Magnetic field interaction with high-permeability planar passive shields

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Abstract
Cuboid shields formed from sheets of materials of high magnetic permeability are used to provide passive magnetic shielding in multiple applications, ranging from fundamental physics to functional neuroimaging. Current-carrying electromagnetic coils are often used in combination with passive shielding, since coils placed inside or outside the shield can be used to cancel unwanted residual magnetic fields, or to produce desired bias fields inside the shield. Understanding the effect of the high-permeability material on the time-varying fields produced by coils is important for implementing this hybrid active and passive shielding approach. Here, we use analytic expressions describing the interaction of magnetic fields with an infinite plane of high relative permeability (μr≫1) to characterize the behavior of shielding systems using coils attached to the inner or outer face of a cuboid shield in the regime (length scales of 0.01–1 m over a frequency range of 0–500 Hz) that is relevant for biomagnetic measurements. Theoretical predictions were compared with results from numerical simulations and experimental measurements made using a 55-cm-side cubic shield formed from 1.5-mm-thick sheets of MuMetal. The results confirm that the fields produced by internal coils are well predicted by the method of mirror images, with maximum deviations from perfect reflection of ∼2% in the magnitude and −0.02 rad in the phase of the field measured at ∼16 cm from an 8.5-cm-diameter circular coil driven at 440 Hz. The analytic expression indicates that the spatial variation of the shielded field from an external coil can be simply calculated from the two-dimensional Fourier transform of the field from the coil. This was confirmed by simulations and experimental measurements made using an 8.5-cm-diameter coil driven at frequencies of 0.1, 1, 5, and 10 Hz. This simple Fourier relationship facilitates the design of external coils that generate target-field variations inside the shield. Successful implementation of this approach was demonstrated experimentally using an external coil that was designed to produce the same field inside the shield as an unshielded 8.5-cm-diameter coil. Numerical simulations also confirmed the prediction that the shielded field distant from a circular coil is equivalent to that from a magnetic monopole located at the coil center. These findings offer new insight for the design and implementation of magnetic shielding using active and passive systems, particularly for measurements of biomagnetism.
Funding Information
  • Engineering and Physical Sciences Research Council (EP/Z533166/1)
  • Innovate UK (104604)

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