Download Dynamic Demagnetization Computation of Permanent Magnet

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:
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⎤
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.
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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]
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