CONTROLLER DESIGN FOR SEPIC CONVERTER USING MODEL ORDER REDUCTION

Transcription

1 Controller Design For SPIC Converter Using Moel Orer Reuction CONTROLLR DSIGN FOR SPIC CONVRTR USING MODL ORDR RDUCTION BINOD KUMAR PADHI, ANIRUDHA NARAIN Deartment of lectrical ngineering, Motilal Nehru National Institute of Technology, Allahaba Abstract A SPIC (Single-ne Primary Inuctor Converter) is a DC-DC converter, caable of oerating both in steu or ste-own moe an wiely use in battery-oerate alications. There are two ossible moes of oeration in the SPIC converter: Continuous Conuction Moe (CCM) an Discontinuous Conuction Moe (DCM). This aer resents moeling of a SPIC converter oerating in CCM using the State-Sace Averaging (SSA) technique. SPIC converter consists of two inuctor an two caacitor hence it is fourth orer c-c converter. Design of feeback comensator for fourth orer system is quite comlex. In this aer, moel orer reuction technique is use for controller esign of SPIC converter. First small signal ynamic moel for SPIC converter is obtaine using SSA technique which rovies fourth orer transfer function. Then this fourth orer transfer function is reuce to secon orer using Paé aroximation. Then the comensator is esigne for the reuce orer moel of the SPIC converter. Result shows that the comensator esigne for reuce orer moel gives the quite satisfactory resonse with the original system. Keywors- SPIC Converter, CCM, State-Sace Averaging, Moel Orer Reuction, Paé-Aroximation, Comensator I. INTRODUCTION The switche moe c-c converters are the ower electronic systems that convert one level electrical voltage to another level of electrical voltage by the hel of switching action. These are extensively use in battery oerate ortable electronic equiment an system because of its greater efficiency, smaller size an lighter weight [, 5]. The SPIC converter is a tye of c-c converter an is caable of roviing a non-inverte outut voltage which is either greater than, less than or equal to the inut voltage an wiely use in battery oerate equiments. The outut of the SPIC converter is controlle by the uty cycle of the control transistor. The SPIC converter has two moes of oeration one is Continuous Conuction Moe (CCM) an the other one is Discontinuous Conuction Moe (DCM). Here the SPIC is oerate in CCM. SPIC converter has excellent roerties like caacitive energy transfer, full transformer utilization, excellent transient erformance an goo steay-state erformances such as wie conversion ratio, continuous current at inut an caacitor voltage. The ynamic resonse, however, is affecte by the fourth orer characteristic, which generally calls for close-loo banwith limitations in orer to ensure large-signal stabilization. Moreover, stability may require big energy transfer caacitors in orer to ecoule inut an outut stages. The robust multivariable controllers coul be use to otimize the converter ynamics an ensuring the correct oeration in any working conition however this involves consierable comlexity of both theoretical analysis an control imlementation. So in orer to remove these ifficulties first we reuce the orer of transfer function of SPIC converter then esign the controller. The SPIC converter is mae u of two caacitors, two inuctors, a ower switch an a ioe thus it is fourth orer non-linear system an in this aer the equivalent series resistances (SR) of the inuctors an caacitors are consiere. For the feeback control esign linear moel is neee. The linear moel of the converter is erive by the relacement of switch an ioe of converter by small signal average switch moel [7]. In this aer the esire transfer function is obtaine using state sace averaging technique [,, 6, an 9]. This aer resents the moeling an control of SPIC converter oerating in continuous conuction moe. In continuous conuction moe, inuctor current never falls to zero uring one switching erio. The SSA technique is use to fin small signal linear moel an its various forms of transfer functions. Deening on control-to-outut transfer function, the PWM feeback controller [8-9] is esigne to regulate the outut voltage of the SPIC converter. The higher orer system increases the comlexity of the controller. So, in orer to remove these ifficulties the higher orer system is reuce to n orer system by using moel orer reuction technique [-5]. In this aer the Paé aroximation [] moel reuction technique is use to reuce the higher orer system. This aer is organize as follows: SSA Technique is given in section II. Moeling of SPIC converter by SSA Technique is shown in section III. Control Strategy is shown in section IV an Conclusion in section V. ASAR International Conference, Bangalore Chater- 3, ISBN:

2 II. SSA TCHNIQU Controller Design For SPIC Converter Using Moel Orer Reuction State sace moeling is a technique that escribes a given system using a system of linear ifferential equations. The ower stage of close loo system is a non-linear system. The non-linear systems are usually ifficult to moel an are also ifficult to reict the behavior of the non-linear system. So, it is better to aroximate the non-linear system to a linear system. For the linearize ower stage of c-c converter Boe lot can be use to etermine suitable comensation in feeback loo for esire steay state an transient resonse. For this the state sace averaging technique is use. In c-c converter oerating in CCM has two circuit states: one when the switch is turne on an other when the switch is turne off. During switch on: X A X BV < t < T V C X V () During switch off: X A X B V < t < (-)T V C X V () To rouce an average escrition of the circuit over a switching erio, the equations corresoning to the two foregoing states are time weighte an average, resulting in the following equations: X [ A A ( )] X [ B B ( )] V (3) V [ C C ( )] X [ ( )] V (4) III. MODLING OF A SPIC CONVRTR BY SSA TCHNIQU The SPIC converter shown in Fig. (a) contains two caacitors C an C with equivalent series resistors r C an r C resectively, two inuctors L an L with equivalent series resistors r L, r L resectively, a MOSFT switch Q an a ioe D. The resistor R is reresents the loa. The SPIC converter exchanges the energy between the caacitors an inuctors in orer to convert from one voltage to another. The amount of energy exchange is controlle by the control transistor i.e. MOSFT. A SPIC is sai to be in CCM if the current through the inuctor L never falls to zero. In CCM, the converter has two states. During the first state i.e. when Q is turne on (Fig. (b)) L is charge by the source V an L is charge by the caacitor C. Hence current i L an i L increases linearly. During the secon state i.e. when Q is turne off (Fig. (c)) L an L are in a ischarging hase an release the store energy to the caacitors an loa resectively. Hence i L an i L ecreases linearly. In ieal SPIC converter the SRs are zero. For the ieal converter the relationshi between the V an V is given by: ASAR International Conference, Bangalore Chater- 3, ISBN: V o (5) V Where is the uty cycle of the switch. This equation shows that by controlling the uty cycle of the switch the outut voltage Vo can be controlle an outut voltage can be higher or lower than or equal to the inut voltage V. The uty cycle of the SPIC converter can be varie uring oeration by using a controller an the circuit can also be mae to reject isturbances []. A. State Sace Descrition The state sace equations for SPIC converter uring switch on an off are During switch ON: il rl il V (6) t L L i L ( rc rl ) il VC (7) t L L VC il (8) t C VC VC (9) t C( R rc ) RVC () V rc During switch OFF il VC V SiL SiL S3V C () t L L il CRiL ( R rl ) il RVC () t CL L LrC VC il (3) t C VC RiL RiL VC (4) t CrC CrC C( R rc ) RVC V RiL RiL (5) rc Where CR C ( rl rc ) (6) S LC LrL L rl LR S (7) L R R (8) S3 RL An states of the SPIC converter are i L, i L, V C, V C. The average matrices for the steay-state an linear small-signal state-sace equations can be written accoring to above equations. rl L ( rc r ) (9) L L L A C C ( R rc )

3 Controller Design For SPIC Converter Using Moel Orer Reuction S S S3 L CR R r L R CL L LrC A C R R CrC C rc C ( R rc ) () Fig. (c) SPIC Converter when switch is OFF. Fig.. Oeration of SPIC Converter in CCM. L () B B B R C () r C R (3) C R R rc [ ] (4) B. Fining Transfer Function With the state sace matrices efine above, the control to outut transfer function can be calculate as: G v C ( S I A ) B (5) Where A A A ( (6) B B B ( (7) C C C ( ) (8) ( ) (9) B ( A A ) X ( B B ) V (3) Outut to inut transfer function G vv C ( S I A ) B (3) X A B V (3) Fig. (a) SPIC Converter. IV. CONTROL STRATGY C. PWM Feeback Control The SPIC converter with PWM feeback control is shown in the Fig. (a). The outut voltage V is comare with the reference voltage V ref. The error voltage V e between outut voltage an reference voltage is asse through the comensator G c (s) to generate a control signal V C an comare with the saw-tooth voltage of amlitue V M by using the PWM comarator. Finally the PWM comarator converts the control signal into a waveform that rives the MOSFT switch. As eicte in Fig. (b), the MOSFT switch 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 reson by ajusting Vc an then uty cycle of the MOSFT until V is again equal to V ref. Fig. 3 shows a small-signal block iagram of the converter of Fig. (a). The ower stage transfer functions are reresente by G v (s) which is erive earlier. The transfer function of the PWM comarator can be erive from the wave form in Fig. (b). It is given by: FM (33) VM Where V M is the amlitue of saw-tooth waveform an G C (s) is a controller or comensator. From Fig 3 the oen loo transfer function can be efine as: T ( s) G ( s) G ( s) F (34) C v M The loo gain T(s) is efine as the rouct of the small signal gain in the forwar an feeback aths of the feeback loo. It is foun that the transfer function from a isturbance to the outut is multilie by the factor /(+T(s)). So the loo gain magnitue T is a measure of how well the feeback system works. Fig. (b) SPIC Converter when switch is ON. Fig. (a) SPIC converter with PWM feeback control. ASAR International Conference, Bangalore Chater- 3, ISBN:

4 Controller Design For SPIC Converter Using Moel Orer Reuction Fig. (b) Waveform of PWM Comarator. Fig. 3.Small-signal block iagram of SPIC converter with PWM feeback control. D. xamle TABL I. Converter Parameters Circuit Parameters Values Inut Voltage V V Outut Voltage V 5 V Switching frequency khz Loa R Ω PWM Gain F M /7 L µh r L mω L µh r L mω C 8µF r C 3 mω C 3µF r C mω Outut rile 5% The transfer function of the converter is obtaine from (5) is as follows: s.496 s.56 s.3 Gv ( s) s s 8.88 s.9 s 3.5 (35) This is a fourth orer transfer function. It has two air of comlex ole an three zeros (one air of comlex zero an one real zero). Zeros an oles of the converter are as given as: Poles are: i i i i Zeros are: i i. Moel Orer Reuction Using Paé-Aroximation metho [], the reuce orer transfer function of the converter is obtaine as follows: 6.37s.58 GRv ( s) 5 s 34.s (36) This is a n orer transfer function. It has one air of comlex ole an one real zero. Poles an zeros of reuce system are: Poles are: i i Zeros are: Fig. 4, Fig. 5 an Fig. 6 clearly shows that the ste resonse an Boe lot of the reuce system closely aroximates with the ste resonse an boe lot of the original system. Integral Square rror (IS) between original system an reuce orer system is Then the next objective is to esign the controller for the reuce orer converter. A m litu e Ste Resonse Time (sec) Original System Reuce moel Fig. 4. Ste resonse of oen loo original system an reuce moel. A m litu e Ste Resonse Time (sec) Original System Reuce moel Fig. 5. Ste resonse of close loo original system an reuce moel. ASAR International Conference, Bangalore Chater- 3, ISBN:

5 M a g n itu e ( B ) P h a s e ( e g ) Controller Design For SPIC Converter Using Moel Orer Reuction Boe Diagram Frequency (ra/sec) Original System Reuce Moel Fig. 6. Boe lot of oen loo original system an reuce moel. M a g n it u e ( B ) P h a s e ( e g ) Boe Diagram Gm = 7.99 B (at.9e+3 ra/sec), Pm = 3.3 eg (at.68e+3 ra/sec) Frequency (ra/sec) Fig. 7. Boe lot of uncomensate oen loo system which has gain margin 7.98B an hase margin 3.3 eg. Fig. 7 shows that the Boe lot of the system without comensator has hase margin of 3.3 eg which is not sufficient for a stable system. Hence a comensator is esigne to obtain the suitable hase margin. F. Feeback Loo Comensation In this aer voltage-moe linear average feeback controllers [9-] for c c converter is esigne in frequency omain. The main objective of the controller esign is to obtain stable oeration of the converter by varying the uty cycle. Following oints are taken care while esigning of the comensator. First the average mathematical moel is accurate u to one tenth of switching frequency. Here the switching frequency is taken as khz therefore the banwith ( B cross over frequency of close loo system) shoul be near khz. Seconly high gain at low frequency region rovies goo outut voltage regulation. An hase margin etermines the transient resonse to suen change in inut voltage. The suitable hase margin is in between 45 to 6 egree. G. Stes For Comensator Design Ste : Select a resistor value for R. Ste : Select α calculate the comensator s maximum hase frequency w m using the equation w m w c (37) Where w c is the esire cross-over frequency. Ste 3: Calculate the ifference between the zero's frequency an ole's frequency using the equation ( ) cot( c ) (38) Where φ m is the esire hase margin an φ P is the control lant gain. Ste 4: Calculate the zero's frequency z an ole's frequency using the following equations: (39).5( z.5( 4 m 4 m m ) ) (4) Ste 5: Calculate the comensator s constant gain G c ( ) using the equation (4) c G G c ( ) Ste 6: Calculate C using the equation z C RG (4) Ste 7: Calculate C using the equation C C GR (43) Ste 8: Calculate R using the equation R z C (44) Ste 9: Plot the loo Boe lot an verify the hase margin. Ste : Check the gain margin. If the gain margin is not satisfie, ajust an go back to ste to reesign the comensator. Using the stes for comensator esign the comensator is esigne whose transfer function is: s G c (45) s s z ASAR International Conference, Bangalore Chater- 3, ISBN:

6 Controller Design For SPIC Converter Using Moel Orer Reuction an therefore, the overall oen-loo transfer of the reuce orer moel with comensator is s.889 s 8.9 TR ( s) s.774 s.64 s 93. s (46) An similarly the oen loo transfer function for the original system with comensator is s.865 s 8.55 s.4 s6.87 T( s) s.774 s.89 s 6876 s.334 s.499 s6.87 (47) Fig. 8 shows that the Boe lot of oen loo original system with comensator which has gain margin of.87 B an hase margin of 53. eg an Boe lot of oen loo reuce moel with comensator which has gain margin of.87 B an hase margin of 53 eg. Fig. 9 shows that ste resonses of comensate reuce orer moel closely aroximates with the ste resonse of comensate original system. M a g n i t u e ( B ) P h a s e ( e g ) Boe Diagram Gm =.87 B (at 67 ra/sec), Pm = 53 eg (at 43 ra/sec) Gm =.87 B (at 66 ra/sec), Pm = 53. eg (at 43 ra/sec) Original System with Comensator Reuce moel with Comensator Frequency (ra/sec) Fig. 8. Boe lot of original system an reuce moel with comensator. A m litu e Ste Resonse Original System with Comensator Reuce moel with comensator Time (sec) Fig. 9. Ste resonse of original system an reuce moel with comensator. CONCLUSION This aer eals with moeling an control of SPIC converter oerating in continuous conuction moe (CCM). The state sace averaging technique is alie to fin out the linear moel of SPIC converter an the esire transfer function in terms of uty ratio to outut voltage (G v ) is obtaine which is a fourth orer transfer function. Designing a comensator for the fourth orer system is very ifficult. Therefore, fourth orer transfer function of SPIC converter is reuce to secon orer an it is foun that ste resonse of reuce orer moel closely follow the original system. The comensator esigne for secon orer system gives quite satisfactory resonse with the original system. RFRNCS [] R. W. rickson an D. Makimovic, Funamental of Power lectronics, n e., Kluwer Acaemic Publishers,. [] R. D. Milebook an S. Cuk, A General Unifie Aroach to Moelling Switching Converter Power Stages, International Journal of lectronics, vol. 4,. 5-55, June 977. [3] V. Vorerian, Analysis of the Seic converter by Dr. Vatché Vorérian, Riley ngineering Inc, 6. [4] R. Riley, Analyzing the seic converter, Power Systems Design uroe Magazine,. 4-8, November 6. [5] M. H. Rashi, Power lectronics Hanbook: Devices, Circuits, an Alications, n e, lsevier Inc, 7. [6]. Vuthchhay, P. Unnat, an C. Bunlaksananusorn, "Moeling of a seic converter oerating in continuous conuction moe," 6th International Conference on lectrical ngineering/lectronics, Comuter, Telecommunications an Information Technology 9 (CTI-CON 9), , May 9. [7] A. Hren an P. Slibar, "Full Orer Dynamic Moel of SPIC converter," Proc. of the I International Symosium on Inustrial lectronics, , June 5. [8] A. J. Forsyth an S.V. Mollov, "Moelling an control of DC- DC converters," I Power ngineering Journal,. 9-36, 998. [9]. Vuthchhay, C. Bunlaksananusorn, an H. Hirata, "Dynamic Moeling an Control of a Zeta Converter," International Symosium on Communications an Information Technologies 8 (ISCIT 8), Oct. 8. [] B. C. Kuo, Automatic Control Systems, 7th e., Prentice Hall Inc, 995. [] A. Chuinar, T. Chairet, " Feeback comensation techniques to imrove inut isturbance resonse in the Cuk converter," June. 9. [] Y. Shamash, "Stable reuce-orer moels using Paé tye aroximations," I Transactions automatic control, vol.9, no.5, , Oct 974. [3] C.B Vishwakarma, R. Prasa, System reuction using Moifie Pole Clustering an Pae Aroximation, National systems conference, NSC 8, December [4] V. Krishnamurthy an V. Seshari, Moel Reuction Using the Routh Stability Criterion, I Transactions On Automatic Control, VOL. AC--3, NO. 4, AUGUST 978. [5] Vimal Singh, Dinesh Chanra, an Haranath Kar, Imrove Routh Paé Aroximants: A Comuter-Aie Aroach, I Transactions On Automatic Control, Vol. 49, No., February 4. ASAR International Conference, Bangalore Chater- 3, ISBN: