Eddy currents in a gradient coil,

modeled by rings and patches

J.M.B. Kroot*, S.J.L van Eijndhoven, and A.A.F. van de VenDept. of Mathematics and Computer Science

Eindhoven University of TechnologyDen Dolech 2, 5600 MB, Eindhoven, Netherlands

E-mail: J.M.B.Kroot@tue.nl

Abstract: The gradient coils of an MRI-scanner can be modeled by a surfacecurrent on a cylinder. A good approximation of the surface current in the z-coil isobtained from a set of rings and patches on the cylinder. In this paper, an integralequation is derived for the current distribution. To solve this integral equation,the Galerkin method is applied, using global expansion functions. We show thatLegendre polynomials are an appropriate choice for the expansion functions. Theyprovide fast convergence. So, only a restricted number of Legendre polynomialsis needed. Moreover, an analytical solution is derived, which results in efficientsimulations.

1 Introduction

Magnetic Resonance Imaging (MRI) is an imaging technique that plays an importantrole in the medical community. It provides images of cross-sections of a body; seee.g. [1]. The selection of a slice is realized by the gradient coils. A gradientcoil consists of copper strips wrapped around a cylinder. Due to mutual magneticcoupling, eddy currents arise which affect the quality of the image. For the designof gradient coils, finite element packages are used. However, these packages cannotsufficiently describe the qualitative behavior of the currents, relating the geometryto typical phenomena, such as edge effects, mutual coupling and heat dissipation.More analytical techniques are required.

In this paper, we focus on the z-coil, which has the function to create a gradientin the magnetic field in the axial direction of the scanner. In [2], a parallel setof plane conducting strips is used to model the z-coil. In [3] and [4], the z-coil ismodeled as a set of circular loops of strips. In both models, the current flows inthe length direction of the strips only. In an MRI-scanner, the z-coil is embeddedin a system of more coils and magnets, which induce additional eddy currents, thatflow also in the width direction of the strips. We model this dependence by placingrectangular patches between a set of rings.

The overall aim is to approximate the electric current distribution in a set ofrings and patches, using global expansion functions. To find appropriate expansionfunctions, we need to investigate the behavior of the kernel function in the integralequation that follows from the model.

©2005 ACES

2 Model definition

We consider a set of Nr coaxial rings, and Np rectangular patches. All these conduc-tors are on the same imaginary cylinder Sc = {(R, φ, z) | φ ∈ [−π, π],−∞ < z < ∞}.The geometry is depicted in the figure below. Each ring or patch is of uniform widthand has thickness h. A source current is applied to the rings, which is time harmonic

j r

z

x

y

j 1( i , 1 )

j 0( i , M )

j 1( i , M )j 0

( i , 2 )

j 1( i , 2 )

j 1( i , j )j 0

( i , j )

j 0( i , 1 )

z 1z 0 ( 1 ) ( 1 ) z 1z 0 ( 2 ) ( 2 ) z 1z 0 ( i ) ( i ) z 1z 0 ( N ) ( N )

with frequency ω. For frequencies ω < 103 rad/s, a quasi-static approach is allowedand the penetration depth δ =

√2/µσω is much larger than h. Consequently, we

can replace the current density J (in A/m2) by the current per unit of length j (inA/m), according to j = hJ. We write j = jφ(φ, z)eφ + jz(φ, z)ez.

The strips occupy the surface S∪ = Sr + Sp in space, described in cylindricalcoordinates by (see also Figure):

Sr =

Nr∑n=1

S(r)n , S

(r)n = {(R, φ, z) | φ ∈ [−π, π], z ∈ [z

(n)0 , z

(n)1 ]}, (1)

Sp =

Ni∑n=1

S(p)n , S

(p)n = {(R, φ, z) | φ ∈

M∑m=1

[φ(n,m)0 , φ

(n,m)1 ], z ∈ [z

(n)0 , z

(n)1 ]}. (2)

So, S∪ ⊂ Sc. The total current consists of the prescribed source current js and theinduced eddy current je, such that j = js + je. Moreover, the normal component ofthe current at the edges has to be zero, i.e. jφ(φe, z) = jz(φ, ze) = 0, where φe andze are the values of φ and z at the edges, respectively. Finally, j is free of divergence,i.e. ∇ · j = 0.

We use Maxwell’s theory specified on the geometry of our model and express allfields in terms of the vector potential A. For the dimension analysis, the distancesare scaled by the radius of the cylinder, and the current is scaled by the averagecurrent through all rings. The dimensionless φ- and z-component of A can bewritten in an integral form, according to

Aφ(1, φ, z) =1

4π

∫S∪

cos(φ − θ)jφ(θ, ζ)√(z − ζ)2 + 4 sin2(φ−θ

2 )dθ dζ, (3)

Az(1, φ, z) =1

4π

∫S∪

jφ(θ, ζ)√(z − ζ)2 + 4 sin2(φ−θ

2 )dθ dζ. (4)

Introducing the characteristic parameter κ = hσµωR, we obtain from Ohm’s law

jφ(φ, z) − iκAφ(1, φ, z) = jsφ(φ, z), (5)

jz(φ, z) − iκAz(1, φ, z) = jsz(φ, z). (6)

3 Solution procedure

In this section, we explain how we solve jφ(φ, z) from (3) and (5). Note that jz(φ, z)then directly follows from the conditions that the current is free of divergence andhas a zero normal component at the edges of the strips. The Galerkin method isapplied, for which we have to choose appropriate basis functions. In operator form,(5) is written as jφ − iκKjφ = js

φ. The kernel function, represented by Kφ(φ, z), canbe expressed by a Fourier cosine series (see e.g. [5])

Kφ(φ, z) =cos(φ)

4π

√z2 + 4 sin2(φ

2 )=

1

4π2[Q 1

2

(χ) +

∞∑k=1

cos(kφ)(Qk− 3

2

(χ) + Qk+ 1

2

(χ))].

(7)Here, Qk−1/2 is the Legendre function of the second kind of half-integer degree (forproperties, see e.g. [6]), and χ = (2 + z2)/2. The behavior of each term in the seriesof (7) around the point z = 0 is given by

Qk− 1

2

(χ) ≈ 1

2(−2γ + ln 4 − 2Γ(0)(

2k + 1

2) − 2 ln |z|) + O(z), (8)

for k ≥ 0, establishing that the singularity is logarithmic in the z-direction. Here,γ is Euler’s constant and Γ(0) is the polygamma function.

The basis functions we use, are global, i.e. they are valid on the completerings/patches. We consider the current distribution on each ring and each patchas a Fourier series in the φ-direction, and use an expansion of scaled and shiftedLegendre polynomials in the z-direction. We write

jφ(φ, z) =∞∑

k=0

∞∑n=0

(αkn cos kφ + βkn sin kφ)Pn

(z − c(s)

d(s)

), (9)

for every strip s, where c(s) is the center of the strip and d(s) is half the width of thestrip. The currents are projected on the basis functions cos(kφ) and sin(kφ), andwe exploit the orthogonality of these functions. Consequently, every inner productin the Galerkin method, of two basis functions from the two strips s1 and s2, isreduced to a double integral of the form

d(s1)

d(s2)

∫ 1

−1

∫ 1

−1f(χ, k)Pn(ζ)Pn′(z) dζ dz, (10)

where f(χ, k) is the k-th term in (7), with χ = 1+(d(s1)z−d(s2)ζ + c(s1) − c(s2))2/2.The integral in (10) has a logarithmic singular integrand when the two rings/patchescoincide. In that case, we split off the logarithmic part, and use∫ 1

−1

∫ 1

−1Pk(z)Pk′(z) log |z − ζ| dζ dz (11)

=

8

(k + k′)(k + k′ + 2)[(k − k′)2 − 1], if k + k′ > 0 even ,

0, if k + k′ odd ,

4 log 2 − 6, if k = k′ = 0 .

The remaining part is regular and is solved numerically.

4 Results

Considered are four rings with one patch in the middle. The length of the patch isa quarter of the circumference of the cylinder. The first figure shows the amplitudeof the scaled eddy current je

φ(0, z), when an in-phase source current through thefour rings is applied. Edge-effects occur, which become stronger as the frequency isincreased. In the second figure, stream lines of the current in the patch are shown.We observe two eddies in the patch.

0 0.1 0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

0.12

z/R

|j φe (0,z

)|

ω = 100ω = 50ω = 25ω = 12.5

0.18 0.2 0.22 0.24 0.26 0.28 0.3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

z/R

φ5 Conclusions

The mathematical model of a set of rings and patches is given by an integral equationwith logarithmic singular kernel function. In the φ-direction, trigonometric basisfunctions are used. Only the basis functions of the same order in the rings and thepatches show a mutual coupling, others do not. Appropriate choices for the basisfunctions in the axial direction are the Legendre polynomials. An analytical solutionfor the singular part is derived and only a few Legendre polynomials are needed dueto fast convergence. The regular part is computed numerically and has a minimalcontribution to the complete solution.

References

[1] M.T. Vlaardingerbroek and J.A. den Boer, Magnetic Resonance Imaging. Berlin: Springer(1999).

[2] T. Ulicevic, J.M.B. Kroot, S.J.L. van Eijndhoven and A.A.F van de Ven, Current distribution

in a parallel set of conducting strips. Journal of Engineering Mathematics, accepted (2005).

[3] J.M.B. Kroot, Current Distribution in a Gradient Coil, Modeled as Circular Loops of Strips.

Eindhoven: Final report of the postgraduate program Mathematics for Industry, EindhovenUniversity of Technology (2002).

[4] J.M.B. Kroot, S.J.L. van Eijndhoven and A.A.F. van de Ven, Eddy currents in a gradient coil,

modelled as circular loops of strips. Beijing: 2004 3rd international conference on computa-tional electromagnetics and its applications, pp 5-8 (2004).

[5] H. S. Cohl and J.E. Tohline, A compact cylindrical Green’s function expansion for the solution

of potential problems. The astrophysical journal, Vol 257, pp 86-101 (1999).

[6] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. New York: Dover(1972).