ELECTRONICS AND ELECTRICAL ENGINEERING ISSN 139 115 011. No. (108) ELEKTRONIKA IR ELEKTROTECHNIKA ELECTRICAL ENGINEERING T 190 ELEKTROS INŽINERIJA Generalized Mahemaical Model of Conrolled Linear Oscillaing Mecharonic Device A. Senulis D. Eidukas Deparmen of Elecronics Engineering Kaunas Universiy of Technology LT-51368 Kaunas Lihuania phone: +370 614 834 68 e-mail: audriussenulis@yahoo.com E. Guseinoviene Deparmen of Elecrical Engineering 915 +370 46 39 89 54 e-mail: eleonora.guseinoviene@ku.com Inroducion The generalized mahemaical model of conrolled linear oscillaing device aspecs presened in his paper allows analyzing he sysem as a linearized auomaic conrol sysem evaluaing wo-mass mechanical sysem assuming he equivalen linear load and nonlinear elecrical sysem elemens. There are several ways of building mahemaical models for such devices and here conrol sysems: Using magneic field equaions [1 ]; Analyzing he conrol of oscillaion coordinae in real ime [3]; Presening a differenial forms of he sysem bu no all he variables are ime-dependen [3 4 6]; Evaluaing he model by heir supply volage usage and heir effec [4]; The usage of auomaic conrol sysems in elecropneumaic sysems [5]; Mahemaical models by analyzing he magneical circui using finie elemens [ 7 9]. Modeling of faul analysis of he moving par of he mecharonic device [10]; There are a lo of presenaions of he separae pars or more complex models which involve one or several pars of he physical model. The paper presens concep of building linearized mahemaical model for such devices. General sysem overview and main assumpions The analysis of he conrolled double-sided linear oscillaing mecharonic device invesigaes he double mass double-sided springless linear oscillaing elecrical moor-compressor (Fig. 1). The invesigaion of he sysem could be spli in hree main subsysems analysis: Elecrical subsysem analysis; Conrol subsysem analysis; Mechanical subsysem analysis. Fig. 1. Block diagram of linear oscillaing mecharonic device wih conrol sysem srucure as a conrolled double-sided springless linear oscillaing elecrical moor-compressor The sysem mahemaical model could be realized by using he mahemaical model of such mecharonic device. Due o he oscillaing origin of he sysem here wo main regimes o be invesigae: a) ransien processes (saring he sysem change of he load or ask signals) b) quasisaionary processes (when he ransien processes are over) which is analyzed in he paper. The assumpions for he mahemaical model building are: All he variables in he model are ime-dependen; The mechanical par is a wo mass sysem wih linear equivalen siffness and damping properies; The analysis of he sysem could be presened by creaion of he auomaic conrol sysem and using ransfer funcion of he subsysems or heir pars; The real elecrical subsysem is no linear (conaining: inducance nonlineariy sauraion which is presened in he paper; hysrisor nonlineariy) bu is convered o he equivalen linear subsysem; The mahemaical model of elecrical subsysem esimaes hese parameers losses in he windings losses in he magneic circui ime-dependen winding inducances and muual inducance direc volage drop of 55
hyrisors assumed V. The negleced parameers volage drop in he curren feedback resisor r fbk ; The exciing force of he sysem is nonlinear bu he analysis is simplified by using superposiion principal and analyzing he sysem effec from each force effecing harmonic separaely and finally summarizing all he resuls; The oscillaions h() of he sysem is linear and conains 1 s harmonic only and a consan componen (1) which appears only due o he exisence of addiional consan exernal force he oscillaion ampliude is consan in quasi-saionary processes [6 8 11] The assumpions menioned above le s o build he equivalen elecrical circui presened in he Fig. 3. This circui is similar o he presened in papers [13] bu here was no muual inducance esimaion and he inducances were oscillaion-dependen [8 9 1 13]. h( ) H0 H ' m sin( h) (1) here h() oscillaion coordinae; H 0 consan oscillaion par; H m oscillaion ampliude (when H 0 = 0 H m =H m ); h oscillaion phase; oscillaion frequency; The conrol sysem is digial and he conrolled algorihm is based on conrolling he oscillaion of he sysem only by analyzing he oal elecrical sysems harmonics parameers (harmonic ampliudes and harmonics phases relaive o volage phase). The conrol sysem consiss of oal curren feedback harmonic analyzer; A/D and D/A converers microconroller wih conrol algorihm and impulse generaor. Some pars already could be involved in he microconroller archiecure. Finalizing he mahemaical model could be presened in he ime-dependen differenial form or algebraic form and furher usage of Laplace ransformaions and z ransformaions could be applied. The analysis mahemaical model of elecrical subsysem The analysis of he mahemaical model consiss of building an equivalen elecrical circui of he oscillaing mecharonic device. There are several possibiliies of realizing his circui which are presened in Fig.. The figure shows ha here are wo ways o analyze he variaion of he inducances or magneic conduciviy on he basis of oscillaion ampliude h [8 11 1] or ime. The mahemaical model presened here will be build by using he varian shown in Fig. a) and using imedependen variables. a) b) c) d) Fig.. Differen ypes of equivalen winding schemes of linear oscillaing mecharonic device: a) losses in winding and magneic circui oscillaion-dependen or ime-dependen inducance and muual inducance are esimaed; b) muual inducance is negleced; c) losses in magneic circui is negleced; d) muual inducance and losses in magneic circui is negleced Fig. 3. Equivalen elecrical diagram of conrolled double-sided linear oscillaing mecharonic device wih he hysrisor volage converer The primary sysem () build by using Kirchhoff s Laws is presened below: i () i11() i1() A i11() il1() i1() B i1() il() i() C dl ( 1( i ) 1( )) ( ( ) ( )) () 11 11() L dmil u r i d d uthyr1() I dl ( ( i ) ( )) ( ( ) 1( )) () 1 1() L dmil u r i d d uthyr () II dl ( 0 r1i1( ) 1() il1()) dmi ( () L()) III d d dl ( ( i ) ( )) ( ( ) 1( )) 0 ( ) L dmil r i IV d d () here i() oal curren of he circui; i 11 () and i 1 () currens of he each branch of he circui; i L1 () and i L () inducive currens of he each branch; i 1 () and i () currens in braches esimaing he magneic losses; L 1 () L () M() ime-dependen inducance of each circui branch and muual inducance; r 11 and r 1 resisance of each winding; r 1 and r equivalen resisance for esimaing magneic losses; u() supply volage u()=u m sin(); u Thyr1 () and u Thyr () volage drop on hyrisors of each branch. The analysis of he sysem () can be simplified by making hese noices: 56
a) The firs equaion is independen for making ime dependen differenial equaion and can be calculaed separaely; b) The muual inducance effec is negaive. The differenial form of he elecrical subsysem afer rearrangemens would be: d ( ) ( ) ( ) ( ) 1 d ( ) r 11r1 dl 1 L1 L M L il1( ) d d r11 r1 d dm ( ) r i ( ) 1 L ( u( ) uthyr1( )) d r 11 r (3) 1 d ( ) ( ) ( ) ( ) 1 d ( ) dm M L L L il1( ) d d d r ( ) 1r dl r i ( ) L ( u( ) uthyr( )). r1 r d r1 r The equaion sysem (3) in he marix form would be: dx( ) M( ) A( ) x( ) b( ) (4) d here he marices of he equaion (4) are: il1( ) x ( ) ( ) (5) il here L max inducance of winding when h()=h m ; L min inducance of winding when h()=-h m. Assuming he equaion (1) equaions (9) and (10) could be rewrien in ime-dependen form (Fig. 4 b): H 0 ( ) H ' ( ) sin( ) 1( ) 0 sin m h L L K (1) H m H0() H' m() sin( h) L() L0 Ksin. (13) (1) and (13) are suiable for ransien and quasisaionary regimes of he oscillaing mecharonic device and so he ampliudes H 0 () and H m () are ime variables bu when analyzing quasi-saionary hese parameers are consan. The oscillaion ampliude H m is he maximum limied oscillaion value which is consan in any process and H 0 par only exis due he consan exernal force if no presen he par H 0 is negleced. The oscillaion ampliude H m is real oscillaion ampliude which depends on he firing angle of he hyrisors supply volage value and migh be equal or less hen H m. L1 ( ) M ( ) M ( ) ( ) ( ) (6) M L r11r1 ( ) dm ( ) 11 1 ( ) r r d d A (7) dm ( ) r1r dl ( ) d r1 r d u ( ) uthyr1( ). u( ) u ( ) r1 r ( ) 11 r b 1 (8) r Thyr r1 r The analysis of he differenial equaion sysem has several aspecs he nonlineariies of inducances and hyrisors influencing he soluion. The inducance could be presened in he oscillaing ampliude dependen sine form wih consan par [6] (Fig. 4 a in figure he oscillaion ampliude is relaive): h( ) 1 ( ( )) 0 sin L h L K (9) H m h( ) ( ( )) 0 sin L h L K (10) H m here L 0 consan inducance par assumed in oscillaion cener. The coefficien K is equal: Lmax Lmin K (11) sin 5 4 L1() 3 L() 1 a) 0 0 0.01 0.0 0.03 0.04 b) Fig. 4. The graphics of moor winding inducance dependence on: a) oscillaion coordinae; b) ime The oher par of equaion (3) which is nonlinear is he derivaive of he inducances of he windings which equal o: ( ) K H ( ) ' ( ) sin( ) cos 0 H m h d H m dh0( ) dh ' m ( ) sin( h) d d H ' m ( ) cos( h) (14) 57
dl ( ) ( ). (15) d d Analyzing he quasi-saionary process he equaions (1)-(15) could be rewrien: L 1( ) L0 K sink( k( H ' m ) sin( h) (16) L ( ) L0 K sink( k( H ' m ) sin( h) (17) () Kk( H' m ) d cos kh ( 0) kh ( ' m)sin( h) cos( h) (18) dl ( ) Kk( H ' ) m d cosk( k( H ' m ) sin( h) cos( h). (19) The coefficiens k(h 0 ) and k(h m ) represen he relaive consan par of oscillaions H 0 and oscillaing ampliude H m of he oscillaing mecharonic device H ' ( ' ) m H k H m k( H ) 0 0. (0) H m These coefficiens (0) mus obey he rules which esimae he limiaion of oscillaion of he moving par of oscillaing mecharonic device k( H 0 ) k( H ' m ) 1or k( H ' m ) 1 if k( 0. (1) The dependencies (18) and (19) are presened in Fig. 5 b where in Fig. 5 a) he inducance derivaes dependen from he relaive oscillaing ampliude are represened. The coefficiens (0) in Fig. 5 are assumed: k(h 0 ) = 0 and k(h m ) = 1. The reducion of k(h m ) or k(h 0 )+k(h m ) leads o more linear elecrical sysem in he accordance o he inducance variaions of he windings. h ( ()) dlh ( ()) 00 100 0 100 00 1 0.5 0 0.5 1 h () () dl() 1000 500 0 500 1000 0 0.01 0.0 0.03 0.04 a) b) Fig. 5. The dependence of moor winding inducance derivaive on: a) oscillaion coordinae; b) ime The muual inducance and is derivaive which are presened in he equaions () (3) (6) and (7) are no deailed in his paper because of he variey of he consrucions of he magneic sysem of oscillaing mecharonic i would require more invesigaion in he fuure. As i was shown above in he equaions (16)-(17) he ime dependen equaions are nonlinear and heir 58 Laplace ransformaion could be produced only by changing he equaions he equivalen which could be ransformed. The pars of menioned equaions should be replaced are: k sin ( ) sink( k( H ' m )sin( h) () k cos( ) cosk( k( H ' m )sin( h). (3) The equaions () and (3) values according (1) varies in range [-1; 1]. The equaions could be replaced using known rigonomeric funcions expansion o series: sin( ) z 1 1 z z k k cos( ) z 1 0 1 z k k (4) (5) here z he angle or is equaion; k number of series members. The equaions () and (3) afer he analysis and selecion of how many members of he equaions (4) and (5) o ake ino accoun will ake he shape: ksin ( ) k( k( H ' m )sin( h) 3 k( H ) ( ' )sin( ) 1 0 k H 1 m h k k 1 kcos( ) 1 k 0 (6) k( k( H ' )sin( ). 1 m h (7) k Afer he composing of he equaions (16)-(19) and equaions (6) and (7) he new formulas of inducance and derivaives of inducances become linear and he Laplace ransformaion could be realized. The nonlineariy of hryrisors depends on he firing angle of he hyrisors and he dissipaion of he magneic field of he analyzed branch which could be represened by currens i 1i (). If he resisors r i would no be aken ino accoun he represenaive currens would be i Li (). Afer he linearizaion of he hyrisor nonlineariy all he sysem (3) equaions ransformed using Laplace ransformaion. The aspecs of conrol subsysem The conrol subsysem conen was menioned above and more deailed signal disribuion in he feedback is presened in Fig. 6. Fig. 6. Feedback signal disribuion of conrol sysem
The microconroller realizes he conrol algorihm of oscillaion ampliude by using he oal curren i() firs five odd harmonics ampliudes and phases and also he DC par if presen (8) i( ) I0 Imn sinn in (8) n135 here I 0 DC of oal curren; I mn n h oal curren harmonics ampliude; n harmonics order; in -n h oal curren harmonics phase. The 5 h order of he harmonics is analyzed assuming he higher harmonics are very small [13]. The harmonic analyzer would work using FFT and algorihm for he curren phase exracion. The analysis mahemaical model of mechanical subsysem harmonics. The esimaed higher harmonics are 3 rd and 5 h because he oal curren (8) equaion he highes order of he harmonics are also 5 h Fexcie( ) F0 Felm( ) F0 Felm.0 Felm. mn sinn Felm. n (30) n135 here F elm.0 consan par of elecromagneic force; F elm.mn n h elecromagneic force harmonics ampliude; n harmonics order; Felm.n - n h elecromagneic force harmonics phase. The (9) sysem equaion could be rewrien only respecively o he oscillaion coordinaes h 1 () and h () Due o he lineariy of he mechanical sysem differenial equaion sysem could be replaced wih one using Laplace ransformaion: The analysis of he mechanical linear wo-mass sysem is simpler ask han elecrical. The Fig. 7 represens he linear mechanical sysem. m1 p H1( ( R1 R ) ph1( R ph ( ( C1 C ) H1( C H ( 0 m p H ( R ph1( R ph ( C H1( C H ( Fexcie (. (31) Afer analysis of he sysem such ransfer funcions could be exraced: a) H ( ) ( ) 1 p W H 1 H p H( ( R p C) m1 p ( R1 R) p ( C1 C) (3) b) Fig. 7. Equivalen mechanical scheme of double-mass mecharonic device: a deailed b simplified The wo masses sysem represen: 1 saor moving par. Mechanical subsysem parameers are: masses m 1 and m damper properies - siffness C 1 and damping coefficien R 1 equivalen air spring siffness C and damping properies of fricion esimaed by damping coefficien R. Analyzing he simplified equivalen mechanical sysem (Fig. 7 b) he ime-dependen differenial equaion sysem would be: m1a 1( ) ( R1 R) v1( ) Rv( ) ( C1 C) h1 ( ) Ch ( ) 0 ma( ) Rv1 ( ) Rv ( ) Ch1 ( ) Ch ( ) Fexcie( ) (9) here a 1 () a () acceleraions of oscillaions of saor and moving par respecively m/s ; v 1 () v () velociies of oscillaions of saor and moving par respecively m/s; h 1 () h () oscillaion coordinaes of saor and moving par respecively m; F excie () exciing force of he moving par of mecharonic device N. The exciing force in general form consiss of consan exernal force F 0 and elecromagneic force F elm () of he oscillaing mecharonic device. The elecromagneic force also consiss of consan par main and higher odd 59 H ( ) ( ) 1 p WH1 p Fexcie( b' 1 p b' 0 4 3 a4 p a3 p a p a1 p a0 H ( ) ( ) p WH p Fexcie( b p b1 p b 0 4 3 a4 p a3 p a p a1 p a0 (33) (34) WV 1( pwh1( WA1 ( p WH1( (35) here he coefficiens are: a 0 =C 1 C a 1 =C 1 R +R 1 C a =C 1 m +C (m 1 +m )+R 1 R a 3 =R 1 m +R (m 1 +m ) a 4 =m 1 m b 0 =C b 1 =R b 0 =C 1 +C b 1 =R 1 +R b =m 1. The oher ransfer funcions can also be exraced and analyzed in search of ampliude and phase response of oscillaion ampliude of oscillaing mecharonic device. Conclusions The conclusions of his mahemaical invesigaion of he mahemaical model of linear oscillaing mecharonic devices are: The mahemaical model of oscillaing mecharonic device could be spli in hree subsysems o analyze elecrical conrol and mechanical behavior of mecharonic device;
The presened linearizaion of nonlinear moor winding inducances L() and heir derivaives dl()/d allows usage of Laplace ransformaion for elecrical subsysem behavior analysis of mecharonic device; The conrol is based on using oal curren feedback wih harmonic analyzer which exracs values of he ampliudes and phases of he firs five odd harmonics and as well as define he microconroller based conrol algorihm of he sysem; The mechanical subsysem is being assumed linear wo mass sysem and he assumpion allows o analyze ransfer funcions of mechanical subsysem and simplifies modeling of he mecharonic sysem; The linearizaion of hyrisor nonlineariy is a furher research ask. References 1. Dynamic Models of Conrolled Linear Inducion Drives // Elecronics and Elecrical Engineering. Kaunas: Technologija 005. No. 5(61). P. 3 7.. Wang J. Howe D. Lin Z. Design Opimizaion of Shor Sroke Single Phase Tubular Permanen Magne Moor for Refrigeraion // Applicaions IEEE Transacions on Indusrial Elecronics 010. Vol. 57. No. 1. P. 37 334. 3. Brazaiis V. Conrol of Oscillaion Ampliude of Oscillaing Moors // Elecronics and Elecrical Engineering. Kaunas: Technologija 010. No. 3(99). P. 77 8. 4. Janknas V. Eidukas D. Guseinovien E. Cirauas V. Invesigaion of Supply Posibiliies of Mecharonic Acuaor // Elecronics and Elecrical Engineering. Kaunas: Technologija 009. No. 5(93). P. 5 8. 5. V. A. Grigaiis A. Research of Adapive Force Conrol Loop of Elecropneumaic Acing Sysem // Elecronics and Elecrical Engineering. Kaunas: Technologija 007. No. 7(79). P. 7 10. 6. Oscillaion cenre conrol of he oscillaing moor compressor // Elecronics and Elecrical Engineering. Kaunas: Technologija 004. No. 7(56). P. 66 69. 7.. Adequacy of Mahemaical and Physical Model of Oscillaing Mecharonic Device // Elecronics and Elecrical Engineering. Kaunas: Technologija 008. No. 7(87). P. 69 7. 8. Kudarauskas S. Inroducion o Oscillaing Elecrical Machines. 004. P. 183. 9. Heo K. B. Lee H. K. Lee C. W. Hwang M. G. Yoo J. Y. Jeon Y. H. Conrol of a linear compressor // Proc. of Inern. Conference on Compressors and heir Sysems. UK London 003. P. 493 500. 10. Gu F. Shao Y. Hu N. Naid A. Ball A.D. Elecrical moor curren signal analysis using a modified bispecrum for faul diagnosis of downsream mechanical equipmen // Mechanical Sysems and Signal Processing 011. No. 5. P. 360 37. 11. Senulis A. GuseinovienE. Janknas V. UrmonienL. Andziulis A. Didžiokas R. Experimenal Invesigaion of Oscillaion Cener Displacemen of Oscillaing Pulsaing Curren Moor and Springless Compressor Drive // Elecronics and Elecrical Engineering. Kaunas: Technologija 007. No. 7(79). P. 63 66. 1. Compuer Aided Design on Oscillaing Drives // Elecronics and Elecrical Engineering. Kaunas: Technologija 006. No. 6(70). P. 19. 13. Senulis A.. Conrol Possibiliies of Oscillaing Elecrical Moor Using FFT Analysis Daa // JVE Journal of Vibroengineering. Kaunas: Vibromechanika 010. No. 1(1). P. 8 88. Received 010 1 1 A. Senulis D. Eidukas E. Guseinoviene. Generalized Mahemaical Model of Conrolled Linear Oscillaing Mecharonic Device // Elecronics and Elecrical Engineering. Kaunas: Technologija 011. No. (108). P. 55 60. This paper presens he general mahemaical model of double-sided splingless linear oscillaing mecharonic device oscillaing moor-compressor which is supplied by hyrisor volage converer. The model presens possibiliies o evaluae losses of he windings and losses in he magneic circui nonlinear inducances muual inducances special curren feedback using harmonic analysis for he conrol oscillaions no only of he moving par bu also he saor oscillaions. Also using his model such parameers can be calculaed oscillaion cenre drif oscillaing coordinaes and velociies of mover and saor all currens and volages elecromagneic force harmonics of each ime-dependen parameer. Ill. 7 bibl. 13 (in English; absracs in English and Lihuanian). A. Senulis D. Eidukas E. Guseinovien modelis // Elekronika ir elekroechnika. Kaunas: Technologija 011. Nr. (108). P. 55 60. Sraipsnyje apariamas apibendrinasis švyuojamojo kompresorinio variklio a ir sinius indu principas yra harmonine analize o be ir saoriaus švyavimai. Naudojan : koordinaes iek judžiosios dalies iek saoriaus s elekromagnein aip pa kiekvieno paramero harmonikas. Il. 7 bibl. 13 60