Optimal parameters of a nonmagnetic conducting cylindrical double-shell shield rotating in a static magnetic field

Abstract

This paper first presents an analytical model for calculating the shielding effectiveness of static magnetic fields using a rotating double-shell cylindrical shield with electromagnetically thin layers made of non-magnetic conducting material. Then a procedure is given for finding the optimum distance between the shells and their thickness for maximum mass reduction compared to a thick single shield for a given shielding factor.

Keywords

Computational electromagnetics optimization methods shielding modelling and methods

1. Introduction

Physics experiments, e.g. high sensitivity magnetic measurements, often require protection from the influence of an external static magnetic field. A major static magnetic field shielding mechanism is flux shunting by using ferromagnetic materials. Passive magnetic shielding systems, either single or multilayer, typically use a concentric array of thin shells of a high magnetic permeability material to redirect magnetic field lines around a region of interest [1–7]. Generally speaking, multi-layer shielding can achieve better shielding factors than a single layer shield. The phenomenon of magnetic shielding has been discussed at length in the literature and it still remains a topic of lively research interest. The cited references should only be taken as illustrative examples. The above-quoted work [7] contains a list of over 100 literature items in this subject area.

In some situations, however, shields made of ferromagnetic materials cannot be used. A simple example can be given. A turbine generator which has a superconducting field winding requires a shielding system around the superconducting field winding, both for thermal and for magnetic effects [8]. The field winding must be isolated from static and rapidly time-varying magnetic fields, and this calls for effective electrical shielding. Attached to the rotor between the field and the armature are one or more electromagnetic shields. It is obvious that the shielding system must be made of nonmagnetic materials in order to allow the generator to work. The prime function of the shields is to keep the field winding superconducting. These shields through their induced eddy currents limit the magnitude and the time rate of change of induced fluxes and currents at the field winding. In [9] a construction with independently rotatable (free) outer shield was proposed. In the event of a fault, the large alternating torques will simply accelerate and decelerate the freely rotating shield, and will not be transmitted to the rotor, shafts, couplings, and turbine. During normal operation, the freely rotating shield would behave as a drag-cup induction motor with no load, and follow the field winding at very nearly synchronous speed.

Instead of magnetic shields, to produce a magnetic-field-free region, rotating shields made of non-magnetic materials like copper or aluminum can be used.

When such shields rotate in a static magnetic field, eddy currents induced within the conducting material tend to exclude the static magnetic field from the interior of the shield. As a result, the region inside the shielding system has a reduced local magnetic field. A closed-form solution for the shielding factor in case of a rotating cylindrical shield of any thickness is given in [10]. This paper is theoretical in nature and first presents an analytical expression for the shielding factor for a double-shell shield rotating in a static magnetic field, based on the approximate electromagnetically-thin model of the rotating shell described in [11]. Analytical solutions, possible in the case of some idealized models, are useful for a qualitative presentation of the role of various system parameters. Such an analytical solution is obtained in the present study. Then, the solution of the optimization problem is presented. A single shield of a specified thickness for a given value of the shielding factor is replaced by a double-shell shield in such a way that the maximum weight reduction of the shielding system is achieved.

2. Double-shell shield rotating in a static magnetic field

Consider two thin concentric cylindrical non-magnetic metal shells rotating with constant angular velocity around a common axis in the presence of an externally applied transverse uniform static magnetic field of induction B₀. The shells are assumed to be surrounded by a medium of zero conductivity and permeability μ₀. The internal (external) shell has a radius R_i (R_e). The physical properties of the shells can be described by permeability μ₀, conductivity σ and angular velocity 𝛺. The shells are assumed to have the same thickness h, and together they form a double-shell shield. We consider the shells to be sufficiently long compared to their diameters so that end effects may be ignored. Because the shells are cylindrical, it is natural to use cylindrical coordinates (r, φ) in solving the problem. The z-axis is taken to lie along the axis of the shells and the angle φ is measured with respect to the x-axis, which is parallel to the external magnetic flux density B = B₀ 1_x. The shells are rotated in an anticlockwise direction. The geometry for the calculations is shown in Fig. 1.

Fig. 1.

Double-shell shield rotating in a transverse uniform static magnetic field

In theoretical considerations, it is assumed that the thickness h of the shells is infinitely small, h → 0, the electrical conductivity of the shells is infinite, σ → ∞, but the limit of their product has a finite value. In numerical calculations, the product h σ of the actual values of thickness and conductivity replaces this finite value. In “practice”, such an approach can be used for the shells of any thickness as long as they are thinner or at most comparable to the diffusion skin depth δ_𝛺, ${\delta}_{{\Omega}}=∼\sqrt{2/({\mu}_{0}{\Omega}{\sigma})}$ ([6]).

In the rotating metal shells a steady space distribution of currents parallel to the z-axis will be induced. The magnetic flux density obeys 𝛻 ⋅ B = 0, so that B can be deduced from a magnetic vector potential A via B = 𝛻 × A. In the steady state, the magnetic field H within current-free air regions around the rotating shells satisfies 𝛻 × H = 0, and is related to the magnetic flux density by the constitutive relation B = μ₀H (μ₀ = 4π⋅10⁻⁷ H/m). Hence, within air regions 𝛻 × (𝛻×A) = 0. Thus, we seek solutions to vector Laplace’s equations in the three air regions, taking into account the behavior of the magnetic field at infinity (very far from the shield), as well as the matching conditions at the cylindrical boundaries between the air regions. The three air regions are labeled I, II and III as shown in Fig. 1. Since the currents induced in the shells are axial, the magnetic vector potential A has only a z-component, A = A (r, φ) 1_z, and in polar coordinates the vector Laplace equation simplifies to the scalar equation: $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial ^{2}A(r,{\varphi})}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A(r,{\varphi})}{\partial r}}+{\displaystyle \frac{1}{r^{2}}}{\displaystyle \frac{\partial ^{2}A(r,{\varphi})}{\partial r^{2}}}=0. & & \displaystyle\end{eqnarray}$ (1) Once the magnetic vector potential has been determined, it can be used to compute any physically observable electromagnetic quantity. For mathematical convenience, we use the periodicity of the magnetic vector potential with respect to the φ-variable: $\begin{eqnarray} \displaystyle A(r,{\varphi})=\Im m[\text{}\underline{A}(r)\exp (j{\varphi})]. & & \displaystyle \end{eqnarray}$ (2) Now, we can rewrite ((1)) as the ordinary second-order differential equation: $\begin{eqnarray}\displaystyle \frac{\text{d}^{2}\text{}\underline{A}(r)}{\text{d}r^{2}}+\frac{1}{r}\frac{\text{d}\text{}\underline{A}(r)}{\text{d}r}-\frac{1}{r^{2}}\text{}\underline{A}(r)=0 & & \displaystyle\end{eqnarray}$ (3) with the general solution in the form of a linear combination of the functions f₁(r) = r and f₂(r) = r⁻¹.

Since the magnetic vector potential must be finite at the origin, the radial function is only r in region I. Very far from the rotating shells A_III(r → ∞, φ) = B₀rsinφ, or in a complex notation A _III(r →∞) = B₀r, which means that the distorting effect of the rotating shells extends over a finite region, beyond which the magnetic field remains essentially uniform. One can immediately verify that the magnetic vector potential A = B₀rsinφ 1_z corresponds to the magnetic induction B = B₀1_x by calculating B = 𝛻 × A and observing the result. The solutions for A (r) in each region then become: $\begin{eqnarray}\displaystyle \text{}\underline{A}_{\text{I}}(r)=\text{}\underline{a}r, & & \displaystyle\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{}\underline{A}_{\text{II}}(r)=\text{}\underline{b}r+\text{}\underline{c}r^{-1}, & & \displaystyle\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{}\underline{A}_{\text{III}}(r)=B_{0}r+\text{}\underline{d}r^{-1}. & & \displaystyle\end{eqnarray}$ (6)

In these equations we have introduced four complex constants, a , b , c , and d , which must be determined by the matching conditions at the two interfaces of the three regions given above. The immediate goal, however, is just the determination of the constant a , since we need to evaluate the magnetic field in the region I.

The normal component of the magnetic flux density B must be continuous at the boundaries between the regions or, equivalently and more simply, A must be continuous there, which requires: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{A}_{\text{I}}(R_{i})=\text{}\underline{A}_{\text{II}}(R_{i})\quad \Rightarrow \quad R_{i}\text{}\underline{a}=R_{i}\text{}\underline{b}+R_{i}^{-1}\text{}\underline{c},\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{A}_{\text{II}}(R_{e})=\text{}\underline{A}_{\text{III}}(R_{e})\quad \Rightarrow \quad R_{e}\text{}\underline{b}+R_{e}^{-1}\text{}\underline{c}=B_{0}R_{e}+R_{e}^{-1}\text{}\underline{d}.\end{eqnarray}$ (8) The motionally induced currents in the shells can be viewed as surface currents on the cylinder surfaces with radii R_i and R_e: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{K}_{i}(R_{i},{\varphi})=K_{i}(R_{i},{\varphi})\mathbf{1}_{z}=h{\sigma}({\Omega}R_{i}\mathbf{1}_{{\varphi}}\times \mathbf{B})=h{\sigma}[{\Omega}R_{i}\mathbf{1}_{{\varphi}}\times ({\nabla}\times \mathbf{A})]=-h{\sigma}{\Omega}\frac{\partial A(R_{i},{\varphi})}{\partial {\varphi}}\mathbf{1}_{z},\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{K}_{e}(R_{e},{\varphi})=K_{e}(R_{e},{\varphi})\mathbf{1}_{z}=-h{\sigma}{\Omega}\frac{\partial A(R_{e},{\varphi})}{\partial {\varphi}}\mathbf{1}_{z},\end{eqnarray}$ (10) or in complex form: $\begin{eqnarray}\displaystyle \text{}\underline{K}_{i}(R_{i})=-jh{\sigma}{\Omega}\text{}\underline{A}(R_{i}),\quad \text{}\underline{K}_{e}(R_{e})=-jh{\sigma}{\Omega}\text{}\underline{A}(R_{e}). & & \displaystyle\end{eqnarray}$ (11) Application of Ampere’s law to circuits that just enclose the current sheets on the surfaces r = R_i and r = R_e leads to well-known relations between the field discontinuity across the sheets and the surface current densities (H is the magnetic field intensity vector): $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{1}_{r}\times [\mathbf{H}_{\text{II}}(R_{i},{\varphi})-\mathbf{H}_{\text{I}}(R_{i},{\varphi})]=\mathbf{K}_{i}(R_{i},{\varphi}),\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{1}_{r}\times [\mathbf{H}_{\text{III}}(R_{e},{\varphi})-\mathbf{H}_{\text{II}}(R_{e},{\varphi})]=\mathbf{K}_{e}(R_{e},{\varphi}).\end{eqnarray}$ (13) These last two equations may be written in terms of the magnetic vector potential in complex form: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{1}{{\mu}_{0}}\left[\frac{\text{d}\text{}\underline{A}_{\text{II}}(R_{i})}{\text{d}r}-\frac{\text{d}\text{}\underline{A}_{\text{I}}(R_{i})}{\text{d}r}\right]=jh{\sigma}{\Omega}\text{}\underline{A}_{\text{I},\text{II}}(R_{i})\quad \Rightarrow \quad \text{}\underline{b}-R_{i}^{-2}\text{}\underline{c}-\text{}\underline{a}=jh{\sigma}{\Omega}{\mu}_{0}R_{i}\text{}\underline{a},\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{1}{{\mu}_{0}}\left[\frac{\text{d}\text{}\underline{A}_{\text{III}}(R_{e})}{\text{d}r}-\frac{\text{d}\text{}\underline{A}_{\text{II}}(R_{e})}{\text{d}r}\right]=jh{\sigma}{\Omega}\text{}\underline{A}_{\text{II},\text{III}}(R_{e})\quad \Rightarrow \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{156.0pt}\Rightarrow \quad B_{0}-R_{e}^{-2}\text{}\underline{d}-\text{}\underline{b}+R_{e}^{-2}\text{}\underline{c}=jh{\sigma}{\Omega}{\mu}_{0}(B_{0}R_{e}+R_{e}^{-1}\text{}\underline{d}).\end{eqnarray}$ (15) We now have four algebraic equations: (7), (8), (14) and (15), with four unknowns: a , b , c and d . Their solution is: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{a}=B_{0}\frac{\tan {\beta}}{\tan {\beta}_{i}+\tan {\beta}_{e}}\frac{1}{1+j\tan {\beta}},\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{b}=B_{0}\frac{\tan {\beta}}{\tan {\beta}_{i}+\tan {\beta}_{e}}\frac{\tan {\beta}_{i}-j}{\tan {\beta}-j},\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{c}=B_{0}\frac{\tan {\beta}}{\tan {\beta}_{i}+\tan {\beta}_{e}}R_{i}^{2}\tan {\beta}_{i}\frac{1}{-\tan {\beta}+j},\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{d}=R_{e}^{2}\text{}\underline{b}+\text{}\underline{c}-B_{0}R_{e}^{2},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \tan {\beta}_{i}=hR_{i}/{\delta}_{{\Omega}}^{2},\quad \tan {\beta}_{e}=hR_{e}/{\delta}_{{\Omega}}^{2},\quad \tan {\beta}=\frac{\tan {\beta}_{i}+\tan {\beta}_{e}}{1-\tan {\beta}_{i}\tan {\beta}_{e}[1-(R_{i}/R_{e})^{2}]}.\end{eqnarray}$ (19) We can now write the expressions for the magnetic vector potential in each region. For example, in region I (r < R_i) the magnetic vector potential written in Cartesian coordinates (x, y) has the following form: $\begin{eqnarray}\displaystyle A_{\text{I}}\left(x,y\right)=B_{0}\frac{\sin (2{\beta})}{2(\tan {\beta}_{i}+\tan {\beta}_{e})}(y-x\tan {\beta}). & & \displaystyle\end{eqnarray}$ (20) The magnetic vector potential is useful in plotting magnetic field lines. In our case, it is simply constant on a field line. The magnetic field lines inside the inner shell are given by the equation: $\begin{eqnarray}\displaystyle y-x\tan {\beta}=\text{const}, & & \displaystyle\end{eqnarray}$ (21) which represents a family of parallel straight lines, where tan 𝛽 is the slope of the lines, and 𝛽 is the angle between the lines and positive x-axis.

The magnetic field reduction by the rotating double-shell shield surrounding the usable space (region I) can be characterized by the shielding factor defined as in [10] and [11]: $\begin{eqnarray}\displaystyle S(\%)=\frac{B_{0}-B_{\text{I}}}{B_{0}}\times 100, & & \displaystyle\end{eqnarray}$ (22) where B₀ is the magnetic flux density without a shield and B_I is the magnetic flux density at a given point with the shield in place.

In general, the shielding factor is a function of the position at which it is calculated. However, the magnetic vector potential (20) describes a uniform inner field in region I. It can be easily shown that the field B_I and the shielding factor can be calculated from: $\begin{eqnarray}\displaystyle B_{\text{I}}=|\text{}\underline{a}|=B_{0}\frac{\sin {\beta}}{\tan {\beta}_{i}+\tan {\beta}_{e}},\quad S(\%)=\left(1-\frac{\sin {\beta}}{\tan {\beta}_{i}+\tan {\beta}_{e}}\right)\times 100. & & \displaystyle\end{eqnarray}$ (23)

3. Cylindrical shield of any thickness rotating in a static magnetic field

It is obvious that a thicker shield achieves a higher shielding factor. We will show in Section IV that a double-layer shield, consisting of two thin layers, can achieve a shielding factor comparable to a thick single shield with a large material savings. Although the shielding factor for a non-magnetic conducting cylindrical shield of any thickness rotating in a uniform, static magnetic field can be found in [10], it seems worthwhile to make the solution available here as well (Fig. 2).

Fig. 2.

Thick cylindrical shield rotating in a transverse uniform static magnetic field.

The shielding factor is given by ([10], Fig. 2): $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle S(\%)=\left(1-2\frac{R_{2}}{R_{1}}\frac{1}{|\text{}\underline{M}|}\right)\times 100,\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \text{}\underline{M}=\text{}\underline{x}_{1}\text{}\underline{x}_{2}[\text{K}_{2}(\text{}\underline{x}_{1})\text{I}_{0}(\text{}\underline{x}_{2})-\text{I}_{2}(\text{}\underline{x}_{1})\text{K}_{0}(\text{}\underline{x}_{2})],\quad \text{}\underline{x}_{1,2}=\frac{(1+j)R_{1,2}}{{\delta}_{{\Omega}}},\end{eqnarray}$ (24) where I₀(⋅), I₂(⋅), K₀(⋅), and K₂(⋅) are the modified Bessel functions.

4. Optimization problem

In this section, the following hypothesis is explored: a single cylindrical thick shield rotating at a constant angular velocity in a constant magnetic field perpendicular to the shield axis (Section 3.) can be replaced by a thin double-shell shield rotating at the same angular velocity (Section 2.) while reducing the mass of the material used (Fig. 3). It is assumed that the size of the shielded region as well as the assumed shielding effectiveness are identical for both types of shields. Under the above assumptions, the optimization problem is to find the parameters of the double-shell shield that provide the greatest reduction in material weight relative to the single shield defined as $\begin{eqnarray}\displaystyle {\delta}_{m}^{(\min )}(\%)=\min _{h,R_{e}}\left\{\frac{m_{2}-m_{1}}{m_{1}}\right\}\times 100, & & \displaystyle\end{eqnarray}$ (25) where m₁ and m₂ are the masses per unit length of the single and double-shell shields, respectively.

Fig. 3.

Model of equivalent double shield replacing single shield with a preset shielding efficiency.

First, the thickness of a single shield (h₀) made of non-magnetic material of specified electrical conductivity σ providing prescribed shielding effectiveness S is determined for a given shielding region (R₁) and rotation angular velocity 𝛺, e.g. R₁ = 100 mm, 𝛺 = 314 rad/s, σ = 58.5 MS/m (Cu), and S = 90%. Assuming a shielding efficiency of S = 90% indicates a tenfold reduction of the magnetic flux density inside the shield compared to the magnetic flux density B₀ of the external field. The thickness of a single shield is determined using Eq. (24). For the assumed values of the input parameters, it is equal to h₀ = 8.4 mm.

Next, the distribution of the shielding factor S of the double-shell shield as a function of the outer shell radius (R_e) and shells thickness h is determined. In the obtained distribution S (R_e, h), the path corresponding to the present value of the shielding factor for the single shield, i.e. S = 90%, is found (Fig. 4).

Fig. 4.

Distribution of the shielding factor for the double-shell shield together with the path corresponding to the factor S = 90% (red line).

Along the found path, which defines the set of parameters of the double-shell shield with a given shielding effectiveness (S = 90%), the values of the double shield mass reduction ratio (SMR) are calculated according to formula (25). By finding the minimum SMR, the parameters of the optimal double-shell shield providing the greatest mass reduction compared to a single thick shield are obtained (Fig. 5).

Fig. 5.

Distribution of the double shield mass reduction factor together with the minimum found.

Table 1

Mass reduction for various double shield systems

System	R_e (mm)	h (mm)	δ_m(%)
1	110	3.87	5.36
2	120	3.40	13.19
3	130	3.05	19.19
4	140	2.79	22.62
5	150	2.59	25.32

Fig. 6.

Magnetic field flux density distributions in the vicinity of shields with shielding efficiency S = 90% rotating with angular velocity 𝛺 = 314 rad/s determined analytically (a, b) and numerically by finite element method (c, d).

For the given sample input parameters, the found optimal double-shell shield (R_e = 242.7 mm,h_o = 1.74 mm) provides a 31.5% reduction in material mass compared to a single shield with identical shielding effectiveness. However, it should also be noted that there is always a reduction in the mass of the shields in a two rotating shield system compared to a single shield. Table 1 shows the shield parameters and corresponding mass reductions for systems with small outer shield radii while preserving the same shielded area (R₀ = 100 mm).

In order to verify the optimization process, the magnetic flux density distributions around the found shields were first determined analytically based on the formulas in Sections I and II. Then, the shields were simulated within the COMSOL Multiphysics^® program [12] using the finite element method. The magnetic field line distributions presented in Fig. 6 show the full agreement between the analytical calculations and the numerical simulations. The magnetic field lines correspond to the isolines of the magnetic vector potential, more precisely its imaginary part. The colors of the lines illustrate the potential values in the ranges ±0.2825 Wb/m and ±0.2249 Wb/m for Fig. 6a and 6b, respectively (blue color indicates the minimum and red color the maximum of the potential value).

It may also be noted that the magnetic field in the shielded region is uniform with a constant slope of the field lines, which can be easily determined analytically from the value of the coefficient a in the shielded region I, see Eq. (1). The inclination angle 𝛽 of the magnetic field lines for the analyzed thick single shield is about 99° and for the optimal thin double-shell shield about 136°.

5. Conclusions

A mathematical model for calculating the shielding effectiveness of static magnetic fields using a rotating double-shell cylindrical shield with electromagnetically thin layers made of non-magnetic conducting material is presented. Closed-form expressions for the shielding factor and magnetic field distribution were obtained. It is then shown that an optimum double-shell shield consisting of two thin layers allows a large saving of material compared to a thick single shield while maintaining the same value of the shielding factor.

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