Eddy currents in a gradient coil, modeled as circular loops of strips

As a model for the z-coil of an MRI-scanner a set of circular loops of strips, or rings, placed on one cylinder is chosen. The current in this set of thin conducting rings is driven by an external source current. The source, and all excited fields, are time-harmonic. The frequency is low enough to allow for an electro-quasi-static approach. The rings have a thin rectangular cross-section with a thickness so small that the current can be assumed uniformly distributed in the thickness direction. Due to induction, eddy currents occur resulting in an edge-effect. Higher frequencies cause stronger edge-effects. As a consequence, the resistance of the system increases and its self-inductance decreases. The Maxwell equations imply an integral equation for the current distribution in the rings. The Galerkin method with Legendre polynomials as global basis functions is applied. This method shows fast convergence, so only a very restricted number of basis functions is needed. The general method is worked out for N (N ≥  1) rings, and explicit results are presented for N = 1, N = 2 and N = 24. The derived integral equation and the numerical results of the simulations show that sets of circular rings and plane strips describe the same electromagnetic behavior, thus demonstrating that inductance effects are local.

[1]  K. Atkinson,et al.  A survey of numerical methods for the solution of Fredholm integral equations of the second kind , 1977 .

[2]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[3]  B. Hargreaves,et al.  Variable‐rate selective excitation for rapid MRI sequences , 2004, Magnetic resonance in medicine.

[4]  D. Doddrell,et al.  Gradient-Coil Design by Simulated Annealing , 1993 .

[5]  T. Teramoto,et al.  Numerical analysis on fracture behavior of a single-edge-cracked plate subjected to electromagnetic force , 1998 .

[6]  B. Rutt,et al.  Acoustic noise analysis and prediction in a 4‐T MRI scanner , 2004 .

[7]  Attay Kovetz,et al.  The principles of electromagnetic theory , 1990 .

[8]  R. Turner.  A Target Field Approach To Optimal Coil Design , 1986 .

[9]  Kendall Atkinson,et al.  The Numerical Solution of Integral Equations of the Second Kind: Solving multivariable integral equations , 1997 .

[10]  R. L. Stoll.  The analysis of eddy currents , 1974 .

[11]  H. Siebold.  Gradient field coils for MR imaging with high spectral purity , 1990 .

[12]  R. Bowtell,et al.  Gradient coil design using active magnetic screening , 1991, Magnetic resonance in medicine.

[13]  Jianming Jin.  Electromagnetic Analysis and Design in Magnetic Resonance Imaging , 1998 .

[14]  J W Carlson,et al.  Currents and fields of thin conductors in rf saddle coils , 1986, Magnetic resonance in medicine.

[15]  D. Tomasi.  Stream function optimization for gradient coil design , 2001, Magnetic resonance in medicine.

[16]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[17]  Jmb Jan Kroot.  Analysis of eddy currents in a gradient coil , 2006 .

[18]  van Stef Stef Eijndhoven,et al.  Current distribution in a parallel set of conducting strips , 2005 .

[19]  A. Jasiński,et al.  Apparatus for generation of magnetic field gradient waveforms for NMR imaging , 1988 .

[20]  S. Vossen.  A two-wire antenna system for detecting objects in a homogeneous dielectric half space , 2003 .

[21]  C-N. Chen,et al.  Biomedical magnetic resonance technology , 1989 .

[22]  M. T. Vlaardingerbroek,et al.  Magnetic Resonance Imaging: Theory and Practice , 1996 .

[23]  G. Peeren.  Stream function approach for determining optimal surface currents , 2003 .

[24]  Emmanuel Simeu,et al.  Modeling and control of an eddy current brake , 1996 .

[25]  Mark S. Mirotznik,et al.  Numerical evaluation of heating of the human head due to magnetic resonance imaging , 2004, IEEE Transactions on Biomedical Engineering.

[26]  L. Dupré,et al.  Magnetic field computation for optimized shielding of induction heaters , 2004 .

[27]  R. Turner,et al.  Passive screening of switched magnetic field gradients , 1986 .

[28]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[29]  J. Reade.  Asymptotic Behaviour of Eigen-Values of Certain Integral Equations , 1979 .

[30]  J. Hyde,et al.  Coil optimization for MRI by conjugate gradient descent , 1991, Magnetic resonance in medicine.

[31]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[32]  D. Hoult,et al.  Magnet field profiling: Analysis and correcting coil design , 1984, Magnetic resonance in medicine.

[33]  B. Suits,et al.  Improving magnetic field gradient coils for NMR imaging , 1989 .

[34]  P. Mansfield,et al.  Controlled E-field gradient coils , 2003, Magnetic Resonance Materials in Physics, Biology and Medicine.

[35]  D. Harberts.  Dissipation and ringing of CRT deflection coils , 2001 .

[36]  P. Mansfield,et al.  Active acoustic control in gradient coils for MRI , 2001, Magnetic resonance in medicine.

[37]  C. D. Castilho,et al.  Field ion energy deficit calculations for liquid metal ion sources , 1986 .

[38]  L. Forbes,et al.  Electromagnetic fields in the human body due to switched transverse gradient coils in MRI. , 2004, Physics in medicine and biology.

[39]  E. Kriezis,et al.  Eddy currents in linear conducting media , 1984 .

[40]  Yu. A. Brychkov,et al.  Integrals and series , 1992 .

[41]  Paul F. Byrd,et al.  Handbook of elliptic integrals for engineers and scientists , 1971 .

[42]  J. Swinburne.  Electromagnetic Theory , 1894, Nature.

[43]  Stuart Crozier,et al.  Caluculating Current Densities and Fields Produced by Shielded Magnetic Resonance Imaging Probes , 1997, SIAM J. Appl. Math..

[44]  W. R. Smythe.  Static and Dynamic Electricity , 1989 .

[45]  Stuart Crozier,et al.  A methodology for current density calculations in high-frequency RF resonators , 1997 .

[46]  P. Mansfield,et al.  Multishield active magnetic screening of coil structures in NMR , 1987 .