How can you increase the magnetism of a permanent magnet?

2International Bureau of Weights and Measures (BIPM), Pavillon de Breteuil, F-92312 Sèvres Cedex, France

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Stephan Schlamminger

1National Institute of Standards and Technology, Gaithersburg, MD 20899, United States of America

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1National Institute of Standards and Technology, Gaithersburg, MD 20899, United States of America

2International Bureau of Weights and Measures (BIPM), Pavillon de Breteuil, F-92312 Sèvres Cedex, France

Email: moc.anis@gnosihseel

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Abstract

All present watt balances employ permanent magnet systems using a yoke with high permeability as flux return. Very often these systems are built with vertical and azimuthal symmetries. In its simplest form, the air gap is defined as the radial distance between an inner and outer yoke with the same height. This design leads to sloped field lines away from the plane of vertical symmetry. In order to suppress this vertical magnetic field, we propose two modified magnet constructions: (1) adding a permanent magnet in the outer yoke, and (2) decreasing the height of the outer yoke. Finite element method simulations show that, with reasonable optimization, either proposal can lower the vertical magnetic field by about one order of magnitude.

Keywords: watt balance, alignment, magnetic field

1. Watt balance magnet

The watt balance [1] is an instrument for realizing a unit of mass, the kilogram [2], in the revised International System of Units (SI) [3]. A summary of the worldwide development of watt balance experiments can be found in [4–6]. Traditionally, two measurement modes are used in the operation of a watt balance: in the weighing mode, the weight of a test mass, mg, is balanced by an electromagnetic force BlI = mg, where m, g, B, l and I denote the mass of the test mass, the local gravitational acceleration, the magnetic flux density at the coil position, the coil wire length and the current through the coil, respectively. In the velocity mode, the coil is moved with a velocity v through the same magnetic field, inducing an electromotive force (EMF) U, and the Bl can be obtained as Bl = U/v. An elimination of the factor Bl in both modes yields a virtual power balance equation, mgv = UI, by which the mass m is linked to the Planck constant h via measurements of the voltage, the resistance, the time and the length, see e.g. [7] for details.

An assumption of the watt balance experiment is that the geometric factor Bl (integral of the magnetic flux density along the coil wire path) is the same in weighing as it is in velocity mode. Since the exact coil position is hard to control in velocity mode, it is important to design a magnet whose Bl is insensitive to the exact location (x, y, and z) and orientation (θx, θy, and θz) of the coil. For example, if the horizontal magnetic flux density Br is proportional to 1/r (r=x2+y2 is the radial distance), the Bl is independent of x and y. Gauss’ law links the vertical component of the magnetic flux density to the horizontal component and as a consequence Br ∝ 1/r can be achieved with Bz = 0 in the entire field volume. With these two requirements, the geometric factor is independent to the coil horizontal movement, wire deformation [8] and yoke misalignment [9]. If Bz does not vanish, Bl will depend on the remaining degrees of freedom of the coil (x, y, θx, θy), and in this case the mechanical system must be very well designed to avoid coil motion along these degrees of freedom during the velocity mode. Conversely, a magnet with a geometric factor that is independent of these coordinates allows the use of a simpler mechanical system, effectively relaxing the requirements for aligning the coil to the magnetic field.

Recently, permanent magnets with high permeability yokes have been the preferred magnet system in watt balances at various laboratories [10–17]. A popular design is the BIPM type magnet, pioneered by the group working at the Bureau International des Poids et Mesures (BIPM). This design employs two opposing permanent magnets, shown in figure 1(a). A detailed construction of such magnets can be found in [14]. Generally, the magnetic flux of the permanent magnets is guided by yokes horizontally through an air gap, where the coil is located. The air gap has the shape of a cylindrical shell. Its inner and outer boundaries are given by the inner and outer yokes. Typically, the heights of the inner yoke and outer yoke are the same.

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Figure 1.

The upper subplot shows three different designs for watt balance magnets. In (a)–(c), only the right half cross section is shown. In these plots, PM denotes the magnetic material and the red arrow denotes the magnetization direction. The red dotted line shows the main flux flow path. (a) is a conventional construction with two permanent magnets and equal heights of the inner and outer yoke, i.e. Hi = Ho. (b) is the modified construction with four permanent magnets and identical heights for both yokes. (c) is a construction with two permanent magnets and a lower height of the outer yoke, i.e. Hi > Ho. The lower subplots show the magnetic flux lines at the upper end of the air gap. (d)–(f) correspond to the designs (a)–(c).

2. Problem

It has been shown in [18] that a magnet design depicted in figure 1(a) will create a noticeable vertical component of the field Bz, which has to be considered for the coil alignment. Figure 2 shows the contribution of rBr(r, z)/rc, where rc is the radius of the center of the air gap and Bz(r, z) based on simulations using the finite element method (FEM) of a BIPM type magnet used in the fourth generation watt balance experiment at the National Institute of Standards and Technology (NIST-4) [14]. The vertical magnetic component Bz in the air gap center is about 3 mT at |z| = 50 mm, i.e. 0.6% of the horizontal magnetic component. Bz(r) reaches a maximum value near the center of the air gap r = rc, see figure 2(b), which is different from the field contribution of a classic two parallel plate, where Bz ≈ 0 in the center of the gap center. This difference is caused by the asymmetry of inner and outer yoke–air boundaries. Figure 1(a) shows the inner/outer yoke-air boundary in pink/green respectively, while the magnet–air surface is highlighted in red. At least two asymmetries are apparent: first, the parallel boundary (red air-magnet surface) exists only on the surface of inner side. Second, the outer yoke–air particular boundary is much longer than that of the inner yoke side. Together these asymmetries will create a sloped magnetic flux towards the ends of the air gap and as a result produce a vertical field component Bz(r,z) as shown in figure 2(b).

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Figure 2.

Calculation results of magnetic flux density distribution of figure 1(a). (a) is the 3D magnetic flux density distribution of rBr(r, z)/rc (rc is the radius of the air gap center) while (b) shows the 3D magnetic flux density distribution of Bz(r, z). The focused region is z ∈ (−50 mm, 50 mm), r ∈ (200 mm, 230 mm). (c) and (d) are 2D plots of (a) and (b) in mT. The height and width of the air gap is H = 170 mm and δ0 = 30 mm in NIST-4 magnet.

3. Solutions

We propose two different methods to suppress the unwanted vertical flux density Bz. The first approach is to add a second pair of permanent ring magnets to restore the symmetry of the air-yoke and air-magnet boundaries. The second is to change the heights of the inner and outer yokes.

3.1. Approach 1: adding permanent magnets

Figure 1(b) shows the construction of a magnet system that has permanent magnets added to the outer yoke. This will divide the outer yoke into three separate parts, the upper, middle and lower segments. Since the upper and lower segments perform as the common yoke of both inner and outer yokes, it is possible to adjust the inner yoke and the middle segment of the outer yoke to make the air gap magnetically symmetrical. A simple estimation of the step widths a and b is based on an assumption that the ratio of the flux through both steps to the flux through the corresponding magnets is the same on both sides of the air gap. This requires

ab≈lalb,

(1)

where la and lb are the widths of the inner and outer permanent magnets, shown in figure 1(b). In order to verify equation (1), a FEM simulation has been performed. Two additional permanent magnets are set at the same height of the original pair. a = 30 mm, b = 14 mm, la = 120 mm and lb = 56 mm. The simulation result is shown in figure 3. The vertical component of the field in the entire air gap region has been greatly reduced. The improvement can be quantified with a figure of merit, the average absolute value (aav) of the vertical flux density Baav defined as

Baav:=∫z1z2∫riro|Bz(r,z)|drdz∫z1z2∫rirodrdz,

(2)

where the integrals are evaluated over the nominal extent of the air gap, accessible by the coil, given by r ∈ (ri, ro) and z ∈ (z1, z2). The Baav of the two designs shown on the left of figure 1 are calculated to be 0.44 mT and 0.066 mT in the region z ∈ (−50 mm, 50 mm), r ∈ (200 mm, 230 mm). By using additional permanent magnets, the average absolute value of the vertical flux density is reduced to about 15% of the original value while the peak value of Bz is reduced by more than an order of magnitude.

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Figure 3.

Calculation results of the magnetic flux density distribution of a construction shown in figure 1(b). (a) is the 3D magnetic flux density distribution of rBr(r, z)/rc while (b) shows the 3D magnetic flux density distribution of Bz(r, z). (c) and (d) are 2D plots of (a) and (b) in mT with a = 30 mm, b = 14 mm, la = 120 mm and lb = 56 mm.

Adding a permanent magnet into the outer yoke also has a significant disadvantage: the shielding of the yoke is compromised. In the original design, the coil is completely enclosed in iron. Magnetic fields from the outside generated by alternating currents cannot reach the coil. This is no longer true after the set of outer ring magnets have been added.

3.2. Approach 2: lowering outer yoke height

The second method to suppress the vertical field component is based on a compensation idea. By creating an additional magnetic flux pointing to the center of the outer yoke, the original flux pointing to the end of the outer yoke can be compensated, yielding a flat Bz profile. One approach is to shorten the height of the outer yoke, i.e. by designing Hi > Ho where Hi and Ho denote the height of the inner and outer yokes, shown in figure 1(c). This idea was implemented in the original BIPM design, see [19]. Another FEM is used to test this idea. For this FEM, all other parameters are kept the same as in figure 1(a) except Ho which is set as a variable. From the simulation, an optimal height of Ho = 151 mm has been found yielding a Baav = 0.046 mT, about an order of magnitude smaller than the original design with Ho = 170 mm. The results of the FEM are shown in figure 4.

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Figure 4.

Calculation results of the magnetic flux density distribution of a construction shown in figure 1(c). (a) is the 3D magnetic flux density distribution of rBr(r, z)/rc while (b) shows the 3D magnetic flux density distribution of Bz(r, z). (c) and (d) are 2D plots of (a) and (b) in mT. Hi = 170 mm, Ho = 151 mm.

A good way to obtain the ideal height of Ho would be to find an analytic solution in the air gap region. A calculation using an equivalent electrostatic field, shown in figure 5(a), has been attempted. The boundary condition has been met, however it is still difficult to give an analytic formula for Ho as a function of the air gap dimensions.

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Figure 5.

(a) Presents an electrostatic field problem with similar solutions of the magnetic contribution in the air gap of watt balance magnets. The image axis is r = rm where rm is the outer radius of the permanent magnet. (b) Shows a comparison of three different Br(r, z) − Ba profiles, where Ba is the mean value of Br(r, z), with r = rl, rc and rr before and after modifications.

Short of an analytic equation, the best outer yoke height can be found by iteration. The magnet system of a watt balance can be assembled and disassembled without major change. The achieved flatness of the field magnetic field can be measured and adjusted. One way to measure the field flatness is by using a field sensing coil made up by three stacked pairs of gradient coils with mean radii rl, rc and rr [14] to measure the gradient of Br(rl, z), Br(rc, z) and Br(rr, z) profiles. Figure 5 shows the simulated results of an example with rl = 209 mm, rc = 215 mm and rr = 221 mm, before and after modification of the outer yoke height Ho. It can be seen with a well optimized Ho (lower graph) that the gradients of three profiles Br(rl, z), Br(rc, z) and Br(rr, z) are comparable, while in a bad case (upper graph), the three magnet flux densities differ by about 1 mT. Therefore, the gradient consistency of the three Br(z) profiles can be used as a figure of merit to find the best Ho value.

4. Discussion

Both solutions presented here require small deviation from the BIPM type magnet design. The first solution requires a substantial change, i.e. the addition of two ring magnets in the outer yoke. This solution will add to the manufacturing cost and compromise the shielding property of the magnet system. The second solution, which only requires a change of the height of the outer yoke, was already incorporated in the original BIPM magnet design.

4.1. Magnetic field strength

In these magnet systems, the magnetic flux through the air gap is determined by the magnetomotive force and the reluctance of three different parts, the air gap, the permanent magnet and the yoke. With an up-down symmetrical construction, only half of the magnet needs to be considered. The magnetic equation of either half can be written as

ϕ(δ0μ0S0+δmμ0Sm+δyμSy)=Hmδm,

(3)

where ϕ is the magnetic flux through the upper/lower air gap. δ0, δm, δy denote the air gap width, the permanent height and the yoke average length along flux lines, and S0, Sm, Sy are the sectional area of the upper/lower air gap, the permanent and the yoke. The permeability of the vacuum is given by µ0 and μ is the permeability of the yoke, µ = µrµ0. Hm is the magnetic field strength of the permanent magnet in the vertical direction. Since µ≫µ0, the magnetic flux or Br is mainly determined by the reluctance of the air gap and the permanent magnet. Modifying the height of the outer yoke reduces S0 and Sy slightly, resulting in a very small increase in the magnetic flux in the gap. To calculate the effect of adding a magnet ring on the outer yoke in one half of the magnet, we make the following simplifications, S0 ≈ Sm and δ0 ≈ δm. This approximation is justified, since the design of most magnet systems is such that the area and thickness of the permanent magnetic material and the gap are similar. Hence equation (3) simplifies to

ϕ(2δmμ0Sm)≈Hmδm.

(4)

With this simplification, the magnetic flux seems to be independent of the thickness of the magnet. However, by introducing a permanent magnet ring in the outer yoke, the magnet thickness will be doubled, while the reluctance increases by 50%, i.e.

ϕ(3δmμ0Sm)≈2Hmδm.

(5)

Hence the magnetic flux through the air gap will increase by approximately 17%, i.e. the difference between 2/3 and 1/2.

4.2. Adjusting the radial magnetic flux profile

Very often the magnetic field profile, Br(z), of the manufactured magnet differs from the desired field profile. Several shimming methods can be used to modify the field profile. For example, the magnet can be disassembled and the yoke can be precision ground to change the field profile. This is usually done by increasing the gap size in regions where the field is larger. Another method, described in [14], takes advantage of the hysteresis properties of the yoke. The idea is to shift the yoke BH working point by partially changing the main flux path through the yoke based on the hysteresis feature of the yoke material. A BH point shift can increase the reluctance of the yoke, i.e. δy/(µSy), so that the magnetic flux density can be reduced in the original stronger part. Since the percentage of the yoke reluctance and the total magnetic reluctance remains more or less of the same order (see the analysis in section 4.1), these methods can be also used with both modifications proposed here.

4.3. Magnetic field change in the weighing mode

In the weighing mode of a watt balance, the flux generated by the current-carrying coil will be added to the flux produces by the permanent magnet system. This flux yields an additional force in the measurement, which is known as the reluctance force. Note that the majority of the additional coil flux will go directly through the air gap (twice), the inner yoke and the outer yoke. The magnetic equations for the additional flux can be written as

ϕc(δ02μ0πrc(H/2+z)+δ02μ0πrc(H/2−z))=NI,

(6)

where ϕc is the additional flux generated by the coil, rc the center radius of the air gap, H the full yoke height, z the coil vertical position, and N the coil number of turns. Based on equation (6), the reluctance force can be solved as

fr=I22∂(Nϕc/I)∂z=−2πμ0I2N2rczδ0H.

(7)

It can be seen from equation (7) that the reluctance force is determined by the air gap dimension and the ampere turns squared. The modification proposed in figure 1(b) leaves the air gap unchanged. Hence, the reluctance force will be identical to the original design. For the other solution, shown in figure 1(c), the outer yoke is shortened. This will increase the reluctance force by a few percent.

5. Summary

Yoke-based permanent magnet systems are widely used in watt balances. In a design where the inner and outer yokes have the same height, a large vertical component of the field can arise. This vertical component is caused by an asymmetry of the magnet–air boundary between the inner and outer yoke. A reduction of the vertical component of the magnetic field is desired for several reasons: this reduction decreases the alignment error, increases the parameter space for the coil radius, and extends the region of the flat Br(z) profile.

Two solutions to minimize the vertical flux density have been proposed: (1) adding a permanent magnet in the outer yoke, and (2) decreasing the height of the outer yoke. FEM simulations show the effectiveness of both solutions. The vertical component can be reduced by almost one order of magnitude by either solution.

The ideas to flatten the magnetic field were analyzed using the BIPM magnet design. This analysis can, without loss of generality, be applied to other magnet designs. For example, in the magnet designed by the National Physical Laboratory in the United Kingdom [20], the vertical component of the magnetic flux in the gap can be reduced by decreasing the height of the inner yoke or adding an additional permanent magnet ring in the inner yoke.

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