Swissmetro: design methods for ironless linear transformer

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1 Swissmeto: design methods fo ionless linea tansfome Nicolas Macabey GESTE Engineeing SA Scientific Pak PSE-C, CH-05 Lausanne, Switzeland Tel (+4) , Fax. (+4) , nicolas.macabey@geste.ch Keywods Swissmeto, contactless enegy tansfe, design method, linea tansfome Abstact Among the vaious contactless electomechanical equipments used in high speed maglev systems, the function of enegy tansfe is, in pactice, not easy to implement. The two main classes of components ae the linea geneatos and the linea tansfomes. The latte have the advantage of being insensitive to the vehicle speed. They also pesent inteesting potential fo the industial applications. Nevetheless, if the ionless linea tansfome has a paticulaly simple stuctue, its design is not tivial. This pape gives some infomation about the methods to use, to design and calculate a multipimay linea tansfome, consideing vaious extenal constains.. Intoduction.. Geneal Maglev systems involve contactless electomechanical devices. This is usual fo populsion, levitation and guidance functions but, in the case of a completely flying maglev vehicle, the enegy tansfe shall also be fictionless. Guaanteeing this function without any influence of the speed of the vehicle is paticulaly difficult and is limited to the linea tansfome appoach. The potential and limits of this appoach has been descibed in a pevious pape []. The pesent aticle deals with the modelling, calculation and design methods fo ionless linea tansfomes..2. Constaints of the envionment Even if the ionless vaiant of linea tansfome is limited by seveal factos (see [] fo details), an ionless specification could be imposed eithe by a lage aigap o by constaints of mass (vehicle). In the domain of high speed systems (maglev tains), classical solutions of enegy tansfe (fiction) ae limited by electomagnetic pollution as well as by the oveall dimensions (Swissmeto). Inteesting vaiants in the field of inductive contactless solutions belong to the domains of linea geneatos and linea tansfomes. In linea tansfome systems, accoding to the specific envionment, the most fequently used solutions ae eithe ionless stuctues o systems with an ion coe located at the level of the seconday winding (the winding in motion). Moeove, the envionment of the system can impose constaints on the dimension of the tansfome, the value of the aigap, the invisibility of the system, the acceptable electomagnetic emission aeas, etc..3. Definition of the main paametes Thee is no univesal method, available in any case, to design an ionless linea tansfome. Accoding to the global envionment of the system, diffeent paametes will have to be selected as pioity holde. These main paametes can be the mass, the efficiency, the pimay voltage, the length of the windings, the volume, etc. This pape develops the methods fo the calculation and pesents stategies fo inductance calculation and multi-pimay achitectue. An analyse of the geometical chaacteistics of the tansfome ensues the choice of the technique and the adequate model of calculation.

2 2. Calculation of mutual inductances The calculation of the mutual inductances is based on the solving of the integal foms of the Maxwell equations. 2.. Neumann s fomula Neumann s fomula is used in the calculation of the mutual inductances which exists between two loops (one tun coils). y a 2 C2 2d dl2 h C dl ε z a d d 2 b b 2 c 2 c Figue Geomety between two loops (one tun coils) espectively cicula and ectangula 2e x This method takes into account both the finite length of the wie and the longitudinal non-unifomity of the linked magnetic field. The geneal fomula is: M = µ dl 2 dl 4 π 2 {} c 2 c Fo two ectangula coils, one fixed and one in motion, the use of Neumann's fomula involves the decomposition into segments of a staight line (the sides and the extemities). The fomula is applied to pai of segments. The coodinates et 2 coespond espectively to the pimay and seconday staight segments of coils. The integation contou-lines l et l2 ae consequently the pimay and seconday coils. These contou-lines ae coveed in the following diection: fo l (pimay): fo l2 (seconday): d l = dx e x + dz e z dx e x dz e z {2} a b c d d l 2 = dx 2 e x + dz2 e z dx 2 e x dz2 e z {3} a 2 b 2 c 2 d 2 Fo the mutual inductance which links the sides a and a2, the tems of equation {}become: = x e x d e z 2 = ( x 2 + ε) e x + h e y e e z {4} 2 = dl = dx e x ( x 2 x + ε) 2 + h 2 + ( d e) 2 {5} dl 2 = dx 2 e x {6}

3 M a a 2 = µ dl 2 dl 4 π 2 c 2 c = µ l l dx 2 dx 4 π ( x 2 x + ε) 2 {7} + h 2 + ( d e) 2 l l The total mutual inductance between the two coils is the sum of the mutual inductances of the segments: M 2 = M a a 2 + M b b 2 + M c c 2 + M d d 2 + M a c 2 + M b d 2 + M c a 2 + M d b 2 {8} 2.2. The flux method The flux method consides the sides of the ectangula coil as longitudinally unifom and infinite. Thus, it applies to the calculation of linea mutual inductance between long ectilinea coils. The basic concept of this method consists in calculating the magnetic field by integating Poisson's vectoial equation outside the wie: H = e z J z {9} Outside the conducto, whee the cuent density is zeo, the magnetic field becomes: H ϕ = I 2 πρ {0} The mutual inductance then becomes: L 2s = µ 0 π 2e 0 d e + x d e + x ( ) 2 + h 2 dx {} 2 e α H secondaie h α x 2 d pimaie Figue 2 Coss-section The wies ae actually of finite length and the magnetic field along them is non-unifom. Close to the conducto (length 2d), oiented by the axe z, the magnetic field is (accoding to Biot-Savat s law): H ( ) = I e 4 πρ ϕ d z ρ 2 + ( d z) 2 + d + z ρ 2 + ( d + z) 2 {2} This fomula shows that the amplitude of the pue azimuthal magnetic field is a function of the length of the conducto (2d). When this length (2d) goes to infinity, {2} simplifies in {0}. Theefoe, the use of the simplified fomula {0} leads to an inaccuacy, which is invesely popotional to the length of the conductos.

4 3. Calculation of the main inductances This section is devoted to the calculation of the main inductances of ectangula coils. Vaious methods ae consideed accoding to thei ability to conside the following aspects: Finite length of the conductos; Modelling with and without extemities; Coupling between tuns of a single winding. 3.. Complete model The fist calculation is based upon the following hypotheses: Calculation of the main inductance of a ectangula coil (one tun coil, N=); The extenal component of the inductance integates the finite chaacteistic of the side of the coil; The intenal component of the inductance is based upon an expession of the magnetic field linked to ectilinea and infinite wie. By Biot-Savat s fomula, the extenal component of a single tun coil main inductance is: L e = µ 0 π {2 + l ( ) ( l ) 2 + ( 2d ) 2 ( l ) ( 2d ) ln + 2 ( ) ln l + ( 2d ) 2 + ( l ) 2 + 2d 2d ( ) ln 2d + l ( )2 + 2d l ( ) 2 + ln l ( l ) 2 + ln 2d ( 2d ) 2 + ln + ( l ) ln ( 2d ) 2 l 2d } This extenal component is a function of the section of the wie ( is the adius). {3} The intenal component of the main inductance is calculated thanks to the magnetic enegy fomulas: L i = µ l {4} 8π Deepe investigations in intenal component of the main inductance show that the hypothesis of infinite length is widely valid. The sole estiction appeas when the diamete of the wie is close to its length. In that vey paticula case the flux is: d ix Φ i = B ds = µ 0 4 π 2 d y S y= d x=0 x 2 + (d y) 2 + d = µ 0 y= d x=0 I x2 2 x d y 4 π 2 x 2 + (d y) 2 + d + y x 2 + (d + y) 2 dx dy d + y x 2 + (d + y) 2 dx dy {5} Finally, the main inductance is the sum of the two components (intenal and extenal): L = L i + L e {6}

5 3.2. Linea main inductance When the geomety of the coils is such that the value of the length is way lage to the value of the width, then the linea appoximation is valid. c x 2 d Figue 3 Coss section of a linea main inductance The well known associated fomula is: L = N 2 µ 0 π 2 + ln 2 d {7} c 3.3. Main inductance of a N tuns ectangula coil The two ways to calculate the main inductance ae: To add the sum of the main inductances linked to each tun to the sum of the mutual inductances between tuns of a single winding; To detemine the position of an equivalent cental tun and to multiply its main inductance value (L eq ) by the squae of N, the numbe of tuns. N N L bob = L i + M i, j L bob = N 2 L eq {8} i= i, j = The atio of the winding section vesus the oveall dimension is elevant to choose a calculation method. 4. Results: sensitivity analysis 4.. Mutual inductance Finite length The fist esult illustates the influence of the finite length of the coil. Fo a pimay coil of one mete long and a seconday length, which vaies fom 0. m to 0 m, the diffeence between the flux and Neumann s methods is (width = 0.3 m) given in the following figue 2 Difféence [%] l2 [m] Figue 4 Diffeence the flux and Neumann s methods

6 The maximum eo occus when the lengths of the coils ae equal. Thus, when the diffeence between the lengths inceases, the eo is educed. A naow stuctue will impose a geate length diffeence between the coils in ode to educe the eo of the flux method. Extemities The table below illustates the influence of consideing the extemities of the coils (2 coils of one mete long, 0.3 m in width, aigap 5 mm, pefectly ovelapped): method Flux_sides Flux_sides + extemities Neum_sides Neum_sides + extemities Flux2D_ sides Measued value L2 [mh], Nomed eo [%] Table 5 Diffeence between theoy and measues, with and without extemities Conclusion The flux method can be used when both the diffeence between pimaies and seconday is impotant (facto of 3) and the diffeence between the length and the width exceed 2. It is also elevant when the pimay and seconday lengths ae simila and the atio between length and width supass 4. When the stuctue of the windings is long, the extemities ae impotant in Neumann s fomula if the atio between the pimay and seconday lengths is infeio to 5 (the extemities ae negligible beyond this value). In the flux method to conside the extemities epesents a souce of eo in most cases Main inductances Fist of all, it is impotant to notice that, in the case of the main inductance, it is difficult to establish geneal ules due to the lage numbe of factos, to thei imbication and to thei possible mutual eo compensation. Table 6 shows fo a coil ( m long, 0.3 m width, wies adius of.75 mm and N=4), the value of the main inductance calculated by vaious methods, as well as measued. inductance [mh] Nomed [%] eo Llin Leq L (tun by tun) Lmeasued Table 6 Compaison between theoetical and measued values The sophisticated method, consisting in consideing each tun (tun by tun), yields a high level of accuacy compaed with the appoach based on an equivalent tun centeed in the winding. The good pefomance of the simplest method (linea inductance L lin ) is due to an eo compensation effect (neglect of the extemities and equivalent tun). With a ough estimate, the simple linea appoach is valid fo long stuctues and a low numbe of tuns. When both the winding geomety is close to a squae and the numbe of tuns is athe high, the sophisticated method gives excellent esults (calculations tun by tun, application of Biot-Savat s fomula fo both intenal and extenal components of the inductance; calculation of the mutual tem by the Neumann s fomula). Some configuation could cate poblems in tems of numeical calculations (convegence difficulties). Moe complex decision pocedues and tees ae given in [2].

7 5. Design and calculation of a multi-pimay linea tansfome 5.. The tansfome model Ref [] intoduces a multi-pimay tansfome model based on voltage equations. It also shows the conditions when the mutual inductances between pimay can be neglected. It also demonstates the lack of sensitivity of linea tansfomes vesus speed. The powe distibution between the vaious pimaies is illustated in [3]. To compensate the eactive powe o to tune the cicuity fo a HF supply by fequency convete, it is necessay to connect an input capacito in paallel with the two pimaies (see Fig. 7). C 2 capacito allows an exchange of eactive powe (magnetisation of the tansfome). L C 2 I 2 I in I L 22 U 2 U C L R ch Figue 7 Electic scheme of the multi-pimay tansfome The expession fo the input impedance Zin is: Z in = Z eq + Z C {9} whee Zeq=Req+jXeq is the impedance coesponding to two pimaies and the seconday efeed to the pimaies; and, Z C= jx C is the impedance of the capacito C. C is calculated so that the eactance X C the imaginay pat of Zin cancel each othe X C = R 2 2 eq + X eq {20} X eq Ref [] analyses the influence of espectively the fequency, the type of wie, the aigap, the capacities at the pimay and seconday, the position of the seconday, the speed, the coupling between contiguous pimaies and the numbe of tuns CAD softwae and compaison theoy - measuement Based upon the specifications of the seconday, the CAD softwae enables to design the tansfome by calculating the pimay main values (voltage, cuent, efficiency, powe facto). It also evaluates the sensitivity of the tansfome chaacteistics to the main paametes (geometical, electical,...). The paametic cuves showed below illustate the esults obtained on an enegy tansfe test bench of low powe (two m long pimaies, a 0.9 m long seconday, width of the tansfome 0.3 m). The validation of the method has been done on a low voltage and high fequency (00 khz) tansfome bench designed fo electic vehicles application. The stuctue of the test bench is constituted by two contiguous pimaies connected in paallel to a fequency convete. The seconday winding, connected though a ectifie to a load esistance, slides ove the pimaies.

8 Fo a tansfome with the following paametes: l(a) = l(b) = m, δ = 55 mm, width = 300 mm (δ is the aigap) N=4, N2=4 U=350 V, U2=25 V The calculated and measued values of the powe as a function of seconday capacito fo educed voltage supply (dc 50 V) ae, fo Litz wies: P [W] P_mes P2_mes P_calc P2_calc C2 [nf] Figue 8 P=f(C 2 ) The theoetical and measued evolution of the seconday voltage vesus the aigap is given below: U2 [V] U2_calc [V] U2_mes [V] Figue 9 Seconday voltage vesus aigap value The concodance between the chaacteistics calculated by the pogam and measued shows that these calculation methods ae valid and the esults puts to light the inteesting potential of this enegy tansfe technique. The best configuation leads to an efficiency of 94% fo the tansfome and 95% fo the fequency convete. This coesponds to a global efficiency close to 89%. delta [mm]. 6. Conclusion This pape descibes the best way to model and calculate a ionless linea tansfome compising vaious pimay windings. This method and its CAD softwae have been successfully applied in vaious industial and tanspot applications ove the last yeas. The linea tansfome fo the Swissmeto application was designed by these method and tools. [] N. Macabey, A. Cassat, Swissmeto and othe applications Potentials and limits of enegy tansfe system by linea tansfome, Maglev 2000, Rio. [2] N. Macabey, Alimentation et guidage linéaie sans contact, Ph D Thesis, Swiss Fedeal Institute of Technology, Lausanne, 998 [3] M. Jufe, N. Macabey, M. Peottet, Modelling and test of contactless inductive enegy tansmission, Mathematics and computes in simulation 46, published by Elsevie Science B.V., 998