 31 I I I 0.00 0.33 tn 0.67 1.oo Figure 7. Source voltage VS-N, and (a) rectifier current ir-n and rectifier current fundamental ir(l)-n, and source current is-n and source current fundamental is(1)-n for the minimum-harmonic unity-displacement-power-factor condition with LO-N = 0.030 and CI-N = 0.12. Equation (29) suggests a rectifier-design strategy of selecting LO-N to produce minimum rectifier current harmonics and selecting CI-N to correct the resulting displacement power factor to unity. Figure 6 shows that minimum rectifier current harmonics and maximum purity factor IR(I)-N/IR-N = 0.94 occur for LO-N = 0.030. For this value of Lo-N, the rectifier current fundamental phase angle is @R(~)-N = -37", the displacement power factor is COS@R(~)-N = 0.80, and the rms rectifier current fundamental is IR(I N 1.25. The overall power factor is a poor PF = 0.75. AvaGation of (29) with these values, with VS-N = 1 and Ts-N = 1, shows an input capacitor CI-N = 0.12 is required to achieve unity displacement power factor, resulting in an overall power factor PF = 0.905. Figure 7(a) shows time waveforms of the source voltage VS-N and rectifier current ir-n, and illustrates the rectifier current fundamental ir(1 -N for rectifier operation with LO-N = 0.030. For this value of 10-N the rectifier is operating in the DCM II region. Figure7 shows time waveforms of the source voltage VS-N, capacitor current ic1-n and source c mnt is-n, and illustrates the source current fundamental is(i)-n for LO-N = 0.030 and CI-N = 0.12. The overall PF = 0.905 achieved in this operating condition - without the use of a near-infinite LO-N - exceeds the overall PF = 0.900 possible with a near-infinite b~. As a design example, recall the 1200-W 120-V 60-Hz rectifier for which the normalization references are VS(nom) = 120 V, IS(nom) = 10 A, and TS(nom) = 16.67 ms. Substitution of the normalization references and LQ-N = 0.030 into design relationship D.15 gives = 6 mh. Substitution of the normalization references and CI-N = 0.12 into D. 14 gives CI = 167 pf. Equations D.14 and D.15 indicate that CI and decrease as frequency increases. The same rectifier design c 20

carried out for a 400-Hz aircraft system gives LO = 900 ph and CI = 25 pf. The same rectifier design carried out for a 20-kHz all-electric aircraft or the Space Station system gives Lo = 18 ph and CI = 0.50 pf. This short trade study illustrates the ease with which a single-phase rectifier is designed for minimum line-current harmonics and maximum power factor. THREE-PHASE RECTIFIER This section presents curves of current fundamental and current harmonics, displacement power factor, purity factor, and overall power factor as a function of normalized output filter inductance b.~ for the three- hase-rectifier shown in Fig. 2. This problem has largely geen treated by previous investigators [11][ 121, and these results are presented, for completeness, verification of prior work, and for companson with the single-phase rectifier. Figures 8(a),, (c), and (d) show time waveforms of lineto-neutral source volta e vs N. per-phase rectifier current ir-n, and illustrate the rectiker cirrent fundamental ir 1 N for four values of output filter inductor LO-N = 0.0024, 6.bbSp, 0.10, and 1.0. These waveforms are for phase 1 as defined by (13), but apply to phases 2 and 3, as defined by (14) and (15), if phase-shifted by 2x/3 and4x 3 res ectively. The three-phase rectifier also exhibits DCId i, D& 11, and CCM operation, as illustrated in Figs. 8(a),, and (c), respectively. Figure (d) illustrates the classical CCM waveforms for near-infinite output filter inductance and nearzero output-filter-inductor ripple current. As with the singlephase rectifier, a comparison of Fig. 8(d) to Figs. 8(a),, and (c) reveals the significant error that results from assuming a finite LO-N to be a near-infinite inductance. The description of the three modes of operation in the revious section for the sin le phase rectifier is applicable to tte three-phase rectifier, witi the exception that three-phase rectifier has six odd-even two-diode pairs and six conduction intervals as opposed to the two odd-even two-diode airs and two conduction intervals for the single-phase rectiier. The boundarv between DCM I and I1 occk for LQ-N = 0.0050, and the 6oundary between DCM I1 and CCM occurs for LO-N = 0.0083. Figure 9(a) shows the rms values - IR 1 N, IR(s)-N, IR(~)-N, IR(~~)-N, and IR(~~)-N - and Figure d(k) shows the phase angles -@R(l)-N,@R(5)-N,@R(7 N @R 11) N. and @R(13)-N- of the current fundamental, and'tde fi!th,-seventh, eleventh, and thirteenth current harmonics, respectively, of the rectifier current ir-n as a function of b-~. Figure 10 shows the relationship between the displacement power factor cos@g(l N, the punty factor IR(l)-N/1.N, and the overall power factor df as a function of b-~. fn contrast to the singlephase rectifier, the current fundamental phase angle @R 1) N is near zero re ardless to the value of b-~. The faigest @R(I)-N = -15' and the worst displacement power factor COS@R(~)-N = 0.98 occur for LO-N = 0.0055. An added input capacitor would only marginally improve the overall ower factor for the three-phase rectifier. The rectifier current K armonics and the urity factor are the dominant influences on the overall power Lctor. As shown by Fig. 9(a), the current harmonics are high for small LO-N values and, except for a small section of IR(~~)-N, decrease uniformly for lar er LO-N values, reaching a minimum at LO-N = 0.10. Tierefore, the urity factor and the overall power factor are low for small L-N values and increase to a maximum IR(I)-N/IR.N = PF = 0.96 for LO-N = 0.10. A larger value of LO-N does not significantly improve dis lacement ower factor or overall power factor. Therefore, EO-N = 0.18 is a reasonable approximation to a near-infinite out ut filter inductance. As a design example, consider a 12OE-W three- hase rectifier connected to a 208/120-V 60-Hz source. #he normalization references are V S(nom) = 120 V, IS(nom) = 10 A, and TS(nom) D1&4 - D 1&6 I 0.00... I 0.33.. tn. I 0.67. ". I 1.oo Figure 8. Three-phase rectifier waveforms: source voltage VS-N, rectifier current ir-n, and rectifier current fundamental ir(i)-n for: (a) discontinuous conduction mode I (DCM I) for LO-N = 0.0024, discontinuous conduction mode II (DCM II) for LO-N = 0.0064, (c) continuous conduction mode (CCM) for L0-~=0.10, and (d) CCM for near-infinite LO-N = 1.0. Diode conduction intervals are indicated by - for ON and for OFF. 21

0.0050 0.0083 0-N (a) 0.0050 0.0055 0.0083 0-N Figure 9. Harmonics of three-phase rectifier per-phase current ir-n: (a) normalized rms values - IR(I)-N, IR(s)-N, IR(~)-N, IR(~~)-N, and I~(i3) - and phase angles -@R(I)-N, $R(5)-N9 $R(7)-N1 ~R(I~)-N, and $ ~(i3) - as a function of output filter inductance b-n. 0. 6 1 7 m - m....... a lo- 0.0050 0.0055 0.0083 0-N loo Figure 10. Displacement power factor COS$R(I)-N, punty factor IR(I)-N/IR-N, and overall power factor PF as a function of output filter inductance b-~ for the three-phase rectifier. 0 0 = 16.67 ms, which are the same as those for the sin le hase rectifier. Substitution of the references and LO-N = 8.18 into D.15 gives Lo = 20 mh. SUMMARY Classical rectifier analysis based on near-infinite output filter inductance and zero-ripple filter inductor current becomes less satisfacto as line-current harmonics and power factor issues increasingy concern power systems and power electronics engineers. This paper provides quantitative design data for line-current harmonics and power factor for single and threephase rectifiers for realistic design situations with finite inductance and appreciable current ripple. This data will provide a reference for designers of new equipment, and for the evaluation of harmonic and power factor problems within existin equipment. The maximum power factor for a singlephase kite-inductance rectifier without an input power factor correction capacitor is 0.76. However, this operating condition does not result in minimum rectifier line-current harmonics. Design of the rectifier for minimum line-current harmonics and the addition of an input wer factor correction capacitor im rove the power factor to r905 without the need for a near-ininite inductor. This is a distinct advantage over the 0.900 power factor possible with an infinite output filter inductance. The maximum power factor for a three-phase rectifier is 0.96, and does occur for an infinite output inductance. This operating condition also results in minimum line-current harmonics. Power factor is not significantly im roved, nor are line-current harmonics significantly redced, for output filter inductances larger than a specific threshold value. REFERENCES G. L. Park and E. G. Strangas, Ac Power-Electronic System Interfacing Concerns, Power Conversion and Intelligent Motion, pp. 30-35. November 1988. W. W. Burns and J. Kociecki, Power Electronics in the Minicomputer Industry. Proceedings of the IEEE, vol. 76, no. 4. pp. 311-324, April 1988. I. Schaefer. v r - and De- John Wiley & Sons., New York, N.Y.. 1965. The IEEE, IEEE Guide for Harmonic Control and Reactive Compensation of Static Power Converters. IEEE/ANSI Standard 519-1981. 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. Record of the 1983 International Telecommunications Energy Conference (INTELEC 83). pp. 374-381. Tokyo, Japan, October 1983. E. B. Sharodi and S. B. Dewan. Simulation of the Six-Pulse Bridge Converter with Input Filter. Record of the 1985 Power Electronics Specialists Conference (PESC 85). pp. 502-508, Toulouse, France, June 1985. G. E. Forsythe, M. A. Malcolm and C. B. Moler. Prentice-Hall. Inc., Englewood Cliffs, New Jersey, chapter 6. F. C. Schwarz, Time-Domain Analysis of the Power Factor for a Rectifier-Filter System with Over- and Subcritical Inductance. IEEE Transactions on Industrial Electronics and Control Instrumentation, vol. IECI-20, no. 2, May 1973. S. B. Dewan. Optimum Input and Output Filters for Single-phase Rectifier Power Supply. IEEE Transactions on Industry Applications, vol. IA-17. no. 3. pp. 282-288, Mayllune 1981. California Institute of Technology, Power Electronics Group, Input- Current Shaped Ac-to-Dc Converters, Final Report, NASA-CR- 176787, Prepared for NASA Lewis Research Center, 49 pages, May 1986. M. Gratzbach. B. Draxler and J. Scharner. Line Harmonics of Controlled Six-Pulse Bridge Converters with DC-Ripple. Record of the 1987 IEEE Industry Applications Society Annual Meeting. Part I, pp. 941-945. Atlanta, Georgia. October 1987. S. W. H. De Haan. Analysis of the Effect of Source Voltage Fluctuations on the Power Factor in Three-phase Controlled Rectifiers. IEEE Transactions on Industry Applications, vol. IA-22, no. 2. pp. 259-266, MarcNApril 1986. 22

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