X-Jun Yang, Chen Yao, Nng-Yun Zhang Research on Two-channel Interleaved Two-stage Paralleled Buck DC-DC er for Plasma Cuttng Power Supply XI-JUN YANG, CHEN YAO, NING-YUN ZHANG, HAO QU, HOU-JUN TANG, FREDE BLAABJERG 2 Key Lab. of Power Transmsson and Converson Control (Mnstry of Educaton), 2 Dept. of Energy Technology Shangha Jao Tong Unversty, 2 Faculty of Engneerng and Scence, Aalborg Unversty Shangha Chna 200240, 2 Aalborg Denmark 9220 P.R. CHINA, 2 DENMARK youngxun@63.com Abstract: - Buck DC-DC converter s an mportant and typcal power electronc converter wth relatvely smple topology and rch study contents. Nowadays, t has been wdely used as SMPS and hgh power DC power supples. As for hgh power plasma cuttng machne, mult-channel nterleaved and mult-stage paralleled buck DC-DC converter s the frst choce. When desgned as a constant DC current supply, t s characterstc of hgh current precson, good stablty and relablty, and t can be desgned wth current lmtaton, power equlbrum, constant swtchng frequency, zero output current steady-state error and phase shft drvng. In the paper, a two-channel nterleaved, two-stage paralleled buck DC-DC converter s analyzed and desgned by usng sldng mode control (SMC) and smulated by means of MATLAB/SIMULINK. The basc prncple of sldng mode control s also revewed. Then the expermental setup of plasma cuttng machne (PCM) DC current supply s mplemented on the bass of the theoretcal analyss, whch outputs the rated current of 260A and the rated DC voltage of 50V, and the swtchng frequency s selected as 0kHz. The ganed results prove the desgned sldng mode control. Key-Words: - Plasma cuttng power supply, buck DC-DC converter, Two-channel nterleaved, Two-stage paralleled, Sldng mode control, Current lmtaton, Constant swtchng frequency, Output voltage steady-state error, Phase shft drvng Introducton Cutter and welder s well known as the talor of steel ndustry, wdely used as key technologes n the felds of brdge, constructon, machnery, shppng, ralway, metallurgy, petrochemcal, water power generaton, home applance, contaner, boler pressure vessel, power generaton equpment ndustry, etc. Cuttng s the frst procedure of metal weldng, ready for subsequent processng and weldng. The metal cuttng ncludes hot cuttng and cold cuttng, and the former s the most wdely used, ncludng flame cuttng machne, plasma cuttng machne (PCM) and laser cuttng machne. Plasma cuttng machne has a long range of applcatons, due to hgh cuttng speed, hgh cuttng thckness and better cuttng qualty, whch s near laser cuttng. The generaton of plasma arc column needs plasma DC current power supply. Plasma power supply has gone through three technologcal nnovatons, they are, slcon rectfer DC power supply, thyrstor rectfer DC power supply, and IGBT DC-DC power supply [-5]. The last type ncludes the solaton DC-DC converter (below 20kW) and the non-solaton buck DC-DC converter (above 20kW). Mult-channel nterleaved, mult-stage paralleled buck DC-DC converter s the frst canddate for hgh power PCM, because such converter can support hgh DC current (above 30A per sub-module) and has smple power stage and control stage, smple desgn, low overall cost. As a swtchng mode power supply (SMPS), buck DC-DC converter has a lot of control methods. Due to nonlnear and tme-varyng property, nonlnear control approaches should be consdered n the frst place, though t s a challengng task to get hgh-performance control. The controller should brng about satsfactory statc and dynamc performances and make the system rugged and stable n the presence of nput voltage dsturbance, load power change and system parameter varaton. Sldng mode control (SMC) s one of the control approaches based on varable structure systems (VSS), defned as a system where the crcut topology s ntentonally changed accordng to a certan control rule, n order to mprove the system E-ISSN: 2224-266X 26 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang behavor n terms of response speed, global stablty, and robustness. VSS nvolves a certan number of ndependent sub-topologes, whch are determned by the status of nonlnear elements (swtches). The global dynamcs of VSS s qute dfferent from each sub-topology. There are many references provde systematcal analyses of VSS and sldng mode control [6,7]. Sldng mode control was employed to the feld of DC-DC converter earler [8], and a great deal of work was done to mprove sldng mode control effectveness n terms of current lmtaton [9-2], power equlbrum [3,4], reduced or constant swtchng frequency [5-6], zero output current steady-state error and phase shft drvng [4,7,8]. In the paper, manly on the bass of references [4, 7, 8], sldng mode control (SMC) of three-channel nterleaved, two-stage paralleled buck DC-DC converter s analyzed n theory, smulated by means of MATLAB/SIMULINK and mplemented n practce wth current lmtaton, constant swtchng frequency and output voltage steady-state error, whch outputs rated current of 260A and rated DC voltage of 50V. The paper s organzed as below: In secton 2, ntroducton to PCM platform and ts block dagram, topology of mult-channel nterleaved, mult-stage paralleled buck DC-DC converter, the VA characterstcs of plasma cuttng power supply; In secton 3, state space model of buck DC-DC converter,ntroducton to SMC, SMC applcaton to DC-DC ers,selecton of sldng lne; In secton 4, practcal ssues of nterleaved and paralleled buck converters, ncludng current lmtaton, power equlbrum or current sharng, constant swtchng frequency, output current steady-state error. In secton 5, smulaton by means of MATLAB/ SIMULINK, desgn and mplementaton of plasma cuttng machne (PCM) DC current supply. 2 Specfcatons of PCM 2. Introducton to PCM platform Plasma cuttng machnes ncludes two mportant parts: mechancal structure and electrcal control, where the latter ncludes plasma cuttng power supply, computer numercal control system (CNC) and arcng-box, control box, water coolng systems, and other necessary parts, such as valves and the torch. Typcal plasma cuttng machne system s shown n Fg. []. Fg. Block dagram of the PCM system In Fg., system components nclude A-power supply; B-gnton console; C-off-vale assembly; D-gas console; E-torch. Cables and hoses nclude -Plot arc lead; 2-Negatve lead; 3-Ignton console power cable; 4-Ignton console coolant hoses; 5-Gas control cable; 6-Gas power cable; 7 -Gas console to off-valve hose and lead assembly; 8 - CNC nterface cable; 9 - Optonal CNC nterface cable for systems wth multple power supples; 0-Torch lead assembly; -work lead; Supply gas hoses nclude 2 - Oxygen; 3 - Ntrogen or argon; 4-Ar; 5-Argon-hydrogen (H35) or ntrogen-hydrogen (F5); Man power cable ncludes 6-Customer-suppled power cable. Plasma cuttng machne s an ntellgent mechatronc equpment, ntegrated wth power flow, nformaton flow and control flow. Plasma cuttng power source s one of the key components, manly determnng the qualty of cuttng current, such as stablty, precson and speed of response, and then determne the workpece cuttng qualty. Prncple of plasma cuttng s: to make use of the hgh temperature of plasma arc column to make the metal workpece molten and cut (evaporated), E-ISSN: 2224-266X 262 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang and to get rd of molten metal to form cut va the plasma's momentum, sutable for cuttng stanless steel, cast ron, copper, alumnum and other nonferrous metal tube and metal sheet, the thckness up to 35mm or more. Plasma arc beam tself has good moblty and dffusvty, good electrcal conductvty and thermal conductvty, wth ts hundreds of tmes of the specfc heat capacty at hgh temperature (proportonal to the temperature). Actually, plasma cuttng power supply s a knd of DC voltage to DC current converter. At present, plasma current goes up to 000A. Take plasma current of 260A for nstance, the man electrcal parameters are lsted below: AC nput voltage (V): three-phase 380±5%; output DC current (A): 260; no-load output voltage (V): 30; workng voltage (V): 50~70; step-down transformer (V/VV): 380/220/220, Y/YY connecton, or to use more secondary wndngs, n order to support greater power output and mprove the qualty of the mans current; rated power factor: not less than 0.92; work duty rate (45.5kW,40 c): 00%, EMC regulaton: EN6000-2-2/2. 2.2 VA characterstcs of plasma power supply Consderng the statc characterstc of plasma arc and the varaton of arc length durng cuttng process, n order to keep the power supply stable, when arc voltage fluctuates, the current should be unchanged or changed slghtly, the output of the plasma power source should have the sharp or vertcal droop characterstc. For plasma cuttng machne, except that PMSM based servo drver s responsble for horzontal movement of the torch, and DCM based servo drver s responsble for vertcal movement of the torch, whch mantans the length of arc wthn a reasonable range and keep the cuttng current constant ndrectly. As for plasma power source, the external characterstc should be ntersected wth volt-ampere characterstc of plasma arc, shown n Fg.2. The condton of stable operaton pont can be expressed as below [4,7,8] : u = u2 du du dt dt 2 () where u s the power supply voltage; u 2 s the arc voltage. The drop rate of external characterstc of power supply should be faster than that of volt-ampere characterstcs of the plasma arc, or else the arc wll put out easly. In the vcnty of current settng, the output voltage has a quck and obvous change than that of output current, apparently showng as a DC current source. The current settng s determned by the materal and thckness of the cut workpece. Fg.2 -external characterstc of power supply; 2-volt-ampere characterstcs of the plasma arc 2.3 Topology of buck DC-DC converter Non-solated buck DC-DC converter and solated nverter-rectfer can be used as power supply of plasma cuttng machne. To accommodate the requrements of cuttng thcker work pece, hgh power output power supply s nevtable. Three-channel nterleaved two-stage paralleled buck DC-DC converter s employed as plasma cuttng power supply, shown n Fg.3, whch can support output DC current up to 750A and selected as the concerned topology n the paper. Fg.3 Three-channel nterleaved two-stage paralleled buck DC-DC converter E-ISSN: 2224-266X 263 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang Sngle-channel two-stage paralleled buck DC-DC converter can output DC current of 260A, whch s the commonly used plasma cuttng power supply at present, shown n Fg.4. Fg.4 Sngle-channel two-stage paralleled buck DC-DC converter As for Fg.3, defne S s bnary logc swtchng functon for reverse conducton swtch (RCS), where =, 2, 3, 4, 5 and 6. When the S s swtched on, S =, or else S =0. Accordng to reference [6], plasma power supply produces the arc column, lke constant current supply. The approxmate voltage across the plasma arc column can be expressed as uo = k o (20k 2L 2)=k o E o (2) where u o s the output voltage, that s, the voltage between cathode and workpece; o s the output current or load current; L 2 s the dstance between cathode and workpece; k s a small constant, relevant to load current; k 2 s a constant, relevant to plasma arc column. In the steady state, neglectng the rpple, the average value of o s I o. The ampltude of I o depends on the dfferent torch and workpece. Three-channel nterleaved, two-stage paralleled buck DC-DC converter can be equvalent to sx-stage paralleled buck DC-DC converter. In Fg.3, for PCM, R 3 and C consttute a seres LC flter, R 4 and C 2 consttute a parallel LC flter, R 5 s a resstance wth hgh value. The actual flter s more complcated. Each smoothng reactor has ts own reversely-paralleled RCD as absorber. When output current s 260A, the rated output voltage s 50V or so, and the equvalent resstance s roughly 0.58Ω. Practcally, R 3 =0Ω/50W, R 4 =00kΩ/25W, R 5 =0k//0k//0k/25W, C =350µF/450V, C 2 =0.22µF/20V, therefore R 4, R 5 and C 2 can be gnored, even more, R 3 branch can also be omtted. The control over the buck DC-DC converter s actually to control the current through L 7 and voltage across C, to control the currents through reactor L -L 6. 3 Prncples of SMC 3. State Space Model of buck DC-DC converter Assumng that dstrbuton resstance of smoothng reactor s R L, the forward conducton voltage of power swtch s near zero. The swtchng frequency s f s, carrer frequency s f c, then the current rpple frequency of L 7 s 6f s. L 7 s used to flter hgh frequency current rpple, desgned wth low nductance. For the convenence, L 7 can be omtted. For the sx RCSs, gven the duty cycle of the th RCS n the k th swtchng perod s d cs, then, t t t d T k k cs s S = 0, tk d csts t t(k) t = t T,k = 0,,2,... (k) k s (3) where T s s the swtchng perod, t k s the ntal tme of the k th swtchng perod for the th RCS. Smlarly, t (k) can be dealt n the same manner. Choose reactor currents ( L L2 L3 L4 L5 L6 ) and output voltage (u o ) as the state varables. Let Un= Un2 = U. n Accordng to KVL, dl dl7 L =-L7 -RLL - uo dcsu (4) dt dt 6 6 6 dl7 dl 6L 7 =- L - RLL -6 uo dcsu (5) dt = dt = = Accordng to KCL, 6 c o o o o C du = L - u E = L7 - u E (6) dt = RL RL RL RL Due to duc uo = R3C u, then c dt R L R 3 duc Eo C = L7 - uc (7) R L dt R L R L The state space equatons of the sx-stage paralleled buck DC-DC converter are rewrtten as below. E-ISSN: 2224-266X 264 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang d dt duc dt L d R L 7 L7 L =- - L - uout dcsu n L dt L L L R = - u C (R R ) C (R R ) L L7 L 3 L 3 E o C (R L R 3 ) [4, 7, 8] 3.2 Introducton to SMC c (8) The frst substructure, referred as substructure I, s gven by the followng equatons: x = x2 x = K x 2 (9) where the egenvalues are complex wth zero real parts. The phase traectores are crcles, as shown n Fg. 5 and the system s stable margnally. The second substructure, referred as substructure II, s gven by x = x2 x = K x 2 (0) In ths case, the egenvalues are real wth opposte sgns. The correspondng phase traectores are shown n Fg.5, and the system s unstable. Only one phase traectory, namely x2 = qx Kx converges towards the orgn, whereas all other traectores are dvergent. Fg.5 Phase-planes correspondng to substructures I and II Parttonng the phase-plane n two regons, shown n Fg.6, as follows: Regon I: x (x 2 cx )<0 Substructure I Regon II: x (x 2 cx )>0 Substructure II Fg. 6 Sldng regme. (a) deal swtchng lne; (b) swtchng lne wth hysteress; (c) unstable sldng mode In Fg.6, c>0 s lower than q. The swtchng boundares are the x 2 axs and the lne x 2 cx =0. The system structure changes whenever the system representatve pont (RP) enters a regon defned by the swtchng boundares. The mportant property of the phase traectores of both substructures s that, n the vcnty of the swtchng lne x 2 cx =0, phase traectores wll converge to the swtchng lne. The mmedate consequence of ths property s that, once the RP hts the swtchng lne, the control law ensures that the RP does not move away from the swtchng lne. Fg.6a shows a typcal overall traectory startng from an arbtrary ntal condton P 0 (x 0, x 20 ): after the ntervals correspondng to traectores P 0 P (substructure I) and P P 2 (substructure II), the fnal state evoluton les on the swtchng lne (n the hypothess of deal nfnte frequency commutatons between the two substructures). Ths moton of the system RP along a traectory s called the sldng mode, on whch the structure of the system changes and whch s not part of any of the substructure traectores. The swtchng lne x 2 cx =0 s called the sldng lne. When sldng mode exsts, the resultant system performance s entrely dfferent from that dctated by any of the substructures of the VSS, and can be made ndependent of the propertes of the substructures, dependent only on the control E-ISSN: 2224-266X 265 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang law (n ths case, the boundary x 2 cx =0). In ths case, for example, the dynamc s of the frst order wth a tme constant equal to /c. The ndependence of the closed-loop dynamcs on the parameters of each substructure s not usually true for more complex systems, but n these cases t has been proved that the sldng-mode control shows better robustness than other control technques. For hgher-order systems, the control rule can be wrtten n the followng manner: σ = f ( x, x,..., x ) = c x = 0 () N 2 N = where N s the system order and x are the state varables. Ths wll result n a partcularly smple mplementaton n SMPS, f a lnear combnaton of state varable n Eq.3 s employed. When the swtchng boundary sn t deal, that s, the commutaton frequency between the two substructures s lmted, and the system RP traectory s as shown n Fg.2b. Of course, the wdth of the hysteress band besde the swtchng lne determnes the swtchng frequency between the two substructures. Consderng the followng generalzed system wth scalar control [6, 7] : [4, 7, 8] ers The general sldng mode control scheme of DC-DC converters s shown n Fg.7, where u and u o are the nput and output voltages, respectvely, whle L and u C (=,, r, =r,, N-) are the nternal state varables of the converter (nductor currents and capactor voltages). Swtch S accounts for the system nonlnearty and ndcates that the converter may assume only two lnear sub-topologes, each assocated to one swtch status. All DC-DC converters havng ths property (ncludng all sngle-swtch topologes, plus push-pull, half and two-level full-brdge converters) are represented by the equvalent scheme of Fg.7. The above condton also mples that the sldng mode control s vald only for contnuous conducton mode (CCM) operaton. x = f ( x, t, u) (2) where x s a column vector; f s a functon vector, wth dmenson of N, and u s an element whch determnes the system moton (control law). Consderng that the functon vector f s dscontnuous on a surface σ(x, t)=0. The above equaton can be rewrtten as follow. f ( x, t, u ) for σ 0 x = f ( x, t, u) = f ( x, t, u ) for σ 0 (3) where the scalar dscontnuous nput u s gven by u for σ ( x) > 0 u = (4) u for σ ( x) < 0 The system s n sldng mode f ts representatve pont moves on the sldng surface σ(x, t)=0. 3.3 SMC applcaton to generc DC-DC Fg.7 Prncple scheme of a SM controller appled to DC-DC converters As for the scheme n Fg.7, accordng to the general sldng mode control theory, all state varables are sensed, and the correspondng errors (defned by the dfference to the steady-state values) are multpled by proper gans c and added together to form the sldng functon σ. Then, hysteretc block HC mantans ths functon near zero, so that N (5) = σ= c ε Observe that Eq.5 represents a hyperplane n the state error space, passng through the orgn. Each of the two regons separated by ths plane s assocated, by block HC, to one converter sub structure. If assumng (exstence condton of the sldng mode) that the state traectores near the E-ISSN: 2224-266X 266 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang surface are drected towards the sldng plane, the system state can be enforced to reman near (fall nto) the sldng plane by proper operaton of the converter swtches. Sldng mode controller desgn requres only proper selecton of the sldng surface Eq.9, that s, coeffcents c, to make sure exstence, httng and stablty condtons. From a practcal pont of vew, the selecton of the sldng surface s not dffcult f t s a second-order converters. In ths case, the above condtons can be verfed by smple graphcal technques. However, for hgher-order converters, lke Cúk and SEPIC converters, the more general approach must be rearranged. One of the maor problems of the general scheme n Fg.7 s that nductor current and capactor voltage references are dffcult to evaluate, because they generally depend on load power demand, supply voltage and load voltage. It s true for all basc topologes, except the buck DC-DC converter, whose dynamc equatons can be expressed n canoncal form. Thus, for all converters, except the buck DC-DC topology, some provsons are needed to estmate the references, whch wll strongly affects the closed-loop dynamcs. 3.4 SMC applcaton to buck DC-DC ers [4, 7, 8] The most mportant features of the sldng mode regmes n VSS s the ndependence of responses to system parameter varaton wth the only constrant of canoncal form descrpton of the system. The buck DC-DC converter s partcularly sutable for the sldng-mode control, ust because the controllable states (output voltage and ts dervatve) are all contnuous and accessble for measurement. In order to guarantee the dynamc power equalzaton (current sharng) among the paralleled sub-converters, the th reactor current can be represented by the followng expresson. 6 = ( ); =, 2,3,4,5,6 L n- = L (6) To satsfy the above condtons, the dscontnuous control law d cs of the th sub-converter must be swtched wth a sldng surface, and the sldng surface should nclude the nformaton about the current of the th paralleled sub-converter. It s more convenent to use the system descrpton, whch nvolves the output error and ts dervatve, that s, x = uo Uo (7) dx Uo C x2 = = = dt dt C Consderng state varables x and x 2, and a contnuous conducton mode (CCM) operaton, the system equatons can be wrtten as x = x2 (8) x x2 x2 = ( U ) Uo u LC RC LC where u s the dscontnuous nput, u= means reverse conducton swtch s swtched ON, and u=0 means swtched OFF. Rewrte the above equaton n state space form. x = Ax Bu D (9) where 0 0 0 A =, B, D = = U Uo LC RC LC LC Practcally, the dampng factor of ths second-order system s less than, resultng n complex conugate egenvalues wth negatve real part. The phase traectores correspondng to the substructure u= are shown n Fg.8a for dfferent values of the ntal condtons. The equlbrum pont for ths substructure s 2eq 0 x = U U, eq o x =. Wth u=0 the correspondng phase traectores are reported n Fg.8b and the equlbrum pont for ths second substructure s xeq = U o, x 2eq = 0. The boundary of ths regon can be derved from the constrant L =0 and s gven by the equaton: o x = 2 x RC U (20) RC whch corresponds to the straght lne wth a E-ISSN: 2224-266X 267 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang negatve slope equal to /RC and passng through the pont ( U, 0) shown n dashed lne n Fg.8b. o In the same fgure, the lne x = U s also o drawn, whch defnes another physcally naccessble regon of the phase-plane, namely the regon n whch u o <0. the same tme, at least n a small regon around the system equlbrum pont. By usng ths control law, on both sdes of the sldng lne, the phase traectores of the correspondng substructures are drected toward the sldng lne. From Eq. (23), t s easy to see that the output voltage dynamcs n sldng mode s smply gven by a frst-order system wth tme constant equal to /c. Typcal waveforms when c =0.8/RC are shown n Fg. 9. Fg.8 (a) Phase traectores correspondng to u=; (b)phase traectores correspondng to u=0; (c)subsystem traectores and sldng lne [4, 7, 8] 3.5 Selecton of the Sldng Lne It s convenent to select the sldng surface as a lnear combnaton of the state varables, because the real control system s smple to desgn and equvalent control can be used to descrbe the system dynamc n sldng mode. Thus, sldng surface s gven T ( x) cx x2 C x 0 σ = = = (2) where C T =[c, ] s the vector of sldng surface coeffcents, and coeffcent c 2 was set to wthout loss of generalty. As shown n Fg.8c, ths equaton descrbes a lne n the phase-plane, whch passes through the orgn, representng the stable operatng pont of the converter (zero output voltage error and ts dervatve). ( x x) 0, (22) σ = ( x) cx x 0 σ = = (23) whch completely descrbes the system dynamc n sldng mode. If the exstence and reachng condtons of the sldng mode are satsfed, a stable system wll be obtaned by choosng a postve c. Fg.8c reveals the great potentaltes of the phase-plane representaton for second-order systems. When choosng the followng control law: 0 for σ ( x) > 0 u = (24) for σ ( x) < 0 the exstence and reachng condtons are satsfed at Fg.9 (a) Phase traectores for two dfferent ntal condtons (c = 0.8/RC); (b) Tme responses of normalzed output voltage u on and output current LN (c = 0.8/RC) (ntal RP n P) The ncrease of c value causes a reducton of sldng-mode exstence regon. The sldng lne coeffcent c also determnes the system dynamc response n sldng mode, snce the system dynamc response results are of frst order wth a tme constant τ=/c. Therefore, when τ<rc, the resultant hgh response speed wll narrow the exstence regon of the sldng mode. Ths can cause overshoots and rngng durng transents. 4 Prncples of SMC [4, 7, 8] 4. Current Lmtaton Accordng to reference [4, 7, 8], a fast output voltage dynamc causes overshoot of the nductor current L. The frst part of the transent response depends on the system parameters, and only when the system representatve pont (RP) hts the sldng lne at a pont belongng to the exstence regon, the system dynamc s dctated by the sldng equaton. For the buck DC-DC converter, t s actually dependent not the converter parameters but the sldng coeffcent c. The large nductor current brngs about lots of shortcomngs, ncludng magnetc core saturaton, over-current protecton, and over hgh surgng E-ISSN: 2224-266X 268 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang voltage, etc. Current lmtaton can be easly ncorporated nto the sldng-mode controller by means of proper modfcaton of the sldng lne. For buck DC-DC converter, current lmtaton can be mplemented by forcng the system RP on the new lne: ILmax Uo x2 = x (25) RC C RC Thus, the global sldng lne conssts of two peces: σ '( x) RC C RC c x x ILmax U o x x2- = 2 (26) The phase plane traectores for a buck DC-DC converter wth nductor current lmtaton and wth c =2/RC are shown n Fg.0, and the correspondng normalzed nductor current transent behavor s shown n Fg.0. It s nterestng to note that Eq. (26) gves an explanaton of why the fastest response wthout overshoots s obtaned for c =/RC. In fact, f c =/RC and I Lmax = U /R, the two peces of the o sldng lne σ become a sngle lne and thus the nductor current reaches ts steady-state valueu /R, wthout overshoot. u= 0 x2 u=0 x LN Lmax LO o range of varaton becomes too hgh. One possble soluton to the problem s to use a varable hysteress band. For example, to use a PLL (phase locked loop). Another smple approach s to nect a sutable constant frequency sgnal w nto the sldng functon as shown n Fg. [0]. Fg. Reduced-order sldng-mode controller wth nductor current lmtaton, constant swtchng frequency, and wthout output voltage steady-state error In the steady state, f the ampltude of w s predomnant n σ f, then a commutaton occurs at any cycle of w. Ths also allows converter synchronzed to an external trgger. Instead, under dynamc condtons, error terms x and x u ncrease, w s overrdden, and the system retans the excellent dynamc response of the sldng mode. Smulated waveforms of ramp w, and σ PI, σ f sgnals are reported n Fg.2. (x)=0 Fg.0 (a) Phase traectores for a buck DC-DC converter wth nductor current lmtaton (c=2/rc); (b) Tme responses of normalzed output current LN (c =2/RC) 0 W 0 PI 0 S r S e [4, 7, 8] 4.2 Sldng-Mode Control Implementaton Compared wth the current control, some aspects of the sldng mode approach should be mproved. The frst problem s that the swtchng frequency depends on the rate of change of functon σ and on the ampltude of the hysteress band. Snce σ s a lnear combnaton of state varable errors, t depends on actually the produced currents and voltages, and ts behavor may devate from predcton. Ths system wll be out of control, f the f 0 Fg. 2 Schematc waveforms of ramp w, and σpi, σf sgnals But the ampltude of ramp sgnal w s worthy of further consderaton. When selected, the slope of functon σ PI and the ampltude of hysteress band should be determned so that functon σ f hts the lower part of the hysteress band at the end of the ramp, as can cause the commutaton. From the analyss of the waveform shown n Fg.2, we can fnd that the slope S e of the external ramp must satsfy the followng nequalty. E-ISSN: 2224-266X 269 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang B Se > S (27) r δ T where B represents the hysteress band ampltude and S r s the slope of functon σ PI durng the swtchng-on tme. Note that, n the presence of an external ramp, sgnal σ PI must have a nonzero average value to accommodate the desred converter duty cycle (see Fg.2). Trangular dsturbng sgnal w s not the only waveform that can be used. Pulse sgnal can also be employed alternatvely [6]. s The second problem rests wth steady state error of the output voltage. When the nductor current reference comes from a low-pass flter, naturally the current error leads to zero average value n steady state. Therefore, f the sldng functon has nonzero average value, due to the hysteretc control or due to the added ramp sgnal w, the steady-state output voltage error certanly appears. Ths problem can be solved by ntroducng a PI regulaton on sldng functon to elmnate ts DC value (see Fg.). In practce, the ntegral acton of ths regulator s enabled only when the system s on the sldng surface. In ths way, the system behavor durng large transents sn t affected, when σ can be far from zero, stll mantanng the large-sgnal dynamc characterstcs of sldng-mode control. A general-purpose sldng-mode controller scheme that ncludes the aforementoned mprovements and a possble mplementaton of current lmtaton by means of another hysteretc comparator and an AND port, s shown n Fg.. 4.3 SMC of paralleled n-buck DC-DC converters [4, 7, 8] The new sldng mode surface s defned n functon of <u o, c, L > and expressed by: σ ( ev, ev, e ) = C αev βe (28) C The dscontnuous control law s defned by: E; f σ<0 v = (29) 0; f σ>0 where v s the th component of control; σ s the th component of the n sldng surfaces; e s the th current error defned by n e = ( n ) L Lk (30) n k = k where =,2,, n; n s the number of the paralleled buck DC-DC converter; β >0, a constant gan. Fg.3 shows the block dagram of the SMC n strategy, where ref = Lk n. k= Fg.3 Block dagram of SMC over sx paralleled buck DC-DC converter The goal of VSS control through sldng mode scheme s to force the system to reach a prescrbed deal sldng surface. The exstence condton for a sldng mode mples that the state traectores on ether sde of the surface must be drected towards the sldng surface s =0 at least n an nfntesmal neghborhood. Mathematcally the exstence condton for sldng modes to occur on th surface may be stated by [4]: Lmσ < 0 < Lmσ σ 0 σ 0 or by the equvalent nequalty: Lmσ σ < 0 σ 0 (3) (32) The condton (33) s referred as the exstence condton for sldng regme on the surface σ =0. 4.4 DESIGN of SMC [4, 7, 8] The control desgn s obtaned from the above equatons and nequalty (27). The procedure of desgn s summarzed as follows: # Step : σ calculaton for =,2,...,n: E-ISSN: 2224-266X 270 Volume 4, 205
N A B C A B C Vabc Iabc a b c Dscrete, Ts = e-006 s. a3 b3 c3 a3 b3 c3 A B C A- B- C- A B C A- B- C- A B C A B C - - IGBT S3 g m C E FRD IGBT S6 g C m E FRD4 L3 L6 L7 - R5 C3 RL R0 C4 R4 20 9.6 7.5 - Mux u()u(2)u(3) s - v PWM L uo PWM2 L2 WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS X-Jun Yang, Chen Yao, Nng-Yun Zhang σ = C α e β e C v #Step 2: exstence condton calculaton. (33) From equaton (33), the condtons for sldng mode to occur on ether sde of s = 0 are: If σ < 0, then υ = E, =, 2,, n; If σ > 0, then υ = 0, =, 2,, n. The deal SMC operates at an nfnte swtchng frequency that can t be mplemented n practcal systems. In order to reduce the swtchng frequency, several reducton methods, well- known n power electroncs, can be employed. Among them the hysteress control approach [4] s used. The acton controller s gven by: E f σ < σ v = (34) 0 f σ > σ where σ establshes a symmetrcal neghborhood to the swtchng surface and lmts the maxmum swtchng frequency. 5 Smulaton and Implementaton 5. Smulaton of SMC based PCM power supply Accordng to power stage (n Fg. 3 and 4), and also after ntegratng SMC (n Fg.7, and 3), two new SMCs are desgned wth the followng features: current lmtaton (as rated current), power equlbrum or current sharng, constant swtchng frequency (0kHz), output current steady-state error, phase-shft drvng and PID control. At frst the SMCs are smulated thoroughly by means of MATLAB/ SIMULINK accordng to the practcal workng condtons of PCM, whch s shown n Fg.4, ncludng power stage and control stage wth ode23(mod. stff/trapezodal) for solver, dscrete for smulaton type, e-6s for sample tme, and K P =0., K I =0. for the PI regulator. c PWM L uo L uo L2 PWM2 LL2 L2 50 -K- -K- Xu (a) Power stage (b) Control stage (Based on Fg.7 and 3).0260/2 50 -K- -K- 0.000s.0260/2 Xu (c) Control stage 2 (Based on Fg. and 3) Fg.4 Smulaton crcut of SMC on two paralleled buck DC-DC converter The man parameters are lsted below: There-phase nput AC voltage s 380V/50Hz; there-phase output AC voltage of the step-down transformer wth two separated wndng s 220V/50Hz, Y/Y ; For the buck DC-DC sub-converter, the rated output current s 30A, and the rated output voltage s 50V; the peak to peak current rpple s less than 5A; the swtchng frequency s 0kHz; the nomnated nductance of reactor s 2.2mH/30A/35T, made of slcon steel. The selected nductances of reactors are.98mh and 2.42mH, respectvely; the mans sde LC flter s characterstc of nductance of 0.5mH/75A and capactance of 2.2µF; the total electrolyss capactor bank s of 4x6800µF/450V. S S X 0.000s.0260/2 0.000s.0260/2 X X S PID(s) S2 PID(s) Consderng plasma cuttng s more complcated, and t s requred to start the buck S PID(s) S PID(s) Sgma PI W Sgma PI W - Sgma PI W Sgma PI W Delta f Delta f HC HC Delta f Delta f HC HC HC HC NOT NOT - AND AND - NOT NOT PWM PWM2 E-ISSN: 2224-266X 27 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang DC-DC converter after power-up, thus the ntal voltage value of electrolyss capactor s set 30V. Under the rated operaton condtons, the waveforms of entre output voltage and output current are shown n Fg.5. Those of output current for the two sub-converters n Fg.6. As we can see that the maxmum output current s lmted, the power dstrbuton s equal between two sub-converters due to the same average current, and output current s bascally free of error n steady-state and wthout obvous overshoot n dynamc-state. In addton, the swtchng frequency s constant, and phase-shft drvng and PID control are employed. The resultant sldng traectory of SMC s shown n Fg.7. The representatve pont reaches the orgn from the ntal condton (-50, 0) through near 7ms after the start of smulaton. Fg.7 Smulated sldng traectory of SMC 5.2 Implementaton of SMC based PCM power supply After theoretcal analyss and smulaton analyss, the fully-dgtal controlled plasma cuttng power supply and complete PCM are developed successvely, ncludng computer numercal controller (CNC), torch heght controller (THC), gnton console, gas console, water cooler, off-vale assembly, and so on. Several PCM sets are put nto test on the market. The desgn specfcatons are the same as those n smulaton on the large, except that the real nductances of reactors are about 2.5mH and 2.30mH, respectvely. The desgn of power crcut s n the form of power module, easy to assemble and dsassemble. One leg module SKM300GB063D s selected as IGBT and FRD, F28335 s used as the kernel controller, whch supports floatng pont operaton. The swtchng part of buck DC-DC converter s shown n Fg.7. Fg.5 Smulated waveforms of entre output voltage and output current for buck DC-DC converter L2 L Fg.7 Power crcut of two-stage paralleled buck DC-DC converter t (5ms/dv) Fg.6 Smulated waveforms of output current for each buck DC-DC converter After long tme repeatedly regulaton, the two paralleled buck DC-DC converter shows a satsfactory expermental results, and the work-pece cuttng qualty s hgh satsfactorly. When operatng at the output current of 260A, the tested waveforms of output voltage and output current for are shown Fg.8, the total peak to peak current rpple s less than 5A, and the steady state error s not large than A. E-ISSN: 2224-266X 272 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang Fg.8 Tested waveforms of output voltage and output current for buck DC-DC converter 6. Conclusons The DC current supply of plasma cuttng machne s the typcal applcaton of hgh power buck DC-DC converter. In order to mantan and mprove the whole power performance of the DC current supply, by usng sldng mode control, the mult-channel nterleaved mult-stage paralleled buck DC-DC converter s nvestgated n the paper, ncludng theoretcal analyss, smulaton analyss and expermental verfcaton. An sngle-channel nterleaved two-stage paralleled buck DC-DC converter s desgned and mplemented wth rated output voltage of 50V and rated output current of 260A, whch s powered by three-phase AC power supply and step-down transformer. Accordng to references [4, 7,8], two new sldng mode control (SMC) are made out and employed together, whch features current lmtaton, power equlbrum, constant swtchng frequency, zero output current steady-state error and phase shft drvng. Consequently, smulaton and expermental results show that the proposed control strategy can make the desgned DC current supply exhbt a good dynamc and statc performance under severe load and dsturbance condtons, and the hgh current precson, good stablty and relablty brng about satsfactory cuttng qualty. Acknowledge The authors acknowledge the Natonal Natural Scence Foundaton of Chna for the supports of key and general proects of natural scence foundaton (U360203, 527756). References [] HPR260XD Man Gas IM (806349). Revsed edton [M], Hypertherm, 2009.. [2] Ruxang Hao, Qongln Zheng, Xaofe You, Wene Guo, Fe Ln, Characterstc analyss and expermental research on hgh-power plasma arc heater power supply, Transacton of Chna Electrotechncal Socety, 2007, Vol. 22, Sup., pp. 78-82. (n Chnese) [3] Jan Wu, Research on hgh-performance plasma cuttng power supply system, A thess n Electrcal Engneerng for the Degree of Master of Engneerng, Nanng Unversty of Aeronautcs and Astronautcs. March, 200. (n Chnese) [4] Yaolng Chen, IGBT nverted plasma cuttng power supply, A thess n Electrcal Engneerng for the Degree of Master of Engneerng. Lanzhou Unversty of Scence and technology, May, 2008. (n Chnese) [5] Baoq Lu, Shanxu Duan, Xun L, An mproved double closed loop control strategy for ar plasma cuttng converter, Przeglad Elektrotechnczny (Electrcal Revew), 202, Vol. 88, No. 2, pp.278-282. [6] V. I. Utkn, Sldng modes and ther applcaton n varable structure systems, MIR publshers, Moscow, 978. [7] U. Itks, Control Systems of Varable Structure, John Wley & Sons, New York, 976. [8] R. Venkataramanan, A. Sabanovc, S. Cúk, Sldng-mode control of DC-to-DC converters, IECON Conf. Proc., 985, pp. 25 258. [9] K. Sr, C. Q. Lee, Current dstrbuton control of converters connected n parallel, Proceedngs of the IEEE IAS, Oct., 990, pp.274-280. [0] C. Q. Lee, K. Sr, T. F. Wu, Dynamc current dstrbuton control of a parallel connecton converter system, IEEE PESC 9, June 99, Cambrdge, MA, pp. 875-88. [] R. Wu, T. Kohama, Y. Kodera, T. Nnomya, F. Ihara, Load-current-sharng control for parallel operaton of DC-to-DC ers, IEEE PESC 93, June 993, pp. 0-07. [2] P. F. Donoso-Garca, P. C. Cortzo, B. R. Menezes, M, A. S. Mendes. Sldng mode control for current dstrbuton n DC-DC converters connected n parallel, IEEE-PESC 96, 996, Italy, pp. 53-58. [3] I. Batarseh, k. Sr, H. Lee, Investgaton of the output droop characterstc of parallel-connected DC-DC converter, IEEE-PESC 94, June 994, Tawan, Chna, pp.342-35. [4] P.F. Donoso-Garca, P.C. Cortzo, B.R. De Menezes, M.A. Severo Mendes, Sldng-mode control for current dstrbuton n parallel-connected DC-DC converters, IEE Proceedngs Electrc Power Applcatons, Vol.45, No.4, Jul 998, pp. 333-338. [5] J. Fernando Slva, Sona S. Paulo, Fxed frequency sldng mode modulator for current mode PWM nverters, IEEE PESC 93, 993, pp. 623 629. [6] Fo. B. J. Cardoso, B. R. Menezes, A. F. Morera, P. C. Cortzo, Analyss of swtchng frequency reducton methods appled to sldng-mode controlled DC-DC E-ISSN: 2224-266X 273 Volume 4, 205
X-Jun Yang, Chen Yao, Nng-Yun Zhang converters, IEEE PESC'92, June 992, pp.403-40. [7] Tmothy L. Skvarenna, The power electroncs handbook. Purdue Unversty, Chapter 8: sldng-mode control of swtched-mode power supply, West Lafayette, Indana, 2002, CRC Press. [8] Mousum Bswal, Control technques for dc-dc buck converter wth mproved performance, A thess submtted n partal fulfllment of the requrements for the degree of master of technology (research), Electrcal Engneerng, Natonal Insttute of Technology Rourkela, March 20. [9] - [20] ---------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------- E-ISSN: 2224-266X 274 Volume 4, 205