The simulation analysis of the bridge rectifier continuous operation in AC circuit

Transcription

1 Computer Appliations in Eletrial Engineering Vol. 4 6 DOI 8/j The simulation analysis of the bridge retifier ontinuous operation in AC iruit Mirosław Wiślik, Paweł Strząbała Kiele University of Tehnology 34 Kiele, al. Tysiąleia Państwa Polskiego 7, e mail: wislik@tu.kiele.pl, pstrzabala@tu.kiele.pl The paper deals with a nonlinear model of the full wave bridge retifier in AC iruit with RC load for non rigid supply system. The mathematial model of the AC iruit for this load was worked out using dimensionless variables. The model of the iruit was reated and analyzed with MATLAB/Simulink. Analysis of harmoni distortions of the urrents and voltages in onsidered iruit was arried out. Indies of power quality are desribed. The urrent voltage harateristis for different values of apaitive load are presented. The balanes ative and reative powers in the iruit were performed. KEYWORDS: nonlinear load, bridge retifier, modelling, ative and reative power. Introdution Harmonis are one of the main problems of the widely understood power quality. Whereas nonlinear load are the main soure of their propagation in the power system. The energy is supplied by the first harmoni from soure to the load. In nonlinear load this energy is onvert to higher harmonis energy and returned to the supply system. Numbef nonlinear loads onstantly is growing and aording to [] estimated that over the next years, more than 6 % of all loads will onstitute the nonlinear load. Therefore, assessment of interation of nonlinear load and power system is important issue. The interation may be onsidered using omputer simulation and redible mathematial models that desribe the phenomena ourring in the proess of energy transfer from the produer to the onsumer. For the assessment of interation of nonlinear loads and power system i.e. the propagation of harmonis and reative power a mathematial model of suh iruit was reated. The analyzed iruit diagram is presented in Figure. The iruit ontains the full wave bridge retifier loaded by the apaitor C and the resistane of the load R O. The retifier is supplied by sinusoidal voltage with the angular frequeny, through the series onneted indutane L and the resistane R S 3

2 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier... whih model the series equivalent impedane of the power supply system and other elements. Fig.. The analyzed AC iruit with bridge retifier The indutane L is the sum of the indutane of the power supply system and additional series indutane by AC side of retifier. This diagram differs from the ones known from the literature [, 3] in whih the additional indutane L is onsidered, but on the DC side of the retifier. Both ontinuous and disontinuous modes operations of the bridge retifier are possible for proposed iruit model.. Mathematial model of the iruit The model of the AC iruit with the bridge retifier in Figure an be desribed with following equations: dis L E sin( t) Is ( Rs Rd ) ( Ud U ) sign( Is) () dt du C Is U () dt Ro where: I S represents the instantaneous value of the urrent flowing from the soure to the retifier iruit, L, R S are respetively the indutane and the resistane of the supply system, R d is the series resistane of the diode in ondution state, U d is the voltage drop aross of the diode in ondution state, U C the output voltage of the bridge retifier, E the supply voltage, R O resistane of the load. The mathematial model of the retifier (the bridge retifier) was reated using the signum funtion of the urrent I S for whih a proportion oeffiient is the sum of the voltage on the apaity C and voltage drops on two ondutive diodes. With the bridge retifier put out the absolute value of the input urrent I S of the retifier filtered on parallel onneted: the apaitor and the resistane of the load. The numbef parameters (R d, R S, R O, L, C, ω, t, E, U d ) of suh model is signifiant and in effet the analysis of interations is ompliated. For the 4

3 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier... purpose of the simplifiation of the analysis, saling of the time and the referene variables were used: E t; X L; Im ; Y C (3) X Using the dimensionless variables: Is Ud U is ; ud ; u (4) I E E m Rs Rd Ro rz ; ro ; XY LC; () L L equations ( and ) an be written as: dis sin( ) is rz ( ud u ) sign( is ) d (6) du ( is ro u ) d r (7) o Use of transformation aused signifiantly redution the numbef input parameters. Their initial number was 9, and after transformation the model onsists of parameters (,, r z, u d, τ). 3. The model of the iruit in Simulink The operational diagram of onsidered iruit with Figure was modelled in Simulink by the use of dimensionless equations 6 and 7. This model presents the Figure. Fig.. The operational diagram in Simulink

4 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier... In the diagram adders, gain bloks and integrators may be distinguish. The instantaneous values voltages and urrent were led out to MATLAB workspae using output ports. On the basis of these values the harmoni analysis of urves voltages and urrent was arried out and balanes ative and reative power were performed. The simulation of the model and the analysis of results was arried out with input parameters,, u d and r z. The Figure 3 shows voltages and urrent waveforms for the sinusoidal funtion sin( ) supply and input parameters =, = 6, u d =.6 and r z =... u z i s u u p - - Fig. 3. The ourses of the voltages and urrent in iruit from Figure. Where: u z the supply voltage, u the output voltage of bridge retifier, u p the voltage on bridge retifier by AC side, i s the urrent flowing in the power supply iruit In the figure the dimensionless instantaneous waveforms of supply voltage u z, urrent i s flowing in supply iruit, voltage u p on the retifien the AC side and the output voltage u are presented. The urrent i s has negative phase shiftment in relation to the supply voltage u z. The variables and express respetively dimensionless value of resistive and apaitive load of the retifier bridge in referene to reatane L. For onstant input parameters =, u d =.6, r z =. and different values of apaitive load the urrent voltage u p (i s ) harateristis were omputed. The harateristis are presented in Figure 4. The harateristis show the dependene of the voltage u p on the retifier by AC side in funtion of the urrent i s. On these harateristis flutuations of the output voltage u in funtion of dimensionless apaity are observed. 6

5 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier = = u p i s = = = i s Fig. 4. The urrent voltage u p (i s ) harateristis for =, r z =., u d =.6 and different values The lowest level of these flutuations is observed for =. When for suh ase we will onsider different values of the resistane then harateristis u p (i s ) are suh as in Figure. The inrease of the voltage on the retifier u p and the dereasing value of the urrent i s in supply iruit are observed for growing values of parameters. = = = = u p =.7 = = i s Fig.. The urrent voltage u p (i s ) harateristis for =, r z =., u d =.6 and different values 7

6 .3.3 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier... One of the fundamental output quantities of the retifier iruit is the mean value of the output voltage u s. Assuming onstant values of u d =.6 and r z =. the mean values of the dimensionless output voltage u s was obtained in funtion and. The mean value of the output voltage u s is presented respetively in Figures 6 and 7. u s Fig. 6. The surfae plot of the output voltage mean value u s for varying load and Fig. 7. The ontour plot of the output voltage mean value u s for varying load and It an be noted that voltage u s for >.3 has maximum, whih value inrease together with more and more. Using harateristis from Figure 6 and 7 it an be determined for whih value of apaity the voltage u s has the maximal values. For that purpose helpful is the graphs in Figure 8, where the 8

7 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier... dependene of the voltage u s in funtion of the load resistane for seleted values is presented. u s.3 = = = = = Fig. 8. The maximum values of the mean output voltage u s in funtion of the resistane load, for different values of apaity Flutuations of the output voltage u for different values and are shown in Fig. 9 and. It an be seen that the voltage u t is getting lower for inreasing values of and establishes on the onstant level in the wide range of load variations..8 u t Fig. 9. The surfae plot of the output voltage flutuations u t in the funtion of and 9

8 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier Fig.. The ontour plot of the output voltage flutuations u t in the funtion of and 4. Harmonis and indiators of power quality For urrent and voltages from figure 3 the harmoni analysis was performed. For periodial waveforms the harmoni analysis was arried out using Fourier series. For this purpose the iruit model in Figure reated in Simulink was ontrolled by the MATLAB sript. The instantaneous values of urrents and voltages were olleted, and then their disrete Fourier transformation was arried out. The Figure shows that the voltage on the retifier u p has the highest harmonis ontents. The values of these harmonis are harateristi for suh nonlinear load. The amplitudes of the individual harmonis derease with an inrease theirders. u p - u z - i s harmoni frequeny Fig.. The amplitudes of the real part (blak) and imaginary (white) of harmonis of voltages and urrent in Figure 3 6

9 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier... Harmonis ontents in the urrent is not large. There is observed signifiant value fundamental harmoni, small value of the third harmoni and minimal values of the remaining harmonis. For the assessment of the shape of the voltage and urrent waveforms is used the distortion fator harmoni, whih is defined aording to the IEEE 9 formula [4]: THDi 3 4 i i i 3 i i i where: i n RMS values suessive harmonis. Total harmoni distortion values for voltages and urrent i s of the iruit are shown in Table. Table. The values of THD oeffiient for voltages and urrent i s in iruit from Figure (8) THDu z THDu p THDi s THDu..9 THD oeffiient values in table signifiantly depend on the hanging parameters and. The influene hanges of load apaitane and resistane of the retifien the THD oeffiient of the urrent i s is presented in Figure and 3. THDi s Fig.. The surfae plot of the urrent i s THD oeffiient in funtion of and 6

10 . M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier Fig. 3. The ontour plot of the urrent i s THD oeffiient in funtion of and The shape of the graph demonstrates that values THD of the urrent i s depend both on parameters and. For onsidered values of parameters and the maximum value THD is % and appears when >.9. The value of the power fator for onsidered iruit in the funtion of and, for onstant values u d =.6 and r z =. is presented in Figures 4 and. For >.98 and = the power fator ahieves the maximum value whih is equal.8. In suh ase the iruit has the reative power least value..8 PF r o Fig. 4. The surfae plot of the power fator PF in funtion of and 6

11 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier Fig.. The ontour plot of the power fator PF in funtion of and. Power balane Using the Simulink model of the measuring system proposed in [] the instantaneous values of the ative and reative powen individual elements of the iruit were designated. The operational diagram of the model is presented in Figure 6. The instantaneous values of the ative and reative power are designated as produts of the multipliation of the vetof voltages drops on the iruit elements by the urrent i s and urrent derivative di s / dτ. The respetively instantaneous values vetors of the ative and reative power are averaged on digital filters MeanV and MeanV over the period. The results are shown on displays and led out to ports P and Q. Fig. 6. The diagram of the ative and reative power measuring system. Parameters of simulation: =, =, u d =.6 and r z =.; p z, q z ; p L, q L ; p s, q s ; p p, q p denote respetively ative and reative power supply soure, indutane, resistane r z and retifier 63

12 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier... The graphs of ative and reative power for onstant values =, u d =.6, r z =. and different are plaed in the Figures 7 and 8. Balanes of the power in the iruit are fulfilled. Interesting phenomenon an be observed for reative power. The retifier has a apaitive reative power. The value of this power is muh smaller than the indutive reative power.,3,, p p z p L p s p p -, -, -, Fig. 7. Ative powef the elements of the iruit for different value of q q z q L q s q p Fig. 8. Reative powen the individual elements of iruit for different value of In the Figure 8 is shown that for = apaitive reative power retifier has a value of q p =.7, whih provides maximal value of reative power ompensation of indutive reative power in iruit by the retifier. 64

13 M. Wiślik, P. Strząbała / The simulation analysis of the bridge retifier Conlusion Presented model of the iruit onstitutes the good and onvenient model for analysis of influene of the non linear load on mains and other loads onneted to the network. Charateristis urrent voltage observed on terminals AC the bridge are ambiguous, the reative powef the retifier is non zero and has the negative sign like the power soure or apaitive reative power. For suh ase some ompensation of the reative indutive power is observed in the iruit. Referenes [] Das J. C., Power System Harmonis and Passive Filter Designs, John Wiley & Sons, New Jersey,. [] Dokić B. L., Blanusa B., Power Eletronis: Converters and Regulators, Third Edition, Springer,. [3] Mohan N., Undeland T. M., Robbins W. P., Power Eletronis: Converters, Appliations, and Design, John Wiley & Sons, 3. [4] Wiślik M., Harmonizne w obwodzie prądu przemiennego z obiążeniem nieliniowym i kompensają moy biernej, Computer Appliations in Eletrial Engineering, Poznań,, s [] Wiślik M., Bilanse moy w obwodzie prądu przemiennego z odbiornikiem nieliniowym, Przegląd Elektrotehnizny, 4, nr., 8. (Reeived: , revised: 8.. 6) 6