Theory and mitigation of motional eddy current in high-field eddy current shielding

Abstract

Eddy current shielding by a Faraday cage is an effective way to shield alternating-current magnetic fields in scientific instrumentation. In a strong static magnetic field, however, the eddy current in the conductive shield is subject to the Lorentz force, which causes the shield to vibrate. In addition to mechanical issues (e.g., acoustic noise), such vibration induces motional eddy current in the shield that can dominate the original, electromagnetic eddy current to undermine the conductor's shielding capability. In this work, we investigate a method to control motional eddy current by making cut-out patterns in the conductor that follow the electromagnetic eddy current image. This effectively limits the surface current of the plate to a single mode and prevents the proliferation of uncontrolled motion-induced surface currents that disrupts eddy current shielding. After developing a comprehensive theory of magneto-mechanical interaction in a conductive plate, the proposed method was tested on a flat-geometry testbed experiment inside a 3 T magnetic resonance imaging (MRI) magnet. It was found that the magnetic field generated by the motional eddy current was much more localized in space and frequency for a patterned-copper shield compared to a solid copper. The magnetic field of the patterned shield could be accurately predicted from the impedance measurement in the magnet. Implications of our results for improved shielding of gradient fields in high-field MRI are discussed.

FIG. 1.

Definition of the coordinate system and variables. A static magnetic field ${\overrightarrow{B}}_0=B_0\hat{z}$ is applied parallel to a conductive plate in the zx plane whose vibration is characterized by the vertical displacement $u(z,x)\hat{y}$. The eddy current stream function and the corresponding surface current density are denoted by T and $\overrightarrow{j}$, respectively.

FIG. 2.

Eddy image current and its discretization. (a) In the high-frequency limit, the eddy current in a conductive plate (gray horizontal bar) prevents the penetration of an applied magnetic field generated by a driving coil (black-outlined horizontal bar). Such eddy current, whose stream function is denoted as $T_{eddyi}$, is analogous to the DC screening current in a superconducting plate. (b) Illustration of $T_{eddyi}$ and corresponding surface current map. (c) In the proposed passive shield, $T_{eddyi}$ is discretized into a closed-loop coil with current $i_0$. The black lines indicate cutouts on the conductor and the current flows from the red dot in the middle (P₁) to the blue dot at the bottom (arrow, P₂). The two dots are shunted by a copper bridge (not shown) to close the loop and cancel the turn-to-turn radial current.

FIG. 3.

Illustration of motional inductance and onset of magneto-mechanical resonance. (a) At low frequencies $(\omega \ll {\omega }_{mech})$, the Lorentz force $f_L$ acting on an oscillating current $I(\omega )$ tilts the plate towards and away from ${\overrightarrow{B}}_0$ in phase with the current. When the current is positive, the tilt of the normal vector $\hat{n}$ is toward ${\overrightarrow{B}}_0$. Such tilt results in positive (locally upward) flux of ${\overrightarrow{B}}_0$ threading the loop, making motional inductance positive. (b) At high frequencies $(\omega \gg {\omega }_{mech})$, the Lorentz force and angular tilt of $\hat{n}$ are 180° out of phase; when the current is positive, the tilt is still away from ${\overrightarrow{B}}_0$ trying to catch up. In this case, ${\overrightarrow{B}}_0$ threads the current in the negative direction, or locally downwards, and the motional inductance is negative. If the negative motional inductance cancels the positive self-inductance, the loop is at magneto-mechanical resonance. Then, the current induced by an applied oscillating magnetic field (not shown) is impeded only by the resistance of the loop. $k_H$ denotes the spring constant.

FIG. 4.

Experimental setup. (a) Driving (yellow arrow) and sensing (white arrow) coils. (b) Solid copper plate placed between the driving (hidden below the plate) and sensing (white arrow) coils. (c) Patterned copper during impedance measurement. The two ends of the spiral pattern were shorted during the FRF measurements. (d) Setup in a whole-body 3 T magnet with sensing coil indicated by the white arrow. The copper plate was mounted on an aluminum frame fixed against the magnet end flanges, while the sensing coil was hanging from a wooden pole supported from the floor of the room.

FIG. 5.

Frequency response functions for unshielded baseline setup (a) and setup with solid copper plate between the driving and sensing coils (b) outside the magnet. Each box contains a current-to-voltage FRF spectrum on the axis defined at the top left corner. Nineteen FRFs obtained at different z locations of the sensing coil are presented. Note fourfold larger vertical scale in (a). Solid copper outside the magnet provides nearly perfect shielding.

FIG. 6.

Frequency response functions for solid (a) and patterned (b) copper shields inside a 3 T magnet. Compared to Fig. 5(b), shielding is severely disrupted at 3 T with numerous magneto-mechanical resonance peaks appearing as a result of motional eddy current. Patterned shield makes the leakage field peaks more localized in space and frequency than solid copper.

FIG. 7.

Root-mean-square of the measured pickup-coil FRF over 0–3 (a) and 1–3 kHz (b) as a function of z. Large difference between solid copper shields in (red) and out of (black) the magnet highlights leakage field amplification [arrow in (a)] due to motional eddy current. Patterned-copper shield in the magnet (blue) makes the leakage field more concentrated in frequency and space [arrows in (b)] compared to solid copper.

FIG. 8.

Measured (a) and calculated (b) pickup coil voltage FRF spectra for the patterned-copper shield. Measurement was taken with the sensing loop at z = 0. Calculation is based on the impedance of the copper spiral pattern open-circuited and measured in the magnet. Major motional eddy current peaks are well reproduced in the calculated spectrum [arrows in (b)].