THE MULTI INPUT-MULTI OUTPUT STATE SPACE AVERAGE MODEL OF KY BUCK-BOOST CONVERTER INCLUDING ALL

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1 THE MUTI INPUT-MUTI OUTPUT STATE SPACE AVERAGE MODE OF KY BUCK-BOOST CONVERTER INCUDING A OF THE SYSTEM PARAMETERS Mohammad Reza Modabbernia 1, Seyedeh Shia Nejati 2 & Fatemeh Kohani Khoshkbijari 3 1 Electrical Engineering Group, Technical and Vocational Uniersity, Rasht, Iran 2 Electrical Engineering Group, Sardar Jangal Higher Education Institute, Rasht, Iran 3 Sama Technical and Vocational Training llege, Islamic Azad Uniersity, Rasht, Iran ABSTRACT In this paper a complete multi input-multi output state-space aerage model for the KY buck-boost conerter is presented. The introduced model includes the most of the regulator s parameters and uncertainties. In modeling, the load current is assumed to be unknown, and it is assumed that the inductor, capacitor, diode and regulator actie switches are non ideal and hae a resistance in conduction condition. Some other non ideal effects look like oltage drop of conduction mode of the diode and actie switches are also considered. After presenting the complete model, the KY buck-boost conerter Benchmark circuit is simulated in PSpice and its results are compared with our model simulation results in MATAB SIMUINK. The results show the merit of our model. KEYWORDS: KY buck-boost conerter, aerage model, SMPS, SIMUINK, PSpice. I. INTRODUCTION In many applications such as portable deices, personal computers, car equipments, etc., there is a main supply that must be conerted to some other smaller or greater oltages. In these applications buckboost conerters are ery efficient. Recently, a new circuit was introduced for buck-boost conerter by Hwu based on the KY conerter structure [1]. This regulator has a good transient response and its performance is look like buck conerter without any right half plan zeros [2]. One of the other adantages of this conerter is its continuous conduction mode (CCM) performance, which decreases the output oltage ripple [1-2]. The topology of DC-DC conerters consists of two linear (resistor, inductor and capacitor) and nonlinear (diode and actie switches) parts. Because of the switching properties of the power elements, the operation of these conerters aries by time. Since these conerters are nonlinear and time ariant, to design a linear controller, we need to find a small signal model basis of linearization of the state space aerage model about an appropriate operating point of it. The small signal analysis and modeling in frequency domain for DC-DC conerters are carried out by references [3-5]. A complete model with all of the conerter parameters (such as turn-on resistance of the diode and actie switches, resistance of inductor and capacitor, and unidentified load current that it can receie from the conerter) is the main step in designing a non conseratie robust controller for the regulators [6-7]. The essential of KY conerters and their deriaties are introduced by Hwu in 29 [8], but a model that consists of the aforementioned parameters was not presented yet. The aerage model of KY buck-boost conerter is presented in [1-2] without concerning the deiation of input capacitance (C). A model for KY and second-order-deried KY conerter is presented in [8-9]. In [1], a model for KY Boost conerter is introduced. Inerse KY conerter and its model are demonstrated in [11]. The transfer function of negatie-output KY buck conerter and the steady state model of KY oltage-boosting conerter with leakage inductance and without leakage inductance are 862 Vol. 6, Issue 2, pp

2 introduced in [12-13] respectiely. In these references the model of conerters were calculated by minimum parameters and all of the switches and diodes assumed are ideal. Their on state resistance and oltage drop are neglected and there are not any parasitic resistance for capacitances and inductors. This paper is organized in seen parts. On the basis of state space aerage method [3], we first obtain the state space equations of a KY buck-boost regulator in turn on and turn off modes by considering all the system parameters such as an inductor with resistance, a capacitor with resistance, a diode and switches on mode resistance and oltage drop, a load resistance and unidentified load current in section II. Then in section III, the state equations are linearized around circuit operation point (input DC oltage and current ersus output DC oltage) in section IV. The coefficients of state space equations will therefore be dependent on the DC operating point in addition to the circuit parameters. At the end the duty cycle parameter d (control input) is extracted from the coefficients and introduced as an input. This work was introduced for the Boost and Buck-boost conerters in [14-15] respectiely. The effects of parasitic resistances, on state oltage drop of switches and the deiation of load current can be studied with this completed model. In section V, by neglecting the parasitic resistances of the regulator s elements ( rm rd r rc r ), the steady state aerage model of KY buck-boost conerter will be simplified. Anybody can use this simple model to design a linear controller for the conerter and then utilize the complete model for analyzing the robustness of his or her controller [16-17]. In section VI, the KY buck-boost conerter Benchmark circuit is simulated in PSpice and its results are compared with our complete model simulation results in MATAB. The simulations were done in three scenarios. The results are so closed to each other. Finally, in Section VII, some suggestions are presented for future works. II. KY BUCK-BOOST CONVERTER STATE EQUATIONS FOR ON-OFF TIME SWITCHING In modeling of the state space, the state ariable which principally are the elements that store the energy of circuit or system (capacitance oltage and inductor current) hae significant importance. In an electronic circuit, the first step in modeling is conerting the complicated circuit, into basic circuit in which the circuit lows can be established. In switching regulators, there are two regions; the on region and off region. The on time denoted by dt, and the off time is denoted by dt (1 d) T, in which is the period of steady state output oltage. Fig.1 shows a KY buck-boost conerter. The switch is turned on (off ) by a pulse with a period of T and its duty cycle is d. Therefore we can represent the equialent circuit of the system in two on and off modes with dt and dt seconds respectiely, by T Fig.2 and Fig.3. By considering i, and as our state ariables ( x [ i C ] ) and writing the KV for the loops of Fig.2, we will hae: C ' Figure 1. KY Buck-boost regulator circuit 863 Vol. 6, Issue 2, pp

3 Figure 2. Equal circuit of KY Buck-boost regulator in on times Figure 3. Equal circuit of KY Buck-boost regulator in off times i i O x A1 x B1 u i x C u M1 y y C1 x D1 u O M 2 A 1 G D m C r 2r r R r 1 R R r 1 C R 1 R r R r B 1 1 R r 2 R R r C o C D 1 1 R r 1 R R r R r (1) (2) (3) (4) (5) 864 Vol. 6, Issue 2, pp

4 Also for off time or d' T seconds the KV equations from Fig.3 are gien by (2). B 2 i i O x A2 x B2 u i x C u M1 y y C2 x D 2 u O M 2 A 2 m G D r r R r R R r 1 rd rm rc C R 1 R r R r C o R r rd rm rc C rd rm rc C rd rm rc C R R r C 2 R r 1 R R r (6) (7) (8) (9) D 2 R r The set of state equations (1) to (1) shows the state of KY buck-boost conerter in the on and off time of switches. We can combine these two set of equations as following [5]: (1) 1 AP A1d A2 1 d x AP x B P u B p B1d B 2 1 d y C x D u C C d C d P P p 1 2 p D D d D d (11) By substituting equations (1) to (1) we can obtain coefficients of A P to D P. A P r rm R r rm rc d d R R r d d C rd rm rc R 1 R r R r C o (12) 865 Vol. 6, Issue 2, pp

5 B P R r d 2d d d d d rd rm rc C rd rm rc C rd rm rc C R R r C D P P R r 1 R R r R r (13) (14) (15) III. INEARIZATION OFF STATE EQUATIONS AROUND OPERATING POINT The results presented in section II are acceptable when the circuit time constant is much larger than the period of switching.. If the duty cycle be a constant alue (d = D), the state equations in (11) will become linear. For regulating the oltage on a desired alue, we hae to change the alue of D by a controller. In general, the state equations of (11) are nonlinear and we hae to linear them around an operating point (D). When the system is in equilibrium and the duty cycle is on its nominal alue (D), then we can obtain the system state alues in equilibrium points ( X [ I VC V ]) and the DC outputs alues. VG I O I 1 x AP x B d D p u X A d D P Bp VM 1 VC with d D VM 2 V V D I Y Cp X D, d D p U Y with d D d D V O Where X was calculated from equation (16). Finally for linearization of the system, on basis of classic method, we diided our ariables into two parts. The first part is static part (a fixed DC leel), and the second part is a small amplitude that modulates the DC leel. On this basis, the ariables in the state equations can be defined as follows: In which Y I V x() t X xˆ u() t U uˆ d() t D dˆ y() t Y yˆ ' [ C ] and U V ' G IO VM 1 VM 2 VD O, X I V V are the nominal alues of the DC outputs, state ariables and no controllable inputs respectiely. Each of them has small ariations (denoted with ^) around nominal alues. By substituting equations (18) in (11) and assumed that the duty cycle d has also ariation (d= D + ), we will hae ˆd ˆd ' (16) (17) (18) 866 Vol. 6, Issue 2, pp

6 X xˆ A ˆ P xˆ BPuˆ A1 A2 X B1 B2 U d X Y yˆ C ˆ ˆ ˆ P x DPu C C X D D U d Y ˆ xˆ AP xˆ BPuˆ E d, E A1 A2 X B1 B2 U yˆ CP xˆ DPuˆ (19) (2) IV. STATE SPACE AVERAGE MODE An important point in the set equation (2) is that A P and C P are related to d'=1-d. Since d = D + then A P and C P are related to. It can be shown that with good approximation this dependence is negligible. By sub situation A P, B P, C P and D P by their equialents in term of d, A 1, B 1, C 1 and D 1 we will obtain: ˆd ˆd ˆ xˆ A d A d xˆ B d B d uˆ E d yˆ C1 d C2 d xˆ D1 d D2 d uˆ (21) d = D + ˆd therefore, we hae for the first aboe equation. 1 1 xˆ A D A D xˆ B D B D uˆ E dˆ A A dˆ xˆ B B dˆ uˆ (22) ˆd Since, û and product is ery small and we can neglect terms such as dxand ˆx denotes small ariation of the duty cycle, input and state of system respectiely, their ˆ ˆ du ˆ ˆ xˆ Axˆ B uˆ E dˆ (23) ˆ ˆ dx ˆ ˆ du In the same manner, the effect of and in second equation of (21) is negligible. Therefore we can represent the KY buck-boost regulator state equations like this: i i ˆ O xˆ A xˆ Buˆ E d ˆ i x C uˆ M1 yˆ yˆ C xˆ Duˆ O M 2 G D r rm R r rm rc D D R R r D D A C rd rm rc R 1 R r R r C o. (24) (25) 867 Vol. 6, Issue 2, pp

7 R r D 2 D D D D D B rd rm rc C rd rm rc C rd rm rc C R R r 1 C R R r R r D R r (26) (27) (28) E can be calculated with equation (2). V. A SPECIA CASE By neglecting the parasitic resistances of the regulator s elements ( rm rd r rc r ), the steady state aerage model of KY buck-boost conerter will be simplified. During the off state of and Mosfets ( dt (1 D) T T off ), the oltage of input capacitance will be constant ( VC VG VM 1 VD ). This capacitance is charged rapidly and saed its oltage during the time dt interal. In this situation, one of the state of the conerter ( C ) was neglected and the steady state aerage model of KY buck-boost conerter will be replaced by equations set (29). M 4 O x A x Bu E d i i x u M1 y y C x Du O M 2 D 1 2D 2D 1 2D D 1 A B C RC o C o VG VM 1 VM 2 VD D E Applying the laplace transform to model equations (29) yields 12 transfer functions which the following output oltage and inductor current to duty-cycle (d) and input oltage transfer functions hae been shown: i G ( C) (29) M Vol. 6, Issue 2, pp

8 o d 1 C o S 2V 2V 2V V 2 G M1 M 2 D 1 1 S R (3) o G S 2 2D C o 1 1 S R (31) i d 1 1 2VG 2VM 1 2VM 2 VD S RC S S R o (32) i G S 2 2D 1 S R 1 1 S R (33) VI. SIMUATION WITH PSPISE AND MATAB To show the accuracy of our model, we simulate the KY buck-boost benchmark circuit with PSpice and then compare its consequences with the simulation results of presented model in MATAB SIMUINK. Fig. 4 and Fig. 5 show the KY buck-boost benchmark circuit in PSpice and Fig. 6 and Fig. 7 show its equialent model in SIMUINK respectiely. The simulations were performed under the following conditions: = 1 mh, C =1 mf, C O = 1 F, R =1 Ω, r m=r d=r C =.1Ω, r =.2 Ω and V G = 12 V. The switching frequency is 5 khz and arious cases of simulation hae been considered. Figure 4. The KY buck-boost benchmark circuit in PSpice with Switch 869 Vol. 6, Issue 2, pp

9 Figure 5. The KY buck-boost benchmark circuit in PSpice with Mosfet Figure 6. The KY buck-boost benchmark circuit in SIMUINK Figure 7. Equialent model of KY Buck-boost regulator in SIMUINK 87 Vol. 6, Issue 2, pp

10 6.1. Analog switches with 1V forward oltage drop and disturbance in output current In this scenario, Fig. 4 with PSpice analog switches has been used. The resistance of switches and their forward oltage drop are r m=.1ω and V m1= V m2 = 1V respectiely. Also, the diode on state resistance and its forward oltage drop has been considered.1ω and.9v. The output current is I O=2A and there is a 1A sudden rise in it. The simulation results with D=.4 were shown by Fig. 8 and Fig. 9 in PSpice and MATAB respectiely. The regulator works look like a Buck conerter because its duty cycle is D=.4, therefore, its output oltage will be 8.35V and 8.537V in PSpice and MATAB respectiely. In table I, the results of these two simulations hae been compared with each other. Figure 8. PSpice Output oltage and oad Current with I O = 2 A, V D =.9 V, V M = 1 V and 1A sudden rise in I O Figure 9. MATAB Output oltage and oad Current with I O = 2A,V D =.9V,V M = 1V and 1A sudden rise in I O TABE 1. COMPARING THE RESUTS WITH I O = 2 A, V D =.9 V, V M = 1 V AND 1A SUDDEN RISE IN I O Steady State Output Voltage Steady State Output Current PSpice V A MATAB V A 871 Vol. 6, Issue 2, pp

11 6.2. Analog switches with 1V forward oltage drop and disturbance in output oltage In this scenario, Fig. 4 with PSpice analog switches has been used. The resistance of switches and their forward oltage drop are r m=.1ω and V m1= V m2 = 1V respectiely. Also, the diode on state resistance and its forward oltage drop has been considered.1ω and.9v. The output current is I O=A. The simulation results with D=.8 and a 12V sudden rise in input oltage were shown by Fig. 1 and Fig. 11 in PSpice and MATAB respectiely. The regulator works look like a Boost conerter because its duty cycle is D=.8, therefore, its output oltage will be V and 14.77V in PSpice and MATAB respectiely. In table 2, the results of two simulations hae been compared with each other. Figure 1. PSpice Output oltage and oad Current with I O = A, V D =.9 V, V M = 1 V and 12V sudden rise in input Voltage Figure 11. MATAB Output oltage and oad Current with I O = A, V D =.9 V, V M = 1 V and 12V sudden rise in input Voltage 872 Vol. 6, Issue 2, pp

12 TABE 2. COMPARING THE RESUTS WITH I O = A, V D =.9 V, V M = 1 V AND 12V SUDDEN RISE IN INPUT VOTAGE Steady State Output Steady State Output Current Voltage PSpice A 1.47 A MATAB V A V and 1A disturbances in the input oltage and load current with IRF45 and IRF913 Mosfets If we consider three IRF45 n-mosfet instead of M 1, M 2 and M 4 switches, and one IRF913 p-mosfet instead of M 3 switch, we will hae a practical simulation in PSpice. The results of simulation with I O = A, 12V sudden rise in input oltage and 1A pulse disturbance in output current were shown by Fig.12 and Fig. 13 in PSpice and MATAB respectiely. In table 3, the results of these simulations hae been compared with each other. Figure 12. PSpice Output oltage and oad Current with Real Mosfet and Diode. There are a 12V and 1A disturbances in input oltage and load current respectiely Figure 13. MATAB Output oltage and oad Current with I O = A, V D =.9 V, V M = 1 V. There are a 12V and 1A disturbances in input oltage and load current respectiely iely 873 Vol. 6, Issue 2, pp

13 VII. TABE 3. COMPARING THE RESUTS WITH V D = V M = 1 V. THERE ARE A 12V AND 1A DISTERBANCES IN FUTURE WORK INPUT VOATGE AND OAD CURRENT RESPECTIVIY Steady State Output Voltage Steady State Output Current PSpice V.584 A MATAB V.6595 A nerting this complete model to the P-Δ-K configuration of μ-theorem. With this configuration, any linear controller can be analyzed by μ-synthesis theorem. This work was done in [16] for the boost conerter. Design a precise controller that can satisfy robust stability and robust performance of the KY buckboost conerter in the presence of all the conerter parameters. This work was done in [17] for the boost conerter. VIII. CONCUSION There are a lot of parameters in KY buck-boost conerters. These are capacitances and their resistance, inductance and its resistance, resistance of diode and actie switches and their conductie oltage drop, resistance and current of load and uncontrollable input oltage. In this paper, an aerage model with multi-input multi-output is presented for KY buck-boost conerter with all of the aboe parameters. By neglecting some of them, this complete model can be easily conerted to any other simple model. The simplified steady state aerage model of KY buck-boost conerter with ( rm rd r rc r ) was presented in the paper. Based on our complete aerage model a SIMUINK block was presented to simulate the performance of the conerter. Anybody can use it to ealuate the performance of its controller which was designed for the conerter. Finally, the KY buck-boost conerter Benchmark circuit is simulated in PSpice and its results are compared with our model simulation results in MATAB. The results are so closed to each other. REFERENCES [1] K. I. Hwu, and Y. T. Yau, (AUGUST 29) Two Types of KY Buck Boost nerters, IEEE Transaction on Industrial Electronics, Vol. 56, No. 8. [2] Marc Aoun, Michel El-Maalouf, Najib Rouhana, Hadi Y. Kanaan and Kamal Al-Haddad,( 2-4 May 212) Aerage Modeling and inear ntrol of a Buck-Boost KY nerter, Proceedings of the 5th International Symposium On mmunications, ntrol and Signal Processing, Rome, Italy. [3] R.D. Middlebrook, and R S. Cuk, (1976) A General unified Approach to Modeling switching conerter power stages," IEEE PESC, Record, PP [4] J R. B. Ridley, (1991) A New ntinuous-time Model for Current Mode ntrol, IEEE Transaction On Power Electronics, Vol. 6, No. 2, PP [5] V. Vorperian, Simplified Analysis of PWM nerters Using the Model of the PWM Switch,Parts I (CCM) and II (DCM), Trans. On Aerospace and Electronics systems, ol. 26, no. 3 May 199. [6] A. Towati,( 28) Dynamic Analysis QFT based Robust ntrol Design of Switched Mode Power nerters, Doctoral Thesis, Helsinki Jniersity of Technology. [7] G. F. Wallis and R. Tymerski, (2) Generalized approach for - synthesis of robust switching regulators, IEEE Transaction On aerospace and electronic systems, Vol.36, No.2. [8] K. I. Hwu, and Y. T. Yau, (JANUARY 29) KY nerter and Its Deriaties, IEEE Transaction On Power Electronics, VO. 24, NO. 1. [9] K. I. Hwu, and Y. T. Yau, (Noember-December 21 ) Topology Exchange Between KY nerter and Its Deriatie Based on Duty Cycle International Reiew of Electrical Engineering (I.R.E.E.), Vol. 5, No. 6. [1] K. I. Hwu, and Y. T. Yau, (NOVEMBER 21) A KY Boost nerter, IEEE Transaction On Power Electronics, VO. 25, NO. 11. [11] K. I. Hwu, and Y. H. Chen, (July 5-8, 29) Bidirectional ntrol of Inerse KY nerter, IEEE International Symposium on Industrial Electronics, Seoul, Korea (ISlE 29). 874 Vol. 6, Issue 2, pp

14 [12] K. I. Hwu, and Y. T. Yau, (May-June 211) Negatie-Output Soft Switched KY Buck nerter, International Reiew of Electrical Engineering (I.R.E.E.), Vol. 6, No. 3. [13] K. I. Hwu, and Y. T. Yau, (MAY 21) Voltage-Boosting nerter Based on Charge Pump and upling Inductor With Passie Voltage Clamping, IEEE Transaction on Industrial Electronics, Vol. 57, No. 5. [14] M. R. Modabbernia, A. R. Sahab, M. T. Mirzaee, and K. Ghorbani, (212) The State Space Aerage Model of Boost Switching Regulator Including All of the System Uncertainties, Adanced Materials Research, Vol , pp [15] M. R. Modabbernia, F. Kohani, R. Fouladi, and S. S. Nejati, (Februry 213) The State Space Aerage Model of Buck-Boost Switching Regulator Including All of the System Uncertainties, International Journal on mputer Science and Engineering (IJCSE), Vol. 5, No. 2, pp [16] M. R. Modabbernia, A. R. Sahab, and Y. Nazarpour, (Noember 211) P-Δ-K Model of Boost Switching Regulator with All of The system Uncertainties Based On Genetic Algorithm, International Reiew of Automatic ntrol, Vol. 4, No. 6, pp [17] M. R. Modabbernia, A. Gholami Pastaki, and Y. Nazarpour, (December 212) Robust control of Boost Switching Regulator with All of The system Uncertainties Based On -Synthesis International Reiew on Modelling and Simulation, Vol. 5, No. 6, pp AUTHORS Mohammad Reza Modabbernia was born in Rasht, IRAN, in He receied the B.S. degree in Electronics Engineering and M.S. degree in control engineering from KNT, the Uniersity of Technology, Tehran, IRAN in 1995 and 1998 respectiely. He is the staff member of Electronic group of Technical and Vocational uniersity, Rasht branch, Rasht, IRAN. His research interests include Robust ntrol, Nonlinear ntrol and Power Electronics. Seyedeh Shia Nejati was born in Bandar-Anzali, Iran in She receied her B.Sc. degree in electrical engineering from Guilan Uniersity, Iran, in 25 and the M.Sc. degree in Electrical Engineering from Tabriz Uniersity, Iran, in 28. Her research interests include optical fiber, optoelectronics and optical deices. She is a uniersity lecturer at higher education Institute of Sardar Jangal, Rasht, Iran. Fatemeh Kohani Khoshkbijari was born in Rasht, Iran in She receied her B.Sc. degree in electrical engineering from Islamic Azad Uniersity of ahijan, ahijan, Iran, in 25 and the M.Sc. degree in Electrical Engineering from Islamic Azad Uniersity Tehran South Branch, Tehran, Iran, in 28. As a noel research in nanotechnology, her M.Sc. thesis was awarded by Iranian Nanotechnology Initiatie uncil. She is a uniersity lecturer at Sama Technical and Vocational Training llege, Islamic Azad Uniersity, Rasht Branch since 29. Her research interests include ow Power Design, Deice Physics, Process/Deice Design, CAD Deelopment for Process and Deice Design, Simulation, Modelling and Characterization of Nanoscale Semiconductor Deices. 875 Vol. 6, Issue 2, pp