Calculation of Uncharacteristic Harmonics Generated by Three-Phase Diode-Bridge Rectifier with DC Filter Capacitor

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1 Paper Calculation of Uncharacteristic Harmonics Generated by Three-Phase Diode-Bridge Rectifier with DC Filter Capacitor Member Masaaki Sakui (Toyama University) Member Hiroshi Fujita (Toyama University) Three-phase diode-bridge rectifiers are used to convert the AC input into DC voltage. A large capacitor is connected as a filter on the DC side, since in most applications, a low ripple in the DC output voltage is desirable. Unbalance in the supply conditions of a three-phase diode-bridge rectifier with a DC filter capacitor gives rise to uncharacteristic harmonics on both the AC and DC sides. This paper proposes a simplified method for calculating the uncharacteristic harmonic currents generated by the three-phase diode-bridge rectifier under unbalance, taking into account the effects of both the AC and DC side impedances. Analytical expressions for the harmonic currents are derived using the switching functions with overlap period and the AC circuit theory. Therefore, the proposed method can be executed only with algebraic computation and its accuracy is quite high. The validity of the proposed method is demonstrated by comparison with the results of time simulation. Key words : Bridge rectifier, Uncharacteristic harmonics, Fourier series 1. Introduction The three-phase diode-bridge rectifier with a DC filter capacitor is often employed in the power electronic applications, such as switching DC power supplies, AC-motor drives and so on. This rectifier circuit will generate uncharacteristic current har monics on both the AC and DC sides, if the power supply conditions are unbalanced. Two basically different methods have been proposed to calculate the uncharacteristic har monics generated by the three-phase rectifier under unbalanced conditions. The first uses frequency domain analysis, which is based on the switching functions(1) `(3). This method is convenient, but does not take into account the effect of the AC side impedance. Often in three-phase rectifier, to improve the line current waveform, filter inductor is added on the AC side. In this case, the AC side impedance results in a higher value and therefore must be considered. The second method is time simulation, which is versatile and has the advantage of the calculation accuracy""). But, it is rather tedious and requires a long computing time for solving differential equations using a numerical solution. This paper proposes a new method that makes it possible to easily calculate the uncharacteristic harmonic currents of a diode-bridge rectifier with a DC filter capacitor under unbalanced power supply conditions. It takes into account the effects of both the AC and DC side impedances. Analytical expres sions for the harmonic currents on both the AC and DC sides are derived using the frequency domain method and the rectifier switching functions. There fore, all the calculation are conducted only by alge braic calculations, and its accuracy is sufficient for practical usage. Also, unbalances in voltage

2 magnitude, phase angle and commutation reactance can be easily incorporated into the calculation. The validity of the proposed method is demonstrated by comparison with the results of time simulation. 2. Theoretical basis for harmonic calcula tion Fig. 1 shows three-phase diode-bridge rectifier circuit with a filter capacitor on the DC side. The analysis of the circuit in Fig. 1 is carried out with the following assumption : (1) the diodes are ideal ; (2) DC side current id is not interrupted ; and (3) the AC system is not resonant. Following the method proposed earlier(6), the general expressions for the DC and AC side currents under unbalanced conditions in Fig. 1 are derived below. DC side current id, which consists of the DC component Id and the AC component id, can be represented as (1) where, m : order of harmonics, Id.,/3.: rms value and phase angle of the m-th harmonic component of DC side current, respectively. For a given Id, the circuit for the AC'component of the DC side current in Fig. 1 can be represented by the equivalent circuit as shown in Fig. 2(6), as seen from the DC side. In Fig. 2, edm is the m-th harmonic component of open-circuit output DC voltage. In addition, Zm and Zom are the impedances of the DC and AC sides for the m-th harmonic component as viewed from the terminals a and b in Fig. 1, respectively. Clearly, Fig. 2 represents a Thevenin's model. Therefore, the m-th harmonic component idm of the DC side current id coincides the 'current flowing through the circuit in Fig. 2 and is given as follows : If the filter reactance (2) X, is infinitely large, the value of impedance Zm is infinite and terminals a and b are open-circuited. Therefore, the open-circuit output DC voltage edm in Eq. (2) is obtained easily from the assumption that the DC side current id is completely smooth. Also, knowing idm in Eq.(2 ), Idm and Nm can be easily obtained. Therefore, for a given Id, the DC side current id is determined from Eq.(1). Similarly, the AC side current can be analyzed decomposing it into two currents. More precisely, the respective AC side currents consist of two AC currents produced by the DC component Id and the AC component id, of id. Therefore, they expressed as fo11owe: (3) where iaa, iob, ioc : AC currents of the respective phases produced by the DC component Id of id, ia, Fig. 1. Three-phase diode-bridge rectifier circuit. ib, ic : AC currents of the respective phases in duced in the AC side by the AC component id, of id. Analytical expressions for the currents id, ia, iy and i~ are derived in Sections 3 and DC side harmonic calculations The three unbalanced supply phase voltages are represented by Fig. 2. Equivalent circuit for calculating the DC side harmonic current with an average value Id. (4) 588 T. IEE Japan, Vol. 113-D, No. 5, '93

3 As described in the previous section, the m-th component edm of open-circuit output DC voltage in Eq. (2) can be obtained using the assumption that the DC side current is ripple-free. Under this assumption, the output DC voltage ed is expressed in terms of rectifier switching functions Sa, Sb and S., as shown in Fig. 3(6', as follows : ed=saea+sbeb+scec c c(5) Each rectifier switching function can be represented by the complex form of Fourier series as follows : The angles /l,, A2 and A3 in Eq. ( 6), which are the starting and terminating points for the rectifier switching functions, are determined so as to satisfy (6) Also, the overlap angles u,, uz and U3 are given as follows (see Appendix 1) u1 Therefore, substituting Eqs( 4) and (6) into Eq. (5) gives (7) where Fig. 3. Voltage waveforms and rectifier switching functions. Therefore, from Eq. ( 7), the m-th harmonic volt age edm in Eq. ( 2) can be determined. For the m-th component of the DC side voltage, impedance Zm of the DC side as viewed from the terminals a and b in Fig. 1 is obtained using the AC circuit theory as follows :

4 where Z1=jmXl, Z2=-jXf/m, Z3=Rd+jmXd, M>0. On the other side, it is impossible to determine accurately impedance Zom of the AC side as viewed from the same terminals due to the presence of the rectifier. Now, we present an approximate solution. By calculating the average of the overlap angles and the commutation reactances, impedance Zon, for the m-th component is approximately obtained by (see Appendix 2) (9) where m>0. From Eqs(1), (2) and (7) `(9),the AC com ponent id, of id can be expressed as follows : (10) Therefore, the rms value of the m-th harmonic current of id, is given by (11) The proposed method is valid only for a continu ous DC current. In general case, the first (sixth) harmonic component of the DC current is relatively large compared to all other harmonic components. However, if the filter component values on the DC side are not properly selected, the harmonic reso nance conditions occur, which result in higher harmonic components. Therefore, the range of the harmonic current Id. is expressed as approxi matelv(6) 4. AC side harmonic calculation (12) Since the DC component Id of id is constant, it can be considered that the AC side current produced by Id is equivalent to that for the case id=ld (idr=0). Therefore, the AC current ioa in phase a for Id can be expressed as follows (see Appendix 3) : (13) Expressions for the currents iob and is of phases b and c can be easily obtained by replacing the suffixes (a, b, 1, 2) in Eq.(13) with (b, c, 2, 3) and (c, a, 3, 1), respectively. AC current 4ia is induced in the phase a by the AC current id, of id. The rectifier switching func tions in Eq. ( 6) are related to the conduction state of the diodes in each phase. Therefore, 4iQ can be expressed by the product of current id, and rectifier switching function Sa as follows(6): where (14) Therefore, the AC side current iq which corresponds to the DC side current id is derived from Egs(3 ), (13) and (14) as follows: (15) The rms value of the n-th harmonic current of is is given by (16) 590 T. IEE Japan, Vol. 113-D, No. 5, '93

5 Also, the expressions for the other two phase currents ib and is can be easily derived from the replacement of the suffixes, as described previously. 5. Calculation results To test the calculation accuracy of the proposed method, the results obtained by the proposed method are compared with those obtained by the simulation method(4), which uses the time domain analysis. In this simulation method, no approxima tion are made of, for example, the effects of the overlap and the current ripple. Therefore, a check on the proposed method can be made using the simulation method. For case of unbalanced system conditions, a maximum imbalance of }5% was introduced in the values of voltage magnitude and the commutation reactance as well as }2 K voltage phase imbalance. The system data (in per unit) for the unbalanced conditions are given below : The calculated magnitudes of harmonic currents generated on the DC and AC sides are shown in Tables 1 and 2. In Table 2, 1, is the rms value (-??_)Id of fundamental component of the line Table 1. Comparison of DC side harmonic currents. Table 2. Comparison of AC side harmonic currents.

6 current in an ideal case. These tables indicate that the results obtained by the proposed method agree very well with those obtained by the simulation method even for the case of considerable unbalance. Also, it should be notes that for the same value of Id, increased AC side reactances reduce the generations of both character istic and unchracteristic harmonic currents. There appears somewhat large difference in some places for XJd/Eo=0.02. However, the error for the per centage harmonic current content is less than 2.6% and 6.2% for the DC and AC side harmonics, respec tively, and therefore the proposed method is sufficiently accurate from practical viewpoints. The error may be due to the fact that the impedance Zom cannot be accurately calculated taking into account the effect of overlap angles and the effect of unbal. ance is neglected in the determination of this impedance. The overlap angle increases with Xald/Ea, and therefore the error of Idm also natu rally increases. In this case, however, the value of Id./Id becomes small, and the DC current becomes almost flat. Hence, the effect of the error of Idm is not very great, and the calculation accuracy is good when Xald/Ea is large. The simulation method takes 4.2 sec to evaluate the harmonic currents for the above case at the simulation time of 5 cycles, using a IBM 3081-KX4 computer. In contrast, the proposed method only takes 0.1 sec to produce the same information. Furthermore, the validity of the proposed method for the balanced conditions is demonstrated in Ref Conclusions In this paper, we have proposed a new method to calculate easily both characteristic and unchar acteristic harmonic currents under unbalanced oper ating conditions. The proposed method takes into account the DC current ripple component and the overlap angles. Analytical expressions for the har monic currents on both the DC and AC sides are derived using the frequency domain analysis and the rectifier switching functions. All the calculations are conducted only by algebraic calculation with high accuracy. Therefore, neither a complicated algorithm nor a long computing time are required. The validity of the proposed method is demonstrat ed by comparison with the results obtained by the time simulation (Manuscript method. received June 24, '92, References revised Sep. 16, '92) (1) R. Bonert & S. B. Dewan : "Line unbalance effects in a three-phase rectifier with L, C. filter", Conf. Rec. IEEE- IAS, p. 479 (1979) (2) M. H. Rashid & A. Maswood : "Analysis of three-phase AC-DC converters under unbalanced supply conditions", IEEE Trans. Industr. Applic., IA-24, No. 3, 449 (1988) (3) P. N. Enjeti & P. D. Ziogas : "Analysis of a static power converter under unbalance: A novel approach", ibid, IE 37, No. 1, 91 (1990) (4) J. S. C. Htsui & W. Shepherd : "Method of digital compu tation of thyristor switching circuits", Proc. Instn Elect, Engm, 118, No.8, 993 (1971) (5) A. M. Sharaf, Y. Yao & R. Wu : "Thyristor converter modulated impedance under unbalanced system", Electr. Power Syst. Res., 11, 59 (1986) (6) M. Sakui, H. Fujita & M. Shioya : "A method for calculat ing harmonic currents of a three-phase bridge un controlled rectifier with DC filter", IEEE Trans. Ind. Electron., IE-36, No. 3, 434 (1989) Appendix 1. Derivation of overlap angles Consider the situation where the current is trans ferred from phase c to phase a and the current c ommutation begins at O=ă1. The commutation current i for phase a is given by (Al) Therefore, the overlap angle u, is found by setting i=id at ľ=Ď1+u1 in Eq.(a1). In the same manner, the overlap angles u2 and u3 can be obtained. 2. Derivation of Zom Since the commutation reactance is not extremely different in each phase, we consider the effect of each commutation reactance on the average for determining Zom. The average commutation reactance Xe for one phase is given by (A2) By using Xe, the reactance Xt of the AC side as viewed from the terminals a and b in Fig. 1 is given 592 T. IEE Japan, Vol. 113-D, No. 5, '93

7 by (3/2)Xe for the overlap periods and by 2Xe for the non-overlap periods. Therefore, the average value of X over a half cycle (7r rad) is expressed as follows : Therefore, the complex Fourier series expansion of the current ioa gives Eq. (13). Masaaki Sakui (Member) (A3) Expressing Eqs (A2) and (A3) in terms of impedan ce for the m-th harmonic component gives Eq. ( 9). 3. Fourier series of ioa AC current ioa in phase a is produced by the smoothed DC current being equal to the average current Id. From Eq. (A 1), iaa is represented by a symmetric waveform as described below : (i) (ii) (iii) (iv) He received the B. E. and M. E. degrees from Toyama University in 1972 and 1974, respectively, and the Dr. Eng. degree from Tokyo Metropolitan University in Since 1974, he has been with Toyama University as Research Assistant, Instructor and Associ ate Professor. He is a member of the IEEE. Hiroshi Fujita (Member) He received the B. E. degree from Toyama University in 1954, and the Dr. Eng. degree from Tokyo Metropolitan University in He joined the Meidensha Electric Mfg., Co., Ltd., in Since 1963, he has been with Toyama University as Research Assis tant, Associate Professor and Professor. He is a member of the IEEE.