Approximate expressions for the magnetic field created by circular coils inside a closed cylindrical shield of finite thickness and permeability

Abstract The common approach to create a uniform magnetic field in the μ T range is placing a coil set inside a magnetic shield. One of the main obstacles to achieving the requested uniformity can be a field distortion caused by the shield. We derive an approximate expression for the magnetic field generated by circular coils enclosed in ferromagnetic cylinders, the most used shield geometry. The main advantage of the expression over the known approximation is considering the finite permeability and the finite thickness of the cylindrical shield. To model the cylinder end caps, we propose the finite thickness mirror method, which extends the method of mirror image to the case of finite thickness. The cylinder barrel is modeled as an infinite tube and its effect on the field is analytically solved. The error of the expression and its dependency on the observation point and the shield geometry are discussed. Compared with finite-element simulations our analytic method is consistent within a relative error of less than 0.3% for practical shielding setups. The expression enables the rapid optimization of shield-coupled coil sets with a custom-designed objective function. To show the applicability of our method, we present simulations for the direct selection of the optimum spacing of a coil pair to match the given dimensions of a cylindrical shield.