1. Introduction
The paper discusses the challenge of computing the mutual inductance of conventional coils, which has been a topic of research since the time of Maxwell [
1]. While analytical solutions in the form of elementary functions exist for linear coils, more complex configurations, such as circular and elliptical coils require solutions in terms of elliptic integrals, Bessel and Struve functions, and hypergeometric convergent series [
2,
3,
4,
5,
6,
7,
8]. Numerical methods and commercial software packages are also available, but there is interest in developing analytical and semi-analytical methods for more efficient computations. Reviewing the corresponding literature in physics and electromagnetics, as well as in scientific papers in engineering, one cannot find the calculations of self-inductance and mutual-inductance of the coils of the conical form very often. Recently, in [
9], the calculation of the self-inductance of a thin sheet inductor is obtained in the semi-analytical form. Using the same reasoning, in this paper, a semi-analytical formula for calculating the mutual inductance between a thin conical sheet and a filamentary circular loop is given. Coils are coaxial and placed in the air. The thin conical sheet inductor is made of a thin wire with a cross-section that is practically negligible. This assumption also applies to the circular loop.
The new presented method is based on complete elliptical integrals of the first, second, and third kind, along with a term to be solved by numerical integration. As the special case of this new developed formula is the formula for calculating the mutual inductance between the cylindrical solenoid and the filamentary circular loop, the practical applications of the previous mentioned configuration could be interesting in many domains of electromagnetics such as the excitation coil used in electromagnetic-levitation melting, the production of magnetic field homogeneity and the high magnetic field under the cone, which can be used to magnetize a magnetic sheet [
10,
11,
12,
13]. The calculation of the previously mentioned coils is useful for conical inductors which are of ideal form for ultra-broadband applications up to 40 GHz since the conical shapes limit the effects of stray capacitances and effectively substitute a series of narrow-band inductors, creating the high impedance over a very wide bandwidth and in wireless power transfer systems that utilize conical inductors [
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24]. In all cases, either the regular or the singular are explained with precise explications. The validation of the presented method is performed using the single and double integration as well as the semi-analytical formula. The Mathematica files with implemented formulas are available upon request.
2. Basic Formulas
Let us consider a thin conical sheet and a circular loop as shown in
Figure 1. The thin conical sheet has the radii of basis
. and the axial positions
, with the number of sheets turns
N. The circular loop has the radius
R and axial position
.
Let us begin the complete analysis for the first case
,
Figure 1.
The mutual inductance between the thin conical sheet inductor and the circular loop can be calculated by the following.
with
Equation (3) is obtained from the Neumann formula for the mutual inductance, derived from the magnetic vector potential [
7] which is applicable for two coils. By inspection for
, Equation (3) becomes [
8].
Even though
one must use
. By the simple verification for
the thin conical sheet degenerates to the thin cylindrical solenoid. For this case, the Formula (3) is the formula for calculating the mutual inductance between the thin cylindrical solenoid and the circular loop [
8].
Introducing the substitution,
in (3) one has the following.
where
Let us calculate the first integral with respect to
z in (5).
with
Thus, the solution of this integral is obtained in the close form.
Now, the Formula (5) can be given as follows.
where
Let us solve the following integrals in (9) and (10).
Using [
25] one obtains the following.
or
This solution is obtained in the close form where
are the complete elliptic integrals of the first and second kind [
25,
26].
Similarly, the solution of the next integral,
is also obtained in the close form as follows,
with
The next integral is
or
where
This integral does not have the analytical solution, so it must be solved numerically. The kernel function of this integral is the continuous function of the interval of integration.
Let us solve this integral by partial integration (
Appendix A).
Similarly,
can be obtained as follows (
Appendix A),
where
Finally, from (9), (14), (16), (19) and (21) the mutual inductance between the thin conical sheet and the circular loop can be calculated in the semi-analytical form as follows,
where
All parameters in (23) can be found in
Appendix A.
Thus, the general solution of Equation (22) with (23) is expressed by the complete elliptic integrals K(k), E(k) and Π(h, k) as well as one term J0 which must be solved numerically.
2.1. Singular Cases
The Equation (22) with (23) can be directly applied for the general cases. It is possible to have in (23) four singular cases so that one must do some corrections to overcome these problems. The singular cases appear when coils are in contact or overlap. In these cases, some parameters obtain the values for which the elliptical integrals begin infinite, so they must be considered particularly.
2.1.1.
From (23), one can see that for
the singular case appears and
so that so that the following is true.
In (23), the singularity appears for
and
so that so that the following is true.
From (24) and (25), it is obvious that the complete elliptic integrals of the third kind and will vanish.
Thus, (23) begins as the following.
2.1.2.
From (23), one can see that for
the singular case appears and
so that so that the following is true.
In (23) the singularity appears for
and
so that so that the following is true.
From (27) and (28), it is obvious that the complete elliptic integrals of the third kind and will vanish.
Thus, for this case we use (26).
2.1.3.
From (23) one can see that case that for
the singular case appears and
so that so that the following is true.
In (23) the singularity appears for
and
so that so that the following is true.
From (34) and (35), one can see that the complete elliptic integrals of the third kind and will vanish.
Thus, (23) begins as the following.
2.1.4.
From (23) one can see that case that for
the singular case appears and
so that so that the following is true.
In (23) the singularity appears for
and
so that so that the following is true.
From (32) and (33), one can see that the complete elliptic integrals of the third kind and will vanish. Thus, for this case we use (31).
From this detailed analysis, all partial singular cases can be found in the previous discussed cases.
2.2. Case
Now, let us consider a thin conical sheet and a circular loop as shown in
Figure 2. The thin conical sheet has the radii of basis
and the axial positions
, with the number of sheets turns
N. The circular loop has the radius
R and radial position
.
In this case of , one can use the same reasoning as in the previous case.
The mutual inductance can be calculated as follows.
with
Introducing the substitution,
in (36) one has the following.
with
The mutual inductance can be calculated by the double integration (38). Following the procedures as in
Section 2.1, after the first integration over the variable
z, one has the following.
where
Finally, using the same procedures as in
Section 2.1 after the second integration the mutual inductance in the semi-analytical form is given as follows.
with
All parameters in (42) can be found in
Appendix B.
Thus, the general solution of Equation (41) with (42) is expressed by the complete elliptic integrals K(k), E(k) and Π(h, k) as well as one term J00 which must be solved numerically.
2.3. Singular Cases
It is possible to have in (42) four singular cases. One must do some corrections to overcome these problems.
2.3.1.
From (42), one can see that for
the singular case appears and
so that so that the following is true.
For
in (42),
so that so that the following is true.
From (43) and (44), it is obvious that the complete elliptic integrals of the third kind and will vanish.
Thus, (42) begins as follows.
2.3.2.
From (42) one can see that case that for
the singular case appears and
so that so that the following is true.
In (42) the singularity appears for
and
so that so that the following is true.
Thus, for this case we use (45).
2.3.3.
From (42) one can see that for
the singular case appears and
so that so that the following is true.
In (47) the singularity appears for
and
so that so that the following is true.
From (48) and (49), it is obvious that the complete elliptic integrals of the third kind and will vanish.
Thus, (42) begins as follows.
2.3.4.
From (42) one can see that case that for
the singular case appears and
so that so that the following is true.
In (42) the singularity appears for
and
so that so that the following is true.
From (51) and (52), it is obvious that the complete elliptic integrals of the third kind and will vanish.
Thus, for this case we use (50).
Finally, all partial singular cases can be found in the previously discussed cases.
2.4.
This is the special case when the thin conical sheet degenerates to the thin cylindrical solenoid (
). The mutual inductance between the thin cylindrical solenoid and the circular loop is given in [
8].
where
is the Heuman Lambda function [
26],
R1 is the radius of the thin cylindrical solenoid,
z1 and z2 are the axial positions of the thin cylindrical solenoid,
R is the radius of the circular loop,
zQ is the axial position of the circular loop,
N is the number of turns of the thin cylindrical solenoid.
3. Numerical Validation
To verify the validity of the new presented formula for calculating, the following set of examples is presented. In each calculation one begins with the exact basic formula in the form of the double integral. The verification of this integral is verified with the expressions after the first and second integration. Thus, these analytical and semi-analytical expressions validate the presented new formulas. The special case is when the given configuration, thin conical sheet coil and the circular loop degenerates to the combination, thin cylindrical solenoid, and the circular loop.
Example 1. Calculate the mutual inductance between the thin conical sheet inductor and the circular loop for which R1 = 10 m, r1 = 2 m, z1 =−1 m, z2 = 2 m, N = 1000, R = 5 m, and zQ = 0 m.
This is the case
(
Figure 1). Let us begin with the basic Formula (5) where the mutual inductance is given by the double integration.
The mutual inductance is the following.
Using Formula (9) with (10), the mutual inductance is obtained by the single integration.
Let us use the semi-analytical Formula (22) with (23).
All results are in an excellent agreement.
Example 2. Calculate the mutual inductance between the thin conical sheet inductor and the circular loop for which R1 = 2 m, r1 = 10 m, z1 = −1 m, z2 = 2 m, N = 1000, R = 5 m, and zQ = 0 m.
Using the basic Formula (38) for the double integration, the mutual inductance is the following.
Using Formula (39) with (40), the mutual inductance is obtained by the single integration.
Finally, let us use the semi-analytical Formula (41) with (42) the mutual inductance as follows.
All results are in an excellent agreement.
Example 3. Calculate the mutual inductance between the thin conical sheet inductor and the circular loop for which R1 = 3 m, r1 = 2 m, z1 = 0 m, z2 = 0.2 m, N = 1000, R = 1 m, and zQ = 0.2 m.
This is the case
(
Figure 1). The basic formula for the mutual inductance is given by the double integration (5).
The mutual inductance is the following.
Using Formula (9) with (10) the mutual inductance is obtained by the single integration.
Finally, let us use the semi-analytical Formula (22) with (23) for the mutual inductance which gives the following.
All results are in an excellent agreement. The figures that agree are in bold.
Example 4. Calculate the mutual inductance between the thin conical sheet inductor and the circular loop for which R1 = 2 m, r1 = 3 m, z1 = 0 m, z2 = 0.2 m, N = 1000, R = 1 m, and zQ = 0.2 m.
This is the case
The basic formula for the mutual inductance is given by the double integration (38) which gives the following.
Using Formula (39) with (40), the self-inductance is obtained by the single integration.
Let us use the semi-analytical Formula (41) with (42).
All results are in an excellent agreement. The figures that agree are in bold.
Example 5. In this example one calculates the mutual inductance between the thin cylindrical solenoid and the circular loop for which R1 = r1 = 2 m, z1 = 0 m, z2 = 0.2 m, N = 1000, R = 1 m, and zQ = 0.2 m.
In this example, the thin conical sheet inductor degenerates to the thin cylindrical solenoid ().
The exact formula for calculating the mutual inductance between the thin cylindrical solenoid and the circular loop is given by (53) with (54) as follows [
8].
Using the formulas for the double and the single integration either for the case
or
the mutual inductance is, respectively, the following.
Equations (5) and (38) for the double integration and (9) with (10) as well as (39) with (40) for the single integration are not singular for . It is not case for the semi-analytical solutions (22) with (23) or (41) with (42) when they are singular or indeterminate.
However, one can take, for example,
R1 = 2.0000001 m and
r1 = 2.0000001 m Equations (23) and (42), respectively, so that they give the following.
All figures that agree with the exact results are bolded. Even though the results are in the particularly good agreement it is recommended to use the formula (53) with (54) for calculating the mutual inductance between the thin cylindrical solenoid and the circular loop. This formula can be carefully obtained from (23) and (42) in the limit case when and vice-versa.
Example 6. Calculate the mutual inductance between the thin conical sheet inductor and the circular loop for which R1 = 3 m, r1 = 2 m, z1 = 0 m, z2 = 0.2 m, N = 1000, R = 1 m, and zQ = 0.4 m.
In this example one finds that
so that it is the singular case
Section 2.1.
Applying (26), the mutual inductance is the following.
Using the basic Formula (5) for the double integration the mutual inductance is the following.
Using the Formula (9) with (10), the mutual inductance is obtained by the single integration.
All figures, that agree, are in bold. All results are in an excellent agreement.
Example 7. Calculate the mutual inductance between the thin conical sheet and the circular loop for which R1 = 3 m, r1 = 2 m, z1 = 0 m, z2 = 0.2 m, N = 1000, R = 1 m, and zQ = 0.8 m.
In this example one finds that
so that it is the singular case
Section 2.2.
Applying (26), the mutual inductance is the following.
Using the basic Formula (5) for the double integration the mutual inductance is the following.
Using the Formula (9) with (10) the mutual inductance is obtained by the single integration.
All results are in an excellent agreement. The figures that agree are in bold.
All presented results have been obtained by the Mathematica programing. One can see that they are in an excellent agreement so that the potential users can use the presented formulas by their preference.