CMP-843

3-D Eddy Current Torque Modeling

Subhra Paul, Walter Bomela and Jonathan Z. Bird

Laboratory for Electromechanical Energy Conversion and Control University of North Carolina at Charlotte, NC 28223, USA

This paper presents an analytic based eddy current torque analysis procedure. The equations are derived using the second order vector potential and the magnetic rotor is modeled using the magnetic charge sheet concept. The formulation enables the damping and stiffness equations to be derived. The equations are numerically computed and the accuracy is compared against experimental torque and power measurements.

Index Terms— eddy currents, torque, Halbach rotor, maglev, second order vector potential.

I. INTRODUCTION HEN a magnetic source moves in the vicinity of a Wconductive plate, time varying magnetic fields induce eddy currents in the plate which in turn interacts with the source magnetic field to create velocity dependent drag and lift force. The generated drag force can be utilized in applications such as eddy current braking [1] and rotor vibration damping [2]. While the lift force can be utilized to (a) (b) provide suspension for high-speed maglev trains [3]. Fig. 2 The (a) x-y and (b) z-y view of the problem region. Electrodynamic maglev suspension systems typically rely on a II. GOVERNING EQUATIONS null-flux coil guideway topology in order to maximize the lift- A schematic showing the relevant problem regions is to-drag ratio [4]. Another way to avoid this drag and achieve shown in Fig. 2. The rotor velocity in the x, y and z directions integrated suspension and propulsion for the vehicle at low as well as rotational speed, ω , is shown. The length, l and cost is to rotate the magnetic source rather than translationally m width, w, of the plate are assumed to be large enough so that moving it above a conductive plate guideway. This all fields are zero at the edges. The plate is assumed to have electrodynamic wheel (EDW) concept is illustrated in Fig. constant conductivity be nonmagnetic and simply connected. 1(a) [5]. In order to create large lift force a flux-focusing

Halbach rotor, as shown in Fig. 1(b), is used. In this approach A. Conductive Region, ΩII the thrust force is dependent on the slip speed, sl, defined as Utilizing the magnetic vector potential ω sl=ωmro-vx where m, ro and vx are mechanical angular BA= ∇ × (1) velocity, outer radius and translational velocity respectively. the eddy current problems can be formulated as [8] Reitz and Davis [6] and later Langerholc [7] developed dA 2A µ σ force and eddy current torque equations for a coil when there ∇ = 0 (2) dt is only translational motion. This paper presents an analytic where σ = conductivity. Solving (2) leads to a solution based eddy current torque, power, damping and stiffness procedure that is complicated due to the need to solve three analysis procedure for the case when a magnetic rotor is scalar terms and also account for the coupling within the rotated and translationally moved above a conductive plate. magnetic flux density components. This derivation complexity The equations are derived using the second order vector can be avoided by using the SOVP, W, defined as [9, 10] potential[8-10] approach and the magnetic rotor is modeled AW= ∇ × = ∇ ×[W yˆ +y ˆ × ∇ W ] (3) using the magnetic charge sheet concept [11]. a b where Wa and Wb are scalars. Only the transverse electric (TE) potential, Wa, is non-zero when solving eddy current problems in a conductive medium that is infinite in the x-z plane [9]. Therefore, the TE potential exists only normal to the conductive plate as illustrated in Fig. 3. The formulation in terms of TE enables one to think of eddy current problems in terms of reflected and transmitted field components [12]. Substituting (3) into (2) yields

(a) (b) dW ∇2W = µ σ a (4) Fig. 1(a) A 2 pole-pair Halbach rotor rotating and translationally moving a 0 dt above an aluminum plate. Isoline plot of the reflected radial flux density and isosurface plot of the eddy current density is shown, (b) An experimental 2 Utilizing the convective derivative pole-pair Halbach rotor with radial and shunt magnets

CMP-843

∞ ∞ dWa∂ W a = +()v ⋅ ∇W (5) III III jξm x jk n z κ mn () y+ h dt∂ t a Wa = ∑ ∑ Cmn e e e (19) m n and assuming a steady state solution such that =−∞ =−∞ where ξ= 2 πm / l (20) −j() Pωm t m Wa(,,,)(,,) x y z t= Wa x y z e (6) allows (4) to be written as kn = 2π n / w (21)  ∂WWW ∂ ∂  2 2 2 2  a a a  βmn = λ + γmn (22) ∇Wa = −µ0 σ jP ωm W a+ v x + vy + vz  (7)  ∂x ∂ y ∂ z  λ= −0.5vy µ0 σ (23) where P=pole-pairs. Substituting (3) into (1) yields 2 2 2  2 2  2 γmn= κ mn − µ0 σso (24) ∂WWWWa  ∂a ∂ a  ∂ a B =xˆ − +yˆ + zˆ , in ΩII (8) 2 2 2  2 2  κ= ξ + k (25) ∂x ∂ y  ∂x ∂ z  ∂z ∂ y mn m n

so= j( Pw m +ξ m v x + k n v z ) (26) B. Nonconductive Regions, ΩI, ΩIII Since the conductivity is zero the governing equation is [13] IV. SOURCE FIELD 2 ∇Wa = 0 , in ΩI, ΩIII (9) In this paper the TE potential due to a Halbach magnetic and the relationship to B simplifies to rotor is calculated directly from the field created by an W  equivalent fictitious magnetic charge cylinder as shown in Fig. B  ∂ a  = ∇  , in ΩI, ΩIII (10)  ∂y   4. The surface charge density on the cylinder is defined as [11] s jP()θo− ω m t As the magnetic scalar potential, φ , is defined as ρms(r o , θ o , t )= 2 B r ( r o ) e (27) B where = −µ0 ∇ φ (11) PPP+1 + 1 2 the magnetic scalar potential is related to the SOVP by s 2Brem P (1+µ r )( r i − r o) r o 1 Br () r = (28) P2 r 2P 2r 2PP r + 1 1 ∂Wa (1+ )[(1 −µr ) i − (1 + µr ) o ] φ = − , in ΩI, ΩIII (12) µ0 ∂y is the Halbach rotor radial flux density [15]. Brem= remanent flux density, µ = magnet relative permeability. The origin of r C. Boundary Conditions the cylindrical charge sheet is located at (xc,yc,zc)=(0,ro+g,0). The continuity conditions in terms of the TE potential are The scalar potential and magnetic flux density created by this [14] charge cylinder at y=0 can be computed from [11] π w WWWs+ r = II , at y = 0 (13) 2 o /2 a a a so ro ρms(,,)r o θ o t φ (,,,)x y z t = dzo dθ o (29) s r II 4πµ ∫ ∫ r ∂WWW ∂ ∂ 0 0−w /2 MA a+ a = a , at y = 0 (14) o ∂y ∂ y ∂ y 2π w /2 r o ρ(,,)r θ t WWII= III , at y = -h (15) Bso(,,,)x y z t = o ms o o r dz dθ (30) a a 4π ∫ ∫ 3 MA o o 0−w /2 rMA ∂WWII ∂ III o a= a , at y = -h (16) where ∂y ∂ y r MA =(x − ro cos)θ o xˆ + ( y − yc − r o sin)(θ o yˆ + z − zo ) zˆ (31) y Cylindrical magnetic Utilizing (12) the source field in terms of the SOVP is charge sheet s Wa 2π wo /2 so ro W r (xc,yc,zc) A (r ,θ ,z ) W(,,,) x y z t = −  ρ(,,)r θ t a . o o o a  ms o o z r 4π ∫ ∫ MA 0−w /2 r . o W II W II o M (x,y,z) a a x .  ln{rMA + ( y − yc − r o sinθ o )} dz o d θ o  (32) III Wa wo The integration with respect to zo is performed analytically Fig. 3 The source, reflected and Fig. 4. Cylindrical magnetic charge transmitted TE potentials. sheet. whereas integration with respect to θo is accomplished numerically. In order to match the modes with the TE III. GENERAL SOLUTION potentials given by (17)-(19), the source field, as given by (32) Applying the separation of variables method to (7) and (9) , is expressed as the following Fourier series within ΩI and noting that the field must decay when moving away from ∞ ∞ S jξ x jk z κ y jP ω t Ws(,,,) x y z t = mn em e n e mn e− m (33) the conductive surface, the solution within each region is a ∑ ∑ 2 m=−∞ n =−∞ κmn ∞ ∞ r I jξm x jk n z− κ mn y where the Fourier coefficients are determined from Wa = ∑ ∑ Cmn e e e (17) m=−∞ n =−∞ 2 w/2 l /2 κ jξ x jk z S mn Wso x z t e−m e − n dxdz ∞ ∞ mn = ∫ ∫ a ( ,0, , ) (34) β y βy j ξ x jk z lw II II mn II − mn λy m n −w/2 − l /2 Wa = ∑ ∑ [Cmn e+ Dmn e] e e e (18) m=−∞ n =−∞ Substituting (17)-(19), (33) into (13)-(16) and solving yields CMP-843

∞ ∞ ∂Rw (0) jµ σ P r Smn jξ x jk z− jPω t mn 0  W(,,,) x y z t = R() y em e n e m (35) = dmn + µ0 σκmn[ κ mnvy+ s o ]coth (βmn h )/ β mn a ∑ ∑ 2 mn ∂ω d 2  m=−∞ n =−∞ κmn m mn where the reflection function, Rmn(y), is +µ σ[ κ v + s ]{1 − hκcsch2( β h )} (47) 0 mn y o mn mn  v s µ0 σ[] κmn y+ o −κ y R() y = e mn (36) ∂Rw (0) µ σκ mn 2 mn 0 mn  2κmn− µ0 σs o + 2 κ mn β mncoth() βmnh = dmn + λ[κmnv y+ s o]()/coth β mnh β mn ∂v d 2  Substituting (35) into (12) and (10), the reflected scalar y mn potential and magnetic flux density are −λh[]() κ v + scsch2 β h  (48) mn y o mn  ∞ ∞ r Rmn() y jξ x jk z− jPω t 22 φ (,,,)x y z t = S em e n e m (37) and dmn = κmn +γmn + 2κmn β mncoth() β mnh (49) ∑ ∑ mn µ κ m=−∞ n =−∞ o mn Utilizing (39) the angular stiffness constant is obtained from ∞ ∞  ξ  m ∂T 1 ∂2U  Br jξm x jk n z− jPωm t em m  (,,,)x y z t = ∑ ∑ e e e− j xˆ + y ˆ kθ = − = − Re   (50) m=−∞ n =−∞  κmn ∂θ ωm ∂t ∂ θ    ρms (x ,0, z , t )= constant  2 kn 2 ∞ ∞ |S | − j zˆ S R() y (38) lwP mn  mn mn or, kθ = Re[Rmn (0)] (51) κmn  ∑ ∑  µ0 m=−∞ n =−∞ κmn Similarly the vertical stiffness can be computed as V. POWER, TORQUE, STIFFNESS AND DAMPING ∂T lwP ∞ ∞ Within the nonconductive regions the fields are governed by k = −em = |SR |2 Im[ (0)] (52) y ∂y µ ∑ ∑ mn mn magnetostatic equations, therefore the complex energy, Um, 0 m=−∞ n =−∞ can be computed from the magnetic charge [16, 17] TABLE I EXPERIMENTAL SETUP PARAMETERS l/2 w /2 1 * r Quantity Value Unit Um = ρms(,0,,)x z t φ (,0,,) x z t dxdz , at y=0 (39) 2 ∫ ∫ Outer radius, ro 26 mm −l/2 − w /2 Inner radius, ri 9.6 mm This enables the total energy to be determined from the Magnetic Width, wo 52 mm rotor Remanent flux density, Brem 1.42 T interaction of the surface magnetic charge with the reflected Relative permeability, µr 1.108 - field due to the eddy currents. For computation purposes it is Pole pairs, P 2 - most convenient to assume that the source magnetic charge is Conductivity, σ 2.459×107 S/m Width, w 77 mm located on the surface of the conductive plate. In order to Conductive Outer radius mm plate 600 ± 0.58 replicate the source field below the magnetic charge sheet the Thickness, h 6.3 mm source charge density must be twice the normal component of Sheets separation 101 mm the source magnetic flux density [11] which is s VI. EXPERIMENTAL RESULTS ρms(,0,,)x z t= 2 By (,0,,) x z t , at y=0 (40) An experimental vehicle setup with four EDWs has been From (10) and (33) one obtains constructed and is shown in Fig. 5-Fig. 6. The experimental ∞ ∞ jξm x jk n z− jPωm t parameters are given in Table I. In the setup the vehicle has ρms(x ,0, z , t )= 2 ∑ ∑ Smn e e e (41) m=−∞ n =−∞ been kept translationally stationary and the conductive plate The power is computed from the time derivative of (39) on the guideway wheel is rotated to simulate translational ∂U  motion, vx. The EDWs are centered over the width of the P = Re m  (42) conductive plate. The brushless dc motor torque and power are em t  ∂ ρ (x ,0, z , t )= constant ms calculated using Tem=KtIa and Pem=KtIaωm respectively where ∞ ∞ 2 Kt=0.0295Nm/A. The current, Ia and rotational speed, ωm, are lwPω |S mn | P = m Im[R (0)] (43) measured using phase current and Hall Effect sensors. Laser em µ ∑ ∑ κ mn 0 m=−∞ n =−∞ mn displacement sensor is used to measure the air-gap, g. and the torque is then just

TPem= em/ω m (44) Using (44) the damping terms are ∞ ∞ 2   ∂T lwP |S mn | ∂R (0) D = −em = − Im mn  (45) θ ∂ω µ ∑ ∑ κ∂ ω  m 0 m=−∞ n =−∞ mn m  ∞ ∞ 2 ∂T lwP |S mn | ∂R (0)  D = −em = − Im mn  (46) y ∂v µ ∑ ∑ κ ∂v  y 0 m=−∞ n =−∞ mn y  where the derivative terms are given by

Fig. 5. Guideway wheel with an in-line gear reducer and DC braking motor. CMP-843

power both measured and computed are shown in Fig. 8. An excellent agreement was obtained. Equations (43)-(44) were calculated using 31 harmonic components. The torque, vertical damping and stiffness terms as a function of translational velocity and slip speed are shown in Fig. 9-Fig. 10.

Fig. 6. Vehicle setup with four laser displacement sensors to measure air-gap 30 0.1 4000 3000 0.05 20 2000

RPM 0 1000 vx(m/s) 10 0 -0.05 0 5 10 15 20 25 30 13 0 -0.1 11 -20 -10 0 10 20 30 40 50 Slip speed (m/s) 9 Fig. 10. ky (kN) as function of translational velocity, vx and slip speed, sl, for 7 (v ,v )=(0,0) m/s Airgap(mm) y z 5 0 5 10 15 20 25 30 ONCLUSION 0.5 VII. C A 3-D analytic based steady-state eddy current torque, 0.4 damping and stiffness equations have been derived using the 0.3 vx(m/s) SOVP formulation. The accuracy of the developed model has 0.2 been validated against experimental results. 0 5 10 15 20 25 30 Time(s) Fig. 7. RPM, airgap and translational velocity profiles of rear right EDW REFERENCES 400 [1] K. Lee and P. Kyihwan, "Modeling eddy currents with boundary conditions by using Coulomb's law and the method of images," IEEE Trans. Magn., vol. 38, pp. 1333-1340, 2002. 200 Analytic (W) [2] Y. Kligerman, O. Gottlieb, and M. S. Darlow, "Nonlinear Vibration of a Measured Rotating System With an Electromagnetic Damper and a Cubic Power Transfer Power 0 Restoring Force," J. Vibration and Control, vol. 4, pp. 131-144, 1998. 0 5 10 15 20 25 30 [3] H. Nakashima, "The superconducting magnet for the Maglev transport 2 Analytic system," IEEE Trans. Magn., vol. 30, pp. 1572-1578, 1994. Measured [4] K. R. Davey, "Designing with null flux coils," IEEE Trans. Magn., vol. 33, pp. 4327-4334, 1997. 1 [5] J. Bird and T. A. Lipo, "Modeling the 3-D Rotational and Translational Motion of a Halbach Rotor Above a Split-Sheet Guideway," IEEE Torque(Nm) Trans. Magn., vol. 45, pp. 3233-3242, 2009. 0 0 5 10 15 20 25 30 [6] J. R. Reitz and L. C. Davis, "Force on a Rectangular Coil Moving above Time(s) a Conducting Slab " J. Appl. Phys, vol. 43, 1972. Fig. 8. Output power and torque comparison for rear right EDW [7] J. Langerholc, "Torques and forces on a moving coil due to eddy 30 currents," J. Appl. Phys, vol. 44, 1973. 1 [8] W. R. Smythe, Static & Dynamic Electricity, 5th ed., 1989. 0.5 [9] T. P. Theodoulidis and E. E. Kriezis, "Impedance evaluation of 20 rectangular coils for eddy current testing of planar media," NDT & E 0 Int., vol. 35, pp. 407-414, 2002.

vx (m/s) 10 -0.5 [10] C. R. I. Emson, J. Simkin, and C. W. Trowbridge, "Further -1 developments in three dimensional eddy current analysis," IEEE Trans. Mag., vol. 21, pp. 2231-2234, 1985. 0 -1.5 [11] S. Paul, D. Bobba, N. Paudel, and J. Z. Bird, "Source Field Modeling in -20 -10 0 10 20 30 40 50 (a) Air Using Magnetic Charge Sheets," IEEE Trans. Magn., vol. 48, pp. 3879-3882, 2012. 30 0.1 [12] Y. Li, T. Theodoulidis, and G. Y. Tian, "Magnetic field-based eddy- current modeling for multilayered specimens," IEEE Trans. Mag., vol. 20 0.05 43, pp. 4010-4015, Nov. 2007. 0 [13] T. P. Theodoulidis and J. R. Bowler, "Eddy current coil interaction with a right-angled conductive wedge," Proc. Roy. Soc. A, vol. 461, pp. 3123- vx (m/s) 10 -0.05 3139, October 8, 2005 2005. [14] T. P. Theodoulidis, N. V. Kantartzis, T. D. Tsiboukis, and E. E. Kriezis, -0.1 0 "Analytical and numerical solution of the eddy-current problem in -20 -10 0 10 20 30 40 50 spherical coordinates based on the second-order vector potential (b) formulation," IEEE Trans. Magn., vol. 33, pp. 2461-2472, 1997. Slip speed(m/s) [15] Z. P. Xia, Z. Q. Zhu, and D. Howe, "Analytical magnetic field analysis Fig. 9. (a) Torque (Nm) and (b) Dy (Ns) as function of the translational of Halbach magnetized permanent-magnet machines," IEEE Trans. velocity, vx and slip speed, sl, for (vy,vz)=(0,0)m/s Magn., vol. 40, pp. 1864-1872, 2004. [16] E. P. Furlani, Permanent magnet and electromechnaical devices The variation in rotational speed, air-gap and translational materials, analysis, and applications. San Diego: Academic Press, 2001. speed are shown in Fig. 7. The corresponding torque and [17] O. D. Jefimenko, Electricity and Magnetism. New York: Meredith Publishing Co., 1966.