Tools for Inestigation of Dynamics of DD onerters within Matlab/Simulink Riga Technical Uniersity, Riga, Latia Email: pikulin03@inbox.l Dmitry Pikulin Abstract: In this paper the study of complex phenomenon in buck conerter under oltage mode control, operating in discontinious current mode, within Matlab/Simulink simulation enironments is proided. To perform simulations different types of models are used: based on discretetime maps, differential equations and real elements (including different nonidealities). The main goal of this paper is to detect the ability of arious Matlab/Simulink models to identify and to explore different types of complex behaiour, such as chaos and bifurcataions, in switchmode DD conerters, as system parameters are changed, as well as to estalish the possibilities of each model in this kind of inestigation. Simulations are carried out by means of Matlab/Simulink simulation enironment that proides wide range of blocks and elements for complete inestigation procedure, including the implementation of all types of mentioned models and appropriate result data postprocessing and isualization. The erification of accuracy of deeloped models is based on the detection of Feigenbaum numbers. All models with definite leel of precision are able to reeal that under certain circuit parameters period doubling route to chaos is obsered. Keywords: Bifurcation diagram, buck conerter, chaos, simulation tools, subharmonics.. Introduction Modern electronic deices are equipped with switch mode DD conerters, which hae seeral adantages oer linear ones: smaller weight, size and much higher efficiency. It is well known, that these conerters hae always been designed to operate in the specific type of periodic operation, known as period mode, in which all waeforms repeat at the same rate as driing clock. Howeer, under certain conditions, the circuit may operate in the periodn regime (in which the periods of all waeforms are exactly n times larger then that of driing clock), or in randomlike fashion chaotic regime (in which all waeforms become aperiodic). As some circuit parameters are slightly aried, the operation of conerter can toggle between different operating regimes, sometimes in abrupt manner. Such a phenomenon, where one mode fails to operate and other picks up, is termed as bifurcation. The existence of this type of nonlinear phenomenon shows that een if the conerter is appropriately designed to work in period regime, it could fail to operate as expected, if some parameters are aried, causing it to assume another, undesirable mode of operation.
Pikulin, D. During the last three decades a significant amount of work in analysis of different chaotic effects and control of nonlinear phenomena (such as chaos and bifurcations) in switch mode D D conerters has been reported by power electronics specialists in course of studying and designing power electronic circuits [,,3]. These researchers established that the D D conerters under feedback control may exhibit arious types of nonlinear phenomena, such as period doubling, Hopf bifurcation, quasiperiodic and chaotic operation. The already mentioned findings highlight that the inestigation of these phenomena is important and ital, if reliable power supplies are to be designed. Subharmonic and chaotic operation modes could be aoided and the reliable DD conerters could be designed only if the specialists hae some background knowledge in nonlinear dynamics. Unfortunately the linear methods used by power electronics specialists alone hae limited possibilities in analyzing nonlinear phenomena in switch mode DD conerters. The modern simulation software, proiding designers and researchers with powerful tools for analysis of power electronic circuits, could help to oercome the mentioned drawback. In this paper the simulation aspects of power electronic conerters are addressed and different simulation approaches are ealuated and compared. The paper is organized as follows. The second section of the paper proides the information about the system under test the oltagemode controlled D D buck conerter operating in discontinuous inductor current mode (DM). This section also presents the brief description of the Matlab/Simulink models used to inestigate the period doubling route to chaos. In Section 3 the analysis of simulation results is proided. Finally concluding comments are presented in the Section 4.. Models of Voltage Mode ontrolled Buck onerter.. Basic operation principles of buck conerter The circuit diagram and all parameters of the oltagemode controlled D D buck conerter under test are shown in the Figure. It is assumed that the conerter operates in the DM, so that the inductor current falls to zero for the part of switching period. The feedback loop keeps track of the output oltage ariation and adjusts accordingly the alue of the duty cycle. In this control scheme the difference between output oltage and reference oltage is processed by the compensation network, which generates a control signal con : t) g( V ), () con( ref c where g(.) the function determined by the compensation network and including also the parameter k small signal feedback gain.
Tools for Inestigation of Dynamics of DD onerters within Matlab/Simulink S L 08uH E 33V D i L R uf.5ω + c 5V PWM Vcon Vramp f=3khz g(.) k=0...0.6 + Figure. Voltage mode controlled buck conerter The control signal () efficiently determines the alue of duty cycle in order to minimize the output oltage ripples and achiee fast dynamic response. The obtained signal con is compared with periodic ramp signal, to generate pulsewidth modulated signal that dries the switch S... Switched statespace (SSS) model The first simulation model under test is the switched statespace model and it is based on the precise system descriptie differential equations. Depending on the state of the switch S and the alue of inductor current, there are three possible topologies that are described by the following statespace equations: d il dt R Switch is ON () dil E dt L d dt di L dt d dt dil dt L 0 It could be seen, that the DD conerter is the example of hybrid system, which includes the discrete eent (DE) and the continuous eent (E) systems. The DE system processes eents that receie and emit control signals, but the E system eoles continuously in time, in agreement with some physical laws and based on signals receied from DE system. To simulate the dynamics of the conerter, the system is separated into two subsystems: the continuous time system implements the equations (), (3), (4) and the discrete eent system proides switching between different topologies. i L R R Switch is OFF and i L >0; Switch is OFF and i L =0. Vref (3) (4)
a k i + Pikulin, D. BUK ONVERTER FEEDBAK ONTROL BLOK /R Vref [ON] Vref 0. / s V V K IL V_er IL V_saw ONTROL ON /R RAMP OMPARATOR _ONTROLLER 0. ONTROL [ON] /L s IL M entry: ONTROL=; [V_er>=V_saw] [IL<=0&&V_er<V_saw] [V_er<V_saw&&IL<=0] [V_er<V_saw&&IL>0] M entry: ONTROL=; E 0 [ON] DM entry: ONTROL=3; [V_er>=V_saw] ON Figure. Simulink block diagram of the SSS model and the control block The oltage feedback circuit block includes the sawtooth generator, reference oltage source and the logic block which produces the output of the control block. As buck conerter is operating in DM and there are three possible topologies, the signal proided by control block is in the range [,,3]. The most suitable way for the implementation of the logic block is the use of Simulink Stateflow chart, representing the DE system, which makes the transition from one state to another prescribed state, proided that the condition defining this change is true. The control signal is than fed to the MULTIPORT SWITH block that operates as multiplexer and accordingly to the control signal, passes to the output the appropriate input signal, thus defining the topology of the conerter. Figure shows the switched statespace model of the buck conerter and the appropriate control block..3. Power System Blockset (PSB) model The next model under test is the DM buck conerter implemented by means of Power System Blockset which allows creating the simulation model SWITH ontinuous powergui g m E IGBT + i IL tik V fcn y V_sample m Diode LOAD + V_LOAD V E node 0 node 0 node 0 node 0 node 0 DIODE PWM Figure 3. PSB schematic diagram of the oltage mode controlled buck conerter
Tools for Inestigation of Dynamics of DD onerters within Matlab/Simulink of DD conerter, based on the precise circuit, composed from the real element models. As it is possible to integrate the PSB with standard Simulink blocks, there is the possibility to implement complex control circuitry. Figure 3 shows that in the PSB model the physical structure of the circuit is retained in the contrast to the block diagram of the SSS model. It should be noted that the design based on PSB would sae significant amount of time in model building as it doesn t require any additional mathematical or logical preprocessing. On the other hand it works a number of times slower, than the SSS model, as it takes a lot of time for Simulink to obtain the difference equations, based on the circuit diagram created..4. The model based on iteratie mappings Although switch mode DD conerters are modeled using switched statespace models, discretetime models are also found to be useful in analytical inestigation of nonlinear phenomena [4]. The most widely used discretetime model for DD conerters is called iteratie map (F) that proides the relationship that links the state ariable samples: x n F( xn ) The iteratie maps are used as analytical tools for locating bifurcation points and constructing bifurcation diagrams..k.tse [5] has deried one dimensional map in closed form for the capacitor oltage of the buck conerter, operating in DM under oltage mode control:, n ( H ( D k( Vref ))) E( E, n), n (5) where D ( ) V ; E( E V ), n T T ; R R T ; L H ( x), n 0, _ if _ x, _ if _ x 0 x, otherwise. D steadystate duty cycle; H(.) accounts to limit the range of duty cycle between 0 and ;V ref reference oltage; k small signal gain. The iteratie map shown allows ealuating the dynamics of buck conerter, when one or more system parameters are changed. The drawback of this mapping is that parasitics of the inductor, diode, switch and capacitor are not taken into account. In order to facilitate the inestigation of buck conerter operating in DM by means of iteratie maps, the Matlab Graphical User Interface (GUI) was designed. The main window of this GUI (see. Figure 4) is diided into seeral logical regions. At the beginning the user has to select the type of the conerter (buck or boost) and define the input data. After the confirmation of parameters, the user is able to generate iteration diagram, bifurcation diagram or the graph of Lapuno exponents in the selected range of parameter k alues. The bifurcation diagram, based on the iteratie mapping, was generated by means of this GUI..
Pikulin, D. 3. Simulation Results Figure 4. The designed GUI One of the most conenient ways to study the dynamics of the system is to obtain a summary chart of the different types of its behaior when some parameters are aried. The simplest case of bifurcation diagram, when there is the ariation of only one parameter small signal feedback gain k, is obsered in this inestigation. Thus the bifurcation diagram consists of an xy plot, where the samples of output oltage are plotted against the chosen bifurcation parameter k. This bifurcation diagram clearly shows the behaioral change of the conerter within the parameter range of interest. a a a 3 a n Figure 5. Bifurcation diagram obtained from SSS model
Tools for Inestigation of Dynamics of DD onerters within Matlab/Simulink The general appearance of the bifurcations diagrams generated by all models under test are ery similar, so the Figure 5 shows an example diagram obtained only from iteratie map. The obtained bifurcation diagrams show that, as the alue of k is aried, the behaior of the system changes from stable period to chaotic mode of operation. More precisely, the period doubling is the characteristic route for the oltage mode controlled buck conerter to follow it changes from periodic motion to complex aperiodic motion. It is well known, that Feigenbaum reealed, that there is uniersal solution common to all systems undergoing period doubling and found the coefficient of geometric conergence that allows linking the parameter alues of system, at which the period doubling occurs. This coefficient has now been named Feigenbaum constant 4.66906... The aboe mentioned uniersal property gies us a possibility to ealuate simulation results, calculating the alue of Feigenbaum number for each model and comparing it to Feigenbaum constant. The model, in which the difference between the calculated Feigenbaum number and the Feigenbaum constant is the smallest, is the most precise. The comparison of different models, taking into account the accuracy of calculated Feigenbaum numbers and time required for the simulation, as well as some adices for most suitable application areas for each model are summarized in the Table. Table. The comparison of different simulation models Model Simulation speed Difference from Feigenbaum constant δ,% The most suitable application SSS model 0.05% Theory/ Experiments PSB model 3.09% Experiments Iteratie map 4.8% Theory 4. onclusions This inestigation includes the ealuation of different simulation approaches in order to create the models of switchmode DD conerters. The suitability of these models for simulation of complex behaior of buck conerter was considered. As it has been mentioned in Section 3, all models allowed distinguishing the ariety of complex behaior exhibited by buck conerter, as the small signal feedback gain was aried. Simulation results were compared with each other, obtaining the bifurcation diagrams, calculating the Feigenbaum number for each model and ealuating simulation times. The results summarized in the Table demonstrate the good agreement of Feigenbaum number, obtained from the bifurcation diagram of SSS model with the Feigenbaum constant, so it could be concluded, that this
Pikulin, D. model proides the most accurate description of nonlinear phenomena in buck conerter under study. The drawback of this model is that the inclusion of non ideal element parameters would lead to noticeable complexity of the system. As this model is based on the precise differential equations, it could be used in order to erify some analytical calculations. The fact, that the alidity of the iteratie mapping relies on the leel of approximation, and on the number of elements of Taylor series used to obtain the simplified solutions of switch state space equations, explains the big difference between δ and Feigenbaum number obtained from this model. Despite the mentioned drawback, the designed GUI proides practical tool for inestigation of complex behaior of buck conerter, operating in DM. The Feigenbaum number obtained from the PSB model differs from δ, as this model is based on equations generated automatically by Simulink enironment, and also includes some additional parasitics. In order to obtain clear bifurcation diagrams, depicting the real behaior of the system, sufficiently large number of data points should be generated and the initial transients after the change of bifurcation parameter should be discarded. These preconditions determine the simulation speed and the resolution of the obtained bifurcation diagram. As PSB model takes into account parasitics of real elements, transient times are really noticeable and this model proides the lowest simulation speed and resolution of bifurcation diagram. On the other hand, the fact that this model is capable of taking into account the parasitics of real elements makes it possible to use this model as the ealuation tool for experimental results. References [] Tse.K. omplex Behaiour of Switching Power onerters. R Press, 004. 6p. [] Banerjee S., Verghese G. Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, haos, and Nonlinear ontrol. New York: IEEE Press, 00.44p. [3] Hasler M. Electrical circuits with chaotic behaior // Proceedings of IEEE, ol. 75, no. 8, August 987, pp. 009 0. [4] Hammil D.., Deane J.H.B., Jefferies D.J. Modeling of chaotic dc/dc conerters by iteratie nonlinear mappings // IEEE Transactions on ircuits and Systems Part I. Vol.35, no.8,99, pp. 536. [5] Tse.K. haos from a buck regulator operating in discontinuous conduction mode // International Journal of ircuit Theory and Applications, ol., 994, pp. 63 78.