Method for calculating permeability of circular cross-section ferrite core in toroidal inductor

Abstract

The application relates to the technical field of permeability calculation, and discloses a permeability calculation method for a circular cross-section ferrite magnetic core in a toroidal inductor; when the magnetic field intensity is calculated, high-frequency analysis is carried out based on Maxwell equations, the characteristic that the magnetic field in the circular cross-section ferrite magnetic core in the toroidal inductor is uniform is used, and a virtual variable Bessel equation is derived based on a cylindrical coordinate system, and the magnetic field intensity calculation model is obtained by solving the equation, so that the limitation that the traditional measurement method needs to depend on a measuring device is broken. When the electric field intensity is calculated, the electric field intensity calculation model is obtained based on the relationship between the electric field intensity and the magnetic field intensity in the Maxwell equations. Based on the calculated magnetic field intensity and electric field intensity, the complex power of the toroidal inductor is calculated, and then the permeability prediction model is obtained in combination with an equivalent circuit model. The application is based on theoretical model calculation, is not limited by specific frequency bands and measuring instruments, is also applicable in a high-frequency range, and the predicted permeability is more accurate.

Description

Technical Field

[0001] This invention relates to the field of magnetic permeability calculation technology, and in particular to a method for calculating the magnetic permeability of a circular cross-section ferrite core in a toroidal inductor. Background Technology

[0002] In switching power converters, ferrite cores are widely used in high-frequency transformers and inductors. Miniaturization of these devices necessitates reducing the core's size. To ensure these miniaturized cores possess the same power handling capability as their original counterparts, the operating frequency inevitably needs to be increased. However, increasing the operating frequency leads to higher core losses, which directly impact the efficiency and performance of magnetic devices. Therefore, accurately determining core losses is crucial for circuit designers.

[0003] In evaluating core losses, theoretical analysis methods are more effective and important. To theoretically evaluate core losses, researchers need to obtain the permeability value of the ferrite core beforehand. The permeability value is a key input parameter for theoretical evaluation and optimized design of ferrite devices. Manufacturer datasheets typically provide initial permeability, while the actual permeability value will vary with operating frequency and core size. Therefore, determining the actual permeability value for a specific magnetic device is crucial.

[0004] Currently, several permeability measurement devices are available on the market, such as impedance analysis, transmission line method, resonant cavity method, vibrating sample magnetometer, superconducting quantum interference device, magneto-optical effect method, and magnetic resonance method. These methods all have significant limitations, rely on expensive instruments, exhibit high-frequency uncertainties, and are only applicable to specific frequency bands, thus possessing certain limitations.

[0005] Existing methods for calculating magnetic permeability without the aid of measuring instruments are applied to toroidal ferrite cores with square cross-sections. However, the presence of sharp edges in square cross-sections can lead to magnetic field concentration or non-uniformity at these points. Therefore, calculation methods based on the non-uniformity of the magnetic field in square cross-section cores cannot be directly applied to the permeability calculation of circular cross-section cores, which have a uniform magnetic field distribution. In rotating equipment such as generators, where the core needs to rotate or require a omnidirectional magnetic field, circular cross-section cores must be used. Therefore, a more accurate method for calculating the permeability of circular cross-section cores is needed. Summary of the Invention

[0006] Therefore, the technical problem to be solved by the present invention is to overcome the problem that the existing technology cannot calculate the permeability of a magnetic core with a circular cross-section.

[0007] To solve the above-mentioned technical problems, the present invention provides a method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor, comprising:

[0008] High-frequency analysis was performed on a circular cross-section ferrite core to obtain the corresponding Maxwell's equations.

[0009] Based on cylindrical coordinates and Maxwell's equations, an initial expression for the magnetic field strength with respect to the radius is obtained, and curl is calculated to obtain the corresponding curl expression.

[0010] The curl expression is recalculated to obtain the quadratic curl expression. Combined with Maxwell's equations, the complex expression for the quadratic curl is obtained.

[0011] Based on Maxwell's equations and the complex expression for quadratic curl, magnetic field parameters are constructed; based on the complex expression for quadratic curl and magnetic field parameters, the imaginary argument Bessel equation for magnetic field strength is constructed and solved to obtain the exact solution, which serves as the calculation model for magnetic field strength.

[0012] Based on Maxwell's equations, the correlation between electric field strength and magnetic field strength is obtained; based on the correlation and magnetic field strength calculation model, the electric field strength calculation model is obtained.

[0013] The specifications of the circular cross-section ferrite core and the effective value of the supply current of the toroidal inductor are input into the magnetic field strength calculation model and the electric field strength calculation model to obtain the magnetic field strength and electric field strength at the radius r on the circular cross-section ferrite core.

[0014] The complex power of the toroidal inductor is calculated by integrating the magnetic field strength and electric field strength at each radial position on the circular cross-section ferrite core.

[0015] Based on the specifications of the circular cross-section ferrite core, the effective value of the supply current and the complex power of the toroidal inductor, the true permeability of the toroidal inductor is calculated.

[0016] Preferably, high-frequency analysis is performed on the circular cross-section ferrite core to obtain Maxwell's equations for the quasi-static magnetic system, including:

[0017] Ampere's circuital law:

[0018] Faraday's law of electromagnetic induction:

[0019] Gauss's Law for Magnetic Fields:

[0020] Gauss's Law for Electric Fields:

[0021] Where H represents magnetic field strength, E represents electric field strength, B represents magnetic flux density, D represents electric displacement vector, and J represents magnetic flux density. c ρ represents current density, j represents the imaginary unit, ω represents angular frequency, and ρ represents the current density. v This represents charge density.

[0022] Preferably, an initial expression for the magnetic field strength with respect to the radius is obtained based on cylindrical coordinates and Maxwell's equations, and curl is calculated to obtain the corresponding curl expression, including:

[0023] The initial expression for the magnetic field strength H with respect to radius r is obtained based on cylindrical coordinates and Maxwell's equations, and is expressed as:

[0024] H=H(r)e z ;

[0025] The curl is calculated based on the initial expression to obtain the corresponding curl expression, which is expressed as follows:

[0026]

[0027] Where H(r) represents the magnetic field strength at radius r on a circular cross-section ferrite core, e z e represents the unit vector along the z-axis. θ This represents the unit vector along the θ-axis.

[0028] Preferably, the curl expression is recalculated to obtain a quadratic curl expression. Then, using Maxwell's equations and the law of electromagnetic induction, a complex quadratic curl expression is obtained, including:

[0029] The curl expression is recalculated to obtain the quadratic curl expression, which is expressed as:

[0030]

[0031] Based on the expression for quadratic curl and Maxwell's equations, the complex expression for quadratic curl is obtained as follows:

[0032]

[0033] Where μ represents the initial permeability, σ represents the conductivity, and ε represents the dielectric constant.

[0034] Preferably, magnetic field parameters are constructed based on Maxwell's equations and the complex expression for quadratic curl; the imaginary argument Bessel equation for magnetic field strength is constructed based on the complex expression for quadratic curl and the magnetic field parameters, including:

[0035] Based on Maxwell's equations and the complex expression for the second curl, the magnetic field parameter k is constructed as follows:

[0036]

[0037] Based on the complex expression for the quadratic curl and the magnetic field parameters, the Bessel equation for the imaginary argument of the magnetic field strength is constructed as follows:

[0038]

[0039] Preferably, solving the imaginary argument Bessel equation to obtain an exact solution serves as the magnetic field strength calculation model, including:

[0040] The general solution of the Bessel equation with imaginary arguments, including undetermined coefficients c1 and c2, is expressed as:

[0041] H(r) = c1I0(kr) + c2K0(kr);

[0042] I0(kr) is the zeroth-order imaginary argument Bessel function of the first kind, expressed as:

[0043]

[0044] K0(kr) is a zero-order imaginary argument Bessel function of the second kind, expressed as:

[0045]

[0046] When the radius r of the circular cross-section ferrite core is 0, K0(0) = ∞, c2 = 0, and the general solution of the imaginary argument Bessel equation simplifies to:

[0047] H(r) = c1I0(kr);

[0048] When the radius r = R of a circular cross-section ferrite core,

[0049] Substituting the boundary condition r = R into H(r) = c1I0(kr), we obtain the undetermined coefficients.

[0050] Based on the undetermined coefficients c1 and c2, the exact solution of the imaginary argument Bessel equation is obtained, expressed as:

[0051]

[0052] The magnetic field strength calculation model is expressed as follows:

[0053] Where Γ represents the gamma function, and R1 and R2 represent the outer and inner circumference radii of the circular cross-section ferrite core, respectively.

[0054] Preferably, the specifications of the circular cross-section ferrite core include: the average circumference of the circular cross-section ferrite core, the number of turns of the core winding, the area of ​​the circular cross-section, and the outer and inner circumference radii.

[0055] Preferably, based on Maxwell's equations, the correlation between electric field strength and magnetic field strength is obtained; based on the correlation and the magnetic field strength calculation model, an electric field strength calculation model is obtained, including:

[0056] Based on Maxwell's equations, the relationship between electric field strength and magnetic field strength is obtained, expressed as:

[0057]

[0058] Based on the correlation formula and the magnetic field strength calculation model, the electric field strength calculation model is obtained, which is expressed as:

[0059]

[0060] Where I1(kr) represents the first-order imaginary argument Bessel function of the first kind.

[0061] Preferably, the complex power of the toroidal inductor is calculated by integrating the magnetic field strength and electric field strength at each radial position on the circular cross-section ferrite core, as expressed in:

[0062] S=∫∫∫σ|E| 2 dV+jω(∫∫∫μ|H| 2 dV-∫∫∫ε|E| 2 dV).

[0063] Preferably, obtaining the permeability prediction model includes:

[0064] The toroidal inductor is equivalently represented as a circuit consisting of a resistor and an inductor connected in series. Based on the equivalent resistance and inductance in this equivalent circuit, an equivalent permeability calculation model is obtained, expressed as: μ c =μ′-jμ″;

[0065] Based on the relationship between the equivalent inductance L and the real part μ′ of the permeability, the expression for the real part is obtained as follows:

[0066]

[0067] Based on the relationship between the equivalent resistance R and the imaginary part μ” of the permeability, the expression for the imaginary part is obtained as follows:

[0068]

[0069] The power of a toroidal inductor satisfies: S = I 2 (R+jωL);

[0070] Based on the real and imaginary expressions and the conditions satisfied by the power of the toroidal inductor, the real part Re(S) and imaginary part Im(S) of the complex power are obtained, expressed as: Re(S) = I 2 R,Im(S)=I 2 ωL;

[0071] Substituting the real and imaginary parts of the complex power into the equivalent permeability calculation model, we obtain the permeability prediction model, which is expressed as:

[0072] Where, μ c denoted by , S represents the complex power of the toroidal inductor under test, Im(S) and Re(S) represent the imaginary and real parts of the complex power, respectively, and j represents the imaginary unit; l, N, and A represent the average circumference, number of turns, and cross-sectional area of ​​the circular cross-section ferrite core, respectively, and ω and I represent the supply frequency and effective value of the supply current, respectively.

[0073] Compared with the prior art, the above-described technical solution of the present invention has the following advantages:

[0074] The permeability calculation method for a circular cross-section ferrite core in a toroidal inductor described in this invention considers the uniform magnetic field distribution within the circular cross-section ferrite core during the calculation process. This makes the constructed magnetic field strength calculation model and electric field strength calculation model more applicable to circular cross-section ferrite cores, resulting in a more accurate permeability calculation. When calculating the magnetic field strength, this invention employs high-frequency analysis based on Maxwell's equations. Leveraging the uniform magnetic field distribution within the circular cross-section ferrite core of the toroidal inductor, and using cylindrical coordinates, it derives and solves the imaginary argument Bessel equation to obtain the magnetic field strength calculation model, overcoming the limitation of traditional measurement methods that rely on measuring devices. When calculating the electric field strength, it uses the relationship between electric and magnetic field strengths in Maxwell's equations to obtain the electric field strength calculation model. This invention calculates the magnetic and electric field strengths by substituting the core's specifications into the calculation model. Based on the calculated magnetic and electric field strengths, the complex power of the toroidal inductor is calculated, and then combined with the equivalent circuit model, a permeability prediction model is obtained. This invention is based on theoretical model calculations and is not limited by specific frequency bands or measuring instruments, and is also applicable in the high-frequency range. Furthermore, based on the uniform magnetic field characteristics of circular cross-section ferrite cores, the magnetic field strength and electric field strength are calculated, making the permeability prediction model fully applicable to circular cross-section ferrite cores, resulting in more accurate permeability predictions. In addition, based on the accurate permeability of circular cross-section ferrite cores, core losses are evaluated to improve the working efficiency and performance of magnetic devices. Attached Figure Description

[0075] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings, wherein:

[0076] Figure 1 This is a flowchart of the steps for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor provided by the present invention.

[0077] Figure 2 (a) is a schematic diagram of a toroidal inductor; Figure 2 (b) is the equivalent circuit diagram of the toroidal inductor. Detailed Implementation

[0078] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0079] Reference Figure 1 The flowchart shown illustrates the steps of the method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor provided by this invention. The specific steps include:

[0080] S101: High-frequency analysis of a circular cross-section ferrite core to obtain Maxwell's equations for a quasi-static magnetic system, including:

[0081] Ampere's circuital law:

[0082] Faraday's law of electromagnetic induction:

[0083] Gauss's Law for Magnetic Fields:

[0084] Gauss's Law for Electric Fields:

[0085] Where H represents magnetic field strength, E represents electric field strength, B represents magnetic flux density, D represents electric displacement vector, and J represents magnetic flux density. c ρ represents current density, j represents the imaginary unit, ω represents angular frequency, and ρ represents the current density. v This represents charge density.

[0086] S102: Based on cylindrical coordinates and Maxwell's equations, obtain an initial expression for the magnetic field strength with respect to the radius, and perform curl calculations to obtain the corresponding curl expression, including:

[0087] S102-1: Based on cylindrical coordinates and Maxwell's equations, the initial expression for the magnetic field strength H with respect to radius r is obtained, expressed as: H = H(r)e z ;

[0088] S102-2: Calculate the curl based on the initial expression to obtain the corresponding curl expression, which is expressed as:

[0089] Where H(r) represents the magnetic field strength at radius r on a circular cross-section ferrite core, e z e represents the unit vector along the z-axis. θ This represents the unit vector along the θ-axis.

[0090] S103: Calculate the curl expression again to obtain the quadratic curl expression. Combine this with Maxwell's equations to obtain the complex quadratic curl expression, including:

[0091] S103-1: Calculate the curl expression again to obtain the secondary curl expression, which is as follows:

[0092]

[0093] S103-2: Based on the expression for quadratic curl and Maxwell's equations, the complex expression for quadratic curl is obtained, expressed as:

[0094]

[0095] Where μ represents the initial permeability, σ represents the conductivity, and ε represents the dielectric constant.

[0096] S104: Based on Maxwell's equations and the complex expression of quadratic curl, construct the magnetic field parameters; based on the complex expression of quadratic curl and the magnetic field parameters, construct and solve the Bessel equation for the imaginary argument of the magnetic field strength to obtain the exact solution, which serves as the calculation model for the magnetic field strength.

[0097] S105: Based on Maxwell's equations, obtain the correlation between electric field strength and magnetic field strength; based on the correlation and magnetic field strength calculation model, obtain the electric field strength calculation model.

[0098] S106: Input the specifications of the circular cross-section ferrite core and the effective value of the supply current of the toroidal inductor into the magnetic field strength calculation model and the electric field strength calculation model to obtain the magnetic field strength and electric field strength at the radius r on the circular cross-section ferrite core.

[0099] S107: Integrate the magnetic field strength and electric field strength at each radial position on the circular cross-section ferrite core to calculate the complex power of the toroidal inductor, expressed as:

[0100] S=∫∫∫σ|E| 2 dV+jω(∫∫∫μ|H| 2 dV-∫∫∫ε|E| 2 dV);

[0101] S108: Based on the specifications of the circular cross-section ferrite core, the effective value of the supply current and the complex power of the toroidal inductor, calculate the true permeability of the toroidal inductor.

[0102] Specifically, in step S104, obtaining the magnetic field strength calculation model includes:

[0103] S104-1: Based on Maxwell's equations and the complex expression for the second curl, the magnetic field parameter k is constructed as follows:

[0104] S104-2: Based on the complex expression for the second curl and the magnetic field parameters, the imaginary argument Bessel equation for the magnetic field strength is constructed, expressed as:

[0105] S104-3: Obtain the general solution of the Bessel equation with imaginary arguments containing undetermined coefficients c1 and c2, expressed as:

[0106] H(r) = c1I0(kr) + c2K0(kr);

[0107] I0(kr) is the zeroth-order imaginary argument Bessel function of the first kind, expressed as:

[0108]

[0109] K0(kr) is a zero-order imaginary argument Bessel function of the second kind, expressed as:

[0110]

[0111] S104-4: When the radius r of the circular cross-section ferrite core is 0, K0(0) = ∞, c2 = 0, the general solution of the imaginary argument Bessel equation is simplified to: H(r) = c1I0(kr);

[0112] S104-5: When the radius r of a circular cross-section ferrite core is R,

[0113] S104-6: Substitute the boundary condition r = R into H(r) = c1I0(kr) to obtain the undetermined coefficients.

[0114] S104-7: Based on the undetermined coefficients c1 and c2, obtain the exact solution of the imaginary argument Bessel equation, expressed as:

[0115] Therefore, the magnetic field strength calculation model is expressed as:

[0116] Where Γ represents the gamma function, and R1 and R2 represent the outer and inner circumference radii of the circular cross-section ferrite core, respectively.

[0117] The specifications of the circular cross-section ferrite core include: the average circumference of the circular cross-section ferrite core, the number of turns of the core winding, the area of ​​the circular cross-section, and the outer and inner circumference radii.

[0118] Specifically, in S105, the acquisition of the electric field intensity calculation model includes:

[0119] S105-1: Based on Maxwell's equations, the relationship between electric field strength and magnetic field strength is obtained, expressed as:

[0120] S105-2: Based on the correlation formula and the magnetic field strength calculation model, the electric field strength calculation model is obtained, expressed as:

[0121] Where I1(kr) represents the first-order imaginary argument Bessel function of the first kind.

[0122] The permeability calculation method for a circular cross-section ferrite core in a toroidal inductor described in this invention considers the uniform magnetic field distribution within the circular cross-section ferrite core during the calculation process. This makes the constructed magnetic field strength calculation model and electric field strength calculation model more applicable to circular cross-section ferrite cores, resulting in a more accurate calculated permeability. When calculating the magnetic field strength, this invention employs high-frequency analysis based on Maxwell's equations. Leveraging the uniform magnetic field distribution within the circular cross-section ferrite core of the toroidal inductor, and using cylindrical coordinates, it derives and solves the imaginary argument Bessel equation to obtain the magnetic field strength calculation model, overcoming the limitation of traditional measurement methods that rely on measuring devices. When calculating the electric field strength, it uses the relationship between electric and magnetic field strengths in Maxwell's equations to obtain the electric field strength calculation model.

[0123] Specifically, in step S108, the true permeability of the toroidal inductor is calculated, including:

[0124] S108-1: The toroidal inductor is equivalent to a circuit consisting of a resistor and an inductor connected in series. Based on the equivalent resistance and inductance in the equivalent circuit, the equivalent permeability calculation model is obtained, expressed as: μ c =μ′-jμ″;

[0125] S108-2: Based on the relationship between the equivalent inductance L and the real part μ′ of the permeability, the expression for the real part is obtained, which is:

[0126] S108-3: Based on the relationship between the equivalent resistance R and the imaginary part μ” of the permeability, the expression for the imaginary part is obtained, which is:

[0127] S108-4: The power of a toroidal inductor satisfies: S = I 2 (R+jωL);

[0128] Based on the real and imaginary expressions and the conditions satisfied by the power of the toroidal inductor, the real part Re(S) and imaginary part Im(S) of the complex power are obtained, expressed as: Re(S) = I 2 R,Im(S)=I 2 ωL;

[0129] S108-5: Substituting the real and imaginary parts of the complex power into the equivalent permeability calculation model, we obtain the permeability prediction model, expressed as:

[0130] S108-6: Substitute the specifications of the circular cross-section ferrite core, the effective value of the supply current of the toroidal inductor and the complex power into the permeability prediction model to obtain the true permeability of the toroidal inductor.

[0131] Where, μ c denoted by , S represents the complex power of the toroidal inductor under test, Im(S) and Re(S) represent the imaginary and real parts of the complex power, respectively, and j represents the imaginary unit; l, N, and A represent the average circumference, number of turns, and cross-sectional area of ​​the circular cross-section ferrite core, respectively, and ω and I represent the supply frequency and effective value of the supply current, respectively.

[0132] This invention calculates the magnetic and electric field strengths by substituting the core's specifications into a computational model. Based on these calculations, the complex power of the toroidal inductor is calculated. Combined with an equivalent circuit model, a permeability prediction model is obtained. This invention, based on a theoretical model, is not limited by specific frequency bands or measuring instruments and is applicable in the high-frequency range. Furthermore, by leveraging the uniform magnetic field of a circular cross-section ferrite core, the calculation of magnetic and electric field strengths ensures that the permeability prediction model is fully applicable to circular cross-section ferrite cores, resulting in more accurate permeability predictions.

[0133] Based on the above embodiments, in this embodiment of the invention, the permeability of the toroidal ferrite core is calculated using the permeability calculation method for the circular cross-section ferrite core in the toroidal inductor provided by the present invention, specifically including:

[0134] S201: Using Maxwell's equations, calculate the magnetic field and electric field distribution inside the toroidal ferrite core;

[0135] The Maxwell's equations for the magnetoquasistatic system are expressed as:

[0136] Assume that the magnetic field intensity distribution on the surface of the toroidal ferrite core is uniform, and the cylindrical coordinate system is used for analysis. The unit vectors in the three directions of the cylindrical coordinate system are e r 、e θ 、e z .

[0137] Assume that the magnetic field intensity is only a function of the radius r and is independent of θ. Therefore, the magnetic field intensity is expressed as: H = H(r)e z ;

[0138] Therefore,

[0139]

[0140] Take the curl of the formula in the Maxwell's equations for the magnetoquasistatic system, and combine it with to obtain:

[0141] where μ is the initial permeability and σ is the conductivity.

[0142] Let Calculate the Bessel equation of imaginary argument with respect to H, which is expressed as:

[0143]

[0144] Its general solution is: H(r) = c1I0(kr) + c2K0(kr);

[0145] where,

[0146] When the radius r of the circular cross-section ferrite core is r = 0, and K0(0) = ∞, it can be inferred that c2 = 0.

[0147] In the region of the core r < R, the magnetomotive force is affected not only by the excitation current but also by the eddy current and displacement current. However, at the outer peripheral interface of the core where r = R, the magnetic field intensity is determined only by the excitation current.

[0148] Assume that the magnetic field on the outer peripheral interface of the core is uniform, and it will satisfy the following relationship:

[0149] where R1 and R2 are the outer radius and inner radius of the core respectively;<0000N is the average radius of the magnetic core; N is the number of turns in the toroidal inductor winding; and I is the effective value of the supply current.

[0150] The coefficient c1 is represented as:

[0151] Therefore, the exact solution to the imaginary Bessel equation, i.e., the expression for the magnetic field strength, is:

[0152]

[0153] Where H(R) is the magnetic field strength at the outer periphery of the magnetic core; R1 and R2 are the outer and inner circumferences of the magnetic core, respectively; N is the number of turns in the toroidal inductor winding; I is the effective value of the supply current; and R is the radius of the magnetic core cross-section. I0(kr) and I0(kR) are zero-order imaginary argument Bessel functions of the first kind.

[0154] Combining the two expressions for magnetic induction intensity, and J c =σE, D=εE, from which the following expression can be derived:

[0155]

[0156] Further calculation yields the expression for the electric field intensity distribution in the magnetic core, which is expressed as:

[0157]

[0158] Where σ is the conductivity and ε is the dielectric constant. I1(kr) is the first-order imaginary argument Bessel function of the first kind.

[0159] S202: Calculate the complex power of the toroidal inductor;

[0160] The expression for calculating the complex power of a toroidal inductor is:

[0161] S=∫∫∫σ|E| 2 dV+jω(∫∫∫μ|H| 2 dV-∫∫∫ε|E| 2 dV);

[0162] Where μ is the initial permeability coefficient, ε is the dielectric constant, and σ is the conductivity.

[0163] Substituting the expressions for magnetic field strength and electric field strength into the above expression for calculating the complex power of the toroidal inductor, the complex power of the toroidal inductor can be calculated.

[0164] S203: Calculate the true permeability coefficient;

[0165] Reference Figure 2 As shown in (a), this is a schematic diagram of a toroidal inductor; refer to Figure 2As shown in (b), this is the equivalent circuit diagram of a toroidal inductor; according to circuit concepts, a toroidal inductor can be equivalently represented as a resistor and an inductor connected in series; the true permeability μ c The real part μ′ and the imaginary part μ″ of =μ′-jμ″ can be calculated based on the equivalent resistance and inductance in series. The calculation method and model are shown below:

[0166] The equivalent resistance connected in series in the equivalent circuit has the following relationship with the imaginary part of the permeability:

[0167] The inductance connected in series in the equivalent circuit has the following relationship with the real part of the permeability:

[0168] Where N is the number of turns, l is the average circumference of the magnetic core, A is the cross-sectional area, and ω is the power supply frequency.

[0169] And the complex power supplied to the toroidal inductor is given by the formula: S = I 2 (R+jωL);

[0170] Therefore: Re(S) = I 2 R, Im(S) = I 2 ωL;

[0171] Substituting the relationships between the equivalent resistance and inductance in series in the equivalent circuit and the imaginary and real parts of the permeability into the complex power relationship of the toroidal inductor, we obtain a predictive model for the real and imaginary parts of the permeability, expressed as:

[0172] This leads to the prediction model for magnetic permeability, expressed as:

[0173]

[0174] Where S is the total complex power of the inductor, N is the number of turns, l is the average circumference of the magnetic core, A is the cross-sectional area, and ω is the power supply frequency.

[0175] Substitute the complex power of the toroidal inductor calculated in step S202 into the permeability prediction model to obtain the true permeability of the toroidal ferrite core in the toroidal inductor.

[0176] This invention proposes an accurate method for predicting the complex permeability of toroidal ferrite cores with circular cross-sections. Based on thorough theoretical analysis and derivation, it has strong theoretical support. Furthermore, this prediction method provides a specific mathematical model; the measurer only needs to input the geometric dimensions of the core into the mathematical model to calculate the complex permeability. Compared to traditional, expensive measuring equipment, the most prominent features of this method are its simplicity, convenience, and accuracy. It can also provide the variation of complex permeability with frequency, remaining applicable in the high-frequency range. This will contribute to the optimized design of inductors and has promising applications in the fabrication of toroidal inductors in the fields of electronic information and electronic circuits.

[0177] The permeability calculation method for a circular cross-section ferrite core in a toroidal inductor described in this invention considers the uniform magnetic field distribution within the circular cross-section ferrite core during the calculation process. This makes the constructed magnetic field strength calculation model and electric field strength calculation model more applicable to circular cross-section ferrite cores, resulting in a more accurate permeability calculation. When calculating the magnetic field strength, this invention employs high-frequency analysis based on Maxwell's equations. Leveraging the uniform magnetic field distribution within the circular cross-section ferrite core of the toroidal inductor, and using cylindrical coordinates, it derives and solves the imaginary argument Bessel equation to obtain the magnetic field strength calculation model, overcoming the limitation of traditional measurement methods that rely on measuring devices. When calculating the electric field strength, it uses the relationship between electric and magnetic field strengths in Maxwell's equations to obtain the electric field strength calculation model. This invention calculates the magnetic and electric field strengths by substituting the core's specifications into the calculation model. Based on the calculated magnetic and electric field strengths, the complex power of the toroidal inductor is calculated, and then combined with the equivalent circuit model, a permeability prediction model is obtained. This invention is based on theoretical model calculations and is not limited by specific frequency bands or measuring instruments, and is also applicable in the high-frequency range. Furthermore, based on the uniform magnetic field characteristics of circular cross-section ferrite cores, the magnetic field strength and electric field strength are calculated, making the permeability prediction model fully applicable to circular cross-section ferrite cores, and the prediction of permeability is more accurate.

[0178] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0179] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0180] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0181] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0182] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor, characterized in that, include: High-frequency analysis was performed on a circular cross-section ferrite core to obtain the corresponding Maxwell's equations. Based on cylindrical coordinates and Maxwell's equations, an initial expression for the magnetic field strength with respect to the radius is obtained, and curl is calculated to obtain the corresponding curl expression. The curl expression is recalculated to obtain the quadratic curl expression. Combined with Maxwell's equations, the complex expression for the quadratic curl is obtained. Based on Maxwell's equations and the complex expression for quadratic curl, magnetic field parameters are constructed. Based on the complex expression of the second curl and the magnetic field parameters, the Bessel equation of the imaginary argument of the magnetic field strength is constructed and solved to obtain the exact solution, which serves as the calculation model for the magnetic field strength. Based on Maxwell's equations, the correlation between electric field strength and magnetic field strength is obtained; based on the correlation and magnetic field strength calculation model, the electric field strength calculation model is obtained. The specifications of the circular cross-section ferrite core and the effective value of the supply current of the toroidal inductor are input into the magnetic field strength calculation model and the electric field strength calculation model to obtain the radius of the circular cross-section ferrite core. The magnetic field strength and electric field strength at that location; The complex power of the toroidal inductor is calculated by integrating the magnetic field strength and electric field strength at each radial position on the circular cross-section ferrite core. Based on the specifications of the circular cross-section ferrite core, the effective value of the supply current and the complex power of the toroidal inductor, the true permeability of the toroidal inductor is calculated.

2. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 1, characterized in that, High-frequency analysis of a circular cross-section ferrite core yielded Maxwell's equations for a quasi-static magnetic system, including: Ampere's circuital law: ; Faraday's law of electromagnetic induction: ; Gauss's Law for Magnetic Fields: ; Gauss's Law for Electric Fields: ; in, Indicates magnetic field strength. Indicates electric field strength. Indicates magnetic flux density. Represents the electric displacement vector. Indicates current density, Represents the imaginary unit. The power supply frequency indicates the power supply current. This represents charge density.

3. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 2, characterized in that, Based on cylindrical coordinates and Maxwell's equations, an initial expression for the magnetic field strength with respect to the radius is obtained, and curl is calculated to obtain the corresponding curl expression, including: Magnetic field strength is obtained based on cylindrical coordinates and Maxwell's equations. Regarding radius The initial expression is represented as: ； The curl is calculated based on the initial expression to obtain the corresponding curl expression, which is expressed as follows: ； in, Indicates the radius of the circular cross-section ferrite core Magnetic field strength at that location express Unit vector along the axis, express Unit vector along the axis.

4. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 3, characterized in that, The curl expression is recalculated to obtain the quadratic curl expression. Combining this with Maxwell's equations and the law of electromagnetic induction, the complex expression for the quadratic curl is obtained, including: The curl expression is recalculated to obtain the quadratic curl expression, which is expressed as: ； Based on the expression for quadratic curl and Maxwell's equations, the complex expression for quadratic curl is obtained as follows: ； in, Indicates the initial permeability. This indicates electrical conductivity.

5. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 4, characterized in that, Based on Maxwell's equations and the complex expression for quadratic curl, magnetic field parameters are constructed. Based on the complex expression for quadratic curl and magnetic field parameters, the imaginary argument Bessel equation for magnetic field strength is constructed, including: Based on Maxwell's equations and the complex expression for quadratic curl, magnetic field parameters are constructed. , represented as: ； Based on the complex expression for the quadratic curl and the magnetic field parameters, the Bessel equation for the imaginary argument of the magnetic field strength is constructed as follows: 。 6. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 5, characterized in that, Solving the imaginary Bessel equations to obtain exact solutions serves as a model for calculating magnetic field strength, including: Obtaining the undetermined coefficients of the Bessel equation with virtual argument and The general solution is expressed as: ； The expression for the zeroth-order imaginary argument Bessel function of the first kind is: ； The Bessel function is a zero-order imaginary argument of the second kind, expressed as: ； ； When the radius of a circular cross-section ferrite core hour, , The general solution of the imaginary argument Bessel equation, simplified, is: ； When the radius of a circular cross-section ferrite core hour, , ; Will Substitute the boundary conditions In the process, obtain the undetermined coefficients. ; Based on undetermined coefficients and To obtain the exact solution to the imaginary argument Bessel equation, it is expressed as: ； The magnetic field strength calculation model is expressed as follows: ; in, Represents the gamma function. and These represent the outer and inner circumference radii of a circular cross-section ferrite core, respectively. Indicates the number of turns in the winding of a toroidal inductor. Let be the radius of the magnetic core cross-section. This is the effective value of the supply current.

7. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 6, characterized in that, The specifications of a circular cross-section ferrite core include: the average circumference of the circular cross-section ferrite core, the number of turns in the core winding, the area of ​​the circular cross-section, and the outer and inner circumference radii.

8. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 6, characterized in that, Based on Maxwell's equations, the correlation between electric field strength and magnetic field strength is obtained; based on the correlation and the magnetic field strength calculation model, the electric field strength calculation model is obtained, including: Based on Maxwell's equations, the relationship between electric field strength and magnetic field strength is obtained, expressed as: ； Based on the correlation formula and the magnetic field strength calculation model, the electric field strength calculation model is obtained, which is expressed as: ； in, Let represent the first-order imaginary argument Bessel function of the first kind.

9. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 8, characterized in that, The complex power of the toroidal inductor is calculated by integrating the magnetic field strength and electric field strength at each radial position on the circular cross-section ferrite core, and is expressed as follows: ； in, This represents the dielectric constant.

10. The method for calculating the permeability of a circular cross-section ferrite core in a toroidal inductor according to claim 9, characterized in that, The acquisition of the permeability prediction model includes: The toroidal inductor is equivalent to a circuit consisting of a resistor and an inductor connected in series. Based on the equivalent resistance and inductance in this equivalent circuit, the equivalent permeability calculation model is obtained, which is expressed as: ''; Based on equivalent inductance With the real part of permeability The relationship between the two expressions, to obtain the real part, is expressed as: ； Based on equivalent resistance With the imaginary part of permeability The relationship between '', obtaining the imaginary part expression, is represented as: ； The power requirement of the toroidal inductor is: ; Based on the real part expression, the imaginary part expression, and the conditions satisfied by the power of the toroidal inductor, the real part of the complex power is obtained. With the imaginary part , is represented as: , ; Substituting the real and imaginary parts of the complex power into the equivalent permeability calculation model, we obtain the permeability prediction model, which is expressed as: ; in, Indicates permeability, This represents the complex power of the toroidal inductor under test. and Let these represent the imaginary and real parts of the complex power, respectively. Represents the imaginary unit; , and These represent the average circumference, number of turns, and cross-sectional area of ​​a circular cross-section ferrite core, respectively. and These represent the supply frequency and the effective value of the supply current, respectively.

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