Controller Design for Cuk Converter Using Model Order Reduction

Transcription

1 n International Conference on Power, Control an Embee Systems Controller Design for Cuk Converter Using Moel Orer Reuction Brijesh Kumar Kushwaha an Mr. Aniruha Narain Abstract: Cuk converter contain two inuctor an two capacitor hence it is fourth orer c-c converter. It provies output voltage both higher as well as lower than the input voltage. Design of feeback compensator for fourth orer system is quite complex. In this paper, moel orer reuction technique is use for controller esign of Cuk converter. First small signal ynamic moel for Cuk converter using state space analysis (SSA) is obtaine which provies fourth orer transfer function. Then this fourth orer transfer function is reuce to secon orer using Pae approximation. Inex Terms- Cuk Converter, Moel Orer Reuction, Compensator, State-Space Averaging. I. INTRODUCTION The Ćuk converter is a type of c-c converter that has an output voltage magnitue that is either greater than or less than the input voltage. Cuk converters has excellent properties like capacitive energy transfer, full transformer utilization an goo steay-state performances such as wie conversion ratio, smooth input an output currents. The ynamic response, however, is affecte by the fourth orer characteristic, which generally calls for close-loop banwith limitations in orer to ensure large-signal stabilization. Moreover, stability may require big energy transfer capacitors in orer to ecouple input an output stages. For the purpose of optimizing the converter ynamics, while ensuring correct operation in any working conition, robust multivariable controllers coul be use. These, however, may involve consierable complexity of both theoretical analysis an control implementation. To remove this ifficulty first we reuce the orer of transfer function of Cuk converter then esign controller. The Cuk converter is mae up of two capacitors, two inuctors, a power switch an ioe thus it is fourth orer non-linear system. For the feeback control esign linear moel is neee. The linear moel of the converter is erive by the replacement of switch an ioe of converter by small signal average switch moel [7]. The esire transfer function is obtaine using state space averaging technique. This paper presents Cuk converter operating in continuous conuction moe (CCM). In continuous conuction moe inuctor current never falls to zero uring one switching perio. The state space averaging technique (SSA) [3] is use to fin small signal linear moel an its various forms of transfer functions. Depening on control-to-output transfer function, the PWM feeback controller [] is esigne to regulate the output voltage of the Cuk converter. This transfer function is foun to have two pair of complex pole in left half plane an three zeros in RHP. RHP zero is unesirable for controller esigning because it provies extra 9 phase lag. Also, higher orer system increases controller complexity, to remove these ifficulties moel orer reuction technique is use. In this paper Pae approximation [3] metho is use for moel orer reuction. The reuce orer system has one RHP zero an one pair of complex pole. II. SSA TECHNIQUE The power stage of close loop system is a non-linear system. Since non-linear system is ifficult to moel an their behavior is also ifficult to preict, therefore, it is common practice to approximate non-linear system to a linear system. For linearize power stage of c-c converter Boe plot can be use to etermine suitable compensation in feeback loop for esire steay state an transient response. For this state space averaging technique is use. In c-c converter operating in continuous conition moe, there exit two states one when switch is on an other when switch is off. During switch on; X AX + BV < t < T During switch off; X AX + BV < t < (-)T V CX uring T V CX uring(-) T To prouce an average escription of the circuit over a switching perio, the equations corresponing to the two foregoing states are time weighte an average, resulting in the following equations- X [ A + A( )] X + [ B + B( )] V V [ C + C ( )] X III. SYSTEM ANAYSIS Cuk converter is a switching regulator which yiels a variable output voltage from a constant c supply. The statespace average moel use to erive the steay-state an the ynamic moels of the Cuk converter base on its state space average moel [3] //$3. IEEE

2 Fig..a Cuk converter Fig..b Cuk converter when switch is ON i C + - Vc C the uty cycle uring operation, the circuit can also be mae to reject isturbances. B.State space equation of Cuk converter The state space equation for Cuk converter uring switch on an off are During switch is ON i ri + t () i VC ( r + + ) i VC ( ) t R+ C () vc i t C (3) vc Ri VC t ( + R) C ( R + ) C (4) During switch off i ( r + ) i VC + t (5) ( ) VC i ( r + R ) ic R + + t (6) Fig..c Cuk converter when switch is OFF Fig. Operation of Cuk converter in CCM A. Moeling of Cuk converter by state space technique The Cuk converter contains two inuctors an with equivalent series resistances r, r respectively, two capacitors C an C with equivalent series resistances r C an r C respectively, switch (MOSFET) S an a ioe D as shown in fig.a. The resistance R is representing the loa. The input voltage V is fe into the circuit via inuctor. When switch is on as shown in fig.b, current i buils the magnetic fiel of the inuctor in the input stage. The ioe D is reverse biase, an the energy issipates from the storage elements in the output stage. When the switch is turne-off as shown in fig..c, the inuctor tries to maintain the current flowing through it by reversing polarity an sourcing current as its magnetic fiel collapses. It thus provies energy to the output stage of the circuit via capacitor C. Sum of both currents i an i must be zero in the steay state, with the assumption that voltage v C is essentially constant (given that the voltage across a capacitor cannot change instantaneously an the switching spee of the circuit is high). This provie fallowing energy conservation relation- Vo Where is the uty cycle of the switch. This equation shows that by controlling the uty cycle of the switch output voltage Vo can be controlle an output voltage can be higher or lower than the input voltage V. By using a controller to vary VC i t C (7) VC Ri VC t ( r + R) C ( R + r ) C (8) C C The average matrices for the steay-state an linear small-signal state-space equations can be written accoring to above equations. r ( ) ( r + + R) R+ r C A C C R ( + R) C ( + R) C r + ( ) ( r + R) R+ r C A C C R ( + R) C ( + R) C (9) ()

3 B B B E E E ( ) ( ) C C C C. Transfer function With the state space matrices efine above, the control -to - output transfer function can be calculate as G v C ( SI A) B + E () Where B ( A A) X + ( B B) () Output to input transfer function Gvg C( SI A) B (3) X CA V (4) IV. CONTRO OF CUK CONVERTER A. PWM feeback control Fig. (a) shows a Cuk converter with PWM feeback control [8]. The output voltage V, is fe back an compare with the reference voltage, Vref. Which prouces error voltage V e which is applie to the compensator, G c (s), which prouces the control voltage, V c, to compare with the saw tooth voltage of amplitue V M at the PWM comparator. As epicte in Fig. (b), the MOSFET is turne on when V c is larger than V saw, an turne off when V c is smaller than V saw. If V is change, feeback control will respon by ajusting Vc an then uty cycle of the MOSFET until V is again equal to Vref Fig..a Cuk converter with PWM control Fig..b waveform of PWM comparator Fig. Fig. 3 shows a small-signal block iagram of the converter of Fig. (a). The power stage transfer functions are represente by Gv which is erive earlier. The transfer function of PWM comparator is given by- F M V M Where V M is the amplitue of sawtooth voltage. From Fig 3 the open loop transfer function can be efine as- T ( s) G ( s) G ( s) F c v M Fig.3 The loop gain T(s) is efine as the prouct of the small signal gains in the forwar an feeback paths of the feeback loop. It is foun that the transfer function from a isturbance to the output is multiplie by the factor /(+T(s)). So the loop gain magnitue T is a measure of how well the feeback system works. B. Example[] Parameters of the Cuk converter: Input Voltage Vin Volts Output Voltage 4 Volts Switching frequency khz oa Ohm PWM Gain / μh.mh C 3.7μF C 984μF Output ripple 5% The transfer function of the converter is obtaine from () is as follows:

4 G v s s.3 s s s s s+. This is the fourth orer transfer function. It has two pair of complex pole an three zero in the RHP. Zeros an poles of the converter are as given as: Poles i i i i Zeros are i i C. Moel Reuction: Using Pae-Approximation metho [3], the reuce orer transfer function of the converter is obtaine as follows: G v 5s s + 7s + 46 Poles an zeros of reuce system- Poles i i Zero (5 ) omain. The main objective of the controller esign is to obtain stable operation of the converter by varying the uty cycle. Following points are taken care while esigning of the compensator. () The average mathematical moel is accurate up to one tenth of switching frequency. Here the switching frequency is taken as khz therefore the banwith ( B cross over frequency of close loop system) shoul be near khz. () High gain at low frequency region provies goo output voltage regulation. An phase margin etermines the transient response to suen change in input voltage. The suitable phase margin is between 45 to 6 egree. Fig.5 shows Boe plot without compensator that has both gain margin an phase margin negative. To make the gain margin an phase margin positive suitable poles an zeros of compensator is selecte. Fig.5 shows complex pair of pole that occurs at 64 ra/sec which provie 8 phase lag. To overcome, this problem two zeros is ae at 64 ra/sec in compensator. To minimize the effect of noise at high frequency one pole is ae at high frequency. To provie goo output regulation one pole is ae at very low frequency. With these consierations the esigne compensator is s 4( + ) G 64 c s s( + ) 5 an therefore, the overall open-loop transfer is T ( s).494s + 793s +.74 s s +.s + 9.s + 46s Step Response Gv Gv Magnitue (B) 5-5 Boe Diagram Gm -.9 B (at 58 Hz), Pm -7.4 eg (at 485 Hz) Amplitue 8 6 Phase (eg) Time (sec) Fig. 4 Step response of reuce an full orer system Integral Square Error (ISE) between original system an reuce orer system is.349 D. Feeback controller esign In this paper voltage-moe linear average feeback controllers for c c converter is esigne in frequency Frequency (Hz) Fig.5 Figure 5 shows the Boe plot of uncompensate open loop system which has gain margin -.9B an phase margin eg.

5 Magnitue (B) Phase (eg) Boe Diagram Gm 6.3 B (at 9.5e+3 Hz), Pm 5.9 eg (at.58e+3 Hz) Frequency (Hz) Fig.6 Figure (6) shows Boe iagram of compensate open loop system..9.8 Step Response [5] R. Riley, "Analyzing the sepic converter," Power Systems Design Europe Magazine, pp. 4-8, November 6. [6] A. Hren an P. Slibar, "Full orer ynamic moel of sepic converter," Proc. of the IEEE International Symposium on Inustrial Electronics, pp , June 5. [7] E. Vuthchhay, P. Unnat, an C. Bunlaksananusorn, "Moeling of a sepic converter operating in continuous conuction moe," 6th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications an Information Technology 9 (ECTI-CON 9), pp , May 9 [8] V. Vorperian, "Simplifie analysis of PWM converters using moel of PWM switch, Part I an Part II: Discontinuous conuction moe," IEEE trans. on Aerosp. Electron. Syst., July 99. [9] E. Vuthchhay, C. Bunlaksananusorn, an H. Hirata "Dynamic Moeling an Control of a Zeta Converter," International Symposium on Communications an Information Technologies 8 (ISCIT 8), Oct. 8. [] A. J. Forsyth an S.V. Mollov, "Moelling an control of DC-DC converters," IEEE Power Engineering Journal, pp. 9-36, 998. [] B. C. Kuo, Automatic Control Systems, 7th e., Prentice Hall Inc, 995. [] A. Chuinar,T. Chairet, "Feeback compensation techniques to improve input isturbance response in the Cukconverter," June. 9 [3] Shamash, Y, "Stable reuce-orer moels using Paé- type approximations," Automatic Control, IEEE Transactions, vol.9, no.5, pp , Oct Amplitue Time (sec) Fig.7 Step response of original system with compensator. Conclusion This paper eals with moeling an control of Cuk converter operating in continuous conuction moe (CCM). The state space averaging technique is applie to fin out the linear moel of Cuk converter an the esire transfer function in terms of uty ratio to output voltage (G v ) is obtaine which is a fourth orer transfer function. Designing a compensator for the fourth orer system is very ifficult. Therefore, fourth orer transfer function of Cuk converter is reuce to secon orer an it is foun that step response of reuce orer moel closely follow the original system. The compensator esigne for secon orer system gives quite satisfactory response with the original system. REFERENCES [] D. W. Hart, Introuction to Power Electronics, Prentice Hall, Inc., 997. [] R.W. Erickson an D. Maksimovic, Funamentals of Power Electronics, n e., Kluwer Acaemic Publishers,. [3] R. D. Milebook an S. Cuk, "A General Unifie Approach to Moeling Switching-Converter Power Stages" International Journal of Electronics, vol. 4, pp. 5-55, June 977. [4] V. Vorpérian, "Analysis of the Sepic converter by Dr. Vatché Vorpérian," Riley Engineering Inc, 6.