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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 20091593
A Novel Approach for Calculations of Helical
Toroidal Coil Inductance Usable in Reactor Plasmas
Mohammad Reza Alizadeh Pahlavani, and Abbas Shoulaie
Abstract—In this paper, formulas are proposed for the self- and
mutual-inductance calculations of a helical toroidal coil by direct
and indirect methods at superconductivity conditions. The direct
method is based on the Neumann’s equation, and the indirect
approach is based on the toroidal and the poloidal components
of the magnetic flux density. Numerical calculations show that the
direct method is more accurate than the indirect approach at the
expense of its longer computational time. Implementation of some
engineering assumptions in the indirect method is shown to reduce
the computational time without loss of accuracy. Comparison
between the experimental, empirical, and numerical results for
inductance, using the direct and the indirect methods, indicates
that the proposed formulas have high reliability. It is also shown
that the self-inductance and mutual inductance could be calcu-
lated in the same way, provided that the radius of curvature is
greater than 0.4 of the minor radius and that the definition of the
geometric mean radius in the superconductivity conditions is used.
Plotting contours for the magnetic flux density and the inductance
show that the inductance formulas of the helical toroidal coil could
be used as the basis for coil optimal design. Optimization target
functions such as maximization of the ratio of stored magnetic
energy with respect to the volume of the toroid or the conductor’s
mass, the elimination or the balance of stress in certain coordinate
directions, and the attenuation of leakage flux could be considered.
Index Terms—Behavioral study of magnetic flux density com-
ponents, helical toroidal coil, inductance calculation, reactor plas-
mas, semitoroidal coordinate system, superconductor.
I. INTRODUCTION
R
and plasma reactors such as the Tokamak has suggested the
use of advanced coil with a helical toroidal structure [1]–[4].
The main reason for this suggestion is the ability to implement
special target functions for this coil in comparison with other
structures such as the toroidal, the solenoid, and the spherical
coils [5], [6]. The structure of this coil is shown in Fig. 1. In
this coil, the ratio of the major to the minor radius (A = R/a),
the number of turns in a ring (N), and the number of rings in
a layer (υ) are called aspect ratio, poloidal turns (or the pitch
number), and helical windings, respectively. For example, the
coil in Fig. 1 is composed of five helical windings (υ = 5) with
nine poloidal turns (N = 9). The inductance formulas show
ECENT research work on superconducting magnetic en-
ergy storage (SMES) systems, nuclear fusion reactors,
Manuscript received January 29, 2009; revised May 7, 2009. First published
July 17, 2009; current version published August 12, 2009.
The authors are with the Department of Electrical Engineering, Iran Univer-
sity of Science and Technology, Tehran 16846, Iran (e-mail: MR_Alizadehp@
iust.ac.ir; shoulaie@iust.ac.ir).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPS.2009.2023548
Fig. 1.
nine turns.
Structure of a monolayer helical toroidal coil with five rings of
that parameters a, R, and N of the helical toroidal coil can be
used as design parameters to satisfy special target functions.
With respect to the fact that each ring of the coil generates
both toroidal and poloidal magnetic fields simultaneously, the
coil can be regarded as a combination of coils with toroidal
and solenoid fields. Furthermore, the coil can be designed in
a way to eliminate the magnetic force component in both the
major and minor radius directions. These are called force- and
stress-balanced coils, respectively. In addition, the coils that
utilize the virial theorem to balance these two force components
are called virial-limited coils [7]–[9]. In some applications,
helical toroidal coils are used in a double-layer manner with
two different winding directions (respectively with different or
the same current directions in each layer) to reduce the poloidal
leakage flux being compared to the toroidal leakage flux or
vice versa. In this paper, the investigation is focused on the one-
layer helical toroidal coil.
In general, any simulation program that simultaneously
solves equations, the particle position, and its velocity can be
called a particle-in-cell (PIC) simulation. The name PIC comes
from the way of assigning macroquantities (like density, current
density, and so on) to the simulation particles. Inside the plasma
community, PIC codes are usually associated with solving
the equation of motion of particles with the Newton–Lorenz’s
force. PIC codes are usually classified depending on the di-
mensionality of the code and based on the set of Maxwell’s
equations used. The codes solving an entire set of Maxwell’s
equations are called electromagnetic codes, while electrostatic
ones solve just the Poisson equation. Some advanced codes are
able to switch between different dimensional and coordinate
systems and use electrostatic or electromagnetic models. PIC
0093-3813/$26.00 © 2009 IEEE
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1594IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009
Fig. 2. Semitoroidal coordinate system.
simulation starts with an initialization and ends with the output
of results. This part is similar to the input/output routines
of any other numerical tool. Usually, the numerical methods
based on the PIC simulation are obtained from the solution
of partial differential–algebraic equations, for example, by the
fourth-order Runge–Kutta method. Considering the number
of particles which are on the order of 1010, the simulations
based on PIC methods take a long time to solve the aforemen-
tioned equations. Usually, in order to resolve the time issue
in the PIC methods, special computers may be employed. On
the other hand, this paper used Biot–Savart equation and the
mathematical equations of the current path in the conductor
of the helical toroidal coil in order to obtain the magnetic flux
density components. The numerical integrals resulting from the
Biot–Savart equations are solved using the extended three-point
Gaussian algorithms. This method has the least error among
all numerical integration methods. In addition, the method used
by the authors, contrary to the PIC method, does not need any
special computers to solve the equations.
In Section II of this paper, the semitoroidal coordinate sys-
tem, the longitudinal components of a ring element, and the
ring radius of curvature are briefly discussed. In Sections III
and IV, the formulas of the magnetic flux density compo-
nents are presented and the behaviors of these components are
simulated, respectively. In these sections, the variations of the
ring radius of curvature with respect to the toroidal angle for
the geometric parameters of the coil are also investigated. In
Sections V and VI, the formulas for the self-inductance and
mutual inductance of the helical toroidal coils using the direct
and indirect methods are developed. Finally, in Section VII, the
experimental inductance measurements are compared with the
direct and indirect method simulation.
II. LONGITUDINAL COMPONENTS AND RADIUS OF
CURVATURE OF A RING
The semitoroidal coordinate system is orthogonal, 3-D, and
rotational (see Fig. 2). Longitudinal components of the ring
element in this coordinate system can be defined by (1). Table I
presents the dot products of the unit vectors for the Cartesian
and the semitoroidal coordinates using the projection of these
vectors on the planes ϕ = const and z = 0
dl =? aθadθ +? aρdρ +? aϕ(R + acosθ)dϕ
=? axdlx+? aydly+? azdlz.
(1)
TABLE I
DOT PRODUCT OF THE UNIT VECTORS FOR CARTESIAN AND
SEMITOROIDAL COORDINATE SYSTEMS
Using Table I, the longitudinal components of the ring ele-
ment in the Cartesian coordinate system can be defined by (2).
As the ring’s geometric loci is in the form of ρ = const, its
differential dρ = 0 can be replaced in (2). The relation between
θ and ϕ for a ring of N turns can be expressed as (3) in which
θ0is the poloidal angle of the ring at plane ϕ = 0
dlx= dl ·? ax
= −asinθcosϕdθ − sinϕ(R + acosθ)dϕ
+ cosθcosϕdρ
(2)
dly= dl ·? ay
= −asinθsinϕdθ + cosϕ(R + acosθ)dϕ
+ sinϕcosθdρ
dlz= dl ·? az= acosθdθ + sinθdρ
θ = Nϕ + θ0.
(3)
If the positional vector of each longitudinal element of the
ring with respect to the origin of Cartesian coordinate system
is defined as (4), then the tangent vector to the longitudinal
element?Tυ(ϕ) and the radius of curvature of the ring can be
definedas(5)and(6),respectively.Intheseequations,f?
are the first and second derivatives of function fiwith respect
to ϕ, respectively
iandf??
i
?Pυ(ϕ) =f1? ax+ f2? ay+ f3? az
υ(ϕ)?????P?
???
=f2
4
(4)
?Tυ(ϕ) =?P?
υ(ϕ)
???
= (f?
1? ax+ f?
????P?
(f??
2? ay+ f?
?????T?
3? az)??
f4
(5)
Pc(ϕ) =
υ(ϕ)
??
υ(ϕ)
???
1f4− f?
1f5)2+ (f??
2f4− f?
2f5)2
+(f??
2f4− f?
2f5)2?0.5
=a · g(A,N)
(6)
where
f1= (R + acos(Nϕ + θ0))cosϕ
f2= (R + acos(Nϕ + θ0))sinϕ
f3=asin(Nϕ + θ0)
f4=f?2
1+ f?2
2+ f?2
3
f5=f?
1f??
1+ f?
2f??
2+ f?
3f??
3.
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ALIZADEH PAHLAVANI AND SHOULAIE: APPROACH FOR CALCULATION OF HELICAL TOROIDAL COIL INDUCTANCE 1595
III. MAGNETIC FLUX DENSITY COMPONENTS
In this section, each infinite surface to which the normal
vector? aϕis perpendicular is called the toroidal plane, and each
infinite surface (with conical shape) to which the normal vector
? aθ is perpendicular is called poloidal cone. Therefore, each
toroidal plane could be represented with the equation ST: ϕ?=
const&π + const, and each poloidal cone could be represented
with the equation SP: θ?= const&π + const. It is noted that
the apex of the cone for each poloidal cone is placed on the
z-axis and that the surfaces of all poloidal cones include
the geometric loci of origins for the semitoroidal coordinate
systems. If the Cartesian coordinates of any point β on each
toroidal or poloidal cone are xβ, yβ, and zβ, and the Cartesian
coordinates of any point α on the ring with characteristics
of θo, a, R, and N are presented by xα, yα, and zα, then
replacing (2), (3), and (7) in the Biot–Savart equation results
in the Cartesian form of the magnetic flux density components
of (8). In addition, it is possible to change these components
into toroidal component of the magnetic flux density with (10)
by defining point β on each toroidal plane using (9)
xα=(R + acosθ)cosϕ
yα=(R + acosθ)sinϕ
zα=asinθ
(7)
Bx=(μ0I/4π)
2π
?
0
2π
?
0
2π
?
0
(f7/f6)dϕ
By=(μ0I/4π)(f8/f6)dϕ
Bz=(μ0I/4π)(f9/f6)dϕ
(8)
where
f6=
?
(xβ(R + acos(Nϕ + θ0))cosϕ)2
+ (yβ(R + acos(Nϕ + θ0))sinϕ)2
+(zβ− asin(Nϕ + θ0))2?3/2
f7= [(zβ− asin(Nϕ+θ0))
· ((R+acos(Nϕ+θ0))cosϕ
−aN sin(Nϕ+θ0)sinϕ)
− (yβ−(R+acos(nϕ+θ0))sinϕ)
·(aN cos(Nϕ+θ0))]
f8= [(xβ− (R + acos(Nϕ + θ0))cosϕ)
· (aN cos(Nϕ + θ0)) + (zβ− asin(Nϕ + θ0))
·((R + acos(Nϕ + θ0))sinϕ
+ aN sin(Nϕ + θ0)cosϕ)]
f9= [(yβ− (R + acos(Nϕ + θ0))sinϕ)
· (sinϕ(R + acos(Nϕ + θ0))
+ aN sin(Nϕ + θ0)cosϕ)
− (xβ− (R + acos(Nϕ + θ0))cosϕ)
·(cosϕ(R + acos(Nϕ + θ0))
− aN sin(Nϕ + θ0)sinϕ)].
Moreover, the poloidal and the radial components of the
magnetic flux density are represented by (12) and (13), respec-
tively, by defining the point β on each poloidal cone using (11).
It is noted that the range of variations of ρ on each poloidal
cone is defined as 0 ≤ ρ ≤ −R/cosθ?for π/2 ≤ θ?≤ 3π/2
and 0 ≤ ρ ≤ ∞ for −π/2 ≤ θ?≤ π/2
xβ=γ cosξ cosϕ?,
0 ≤ γ ≤ ∞
yβ=γ cosξ sinϕ?,
0 ≤ ξ ≤ 2π
zβ=γ sinξϕ?= const&π + const
(9)
Bϕ= − Bxsinϕ?+ Bycosϕ?
xβ=(R + ρcosθ?)cosϕ?
(10)
yβ=(R + ρcosθ?)sinϕ?
zβ=ρsinθ?
(11)
Bθ= − Bxsinθ?cosϕ?− Bysinϕ?sinθ?+ Bzcosθ?
Bρ=Bxcosθ?cosϕ?+ Bysinϕ?cosθ?+ Bzsinθ?.
(12)
(13)
IV. BEHAVIORAL STUDY OF THE MAGNETIC
FLUX DENSITY COMPONENTS
In this section, the behavior of the magnetic flux density
components for one ring or several rings is simulated us-
ing MATLAB. The numerical integrations in Sections III, V,
and VI are performed using the extended three-point Gaussian
algorithms [10]. This method has the least error among all
numerical integration methods. In the sketching of the magnetic
flux density components, it is assumed that
Δγ =Δρ = 0.05 [m]
Δθ?=Δϕ?= Δξ = 0.02 [rad]
I =1 [kA]
ρmax=R
γmax=2R
Δϕ =π/150 [rad].
Also in the calculation of the integral, the corresponding
range is divided into n = 300 equal segments in order to
reach the integration error of the derivative order 2n or 600
of the function under integration. It is obvious that, if the
integration range is divided into more parts, the integration
error will decrease further, but this comes at the cost of longer
computational time. In Figs. 3–5, the contours for magnetic flux
density,the toroidal,the poloidal, and the radial components are
shown. It can be inferred that, as we approach the conductor’s
geometric loci, the amplitude of the magnetic flux density
becomes stronger. In these figures, the boundary of surface S0,
from which the magnetic flux density components only enter or
exit, is marked with the number 0. Figs. 3(c) and (h), 4(c) and
(f), and 5(e) and (f) show the behavior of the magnetic flux
density components for υ ring of N turns (total of Nυ turns) in
comparison with a single Nυ turn ring. The comparison of the
results for these two windings indicates that the magnetic flux
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1596IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009
Fig. 3. Contours of toroidal component for magnetic flux density.
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ALIZADEH PAHLAVANI AND SHOULAIE: APPROACH FOR CALCULATION OF HELICAL TOROIDAL COIL INDUCTANCE1597
density components are different for these two case studies, and
the reason for this difference should be sought in the winding
types of these coils. Moreover, Fig. 3(a), (b), (d), and (e) shows
that, by rotation of the ring, Bϕmaximum geometric loci is
altering at θ = Nϕ?+ θ0. From Fig. 3(a), (c), and (g), it can be
observed that the amplitude Bϕis increased with increasing υ
and N (increasing N is more efficient than υ) and that the leak-
age flux percentage of Bϕfrom the coil’s section will decrease.
According to Fig. 3(a), (b), (d), and (e), the surface area S0on
different toroidal planes is different. Furthermore, part of Bϕ
is passing outside the coil’s section, and the maximum of Bϕ
on different toroidal planes is altering for the same parameters
a, R, and N. Symmetry of the right half of Bϕwith the left
half could be seen for the even values of N, and the asymmetry
of the right half of Bϕwith the left half is apparent for the
odd or fractional values of N [see Fig. 3(a), (b), (d), and (e)]. In
Fig. 3(f) and (g), it is shown that, in marginal conditions of N =
0 or N ? 1, the coil behavior is approaching a pure solenoid or
a toroidal coil, and the amplitude of Bϕon each toroidal plane
is decreased and increased, respectively. In Fig. 4, contours
are plotted for Bθ on the poloidal cone of z = 0. Moreover,
Fig. 4(a) and (b) shows that, by the rotation of the ring, Bθ, the
maximumgeometriclociisalteringatϕ = −θ0/N.Inaddition,
the maximum of Bθis independent of θ0. From Fig. 4(a) and
(c)–(f), it is inferred that the surface area S0is continuously
changing and an increase in N increases the area of this surface.
In other words, a percentage of Bθ is passing through the
surface of the helical toroidal coil. In Fig. 4, the number of the
rose leaves of the poloidal magnetic field is Nυ, and there are
always Nυ global maximums and minimums for Bθon each
poloidal cone. Fig. 4(a) and (c)–(f) shows that the amplitude
of Bθ is increased and decreased with increasing υ and N,
respectively. In other words, increasing υ increases the density
of Bθamplitude maximum in the range a ≤ ρ ≤ R. In Fig. 4(c)
and (e), it is shown that, in the marginal conditions of υ ? 1 or
N ? 1, the coil behavior approaches the pure solenoid and the
pure toroidal coils, respectively. Furthermore, the amplitude of
Bθis increased and decreased, respectively. It can be inferred
from Fig. 4(g) that, in the marginal conditions of N = 0, the
behavior of Bθfor the helical toroidal coil is similar to one ring,
and the amplitude of Bθis positive inside the ring and negative
outside it.
In Fig. 5, the contours of the magnetic flux density of the
radial component are shown. Fig. 5(a) and (b) shows that
the Bρ maximum on the different toroidal planes with the
same values of a, R, and N is changing and occurs at θ =
Nϕ?+ θ0. Fig. 5(a), (e), and (f) shows that the amplitude of
Bρis increased with increasing υ and N (increasing υ is more
efficient than N), and there is always Nυ global maximums
and minimums for Bρon each toroidal plane. It can be inferred
from this figure that, due to its radial form, this component
has no magnetic linkage with the ring. In Fig. 5(d) and (f),
the amplitude of Bρin the marginal conditions of N = 0 and
N ? 1 on each toroidal plane is observed. In the same figures,
the symmetry of this component with respect to poloidal cone
z = 0 in the marginal condition of N = 0 is noticed.
In Fig. 6, the variations of the ratio radius of curvature of
the ring to the coil minor radius with respect to the toroidal
angle for parameters A and N are sketched. It can be inferred
from Fig. 6 that the ratio of the radius of curvature to the minor
radius increases with increasing the value of the aspect ratio
in constant poloidal turns. In addition, the minimum of this
value occurs at the aspect ratio of one for poloidal turn numbers
of 0.4.
V. DIRECT CALCULATIONS OF SELF-INDUCTANCE
AND MUTUAL INDUCTANCE
Equation (14) is introduced for the calculation of the mutual
inductance between two rings with characteristics θ0i, ai, Ri,
Niand θ0j, aj, Rj, Njbased on the Neumann’s equation
Mij= MN
ij= (μ0/4π)
2π
?
0
2π
?
0
(f11/f10)dϕidϕj,i ?= j
(14)
where
f10=
?
(ajsin(Njϕj+ θ0j) − aisin(Niϕi+ θ0i))2
+ ((Rj+ ajcos(Njϕj+ θ0j))sinϕj
− (Ri+ aicos(Niϕi+ θ0i))sinϕi)2
+ ((Rj+ ajcos(Njϕj+ θ0j))cosϕj
−(Ri+ aicos(Niϕi+ θ0i))cosϕi)2?0.5
f11= [(aiNisin(Niϕi+ θ0i)cosϕi
+sinϕi(Ri+ aicos(Niϕi+ θ0i)))
· (ajNjsin(Njϕj+ θ0j)cosϕj
+sinϕj(Rj+ ajcos(Njϕj+ θ0j)))
+ (−aiNisin(Niϕi+ θ0i)sinϕi
+cosϕi(Ri+ aicos(Niϕi+ θ0i)))
· (−ajNjsin(Njϕj+ θ0j)sinϕj
+cosϕj(Rj+ ajcos(Njϕj+ θ0j)))
+ aiNicos(Niϕi+ θ0i) · ajNjcos(Njϕj+ θ0j)].
According to classical electrodynamics, if the radius of the
curvature is larger than the dimensions of the transverse section
of the conductor (i.e., the diameter of the conductor’s cross
section is smaller than 0.4 of the minor radius of the helical
toroidal coil), the equation of the mutual inductance between
the two rings can be used to calculate the self-inductance of
the ring. In this condition, the minimum distance between the
two corresponding points in each ring is assumed to be equal
to the geometrical mean radius of the conductor’s cross section.
Thegeometrical meanradius oftheconductor’s crosssectionis,
in fact, the efficient radius of the conductor that the magnetic
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1598IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009
Fig. 4.Contours of the poloidal component of the magnetic flux density.
Fig. 4.
density.
(Continued.) Contours of the poloidal component of the magnetic flux
field is unable to influx. In nonsuperconductivity conditions,
the geometric mean radius of the conductor’s cross section
with radius r is defined as rm= re−0.25. In superconductivity
conditions, the geometric mean radius is assumed to be rm= r,
because the magnetic field cannot influx the conductor’s cross
section. Therefore, (15) with its assumptions can be used to
calculate the self-inductance of the ith ring with characteristics
θ0i, a, R, and N
Lii=Mii,
Ri=Rj= R,
θ0i=θ0i[rad],
i = 1,...,υ
ai= aj= a,
θ0j= θ0i+ (rm/a) [rad].
Ni= Nj= N
(15)
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ALIZADEH PAHLAVANI AND SHOULAIE: APPROACH FOR CALCULATION OF HELICAL TOROIDAL COIL INDUCTANCE1599
Fig. 5. Contours of the radial component of the magnetic flux density.
VI. CALCULATIONS OF SELF-INDUCTANCE AND MUTUAL
INDUCTANCE USING INDIRECT METHOD
In this section, the mutual and self-inductances of the helical
toroidal coil are proposed by the indirect method or the mag-
netic flux density components. The self-inductance of one ring
or the mutual inductance between two rings is proportional to
the surface integral of the magnetic flux density components
of one or two rings. The radial component of the magnetic
flux density is perpendicular to the ring geometric loci and
does not have magnetic link to one ring or two rings. Conse-
quently, this does not affect the calculation of the mentioned
inductances. In other words, the only effective components
in the calculation of the self-inductance and mutual induc-
tance are the magnetic flux density of toroidal and poloidal
components.
A. Calculation of Mutual Inductance via Indirect Method
In this section, the mutual inductances of the ith and the
jth rings with characteristics of θ0i, ai, Ri, Niand θ0j, aj,
Rj, Nj, with incorporation of some engineering assumptions,
are calculated. The pure flux of field without divergence on
each closed surface is zero. The magnetic field entering into
each toroidal or poloidal cone could be calculated as of surface
integration of the absolute values of Bθand Bϕon these planes.
Consequently, it is possible to estimate the values of these
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1600IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009
Fig. 6.
toroidal angle.
Variations of radius of curvature of the ring with respect to the
integrals with the linking flux of the ith ring using the following
equations:
ψT(ϕ?) =0.5
?
ST
?
SP
⎛
⎝
|Bϕ|dsϕ= 0.5
∞
?
0
2π
?
0
|Bϕ|ρdθ?dρ
(16)
ψP(θ?) =0.5
|Bθ|ds
=0.5
⎜
2π
?
0
−Ri/cosθ?
?
0
|Bθ|(Ri+ ρcosθ?)dρdϕ?
+
2π
?
0
∞
?
0
|Bθ|(Ri+ ρcosθ?)dρdϕ?
⎞
⎠.
⎟
(17)
On the one hand, these equations indicate that ψT and ψP
are functions of ϕ?and θ?, respectively. On the other hand,
the toroidal and the poloidal mutual inductances are the result
of ψT and ψP generated by the ith ring linking with the jth
ring. Therefore, it is assumed that the toroidal and the poloidal
mutual inductances are proportional to the minimum of ψTand
ψP generated by the ith ring linking with the cross section of
STMand SPM. These surfaces could be stated as (18) and (19).
It is important to mention that the linking magnetic fluxes with
the cross section of STMand SPMlink Njand one ring and are
defined as (20) and (21), respectively. Based on the definition
of STM, it is necessary to apply the definition of point β based
on (11) for replacement of Bϕin (16) and (20)
STm: 0 ≤ ρ ≤ aj,
⎧
⎪
⎪
ψTm(ϕ?)
0 ≤ θ?≤ 2π,ϕ?= const,π + const
(18)
SPm:
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎩
⎛
⎝
⎛
aj≤ ρ ≤ −Rj/cosθ?
π/2 ≤ (θ?= const,π + const) ≤ 3π/2
0 ≤ ϕ?≤ 2π
or
aj≤ ρ ≤ ∞
−π/2 ≤ (θ?= const,π + const) ≤ π/2
0 ≤ ϕ?≤ 2π
⎞
⎠
⎞
⎝
?
STm
⎠
(19)
= Nj
|Bϕ|ds = Nj
aj
?
0
2π
?
0
|Bϕ|ρdθ?dρ
(20)
ψPm(θ?)
=
?
?
0
SPm
|Bθ|ds
=
2π
−Rj/cosθ?
?
aj
|Bθ|(Rj+ ρcosθ?)dρdϕ?.
(21)
Toroidal angle ϕ?= ϕopand poloidal angle θ?= θopcould
be achieved by the derivation of (20) and (21), which minimizes
ψTMand ψPM. Then, the toroidal and poloidal mutual induc-
tances could be defined as (22) and (23). As it was mentioned
before, ψT and ψP must be particularly independent of the
variations of ϕ?and θ?. In other words, the variations of ψTand
ψPmust be negligible in comparison with the variations of ϕ?
and θ?. Therefore, in order to be able to simplify the equations,
the assumptions of ϕop= 0 and θop= π are important. By
implementing these assumptions, the toroidal and the poloidal
mutual impedances can be defined according to (24) and (25)
MijT=ψTm(ϕ?= ϕop)/I
MijP=ψPm(θ?= θop)/I
MT=ψTm(ϕ?= ϕop= 0)/I
MP=ψPm(θ?= θop= π)/I.
(22)
(23)
(24)
(25)
B. Calculation of Self-Inductance via Indirect Method
In this section, the self-inductance of the ring with charac-
teristics of θ0i, ai, Ri, and Niis calculated with the following
assumptions.
1) ψTand ψPare averaged.
2) Leakage fluxes are negligible.
3) ψTand ψPare independent of the variations of ϕ?and θ?,
or assuming ϕ?= ϕmean= 0 and θ?
calculations are simplified.
op= θmean= π, the
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ALIZADEH PAHLAVANI AND SHOULAIE: APPROACH FOR CALCULATION OF HELICAL TOROIDAL COIL INDUCTANCE1601
Based on the first assumption, the toroidal and the poloidal
fluxes linking with the ring are defined as the average of the
entering fluxes into all toroidal and poloidal cones according to
(26) and (27), respectively. In other words, the toroidal angle of
ϕ?= ϕmeanand the poloidal angle of θ?= θmeanare obtained
from the average values of the toroidal and the poloidal fluxes
of all toroidal and poloidal cones. Incorporating the second
assumption results in (28) and (29). Implementing the third
assumption gives (30) and (31).
Comparing (24) and (25) with (30) and (31) shows that, if the
two rings i and j are in the same layer and Ni= Nj= N, then
MP= LP and MT= LT. Based on the fact that the current
is the same for LiiP and LiiT, it can be inferred that the
self-inductance of the ring i, Lii, could be obtained from
the summation of LiiP and LiiT. In addition, with respect to
the fact that the currents for MijP and MijT are equal, it can
be inferred that the mutual inductance of the rings i and j, Mij,
could be obtained from the summation of and MijT
LiiT(ϕ?=ϕmean) = NiψT(ϕ?= ϕmean)/I
LiiP(θ?=θmean) = ψP(θ?= θmean)/I
LiiTm=NiψTm(ϕ?= ϕmean)/INj
LiiPm=ψPm(θ?= θmean)/I
LT=NiψTm(ϕ?= ϕmean= 0)/INj
LP=ψPm(θ?= θmean= π)/I
Mij=MijT+ MijP,
Lii=LiiT+ LiiP,
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
i ?= j
i = 1,...,υ.
Electrical circuit analysis shows that the inductance matrix
of a monolayer helical toroidal coil could be obtained using
filament method as in (34), and the inductance of one layer,
assuming Lii= Mii, could be obtained by multiplying υ2by
Liior Mij
⎡
⎣
Lcoil=υ2(Lii= Mij).
Lcoil=
⎢
L11
M21
···
Mυ1
M12
L22
···
Mυ2
···
···
···
···
M1υ
M2υ
···
Lυυ
⎤
⎦
⎥
υυ
(34)
(35)
VII. COMPARISON OF EXPERIMENTAL, EMPIRICAL,
AND NUMERICAL RESULTS
In this section, the behavioral study of the inductance char-
acteristics of the ring is simulated using MATLAB. In addition,
the empirical results and experimental results of the induction
measurements are compared with their corresponding numer-
ical values. The empirical results for the mutual inductance
between two flat rings of radius 20 and 25 cm with center-
to-center distance of 10 cm are reported as 0.24879 μH [11].
The parameters of the helical toroidal coil with geometrical
calculations could be obtained as (36) to adapt this problem
with equations mentioned in Section V
N =N1= N2= 0 [turns]
R =R1= R2= 22.5 [cm]
a =a1= a2=
θ01= − tan−1(2) [rad]
θ02=π − tan−1(2) [rad].
√125/2 [cm]
(36)
Fig. 7.Comparison between numerical and empirical results [11].
Fig. 8.
numerical and empirical results [11].
(a) Simulation time versus n. (b) Error percentage versus n between
The convergence diagram, the simulation time, and the error
percentage versus n for numerical and empirical results are
shown in Figs. 7 and 8, respectively. In Fig. 7, it is noticed that
the optimal value of n is 5. In Fig. 8, it is shown that increasing
n to reduce error increases the simulation time in a para-
bolic form.
The empirical (37) has been used to calculate self-inductance
of a flat ring of radius τ and the radius of the conductor’s cross
section of r [12], [13]. For example, the self-inductance of a
flat ring with a radius of 200 cm and a radius of the conductor’s
cross section of 1 cm is 14.1441 μH. The parameters of the
helical toroidal coil with geometrical calculations could be ob-
tainedas(38)toadaptthisproblemwithequationsmentionedin
Section V. The convergence diagram, simulation time, and the
error percentage versus n for numerical and empirical results
are shown in Figs. 9 and 10, respectively. It can be seen from
Fig. 10 that the optimal value of n for the target function error
of less than 0.0015 is 500
L =μ0τ [Ln(8τ/r) − 1.75] [H]
R + a =200 [cm]
θ01=0 [rad]
θ02= re−0.25/a [rad].
(37)
r = 1 [cm]
(38)
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Page 10
1602IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009
Fig. 9. Comparison between numerical and empirical results [12], [13].
Fig. 10.
numerical and empirical results [12], [13].
(a) Simulation time versus n. (b) Error percentage versus n between
TABLE II
COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS OF
INDUCTANCE IN HELICAL TOROIDAL COIL
The experimental result of helical toroidal coil inductance
with geometric characteristics shown in Table II is 17 mH
[6]. It can be observed that the experimental results are in
good agreement with those obtained from (35), with an error
of less than 2.3%. The error can be due to the measurement,
value of n, assumptions made in Section VI, and Mij= Mji.
The convergence diagram, the simulation time, and the error
percentage versus n for numerical and experimental results are
shown in Figs. 11 and 12, respectively. It can be inferred from
Figs. 7, 9, and 11 that the optimal value of n in the indirect
Fig. 11.Comparison between numerical and experimental results [6].
Fig. 12.
numerical and experimental results [6].
(a) Simulation time versus n. (b) Error percentage versus n between
method is nearly half of that in the direct method. As a result,
the simulation time in the indirect method is nearly half of
that in the direct method. Therefore, one can conclude that the
equations presented for the inductance calculations, accepting
calculation error of 2.3% (acceptable engineering error), with
the assumptions in the indirect method being compared to that
in the direct method, have higher reliability with less simula-
tion time.
VIII. CONCLUSION
Helical toroidal coils are superior to other coils and are
extensively used in the SMES systems, nuclear fusion reactors,
and plasma research work. Considering the complexity of the
coils and the fact that not much investigation has been carried
out in this field, this area of research is still open for much more
academic work to come. On the other hand, the calculation
of inductance for these types of coils can be an index to
determine the behavior of the transient state, determination of
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Page 11
ALIZADEH PAHLAVANI AND SHOULAIE: APPROACH FOR CALCULATION OF HELICAL TOROIDAL COIL INDUCTANCE1603
the electrical equivalent circuit, and estimation of values for
electrical elements of the coil equivalent circuit [14]–[24].
In this paper, the inductance of the helical toroidal coil is
calculatedinthedirectandindirectmethods,andtheinductance
characteristics and the magnetic flux density are simulated in
MATLAB. Comparison of the experimental, empirical, and
numerical results shows that the equations for inductance cal-
culations have great reliability and that dividing the inductance
for one ring into two toroidal and poloidal components with
incorporation of some engineering assumptions simplifies the
equationsanddecreasesthecomputationaltimewithouttheloss
of accuracy.
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Mohammad Reza Alizadeh Pahlavani was born
in Iran in 1974. He received the B.Sc. and M.Sc.
degrees in electrical engineering from Iran Univer-
sity of Science and Technology (IUST), Tehran,
Iran, in 1998 and 2002, respectively, where he is cur-
rently working toward the Ph.D. degree in electrical
engineering.
His current research interests include electro-
magnetic systems, power electronics, and electrical
machines.
Abbas Shoulaie was born in Iran in 1949. He
received the B.Sc. degree from Iran University of
Science and Technology (IUST), Tehran, Iran, in
1973, and the M.Sc. and Ph.D. degrees in electrical
engineering from U.S.T.L, Montpellier, France, in
1981 and 1984, respectively.
He is currently a Professor with the Department
of Electrical Engineering, IUST. He is the author of
more than 100 journals and conference papers in the
field of power electronics, electromagnetic systems,
electrical machines, liner machines, and HVDC.
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