A fingerprint-shaped vector magnet

Abstract

This application belongs to the field of magnets and specifically discloses a fingerprint-shaped vector magnet; specifically, it includes: at least one fingerprint-shaped coil with two centrally symmetrical lobes, and / or a multi-turn cylindrical coil; the fingerprint-shaped coil and the multi-turn cylindrical coil are both distributed on concentric cylindrical surfaces, and the radii of the fingerprint-shaped coil and the multi-turn cylindrical coil are different, so that the fingerprint-shaped coil and the multi-turn cylindrical coil are nested with each other; the multi-turn cylindrical coil is used to generate a uniform magnetic field along the axis of the cylindrical surface under the action of current; and the fingerprint-shaped coil is used to generate a uniform magnetic field perpendicular to the axis under the action of current. This application facilitates magnetic field regulation and has high control accuracy.

Description

Technical Field

[0001] The present application belongs to the field of magnets, and more specifically, relates to a fingerprint-shaped vector magnet. Background Art

[0002] Vector magnetic fields have broad and important applications in cutting-edge physics research. For example, in the study of topological states, topology is linked to cutting-edge physics concepts such as quantum computing and quantum geometry. Topological superconducting states, whose properties are combined with superconductivity, can be used in non-dissipative quantum computers. Vector magnetic fields provide a powerful tool for studying topological phase transitions and topological transport. Specific areas include studying the transport properties of the spin quantum Hall effect and the manipulation of the collapse of the quantum anomalous Hall effect by multidimensional magnetic fields. In the study of electron strong correlation mechanisms, vector magnetic fields can be used to control the speed of continuous strongly correlated phase transitions based on carrier filling in two-dimensional moiré superlattices or to achieve discontinuity in such phase transitions, as well as to decouple the electron-phonon interaction in the strongly correlated phase transition mechanism of VO2 at near room temperature. In the field of electron spin control, vector magnetic fields can regulate magnetic field strength from multiple directions, synthesizing external fields in arbitrary directions. This overcomes the previous disadvantage of a single magnetic field not fully matching the device construction direction, and is a powerful means to achieve precise control of the spin torque of spintronic devices.

[0003] In order to apply a vector magnetic field to a sample or device, the sample stage or the rotating magnetic field can usually be rotated. Compared with sample stage rotation, magnetic field rotation does not require changing the spatial position of the sample stage, does not introduce other variables besides magnetic field rotation, has stronger robustness, and can provide more possibilities for the application of vector magnetic fields. There are usually two ways to rotate the magnetic field: one is to achieve the purpose of rotating the magnetic field by rotating the magnet itself, usually a rotating permanent magnet. This method has problems such as low control accuracy, low magnetic field strength, and poor magnetic field uniformity; the other is to achieve the purpose of generating a rotatable magnetic field by controlling the current of the magnet coil, that is, a vector magnet. However, existing vector magnets have problems such as a fixed magnet coil structure, poor central magnetic field uniformity, and the magnetic field configuration is limited to one type of configuration, and the coil design cannot be flexibly changed according to the user's special needs. Summary of the Invention

[0004] In response to the defects of the existing technology, the purpose of this application is to provide a fingerprint-shaped vector magnet, which aims to solve the problems in the existing vector magnets, such as the fixed structure of the magnet coil, poor uniformity of the central magnetic field, the magnetic field shape being limited to one type of shape, and the inability to flexibly change the coil design according to the user's special needs.

[0005] To achieve the above objectives, in a first aspect, the present application provides a fingerprint-shaped vector magnet, comprising: at least one fingerprint-shaped coil with two centrosymmetric lobes, and / or a multi-turn cylindrical coil;

[0006] The fingerprint-shaped coil and the multi-turn cylindrical coil are both distributed on concentric cylindrical surfaces, and the radii of the fingerprint-shaped coil and the multi-turn cylindrical coil are different, so that the fingerprint-shaped coil and the multi-turn cylindrical coil are nested with each other;

[0007] The multi-turn cylindrical coil is used to generate a uniform magnetic field along the axis of the cylindrical surface under the action of current;

[0008] The fingerprint-shaped coil is used to generate a uniform magnetic field perpendicular to the axis direction under the action of current;

[0009] The uniform magnetic fields in different directions generated by the multi-turn cylindrical coil and / or fingerprint-shaped coil constitute a fingerprint-shaped vector magnetic field.

[0010] Further preferably, a plurality of fingerprint-shaped coils are provided, and the axis connecting the center points of the two centrally symmetrical fingerprint-shaped coils is the direction in which the fingerprint-shaped coil generates a uniform magnetic field; the angle between the directions of the uniform magnetic fields generated by the fingerprint-shaped coils is an arbitrary angle, but is perpendicular to the direction of the uniform magnetic field generated by the multi-turn cylindrical coil.

[0011] Further preferably, the fingerprint-shaped coil includes an X coil and a Y coil; the central magnetic fields generated by the X coil and the Y coil are in perpendicular directions; the multi-turn cylindrical coil is a Z coil; the X coil, the Y coil, and the Z coil are used to generate uniform magnetic fields along the X-axis, the Y-axis, and the Z-axis directions, respectively; the Z coil is located at the innermost side, the Y coil is located in the middle, and the X coil is located at the outermost side.

[0012] Further preferably, the X coils, the Y coils and the Z coils are combined in pairs to form a two-dimensional fingerprint-shaped vector magnet structure.

[0013] Further preferably, the fingerprint-shaped coil and the multi-turn cylindrical coil are connected to the same power supply and energized at the same time, or are connected to different power supplies and energized in different orders.

[0014] Further preferably, the fingerprint-shaped coil and the multi-turn cylindrical coil are wound by conventional conductive cables or superconducting cables.

[0015] Further preferably, the fingerprint-shaped coil and the multi-turn cylindrical coil share the same supporting frame, or are separately separate.

[0016] In a second aspect, the present application provides a method for obtaining a fingerprint-shaped vector magnet, comprising the following steps:

[0017] Step S1: Select several target points in a uniform magnetic field region, use the Biot-Savart law to express the magnetic field intensity at the target points using the stream function of the current density on the cylindrical surface, and then establish a target function of the sum of the squares of the deviations between the actual magnetic field intensity at the target points and the theoretical magnetic field intensity;

[0018] Step S2: Introduce a penalty term into the objective function to form an objective functional;

[0019] Step S3: According to the regularization theory, the stream function coefficient matrix is ​​solved based on the target functional;

[0020] Step S4: Obtain the stream function of the current density on the cylindrical surface according to the obtained stream function coefficient matrix, and discretize the stream function of the current density on the cylindrical surface to obtain the position of each turn of the conductor; wherein the center position of each turn of the conductor coincides with the stream function contour line.

[0021] Further preferably, the current flowing into the coil wire is:

[0022]

[0023] in, is the stream function of the current density on the cylindrical surface; N is the number of wire turns obtained by discrete calculation; is the current flowing into the coil wire.

[0024] Further preferably, the penalty items include: coil curvature, coil torque, leakage magnetic field distribution, maximum current density distribution and coil spacing.

[0025] Further preferably, the objective function is:

[0026]

[0027] in, is the objective function; M The number of target points; Q The front of the stream function expansion Q item; For the q The stream function coefficients to be solved; is the position coordinate of the target point; is the magnetic field strength of the target point; Through the m The spatial relationship between the target point and the cylindrical surface is calculated and obtained as follows:

[0028]

[0029] In general, the above technical solutions conceived by this application have the following beneficial effects compared with the existing technologies:

[0030] The present application provides a fingerprint-shaped vector magnet, in which a plurality of fingerprint-shaped coils with two central symmetries and multi-turn cylindrical coils can be provided. The multi-turn cylindrical coils generate a uniform magnetic field along the axis of the cylindrical surface, and the fingerprint-shaped coils generate a uniform magnetic field perpendicular to the axis. A multi-dimensional vector magnet can be formed, and the field strength of the uniform magnetic field in each direction can be controlled according to the magnitude of the current passed through the control coil. The electrical control adopted can excite the coils separately. Therefore, the magnetic field regulation is convenient and the control accuracy is high.

[0031] The present application provides a fingerprint-shaped vector magnet. Since the radii of the multi-turn cylindrical coil and the fingerprint-shaped coil can be different, the fingerprint-shaped coil and the multi-turn cylindrical coil are nested with each other to form a fingerprint-shaped vector magnet with a nested structure. The nesting of the cylindrical surface structure makes the magnet small in size and has little space restriction.

[0032] The present application provides a fingerprint-shaped vector magnet, in which a fingerprint-shaped coil is used to generate a uniform magnetic field perpendicular to the axis direction. This fingerprint-shaped coil is not easily deformed in practical applications and has good magnetic field stability.

[0033] The present application provides a fingerprint-shaped vector magnet, in which, since the target field method theory and regularization theory are used for coil design, the actual magnetic field strength of the target point can be set according to actual needs, and the objective function is constructed in combination with the magnetic field strength expression at the target point. The coil constructed based on this meets the actual magnetic field limitation. At the same time, based on the above objective function, penalty terms are introduced, and the penalty terms include coil curvature, coil torque, leakage magnetic field distribution, maximum current density distribution and coil spacing, so that the obtained coil meets but is not limited to achieving requirements such as high magnetic field uniformity, low leakage magnetic field, special magnetic field configuration and small coil torque. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 This is the fingerprint-shaped structure of the X-coil provided in the embodiment of the present application;

[0035] Figure 2 It is the fingerprint-shaped structure of the Y coil provided in the embodiment of the present application;

[0036] Figure 3 The embodiment of the present application provides a non-uniform multi-turn cylindrical coil structure of the Z coil;

[0037] Figure 4 The fingerprint-shaped vector magnet structure is composed of the X coil, Y coil and Z coil provided in the embodiment of the present application;

[0038] Figure 5 This is a flow chart of the method for obtaining a fingerprint-shaped vector magnet provided in an embodiment of the present application. DETAILED DESCRIPTION

[0039] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0040] The term "and / or" as used herein describes an association between related objects, indicating that three possible relationships exist. For example, "A and / or B" can represent: A exists alone, A and B exist simultaneously, or B exists alone. The symbol " / " as used herein indicates that the related objects are in an "or" relationship, for example, A / B means either A or B.

[0041] In the embodiments of this application, words such as "exemplary" or "for example" are used to indicate examples, illustrations, or descriptions. Any embodiment or design described as "exemplary" or "for example" in the embodiments of this application should not be interpreted as being preferred or advantageous over other embodiments or designs. Rather, the use of words such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner.

[0042] In the description of the embodiments of the present application, unless otherwise specified, “plurality” means two or more.

[0043] The embodiments of the present application are described below in conjunction with the drawings in the embodiments of the present application.

[0044] In the embodiments of this application, the XYZ coordinate system is a Cartesian coordinate system, where the X-axis is the direction of the center axis of the vector magnet X coil, the Y-axis is the direction of the center axis of the vector magnet Y coil, and the Z-axis is the direction of the center axis of the vector magnet Z coil. For ease of description, the X-axis, Y-axis, and Z-axis in this embodiment are merely used to distinguish the three coordinate axes. In actual use, these names should not be limiting, and the names of the coordinate axes can be defined as needed.

[0045] like Figures 1 to 4 As shown, the fingerprint-shaped vector magnet structure provided in the embodiment of the present application includes an X coil, a Y coil and a Z coil;

[0046] The X-coil is distributed on a cylindrical surface with its axis along the Z-axis. The two-pole coil forms a two-lobed centrosymmetrical fingerprint shape. In terms of spatial position, the X-coil is located at the outermost side of the vector magnet and is used to generate a uniform magnetic field Bx along the X-axis in the center area of ​​the coil. The axis connecting the center points of the two-pole coils is the central axis of the X-coil, that is, the direction of the uniform magnetic field along the X-axis.

[0047] The Y coil is distributed on another cylindrical surface with its axis along the Z axis. The shape of its two-pole coil is also a two-lobed centrosymmetrical fingerprint. In terms of spatial position, the Y coil is located inside the X coil and is used to generate a uniform magnetic field By along the Y axis in the center area of ​​the coil. The axis connecting the center points of the two-pole coils is the central axis of the Y coil, that is, the direction of the uniform magnetic field along the Y axis.

[0048] The Z coil is distributed on the third cylindrical surface along the Z axis. The coil is a multi-turn cylindrical coil, and the wire is unevenly distributed along the Z axis, that is, the spacing between each turn of the wire is not necessarily the same. In terms of spatial position, the Z coil is located inside the Y coil and is used to generate a uniform magnetic field Bz along the Z axis in the center area of ​​the coil;

[0049] Therefore, the three cylindrical surfaces in the Z-axis direction where the X coil, Y coil and Z coil are distributed are all concentric, but the radii of the three cylindrical surfaces are different, so that the vector magnet structure presents a structure in which three groups of coils are nested with each other in space.

[0050] The above three sets of coils together form an XYZ three-dimensional fingerprint-shaped vector magnet structure;

[0051] It should be noted here that the relative positional relationship between the X coil, Y coil, and Z coil is not limited to the situation where the Z coil is located at the innermost side, the Y coil is located in the middle, and the X coil is located at the outermost side;

[0052] In some embodiments of the present application, the X coil, the Y coil, and the Z coil may be combined in pairs to form a two-dimensional fingerprint-shaped vector magnet structure. For example, the X coil and the Z coil may be combined to form an XZ two-dimensional fingerprint-shaped vector magnet structure.

[0053] In some embodiments of the present application, the X coil, Y coil, and Z coil may be energized simultaneously or in different sequences; wherein the greater the current flowing through the coil, the greater the strength of the uniform magnetic field generated by the corresponding coil;

[0054] In some embodiments of the present application, the X coil, Y coil, and Z coil may be powered by the same power supply, or may be powered by different power supplies with different currents to generate multi-angle vector magnetic fields.

[0055] In some embodiments of the present application, the X coil, the Y coil, and the Z coil may be made of conventional conductive cables or superconducting cables;

[0056] In some embodiments of the present application, the X coil, Y coil, and Z coil may share the same support frame, or may be separated separately and finally assembled into a vector magnet. The specific structure of the support frame is not limited and may be adopted according to actual conditions.

[0057] In some embodiments of the present application, the magnetic field direction generated by the Z-axis coil needs to be perpendicular to the magnetic field directions generated by other coils, and the magnetic field direction angles between other coils can be non-perpendicular to adapt to magnets with special magnetic field requirements, which can be determined specifically according to actual requirements.

[0058] To sum up, the present application adopts a multi-turn cylindrical coil to generate a uniform magnetic field along the axial direction of the cylindrical surface, and the uniform magnetic field perpendicular to the axial direction is realized by using a two-petal centrosymmetrical fingerprint-shaped coil; it should be pointed out here that the fingerprint-shaped coils that generate a uniform magnetic field perpendicular to the axial direction are not limited to being perpendicular to each other, and can be at any angle, and multiple groups of fingerprint-shaped coils are set to be nested to form uniform magnetic fields in multiple directions perpendicular to the axial direction.

[0059] The flowchart of the method for obtaining the X coil, Y coil and Z coil provided in the embodiment of the present application is as follows: Figure 5 As shown, the acquisition methods of the three sets of coils are the same, but due to different target fields, the acquired X coils and Y coils are fingerprint-shaped coils, while the Z coil is a multi-turn cylindrical coil. The specific steps are as follows:

[0060] Step 1: Construct the stream function of the current density on the cylindrical surface

[0061] In length 2 L , on the cylindrical surface with radius a, a cylindrical coordinate system is established with the center point of the cylinder as the origin ( ), the stream function expression of the current density on the cylindrical surface is constructed as:

[0062]

[0063] in, k is an integer. When designing the Z coil k is 0, otherwise it is 1; c is an arbitrary constant, which can be taken as a for the convenience of calculation; is the stream function coefficient to be solved; usually the stream function expansion takes the first Q Just add the items together, Q The value of depends on the conductor parameters used in engineering practice; when the conductor is a normal conductor, the conductor parameters are the cross-sectional area and resistivity; when the conductor is a superconducting conductor, the conductor parameters are the cross-sectional area and critical current; Q The larger the value, the more accurate the stream function coefficient calculated by the stream function of the current density on the cylindrical surface, but the more difficult it is to discretize the current density; ρ is the radial distance of the cylindrical surface; ϕ is the azimuth of the cylindrical surface; ; z is the vertical coordinate of the cylindrical surface;

[0064] From this we can get the current density vector on the cylindrical surface With stream function The relationship is:

[0065]

[0066]

[0067] in, is the current density vector in the cylindrical coordinate system Directional component; is the Z-direction component of the current density vector in the cylindrical coordinate system;

[0068] Step 2: Establish an objective function related to the current density on the cylindrical surface

[0069] Taking the X coil as an example, since it is necessary to obtain the central magnetic field strength and direction, M target points are selected in the uniform field intensity area ( ), using the Biot-Savart law The magnetic field intensity at the target point is expressed using the current density function on the cylindrical surface mentioned above;

[0070]

[0071] Among them, matrix U is the coefficient matrix to be solved; matrix B is the magnetic field strength matrix of the target point. For example, when calculating the Bx magnetic field strength of the X coil, the element values ​​in matrix B should all be the Bx magnetic field values ​​actually required; matrix D is a known matrix calculated based on the spatial relationship between the target point and the cylindrical surface. It is calculated as follows:

[0072]

[0073] The objective function at this time is the sum of the squares of the deviations between the target field point and the ideal magnetic field strength at that point, and the expression is:

[0074]

[0075] Step 3: Add penalty function to the objective function

[0076] According to the needs of vector magnet acquisition, penalty terms corresponding to various indicators are added to the objective function to form an objective functional to control the corresponding parameters. The importance of each penalty term is characterized by the penalty function weight. The selection of penalty terms includes but is not limited to coil curvature, coil torque, leakage magnetic field distribution, maximum current density distribution, and coil spacing, etc.

[0077] The specific penalty terms used in the vector magnet coil design can be determined according to the user's special needs. At the same time, the weight of each penalty function can also be formulated to represent the importance of each penalty term.

[0078] use Represents each penalty item, using Represents the penalty function weight, then the objective functional becomes:

[0079]

[0080] Step 4: Solve the matrix equation to obtain the stream function coefficients

[0081] According to the regularization theory, the stream function coefficient matrix can be obtained The regularization solution is:

[0082]

[0083] Step 5: Discretization of current density

[0084] According to the above steps, the stream function of the current density on the cylindrical surface of the coil is obtained. And its distribution diagram, the position of each turn of wire can be discretized by using the properties of the stream function; let the number of turns of wire obtained by discretization be N, and the current flowing into the coil wire be , the relationship between the two is:

[0085]

[0086] The center position of each turn of the conductor and the stream function contour line coincide:

[0087]

[0088] The specifications of the conductors should be set according to the actual project. For example, the conductor width should ensure that the conductors do not overlap with each other, and a certain insulation width should be left between the conductors. The current carrying capacity corresponding to the conductor cross-sectional area should meet the current required to generate a vector magnetic field. For conventional conductive cables, in order to reduce the skin effect of the current in the conductor, the conductor cross-sectional area should not be too large. In order to make the resistance and self-inductance of the coil as small as possible, reduce the heating problem of the coil and increase the switching speed of the coil, the conductor cross-sectional area should not be too small. For superconducting cables, it is usually necessary to select from a limited number of cable types with specified specifications. Therefore, in the face of actual engineering needs, the above conditions should be comprehensively considered, and suitable conductor parameters should be selected to discretize the current density. Then, the turns of wire should be connected in series to obtain the actual discrete conductor coil style.

[0089] In summary, compared with the prior art, this application has the following advantages:

[0090] The magnet is small in size, has little space restriction and high excitation efficiency;

[0091] Since electric control is adopted and three sets of coils are excited separately, the magnetic field can be easily controlled and the control accuracy is high;

[0092] The magnetic coil is fingerprint-shaped, not easily deformed, and has good magnetic field stability;

[0093] Since the target field method theory and regularization theory are used for coil design, the magnetic field distribution can be formulated according to user needs, including but not limited to achieving high magnetic field uniformity, low leakage magnetic field, special magnetic field configuration and small coil torque.

[0094] It is easy for those skilled in the art to understand that the above is only a preferred embodiment of the present application and is not intended to limit the present application. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present application should be included in the scope of protection of the present application.

Claims

1. A fingerprint-shaped vector magnet, characterized in that: include: At least one fingerprint-shaped coil with two centrosymmetric lobes and a multi-turn cylindrical coil; The fingerprint-shaped coil and the multi-turn cylindrical coil are both distributed on concentric cylindrical surfaces, and the radii of the fingerprint-shaped coil and the multi-turn cylindrical coil are different, so that the fingerprint-shaped coil and the multi-turn cylindrical coil are nested with each other; The multi-turn cylindrical coil is used to generate a uniform magnetic field along the axis of the cylindrical surface under the action of current; the fingerprint-shaped coil is used to generate a uniform magnetic field perpendicular to the axis under the action of current; The uniform magnetic fields in different directions generated by the multi-turn cylindrical coil and the fingerprint-shaped coil constitute a fingerprint-shaped vector magnetic field; Among them, the multi-turn cylindrical coil and fingerprint-shaped coil are designed using the target field method theory and regularization theory; The method for obtaining a fingerprint-shaped coil and a multi-turn cylindrical coil specifically comprises the following steps: Step S1: Select several target points in a uniform magnetic field region, use the Biot-Savart law to express the magnetic field intensity at the target points using the stream function of the current density on the cylindrical surface, and then establish a target function of the sum of the squares of the deviations between the actual magnetic field intensity at the target points and the theoretical magnetic field intensity; Step S2: Introduce a penalty term into the objective function to form an objective functional; Step S3: According to the regularization theory, the stream function coefficient matrix is ​​solved based on the target functional; Step S4: Obtain the stream function of the current density on the cylindrical surface according to the obtained stream function coefficient matrix, and discretize the stream function of the current density on the cylindrical surface to obtain the position of each turn of the conductor; wherein the center position of each turn of the conductor coincides with the stream function contour line.

2. The fingerprint-shaped vector magnet according to claim 1, characterized in that: A plurality of fingerprint-shaped coils are provided, and the axis connecting the center points of the two centrally symmetrical fingerprint-shaped coils is the direction in which the fingerprint-shaped coil generates a uniform magnetic field; the angle between the directions of the uniform magnetic fields generated by the fingerprint-shaped coils is an arbitrary angle, but is perpendicular to the direction of the uniform magnetic field generated by the multi-turn cylindrical coil.

3. The fingerprint-shaped vector magnet according to claim 1 or 2, characterized in that: The fingerprint-shaped coil includes an X coil and a Y coil; the central magnetic fields generated by the X coil and the Y coil are perpendicular to each other; the multi-turn cylindrical coil is the Z coil; the X coil, the Y coil, and the Z coil are used to generate uniform magnetic fields along the X axis, the Y axis, and the Z axis, respectively; the Z coil is located on the innermost side, the Y coil is located in the middle, and the X coil is located on the outermost side.

4. The fingerprint-shaped vector magnet according to claim 3, characterized in that: The X coil, the Y coil and the Z coil are combined in pairs to form a two-dimensional fingerprint-shaped vector magnet structure.

5. The fingerprint-shaped vector magnet according to claim 1 or 2, characterized in that: The fingerprint-shaped coil and the multi-turn cylindrical coil are connected to the same power supply and energized at the same time, or are connected to different power supplies and energized in different sequences.

6. The fingerprint-shaped vector magnet according to claim 5, characterized in that: The fingerprint-shaped coil and the multi-turn cylindrical coil are wound by conventional conductive cables or superconducting cables.

7. The fingerprint-shaped vector magnet according to claim 1 or 6, characterized in that: The fingerprint-shaped coil and the multi-turn cylindrical coil share the same supporting frame, or are separate from each other.

8. The fingerprint-shaped vector magnet according to claim 1, characterized in that: The stream function expression of the current density on the cylindrical surface is: The relationship between the current flowing into the coil wire and the number of turns of the wire is: in, is the stream function of the current density on the cylindrical surface; N is the number of wire turns obtained by discrete calculation; is the current flowing into the coil wire; is the azimuth of the cylindrical surface; ρ is the radial distance of the cylindrical surface; is the stream function coefficient to be solved; ; c is an arbitrary constant; k is an integer; z is the vertical coordinate of the cylindrical surface; L is half the length of the cylinder; Represents the maximum value; Represents the minimum value; The center position of each turn of the conductor and the stream function contour line The expression for coincidence is: 。 9. The fingerprint-shaped vector magnet according to claim 1, characterized in that: Penalties include: Coil curvature, coil torque, leakage magnetic field distribution, maximum current density distribution and coil spacing.

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