Optimization of PMSMs Considering Multi-Harmonic Current Waveforms: Theory, Design Aspects, and Experimental Verification

This article is about investigating the potential of multi-harmonic design of permanent magnet synchronous machines. Thereby, multi-harmonic refers to the temporal periodic characteristics of linked fluxes and currents of the stator phases in order to ensure high-efficient operation with very low torque ripple. At first, a suitable model allowing for multi-harmonic currents evaluation is derived. Consequently, an interior-rotor IPM machine is studied. In order to have a fair comparison with conventional PMSMs featuring sinusoidal currents, an optimization scenario is defined where the same outer dimensions as well as magnet and copper volume for both design variants are considered. The use of multiple current harmonics requires a dedicated strategy for evaluating machine designs, as the conventional $dq$-grid analysis cannot be utilized. The introduced approach is applied and results of the optimization are illustrated. Prototypes were manufactured and machine designs were subsequently evaluated through experiments. A valuable lesson learned during practical implementation regarding rotor angle determination is highlighted. Measurement results reveal that adding more degrees of freedom in terms of multiple harmonics can give significant improvements for performance measures efficiency and torque ripple. This particularly holds if the harmonics are already considered throughout the machine design and optimization process, rather than only at post-processing stage.


I. INTRODUCTION
O PTIMIZATION of electric machines is an important field, both in academia and industry. Electric machines and drives are optimized for multiple reasons, e.g., to fulfill national or transnational requirements regarding motor efficiency [1]. In [2] a comprehensive review of the current state-of-the-art was presented. Due to its superior properties, the permanent magnet synchronous machine (PMSM) is widely used. Typically, a slotted stator holds concentrated windings. In case of fractional slot configurations, the air gap magnetomotive force (mmf) due to stator currents is far off sinusoidal. Moreover, the applied lowloss highly-utilized ferromagnetic material shows significantly nonlinear characteristics. Thus, even though often neglected, a PMSM typically features nonlinear characteristics with multiple load-dependent harmonics of considerable magnitudes in both the air gap field and, consequently, the linked fluxes. This raises particular challenges for applications where low cogging torque and torque ripple are essential. Besides maximizing the efficiency and minimizing torque ripple, usually additional objectives like cost have to be considered. Consequently, multiobjective optimization scenarios are evermore tackled [3], [4], [5], [6].
Cogging torque and torque ripple minimization are treated by many authors throughout the design stage [7], [8], [9], [10], [11], [12], [13]. The applied techniques, e.g., skewing, follow a decrease of efficiency and thus one has to consider a tradeoff regarding the maximization of efficiency while minimizing the torque ripple and cogging torque. Other researchers focused on minimizing torque ripple by optimizing and applying particular current waveforms for already available machine designs [14], [15], [16]. This requires dedicated current control techniques, but in turn follows (more) smooth torque characteristics. One technique frequently applied is to use a multiple reference frame based approach [17]. Current harmonics are particularly impressed, which is in contrast to another often applied procedure that aims at eliminating higher harmonics of the currents [18]. Moreover, current harmonics are also used to determine specific machine parameters, like winding resistance(s) [19]. Some authors further consider alternative objectives for optimizing current waveforms, e.g., the minimization of the DC-link current ripple or the acoustic noise [20].
With any of these measures or even a combination of techniques, usually a low cogging torque and torque ripple can be achieved while guaranteeing a decent efficiency level.
The classical machine design process, however, typically starts with solving an optimization problem where standard well established current waveforms are focused. Fig. 1  atypical currents in general or doing a current waveform optimization as post-processing action to a standard optimization problem follows a good but highly likely not the best solution regarding the tradeoff efficiency versus torque ripple. Achieving the best solutions available in the multi-dimensional design space regarding geometrical parameters, air gap flux and current harmonics can thus not be guaranteed with that sequential approach. Consequently, the goal of this work is to incorporate the current waveform optimization to the optimization run, where usually only the change of the geometry for sinusoidal currents is considered. This shall facilitate unlocking the full design potential and, subsequently, finding the global optimum.
In [21] optimal current waveform was considered during optimization. The optimization was done on a regression model and torque ripples smaller 10 percent were accepted. In [22], an SRM optimization was considered. By contrast to previous work, this study gave a direct comparison of machine designs geometrically optimized for sinusoidal currents and designs where further degrees of freedom in terms of additional non-zero current harmonics were taken into account. In contrast to the work presented here, the authors used an analytical method to replace computationally intensive finite element simulations. In [23], a design study was performed on average torque improvement through considering changes in the rotor shape by means of a harmonic definition, while sine-shaped currents were applied. It was found that the output torque can be increased by more than 6% with no change in the utilized permanent magnet mass and practically no degradation in the torque rippled performance. Another study was performed for five-phase machines in [24], where selected rotors featuring surface mounted permanent magnets and harmonic-based shape definition were investigated with regard to higher harmonic current impression. It turned out that the torque performance can be improved, especially with the additional degree of freedom of non-zero third-harmonic current components, which is not available in standard three-phase machines.
The here considered work is about optimizing the performance of an industrial fractional horse power motor for general purpose application. Thereby, in contrast to standard optimization problems, additional degrees of freedom with regard to both harmonics in the geometric design and in the stator currents were specified. The final design to be selected was obtained through a multi-objective optimization problem based on results acquired through finite element simulations. It was compared with a design featuring sine-shaped currents and optimized using the traditional approach. As material cost and machine volume are essential in practical applications, the same magnet and copper volume, outer diameter and active axial length were considered in the optimization presented within this work. Moreover, the same materials as well as the same DC bus voltage are utilized.
The outer rotor contour of the design featuring the definition through harmonics is optimized by using a polynomial Bézier curve representation. This allows a more flexible continuous definition of the geometry. With reference to the work originally presented in [25], this work is subdivided as follows: Section II gives a brief introduction about the torque in rotary synchronous machines for multi-harmonic currents. Section III is about the considered multi-stage design process for optimizing the multi-harmonic design and illustrates the results of the multi-objective optimization scenario. This will be followed by a detailed analysis of the obtained design in Section IV. While the optimization problem featured a single-load point analysis, here load-dependent optimal current waveforms are derived. For both, an optimal conventional design optimized for sine-shaped currents, and the optimal multi-harmonic counterpart, a prototype was manufactured and the respective comparison of the measured performance is illustrated in Section V. Section V-B provides a valuable lesson learned during practical implementation with regard to rotor angle determination. This is followed by an interpretation and discussion of the results and a conclusion in Section VI.

II. MACHINE TORQUE FOR MULTI-HARMONIC CURRENT IMPRESSION
The optimization of machines with sine-shaped currents of fundamental order is well-known. Usually, a grid of current vectors represented in the dq-reference frame is analyzed for any design under consideration, and then the optimal current vector for particular load requirements is determined. The reader is referred to explanations presented in [26].
In case of the multi-harmonic design, incorporating the current waveform analysis and optimization is not straight forward. A detailed discussion of linked flux harmonics, current harmonics, and respective torque harmonics was given in [27]. For a better understanding, the there derived key aspects are adapted and applied for the particular machine design configuration investigated here.
The total electromagnetic torque can be defined as where t pm gives the cogging torque, and t i s ,pm the torque component due to the interaction of the stator current and permanent magnet magnetomotive forces. t i s represents the reluctance torque due to the saliency of the arrangement. The set of potential cogging torque harmonics n pm,torque can be derived based on the symmetry of the machine design. With regard to the electric rotor angle α el , they can be defined as: N + is the set of positive natural numbers, i.e. {1, 2, 3, . . .}.
In case of the here considered symmetric three-phase machine designs, this follows that only multiples of order 6 exist for n pm,torque . The two remaining torque components t i s ,pm and t i s can be described by the linked fluxes and the currents of the m = 3 stator phases. As the motor shall operate in both directions of rotation, a rotor with symmetric poles is considered. Thus, the linked fluxes due to the permanent magnet excitation can only feature harmonics of odd order regarding the electric rotor angle α el . Due to the specific slot-/pole-/winding configuration of the considered machine, the winding factors for harmonic orders that are multiples of 3 regarding α el are zero. The linked fluxes due to the permanent magnet excitation thus comprise harmonics of orders n pm,f lux = {1, 5, 7, 11, 13, . . .}. ( It is known that only harmonics of same order regarding linked fluxes and currents can develop a mean torque component for net energy conversion. Besides, a dedicated analysis reveals that torque ripple harmonics of order 6 k, with k ∈ N, can be developed by harmonics in current i μ and linked flux i ν that fulfill either 6 k = μ + ν or 6 k = |μ − ν|. For instance, a torque ripple of order 6 can be caused by a current harmonic of fundamental order (μ = 1) and a linked flux harmonic of higher order ν = 5 or ν = 7, respectively. Additionally, a change of the harmonic orders for the current and linked flux follows the same result. According to these findings, only current harmonics of same order as for the linked fluxes are considered n pm,f lux = n current for the upcoming investigations. The harmonic orders of the total flux linkage of the stator coils and respective phases n flux , embracing both the linked fluxes due to the permanent magnet mmf as well as due to the stator currents mmf, thus feature the same set of harmonics n pm,f lux = n flux . The flux linkage can be represented as complex Fourier series with regard to the rotor angle α el . By taking into account the phase shifts of a symmetric three-phase system, this follows: The currents can be expressed in similar way: The net torque and its harmonics can now either be derived by evaluating the interaction of those two components in the present form or by considering a representation in the dq-reference frame or multiple reference frames [17]. A detailed analysis follows that, for reluctance torque and torque due to the interaction of permanent magnets and stator currents, only torque harmonics of order appear. Thus, the same harmonics as for the cogging torque, except the additional one of order zero, i.e. the net torque for energy conversion, are potentially generated. These torque harmonics are the result of any current harmonic multiplied by any of the linked flux counterparts. Strictly speaking, the (partial) derivative of the linked flux with regard to the rotor angle α el needs to be considered. However, while the derivative changes the magnitude of the harmonics, it does not change their harmonic order. For a standard machine, only current harmonics of fundamental order are considered, while the linked fluxes typically feature multiple harmonics of load-dependent non-zero magnitude. The interaction of higher-order linked flux harmonics with the fundamental current harmonic as well as the cogging effect thus cause torque ripple. For instance, the linked flux harmonics of order 5 and 7 together with the fundamental current harmonic cause a torque harmonic of order 6. Besides, an additional cogging torque component of same order highly likely is present. To cancel out this torque harmonic, proper current harmonics of order 5 and 7 in terms of their magnitude and phase shift need to be impressed.
Theoretically, all further higher-order torque harmonics, i.e. orders 12, 18, etc., can be perfectly canceled out. In practice, multiple limitations arise, e.g., the precision of the rotor angle measurement, the dynamics of the electronics and control, the accuracy of the derivation of higher harmonics based on finite element simulations, etc. Consequently, the here presented study focuses on additional current harmonics of order 5 and 7. Finally, it turned out that the magnitudes of the torque harmonics significantly decreased with increased harmonic order. Thus, good results were achieved even with this simplification. A more detailed explanation about the presented multi-harmonic theory of electric machine's performance can also be found in [28].

III. OPTIMIZATION PROBLEM
The optimization was done for a fractional horse power PMSM. The characteristic data of the interior rotor design can be found in Table I. To ensure a fair comparison of all designs and, in particular, of designs featuring either sine-shaped or multi-harmonics current waveforms, constraints were defined for the optimization problems, i.e. : r same materials to be applied r same or less permanent magnet mass for the multiharmonic design r same or less copper mass for the multi-harmonic design r same torque and speed requirements r same topology in terms of number of phases, slots, poles, and magnet shape (rectangular cross section) r same minimum air gap width r same DC bus voltage Particular focus is laid on defining a suitable but computationally reasonable analysis of designs under investigation for the optimization process. Essential steps are listed below: r CAD-based geometry definition / preprocessing r No load simulation r Definition and simulation of stator phase currents of different magnitude for the fundamental harmonic r Estimation of the fundamental current harmonicî 1 for obtaining the rated torque t r r Simulation of a three-dimensional current grid, i.e. regarding the magnitudes i 1 , i 5 , and i 7 r Current waveform optimization for maximum efficiency and minimum torque ripple

A. CAD-Based Geometry Definition
The definition of the rotor outer contour shall be described in more detail here. Fig. 2 gives the definition of a rotor pole. As the treated PMSM should similarly perform for both clockwise and counterclockwise operation, symmetric poles are considered. The saturation bar area is left unchanged, and only the bold part of the contour is modified. Fig. 3 gives the definition of the contour by using a polynomial Bézier curve representation. The points defined as c i and c i,s give both the desired points along the curve that are varied during optimization, and their symmetric counterparts, respectively. In order to have a smooth rotor outer contour instead of a piecewise linear interpolation,  the control points p i are computed and are applied using the Bézier curve based approach. This further allows for easily computing any required intermediate point. Besides, the definition of the rotor contour is more flexible than for standard parametrized definitions, e.g., a rotor contour with eccentric rounding, etc.

B. No-Load Simulation / Simulation for Fundamental Current Harmonic Estimation
Starting with a no-load simulation, the information about the respective no-load fluxes and corresponding phase shifts, and the cogging torque, can be derived. Moreover, this gives the basis for setting up the stator phase currents to be analyzed in the next step. Here, currents of fundamental order with different magnitudes that are in phase to the fundamental component of the no-load Back-EMF were evaluated.
C. Estimation of| i| 1 for Obtaining the Rated Mean Torque t r Fig. 4 gives an exemplary result of the mean torque t em versus current magnitude i 1 characteristics. In addition, the estimated current magnitude| i| 1 for achieving rated torque is illustrated. While the depicted exemplary characteristics show an approximately linear behavior, more nonlinear counterparts were also observed during optimization.

D. Simulation of Current Grid in Three Dimensions
In the next step, a three-dimensional current grid is investigated. It comprises the variation of |i| 1 , |i| 5 , and |i| 7 , i.e. the current magnitudes of the first, fifth, and seventh order harmonics. Based on the initial results, a regular grid is defined, as can be seen in Fig. 5. The maximum values for the magnitudes of the higher harmonics of the current are specified based on (i) the ratios of the corresponding magnitudes of the higher harmonic components of the linked fluxes with regard to the fundamental component of the linked flux, and (ii) the current magnitude for achieving rated torque when only the fundamental harmonic is utilized. All relevant information is obtained based on the initial simulations with non-zero fundamental current harmonic only. The variation of the fundamental current magnitude is also done in accordance with the initial results illustrated in Fig. 4.
Obviously, the magnitudes of the linked flux harmonics as well as their phase shifts generally are subject to change with changing fundamental current magnitude. Fig. 6 gives the change of the phase shift of the higher harmonics with regard to the magnitude of the fundamental component |i| 1 over the full permissible range of the stator current due to thermal limits for an exemplary design. Thereby, arg(û) x defines the phase shift of the x − th harmonic of the Back-EMF of one stator phase. |i| 1 is only varied little here after determining a first estimate for the fundamental component based on the initial simulations, and the higher harmonics themselves do not impact the linked fluxes, i.e. the magnitudes and phase shifts of any component, very much due to their comparably small magnitudes. Hence, only currents that are in phase to the initially determined linked fluxes are considered here. The computational cost would significantly increase if also changes of the phase shifts of the three selected current harmonics besides the variation of their magnitude would have been considered during optimization. Considering a six-dimensional grid for the variation of the three stator current harmonic magnitudes and corresponding phase shifts of only three steps each, the latter would already require a total of 3 6 = 729 simulations per design variant. Thus, the adopted approach is considered as a good compromise regarding accuracy versus simulation effort.

E. Current Waveform Optimization
Based on the obtained results for the three-dimensional current grid defined in Section III-D, the optimal current waveform for minimum torque ripple at maximum possible efficiency is determined. For deriving the efficiency, the copper losses are taken into account. By contrast, the iron losses are not considered. The reason for not adding the iron losses is that they were comparably small for the machine design obtained for sine-shaped currents. The design with optimized current waveforms might have relatively more higher harmonic content in the air gap field and thus losses in the ferromagnetic components. Nevertheless, the higher harmonics of the current could also have a positive effect on the harmonics, i.e. such that some air gap field harmonics are (partly) canceled out. Thus, some doubts arose if higher iron losses will be obtained for the new design or not. As the accurate iron loss calculation in case of multiple flux density harmonics is non-trivial and still an open research topic [29], [30], [31], it was decided to not consider the iron losses for efficiency determination.
Theoretically, an infinite number of combinations of |i| 1 , |i| 5 , and |i| 7 can be found to achieve a desired rated torque t r . |i| 1 = f(|i| 5 , |i| 7 ) can be utilized to find all potential solutions within the  defined current grid. Subsequently, also the Joule losses and thus an indirect measure for the efficiency can be defined in terms of the higher harmonics of the stator currents p loss = f (|i| 5 , |i| 7 ) for given torque requirement. Exemplary characteristics are illustrated in Fig. 7. Consequently, the loss minimum and thus the maximum efficiency can be derived. Besides the losses, the corresponding torque ripple t ripp,pp is analyzed and considered for the optimization scenario.

F. Definition of the Optimization Problem
The optimization problem is specified through the design parameters and objectives given in Tables II and III. The rotor definition consists of eight parameters, i.e., the rotor outer diameter, magnet height, rotor eccentricity and the pole angle. The latter describes the angle between the saturation bar center between two poles and the position where the rotor contour shaping is started. The overall pole is defined to be symmetric. Additionally, the relative positioning of four equally spaced Bézier control points is considered for optimization, while another fifth point, the starting point, is fixed to be of zero value, cf. Fig. 3 for more details. As the goal is to compare designs of same magnet volume for a fair comparison, the magnet width is adjusted according to a change of the magnet height. Thus, the magnet width is not considered as independent freely to vary design parameter. No further changes were applied, as the axial length and generally the overall outer dimensions of the machine designs are kept constant. Apart from optimizing the rotor, two more design parameters were included for varying the stator tooth and slot shape.
Two objectives were focused here, the losses and torque ripple at rated load that can be achieved. Both quantities shall be minimized during optimization. These two objectives represent key quantities of electric machines. Minimizing the losses for given load point specifications is equal to maximizing the efficiency. Additionally, machine designs are favored that feature low vibration and corresponding noise radiations. As individual per design modeling and direct evaluation of vibration and noise is very time consuming and still under investigation, mostly the minimization of the torque ripple is considered for optimization problems instead. This quantity typically correlates with the noise and vibration performance of machine designs. As the overall dimensions and the magnet volume were kept constant, no objective referring to the design variants' (material) cost was introduced.
Any design under investigation was characterized through 34 different stator current profiles, all evaluated through individual finite element simulations analyzed over a full electrical period.

G. Evaluation of the Optimization Problem
The genetic algorithm SPEA2 [32] was applied for this work. It is a generation-based evolutionary algorithm suitable for multi-objective optimization problems. It features several major components that such algorithms typically embrace, i.e., selection of previously evaluated individuals for recombination, and techniques for the crossover operation and mutation to avoid getting stuck in local optima. It additionally includes a density estimation regarding the position of the design variants inside the objective space, in particular for the non-dominated configurations, to prioritize among the individuals such that a good spread of the solutions is achieved. For the present work, the crossover and mutation probability were set to 0.8 and 0.4, respectively, while the value 20.0 was assigned to the corresponding distribution indices. A binary tournament selection facilitates specifying new design variants to be investigated. The optimization included an initial Latin-Hypercube-sampling based determination of 300 design variants to screen the available design space. This was followed by 3400 more designs to be evaluated that were defined by applying the optimization algorithm. The population size was set to 150. The analysis was carried out by running the optimization on a host computer while making use of a computer cluster with approximately 300 cores. The entire optimization was automatized using SyMSpace [33]. Fig. 8 gives the final Pareto front of the optimization. As can be observed and was already discussed above and highlighted in the introduction in Section I, the minimization of the losses, which equals the increase of the efficiency, and the minimization of the torque ripple are conflicting objectives. Thus, a tradeoff has to be found. Accordingly, a design featuring about 12.8 W losses and slightly more than 5% torque ripple was selected.  In Fig. 9 the finally obtained design for the multi-harmonic current optimization problem is presented. The stator winding is realized through concentrated coils. According to the optimization result, a wire diameter of d wire = 0.8 mm was selected, following 20 turns per coil. The design parameters for the selected configuration are equal to:

IV. OPTIMAL DESIGN
After the final design had been selected, it was further analyzed by additional finite element simulations and nonlinear current waveform optimization. As now only a single design is treated, significantly more different current vectors can be considered for a detailed modeling and evaluation.
Taking into account the wye-connection and the therefrom resulting constraint for the three phase currents i ph,3 = −i ph,1 − i ph,2 , different combinations of constant currents i ph,1 and i ph,2 are now investigated for an entire electrical period by FE-simulations, while i ph, 3 is adapted such that the constraint is fulfilled. Considering the same maximum current magnitude for all three phase currents and the same discretization levels, the analyzed current vectors were defined, as illustrated in   Fig. 10. Consequently, optimal current vectors can be derived for any load torque requirement, while the optimization problem focused on the rated torque. Fig. 11 gives the optimal stator current for phase 1 for zero torque ripple starting from t em = 0Nm up to t em = t r (rated torque).

A. General
Prototypes for the two different machine designs, the conventional one optimized for sine-shaped currents, and the one optimized for additional higher order current harmonics, as presented in detail here, were manufactured. Fig. 12 gives a photograph of the cross section of the optimal motor design that was the result of the new multi-harmonic design approach. Both the rotor and the stator featuring the tooth-wound coils can be seen. The white domains at the stator teeth and the stator yoke belong to plastics-based rapid prototyping caps mounted at the axial ends of the stator core to improve the winding process and avoid insulation faults of the winding materials.
A suitable current control had to be found [34] for the multiharmonic design, which was realized based on an iterative learning control (ilc) based approach. Measurements were carried out for the two machines for the entire torque-/speed-range starting close to zero speed and zero torque up to the rated speed and  rated torque. Fig. 13 gives the utilized test rig and a description of the relevant components. For both, the conventionally optimized design as well as the multiple-harmonics design, all the auxiliary components like the housing and the bearings of the machine, the rotor position sensing etc. were implemented the same way, as the aim was a fair comparison of the two designs.

B. Lesson Learned -Rotor Angle Determination
During the practical implementation and measurements, the optimal current angle for maximum efficiency for the machine design featuring sine-shaped currents was found by minimizing the overall losses via changing the current angle for given load torque and speed. This was done in order to definitely find the best current angle and have an utmost fair comparison with the newly designed machine for optimized current waveforms. In contrast, for the machine design controlled by the optimized current waveforms, no such measurement-based optimization was applied, because the correct phase of the higher order current harmonics to reduce torque ripple and thus vibration was prioritized.
No satisfying results were obtained at first both regarding efficiency and vibration for the machine with multi-harmonic currents. In particular, the impression of higher order current harmonics even frequently followed an increase in vibration, and efficiency levels were not as high as expected. A detailed analysis revealed the origin for these effects. The overall delay time when determining the rotor angle, mainly caused by the rotary encoder, was about t delay = 50 μs.
Considering a rated speed n r = 4000 rpm for this machine design with number of poles 2p = 8, the period of the fundamental current harmonic T 1 can be determined by As harmonics of 5th and 7th order were considered for stator current waveform optimization, their periods can be computed by using which followed T 5 = 750μs and T 7 = 536μs, respectively. Considering the delay time for determining the rotor angle t delay , a phase shift for the 1st, 5th, and 7th harmonic of 4.6 • , 23 • , and 32 • can be determined. Thus, especially the torque ripple cancellation, but also the maximum efficiency operation did not work properly because of this time delay. After a simple compensation through a time shift of the rotor angle, it was possible to successfully carry out the measurements with satisfying results. The same procedure was also applied for the machine design featuring sine-shaped currents. However, due to the anyway applied current angle optimization during measurements for that arrangement, as illustrated at the beginning of this subsection, no significant differences in experimental results were obtained for that machine design. Authors hope that the here provided information will help other researchers to minimize problems or dissatisfying results when measuring their prototypes.

C. Torque Ripple Comparison
The results for the torque ripple harmonics for rated mean torque are presented in Fig. 14. More detailed results were initially presented and discussed by the authors in [26], [34], including particular current waveforms and their effect on the torque output. As can be observed, the multi-harmonic machine features a much lower 6-th order torque ripple harmonic. Even though for the motor optimized for sine-shaped currents a magnitude of only 2.5% with regard to the rated torque was observed, the new design with optimized current waveforms supersedes it by having a torque ripple magnitude lower than 1%. For order 12, the comparison does not show significant differences. This comes as no surprise, as this and subsequent higher harmonics cannot be actively controlled when dealing with current harmonics of order 1, 5, and 7, as discussed in Section II. As can be observed, a small torque ripple of order 2 is also present for both designs. It is highly likely caused by the auxiliary machine used for testing, as it is approximately of same size for both measurement results, and, assuming ideal symmetric designs for the treated PMSMs, this harmonic theoretically is not possible to appear.
It would even be possible to obtain better results when performing a measurement-based current waveform optimization. In [26], the static torque as function of the phase currents was determined for the multi-harmonic design through a dedicated measurement setup and a particularly developed measurement approach. An auxiliary machine was used to hold the rotor of the device under test at a certain rotor angle position. Then, different current vectors were applied and the characteristics were analyzed by a torque transducer. A full (mechanical) rotor revolution was considered to discover potential differences among the ideally symmetric arrangement with symmetry angle of 90 • .
Due to manufacturing tolerances, eccentricity, etc., differences between simulated and measured torque responses are likely to occur. This can follow (i) differences in the three optimal waveforms for the three stator phase currents and (ii) current harmonics that are not present in case of the assumed ideal arrangement, featuring a four-fold symmetry for this design. Fig. 15 gives a comparison of results from simulation and from measurement regarding the power-optimal current waveforms for zero torque ripple at rated load. As can be observed, the ideal  currents determined through measurements contain additional harmonics. This includes non-negligible magnitudes for orders that arise due to 'differences of the four electric periods' that are passed through for an entire rotor revolution, as p=4. Besides, small differences for each phase were observed. The rms value of the phase current i s,rms and the associated current density j s,rms for the ideal excitation based on simulation results are equal to: i s,rms = 12.77 A and j s,rms = 6.35A/mm 2 , respectively. The relative changes in obtained rms values for the phase currents and current densities, according to their calculation based on the measurement results, are within one percent and thus negligible.

D. Efficiency Map Comparison
Besides torque ripple, the comparison of the efficiency is of major interest. Particularly, as the accurate iron loss computation in presence of considerable higher-order harmonic content is still an open research topic, the verification on the test rig is of utmost importance and gives the fairest comparison.
The obtained results are presented in Figs. 16 and 17, where Fig. 16 characterizes the motor with sine-shaped currents, while Fig. 17 illustrates the performance of the motor with optimized current waveforms. The comparison is done for the entire torque-/speed-range instead of only evaluating a single load point.
The motor with optimized current waveforms showed approximately 3.5% higher efficiency for rated operation. Moreover, the efficiency was increased over the entire torque-/speed-range, and in a wide reach it was enhanced by significantly more than 3.5%. This is indicated by the area of efficiency equal or higher than 88% for both machines. As a consequence, considering non-sinusoidal current waveforms can follow significant performance improvements.
It is important to once more remark that the multi-harmonic design features same overall dimensions, same amount of copper and same permanent magnet volume of same block-shaped structure. Hence, a fair comparison was made in terms of manufacturing cost. Moreover, the authors are convinced that the additional effort for developing the suitable control structure for higher harmonic current impression would not take significant effect on the overall production cost. There was no need for a different position sensor and the same DC bus voltage was applied.
Very often, higher harmonics are not considered because of the rule of thumb saying 'higher harmonics in the current require higher supply voltage'. The conclusion is drawn based on the fact that for a current harmonic of increased frequency, the term specifying the correspondingly induced voltage, that is proportional to the rated change of the current, becomes higher. However, this is only true when single harmonics of same magnitude are compared. By contrast, when analyzing a multi-harmonic current, the overall current waveform and its derivative must be analyzed. Adding higher harmonics with comparably smaller magnitude and simultaneously changing the fundamental harmonic can follow both, higher or lower voltage requirements and thus needs to be particularly addressed and optimized. Besides, higher order current harmonics might (at least partly) cancel out initially present higher order flux harmonics, which also could be beneficial regarding inverter voltage requirements.

VI. CONCLUSION
The work presented here is about optimizing permanent magnet synchronous machines by simultaneously taking into account non-sinusoidal adapted current waveforms and geometry modifications within the optimization problem. Often, current waveform optimization is considered during post-processing, after the optimization of the geometry has been completed. However, it needs to be taken into account at design level in order to find the overall (globally) best solution.
At first, basic theory about current and linked flux and respective torque harmonics was presented here. Taking into account the typical three-phase symmetric electric machine characteristics, this allowed to constrain the considered current harmonics for optimization to orders 1, 5, and 7.
An exemplary optimization problem for comparing results when considering either sine-shaped or optimized currents has been analyzed. The problem was solved by utilizing a particular software framework with automated design generation, preprocessing, definition and evaluation of finite element simulations, and post-processing tasks for determining and comparing the designs' performances. In total, several thousand design candidates were investigated. The optimization problem was solved by applying an evolutionary algorithm and making use of a computer cluster. A fair comparison was ensured by considering same material types, same or less material volumes, same DC bus voltage, and same overall dimensions for the multi-harmonic machine design.
Prototypes of optimal solutions have been manufactured and compared in terms of efficiency and torque ripple. The same digital signal processor was applied with two different control strategies for the current impression, a standard dq-current control for the conventionally obtained design with sinusoidal currents and an iterative learning control (ilc) based approach for the multi-harmonic design. A lesson learned regarding the determination of the rotor angle and its impact on the control of optimized current waveforms was presented as help for other researchers.
As could be observed, adding more degrees of freedom for the geometry definition and the current waveform and simultaneously optimizing both can follow machine designs featuring significant improvements in both the efficiency and the torque ripple at once. Future work should be about investigating the impact of tolerances on torque ripple harmonics and the consequently required change of current waveforms for compensating inevitably arising non-idealities observed during manufacturing.