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Calculation of Mutual Inductance between Circular and Arbitrarily Shaped Filaments via Segmentation Method

In this article, two analytical formulas for the calculation of mutual inductance between a circular filament and line segment arbitrarily positioning in the space are derived by using Mutual Inductance Method (MIM) and Babic's Method (BM), respectively. Using the fact that any curve can be interpolated by a set of line segments, a method for calculation of mutual inductance between a circular filament and filament having an arbitrary shape in the space is proposed based on the derived analytical formulas. The derived two formulas and the proposed method (Segmentation Method) were numerically validated by using FastHenry software and reference examples from the literature. In particular, the proposed method was successfully applied to the calculation of mutual inductance between the circular filament and the following special curves such as circle, circular arc, elliptic arc, ellipse, spiral, helices and conical helices. All results of the calculation are in good agreement with the reference examples.


I. INTRODUCTION
A NALYTICAL and semi-analytical methods in the calculation of self-and mutual-inductances of conducting elements of electrical circuits have played an important role in development of power transfer, wireless communication, sensing and actuation and have been applied in a broad fields of science, including electrical and electronic engineering, medicine, physics, nuclear magnetic resonance, mechatronics and robotics, to designate the most prominent.
Although, a number of efficient numerical methodes implemented in the commercially developed software are available, analytical and semi-analytical methods allow to obtaining the result of calculations in the form of a final formula with a finite number of input parameters, which when applicable may significantly reduce computation effort. Providing the direct access to a calculational formula for a user in such methods facilitates mathematical analysis of obtained results of calculation and opens an opportunity for their further optimization.
Kalantarov and Zeitlin showed that the calculation of mutual inductance between a circular primary filament and any other secondary filament having an arbitrary shape and any desired position with respect to the primary filament can be reduced to a line integral [6,page 49]. Adapting this result, Poletkin derived the analytical formulas for calculating the mutual inductance between two circular filaments having any desired position with respect to each other in work [12], as an alternative to Grover's and Babic's expressions reported in works [2] and [11], respectively.
Moreover, it was mentioned in work [12] that the obtained formula for the treatment of the singular case, when the circular filaments are mutually perpendicular, can be applied also to the calculation of the mutual inductance between a circle and line segment after taking a minor modification of the original formula. Using this fact and an alternative approach developed by Babic [11], two analytical formulas for calculation of mutual inductance between a circular filament and line segment arbitrarily positioning in the space are derived by means of MIM and BM, respectively.
Since, any curve can be interpolated with a desired accuracy by a finite number of line segments, a segmentation method for calculation of the mutual inductance between the primary circle and a filament having an arbitrary shape is proposed and successfully developed based on the two derived formulas for calculation of the mutual inductance between the circle and line segment. It is shown that for calculation of mutual inductance by means of the segmentation method, a set of points belonging to the arbitrarily shaped filament as the input data is only needed. The developed methodology is successfully applied to the calculation of mutual inductance between the circular filament and the following special curves such as circle, circular arc, elliptic arc, ellipse, spiral, helices and conical helices. All results of calculation are in a good agreement with the reference examples and successfully validated by the FastHenry software [46] .

II. MUTUAL INDUCTANCE BETWEEN CIRCULAR FILAMENT AND LINE SEGMENT
Let us consider the filament system consisting of a primary circular filament having a radius of R p and a secondary filament represented as a line segment having a length of L s , which is arbitrarily positioned in the space with respect to the primary circle as shown in Fig. 1. A coordinate frame (CF) denoted as XY Z is assigned to the primary circle in a such way that the Z axis is coincident with the circle axis and the XOY plane of the CF lies on circle's plane, where the origin O corresponds to the centre of primary circle. In turn, the xyz CF is assigned to the secondary filament so that its origin C is attached to one of its endpoints. The axes of the xyz CF are parallel to the axes of XY Z, respectively.
The linear position of the secondary filament with respect to the primary one is defined by the coordinates of the origin C (x C , y C , z C ). The angular position of the line segment is defined by the angle η and θ corresponding to the angular rotation around the z-axis of the plane passing through the z-axis and line segment, and then the rotation of the segment around an axis, which is perpendicular to this segment-z plane and passing through the origin C, respectively, as shown in Fig. 1.

B. Babic's Method (BM)
Let us consider again the calculation scheme shown in Fig.  1, but in a difference with MIM the line segment is given by coordinates of its two endpoints, namely, p 0 (x 0 , y 0 , z 0 ) and p 1 (x 1 , y 1 , z 1 ).
Then, the coordinates of an arbitrary point S (x s , y s , z s ) of the secondary line segment can be written parametrically as follows: x s = q¯ + x 0 , y s = r¯ + y 0 , z s = s¯ + z 0 ,¯ ∈ [0, 1], (7) where q = x 1 − x 0 , r = y 1 − y 0 , s = z 1 − z 0 . Hence, the differential element of the line segment becomes While, parametric coordinates of an arbitrary point P (x p , y p , z p ) of the primary circular filament can be defined by the following expressions: The differential of the primary circular filament is given by Accounting for equations (10) and (8), the mutual inductance between circular filament and the line segment can be calculated by where can be expressed via the complete elliptic functions of the first and second kind K(k) and E(k), respectively, as follows where Hence, replacing the integral (13) in (11) by (14), the final formula for calculation of mutual inductance becomes The input parameters for Babic's and MI method are related by the following relationships: Worth noting that according to (2) and (16) the mutual inductance is equal to zero, when the line segment crosses the Z axis or is located on a line crossing the Z axis.

III. SEGMENTATION METHOD
The secondary filament having an arbitrary shape and the primary circular filament of a radius R p are considered as shown in Fig. 2. The secondary filament is given, for instance, by a 3D parametric curve σ = σ( ), where is the parameter defined within a finite interval [ϕ 0 , ϕ 1 ], and the following inequality It is assumed that the curve is sampled by n points, so that we have p 0 (h 0 ), p 1 (h 1 ), p 2 (h 2 ), . . . , p n−1 (h n−1 ), p n (h n ). The points can be defined in the following way as (3 × 1)-column-matrices, where x σ , y σ and z σ are the coordinates of the σ-curve and and so h 0 = ϕ 0 . Using the set of points (18) and (19), the σ-curve can be interpolated by line segments as shown in Fig.  3. The length of each line segment can be calculated by while the angles and dimensionless coordinates of i-th line segment become and respectively. Substituting (20), (21) and (22) into (2), the mutual inductance M i between the circular filament and i-th line segment can be calculated. Hence, performing summation of all n terms of M i , the formula for calculation of the mutual inductance between the arbitrary shape and the primary circular filament can be written as follows Alternatively, the calculation can be performed by using Babic's formula (16), namely, Thus, interpolating the parametric σ-curve by n line segments, the calculation of the mutual inductance between the arbitrary shape and the primary circular filament can be performed by means of using developed formulas, namely, (2) and (16), which are applied n times to each interpolating line segment followed by summation executed by (23) and (24) for all obtained calculation results. The Matlab files with the implemented formulas (2), (16), (23) and (24) are available from the authors as supplementary materials to this article.

IV. EXAMPLES OF CALCULATION. NUMERICAL VERIFICATION
In this section the developed formulas (2) and (16), as well as the proposed segmentation method based on (23) and (24)  In this section, the mutual inductance between a regular polygon of n sides, which is arbitrarily positioned in the space with respect to the circle of radius R p is calculated by segmentation method based on (23) and (24). It is assumed that the polygon is inscribed in a circle of a radius R σ lying on the xy plane as shown in Fig. 4. The parameter is defined within the finite interval [0, 2π]. This interval determines the distribution of the polygon vertices on the xy plane. In particular, n-th vertex within such interval of the parameter is located on the x axis. The linear misalignment of the polygon is defined by coordinates of point C corresponding to the centre of the secondary circle, while the angular misalignment can be defined by Grover's angles denoted by θ and η [12].
Then, input points can be calculated by The primary circle has a radius of R p =16 cm. The triangular filament is inscribed in a circle having a radius of R σ =10 cm. The centre of the secondary circle has the following coordinates: x C = 0, y C =4.3301 cm, and   Example 9 : The primary circle has a radius of R p =40 cm. The triangular filament is inscribed in a circle having a radius of R σ =10 cm, which is perpendicular to the primary circle that Grover's angles of η =0°and θ =90.0°. The centre of the secondary circle has the following coordinates: x C = 0, y C =20 cm, and z C =10 cm as shown Fig. 7

C. Mutual inductance between two circular filaments
In this section, the method of segmentation is applied to the calculation of mutual inductance between two circular filaments arbitrarily orientated in the space with respect to each other. Similar to Sec.IV-B, the secondary circle is lying on the xy plane. The linear misalignment is defined by coordinates of point C corresponding to the centre of the secondary circle, and the angular misalignment is defined by Grover's angles denoted by θ and η [12]. The results of the calculation are compared with results obtained by means of analytical formulas developed by Babic in work [11] and Poletkin in [12].
The accuracy of interpolation of a circle by line segments can be measured by the relative error ε, which can be estimated as the deference between the radius of the circle and a length of a line joining a centre of the circle and mid point of a line segment divided by the radius: Form (27) follows the obvious conclusion that the larger the number of line segments the smaller the error. The required number of line segments for the given relative error can be also evaluated by (27) and presented, for instance, in the logarithmic scale as shown in Fig. 8. As seen from Fig. 8, to interpolate a circle with the relative error, for instance, less than 10 −3 , the number of line segments must be more than 100.
For generating input points, equation (25) can be used with the only difference that the number of line segments is being chosen in according to the desired accuracy of interpolation of a circle, which in turn effects on the accuracy of calculation of mutual inductance.
Indeed, the accuracy of calculation of mutual inductance can be measured also by the relative error, which is defined as follows where M is the mutual inductance calculated by Babic's or Poletkin's formula as mentioned above. Clearly that it is dependent on the number of line segments. To exam this point let us consider the following geometrical arrangement taken from Babic's work [11, page 3597, Example 12], as an illustrative example, in which two circles with radii R p =16.0 cm and R σ =10.0 cm and the centre of the secondary σ-circle is located at x C = 0, y C =4.3301 cm, z C =17.5 cm and the angle, θ of 60.0°, but the η angle is chosen to be 45.0°as shown in Fig. 10. Performing calculation for different numbers Fig. 10. Arrangement of circles in the illustrative example.  Using the obtained results shown in the above table, the relative error ε σ as a function of the number n of line segments can be defined, namely, ε σ = ε σ (n). Moreover, both functions ε σ = ε σ (n) and n = n(ε) can be joined into one function exhibiting the dependency of the relative error of calculation mutual inductance on the relative error of interpolation of a circle as shown in Fig. 9. Although, Fig. 9 shows the evaluation of the error for the particular arrangement. However, the result is surprisely consistence and can be used for the preliminary estimation of the error of calculation in other arrangements as shown below. For further calculation, the number of line segments is set to 200 that, according to Fig. 9, the error, ε σ , is estimated to be around 0.00015.
Example 12: The primary and secondary circles have radii of R p =5 mm and R s =1 mm, respectively. The centre of the secondary circle is located at x C = 3 mm, y C =1 mm, z C =0.5 mm and its angular misalignment are defined by θ = 57.6885°and η = 108.4349°as shown in Fig. 11. The result of calculation is Example 13: The primary and secondary circles have radii R p =40 cm and R s =5 cm, respectively. The centre of the secondary circle is located at x C = 10 cm, y C =15 cm, z C =0.0 cm. The angular misalignment of the secondary filament is defined as follows θ = 74.4986°and η = 123.6901°.   R p =40 cm and R s =10 cm, which are mutually perpendicular to each other that angles of η =0 and θ =90.0°. The centre of the secondary circle has the following coordinates: x C = 0, y C =20 cm, and z C =10 cm as shown in Fig. 12 The results show the small error, which has the chaotic nature. Indeed, changing the angle η in a range of 0 < η ≤ 360 o the chaotic distribution of the error is observed as shown in Fig. 13. Analysis of the figure depicts that the error does not exceed of 4 × 10 −15 nH. Worth noting that varying the number of line segments, n, alters the error distribution, but keeps its value below the observed maximum.
Example 16 (Example 22, Poletkin's article [12]) : Now we again consider mutually perpendicular circles having the same radii as in Example 14, but in this case the centre of the secondary circle occupies a position on the XOY -surface M, nH η , degree Then, again let us rotate the η angle in a range 0 < η ≤ 360 o , the calculation of mutual inductance demonstrates the chaotic distribution of the small error of calculation, which does not exceeded of 15 × 10 −15 nH as shown in Fig. 14. The error is continually distributed without interruption of its continuity at η =90°and 270°in comparing with the application of Babic's and Grover's formula (please, see Example 22, in article [12]). Thus, the segmentation method for the calculation of mutual inductance between two circular filaments arbitrarily orientated in the space with respect to each other does not suffer from the singularity case, when two circles are mutually perpendicular.

D. Mutual inductance between circle and a filament of arbitrary shape
Using the proposed methodology based on (23) and (24), the segmentation method is applied to the calculation of the mutual inductance between the primary circle and filaments arbitrary positioning in the space and having different shapes, which can be defined, for instance, by parametric equations of special curves to generate the input set of points (18). In particular, the following curves such as circular and elliptic arcs, ellipse, spiral, helices and conical helices are considered in the proceeding section.
Beginning with the calculation of mutual inductance between the circle and elliptic arc arbitrary positioning in the space, the secondary filament can be defined as a part of ellipse lying on the xy plane. The ellipse is characterized by its semimajor axis A σ directed along the x-axis and semi-minor B σ along the y-axis. The parameter is defined within the finite interval [ϕ 0 , ϕ 1 ] as shown in Fig. 15. The linear misalignment of the arc is defined by coordinates of point C corresponding to the centre of the ellipse, while the angular misalignment can be determined by Euler's angles denoted by α, β and γ. The following relationship between Grover's and Euler's angles is true [12]: sin β = sin η sin θ; cos β sin α = cos η sin θ.
According to Fig. 15, the input points can be calculated as follows   It is clear that an elliptic arc can be transformed into the other curves such as a circular one (A σ = B σ ,    For the case of the calculation of mutual inductance between the circle and a spiral, the secondary filament is lying on the xy plane and its geometry can be characterized by the inner radius R σ , the number of turns t and the distance between the turns d as shown on Fig. 18. The boundaries of the finite interval of the parameter are estimated as follows Similar to the elliptic arc, the linear misalignment of the spiral is defined by coordinates of point C, and the angular misalignment is defined by Euler's angles as shown in Fig.  18. The input points are generated by the following equation: where cosine matrices are determined by Eq. (31).

Example 20 (spiral):
The primary circle has a radius of R p =0.5 m. The spiral is defined by the following parameters: the inner radius R σ is 0.05 m, the distance d is 0.04 m and the number of turns t is 9. The centre C is located at x C = 0.6 m, y C =0.1 m, and z C =0.7 m. The angular misalignment is defined as follows α = 45.0°, β = −45.0°, but γ is changed in a range from 0 to 350.0°. The number of line segments n is chosen to be 900 (It is recommended to use at least 100 line segments per turn to have a reasonable accuracy). The results  Further, the calculation of mutual inductance between the circle and a helices is considered. The geometry of helices can be defined on the xyz CF by its radius R σ , the number of turns t and the pitch p as shown in Fig. 19. The boundaries of the finite interval of the parameter are Similar to previous cases, the linear misalignment of the helices is defined by coordinates of origin C of the xyz CF, and the angular misalignment is defined by Euler's angles as shown in Fig. 19. The following equation can be used to generate the input points: Example 21 (helices): The primary circle has radius of R p =0.9 m. The helices is defined by the following parameters: the radius R σ is 0.6 m, the pitch p is 0.05 m and the number of turns t is 4. The origin C is located at x C = 0.3 m, y C =0.2 m, z C =0.5 m. The angular misalignment is characterized by α = 54.7356°, β = 0.0°and γ is taken from a range from 0 to 335.0°. The number of line segments n is  Finally, the calculation of mutual inductance between the circle and a conical helices is carried out. The geometry of helices can be defined on the xyz CF by its radius R σ , the number of turns t, the pitch p and the distance between turns d as shown in Fig. 20. The boundaries of the finite interval of the parameter are The linear misalignment of the conical helices is defined by coordinates of origin C of the xyz CF, and the angular misalignment is defined by Euler's angles as shown in Fig.  20. The following equation generates the input points: Example 22 (conical helices): The primary circle has radius of R p =0.9 m. The conical helices is defined by the following parameters: the radius R σ is 0.0 m, the pitch p is 0.1 m, the number of turns t is 4 and the distance d is 0.04 m. The origin C is located at x C = 0.3 m, y C =0.2 m, z C =0.5 m. The angular misalignment is characterized by α = 135°, β = 0.0°a nd γ is taken from 0 to 335.0°as shown in Fig. 21. The number of line segments n is chosen to be 400. The results of calculation are

V. CONCLUSION
In this article, two novel formulas for calculation of mutual inductance between a circular filament and line segment arbitrarily positioning in the space were derived by means of mutual inductance method and Babic's method, respectively. These two formulas were expressed via the integral relationships, whose kernels include the elliptic functions of the first and second kinds. The two formulas are successfully validated to each other mutually and numerically via the FastHenry software.
Using the fact that any curve can be interpolated with a desired accuracy by a finite number of line segments, a segmentation method for calculation of the mutual inductance between the primary circle and a filament having an arbitrary shape is proposed and successfully developed based on the two derived formulas. It is shown that for calculation of mutual inductance between the primary circle and the secondary filament of an arbitrarily shape by means of segmentation method based on (23) and (24), the set of points belonging to the secondary filament as the input data is only needed.
The proposed segmentation method is successfully applied to the calculation of the mutual inductance between the primary circle and filaments arbitrary positioning in the space and having different shapes such as circular and elliptic arcs, ellipse, spiral, helices and conical helices. All results of calculation are in a good agreement with the reference examples and successfully validated by the FastHenry software.
Considering a limit case, when the radius of the primary circle is trending to zero, the developed methodology can be used for calculation of the magnetic flux density and its gradient generated by a current-carrying arbitrarily shaped filament. As a result, the force and stiffness between two arbitrarily shaped current-carrying filaments can be calculated by using developed methodology. This matter can be considered as our future work.