* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Immunity-aware programming wikipedia, lookup

Power engineering wikipedia, lookup

Control system wikipedia, lookup

Mercury-arc valve wikipedia, lookup

Time-to-digital converter wikipedia, lookup

Electrical ballast wikipedia, lookup

History of electric power transmission wikipedia, lookup

Three-phase electric power wikipedia, lookup

Pulse-width modulation wikipedia, lookup

Current source wikipedia, lookup

Stray voltage wikipedia, lookup

Power inverter wikipedia, lookup

Power MOSFET wikipedia, lookup

Electrical substation wikipedia, lookup

Resistive opto-isolator wikipedia, lookup

Schmitt trigger wikipedia, lookup

Voltage optimisation wikipedia, lookup

Voltage regulator wikipedia, lookup

Distribution management system wikipedia, lookup

Alternating current wikipedia, lookup

Variable-frequency drive wikipedia, lookup

Analog-to-digital converter wikipedia, lookup

Mains electricity wikipedia, lookup

Television standards conversion wikipedia, lookup

Opto-isolator wikipedia, lookup

Integrating ADC wikipedia, lookup

Switched-mode power supply wikipedia, lookup

Transcript

4th Power Electronics, Drive Systems & Technologies Conference (PEDSTC2013), Feb l3-14, 2013, Tehran, Iran Advanced Non-Inverting Step up/down Converter with LQR Control Technique l 2 3 Hassan Oehghani , Ali Abedini , Mohammad Tavakoli Bina 123 , , K.N. Toosi University of Technology 123 , , Tehran, Iran 2 3 I [email protected], [email protected], [email protected] Abstract-This paper proposes a new topology of non inverting in this paper, is LQR to control the proposed converter. LQR buck-boost converter. This topology is composed of a boost control technique has the advantages of more robustness and converter which is followed by a buck converter through a faster magnetic coupling. A series resistor with capacitor is considered in the boost part to enhance the dynamic of the converter. The more accuracy. A control algorithm PI or PID conventional mathematical model of the converter. This modeling is carried and AC out considering losses in the windings, power mosfets and models of the converter are obtained to analysis the dynamic of the converter with to proposed converter structure. Section III presents DC and AC higher bandwith and faster response compared to conventional Non-ideal DC compared This paper is organized as follows: Section II introduces advantages of proposed topology are smaller capacitors size, non inverting buck-boost topologies. dynamics controllers[3]. diodes. In Section IV elements of the converter is designed step is by step. Section V introduce control algorithm for the buck developed based on LQR method to regulate the output voltage boost converter. In Section VI a sample buck-boost converter is of the converter. This controller is more robust compare to simulated in frequency domain and time domain. Also, The conventional controller. PSCAD/EMTDC software is used to evaluate and verify mathematical model results and simulated converter circuit model. PSCADIEMTDC is used for simulation. The last section losses and efficiency are calculated. presents the conclusion of this paper. Keywords-Cascade buck-boost converter; LQR control; RHP zeros;Losses; Efficiency. I. II. Fig. I shows the proposed converter circuit diagram. This INTRODUCTION converter is consisted of a boost converter cascaded with a Nowadays, applications of DC-DC converters have been increased significantly because they are widely used buck converter. A transformer with turns ratio of 1 is in embedded into the boost part, so that the secondary side of renewable energy systems such as solar systems, fuel cells, transformer is in series with inductor L of the buck part. Also, battery chargers and etc[1-8]. a damping resistor Rd is in series with the capacitor C in the In many applications due to the large variation of input voltage, step up-down converters must be used. boost part. Co and Ro show the filter capacitor and load resistor These in the buck part, respectively. Vg shows the input voltage. Lm converters are divided into single inductance i.e. conventional is the magnetizing inductance of the transformer, Ron displays buck-boost converter, or two windings Converter i.e. cascade the on resistance of the power mosfets Ql and Qz, and VD buck boost groups. Single winding are used to reduce the cost shows the forward voltage drop of the diodes 01 and O2. RLillustrates ohmic resistance of the transformer and output and size of the circuit. However two windings converters are more for high voltage applications where the size of capacitors is more important. Other advantages of two inductor. windings Boost Part 1'-'-'-'--;'-'-" converters are the continuous input and output currents, less i EMI noises, and better control of input and output currents compared to single winding converters[ I]. But the disadvantages converters are RHP of zeros response[8,9]. A solution to conventional which restrict step up-down the controller L{ bandwidth a new converter that and more topology composed of efficiency[1,9]. of non a boost inverting This i Co paper buck-boost converter which � .-.-.-.-.-.-. �� .-.-.-.-.-.-.-.-.-. is Figure 1. followed by a buck converter through a magnetic coupling in order to removing RHP zeros. Also, a series resistor with a capacitor is considered in the boost part to enhance the dynamic of the converter. The control technique which is used 978-1-4673-4484-5/13/$31.00 ©2013 IEEE ,i L resistor with capacitor. Moreover this solution helps to gain proposes I Buck Part r'-'-'-'-'-'-'-'-'-" + remove RHP zeros, is using a magnetic coupling between two windings along with a series higher STRUCTURE 254 Proposed buck-boost converter topology Ro Vo III. AC AND DC MODELING (3) Assume Ts as switching period, QJ and Q2 conduct during Where in the above terms R 1 and R 2 are as follows: dJ(t)Ts and d2(t)Ts intervals, respectively, while DJ and D2 are off in that times. Fig. modes of the 2 and Fig. 3 converter, represent the boost and buck respectively. Assume that the )Ron +(�) Rd R1 = (� Dz-D'l Dz-D'l (D2D'2) Rd +Rl (4) R2 = (D, 1 +D, 2 +D1D2 �) Ron + D1 + converter operates in continuous conduction mode(CCM), and switching frequency is much higher than the converter natural frequencies[8], the nonlinear state differential equations are as � follows: Considering x as the ac state vector of the converter, then: dILm(t) dt vg(t)-d1 (t) Ron h(t)+tLm(t) -d� (t)[VC(t)+Rdhm(t)+vDl-d� (t)RdIL(t) Lm vg(t)-Vo(t)-Rlh(t)-d (t) Ron lL(t)+hm(t) +RdlL(t)-vc(t)1 1 I d L(t) _ [ ( [ ( = � (5) )l the small signal model of the converter is equation (6). ) L x ) [ ( +----------------���---=��----------� dCit = + B1dl + B2d2 + B3Vg + B4rO -d� (t)(Ron[L(t)+VD)-d� (t) Rd h(t)+lLm(t) +vC(t)+VD-RonlL(t)1 L A Ax A (6) Where ,B1, B2, B3 and B4 are as follows: c dvo(t) dt h(t) - - A= Vo(t) --- (1) -(D1Ron+D'lRd) Lm -(D1Ron+D' ZRd) -(D1Ron+D' ZRd) Lm -(DzRon+(D1+D'z)Rd+Rl) D'l Lm Dz-D'l D'l C D'z-D1 C 0 L + Ro 1 0 Vo B1 = Figure 2. L The boost mode ofthe converter 1 L 0 Co o L o -1 Ro Co [(Vc+RdlLm+VD-Ronlg) (VC+VD-RonlLm-Rdh) -lg O ] T L C Lm T [Rd1L (VC+VD-Ron1L+Rd1g) -IL O ] B2 = Lm B3 + C L = [2..� Lm L 0 O ] T Vg Ro Vo (7) Figure 3. In the above terms,ILm + IL = Ig, is the converter input The buck mode ofthe converter current. Fig. 4 and Fig. 5 illustrate DC and ac small signal transformer model of the proposed converter, respectively. In Where Zj is mean of the Zj in the switching period T" the ac model, a and f3 are as follows: a = Vc + VD -RonIg +D'lRdIg -D2Rdh where Zj is any variable i.e. vg,iL,d1,... : (2) Where the Zj is the DC part of Zj' and Zj is the ac part of it, around Zj. By substituting (2) in (1) f3 = Vc and separating the ac part + VD - RonfL + D'lRdfg - D2RdfL (8) The voltage conversion ratio is gained as follow: from the DC part, the operating point of the converter is gained as following: (9) Fig. IL m = 6 shows M as function of D1+D2 , as shown M is a continuous curve between the boost and buck modes of (DD,z_l ) h operation. 1 255 In the last equations, £::"iLm,Boost,p-p and £::"iLm,Buck,p-p are peak to peak magnetizing inductor current ripples in the boost Ro and buck modes respectively. Lm is determined with regard to the maximum allowed of the input current ripple. Figure 4. Output Capacitor(Co) Selection DC transformer model of the converter Co Capacitor is calculated as follow: Co CO Where C Figure 5. l and CO = l:J.iL 0, Max{Col' CO2} = (14) are: 2 BoostP-P Ts 8l:J.vo BoostP-P £::"VO,Boost, £::"VO,BUCk, In above equations, P-P and P-P are peak to peak output capacitor voltage ripples in the boost and ac transformer model of the converter buck modes respectively. COl and CO2 are determined with regard to the maximum allowed output voltage ripples. Input Capacitor(C) and Damping Resistor(RcJ Selecting values of L, Lm and C o is accomplished with regard to desired peak to peak their current and voltage ripples. Rd But C and determine the location of the converter zeros and converter dynamic. Therefore they must be selected so that the zeros are in the left half plane(LHP) with good dynamic converter response. The output transfer function of the converter based on Fig. Figure IV. 6. Voltage conversion ratio M, as function of 3 0,+02 is this: (16) DESIGNING ELEMENTS OF THE CONVERTER Output Inductor(L) Selection Where: inductor L is calculated as follow: (10) Where L1 and L2 are: 2 bo D'1 Ro + D1D'lRd 2 2 b1 D1 Lm + D'1 L + D1D'lRdRoCo + D'lRdD'lRd + D1D'lRd2C 2 b2 D12LmRoCo + D\ LRoCo + D1D'lRd2C + D1LmRdC + D'lRdLC + LmCRo b3 LmLC + D1LmRdCRoCo + D'lRdCLRoCo b4 LmLCRoCo (20) = = = In the above equations, £::"iL,Boost,P-P and £::"iL,BUCk,P-P are peak to peak output inductor current ripples in the boost and = = buck modes respectively.In the each mode, with regard to To eliminate RHP zeros the following conditions must be maximum value of these current ripples, minimum value for L1 met: and L2must be selected separately, in order to obtaining CCM conditions for current of the output inductor L. ao Magnetizinglnductor(LmJ Selection Having Magnetizing inductor Lmis gained of: Lm Where Lm 1 and Lm 2 Lm = = = (12) Having using (Vg-RonIg)D1Ts (l1igBoostP P -l1iLBoostP -p) (Vg-VD-RdI Lm -Vc DzTs ) (l1i9BuCkP P -l1iLB CkP P - ) U > 0 and a2 al > 0, a2 > 0 > 0, with respect to (21) (3) must be following condition satisfied: are: l L z m Max{Lml,LmJ ao > 0, (3), a Rd=< Ro 1 > 0, since 10 IL, by replacingit in (22) (18), and then: (23) (13) 256 Considering where a1 = (23), as shown in Fig. 0, then: I a Critical = 7, IO Critical R dCVg Rd 2 C +E.!..L D'1 m v. is defmed as LQR control CONTROLALGORITHM techniqueis used to design the control algorithm of the proposed converter. The theory of optimal (24) control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is To have all zeros in LHP, the following conditions must be described by a quadratic functional is called the LQ problem. met: One of the main results in the theory is that the solution is provided by the linear-quadratic regulator (LQR)[lO]. If the state equations of the system are these: x =Ax + Bu Ro y=Cx +Du 1 2 (27) 2 According By using LQR: (27) must: u = -Kx Then: J = (29) By taking derivative of opt (24) (32) Where matrix K must be selected so that, J is minimized: (28) equaling to zero, Rd (31 ) f (xTQx + uT Ru)dt (33) Where J is the cost function which must be minimized and Q(state weighting matrix)? 0 and R(inputweighting matrix) with respect to Rd and > O. is achieved as follow: K is gained as follows: (30) (34) Fig. 8 illustrates IO as function of the Rd for different Critical values of the duty cycle D1. And P is obtained via the following algebraic ricatti equation: Alp + PA + -PSR -ISlp + Q = 0 12XIO" (35) Fig. 9 illustrates the control block diagram of the converter. As shown, because the system type is zero, an integrator must be used to have zero steady-state error. OJ. ·�O�-.7--.'!;--;!;----7:--+'5 -t" .-----.--.:,-� --, ,,, 1 -----16c'-------;' lo(Ampere) Figure 7. Coefficient a1 as function ofIo Figure 9. control block diagram for converter [ Q and R are selected as: Q= 0 �m Q 0 Q1L 0 0 0 0 Qfjc 0 0 - - dz (36) And K is this: DO��O.� 5 -�-�1.5'-�--+".5'--+-�--+-�'�5 -� Rd(Ohm) Figure 8. d, 0 �l Q. R [R� R� ] 'OCritical as function of the Rd for different values of the duty cycle (37) Dl and showing Rdopt in the each them Thus the closed-loop system representation is given by: 257 Ac = (38) A - B1Kl - B2K2 VI. [n this paper a study has been 1 0 VD -� -- r---.- 1! (a) SIMULAT10NRESUL TS case 50.50 _ Where in that, Ac is new state matrix of the converter. 49.50 accomplished. 00 50 40 30 20 10 0 Simulation of the buck-boost converter has been considered for the design parameters which are mentioned in table I. By using equations given previously in the paper the elements values are obtained in table [I. There are 3 columns in table II: boost I VQ2 -- 1 12. a -,---" iQ ::.:-- A----- A-----'-- -='---:; mode, buck mode and fmal selection. The boost and buck " modes columns show the obtained value of the elements in the � (e) � boost and buck modes respectively, and the fmal selection is made based on the worst case conditions. Table III shows input O. parameters of the converter shown in Fig. 1. Assuming as follows: Q1Lm = 0, = 0, a = QIL R 1 Qvc = 0, Raz = 1 Qvo = 0.03 time (5) G.G22515 (39) K is gained as follows: Kll = 0, K12 K21 = 0, K22 = = K13 = 0, K14 = 0.1062 0.0340, K23 = 0, K24 = 0.1242 Dl=O,Dz= 1&wc=31400 K[ = 0.022535 0. .0022540 0.022545 ().G22555 0..0.22550 voltage OfQ2 with gray color; (c):current OfQl with black color and current (40) = 50.50 31400) . = 1 = -� -- r-- � (a) ° db 49.50 ,,,, Jb Db l� i (CI :V1r1 � (41) 8102 r: OfQ2 with gray color; (d): ig with black color and iL with gray color. order that cut off frequency of the open loop transfer function be a twentieth of the switching frequency(wc 0 ().G2253G mode(Vg=35V)(a):output voltage; (b):voltage OfQl with black color and K( as an integrator coefficient according Fig. 9, is achieved in CJW) 0..0022525 Figure 10. Voltgaes and currents ofthe converter in the boost 0.0291, I�Gv aUw)1 0.022520. Fig. 10 and Fig. 11 illustrate Voltgaes and currents of the converter in the boost(V g=35V) and buck(V g=65V) modes, respectively, A, Converter analysis results in MATLAB software In this section, the time and frequency domain performance of the proposed converter is investigated. Fig. 12 illustrates 70 step response of the proposed converter in the boost and buck modes. As shown, the system responds quickly with little CI) �'" overshoots, and zero steady-state errors and the short settling times in the both modes, but the overshoot and settling time in (d) 6.0 5.0 40 3.0 2.0 1.0 ;--"'-----�.----- --�<___"=-----�.----- --� time (5) O.GnS15 the boost mode is more than the buck mode, Fig, 13 illustrates O.G2252G (lOnS25 G.(}2253D o. a 22S35 0- . 022:540 O.(}22545 O.(}2255G ()'021555 bode diagram of the proposed converter in the boost and buck modes. As shown, Desirable phase margin(about 60°), and cut Figure II. Voltgaes and currents ofthe converter in the buck off frequency(about 5kHz), is obtained with used control mode(Vg=65V)(a):output voltage; (b):voltage OfQl with black color and algorithm. H voltage OfQ2 with gray color; (c):current OfQl with black color and current OfQ2 with gray color; (d): ig with black color and iL with gray color. Losses and efficiency calculation of converter The conduction losses of the converter regardless of the equivalent series resistance (ESR) of the capacitor is obtained PLOSSSWQl as following equation: 2 2 2 R1h,rms + Ron(!Ql,rms + IQz,rms ) + 2 RdIc,rms + VD(!Dvrms + IDz,rms) PLoss,cond converter switching losses which is due to PonswQl +POffswQl Q I,minIQ bmintr + V:Q I,maxIQ I,maxtf)Boost &(V: 2 = (42) The = = Mode (43) PLoss,SwQz - Pon,SwQz +Poff,SwQz - � the switching of the power MOSFETS, are obtained as follows: (VQz,minIQz,mintr + VQz,maxIQz,maxtf) Buck ModJ44) Total losses of the converter is as follows: PLosses 258 = PLoss,Cond +PLoss,Sw (45) And finally, efficiency of the buck-boost converter is: __ _ .PS ,-ns _._�_ o� P R_ --, 1.4,---_�-___,_--,_____ St�.'---,-----, ----, 1.2 Pin - PLosses 7]Converter == Where Pin VgIg,rms' Fig. = (46) Pin 14 represents the conduction and switching losses, and Fig. 15 represents the efficiency of the converter as functions of the input voltage, in the boost and buck modes. 2.5 1.5 Time (sec) TABLE1. 3.5 THE CONVERTER DESIGN PARAMETERS Vg = 3SV-6SV Input voltage range Va = SOV Reference value of the output voltage Figure 12. The Step responses of the proposed converter in the boost and buck modes llvop_p = � !1. 0.16Va peak to peak output voltage ripple or------------- __� � ·50 � :el-100 po•max = Mgp_p = 9A peak to peak input current ripple lliLp_p = 4A peak to peak inductor L current ripple SOOW, Po.rated = 2S0W Nominal and Maximum output power fs = :.: �"""'....-......I .270 L,-����������-��...::::'t � � � � � 100kHZ Switching frequency Frequency (Hz) THE CONVERTER CIRCUIT ELEMENTS VALUES TABLEII. Figure 13. Bode diagram of the converterin the boost and buck modes Element 20 ,----.---,----,---� Boost mode Magnetizing inductor Output inductor Output capacitor Input capacitor '8 16 '4 4 Lmi Input Voffage(Volt) Lm, = L2 = 27.8J.lH ( = ( = 62.5J.lF 0, (, = 17.IJ.lF 62.5J.lF ( = 0 (= 66J.lF 16J.lF VALUES AND TYPE OF THE CONVERTER CIRCUIT ELEMENTS Symhol(type) Quantity Rd 0.8 il Damping resistor Transformer and output inductor ohmic resistance input voltage Lm= 25J.lH L= 30J.lH (2 = 8.3J.lF Element Figure 14. The converter conduction and switching losses as function of the Final selection 18.47J.lH L, = 26.32J.lH 0, TABLElll. ��5--��--�--�-�· 23. 86J.lH = Buck mode 70 mil RL Q,&Q2 Power mosfet IRFB4115PbF D,&D2 Schottky diode 40CPQ080G tr = tf = 73ns 39ns Ron = 10mil VD=O.6V C. Converter simulating results in PSCAD/EMTDC software Simulating is accomplished with considering 1 microsecond delay time to make gate signals Ul and U2. Fig. 16 shows the case which the input voltage changes from 35V to 65V, with frequency of 50Hz, at nominal load. This test is done to ��5--�� �---�=---�5=O---=55�--� 60�--�65 evaluate the dynamic stability of the proposed converter, and Input Voltage(Volt) respond of the designed control to such small signal disturbances. As shown, in the both boost and buck modes, Figure 15. The converter efficiency as function ofthe input voltage for the nominal and maximum loads gates signal 259 Ul and U2 are made correctly, so that the output voltage is regulated at SOY in the allowed ripple range. Fig. 17 and Fig. 18show the casesin which the converter is on the minimum input voltage(3SY) in the boost mode and the maximum input voltage(6SY) in the buck mode, respectively. Suddenly, load resistor changes from 1012 to 512 and after a short time returns to the original state. These tests done to evaluate transient stability of the proposed converter to these large signal perturbations. As shown in the both modes, the designed controller regulates the output voltage in the reference value,among allowed ripple range with appropriate overshoot and settling time similar to step response of Fig. 12. I. CONCLUSION A new non-inverting step up/down converter topology is discussed here. LQR technique is applied to regulate the output voltage of the converter. The DC model and dynamic model of the converter are gained to compare the performance of the Figure 16. Changing the input voltage at nominal load, and keeping output converter with conventional converter. The Dynamic model showed that there is no any RHP zero in the transfer function of the proposed converter. voltage in the allowed range for converter dynamic stability evaluation; (a):input voltage; (b):output voltage; (c):gate signal Simulation results also match closely with the responses of mathematical model gained in MATLAB. "'u 5 � (a) efficiency has not been scarified in this converter compared to � (Q (b:' 0 � -Yo REFERENCES Carlos RestrepoO, Javier Calvente, Angel Cid-Pastor, Abdelali El AO.._ ...,.- � UI ________ 1 .5. ,-==---- ------------------' 1.00 -0.50 .5. � ..... •. 5. •••• VOL. 26, NO. 9, SEPTEMBER 2011 (d) Erik Schaltz, and Peter Omand Rasmussen, Alireza Khaligh, "Non Inverting Buck-Boost Converter for Fuel Cell Applications," IEEE 0.00 -0.50 1. Calvente, L. Martinez-Salamero, P. Garces, R. Leyva, andA. Capel, lmer.) u2 �"5.,--"'----===== = ======� == ... .. .. . .. 1 ,OO � Trans. Power Electron., vol. 24, no. 4, pp. 1002-1015, Apr. 2009. [3] A 48_SOrv- (e) Aroudi, and Roberto Giral, "A Non-Inverting Buck-Boost DC-DC Switching Converter with High Efficiency and Wide Bandwidth,"IEEE [2] Ul 4_ 5 g TRANSACTIONS ON POWER ELECTRONICS, (d):gate signal Tra nsient Siabilly assessment in the Boost Mode Finally the losses calculation shows that the conventional converter. [1] Ul; . - -. 046 •.� 4 40� 2 -0. 04. � 4 .� •. � 38 .� 36-.� 0 -� 0 4 003 • -•. �.3<I � 28 -O0 0 O 0 OO32-- "DSP-Based Implementation of an LQR With Integral Action for a Three-Phase Three-Wire ShuntActive Power Filter,"in Proc. 32nd IEEE [4] Annu. Power Electron. Specialists Conf., PESC, vol. 4, 2001, pp. 1994- Figure 17. Converter transient stability evaluationin the boost mode(Vg=35V) 1999. (a): load current ; (b): output voltage; (c): gate signal T. Crocker and N. Cooper, "Fast and furious [electric dreams]," Power (d):gate signal Ul Tran sient Stability assessment in the Buck Mode Engineer, vol. 18, no. 4, pp. 39 -41, Aug. 2004. [5] Ul; P.-C. Huang, W.-Q.Wu, H.-H.Ho, and K.-H. Chen, "Hybrid buck-boost feedforward and reduced average inductor current techniques in fast line transient and high-efficiency buck-boost converter," IEEE Trans. PowerElectron., vol. 25, no. 3, pp. 719-730, Mar. 2010. [6] : O. Mourra, A. Fernandez, and F. Tonicello, "Buck boost regulator(B2R) g (b for spacecraft solar array power conversion," in Proc. 25th IEEEAppl. Power Electron. Conf. Expo., APEC, Feb. 2010, pp. 1313-1319. [7] Bachir Kedjar, and Kamal AI, Fellow, IEEE, 4S. S0 "DSP-Based Three-Wire Shunt Active Power Filter ," IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 8, AUGUST 2009 [9] J. Calvente, L. Martinez-Salamero, P. Garces, and A. Romero, "Zero 1. 00 5. •. . .. . -Vo ..... "'-- C r---v �- ,, -- ---------_ ul -1--------------------- -ll.SO Chapman & Hall, 1997. Robert W.Erickson; Fundamentals of Power Electronics, 2nd Edition, 1 . 5. (e) Implementation of an LQR With Integral Action for a Three-Phase [8] ' S• 1 . 5. (d) dynamics-based design of damping networks for switching converters," J!� j::JIIIII.11 time(s� IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 4, pp. 1292-1303, Oct. I u2 0.0032 O.D034 O.OD36 O.OG38 0.0040 11.0042 0.0044 D.DQ46 0.0048 2003 [10] en.wikipedia.orglwiki/Linear-quadratic_regulator Figure 18. Converter transient stability evaluation in the buck mode(Vg=65V) (a):load current ; (b):output voltage; (c):gate signal 260 Ul; (d):gate signal Ul