Electromagnetic-based evaluation of different Halbach array topologies with gap consideration for the permanent magnet synchronous machines


This paper investigates the influence of various Halbach arrays permanent magnets (PMs) on the electromagnetic performance of a radial flux machine with outer rotor topology. The static analysis is performed using a 2-D finite-element analysis (FEA) of six different machines with different Halbach array-based orientations. The gap consideration between PM segments is considered for the first time. The aim of the study is to find the most suitable magnetization topology determination for the PMSMs which brings maximum airgap flux density, coenergy, output torque turning, back-EMF, output electromagnetic power, and minimum corresponding harmonics, cogging torque and the likelihoods of saturation. Additionally, a comparative study of a continuous distribution that commercially is called polar anisotropic will be entirely discussed. The proposed model is verified by FEA, together with its experimental investigation for the small wind power generation application in the urban area. In addition, commercial and environmental issues of the project have been highly considered to reduce \(\hbox {CO}_{2}\) emissions as the part of green power generation mission.

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\(L_{\mathrm{d}}\) :

d-axis inductance (mH)

\(L_{\mathrm{q}}\) :

q-axis inductance (mH)

\(M_{\mathrm{p}}\) :

Ripple torque (%)

\(T_{\mathrm{r}}\) :

Rated torque (N \(\cdot \) m)

\(T_{\mathrm{c}}\) :

Cogging torque (N \(\cdot \) m)

\(W_{\mathrm{c}}\) :

Coenergy (J)

\(\psi \) :

Flux linkage (mVs)

\(P_{\mathrm{o}}\) :

Output power (W)

\(\alpha _{\mathrm{pa}}\) :

The relative pole arc coefficient

\(D_{\mathrm{si}}, D_{\mathrm{so}}\) :

Inner and outer stator diameters (mm)

\(L_{\mathrm{m}}\) :

Thickness of permanent magnet (mm)

\(D_{\mathrm{ri}}, D_{\mathrm{ro}}\) :

Inner and outer rotor diameters (mm)

A :

Cross-sectional area of a counter \((\hbox {mm}^{2})\)

\(S_{\mathrm{d}}\) :

Effective slot depth (mm)

\(P_{\mathrm{arc}}\) :

PM arc (\(^{\circ }\hbox {e}\))

\(l_{\mathrm{i}}\) :

The initial segment number per pole

\(n_{\mathrm{m}}\) :

Minimum speed (rpm)

\(P_{\mathrm{EM}}\) :

EM output rated power (W)

\(n_{\mathrm{r}}\) :

Rated speed (rpm)

\(S_{\mathrm{w}}\) :

Slot width (mm)

\(M_{\mathrm{r}}, M_{\theta }\) :

The radial and tangential components of the airgap magnetic flux density (T)

\(W_{\mathrm{p}}\) :

The pole pitch

\(W_{{n\mathrm{pa}}}\) :

The pole arc of the \(n\hbox {th}\) magnet pole

\(W_{\mathrm{m}}\) :

The pole arc of mid-magnet


The fundamental amplitude of the radial component of airgap flux density (T)

\(B_{\mathrm{radial}}^h\) :

The \(i\hbox {th}\) harmonic amplitude of the radial component of airgap flux density (T)

\(J_{\mathrm{c}}\) :

Current density \((\hbox {A/mm}^{2})\)

\(l_{\mathrm{s}}\) :

Active stack stator length

m :

Stator number of phases

N :

Number of turns per phase

2pp :

Number of poles and poles pair

\(Q_{\mathrm{s}}\) :

Total number of the stator slots

\(W_{\mathrm{ph}\hbox {-}\mathrm{s}}\) :

Width of phase separator (excluding legs) (mm)

\(L_{\mathrm{a}\hbox {-}\mathrm{pm}}\) :

Airgap between PMs (mm)

\(\delta _{\mathrm{g}}\) :

Airgap length between stator and rotor (mm)

\(\delta _{\mathrm{gs}}\) :

Sides gap between each PM’s segment (mm)

\(\delta _{\mathrm{gm}}\) :

Middle gap between each PM’s segment (mm)

\(\mu _\mathrm{0}, \mu _{\mathrm{r}}\) :

The permeability of airgap, and the relative recoil permeability of PM

\(A_\mathrm{1}, A_\mathrm{2}\) :

The distance between PMs, and the width segment

\(B_{\mathrm{g}}\) :

Airgap magnetic flux density (T)

\(H_{\mathrm{g}}\) :

Airgap magnetic flux intensity (At/m)

\(H_{\mathrm{PM}}\) :

Magnetic flux intensity produced by magnets (At/m)

M :

The amplitude of magnetization vector in polar coordination (T)

\(\theta _{{i}}\) :

The angle between the center of the \(i\hbox {th}\) segment

\(R_\mathrm{0}, R_\mathrm{1}\) :

Inner and outer radii of the magnet (mm)

\(T_{\mathrm{ph}}\) :

The number of series turns per phase

\(k_\mathrm{w1}\) :

The fundamental harmonic winding factor

\({\varPhi }_\mathrm{m1}\) :

The fundamental magnet flux per pole (Wb)

\(\omega _{\mathrm{m}}\) :

Mechanical angular velocity (mech. Rad/s)

e :

Instantaneous EMF (V)


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Corresponding author

Correspondence to M. R. Barzegaran.



Machine design key parameters can be presented by the following Table 6.

Table 6 Design of the machine geometry

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Asef, P., Perpina, R.B., Barzegaran, M.R. et al. Electromagnetic-based evaluation of different Halbach array topologies with gap consideration for the permanent magnet synchronous machines. Electr Eng 100, 1847–1856 (2018). https://doi.org/10.1007/s00202-017-0656-6

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  • Halbach array
  • Permanent magnet synchronous machines
  • Electromagnetics
  • Gap consideration
  • FEA