Mathematical Analysis of Tubular Linear Motor

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Mathematical analysis of tubular linear motor

Krzysztof Falkowski, Maciej Henzel, Paulina Mazurek

Faculty of Mechatronics and Aerospace, Military University of Technology, Poland

Abstract: The paper deals with mathematical analysis of linear (flaps, rudders, elevator, ailerons). On aircraft are used electric motor. There are characterized modern trends in aviation, electric, pneumatic and hydraulic actuation systems. especially in aircraft actuator field. There are also described Electric actuators ensure operating flexibility of aircraft operation principles of linear motor and transformation from systems by reduction of hydraulic component, limitation of spare parts and tools, minimalize of diagnostic mistakes cylindrical rotary motor to tubular linear motor. There are pre- by built-in test function. Electric data transfer is more sented motion equation of linear motor and formula derivation of practical and ensure flexibility in case of modification [8]. mutual inductances matrix. In three phase power supply mutual Main advantage of electric actuator is reduction of air- inductances are depend on slider linear movement. craft operating costs (reduce fuel costs by minimization aircraft weight and minimalize number of maintenance Keywords: actuator, linear electric motor, More Electric Aircraft services). Linear motors are electro-mechanic converters, which

converse received electric power to mechanical energy. owadays in aviation trends occur explicit tends to use This is alternative solution for hydraulic and pneumatic electric equipment instead of device powered N drives. Linear motor output is linear movement. This by pneumatic and hydraulic systems. This concept are system does not require gear, couplings etc. It have better named “More Electric Aircraft” and give a wide range dynamic parameters (acceleration, velocity, breaking) of opportunity to increase device reliability, vulnerability, and operating reliability than rotary drives. So, the struc- maintainability and flight safety. Thanks to this concept ture of linear motor is simply. That motor consists new aircraft produce less noise and pollution. Their of electric stator and slider with permanent magnets. weight, fuel consumption and utilization costs are re- Linear actuator are modern solution of actuation systems duced. components. They allowed to quick movement and char-

acterize operating precision. In most solutions are used air

bearings, which eliminate friction forces. 1.Introduction In the last years, the aircraft systems are changed 2.Mathematical model ofaircraft by the equivalent electrical systems, e.g. pneumatic anti- actuator with linear electric drive icing system are eliminated by electric heating mats, hy- As an example of linear electric drive will be presented draulic actuators are changed by electric servo-actuators, tubular linear electric motor mechanical control systems are evaluated to fly-by-wire systems. This on-board revolution are named “More Elec- 2.1. Tubular linear motor tric Aircraft” (MEA) technology. Advantages of this solution were integration of distribution system, energy stor- Linear electric motor is an electrical machine, which con- age, actuators and control systems. The MEA technology verts electric energy to mechanic energy of progressive allow to get greater precision, rapidity and reliability movement without any additional elements (gears, cou- of systems. This idea have an effect on increase of mainte- plings, etc.). In turn, tubular motor mostly has cylindrical nance susceptibility, ensure systems flexibility during active surfaces, which primary winding coils are located modification and reduce operation costs and system in perpendicular plane to movement direction. [1] weight. Flat linear motor arise through cross-cut rotary elec- The More Electric Aircraft concept are used in passen- tric motor and flat drop-down packets (active flat surfaces ger aircraft Boeing 787, Airbus A380, multipurpose fighter are separate by flat air gap – fig. 1a–b). The cross-cut is F-35 and in unmanned aircraft Predator and Global made by semi-plane limited by motor pivot Or. While Hawk. There are applied e.g. anti-icing systems, hydraulic tubular motor arise through wind up flat motor around systems, environmental control systems, aircraft engines, perpendicular pivot Ot to rotor motor pivot (tangential electrical power and actuation systems. [4, 8] coordinate of rotary motor become linear coordinate of Actuator is a component of control system, which tubular motor oriented along him pivot, linear coordinate converse control signal to physical process, e.g. movement, oriented along rotary motor pivot become tangential coor- rotation, moment or force. In that system input signal dinate of tubular motor – fig. 1b–c). Air gap of tubular are generated by flight control computer. Whereas output motor is also cylindrical, but moving member move along signals are movement motion signals of aircraft surfaces pivot (linear movement). [1]

508 a) b) b – friction factor, k – spring rate, F – electromagnetic force,

Fzew – external force, d2x/dt2– acceleration, dx/dt – linear velocity, x – linear movement. c) 2.2.1. Electric equation Equation (1) for the three phase system can be expressed by

FV F VFV Fψ V G W G W (3) G W = G W + ψ G W G WG W G W Fig. 1. G W G W Transformation from cylindrical motor to tubular motor; HG XW H XHG XW ψ where: H X a) cylindrical motor, b) linear flat motor, c) tubular linear j – resistance of j-coil. motor Whereas linkage magnetic flux on a, b and c phase can Rys. 1. Ewolucja silnika cylindrycznego w silnik tubowy; a) silnik be presented by [7] cylindryczny, b) silnik liniowy płaski, c) silnik liniowy tu-

bowy Fψ V F VFV Fψ V G W G W (4) ψ = G WG W + ψ G W G WG W G W G W G W ψ HG XWHG XW ψ H X H X where:

Li – self-inductance of i-coil,

Mij –mutual inductance of j-coil located in magnetic flux generated by k-coil of electric circuit.

On equation (4) 1ma, 1mb and 1mc represents linkage magnetic fluxes from permanent magnets [6] F π V Fψ V G τ W Fig. 2. Magnetic flux from permanent magnets in tubular linear G W G π π W ψ =ψ G − W motor G W τ G W G W Hψ X π π (5) Rys. 2. Strumień magnetyczny od magnesów trwałych w tubo- G + W where: G τ W wym silniku liniowym H X 7 – pole pitch. Main components of linear motor are inductor and slider (equivalent of stator and rotor in rotary machine). Based on (4–5) equations linkage magnetic flux deriva- Principle of operation that kind machines based on influ- tive can be expressed by ence between magnetic field generated by inductor wind- Fψ V F V F π V ing and magnetic field derive from magnets located on G W G W G W G W F V τ slider. Range of slider movement is limited by race length. ψ G W π G π π W G W G WG W = − ψ G − W G W G WG W τ G τ W G W (6) G ψ W H XG W π π G + W G W G W G W G W H X H τ X 2.2. Mathematical analysis of tubular linear motor H X Electric equation and motion equation of linear motor, can be express by Eq. (1) and Eq. (2). Finally equation (3) can be presented by

F V = + ψ (1) G W F V F VF V F V G W G W = G WG W + G WG W G W G WG W G WG W G W G WG W G W H X H XH X H XG W (2) G W + + + = H X F π V (7) where: G τ W G W ?(t) – column matrix of linkage fluxes on mu-phases, π π π − ψ G − W R – quadratic diagonal matrix of resistances (rank τ G τ W G π π W of matrix is mu), + HG τ XW u(t) – column matrix of phase voltages with mu-rows,

i(t) – column matrix of phase currents with mu-rows, m – mass of mobile elements, Magnetic flux from permanent magnets 1m can be calcu-

mu – number of phases, lated by

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ψ = μ τ (8) π (15) + δ = ζ − π δ where:

lm – permanent magnets thickness, –air gap length, S –permanent magnet area,

Br –remanent flux density of the permanent magnets

(for neodymium magnets Br=1.3 T).

Resistance of j-coil from equation (7) can be calculated by = ρ (9) where: 4–coil material resistivity (for copper 1.78\10-8 Ωm),

lz – length of j-coil, s – wire section.

Self-inductance of j-coil from equation (7) can be calculat- Fig. 3. Scheme of geometric parameters designations of motor ed by Rys. 3. Schemat oznaczeń parametrów geometrycznych silnika ∂ψ = ∂ (10) 2.2.2. Motion equation of linear tubular motor

Whereas mutual inductance of j-coil located in magnetic Electromagnetic force F from equation (2)for three phase flux generated by k-coil of electric circuit system, which act on slider in direction x is described

∂ψ by [7]: = (11) ∂

C F V F VF V F V Fψ VS where: D T ∂ D G W G WG W G W G WT = + ψ = ?jk – magnetic flux linkage from j-coil, induce by k-coil ∂ D G W G WG W G W G WT G W G WG W G W G W D H X H XH X H X Hψ XT current.[3] E U Magnetic flux of harmonic 1j induction associate F π V G − W (16) F V τ with windings current ij [5] π G π π W ψ G WG− − W G W τ τ G τ W μ τ ζ G W ψ = ζ O = (12) H XG π π W G− + W −τ π δ H τ X

? Whereas magnetic flux linkage jk associate with j-windings generate by k-windings influence Equation (2) for the three phase system allowing of equa-

τ τ tion (16) can be express by μ ψ = O = (13) π −τ δ F π V G − W τ I ζ Fπ π VY F V τ L L G W O B Jσ G + − WZ π G W π π (17) −τ L Hτ XL + + + = ψ G− − W K [ τ G WG τ W G W H X π π W where: G− + HG τ XW i= kckns, =sinv(4/2), 2.2.3. Coefficient evaluate lj – ideal length of slot, Coefficient from chapter 2.2. can be calculate: )0 – vacuum magnetic permeability, − friction factor b: p – number of poles,

kc – Carter`s coefficient (increase air gap coefficient), = (18) kns –saturation coefficient, μ δ v – number of pole harmonic, where: #v– windings coefficient, k-– coefficient of internal friction (for air x – coordinates. k) = 0,000018 Pa·s),

St – friction surface, Where after take into account only fundamental compo- – air gap between parallel friction surfaces, nent v=1 0,5 << (St) ; μ τ − windings coefficient #v , = (14) ζ π δ

510 ζ = ζ ζ ζ (19) On the average Carter’s coefficient is situated in range from 1.05 to 1.1 for half-closed slots and from 1.2 to 1.3

− group coefficient #qv, for opened slots. Whereas saturation coefficient average from 1.05 to 1.6. C π S Parameters of designed tubular linear motor were D T D T presented in tab. 1, whereas calculated parameters of ζ = E U C π S (20) tubular linear motor were shown in tab. 2. D T E U

Tab. 2. Calculated parameters of designed tubular linear motor where: Tab. 2. Obliczone parametry zaprojektowanego liniowego silnika q – number of slots for pole and phase, tubowego Q – number of slots for phase (in three phase windings Value of pa- Q = 2q), Parameter Unit rameter

Ra=Rb=Rc 0.762 Ω − pitch factor #sv, La=Lb=Lc 0.0906 H

C π S Mab= Mba=Mcb=Mbc=Mac=Mca 0.0452 H ζ = D T (21) -3 2 E τ U ZI 0.868\10 T*m where: y – coil span calculated in slots 3.Summary

− bevel factor #skv, Paper presents modern trends in aircraft control system,

C γ S D T especially in electric actuator. Tubular motor (22) ζ = E U is an example of solution developed according to More γ Electric Aircraft concept. where: There are characterized electric actuators, linear motor " – angle of slot bevel relative rotor generating line. and operation principle of tubular linear motor. There are

also presented mathematical model of linear motor. That Tab. 1. Parameters of designed tubular linear motor model was created for first pole harmonic with assumption Tab. 1. Parametry zaprojektowanego liniowego silnika tubowego shaft length is infinite (omission of marginal phenomena) Value of and air gap length is uniform. Parameter Unit parameter Presented mathematical model is base to create simu- Number of phases 3 - lation model, which can be verify in experimental re- Number of coils per search. 240 - phases Slider weight 1.170 kg Acknowledgment Saturation coefficient 1,1 - The work was financed in part from the government sup- Carter’s coefficient 1.075 - port of scientific research for years 2010–2013, under grant Ideal length of slot 295.16 mm No. O N509 165238. Slot opening 4 mm The work was supported by Innovative Economy Number of poles 8 - Programme, No. POIG.02.02.00-14-022/09. Number of slots 12 - Inside diameter of 25.5 mm Bibliography stator 1. Pawluk K., Szczepański W., Linear electric motor, Friction coefficient 0.01 - WNT, Warsaw 1974. (in Polish) Air gap length 0.5 mm 2. Gieras J., Induction linear motor, WNT, Warsaw Slots angle 90 ° 1990. (in Polish) Length of coils 33600 mm 3. Turowski J., Electromagnetic calculation of machines Windings coefficient 0.76 Permanent magnet and electric devices components, WNT, Warsaw 602.88 mm2 area 1982. (in Polish) Permanent magnets 4. Mazurek P., Review of bearingless constructions 6 mm thickness for aviation application, “Advances in chemical and Pole pitch 33 mm mechanical engineering”, Gdańsk 2012 ISBN 978-83- Additional waste 88579-97-4, Volume II, 41–50. 2.6 - coefficient 5. Turowski J., Alternating currents electric machines, Conductor conductivi- Publishing House of Technical University in Łódź, 59770000 S/m ty Łódź 1973. Wire section 0.785 mm2 6. Luis R., Quadrrado J.C., On PM Tubular Linear Winding temperature 40 oC Synchronous Motor Modelling, FS’05 Proceedings of the 6th WSEAS international conference on Fuzzy

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systems, World Scientific and Engineering Academy Paulina Mazurek, MSc and Society (WSEAS), Wisconsin, USA 2005 Paulina Mazurek graduated 7. M. Henzel, K. Falkowski, and P. Mazurek, The math- of Military University of Technology. ematical and simulation model of the bearingless mo- She has worked in Military University tor with permanent magnets, PAR 2/2012, ISSN: of Technology since 2011. Her areas 1427-9126, 450–455, (in Polish). of interest are aircraft control 8. Henzel M., Falkowski K., Żokowski M., The analysis and actuation system and of the control system for the bearingless induction bearingless machines. She is author electric motor, “Journal of Vibroengineering”, ISSN or co-author of many articles about 1392-8716, Volume 14, ISSUE 1, March 2012. new trends in on-board systems, modern control methods and bearingless solutions. Analiza matematyczna tubowego liniowego e-mail: [email protected] silnika elektrycznego

Streszczenie: W artykule przedstawiono analizę matematyczną liniowego silnika elektrycznego. Scharakteryzowano tutaj nowe trendy rozwoju lotniczych układów wykonawczych. Opisano również zasadę działania silnika liniowego. W artykule zaprezen- towano równania ruchu silnika liniowego oraz wyprowadzenie macierzy indukcyjności własnych jego uzwojeń. W przypadku zasilania prądem trójfazowym indukcyjności wzajemne zależą od ruchu biegnika, dlatego przedstawiona analiza jest istotna.

Słowa kluczowe: lotniczy układ wykonawczy, liniowy silnik elektryczny, More Electric Aircrat

Krzysztof Falkowski, PhD Krzysztof Falkowski graduated Mili- tary University of Technology. He received PhD title in 1999. He does research about magnetic suspen- sions, magnetic bearings and bearingless electric motors. He is author or co-author of many articles about magnetic levitation phenomena. He is organizer of Magnetic Suspension Workroom of Aircraft Engines Laboratory in Military Uni- versity of Technology. e-mail: [email protected]

Maciej Henzel, PhD Maciej Henzel graduated of Military University of Technology. He received PhD title in mechanics disci- pline and control systems specializa- tion in 2004. He works in Military University of Technology since 1998. He does research aircraft control and actuation systems, measure- ment systems and bearingless machines. He is author or co-author of many articles about new trends in on-board systems, modern control methods and bearingless drives. e-mail: [email protected]

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