Equivalent Circuit Model for the Insulated Core Transformer

Equivalent Circuit Model for the Insulated Core Transformer 1 first-author 2 corresp Corresponding author, leicao@hust.edu.cn Corresponding author, leicao@hust.edu.cn leicao@hust.edu.cn 2 1 1 China Electric Power Research Institute, Nanjing 210003, China China Electric Power Research Institute, Nanjing 210003, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China The design of the insulated core transformer (ICT) needs to consider the flux leakage effects. An equivalent linear circuit model is proposed based on the principle of duality. It is composed by two types of leakage inductances: conventional leakage between windings and special leakage introduced mainly by the insulation gaps. The values of leakage inductances depend on the dimensions of the core, gaps, or windings and the property of magnetic materials. The circuit allows for quantitatively evaluating influences of ICT internal parameters on its output properties. The winding self and mutual inductance matrix is mathematically converted to derive the inductance formula. As an example, the leakage parameters of a six-stage two dimensional (2D) ICT are calculated and analyzed. High voltage power supply Insulated core transformer Leakage inductance Equivalent circuit 1 Introduction 1 Industrial irradiation has become the fundamental application area of electron beam accelerators in recent years [ 1 1 - 4 4 1-4 ]. A high voltage generator is the kernel component in the power supply for accelerating electrons. The direct current (DC) voltage source based on insulated core transformer (ICT) technology has advantages over others, from several hundred kV to a few MV. It was invented by Van de Graaff to convert industrial frequency (50 Hz) alternating current (AC) voltage to high DC voltage [ 5 5 ]. The magnetic core of this step-up transformer is divided into several electrically insulated and magnetically coupled sections by thin insulation layers (usually teflon or mica sheets). Each section has its own secondary winding and all the windings are arranged in a top-bottom configuration. The AC output from each winding is converted into DC voltage by a double rectifier circuit, and all the output terminals are connected in series to generate high voltage (HV) with the overall efficiency higher than 80%. Because the insulation requirement for each section is never more than the localized DC output of that section, the creepage distance of insulation in an ICT is smaller than in a conventional HV transformer. To further reduce the size, the planar ICT based on printed circuit boards (PCBs) at mid-frequency was proposed [ 6 6 , 7 7 ]. However, the introduced insulation gaps result in magnetic flux leakage in an ICT, because the gaps ( μ r =1) have much higher reluctances compared to the magnetic core. This effect will cause a drop in magneto motive force (MMF) and, accordingly, the output voltages from the secondary windings. The gap inductance is approximated by the reluctance model [ 7 7 ]. <math display="block" id="M1"><mrow><mi>L</mi><mo>=</mo><mfrac><mrow><msup><mi>N</mi><mn>2</mn></msup></mrow><mi>R</mi></mfrac><mn>,</mn><mi>R</mi><mo>=</mo><mfrac><mi>l</mi><mrow><msub><mi>μ</mi><mtext>r</mtext></msub><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>A</mi><mtext>s</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where N is the number of turns of the primary winding, l is the dielectric thickness, μ r is the relative permeability of the insulation material, and A s is the effective cross-section area. Fringing flux at the gap periphery makes A s larger than its physical area [ 8 8 ]. For the ICT with multiple insulation gaps, the equivalent gap inductance seen at each winding terminal will deviate from Eq. (1) and vary from one gap to another. In addition, the leakage inductances between adjacent windings also reduce the magnetic flux more or less, depending on the locations and dimensions of the windings. Ideally, the leakage inductance between winding w1 and w2 can be estimated as[ 9 9 ]. <math display="block" id="M2"><mrow><msub><mi>L</mi><mrow><mtext>w1w2</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>S</mi><msup><mi>N</mi><mn>2</mn></msup></mrow><mrow><msup><mi>W</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>H</mi><mtext>a</mtext></msub><mo>+</mo><msub><mi>H</mi><mrow><mtext>w1</mtext></mrow></msub><mo>/</mo><mn>3</mn><mo>+</mo><msub><mi>H</mi><mrow><mtext>w2</mtext></mrow></msub><mo>/</mo><mn>3</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where S is the winding cross-section area. W is the winding width. H w1 and H w2 is the height of winding w1 and w2, respectively. H a is the height of air region between w1 and w2. Precise calculation of the winding leakage inductance requires knowing the magnetic field distribution [ 10 10 ]. Gaps between windings make the analytical calculation difficult in an ICT. Transformer models are needed to accurately analyze and mitigate the above mentioned leakage effect. Generally, the modeling methods can be classified into four groups. The first type is the finite element method (FEM). It is most accurate to simulate the transient response of three-dimensional (3D) transformers, but the method is time-consuming. It is used to validate the equivalent circuit model in this paper. The second group utilizes the impedance or admittance matrix, as in the electromagnetic transients program (EMTP) [ 11 11 ]. This black-box approach can be easily implanted, but the topology details of the transformers are missing. The third kind is based on the coupling between magnetic (transformer) and electrical (external circuit) equations taking into account core dimensions and material characteristics [ 12 12 ]. The 2D transformer model was developed, but a full 3D transformer structure has not been considered. The last group tries to establish the equivalent electrical circuit of transformers based on the principle of duality [ 13 13 - 15 15 13-15 ]. The model can be realized in commercial circuit software. The transformer model of the last group with multiple insulation layers (several millimeter thicknesses) in an ICT has not been reported in the literature. This Letter investigates the first time leakage phenomena for the design of a multi-winding ICT based on its physical nature. The leakage inductances between windings and the leakage inductances due to the insulation gaps are calculated numerically by winding a mutual inductances matrix. The unique feature of the leakage inductance model is to facilitate the analysis of the internal parameters and the steady-state response of an ICT. Section 2 gives a detailed process to construct the electrical circuit of a general multi-winding ICT based on the principle of duality. In Sect. 3, the method to calculate each inductance in the equivalent circuit is demonstrated by converting the winding inductances matrix according to the terminal output characteristics. The typical example of a six-stage 2D ICT structure is analyzed by the proposed electrical circuit model in Sect. 4. 2 Equivalent Circuit Model 1 Figure 1 Figure 1 shows the ideal magnetic flux distribution in the multi-winding ICT with three core legs and top/bottom yokes. For the sake of simplicity, the rectifier circuits are not shown. Besides one primary winding (p) and n secondary windings (s1 ∼ s n ), there are n gaps (g1 ∼ g n ) located between adjacent cores in the central leg. The primary winding with larger cross section is at the bottom, while secondary windings are arranged accordingly at the upper sections. Due to the structural symmetry, one half of the geometry is used to illustrate the magnetic flux in the core and gaps ( ϕ mp , ϕ m1 ∼ ϕ mn ) and the leakage flux in the air (windings) region ( ϕ s1 ∼ ϕ sn ). The single-phase structure is sufficient to understand the leakage effects in an ICT. The leakage between windings on different core legs can be calculated using the same principle. According to the flux paths in Fig. 1 Fig. 1 , the reluctance model is described in Fig. 2 Fig. 2 . R m is the magnetizing reluctance. R s1 ∼ R sn are the leakage reluctance between windings. R g1 ∼ R gn are the reluctance associated principally with the gaps, and also to the surrounding magnetic cores. N p I p , N 1 I 1 ∼ N n I n are the assumed infinitely small sources positioned at the center of the corresponding winding. Based on the principle of duality [ 15 15 ], the equivalent electrical circuit model is shown in Fig. 3 Fig. 3 in the form of π network. Both the magnetizing inductance, L m (in parallel with the primary winding as in a traditional transformer), and the inductances L g1 ∼ L gn (in parallel with the left side of the ideal transformer representing secondary windings) are influenced by the core properties, as well as gap dimensions. For the sake of simplicity, we call the special inductance L g1 ∼ L gn in this Letter to distinguish from the pure gap inductance in Eq. (1). As discussed in Ref. [ 13 13 ], the winding leakage inductances, L s1 ∼ L sn , are mutually coupled with each other, due to the overlap of adjacent magnetic flux, Φ si and Φ si+1 ( i =1,2 ..., n -1). It was found in Ref. [ 16 16 ] that the equivalent π network of a multi-winding transformer without gaps is physically based, which can be mathematically converted into the form of a T network [ 17 17 ]. If there are no gaps, the circuit in Fig. 3 Fig. 3 returns to the case in Ref. [ 14 14 ]. In order to design an ideal ICT without any flux leakage, L s1 ∼ L sn should be as small as possible (ideally zero), while L g1 ∼ L gn should be as large as possible (ideally infinite). In this situation, the secondary windings will have the same output voltage. The evaluation of the leakage phenomena in a real ICT depends on the calculation of leakage inductances, which relates critically to the magnetic field distribution in the interested computation regions. However, the real flux pattern with both vertical and horizontal components will deviate from the assumed ideal case in Fig. 1 Fig. 1 , making the direct analytical calculation of the different reluctance or inductance components difficult via Eq. (1) and (2). The reluctance will have complicated components connected in series or parallel form. The calculation of each component is not necessary because we only care about the equivalent reluctances/inductances seen at and between windings terminals. A numerical method is preferred to determine the leakage inductances in the next section. 3 Inductance Determination 1 Generally, an inductance can be accurately calculated by the magnetic energy storage formula[ 9 9 ] <math display="block" id="M3"><mrow><mi>L</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mtext>r</mtext></msub><msub><mi>μ</mi><mn>0</mn></msub></mrow><mrow><msup><mi>I</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∭</mo></mrow></mstyle><msup><mi>H</mi><mn>2</mn></msup><mtext>d</mtext><mi>v</mi><mn>,</mn></mrow></math> where H is the magnetic field intensity inside the volume of an inductor material with a relative permeability μ r , and I is the current flowing through the inductor. Existing methods for characterizing the terminal output properties of a multi-winding transformer are usually based on the winding self and mutual inductances matrix ( M w ). Its elements are normally calculated through expression Eq. (3) in magneto-static FEM simulations. For the ICT structure, its output voltages obtained by the equivalent circuit in Fig. 3 Fig. 3 should be the same as the mathematical results predicted by M w . <math display="block" id="M4"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>V</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>ω</mi><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>w</mi></msubsup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>I</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mn>1</mn></mrow></msub><mn>,</mn></mrow></math> where ω =2 πf is the angular frequency. For the sake of computation convenience, the matrix is written in 2× 2 block matrix form, <math display="block" id="M5"><mrow><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>w</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mtable columnalign="left"><mtr><mtd><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mn>1</mn><mo>×</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mi>ω</mi><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mtext> </mtext><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mn>1</mn><mo>×</mo><mi>n</mi></mrow><mrow><mo stretchy="false">(</mo><mi>ω</mi><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mi>n</mi><mo>×</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mi>ω</mi><mn>3</mn><mo stretchy="false">)</mo></mrow></msubsup><mtext> </mtext><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><mrow><mo stretchy="false">(</mo><mi>ω</mi><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup></mtd></mtr></mtable><mo>)</mo></mrow><mn>.</mn></mrow></math> By equating the voltage-current relationship at each winding terminal, the magnetizing inductance and gap associated leakage inductances are expressed by <math display="block" id="M6"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><msub><mi>L</mi><mi>m</mi></msub><mn>,</mn></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mn>1</mn></mrow></msub><mn>,...,</mn><msub><mi>a</mi><mi>n</mi></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mi>n</mi></mrow></msub><msub><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mo stretchy="false">(</mo><mn>1,</mn><mo>−</mo><msub><mi>b</mi><mn>1</mn></msub><mn>,...,</mn><mo>−</mo><msub><mi>b</mi><mi>n</mi></msub><msub><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msup><mrow><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>ω</mi></msubsup><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where a n = N p / N n and b n = 1/ a n . The winding leakage inductance matrix is <math display="block" id="M7"><mrow><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><mi>l</mi></msubsup><mo>=</mo><mstyle mathvariant="bold" mathsize="normal"><mi>B</mi></mstyle><msup><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>D</mi></mstyle><mo>+</mo><mstyle mathvariant="bold" mathsize="normal"><mi>C</mi></mstyle><msup><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mi>b</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>,</mn></mrow></math> where <math display="block" id="M8"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mi>b</mi></msub><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>ω</mi><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>ω</mi><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><msup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>ω</mi><mn>2</mn><mo stretchy="false">)</mo></mrow></msup><mo>/</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mo stretchy="false">(</mo><mi>ω</mi><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mn>.</mn></mrow></math> <math display="block" id="M9"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>B</mi></mstyle><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><mo>−</mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><msub><mi>a</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><mo>−</mo><msub><mi>a</mi><mn>3</mn></msub></mrow></mtd><mtd><mo>⋯</mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mtd><mtd><mrow><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M10"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>C</mi></mstyle><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>=</mo><mi>ω</mi><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><msub><mi>b</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>b</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>b</mi><mn>3</mn></msub></mrow></mtd><mtd><mo>⋯</mo></mtd><mtd><mrow><msub><mi>b</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><msub><mi>b</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>b</mi><mn>3</mn></msub></mrow></mtd><mtd><mo>⋯</mo></mtd><mtd><mrow><msub><mi>b</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mi>b</mi><mn>3</mn></msub></mrow></mtd><mtd><mo>⋯</mo></mtd><mtd><mrow><msub><mi>b</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mrow><msub><mi>b</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mi>b</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M11"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>D</mi></mstyle><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mn>1</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mn>2</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mn>3</mn></mrow></msub></mrow></mtd><mtd><mo>⋯</mo></mtd><mtd><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mi>n</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mn>2</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mn>3</mn></mrow></msub></mrow></mtd><mtd><mo>⋯</mo></mtd><mtd><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mi>n</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mn>3</mn></mrow></msub></mrow></mtd><mtd><mo>⋯</mo></mtd><mtd><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mi>n</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mi>n</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>/</mo><msub><mi>L</mi><mrow><mi>g</mi><mi>n</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> The leakage inductances, L s 1 ∼ L sn , are the diagonal elements of matrix <math id="M12"><mrow><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><mi>l</mi></msubsup></mrow></math> and the mutual inductances, M i , j ( i , j =1,..., n , i i ), correspond to the non-diagonal elements. All the leakage inductances formulas have been derived based on the winding matrix M w . In the next section, a six-stage ICT will serve to demonstrate the validity of the equivalent circuit model. 4 Example of a Six-stage ICT 1 Figure 4 Figure 4 shows one half structure of the studied ICT with the indicated dimensions. It has six secondary windings with the same dimensions. The number of turns is 50 for the primary winding, and 1000 for all the secondary windings. The thickness of the insulation layer is 2 mm. A sinusoidal voltage source with a peak amplitude of 500 V and a frequency of 50 Hz is connected directly to the primary winding. Because of the linear inductances in the equivalent circuit, we assume that the magnetic core behaves linearly. The influences of core saturation on the leakage inductances will also be considered. By manipulating the inductance matrix of the seven windings (p, s1∼s6), the calculated inductances per unit depth in the equivalent circuit are listed in Table 1 Table 1 and Table 2 Table 2 . The winding leakage inductances (several mH) are much smaller than the magnetizing inductance and gap associated leakage inductances (several tens of mH). The coupling between different winding leakage inductance is weak. 1 1 L 1 1 Equivalent circuit (H) 1 1 Eq. (1) (H) 1 1 Core saturation (mH) 1 1 Without gap (H) 1 1 L m 1 1 0.5914 1 1 – 1 1 2.3653 1 1 0.9540 1 1 L g1 1 1 0.2190 1 1 0.2827 1 1 11.3844 1 1 5.6965 1 1 L g2 1 1 0.2734 1 1 0.2827 1 1 10.6888 1 1 4.6990 1 1 L g3 1 1 0.2699 1 1 0.2827 1 1 10.5646 1 1 4.5876 1 1 L g4 1 1 0.2711 1 1 0.2827 1 1 10.2020 1 1 4.4150 1 1 L g5 1 1 0.2558 1 1 0.2827 1 1 9.9503 1 1 4.2680 1 1 L g6 1 1 0.5101 1 1 0.2827 1 1 4.6460 1 1 2.0620 1 1 L&M 1 1 Equivalent circuit (mH) 1 1 Eq. (2) (mH) 1 1 Core saturation (mH) 1 1 Without gap (mH) 1 1 L s1 1 1 4.6429 1 1 4.2257 1 1 3.8291 1 1 4.6079 1 1 L s2 1 1 3.1973 1 1 2.1589 1 1 2.9020 1 1 3.2028 1 1 L s3 1 1 3.1975 1 1 2.1589 1 1 2.9082 1 1 3.2028 1 1 L s4 1 1 3.2132 1 1 2.1589 1 1 2.9265 1 1 3.2029 1 1 L s5 1 1 3.2649 1 1 2.1589 1 1 2.9760 1 1 3.2532 1 1 L s6 1 1 3.5284 1 1 2.1589 1 1 3.0613 1 1 3.5286 1 1 M s1,s2 1 1 0.6866 1 1 – 1 1 0.5073 1 1 0.6719 1 1 M s1,s3 1 1 0.4491 1 1 – 1 1 0.2798 1 1 0.4481 1 1 M s1,s4 1 1 0.2060 1 1 – 1 1 0.1309 1 1 0.1986 1 1 M s1,s5 1 1 0.0988 1 1 – 1 1 0.0726 1 1 0.1239 1 1 M s1,s6 1 1 0.0411 1 1 – 1 1 0.0404 1 1 0.0240 1 1 M s2,s3 1 1 0.3987 1 1 – 1 1 0.3642 1 1 0.3984 1 1 M s2,s4 1 1 0.3181 1 1 – 1 1 0.2148 1 1 0.3240 1 1 M s2,s5 1 1 0.1339 1 1 – 1 1 0.0929 1 1 0.1243 1 1 M s2,s6 1 1 0.0507 1 1 – 1 1 0.0464 1 1 0.0493 1 1 M s3,s4 1 1 0.3938 1 1 – 1 1 0.3548 1 1 0.3984 1 1 M s3,s5 1 1 0.3023 1 1 – 1 1 0.2079 1 1 0.2990 1 1 M s3,s6 1 1 0.1080 1 1 – 1 1 0.0828 1 1 0.1242 1 1 M s4,s5 1 1 0.3648 1 1 – 1 1 0.3401 1 1 0.3482 1 1 M s4,s6 1 1 0.2514 1 1 – 1 1 0.1881 1 1 0.2488 1 1 M s5,s6 1 1 0.2504 1 1 – 1 1 0.2882 1 1 0.2478 The third column is the calculated gap inductances by Eq. (1) and winding leakage inductances by Eq. (2). Compared with the values of L g1 ∼ L g6 in Table 1 Table 1 , it is obvious that Eq. (1) cannot exactly predict the gap associated leakage inductances seen at secondary winding terminals. The relative difference lies between 22% and 80%. If the fringing field effect is considered, the pure gap inductance will increase by about 3% [ 8 8 ]. This deviation mainly comes from the additional large core inductance not considered in Eq. (1). Indeed, if there are no gaps (insulation layers are replaced by magnetic cores), the calculated inductances L g1 ∼ L g6 are much higher, which is the solo contribution of magnetic cores, as shown in the last column of Table 1 Table 1 . It should be pointed out that the gap associated inductances at secondary windings are different, while most previous publications concerning ICT problems assume the same value [ 6 6 , 7 7 , 18 18 ]. On the other side, the analytic Eq. (2) underestimates the winding leakage inductance by 8.99% ( L s1 ), 32.48% ( L s2 and L s3 ), 32.81% ( L s4 ), 33.87% ( L s5 ), and 38.81% ( L s6 ). The differences come from the assumptions in Eq. (2): the magnetic flux flows in the horizontal direction in the winding region; the magnetic field is trapezoidal along the direction of winding height; the total amp-turns are zero between two adjacent windings. It is noted from the last column in Table 2 Table 2 that the gaps have negligible influences on the winding leakage inductances. These results confirm the effectiveness of the equivalent circuit model, and specially, the mutual inductances between winding leakage inductances can also be determined. In the worst case, when the magnetic core is highly saturated, the leakage inductances in the equivalent circuit are listed in the fourth column (note the different units in Table 1 Table 1 ). The inductances L g1 ∼ L g6 are significantly reduced and become comparable with the winding leakage inductances in Table 2 Table 2 . Therefore, the magnetic flux in the ICT is seriously leaky. As a result, its output properties will deteriorate. The peak amplitudes of the no-load output voltages from the six stages are listed in Table 3 Table 3 . The equivalent circuit is implanted in an electrical circuit simulator and an independent transient FEM simulation are performed according to the ICT structure in Fig. 4 Fig. 4 . The results from the equivalent circuit method (second column) are verified by the FEM simulation (third column). The voltage difference between the first and sixth stage reaches to 0.6082 kV. However, if the gaps are not present, the output voltages are uniform and the maximum difference is only 0.0788 kV in the last column of Table 3 Table 3 . As predicted, the voltages decrease and are highly inhomogeneous if the core saturates (fourth column). The excitation current passing through the primary winding increases up to 27 times the normal value. The large current will probably harm the ICT. 1 1 Stage 1 1 Equivalent circuit (kV) 1 1 FEM simu. (kV) 1 1 Core saturation (kV) 1 1 Without gap (kV) 1 1 V s1 1 1 9.2887 1 1 9.2889 1 1 5.1574 1 1 9.9464 1 1 V s2 1 1 9.0242 1 1 9.0243 1 1 3.3321 1 1 9.9218 1 1 V s3 1 1 8.8487 1 1 8.8492 1 1 2.3012 1 1 9.9037 1 1 V s4 1 1 8.7368 1 1 8.7367 1 1 1.6653 1 1 9.8895 1 1 V s5 1 1 8.6790 1 1 8.6790 1 1 1.2511 1 1 9.8779 1 1 V s6 1 1 8.6805 1 1 8.6802 1 1 0.9689 1 1 9.8676 One method to eliminate the gap associated flux leakage in an ICT is to add compensation capacitances at the output terminals of the secondary windings [ 6 6 , 7 7 , 18 18 ]. The values of capacitances are calculated by <math display="block" id="M13"><mrow><msub><mi>C</mi><mi>n</mi></msub><mo>=</mo><msubsup><mi>a</mi><mi>n</mi><mn>2</mn></msubsup><mo>/</mo><mo stretchy="false">(</mo><msup><mi>ω</mi><mn>2</mn></msup><msub><mi>L</mi><mrow><mtext>g</mtext><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where a n = 50/1000. At the primary winding, the magnetizing inductance is usually compensated with a tuning capacitance ( C m =1/( ω 2 L m )), as in a traditional transformer to reduce further the reactive power. When the compensation capacitances C n are positioned at the terminals of the secondary windings of the ICT, the output voltages become homogeneous and approach 10 kV, which is the secondary output of an ideal transformer with the turn ratio N p : N s = 1: 20 ( V p = 500 V) without any leakage. These results are also confirmed by the traditional method with the winding inductance matrix. It should be stated that the circuit in Fig. 4 Fig. 4 can also be used to study the steady-state response of a 3D ICT structure in case the secondary windings are loaded, short-circuited, and open-circuited. 5 Conclusion 1 In summary, a physically equivalent circuit was proposed to study the flux leakage effects in the insulated core transformer. The circuit can be implanted feasibly in electrical circuit simulators. The circuit parameters were determined exactly by the traditional winding self and mutual inductance matrix from magneto-static FEM simulations. The major contribution of flux leakage is from the insulation gaps. Although the equivalent circuit presented in this work is for ICT without losses (insulation layer, winding and core losses), it could be slightly modified to accommodate the loss nature of a 3D ICT. References Zimek Z , Przybytniak G and Nowicki A . Optimization of electron beam crosslinking of wire and cable insulation . 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DOI: 10.1063/1.4884340 http://doi.org/10.1063/1.4884340 A combined compensation method for the output voltage of an insulated core transformer power supply 10.1007/s41365-016-0060-3 Supported by National Natural Science Foundation of China (No. 11305068), and the "2011 project" organized by Hubei Collaboration Innovation Center of Non-power Nuclear Technology. 2015-06-26 2015-07-30 2015-09-08 2016-05-12 2016-06-20 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2016 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-016-0060-3 http://dx.doi.org/10.1007/s41365-016-0060-3 . 2016 Nuclear Science and Techniques 03 27