Line Current Characteristics of Three-Phase Uncontrolled Rectifiers Under Line Voltage Unbalance Condition

Transcription

1 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 6, NOVEMBER Line Current Characteristics of Three-Phase Uncontrolled Rectifiers Under Line Voltage Unbalance Condition Seung-Gi Jeong, Member, IEEE, and Ju-Yeop Choi Abstract Three-phase uncontrolled rectifiers with capacitive filters are highly sensitive to line voltage unbalance, drawing significantly unbalanced line current even under slightly unbalanced voltage condition. This paper presents an analysis of this unbalance amplification effect for an ideal rectifier circuit without acand dc-side inductors. A novel way of ling the voltage unbalance is established by introducing a deviation voltage superimposed on balanced three-phase line voltages, which allows simplified analysis of the rectifier operation. With proper approximations, closed-form expressions for symmetrical components of the fundamental line current and correspondingly the current unbalance factor are derived in terms of the voltage unbalance factor, filter reactance, and load current. The equivalent rms value of the third harmonic current is also derived. The analysis clearly shows high sensitivity of the current unbalance to the voltage unbalance, and provides a basic guideline for rectifier circuit design. The validity of the analysis is confirmed by simulation. Index Terms Diode rectifier, harmonics, power quality, unbalance, unbalance factor, uncontrolled rectifier. I. INTRODUCTION AS VAST majority of rn power converters are supplied from ac lines via uncontrolled rectifiers, the input line current characteristics of uncontrolled rectifiers have significant impacts on power quality. The rectifiers, controlled or uncontrolled, are not friendly to supply lines due to their nonlinear nature, producing considerable harmonic currents in distribution networks that have drawn much attentions [1] [7]. The situation is more severe for uncontrolled rectifiers usually accompanied by large smoothing dc capacitors. The capacitor at the output of the rectifier tends to enhance the nonlinearity of the rectifier operation by shortening the conduction period of rectifying devices, resulting in pulse-shaped line current rich in harmonics. In three-phase rectifiers, not only the harmonics but also the unbalance of input current is one of primary concerns in field applications. As the smoothing capacitor keeps dc voltage very close to the peak of line-to-line voltage, the current through the rectifier flows over brief periods to charge the capacitor in the vicinity of line voltage peaks. Therefore, a slight variation of Manuscript received December 26, 2001; revised July 15, Recommended by Associate Editor F. Blaabjerg. This work was supported by the Academic Financial Support Program, Kwangwoon University. The authors are with the Department of Electrical Engineering, Kwangwoon University, Seoul , Korea ( [email protected]; [email protected]). Digital Object Identifier /TPEL the voltage peak in some phase(s), or a small voltage unbalance may cause a significant variation in line current waveforms, that is, highly unbalanced line currents. Similar unbalance amplification is observed in induction motor drives where negative-sequence impedance is much smaller than positive-sequence impedance, yielding the current several times more unbalanced than the line voltage. Uncontrolled rectifiers, however, are far more sensitive to the voltage unbalance due to their nonlinear behavior under unbalanced voltage supply. As a result, it is not uncommon to encounter rectifier currents significantly unbalanced even when the supply voltage unbalance is well below a usually acceptable level. Unbalanced rectifier currents cause detrimental effects including: Uneven current distribution over the legs of the rectifier bridge that increases the conduction loss and may cause failure of rectifying devices; Increased rms ripple current in the smoothing capacitor; Increased total rms line current and harmonics, in particular, noncharacteristic triplen harmonics that do not appear under balanced condition [5] [7]. As most industrial and commercial power lines are more or less unbalanced, care should be taken in designing and installing uncontrolled rectifiers to keep the current unbalance within an acceptable level, and to avoid the above undesirable effects. Although the effects have been well recognized in field applications [5], [8] [10], only a few works have been devoted to the theoretical investigation of the input current characteristics of uncontrolled rectifiers under unbalanced voltage condition. Some analytical approaches can be found in [1], [6], [7] and [11], but they assume continuous dc current to examine noncharacteristic harmonics, which seems not to be valid in frequently encountered discontinuous current operations. This paper is intended to establish a basic relationship between the voltage unbalance and the current unbalance of a three-phase uncontrolled rectifier. To attain a better physical understanding, an analytical approach is attempted. The effects of ac-side and dc-side inductors are neglected, to ease the analysis and also to give a worst-case figure of current unbalance. II. REPRESENTATION OF VOLTAGE UNBALANCE The most popular way of representing the voltage unbalance is the maximum percent deviation of line voltages from their average value. Although this definition is simple and useful in field applications, it is not suitable for analytical purpose or not adequate to properly represent various unbalance con /02$ IEEE

2 936 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 6, NOVEMBER 2002 ditions. Therefore, in the following analysis, the voltage unbalance factor defined with symmetrical components is used, which is expressed in a complex form where (1) (2) It should be noted that (2) is defined with line-to-line voltages instead of phase voltages, considering that the rectifier behavior is more effectively described with line-to-line voltages. The consequence is the positive- and negative-sequence voltages times as large as those with phase voltages, and trivially zero zero-sequence voltage. In order to evaluate current unbalance for given voltage unbalance condition, the line voltages should be determined in terms of the voltage unbalance factor. This requires an inverse relationship of (1) and (2). The relationship can be derived by introducing the voltage deviation superimposed on a balanced set of line voltages where is taken as a reference phasor fixed on the real axis. Substituting (3) into (2) results in the positive-sequence and negative-sequence voltage as Substituting (4) and (5) into (1) yields the deviation voltage in terms of the voltage unbalance factor With the above equation, the line voltages in (3) can be determined for given unbalance factor, and associated symmetrical components are given by If the voltage unbalance factor is given by its magnitude, as usually is the case, the symmetrical components and corresponding line voltages cannot be uniquely determined but appear as phasor loci over a range of the phase angle of. With some algebraic manipulations, it can be shown that (3) (4) (5) (6) (7) (8) (9) Fig. 1. Phasor loci of symmetrical components and line voltages for given voltage unbalance factor. Therefore, for given unbalance factor, the phasor of the negative-sequence component lies on a circle centered at the position shifted from the origin along the real axis. As can be seen in (5), the deviation voltage is proportional to the negative-sequence voltage and the phasors of the unbalanced line voltages also draw circles as illustrated in Fig. 1. In Fig. 1, phasor loci are drawn for 20% voltage unbalance factor. In normal operating conditions, however, the voltage unbalance is usually restricted to be less than 2 3%. Under these circumstances, (6) (8) reduces to (10) (11) (12) Above equations indicate that the phasor locus of the negative-sequence voltage is centered very close to the origin and its radius is proportional to the voltage unbalance factor. Accordingly, the loci of line voltage phasors are approximately centered at their balanced phasor locations and their radius,,is also proportional to the voltage unbalance factor. And the phase angle of the voltage deviation is related to the argument of the complex voltage unbalance factor with (13) These approximations greatly simplify the descriptions and analysis of the rectifier operation in the following. III. OPERATING MODES OF RECTIFIER UNDER UNBALANCED VOLTAGE CONDITION Fig. 2 shows a three-phase uncontrolled rectifier circuit with a capacitor filter. For simplicity, the load is supposed to be a con-

3 JEONG AND CHOI: THREE-PHASE UNCONTROLLED RECTIFIERS 937 Fig. 2. Three-phase uncontrolled rectifier. Fig. 4. Unbalanced line voltage phasors that appear at the rectifier output. Fig. 3. Rectifier waveforms in balanced operation. stant current sink of, and the effects of line side inductance are not considered. Fig. 3 shows the voltage and current waveforms of the rectifier circuit under balanced condition. In the rectifier output, six line-to-line voltages appear sequentially as indicated in the figure. In the vicinity of peak of each line voltages, a brief charging period appears and is followed by a discharging period that approaches to 60 as the capacitor becomes large and/or the load current decreases. Thus the voltage droop during the discharging period is approximately given by (14) where is the reactance of the filter capacitor at line frequency. When balanced, represents the peak-to-peak ripple of the rectifier output, to give the ripple factor defined as (15) which is referred to the voltage droop ratio in this paper. The voltage droop ratio represents combinational effects of the load current and filter capacitance in describing the behavior of the rectifier under unbalanced voltage condition. When unbalanced, the peaks of the line voltages are not uniform, but fluctuate periodically over a cycle. Fig. 4 shows the Fig. 5. Variation of line-to-line peak voltages in time axis. deviated six line voltages where and are deviated by while and deviated by. In time axis, the variation of the peak voltages over a half cycle appears as shown in Fig. 5. It is clear from Fig. 4 that the same variation repeats every half cycle. Fig. 5 assumes a small voltage unbalance factor so that the center of the circles coincides the peak of the reference balanced voltages. From (10), the radius of the circles is given by (16) Therefore, the unbalanced peak voltages vary in the range of according to the phase angle of the deviation voltage,. The position of the peak voltages in time axis also varies as indicated by in Figs. 4 and 5. However, the effect of the phase shift is not significant and neglected to avoid complication. Under the assumption of small charging period, the variation of the output voltage of the rectifier over a cycle can be represented as a pulse train shown in Fig. 6. In view of Fig. 5, the peak line voltages are given by (17)

4 938 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 6, NOVEMBER 2002 And the condition for the of operation shown in Fig. 6(c) can be written down as (23) (24) The conditions for all seven s can be established in a similar way. With (14) (19), it can be shown that the conditions reduce to the following relationships between the voltage droop ratio, voltage unbalance factor, and the phase angle of the deviation voltage (or the argument of the complex voltage unbalance factor) Fig. 6. Operation s under unbalanced condition. (a) Six-pulse. (b) Four-pulse. (c) Two-pulse. (18) (19) where. Fig. 6 shows three distinct s of operation. In Fig. 6(a), all six peak voltages appear in the rectifier output, causing six charging period over a cycle, which is referred to the six-pulse of operation. In Fig. 6(b), two peak voltages are submerged below the rectifier output voltage profile resulting in the four-pulse of operation. If one of the three line voltages is significantly larger than the others, then only two charging periods appear over a cycle as in Fig. 6(c), resulting in the twopulse of operation. The four-pulse is further divided into three different operating s according to which line voltages appear in the rectifier output. For instance, Fig. 6(b) shows the case of and active in the rectifier output, which is referred to the in the following. The and the are defined in a similar manner. The two-pulse is also divided into three s:,, and according to which one of,, and appears active in the output. Therefore, there are seven different s of operation in total. Which the rectifier operates in depends on the magnitude and phase angle of the deviation voltage, and the voltage droop during the discharging period ( ). For example, by examining Fig. 6(b), the condition for of operation is given by (20) (21) (22) (6-pulse) and and (25) and and (26) and and (27) and and (28) and (29) and (30) and (31) where (32) (33) (34) Fig. 7 shows the operation s in the plane of complex voltage unbalance factor. Six solid lines divide the whole area into seven regions that correspond to seven operation s described above. The triangular region at the center represents the 6-pulse given by (25). The circle inscribing the triangle corresponds to, or. For the voltage unbalance factor less than this value, the rectifier operates in 6-pulse regardless of or. And above this value, the rectifier operates in 6-pulse or 4-pulse, depending on the phase of the deviation voltage. This holds until the voltage unbalance factor reaches or indicated by the circle circumscribing the triangle. Further increase of voltage unbalance will make the rectifier to operate in two-pulse or four-pulse. The larger the voltage unbalance factor is, the narrower the angular region of four-pulse. But at the angles of and 180 the rectifier never reaches the two-pulse. And at the angles of, 30 and 150, the four-pulse never occurs and the rectifier gets into the two-pulse from six-pulse directly.

5 JEONG AND CHOI: THREE-PHASE UNCONTROLLED RECTIFIERS 939 Fig. 7. Operation s in the complex unbalance factor plane. IV. LINE CURRENT UNBALANCE IN 6-PULSE MODE The waveform of in Fig. 3 is composed of charging current pulses. Assuming short charging periods, the current-time area of each current pulse can be approximated to (35) where is the increment of the output voltage during the charging period. To simplify the analysis the current pulses are approximated to square pulses that have the same area with the actual current pulses. With this approximation, the dc and line side current waveforms in six-pulse can be represented as shown in Fig. 8. The subscripts of six current pulses over a cycle indicate associated active line voltages that appear at the output. Due to half-cycle symmetry, the current pulses and are of the same area,, and so on. The areas of the current pulses are obtained with (35), and by substituting (17) (19) (36) (37) (38) In line current waveforms, a pair of equal-area current pulses, for instance, and, appears as an alternating pulse current over a cycle. Its Fourier coefficient is given by (39) where and represent the magnitude and width of the current pulse, respectively. If in (39) is sufficiently small, the Fourier Fig. 8. Square pulse approximation of line currents for six-pulse. coefficient is approximated to (40) Numerical study shows that the above approximation is valid for appreciable range of the voltage drooping ratio for the harmonics of relatively low frequency (see the Appendix). With (36) (38) and (40), the rms magnitude of the fundamental components of three pulse pairs are given by (41) (42) (43) As each phase of the line currents comprises two current pulse pairs, its fundamental component is the sum of the contributions of the current pulse pairs. Accounting for the phase of the current pulse pairs with respect to the reference phasor, the phasors of three-phase fundamental line currents are given by (44) (45) (46) The above expressions give the symmetrical components of the line currents (47) (48)

6 940 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 6, NOVEMBER 2002 and the unbalance factor of fundamental current (49) By substituting (41) (43) into (47) (49), the symmetrical components and current unbalance factor are represented in terms of the line voltage deviation or the voltage unbalance factor. The positive-sequence current reduces to or, with (14), The negative-sequence current is or with (16) (50) (51) (52) (53) Above expressions clearly show the effects of three parameters the voltage unbalance factor (or the magnitude of deviation voltage), the load current, and the reactance of the capacitor filter. The load current is reflected to only the positive-sequence current while the voltage unbalance factor has an effect on the negative-sequence current only. The negative-sequence current is proportional to the voltage unbalance factor and inversely proportional to the filter reactance. With (12), the magnitude of negative-sequence current is related to the negative-sequence voltage as (54) which shows that the negative-sequence impedance of the rectifier is solely determined by the capacitive filter reactance. The expression of the current unbalance factor is, from (50) and (52), given by (55) With the voltage droop ratio defined in (15), above equation can be written as (56) As the voltage droop ratio,, i.e., the output voltage ripple factor in balanced case is usually limited to a few percent, (56) indicates that the current unbalance is several tens times as large as the voltage unbalance, clearly showing the unbalance amplification effect. Another important result of the voltage unbalance is the third harmonic components existing in line current waveforms. Assuming again sufficiently short duration of charging current pulses, (40) still holds so that the rms magnitude of third harmonic component in a current pulse pair is approximately the same as that of the fundamental component, giving, and so on. Therefore, the expressions in (41) (43) are still valid for evaluating third harmonic currents. The magnitude of third harmonics contained in line current waveforms in Fig. 8 can be obtained by replacing in (44) (46) with,togive (57) (58) (59) One way of representing unbalanced line currents is the equivalent rms value that accounts for total Ohmic losses in three-phase transmission lines [12]. Applying this concept, the equivalent third harmonic current reduces to, by substituting (41) (43) and (16) (60) (61) It is interesting that (61) appears to be times as large as the magnitude of the fundamental negative-sequence current in (53). As a natural consequence, the equivalent third harmonic current normalized with respect to the positive-sequence current is times the current unbalance factor V. LINE CURRENT UNBALANCE IN 4-PULSE AND 2-PULSE MODE (62) Fig. 9 shows a four-pulse of operation. Shown is the where the current waveforms of phase and phase is similar to that of a single-phase rectifier whereas only the phase carries four pulses per cycle. In this case, the pulse areas are given by (63) (64) which gives the magnitude of fundamental component of current pulse pairs (65) (66) Comparing Fig. 9 with Fig. 8 shows that line current phasors and their symmetrical components in can be obtained by simply removing terms from six-pulse case. Therefore (67) (68)

7 JEONG AND CHOI: THREE-PHASE UNCONTROLLED RECTIFIERS 941 Fig. 9. Line currents in four-pulse. Similar procedure applies for and and it can be shown that the positive-sequence current is given by (69) which is the same as (51) and invariably holds for all 4-pulse s. The negative-sequence current, however, depends on not only the magnitude but also the phase of the deviation voltage as given below (70) (71) (72) The above expressions give the magnitude of the negative-sequence current (73) Fig. 10. Line currents in two-pulse. Therefore, from (69) and (73), the current unbalance factor is given by (75) The third harmonic components can be obtained in a similar way of six-pulse. The equivalent rms value of the third harmonic current is (76) where is given in (73). As in six-pulse case, the equivalent third harmonic current in (76) is times as large as the magnitude of the negative-sequence current given in (73), thus the relationship in (62) holds again. The two-pulse of operation is illustrated in Fig. 10. The figure assumes the - where is the largest among the line voltages. In this case, the phase does not conduct and the rectifier operates like a single-phase rectifier connected across the phase lines and. There is only one active current pulse pair that has the pulse area given by (77) Thus the magnitude of the fundamental component of the current pulse pair is where (78). (74) or, by substituting (14) (79)

8 942 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 6, NOVEMBER 2002 Fig. 11. Phasor of the positive-sequence current and phasor loci of the negative-sequence current. Comparing Fig. 10 with Fig. 8 shows that line current phasors and their symmetrical components in ab can be obtained by removing and terms from six-pulse case. Therefore, (80) (81) The positive-sequence current in (80) reduces to the same expression as those in six-pulse and four-pulse, which holds for and as well. The negative-sequence current is equal to the positive-sequence current in its magnitude, independent of the degree of the voltage unbalance. But its phase angle depends on the phase of the deviation voltage. For completeness, the negative-sequence current in all three two-pulse s is given as And it is apparent that the current unbalance factor is (82) (83) (84) (85) It can readily be shown that the equivalent third harmonic current is still times the magnitude of the negative-sequence current, to give (86) Fig. 12. Current unbalance factor versus complex voltage unbalance factor. VI. CHARACTERISTICS OF UNBALANCED LINE CURRENT With the expressions of symmetrical components of the line currents given above, the phasor loci for various voltage unbalance factor are depicted in Fig. 11. The positive-sequence current is fixed in the complex plane as long as the dc load current is kept constant. The negative-sequence current is always equal to or less than the positive-sequence current in magnitude. Thus its phasor locus never exceeds the circle drawn with dashed lines. The negative-sequence current is further constrained within the triangle circumscribed by the circle. Three corners of the triangle represent the phasor locations in two-pulse given by (82) (84), and the sides of the triangle represent the phasor locus in four-pulse given by (70) (72). The inside region of the triangle represents six-pulse. In six-pulse operation, the negative-sequence current leads the positive-sequence current by the phase angle of the deviation voltage, drawing a circle for given magnitude of the voltage unbalance factor. For a voltage unbalance factor less than, the circle is well within the triangle and the rectifier operates in six-pulse all over the phase angle of the deviation voltage. When, the locus inscribes the triangle and its radius is one-half the magnitude of the positive-sequence current, resulting in 50% current unbalance factor. For, the negative-sequence phasor lies on a side of the triangle in four-pulse or on a circle in six-pulse, depending on its phase angle. For, the six-pulse no more appears. The phasor is fixed at one of three corners of the triangle in two-pulse and moves along the sides of the triangle in four-pulse. Fig. 12 shows the variation of the current unbalance factor versus the complex voltage unbalance factor. The figure clearly shows a distinction between the six-pulse, four-pulse, and twopulse that appear as the corn-shaped part at the center, three valleys, and three flat areas at the top, respectively. The behavior of the current unbalance factor can be more clearly observed in Fig. 13 that shows constant angle plots drawn

9 JEONG AND CHOI: THREE-PHASE UNCONTROLLED RECTIFIERS 943 Fig. 14. Comparison between numerical results and theoretical predictions for line current unbalance factor. Fig. 13. Current unbalance factor vs. voltage unbalance factor ( =2%). for the voltage unbalance factor angle from 60 to 60. The plots for the other angles can be obtained considering 120 symmetry. As the voltage unbalance factor increases, the current unbalance factor linearly increases from 0 to 50% and then varies in a rather complicated manner until it reaches to 100%. An exception is the case of ( ) where the current unbalance factor remains at 58% ( ). This is because, as can be observed in Fig. 7, the rectifier stays at four-pulse for and never gets into two-pulse at this angle. Another interesting case is at where four-pulse does not exist and the current unbalance factor increases linearly all the way to 100%. The plot for the per unit equivalent third harmonic rms current is the same as Fig. 13 except that it is scaled up by times. To confirm the validity of the analytical results given above, simulation is carried out with MATLAB/SIMULINK. Line current waveforms of a three-phase uncontrolled rectifier are obtained by simulation for given complex voltage unbalance factor and voltage droop ratio. Assuming a resistive load, the capacitance of the filter is determined to meet the given voltage droop ratio, or to have an angular time constant given by (87) for an arbitrarily chosen load resistance. With simulated current waveforms, the fundamental current unbalance factor and the equivalent third harmonic rms current are calculated. Fig. 15. Comparison between numerical results and theoretical predictions for normalized equivalent rms of third harmonic current. The results are plotted in Fig. 14 for fundamental current unbalance factor and Fig. 15 for normalized equivalent third harmonic current, along with analytical predictions. For visual clarity, only the results for and are shown. It is obvious that theoretical curves generally show good agreement with simulation results even to the range of relatively large voltage unbalance factor up to 10% in spite of the assumption of small voltage unbalance imposed on the analysis. As the voltage droop ratio becomes larger the analytical results show larger errors due to increased charging periods. This is more considerable in Fig. 15 as the third harmonics in line currents are more sensitive to the current pulse width than fundamental components. However, the error seems to be generally acceptable as the voltage droop ratio is not expected to be greater than 5% in most uncontrolled rectifier applications.

10 944 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 6, NOVEMBER 2002 VII. CONCLUSION This paper gives an analysis of the line current characteristics of an uncontrolled rectifier under unbalanced line voltages. It is shown that, for usual voltage unbalance of a few percent, the line voltage phasors can be represented by circular loci centered at balanced phasor locations, which greatly simplifies the representation of the output voltage variation of the rectifier in time axis. The conditions of seven distinct operating s of the rectifier are established in terms of the voltage unbalance factor and voltage droop ratio that reflects combinational effects of the filter reactance and load current. In each of operation, analytical formulas of symmetrical components of fundamental line currents and current unbalance factor are derived. The negative-sequence current appears to be decoupled from the load current by the capacitor filter that solely determines the negative-sequence impedance of the rectifier. Thus a large capacitor in the rectifier output causes a large negative-sequence current and correspondingly a high current unbalance factor even for a small voltage unbalance. On the other hand, the positive-sequence current is not affected by the voltage unbalance but is determined by the load current only. The third harmonic current is also evaluated and, interestingly, its equivalent rms value turns out to be a constant multiple of the magnitude of the negative-sequence current. The analysis is based on the following assumptions. 1) The voltage unbalance is relatively low. 2) Capacitor charging period is very short. 3) Effects of line-side reactance are negligible. The first assumption seems to be valid in most practical situations as the voltage unbalance factor is usually restricted within 2 3%. The second assumption implies that the filter capacitor is sufficiently large and/or the load current is small. Therefore, the analysis is valid only when the voltage droop ratio, or the ripple factor in balanced case, is relatively low. Numerical studies show that the analytical predictions for the voltage droop ratio of less than 5% are largely acceptable. The third assumption can be met if the capacity of the rectifier is far smaller than the short circuit capacity of the supply network at the rectifier terminals. The line inductances normally encountered in distribution network, however, may cause the current unbalance to be considerably less sensitive to the voltage unbalance than predicted in this paper. Nevertheless, the approach made in this paper can be justified in that it provides a figure of worst-case, theoretical maximum of current unbalance in an analytical manner, providing an insight on the line current characteristics of uncontrolled rectifiers fed from unbalanced supply. And proposed formulas are expected to be useful in designing the supply system and selecting filter capacitor for a rectifier, or assessing the impact of the voltage unbalance on power quality of an existing network feeding rectifier loads. APPENDIX SQUARE PULSE APPROXIMATION OF LINE CURRENT PULSE PAIR The waveform at the top of Fig. 16 shows the capacitor current of the rectifier supplying constant load current. The Fig. 16. Square pulse approximation of a current pulse pair. second waveform shows associated line current pulse pair comprising two current pulses of opposite polarities apart from each current other by a half cycle. Under balanced condition, each pulse area should be (A1) If the current pulse duration is sufficiently small, then it can be approximated to a square pulse of the same area, as shown in the last waveform in Fig. 16. This square pulse pair has the harmonic amplitude or, from (A1) (A2) (A3) This constant harmonic amplitude implies that the pulse is approaching a delta function of infinitesimal duration ( ). Thus the approximation applies to only low order harmonics for actual current pulses that have a finite duration. To examine the validity of this approximation, the harmonic amplitudes of the actual current pulse pair are evaluated. It can be shown that the line current pulse pair in Fig. 16 is given by where for for otherwise (A4) (A5)

11 JEONG AND CHOI: THREE-PHASE UNCONTROLLED RECTIFIERS 945 Fig. 17. Normalized harmonics of line current pulse pair versus voltage drooping ratio. or, with the definition of the voltage droop ratio in (15) (A6) To calculate harmonics, the angles and should be determined. It can readily be shown that, from (A7) while is given by a solution of the nonlinear equation for the current pulse area or (A8) With and determined for given voltage droop ratio, the harmonic amplitudes are calculated for the current pulse pair in (A4). Fig. 17 shows harmonic amplitudes normalized with respect to versus the voltage droop ratio. The figure clearly shows that low order harmonics, say, up to 5th harmonics, remain constant near over an appreciable range of voltage drooping ratio, which confirms the validity of (A3). Although the result is given for balanced condition, it implies that the approximation (A2) is valid regardless of voltage unbalance. REFERENCES [1] M. Sakui, H. Fujita, and M. Shioya, A method for calculating harmonic currents of a three-phase bridge uncontrolled rectifier with DC filter, IEEE Trans. Ind. Electron., vol. 36, pp , Aug [2] A. W. Kelley and W. F. Yadusky, Rectifier design for minimum line-current harmonics and maximum power factor, IEEE Trans. Power Electron., vol. 7, pp , Apr [3] D. E. Rice, A detailed analysis of six-pulse converter harmonic currents, IEEE Trans. Ind. Applicat., vol. 30, pp , Mar./Apr [4] S. Hansen, P. Nielsen, and F. Blaabjerg, Harmonic cancellation by mixing nonlinear single-phase and three-phase loads, IEEE Trans. Ind. Applicat., vol. 36, pp , Jan./Feb [5] G. A. Conway and K. I. Jones, Harmonic currents produced by variable speed drives with uncontrolled rectifier inputs, in Proc. IEE Colloq. Three-Phase LV Ind. Supplies: Harmonic Pollution Recent Develop. Remedies, 1993, pp. 4/1 4/5. [6] M. Grötzbach and J. Xu, Noncharacteristic line current harmonics in diode rectifier bridges produced by network asymmetries, in Proc. Euro. Power Electron. Conf., 1993, pp [7] M. Bauta and M. Grötzbach, Noncharacteristic line harmonics of AC/DC converters with high DC current ripple, IEEE Trans. Power Delivery, vol. 15, pp , July [8] E. R. Collins and A. Mansoor, Effects of voltage sags on AC motor drives, in Proc. IEEE 1997 Annual Textile, Fiber, Film Ind. Tech. Conf., 1997, pp [9] A. Fratta, M. Pastorelli, A. Vagati, and F. Villata, Comparison of power supply topologies for industrial drives, based on safety and EMC European standards or directives, in Proc. 8th Meditarranean Electrotech. Conf., 1996, pp [10] D. A. Rendusara, A. V. Jouanne, P. N. Enjeti, and D. A. Paice, Design considerations for 12-pulse diode rectifier systems operating under voltage unbalance and pre-existing voltage distortion with some corrective measures, IEEE Trans. Ind. Applicat., vol. 32, pp , Nov./Dec [11] Y. Wnag and L. Pierrat, Probabilistic ling of current harmonics produced by an AC/DC converter under voltage unbalance, IEEE Trans. Power Delivery, vol. 8, pp , Oct [12] A. E. Emanuel, The Buchholz Goodhue apparent power definition: The practical approach for nonsinusoidal and unbalanced systems, IEEE Trans. Power Delivery, vol. 13, pp , Apr Seung-Gi Jeong (M 88) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1982, 1984, and 1988, respectively. Since 1987, he has been a member of the faculty of the Department of Electrical Engineering, Kwangwoon University, Seoul, where he is currently a Professor. He has been engaged in the research and development in the areas of control of static power converters, industrial drives, and active power filters. Currently, he is interested in the issues on power quality and generalized power theories. He published more than 30 papers. Dr. Jeong received three Best Paper Awards from the Korean Institute of Electrical Engineers (KIEE) and Korean Institute of Power Electronics (KIPE), the Chunkang Academic Award from the Chunkang Memorial Foundation, and the Jang Young Shil Award from the Korea Industrial Technology Association. Ju-Yeop Choi received the B.S. degree from Seoul National University, Seoul, Korea, in 1983, the M.S. degree from University of Texas at Arlington, in 1988, and the Ph.D. degree from the Virginia Polytechnic Institue and State University, Blacksburg, in 1994, all in electrical engineering. He had been with the Intelligent System Control Research Center, Korea Institute of Science and Technology, Seoul, from 1995 to Since 2000, he has been an Assistant Professor in the Department of Electrical Engineering, Kwangwoon University, Seoul. His primary research interests are in switching converter, soft switching, and photovoltaic systems.