Magnetic shielding property for cylinder with circular,square,and equilateral triangle holes∗

Si-Yuan Hao(郝思源) Xiao-Ping Lou(娄小平) Jing Zhu(祝静)Guang-Wei Chen(陈广伟) and Hui-Yu Li(李慧宇)

1Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument,Beijing 100192,China

2Beijing Laboratory of Optical Fiber Sensing and System,Beijing Information Science&Technology University,Beijing 100016,China

3Beijing Key Laboratory of Optoelectronic Measurement Technology,Beijing Information Science&Technology University,Beijing 100192,China

Keywords: SERF atomic magnetometer,magnetic shielding,holes,finite element analysis(FEA)

1. Introduction

High sensitivity magnetometry techniques have gained more attention in many fields of basic research and applications, such as physics, biology, neuroscience, materials science, and geology.[1,2]The common magnetic sensors include fluxgate magnetometer, proton precession magnetometer,superconducting quantum interference devices(SQUIDs),and optically atomic magnetometers(OPAMs)working in the spin-exchange relaxation free(SERF)regime.[3]

SQUIDs and SERF magnetometer are two of the most sensitive magnetic sensors used in weak magnetic field detection. SQUIDs are demonstrated an ultrahigh sensitivity of 1 fT/Hz1/2,[4]but operate in a low temperature environment causing high maintenance charge and manufacturing costs. Compared with SQUIDs,the sensitivity of SERF magnetometer has reached 0.16 fT/Hz1/2in the low-frequency range.[5]Due to the advantages of low power consumption,non-cryogenic operation and high spatial resolution, SERF magnetometer has become one of the most potential weak magnetic field detection devices.[6]However, all the weak magnetic measurements based on SERF magnetometer are performed in magnetically shielded rooms for providing a nearly zero magnetic field circumstance.[7]The sensitivity of SERF magnetometer theoretically is 0.01 fT/Hz1/2, and the ambient magnetic field noise is the key factor limiting the augment of sensitivity in laboratory(0.16 fT/Hz−1/2).[8]Thus,high-performance magnetic shielding system plays an important role in suppressing the ambient magnetic field noise and increasing ultrahigh sensitive magnetic measurement.

Compared with air,high magnetic permeability(µ)materials with small magnetic resistance,provide magnetic circuit for the Earth’s magnetic field, resulting in high-performance magnetic shield inside. Cylinder is one of the most widely used structures, with advantages of simple manufacture and high shielding performance.[9,10]Hole located on the surface of shielding cylinder is necessary to keep light and wire of equipment passing through, resulting in more leakage fluxes and less shielding ability. Therefore, shielding property depended on hole structure is an important problem,which was studied by many scholars. Maet al.[11]explored the effect on shield ratio when holes and gaps exist, simultaneously. They quantitatively analyzed magnetic shielding ratio for a range of hole diameters and gap thicknesses in the transverse and longitudinal direction,using the finite element method. Besides,the relationship between shielding factor of a spherical shell and the diameter of hole was reported by Xuanet al.[12]

In this paper,the relationship between shielding property and the shape of hole structure was investigated by finite element analysis (FEA). There are circular, square, and equilateral triangle shapes. In addition, the anisotropy of shielding property was evaluated by analyzing the residual field on theX,Y,andZdirections,individually.

2. Shielding principle and simulation method

Shielding factors (S) and shielding effectiveness (SE)are used to assess the magnetic shielding performance of cylinder.[13]The definitions are given as follows:

whereBiandBoare the magnetic fields in the same position with and without magnetic shield,respectively.

Material and structure are two key factors for improving shielding performance of cylinder.[14]On the one hand,the Earth’s magnetic field is one of the typical static magnetic fields. Magnetic shielding cylinder is made of ferromagnetic material with high permeability. The permeability of a ferromagnetic material is several thousand times greater than that of air,and,the reluctance of a cavity is much greater than that of a ferromagnetic material.Therefore,the vast majority of the external magnetic induction lines will pass along the walls of the ferromagnetic material, and entering the cavity with very little fluxes.

On the other hand,multi-layer shielding cylinder is used to screen out the residual magnetic flux leaking into the cavity again and again to achieve better shielding effectiveness.Structure parameters include innermost layer radius (R) and length(L),single layer thickness(T),radial gap(∆R)and axial gap (∆L) between layers, and number of layer (n). The axial shielding factor is proportional ton,µ,T, ∆Rand inversely proportional toR,L,and ∆L.[15]The model of axial shielding factor was optimized by nonlinear programming quadratic line(NLPQL) algorithm, and a combination of optimum parameters was obtained.[16]

Because of the economic efficiency and the experimental error, FEA method was selected to simulate the Earth’s magnetic field and establish the three-dimensional model of shielding cylinder. The diagrams of the external and internal structures of the shielding cylinder are shown in Figs. 1(a)–1(b),which is made up of the curving tube body and the planar cover. In this research, structure parameters of shield on the basis of Ref.[16]are shown in Table 1,andR,L,∆R,∆L,T,andnare equal to 11.0 cm,33.0 cm,1.6 cm,2.0 cm,0.2 cm,and 5.0, respectively. Permalloy material is selected and its relative permeability is approximately equal to 2×104. The three-dimensional model of shielding cylinder with square,circular, and equilateral triangle holes are established and illustrated in Figs. 1(c)–1(e), separately. Because, the small vacuum vapor of the SERF magnetometer is located at center position of the shield,two holes are designed and placed at the center of cover and body,separately.

Table 1. Parameters of the magnetic shielding cylinder.

Fig.1. (a)External and(b)internal structures of the cylinder. The three-dimensional model of the shielding cylinder with square(c),circular(d),and equilateral triangle(e)holes.

Due to the structural anisotropy of shielding cylinder,there are significant differences about shielding properties between axis(Zdirection)and radial direction(Xdirection andYdirection). TheSEof radial direction is much more than that of axial direction.[17]Regarded as a uniformly distributed field, the geomagnetic field intensity vector is approximately equal to 50.0µT.HX,HY,andHZrepresent the orthogonal decomposition of the magnetic field vector on theX,Y, andZaxes. Based on the superposition theorem,HX,HY,andHZare set as 26.7µT in this paper to assess the shielding property in different directions,individually.

3. Results and discussion

The magnetic field in center position of cylinder is named“the central magnetic field(M)”,which is used to evaluate the shielding property of cylinder with various shape holes. The size of hole area(projected area)is an important factor in magnetic field transmission on the surface of cylinder.[18]The relationship of hole area(Shole),M,and hole shapes inX,Y,andZdirections is simulated and illustrated as shown in Fig.2.MC,MS,andMETare theMcontributed by cylinders with circular,square,and equilateral triangle holes,respectively.

The values ofSholerange from 0 to 150 cm2,and are limited byRequaling to 156.32 cm2. With the increase ofShole,the values ofMC,MS, andMETincrease with various rates.Smaxis defined as the maximum ofSholeon condition of low speed. WhenSholeis less thanSmax,MC,MS, andMETrise slowly as shown in the inset of Figs. 2(a)–2(c). The growth rate increases rapidly whenSholeis greater thanSmax. Therefore,Smaxis the optimum hole area, which is marked in the inset of Fig. 2. TheSmaxinX,Y, andZdirections are equal to 2.56, 23.00, 7.29 cm2, andMC,MS, andMETare equivalent to 1.953×10−2nT, 1.505×10−2nT, and 0.809 nT, respectively. Thus,SandSEinX,Y, andZdirections are equal to 1.367×106and 122.715, 1.774×106, and 124.986,and 3.302×105and 110.370 atShole=Smax,separately.

The effect of various hole shapes on shielding property is limited byShole.The critical area(SH)inX,Y,andZdirections are equal to 8.41, 23.00, 14.44 cm2respectively, as shown in Figs.2(a)–2(c). WhenSholeis less thanSH, the cylinder with three shapes of holes obtains the same remanent magnetization inside,indicating that the shielding property is unaffected by the shape of the hole. On the condition ofShole＞SH,MC,MS,andMETincrease in an order ofMC＞MS＞MET,indicating that the types of the hole shape have obvious effect on the shielding property.

In addition, the anisotropy of the shielding property is evaluated by analyzing the residual fields on theX,Y, andZdirections,individually. As is shown in Figs.1(c)–1(e),HZis perpendicular to the hole in cover of cylinder but horizontal to the hole in body. The relationship betweenHXand hole in tube body and cover is contrary compared withHZ.HYis horizontal to the two holes designed in this paper. (θcover,θbody)X,(θcover,θbody)Y,and(θcover,θbody)Zrepresent theθbetween the magnetic field in theX,Y, andZdirections and holes in cylinder cover and body,and are equal to(0◦,90◦)X,(0◦,0◦)Y,and(90◦,0◦)Z. The values ofMinX(MX),Y(MY),andZ(MZ)direction is in an order ofMZ ＞MX ＞MYunder the sameSholevalue and hole shape condition,as demonstrated in Fig.2. Therefore,cylinder with various hole structures shows good shielding forY-direction magnetic field becauseHYis horizontal to the two position holes.

Fig.2. The relationship between hole area,hole shapes,and M in X (a),Y (b),and Z(c)directions.

Fig. 3. The magnetic field distribution at Z, X, and Y axes of cylinder with circular, square, and equilateral triangle hole shapes on the condition of Shole ≈3.14 cm2 [(a)–(i)]and Shole ≈78.54 cm2 [(j)–(r)].

The right-angle structure in axial direction does not guide flux well, resulting in more magnetic flux leakings into the shielded region, worsening the shielding effectiveness.[19]Compared with the planar cover,there is a better shielding effectiveness of curving tube body, resulting in biggerMZthanMX. The axial shielding factor of cylinder is always smaller than the radial shielding factor.[20]The shielding property of cylinder was optimized by finite element analysis. The radial shielding factor was 20.6 times of the axial shielding factor.[21]Therefore, there is better shielding effectiveness in the radial direction, compared with axis direction, under the same hole shape and size condition.

Figure 3 shows the magnetic field distribution for cylinder with different hole shapes on the condition ofShole≈3.14 cm2(＜SH) andShole≈78.54 cm2(＞SH).SRefrenceis defined as the area where the magnetic field value is less than or equal to 1.326 nT,marked by the black dotted line. Under the same magnetic field circumstance, the value ofSRefrenceis bigger,the shielding capability is better. TheSC,SS, andSETrepresentSRefrencein cylinder with circular, square, and regular triangle holes. On the condition ofShole≈3.14 cm2,the hole shapes have little impact on the magnetic field distribution on the surface of cylinder,resulting inSC=SS=SET,as demonstrated in Figs.3(a)–3(i). By contrast,under the circumstance ofShole≈78.54 cm2,there is the same relationship amongSC,SS,andSETin cylinder cover and body,that isSC＜SS＜SET,as exhibited in Figs. 3(j)–3(r). Thus, the sequence of the shielding ability of cylinder with various holes is equilateral triangle＞square＞circular.

The sketch ofMdependence onSholeis shown in Fig.4.The purple, orange, and green curves representMC,MS, andMETas a function ofShole,respectively. Magnetostatic shielding obtained by shunting the magnetic flux and diverting it away from a shielded region. Compared with air, highpermeability material shows low magnetoresistance. It is realized by the introduction of ferromagnetic materials with high magnetic permeability, which creates a preferential path for the magnetic field lines. Higher leakage flux is the reason of thatMC,MS,andMETaugments withSholeincreasing.[22]The relationship betweenMandSholeis in accordance with the result reported by Liet al.[23]In addition, on the condition ofShole＜SH,various hole shapes have no effect on the shielding ability. Thus, high-permeability material is the major magnetic path. Under the condition ofShole＞SH, the sequence ofMis equilateral triangle＜square＜circular, indicating that the order of the shielding ability is circular＜square＜equilateral triangle. With the increase ofShole,there are more leakage fluxes shown in air,and shielded pass through air and high-permeability material.

Fig.4. The sketch of MC,MS,and MET dependence on Shole.

Magnetoresistance (Zm), the resistance of a magnetic path,is determined by the formula[24]

whereFandΦare the magnetomotive force and magnetic flux. Under the condition of a uniform magnetic circuit,Φis written as

whereµ,H,andSare the permeability of material,magnetic intensity,and cross-sectional area,respectively.Zmis written as

whereLis the length of magnetic circuit.

ZC,ZS,andZETrepresent magnetoresistances of cylinder with circular,square,and equilateral triangle hole,separately.LC,LS,andLETrepresent the shortest lengths of the magnetic circuit of circular, square, and regular triangle holes, respectively. The mechanism of the model is proposed and exhibited as illustrated in Fig. 4. The order ofLC,LS, andLETisLET＜LS＜LC, as shown in Fig. 4. On the same condition of cross-sectional area(S)and magnetic permeability(µ),the relationship betweenZC,ZS, andZETisZC＜ZS＜ZET, resulting inMC＞MS＞MET. Thus, the order of the shielding property is circular＜square＜regular triangle. There is the most excellent shielding property of cylinder with equilateral triangle hole structure.

4. Conclusion and perspectives

In summary, based on FEA method, the effect of hole shapes on shielding effectiveness of cylinder was investigated in this research. The selected shapes are square,circular,and equilateral triangle. Because of the structural anisotropy of cylinder,Mdependence onSholeinZ,Y,andXdirections was explored, and there is better shielding effectiveness in the radial direction,compared with the axis direction.

In addition,with the increase ofShole,there are more leakage fluxes,resulting in augmentsMC,MS,andMET. However,whenSholeexceedsSH, the sequence of the shielding ability is equilateral triangle＞square＞circular, due to the magnetoresistance of the leakage flux in air dielectric. The relationship betweenZC,ZS,andZETisZC＜ZS＜ZET,resulting inMC＞MS＞MET. Therefore, there is the most excellent shielding property of cylinder with equilateral triangle hole structure.

Finally,the growth rate ofMC,MS,andMETis different at variousShole. WhenSholeis less thanSmax,MC,MS,andMETrise slowly. On the condition ofShole＞Smax,the growth rate increases rapidly. Therefore,Smaxis identified as the optimum hole area.