1 Introduction

Current-carrying coils are basic elements for constructing electromagnetic magnet systems such as linear actuators and motors^{[1, 2]},or even higher field magnets from high temperature superconductor wires or tapes^{[3, 4, 5]},for example,the magnetic resonance imaging (MRI)^[6] or nuclear magnetic resonance (NMR)^[7] magnet,and the international thermonuclear experimental reactor (ITER) magnet system^{[8, 9, 10]}. For the design and safety operation^[11] of these magnet systems with current-carrying coils,it is important to accurately calculate the magnetic field of a current-carrying coil and the magnetic force between two coils.

So far,some important studies have been accomplished on calculating of the magnetic force between two current-carrying coils. For a current-carrying coil,since the distribution of magnetic field is firstly required for calculating the magnetic force on the current unit,previously researchers made great effort mainly on calculating the magnetic field of current-carrying coils. Some of them presented the analytic expression of the magnetic field of a current-carrying coil (or a permanent/electromagnetic magnet from coils) with a circular^[12],rectangular^[13] cross section,and some made the numerical simulation of the distribution of its magnetic field^[14]. On these bases,the magnetic force between two current-carrying coils has been further studied. Babic and Akyel^[15] and Robertson et al.^[16] presented the analytic expression of the magnetic force between two thin coaxial coils and between a coaxial magnet and a thin coil. Furthermore, for the two coils with a limited horizontal misalign (whose axes are paralleled),Kim et al.^[17] and Ren^[18] calculated the magnetic force between each other. In fact,the misalignment in assembly design of the system and shift from the original position due to the imbalance of magnetic force in current-carrying operation,in some extent,are unavoidable. These factors can make current-carrying coils to be located at an arbitrary position and orientation between each other in the three-dimensional space,not just as the cases mentioned above in which two coils are the coaxial or translationally misaligned. Thus,for the practical design and safety operation of the magnet systems with current-carrying coils,it should be extremely significant to accurately calculate the magnetic force between the two coils with non-coplanar axes (actually these two coils have arbitrary spatial position and orientation between each other).

In this paper,based on Biot-Savart’s law,for the two current-carrying coils with noncoplanar axes,the analytic expressions of the magnetic field of a coil,and the magnetic force between each other are presented. On the verification of our approach,the variation of the magnetic force between the two current-carrying coils is comprehensively investigated. 2 Analytic calculation of magnetic force 2.1 Geometric configuration of problem

As shown in Fig. 1,Coils 1 and 2 with the symmetry axes I₁ and I₂,which are in noncoplanar, are at arbitrary relative positions. Without loss of generality,Coil 1 is fixed at the origin of coordinate (i.e.,the point O) and located in the xy-plane. The location and the orientation of Coil 2 can be determined by the coordinate of its center O′(xO′,yO′,zO′) and the angles of the plane of Coil 2 referring to the x- and y-axes (specifically noted below),respectively. The radii of Coils 1 and 2 are,respectively,r₁ and r₂,and the radius ratio is introduced as μ = r₁/r₂ (μ > 1). The current intensities of Coils 1 and 2 are I₁ and I₂,respectively.

In the O-xyz coordinate system,for the point P (x_P,y_P,z_P ) on Coil 1,the coordinates are given as

in which θ is the rotation angle referring to the x-axis.

For the point P′ (x_P′,y_P′,z_P′) (its vectro is noted as) on Coil 2 in the coordinate system,we can obtain

in which and A and B are the rotation operations of the angles α and β,which are the angles of the plane of Coil 2 referring to the x-axis and the y-axis,respectively.

On above the geometric model,for rationally establishing the approach of the calculations of the magnetic field of a current-carrying coil and the magnetic force between the two coils, some basic assumptions have to be given as follows:

(i) The current in the coils keeps to be steady. Resistance loss of current-carrying coils is not taken account here. The magnetic coupling between two coils is not considered. Hence,the effect of the variation of the current in one coil on the other coil is neglected here.

(ii) The distance between the centers of the two current-carrying coils cannot be infinitely close to each other. Physically,the distance between the centers of two circle coils can be zero. However,mathematically,it may cause serious singularity in calculation.

(iii) The elastic deformation and dynamic behavior of current-carrying coils are not further taken into account in this study. Based on the geometric model shown in Fig. 1 and the above necessary assumptions,according to the rule of Ampere circulation and Biot-Savart’s law,the magnetic force of the coils with non-coplanar axes can be analytically obtained. For this aim,there are two basic steps of the spatial distribution of the magnetic field of Coil 1 and the magnetic force of Coil 2 in the magnetic field from Coil 1. 2.2 Magnetic field of Coil 1

For simplicity,the magnetic field of the point P′ (r_P′,θ_P′,z_P′) on Coil 2 induced from Coil 1 can be given in the cylindrical coordinate system as^[17]

in which where K(k) and E(k) are,respectively,the complete elliptic integrals of the first and second kinds,and μ0 (the permeability of free space) is 4π × 10−7. In the O-xyz coordinate system, the magnetic field at the point P′(x_P′,y_P′,z_P′ ) can be transformed as in which rP′ = . 2.3 Magnetic force between two coils

The magnetic force of the current unit I₂dl of Coil 2 in the magnetic field induced from Coil 1 can be given by

in which dl = . Therefore,the projection of the x-,y-,and z- axes of the current unit I₂dl can be written by (I₂dl)x,(I₂dl)y,and (I₂dl)z. Then,Eq. (7) in cylindrical coordinates can be developed as in which e_i,e_j,and e_k are the unit vectors of the x-,y-,and z-axes. Thus,the magnetic force between Coils 1 and 2 can be obtained as 3 Results and discussion 3.1 Verification of approach

To veriF_y the approach presented above,the case of the two current-carrying circle coils with parallel axes is considered,and the magnetic force between the coils will be further compared with those in the published work^[17]. In this case,the planes of the two coils are parallel. The coordinate z_O′ of Coil 2 is fixed,and the misaligned distance is limited just in the xy-plane. For simplicity,as shown in Fig. 2,the misaligned distance is just along the y-axis,noted as Δy. Then,the coordinate of the point P′ can be written as

After Eq. (10) is substituted into Eq. (4),the magnetic field of Coil 1 can be given by

in which

According to Eq. (8),dF_y can be given as (I₂dl)zB_x−(I₂dl)xB_z = (I₂r₂sin θ)Bzdθ,and dF_z along the z-axis is (I₂dl)xB_y − (I₂dl)yB_x = −(r₂ sin θy_P′/r_P′ + r₂ cos θx_P′/r_P′ )Brdθ. Thus,the y- and z-components of the magnetic forces (F_y and F_z) can be,respectively,given as

When Coils 1 and 2 have N₁ and N₂ turns,respectively,the magnetic force between the coils can be further obtained as^[17]

In calculation,r₁ and r₂ are chosen as 0.5m and 0.3m,respectively. The current intensities I₁ and I₂ are given as 10 A and 20 A,respectively. If zO′ is kept as 0.2m,we can obtain the variation of the magnetic force with the misaligned distance Δy. The calculation results of the magnetic forces F_y and F_z are listed in Tables 1 and 2,respectively.

As listed in Tables 1 and 2,though the calculation approach is somewhat different from that in Ref. ^[17],the calculation results are close to each other. The maximum error is 1.038% for the radial magnetic force and about 0.026% for the axial magnetic force. Thus,the comparison of the magnetic force verifies that our approach in this paper is exact and achieves enough accuracy.

It is noted here that,even Δy is zero (the first line and the first column in the two tables), it does not mean that the distance of the center of Coil 2 away from that of Coil 1 is zero. For the distance z_O′ is always kept as a nonzero constant. 3.2 Results and discussion

Based on the geometric model shown in Fig. 1 and applying the analytic expression of the magnetic force between two current-carrying coils,we can further investigate the variation of the magnetic force between the two coils with the spatial relative position and the orientation of each other,which is very important not only for the design,but also for the safety operation of the related electromagnetic equipment with current-carrying coils. In the calculation below, r₁ is chosen as 60mm (r₂ can be determined by μ = r₁/r₂),and the current intensities I₁ and I₂ are given as 10 A and 20 A,respectively. Coil 1 is still supposed to be fixed at the origin of coordinate system O-xyz. As a result,the calculation results of the variation of the magnetic force between the two coils with the spatial relative position and the orientation of each other have been obtained as shown in Figs. 3-6.

The spatial variation of the components F_x,F_y,and F_z of the magnetic force with the coordinate of the centre of Coil 2 in the xy-plane is shown in Fig. 3. It is found that the variation of the components F_x and F_y with the coordinates xO′ and yO′ is almost monotone, and it ranges from positive to negative,while the component F_z keeps always negative,its value reaches the maximum at the distance about 2 mm from the center of Coil 1 (i.e.,the origin of coordinate),and then decreases with the distance. Within a same spatial range,generally,the magnitude of the component F_z is always much larger than those of the components F_x and F_y. At a given height (for example,here z_O′ is 30 mm),the misaligned distance in the xy-plane results in considerable in-plane magnetic forces (i.e.,the components F_x and F_y). In fact,these in-plane magnetic forces result in the restoring force,usually defined as F_r as , which can make Coil 2 back to its original position. Figure 4 shows that the variation of the restoring force with the distance between the centers of the coils in the xy-plane (defined as It is found that the restoring force always has a peak value with the distance. Before the peak,the restoring force increases with the distance,while decreases after the peak. Besides,under the given size of Coil 1,the larger the radius ratio is,the smaller the restoring force is. These results should be very helpful for safety control of the current-carrying coils to be reset in electromagnetic systems.

For the given angles of α and β (here α = π/20 and β = π/10),the magnetic force with the coordinates y_O′ and z_O′ of the center of Coil 2 has been calculated and shown in Figs. 5 and 6, respectively. For simplicity,the x-coordinate of the center of Coil 2 is also assumed to be zero, that is,the movement of the center of Coil 2 is limited just in the yz-plane. As shown in Fig. 5, the components F_y and F_z are much larger than the component F_x,and the latter is close to zero. Obviously,the components F_y and F_z dominate. With the y-coordinate increasing,the component F_y decreases,while the component F_z increases in this case. In fact,practically,the restoring force (related to F_x and F_y) increases,while the magnitude of F_z decreases. Bedsides, once the size of Coil 1 is given,the larger the radius ratio μ is,the smaller the magnitude of the magnetic force is (here F_y and F_z are always minus). As shown in Fig. 6,under these given parameters,the component F_x is always positive,while the components F_y and F_z are negative. Just from their magnitudes,comparatively,the components F_y and F_z are larger,and with the z-coordinate increasing,all the components F_x,F_y,and F_z decrease. Furthermore,similarly as in Fig. 5,the larger the radius ratio is,the smaller all the components of the magnetic force are.

To further investigate the variation of the magnetic force with the orientation of Coil 2,for simplicity,we set the center of Coil 2 at a given height,and on the symmetry axes I₁ of Coil 1, and then just make the orientation of the plane of Coil 2 vary referring to the xy-plane. As a result,the variations of the magnetic force with the angles α and β are obtained and shown in Fig. 7. Compared with the results in Fig. 3,the variation of the magnetic force with the angles α and β is more distinct. The variations of the components F_x and F_y are symmetric referring to the angles α and β,respectively,and for the component F_z,it is centrosymmetric. As shown in Figs. 7(a) and (b),for the component F_x,it reaches zero when the angle β is equal to zero, and it increases with the angle β. The similar variation is for the component F_y with the angle α. The difference is that the component F_x varies not only with the angle β but also with the angle α. It can reach the extreme point (positive and negative) when the angle α is equal to zero. For the component F_y,it varies only with the angle α,almost not with the angle β. For the component F_z,it can reach the negative extreme point when both the angles α and β are zero,and monotonously increases when the angles α and β increase.

For a practical electromagnetic system with current-carrying coils,in the initial assembly and operation,the plane of the coil referring to others,usually has a misaligned angle,not always parallel with each other (here it means α = β = 0). To control the coil to be reset back to the original orientation,the moment on the coil from the magnetic force,which has the mechanical effect of rotation,is very important. As shown in Fig. 1,the moment M of Coil 2 from Coil 1 can be calculated as

in which d_P′ is the vector of the point P′ on Coil 2 in the O-xyz coordinate system,dF is the magnetic force of the current unit of Coil 2 in the magnetic field from Coil 1. To characterize the rotation effect of the magnetic force on Coil 2 around the x- and y-axes,we also define the restoring moment M_r as in which M_x and M_y stand for the x- and y-components of moment,respectively.

The variation of the restoring moment M_r with the angles α and β is calculated as shown in Fig. 8. Obviously,the variation of the moment M_r is symmetrical with the angles α and β. Specifically,the variation of the moment M_r has a saddle shape about the angle β. When both the angles β and α are close to zero,the moment M_r arrives to the minimum extrem point. With the angle β getting bigger,the moment M_r accordingly increases. Meanwhile,when the angle α is zero,M_r reaches the maximum extreme point,and when the angle α gets bigger,M_r decreases sharply.

4 Conclusions

For two current-carrying coils with non-coplanar axes,based on Biot-Savart’s law,the analytic expression of the magnetic force is presented. The comparison of the magnetic force verifies that the approach in this paper has enough exactness. Furthermore,the complete and deep investigation of the variation of the magnetic force shows that the magnetic force between the two coils is closely related to the size,the relative position,and the orientation of each other. Besides the statical magnetic force,for two current-carrying coils with non-coplanar axes,hopefully,the elastic deformation and dynamic behaviours of the coils due to the magnetic force should be further studied. These studies including that in this paper will be helpful for the design and safety operation of related electromagnetic equipment with current-carrying coils.