L Barrientos

M Borja-Lloret

A Etxebeste

E Muñoz

J F F Oliver

A Ros

J Roser

C Senra

R Viegas

G Llosá

Performance evaluation of MACACO II Compton camera

Keywords: Compton camera, Hadron therapy, LaBr 3, Silicon photomultipliers

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Compton cameras (CCs) have gained renewed interest in different areas including medical applications, such as hadron therapy treatment monitoring [1] or molecular imaging in nuclear medicine [2]. In recent years Compton cameras have been developed using different types of detectors, e.g. scintillator crystals coupled to SiPMs [3], solid state [4] or gaseous detectors [5]. In particular, SiPMs are being used increasingly on CCs due to their easy handling [6].

The IRIS group at IFIC-Valencia has developed a second version of a multilayer Compton camera, MACACO (Medical Applications CompAct COmpton camera) [7]. The system is composed of three detector layers of LaBr 3 continuous scintillator crystals coupled to SiPMs arrays. This version features new detectors which mainly present an enhanced energy and angular resolution. The work presented here has several objectives that go beyond the tests carried out with the first prototype. First, a system characterization has been carried out in order to assess the performance improvement (section 3.1). Then, a set of simulations have been performed employing the new GATE module CCMod which allows to reproduce accurately the experimental results obtained with Compton cameras [8]. The experimental results have been employed to validate the simulations (section 3.2). Finally, reconstructed images from the experimental data acquired in two and three plane modes have been included as an indicator of the MACACO II performance (section 3.2.4).

Although the primary objective of the developed system is hadron therapy treatment monitoring, the work presented here is generic and suitable for different applications.

Material and methods

Experimental prototype description

MACACO II is composed of three parallel detector layers. The first and second detectors are conformed by LaBr 3 continuous scintillator crystals from Saint Gobain with dimensions of 25.8 x 25.8 mm 2 and 5 mm thickness coupled to MPPC arrays S13361-3050AE-08 (with TSV technology) from Hamamatsu Photonics. Each array has 64 (8 x 8) channels of 3 x 3 mm 2 . The third layer is composed of a 32.4 x 36 x 10 mm 3 monolithic LaBr 3 crystal, coupled to four SiPM arrays (S11064050P(X1)) also from Hamamatsu Photonics with 16 (4 x 4) channels of 3 x 3 mm 2 . The micro pixel pitch is 50 µm for all three detectors.

The intrinsic spatial resolution of the detectors is 1.2 mm. This value has been determined and reported in previous works for the first and second detector layers [7] and for the third one [9]. Both crystals and photodetectors are confined in a black plastic holder that prevents ambient light from reaching the photodetectors.

The signals from the 64 SiPMs channels of each detector are read out by an ASIC VATA64HDR16 from Ideas [10]. All detector components are attached to a printed circuit board (PCB) designed to provide mechanical support, SiPMs bias voltage and to route the ASIC output signals to the data acquisition (DAQ) system. A common bias voltage is provided to each detector by a power supply specifically made for SiPMs [11]. The bias voltage of each SiPM pixel is individually adjusted by means of an 8-bit input DAC (digital-to-analog converter), allowing to modify the voltage in 256 steps, with a maximum range of 1.5 V. Each PCB is connected to a custom DAQ board that manages the readout process and controls the output signals digitalization and the ASICs configurations. Figure 1 shows a schematic drawing of the experimental setup. The boards are connected to the computer via Ethernet and driven by the DAQ software that allows to select the different acquisition modes (singles and coincidences) and set the ASIC configurations (threshold, DACs). In singles mode, the readout of each detector is activated internally in the DAQ board by the detector trigger. In coincidences mode, an external trigger is employed to start the readout process. Upon arrival of a detector trigger, each DAQ board outputs a trigger signal (fixed width of 50 ns) that is fed to an external coincidence board. The coincidence board is based on a Virtex 5 FPGA evaluation board; it outputs a signal when any two or all three detectors trigger signals overlap. The output signal is sent back to the three DAQ boards and all three detectors are read out. The measurements presented in this work were performed inside a Dycometal CCK climatic chamber that allows to control the temperature with an accuracy of 0.2 • C. Figure 2 shows the system inside the climatic chamber. The determination of the optimum bias voltage was performed in two steps. In a first step, the voltage that provides the best energy resolution without applying DAC corrections was determined (54 V in case of the first and second detectors and 72 V for the third one). In a second step, a finer sweep around that value in steps of 0.1 V was performed. At each step, the optimum values of the voltage for each SiPM pixel were selected. The DACs voltage equalization per SiPM pixel was done by placing the source at 7 cm from the first detector to ensure that the detector was uniformly irradiated and to determine the differences in SiPM pixel response [10].

Since the system response is linear at the energies under study, a linear fit was used in all cases, to calibrate the energy taking the 511 and 1275 keV 22 Na energy peaks as references. After calibrating the data, the detectors energy resolution was determined by performing a Gaussian fit of the 511 and 1275 keV photopeaks.

System characterization

Experimental data have been acquired with the telescope in time coincidences mode employing either two or three planes. Several configurations with different inter-plane distances were set for different studies including angular resolution and coincidence detection efficiency which were compared with simulated data (see section 3). The configurations used in this work are listed in table 1. A 22 Na point-like source has been employed and placed at a distance of 40 mm in the z-axis with respect to the origin for all cases. In order to calibrate the energy spectra when measuring in coincidences, data were also acquired in singles mode. The angular resolution measure (ARM) has been determined from the first two telescope planes as the difference between the Compton angle θ c , calculated from the Compton scattering equation (Eq. 1) and the geometrical angle θ g (calculated from the interaction positions of the photons in the two crystals and the source position).

cos θ c = 1 - m e c 2 Ẽ1 E 0 (E 0 -Ẽ1 ) (1)
Where E 0 is the initial energy of the incoming photon and Ẽ1 corresponds to the energy deposited at the first interaction. Figure 3 shows a schematic diagram of a two-interaction event on a Compton camera, where both Compton and geometrical angles are represented.

In our case, the determination of these interaction positions in the continuous crystals is estimated using a model of scintillation light transportation [12] which takes into account the contribution of the diffuse reflections in the crystal surfaces. The ARM has been determined by measuring with a 22 Na point-like source in configuration M6, which provides a balance between coincidence detection efficiency and spatial resolution [13].

The coincidence detection efficiency was measured at 1275 keV for two and three planes for the configurations M1, M2 and M4 to M8. Energy cuts have been set in order to select the events with total energy deposition over 600 keV and under 1350 keV in all cases. This energy was selected in order to avoid contamination from other energies, which is unavoidable with the 511 keV contribution.

Imaging tests

Coincidences data were acquired with two and three interaction events for configurations M6 and M3 respectively keeping the 40 mm distance from the 22 Na source to the origin, and placing it at different positions in the x-y plane. In the case of two planes, the source positions were 0 (centered), +5 and +9 mm in the x-axis direction (Figure 1). In the case of three detector layers, data were measured only for the case in which the source was positioned at the center.

The List Mode Maximum Likelihood Expectation Maximization (ML-EM) algorithm [14,15] was used for the reconstruction, and a maximum number of 10 iterations was chosen. For all images a Field-of-View (FOV) of 101 x 101 x 51 mm 3 has been used with a voxel size of 1 mm 3 .

Finally, and in order to reduce the noise in the images, a median filter was employed after the final iteration.

Simulations

The new prototype has been modeled using GATE v8.2, a Monte Carlo simulation toolkit based on Geant4 [16]. GATE offers the user several features for easy modeling of complex sources and electronics read-out [17].

In order to facilitate the comparisons between experimental and simulated data, the CCMod actor [8] has been employed. GATE and CCMod simplify the simulation of Compton cameras allowing realistic comparisons between different prototypes in a versatile environment designed for medical physics simulations where dosimetry calculations can be performed in the same framework.

Point-like source simulations

Simulations have been carried out with the aim of reproducing the performance of the Compton telescope prototype. The detectors have been modeled as LaBr 3 scintillator crystals with a density 5.08 g/cm 3 . Other elements that are part of the detector (black holders, SiPMs, aluminum housing and PCBs) were also simulated, whereas the acquisition boards (DAQs), the structure supporting the detector and the climatic chamber were not considered in the simulation.

The radioactive decay process and the QGSP BIC HP EMZ physics list [18,19] were used in the simulations.

A simulated spherical 22 Na source with a radius of 0.01 mm, activity of 370 kBq and a halflife of 2.6 years was simulated. It was contained in a cylindrical plastic volume similar to the encapsulation of the 22 Na source used in the laboratory.

In order to reproduce the experimental response of each detector layer, the energy resolution was set to 5.6 %, 6.6 % and 7.2 % FWHM at 511 keV in the simulation for the first, second and third layers respectively (see section 3).

The measured singles in the different layers of the system were externally sorted into coincidences applying a 50 ns time coincidence window (TCW). The events selection process and image reconstruction are similar to the experimental case.

In order to minimize the discrepancies between the simulated and experimental two planes coincidence spectra, a global energy threshold of 85 keV was also applied to the total energy deposition in each layer as in [8] but after coincidences. For the three plane simulations, a general energy threshold of 20 keV has been applied in order to avoid low energy events and not affect the 511 keV photopeaks.

Different geometrical configurations were simulated in order to reproduce the experimental measurements. Simulated data from the first two-planes in configuration M6 (table 1) were employed for the determination of the ARM and the comparison with experimental spectra. Configurations M5 to M8 were used for coincidence detection efficiency studies. When using three planes, configurations M1, M2 and M4 have been used to obtain the coincidence detection efficiency.

Results

Detector characterization

The best energy resolution achieved with the experimental data was obtained applying a bias voltage of 53.8 V to the first and second detectors, which correspond to 2.2 V lower than the recommended operation bias voltage range given by the manufacturer (≈ 56 V). Degradation of the measured spectra for voltages higher than 53.8 V was observed due to ASIC saturation. The voltage for the third detector was found to be 72.1 V, 1.1 V higher than the recommended one.

The energy resolution obtained in such conditions was 5.6 %, 6.6 % and 7.2 % FWHM at 511 keV and 4.8 %, 5.2 % and 5.5 % FWHM at 1275 keV, for the three detectors, respectively.

Coincidence studies

The results shown in this section include both the experimental measurements and the simulated data.

Detector performance

Figure 4 shows experimental and simulated coincidence results in two and three plane mode with a 22 Na source. Figures 4a and4b show the spectra resulting from summing the energies deposited by the photons in two detectors in the two plane case and in three detectors in the three plane case. In both cases, the 511 and 1275 keV photopeaks can be clearly observed. The sum peak of 1786 keV, produced by the simultaneous absorption of both 22 Na energies, can also be distinguished. In addition, for the two-plane case, Figure 4c shows the experimental deposited energies on the second detector ( Ẽ2 ) versus the first detector ( Ẽ1 ). The diagonal lines denote a coincidence produced by a Compton-scatter interaction on the first plane and a photoelectric absorption on the second one, while the horizontal and vertical lines represent events produced by two different photons (randoms), where one is completely absorbed. Figure 4d shows the simulated deposited energies on the second detector ( Ẽ2 ) versus the first detector ( Ẽ1 ) with coincidences data.

Angular resolution

The ARM has been calculated from both experimental and simulated results employing data from 1275 keV photons. The distribution peak has been fitted with a Gaussian function.

Detection efficiency

The experimental and simulated results of the coincidence detection efficiency with two and three planes are shown in this section.

Reconstructed Images

22 Na point-like source images have been reconstructed employing the coincidence data acquired with two and three planes in configuration M6 and M3 respectively (table 1). Figure 7a shows

x-y images reconstructed from two plane mode data for three different source positions. Each source was reconstructed using 44,000 events per position. The left one corresponds to the origin position relative to the detector center, whereas the other sources correspond to displacements of 5 and 9 mm in the x-axis direction, respectively. The three images have been reconstructed

independently and plotted in the same image. Figure 7b shows line profiles along the x-axis through the maximum intensity point of the reconstructed images. The determination of the source positions was obtained by calculating the mean of the fit of a Gaussian function over the profiles. The mean of the fitted profile on the right side is located at 3.99 mm from the middle one, and the mean of the latter is located at 4.98 mm from the left one, values that are in accordance with the source displacements described in section 2.2.3. Figure 7c shows an image of the source reconstructed in three plane mode centered on the detector. The number of events used in the image reconstruction was of 5,517. Figure 7d shows the line profile in the x-axis from image 7c.

The FWHM of the reconstructed images and the mean values for two interactions are shown in table 2. Varying the geometrical configurations can improve the efficiency to some extent (at the price of degrading the spatial resolution [13]), but not significantly. The use of larger detectors, of size 20 x 20 x 0.5 cm 3 would allow to improve the coincidence detection efficiency by at least one order of magnitude (≈ 10 -2 ) at 1275 keV.

The GATE CCMod actor has shown to be a useful tool in the simulation of Compton cameras.

The simulations reproduce quite accurately the experimental data in terms of energy deposition and efficiency. In spite of the small discrepancies in the obtained graphs, mostly due to random and backscatter events produced by non-simulated passive material interactions, they can be reliably employed for the simulated performance studies.

Conclusions and future work

The new system shows a significantly improved performance when compared to the previous prototype. In spite of the significant progress, mainly in terms of energy and angular resolution, the system is still not able to reach the necessary performance for its final application and, therefore possible improvements are being assessed. Our time coincidence window (50 ns) imposes an important hardware limitation that must be solved in the future, since this contributes to degrade the quality of reconstructed images by presence of random events. Future tests will include the evaluation of different SiPMs from several manufacturers to enhance the energy resolution and the improvement of the readout electronics. In addition, the MADDAQ board currently employed is being replaced by the AliVATA readout system, that will allow us to operate the three system detectors with just one board and improve the readout speed.

Figure 1 :

 1 Figure 1: MACACO II setup scheme in coincidences mode (not to scale). The origin of the coordinate system is located on the center of the first detector surface.

Figure 2 :

 2 Figure 2: Image of MACACO II inside the Dycometal CCK climatic chamber.

2. 2 .

 2 Experimental measurements 2.2.1. Detector characterization The detector characterization includes optimized bias voltage determination, calibration and energy resolution measurement. The acquisition of experimental data has been done keeping the temperature stable at 20 • C for each measurement to avoid variations in the SiPMs response. The employed radioactive isotope was a 22 Na point-like source with an activity of 370 kBq which allows us to study the detector response at two different energies: 511 and 1275 keV.

Figure 3 :

 3 Figure 3: Geometrical configuration M6 has been used to estimate ARM.

Figure 4 :

 4 Figure 4: Comparison between the experimental measurements and GATE Monte Carlo simulations for the three planes Compton telescope. The 22 Na sum spectra are shown for (a) two interaction events. (b) Three interaction events. Deposited energy plots on a two plane Compton telescope. (c) Experimental Ẽ2 versus Ẽ1 deposited energy. (d) Simulated deposited energy ( Ẽ2 versus Ẽ1 ).

Fig 5 shows

 5 the measured and simulated angular error distribution. The obtained ARM was 7.5 • and 8.0 • FWHM for the simulated and experimental cases, respectively.

Figure 5 :

 5 Figure 5: Simulated and experimental angular error distribution. The obtained ARM was of 7.5 • and 8.0 • FWHM at 1275 keV for the simulated and experimental cases, respectively.

Figure 6

 6 shows such comparison at 1275 keV for different inter-plane distances and with the gamma source at a distance of 40 mm in the z-axis from the origin. In order to disentangle the intrinsic coincidence efficiency from the acceptance effect due to the plane separation, the solid angle subtended by the second detector from the center of the first detector was calculated for each inter-plane distance in the two-plane case. Taking into account such geometrical factors in each case and averaging the results for all cases, an experimental intrinsic coincidence efficiency of 5.5 x 10 -2 has been obtained.

Figure 6 :

 6 Figure 6: Coincidence detection efficiency at 1275 keV as a function of the distance between the detectors for (a) Two interaction events. (b) Three interactions events.

Figure 7 :

 7 Figure 7: Point-like source reconstructed images at 1275 keV. (a) Two-layer Compton telescope images at different positions. (b) Horizontal line profile plot of the reconstructed images with two interaction events. (c) Three-layer Compton telescope image. (d) Horizontal line profile through the peak of the plot with three interaction events.

Table 1 :

 1 Experimental arrangements employed in the data acquisition for different inter-plane distances.

		Plane 1 -Plane 2 Plane 2 -Plane 3
		(mm)	(mm)
	Configuration M1	30	30
	Configuration M2	40	40
	Configuration M3	50	40
	Configuration M4	50	50
	Configuration M5	40	-
	Configuration M6	50	-
	Configuration M7	70	-
	Configuration M8	80	-

Table 2 :

 2 FWHM (mm) of the x and y profiles and mean values of the reconstructed images.

		2-Plane Compton telescope
	Position	FWHM x	FWHM y	Mean x
		(mm)	(mm)	(mm)
	Left	3.49	3.74	51.66 ± 0.17
	Middle	3.76	3.88	56.64 ± 0.05
	Right	3.59	3.71	60.63 ± 0.12
		3-Plane Compton telescope
	Position	FWHM x	FWHM y	Mean x
		(mm)	(mm)	(mm)
	Middle	4.81	4.95	-

Acknowledgments

This work has received funding from the Spanish Ministerio de Ciencia e Innovación (FPA2017-85611-R), from Generalitat Valenciana (AICO/2019/070), from PIC 2018 programme (CSIC 2018FR0032 and CNRS IEA 08272) and from European Commission H2020 ENSAR2-MediNet (project number 654002). The simulation developments were also supported by the LABEX PRIMES (ANR-11-LABX-0063) and SIRIC LYriCAN Grant INCa-INSERM-DGOS-12563. Group members are supported by Ramón y Cajal, FPU, UVEG Atracció de Talent and GV SEJI-GENT and Grisolía contracts.

A second version of a Compton telescope prototype has been developed with significantly improved performance when compared with its predecessor [20]. The experimental evaluation tests show a substantial enhancement in terms of energy resolution (5.6 % FWHM at 511 keV) when compared to the 7-10 % obtained in the previous work [13]. The system imaging capabilities tested with point-like sources, show improved images in terms of FWHM for the two acquisition modes: a FWHM of 3.49 mm was obtained for two-layer events, and 4.81 mm FWHM for threelayer events for 1275 keV gamma rays when measuring at 40 mm distance from the source to the first detector. In terms of angular resolution, an ARM of 8.0 • FWHM at 1275 keV has been obtained. Although this value is considered insufficient for medical use [21], the implementation of new detectors with improved energy and/or spatial resolution could enhance this quantity. Finer segmentation of the SiPMs would allow improving the spatial resolution [22]. However, this would increase the number of photodetector channels and the system complexity. Energy resolution in our system could potentially be further improved, since the new detectors are being operated below the recommended operating voltage in order to avoid electronics saturation. Alternative solutions such as ASICs with larger dynamic range or lower gain photodetectors are being considered.

A direct comparison with other developed systems is not a trivial task, given the broad variety of materials utilized as detectors on Compton cameras, the different energies and geometrical configurations employed. In spite of the above, in an attempt to compare our results with other published work, the published results obtained at the same energy have been selected. As expected, the energy resolution of our system falls behind the results with solid-state detectors [23,24], but it is superior to the systems employing GAGG scintillator crystals coupled to SiPMs, which quote energy resolution of 15.06 % FWHM [START_REF] Yoshihara | IEEE/SICE International Symposium on System Integration (SII)[END_REF] and 9.7 % FWHM [START_REF] Shimazoe | IEEE Nuclear Science Symposium and Medical Imaging Conference[END_REF] at 511 keV.

Another factor to consider is the small size of the detectors that limits the detection efficiency.

We report total experimental coincidence detection efficiencies of 1 x 10 -3 and 5 x 10 -5 for two and three interaction modes respectively, with inter-plane distances of 50 mm at 1275 keV in configuration M6 (table 1) and slightly smaller detectors than the first prototype. The reported values vary with detector size and inter-plane distance. In addition, an experimental intrinsic coincidence efficiency of 5.5 x 10 -2 was obtained by separating the geometrical component from the total coincidence detection efficiency in the two-plane case. Some examples from other groups at the same energy include a total coincidence detection efficiency of 1.7 x 10 -5 [START_REF] Steinbach | [END_REF] employing solid-state detectors larger than those of MACACO II, or 2 x 10 -4 [START_REF] Yoshihara | IEEE/SICE International Symposium on System Integration (SII)[END_REF] with GAGG crystals.


Duc Tan Vu

Ngac Ky Nguyen

Eric Semail

Torque Ripple Eliminations for Multiphase Nonsinusoidal Permanent Magnet Synchronous Machines

Keywords: Multiphase machine, seven-phase PMSM, nonsinusoidal back-EMF, torque ripple elimination, ADALINE, artificial intelligence

This paper is to propose a control scheme, a combination of the classical field-oriented control (FOC) technique and artificial intelligence (AI), to obtain constant torques in multiphase non-sinusoidal permanent magnet synchronous machine (PMSM) drives. Higher torque density, easier fabrication, and lower costs are several advantages of non-sinusoidal back electromotive force (back-EMF) machines over sinusoidal ones. However, multi-harmonics existing in back-EMFs possibly generate torque ripples, reducing torque quality of the drive. Therefore, in this paper, an adaptive linear neuron (ADALINE), a simple type of AI, is combined with the classical FOC technique to eliminate these torque ripples. The proposed control scheme is validated by numerical results with a seven-phase PMSM. In addition, these results are compared with an existing strategy to prove its effectiveness.

I. INTRODUCTION

Multiphase machines have been becoming a favorable choice in applications that require high functional reliability. An increase in the number of phases (n>3) gives multiphase machines more degrees of freedom for design and control. Therefore, fault tolerance, low power per phase rating, and low-ripple torque can be several advantages of multiphase machines over conventional three phase machines [START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF][START_REF] Barrero | Recent Advances in the Design, Modeling, and Control of Multiphase Machines Part I[END_REF].

The design requirement is relaxed as the number of phases is high. Indeed, according to the multi-reference frame theory [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF], a n-phase symmetrical machine is characterized by (n+1)/2 (if n is odd) and (n+2)/2 (if n is even) characteristic planes, known as reference frames. One frame is associated with a group of harmonics. A constant torque can be ideally obtained when only a single harmonic of currents and back-EMFs exists in each reference frame (except the zerosequence frames). Therefore, an increase in the number of phases results in more reference frames, permitting to have more harmonics in back-EMFs. Consequently, in a welldesigned multiphase machine, the torque generated by nonsinusoidal back-EMFs can be constant by imposing constant d-q currents in each reference frame. Therefore, the FOC technique with classical proportional integral (PI) controllers that have been popular in industry are facilitated. This characteristic is not available in a three-phase machine. Indeed, a three-phase machine with two reference frames (including one zero-sequence frame) requires the classical constraints on sinusoidal back-EMFs and sinusoidal currents. Therefore, a multiphase machine leads to fewer constraints on design than a three-phase machine.

However, due to the imperfection of machine manufacture, non-sinusoidal back-EMFs could contain more than one harmonic per frame, conflicting the multi-reference frame theory. Negative impacts of these harmonics on current control have been discussed and solved in [START_REF] Vu | Eliminations of Low-frequency Current Harmonics for Five-phase Open-end Winding Non-sinusoidal Machine Drives applying Neural Networks[END_REF][START_REF] Vu | Current Harmonic Eliminations for Seven-phase Non-sinusoidal PMSM Drives applying Artificial Neurons[END_REF]. The approach is similar to [START_REF] Wang | Current Harmonics Suppression Strategy for PMSM with Non-Sinusoidal Back-EMF Based on Adaptive Linear Neuron Method[END_REF] for a high-speed three-phase machine drive. Nonetheless, even when current control quality is guaranteed, torque ripples still exist due to the interaction between currents and back-EMF harmonics. The vectorial approach [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF], known as maximum torque per ampere (MTPA) for all phase numbers, all types of back-EMFs, and all operating modes, can be a solution to eliminate torque ripples. Time-variant current references for control by this approach require a controller whose bandwidth increases with the rotating speed. Therefore, hysteresis controllers have been chosen in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF], leading to inevitable variable switching frequencies, high switching losses, and electromagnetic compatibility problems. Another approach using ADALINEs can be found in [START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF] in which optimal currents for non-sinusoidal multiphase PMSMs. However, the entire optimal torque is used in the learning process, making all current references have highly oscillating waveforms as in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF]. An artificial Gaussian-type radial basis function neural network has been applied to control the speed of a 5-phase drive without knowledge of accurate machine models or parameters [START_REF] Guo | Model Reference Adaptive Control of Five-Phase IPM Motors Based on Neural Network[END_REF]. However, this neural network (multi-layers) is more complicated than ADALINEs, requiring more learning time. In addition, hysteresis-type controllers are used for the control of timevariant current references, resulting in unexpected effects as in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF]. Several control techniques, such as model predictive control [START_REF] Li | An Improved Model Predictive Direct Torque Control Strategy for Reducing Harmonic Currents and Torque Ripples of Five-Phase Permanent Magnet Synchronous Motors[END_REF] or pulse width modulation (PWM) carrier phase shift [START_REF] Wang | Selective Torque Harmonic Elimination for Dual Three-Phase PMSMs Based on PWM Carrier Phase Shift[END_REF], have been applied to reduce torque ripples. However, torque ripples caused by back-EMF harmonics cannot be eliminated.

In this paper, a novel control scheme for non-sinusoidal back-EMF machines is proposed to obtain constant torques regardless of back-EMF waveforms. The proposed scheme is a combination of the classical FOC technique and a simple ADALINE. Thanks to self-learning and easy implementation, ADALINE is chosen in this study. Importantly, the ADALINE takes advantage of harmonic components in the torque to avoid the calculation burden, a long learning time, and highly fluctuating current references. Basically, these torque harmonics are multiples of the number of phases. The effectiveness of the proposed control scheme is shown by numerical results with a 7-phase PMSM drive. However, the proposed control scheme can be applied to every electric machine with an arbitrary number of phases and back-EMF characteristics, especially in real-time industrial systems. This paper is organized as follows: modeling of a sevenphase machine will be described in section II; impacts of harmonic components of back-EMFs will be shown in section III; section IV is to describe control schemes; numerical results will be presented in section VI.

II. MODELING OF A 7-PHASE MACHINE

The following assumptions are considered to model the machine: seven phase windings of the machine are equally shifted and wye-connected; non-salient rotors are used; the 1 st , 3 rd , and 9 th harmonics of back-EMFs account for highest proportions while other harmonics such as the 13 th , 19 th , and 11 th are tiny but unneglectable; the magnetic circuit saturation is not considered in calculations of back-EMFs and fluxes. Thus, the voltage and electromagnetic torque can be expressed in natural frame as follows:

  s di v R i+ e dt  L (1) T em ei T Ω  (2)
with

  T A B C D E F G v v v v v v v v    T A B C D E F G i i i i i i i i    T A B C D E F G
e e e e e e e e  where v, i, and e are the 7-dimensional vectors of phase voltages, phase currents and back-EMFs, respectively; Rs is the resistance of the stator winding of one phase; [L] is the 7by-7 stator inductance matrix; Tem is the electromagnetic torque of the machine; Ω is the rotating speed of the rotor. To facilitate the classical FOC technique, the classical 7-by-7 transformation matrices, Clarke   T and Park   P , are applied to convert parameters of the machine from natural frame into decoupled d-q frames. For example, the transformation for currents is given by:

   dq ii  PT (3) with   T 1 1 9 9 3 3 d q d q d q z dq i i i i i i i i 
where the selection of harmonic components of currents for control in each d-q frame depends on the main back-EMF harmonic in this frame. A 7-phase machine can be mathematically decomposed into 3 two-phase fictitious machines (FM1, FM2, FM3) and 1 zero-sequence machine (ZM) with corresponding decoupled reference frames (d1-q1, d9-q9, d3-q3, and z) [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF]. A fictitious machine and its reference frame are associated with a given group of harmonics as described in TABLE I. As the wye-connected stator windings, zero-sequence current iz in (3) is always equal to zero. Ideally, in the FOC-based control scheme, 6 constant d-q currents (id1, iq1, id9, iq9, id3, iq3) can be simply controlled by six classical PI controllers.

According to the multi-reference theory presented in [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF], a constant torque can be generated by constant d-q currents if each fictitious machine (FM1, FM2, or FM3) contains only one harmonic of back-EMF, for example the 1 st , 9 th , and 3 rd harmonics. However, the imperfect machine design at low costs possibly results in undesired back-EMFs with more harmonic components. Without loss of generality, to see impacts of multi-harmonics in back-EMFs, it is assumed that there are two associated harmonics per reference frame as follows: the 1 st and 13 th associated with FM1 (d1-q1); the 9 th and 19 th associated with FM2 (d9-q9); the 3 rd and 11 th associated with FM3 (d3-q3). The 11 th , 13 th , and 19 th harmonics are called unwanted back-EMF harmonics in this study.

III. IMPACTS OF UNWANTED BACK-EMF HARMONICS

A. Impacts of unwanted back-EMF harmonics on current control

The impacts of unwanted harmonic components in back-EMFs on current control have been investigated in [START_REF] Vu | Eliminations of Low-frequency Current Harmonics for Five-phase Open-end Winding Non-sinusoidal Machine Drives applying Neural Networks[END_REF][START_REF] Vu | Current Harmonic Eliminations for Seven-phase Non-sinusoidal PMSM Drives applying Artificial Neurons[END_REF][START_REF] Wang | Current Harmonics Suppression Strategy for PMSM with Non-Sinusoidal Back-EMF Based on Adaptive Linear Neuron Method[END_REF]. Indeed, unwanted back-EMF harmonics generate unwanted harmonics in d-q currents that low bandgap controllers such as PI cannot filter, resulting in low current control quality (unwanted harmonics in phase currents). Therefore, PI controllers combined with four ADALINEs have been used for 5-phase machines. Meanwhile, six ADALINEs with PI controllers have been applied to 7-phase machines. The effectiveness of the combination of PI controllers and ADALINEs has been proven in transient and steady states. Therefore, the impacts of unwanted back-EMF harmonics are eliminated.

B. Impacts of unwanted back-EMF harmonics on torque generation

When unwanted harmonics in d-q currents are eliminated by using the above combination of PI controllers and ADALINEs, all d-q currents are well controlled. In other words, phase currents only contain the imposed harmonics, for example 1 st , 3 rd , and 9 th in the considered drive. However, torque ripples still exist. Indeed, the imposed current harmonics (1 st , 9 th , and 3 rd ) interact with unwanted back-EMF harmonics (13 th , 19 th , and 11 th ) to generate torque ripples with frequencies of 14θ and 28θ (θ is the electrical position) as described in TABLE II. Specifically, the torque in FM1 has a frequency of 14θ because the 1 st harmonic of currents interacts with the 13 th harmonic of back-EMFs. Similarly, torques in FM2 and FM3 have frequencies of 28θ and 14θ, respectively. Finally, the electromagnetic torque of the considered machine Tem, the sum of the three torques generated by the three fictitious machines, has frequencies of 14θ and 28θ. It is noted that these harmonic orders are multiples of the number of phases. By the same way, torque ripples at main frequencies of 6θ and 10θ can be observed in 3-and 5-phase machines, respectively. Therefore, these torque ripples need to be eliminated to guarantee smooth torques.

IV. TORQUE RIPPLE ELIMINATIONS

A. The vectorial approach

The vectorial approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF], known as maximum torque per ampere (MTPA), is to obtain constant torques regardless of the presence of multi-harmonics in back-EMFs of multiphase machines. Current references in natural frame can be generally calculated as follows:

2 em_ref ref T Ω ie e  (4)
where iref is the 7-dimensional vector of phase current references (natural frame); Tem_ref is the constant torque reference; e is the norm of the back-EMF vector e in natural frame.

It is noted that current references in (4) are linearly proportional to back-EMFs. Therefore, when back-EMFs contains unwanted harmonics, currents in d-q frames are timevariant. Phase current references in (4) have been directly controlled by hysteresis controllers as in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF], resulting in inevitable variable switching frequencies, high switching losses, and electromagnetic compatibility problems. Similarly, highly oscillating current references are generated by the approach in [START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF] with ADALINE. For the sake of simplicity, the vectorial approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF] is used to compare with the proposed control scheme in this study.

B. Proposed control scheme with adaptive linear neurons

The principle of the proposed control scheme is to add compensating currents to the existing constant current references. These compensating currents are directly determined from harmonic components of the electromagnetic torque as described in TABLE II. Amplitudes of these torque harmonics depend on the back-EMF harmonic distribution. Meanwhile, phase angles of these torque harmonics depend on the drive (parasitic time delay, for example). These phase angles may change in the real-time operation of the drive. Accordingly, an additional torque, called compensating torque Tem_com, needs to be adaptively determined. This torque is expected to be equal to the difference between the constant torque reference Tem_ref and the electromagnetic torque Tem. To effectively determine the compensating torque, the control structure with only one ADALINE is proposed in Fig. 1. As previously discussed, the ADALINE is chosen because it possesses some advantages such as self-learning and easy-to-implement. The inputs of the ADALINE are harmonic components of the torque as described in TABLE II. The output of the ADALINE, the compensating torque Tem_com, is the weighted sum of the inputs as given by:

where (w1_tor, w2_tor) and (w3_tor, w4_tor) are weights of two harmonics 14θ and 28θ, respectively.

In this study, Least Mean Square rule is applied to update these harmonic weights. In Fig. 1, the weights of harmonic components are updated by using learning rate η, corresponding harmonics (14θ and 28θ), and torque error Terror. The learning rate is required to be between 0 and 1 to guarantee the stability of the system. Its value depends on the input and output signals, and especially on the calculation power of processors used in the drive. Torque error Terror, the difference between torque reference Tem_ref and electromagnetic torque Tem, is given by:

The compensating currents icom are transformed into d-q frames idq_com and added to constant current references idq_ref. Then, the total current references 𝑖 𝑑𝑞_𝑟𝑒𝑓 * are used for control to create constant electromagnetic torques in the 7-phase drive. The total current references in d-q frames are no longer time-constant due to the compensating currents. It is worth noting that the proposed approach in Fig. 1 aims at learning only pulsating components of the torque to find compensating currents. Therefore, d-axis current references that do not generate torques are almost zero, facilitating current control with classical PI controllers. Meanwhile, the approach in [START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF] considers the entire torque reference in the learning process, resulting in highly oscillating d-q current references. Consequently, these d-q currents in [START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF] are like time-variant current references in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF]. Moreover, in the proposed control scheme, the ADALINE has only to find the amplitudes of considered harmonics instead of the entire harmonic signals.

In other words, the knowledge of torque harmonics has been significantly used to reduce the number of calculations with therefore an expected short learning time. This proposed structure in Fig. 1 can be generally applied to every electric machine with different back-EMF characteristics and different numbers of phases. In those cases, torque harmonic components of which orders are multiples of the number of phases need to be determined.

V. NUMERICAL VERIFICATION

A. Parameters of the considered drive

The control schemes presented in the previous section are verified with MATLAB Simulink. Parameters of an experimental electric drive described in TABLE III are used Fig. 1. The proposed control scheme using an ADALINE to eliminate torque ripples (implemented in the three rotating (d-q) frames). for simulations in this section. The 7-phase PMSM is supplied by a 7-leg voltage source inverter (VSI). The PWM strategy with a 3-level single-modulation is applied to generate switching signals of inverter switches at a frequency of 10 kHz. The 7-phase PMSM has a special concept. Specifically, this machine has an axial flux with double rotors. Different configurations of the double rotors result in several back-EMF waveforms. For example, when the two rotors have different numbers of poles and spatially shifted an angle of 7 degrees, the back-EMF waveform and its harmonic spectrum are presented in Fig. 2. Besides the 1 st harmonic of the back-EMFs, the 9 th and 3 rd harmonics account for the highest proportions (12.5 and 32.3% of the 1 st harmonic, respectively). Each of these harmonics presents in one fictitious machine. Several minor harmonics in fictitious machines can be described as follows:

1) In FM1: besides the 1 st harmonic, the 13 th harmonic is equal to 5% of the 1 st harmonic.

2) In FM2: besides the 9 th harmonic, the 19 th harmonic is equal to 2% of the 1 st harmonic, and approximately equal to 16% of the 9 th harmonic.

3) In FM3: besides the 3 rd harmonic, the 11 th harmonic is equal to 10.3% of the 1 st harmonic, and approximately equal to 32% of the 11 th harmonic.

4)

In ZM: the 7 th and 21 st harmonics are equal to 9.4 and 3.2% of the 1 st harmonic. Due to the wye-connected stator windings, these harmonics have no effects on currents and torques.

B. Three operating stages for comparisons

To compare the vectorial approach and the proposed control scheme, an operation of the drive is considered with 3 stages as described as follows:

1) Stage 1: Time-constant d-q current references idq_ref are used to generate an average torque equal to torque reference Tem_ref. Only three main harmonics of back-EMFs (1 st , 9 th , 3 rd ) are considered to calculate the current references by using (3)-( 4). The classical FOC technique with 6 PI current controllers is applied.

2) Stage 2: Without changing the control scheme from stage 1, the vectorial approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF] is applied to define new d-q current references (time-variant), as calculated in ( 3)-( 4), to theoretically generate constant torques. In other words, all harmonics of back-EMFs are used to calculate current references.

3) Stage 3: The proposed scheme in Fig. 1, a combination of the classical FOC technique and a simple ADALINE, is applied. In this case, compensating d-q currents are added to the constant d-q current references from stage 1 to obtain the total current references 𝑖 𝑑𝑞_𝑟𝑒𝑓 * for control.

C. Numerical results

Fig. 3 demonstrates the three operating stages at two speeds to see the effectiveness of the proposed control scheme in eliminating torque ripples. Specifically, torque, current control performances, and weight convergence are presented. Speeds of 10 rad/s and 40 rad/s are chosen due to the considered DC-bus voltage of 300 V. If the rotating speed is higher, the effectiveness of the proposed control scheme is more impressive over the vectorial approach. In Fig. 3, four weights (w1_tor, w2_tor, w3_tor, w4_tor) with a learning rate of 0.001 rapidly converge within one third of the electrical period or even faster. If the learning rate is increased, the convergence will be faster and vice versa but a higher overshoot even an instability may appear.

At a speed of 10 rad/s, stage 1 with a ripple of 10.7% at the main frequency 14θ as described in Fig. 3a. This ripple represents the impact of unwanted back-EMF harmonics on torque generation. In stage 2, new current references, created by using the vectorial approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF] and (3)-( 4), generate a torque ripple of 1.7%. The torque is reduced to 1.2% by using the proposed control scheme as shown in stage 3. It is noted that the effectiveness of the proposed control scheme in stage 3 is not clear due to the low speed. Additionally, current references in stages 2 and 3 are not constant. Specifically, 6 dq current references in stage 2 are time-variant with high amplitudes. Meanwhile, these currents of the proposed control scheme in stage 3 have much lower amplitudes, especially daxis currents. This important characteristic affects current control quality at higher speeds. Indeed, at a speed of 40 rad/s, as shown in Fig. 3b, stage 3 using the vectorial approach has a ripple of 6.7% while the torque ripple in stage 3 is only 1.6%.

Phase currents in the three operating stages at 40 rad/s are shown in Fig. 4a. Current waveforms are like the back-EMF waveform (see Fig. 2). The root mean square (RMS) currents are similar in the three stages at about 5.1 A, the rated RMS current of the considered machine. Therefore, copper losses in the three stages are alike. However, a high peak phase current (11 A) can be seen in stage 2 while these currents in stages 1 and 3 are respectively equal to 9.2 A and 9.3 A. Phase voltage references at 40 rad/s are shown in Fig. 4b. It is noted that a high peak phase voltage (139.3 V) can be seen in stage 2 while these currents in stages 1 and 3 are respectively equal to 91.8 V and 106.5 V. In other words, the constant-torque region with the proposed control scheme will be larger. 3. With Tem_ref=33.5 Nm, torques, weights, and d-q currents in three operating stages at 10 rad/s (a), and at 40 rad/s (b), in which stage 1 considers constant d-q current references, stage 2 applies the vectorial approach with time-variant d-q current references, and stage 3 uses the proposed control scheme with ADALINE (η=0.001) (one period refers to one electrical period).

(a) (b) Fig. 4. At 40 rad/s with Tem_ref=33.5 Nm, phase currents (a), and phase voltage references (b) in the considered three operating stages in which stage 1 considers constant d-q current references, stage 2 considers the vectorial approach with time-variant d-q current references, and stage 3 uses the proposed control scheme with ADALINE (η=0.001).

Dynamic performances with variations of the rotating speed and torque reference are described in Fig. 5. The rotating speed suddenly varies from 10 to 25 rad/s and then returns to 5 rad/s in Fig. 5a. It is noted that the high torque quality is almost guaranteed at these different speeds when the torque ripple varies from 1 to 1.4%. In addition, the four weights converge within about one third of the electrical period even faster. When the rotating speed is fixed at 25 rad/s, the torque reference suddenly decreases from 33.5 to 13.5 Nm then goes up to 23.5 Nm as described in Fig. 5b. The torque ripple varies from 1.4 to 3.4%. The increase in the torque ripple is caused by the decrease in the average torque but not by the control quality. The weights converge within one fifth of the electrical period even faster.

VI. CONCLUSIONS

In this paper, a generic control scheme for non-sinusoidal back-EMF of n-phase machines has been proposed and numerically verified with n=7. Thanks to the proposed control scheme, a low-cost machine with an imperfect back-EMF can generate constant torques. Especially, the classical FOC technique that is popular in industry continues to be used. Only a simple ADALINE is added. Thanks to self-learning, easy implementation, ADALINE has been chosen. Especially, the use of knowledge of only torque harmonic orders 2n and 4n in a n-phase drive enables to reduce the calculation burden, enhancing the applicability of the proposed control scheme in real-time applications. Numerical results have proven the effectiveness and advantages of the proposed control scheme over the existing approaches.

  introduced in[START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF], compensating currents in natural frame icom, calculated from compensating torque Tem_com in (5) and back-EMFs e, can be expressed by:

Fig. 5 .

 5 Dynamic performances with the proposed control scheme using ADALINE with η=0.001 in terms of variations of the rotating speed with Tem_ref=33.5 Nm (a), and of the torque reference at 25 rad/s (b) (one period refers to one electrical period).

TABLE I .

 I HARMONIC CHARACTERIZATION OF FICTITIOUS MACHINES AND CORRESPONDING REFERENCE FRAMES (ONLY ODD HARMONICS)

	Fictitious machine	Frame Associated harmonic	m	0
	1 st machine (FM1)	d1-q1	1, 13, …, 7m ± 1
	2 nd machine (FM2)	d9-q9	5, 9, 19 …, 7m ± 2
	3 rd machine (FM3)	d3-q3	3, 11, …, 7m ± 3
	Zero-sequence machine (ZM) z	7, 21, …, 7m

TABLE II .

 II TORQUE RIPPLES GENERATED BY UNWANTED BACK-EMF HARMONICS IN FICTITIOUS MACHINES OF A 7-PHASE MACHINE.

	Fictitious machine	Current harmonics	Unwanted back-EMF harmonics	Harmonics in Tem
	FM1	1	13	14θ
	FM2	9	19	28θ
	FM3	3	11	14θ

TABLE III .

 III ELECTRICAL PARAMETERS OF THE 7-PHASE PMSM DRIVE.

	Parameter	Unit	Value
	Stator resistance Rs	Ω	1.4
	Self-inductance L	mH	14.7
	Mutual inductance M1	mH	3.5
	Mutual inductance M2	mH	-0.9
	Mutual inductance M3	mH	-6.1
	1 st harmonic of the speed-normalized back-EMF	V/rad/s	1.27
	3 rd harmonic of back-EMF over 1 st harmonic	%	32.3
	9 th harmonic of back-EMF over 1 st harmonic	%	12.5
	Number of pole pairs p		3
	Rated RMS current of the 7-phase PMSM	A	5.1
	DC-bus voltage VDC	V	300
	PWM frequency	kHz	10

Fig. 2. Speed-normalized back-EMF and its harmonic spectrum of the considered 7-phase PMSM.

ACKNOWLEDGMENT

This work has been achieved within the framework of CE2I project. CE2I is co-financed by European Union with the financial support of European Regional Development Fund (ERDF), French State and the French Region of Hautsde-France.