Synopsis

Keywords: Magnets (B0), Magnets (B0), magnetic damping

The motion of a conductive object is profoundly affected by the presence of a strong magnetic field. A conductive loop falling inside an MRI magnet can serve as an analytically solvable model to explore magnetic damping. We present a pair of magneto-mechanical coupled equations to describe its motion and verify their solution with experimental data. A simple formula for the loop's falling time was obtained, which indicates that magnetic damping is proportional to B₀².

Introduction

It has been suggested that magnetic damping may substantially reduce vibration of conductive materials in high-field MRI [1-3]. While magnetic damping and stiffening are familiar phenomena to anyone who tried to move a metal in a clinical MRI magnet, the exact mechanism of magnetic disturbance of mechanical motion has rarely been articulated in the MRI literature. There are, for example, differing views on how magnetic damping scales with B₀, i.e., linearly [1] vs. quadratically [2]. Here, a simple magneto-mechanical coupled system, a conductive loop falling on the table of an MRI magnet, is presented as a model to demonstrate the interplay between mechanical (angular displacement θ) and electromagnetic (current I) degrees of freedom. An analytical expression for the fall time illustrates B₀² dependence of magnetic damping which is experimentally verified.

Theory

System description. Consider an electrically conductive and flat, single-turn closed loop whose plane can rotate freely about an axis (x-axis) on the horizontal (zx) plane under a uniform static magnetic field B₀ (Fig. 1). The loop has an area A, resistance R, inductance L, and number of turns N = 1. Mechanically it is part of a flat rigid body (e.g. frame) whose total mass and moment of inertia are m and I_i, respectively. The body's center of mass is at distance r from the x-axis. This system has two degrees of freedom: rotation angle θ and electric current I (Fig. 1).
Coupled equations. If the loop is let go at an initial angle θ₀, its gravitational fall will incur time-varying magnetic flux which induces eddy current that slows down the fall via the Lorentz force. This is captured by Eqs.(1-2): Eq.(1) describes angular acceleration under gravitational and magnetic torque, while Eq.(2) describes the Kirchhoff's voltage law with motion-induced electromotive force. Eqs.(3ab) define the initial condition. Eqs.(1-3) can be numerically solved for I(t) and θ(t). If N > 1, A, R, L will scale as AN, RN, LN², while m will increase linearly with an additive constant (mass of the frame).
Approximate solution. An approximate solution Eq.(4) can be found in the overdamped case where d²θ/dt² and dI/dt can be neglected. Eq.(4) predicts that the loop will hit the ground (θ=90deg) at the fall time Eq. (5) provided θ₀ << 1. Eqs.(4-5) turn out to be a remarkably good solution for a wide range of parameters. Figure 3 shows a representative case where cosθ is almost exactly linear in time, in accordance with Eq.(4). Comparing Eq.(4) with the velocity of an overdamped linear motion (v = F/b, b: damping coefficient), we identify (AB₀)²/R as the magnetic damping coefficient.

Experimental Methods

Circular loops with 11.9 cm diameter and N = 2,4,6,8 were wound from an 18-gauge (AWG) copper wire and electrically closed by soldering. Each loop was glued to the lid of a DVD jewel case which was hinged onto its base with low friction. The base was taped to a 3T MRI table (Fig. 4). After moving the table to a desired position, the lid and the loop were released from a near-vertical position (θ₀ ~10deg) and the time for touchdown was measured by a stopwatch. The experiment was repeated at multiple positions for N = 8, and at the isocenter for each N. The measured times were compared with the theoretical fall time Eq. (5) based on the measured weight mg of each lid (+loop) and calculated resistance R of the wire. The static field at different locations was estimated from the vendor-supplied electromagnetic design of the magnet.

Results

Figure 5(a) shows that the measured fall times at the isocenter almost exactly match the theoretical formula, with errors less than 2%. Small uncertainties in estimation of the center of mass, area of the loop, and electrical resistance at the solder joint could be sources of the error. This result strongly supports our theoretical formulation Eqs.(1-2) and the overdamped solution Eq.(5). The B₀ dependence is illustrated in Fig. 5(b) where the measured times agree well with theory in regions of homogeneous B₀ but fall short in the transition region near the bore entrance. The discrepancy is likely because large B₀ gradient creates nonzero net force in the loop, in addition to magnetic torque, which was ignored in Eq.(1). The inset shows that the fall time data follow much more closely the magnet's B₀² profile than B₀, providing strong support of B₀² dependence of magnetic damping.

Discussions and Conclusion

Dramatic slowdown of a falling conductor has been a popular item of demonstration to illustrate the power of an MRI magnet. In this work we have attempted to elucidate its mechanism in terms of motion-induced eddy current and magnetic torque acting as a damping force. Our theory was well supported by experimental data and helps identify factors that govern magnetic damping. In particular, the measured fall times inside and outside of a 3T magnet strongly favored B₀² dependence of magnetic damping. The presented model can provide a benchmark for future research on magneto-mechanical coupling in MR engineering.

Acknowledgements

No acknowledgement found.