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Analysis of Eddy Currents in a Gradient Coil
Magnetic Resonance Imaging (MRI) is an imaging technique that plays an important role in the medical community. It provides images of cross-sections of a body, taken from any angle. The principle of MRI is based on the reaction of the body on a magnetic field. Hydrogen protons are stimulated by a strong external magnetic field and additional radio pulses, resulting in small electromagnetic signals emitted by the protons. The emitted signals are received by an acquisition system and processed to become an image with contrast differences.
The selection of a slice is realized by the so-called gradient coils. A gradient coil consists of copper strips wrapped around a cylinder. Due to mutual magnetic coupling, eddy currents are induced, resulting in a non-uniform distribution of the current. The eddy currents cause a distortion in the desired magnetic field. Spatial non-linearity of the gradient results in blurred images. Moreover, the presence of eddy currents increases the resistance of the coil and, consequently, the power dissipation. In order to reduce the energy costs, the dissipated power has to be minimized. Meanwhile, the self-inductance of the coil decreases due to the eddy currents, resulting in the need for a smaller voltage supply.
For analysis and design of gradient coils, finite element packages are used. However, these packages cannot always provide sufficient insight in the characteristics describing the qualitative behavior of the distribution of the currents, relating the geometry to typical parameters such as edge-effects, mutual coupling and heat dissipation. In this thesis, a detailed analysis of the eddy currents in gradient coils is presented. In particular, the analysis has led to the design of a software tool that simulates the current distribution and the electromagnetic fields inside the scanner quantitatively. Both the analysis and the software tool support the overall design of gradient coils. In the simulation, special attention is devoted to time effects (different frequencies) and spatial effects (space-dependent magnetic fields). Moreover, characteristic quantities of a system are derived, in particular resistance, self-inductance, characteristic frequencies, and linearity of the gradient field. For all these characteristic quantities, their dependence on the frequency of the applied source, the shape of the conductors, the distance between the conductors, and the conductivity is investigated.
For the mathematical model, Maxwell's equations are used, together with
the associated boundary conditions and constitutive equations.
Assumptions to reduce the set of equations are:
The current is transformed into a surface current and the model leads to an integral equation of the second kind for the current density
with the source current, a system parameter, and the united surface of all strips. It is sufficient to consider the -component only. The component in the -direction follows automatically from the law of conservation of charge
We write (1) as
From a numerical point of view, we use the Fourier cosine series of the kernel function Where are the Legendre functions of the second kind of odd-half-integer order, which have a logarithmic singularity in , i.e. .
The model analysis predicts that the induced eddy currents in the strips prefer to flow in the direction opposite to the applied source current. The physical explanation is that eddy currents tend to oppose the magnetic field caused by the source current. The leading integral equation is formulated in terms of the component of the current in this preferred direction, imposed by the source current. The essential behavior of the kernel in this leading integral equation is logarithmic in the coordinate perpendicular to the preferred direction.
To solve the integral equation, the Galerkin method with global basis functions is applied to approximate the current distribution. In the preferred direction, trigonometric functions are chosen to express periodicity. Thus, the current is expanded in Fourier modes and via the inner products a direct coupling between the modes is achieved. In the width direction of the strips, Legendre polynomials are chosen. With this choice, basis functions are found that are complete, converge rapidly and the resulting inner products are easy to compute. Hence, this method is especially dedicated to the problems considered in this thesis.
The simulations show how the eddy currents are characterized by edge-effects. Edge-effects become stronger when the frequency is increased, and the errors in the magnetic field increase accordingly. The eddy currents also affect the resistance and self-inductance of a coil. For every configuration, the resistance increases with the frequency, whereas the self-inductance decreases with the frequency. Furthermore, both quantities show a point of inflection at a characteristic frequency. This characteristic frequency is expressed by an analytical formula.
The software tool has been designed for strips of different types of gradient coils and to model slits in the strips. The implementation makes use of limited memory, this in contrast to numerical packages, in which a lot of elements (memory) is needed to come to a high accuracy. Moreover, a fast approximation of the current distribution is achieved, because of appropriate basis functions and the use of explicit analytical results.Publications J.M.B. Kroot, Analysis of eddy currents in a gradient coil, Scientific computing in electrical engineering, book series mathematics in industry (2007) 9: 221 - 226.
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, Journal of Engineering Mathematics (2006), 57: 333 - 350.
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 (2005) 51: 381 - 400.
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, Proceedings of the 3rd international conference on computational electromagnetics and its applications, Beijing, China, November 1-4, 2004.
J.M.B. Kroot, Analysis of eddy currents in a gradient coil, Proceedings of the 5th international workshop on scientific computing in electrical engineering, Capo d'Orlando, Sicily, Italy, September 5-9, 2004.
J.M.B. Kroot, Analysis of eddy currents in a gradient coil, Proceedings of the 13th European conference on mathematics in industry, June 21 - 25, 2004.
J.M.B. Kroot, Analysis of eddy currents in gradient coils, Proceedings of workshop on optimisation and coupled problems in electromagnetism, Napoli, Italy, September 20-24, 2003.
J.M.B. Kroot, Current distribution in a gradient coil, modelled as circular loops of strips, Master's thesis Postgraduate programme Mathematics for Industry, Eindhoven University of Technology (2002).
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