a novel approach for calculations of helical toroidal coil inductance usable … · 2019-03-05 ·...

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009 1593

A Novel Approach for Calculations of HelicalToroidal Coil Inductance Usable in Reactor Plasmas

Mohammad Reza Alizadeh Pahlavani, and Abbas Shoulaie

Abstract—In this paper, formulas are proposed for the self- andmutual-inductance calculations of a helical toroidal coil by directand indirect methods at superconductivity conditions. The directmethod is based on the Neumann’s equation, and the indirectapproach is based on the toroidal and the poloidal componentsof the magnetic flux density. Numerical calculations show that thedirect method is more accurate than the indirect approach at theexpense of its longer computational time. Implementation of someengineering assumptions in the indirect method is shown to reducethe computational time without loss of accuracy. Comparisonbetween the experimental, empirical, and numerical results forinductance, using the direct and the indirect methods, indicatesthat the proposed formulas have high reliability. It is also shownthat the self-inductance and mutual inductance could be calcu-lated in the same way, provided that the radius of curvature isgreater than 0.4 of the minor radius and that the definition of thegeometric mean radius in the superconductivity conditions is used.Plotting contours for the magnetic flux density and the inductanceshow that the inductance formulas of the helical toroidal coil couldbe used as the basis for coil optimal design. Optimization targetfunctions such as maximization of the ratio of stored magneticenergy with respect to the volume of the toroid or the conductor’smass, the elimination or the balance of stress in certain coordinatedirections, 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 ECENT research work on superconducting magnetic en-ergy storage (SMES) systems, nuclear fusion reactors,

and plasma reactors such as the Tokamak has suggested theuse of advanced coil with a helical toroidal structure [1]–[4].The main reason for this suggestion is the ability to implementspecial target functions for this coil in comparison with otherstructures such as the toroidal, the solenoid, and the sphericalcoils [5], [6]. The structure of this coil is shown in Fig. 1. Inthis 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 ina layer (υ) are called aspect ratio, poloidal turns (or the pitchnumber), and helical windings, respectively. For example, thecoil in Fig. 1 is composed of five helical windings (υ = 5) withnine poloidal turns (N = 9). The inductance formulas show

Manuscript received January 29, 2009; revised May 7, 2009. First publishedJuly 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: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2009.2023548

Fig. 1. Structure of a monolayer helical toroidal coil with five rings ofnine turns.

that parameters a, R, and N of the helical toroidal coil can beused as design parameters to satisfy special target functions.With respect to the fact that each ring of the coil generatesboth toroidal and poloidal magnetic fields simultaneously, thecoil can be regarded as a combination of coils with toroidaland solenoid fields. Furthermore, the coil can be designed ina way to eliminate the magnetic force component in both themajor and minor radius directions. These are called force- andstress-balanced coils, respectively. In addition, the coils thatutilize the virial theorem to balance these two force componentsare called virial-limited coils [7]–[9]. In some applications,helical toroidal coils are used in a double-layer manner withtwo different winding directions (respectively with different orthe same current directions in each layer) to reduce the poloidalleakage flux being compared to the toroidal leakage flux orvice versa. In this paper, the investigation is focused on the one-layer helical toroidal coil.

In general, any simulation program that simultaneouslysolves equations, the particle position, and its velocity can becalled a particle-in-cell (PIC) simulation. The name PIC comesfrom the way of assigning macroquantities (like density, currentdensity, and so on) to the simulation particles. Inside the plasmacommunity, PIC codes are usually associated with solvingthe equation of motion of particles with the Newton–Lorenz’sforce. PIC codes are usually classified depending on the di-mensionality of the code and based on the set of Maxwell’sequations used. The codes solving an entire set of Maxwell’sequations are called electromagnetic codes, while electrostaticones solve just the Poisson equation. Some advanced codes areable to switch between different dimensional and coordinatesystems and use electrostatic or electromagnetic models. PIC

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1594 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009

Fig. 2. Semitoroidal coordinate system.

simulation starts with an initialization and ends with the outputof results. This part is similar to the input/output routinesof any other numerical tool. Usually, the numerical methodsbased on the PIC simulation are obtained from the solutionof partial differential–algebraic equations, for example, by thefourth-order Runge–Kutta method. Considering the numberof particles which are on the order of 1010, the simulationsbased on PIC methods take a long time to solve the aforemen-tioned equations. Usually, in order to resolve the time issuein the PIC methods, special computers may be employed. Onthe other hand, this paper used Biot–Savart equation and themathematical equations of the current path in the conductorof the helical toroidal coil in order to obtain the magnetic fluxdensity components. The numerical integrals resulting from theBiot–Savart equations are solved using the extended three-pointGaussian algorithms. This method has the least error amongall numerical integration methods. In addition, the method usedby the authors, contrary to the PIC method, does not need anyspecial computers to solve the equations.

In Section II of this paper, the semitoroidal coordinate sys-tem, the longitudinal components of a ring element, and thering radius of curvature are briefly discussed. In Sections IIIand IV, the formulas of the magnetic flux density compo-nents are presented and the behaviors of these components aresimulated, respectively. In these sections, the variations of thering radius of curvature with respect to the toroidal angle forthe geometric parameters of the coil are also investigated. InSections V and VI, the formulas for the self-inductance andmutual inductance of the helical toroidal coils using the directand indirect methods are developed. Finally, in Section VII, theexperimental inductance measurements are compared with thedirect and indirect method simulation.

II. LONGITUDINAL COMPONENTS AND RADIUS OF

CURVATURE OF A RING

The semitoroidal coordinate system is orthogonal, 3-D, androtational (see Fig. 2). Longitudinal components of the ringelement in this coordinate system can be defined by (1). Table Ipresents the dot products of the unit vectors for the Cartesianand the semitoroidal coordinates using the projection of thesevectors on the planes ϕ = const and z = 0

dl = �aθadθ + �aρdρ + �aϕ(R + a cos θ)dϕ

= �axdlx + �aydly + �azdlz. (1)

TABLE IDOT 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, itsdifferential dρ = 0 can be replaced in (2). The relation betweenθ and ϕ for a ring of N turns can be expressed as (3) in whichθ0 is the poloidal angle of the ring at plane ϕ = 0

dlx = dl · �ax

= −a sin θ cos ϕdθ − sinϕ(R + a cos θ)dϕ

+ cos θ cos ϕdρ

(2)

dly = dl · �ay

= −a sin θ sinϕdθ + cos ϕ(R + a cos θ)dϕ

+ sinϕ cos θdρ

dlz = dl · �az = a cos θdθ + sin θdρ

θ = Nϕ + θ0. (3)

If the positional vector of each longitudinal element of thering with respect to the origin of Cartesian coordinate systemis defined as (4), then the tangent vector to the longitudinalelement �Tυ(ϕ) and the radius of curvature of the ring can bedefined as (5) and (6), respectively. In these equations, f ′

i and f ′′i

are the first and second derivatives of function fi with respectto ϕ, respectively

�Pυ(ϕ) = f1�ax + f2�ay + f3�az (4)

�Tυ(ϕ) = �P ′υ(ϕ)

/ ∣∣∣�P ′υ(ϕ)

∣∣∣= (f ′

1�ax + f ′2�ay + f ′

3�az)/√

f4 (5)

Pc(ϕ) =∣∣∣∣∣∣�P ′

υ(ϕ)∣∣∣ /

�T ′υ(ϕ)

∣∣∣= f2

4

/ [(f ′′

1f4 − f ′1f5)

2 + (f ′′2f4 − f ′

2f5)2

+ (f ′′2f4 − f ′

2f5)2]0.5

= a · g(A,N) (6)

where

f1 = (R + a cos(Nϕ + θ0)) cos ϕ

f2 = (R + a cos(Nϕ + θ0)) sin ϕ

f3 = a sin(Nϕ + θ0)

f4 = f ′21 + f ′2

2 + f ′23

f5 = f ′1f

′′1 + f ′

2f′′2 + f ′

3f′′3 .


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 normalvector �aϕ is perpendicular is called the toroidal plane, and eachinfinite surface (with conical shape) to which the normal vector�aθ is perpendicular is called poloidal cone. Therefore, eachtoroidal plane could be represented with the equation ST : ϕ′ =const&π + const, and each poloidal cone could be representedwith the equation SP : θ′ = const&π + const. It is noted thatthe apex of the cone for each poloidal cone is placed on thez-axis and that the surfaces of all poloidal cones includethe geometric loci of origins for the semitoroidal coordinatesystems. If the Cartesian coordinates of any point β on eachtoroidal or poloidal cone are xβ , yβ , and zβ , and the Cartesiancoordinates of any point α on the ring with characteristicsof θo, a, R, and N are presented by xα, yα, and zα, thenreplacing (2), (3), and (7) in the Biot–Savart equation resultsin the Cartesian form of the magnetic flux density componentsof (8). In addition, it is possible to change these componentsinto toroidal component of the magnetic flux density with (10)by defining point β on each toroidal plane using (9)

xα = (R + a cos θ) cos ϕ

yα = (R + a cos θ) sin ϕ

zα = a sin θ (7)

Bx = (μ0I/4π)

2π∫

0

(f7/f6)dϕ

By = (μ0I/4π)

2π∫

0

(f8/f6)dϕ

Bz = (μ0I/4π)

2π∫

0

(f9/f6)dϕ (8)

where

f6 =[(xβ (R + a cos(Nϕ + θ0)) cos ϕ)2

+ (yβ (R + a cos(Nϕ + θ0)) sin ϕ)2

+ (zβ − a sin(Nϕ + θ0))2]3/2

f7 = [(zβ − a sin(Nϕ+θ0))· ((R+a cos(Nϕ+θ0)) cos ϕ

− aN sin(Nϕ+θ0) sin ϕ)− (yβ−(R+a cos(nϕ+θ0)) sin ϕ)· (aN cos(Nϕ+θ0))]

f8 = [(xβ − (R + a cos(Nϕ + θ0)) cos ϕ)· (aN cos(Nϕ + θ0)) + (zβ − a sin(Nϕ + θ0))· ((R + a cos(Nϕ + θ0)) sinϕ

+ aN sin(Nϕ + θ0) cos ϕ)]f9 = [(yβ − (R + a cos(Nϕ + θ0)) sin ϕ)

· (sin ϕ (R + a cos(Nϕ + θ0))+ aN sin(Nϕ + θ0) cos ϕ)

− (xβ − (R + a cos(Nϕ + θ0)) cos ϕ)· (cos ϕ (R + a cos(Nϕ + θ0))

− aN sin(Nϕ + θ0) sin ϕ)] .

Moreover, the poloidal and the radial components of themagnetic 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 poloidalcone is defined as 0 ≤ ρ ≤ −R/ cos θ′ for π/2 ≤ θ′ ≤ 3π/2and 0 ≤ ρ ≤ ∞ for −π/2 ≤ θ′ ≤ π/2

xβ = γ cos ξ cos ϕ′, 0 ≤ γ ≤ ∞yβ = γ cos ξ sin ϕ′, 0 ≤ ξ ≤ 2π

zβ = γ sin ξ ϕ′ = const&π + const (9)

Bϕ = − Bx sin ϕ′ + By cos ϕ′ (10)

xβ = (R + ρ cos θ′) cos ϕ′

yβ = (R + ρ cos θ′) sin ϕ′

zβ = ρ sin θ′ (11)

Bθ = − Bx sin θ′ cos ϕ′ − By sin ϕ′ sin θ′ + Bz cos θ′ (12)

Bρ = Bx cos θ′ cos ϕ′ + By sin ϕ′ cos θ′ + Bz sin θ′. (13)

IV. BEHAVIORAL STUDY OF THE MAGNETIC

FLUX DENSITY COMPONENTS

In this section, the behavior of the magnetic flux densitycomponents 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 Gaussianalgorithms [10]. This method has the least error among allnumerical integration methods. In the sketching of the magneticflux 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 correspondingrange is divided into n = 300 equal segments in order toreach the integration error of the derivative order 2n or 600of the function under integration. It is obvious that, if theintegration range is divided into more parts, the integrationerror will decrease further, but this comes at the cost of longercomputational time. In Figs. 3–5, the contours for magnetic fluxdensity, the toroidal, the poloidal, and the radial components areshown. It can be inferred that, as we approach the conductor’sgeometric loci, the amplitude of the magnetic flux densitybecomes stronger. In these figures, the boundary of surface S0,from which the magnetic flux density components only enter orexit, 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 fluxdensity components for υ ring of N turns (total of Nυ turns) incomparison with a single Nυ turn ring. The comparison of theresults for these two windings indicates that the magnetic flux



Fig. 3. Contours of toroidal component for magnetic flux density.



density components are different for these two case studies, andthe reason for this difference should be sought in the windingtypes of these coils. Moreover, Fig. 3(a), (b), (d), and (e) showsthat, by rotation of the ring, Bϕ maximum geometric loci isaltering at θ = Nϕ′ + θ0. From Fig. 3(a), (c), and (g), it can beobserved that the amplitude Bϕ is increased with increasing υand N (increasing N is more efficient than υ) and that the leak-age flux percentage of Bϕ from the coil’s section will decrease.According to Fig. 3(a), (b), (d), and (e), the surface area S0 ondifferent 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 parametersa, R, and N . Symmetry of the right half of Bϕ with the lefthalf could be seen for the even values of N , and the asymmetryof the right half of Bϕ with the left half is apparent for theodd or fractional values of N [see Fig. 3(a), (b), (d), and (e)]. InFig. 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 ora toroidal coil, and the amplitude of Bϕ on each toroidal planeis decreased and increased, respectively. In Fig. 4, contoursare 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, themaximum geometric loci is altering at ϕ = −θ0/N . In addition,the maximum of Bθ is independent of θ0. From Fig. 4(a) and(c)–(f), it is inferred that the surface area S0 is continuouslychanging and an increase in N increases the area of this surface.In other words, a percentage of Bθ is passing through thesurface of the helical toroidal coil. In Fig. 4, the number of therose leaves of the poloidal magnetic field is Nυ, and there arealways Nυ global maximums and minimums for Bθ on eachpoloidal cone. Fig. 4(a) and (c)–(f) shows that the amplitudeof Bθ is increased and decreased with increasing υ and N ,respectively. In other words, increasing υ increases the densityof Bθ amplitude maximum in the range a ≤ ρ ≤ R. In Fig. 4(c)and (e), it is shown that, in the marginal conditions of υ � 1 orN � 1, the coil behavior approaches the pure solenoid and thepure toroidal coils, respectively. Furthermore, the amplitude ofBθ is increased and decreased, respectively. It can be inferredfrom Fig. 4(g) that, in the marginal conditions of N = 0, thebehavior of Bθ for the helical toroidal coil is similar to one ring,and the amplitude of Bθ is positive inside the ring and negativeoutside it.

In Fig. 5, the contours of the magnetic flux density of theradial component are shown. Fig. 5(a) and (b) shows thatthe Bρ maximum on the different toroidal planes with thesame values of a, R, and N is changing and occurs at θ =Nϕ′ + θ0. Fig. 5(a), (e), and (f) shows that the amplitude ofBρ is increased with increasing υ and N (increasing υ is moreefficient than N ), and there is always Nυ global maximumsand minimums for Bρ on each toroidal plane. It can be inferredfrom this figure that, due to its radial form, this componenthas no magnetic linkage with the ring. In Fig. 5(d) and (f),the amplitude of Bρ in the marginal conditions of N = 0 andN � 1 on each toroidal plane is observed. In the same figures,the symmetry of this component with respect to poloidal conez = 0 in the marginal condition of N = 0 is noticed.

In Fig. 6, the variations of the ratio radius of curvature ofthe ring to the coil minor radius with respect to the toroidal

angle for parameters A and N are sketched. It can be inferredfrom Fig. 6 that the ratio of the radius of curvature to the minorradius increases with increasing the value of the aspect ratioin constant poloidal turns. In addition, the minimum of thisvalue occurs at the aspect ratio of one for poloidal turn numbersof 0.4.

V. DIRECT CALCULATIONS OF SELF-INDUCTANCE

AND MUTUAL INDUCTANCE

Equation (14) is introduced for the calculation of the mutualinductance between two rings with characteristics θ0i, ai, Ri,Ni and θ0j , aj , Rj , Nj based on the Neumann’s equation

Mij = MNij = (μ0/4π)

2π∫0

2π∫0

(f11/f10)dϕidϕj , i �= j

(14)

where

f10 =[(aj sin(Njϕj + θ0j) − ai sin(Niϕi + θ0i))

2

+ ((Rj + aj cos(Njϕj + θ0j)) sinϕj

− (Ri + ai cos(Niϕi + θ0i)) sinϕi)2

+ ((Rj + aj cos(Njϕj + θ0j)) cos ϕj

− (Ri + ai cos (Niϕi + θ0i)) cos ϕi)2]0.5

f11 = [(aiNi sin(Niϕi + θ0 i) cos ϕi

+ sin ϕi (Ri + ai cos(Niϕi + θ0i)))

· (ajNj sin(Njϕj + θ0j) cos ϕj

+ sin ϕj (Rj + aj cos(Njϕj + θ0j)))

+ (−aiNi sin(Niϕi + θ0i) sin ϕi

+ cos ϕi (Ri + ai cos(Niϕi + θ0i)))

· (−ajNj sin(Njϕj + θ0j) sin ϕj

+ cos ϕj (Rj + aj cos(Njϕj + θ0j)))

+ aiNi cos(Niϕi + θ0i) · ajNj cos(Njϕj + θ0j)] .

According to classical electrodynamics, if the radius of thecurvature is larger than the dimensions of the transverse sectionof the conductor (i.e., the diameter of the conductor’s crosssection is smaller than 0.4 of the minor radius of the helicaltoroidal coil), the equation of the mutual inductance betweenthe two rings can be used to calculate the self-inductance ofthe ring. In this condition, the minimum distance between thetwo corresponding points in each ring is assumed to be equalto the geometrical mean radius of the conductor’s cross section.The geometrical mean radius of the conductor’s cross section is,in fact, the efficient radius of the conductor that the magnetic



Fig. 4. Contours of the poloidal component of the magnetic flux density.

Fig. 4. (Continued.) Contours of the poloidal component of the magnetic fluxdensity.

field is unable to influx. In nonsuperconductivity conditions,the geometric mean radius of the conductor’s cross sectionwith radius r is defined as rm = re−0.25. In superconductivityconditions, the geometric mean radius is assumed to be rm = r,because the magnetic field cannot influx the conductor’s crosssection. Therefore, (15) with its assumptions can be used tocalculate the self-inductance of the ith ring with characteristicsθ0i, a, R, and N

Lii =Mii, i = 1, . . . , υ

Ri =Rj = R, ai = aj = a, Ni = Nj = N

θ0i = θ0i [rad], θ0j = θ0i + (rm/a) [rad]. (15)



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 helicaltoroidal coil are proposed by the indirect method or the mag-netic flux density components. The self-inductance of one ringor the mutual inductance between two rings is proportional tothe surface integral of the magnetic flux density componentsof one or two rings. The radial component of the magneticflux density is perpendicular to the ring geometric loci anddoes not have magnetic link to one ring or two rings. Conse-quently, this does not affect the calculation of the mentionedinductances. In other words, the only effective componentsin the calculation of the self-inductance and mutual induc-

tance are the magnetic flux density of toroidal and poloidalcomponents.

A. Calculation of Mutual Inductance via Indirect Method

In this section, the mutual inductances of the ith and thejth rings with characteristics of θ0i, ai, Ri, Ni and θ0j , aj ,Rj , Nj , with incorporation of some engineering assumptions,are calculated. The pure flux of field without divergence oneach closed surface is zero. The magnetic field entering intoeach toroidal or poloidal cone could be calculated as of surfaceintegration of the absolute values of Bθ and Bϕ on these planes.Consequently, it is possible to estimate the values of these



Fig. 6. Variations of radius of curvature of the ring with respect to thetoroidal angle.

integrals with the linking flux of the ith ring using the followingequations:

ψT (ϕ′) = 0.5∮

ST

|Bϕ|dsϕ = 0.5

∞∫0

2π∫0

|Bϕ|ρdθ′dρ (16)

ψP (θ′) = 0.5∮

SP

|Bθ|ds

= 0.5

⎛⎜⎝

2π∫

0

−Ri/ cos θ′∫

0

|Bθ|(Ri + ρ cos θ′)dρdϕ′

+

2π∫

0

∞∫

0

|Bθ|(Ri + ρ cos θ′)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 resultof ψT and ψP generated by the ith ring linking with the jthring. Therefore, it is assumed that the toroidal and the poloidal

mutual inductances are proportional to the minimum of ψT andψP generated by the ith ring linking with the cross section ofSTM and SPM. These surfaces could be stated as (18) and (19).It is important to mention that the linking magnetic fluxes withthe cross section of STM and SPM link Nj and one ring and aredefined as (20) and (21), respectively. Based on the definitionof STM, it is necessary to apply the definition of point β basedon (11) for replacement of Bϕ in (16) and (20)

STm : 0 ≤ ρ ≤ aj , 0 ≤ θ′ ≤ 2π, ϕ′ = const, π + const

(18)

SPm :

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

⎛⎝ aj ≤ ρ ≤ −Rj/ cos θ′

π/2 ≤ (θ′ = const, π + const) ≤ 3π/20 ≤ ϕ′ ≤ 2π

⎞⎠

or⎛⎝ aj ≤ ρ ≤ ∞

−π/2 ≤ (θ′ = const, π + const) ≤ π/20 ≤ ϕ′ ≤ 2π

⎞⎠

(19)

ψTm(ϕ′)

= Nj

∮

STm

|Bϕ|ds = Nj

aj∫

0

2π∫

0

|Bϕ|ρdθ′dρ (20)

ψPm(θ′)

=∮

SPm

|Bθ|ds

=

2π∫

0

−Rj/ cos θ′∫aj

|Bθ|(Rj + ρ cos θ′)dρdϕ′. (21)

Toroidal angle ϕ′ = ϕop and poloidal angle θ′ = θop couldbe achieved by the derivation of (20) and (21), which minimizesψTM and ψPM. Then, the toroidal and poloidal mutual induc-tances could be defined as (22) and (23). As it was mentionedbefore, ψT and ψP must be particularly independent of thevariations of ϕ′ and θ′. In other words, the variations of ψT andψP must 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. Byimplementing these assumptions, the toroidal and the poloidalmutual impedances can be defined according to (24) and (25)

MijT =ψTm(ϕ′ = ϕop)/I (22)MijP =ψPm(θ′ = θop)/I (23)

MT =ψTm(ϕ′ = ϕop = 0)/I (24)MP =ψPm(θ′ = θop = π)/I. (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 Ni is calculated with the followingassumptions.

1) ψT and ψP are averaged.2) Leakage fluxes are negligible.3) ψT and ψP are independent of the variations of ϕ′ and θ′,

or assuming ϕ′ = ϕmean = 0 and θ′op = θmean = π, thecalculations are simplified.



Based on the first assumption, the toroidal and the poloidalfluxes linking with the ring are defined as the average of theentering fluxes into all toroidal and poloidal cones according to(26) and (27), respectively. In other words, the toroidal angle ofϕ′ = ϕmean and the poloidal angle of θ′ = θmean are obtainedfrom the average values of the toroidal and the poloidal fluxesof all toroidal and poloidal cones. Incorporating the secondassumption results in (28) and (29). Implementing the thirdassumption gives (30) and (31).

Comparing (24) and (25) with (30) and (31) shows that, if thetwo rings i and j are in the same layer and Ni = Nj = N , thenMP = LP and MT = LT . Based on the fact that the currentis the same for LiiP and LiiT , it can be inferred that theself-inductance of the ring i, Lii, could be obtained fromthe summation of LiiP and LiiT . In addition, with respect tothe fact that the currents for MijP and MijT are equal, it canbe 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 (26)LiiP (θ′ = θmean) = ψP (θ′ = θmean)/I (27)LiiTm =NiψTm(ϕ′ = ϕmean)/INj (28)LiiPm =ψPm(θ′ = θmean)/I (29)

LT =NiψTm(ϕ′ = ϕmean = 0)/INj (30)LP =ψPm(θ′ = θmean = π)/I (31)Mij =MijT + MijP , i �= j (32)Lii =LiiT + LiiP , i = 1, . . . , υ. (33)

Electrical circuit analysis shows that the inductance matrixof a monolayer helical toroidal coil could be obtained usingfilament method as in (34), and the inductance of one layer,assuming Lii = Mii, could be obtained by multiplying υ2 byLii or Mij

Lcoil =

⎡⎢⎣

L11 M12 · · · M1υ

M21 L22 · · · M2υ

· · · · · · · · · · · ·Mυ1 Mυ2 · · · Lυυ

⎤⎥⎦

υυ

(34)

Lcoil = υ2(Lii = Mij). (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 inductionmeasurements are compared with their corresponding numer-ical values. The empirical results for the mutual inductancebetween 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 geometricalcalculations could be obtained as (36) to adapt this problemwith equations mentioned in Section V

N = N1 = N2 = 0 [turns]R = R1 = R2 = 22.5 [cm]a = a1 = a2 =

√125/2 [cm]

θ01 = − tan−1(2) [rad]θ02 = π − tan−1(2) [rad]. (36)

Fig. 7. Comparison between numerical and empirical results [11].

Fig. 8. (a) Simulation time versus n. (b) Error percentage versus n betweennumerical and empirical results [11].

The convergence diagram, the simulation time, and the errorpercentage versus n for numerical and empirical results areshown in Figs. 7 and 8, respectively. In Fig. 7, it is noticed thatthe optimal value of n is 5. In Fig. 8, it is shown that increasingn to reduce error increases the simulation time in a para-bolic form.

The empirical (37) has been used to calculate self-inductanceof a flat ring of radius τ and the radius of the conductor’s crosssection of r [12], [13]. For example, the self-inductance of aflat ring with a radius of 200 cm and a radius of the conductor’scross section of 1 cm is 14.1441 μH. The parameters of thehelical toroidal coil with geometrical calculations could be ob-tained as (38) to adapt this problem with equations mentioned inSection V. The convergence diagram, simulation time, and theerror percentage versus n for numerical and empirical resultsare shown in Figs. 9 and 10, respectively. It can be seen fromFig. 10 that the optimal value of n for the target function errorof less than 0.0015 is 500

L = μ0τ [Ln(8τ/r) − 1.75] [H] (37)R + a = 200 [cm] r = 1 [cm]

θ01 = 0 [rad] θ02 = re−0.25/a [rad]. (38)



Fig. 9. Comparison between numerical and empirical results [12], [13].

Fig. 10. (a) Simulation time versus n. (b) Error percentage versus n betweennumerical and empirical results [12], [13].

TABLE IICOMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS OF

INDUCTANCE IN HELICAL TOROIDAL COIL

The experimental result of helical toroidal coil inductancewith geometric characteristics shown in Table II is 17 mH[6]. It can be observed that the experimental results are ingood agreement with those obtained from (35), with an errorof 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 errorpercentage versus n for numerical and experimental results areshown in Figs. 11 and 12, respectively. It can be inferred fromFigs. 7, 9, and 11 that the optimal value of n in the indirect

Fig. 11. Comparison between numerical and experimental results [6].

Fig. 12. (a) Simulation time versus n. (b) Error percentage versus n betweennumerical and experimental results [6].

method is nearly half of that in the direct method. As a result,the simulation time in the indirect method is nearly half ofthat in the direct method. Therefore, one can conclude that theequations presented for the inductance calculations, acceptingcalculation error of 2.3% (acceptable engineering error), withthe assumptions in the indirect method being compared to thatin the direct method, have higher reliability with less simula-tion time.

VIII. CONCLUSION

Helical toroidal coils are superior to other coils and areextensively used in the SMES systems, nuclear fusion reactors,and plasma research work. Considering the complexity of thecoils and the fact that not much investigation has been carriedout in this field, this area of research is still open for much moreacademic work to come. On the other hand, the calculationof inductance for these types of coils can be an index todetermine the behavior of the transient state, determination of



the electrical equivalent circuit, and estimation of values forelectrical elements of the coil equivalent circuit [14]–[24].

In this paper, the inductance of the helical toroidal coil iscalculated in the direct and indirect methods, and the inductancecharacteristics and the magnetic flux density are simulated inMATLAB. Comparison of the experimental, empirical, andnumerical results shows that the equations for inductance cal-culations have great reliability and that dividing the inductancefor one ring into two toroidal and poloidal components withincorporation of some engineering assumptions simplifies theequations and decreases the computational time without the lossof accuracy.

REFERENCES

[1] U. Zhu, Z. Wang, X. Liu, G. Zhang, and X. H. Jiang, “Transient behaviorresearch on the combined equipment of SMES-SFCL,” IEEE Trans. Appl.Supercond., vol. 14, no. 2, pp. 778–781, Jun. 2004.

[2] M. Fabbri, D. Ajiki, F. Negrini, R. Shimada, H. I. Tsutsui, andF. Venturi, “Tilted toroidal coils for superconducting magnetic energystorage systems,” IEEE Trans. Magn., vol. 39, no. 6, pp. 3546–3550,Nov. 2003.

[3] X. Jiang, X. Liu, X. Zhu, Y. He, Z. Cheng, X. Ren, Z. Chen, L. Gou,and X. Huang, “A 0.3 MJ SMES magnet of a voltage sag compensationsystem,” IEEE Trans. Appl. Supercond., vol. 14, no. 2, pp. 717–720,Jun. 2004.

[4] L. Chen, Y. Liu, A. B. Arsoy, P. F. Ribeiro, M. Steurer, and M. R. Iravani,“Detailed modeling of superconducting magnetic energy storage (SMES)system,” IEEE Trans. Power Del., vol. 21, no. 2, pp. 699–710, Apr. 2006.

[5] T. Hamajima, T. Yagai, N. Harada, M. Tsuda, H. Hayashi, and T. Ezaki,“Scaling law of fringe fields as functions of stored energy and maximummagnetic field for SMES configurations,” IEEE Trans. Appl. Supercond.,vol. 14, no. 2, pp. 705–708, Jun. 2004.

[6] S. Nomura, N. Watanabe, C. Suzuki, H. Ajikawa, M. Uyama, S. Kajita,Y. Ohata, H. Tsutsui, and S. Tsuji-Iio, “Advanced configuration of su-perconducting magnetic energy storage,” Energy, vol. 30, no. 11/12,pp. 2115–2127, Aug./Sep. 2005.

[7] H. Tsutsui, S. Kajita, Y. Ohata, S. Nomura, S. Tsuji-Iio, and R. Shimada,“FEM analysis of stress distribution in force balanced coils,” IEEE Trans.Appl. Supercond., vol. 14, no. 2, pp. 750–753, Jun. 2004.

[8] S. Nomura, R. Shimada, C. Suzuki, S. Tsuji-Iio, H. Tsutsui, andN. Watanabe, “Variations of force-balanced coils for SMES,” IEEE Trans.Appl. Supercond., vol. 12, no. 1, pp. 792–795, Mar. 2002.

[9] H. Tsutsui, S. Nomura, and R. Shimada, “Optimization of SMES coilby using virial theorem,” IEEE Trans. Appl. Supercond., vol. 12, no. 1,pp. 800–803, Mar. 2002.

[10] R. H. Pennington, Introductory Computer Methods and NumericalAnalysis, 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 forinductance calculation,” Can. J. Phys., vol. 75, no. 6, pp. 357–362, 1997.

[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 identificationof parameters for lumped SMES coil models using the least squaresalgorithm,” (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 SMEScoil 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 inhelical 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 ofmagnetic 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 ofhelical toroidal coil model using Kalman filter,” (in Persian), presented atthe 23rd Int. Power System Conf., Tehran, Iran, Nov. 2008, Paper 98-F-PQA-0202.

[20] M. R. Alizadeh Pahlavani and A. Shoulaie, “Elimination of compressivestress 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 identificationof 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 helicaltoroidal coil inductance for the superconductor magnetic energy storagesystems,” (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 ofhelical 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 IranianConf. Elect. Eng., Tehran, Iran, May 2009.

Mohammad Reza Alizadeh Pahlavani was bornin 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 electricalengineering.

His current research interests include electro-magnetic systems, power electronics, and electricalmachines.

Abbas Shoulaie was born in Iran in 1949. Hereceived the B.Sc. degree from Iran University ofScience and Technology (IUST), Tehran, Iran, in1973, and the M.Sc. and Ph.D. degrees in electricalengineering from U.S.T.L, Montpellier, France, in1981 and 1984, respectively.

He is currently a Professor with the Departmentof Electrical Engineering, IUST. He is the author ofmore than 100 journals and conference papers in thefield of power electronics, electromagnetic systems,electrical machines, liner machines, and HVDC.


a novel approach for calculations of helical toroidal coil inductance usable … · 2019-03-05 ·...

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