A Design Method for a Mutual Inductance Coupled Three-Tuned Foster-Type Filter

Abstract

A design method for a mutual inductance coupled triple-tuned Foster-type filter, comprising: Step1. Initially determine the inductance and capacitance values of each branch in the filter based on the driving point impedance function of the LC one-port network; Step2. Determine the cross-sectional dimensions of the wires of each filtering coil; Step3. Substitute the self-inductance and mutual inductance formulas of the filtering coils into the mutual inductance coupling equations of the circuit, and solve the equations by the Newton-Raphson method to obtain the number of turns, radius, and spacing of each coil; Step4. Verify the effectiveness of the filter design scheme. This device arranges multiple disk coils coaxially, which can greatly reduce the floor area of the filtering device and achieve the function of multi-harmonic filtering. Hollow disk coils are used to improve the utilization rate of the wire, and the self-inductance and mutual inductance of the disk coils are accurately calculated with corresponding formulas; in the design, the coil with a larger size is split into several coaxial small coils to further compress the floor area of the coil group; the mutual inductance coupled triple-tuned Foster-type filter provided by this method fully considers the actual manufacturing details and has high practical value.

Description

Technical Field

[0001] The present invention relates to the technical field of filters, and particularly relates to a design method for a mutual inductance coupled three-tuned Foster type filter. Background Art

[0002] Passive filters are widely used in power systems for harmonic suppression. To avoid the problem of core saturation under large current conditions, power filters usually use air-core coils. Such coils will generate a magnetic field spreading over a large area during operation. To prevent the mutual inductance magnetic flux coupling between each filtering branch, multiple filtering coils need to be arranged independently and maintain a sufficient distance, which will lead to the problems of large floor area of the filtering system and electromagnetic environmental pollution, and increase the installation and layout costs of the filtering system. The mutual inductance coupled three-tuned Foster type filter of this patent is a new type of filtering device, which arranges multiple disk coils coaxially and closely, greatly reducing the floor area while realizing the three-harmonic filtering function.

[0003] Although a similar filtering principle has been proposed in Chinese Patent Document ZL201210334550.8, it is only a conceptual description of the principle of the mutual inductance coupled filter, and does not consider the details in the actual manufacture of the mutual inductance coupled filter, and there is still a large gap from the actual design and manufacture. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a design method for a mutual inductance coupled three-tuned Foster type filter, providing technical guidance for the actual design of the mutual inductance coupled three-tuned filter.

[0005] To solve the above technical problem, the technical solution adopted by the present invention is:

[0006] A design method for a mutual inductance coupled three-tuned Foster type filter, comprising:

[0007] Step1. Combining the voltage level and capacity of the filter, initially determining the inductance and capacitance values of each branch in the filter with the driving point impedance function of the LC one-port network;

[0008] Step2. Performing circuit simulation to obtain the estimated current values on each filtering branch, and determining the wire cross-sectional dimensions of each filtering coil in combination with the current-temperature rise equation of the coil;

[0009] Step3. Substituting the self-inductance and mutual inductance formulas of the filtering coils into the mutual inductance coupling equations of the circuit, and solving the equations with the Newton-Raphson algorithm to obtain the number of turns, radius, and spacing of each coil;

[0010] Step 4: Calculate the corresponding inductance matrix based on the structural dimensions of the actual filter coil array, and then simulate the relevant filter circuit to obtain the frequency-impedance characteristic curve of the filter and the filter technical indicators such as the harmonic current and harmonic voltage distortion rate, and verify the effectiveness of the filter design scheme.

[0011] The filter circuit structure of the above mutual inductance coupled three-tuned Foster type filter is as follows:

[0012] A coil is serially arranged between two stages of the filter , , and a filter capacitor . Between the connection point of the coil and and another stage, a serially connected coil and a filter capacitor are provided. Between the connection point of the coil and and another stage, a serially connected coil and a filter capacitor are provided.

[0013] Any two of the above coils , , , , are mutually inductively coupled, with a total of 10 pairs of mutual inductances.

[0014] The above two mutually inductively coupled coils are disk coils arranged coaxially. The coils , , , , form a magnetic coupling inductance array.

[0015] In the above Step 1, according to Otto Brune's theorem, for a single-port LC network, its driving-point impedance function Z ( s ) must be one of the following two forms:

[0016] ; (1)

[0017] ; (2)

[0018] where ω z1 , ω z2 , … are impedance zeros, ω p1 , ωp2 , … are impedance poles, K is an arbitrary constant;

[0019] The mutual inductance coupled triple - tuned Foster - type filter that filters the 3rd, 5th, and 7th harmonics simultaneously, its driving - point impedance function Z ( s ) is:

[0020] ; (3)

[0021] where ω 0 = 100π; the coefficient K is determined according to the reactive power compensation capacity of the filter; Using the Foster - type circuit to realize the driving - point impedance function formula (3), the uncoupled triple - tuned filter circuit can be obtained, and the values of each circuit element are:

[0022] ;

[0023] The parameter K can be determined according to the voltage level of the filter and the reactive power compensation capacity Q as follows:

[0024] ; (4)

[0025] U is the voltage level of the mutual inductance coupled triple - tuned Foster - type filter. According to the required reactive power compensation capacity Q the filtering parameters K can be obtained, and the circuit parameters of the uncoupled triple - tuned Foster - type filter can be obtained.

[0026] The equivalent condition of the mutual inductance coupled circuit of the above - mentioned mutual inductance coupled triple - tuned Foster - type filter and the uncoupled triple - tuned filter circuit is:

[0027] ; (5)

[0028] If the above formula holds, the mutual inductance coupled circuit of the filter can realize the triple - tuned filtering function.

[0029] For one of the two mutually inductive coupled disk - type coils mentioned above, the number of turns is N 1, the axial thickness is 2 h 1, and the inner and outer radii are respectively r 1, r 2, then the self - inductance solution in Step3 is:

[0030] ; (6)

[0031] where:

[0032] ; (7)

[0033] The dimensionless parameters are: ;

[0034] The self - inductance value of the coil is:

[0035] ; (8)

[0036] The functions included in Equation (6) are:

[0037] m F n ( a , b , x ): Generalized hypergeometric function, J n ( x ): n Order Bessel function, H n ( x ): n Order Struve function, K( x ), E( x ): Complete elliptic integrals of the first and second kind, G = 0.91596559…: Catalan constant.

[0038] For the above two mutually - inductively - coupled disk - type coils with the number of turns being N 1, N 2, the inner radii being r 1, r 3, the outer radii being r 2, r 4, and the axial thicknesses being 2 h 1, 2 h 2, and the distance between the mid - sections of the two coils being z 0, z 0 ≥ h 1 + h 2， Then the mutual - inductance solution between them is:

[0039] ; (9)

[0040] Where:

[0041] ; (10)

[0042] The generalized integrals in Formulas (6) and (9) are calculated by the Gauss - Laguerre quadrature rule.

[0043] When designing the above mutual inductance coupled triple-tuned Foster-type filter, the relationship between the current-carrying capacity and temperature rise of the filter coil needs to be considered, and the following formula is used:

[0044] ; (11)

[0045] Among them, I is the effective value of the current-carrying capacity of the coil; γ 0 is the temperature T of the wire at 0; T 0 = 293K (aluminum) γ 0 = 2.82×10 -8 Ω·m; α = 0.0043K -1 is the temperature coefficient of aluminum; the ambient temperature T ∞ = T sur = 300K, that is, 27°C; h is the convective heat transfer coefficient. For natural convection, that is, when there is no forced ventilation, it can be set h = 20W·m -2 ·K -1 ; ε is the thermal radiation coefficient of the aluminum wire, and it can be set ε = 0.8; σ = 5.670373×10 -8 W·m -2 ·K -4 is the Stefan-Boltzmann constant of blackbody radiation; the others are the geometric parameters of the coil, among which a , b are the long and short sides of the single-strand bare wire; n is the number of wire strands.

[0046] When winding the above coils, considering the current carried by each coil and combining the current-carrying capacity - temperature rise relationship formula (11), the cross-sectional area of each coil wire is taken as:

[0047] ; (12)

[0048] Among them η is the long side of the single-strand aluminum wire; that is to say, the 1st to 5th filter coils are wound with 5 strands, 2 strands, 1 strand, 4 strands, and 1 strand of wire respectively.

[0049] A design method of a mutual inductance coupled triple-tuned Foster filter provided by the present invention. This mutual inductance coupled triple-tuned filter adopts a Foster-type circuit and realizes a magnetic coupling inductance array with coaxially arranged disk coils. In the design, the actual geometric parameters of the disk coils are considered, including the number of turns, inner radius, outer radius, axial thickness, and wire turn size of the coils, and the self-inductance and mutual inductance values are accurately calculated by corresponding formulas. The disk coils adopt a hollow structure, that is, the inner radius of the coil is greater than zero, to improve the utilization rate of the wire and facilitate the assembly and heat dissipation of the coil array. The disk coil with a larger radius is split into coaxially connected small coils to further reduce the radius of the filter coil. According to the current values carried by each coil during operation and combined with the current-carrying-temperature rise equation of the coil, the temperature rise of each coil during operation can be calculated, so as to reasonably select the wire diameter of each coil wire. Brief Description of the Drawings

[0050] The present invention will be further described below in conjunction with the drawings and embodiments:

[0051] Figure 1 It is the mutual inductance coupled triple-tuned Foster filter circuit of the present invention;

[0052] Figure 2 It is the non-coupled triple-tuned Foster filter circuit of the present invention;

[0053] Figure 3 It is the schematic diagram of the coaxial disk coil structure of the present invention;

[0054] Figure 4 It is the structure of the mutual inductance coupled triple-tuned Foster filter coil array of 10kV / 3000kvar in the embodiment;

[0055] Figure 5 It is the frequency-impedance characteristic curve of the mutual inductance coupled triple-tuned Foster filter in the embodiment;

[0056] Figure 6 It is the current waveform on the bus after filtering by the mutual inductance coupled triple-tuned Foster filter in the embodiment;

[0057] Figure 7 It is the current waveform on the bus before filtering. Detailed Embodiments

[0058] The technical solution of the present invention will be described in detail below in conjunction with the drawings and embodiments.

[0059] A design method of a mutual inductance coupled triple-tuned Foster filter includes:

[0060] Step1. Combining the voltage level and capacity of the filter, initially determine the inductance and capacitance values of each branch in the filter with the driving point impedance function of the LC single-port network;

[0061] Step 2: Perform circuit simulation to obtain the estimated current values on each filter branch, and determine the cross-sectional dimensions of the wires of each filter coil in combination with the current-temperature rise equation of the coil.

[0062] Step 3: Substitute the self-inductance and mutual-inductance formulas of the filter coils into the mutual-inductance coupling equations of the circuit, and solve the equations with the Newton-Raphson algorithm to obtain the number of turns, radius, and spacing of each coil.

[0063] Step 4: Calculate the corresponding inductance matrix based on the structural dimensions of the actual filter coil array, and then simulate the relevant filter circuit to obtain the frequency-impedance characteristic curve of the filter and the filter technical indicators such as the harmonic current and harmonic voltage distortion rates, and verify the effectiveness of the filter design scheme.

[0064] The filter circuit structure of the above mutual-inductance coupling three-tuned Foster type filter is as follows:

[0065] A coil is serially arranged between two stages of the filter , , and a filter capacitor . Between the connection point of the coils and and another pole, a serially connected coil and a filter capacitor are provided. Between the connection point of the coils and and another pole, a serially connected coil and a filter capacitor are provided.

[0066] Any two of the above coils , , , , are mutually inductively coupled, with a total of 10 pairs of mutual inductances.

[0067] The above two mutually inductively coupled coils are disk coils arranged coaxially. The coils , , , , form a magnetic coupling inductance array.

[0068] In the above Step 1, according to the Otto Brune theorem, for a single-port LC network, its driving-point impedance function Z ( s ) must be one of the following two forms: ​​​​​​​​​​​​​​​​​​​​

[0069] ; (1)

[0070] ; (2)

[0071] where ω z1 , ω z2 , … are impedance zeros, ω p1 , ω p2 , … are impedance poles, K is an arbitrary constant;

[0072] The driving-point impedance function of the mutual inductance coupled triple-tuned Foster type filter that filters out the 3rd, 5th, and 7th harmonics simultaneously is Z ( s ) is:[[]]

[0073] ; (3)

[0074] where ω 0 = 100π; the coefficient K is determined according to the reactive power compensation capacity of the filter; using the Foster type circuit to implement the driving-point impedance function formula (3), an uncoupled triple-tuned filter circuit can be obtained, as Figure 2 shown. The values of each circuit element in the figure are:[[]]

[0075] ;

[0076] The parameter K can be determined according to the voltage level of the filter and the reactive power compensation capacity Q as follows:[[]]

[0077] ; (4)

[0078] U is the voltage level of the mutual inductance coupled triple-tuned Foster type filter. According to the required reactive power compensation capacity Q the filtering parameters K can be obtained, and the circuit parameters of the uncoupled triple-tuned Foster type filter can be obtained.[[]]

[0079] The equivalent condition between the mutual inductance coupled circuit of the above mutual inductance coupled triple-tuned Foster type filter and the uncoupled triple-tuned filter circuit is:[[]]

[0080] ; (5)

[0081] If the above equation holds, the mutual inductance coupled circuit of the filter can achieve the triple-tuned filtering function.[[]]

[0082] The above two mutually inductively coupled disk coils, one of which has a number of turns of N 1 and an axial thickness of 2 h 1, and inner and outer radii of r 1, r 2, then the exact solution of its self-inductance in Step 3 is:

[0083] ; (6)

[0084] Where:

[0085] ; (7)

[0086] The dimensionless parameter is: ;

[0087] The self-inductance value of the coil is:

[0088] ; (8)

[0089] The functions included in equation (6) are:

[0090] m F n ( a , b , x ): Generalized hypergeometric function, J n ( x ): n Bessel function of order, H n ( x ): n Struve function of order, K( x ), E( x ): Complete elliptic integrals of the first and second kind, G = 0.91596559…: Catalan constant.

[0091] The above two mutually inductively coupled disk coils have numbers of turns of N 1, N 2, inner radii of r 1, r 3, outer radii of r 2, r 4, and axial thicknesses of 2 h 1, 2 h 2, and the distance between the mid-sections of the two coils is z 0, z 0 ≥ h 1 + h 2， Then the exact solution of the mutual inductance between them is:

[0092] ; (9)

[0093] Wherein:

[0094] ; (10)

[0095] The improper integrals in formulas (6) and (9) are calculated by the Gauss-Laguerre quadrature rule.

[0096] When designing the above mutual inductance coupled triple-tuned Foster-type filter, the relationship between the current-carrying and temperature rise of the filter coil needs to be considered, and the following formula is used:

[0097] ; (11)

[0098] Wherein, I is the effective value of the current-carrying of the coil; γ 0 is the temperature T of the wire at 0; T 0 = 293K (aluminum) γ 0 = 2.82×10 -8 Ω·m; α = 0.0043K -1 is the temperature coefficient of aluminum; the ambient temperature T ∞ = T sur = 300K, that is, 27°C; h is the convective heat transfer coefficient. For natural convection, that is, when there is no forced ventilation, it can be set h = 20W·m -2 ·K -1 ; ε is the thermal radiation coefficient of the aluminum wire, and it can be set ε = 0.8; σ = 5.670373×10 -8 W·m -2 ·K -4 is the Stefan-Boltzmann constant of blackbody radiation; the others are the geometric parameters of the coil, wherein a , b are the lengths of the long and short sides of the single-strand bare wire; n is the number of wire strands.

[0099] When winding the above coil, considering the current carried by each coil and combining with the current-carrying - temperature rise relational formula (11), the cross-sectional area of each coil wire is taken as:

[0100] ; (12)

[0101] Whereinη is the long side of the single-strand aluminum wire; that is, the 1st to 5th filter coils are wound with 5 strands, 2 strands, 1 strand, 4 strands, and 1 strand of wire respectively.

[0102] Embodiment:

[0103] The design process of this mutual inductance coupled triple-tuned Foster-type filter is briefly described as follows: First, in combination with the voltage level and capacity of the filter, the inductance and capacitance values of each branch in the filter are preliminarily determined based on the driving-point impedance function of the LC one-port network; the estimated current values on each filter branch are obtained through circuit simulation, and the cross-sectional dimensions of the wires of each filter coil are determined in combination with the current-temperature rise equation of the coil; next, the self-inductance and mutual-inductance formulas of the filter coils are substituted into the mutual inductance coupling equations of this circuit, and the Newton-Raphson method is used to solve this system of equations to obtain the number of turns, radius, and spacing of each coil; the completion of this step will enable us to obtain the actual structural dimensions of this filter; finally, based on the structural dimensions of the actual filter coil array, the corresponding inductance matrix is calculated, and then the relevant filter circuit is simulated to obtain the frequency-impedance characteristic curve of this filter, as well as key filter technical indicators such as the harmonic current and harmonic voltage distortion rates, so as to verify the effectiveness of the filter design scheme.

[0104] 1. Determination of filter circuit parameters

[0105] In the existing similar invention patent ZL201210334550.8, the mutual inductance coupled triple-tuned filter is implemented with a Cauer-type circuit; the disadvantage of this circuit is that the values of several filter inductances vary greatly, resulting in large differences in the sizes of each coil during actual design, which is not conducive to the mechanical stability of the coil group, and coils with too large sizes are also not conducive to cost control during actual manufacturing; in this invention, the mutual inductance coupled triple-tuned filter is implemented with a Foster-type circuit; in the Foster-type filter circuit, the values of each filter inductance are relatively close, making the sizes of each filter coil similar, thus being conducive to the mechanical stability of the coil group and helping to control the manufacturing cost.

[0106] According to Otto Brune's theorem, for a single-port LC network, its driving-point impedance function Z ( s ) must be one of the following two forms:

[0107] ; (1)

[0108] ; (2)

[0109] where ω z1 , ω z2 , … are impedance zeros, ωp1 , ω p2 , … are impedance poles, K is an arbitrary constant.

[0110] Taking the mutual inductance-coupled triple-tuned Foster-type filter that can filter out the 3rd, 5th, and 7th harmonics simultaneously as an example; its driving-point impedance function Z ( s ) is:

[0111] ; (3)

[0112] where ω 0 = 100π. The coefficient K is determined according to the reactive power compensation capacity of the filter. Using the Foster-type circuit to implement the driving-point impedance function (3), the uncoupled triple-tuned filter circuit can be obtained, as shown in Figure 2 ; the values of each circuit element in the figure are

[0113] ;

[0114] The parameter K can be determined according to the voltage level of the filter and the reactive power compensation capacity Q as follows:

[0115] ; (4)

[0116] Taking U = 10 kV voltage level mutual inductance-coupled triple-tuned Foster-type filter as an example, assuming the reactive power compensation capacity of the filter is Q = 3000 kVar, then the filter parameters are solved from equation (4) as:

[0117] ; (5)

[0118] Thus, the circuit parameters of the uncoupled triple-tuned Foster-type filter are:

[0119] ; (6)

[0120] It can be proved that Figure 1 the equivalent condition of the mutual inductance-coupled circuit shown in Figure 2 and the uncoupled circuit shown in

[0121] ; (7)

[0122] As long as equation (7) holds, the mutual inductance-coupled circuit ( Figure 1 ) can achieve the triple-tuned filtering function.

[0123] 2. Self - inductance and mutual - inductance calculation formulas considering the actual size of the coil

[0124] In the existing similar invention patent ZL201210334550.8, the disc - type coil is idealized as a thin coil with infinitesimal axial thickness. Such a simplification has a large difference from the actual coil shape; the calculation formula in this patent will give values that are very different from the self - inductance and mutual - inductance values of the actual coil, resulting in the inability to complete the actual design and manufacture of the mutual - inductance coupling filter. On the other hand, in the above - mentioned patent, the disc - type coil is wound starting from the center point. Such a design has two defects. First, the turns in the central region have a very small radius of gyration, so their contribution to the coil magnetic field is also very weak, actually reducing the wire utilization rate. Second, the disc - type coil wound starting from the center point is not conducive to the heat dissipation and assembly of the coaxial coil array. To sum up, in the present invention, the following formulas will be used to calculate the self - inductance and mutual - inductance of the disc - type coil considering the actual size.

[0125] For Figure 3 the disc - type coil shown, the number of turns is N 1, the axial thickness is 2 h 1, and the inner and outer radii are respectively r 1, r 2, then its exact self - inductance solution is:[[]]

[0126] ;(8)

[0127] where:[[]]

[0128] ;(9)

[0129] The dimensionless parameter is:[[]]

[0130] ;

[0131] The self - inductance value of the coil is

[0132] ;(10)

[0133] The special functions included in formula (8) are:[[]]

[0134] m F n ( a , b , x ): Generalized hypergeometric function;

[0135] J n ( x ): n Bessel function of order

[0136] H n (x ): n Struve function of order

[0137] K( x ), E( x ): Complete elliptic integrals of the first and second kind;

[0138] G = 0.91596559…: Catalan constant.

[0139] For Figure 3 the two coaxial disk coils in N 1, N 2, with inner radii r 1, r 3, outer radii r 2, r 4, and axial thicknesses 2 h 1, 2 h 2, and the distance between the mid-sections of the two coils is z 0, z 0 ≥ h 1 + h 2, then the exact solution of the mutual inductance between them is:

[0140] ; (11)

[0141] Where:

[0142] ; (12)

[0143] Note that the generalized integrals in equations (8) and (11) are calculated using the Gauss-Laguerre quadrature rule.

[0144] 3. Relationship between the current-carrying and temperature rise of the filter coil

[0145] The power filter needs to undertake a certain fundamental reactive power compensation task, and in addition, the harmonic current value is relatively large. The selection of the coil wire diameter must be reasonable to ensure that the current density on the cross-section of the coil wire is within the safe range. Otherwise, the overheating of the coil may affect the safe and stable operation of the equipment. When calculating the temperature rise of the coil based on the current-carrying capacity of the coil, the following formula is used:

[0146] ; (13)

[0147] Where, I is the effective value of the current-carrying of the coil; γ 0 is the resistivity of the wire at temperature T 0; T 0 = 293K (aluminum) γ 0 = 2.82×10 -8Ω·m; α = 0.0043 K -1 is the temperature coefficient of aluminum; ambient temperature T ∞ = T sur = 300 K, that is, 27 °C; h is the convective heat transfer coefficient. For natural convection, that is, when there is no forced ventilation, it can be set h = 20 W·m -2 ·K -1 ; ε is the thermal radiation coefficient of the aluminum wire, and it can be set ε = 0.8; σ = 5.670373 × 10 -8 W·m -2 ·K -4 is the Stefan-Boltzmann constant of blackbody radiation; the others are the geometric parameters of the coil, where a , b are the long and short sides of the single-strand bare wire; n is the number of wire strands.

[0148] 4. Determination of the structure of the mutually coupled coil group

[0149] The Figure 1 circuit topology can be used, combined with the specific inductance and capacitance parameters calculated in Sections 1 and 2, for simulation to obtain the initial estimated values of the harmonic currents on each harmonic branch. Assume that the 3rd, 5th, and 7th harmonic currents are 200 A, 100 A, and 50 A respectively, then the circuit simulation shows that Figure 1 the effective current values on each coil branch in

[0150] ; (14)

[0151] This result provides a basis for the selection of the wire diameter of each coil.

[0152] When winding the coil, a rectangular cross-section is used (for example, in the calculation example of this patent, an aluminum wire with a wire specification of 16.5 mm × 3.85 mm and an insulation thickness of 0.25 mm is used. Considering the current carried by each coil, each turn of the coil will be spliced by several aluminum wires along the short side of 3.85 mm; according to the current-carrying values of each coil in (14), combined with the current-carrying - temperature rise relationship (13) of the coil, the cross-section of each coil wire can be taken as:

[0153] ; (15)

[0154] where η(In this example, 16.5 mm is the long side of a single-strand aluminum wire; that is, the 1st to 5th filter coils are wound with 5 strands, 2 strands, 1 strand, 4 strands, and 1 strand respectively.

[0155] When calculating the self-inductance and mutual inductance of the disc coils, formulas (8) to (12) are used; assuming m The number of turns, inner and outer radii, and axial thickness of the N m , r 0, r m , 2 h m ; m The distance between the mid-sections of the d m coil and the 1st coil is a = 16.5 mm, b = 3.85 mm; and they are spliced together along the short side b to form one turn, that is n m strands; then according to the actual geometric shape of the coil, there is a relationship:

[0156] ; (16)

[0157] During design, all the inner radii of the coils are taken as r 0 = 300 mm; substituting (16) into the equivalent equations (7) and adding two artificial constraint conditions M 12 = M 14 , M 34 = M 45 , and using the self- and mutual-inductance formulas (8) to (12), an underdetermined system of equations with 8 equations and 9 unknowns can be obtained; when solving, the number of turns of a certain coil can be set as a fixed value. For example, setting N 4 = 50, then this system of equations can be made into a well-determined system of equations. This well-determined system of equations is a non-linear system of equations and can be solved by the Newton-Raphson method.

[0158] After solving the equivalent equations, the specific structural dimensions of the mutual inductance-coupled Foster-type triple-tuned filter can be obtained; taking the 10 kV / 3000 kVar mutual inductance-coupled triple-tuned Foster-type filter as an example, its coil array structure is as Figure 4 shown; the parameters of this coil array are as follows:

[0159] Inner radius of each coil (mm): 300

[0160] Number of turns of each coil: N 1 = 34, N 2 = 28, N 3 = 25, N 4 = 50, N 5 = 59

[0161] Coil spacing (mm): d 2 = 456.15, d 3 = 606.22, d 4 = 834.59, d 5 = 201.08

[0162] Number of coil split parts: ν 1 = 2, ν 2 = 2, ν 3 = 2, ν 4 = 3, ν 5 = 3

[0163] Coil split distance (mm): δ = 10

[0164] Number of strands per turn of each coil: η 1 = 5, η 2 = 2, η 3 = 1, η 4 = 4, η 5 = 1

[0165] DC resistance of each coil (mΩ): R 1 = 20.497, R 2 = 39.869, R 3 = 72.955, R 4 = 60.223, R 5 = 295.988.

[0166] Note that in the design, each coil can be split into multiple coaxially connected sub - coils to reduce the radius of the coil array; here, coils 1 - 5 are split into 2, 2, 2, 3, and 3 sub - coils respectively. This design controls the diameter of the entire coil array within 1.054 m.

[0167] The inductance matrix of the mutual - inductance - coupled filter coil group is:

[0168] Table 1 Inductance matrix of 10 kV / 3000 kVar mutual - inductance - coupled filter (mH)

[0169]

[0170] 5. Simulation characteristics of the mutual - inductance - coupled triple - tuned Foster - type filter

[0171] The frequency-impedance curve of the mutual inductance coupled triple-tuned Foster-type filter obtained from the filter simulation circuit is as follows Figure 5 ; It can be seen that the mutual inductance coupled filter designed by this method has three obvious resonance points at 150 Hz, 250 Hz, and 350 Hz, and can achieve the triple-tuned filtering function; the same design method can also achieve a mutual inductance coupled triple-tuned Foster-type filter that can simultaneously filter out 250 Hz, 350 Hz, and 550 Hz.

[0172] In the simulation, the short-circuit capacity of the 10 kV power grid is set to 200 MVA, and the effective values of the third, fifth, and seventh harmonic currents are 200 A, 100 A, and 50 A respectively; according to the above circuit parameters, the effective values of the harmonic currents of each coil in the mutual inductance coupled filter are calculated, and the results are shown in Table 2; the second column in Table 2 is the effective value of the fundamental current, and the third, fourth, and fifth columns give the effective values of the harmonic currents of each filtering coil.

[0173] Table 2 Harmonic current values of each coil in the 10 kV / 3000 kVar mutual inductance coupled triple-tuned filter coil group

[0174]

[0175] Before filtering, the settings of 200 A for the third harmonic, 100 A for the fifth harmonic, and 50 A for the seventh harmonic cause the current waveform on the 10 kV bus to be severely distorted, as shown in Figure 6 shown.

[0176] After adopting the 10 kV / 3000 kVar mutual inductance coupled filter, the current waveform on the bus is as shown in Figure 7 shown; the effective values of the fundamental, 3rd harmonic, 5th harmonic, and 7th harmonic voltages are 5853.88 V, 15.99 V, 34.07 V, and 6.57 V respectively, and the total harmonic distortion rate of the voltage is THD V = 0.65%; the effective values of the fundamental, 3rd harmonic, 5th harmonic, and 7th harmonic currents are 611.48 A, 10.66 A, 13.63 A, and 1.88 A respectively, and the total harmonic distortion rate of the current is THD I = 2.85%, which indicates that this scheme fully meets the national standard "Power Quality - Harmonics in Public Power Grids" (GB / T 14549-93); GB / T 14549-93 stipulates that the total harmonic distortion rate of the 10 kV power grid does not exceed 4.0%, and the upper limit of the effective value of the harmonic current is: 20 A for the third harmonic, 15 A for the fifth harmonic, and 15 A for the seventh harmonic.

[0177] According to the effective values of the currents on each filtering coil and combining with the coil current-carrying-temperature rise equation, the temperatures (°C) of each filtering coil under the working conditions can be obtained: T 1 = 61.5, T 2 = 46.2,T 3 = 36.0, T 4 = 60.8, T 5 = 61.1. It can be seen that the temperature rise of each coil under the working conditions is within the safe range.

Claims

1. A design method for a mutually inductance coupled triple tuned Foster type filter, characterized in that: include: Step 1. Combine the voltage level and capacity of the filter and use the driving point impedance function of the LC single-port network to preliminarily determine the inductance and capacitance values ​​of each branch in the filter; Step 2: Conduct circuit simulation to obtain the estimated current value on each filter branch, and determine the cross-sectional size of each filter coil wire in combination with the coil current-temperature rise equation; Step 3, substitute the self-inductance and mutual inductance formulas of the filter coil into the mutual inductance coupling equations of the circuit, and solve the equations using the Newton-Raphson algorithm to obtain the number of turns, radius, and spacing of each coil; Step 4. Calculate the corresponding inductance matrix based on the structural dimensions of the actual filter coil array, and then simulate the relevant filter circuit to obtain the filter's frequency-impedance characteristic curve, as well as the filter technical indicators of harmonic current and harmonic voltage distortion rate, to verify the effectiveness of the filter design scheme; The filter circuit structure of the mutually inductively coupled three-tuned Foster type filter is as follows: A coil is arranged in series between the two stages of the filter. , , And filter capacitor , coil and A coil is connected in series between the connection point between the two poles. and filter capacitors , coil and A coil is connected in series between the connection point between the two poles. and filter capacitors ; The coil , , , , There is mutual inductance coupling between any two coils, with a total of 10 pairs of mutual inductances; The two mutually inductively coupled coils are coaxially arranged disc coils. , , , , forming a magnetically coupled inductor array; In Step 1, according to Otto Brune's theorem, the driving point impedance function of a single-port LC network is Z ( s ) must be one of the following two forms: ；（1） ；（2） in ω z1 , ω z2 , …is the impedance zero point, ω p1 , ω p2 , …is the impedance pole, K is an arbitrary constant; The driving point impedance function of the mutually inductively coupled three-tuned Foster filter that simultaneously filters out the 3rd, 5th and 7th harmonics is Z ( s )for: ；（3） in ω 0=100π; coefficient K Determined by the reactive power compensation capacity of the filter; using the Foster type circuit to implement the driving point impedance function formula (3), the uncoupled three-tuned filter circuit can be obtained, and the value of each circuit component is: ； parameter K According to the filter voltage level and reactive power compensation capacity Q Determine: ；（4） U The voltage level of the mutual inductance coupled three-tuned Foster filter is based on the required reactive power compensation capacity. Q The filtering parameters can be obtained K , the circuit parameters of the uncoupled three-tuned Foster filter can be obtained; The equivalent conditions of the mutual inductance coupling circuit of the mutual inductance coupling three-tuned Foster type filter and the non-coupled three-tuned filter circuit are: ；（5） If the above formula is established, the mutual inductance coupling circuit of the filter can realize the three-tuned filtering function; The two mutually inductively coupled disc coils have one turn number of N 1, axial thickness is 2 h 1, the inner and outer radii are r 1, r 2, then the self-inductance solution in Step 3 is: ；（6） in: ；（7） The dimensionless parameters are: ; The self-inductance of the coil is: ；（8） The function contained in formula (6) is: m F n ( a , b , x ): generalized hypergeometric function, J n ( x ): n Bessel function of order H n ( x ): n Order Struve function, K( x ), E( x ): Complete elliptic integrals of the first and second kind, G =0.91596559…: Catalan constant.

2. The design method of a mutually inductively coupled triple-tuned Foster type filter according to claim 1, characterized in that: The two mutually inductively coupled disc coils have two turns: N 1, N 2, the inner radius is r 1, r 3, the outer radius is r 2, r 4, axial thickness is 2 h 1, 2 h 2. The cross-sectional distance between the two coils is z 0, z 0≥ h 1+ h 2， The mutual inductance between them is: ； （9） in: ；（10） The generalized integrals in equations (6) and (9) are calculated using the Gauss-Laguerre quadrature rule.

3. The design method of a mutually inductively coupled triple-tuned Foster type filter according to claim 2, characterized in that: When designing the mutually inductively coupled three-tuned Foster filter, the relationship between the current carrying capacity and the temperature rise of the filter coil needs to be considered, using the following formula: ； （11） in, I is the effective value of the coil current; N is the number of turns of the filter coil; γ 0 is temperature T The resistivity of the wire at 0; T When 0=293K, aluminum γ0=2.82×10 －8 Ω・m; α =0.0043K -1 is the temperature coefficient of aluminum; ambient temperature T ∞ = T sur =300K, i.e. 27℃; h is the convection heat transfer coefficient. For natural convection, i.e. when there is no forced ventilation, h =20W·m -2 ·K -1 ; ε is the thermal radiation coefficient of the aluminum conductor, which can be set ε =0.8; σ =5.670373×10 -8 W·m -2 ·K -4 is the blackbody radiation Stefan-Boltzmann constant; the others are the geometric parameters of the coil, among which a , b The long and short sides of a single bare wire; n is the number of wire strands.

4. The design method of a mutually inductively coupled triple-tuned Foster type filter according to claim 3, characterized in that: When the coils are wound, the current carried by each coil is taken into consideration, and combined with the coil current-temperature rise relationship (13), the cross-section of each coil wire is taken as: ；(12) in η is the long side of a single strand of aluminum wire; that is, 5 strands, 2 strands, 1 strand, 4 strands, and 1 strand are used to wind filter coils 1 to 5 respectively. η 1. η 2. η 3. η 4. η 5 is the total length of the long side of the cross section of coil wire No. 1, 2, 3, 4, and 5 respectively.

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