Asymmetrical Flux Density Distribution in Stator Teeth of Surface Permanent Magnet Machines

This work is showing in detail the flux density behaviour in the stator teeth of a synchronous machine. A 3-phase Surface Permanent Magnet (SPM) motor is considered. These motors are widely employed in applications where high efficiency and power densities are required. This paper aims to analytically demonstrate the asymmetrical distribution of the stator teeth flux density. It is shown that this phenomena is depending on the number of slots per pole and per phase in the machine. Finally, a comparison with Finite Element results is given to validate the effectiveness of the proposed model.


I. INTRODUCTION
FOR any machine design, the computation of the iron losses is a very important aspect for improving the machine efficiency and thermal management [1]. This is strictly related to the estimation of the flux density distribution and its behaviour with respect to different harmonic components. Several works in literature are proposing techniques to minimise the iron losses both analytically and via finite element optimisation. In [2], a method for reducing the harmonics due to the permanent magnets is proposed. Therefore, the flux density analysis is essential to estimate and minimise the iron losses at the design stage [3]- [5]. Other works [6], implement complex subdomain models to predict the flux density within the stator core always considering symmetrical behaviour [7]. In this paper, the flux density distribution in different stator teeth has been analysed in detail considering a distributed winding with a single layer and full pitch. Such winding configuration is often chosen in fault-tolerant electrical machines [8], [9]. First, an analytical solution has been implemented in section II, to identify eventual asymmetries in different parts of the stator core. This has been compared with Finite Elements results, presented in section III, for validation. Finally, a summary of the paper outcomes is offered in the conclusions.

II. ANALYTICAL MODEL
The model proposed in this paper is based on the assumption of radial flux density in the air-gap, no slotting effect and linear materials [10]. Under these hypotheses, the magnetic field in the air-gap can be evaluated for each angular position (ϑ) in terms of Fourier series as:  whereH ρ (H ρ =H S,ρ +H P M,ρ ) is the ρ th Fourier series component of the spatial distribution of the magnetic field in the air-gap. The ρ th harmonic contribution produced by the stator currentsH S,ρ can be defined as: where N is the number of turns per phase, q the slots per pole and per phase, δ the air-gap thickness and K aρ is the winding factor for the ρ th field harmonic.ī S ρ is the ρ th current space vector defined by the Clarke transformation of the currents for the three-phase winding (U-V-W) considered: where j is the unity imaginary number (j 2 = −1).
Because the zero sequence current is null, due to the star connection of the three-phase winding, it is possible to write the following relationships among the space vectors: Finally, the rotor magnets contribution, to the ρ th harmonic of field in the air-gap, can be expressed with a good approximation as:H where B r is the remanence flux density, δ P M the magnet thickness, µ P M the magnet permeability, ∆ P M the magnet angular width, and θ r the rotor position in electrical radians. Table I is summarising the main machine parameters. The airgap flux density, for the considered SPM machine, when the rotor is aligned with the magnetic axis of phase U (θ r = 0), is shown in Fig. 1. The figure highlights the two components of the flux: the one produced by the magnets, and the flux generated by the stator currents, when the machine is fed with its rated current (ī S ρ = ji q e jϑr , with iq = 683 A pk ). The total flux density is obtained by the sum of two components. In order to evaluate the flux density in each stator tooth, the proposed model considers that all the flux, under the slot pitch, is crossing the air-gap and flowing through the same tooth. Therefore, the analytical evaluation of the teeth flux is provided for each T th tooth by: with ϑ T = 2πp N slots (T − 1) the angular position of the centre of the T th tooth (N slots is the overall slot number of the stator), N slots the angular pitch between two neighbouring slots (or teeth), L the active length of the machine, and R g the middle radius of the air-gap.
Substituting (1), with (2) and (5), in (6) it is possible to define the T th tooth flux in terms of Fourier series as: with: Under the assumption of the model, the flux density related to one tooth can be evaluated as: Eq. 9 can be rewritten in terms of Fourier series as: with the stator contribution: and the rotor contribution: e jρϑr e −jρϑ T = K P M ρ e jρ(ϑr−ϑ T ) , (12) with K Sρ and K P M ρ constants depending from the harmonic order.
At steady state operation, with an angular speed of the rotor ω (ϑ r = ωt), the flux density in each T th tooth varies in the time according to the following equation: From the first term of (14) it is possible to note that the magnets generate in each tooth infinite time harmonics (at angular frequency ρω). These present the same amplitude in each tooth and they are shifted with a fixed time delay dt = ϑ T ω . Therefore, the behaviour of the teeth is completely symmetrical in time. Instead, the flux density contribution produced by the currents to each tooth is composed by a sum of sinusoidal terms all at the same angular frequency (ω). In this case every component presents a time delay, from one tooth to the consecutive one, which depends on the field harmonic order (space harmonic): dt = ± ρϑ T ω . Therefore, the resulting flux density component is sinusoidal at angular frequency (ω), but it presents a different magnitude and phase depending on the considered tooth. It is worth to note that under the assumption of sinusoidally distributed winding (i.e., sinusoidal flux distribution of the armature field in the air-gap), the latter phenomenon does not appear, and (14) can be simplified as: This highlights that the asymmetrical distribution of flux density in the stator teeth is due to the higher order harmonics of the armature field. In Fig. 2 the expected waveforms of the flux density produced by the magnets and the stator currents are shown, respectively, for three consecutive teeth when the machine has q > 1 (in this case q = 3). The corresponding Fourier spectrum is given in Fig. 3 in terms of time harmonics with respect to the fundamental, with the current and magnet contributions. Whereas the contribution of each permanent magnet spatial field harmonic generates a different time harmonic, the ones produced by the stator result in flux density components at the fundamental frequency. The analytical model, allows to decouple the effect of each stator harmonic contribution generated at the air-gap, by (11) and (13), as shown in Fig. 4. The same analyses are carried out for a machine with one slot per pole and per phase (q = 1): in Fig. 5, flux density waveforms by currents and magnets are shown respectively, Fig. 6 displays the corresponding Fourier spectra and Fig. 7 shows the stator spatial harmonic contributions.   As expected from the analytical model, the only case with the harmonic contributions equally shifted among the teeth is for q = 1.
The angular phase shift of the flux density in one tooth   caused by the ρ th space harmonic (ρϑ T ) can be rewritten considering that ρ is equal to 3n ± 1, with n an even number (n = 2k) and N slot = 6pq. Therefore, it is possible to re-write   the angular phase shift as follows: It can be seen that the first term in (16) is a multiple of 2π only when q = 1, while the second term does not depend from the considered space harmonic. It results that q = 1 is the only solution for which the phase shift is not affected by the harmonic order.

III. FEA VALIDATION AND RESULTS
In order to validate the analytical model, the obtained results have been compared with the flux density values determined by means of FE simulations. The comparison is carried out considering only the radial component of the flux density at the centre of each tooth. This is a good approximation for the analysis of asymmetries among different teeth, neglecting local effects. To compare the FE results with the analytical model, linear materials have been considered [11]. As a case study, two SPM machines, with q = 3 and q = 1 (the parameters are listed in Table I), have been considered. The two machine geometries are shown in 8 and 11 respectively. Firstly, the 36 slots machine (with q = 3) is analysed. Fig. 9 shows the comparison between FE and analytical results for the flux density distribution in all the 36 stator teeth. The magnet and current contributions (presented at the top and bottom, respectively), when the rotor is at its initial position (ϑ r = 0), are considered separately. Fig. 12 displays the same results obtained for the machine with 12 teeth (q = 1). To compare the behaviour for all the rotor positions, Fig. 10 and Fig. 13 show the distribution of flux density in three consecutive teeth for one mechanical revolution of the rotor (ϑ r = 0 : 360), for q = 3 and q = 1, respectively. The results presented are validating the qualitative behaviour expected by the analytical model, i.e., the presence of asymmetries in the flux density distribution among the stator teeth when q > 1.

IV. CONCLUSION
In this work, a detail investigation of the flux density distribution in the teeth of an SPM machine has been carried out via analytical and FE analyses. The results highlight that in a three-phase machine with distributed winding the flux density distribution is not symmetrical when the number of slots per pole and per phase is higher than one (q > 1). This aspect can be taken into account in the early stage of the machine design, when simplified analytical models are used to predict the initial geometrical parameters of the electrical machine. In particular, the presented model can be used to predict the effects of these parameters on the asymmetrical distribution of iron losses and saturations, which are related to the teeth flux density. The behaviour, expected by the simplified analytical model, has been validated against FE results for two examples of SPM machines with different numbers of slots per pole and per phase: q = 3 and q = 1. It is found that the flux density in different teeth is asymmetrical for a machine with q = 3, while it is symmetrical for a machine with q = 1. Fig. 11. SPM machine geometry with q = 1. Fig. 12. Flux density for each tooth when the rotor is aligned with the phase A (ϑr = 0) and the stator currents are controlled to generate the rated torque for the machine with q = 1. Fig. 13. Flux density comparison for three consecutive teeth between FEA and analytical approach with q = 1.