3.1. The Self-Inductance Caused by the Radial Current Density
Introducing the substitution
, in Equation (2) and making the first four integrations in order to the variables
, or
y,
x,
v and
z, (
Appendix A) we obtained the self-inductance
in the following form:
where
where
Expressions for are given as in the Addendum where we calculate the self-inductance for the coil of the rectangular cross-section with radial current in Example 2 in the Addendum.
Thus, the new formula for the self-inductance of the circular coil with the rectangular cross-section and the radial current density can be obtained by Equation (5) using the simple integration of the previous elementary functions. In this paper, we use the Gaussian numerical integration in MATLAB programming and the numerical integration by default in Mathematica programing.
The special case of Equation (5) is the self-inductance of the thin-disk coil with a radial current [
23]. This self-inductance can be obtained from Equation (5) by finding the limit when
, or doing three integrations such as in [
23].
The self-inductance
is obtained in the analytical form as follows:
where
3.2. The Self-Inductance Caused by the Azimuthal Current Density
Introducing the substitution
, in Equation (2) and making the first four integrations in order to the variables
, or
y,
x,
v and
z, (
Appendix B) we obtained the self-inductance
in the following form
where
Expressions for are given as in the Addendum where we calculate the self-inductance for the coil of the rectangular cross-section with azimuthal current in Example 4 in the Addendum.
Thus, the self-inductance of the circular coil of the rectangular cross-section with the azimuthal current density can be obtained by Equation (7) using simple integration of the previous elementary functions.
The special case of this calculation is the self-inductance of the thin-disk coil (pancake) with the azimuthal current [
26,
27]. This self-inductance can be obtained from Equation (7) finding the limit when
or doing the three integration such as in [
26].
The self-inductance
is obtained in the analytical form as follows:
where
and
… is the Catalan’s constant [
26,
27].
The self-inductance is obtained as the combinations of the elementary functions, the elliptical integral of the second kind [
28,
29,
30], and the single integrals (the semi-analytical solution).
In [
27], the new expression for
is given by:
This expression is also very friendly for the numerical integration in Equation (9).
For the full disk (
), the self-inductance is
This formula can be found in [
10,
27,
31].
There is also one special case when
(thin-wall solenoid of radius
). Finding the limit in Equation (7) or solving the three integrals in [
32], we obtain the well-known Lorentz’s formula (1879),
where
and
,
are the elliptic integrals of the first and second kind [
32].
From previous formulas for it is obvious that they have similar terms and all expressions are elementary functions that are very friendly for single numerical integration. The special cases are obtained as the analytical and semi-analytical expressions for these important electromagnetic quantities (6)–(12).
3.3. Asymptotic Behaviors of Disk Coils and Thin-Wall Solenoids
At first, we analyze the disk coil.
For
we have a well-known singular case which gives
For
(inner radius tends toward the outer radius), this case leads to the well-known formula [
31]
where
R is the turn radius,
a is the radius of the circular wire from which the turn is constructed. If the current flows only on the wire surface (due to the skin effect)
Y = 0, and the current flow is homogeneous in the wire, then
Y = 0.25.
For
(the case of a logarithmic singularity), Conway [
31] gives the analogous formula
From Kirchhoff’s formula for the self-inductance of a circular ring of the radius
R and the circular section of the radius
a with one turn [
13], we have
Luo, Y. and Chan, B. [
13] obtained, for this asymptotic case
The asymptotic case for the thin-wall solenoid can be calculated from [
13]
where
is the wall solenoid’s radius and
is its hight.
From this approach, the self-inductance of a thin-wall solenoid in the asymptotic case is obtained for the first time in the literature.
Let us put in Equation (12)
so that the self-inductance of the thin-wall solenoid is
To find the self-inductance of thin-wall solenoid for
the asymptotic behavior of
and
near the singularity at
are given by the following expression [
33]:
The approximations (20) and (21) are the first terms of the convergent series [
28,
29,
30,
33]. We calculate the normalized self-inductance of the extremely short-wall solenoid (
) as
From Equations (19)–(22) we finally have:
For
extremely near at zero, we find using the l’Hospital’s Rule from Equation (23) that the first term tends to
, the second to
and the third to
. Finally, the self-inductance from this range of the parameter
is
This formula has been obtained by the ansatz in [
13].
To our knowledge, the formula (23) appears for the first time in the literature.
Thus, we cover all possible cases with the new formulas and the already well-known or the improved formulas in the calculation of the self-inductance of the previously mentioned circular coils.