Time Domain Transfer Function of the Induction Motor

Transcription

1 Sudies in Engineering and Technology Vol., No. ; Augus 0 ISSN 008 EISSN 006 Published by Redfame Publishing URL: hp://se.redfame.com Time Domain Transfer Funcion of he Inducion Moor N N arsoum Correspondence: N N arsoum, Elecrical Engineering Program, School of Engineering and Informaion Technology, Universiy Malaysia Sabah, UMS sree, oa inabalu 8800 Sabah, Malaysia. nader@ums.edu.my Received: March, 0 Acceped: April, 0 Available online: April, 0 doi:0./se.vi.9 URL: hp://d.doi.org/0./se.vi.9 Absrac Small signal sabiliy of elecrical machines a frequency domain has been shown by oque coefficiens and eigenvalue of moional impedance mari in sae space form. The relaion of damping, synchronizing and oal synchronizing orque coefficiens wih he eigenvalue or he roos of he characerisic equaion of he perurbed machine shows ha he insabiliy occurs a differen modes. Saic mode represened by real roo a over load condiion, and dynamic mode represened by comple roo a he condiion when he oal synchronizing coefficien ehibis zero value wihin he negaive range of he damping orque coefficien. However, small signal insabiliy deails a ime domain are no given in he lieraures. This paper discusses wih figures he ime domain signals of he inducion moor perurbaion variables under huning condiion, and presens he differences observed beween inverse Laplace ransform and Fourier ransform in ime domain response, based on he ransform of he ransfer funcion from he frequency domain, The figures demonsrae and confirm he machine small signal sabiliy performance given in frequency domain. eywords: Laplace, Fourier, Inducion moor, Impulse orque, Perurbaion, Transfer funcion. Inroducion Damping T D, synchronizing T S and oal synchronizing T S orque coefficiens are imporan for small signal sabiliy of elecrical machines because of heir indirec relaion and direc link o he roos of he characerisic equaion of sae space form of he perurbed machines [arsoum N. N 99]. The heorems given in [arsoum, N. N. & Harris, M. R 00] proved ha insabiliy may occur in differen modes. Saic mode represened by real roo a over load condiion a low speed or low frequency, and dynamic mode represened by comple roo a he condiion when he oal synchronizing coefficien ehibis zero value wihin he negaive range of he damping orque coefficien. This insabiliy mode can occur a boh low and high frequency [arsoum N. N 998] depends on machine parameers, paricularly saor resisance, and working condiion of boh consan flu and weakening magneic field. This remark abou he magniude of he ransfer funcion in [arsoum, N. N. & Harris, M. R 00, arsoum, N. N. 00] and is characerisics wih he perurbaion frequency in frequency domain, gives a relaion o sabiliy characerisics in erms of orque coefficiens, damping T D and oal synchronizing T S, in ime domain. I shows he ime response of he perurbaion velociy p he derivaive of he perurbaion displacemen and he relaion o sabiliy and insabiliy of boh real and comple roos [arsoum, N. N. & Harris, M. R 00]. Since he impedance mari of elecrical machines can be represened in hree differen reference frames, saor, roor and synchronous reference frames as well as he rue reference frame [arsoum, N. N 99], he inducion moor under huning condiion is represened in synchronous flu wave reference frame in his paper, and he response of p is observed in his paricular frame. Two ypes of ransformaions are used in his paper, Laplace and Fourier. These will show he similariy in he response of p of he moor in case of sable machine, and he unsimilariy for he unsable machine [arsoum, N. N. & Harris, M. R 00, arsoum, N. N 998]; where Laplace is available o apply on a single operaing condiion, given by he eigenvalue, while Fourier is for evaluaing he response of he ransfer funcion in frequency domain, by considering all he values of from 0 o, for he same operaing condiion [Gibbs, W. J 96]. This maer is discussed in he following secion.. Laplace and Fourier Transforms for Torque Coefficiens In his secion boh ransformaions are applied o obain he response of p for he inducion moor, defined by h in ime domain, for a given uni impulse of orque T m. The ransfer funcion in frequency domain is herefore:

2 Sudies in Engineering and Technology Vol., No. ; 0 Hj = p, which ransformed ino h = p, a T m = uni sep, in ime domain. T m In Laplace ransform, he soluion of h is given by equaion in he appendi, which conains all he eigenvalues for paricular operaing condiion. This is solved in he appendi by means of parial fracion operaion equaion a T m =. This depends on he numerical soluion of he eigenvalue, which are obained from he characerisic equaion, found in he denominaor of equaion in he appendi. I can be seen ha he soluion of h is direcly relaed o he condiion for sabiliy. I is of damped oscillaion for sable machine, increasing wih oscillaion if he machine has an unsable comple eigenvalue, and increasing eponenially if i has an unsable real eigenvalue [Chen, W. H. 96]; while he reain eigenvalue remain sable. These are shown in all of Figsa, bu he behavior is differen in case of applying Fourier ransform shown in all Figs b. Fourier ransform has been defined by [Heading, J 969, Chen, W. H. 96] as: Hj = j he d and h = j Hj e d 0 which can be modified by runcaing range a [Chen, W. H. 96] as follows: In, needs o be sufficienly high h = 0 sin j Hj e d 0, where is he smalles ime sep of ineres, as i discussed in [Heading, J 969, Chen, W. H. 96], so ha he inegraion of Hj needs a sufficienly high value, for converging h0. The way of doing his is o ge he Gibbs oscillaion problem [Gibbs, W. J 96] well away from he main swing frequency, as i appears in Fig8a. The ransformaion equaion has been esablished in [Chen, W. H. 96] for all 0 and used in his resul o apply on: in real and imaginary pars, or: TS jtd Hj T jt T T S D S D Hj S D T T an T T S D in magniude and angle,. For inducion moor, i can be appreciaed from he characerisics of he magniude of ransfer funcion TF = Hj in [arsoum, N. N. & Harris, M. R 00, arsoum, N. N. 00, arsoum, N. N.99], ha if he machine is sable, hen Fourier h is similar o Laplace h when he dominan eigenvalue is lighly damped. I should be noiced ha Laplace ransform is obained by he eigenvalue, i.e. for a single value of, which is he frequency of oscillaion of he dominan roo; while Fourier is concerned wih all values beween 0 and. If he machine works on he sabiliy boundary, where he real par of he dominan roo = 0, hen TF = a = of he same roo of having = 0. Thus, Laplace h will oscillae freely a he sabiliy boundary of he comple roo, bu i is consan a he boundary of he real roo where = 0. While Fourier h does no eis in boh cases, since Hj = a = if comple or a = 0 if real. Similarly, for a machine wih heavily damped pair of eigenvalue, ha Hj 0 as and herefore i conribues very lile o H and consequenly Fourier h0. Noe ha, Hj 0 usually occurs for inducion moor when, so ha h of 0, bu sars o oscillae a high values, only in case of having very high value, and courses Gibbs problem. These purposes are all illusraed in Figs o 8. If he machine is unsable, wheher dynamically in comple roo or saically in real roo [arsoum, N. N. & Harris, M. R 00] or boh ogeher, hen according o he eisence heorems Fourier h ecludes all he unsable eigenvalue and measures T D and T S as he coefficiens of he sum of he remaining sable roos. Obviously, i is rue for Fourier ransform in he highorder sysems, which shows he response for he sable roos only, wihou measuring wha is included on he righ hand side of comple plane wihin he coefficiens of he ransfer funcion, as he eisence

3 Sudies in Engineering and Technology Vol., No. ; 0 heorems sae. Therefore, Fourier h concerns only wih he sable roos, bu if one roo is on he boundary hen Hj does no eis. Figsb show he differences of his mehod corresponding o Figsa. However, his seems unlikely o happen in he secondorder sysem wih a pair of unsable roos. The remaining sable roos are none, and so T S and T D should be in such a way o give Hj = 0 for all. This seems o require T S + jt D = for all values of, which seems impossible. In Figsb, 8b, 8c, is chosen o be per uni pu, and he sep of is 0.00 pu. So he inegraion is of ime sep equals 0. sec. and includes 000 value for each T S and T D. However, in Fig8a, = 0 pu, which is he reason of finding Gibbs oscillaion a high as shown while i is no found in he oher figures 8a, 8b a he same = 000 sec. Figsb show he discree Fourier ransform DFT, in which he inegraion is effecively a summaion of Hj wih an infiniesimal seps in a long range of [Heading, J 969, Chen, W. H. 96] of h and eplain how he characerisics of he coefficiens T D and T S diagnose he sabiliy characerisics in very differen way han he characerisics of Figsa, which are invesigaed by Laplace ransform. This difference appears in h response, when i carries he effecs of he harmonics in ampliude and frequency in some seps of if i is obained by Fourier equaion, while hese harmonics are no involved if i is obained by Laplace equaion, appendi. Harmonic effec appears as some ripples occur wih he fundamenal wave, and can be seen from he cases of heavily damped and real roo on he boundary Figsb, b. In all cases, he magniudes and frequencies are no he same, comparing Laplace fundamenal wih Fourier fundamenal plus harmonics, bu Fourier also shows he cases of sabiliy, insabiliy and boundary of each eigenvalue in differen ways of Laplace. These are shown in all Figsa, b and illusrae ha Fourier h can define all hese condiions, bu does no recognize he insabiliy in h according o an unsable roo. This in fac indicaes ha Fourier is only concerning wih he remaining sable roos in measuring T S and T D coefficiens from = 0 o, as shown in Figsb,6b. Alhough Fourier h shows he case of sabiliy boundary in Figsb, b, which are similar o Figsa, a of Laplace bu are no realy idenical, Fourier funcion in frequency domain Hj a = of he roo on he boundary is infiniy, and Fourier h of ha roo does no eis. u in principle he effec of he roo on he boundary dominaes he soluion of h, since he oher roos are sable and aracive. Thus, he soluion is of consan ampliude of he peak values, as shown in he figures. Figs. show he lighly damping operaion of inducion moor. The parameers in per uni are: R s = 0.0, R r = 0.0, L ls = 0., L lr = 0., M =, S = 0.0, J = 7.0, v = = 0. Figure a. Laplace ransform a uni impuls orque Figure b. Fourier ransform a uni impulse orque

4 Sudies in Engineering and Technology Vol., No. ; 0 Figs. show he heavily damping operaion of inducion moor. The parameers in per uni are: R s = R r = 0.0, L ls = L lr = 0.08, M =, v = 0.09, S = 0.06, J = 6, = 0. Figure a. Laplace ransform a uni impulse orque Figure b. Fourier ransform a uni impulse orque Figs. show he comple roo on sabiliy boundary of inducion moor. The Parameers in per uni are: R s = R r = 0.0, L ls = L lr = 0.08, M =, J = 6, S = 0.06, v = = 0. Figure a. Laplace ransform a uni impuls orque Figure b. Fourier ransform a uni impulse orque

5 Sudies in Engineering and Technology Vol., No. ; 0 Figs. show he real roo on sabiliy boundary of inducion moor The parameers in per uni are: R s = 0.0, R r = 0.0, L ls = L lr = 0., M =., S = 0.07, J = 6.8, v = =.0 Figure a. Laplace ransform a uni impulse orque Figure b. Fourier ransform a uni impulse orque Figs. show he unsable comple roo of inducion moor, The parameers in per uni are: R s = 0.0, R r = 0.0, L ls = L lr = 0., M =, S = 0.00, J =, = 0., v = 0.08 Figure a. Laplace ransform a uni impuls orque Figure b. Fourier ransform a uni impulse orque

6 Sudies in Engineering and Technology Vol., No. ; 0 Figs. 6 show he unsable real roo of inducion moor. The parameers in per uni are: R s = 0.0, R r = 0.0, L ls = L lr = 0., M =., S = 0.07, J = 6.8, =., v =.6 Figure 6a. Laplace ransform a uni impulse orque Figure 6b. Fourier ransform a uni impulse orque Figs. 7 show he unsable real and comple roos ogeher of inducion moor. The parameers in per uni are: R s = 0.0, R r = 0.0, L ls = L lr = 0., M =., S = 0.08, J = 6.8, =.0, v =.6 Figure 7a. Laplace ransform a uni impulse orque 6

7 Sudies in Engineering and Technology Vol., No. ; 0 Figure 7b. Fourier ransform a uni impulse orque Figure 8a. Fourier ransform wih = 0 per uni for sable machine Figure 8b. Fourier ransform wih = per uni for he same case of Fig8a Figure 8c. Fourier ransform wih = for unsable machine, in comple roo 7

8 Sudies in Engineering and Technology Vol., No. ; 0. Conclusion The paper presens some figures on ime domain response of he oupu speed p for an inpu impulse shaf orque of he inducion moor perurbaion variables under huning condiion. I shows he similariy of he response beween wo mehods of ransformaions, Laplace and Fourier in case of small signal sable mode, and he difference on response in wo cases, one in finding an unsable roo, and one in finding a roo on he boundary. This indicaes ha he response of an unsable machine canno be ruly observed by using Fourier ransform. I is, herefore, imporan o pu he machine under es for sabiliy, during he pracical work of design, using he mehod of parameer esimaion wih Fourier ransform. This mus be applied when he machine variables are epressed in synchronous flu wave reference frame no he saor reference frame, since he perurbed variables have a imeinvarian coefficiens. References arsoum, N. N. 99. Equivalen Circuis and Torque Componens in True Variables for he Inducion Moor, AEJ Aleandria engineering journal, Aleandria, Egyp, 0, 8. arsoum, N. N Characerisics and Properies of a Small Signal Elecric Machines Sabiliy, JIE journal of he Insiue of Engineers, India, Elecrical Eng. Division, Calcua, India, 76, 667. arsoum, N. N. 00. Observaion on Perurbaion frequency and Dominan Eigenvalue in he Dynamic Machine Equaion, CCCP conference a Musku, Oman, Feb.60. arsoum, N. N., & Harris, M. R. 00. Theorems of Torque Coefficiens on Sabiliy for Inducion and Relucance Machines, IJEEE inernaional journal of elecric engineering educaion, Mancheser insiue of science and echnology, U, 8, 779. Chen, W. H. 96. The analysis of linear sysems, McGraw Hill com. Inc. Gibbs, W. J. 96. Elecric machine analysis using marices, Piman. Heading, J. 99. Mahemaical mehods in science & engineering, Edward Arnold Ld., London. Appendi The soluion of he oupu perurbaion velociy p in he dynamic machine equaion [arsoum, N. N 998, arsoum, N. N 99] for he inducion moor is h, given by Laplace ransform where h is he response of p in imedomain o a uni impulse of he orque inpu T m. y applying Laplace ransform o he dynamic machine equaion [arsoum, N. N 99] wih T m = and all he iniial values of he perurbaion currens and speed are zero which can be deduced from he moional impedance mari a = 0 where, p = 0 and = 0. Therefore, he Laplace ransform of p is denoed by p, where: p C C C C 0 where is he Laplace symbol, and he coefficiens o and C 0 o C are defined in [arsoum, N. N 998, arsoum, N. N 99] in erms of machine parameers. In paricular operaing condiion, he epression of p can be wrien as follows: p j j j j where, j, j and are he eigenvalue of he inducion moor model [arsoum, N. N 998] a ha operaing poin. Equaion can be splied ino erms by he mehod of he parial fracion as follows: where: p j = X + j = X + j * = X j * = X j * j 8 j * j.

9 Sudies in Engineering and Technology Vol., No. ; 0 9 * denoes he conjugae, and equaion can, herefore, be wrien as: X X p where: y dy c X y by a X y cy d y ay b ] ][ [[ 6 a b 6 c d ] [ ] } { [ y ] [ ] } { [ y Thus, he soluion of p in imedomain is he Laplace inverse of p of equaion, which is equal o h, where: sin cos X e sin cos X e e h. This work is licensed under a Creaive Commons Aribuion.0 License.