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Eddy current effects in plain and hollow cylinders spinning inside homogeneous magnetic fields: Application to magnetic resonance

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10.1063/1.4756948

### Abstract

We present a thorough analysis of eddycurrents that develop in a rectangular cross section toroid rotating in a uniform magnetic field. The slow rotation regime is assumed. Compact expressions for the current density, the total dissipated power, and the braking torque are given. Examination of the topology of current lines reveals that depending upon the relative dimensions of the side and length of the toroid two different regimes exist. The conditions of existence of the two regimes are analytically established. In view of nuclear magnetic resonance(NMR) applications, we derive the angular variation of the magnetic field created by eddycurrents and lay down the formalism necessary for calculating the effect of this field on the NMR spectra of the conductor itself or of a sample co-rotating with the conductor, a situation encountered when dealing with rotating detectors. Examples of calculations for cases of practical interest are presented. The theory is confronted with available data, and we give guidelines for the design of optimized rotating micro-coils.

© 2012 American Institute of Physics

Received 13 July 2012
Accepted 18 September 2012
Published online 15 October 2012

Acknowledgments: The authors would like to thank Dr. C. Hugon and Dr. J. P. Yesinowski for stimulating discussions. This work was supported by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013): ERC Grant Agreement No. 205119.

Article outline:

I. INTRODUCTION

II. GENERALITIES AND LAYOUT OF THE PROBLEM IN THE SLOW ROTATION REGIME

A. Time constants and rotation regimes

B. Electric field in the rotating frame

C. Alternative choices for the electric field in the rotating frame

D. General scheme for the determination of and of the current distribution

E. Stationarity of the eddycurrents distribution in the case of a cylinder

III. CURRENT DISTRIBUTION IN THE CASE OF A HOLLOW CYLINDER

A. Symmetry of the solution

IV. CALCULATION OF THE EDDYCURRENT LOSSES P AND OF THE BRAKING TORQUE Γ_{Z}

A. Variation of P with shape dimensions

B. Case of a small cross section toroid

C. Other particular geometries

D. Two examples taken from the literature in the field of NMR

1. Aguiar *et al.*

2. Yesinowski *et al.*

E. Discussion: Improving performances of rotating micro-coils

V. EDDYCURRENTS TRAJECTORIES

A. Case of a plain cylinder

B. Case of a hollow cylinder

C. Condition of existence of a 2-type of trajectory topology

VI. FIELDS CREATED BY EDDYCURRENTS

A. Angular variation of the fields generated by eddycurrents

B. Case of a sample placed inside a hollow conductor

1. Spatial variation of field

2. Influence on the NMR spectrum

C. Field inside a plain cylinder

VII. CONCLUSION

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2012-10-15

2014-04-23

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