temperature dependent extension of the Jiles-Atherton model is presented. .... data, we know the value of a at a given temperature, consequently, the Eq. (8) ...
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0 and the value -1 when dH/dt SOFT MAGNETIC MATERIALS CONFERENCE, SMM 2011, S03-P0191< factor, k, as functions of temperature. The other parameters: domain coupling α, and reversibility, c, can be represented as functions of precedent parameters. A. Saturation Magnetization The relation of the saturation magnetization to temperature has been considered in [1-2], it can be writing by: T (6) M s (T ) = M s (0° K )(1 - ) β1 Tc where Ms(0°K) is the value of saturation magnetization at 0°K, Tc is the Curie point and β1 is the material dependent critical exponent, a representative of mean field interactions, which can be derived from mean field theory. The measured saturation polarization, Js versus temperature curve was used to identify Js(0°K) and β1 by fitting as shown
saturation magnetization at 0°K corresponds to that given by Hegg [3] for the NiFe 80/20 material. Fig.2 shows the evolution of the saturation magnetization as a function of temperature. The change from the ferromagnetic to the paramagnetic state is perfectly sharp at the Curie point Tc . A. Pinning Factor or Loss Coefficient In soft magnetic materials, the pinning factor can be approximated by the coercivity (k=Hc) [4]. Authors in [1,2] give an analytical law relating the pinning factor k to temperature according to the following equation: -1 T β 2 Tc
(7) k (T ) = k (0°K ) e where k(0°K) is the value of pinning factor at 0°K and β 2 is the critical exponent for pinning constant and is approximated to be β1 2 . Fig.3 shows the variation of k with temperature. The critical exponent is known from (A), and k (0°K ) is estimated from fitting the analytical law of k to the measured data of coercive field, Hc , as indicated in Fig.4.
0.8
25 0.7 Pinning factor,k( A/m)
Saturation polarization,Js, (T)
in Fig.1. The calculated value of β1 is equal to 0.42 and the estimated saturation polarization, Js(0°K) is equal to 1 T. The saturation magnetization, M s (0°K ) = μ 0 J s (0°K ) , where μ 0 is the vacuum permeability= 4π.10− 7 H / m.
2
0.6 0.5 0.4 0.3 0.2
analytical measured
0.1 0 250
300
350
400
450
500
550
600
650
5
Saturation magnetization (A/m)
8
x 10
7 6 5 4
10
5
100
200
300
400
500
600
700
Temperature (°K) Fig. 3. Pinning factor as a function of temperature. 2.5 Pinning factor, k (A/m)
Hegg determined the saturation induction for various temperatures and also extrapolated to absolute zero. He also established with some precision the Curie point curve for the whole series of Iron-Nickel alloys. We can note that our estimated value of polarization saturation at 0°K, therefore
15
0 0
Temperature (°K) Fig. 1. Variation of saturation polarization with temperature measured in NiFe 20/80with Curie temperature of Tc=642°K
20
2
analytical measured Hc
1.5
1
0.5
0 300
350
400
3
450
500
550
600
650
Température (°K)
2
Fig. 4. Variation of coercive field in NiFe 80/20 material with temperature.
1
B. Domain Density The domain density, a, shows a similar exponential decay with temperature, it can be described by:
0 0
100
200
300
400
Temperature (°K)
500
600
700
Fig. 2. Saturation magnetization as a function of temperature in NiFe 80/20 material.
a(T ) = a(0°K ) e
-1 T β3 Tc
(8)
> SOFT MAGNETIC MATERIALS CONFERENCE, SMM 2011, S03-P0191< where a(0°K) is the domain density at 0°K and β 3 is the critical exponent for domain density and is approximated to be β1 2 . From the fitting of model parameters to experimental data, we know the value of a at a given temperature, consequently, the Eq. (8) makes possible the evaluation of a(0°K). Fig.5 indicates the evolution of a with temperature.
3
A. Reversibility Factor As the domain coupling, α , the reversibility factor, c, is expressed by substituting the expressions of a and M S in Eq. (11) [5]: 3a ' c= χ M s in
(12)
Assuming constant initial susceptibility, χ ini , c is given by: '
40
30
20
0 0
100
200
300
400
500
600
700
Temperature (°K) Fig. 5. Domain density as a function of temperature.
C. Domain Coupling The domain coupling, α , which represents the strength of magnetic interaction between domains in an isotropic material can be expressed as [5]: 3a 1 (9) α= - ' M s χ an At high anhysteretic susceptibilities,
χ 'an , the contribution of
the second term to domain coupling is negligible, so, from the expressions of domain density, a (Eq. (8)) and saturation magnetization, M S (Eq. (6)), α can be written by: -2 T β1 Tc
T -β (10) ) Tc where α(0°K) is the domain coupling at 0°K given by the Eq.(11), β1 is the material dependent critical exponent. α(T ) = α(0° K ) e
(1 -
3a(0°K ) M s (0° K ) Fig. 6 shows the variation of α versus temperature. α(0°K ) =
(11)
0.9 0.8 0.7 0.6 0.5 0.4
0
200
300
400
500
600
Fig. 7. Reversibility factor as a function of temperature.
IV. VALIDATION AND COMPARISON Now, we can consider the variation of J-A model parameters with temperature developed previously. After calculating the all parameters at a given temperature (T=310°C for example), the measured hysteresis loop of NiFe 80/20 material at the same temperature is compared with the calculated loop based on J-A theory. As shown in Fig.8, this method gives «inaccurate» hysteresis loops especially around the Curie point (Tc=369°C). 0.4
x 10
0.3 0.2
1.2
0.1 B (T)
1.4
1
0
0.8
-0.1
0.6
-0.2
0.4
-0.3
0.2
-0.4 -80
0 0
100
Temperature (°K)
-4
Domain coupling
(1 -
1
10
1.6
-2 T β1 Tc
T -β1 (13) ) Tc where c(0° K ) = 3a(0° K ) χ ' is the reversibility factor at M s (0° K ) ini 0°K, β1 is the material dependent critical exponent. As shown in Fig.7, c has an upper limit of 1 before or at the Curie point. c(T ) = c(0° K ) e
Reversibility Factor,c, A/m
Domain density, a,(A/m)
50
measured at 310 °C simulated at 310 °C
-60
-40
-20
0
20
40
60
80
H (A/m) 100
200
300
400
500
Temperature (°K) Fig. 6. Domain coupling as a function of temperature.
600
700
Fig. 8. Measured and simulated hysteresis loops of the NiFe 80/20 material at T=310 °C (f = 0.5Hz) .The parameters used in the modeling are: α= 8.1899e-007, Ms=2.8667e+5 A/m, a=0.6010, c=0.3448, k=0.3202 A/m.
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