12th International IGTE Symposium
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Proceedings
Hysteresis measurement simulation by fixed-point method Peter Kis, Amalia Ivanyi University of Pecs, Pollack Mihaly Faculty of Engineering, Dept. of Information Technology, Rokus u. 2, H-7624 Pecs, Hungary E-mail:
[email protected] Abstract: The hysteresis characteristics of a structural steel ring has been measured. The cross-section area of the ring is high enough to obtain eddy currents inside of the ring at relatively low frequency e.g. 5 Hz. The purpose of the paper is to present a nonlinear eddy current field computational method, which is able to consider the dynamic hysteresis effect. The dynamic Jiles-Atherton model is used to represent the material non-linearity and it is combined with the fixedpoint method. The field computation is done by FEM with higher order edge elements. Keywords: eddy currents, fixed-point method, hysteresis, higher order edge finite element method
I. I NTRODUCTION The purpose of this paper to present how can be included the hysteresis operator into the electromagnetic field equations. The EM field equations are discretized by the FEM. After the discretization a nonlinear system of equations is resulted, where both the stiffness matrix and the left-handside (LHS) excitation vector depend on the nonlinear quantity too. The material non-linearity can be handled by several nonlinear iteration methods. The Newton method is the most famous among them, because its convergence rate is very high, only several iteration steps are required to find the correct solution of the nonlinear system of equations [1]. In the case of FEM, the Newton method has a drawback, namely the stiffness matrix must be reassembled in each iteration steps, which is a very time consuming task and in the case of hysteresis it does not work. Instead of the Newton method we are using another method to solve the nonlinear problem, where the reassembling is not required. The polarization or fixed-point method complies the requirement. The polarization method determines a permeability – called optimal permeability – value where from the iteration can be started. Using the constant optimal permeability the stiffness matrix can be assembled once and only the LHS excitation vector is varied during the iteration process. So the reassembling can be avoided by using the polarization method, but the convergence speed of the method is very far from the Newton method, because the contraction factor of the iteration is very closed to one. Therefore usually the over-relaxation technique is used to speed up the method convergence. Several overrelaxation method can be found in the literature [2], [3], but we introduce an other approach, where both the overand under-relaxation is also remitted to achieve the desired tolerance. The fine tuning of the fixed point technique is done by the well known 1D half-space example, where a few speeding up procedure is investigated. The experience in halfspace problem is adapted to the original problem, the ring simulation. The efficiency of the speeding up techniques are illustrated by tables and diagrams.
II. S CALAR
HYSTERESIS MEASUREMENT
This section deals with the computer aided automated magnetic scalar hysteresis measurement on ferromagnetic ring shaped material. A procedure of automated computer aided ferromagnetic hysteresis measurement is presented. Task of computer in the frame of the measurement is the driving all phases of the measurement process, e.g. controlling data acquisition cards, saving the measured data to file and post processing of experimental data. The voltage of measuring coil and the current of excitation coil are measured and the magnetic field intensity H and the magnetic flux density B are calculated according to the Ampere’s and the Faraday’s laws. The main advantage of the digital measurement consists of the measurement driving and the post processing, because the digital data can be handled easily. The scanning rate must be chosen carefully because the down-sampling analog signals are inaccurate in the meantime in the case of over-sampling the size of data files can become needlessly large. The examination is focused on the scalar hysteresis measurement of ring shaped ferromagnetic structural steel (Fig. 1). The measurement requires two coils (excitation and measuring) on the examined material. The components of the measurement are Personal Computer with measuring cards, software and power supply. The automated magnetic hysteresis measurement is controlled by personal computer. The excitation signal can be generated with the help of the computer and the required signals can also be measured. The applied power supply can amplify analog input signal ensuing the required power for excitation. The current of the primary coil is generated by the power supply. The data acquisition and the generation of excitation current can be performed simultaneously, but some common problems can be occurred during the magnetic hysteresis measurement in the digital signal processing part e.g. the question of correct sampling rate, the noises and so on [4]. The measuring arrangement of magnetic hysteresis can be seen in Fig. 2. The investigated ferromagnetic material has ring shape. The magnetic field inside of the toroid (Outer/inner diameters are 60/40 mm, height is 16 mm) is uniform with a scalar value H. The average magnetic moment per unit volume has a scalar value M and it is parallel to H. H is assumed to be equal to the magnetic field of external origin in the material, as the demagnetizing field is approximately equal to zero for the shape under consideration. The primary coil is controlled by the current of the power supply. The secondary coil is used for measuring the induced voltage of the pick-up coil. The amplitude of excitation current can be measured as a voltage on the resistance R. The value of resistance R is regarded to be constant, which is independent of the ambient temperature and the
Proceedings
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12th International IGTE Symposium
be generated by modifying the signal shape of the excitation current. A nonlinear iteration method has been applied in the measurement, which modifies the excitation trough a feedback. The iteration starts with sinusoidal excitation and the differences between the prescribed reference sinusoidal signal Bref and the measured quantities are calculated (∆B = B − Bref ), according to the difference ∆B the signal shape of excitation is modified in each time step. The iteration stops when the maximum value of vector ∆B is small enough. The convergence of the method can be declared fast, approx. 10 cycles are needed for reaching less then 1% maximal difference between the sinusoidal reference signal Bref and the measured one [4].
Figure 1: Picture taken about the measured coil Power Supply
III. 1D EXAMPLE – F ERROMAGNETIC
KIKUSUI PBX-2020
NI-DAQ BNC-2090
Computer
AI Analog control signal Ferromagnetic Excitation material coil
Pick up coil Voltage
Ri(t)
AO
u(t)
R Current
Figure 2. The arrangement of the automated computer aided magnetic hysteresis arrangement
voltage. [5]. The controlling of measurement and post-processing of measured data are carried out in LabVIEW environment. Magnetic flux density and magnetic field intensity are calculated from measured data during the post-processing phase with the following relationships H(t) =
Ne · i(t) , l
1 B(t) = B0 + A · Nm
Zt
(1)
HALF - SPACE
The sketch of the half-space problem can be seen in Fig. 4. The half-space, where z > 0 is conductive ferromagnetic material. A sinusoidal magnetic field is prescribed at z = 0 as a Dirichlet boundary condition. Indeed, eddy currents is induced in the electrically conductive material, which is decreasing by z. The problem can be described by a 1D model, because the field quantities are not changing by x and y. A finite element model of this half-space is very simple, which enables to setup the field computation procedure with hysteresis. Three problems is solved in this example going from the simplest linear case to the computation including the JAM.
σ=0
x 6 if z > 0, then σ ≫ 0 and B = H {H }
µr = 1
z -
u y
@ @ HD (z = 0, t) = Hamp sin(ωt) Dirichlet condition
u(τ )dτ,
(2)
0
where l is the equivalent magnetic length of the ring shaped material, Ne and Nm are the ampere turns of the excitation and the measuring coils respectively, i(t) is the excitation current, B0 is integration constant, A is the cross-section of the material and u(t) is the voltage of the measuring coil. A. Sinusoidal B As it was mentioned before our hysteresis measurement has been driven by sinusoidal current, in the case of toroidal shape magnetic core means sinusoidal H. However there are some disadvantages of this configuration, namely the measured points are not distributed evenly on the hysteresis loop. It means a lot of points are located at the saturation part of the curve and few points are at the high slope part of the characteristics. This problem arises in hysteresis model identification. If the B is sinusoidal, then the distribution of the points becomes favorable. The sinusoidal shape of magnetic flux density can
Figure 4. The ferromagnetic half-space. The z > 0 half-space is the ferromagnetic and electrically conductive material, the z < 0 can be considered as air or vacuum, where the sinusoidal magnetic field is coming from. It is prescribed as a boundary condition at z = 0
A. Solution of the nonlinear problem with Langevin characteristics The half-space problem is solved in this section in the case of non-linear magnetic material. The material characteristics is described by the Langevin function. " ! # H a B(H) = Bs coth − , (3) a H where Bs is the induction at saturation and a is a form parameter, which ensures the desired slope at the origin. The differential relative permeability is the derivative of the equation above with respect to H, The slope of the Langevin characteristics at the origin is µ = 4000µ0
12th International IGTE Symposium
TABLE I: N UMBER OF
- 118 -
REQUIRED
FP
OR
µopt
Hamp
+ + + + + + + + +
– – – – – – – – –
2000 2000 2000 4000 4000 4000 6000 6000 6000
100 1000 10000 100 1000 10000 100 1000 10000
H
Proceedings
FP ITERATION STEPS
WITH DIFFERENT
Number of iteration steps (for two periods, 80 time steps) – – – 474 5194 73176 1294 7836 109559
=100 A/m @ t=35msec
H
amp
µOPT
Notes
divergent divergent not convergent after 37125 steps avg. steps: 5.925 avg. steps: 64.925 avg. steps: 914.7 avg. steps: 16.175 avg. steps: 97.95 avg. steps: 1369.4875
=1000 A/m @ t=35msec
amp
50
500
0.5
−50
−100 0
H [A/m]
1
H [A/m]
1000
H [A/m]
100
0
0
−500
1
2
3
4
5
−1000 0
STARTING VALUE
H4 =10000 A/m @ t=35msec x 10 amp
0
−0.5
1
2
z [mm]
3
4
5
−1 0
1
2
z [mm]
3
4
5
z [mm]
Figure 3: Solutions with Langevin characteristics, Hamp = 100, 1000, 10000 by columns at t=35 ms (Bs = 1 T, a = 66.3 A/m), because the relative permeability should be the same like in linear case. A kind of verification can be carried out, where the characteristics can be considered as linear for small signal excitations. A.1. Formalism
where Nj are nodal shape functions of FEM and it is equal to the test function of H. After solving (6) for H, the new residual can be computed as
The above described material non-linearity is taken into account by the fixed point method, whereas a constant linearization point is assumed (µopt ) and the error of the linearization is corrected by a residual vector R. So the constitutive relationship for the magnetic flux density can be formed as
where H is the value of the magnetic field intensity in the current iteration step. A.2. Solution with fixed point iteration
B = H {H} = µopt (H + R),
(4)
where the operator H {} represents the material characteristics. Substituting (4) into the differential equation ∂ 1 ∂H ∂B =− , (5) ∂z σ ∂z ∂t the linearized fixed point equation is obtained ! ∂ 1 ∂H ∂H ∂R = −µopt + → H = φ2 (R), ∂z σ ∂z ∂t ∂t (6) where R is the residual. The weak form of the non-linear half space problem can be formulated as " # Z Z ∂H 1 ∂Nj ∂H ∂R − dΩ − Nj µopt + dΩ = 0, ∂z σ ∂z ∂t ∂t Ω
R = H {H} /µopt − H
→
R = φ1 (H),
(8)
In this section the half-space problem is solved by the traditional FP iteration method without any speed up technique. Beside the FP algorithm the φ2 represents the solution of the differential equation, ! ∂ 1 ∂H H − Hold R − Rold − − µopt + = 0, (9) ∂z σ ∂z ∆t ∆t where Hold and Rold represent the magnetic field and the residual values at the previous time step respectively. After solving the differential equation (9) based on the previous value of R, the new value of the residual can be computed by φ1 , which is as follows R = H {H} /µopt − H,
(10)
where the nonlinear relationship between the B and H is denoted by the operator H {}. The definition of the error is the following
Ω
(7)
error = ||R − Rprev ||ν ,
(11)
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TABLE II: N UMBER
OF REQUIRED
FP+OR
FP
OR
µopt
Hamp
+ + + + + + + + +
+ + + + + + + + +
2000 2000 2000 4000 4000 4000 6000 6000 6000
100 1000 10000 100 1000 10000 100 1000 10000
12th International IGTE Symposium
ITERATION STEPS WITH DIFFERENT
Number of iteration steps (for two periods, 80 time steps) 1248 2761 34639 397 2771 33587 894 4124 49 812
where ν is the magnetic susceptibility at the current iteration point, !−1 dH {H} . (12) ν(H) = dH The weighting by ν(H) ensures that the error is computed justly at even the places, where the slope of characteristics is much lower. The parameters of the computation are • frequency: f = 50 Hz, 7 • electrical conductance: σ = 1.12 · 10 S/m, • material characteristics is given by the Langevin function B = coth(H/66.3) − 66.3/H • amplitude of the Dirichlet condition at z = 0: Hamp = 1000 A/m, • number of samples per period: N = 40 and two periods are considered (80 time steps). −8 • tolerance of the FP iteration ε = 10 The FP method is convergent if φ1 is a contraction (µopt > µmax /2). As it can be seen in Table I if µopt = 2000, which is lower than the µmax /2, then the method will be divergent. The total number of required FP iteration steps are summarized in Table I. (In fact, this value should be multiplied by two, because there are two FP algorithm in one iteration step.) Only FP algorithm is used to obtain the solutions without over-relaxation (OR). Three different so called optimal relative permeability is used as a starting value. As it is already discussed the µopt = 2000 case is unstable. In the case of µopt = 4000 the slope of the material characteristics is very closed at the origin. The exact slope of the characteristics at the origin is 4000.9, that is why that some iterations steps are required to achieve the prescribed tolerance (in this example ε = 10−8 ). If the optimal permeability is exactly the same like the slope of the material characteristics at the origin and the amplitude of the excitation is small for tracking on the linear part, then no FP iterations are required to achieve the desired tolerance. The problem can be considered as linear, where the optimal permeability is 4000 and the amplitude of the magnetic field intensity is 100 in Table I. That is the explanation of the very small number of iterations. If the amplitude of the excitation is increasing (Hamp =1000,10000), then the characteristics can not be considered as linear any more, because we reach the saturation part. Therefore more iteration steps are required. It is straightforward that less
µOPT
STARTING VALUE
Notes
avg. avg. avg. avg. avg. avg. avg. avg. avg.
steps: steps: steps: steps: steps: steps: steps: steps: steps:
15.6 34.5125 432.9875 4.9625 34.6375 419.8375 11.175 51.55 622.65
iteration is required, where the excitation is crossing zero. The solutions are plotted in Fig. 3. A.3. Solution with fixed point iteration + over-relaxation The main disadvantage of the polarization method is the slow convergent speed. An important improvement can be obtained by over-relaxation (OR). The method is based on the property of the contraction mapping [2]: ||R∗ − φ1 (φ2 (R))||ν ≤
q ||φ1 (φ2 (R)) − R||ν , (13) 1−q
where R∗ is the fixed point of R = φ1 (φ2 (Rprev )), R is any value of the residual and q is the contraction factor. This means that, reducing the distance d = ||φ1 (φ2 (R)) − R||µ , φ1 (φ2 (R)) approaches to R∗ . The problem is that the contraction factor q is very closed to unit. In the case of ferromagnetic material the contraction factor might be 0.999 999. To increase the convergence speed, a suitable factor ω is introduced in order to force a reduction of distance d and accelerate the convergence toward the solution. The numerical over-relaxation procedure has the following steps: 1) Solving Hprev = φ2 (R0 ). 2) Computing Rprev = φ1 (Hprev ). 3) Solving H = φ2 (Rprev ). 4) Computing R = φ1 (H). 5) ∆R = Rprev − R0 , ∆H = H − Hprev , 6) d(1) = ||R − Rprev ||ν ; if d(1) < ε we stop the iteration, otherwise: 7) An over-relaxation factor ω is introduced and a new value of the residual is computed by the following relationship: R′ = R0 + ω∆R
(14)
due to the linearity of φ2 , the modified value of to this value of magnetic field intensity can be obtained by H ′ = Hprev + ω∆H. (15) without solving the FEM equations again. The value of the over-relaxation factor ω can be determined by the minimum search of the functional d(ω) = ||φ1 (H ′ ) − R′ ||ν over the investigated region.
(16)
12th International IGTE Symposium
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TABLE III: N UMBER
OF REQUIRED
FP
OR
µopt
Hamp
+ + +
+ + +
update update update
100 1000 10000
Proceedings
FP+OR
ITERATION STEPS WITH
Number of iteration steps (for two periods, 80 time steps) 395 1696 10765
Using this method, same results have been obtained like in the previous section, where only fixed point method were used without any speed-up procedure. The only difference is in the convergence speed. The method with OR is about two times faster than without OR (Tables I, II). A.4. Solution with fixed point iteration + over-relaxation and µ update
The main purpose of this investigation is to speed up the original fixed point iteration. One possible way is the overrelaxation, but if we consider the Tables I and II, then we can see that the number of required iteration steps depends on the optimal permeability µopt as well. In this case, if the µopt = 4000, which is very closed to the initial slope of the nonlinear curve, then the iteration requires less steps. It is straightforward that the optimal permeability µopt should be chosen as close as possible to the current slope of the nonlinear curve. Of course the slope is changing by marching in time, therefore the permeability values must be updated in each time iteration step. It is important to emphasize that the optimal permeability is untouched during the FP iterations, it is modified only in the time iteration steps. The choice of the optimal permeability is not arbitrary, because the condition of contraction must be fulfilled always. This is the reason that the optimal permeability can not be the exact slope of the nonlinear material characteristics. It this case the optimal permeability must be always greater than the half of the maximal permeability, µopt > µopt /2. Therefore the rule of the choice of the optimal permeability is that the µopt is equal to the current slope of the magnetization curve if it is greater than µopt /2, if not, then µopt = µopt /2 is the correct choice. Furthermore, a more sophisticated procedure is resulted if the optimal permeability is different element by element. The required iteration steps are summarized in Table III. If the amplitude of the magnetic field is low Hamp = 100 A/m and the µopt is very closed to the real slope of the material characteristics (in this case it is 4000), then the path of the nonlinear iteration remains on the linear part of the Langevin function. This is the reason that in the first rows of Tables II and III almost the same values (397 and 395 steps) can be found, because no better optimal permeability value can be found like 4000. The conclusion is that a much faster method has been obtained by using the over-relaxation and the µ update technique and the solution remained same like in the case of the original fixed point method. This method is used in the further computations.
µ UPDATE
Notes
avg. steps: 4.9375 avg. steps: 21.2 avg. steps: 134.5625
B. Solution of the nonlinear problem with Jiles-Atherton model of hysteresis In this section, the half-space problem is solved with hysteresis. It is supposed that the material of half-space is ferromagnetic and electrically conductive. The hysteretic property of the half-space material is described by the rate independent Jiles-Atherton model (JAM) of hysteresis. The problem is discretized by the finite element method (FEM) and the material nonlinearity is taken into account by the fixed point method with over-relaxation and µ update as it is detailed in the previous section. It is important to emphasize that the Langevin function, which is applied to describe the matereial nonlinearity in the previous section, is the anhysteretic curve of the JAM. In fact, the original JAM was two shifted Langevin functions in Jiles’ old papers. By the way, the choice of the anhysteretic curve is arbitrary, not only the Langevin function is allowed. Since the Langevin function has singularity at zero, therefore the Frhlich function is also often used in JAM. The parameters of the JAM have been determined to be similar to the Langevin function. The JAM parameters are Ms = 8.1 · 105 A/m, α = 10−3 , a = 300, c = 0.5, k = 800. The computation details are summarized in Table IV, where the number of required iteration steps are plotted as a function of the time and the number of required iteration steps are shown by different amplitudes respectively. The tolerance of the nonlinear iteration is ε = 10−8 . The solution at t = 35 ms can be seen in Fig. 5. It means that same procedure has been used for solving the problem with hysteresis like in the case of Langevin function. The only assumption is that the operator of the material nonlinearity must be uniform monotone and Lipschitzian. The running time is almost the same in both (Langevin and JAM) cases. One iteration step takes about 0.5 sec. on a portable computer with PentiumM 2 GHz processor and 2 GB of RAM. It means that the total process takes about 25 min. if we consider the worst case (last row) in Table IV. IV. 2D
EXAMPLE
– H YSTERESIS
MEASUREMENT
SIMULATION
The simulation of the ferromagnetic hysteresis measurement has been performed in this section. A regular ring shaped specimen made from structural steel is excited by a current driven coil. The induced voltage is recorded during the measurement and after that the magnetic flux density is computed from it. After introducing the magnetic vector potential B = curl A and after taking into account the linearization of H = H −1 {B} constitutive relation by the fixed point
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TABLE IV: N UMBER
H
FP
OR
µopt
Hamp
+ + +
+ + +
update update update
100 1000 10000
Number of iteration steps (for two periods, 80 time steps) 463 936 2990
=100 A/m @ t=35msec
H
Notes
avg. steps: 5.7875 avg. steps: 11.7 avg. steps: 37.375
=1000 A/m @ t=35msec
amp
1
50
500
0.5
−50
−100 0
H [A/m]
1000
H [A/m]
100
0
0
−500
1
2
3
4
H4 =10000 A/m @ t=35msec x 10 amp
0
−0.5
−1000 0
5
JAM
OF REQUIRED ITERATION STEPS IN THE CASE OF
amp
H [A/m]
12th International IGTE Symposium
1
z [mm]
2
3
4
−1 0
5
1
2
z [mm]
3
4
5
z [mm]
Figure 5: Solutions with JAM, Hamp = 100, 1000, 10000 A/m by columns at t=35 ms
z
method H = νopt B + R the following eddy current equation is resulted curl (νopt curl A + R) + σ
∂A =J ∂t
ferromagnetic ring
B = φ2 (R), (17) where νopt is the reluctivity at the linearization point and R is the residual vector according to the fixed-point method, which can be computed by {B} − νopt B
→
R = φ1 (B).
(18)
The corresponding weak form of the eddy current equation above is as follows Z Z Z ∂A ˜ ˜ ˜ dΩ, curl A (νopt curl A + R) dΩ+ Aσ dΩ = AJ ∂t
Ω
Ω
Ω
(19) ˜ is the test function of A. where A The 2D model of the ferromagnetic ring measurement in cylindrical coordinate system can be seen in Fig. 6 including the enclosure, where the tangential component of the magnetic vector potential prescribed as zero. This enclosure is far enough from the ring to vanish the magnetic field. The magnetic field generated by the excitation coil is prescribed as a surface current on the perimeter of the ring. So that the surface current is Js (t) =
Ne · i(t) , 2πr
(20)
where Ne is the Ampere turns of the excitation coil, i(t) is the current flowing through the coil wires and r is a coordinate variable. R EFERENCES [1] W. Peterson, “Fixed-point technique in computing nonlinear eddy current problems,” COMPEL, vol. 22, no. 2, pp. 231–252, 2003.
Js − surface current Figure 6. The geometry model of the ferromagnetic ring simulation 1.5 simulated measured
simulated measured 1
1
0.5
0.5
B [T]
−1
r
B [T]
R=H
→
0
0
−0.5
−0.5
−1
−1
−1.5 −3000
−2000
−1000
0
H [A/m]
1000
2000
3000
−3000
−2000
−1000
0
1000
2000
3000
H [A/m]
Figure 7. The measured and computed curves with rate independent JAM (at left) and with rate dependent JAM (at right)
[2] F. I. Hantila, G. Preda, and M. Vasiliu, “Polarization method for static fields,” IEEE Trans. Magn., vol. 36, no. 4, pp. 672–675, 2000. [3] L. R. Dupr´e, O. Bottauscio, M. Chiampi, M. Repetto, and J. A. A. Melkebeek, “Modeling of electromagnetic phenomena in soft magnetic materials under unidirectional time periodic flux excitations,” IEEE Trans. Magn., vol. 35, no. 5, pp. 4171–4184, Sept. 1999. [4] P. Kis, M. Kuczmann, A. Iv´anyi, and J. F¨uzi, “Hysteresis measurement in labview,” Physica B, vol. 343, pp. 357–363, 2004. [5] P. Kis and A. Iv´anyi, “Computer aided magnetic hysteresis measurement in LabView environment,” Journal of Electrical Engineering, Bratislava, vol. 53, no. 10/S, pp. 173–176, 2002.
