Calculation method of electromagnetic parameters of circular wire solenoid coil based on equivalent transformation

Abstract

The superconducting magnet coil is composed of several solenoid coils connected in series, and each solenoid can be wound by a wire with a rectangular cross-section or a wire with a circular cross-section. When calculating the magnetic induction intensity generated by the solenoid coil, the traditional approach is to equate each solenoid coil to a large circular ring with a rectangular cross-section and uniform current density distribution. For a solenoid coil wound with a circular wire, if the coil cross-section is equated to a rectangular cross-section tangent to its outer contour, the calculation of parameters such as the magnetic induction intensity will be inaccurate. In response to this problem, the present invention discloses a method for calculating the electromagnetic parameters of a circular wire solenoid coil based on equivalent transformation. The circular wire is converted into a rectangular wire of corresponding size through an equivalent transformation formula, and the size of the circular wire solenoid coil is equivalent to the size of the rectangular wire solenoid coil, so that the electromagnetic parameters such as the magnetic induction intensity generated by the circular wire solenoid coil can be simply and accurately calculated.

Description

Technical Field

[0001] The invention relates to a method for calculating electromagnetic parameters of a solenoid coil, and more specifically to a method for calculating electromagnetic parameters generated by a solenoid coil wound with a circular wire. Background Art

[0002] The solenoid coil is composed of many turns of wire wound on a cylindrical surface. Each turn of wire can be approximately regarded as a circular ring structure. In engineering, solenoid coils have many applications, such as transformers, inductors, superconducting magnet coils, etc. The solenoid coil can be wound with a rectangular cross-section wire or a circular cross-section wire, and the winding process of the two is different. When analyzing the magnetic induction intensity generated by the solenoid coil, the difference between rectangular wire and circular wire needs to be considered. In general literature, the electromagnetic parameter calculation of the solenoid coil is to treat the cross section of the solenoid coil as a rectangular cross-section, and the cross section of the wire is also rectangular, so that each turn of the wire can be regarded as a current-carrying ring, and the centers of all current-carrying rings are evenly distributed, which is convenient for electromagnetic calculation. However, for a circular cross-section coil, a simple equivalence cannot be performed. Directly equating the outermost circle of the coil to a rectangular cross-section will result in uneven current density at the boundary, resulting in inaccurate calculation of electromagnetic parameters. However, this type of problem is rarely involved in the literature, so it is necessary to study the accurate calculation method of the electromagnetic parameters of the solenoid coil wound with a circular wire. Summary of the invention

[0003] Purpose of the invention: In view of the problem that there is no effective algorithm for calculating the electromagnetic parameters of a solenoid coil made of circular conductor in current engineering, the purpose of the present invention is to provide an equivalent transformation method for a solenoid coil wound with a circular conductor to a solenoid coil wound with a rectangular cross-section conductor, and to calculate the electromagnetic parameters of the solenoid coil wound with a rectangular cross-section conductor after the equivalent transformation, so as to realize the accurate calculation of electromagnetic parameters such as magnetic induction intensity.

[0004] Technical solution: To achieve the above-mentioned invention object, the present invention adopts the following technical solution:

[0005] A method for calculating electromagnetic parameters of a circular wire solenoid coil based on equivalent transformation, wherein the method converts a solenoid coil wound with a circular cross-section wire into a solenoid coil wound with a rectangular cross-section wire and then calculates the electromagnetic parameters of the solenoid coil, wherein the method of equivalent transformation is: a cylindrical coordinate system is established with the axis of the solenoid coil as the Z axis, the radius of the wire contained in the solenoid is known to be r, the inner radius of the solenoid is R1, the outer radius is R2, the left axial coordinate is Z1, and the right axial coordinate is Z2. When calculating the electromagnetic parameters of the solenoid coil, the circular cross-section wire is firstly equivalent to a rectangular cross-section wire with a radial height of H and an axial width of W through the following formula:

[0006] H=2βr,W=2r

[0007] The inner radius R'1, outer radius R'2, left axial coordinate Z'1, and right axial coordinate Z'2 of the equivalent rectangular wire wound solenoid coil are obtained by the following formula:

[0008] R′1=R1+(1-β)r

[0009] R'2=R2-(1-β)r

[0010] Z′1=Z1+0.5r

[0011] Z′2=Z2-0.5r

[0012] Where β is a positive real number.

[0013] Furthermore, the value of β is 0.86≤β≤0.88.

[0014] Further, the radial coordinate r of the center of the n-th turn of the l-th layer in the solenoid coil is calculated using the following formula: l,n With the axial coordinate z l,n :

[0015] r l,n =R′1+H·(l+0.5)

[0016] z l,n =Z′1+W·(n+0.5)

[0017] Furthermore, each turn of the wire in the solenoid coil is approximately a current-carrying ring, and the axial magnetic induction intensity generated by each turn of the wire is calculated using the following formula:

[0018] When ρ≠0,

[0019]

[0020] Where μ0 is the magnetic permeability, I is the ring current, r l,n is the radial coordinate of the center of the nth turn of the lth layer of the solenoid coil, ρ is the radial coordinate of the field point, z is the axial coordinate of the field point, and z l,n is the axial coordinate of the plane where the center of the nth turn of the lth layer of the solenoid coil is located, K(k), E(k) are the first and second kind of elliptic integrals, and the parameter k is defined as follows:

[0021]

[0022] When ρ=0:

[0023]

[0024] Furthermore, the radial magnetic induction intensity B generated by each ring is calculated using the following formula:r :

[0025]

[0026] Beneficial effects: The present invention converts a circular cross-section ring into a rectangular cross-section ring of corresponding size through an equivalent transformation formula, and converts the size of a solenoid coil wound with a circular wire into the size of a solenoid coil wound with a rectangular wire. Through this conversion, the existing electromagnetic parameter calculation formula for a solenoid coil wound with a rectangular wire can be applied, so that electromagnetic parameters such as the magnetic induction intensity generated by the circular wire solenoid coil can be simply and accurately calculated. When writing the electromagnetic analysis code for the solenoid coil, the analysis of the circular wire solenoid coil and the rectangular wire solenoid coil can share a set of codes, which facilitates code writing. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 Schematic diagram of the cross section of a circular wire solenoid coil and its equivalent rectangular cross section. DETAILED DESCRIPTION

[0028] This section specifically introduces the present invention in conjunction with specific embodiments.

[0029] The superconducting magnet coil is generally composed of several solenoid coils connected in series. When considering the coil structure, it is only necessary to consider the distribution of the wire or the current on a cross section of the coil. Generally, when performing electromagnetic analysis on a solenoid coil, the cross section of the solenoid is usually regarded as a rectangle, and the wires are evenly distributed in the rectangle. This treatment undoubtedly facilitates electromagnetic calculations. However, when analyzing the electromagnetic parameters of a solenoid coil wound with a circular cross section, such as the magnetic induction intensity, this treatment will cause calculation errors, so a special conversion is required. The conversion process in the present invention is described as follows:

[0030] It is known that the solenoid contains a circular ring with L layers and N turns of metal wire. The axis of the solenoid coil is taken as the Z axis, and any straight line perpendicular to the axis is taken as the R axis to establish a cylindrical coordinate system. The wire radius is r, the inner radius of the solenoid is R1, the outer radius is R2, the left axial coordinate is Z1, and the right axial coordinate is Z2. When calculating the electromagnetic parameters of the solenoid coil, the circular wire is firstly equivalent to a rectangular cross-section wire with a radial height of H and an axial width of W through the following formula:

[0031] H=2βr,W=2r

[0032] Then, the size of the solenoid coil is transformed as follows to obtain the inner radius R'1, outer radius R'2, left axial coordinate Z'1, and right axial coordinate Z'2 of the equivalent solenoid coil wound with a rectangular wire:

[0033] R′1=R1+(1-β)r

[0034] R'2=R2-(1-β)r

[0035] Z′1=Z1+0.5r

[0036] Z'2=Z2-0.5r

[0037] After the above transformation, the solenoid coil wound with a circular cross-section wire can be equivalent to a solenoid coil wound with a rectangular cross-section wire for calculation.

[0038] The following explains the basis for the above transformation. The cross section of a solenoid coil wound with a round wire is as follows: Figure 1 As shown in the figure, when circular wires are used for winding, the centers of the wires are not evenly distributed, and the centers of two adjacent layers differ by a radius in the axial direction. However, further analysis shows that the axial distance between two adjacent circles of wires in the same layer is equal, both 2r; the radial distance between the centers of any two adjacent layers of wires is equal, and the distance is less than 2r. We assume that the radial distance between the centers of two adjacent layers of wires is 2βr, according to Figure 1 The geometric relationship in can be deduced However, in actual winding, considering the engineering error, the value of β may not be exactly equal to Its value range is 0.86≤β≤0.88, and its specific value is determined by the wire material and the tension. For WIC superconducting wires, the wire size may be reduced when the winding tension is large.

[0039] In the same way, the distance between two adjacent turns of wire is not necessarily exactly equal to the wire diameter. When calculating, we replace the diameter of the circular wire with the center axial distance between the two turns of wire during winding, so we only need to correct the radial dimension.

[0040] Next, consider the axial spacing between the wires. In the same layer, the spacing between every two turns of wire is equal, both 2r. However, there is a radial difference of r between the odd-numbered layers and the even-numbered layers. Therefore, we translate the odd-numbered layers to the right by 0.5r and the even-numbered layers to the left by 0.5r. In this way, the centers of all the wires are evenly distributed. According to our calculations, the numerical error caused by this treatment in calculating the magnetic induction intensity can be ignored.

[0041] After the above processing, the solenoid coil wound with a circular wire can be equivalent to the solenoid coil wound with a rectangular wire. The center position of the nth turn of the wire in the lth layer of the solenoid coil can be calculated by the following formula:

[0042] r l,n =R′1+H·(l+0.5)

[0043] z l,n=Z′1W·(n+0.5)

[0044] When calculating the magnetic induction intensity generated by the coil, each circle of wire is approximated as a current-carrying ring, and the axial magnetic induction intensity generated by each circle of wire is calculated using the following formula:

[0045] When ρ=0:

[0046]

[0047] When ρ≠0,

[0048]

[0049] Here μ0 is the magnetic permeability, I is the ring current, r l,n is the radius of the nth turn of the lth layer in the solenoid, ρ is the radial coordinate of the field point, z is the axial coordinate of the field point, and z l,n is the axial coordinate of the plane where the center of the nth turn of the lth layer of the solenoid coil is located, K(k), E(k) are the first and second kind of elliptic integrals, and the parameter k is defined as follows:

[0050]

[0051] The expressions of K(k) and E(k) are as follows:

[0052]

[0053]

[0054] The radial magnetic induction intensity B generated by each ring is calculated using the following formula: r :

[0055]

[0056] In this way, the inner and outer diameters of the coil and the position of each current-carrying ring are determined by the equivalent formula, and the magnetic induction intensity generated by the solenoid coil can be calculated using the above formula. Finally, for the convenience of calculation, we provide the size conversion formula of the solenoid coil composed of round wire and rectangular wire in Table 1.

[0057] Table 1: Conversion relationship between round and rectangular wires for solenoid coils

[0058] Round wire Rectangular wire Number of layers L L Number of turns N N Conductor width 2r 2r Wire height 2r 2βr Solenoid inner radius <![CDATA[R1]]> <![CDATA[R1+(1-β)r]]> Solenoid outer radius <![CDATA[R2]]> <![CDATA[R2-(1-β)r]]> Solenoid left axis coordinates <![CDATA[Z1]]> <![CDATA[Z1+0.5r]]> Solenoid right axis coordinates <![CDATA[Z2]]> <![CDATA[Z2-0.5r]]> Solenoid radial thickness 2(L-1)βr+2r 2βr·L Solenoid axial length (N+0.5)·2r 2Nr

[0059] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the above embodiments do not limit the present invention in any form, and any technical solution obtained by equivalent replacement or equivalent transformation falls within the protection scope of the present invention.

Claims

1. A method for calculating electromagnetic parameters of a circular wire solenoid coil based on equivalent transformation, characterized in that: The method converts a solenoid coil wound with a circular cross-section wire into a solenoid coil wound with a rectangular cross-section wire and then calculates the electromagnetic parameters of the solenoid coil. The equivalent transformation method is as follows: a cylindrical coordinate system is established with the axis of the solenoid coil as the Z axis. It is known that the radius of the wire contained in the solenoid is r, the inner radius of the solenoid is R1, the outer radius is R2, the left axial coordinate is Z1, and the right axial coordinate is Z2. When calculating the electromagnetic parameters of the solenoid coil, the circular cross-section wire is first converted into a rectangular cross-section wire with a radial height of H and an axial width of W through the following formula: H=2βr,W=2r The inner radius R'1, outer radius R'2, left axial coordinate Z'1, and right axial coordinate Z'2 of the equivalent rectangular wire wound solenoid coil are obtained by the following formula: R′1=R1+(1-β)r R′2=R2-(1-β)r Z′1=Z1+0.5r Z′2=Z2-0.5r Where β is a positive real number.

2. The electromagnetic parameter calculation method of a circular wire solenoid coil based on equivalent transformation according to claim 1 is characterized in that: The value of β is 0.86≤β≤0.

88.

3. The electromagnetic parameter calculation method of a circular wire solenoid coil based on equivalent transformation according to claim 1 is characterized in that: The radial coordinate r of the center of the nth turn of the lth layer of the solenoid coil is calculated using the following formula: l,n With the axial coordinate z l,n :

4. The electromagnetic parameter calculation method of a circular wire solenoid coil based on equivalent transformation according to claim 1 is characterized in that: Each turn of wire in the solenoid coil is approximately a current-carrying ring. The following formula is used to calculate the axial magnetic induction intensity generated by each turn of wire: When ρ≠0, Where μ0 is the magnetic permeability, I is the ring current, r l,n is the radial coordinate of the center of the nth turn of the lth layer of the solenoid coil, ρ is the radial coordinate of the field point, z is the axial coordinate of the field point, and z l,n is the axial coordinate of the plane where the center of the nth turn of the lth layer of the solenoid coil is located, K(k), E(k) are the first and second kind of elliptic integrals, and the parameter k is defined as follows: When ρ=0:

5. The electromagnetic parameter calculation method of a circular wire solenoid coil based on equivalent transformation according to claim 1 is characterized in that: Each turn of wire in the solenoid coil is approximately a current-carrying ring. The radial magnetic induction intensity B generated by each ring is calculated using the following formula. r : Where μ0 is the magnetic permeability, I is the ring current, r l,n is the radial coordinate of the center of the nth turn of the lth layer of the solenoid coil, ρ is the radial coordinate of the field point, z is the axial coordinate of the field point, and z l,n is the axial coordinate of the plane where the center of the nth turn of the lth layer of the solenoid coil is located, K(k), E(k) are the first and second kind of elliptic integrals, and the parameter k is defined as follows:

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