Modeling and simulation of air-gapped current transformer based on Preisach Theory

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, 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)

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 ₊.

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)

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)

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.

3.1 Characteristics of saturation

A test circuit is simulated in PSCAD/EMTDC to generate different primary currents, as shown in Fig. 11. The closed-core CT model and the air-gapped core CT model are inputted respectively with data of primary current to compare the saturation characteristics and remanence characteristics. Parameters are set as: current ratio 1: 500, load resistance 4 Ω, coil resistance 1 Ω, the load inductance and leakage inductance of the secondary side amount to 0.8 mH.

Characteristics of saturation are shown in Figs. 12 and 13. A-G fault occurs at 0.1 s, and the fault current contains a significant DC component. The primary current is converted to the secondary side.

The short dashed line in Fig. 12 is the curve of flux density in Tesla which its axis is shown on the right side. The solid line and the long dashed line are, respectively, primary and secondary current respectively in Amperes which their axis are depicted on the left side. It can be seen that the closed-core CT is saturated. The course of saturation can be analyzed from Figs. 12 and 13. In the first cycle after the fault, since the current transformer core has no remanence, the magnetic flux density increases from zero, and does not reach the saturation threshold in the first cycle, so the transformer is not saturated. Starting from the second cycle, due to the role of remanence caused by the first cycle, flux density gradually reaches the saturation threshold and exceed, so the core gets saturated. When primary current crosses zero and increases in reverse, the flux density decreases gradually until saturation is removed. The coordinate of the saturation starting point (0.15, 1.787) for the third cycle is marked in Fig. 12.