2.1 Introduction to Preisach magnetization theory
Preisach magnetization Theory is proposed by the German physicist F. Preisach in 1935. After continuous improving and perfecting, it has gradually formed a guiding theory to the ferromagnetic material modeling [8,13,14,, 12–15].
Preisach Theory holds that the ferromagnetic material consists of numerous magnetic dipoles, each of which has its own saturation characteristics, as shown in Fig. 1. Macroscopic magnetization shown by ferromagnetic material is the accumulation of the microscopic magnetization of all the magnetic dipoles that make up it. Each magnetic dipole has its own positive saturation threshold α and negative saturation threshold β. As the name implies, when the applied magnetic field strength is greater than the positive saturation threshold α, the magnetic dipole enters the positive saturation state. When the applied magnetic field strength is less than β., the magnetic dipole enters the negative saturation state, and when the applied magnetic field strength is between β and α, both state may magnetic dipole be in, depending on the historical change of the magnetic field, which manifests itself as a memory characteristic of ferromagnetic matter.
While using Preisach Theory to calculate the core magnetic flux density, it is only necessary to calculate the difference between the integral of the magnetic dipole distribution density in region S
+ and the integral in region S
−. in the triangle ABC in Preisach diagram as Fig. 2 shows.
$$ B(H)={B}_S\left({\displaystyle {\iint}_{S_{+}}\mu \left(\alpha, \beta \right) d\alpha d\beta -{\displaystyle {\iint}_{S_{-}}\mu \left(\alpha, \beta \right) d\alpha d\beta}}\right) $$
(1)
In Equation (1), μ(α, β) represents the distribution density of magnetic dipoles, and should have the following properties:
-
1)
α < β, μ(α, β) = 0;
-
2)
if α > H
sat
or β < −H
sat
, μ(α, β) = 0
Where, H
sat
is positive saturation field strength;
-
3)
μ(α, β) = μ(−β, −α)
The changes of magnetic field strength H influence the value of flux density B in the following way: when H is greater than H
sat
, S
+
covers the entire triangle area, at which time the core is in a positive saturation state. When H is smaller than − H
sat
, S
−
, covers the entire triangle area, at which time the core is in a negative saturation state. When H decreases, triangle ABC will be swept down by a straight line perpendicular to the β-axis, β = H, then all the magnetic dipoles whose β value is greater than H will enter the negative saturation state, so the swept area is covered by S
−. When H increases, triangle ABC will be swept down by a straight line perpendicular to the α-axis, α = H, then all the magnetic dipoles whose α value is smaller than H will enter the positive saturation state, so the swept area is covered by S
+.
Since it is hard to determine μ(α, β) and to calculate double integral, reference [8] put forward the assumption that variables in μ(α, β) can be separated.
$$ \mu \left(\alpha,\ \beta \right)={\mu}_{\alpha}\left(\alpha \right){\mu}_{\beta}\left(\beta \right) $$
(2)
So double integral is transferred into single integral, and formulas (3, 4 and 5), can be deduced according to the shapes of different Preisach diagrams [16].
$$ {B}_i(H)={\left({B}_d(H)/2+{B}_u(H)/2\right)}^2{\left({B}_d(H)\right)}^{-1} H\ge 0 $$
(3)
$$ \sqrt{B_d(H)}= F\left(- H\right) $$
(4)
$$ {B}_u(H)={B}_d(H)-2 F(H) F\left(- H\right) $$
(5)
Where, B
d
(H), B
u
(H), B
i
(H) is the limiting descending branch, the limiting ascending branch and the virgin curve respectively. According to the symmetric properties of the limiting hysteresis loop:
$$ {B}_u(H)=-{B}_d\left(- H\right) $$
(6)
Assuming that (H
1, B
1) is the local extremum of magnetization trajectory, the descending branch and ascending branch from it can be calculated by:
$$ {B}_{H_1,{B}_1}^u=-{B}_d\left(- H\right)-{B}_d\left({H}_1\right)+{B}_1+2\mathrm{F}\left(-{H}_1\right)\mathrm{F}(H) $$
(7)
$$ {B}_{H_1,{B}_1}^d={B}_d(H)-{B}_d\left(-{H}_1\right)+{B}_1-2\mathrm{F}\left({H}_1\right)\mathrm{F}\left(- H\right) $$
(8)
Where, F(H) can be deduced by formulas (4), (5):
$$ F(H) = {\displaystyle {\int}_H^{H_{sat}}{\mu}_{\alpha}}\left(\alpha \right) d\alpha $$
$$ = \left\{\begin{array}{c}\hfill \left({B}_d(H)/2-{B}_u(H)/2\right)/\sqrt{B_d(H)} H\ge 0\hfill \\ {}\hfill \sqrt{B_d\left(- H\right)}\kern5.25em H<0\hfill \end{array}\right. $$
(9)
The alternating sequence of the magnetization’s local extrema is stored in a descending order and updated in stack form. Formulas (3, 4, 5, 6, 7, 8 and 9) are used to calculate the flux density when the core is not saturated, and when it is saturated, magnetization trajectory will turn into single valued, so it is easy to calculate using curve fitting method.
In conclusion, according to the Preisach Theory, CT core model can be established accurately by measuring the descending branch of limiting hysteresis loop and positive saturation magnetization curve, when structure of CT’s core is fixed and the outside temperature does not change obviously.
2.2 Modeling of closed-core CT based on Preisach theory
CT’s simplified equivalent circuit is shown in Fig. 3. In the figure, i
1 and i
m
are the primary and exciting currents which are both converted to secondary side; also, i
2 is the secondary current, R
2 and L
2 are the total resistance and total inductance of secondary side, and L
m
is the equivalent excitation inductance. The circuit equations are listed in Eqs. (10) and (11):
$$ \left\{\begin{array}{c}\hfill {i}_2{R}_2+{L}_2\frac{d{ i}_2}{ d t}={N}_2 A\frac{d B}{ d t}\hfill \\ {}\hfill {i}_2+{i}_m={i}_1\hfill \end{array}\right. $$
(10)
$$ {N}_2 A\frac{d B}{ d t}+{L}_2\frac{d{ i}_m}{ d t}+{i}_m{R}_2={i}_1{R}_2+{L}_2\frac{d{ i}_1}{ d t} $$
(11)
In Eqs. (10) and (11), N
2 is the number of secondary coil, A is the cross area of iron core, B is magnetic flux density in the core. Discrete Eq. (11) as follow:
$$ {N}_2 A{F}_s\left( B(k)- B\left( k-1\right)\right)+{L}_2{F}_s+{i}_m(k){R}_2 $$
$$ ={i}_1(k){R}_2+{L}_2{F}_s\left({i}_1(k)-{i}_1\left( k-1\right)\right) $$
(12)
F
s
is the sampling frequency. Equation (12) is called the solving equation of CT. Since the magnetic flux density B increases monotonously with the exciting current, the solving equation can be solved in the following way: assume that the flux density and the exciting current at time instant k are the same as at time instant k − 1, which means substituting i
m
(k) = i
m
(k − 1) and B(k) = B(k − 1) to the left side of (12), then calculate the value of the right side, compare it with the left, if left < right, increase i
m
(k), otherwise decrease i
m
(k), till left = right. The solving process is joint with core model, while the core model is air-gapped, air-gapped CT model is got, otherwise closed-core CT model is established. Core model based on Preisach Theory is established as follows.
Using the arctangent function and the hyperbolic function, the descending branch of limiting hysteresis loop of measured closed-core can be fitted. As shown in Fig. 4, the measured CT core is closed, with circular shape, the average magnetic path length l = 0.62 m, sectional area A = 2.508 × 10− 3 m
2.
The non-integer power is used to fit the measured positive saturation curve, as shown in Fig. 5. Using the fitting result, the values of flux density for arbitrary magnetization process can be calculated according to Preisach Theory. The displayed results in Figs. 6 and 7 show the magnetization curve and the waveform of applied magnetic field strength respectively.
2.3 Modeling of air-gapped CT based on Preisach Theory
The air-gapped CT core is shown in Fig. 8, where l
0 is the gap length. Since the air gap length is much smaller than the total length of the magnetic circuit, typically about one thousandth of it, so the following assumptions can be made: the fracture surfaces are parallel and perpendicular to the magnetic field lines; the magnetic field lines at the edge of the gap will not protrude. The total length of magnetic circuit could be set as l, and length of magnetic circuit in core as l
1, magnetic field strength in core as H
1, and magnetic field strength in gap as H
0; hence, the exciting current converted to one turn is.
$$ {i}_e={H}_1{l}_1+{H}_0{l}_0 $$
(13)
Where, H
0 = B/μ
0 and the magnetization curve of air-gapped core could be calculated by:
$$ H= F(B)={H}_1+\frac{H_0{l}_0}{l}= f(B)+\frac{B{l}_0}{\mu_0 l} $$
(14)
In (14), f(B) could be obtained from the inverse function of closed-core magnetization curve. In the case of the descending branch of closed-core’s limiting hysteresis loop known, the descending branch of the limiting hysteresis loop of the air-gapped core with different lengths of air gap can be calculated by (14), as shown in Fig. 9.
In Fig. 9, waveforms a, b, and c are the descending branches of limiting hysteresis loop of the core with air gap ratios of 0.001, 0.002, and 0.003, respectively. As can be seen in Fig. 9, the longer the air gap is, the greater the saturation magnetic field strength will be. In this paper, the iron core model is established by selecting the data of the core with the air gap ratio of 0.001.
As a comparison, Fig. 10 shows the limiting hysteresis loops of closed core and air-gapped core with gap ratio of 0.0005.
