Design and Implementation of Type-II Compensator in DC-DC Switch-Mode Step-up Power Supply

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1 Design and Imlementation of Tye-II Comensator in DC-DC Switch-Mode Ste-u Power Suly Arnab Ghosh, Student Member, IEEE, and Subrata Banerjee, Member, IEEE Deartment of Electrical Engineering, National Institute of Technology, Durgaur 739, India. Abstract DC-DC ower sulies have taken lace a rime role in various fields of engineering alications. Such alications demand steady, regulated source of voltage irresective any changes in line and load. The comensators have taken lace a key role for maintaining the accurate erformance of converters dynamics in DC-DC ower sulies. Some DC-DC ower-sulies like boost (ste-u), buck-boost etc have a righthalf-lane zero i.e. there is a roblem of non-minimum hase, so it is difficult for the PID controller to exhibit good erformance. For this reason a secial kind of classical controller i.e. Tye-II comensator is introduced for obtaining better erformance. This aer describes the designing of a Tye-II comensator for a DC- DC ste-u (boost) ower suly. Keywords switch-mode-ower-suly; ste-u converter; nonminimum hase system; Tye-II comensator designing; I. INTRODUCTION The DC-DC ower sulies have been framed an imortant role almost every domain of engineering alications. The introduction of these kinds of ower sulies has increased the technical and economical qualities with extensive area of alications from a few tens of watts to several hundreds of megawatts like as battery chargers, lato/ersonal comuters, microcontroller/dsp kit, ower systems, telecommunication system, utility grid, sacecraft, high voltage dc transmission, adjustable motor drives and many others [][]. In the early days, the otential dividers (PD s) used to take a key role for control and transfer the dc ower from one level to another level to the load. The oerations of the PD s are simler but its alications have been reduced due to several limitations like as sufficient energy loss, difficulties for steing u the load voltage, roblems in regulations etc. That s why the linear regulators were introduced to make the system energy efficient. The oerating rincile of linear regulators are almost similar to the PD s, is embedded with load regulation features and the series resistance is relaced by solid state device. In linear regulators the solid state device oerates in active zone incurring a significant amount of ower losses across it. So, heat sink is required to dissiate the generated heat. Due to the resence of heat sinks, and linefrequency transformer the size and the weight of the ower suly is bulky in nature and it is not suitable and economical for large ower alications in terms of energy efficiency. But in Switch-Mode Power Sulies (SMPS) the solid state device works like a switch i.e. either comletely on or off. When the switch on, large current flows through it with taking almost zero voltage across it. Similarly for off condition, the voltage across the switch is high with almost zero current through it. In both the cases the total ower losses across the switch is almost zero, so there are no conduction losses in switch-mode ower suly. So, energy efficiency is very high (extended u to 95%) in SMPS. Over the last decades the technical develoments are taken lace by the introduction of different tyes of controller [8-6] for achieving fast dynamic resonses as well as better reliable DC-DC switch-mode ower suly. Actually the dynamic erformance of the ower suly is maintained by the control scheme i.e. nature of the controllers. There are several classical controllers like PI, PID controllers; have been develoed over the years to ensure desired erformance of the converter under secific conditions. Some converters like boost (ste-u), buck-boost, and fly-back have a right-halflane zero (non minimum hase system) [3], so it is difficult for the normal PID controller to exhibit good erformance with load, line variations and arametric uncertainty. For this reason Tye-controllers [4] are best suited. The state-sace averaging [5-7] methods have been used for mathematical modelling of converter. The main objective of this aer is to design Tye-II controller for a ste-u i.e. boost tye ower suly and the closed loo stability analysis of the ower suly in frequency domain. Figure : Circuit diagram of switch-mode ste-u converter associated with Tye-II comensator. II. CONVERTER SMALL SIGNAL MODEL A. Transfer Function of Ste-u Converter The small signal modeling [5-7] of DC-DC switch-mode ste-u converter i.e. boost converter is written in equ () /5/$3. 5 IEEE

2 s s v () s ωz ESR ωz RHP T( s) = G do ds () s s ω oq ωo ( do ) in ( ) ( ω ) = ( rc) where, Gain G V D ; Zero due to ESR rad/sec; z ESR ( ω ) ( ) ( ) RHP Zero D R r Lrad/sec; z RHP load l { } ( ω ) ( ) Natural Freq r D R LCR rad/sec; ( Q) = ω ( r L) { C( R r )} Quality Factor c o l load load o l load c From the equ (), it can be observed that a right-half lane zero (RHP) resent in the converter transfer function. So, there will be an effect of non-minimum hase roblem in converter dynamics. That s why the initial under-shoot is coming in time domain waveform (Figure ) and bode lot shows the sufficient hase-lag at gain crossover frequency (Figure 3). Amlitude Ste Resonse Figure : Time resonse of ste-u converter Bode Diagram Figure 3: Frequency resonse of boost-converter. () TABLE I: PARAMETERS OF BOOST CONVERTER Circuit Comonents Values Inut Voltage V in 5 Volt Outut Voltage V o Volt Inductance L 5 µh Outut Caacitance C 56 µf Inductor Resistance r l mω ESR of Caacitor r c 3 mω Load Resistance R load 5 Ω Switching Period T 5 µs III. TYPE-II COMPENSATOR DESIGN The comensator design is an imortant art for assuring good erformance and regulation of a ower suly. The comensator, the oles-zeros combinations, is used to rovide the classical loo shaing by modifying the gain and hase characteristics of oen-loo frequency resonse as well as ensures the converter robustness. The Tye-II comensator is a lead comensator with a ole at origin. So, this comensator rovides to 9 hase boost with zero steady state error. Even though boost converter having non-minimum hase system, it exhibits a better closed loo erformance utilizing a cascaded Tye-II controller. With roer tuning of this comensator the converter erforms faster resonse, with minimal overshoots and zero steady-state error. A. Mathematical Aroach Tye-II comensator is a air of ole-zero combination with a ole at origin. The exression of comensator transfer function: ( s ωz ) ( s) = () ( s ωo )( s ω ) where the locations of ole and zero have been considered at ω and ω z. The magnitude of comensator transfer function is found jωωz ( ωωz) ( jω ) = = (3) jωω o jωω ( ωωo ) ( ωω) The argument is written as arg( jω) = tan ( ω ωz) tan ( ω ω) π (4) Bode Diagram Figure 4: The Bode diagram of Tye-II comensator

3 The frequency domain resonse of Tye-II comensator is lotted in Figure 4. Here the combined action of the ole-zero creates a localized hase boost (7 ). If the ole and the zero coincident, they erfectly neutralize each other i.e. a flat -db magnitude with hase contribution have found over the frequency. The hase boost starts again if both of the ole and zero have been sited together. That s way the hase lead is to be varied from to 9 by changing the ole-zero osition. The frequency where the maximum hase boost will be occurred can easily obtain by derivate equation (4) d d ( arg( jω )) = ( tan ( f fz) tan ( f f) π ) df df or, = = (5) f fz f f f z f By solving the equation (5), the maximum hase boost is obtained at the geometric means of the ole-zero frequencies: fmax = f fz (6) Ultimately it can be concluded that the maximum hase boost is occurred at geometric mean of ole- zero frequencies. Generally this geometric mean frequency is considered as crossover frequency of the comensator. B. Derivation of k in Tye-II Comensator The k is defined as the ratio of the ole frequency to the zero frequency in Tye-II comensator. This ole-zero combinations rovide an adjustable hase boost from to 9 at the cross over frequency. So equitation should need to find the relation between k and the hase boost of the comensator. Initially both terms are unknown, so there must be two equations: tan hase boost = fc fz tan fc f (7) ( ) ( ) and fc = fmax = f fz (8) So, fz = fc f (9) Now f z is substituted in (7) tan hase boost = f fc tan fc f () ( ) ( ) To solve this equation, k is introduced where k = f /f z. hase boost = tan ( k ) tan k () From the trigonometric formula tan ( k ) tan π = k () By solving k equ. () and () π tan ( k ) = hase boost hase boost π or, k = tan 4 (3) The exression has been known as k factor aroach [4]. So the ole location will be hase boost π f = kf. c = tan fc 4 (4) and the zero location will be fc fc fz = = (5) k hase boost π tan 4 If the crossover frequency and the necessary hase boost are known, the comensator ole-zero locations can be easily found from equ. (4) and (5). C. Mid-Band Gain Adjustment for the Comensator Tye-II comensator is combination of single ole-zero air with an origin ole. Comensator has been described by equ. (). Now rewrite the numerator by factoring s/ω z. s ( ωz s ) ω o ωz s ( s) = = ωz ( s ω )( ) z s o s ω ω ω ω, z s or = Go (6) s ω In this exression, ω o is -db crossover ole and the term G o is called the mid-band gain and G o is equal to ω o /ω z. IV. DESIGN EXAMPLE WITH A TYPE-II Let s assume a ower suly that has a gain deficit of - 8 db at a khz selected crossover frequency. The necessary hase boost is 68. From (4) and (5), a ole can be laced at o tan 68 o f == 45 = 5.4 khz (7) and the zero is laced at f z = = Hz (8) o tan 68 o 45 So the transfer function of the designed Tye-II ( s.3) comensator is given by. s s 334 ( ) V. STABILITY ANALYSIS A. Closed-loo erformances: Adjustment of Comensator Crossover Frequency (f c ) The closed-loo erformances of a ste-u (boost) converter cascaded with Tye-II comensator has been studied with different cross over frequencies (f c ) for a constant gain (G o ). Since crossover frequency directly influences the olezero location of comensator, it has to be judiciously selected to get otimum erformance. From Figure 5 & 6 it is clear that for f c = khz, both the time and frequency resonses are better than f c =3 Hz and f c = khz. It is understood that if the f c is increased beyond certain value than the erformance of the closed loo converter deteriorates. Again for a lower value of f c =3 Hz, the time resonse lot is not imressive and it exhibits more undershoots before reaching final steady-state oint. The resective comarative erformance details are also reorted in TABLE II.

4 Amlitude Ste Resonses for Different Crossover Frequencies.4 fc=3 Hz. fc= Hz fc= Hz B. Closed-loo erformance: Adjustment of Comensator Gain The closed-loo erformances of a ste-u ower-suly have been shown in Figure 7 & 8 with different values of comensator gain for a fixed crossover frequency of Hz. Increasing the gain means the daming ratio ξ will be decreased and consequently the rise time as well as the eak overshoot may be increased. The comarative analysis of erformance details are also reorted in TABLE III. It can be observed (from TABLE III) that the best resonse is found at gain (G o ) for cross-over frequency (f c ) of khz Figure 5: The comarative study of closed-loo boost converter for different cross over frequencies with constant comensator gain Bode Diagram for Different Crossover Frequencies fc=3 Hz fc= Hz fc= Hz Figure 6: Bode Diagram of closed-loo boost converter for different cross over frequencies. TABLE II: COMPARITIVE STUDY OF CLOSED-LOOP PERFORMANCES Crossover Frequency (f c) 3 Hz Hz Hz Maximum Overshoot (M ) % 4.5 %. % Rise time (t r).73 sec.34 sec.5 sec Phase margin (PM) Gain crossover frequency (GCF) 699 rad/sec 39 rad/sec 376 rad/sec Gain margin (GM) 9.6 db 5.8 db.9 db Phase Crossover Frequency (PCF) 5 rad/sec rad/sec 896 rad/sec Comensator Gain (G o) Comensator ( s 366.4) ( s.3) s 44.7 Transfer Function (T ss c) ( ) ss ( 334) ss ( 64648) Closed-Loo Stability Stable Stable Stable ( ) Amlitude Ste Resonses for Different Gains of Comensator.4 Go=5. Go= Go= Figure 7: The comarative study of closed-loo boost converter for different gains of comensator with constant crossover frequency Bode Diagram for Different Gains of Comensator Go=5 Go= Go= Figure 8: Bode Diagram of closed-loo boost converter for different gains of comensator.

5 TABLE-III : COMPARITIVE STUDY OF CLOSED-LOOP PERFORMANCES Comensator Gain (G o) 5 5 Crossover Frequency (f c) khz khz khz Maximum Overshoot (M ) % 4.5 %.6 % Rise time (t r). sec.34 sec.3 sec Phase margin(pm) Gain crossover frequency (GCF) 79 rad/sec 39 rad/sec 597 rad/sec Gain margin (GM).8 db 5.8 db.3 db Phase Crossover Frequency (PCF) rad/sec rad/sec rad/sec Comensator 5( s.3) ( s.3) 5( s.3) Transfer Function (T ss c) ( 334) ss ( 334) ss ( 334) Closed-Loo Stability Stable Stable Stable VI. CONCLUSION This work describes design of a Tye-II comensator for a DC-DC switch-mode ste-u ower suly. The cascade comensated closed loo control system has been simulated in MATLAB. The effect of crossover frequency (f c ) and gain (G o ) on the closed loo stability & erformance of converter has been investigated. These two arameters finally decide the comensator transfer function. The closed loo behavior of ower suly with Tye-II comensation is satisfactory. But for obtaining better results the arameters of comensator have to be otimized by soft-comuting techniques and this may be a art of future work. REFERENCES [] N. Mohan, T. M. Undeland, and W. P. Robbins, Power Electronics, 3 rd ed. New York: Wiley, 3. [] M. H Rashid, Power Electronics Devices, Circuits and Alications, Pearson Publishing, 4. [3] K. Ogata, Modern Control Engineering, 5 th ed. Pearson Education, India,. [4] H. Dean Venable, The k-factor: A New Mathematical Tool for Stability Analysis and Synthesis, Proceedings of Powercon, CA, March - 4, 983. [5] R. D. Middlebrook, and S. Cuk, A General Unified Aroach To Modelling Switching-Converter Power Stages, Proceedings of the IEEE Pwer Electr~nics Secialists Conference, June 976. [6] R. D. Middlebrook, Small-Signal Modeling of Pulse-Width modulated Switched-Mode Power Converters, Proceedings of The IEEE, Vol. 76, No. 4, Aril 988. [7] Y. S. Lee, Comuter-Aided Analysis and Design of Switch-Mode Power Sulies, Marcel Dekker, Inc. Hong Kong [8] C. M. Liaw, S. J. Chiang, C.Y. Lai, K.H. Pan, G.C. Leu, and G. S. Hsu, Modeling and Controller Design of a Current-Mode Controlled Converter, IEEE Transacttons On Industrial Electronics, Vol. 4, No., Aril 994. [9] J. H. Su, J. J. Chen, and D.S. Wu, Learning Feedback Controller Design of Switching Converters Via MATLAB/SIMULINK, IEEE Transactions On Education, Vol. 45, No. 4, November. [] B. Bryant, and M. K. Kazimierczuk, Modeling the Closed-Current Loo of PWM Boost DC DC Converters Oerating in CCM With Peak Current-Mode Control, IEEE Transactions On Circuits And Systems I: Regular Paers, Vol. 5, No., November 5. [] T. Geyer, G. Paafotiou, and M. Morari, Hybrid Model Predictive Control of the Ste-Down DC-DC Converter, IEEE Transactions On Control Systems Technology, Vol. 6, No. 6, November 8. [] K. I. Hwu, and Y. T. Yau, Performance Enhancement of Boost Converter Based on PID Controller Plus Linear-to-Nonlinear Translator, IEEE Transactions On Power Electronics, Vol. 5, No. 5, May. [3] J-K. Kuo and C-F. Wang, "An integrated simulation model for PEM fuel cell ower systems with a buck DC-DC converter," International Journal of Hydrogen Energy, vol. 36, ,. [4] A. Ghosh, Nonlinear Dynamics of Power-Factor-Corrected AC-DC Boost Regulator: Power Converter, Nonlinear Phenomena, Controlling The Nonlinearity, LAP Lambert Academic Publishing,. [5] A. Ghosh, and S. Banerjee, Design of Tye-III Controller for DC-DC Switch-Mode Boost Converter, in 6 th IEEE Power India International Conference (PIICON-4), 5 th -7 th December 4, New Delhi, India. [6] P. Karamanakos, T. Geyer, and S. Manias, Direct Voltage Control of DC-DC Boost Converters Using Enumeration-Based Model Predictive Control, IEEE Transactions On Power Electronics, Vol. 9, No., February 4.