EXPERIMENTAL VERIFICATION OF SINUSOIDAL APPROXIMATION IN ANALYSIS OF THREE-PHASE TWELVE-PULSE OUTPUT VOLTAGE TYPE RECTIFIERS

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1 th INTERNATIONAL SYPOSIU on POWER ELECTRONICS - Ee 9 XV eđunarodni sipoziju Energetska elektronika Ee 9 NOVI SAD, REPUBLIC OF SERBIA, October 8 th - th, 9 EXPERIENTAL VERIFICATION OF SINUSOIDAL APPROXIATION IN ANALYSIS OF THREE-PHASE TWELVE-PULSE PUT VOLTAGE TYPE RECTIFIERS Predrag Pejović, Johann W. Kolar, Vladiir Šviković Faculty of Electrical Engineering, University of Belgrade, Belgrade, Serbia Swiss Federal Institute of Technology, Zürich, Switzerl Abstract: Experiental verification of the sinusoidal approxiation approach in analysis of three-phase twelve-pulse output voltage type rectifiers is presented in the paper. The rectifier operating paraeters: dependence of the output voltage on the output current, the output power, the power factor, the efficiency, are deterined applying sinusoidal approxiation. The s are verified on a low-power experiental rectifier odel. Keywords: AC-DC power conversion, converters, haronic distortion, power conversion haronics, power quality, rectifiers. + p + p + p vy vy v Y ia v A ia v A i A i B v A v B DA DA DA DA DA4 DA6 DB DB DB C I v P v + V. INTRODUCTION In this paper, suitability of the sinusoidal approxiation approach in odeling of the rectifier presented in Fig. is analyzed. The rectifier of Fig. is proposed in [], it consists of three coupling inductors, a line-side interphase transforer, two diode bridges, a filtering capacitor. The line-side interphase transforer splits the input currents provides equal load sharing between the diode bridges. Besides, it provides appropriate phase shift to for twelve-pulse voltages v T T, v T on the basis of v A A A B B, v B. The twelve-pulse voltages v T T, v T are connected to the three-phase power line by the coupling inductors. Current filtering at the output side of the rectifier is not required, causing the rectifier to be of the output voltage type. Actually, the current flowing to the filtering capacitor the load contains very low ripple. The rectifier is siple does not require any controlled switches, neither the high frequency switching. Thus, related electroagnetic interference issues, increased losses, reduced reliability are avoided. The basic structure proposed in [] is extended in [, ] to provide 8-pulse operation the reverse flow p p p L v T i L v T i L v v v ib v B ib v B v T i DB DB4 DB6 Fig.. The rectifier of energy in the recuperation ode. An extension to 4-pulse operation detailed analysis of the rectifier are given in [4]. Although the rectifier structure is siple, it is hard to analyze it. In [], sinusoidal approxiation approach is applied to analyze six-pulse output voltage type rectifiers while they operate in the continuous conduction ode. In [6], it is shown that in the case resistive losses in the rectifier can be neglected, there is a closed-fo analytical solution that covers the rectifier behavior, having the sae coputation coplexity as the sinusoidal approxiation based solution of []. However, resistive losses, i.e. current dependent losses, cannot be covered by such solution.

2 Table. Definitions of index functions k k ( ) ( k) An extension of [6] is given in [7], where a closed-for solution of the rectifier of Fig., is given for the continuous conduction ode, as well as the solution based on the sinusoidal approxiation. Coparison of the solutions is provided, indicating that the difference between the exact the sinusoidal approxiation based solution is lower than in the case of the six-pulse rectifier, due to the twelve-pulse operation thus reduced haronic content of the rectifier voltages currents at the AC side. Again, the exact the sinusoidal approxiation based solution have the sae coputational coplexity. Unfortunatelly, the exact solution cannot take account of resistive losses, dependent on the rectifier currents. In the area where this type of losses is relevant, sinusoidal approxiation should be used, being the only choice, but justified with the good agreeent with the exact solution in the case the resistive losses were absent fro the circuit. Application of the sinusoidal approxiation to the analysis of the rectifier of Fig. is given in this paper, closed-for expressions for the dependence of the output voltage on the output current, as well as the output power, the power factor, the efficiency on the output voltage are given. Experiental results obtained on the rectifier odel are presented copared to the s.. SINUSOIDAL APPROXIATION.. Preliinaries Let us assue that the rectifier is supplied by a three-phase voltage syste vk V sin ωt ( k ) () for k {,, }. To siplify the notation to reduce the nuber of equations that needs to be written to characterize the line-side interphase transforer (actual nuber of equations is nine, cannot be reduced regardless the copact notation), let us define the index functions next ( k) previous ( k), defined for k {,, }. Definitions of these functions are given in Table I, being based on odular arithetic with the base equal to, but shifted for to eet the coon indexing of phases. After the index functions are introduced, equations that characterize the line-side interphase transforer are given by p iak ik i ( k ) () p + p + p + p ibk ik + i ( k ) () p + p + ( v ) p + p vtk vak + vbk A ( k ) v B ( k ) (4) p + p + p + for k {,, }. There is a total of nine such equations in the exped for, six of the relating currents, three of the relating voltages. To provide twelve-pulse wavefors of v T T, v T to cancell out haronic coponents at 6 n ±, n N, the line-side interphase transforer turns ratio according to [] should be set to p.66. ().. Noralization In order to generalize the results, noralization of variables should be introduced. It is convenient to noralize the circuit voltages to the phase voltage aplitude v. V (6) The tie variable should be replaced by the phase angle variable ϕ ωt. (7) In the case the currents are noralized according to ωl j i V (8) governing equations for the inductors reduce fro dik L vk vtk dt (9) to djk k Tk dϕ () for k {,, }. To preserve the for of Oh s law, resistances should be noralized to ρ R. ωl ().. Sinusoidal approxiation Sinusoidal approxiation approach in the circuit analysis assues negligible higher-order haronic coponents of all ac wavefors. In the case of the rectifier of Fig., the approxiation is done around the inductors: their terinal voltages are assued as sinusoidal, as well as their currents. Let us assue that the inductor currents are jk J φ ( k ) () for k {,, }, where φ is the phase shift of the phase currents with regard to the corresponding phase voltages. According to (), (), (), this results in the line-side interphase transforer output currents jak J φ + ( k ) () jbk J φ ( k ). (4)

3 .8 r k.6 r Lk.4 t/out. φ r Tk Fig. 4. The phasor diagra ρ J r k Generalizing the result of [] to two three-phase diode bridges, the rectifier output current is given by J 6 J () ( ) which is used in [7]. Each of the inductors has one terinal connected to a corresponding phase voltage, which is assued as sinusoidal by (). The inductor currents are introduced as sinusoidal by (), which is an approxiation. To coplete the sinusoidal approxiation, wavefors of the line-side interphase transforer input voltages v Tk, k {,, }, should be approxiated by their sinusoidal representations. In [7], detailed analysis is perfored assuing constant output voltage, the wavefors of v Tk are obtained as twelve-pulse staircase, depicted in Fig. in the case of T. It should be noted that the wavefor of Fig. is obtained neglecting losses in the line side interphase transforer the diodes in the diode bridges. Approxiate representation of v Tk wavefors by their fundaental haronics results in Tk T φ ( k ) (6) where 6 (7) T phi [deg] Fig.. The line-side interphase transforer input voltage T, for φ 4 v k + i L RL k + v Lk Fig.. The AC-side equivalent circuit per one phase ( ) which is derived in [7]. It is iportant to underline here that the wavefors of v Tk are synchronized to the corresponding wavefors of i Tk, having the sae phase. Thus, the reaining nonlinear part of the rectifier that contains the line-side interphase transforer, the diode bridges, the filtering capacitor, ay be represented by three resistors of the noralized resistance R E + v Tk R 4 ( ) E T ρ E (8) ωl J J which is the resistance eulated at the line-side interphase transforer input. In this anner, the rectifier is at the AC side represented by three identical linear equivalent circuits shown in Fig. for one phase. This equivalent circuit is an essential result of sinusoidal approxiation, it provides the ost iportant inforation regarding the rectifier behavior. In the equivalent circuit of Fig., it is assued that resistive losses can be odelled by a series resistance R. This resistance is suitable to odel conduction losses in the inductors..4. Solution for the rectifier operation paraeters A phasor diagra that relates noralized voltages currents of the circuit shown in Fig. is given in Fig. 4. According to the noralization (6) r k (9) while according to () r r Lk J k. () Since r J k J () r Tk T () noralized aplitudes of the AC quantities are related as ( T + ρ J ) + J () which provides aplitude of the line-side interphase transforer input voltage fundaental haronic as Since T T J ρj. (4) ( 6 ) () since according to [7] J J (6) ( 6 ) the output voltage the output current are related as ( 6 + ) 6 ( + )( J ) 48 ρ ( + ) J )..4J.47ρJ. (7)

4 This is the ost iportant relation that arises fro the sinusoidal approxiation. All other rectifier operation paraeters are derived applying this equation. Noralized output power is given by P J ( 6 ) ( ) ( ρ ) 6( ) + + ρ ( 6 ) ρ ) (8) reaches its axiu for dp d (9) at ( + 8 ) + ρ ρ + ρ () of P + ρ ρ ax. 4 () The rectifier power factor, which is the sae as the displaceent power factor in the case the sinusoidal approxiation is applied, is obtained fro the phasor diagra of Fig. 4. as PF DPF + ρj where T ( 6 ) () + ρ J ( ) 6 J are related according to (7). The analyses perfored up to this point did not take into account forward voltage drop across the diodes. In cases when the input voltages are low, forward voltage drop significantly affects the results. The forward voltage drop can be incorporated in the analysis knowing that the actual output voltage is lower than predicted for twice the diode forward voltage drop. In the case of the rectifier efficiency, this technique yields J η () ( + D ) J + ρj where D is noralized value of the forward voltage drop, while J J are related by (6).. EXPERIENTAL RESULTS To verify the s, a low-power rectifier odel is built. The odel is designed to operate with the phase voltage aplitude of V V. The coupling inductors are easured at the output current of I 4 A, which is the point close to the axiu of the output power. The easureent is perfored by digital post processing of the voltage current wavefors recorded across the inductors. Average inductance of the inductors is easured as L H, inductances of all three inductors are within ±.% of the average value. Analyzing the rectifier losses at the sae output current, the equivalent per phase resistance is deterined as R. 784 Ω, corresponding to ρ Iout [A] Fig.. Dependence of the output voltage on the output current: crosses experiental data; solid line Pout [W] Fig. 6. Dependence of the output power on the output voltage: crosses experiental data; solid line PF Fig. 7. Dependence of the power factor on the output voltage: crosses experiental data; solid line Dependence of the output voltage on the output current is shown in Fig., where crosses represent the experiental data points, while the solid line presents denoralized of (7). Good agreeent between the results is obtained, except at the low output voltages where both the input currents the 4

5 efficiency [%] Fig. 8. Dependence of the rectifier efficiency on the output voltage: crosses experiental data; solid line THD(i) [%] Fig. 9. Dependence of the input current THD on the output voltage: red phase ; blue phase ; green phase ; black solid line of [7] THD(v) [%] Fig.. Dependence of the input voltage THD on the output voltage: red phase ; blue phase ; green phase output current increase due to the saturation of the coupling inductors. In Fig. 6, dependence of the output power on the output voltage is given. Again, the results are in good agreeent with the s of (8). Dependence of the input power factor on the output voltage is presented in Fig. 7. Excellent agreeent with the s of () are achieved. Dependence of the rectifier efficiency on the output voltage is given Fig. 8. Experiental data are in an excellent agreeent with the s, again. The input current THD cannot be predicted applying sinusoidal approxiation, since the higher order haronics are neglected. Prediction for the input current THD is given in [7] as a result of the exact solution of the rectifier odel. However, the exact solution is available only in the case losses are negligible, i.e. ρ. Experientally obtained values of the input current THD are given in Fig. 9, accopanied by the curve predicted in [7] for the case ρ. The results are in a relatively good agreeent, regarding the fact that the losses are neglected, that the input voltage THD values are as given in Fig., since the available voltages were slightly distorted. Increase of the input current THD at low output voltage levels is due to the saturation of the coupling inductors. 4. CONCLUSIONS Sinusoidal approxiation approach in analysis of three-phase twelve-pulse output voltage type rectifiers is experientally verified. To perfor the analysis based on sinusoidal approxiation, the rectifier input currents are assued sinusoidal, resulting wavefors of the rectifier voltages, priarily the input voltages of the line side interphase transforer, are obtained. The input voltages of the line side interphase transforer are approxiated by their fundaental haronics, which causes all voltages currents at the rectifier AC side to be sinusoidal. An equivalent circuit that represents the rectifier AC side is derived. Analyzing the equivalent circuit, corresponding phasor diagra is derived, which relates quantities that yield to the equation that relates the output voltage the output current. This relation is essential to derive all other rectifier operating paraeters. The output power, the power factor, the rectifier efficiency are deterined. To verify the analytical results, a low-power rectifier odel is built. Regarding the output voltage, the output power, the power factor, the efficiency, the experiental results the s are in an excellent agreeent. Regarding the input current THD, the agreeent between the experiental results the of [7] is relatively good, with soe differences caused by the input voltage THD the saturation of coupling inductors at high output currents.. REFERENCES []. Depenbrock, C. Nierann, A new -pulse rectifier with line-side interphase transforer nearly sinusoidal line currents, Proceedings of the Power Electronics otion Control Conference, Budapest, 99, pp [] C. Nierann, New rectifier circuits with low ains pollution additional low cost inverter for energy recovery, Proceedings of the European Power Electronics Conference, Aachen, 989, pp. -6.

6 []. Depenbrock, C. Nierann, A new 8-pulse rectifier circuit with line-side interphase transforer nearly sinusoidal line currents, Proceedings of the nd International Power Electronics Conference (IPEC), Tokyo, 99, pp [4] P. ysiak, A 4-pulse diode rectifier with coupled three-phase reactor, Journal of the Chinese Institute of Engineers, Vol., No. 7, pp. 97-, 7. [] V. Caliskan, D. J. Perreault, T.. Jahns, J. G. Kassakian, Analysis of three-phase rectifiers with constant-voltage loads, IEEE Trans. Circuits Syst. I, Funda. Theory Appl.ol., no. 9, pp. -6, Sep.. [6] P. Pejović, J. W. Kolar, Exact analysis of threephase rectifiers with constant voltage loads, IEEE Transactions on Circuits Systes-II: Express Briefsol., no. 8, pp , Aug. 8. [7] P. Pejović, J. W. Kolar, Analysis of a three-phase twelve-pulse voltage output type rectifier, subitted for review. 6

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