[雙語(yǔ)翻譯]--外文翻譯--輸入濾波器的相互作用在級(jí)聯(lián)buck變換器中的分析（英文）

1、Analysis of Input Filter Interactions in Cascade Buck ConvertersM. Usman Iftikhar, A. Bilal, D. Sadarnac, P. Lefranc and C. Karimi Ecole Sup´ erieure d’Electricit´ e (Sup´ elec) Plateau de Moulon, 3-rue Jo

2、liot Curie, 91192 Gif-sur-Yvette, France E-mails: {muhammad.usman, ahmed.bilal, daniel.sadarnac, pierre.lefranc, charif.karimi}@supelec.frAbstract—Many applications of switch-mode dc-dc converters require high conversion

3、 ratios (Vout/Vin), thus encouraging the use of n-stages connected in cascade. However, when an input filter is added to an existing cascade converter “black box”, it can cause system instability and performance degradat

4、ion due to the negative dynamic input resistance characteristic of the converter. In this paper, we investigate this interaction of input filters in cascaded buck converters. The problem is treated by using a small-signa

5、l averaged model of the converter in which natural parasitic resistances of the circuit elements are also taken into account. Based on the open-loop transfer function of the system, a design criterion is proposed for the

6、 dimensioning of input filter damping that ensures closed-loop stability of the filter-converter system without over sizing or under sizing the filter. Theoretical results of this paper are validated experimentally.Index

7、 Terms—Input filter interaction, cascade dc-dc converter, control loop stability, small-signal averaged model.I. INTRODUCTIONHigh conversion ratios required in low voltage and high current applications lead to the develo

8、pment of cascade con- verters. Higher dc conversion ratios are particularly needed in modern mainframe computers, aeronautics and telecom- munication appliances, where the input bus voltage (usually 48 V) has to be lower

9、ed to very low voltage levels with the help of load converters. One possible solution to fulfil this requirement can be the use of dc-dc converters with transformers (isolated converters). However, the use of trans- form

10、er results in large switching surges that may damage the switching devices. Moreover, the use of transformer limits the switching frequency of the converter [1]. An alternative approach for realizing larger dc conversion

11、 ratios is cascading of the converters [2]. This scheme mainly uses multistage approach that consists of n-basic converters connected in cascade. Usually, an EMI filter has to be employed at the input of the dc-dc conver

12、ter in order to meet the EMI/EMC requirements. However, a presumably well-designed filter, when married to a switching converter, can interact with its feedback control loop leading to system instability in the worst cas

13、e [3]. This interaction was first explained for a standalone converter by Middlebrook in 1970’s and some solutions to rectify the problem have been proposed [4], [5]. In order to study stability issues in interconnected

14、subsystems, most of the previously published analyses have been based on the minor-loop gain[6]–[9], which is defined as the ratio of the output impedance of the supply subsystem (e.g. EMI filter) and the input impedance

15、 of the load subsystem (dc-dc converter). Moreover, extensive research and discussions have been done on filter- converter interactions with pure resistive loads; however, little attention has been paid to the case where

16、 the load is active or another similar converter. In this paper, we have used the open-loop control-to-output transfer function of the complete system for the analysis of the input filter interactions in a buck converter

17、 which is loaded with another buck converter. It is shown that the closed-loop stability of this cascaded system can be assured by damping the input filter dynamics [10]. The conditions of stability are derived in this w

18、ork for cascade buck converter, through which the lower and upper limits on the required value of damping can be determined quantitatively. A classical PWM voltage- mode control is used for the cascade converters with ea

19、ch stage operating at the same frequency and the switches are synchronized. This control scheme is also suitable for low voltage and high current applications [11]. The rest of the paper is organized as follows: In the n

20、ext section, first of all a generalized small-signal linear model is presented for n-stage cascade buck converters based on the averaged modeling technique [12], through which an open- loop transfer function is then deri

21、ved. Effect of filter poles on the converter transfer function is analyzed in section III and the conditions of stability are derived in the following section. Finally, the experimental results are presented in section V

22、.II. MODELING OF CASCADE BUCK CONVERTERSA simplified schematic of n basic buck converters connected in cascade with an LC input filter is shown in Fig. 1.rLF LF+ vCF –EiLF S1D1rL1 L1+ vC1 –iL1 SnDnrLn Ln+ vCn –iLn+ vo –C

23、F C1 CnR u1 unFig. 1. Cascaded n-buck converters with input filter.A. Nonlinear ModelFor the n-buck cascade converters shown in Fig. 1, a continuous-time low-frequency model is obtained by writing state equations for eac

24、h state from its averaged circuit model and is represented in the following form:978-1-4244-1706-3/08/$25.00 ©2008 IEEE.VCk =? UkVC(k?1) ? rkILk for k = 1, 2, · · · , nE ? rLF ILF for k = 0 (7)From (6

25、) and (7), the steady state output voltage can be expressed as:Vo = UnVC(n?1)1 + rn/R (8)If the deviations in the input voltage are not desired (i.e. ? e(t) = 0), the corresponding column in B is eliminated. Moreover, wh

26、en the same switching signal u(t) is used for each stage then ? v(t) = ? u(t) and the matrix B is further reduced to a column vector.C. Open Loop Transfer FunctionIn order to facilitate analytical evaluation of the effec

27、t of input filter on the cascade converter operation, we have confined our study to two converter stages (i.e. n = 2). By taking the Laplace transform of (4) and using the same nominal control signal U for both stages, t

28、he open-loop control-to-output transfer function can be obtained from the above small-signal model as shown below:G(s) = ? vo ? u (9)= K A4s4 + A3s3 + A2s2 + A1s + A0B6s6 + B5s5 + B4s4 + B3s3 + B2s2 + B1s + B0Where K = E

29、/m and m is defined as below:m = {R + rD2 + rL2 + U 4rLF + U 2(rD1 + rL1)}/R (10)The coefficients Ak and Bk of G(s) depend on the converter family and conduction mode. For 2-stage buck converter in continuous conduction

30、mode these coefficients are found to be as follows:A0 = 2U(R ? U 4rLF + rD2 + rL2)/RA1 = U{((2CF + U 2C1)rLF + C1(rD1 + rL1))(R + rD2+ rL2) ? U 2(L1 + 2U 2LF )}/RA2 = {U(C1(R + rD2 + rL2)(L1 + U 2LF + 2CF LF+ CF rLF (rD1

31、 + rL1)) ? U 2CF L1rLF )}/RA3 = {UCF (C1(L1rLF + LF (rD1 + rL1))(R + rD2 + rL2)? U 2L1LF )}/RA4 = UCF LF C1L1(R + rD2 + rL2)/RB0 = mB1 = {L2 + U 2L1 + (R + rD2 + rL2)(CF rLF + C1(rD1+ rL1 + U 2rLF )) + U 2(CF rLF (rD1 +

32、rL1) + C2R(rD1+ rL1 + rD2 + rL2)) + U 4(LF + RC2rLF )}/RB2 = {C2R(L2 + U 4LF + U 2(L1 + CF rLF (rD1 + rL1))+ CF rLF (rD2 + rL2)) + CF ((U 2L1 + L2)rLF + LF (R+ rD2 + rL2 + U 2(rD1 + rL1))) + C1((rD1 + rL1)(L2+ C2R(rD2 +

33、rL2) + CF rLF (R + rD2 + rL2)) + L1(R+ r2) + U 2(LF (R + rD2 + rL2) + rLF L2 + RC2rLF (rD2+ rL2)))}/RB3 = {CF ((L2 + U 2L1)(LF + RC2rLF ) + U 2C2LF R(rD1+ rL1 + rD2 + rL2)) + C1((rD1 + rL1)(C2R(L2+ CF rLF (rD2 + rL2)) +

34、CF (L2rLF + LF (R + rD2+ rL2))) + U 2(L2(LF + RC2rLF ) + C2RLF (rD2+ rL2)) + L1(L2 + RC2r2) + CF rLF (R + r2))}/RB4 = {C2CF LF R(U 2L1 + L2) + C1(CF (L2LF r1 + L1(L2rLF + LF (R + r2))) + C2R(CF (rD1 + rL1)(L2rLF +LF (rD2

35、 + rL2)) + L2(U 2LF + L1) + L1CF rLF r2))}/RB5 = {C1CF (RC2L2LF r1 + L1(L2(LF + RC2rLF )+ RC2LF r2))}/RB6 = CF C1C2LF L1L2III. EFFECT OF FILTER POLES ON CONVERTER TRANSFER FUNCTIONFig. 2 contains the bode plot showing th

36、e magnitude and phase of open-loop transfer function G(s) for 2-stage buck converter with and without input filter. The circuit parameters used for this simulation are: CF = C1 = C2 = 1μF, LF = 10mH, L1 = 1mH, L2 = 0.1mH

37、, R = 33Ω, U = 0.5, rLF = 0.5Ω and r1 = r2 = 0.75Ω. Continuous lines in Fig. 2 are showing the magnitude and phase plots of G(s) for the cascade buck converter when no filter is present at its input. It can be observed t

38、hat converter stage-1 and stage-2 dynamics cause a phase shift of ?360? and ?180? respectively, at their respective resonant frequencies, thus causing its cumulative phase to approach ?540? at higher frequencies. Thus a

39、right-hand side zero pair which appeared due to cascading of the converters can cause instability if the loop bandwidth is near to or greater than either of the cutoff frequencies f1 or f2, even when no input filter is p

40、resent. Nevertheless, when an LC filter is added at its input, it causes an additional phase shift of ?360? at the resonant frequency fF of this filter, as shown by the dotted line plot in Fig. 2. If the crossover freque

41、ncy of the regulator feedback loop is near to or greater than this resonant frequency of the input filter (which is usually the case in practice), then the loop phase margin will become negative and can cause instability

42、. It can be shown that addition of a second order input filter has introduced an additional complex pole pair and a complex right-half plane zero pair to G(s). These right-hand side zeros are the cause of instability in

43、the closed loop and thus can cause oscillations in the dc circuit. It is also evident from Fig. 2 that the internal losses of the circuit are not sufficient to damp these oscillations. Hence we need to add some damping i