# A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{pm}, is retrieved, and is compared with FEM simulations.

## 2. Halbach PM Disposition and Properties

- -
- the flux density distribution amplitude is reinforced in the upper surface of the array, in the air gap, while it is weakened in the lower surface;
- -
- this different distribution leads to the strengthening of the magnetic field where the tangential force is produced, while it reduces the need for a thick yoke width, lightening the mass and the inertia of the moving part;
- -
- through the increase in the number of segments per pole, the shape of the normal flux density distribution in the air gap becomes almost sinusoidal, which leads to the obtention of more sinusoidal EMF waveforms in the windings, and a smoother force with reduced ripple;
- -
- of course, the increase in the number of segments per pole implies increasing tolerance issues in the Halbach array assembly, and higher costs for the device: this requires to find a trade-off between the performance quality and system complexity.

## 3. Magnetic Field between Smooth Ferromagnetic Surfaces

#### 3.1. Introduction to the Model and Elementary Blocks

- The stator and moving surfaces are supposed to be smooth, and separated by an air gap of uniform width;
- The magnetic field is solved in 2D, lying in the paper plane, and assumed invariant in the direction perpendicular to the paper plane;
- The machine is supposed to extend indefinitely in the direction perpendicular to the paper plane, so that the end effects are negligible;
- The relative recoil permeability (typically in the range 1.05–1.10 for rare earth PM materials) is assumed to be equal to the vacuum permeability (µ
_{r}= 1); thus, at each point inside the air-gap width, it is assumed that**B**(z) = μ_{o}∙**H**(z) occurs, both in air and in the PMs; - The iron permeability is assumed to be infinite, so the method of superimposition can be used;
- The iron parts are perfectly laminated; thus, no eddy current can be induced in the iron by the varying magnetic field;
- The conductors placed inside the air gap extend indefinitely perpendicular to the paper plane;
- The skin effect due to alternating currents flowing in the conductors is neglected.

_{c}

_{0}[25,26]. The use of current sheets, suitably positioned in the space, allows us to model permanent magnets with different directions of magnetization. The problem is then usually solved according to the subdomain method, and the solution is retrieved in a Fourier series.

_{p}= x

_{p}+ jy

_{p}is the complex coordinate where the current-carrying conductor is positioned; and z and z

_{p}are the respective complex conjugates. With the use of this formula, the elementary blocks consisting of current sheets are built, in order to model different magnetization directions.

#### 3.2. Air-Gap Magnetization Due to a Current Sheet Disposed Normally to the Iron Surfaces

_{p}= x

_{p}+ jy

_{p}, inside an air gap between two smooth ferromagnetic surfaces. The current sheet is powered by a current density ∆, equal to the magnet linearly extrapolated coercive force H

_{c}

_{0}, flowing outside the paper plane.

_{p}, according to (1), the infinitesimal magnetic strength air-gap complex vector at the point z can be written as:

_{p}

_{1}= 0 and y

_{p}

_{2}= h

_{cs}, the expression of the magnetic strength

**H**

_{ncs}(z) distribution, due to a current sheet orientated normally to the iron surfaces, is retrieved:

#### 3.3. Air-Gap Magnetization Due to a Current Sheet Disposed Tangentially to the Iron Surfaces

_{p}= x

_{p}+ jy

_{p}, inside an air gap between two smooth ferromagnetic surfaces. The current sheet is powered by a current density ∆ = H

_{c}

_{0}flowing outside the paper plane. If we consider an infinitesimal current sheet element ∆ × dx

_{p}, according to (1), the infinitesimal-magnetic-strength air-gap complex vector in the point z can be written as:

_{p}

_{1}and x

_{p}

_{2}, the expression of the magnetic strength distribution

**H**

_{tcs}(z), due to a current sheet orientated tangentially to the iron surfaces, is retrieved:

#### 3.4. PM Magnetized along an Axis Orientated Normally to the Iron Surfaces

_{m}height and b

_{m}width, centered with respect to the origin, and magnetized normally with respect to the iron surfaces, along the y axis (north PM), is modelled with two current sheets: one positioned in x

_{p}= +b

_{m}/2, and with a negative linear current density (−Δ); the other one positioned in x

_{p}= −b

_{m}/2, and with a positive linear current density (+Δ). For both the current sheets, it holds that y

_{p}

_{2}= h

_{cs}= h

_{m}; thus, via (3), the magnetic strength distribution

**H**

_{PM.n}(z), due to a centered PM, magnetized normally to the iron surfaces, can be expressed as follows:

**H**

_{PM.n}

_{_Δx}(z) =

**H**

_{PM.n}(z − Δx).

**B**

_{PM}

_{.n}= μ

_{o}∙

**H**

_{PM}

_{.n}, corresponding to B

_{y}= Im(

**B**

_{PM}

_{.n}) and to B

_{x}= Re(

**B**

_{PM}

_{.n}), respectively, due to a single normally magnetized PM, obtained via the proposed closed-form, analytical expression (6), compared with a FEM 2D simulation realized with a commercial software [27], used as a benchmark, for b

_{m}= 102 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm. As can be observed, the agreement is excellent, with the curves obtained by means of the proposed closed-form analytical expressions being perfectly superimposed with the benchmark obtained from the commercial FEM solver.

#### 3.5. PM Magnetized along an Axis Tangentially Orientated to the Iron Surfaces

_{m}height and b

_{m}width, centered with respect to the origin, and tangentially magnetized, in a horizontal direction along the x axis (north PM toward the right), is modelled with two current sheets: one positioned in y

_{p}= 0, and with a negative linear current density (−Δ); the other one positioned in y

_{p}= h

_{m}, and with a positive linear current density (+Δ). Both the current sheets are centered at the origin, with x

_{p}

_{1}= −b

_{m}/2; x

_{p}

_{2}= +b

_{m}/2.

**H**

_{PM.t}(z), is expressed as:

**H**

_{PM.t}

_{_Δx}(z) =

**H**

_{PM.t}(z − Δx).

**B**

_{PM}

_{.t}= μ

_{o}∙

**H**

_{PM}

_{.t}, corresponding to B

_{y}= Im(

**B**

_{PM}

_{.t}) and to B

_{x}= Re(

**B**

_{PM}

_{.t}), respectively, due to a single tangentially magnetized PM, obtained via the proposed closed-form, analytical expression (8), compared with a FEM 2D simulation realized with commercial software [27], used as a benchmark, for b

_{m}= 102 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm. In this case, also, the agreement is excellent, with the curves obtained by means of the proposed closed-form analytical expressions being perfectly superimposed with the benchmark obtained from the commercial FEM solver.

#### 3.6. Halbach PM Segment with Magnetization Orientated according to an Angle θ_{pm}

_{pm}with respect to the x axis, it can be decomposed in the form:

_{pm}:

**H**

_{H1_Δx}(z,θ

_{pm}) =

**H**

_{H1}(z − Δx, θ

_{pm}).

**B**

_{H}

_{.1}= μ

_{o}∙

**H**

_{H}

_{.1}, corresponding to B

_{y}= Im(

**B**

_{H}

_{.1}) and to B

_{x}= Re(

**B**

_{H}

_{.1}), respectively, due to a single magnetized PM with orientation θ

_{pm}= 30°, obtained via the proposed closed-form, analytical expression (12), compared with a FEM 2D simulation realized with commercial software [27], used as a benchmark, for b

_{m}= 102 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm. In this case, also, the agreement is excellent, with the curves obtained by means of the proposed closed-form analytical expressions being perfectly superimposed with the benchmark obtained from the commercial FEM solver.

## 4. The Halbach PM Arrays

#### Halbach Array Using the PM Segment Model

_{m}. Figure 10 shows some examples of this type of Halbach array. The PM segments belonging to the same pole are shown with magnetization vectors drawn in red, and the change in the orientation angle between adjacent segments is the same. The number of PM segments per pole is indicated with N

_{sp}: Figure 10 shows some Halbach array dispositions, from N

_{sp}= 1 (which is a degenerate case) to N

_{sp}= 6.

_{m}between adjacent magnetization vectors, equal all along the periphery, is given by:

_{m}-th PM segment (j

_{m}= 1, 2,…, N

_{sp}) equals:

_{m}-th segment, acting alone, produces the following flux density:

_{sp}PM Halbach segments of one pole produces the following flux density distribution in the air gap:

^{k}equals 1 for even k values and −1 for odd k values, corresponding to the north and south pole groups of Halbach segments, respectively.

_{sp}= 6, calculated via a FEM 2D magnetostatic simulation realized with commercial software [27], used as a benchmark. In order to ensure the condition of periodicity, the adopted boundary conditions at the left and right ends are master–slave.

_{y}exhibits a roughly sinusoidal shape, while B

_{x}is limited in amplitude, with a peak value in the order of 10% of the B

_{y}peak value.

_{sp}= 10 segments/pole, each with the same width, calculated via FEM 2D magnetostatic simulation realized with commercial software [27], used as a benchmark. In this case, also, in order to ensure the condition of periodicity, the boundary conditions at the left and right ends are master–slave.

_{sp}implies a flux density distribution improvement, with an almost-sinusoidal shape of B

_{y}, and a greatly reduced amplitude of the component B

_{x}, with a peak value in the order of 5% of the B

_{y}peak value.

_{y}flux density distribution, evaluated on the basis of the proposed closed-form, analytical expression (20) for Halbach arrays with different numbers of segments/pole.

_{y}flux density distribution as a function of the number of segments/pole N

_{sp}in the Halbach array. As expected, the THD decreases with the increase in N

_{sp}. Figure 14b reports the harmonic spectra of the B

_{y}flux density distribution, evaluated on the basis of the proposed closed-form, analytical expression (20), expressed in a percentage with respect to the fundamental, and limited to a harmonic order equal to 40, for the cases of N

_{sp}= 3 (red), N

_{sp}= 5 (green), and N

_{sp}= 7 (blue).

_{sp}, harmonics are present only in correspondence of harmonic orders satisfying the condition:

## 5. Calculation Time Comparison between Analytical and FEM Approaches

- -
- the number of calculated points adopted in the analytical method (500), uniformly distributed within the pole pitch τ, is chosen in a such a way to correctly reproduce the flux density distribution everywhere within τ, for all the considered cases of Table 2;
- -
- the calculation time in the analytical method corresponds to the total time needed to evaluate the x and y flux density components in the chosen number of points;
- -
- the calculation time in the FEM numerical approach corresponds to just the time to solve the field; it does not include the time needed to calculate the x and y flux density components in the chosen number of points.

- -
- the accuracy of FEM analysis depends on the mesh refinement level, while the analytical field calculation in each point is direct, without any discretization refinement;
- -
- the calculation times of the cases 1, 2, and 3 are low (of the order of 0.1 s), while the calculation times of the cases 4 and 5 are higher (of the order of 0.2–0.3 s): in fact, the cases 1 and 2 include 2 current sheets, the case 3 includes 4 current sheets, while the case 4 includes 6 × (1 + 2 × 2) = 30 current sheets; finally, the case 5 includes 10 × (1 + 2 × 2) = 50 current sheets. In fact, generally, the field superposition of an higher number of current sheets requires an higher calculation time.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\stackrel{-}{A}$ | conjugate of the complex quantity A (written in bold) |

b_{m} | peripheral width of the PM [m] |

B_{cs} | flux density due to a current sheet [T] |

B_{x}, B_{y} | x, y components of the B vector in the air gap [T] |

B_{H} | flux density vector of a generic Halbach segment inside an array [T] |

B_{H}_{1} | flux density vector of a Halbach segment centred at the origin [T] |

B_{H}_{1.pole} | flux density due to a group of N_{sp} PM Halbach segments of one pole [T] |

B_{H.multi}_{.pole} | flux density due to multiple peripherally adjacent Halbach arrays [T] |

g | total equivalent air-gap width (g = h_{m} + δ) [m] |

h_{cs} | linear extension of a current sheet [m] |

h_{m} | normal height of the PM [m] |

H_{c} | magnetic strength due to a current I in a concentrated conductor [A/m] |

H_{c}_{0} | PM recoil line linearly extrapolated coercive force [A/m] |

H_{H}_{1} | magnetic strength of a single Halbach array segment [A/m] |

H_{H}_{1_Δx} | magnetic strength due to a Halbach segment, displaced by Δx [A/m] |

H_{ncs} | magnetic strength due to a normally orientated current sheet [A/m] |

H_{tcs} | magnetic strength due to a tangentially orientated current sheet [A/m] |

H_{PM.n} | magnetic strength due to a centered PM, normally magnetized [A/m] |

H_{PM}_{.n_Δx} | magnetic strength due to a normally magnetized PM, displaced by Δx [A/m] |

H_{PM.t} | magnetic strength due to a centered PM, tangentially magnetized [A/m] |

H_{PM}_{.t_Δx} | magnetic strength due to a tangentially magnetized PM, displaced by Δx [A/m] |

$\overrightarrow{M}$ | magnetization vector of a PM [A/m] |

M_{x}, M_{y} | x, y components of the M vector of a PM [A/m] |

N_{sp} | number of Halbach segments per pole [-] |

THD_{B} | THD of the flux density distribution due to a Halbach array [%] |

z = x + jy | complex coordinate of the air-gap point where the field is calculated [m] |

z_{p} = x_{p} + jy_{p} | complex coordinate of the point p of the current carrying conductor [m] |

β_{m} | angular displacement between near-segment magnetization vectors [deg] |

δ | mechanical air gap (distance between iron- and PM-faced surfaces) [m] |

Δ | linear current density of a current sheet distribution [A/m] |

μ_{o} | vacuum permeability [H/m] |

μ_{r} | PM recoil permeability [pu] |

θ_{pm} | orientation of a PM segment magnetization vector, referred to the x axis [deg] |

τ | pole pitch, peripheral extension of one pole [m] |

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**Figure 1.**Example of Halbach PM disposition: each segment magnetization is orientated as rotated by a constant angle with respect to the previous one; the group of segments identified by magenta arrows forms a north pole, while the group of segments identified by blue arrows forms a south pole.

**Figure 2.**Graphical representation of the magnetic field H

_{ncs}(z) evaluated at a point of coordinates z = x + jy in an air gap between two smooth, parallel iron surfaces, generated by a current sheet, normally orientated with respect to the iron surfaces, distributed along the points p, with coordinates z

_{p}= x

_{p}+ jy

_{p}.

**Figure 3.**Graphical representation of the magnetic field H

_{tcs}(z) evaluated at a point of coordinates z = x + jy in an air gap between two smooth, parallel iron surfaces, generated by a current sheet, tangentially orientated with respect to the iron surfaces, distributed along the points p, with coordinates z

_{p}= x

_{p}+ jy.

**Figure 4.**Permanent magnet with h

_{m}height and b

_{m}width, centred with respect to the origin, with magnetization orientated normally to the iron surfaces along the y axis (north PM, magnetization angle θ

_{pm}= π/2 with respect to the x axis), modelled with two current sheets positioned, respectively, in x

_{p}= +b

_{m}/2 and x

_{p}= −b

_{m}/2, with a linear current density, respectively, equal to (−Δ) and (+Δ).

**Figure 5.**Distribution of the (

**a**) y and (

**b**) x components of

**B**

_{PM}

_{.n}= μ

_{o}∙

**H**

_{PM}

_{.n}, due to a single normally magnetized PM, obtained analytically (continuous red line) via (6), and via a FEM 2D simulation (dashed blue line), for b

_{m}= 102 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm.

**Figure 6.**Permanent magnet with h

_{m}height and b

_{m}width, centered at the origin, with magnetization orientated tangentially with respect to the iron surfaces along the x axis (north PM, magnetization angle θ

_{pm}= 0 with respect to the x axis), modelled with two current sheets positioned, respectively, in y

_{p}= 0 and y

_{p}= h

_{m}, with a linear current density, respectively, equal to (−Δ) and (+Δ).

**Figure 7.**Distribution of the (

**a**) y and (

**b**) x components of

**B**

_{PM}

_{.t}= μ

_{o}∙

**H**

_{PM}

_{.y}, due to a single tangentially magnetized PM, obtained analytically (continuous red line) via (8), and via an FEM 2D simulation (dashed blue line), for b

_{m}= 102 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm.

**Figure 8.**A Halbach permanent magnet segment with h

_{m}height and b

_{m}width and magnetization vector $\overrightarrow{M}$, orientated according to the angle θ

_{pm}with respect to the x axis, decomposed in the two PM with magnetization orientated according to the x and y vector components, respectively. The PM magnet orientated according to the x axis is modelled by means of two current sheets positioned, respectively, in y

_{p}= 0 and y

_{p}= h

_{m}, with a linear current density, respectively, equal to (−Δ) and (+Δ), while the PM magnet, orientated according to the y axis, is modelled by means of two current sheets positioned, respectively, in x

_{p}= +b

_{m}/2 and x

_{p}= −b

_{m}/2, with a linear current density, respectively, equal to (−Δ) and (+Δ).

**Figure 9.**Distribution of the (

**a**) y and (

**b**) x components of

**B**

_{H}

_{.1}= μ

_{o}∙

**H**

_{H}

_{.1}, due to a single PM magnetized with orientation θ

_{pm}= 30°, obtained analytically (continuous red line) via (12) and via a FEM 2D simulation (dashed blue line), for b

_{m}= 102 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm.

**Figure 10.**Halbach permanent magnet arrays with N

_{sp}segments/pole ranging from 1 to 6, and magnetization vectors highlighted in red; all segments have the same peripheral width b

_{m}= τ/N

_{sp}and the same angular displacement between adjacent magnetization vectors, equal to β

_{m}= π/N

_{sp}[rad].

**Figure 11.**Field map of a Halbach array with N

_{sp}= 6 segments/pole, each with the same width b

_{m}, calculated via FEM 2D magnetostatic simulation realized with commercial software (numerical periodic solution based on master–slave boundary conditions): b

_{m}= 17 mm; h

_{m}= 10 mm; g = h

_{m}+ δ = 11.5 mm.

**Figure 12.**Distribution of the (

**a**) y and (

**b**) x components of

**B**

_{H}= μ

_{o}∙

**H**

_{H}, due to the Halbach array of Figure 11, with N

_{sp}= 6 PM segments/pole, all with width b

_{m}= τ/N

_{sp}: analytically calculated distribution (continuous red line) obtained via (20); numerically calculated distribution (dashed blue line), based on FEM 2D simulation with master–slave boundary conditions at the left and right ends, b

_{m}= 17 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm. The distributions have been calculated along the line at one-half the air-gap width, as drawn in Figure 11.

**Figure 13.**Field map of a Halbach array with N

_{sp}= 10 segments/pole, each with the same width b

_{m}, calculated via FEM 2D magnetostatic simulation realized with commercial software (numerical periodic solution based on master–slave boundary conditions): b

_{m}= 10.2 mm; h

_{m}= 10 mm; g = h

_{m}+ δ = 11.5 mm.

**Figure 14.**Distribution of the (

**a**) y and (

**b**) x components of

**B**

_{H}= μ

_{o}∙

**H**

_{H}, due to the Halbach array of Figure 13, with N

_{sp}= 10 PM segments/pole, all with width b

_{m}= τ/N

_{sp}: analytically calculated distribution (continuous red line) obtained via (20); numerically calculated distribution (dashed blue line), based on FEM 2D simulation with master–slave boundary conditions, b

_{m}= 17 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm. The distributions have been calculated along the line at one-half the air-gap width, as drawn in Figure 13.

**Figure 15.**Harmonic content of the y component of the flux density distribution in the air gap of a Halbach PM array: (

**a**) THD as a function of the number N

_{sp}of segments/pole; (

**b**) harmonic spectra of the y component of the flux density distribution in the air gap of a Halbach PM array for a number of segments/pole N

_{sp}= 3 (red), N

_{sp}= 5 (green), N

_{sp}= 7 (blue), with τ = N

_{sp}∙ b

_{m}= 102 mm, h

_{m}= 10 mm, g = h

_{m}+ δ = 11.5 mm.

Intel Pentium Processor, CPU N4200, 1.10 GHz, 4 GB RAM | |

PC Operating System: Windows 10, 64 Bit | |

FEM software commercial tool: | Ansys Electronics Desktop 2021 R2 |

Software for the analytical calculation: | PTC MathCad 15.0 |

**Table 2.**Cases for the comparative analysis of the calculation times of the analytical and FEM approaches.

Case N° | Description |
---|---|

1 | Single permanent magnet, centred with respect to the origin, with magnetization orientated normally to the iron surfaces (as depicted in Figure 4 and Figure 5) |

2 | Single permanent magnet centered in the origin, with tangential magnetization with respect to the iron surfaces (as depicted in Figure 6 and Figure 7) |

3 | Single Halbach permanent magnet segment with magnetization vector $\overrightarrow{M}$ orientated according to an angle θ_{pm} = 30° with respect to the x axis (as depicted in Figure 8 and Figure 9) |

4 | Halbach array with N_{sp} = 6 segments/pole, with the same peripheral width b_{m} (as depicted in Figure 11 and Figure 12); two Halbach arrays at the left and at the right of the central array, for a total of five Halbach arrays |

5 | Halbach array with N_{sp} = 10 segments/pole, with the same peripheral width b_{m} (as depicted in Figure 13 and Figure 14); two Halbach arrays at the left and at the right of the central array, for a total of five Halbach arrays |

**Table 3.**Analytical and FEM approach calculation data. Number of calculation sampling points within one pole pitch: 500.

Case N° | Analytical Calculation | FEM Calculation | Δt Ratio | |||
---|---|---|---|---|---|---|

Data | Data | |||||

Calculation Time Δt _{A} [s] | Iterations N° | Energy Error [%] | Mesh Triangles N° | Computational Time Δt_{F} [s] | Comp. Times Ratio Δt _{A}/Δt_{F} [%] | |

1 | 0.12 | 8 | 3.21 × 10^{−4} | 4.99 × 10^{4} | 20 | 0.58 |

2 | 0.07 | 14 | 1.34 × 10^{−4} | 2.40 × 10^{5} | 67 | 0.10 |

3 | 0.08 | 13 | 1.36 × 10^{−4} | 1.86 × 10^{5} | 66 | 0.13 |

4 | 0.20 | 12 | 3.08 × 10^{−4} | 4.95 × 10^{4} | 18 | 1.11 |

5 | 0.27 | 12 | 3.77 × 10^{−4} | 4.80 × 10^{4} | 16 | 1.68 |

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## Share and Cite

**MDPI and ACS Style**

Di Gerlando, A.; Negri, S.; Ricca, C.
A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays. *Magnetism* **2023**, *3*, 280-296.
https://doi.org/10.3390/magnetism3040022

**AMA Style**

Di Gerlando A, Negri S, Ricca C.
A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays. *Magnetism*. 2023; 3(4):280-296.
https://doi.org/10.3390/magnetism3040022

**Chicago/Turabian Style**

Di Gerlando, Antonino, Simone Negri, and Claudio Ricca.
2023. "A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays" *Magnetism* 3, no. 4: 280-296.
https://doi.org/10.3390/magnetism3040022