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Inductance and force calculations of circular coils with parallel axes shielded by a cuboid of high permeability

01 May 2018-Iet Electric Power Applications (The Institution of Engineering and Technology)-Vol. 12, Iss: 5, pp 717-727

Abstract: A detailed analysis is presented for a boundary value problem of circular coils with parallel axes shielded by a cuboid of high permeability. Field solutions are given by establishing a suitable ansatz of the magnetic scalar potential, which can satisfy the boundary conditions on six surfaces of the cuboid without difficulty. Analytic expressions are also given for the self and mutual inductance of shielded circular coils with rectangular cross section. By differentiating the self- and mutual magnetic energy with respect to the centre coordinates of the shielded coils, the total forces exerted on them are further obtained, which consist of self and mutual force components. Finally, the numerical results of the proposed method are compared with those of the finite-element method simulations, and the proposed method proves to be accurate and efficient enough for practical applications.
Topics: Cuboid (61%), Boundary value problem (52%), Inductance (52%), Magnetic energy (51%), Scalar potential (50%)

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IET Electric Power Applications
Research Article
Inductance and force calculations of circular
coils with parallel axes shielded by a cuboid
of high permeability
ISSN 1751-8660
Received on 6th October 2017
Revised 29th January 2018
Accepted on 1st February 2018
E-First on 4th April 2018
doi: 10.1049/iet-epa.2017.0646
www.ietdl.org
Yao Luo
1
, Yuanzhe Zhu
1
, Yue Yu
2
, Lei Zhang
3,4
1
School of Electrical Engineering, Wuhan University, Wuhan, People's Republic of China
2
Department of Electrical Engineering, Tsinghua University, Beijing, People's Republic of China
3
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, People's Republic of China
4
Department of Biological and Agricultural Engineering, Texas A&M University, College Station, Texas, USA
E-mail: sturmjungling@gmail.com
Abstract: A detailed analysis is presented for a boundary value problem of circular coils with parallel axes shielded by a cuboid
of high permeability. Field solutions are given by establishing a suitable ansatz of the magnetic scalar potential, which can
satisfy the boundary conditions on six surfaces of the cuboid without difficulty. Analytic expressions are also given for the self
and mutual inductance of shielded circular coils with rectangular cross section. By differentiating the self- and mutual magnetic
energy with respect to the centre coordinates of the shielded coils, the total forces exerted on them are further obtained, which
consist of self and mutual force components. Finally, the numerical results of the proposed method are compared with those of
the finite-element method simulations, and the proposed method proves to be accurate and efficient enough for practical
applications.
1Introduction
In many cases of electric power applications, a static or slowly
varying magnetic field should be confined to a certain region by
magnetic covers [1–3]. Under such circumstances, the field and
inductance of shielded coils will be changed owing to the existence
of shielding material, which cannot be taken into account by the
Neumann formula for inductance of coils in the free air [4–11].
Specific methods should be applied to this type of boundary value
problem (BVP) to take the reflection effect of boundary into
consideration. Normally, to evaluate the effects of shielding forms
such as a pair of parallel plates, circular hollow cylinder or hollow
sphere, we can refer to the relevant monographs [1–3]. However,
for a shielding form of a cuboid, no results have been found as yet
apart from some formulae published for the special case of this
problem [12]. The absence of this shielding form in the literature is
not due to its insignificance. In fact, the cuboidal shielding is
commonly used, and it is more convenient to manufacture and
install in practice. Moreover, it is not enough to simply employ the
finite-element method (FEM) to calculate the effects of cuboidal
shielding. For instance, using FEM to analyse an ‘inverse problem’
may lead to a rather cumbersome task [13] (the self and mutual
inductance of a coupled coil system must satisfy some equations,
so a direct solution with FEM will be infeasible). However,
analytical solutions, by contrast, are competent at this complicated
situation without the limitations of FEM.
In this work, analytical solutions will be given to the BVP
mentioned in the title as a generalisation of the results for coaxial
coils shielded by cuboid of high permeability [12]. Contrary to the
intuition, this seemingly simple problem cannot be tackled easily
with the routine method of magnetic vector potential (MVP) due to
the presence of all components of the magnetic field. In addition,
the solving process will become very tedious by starting with the
Bessel-integral formulas of MVP of the circular rings [14, 15],
since a complicated coefficient matrix of infinite dimensions will
arise from the boundary conditions on the surfaces of the cuboid.
Fortunately, these difficulties can be surmounted by using the
magnetic scalar potential (MSP). In view of the physical meaning,
a magnetic double layer with constant moment density will be
created by a ring carrying the current I [16], as a consequence, the
MSP will experience a jump quantity I while passing through the
area enclosed by the ring, and retain the continuity in the remaining
area. Accordingly, the coefficients of the ansatz can be determined
with the jump characteristics and orthogonality of the
eigenfunctions [17]. With the MSP, the field, inductance, and
forces between the shielded coils are easy to find. The validity of
the results will be confirmed by the numerical calculations
compared with those given by FEM simulations.
The configuration of this BVP is shown in Fig. 1 with the
notations of geometric parameters. Two circular coils with parallel
axes and centres (x
1
, y
1
, z
1
), (x
2
, y
2
, z
2
), are shielded by a cuboid
with length, width, and height of 2a, 2b, 2l, respectively. The axes
of coils are parallel with the side 2l of the cuboid. The analysis will
begin with the field of a single shielded circular ring, and the
results are developed further to the circular coils with rectangular
cross section.
2Ansatz for MSP of a circular ring shielded by a
cuboid of high permeability
We take central point of the cuboid as the origin of the Cartesian
coordinate system (x, y, z) and let x, y, z-axes parallel to the sides
of 2a, 2b and 2l, respectively. The MSP V(x, y, z) must satisfy the
Laplace equation ΔV = 0 and the magnetic field strength H will be
given by negative gradient of V. Supposing that a circular ring
(Ring 1) with radius r
1
and centre (x
1
, y
1
) is located in the plane z
= ζ
1
and carrying the current I
1
, then by separation of variables an
ansatz can be established as
V x, y, z = I
1
m, n = 1
A
mn
f
1
λ
m
, x
× f
2
λ
n
, y f
3
+
λ
mn
, z, ζ
1
for z > ζ
1
(1a)
and
V x, y, z = I
1
m, n = 1
A
mn
f
1
λ
m
, x
× f
2
λ
n
, y f
3
λ
mn
, z, ζ
1
for z < ζ
1
,
(1b)
IET Electr. Power Appl., 2018, Vol. 12 Iss. 5, pp. 717-727
© The Institution of Engineering and Technology 2018
717

where A
mn
are undetermined coefficients,
m
,
n
and
mn
are
eigenvalues such that
λ
mn
= λ
m
2
+ λ
n
2
.
(2)
In (1a) and (1 b), the axial functions f
3
±
λ
mn
, z, ζ
1
are prepared for
later consideration of the jump feature. Furthermore, the
eigenfunctions f
1
(
m
, x) and f
2
(
n
, y) should be the linear
combination of trigonometric functions, namely
f
1
λ
m
, x = A
1
cos λ
m
x + B
1
sin λ
m
x ,
f
2
λ
n
, y = A
2
cos λ
n
y + B
2
sin λ
n
y .
(3)
The permeability of the cuboid can be regarded as infinity to
simplify the analysis. It follows that the tangential components of
H must vanish on the surfaces, namely
H
y
±a, y, z = 0,
H
x
x, ± b, z = 0,
(4a)
and
H
z
±a, y, z = 0,
H
z
x, ± b, z = 0.
(4b)
Since H = V, it follows that
V ±a, y, z = 0,
V
x, ± b, z = 0.
(4c)
Substituting (3) into (4c) yields the following linear systems for A
1
,
B
1
and A
2
, B
2
, respectively
cos λ
m
a A
1
+ sin λ
m
a B
1
= 0,
cos
λ
m
a A
1
sin λ
m
a B
1
= 0,
(5a)
and
cos λ
n
b A
2
+ sin λ
n
b B
2
= 0,
cos
λ
n
b A
2
sin λ
n
b B
2
= 0.
(5b)
Hence, the existence of non-trivial solutions of (5a) demands
cos λ
m
a sin λ
m
a
cos λ
m
a sin λ
m
a
= 0.
(6)
Thus the eigenvalues are given immediately
λ
m
=
2a
, m = 1, 2, 3,
Consequently, the corresponding eigenfunction f
1
can be found
according to the Sturm–Liouville theory [18]
f
1
λ
m
, x = sin λ
m
a cos λ
m
x
cos λ
m
a sin λ
m
x = sin λ
m
a x .
(7a)
From (5b), we have the following result in a similar manner:
λ
n
=
2b
, n = 1, 2, 3,
and
f
2
λ
n
, y = sin λ
n
b cos λ
n
y
cos λ
n
b sin λ
n
y = sin λ
n
b y .
(7b)
The axial functions f
3
±
λ
mn
, z, ζ
1
of the ansatz (1a) and (1b) should
be treated in a slightly different manner. Considering that the total
field should be separated into the primary and secondary parts, and
the source can be regarded as a magnetic double layer, f
3
±
λ
mn
, z, ζ
1
should be expressed as
f
3
+
λ
mn
, z, ζ
1
= A
3
e
λ
mn
z
+ B
3
e
λ
mn
z
+ e
λ
mn
z ζ
1
for z > ζ
1
,
(8a)
f
3
λ
mn
, z, ζ
1
= A
3
e
λ
mn
z
+ B
3
e
λ
mn
z
e
λ
mn
ζ
1
z
for z < ζ
1
.
(8b)
Moreover, for z = ± l the field components
H
x
= V /x, H
y
= V /y must vanish, that is
A
3
e
mn
+ B
3
e
mn
+ e
l ζ
1
λ
mn
= 0,
A
3
e
mn
+ B
3
e
mn
e
l + ζ
1
λ
mn
= 0.
(9)
[For simplicity, the square brackets [] of cosh[
mn
(l + ζ
1
)],
sinh[
mn
(l + z)], sin[
m
(a − x)], cos[
n
(b − y)], etc., will be omitted
henceforth. Accordingly, the factor inside the round bracket always
belongs to the argument of the preceding function. Unless
otherwise stated, this rule will apply to all of the following
formulae in this work.] Solving (9) gives
A
3
=
cosh λ
mn
l + ζ
1
e
mn
sinh
2
mn
,
B
3
=
cosh λ
mn
l ζ
1
e
mn
sinh
2
mn
.
(10)
We, therefore, have the result of axial functions
f
3
+
λ
mn
, z, ζ
1
=
2sinh λ
mn
l z cosh λ
mn
l + ζ
1
sinh 2
mn
for z > ζ
1
,
(11a)
Fig. 1 Side and plan views of the circular coils with parallel axes shielded
by a cuboid of high permeability
718 IET Electr. Power Appl., 2018, Vol. 12 Iss. 5, pp. 717-727
© The Institution of Engineering and Technology 2018

f
3
λ
mn
, z, ζ
1
=
2sinh λ
mn
l + z cosh λ
mn
l ζ
1
sinh 2
mn
for z < ζ
1
.
(11b)
Let F
R
denotes the area encircled by the circular ring, it follows
that V(x, y, z) will obtain a spring quantity I
1
when passing through
F
R
and maintain the continuity on the remaining area F of plane z
= ζ
1
(see Fig. 2). This statement can be written in the following
form:
V x, y, ζ
1
+ V x, y, ζ
1
=
I
1
on F
R
,
0 on F,
(12)
where V(x, y, ζ
1
+) and V(x, y, ζ
1
−) are obtained by letting z = ζ
1
in
(1a) and (1b), respectively. With (1a) and (1 b) and
f
3
+
λ
mn
, ζ
1
, ζ
1
f
3
λ
mn
, ζ
1
, ζ
1
= 2,
(13)
it can be deduced from (12)
m, n = 1
A
mn
sin λ
m
a x sin λ
n
b y =
1/2 on F
R
,
0 on F .
(14)
Multiplying both sides of (14) by the eigenfunctions, we have
p, q = 1
A
pq
sin λ
p
a x sin λ
q
b y sin λ
m
a x
× sin λ
n
b y =
sin λ
m
a x sin λ
n
b y /2 on F
R
,
0 on F,
(15)
where
λ
p
=
2a
, p = 1, 2, 3,
λ
q
=
2b
, q = 1, 2, 3,
(16)
Next, we integrate the left-hand side of (15) with respect to (x, y)
over the whole rectangle 2a × 2b, and the right-hand side (RHS)
with respect to the local cylindrical coordinates (r, φ) over F
R
, that
is
p, q = 1
A
pq
a
a
sin λ
p
a x sin λ
m
a x dx
×
b
b
sin λ
q
b y sin λ
n
b y dy
=
1
2
0
2π
0
r
1
sin λ
m
a x sin λ
n
b y rdrdφ,
(17)
where
x = x
1
+ rcos φ,
y = y
1
+ rsin φ .
(18)
By means of the orthogonal relation
a
a
sin λ
p
a x sin λ
m
a x dx =
pm
,
b
b
sin λ
q
b y sin λ
n
b y dy =
qn
,
(19)
where δ
pm
, δ
qn
are the Kronecker delta, it follows from (17)
A
mn
=
1
2ab
0
2π
0
r
1
sin λ
m
a x
1
rcos φ
× sin λ
n
b y
1
rsin φ rdrdφ .
(20)
Applying the integrals
0
2π
cos λ
m
rcos φ cos λ
n
rsin φ dφ = 2πJ
0
λ
mn
r
(21)
and
0
2π
cos λ
m
rcos φ sin λ
n
rsin φ dφ
=
0
2π
sin
λ
m
rcos φ cos λ
n
rsin φ dφ
=
0
2π
sin
λ
m
rcos φ sin λ
n
rsin φ dφ = 0
(22)
to RHS of (20), where J
n
(x) is the Bessel function of the first kind
of order n, we thus get the coefficient
A
mn
=
πr
1
abλ
mn
J
1
λ
mn
r
1
sin λ
m
a x
1
sin λ
n
b y
1
.
(23)
Eventually, combining (1a), (1b), (7a), (7b), (11a), (11b), and (23)
gives the solution of MSP
V x, y, z = ±
2πr
1
I
1
ab
m, n = 1
J
1
λ
mn
r
1
sin λ
m
a x
1
× sin λ
n
b y
1
sin λ
m
a x sin λ
n
b y
×
sinh λ
mn
l z cosh λ
mn
l ± ζ
1
λ
mn
sinh 2
mn
,
z > ζ
1
,
z < ζ
1
.
(24)
Moreover, by the definition
H = V,
(25)
we find the expressions of magnetic field immediately
Fig. 2 Global and local coordinates of a circular ring shielded by a
cuboid of high permeability
IET Electr. Power Appl., 2018, Vol. 12 Iss. 5, pp. 717-727
© The Institution of Engineering and Technology 2018
719

H
x
x, y, z = ±
2πr
1
I
1
ab
m, n = 1
J
1
λ
mn
r
1
sin λ
m
a x
1
× sin λ
n
b y
1
cos λ
m
a x sin λ
n
b y
×
λ
m
sinh λ
mn
l z cosh λ
mn
l ± ζ
1
λ
mn
sinh 2
mn
,
H
y
x, y, z = ±
2πr
1
I
1
ab
m, n = 1
J
1
λ
mn
r
1
sin λ
m
a x
1
× sin λ
n
b y
1
sin λ
m
a x cos λ
n
b y
×
λ
n
sinh λ
mn
l z cosh λ
mn
l ± ζ
1
λ
mn
sinh 2
mn
,
H
z
x, y, z =
2πr
1
I
1
ab
m, n = 1
J
1
λ
mn
r
1
sin λ
m
a x
1
× sin λ
n
b y
1
sin λ
m
a x sin λ
n
b y
×
cosh λ
mn
l z cosh λ
mn
l ± ζ
1
sinh 2
mn
.
z > ζ
1
z < ζ
1
.
(26)
Equations (26) lay the foundation for following derivations of
inductance and forces of circular coils.
3Self- and mutual inductance of circular coils
with rectangular cross section shielded by a
cuboid of high permeability
Expressions for the inductance of circular coils are deducible
immediately from the magnetic field given by (26). Supposing that
another circular ring (Ring 2) is placed in the aforesaid field of
Ring 1, with the centre (x
2
, y
2
) and radius r
2
and lying in the plane
z = ζ
2
, therefore the mutual inductance between the Rings 1 and 2
can be found by calculating the magnetic flux passing through the
area encircled by the Ring 2, namely
M =
μ
0
I
1
0
2π
0
r
2
H
z
x
2
+ rcos φ, y
2
+ rsin φ, ζ
2
rdrdφ
=
4μ
0
π
2
r
1
r
2
ab
m, n = 1
J
1
λ
mn
r
1
J
1
λ
mn
r
2
× sin λ
m
a x
1
sin λ
n
b y
1
sin λ
m
a x
2
×
sin λ
n
b y
2
cosh λ
mn
l ± ζ
1
cosh λ
mn
l ζ
2
λ
mn
sinh 2
mn
,
ζ
2
> ζ
1
ζ
2
< ζ
1
.
(27)
When x
1
, y
1
, x
2
, y
2
are all equal to zero, (27) will degenerate into
(19a) and (19b) of [12]. For a circular ring of the infinitesimal
cross section, the self-inductance is divergent [19].
To calculate the self- and mutual inductance for the shielded
circular coils with rectangular cross section from (27), we consider
a pair of circular coils, one has the centre (x
1
, y
1
), inner and outer
radii R
1
, R
2
, axial length 2h
1
and N
1
turns (Coil 1), another has the
corresponding parameters of (x
2
, y
2
), R
3
, R
4
, 2h
2
and N
2
(Coil 2),
and the middle planes of them are set in z = z
1
and z = z
2
,
respectively (see Fig. 1). By evaluating the axial integral with
respect to the arguments ζ
1
, ζ
2
of (27) and using the integral
formula [20]
w
λ
mn
, R
1
, R
2
=
R
1
R
2
rJ
1
λ
mn
r dr
=
π
2λ
mn
R
2
J
1
λ
mn
R
2
H
0
λ
mn
R
2
J
0
λ
mn
R
2
H
1
λ
mn
R
2
R
1
J
1
λ
mn
R
1
H
0
λ
mn
R
1
J
0
λ
mn
R
1
H
1
λ
mn
R
1
,
(28)
where H
n
(x) is the Struve function of order n, we have the
following results:
i. For z
1
+ h
1
z
2
h
2
M = κ
12
m, n = 1
w
λ
mn
, R
1
, R
2
w λ
mn
, R
3
, R
4
× sin λ
m
a x
1
sin λ
n
b y
1
sin λ
m
a x
2
×
sin λ
n
b y
2
v
1
λ
mn
, z
1
, z
2
λ
mn
sinh 2
mn
,
(29a)
where
v
1
λ
mn
, z
1
, z
2
= 4sinh λ
mn
h
1
sinh λ
mn
h
2
× cosh λ
mn
l + z
1
cosh λ
mn
l z
2
/λ
mn
2
and
κ
12
=
μ
0
π
2
N
1
N
2
abh
1
h
2
R
2
R
1
R
4
R
3
is a coefficient relative to the number of turns and geometric
parameters of both coils and cuboid.
ii. For z
1
h
1
z
2
h
2
< z
1
+ h
1
z
2
+ h
2
M = κ
12
m, n = 1
w λ
mn
, R
1
, R
2
w λ
mn
, R
3
, R
4
× sin λ
m
a x
1
sin λ
n
b y
1
sin λ
m
a x
2
×
sin λ
n
b y
2
v
2
λ
mn
, z
1
, z
2
λ
mn
sinh 2
mn
,
where
v
2
λ
mn
, z
1
, z
2
=
1
λ
mn
2
h
1
+ h
2
+ z
1
z
2
λ
mn
sinh 2
mn
+sinh λ
mn
h
1
l + z
1
sinh λ
mn
h
2
l z
2
+sinh λ
mn
h
1
+ l + z
1
sinh λ
mn
h
2
l + z
2
+2sinh h
2
λ
mn
sinh λ
mn
h
1
l z
1
cosh λ
mn
l z
2
.
(29b)
iii. For z
2
h
2
z
1
h
1
< z
1
+ h
1
z
2
+ h
2
(see (29c))
In a similar manner, we deduce that the self-inductance, for a
shielded circular coil with the centre (x
1
, y
1
, z
1
), inner and outer
radii R
1
, R
2
, axial length 2h
1
and N
1
turns, is
L =
μ
0
π
2
N
1
2
abh
1
2
R
2
R
1
2
m, n = 1
w
2
λ
mn
, R
1
, R
2
×
sin
2
λ
m
a x
1
sin
2
λ
n
b y
1
v
4
λ
mn
, z
1
λ
mn
sinh 2
mn
(30)
with the axial function given by
v
4
λ
mn
, z
1
=
1
λ
mn
2
cosh 2λ
mn
(h
1
l) cosh 2
mn
+cosh 2h
1
λ
mn
cosh 2z
1
λ
mn
cosh 2z
1
λ
mn
+ 2h
1
λ
mn
sinh 2
mn
.
720 IET Electr. Power Appl., 2018, Vol. 12 Iss. 5, pp. 717-727
© The Institution of Engineering and Technology 2018

4Magnetic force exerted on circular coils
shielded by a cuboid of high permeability
The magnetic forces between the coils are of practical importance.
Before we commence the specific calculation, some preliminary
discussions should be given for the force calculation of coils with
the presence of magnetic materials. Unlike the force calculation of
coils in the free air, in which we only need to consider the
interaction between the coils, in this BVP the interaction between
coils and that between coil and cuboid should be considered
simultaneously. Hence, if we calculate the force from the magnetic
energy, both mutual and self-magnetic energy (mutual and self-
inductance) should be taken into account to obtain the total force.
To clarify the process of force calculation, we will begin with the
definition of magnetic force. With the aid of the obtained magnetic
field (26) of Ring 1, the mutual magnetic force F
2
m
exerted on
Ring 2 can be obtained immediately. According to the definition of
magnetic force, it follows that [21, 22]
F
2
m
=
I
2
dl
2
× B
1
= I
2
r
2
0
2π
B
z
x
2
+ r
2
cos φ, y
2
+ r
2
sin φ, z cos φe
x
+B
z
x
2
+ r
2
cos φ, y
2
+ r
2
sin φ, z sin φe
y
B
y
x
2
+ r
2
cos φ, y
2
+ r
2
sin φ, z sin φ
+B
x
x
2
+ r
2
cos φ, y
2
+ r
2
sin φ, z cos φ
e
z
dφ,
(31)
where ‘dl
2
is the infinitesimal element of Ring 2, and the local
polar coordinate (r, φ)
x = x
2
+ rcos φ
y = y
2
+ rsin φ
(32)
is introduced to assist the integration. Using the angular integrals
0
2π
sin λ
m
r
2
cos φ cos λ
n
r
2
sin φ cos φdφ =
2πλ
m
J
1
λ
mn
r
2
λ
mn
,
(33a)
and
0
2π
cos λ
m
r
2
cos φ cos λ
n
r
2
sin φ cos φdφ
=
0
2π
cos
λ
m
r
2
cos φ sin λ
n
r
2
sin φ cos φdφ
=
0
2π
sin
λ
m
r
2
cos φ sin λ
n
r
2
sin φ cos φdφ = 0
(33b)
it follows that
0
2π
sin λ
m
a x
2
r
2
cos φ
× sin λ
n
b y
2
r
2
sin φ cos φdφ
= 2πλ
m
J
1
λ
mn
r
2
cos λ
m
a x
2
sin λ
n
b y
2
/λ
mn
.
(34)
We thus obtain the x-component of F
2
m
F
2, x
m
=
4μ
0
π
2
r
1
r
2
I
1
I
2
ab
m, n = 1
λ
m
J
1
λ
mn
r
1
J
1
λ
mn
r
2
× sin λ
m
a x
1
sin λ
n
b y
1
cos λ
m
a x
2
×
sin λ
n
b y
2
cosh λ
mn
l ± ζ
1
cosh λ
mn
l ζ
2
λ
mn
sinh 2
mn
,
ζ
2
ζ
1
ζ
2
< ζ
1
.
(35)
Comparing (27) with (35), it is observed that
F
2, x
m
=
W
m
x
2
= I
1
I
2
M
x
2
,
where W
m
is the mutual magnetic energy. In fact, this is a general
theorem which establishes the relationship between magnetic force
and mutual inductance between two coils [23–27]. Consequently,
the remaining force components F
2, y
m
, F
2, z
m
are also deducible from
differentiating (27) with respect to the corresponding coordinates
y
2
, ζ
2
, and we will not repeat them here due to the space
limitations.
Similarly, by differentiation with respect to the corresponding
centre coordinates of coils in (29a)–(29c), the mutual force F
2
m
exerted on Coil 2 is obtained with no difficulty
i. For z
1
+ h
1
z
2
h
2
F
2, x
m
= I
1
I
2
M
x
2
= κ
12
I
1
I
2
m, n = 1
λ
m
w
λ
mn
, R
1
, R
2
w λ
mn
, R
3
, R
4
× sin λ
m
a x
1
sin λ
n
b y
1
cos λ
m
a x
2
×
sin λ
n
b y
2
v
1
λ
mn
, z
1
, z
2
λ
mn
sinh 2
mn
,
(36a)
M = κ
12
m, n = 1
w
λ
mn
, R
1
, R
2
w λ
mn
, R
3
, R
4
× sin λ
m
a x
1
sin λ
n
b y
1
sin λ
m
a x
2
×
sin λ
n
b y
2
v
3
λ
mn
, z
1
, z
2
λ
mn
sinh 2
mn
,
where
v
3
λ
mn
, z
1
, z
2
=
2
λ
mn
2
h
1
λ
mn
sinh 2
mn
+ sinh λ
mn
h
1
cosh λ
mn
l z
1
sinh λ
mn
(h
2
l z
2
)
+sinh
λ
mn
h
1
cosh λ
mn
l + z
1
sinh λ
mn
(h
2
l + z
2
)
.
(29c)
IET Electr. Power Appl., 2018, Vol. 12 Iss. 5, pp. 717-727
© The Institution of Engineering and Technology 2018
721

Citations
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Yue Yu, Yao Luo1Institutions (1)
Abstract: Recently, the mutual inductance between Bitter coils with rectangular cross-section has been calculated through Bessel function approach or figured out with analytical and semi-analytical formulas. In this study, using the inverse Mellin transform, relevant Bessel integrals will be continued analytically to the complex plane, and then by virtue of contour deformation and residue theorem, they can be expanded to the series containing the hypergeometric functions. In addition, the results obtained by this method are compared with those by finite element method software. Also, the presented approach shows great advantages of high accuracy and very short computational time.

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Yuanzhe Zhu1, Baichao Chen1, Yao Luo1, Runhang Zhu1Institutions (1)
Abstract: Coaxial iron-core coil system shielded by a magnetic screen is a complex model for analytical calculation, because a complicated boundary value problem must be tackled for obtaining inductances of the coils in this situation. Truncated region eigenfunction expansion method is a suitable way to work out such a problem containing complex boundary conditions. Three different models are taken into consideration in this study. Starting with coil system shielded by two infinitely large plates, the basic steps of the proposed method are introduced carefully. Then the relevant techniques and conclusions are applied to the iron-core coils shielded by a fully enclosed cylindrical screen of high permeability and a lidless one. The effectiveness of the proposed method is verified by comparing with finite element method software and experiments. From relevant data, it is evident that the proposed method is much faster and can be easily used for forecast and evaluation of new schemes in device design process.

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References
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254 citations


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113 citations


Journal ArticleDOI
John T. Conway1Institutions (1)
Abstract: A simple method for calculating the mutual and self inductances of circular coils of rectangular cross section and parallel axes is presented. The method applies to non-coaxial as well as coaxial coils, and self inductance can be calculated by considering two identical coils which coincide in space. It is assumed that current density is homogeneous in the coil windings. The inductances are given in terms of one-dimensional integrals involving Bessel and Struve functions, and an exact solution is given for one of these integrals. The remaining terms can be evaluated numerically to great accuracy using computer packages such as Mathematica. The method is compared with other exact methods for the coaxial case, and with a filamentary method for the non-coaxial case. Excellent agreement was found in all cases, and the exact method presented here agrees with another exact coaxial method to great numerical accuracy.

79 citations


Journal ArticleDOI
Slobodan Babic1, Frédéric Sirois1, Cevdet Akyel1, Guy Lemarquand  +2 moreInstitutions (1)
Abstract: This paper presents new analytic formulas for determining the mutual inductance and the axial magnetic force between two coaxial coils in air, namely a thick circular coil with rectangular cross-section and a thin wall solenoid. The mutual inductance and the magnetic force are expressed as complete elliptical integrals of the first and second kind, Heuman's Lambda function and one well-behaved integral that must be solved numerically. All possible singular cases are automatically handled by the proposed formulas. The results of the work presented here have been verified by the filament method and previously published data. The new formulas provide a substantially simple alternative over previously published approaches, which involve either numerical techniques (finite element method, boundary element method, method of moments) or other semianalytic or analytic approaches.

60 citations


Journal ArticleDOI
Slobodan Babic1, Cevdet Akyel1Institutions (1)
Abstract: This paper deals with novel formulas for calculating the mutual inductance and the magnetic force between two coaxial coils in air comprising a circular coil of the rectangular cross section and a thin disk coil (pancake). The mutual inductance and the magnetic force are expressed over the complete elliptical integrals of the first and second kinds, Heuman's lambda function, and three terms that must be solved numerically. All singular cases have been resolved analytically. We also give another approach as comparative method that is based on the filament method where all conductors are approximated by the set of Maxwell's coils. The expressions are obtained over the complete elliptical integrals of the first and second kinds. These new consistent expressions are accurate and simple for useful applications. All results obtained by different approaches are in excellent agreement.

38 citations