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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 INDUCTANCE1595

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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[7] H. Tsutsui, S. Kajita, Y. Ohata, S. Nomura, S. Tsuji-Iio, and R. Shimada,

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N. Watanabe, “Variations of force-balanced coils for SMES,” IEEE Trans.

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[10] R. H. Pennington, Introductory Computer Methods and Numerical

Analysis, 4th ed. New York: Macmillan, 1970.

[11] F. W. Grover, Inductance Calculation Working Formulas and Tables.

New York: Dover, 1946, pp. 77–87.

[12] M. Bueno and A. K. T. Assis, “Equivalence between the formulas for

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[13] H. A. Wheeler, “Formulas for the skin effect,” Proc. IRE, vol. 30, no. 9,

pp. 412–424, Sep. 1942.

[14] M. R. Alizadeh Pahlavani and A. Shoulaie, “Modeling and identifica-

tion of parameters for lumped SMES coil models using genetic algo-

rithms,” (in Persian), in Proc. 15th Elect. Eng. Iranian Conf., Tehran, Iran,

Apr. 2007, pp. 337–343.

[15] M. R. Alizadeh Pahlavani and A. Shoulaie, “Modeling and identification

of parameters for lumped SMES coil models using the least squares

algorithm,” (in Persian), presented at the 22nd Int. Power System Conf.,

Tehran, Iran, Oct. 2007, Paper 98-F-PQA-005.

[16] M. R. Alizadeh Pahlavani and A. Shoulaie, “Conceptual design of SMES

coil with current source inverter type using Kalman filter,” (in Persian),

presented at the 22nd Int. Power System Conf., Tehran, Iran, Oct. 2007,

Paper 98-F-TRN-533.

[17] M. R. Alizadeh Pahlavani and A. Shoulaie, “Behavioral study of stress in

helical toroidal, solenoid and toroidal coils with similar ring structure,”

(in Persian), presented at the 23rd Int. Power System Conf., Tehran, Iran,

Nov. 2008, Paper 98-F-ELM-0434.

[18] M. R. Alizadeh Pahlavani and A. Shoulaie, “Designing solenoidal coil of

magnetic energy storage system with recursive algorithms,” (in Persian),

presented at the 23rd Int. Power System Conf., Tehran, Iran, Nov. 2008,

Paper 98-F-ELM-0105.

[19] M. R. Alizadeh Pahlavani and A. Shoulaie, “State variations estimation of

helical toroidal coil model using Kalman filter,” (in Persian), presented at

the 23rd Int. Power System Conf., Tehran, Iran, Nov. 2008, Paper 98-F-

PQA-0202.

[20] M. R. Alizadeh Pahlavani and A. Shoulaie, “Elimination of compressive

stress in coil advanced structure of superconductor magnetic energy stor-

ing system,” (in Persian), presented at the 23rd Int. Power System Conf.,

Tehran, Iran, Nov. 2008, Paper 98-F-TRN-0245.

[21] M. R. Alizadeh Pahlavani and A. Shoulaie, “Modeling and identification

of helical toroidal coil parameters using recursive least square algorithm,”

(in Persian), presented at the 23rd Int. Power System Conf., Tehran, Iran,

Nov. 2008, Paper 98-F-ELM-0246.

[22] M. R. Alizadeh Pahlavani and A. Shoulaie, “Calculations of helical

toroidal coil inductance for the superconductor magnetic energy storage

systems,” (in Persian), presented at the 23rd Int. Power System Conf.,

Tehran, Iran, Nov. 2008, Paper 98-F-ELM-0106.

[23] M. R. Alizadeh Pahlavani and A. Shoulaie, “Conceptual designing of

helical toroidal coil with voltage source type inverter using classic al-

gorithms,” (in Persian), presented at the 23rd Int. Power System Conf.,

Tehran, Iran, Nov. 2008, Paper 98-F-LEM-0203.

[24] M. R. Alizadeh Pahlavani, H. A. Mohammadpour, and A. Shoulaie,

“Optimization of dimensions for helical toroidal coil with target func-

tion of maximum magnetic energy,” (in Persian), in Proc. 17th Iranian

Conf. Elect. Eng., Tehran, Iran, May 2009.

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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