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ANSYS 2011中国用户大会优秀论文 Dynamic Demagnetization Computation of Permanent Magnet Motors Using Finite Element Method with Normal Magnetization Curves W. N. Fu and S. L. Ho Abstract—A time-stepping finite element method (FEM) to simulate the transient operations of permanent magnet (PM) motors is presented. It uses only the normal magnetization curves to address the effects of irreversible demagnetization of PM materials. An effective algorithm to implement the methodology with nonlinear iteration is presented. A formulation of Newton-Raphson method using the true Jacobian matrix which is applicable to any type of elements and any order of elements with anisotropy materials is derived. A matrix method is introduced to express space vectors and the FEM formulation can be conveniently deduced. Index Terms—Demagnetization, electric motor, finite element method, magnetic field, nonlinear, permanent magnet. I. INTRODUCTION P (PM) motors have many outstanding advantages including simple mechanical structure, high efficiency and high power density when compared to many industrial drives. But PM materials are usually temperature sensitive and they can be easily demagnetized when operated outside their design limits [1]. Time-stepping finite element method (FEM), coupled with electric circuits and mechanical balance equation, has many salient merits such as flexibility to model complex geometries, standardized programming techniques, high accuracy and allowing the inclusion of high-order harmonics. Hence they are widely used to simulate the dynamic operation of electric motors [2]. In PM motors, if the magnetization curve (B-H curve) of PM materials is nonlinear, the traditional method is to move the B-H curve from the second quadrant to the first quadrant. The coercive force Hc is taken as the excitation in the field equation. However, if the PM is demagnetized, the magnetic flux density will not increase along the original B-H curve; it will increase along the recoil curve. Some researchers use linear models for the PM materials [3-4]. The remanence of the element in the right-hand side of the equation is revised ERMANENT MAGNET after the nonlinear iteration and recalculation of that time step is usually required. In this paper a FEM to address the irreversible demagnetization process of PM is presented. Unlike many hysteresis models, such as Preisach model, no complicated data of the materials, other than the normal B-H curves of the PM materials, are used. In the proposed method, which is used to estimate the possible demagnetization in PM motor as an illustration, the status of the operating point at each element is recorded and the nonlinear iteration is directly controlled by the true Jacobian matrix. Roll back computation is not required and hence the convergence speed is fast. A matrix method is introduced to express the space vectors and the FEM nonlinear formulation can be easily deduced [5]. II. MODEL OF PM MATERIALS The model of PM materials is based on its single-valued B-H curve. No additional data are required. The hard axis is assumed to be linear. The nonlinear B-H behavior is described along the easy axis. It is assumed that the initial operating point of the PM is at H = 0 and B =Br (Br is the remnant flux density). When H reduces, the operating point will move along the B-H curve. When H increases, the operating point will move along the recoil curve. The recoil curve is in parallel with the tangential line to the B-H curve at H = 0 and B =Br (see Fig. 1). Fig. 1. A B-H curve of permanent magnet and its operating point. ANSYS 2011中国用户大会优秀论文 ∂N 2 ⎡ ∂N1 ⎢ ∂y ∂y =⎢ ⎢− ∂N1 − ∂N 2 ∂x ⎣ ∂x ∂N3 ∂y ∂N − 3 ∂x ∂N 4 ∂y ∂N − 4 ∂x ∂N5 ∂y ∂N − 5 ∂x ~ ⎡ A1 ⎤ ⎢~ ⎥ ∂N6 ⎤⎢ A2 ⎥ ~ ~ ∂y ⎥⎢ A3 ⎥ = ∇ × NT A ⎥⎢ ~ ⎥ ∂N6 ⎥⎢ A4 ⎥ − ~⎥ ∂x ⎦⎢ A ⎢ ~5 ⎥ ⎣⎢ A6 ⎦⎥ [ ][ ] In the above two equations, [A~] = [A~ Fig. 2. When the operating point moves below the H axis. 1 When the operating point moves below the H axis (B < 0), the original B-H curve is extended so that the simulation can continue (Fig. 2). However a warning will be given to the user because the B-H curve in the second quadrant, as specified by most commercial manufacturers, is not enough to express the properties of PM fully. Hitherto such situation is not allowed in practical applications and users have to revise their designs. III. IMPLEMENTATION IN TIME-STEPPING FEM A. Basic FEM Formulation In order to deduce the FEM formulation, in this paper, a space vector is expressed in a matrix. For example, the magnetic flux density B is expressed in matrix form as: T (1) B = Bx iˆ + By ˆj = [Bx By ] The basic field equation in the regions of air, iron cores, solid conductors and PMs is: ∂A σ ˆ (2) ∇ × (ν∇ × A) + σ − Vb k = J + ∇ × H c ∂t l where, A is the magnetic vector potential, ν is the reluctivity of material and σ is its conductivity, l is the depth of the model in the z-direction, Vb is the voltage of the conductor. By applying the Galerkin method and using the shape function N as the weighting function, ∫∫ (∇ × A) T Ω ⎛ ∂A σ ˆ ⎞ ⋅ν∇ × NdΩ + ∫∫ ⎜ σ − Vb k ⎟ ⋅ NdΩ Ω ⎝ ∂t l ⎠ = ∫∫ J ⋅ NdΩ + ∫∫ (∇ × H c ) ⋅ NdΩ Ω (3) Ω here N is a vector. In two-dimensional (2-D) FEM, supposing the solution domain is on the x-y plane, A and N only have the components in the z direction, (4) A = Akˆ (5) N = Nkˆ The scalar variables A and N are the components in the z direction, respectively. 6 ~ A = ∑ N k ( x, y )Ak = [N1 k =1 [ ~ ⋅ A1 ⎡ ∂A B = ∇× A = ⎢ ⎣ ∂y − N2 ~ A2 ∂A ⎤ ⎥ ∂x ⎦ ~ A3 T N3 ~ A4 N4 ~ A5 N5 ] N6 ] [] ~ T T ~ A6 = [N ] A ~ A3 ~ A4 ] ~ A5 ~ T A6 (8) is the A’s values on each node as an example. Here second order 6-node element is used. For the xy solver: ⎡ i ⎡ ∂A ⎤ j k⎤ ⎢∂ ∂ ∂ ⎥ ⎛ ∂A ∂A ⎞ ⎢ ∂y ⎥ (9) ∇× A = ⎢ j ⎟⎟ = ⎢ ⎥ = ⎜⎜ i − ⎥ A ∂ ∂ ∂ ∂ ∂ ∂ x y z y x ⎠ ⎢− ⎥ ⎢ ⎥ ⎝ 0 Az ⎦⎥ ⎣⎢ ∂x ⎦⎥ ⎣⎢ 0 In the Galerkin method, the weighting function is: T (10) W e = [N1 N 2 N3 N 4 N5 N 6 ] and ⎡ ∂N i ⎤ ⎢ ⎥ ⎡ (∇N i )y ⎤ ⎡(∇ × N i )x ⎤ (11) ∇ × N i = ⎢ ∂y ⎥ = ⎢ ⎥ = ⎢(∇ × N ) ⎥ ∂ N ( ) − ∇ N i i y⎦ i x⎦ ⎣ ⎢− ⎥ ⎣ ⎣⎢ ∂x ⎦⎥ where N i is a vector in the z direction. B. Nonlinear Iteration Formulation PM is characterized for its two distinct directions, the easy axis u and the hard axis v, and the magnetic anisotropy nonlinear iteration formulation needs to be derived. To the FEM problem, the algebraic equation obtained after discretization is [6] (12) [S ] A~ = {P } Its nonlinear iterative formula of the Newton-Raphson method is n (13) [J ]n A~ n +1 − A~ n = {P}n − [S ] A~ Defining ~ (14) f = [S ] A the Jacobian matrix is ⎡ ∂ ( f − P ) ⎤ ⎡ ∂f ⎤ (15) J =⎢ = ~ ~ ⎣ ∂A ⎥⎦ ⎢⎣ ∂A ⎥⎦ From the field equation in the PM region, one has T fi ( A) = ∫∫ ν (B 2 )(∇ × A) ⋅ ∇ × N i dΩ = ∫∫ H T ⋅ ∇ × N i dΩ {} { } {} {} Ω = (6) ~ A2 (7) ∑W k k (Gauss point) Ω ⋅ H T ⋅ ∇ × Ni (16) where Wk is the weighting value at the kth Gauss point. The ~ . Moreover, magnetic vector on node j is denoted as A j ⎡ ∂H T ⎤ (17) Wk ⋅ ⎢ ~ ⋅ ∇ × N i ⎥ k (Gauss point) ⎣⎢ ∂Aj ⎦⎥ To perform element assembly on PM materials, the coordinates are rotated to the u-v coordinates and one has ∂fi ( A) ~ = ∂Aj ∑ ANSYS 2011中国用户大会优秀论文 [ ∂H T ∂ ~ = ~ Hu ∂A j ∂A j Hv ] ′ ν vu ′ ⎤ ν vu′ ⎤ ⎡ν uu T ⎡ν ′ (18) = (∇ × N j ) ⎢ uu ⎥ = (∇N j )v − (∇N j )u ⎢ν ′ ν ′ ⎥ ′ ′ ν ν vv ⎦ vv ⎦ ⎣ uv ⎣ uv Therefore, ′ ⎤ ⎡ ⎤ ⎡ν ′ ν vu ∂f i ( A) Wk ⋅ ⎢ (∇N j )v − (∇N j )u ⎢ uu ⋅ ∇ × Ni ⎥ ~ = ∑ ⎥ ′ ν vv′ ⎦ ∂Aj k (Gauss point) ⎣ν uv ⎣ ⎦ ′ ν vu ′ ⎤ ⎡(∇ × N i )u ⎤ ⎤ (19) ⎡ ⎡ν uu = ∑ Wk ⋅ ⎢ (∇N j )v − (∇N j )u ⎢ν ′ ν ′ ⎥ ⋅ ⎢(∇ × N i ) ⎥ ⎥ k (Gauss point) vv ⎦ ⎣ ⎣ uv v ⎦⎦ ⎣ where [ ⎡ ∂H u ′ ν vu ′ ⎤ ⎢ ∂Bu ⎡ν uu ⎢ν ′ ν ′ ⎥ = ⎢ ∂H ⎢ u vv ⎦ ⎣ uv ⎢⎣ ∂Bv [ ] [ ] ] ∂H v ⎤ ⎡ 2 ∂Bu ⎥ = 12 ⎛⎜ ∂H − ν ⎞⎟ ⎢ Bu ⎥ B ⎝ ∂B ⎠ ⎣ Bv Bu ∂H v ⎥ ∂Bv ⎥⎦ Bu Bv ⎤ ⎡ν 0 ⎤ (20) ⎥+⎢ ⎥ Bv2 ⎦ ⎣ 0 ν ⎦ For linear materials or the situation of frozen ν, ⎞ ⎛ ∂H − ν ⎟ = 0 , so there is no first term in (20). ⎜ B ∂ ⎠ ⎝ In the following the calculation of (20) is described. According to B = Bu2 + Bv2 , one obtains ∂H and ν = H . ∂B B H According to ν = , one has H u = νBu and H v = νBv . B 1 ⎛ ∂H ∂ν ∂B 2 ∂H u ∂ν ⎞ − ν ⎟ Bu2 + ν (21) Bu + ν = 2 Bu + ν = 2 ⎜ = B ⎝ ∂B ∂B ∂Bu ∂Bu ∂Bu ⎠ 2 1 ⎛ ∂H ∂H v ∂ν ∂ν ∂B ⎞ (22) − ν ⎟ Bv2 + ν = Bv + ν = 2 Bv + ν = 2 ⎜ B ⎝ ∂B ∂Bv ∂Bv ∂B ∂Bv ⎠ 1 ⎛ ∂H ∂H u ∂ν ∂ν ∂B 2 ⎞ −ν ⎟ Bu Bv = Bu = 2 Bu = 2 ⎜ B ⎝ ∂B ∂Bv ∂Bv ∂B ∂Bv ⎠ 2 1 ⎛ ∂H ∂H v ∂ν ∂ν ∂B ⎞ −ν ⎟ Bu Bv = Bv = 2 Bv = 2 ⎜ B ⎝ ∂B ∂Bu ∂Bu ∂B ∂Bu ⎠ Here the following relationship is used: ⎛H⎞ ∂⎜ ⎟ ∂ν 1 ⎛ ∂H H ⎞ 1 1 ⎛ ∂H B ∂B ⎞ = ⎝ ⎠ 2 = ⎜ − ⎟ = −ν ⎟ ⎜ ∂B ∂B B ⎝ ∂B B ⎠ 2 B 2 B 2 ⎝ ∂B ∂B 2 ⎠ (23) (24) cross-sectional area of the region occupied by the winding in the solution domain; Nw is the total conductor number of this winding; a is the number of parallel branches in the winding; Rw is the d.c. resistance of the winding; iw and uw are the branch current and voltage of the winding, respectively; p is the symmetry multiplier which is defined as the ratio of the original full cross-sectional area to the solution area. The additional current iad are introduced in regions of solid conductors to ensure the last coefficient matrix of the field - circuit coupled equations is symmetrical [6]. Using the Galerkin method to discretize the field equation and circuit equations in the magnetic field regions, the system equations can be written in matrix format as: ⎡ S11 S12 ⎢0 S 22 ⎢ 0 ⎣⎢ 0 ⎧ dA ⎫ S13 ⎤⎧ A ⎫ ⎡ M11 0 0⎤⎪ dt ⎪ ⎧ 0 ⎫ ⎧QA ⎫ (29) ⎪⎪ di ⎪⎪ ⎪⎪ 1 ⎪⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎥⎥⎨ iw ⎬ + ⎢⎢ M 21 M 22 0⎥⎥⎨ w ⎬ = ⎨− uw ⎬ + ⎨ 0 ⎬ lp dt S33 ⎦⎥⎪⎩iad ⎪⎭ ⎣⎢ M31 0 0⎥⎦⎪ diad ⎪ ⎪⎪ 0 ⎪⎪ ⎪⎩ 0 ⎪⎭ ⎪ ⎩ ⎪ ⎭ ⎩⎪ dt ⎭⎪ Using the backward Euler’s method to discretize the time variable and multiply Δt to the additional equation and the branch equation, one obtains the recurrence formulas: M11 ⎡ ⎢ S11 + Δt ⎢ M 21 ⎢ ⎢ M 31 ⎣⎢ ⎤ S13 ⎥ ⎧ Ak ⎫ ⎧ 0 ⎫ ⎧Q A + M11 Ak −1 ⎫ ⎪ ⎪ Δt ⎪ ⎪ ⎪ Δt ⎪ 0 ⎥⎥ ⎨ iwk ⎬ = ⎪⎨− uwk ⎪⎬ + ⎪⎨ M 21 Ak −1 ⎪⎬ k ⎪ k −1 ⎪ ⎪ lp ⎪ ⎪ ΔtS33 ⎥ ⎪⎩ iad ⎭ ⎪⎩ 0 ⎪⎭ ⎪ M 31 A ⎪ ⎦⎥ ⎩ ⎭ S12 ΔtS 22 + M 22 0 (30) The branch equation of the external circuits is: (31) [Re ]{ie } = {ue } + {Qe } where Re is the matrix of the resistance and Qe is the column matrix associated with sources. Multiplying -Δt/lp to the two sides of (31), one has: ⎡ Δt ⎤ ⎧ Δt ⎫ ⎧ Δt ⎫ (32) − R {i } = − u + − Q ⎢ lp ⎣ e ⎥ ⎦ e ⎨ ⎩ lp e ⎬ ⎨ ⎭ ⎩ lp e ⎬ ⎭ Adding the external circuit equations (32) into (30) gives: (25) C. Circuit Equation Coupling Using Loop Method The electric circuit equations of the windings in the magnetic field domain are coupled with FEM equations. The overall equations are [6]: ∂A d w N w d N ∇ ⋅ (ν∇A) − σ + iad + w w iw = 0 (field equation) (26) ∂t Swap S wap dwNw ∂A Rw 1 (27) − dΩ − iw = − uw (branch equation) S wap ∫∫Ω ∂t lp lp d N ∂A R (28) − w w ∫∫ dΩ + w iad = 0 (additional equation) S wap Ω ∂t lp where; dw is the polarity (+1 or –1) to represent, respectively, the forward paths or return paths of the windings; Sw is the total T11 ⎡ ⎢ S11 + Δt ⎢ M 21 ⎢ ⎢ 0 ⎢ ⎢ M 31 ⎣ 0 S12 ΔtS 22 + M 22 0 0 − 0 Δt Re lp 0 ⎤ S13 ⎥ ⎧ A k ⎫ ⎧ 0 ⎫ ⎧Q A + T11 A k −1 ⎫ ⎪ ⎪ Δt ⎪ ⎪ ⎪ Δt k ⎪ 0 ⎥⎥ ⎪ i wk ⎪ ⎪⎪− lp uw ⎪⎪ ⎪⎪ M 21 A k −1 ⎪⎪ ⎨ k ⎬=⎨ ⎬+⎨ ⎬ 0 ⎥ ⎪ ie ⎪ ⎪− Δt uek ⎪ ⎪ − Δt Qe ⎪ ⎥ ⎪ k ⎪ ⎪ lp ⎪ ⎪ lp ⎪ ⎩ i ad ⎭ ⎪ k −1 ⎪⎭ ΔtS 33 ⎥⎦ ⎩ 0 ⎪⎭ ⎪⎩ M 31 A (33) Using the loop method, the relationship between the branch current ib and the loop current il is: (34) {ib } = [BlbT ]{il } where Blb is the loop-to-branch incidence matrix. The Kirchhoff’s voltage law can be expressed as, (35) [Blb ]{ub } = 0 Substituting these relationships into the system equations, one obtains the final global equations: ANSYS 2011中国用户大会优秀论文 ⎤ M 11 k −1 ⎫ ⎧ S13 ⎥ ⎪Q A + Δt A ⎪ ⎥ k ⎥ ⎧ A ⎫ ⎪⎪ ⎛ M 21 A k −1 ⎞ ⎪⎪ ⎪ ⎪ ⎟⎪ ⎜ 0 ⎥ ⎨ ilk ⎬ = ⎪⎨ Blb ⎜ Δt ⎥ ⎪ k ⎪ ⎪ ⎜ − Qe ⎟⎟ ⎬⎪ lp i ⎠⎪ ⎥ ⎩ ad ⎭ ⎪ ⎝ k −1 ΔtS 33 ⎥ ⎪ ⎪ M 31 A ⎥ ⎪⎭ ⎪⎩ ⎦ (36) where the coefficient matrix is symmetrical. D. The Implementation of Nonlinear Problem The derivative ∂H u ∂B u is dependent on the history of the operating point in each element. The data are stored element by element for all nonlinear PM materials. The direction of magnetization remains as those assigned before. Each finite element of the PM materials uses its own recoil curve in the respective transient simulation. Therefore the worst case of demagnetization for the entire transient simulation is recorded. When restarting the simulation, the mesh and geometry must be identical with those of the previous design and target design. In each object with the same material, the recorded data are Hc and νeq. In each element, the recorded data are: Bmin: the minimum value of Bu (u is the direction of magnetization) in the simulation history. HcEq: the current equivalent Hc (which is dependent on the operating point). Bmin_temp: the temporary value of Bmin. It is needed in nonlinear iteration. HcEq_temp: the temporary value of HcEq. It is needed in nonlinear iteration. Status is a flag used to remember the location of the operating point: ⎧1 : operating point on nonliner curve (37) Status = ⎨ ⎩2 : operating point on recoil curve The procedures in time stepping FEM are: (1) Initialize the data: Bmin = Br, HcEq = νeqBr, Bmin_temp = Br, HcEq_temp = νeqBr, Status = 1. (2) At the beginning of the nonlinear iteration: Bmin_temp = Bmin, HcEq_temp = HcEq; set Status = 2. (3) During the nonlinear iteration, determine the status according to the calculated Bu: (a) if Bu < Bmin, Status = 1, the operating point is on the nonlinear curve; Hc is used; update: Bmin_temp = Bu, HcEq_temp = νeqBu; (b) if Bu ≥ Bmin, Status = 2, the operating point is on the recoil permeability line; the recoil line with νeq and HcEq is used. After the computation of each time step, if Status = 1, update: Bmin = Bmin_temp, HcEq = HcEq_temp. IV. APPLICATION TO PM MOTORS The proposed method is applied to analyze the performance of a PM motor. The rating of the motor is 900 W at 24 V; it has 11 pole pairs, 24 stator slots, and operates at 136 rpm; the rotor has surface mounted ceramic magnet. The transient process is simulated when one phase is suddenly short-circuited when time is 0.25 s. Fig. 3 shows a typical flux plot. Fig. 4 shows the stator phase current. The short-circuit current is about 5.5 times of the rated current. The armature reaction during the short circuit demagnetizes the PM. Fig. 5 shows the change of the average HcEq in the PM. Before the short circuit, its value is about 31500 A/m; after the short circuit, its value is reduced to about 30600 A/m. It can be observed that irreversible demagnetization will happen for this design. Fig. 6 shows that the distribution of Hc of PM is still uniform after the short circuit current reaches steady-state. Fig. 3. The flux plot of the PM motor. Fig. 4. The stator current when one phase is suddenly short-circuited. HcEq (A/m) ⎡ ⎛ S12 0 ⎞ T M11 ⎜⎜ ⎟⎟ Blb ⎢ S11 + Δ t ⎝ 0 0⎠ ⎢ 0 ⎞ ⎢ ⎛ ⎛ ΔtS 22 + M 22 ⎢ B ⎜ M 21 0 ⎞⎟ B ⎜ Δt ⎟⎟ BlbT lb ⎜ lb ⎜ ⎟ − 0 Re ⎢ ⎝ 0 0⎠ ⎜ lp ⎟⎠ ⎝ ⎢ ⎢ M 31 0 ⎢ ⎣ Fig. 5. Changes of average HcEq in the PM when one phase is suddenly short-circuited. ANSYS 2011中国用户大会优秀论文 Fig. 6. The distribution of Hc vector of PM after short circuit (t = 0.8 s). V. CONCLUSION By using a matrix to express a space vector, the nonlinear formulation of FEM can be conveniently deduced. It is applicable to any type and any order of finite elements, and also to anisotropy materials such as PMs studied in this paper. A time-stepping field-circuit coupled FEM to address the irreversible demagnetization process of PM is developed. Only normal B-H curves of the PM materials are used. An effective algorithm to implement the methodology with nonlinear iteration is presented. The status of the operating point at each element is recorded and the nonlinear iteration is directly controlled by the true Jacobian matrix. VI. [1] [2] [3] [4] [5] [6] REFERENCES J.-R. R. Ruiz, J. A. Rosero, A. G. Espinosa and L. Romeral, “Detection of demagnetization faults in permanent-magnet synchronous motors under nonstationary conditions,” IEEE Trans. Magn., vol. 45, no. 7, pp. 2961-2969, July 2009. W. N. Fu, S. L. Ho, H. L. Li and H. C. Wong, “A multislice coupled finite-element method with uneven slice length division for the simulation study of electric machines,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1566-1569, May 2003. Ki-Chan Kim, Kwangsoo Kim, Hee Jun Kim and Ju Lee, “Demagnetization analysis of permanent magnets according to rotor types of interior permanent magnet synchronous motor,” IEEE Trans. Magn., vol. 45, no. 6, pp. 2799-2802, June 2009. S. Ruoho, E. Dlala and A. Arkkio, “Comparison of demagnetization models for finite-element analysis of permanent-magnet synchronous machines,” IEEE Trans. Magn., vol. 43, no. 11, pp. 3964-3968, Nov. 2007. W. N. Fu and S. L. Ho, “Matrix analysis of 2-D eddy-current magnetic fields,” IEEE Trans. Magn., vol. 45, no. 9, pp. 3343-3350, September 2009. W. N. Fu, P. Zhou, D. Lin, S. Stanton and Z. J. Cendes, “Modeling of solid conductors in two-dimensional transient finite-element analysis and its application to electric machines,” IEEE Trans. Magn., vol. 40, no. 2, pp. 426-434, March 2004.