Recent Researches in Circuits and Systems
The Simplest Formulas for Self inductance, Mutual
Inductance and Magnetic Force of Coaxial
Cylindrical Magnets and Thin Coils
Slobodan Babic, Cevdet Akyel, Member IEEE, and Nazim Boudjada
Abstract—— In this paper we give the new simplified
analytical formulas for calculating the mutual inductance
and the magnetic force for coaxial cylindrical magnets and
thin coils. From the mutual inductance formula we give the
analytical expression of the self inductance in the limit case
when two same solenoids overlap. Also we show that the
formula for the mutual inductance between two cylindrical
coils gives numerically in the limit case, when two same
coils overlap, twice self inductance of the overlapped
solenoids. Many examples validate these numerical
verifications which never failed. All possible singular cases
are automatically handled by the proposed formulas. The
results of the presented work have been verified by the
filament method and previously published data. The new
formula provide a substantially simple alternative over
previously published approaches, which involve either
numerical techniques (FEM, BEM, MOM) or other semianalytic or analytic approaches.
in cases where a general formula of the mutual inductance is
available. Between proposed formulas for the magnetic force
the simplest is this one given by Snow [1] where the magnetic
force is expressed over complete and incomplete elliptic
integrals of the first and second kind. Also, in [7] Rusinov
using Garrett’s methods [5]-[6], gave a simplest formula for
the magnetic force calculation between those coils but it is not
evident for potential users probably because of its
typographical error. Recently, many calculations of the
magnetic force between two wall solenoids were published, in
which the software Mathematica was used to obtain formulas
in terms of special functions such as Jacobian elliptic integrals
[8]-[15]. Thus, the simplest formula for calculating the
magnetic force between thin wall solenoids is this one given by
Snow [1] which can be yet simplified. The purpose of this
paper is to present an analytic approach to this problem
obtained by direct integration without using the differentiation
of the corresponding mutual inductance. This analytical
formula is the simplification of this one given by Snow which
is expressed over complete elliptic integrals of the first and
second kind (or corresponding D function) and the Heuman's
Lambda function. The new Snow’s simplified formula
obtained in this paper for the magnetic force provide a fast,
reliable and accurate alternative to numerical methods or
previously published formulas. It is simpler to use, and cover
all possible cases, including those in which singularities are
present, such as when solenoids are in contact.
Keywords— Coaxial cylindrical magnets, magnetic force, thin
wall solenoids.
I.
INTRODUCTION
T
HE calculation of the mutual inductance and the magnetic
force between coaxial thin wall solenoids has been
interesting for many researches in the past and in these
days, [1]-[12]. The magnetic force between two coaxial
current sheets can be obtained from the corresponding mutual
inductance. Since their mutual energy is equal to the product
of their mutual inductance and the currents in the coils, the
component of the magnetic force of attraction or repulsion in
any direction is equal to the product of the currents multiplied
by the differential coefficient of the mutual inductance taken
with the respect to that coordinate. It is evident from [1]-[15],
the magnetic force may be calculated by simple differentiation
II. CALCULATION METHOD
The mutual inductance and the magnetic force between two
thin wall solenoids, (see Fig. 1) can be given by the formula
[8],
µ 0 N1 N 2 R1 R2
M =
S. Babic is with École Polytechnique de Montréal, Département de Génie
Physique (Corresponding author, phone: (514) 340-4711 ex. 4673; fax: (514)
340-5892; e-mail: slobodan.babic@polymtl.ca).
C. Akyel is with École Polytechnique de Montréal, Département de Génie
Électrique (e-mail: cevdet.akyel@polymtl.ca).
Nazim Boudjada is a student of École Polytechnique de Montréal,
Département de Génie Physique (nazimboudjada@gmail.com).
ISBN: 978-1-61804-108-1
44
z2
z4
0
z1
z3
∫ ∫ ∫
cos θ dz I dz II dθ
r
( z 2 − z1 )( z 4 − z3 )
µ 0 I1 I 2 N1 N 2 R1 R2
F =
π
π z2 z4
∫ ∫∫
0 z1 z
3
(1)
( zII − zI )cosθ dzI dzII dθ
r3
( z 2 − z1 )( z 4 − z3 )
(2)
Recent Researches in Circuits and Systems
where
2
=
r
2
R1
( z II - z I ) +
+
μ0 = 4π×10 -7 H/m – the permeability of free space (vacuum),
and r, θ, z are the cylindrical coordinates.
z
=
F
N2
R1
n
n=1
n
(5)
N1I1J 2
a1
n=4
∑ (−1) Ψ
n
n=1
n
(6)
TABLE I
SPECIAL FUNCTIONS USED
z3
z4
n=4
∑ (−1) Ψ
z1
z2
N1
µ0
where J1 = μ0N1I1∕a1 and J2 = μ0N2I2 ∕a2 are the magnetizations
in Tesla.
In the case of a magnet of the length a2 = z4 – z3 with the
magnetization J2 and a coil of the length a1 = z2 – z1 with the
number of turns N1 and the current I1, equation (4) is also
applicable,
and N1, N2 are total number of turns. I1 and I2 are
corresponding currents in solenoids.
z
J1J 2
=
F
− 2 R1 R2 cosθ
2
R2
y
Symbol:
K(k):
E(k):
D(k):
y
x
Fig 1. Two coaxial thin wall solenoids.
Λ0(ε,k):
Integrating in (1) and (2) over zI, zII and θ the mutual
inductance and the magnetic force between two coaxial thin
wall solenoids can be expressed in an analytical form as
follow,
µ 0 N1 N 2 R1R2 n = 4
n−1
∑
=
M
3a1a2
F
=
(−1)
Φn
Table I gives the various special functions used in the analysis
presented here [16], [17], (See Appendix I).
III. SINGULARITY TREATMENT
(3)
n =1
µ0 I1I 2 N1N2
The formulas (3) and (4) cover all possible cases either regular
or singular. We show all possible singular cases and we give
the derivation of the new formula for the self inductance of the
thin wall solenoid.
n=4
∑ (−1) Ψ
n
a1a2
n=1
Special function
Complete elliptic integral of the first kind
Complete elliptic integral of the second kind
D-elliptic integral as the combination of the
elliptic integrals of the first and second kind
Heuman's Lambda function
(4)
n
A) Mutual inductance singularities
where
=
Φ n kn [tn2 − 2( R12 + R22 )]D(kn ) +
3π tn R22 − R12
4 R2 R1
K (kn ) −
[1 − Λ 0 (ε n , kn )]
kn
4 RR
1
Ψ n = t n k n R1R2 D ( k n ) −
D(kn )
4 R1 R2
2
kn =
2
2
( R1 + R2 ) + t n
=
π
4
In the mutual inductance calculation the singularities appear if
tn = 0 and kn2 = 1.
There are two possible cases:
2
sgn(t n ) R22 − R12 [1 − Λ 0 (ε n ,k n )]
a1) z2 = z3 and R1 = R2
K (kn ) − E (kn )
a2) z1 = z4 and R1 = R2
2
kn
, h=
4 R1 R2
( R1 + R2 )
2
, ε n = arcsin
In both cases we obtain for Φn,
1−h
1−kn2
2
a1 =−
z 2 z1 , a2 =−
z 4 z3
z1 + z2
=
, z 22
2
z3 + z4
=
, z 0 z 22 − z11
2
a2) The special case of this calculation (3) is the calculation of
the self inductance of a single layer solenoid (R1 = R2 = R) for
which z1 = z3 and z2 = z4 (z2 – z1 = z4 – z3 = l) and N1 = N2 = N.
We find in the limit for this special case that the self
inductance of a single layer solenoid (thin wall solenoid) is
obtained in an analytical form,
z0 – the axial displacement between centers of the thin wall
solenoids.
The formula (4) is applicable for two coaxial cylindrical
magnets (or coils) of lengths a1 = z2 – z1 and a2 = z4 – z3
respectively,
ISBN: 978-1-61804-108-1
(7)
Obviously we have to take the appropriate n for which Φn will
be calculated by (7).
t1 =−
z 4 z1 , t 2 =−
z 4 z 2 , t3 =−
z3 z 2 , t 4 =−
z3 z1
z11
=
2
Φn =
4 R1 or 4 R2
45
Recent Researches in Circuits and Systems
L
=
2 µ0 N 2 R 2 l
l 2 − 4R2
R
E (k ) − 4 ]
[ K (k ) −
kR
kRl
l
3l
2
(8)
2
=
kn
where
k2 
The same reasoning can be applied if z4 = z1.
4R2
, l  z2  z1  z4  z3
4R2  l 2
IV. EXAMPLES
This formula (8) has been obtained also by L. Lorentz (1879),
[18].
Formula (3) can be used to calculate numerically the self
inductance of the thin wall solenoid of the radius R, the axial
length l and the number of turn N. If we apply in equation (3)
numerical values for two same wall solenoids which overlap
(R2 = R1, z1 = z3, z2 = z4, or z2 - z1 = z4 – z3 = l and N1 = N2) in
the numerically treatment of this equation we obtained that the
obtained value for the ‘mutual inductance’ is exactly the self
inductance of the thin wall solenoid.
To verify the validity of the new formulas, we apply it to the
following set of examples.
Example I:
In [11] the axial magnetic force was calculated between a
magnet of length h2 = z4 – z3 = 15 mm, which is axially
centered within a coil of length h1 = z2 – z1 = 20 mm and
displaced in equal amounts in the positive and negative axial
directions. The axial displacement z between centers of the
magnet and the coil is defined as z0. The first cylinder of the
rayon R1 = 20 mm has N1 = 100 turns with the unit current I1 =
1 A. The second cylindrical magnet of the rayon R2 = 15 mm
is uniformly magnetized with J2 = 1 T.
We verified equation (3) numerically for many different
numeric values, and it didn’t fail in none of them. Obviously,
this formula will drop in the case where l = 0 because of the
natural singularity for which k2 = 1. This is the same
conclusion for the exact formula (8).
B) Magnetic force singularities
b1) If the wall solenoids are concentric (z0 = 0) the magnetic
force is,
F=0
b2) If R1 = R2 = R and z2 ≠ z3 the magnetic force is given by (4)
with,
Ψ n = tn kn R D ( k n )
Also Ψn covers the case R1 = R2 = R and z1 ≠ z4
These cases b1) and b2) are directly included in (4).
b3) The singular cases appear for R1 = R2 = R and z2 = z3 or R1
= R2 = R and z1 = z4 (solenoids in contact by their bases). For
the first possibility we have,
Ψ n = t n k n R D ( k n ),
n = 1, 2, 4
Ψ n = 0,
n=
3
t 2 = z 4 − z 2 = a2 , t3 = 0, t 4 = z 2 − z1 = a1
z1 + z2
, z 22
=
2
z2 + z 4
Example II:
Calculate the magnetic force between two wall solenoids with
radius R1= R2= 60mm, if the upper base of the first coil is in
the same level as the lower bases of the second coil (walls in
contact). The coil dimensions, number of turns and currents
are as follows:
First wall solenoid: z1= 0 mm, z2=10 mm, N1=1, I1=300 A
Second wall solenoid: z3=10mm, z4=20mm, N2=1, I2=300 A.
This is the singular case. Applying (4) the magnetic force is,
, z 0 z 22 − z11
=
2
a1 + a2
a2 −a1
a1 =−
z 2 z1 , a2 =−
z 4 z 2 , a11 =
, a22 =
2
2
z1 + z2
z2 + z 4
z11
=
=
=
, z 22
, z 0 z 22 − z11
2
2
t1 z0 + a11=
, t 2 z 0 + a22=
, t3 0=
, t 4 z 0 − a22
=
ISBN: 978-1-61804-108-1
TABLE II
COMPARISON OF COMPUTATIONAL ACCURACY
Z0- DISPLACEMENT BETWEEN CENTERS OF THE MAGNET AND
THE COIL
z0(mm)
FFilament [8]
FThis Work (4)
FChester Snow [1]
(N)
(N)
(N)
-45.00
0.0965645
0.0965645
0.0965645
-40.00
0.1386765
0.1386765
0.1386765
-35.00
0.2041472
0.2041472
0.2041472
-30.00
0.3085519
0.3085519
0.3085519
-25.00
0.4787795
0.4787795
0.4787795
-20.00
0.7501240
0.7501240
0.7501240
-15.00
1.0477141
1.0477141
1.0477141
-10.00
1.0764197
1.0764197
1.0764197
-5.00
0.7088887
0.7088887
0.7088887
0.00
0.00
0.00
0.00
5.00
-0.7088887
-0.7088887
-0.7088887
10.00
-1.0764197
-1.0764197
-1.0764197
15.00
-1.0477141
-1.0477141
-1.0477141
20.00
-0.7501240
-0.7501240
-0.7501240
25.00
-0.4787795
-0.4787795
-0.4787795
30.00
-0.3085519
-0.3085519
-0.3085519
35.00
-0.2041472
-0.2041472
-0.2041472
40.00
-0.1386765
-0.1386765
-0.1386765
45.00
-0.0965645
-0.0965645
-0.0965645
In Table II, we give the calculation of the magnetic force by
three different methods. Obviously, results obtained by all
methods are in an excellent agreement.
a1 = z 2 − z1 , a2 = z 4 − z 2 , t1 = z 4 − z1 = a2 + a1
z11
=
4R
,
k n 1, 2, 4
=
2
2
4 R + tn
46
Recent Researches in Circuits and Systems
F = - 0.91999064195112512 N
The proposed formulas can be used for a large scale of
practical applications, such as millimeter and submillimeter
sized biomedical telemetric systems (e.g., for implanted,
injected, or ingested devices) and superconducting coils.
The computational time is about 0.34 seconds. Applying the
filament method [12] the magnetic force is,
F = - 0.9199058453845213 N
APPENDIX I
The number of subdivisions was K = m = 2000 and the
computational time about 18.4 seconds, [12]. Results are in an
excellent agreement.
ELLIPTIC INTEGRALS AND HEUMAN’S LAMBDA FUNCTION
a) K(kn) - Complete elliptic integral of the first kind,
Example III:
In this example we validate the new formula (8) for the self
inductance of the thin wall solenoid which is derived
numerically form the mutual inductance (3) with conditions as
explained.
In all examples (See Table III) we take N1 = N2 = N = 100. As
we can see for many different dimension values for R and l
which describe the thin wall solenoid the fraction M ∕ L is in
any case equal to 1. Formulas (3) and (8) give the same results
that are slightly different because of the numerical evaluation
of the special functions (elliptic integrals). The figures which
agree are bolded. Thus, equation (3) can be used in the
numerical estimation of the self inductance of the thin wall
solenoid with R1 = R2 = R, z2 – z1 = z4 – z3 = l and N1 = N2 =
N. The obtained values for the self inductance are exactly the
same as those obtained by the exact formula (8).
π
dθ
2
∫
=
K (kn )
1 − k n sin θ
2
0
b) E(kn) -Complete elliptic integral of the second kind,
π
E (kn )
2
∫
=
1 − k n sin θ dθ
2
0
=
D(kn )
K (kn ) − E (kn )
2
kn
d) F(φ/kn)-Incomplete elliptic integral of the first kind,
ϕ
z2 - z1 =
z4 – z3 =
l(m)
0.57
0.037
2.0
0.63
2
0.83
0.01
Formula (3) (mH)
This work = M
Formula (8) (mH)
Lorentz’s formula =L
0.9735988374182611
9.800380125674628
41.492608385387195
410.0031965660745
8.302625337317276
2.342057100299159
77.71839539666425
0.9735988374182607
9.800380125674663
41.492608385387195
410.0031965660799
8.302625337317274
2.342057100299159
77.71839539664639
F (ϕ /k n )
=∫
0
dθ
1 − k n sin θ
2
2
e) E(φ/kn)-Incomplete elliptic integral of the first kind,
ϕ
E (ϕ /k n )
= ∫ 1 − k n2 sin 2 θ dθ
0
f) Heuman’s Lambda function Λ0(φ/α) can be expressed as
complete and incomplete elliptic integrals of the first and
second kind,
V. CONCLUSION
2
The new accurate formulas for the mutual inductance and the =
Λ 0 (ϕ α )
{K (α ) E (ϕ
magnetic force formula for the system of two coaxial thin wall
π
solenoids in air are derived and presented in this paper. Also
from proposed method it is possible to calculate the self f1) φ = 0, 0 < α ≤ 900
inductance of the thin wall solenoid without using the
corresponding formula. The proposed approach has been
approved by previously published data. The results are f ) α = 0, 0 < φ < 900
2
obtained over complete elliptic integrals of the first, second
kind as and Heuman’s Lambda function. We find that the
simplified Snow’s formula for the magnetic force presented
0
0
here is definitely the simplest formula for calculating the f3) α = 90 , 0 < φ < 90
magnetic force between two wall solenoids. Also the new
derivation of the self inductance from the mutual inductance
0
0
between two same thin wall solenoids which overlap, can be f4) φ = 90 , 0 ≤ α ≤ 90
consider a new formula which appear for the first time in the
literature even though its applications is purely numerical but
it give the exact values of the self inductance.
ISBN: 978-1-61804-108-1
2
c) D(kn) -Elliptic integral as the combination of the first and
second elliptic integrals and their module kn2,
TABLE III
COMPARISON OF FORMULAS (8) and (9)
R2 =
R1 =
R(m)
0.13
0.229
2.0
7.932
0.75
0.25
1
2
47
900 − α ) − [ K (α ) − E (α )]F (ϕ 900 − α )}
0
Λ 0 (0 α ) =
Λ 0 (ϕ 0) =
sin(ϕ )
2
Λ 0 (ϕ 900 ) =
ϕ
π
Λ 0 (900 α ) =
1.
Recent Researches in Circuits and Systems
ACKNOWLEDGMENT
This work was supported by the Natural Science and
Engineering Research Council of Canada (NSERC) under
Grant RGPIN 4476-05 NSERC NIP 11963.
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ISBN: 978-1-61804-108-1
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