# Flux Barrier Design Method for Torque Ripple Reduction in Synchronous Reluctance Machines

*Abstract*—The current publication introduces the flux barrier design method and the concrete design of the synchronous reluctance machine with the reduced torque pulsations. The reviewed method provides the accelerated approach for designing of rotor flux barriers, based on Fast Fourier transforms and simple mathematical expressions. The proposed method has been utilized in the rotor design of synchronous reluctance machine and has shown desired results in reduced torque ripple. The innovative nonsymmetrical geometries for rotor flux barriers created on the basis of proposed flux barrier design method have been implemented and proved as beneficial.

*Keywords—Flux barriers; torque ripple reduction; FEM; synchronous reluctance machines.*

## I. Introduction

The intense research in the area of the synchronous reluctance machines (SynRM) continues since 1970. The reduction of torque ripples is one of the most valuable achievements in designing of the SynRM. There are several publications considering torque ripple minimization by choosing the optimal saliency ratio and the form of the flux barriers [1]. In [2] and [3] the basis for choosing the appropriate number of barriers was described. Publications [4] – [10] show that the torque ripple reduction can be achieved by the suitable selection of the ends positions of flux barriers, shape and number of the barriers, skewing of the rotor and consequently by phase shifting of the flux harmonics.

Some publications provide solutions with satisfactory results for torque ripple reduction, but those solutions require hours of computational time [11] – [13]. The flux barrier design can require a lot of modelling effort. There are some software tools which allow computer-aided design (CAD) of the flux-barriers, but learning process of the respective software tools often requires even more time, than designing process itself. Besides, cooperation and data exchange (export and import) between finite element method (FEM) and CAD based software tools is a time-consuming process and requires knowledge in both software tools. Moreover, the compatibility of software between different software developers is not always present. For this reason, the authors of this publication suggest the new time-saving approach for designing of flux-barriers for the rotor of SynRM based on Fourier transform and the design of the SynRM as an example of practical usage of the new flux barrier design method (FBDM).

## II. Flux Barrier Design

Usually it is possible within the FEM modelling software to draw equation based curves, which describe the dependency of “y” coordinate (parametric equation, time or angle function) from the “x” coordinate (second parametric equation). The polar coordinates r and t can be converted to the Cartesian coordinates x and y by using the trigonometric sine and cosine functions described by the set of equations (1):

$$\begin{array}{r}\{\begin{array}{c}y=r\cdot \mathrm{cos}(t)\\ x=r\cdot \mathrm{sin}(t)& \phantom{\rule{3cm}{0ex}}(1)\\ {x}^{2}+{y}^{2}={r}^{2}\end{array}\end{array}$$r – radius of the imaginary circle (radial coordinate)

t – angular coordinate or polar angle

Due to the fact that a point in the complex plane can be represented by a complex number that is written in Cartesian coordinates, the Cartesian coordinates x and y can be converted to the polar coordinates r and t very easily as well. To do so, the time or angle functions and were represented as complex numbers in complex plane in form of equation (2):

$$z(t)=x(t)+j\cdot y(t).\phantom{\rule{3cm}{0ex}}(2)$$According to Euler’s formula, conversion between Cartesian coordinates and polar coordinates was made. This step allows conversion of the complex numbers into the polar form in the polar coordinate system. With other words, every point of the curve was converted to the phase vector with its specific magnitude and phase and introduced in the polar coordinates. This brings following advantages. By showing the magnitude and phase of the vector the flux barrier can be edited really quickly graphically, whereas the Cartesian coordinates were showing the real and imaginary parts of the phasor that was not giving any impression about the final design of the flux barrier. As a starting point for designing the flux barriers with the new tool, following equations set (3) was chosen:

$$\begin{array}{r}\{\begin{array}{c}y=a\cdot \mathrm{cos}(t)+\mathrm{cos}(2t)\\ & \phantom{\rule{3cm}{0ex}}(3)\\ x=b\cdot \mathrm{sin}(t)\end{array}\end{array}$$The left hand side of the Fig. 2 demonstrates how the design of the flux barrier will look like in the FEM software before the FBDM was used.

The left hand side of the Fig. 2 demonstrates how the design of the flux barrier will look like in the FEM software before the FBDM was used.

The left hand side of the Fig. 3 shows the angular functions y(t) and x(t) in the polar coordinate system after the Cartesian coordinates x and y were converted to the polar coordinates r and t with the FBDM (see Fig. 3).

The simulation results in FEM software have shown the necessity of changing the design of the flux barrier. Beside the FBDM for changing the design of the flux barrier and in order to achieve a better machine performance the expert knowledge is important. This allows machine designers to make the initial rotor geometry in compliance with important scientific works made on optimization of the SynRM and therefore leads to the desired machine performance.

Through the expert knowledge the new design of the flux barrier was made by simply manipulating (as an example in this publication – reducing of magnitude with the logarithmic function) the magnitude of phasors (see Fig. 3 (right)).

The right hand side of the Fig. 2 shows how the changed design of the flux barrier will look like in the FEM software after the FBDM was used.

As soon as the machine designer is satisfied with the new form of the flux barriers, the curve fitting part of FBDM should be particularly observed. As mentioned above, it is necessary to get the angular functions for y(t) and x(t), to be able to benefit from the equation based curve in FEM software tool. To receive the y(t) and x(t) from the new designed equation based curve, the sub function of the FBDM based on fast Fourier transform (FFT) was used. Fig. 4 shows the results after precisely made FFT and the comparison with the source angular functions designed with FBDM.

Due to symbolic representation in FEM, additional symbolic operations was made to get the results ready for import in FEM tool. Fig. 5 shows the exact match between the FFT based function and its import ready symbolic representation.

MATLAB allows representing the resulting curve in a very practical form, as given in (3). The next step will be to apply this output in the FEM tool simply by creating an equation based curve. Fig. 6 shows the interface of the software. A closed line will be created which allows creation of two-dimensional objects. In this way, initial flux barrier geometry was created.

In further steps numerical verification is done, so only FEM tool is required. Thus, the change of the flux barrier form does not require any additional step in some other software package (e.g. CAD software). Time-saving solution is expected. Also software compatibility problems do not have to be considered anymore.

## III. FEM Simulation and verification of FBDM

### A. Parameters of the Model

In order to verify the applicability of the FBDM an electromagnetic transient FEM simulation was made. Although the proposed method can be used for every possible number of rotor poles, to get a basic idea a four-pole rotor design is considered. It is assumed having twenty-four stator slots which are equipped with a conventional integer-slot three-phase distributed winding. The number of slots per pole and per phase is an integer number. Every single stator slot contains a single layer. The coil terminals have a star connected topology. Fig. 7(a) represents the winding arrangement of the investigated machine. The coil sides of the different phases are emphasized with a various color. Non-oriented electrical steel M270-50a with stacking factor of 0.97 is the material of the stator and rotor iron laminations.

At the beginning of the design process a two-dimensional model of the SynRM was made using built-in geometry primitives which is shown in Fig. 7(a). Such primitives allow user to make changes in the amount of flux barriers per pole, number of poles, outer rotor radius and shaft radius, distance between single barriers and barrier thickness, etc. There are also some predefined barrier shapes which are available. Although the variety of the geometry parameters that can be modified is satisfactory for a normal case, unfortunately it is impossible to make a specific change in the barrier design (e.g. to adjust the shape of a single barrier, or to develop radially nonsymmetrical barrier), for instance to avoid a local saturation. General model parameters are given in Table I.

TABLE I. Parameters of the Model

Parameter | Value |

Number of stator slots, [-] | 24 |

Number of rotor poles, [-] | 4 |

Number of turns, [-] | 20 |

Rated torque, [Nm] | 69 |

Rated current, [A] | 30 |

Copper slot fill factor, [-] | 0.4 |

Current density, [A/mm^{2}] | 7.3 |

Air gap length, [mm] | 0.55 |

Stator outer diameter, [mm] | 238 |

Rotor outer diameter, [mm] | 140 |

Stack length, [mm] | 130 |

### B. Simulation Results

Another drawback of the predefined geometry primitives is that the user-specific geometry optimization is hard to implement, especially if some unconventional topology have to be investigated. In this subsection the important simulation results are observed and discussed.

#### 1) Unmodified rotor geometry

Fig. 7(b) represents the normalized value of the magneto-motive force (MMF) with a dominant fundamental working harmonic. At the same time additional space harmonics are also present (e.g. 5^{th}, 7^{th}, etc.). The radial component of the air-gap flux density and normalized values of its harmonic content are presented in Fig. 7(c) and Fig. 7(d), respectively. It can be observed that besides the fundamental wave also harmonics with odd ordinal numbers are present. 11^{th} harmonic has highest amplitude after the fundamental. Those harmonics are caused not only by the winding itself but also by stator slots and rotor flux barriers geometries.

Basically presence of the additional harmonics in the radial component of the air gap flux density results in the increased amount of the iron losses and thus decreased efficiency of the electrical machine. Also the quality of the produced torque will be affected by space harmonics content in the air-gap flux density. By adjusting the geometry of the machine and by using some unconventional solutions it is possible to reduce the amplitude of undesirable space harmonics [14]. The produced torque versus time over one electrical period and under rated load conditions is presented in Fig. 7(e).

Fig. 7. Model with initial rotor geometry design. Simulation results.

Although the torque course looks relatively smooth, there is a high amount of torque ripples which is almost 18.2% with the average torque of 69.4Nm. Torque ripple is a ratio between the peak-to-peak and average torque values over one electrical period which is represented in percent. There are different requirements concerning this parameter depending on the application. In traction drive applications, for instance, torque ripple should not exceed 5% of a peak torque at any rotational speed [15]. This requirement is also taken as a reference in the current work but only for the rated rotational speed and under rated load.

The FFT of the produced torque gives information about its harmonic content. Fig. 7(f) shows zero component of the torque which corresponds to the average torque and also harmonics of different order are present. Those are multiples of six. The 24^{th} harmonic corresponds to the number of stator slots and has the highest amplitude.

#### 2) Rotor geometry made with FBDM

After the model with initial rotor geometry has been simulated, the next step was to implement the proposed FBDM. Modification is applied only to the rotor and the stator is kept the same and other model parameters assumed to be similar with those shown in Table I. Fig. 8 shows the rotor designed with proposed FBDM after optimization process. The number of flux barriers per pole remains the same. It can be seen that a slight asymmetry of the flux barriers, correspondingly to the radial axis dividing the flux barrier in two nonsymmetrical parts, is present. However, the second part of the rotor – which is not shown in Fig. 8 – can be easily made by duplicating the first part around the z-axis. Stochastic parameter optimization method was implemented in order to find an optimal position of the barriers.

At the beginning all barriers were placed in the rotor one by one with curtain distance between them in order to avoid unnecessary magnetic saturation in iron. Then parameter P1 which is basically the position of the rotor barriers ends, barrier thickness (P3) and distance between the barriers and the origin (P2) were adjusted for every barrier. For the final refinement thicknesses of the iron ribs (P4) and also of the outer iron bridge (P5) were optimized in order to achieve nominal mechanical stability of the design. Another final aspect includes the slight adjustment of the P2 for barriers of one pole with respect to each other. For instance, third barrier of the right pole is radially shifted by 0.25mm.

In Fig. 9(a) an example of an optimization process is shown. Parameter value variation was done to find an extremum of the function. Only one parameter was changed at the same time, so that there was no influence on other parameters. Thus, the number of all possible parameter variations was much lower than in case of simultaneously change of all model parameters. Optimization process was finished after the torque pulsation was reduced to 5%. Fig. 9 gives an overview of important simulation results after the rotor geometry modification. There is a change in the radial component of the magnetic flux density inside the air gap due to the change of the rotor geometry. Asymmetrical rotor geometry causes the slight difference in the air-gap flux density (see Fig. 9(b)) which also has additional space harmonics with even ordinal numbers and small amplitudes (see Fig. 9(c)).

Furthermore, the amplitudes of the 3^{rd}, 5^{th}, 7^{th}, 11^{th} and 13^{th} harmonics are increased in comparison with the reference model. Presence of the additional even harmonics effects in a reduction of the torque pulsation and, as mentioned before, also leads to the increased amount of the iron loss. Fig. 9(d) represents the comparison of the produced torque over one electrical period and under the rated load conditions for the reference and modified models. A significant reduction of the ripple can be observed (5% torque ripple after optimization) and at the same time the average torque has kept almost the same value of 69.7Nm. Consequently, the harmonic content of the produced torque is shown in Fig. 9(e).

Fig. 9.Model with modified rotor geometry. Simulation results.

6^{th} harmonic is slightly increased by 1%. The 12^{th} and the 24^{th} harmonics are reduced by 1% and 5%, respectively. The 42^{nd} is not present anymore and others were reduced. Thus, the harmonic distortion of the produced torque has become lower.

The amount of iron loss under rated conditions is increased from 40.8W in case of the reference model to 46.7W in the model with modified rotor geometry (14.5% raise).

An additional reduction of the torque ripples can be achieved by adjusting the current phase angle. Thus, if the current angle is changed from 60 to 55 electrical degrees, the amount of torque ripples is changed from 4.3% to 5%, respectively. Wherein, the current angle of 60 electrical degrees corresponds to the maximum torque per current control strategy. Optimal values of the maximum produced torque, torque pulsation, power factor and iron loss correspond to different values of the electrical current angle, so the control strategy can be adjusted for different purposes (see Fig. 10).

#### 3) Application of the FBDM to the concentrated winding

In order to prove the applicability of the proposed method on different winding topologies, the FBDM has been implemented in the optimization process of the SynRM equipped with concentrated winding. Twelve stator slots and four rotor poles topology was chosen and double-layer integer-slot concentrated excitation winding is considered. The number of slots per pole and per phase equals one. Each stator slot is equipped with two layers from different phases and the end-windings do not overlap. Aforementioned winding is characterized by a relatively low fundamental winding factor of 0.5 which results in increased copper loss for the same amount of produced torque in comparison with the previous winding. However, the low harmonic distortion of the MMF (Fig. 12(a)) leads to reduction of eddy-current and hysteresis losses in the stator and rotor iron. Fig. 11 represents the final model geometry after the optimization process.

The same stochastic optimization process was applied to the model with concentrated stator winding, in order to reduce the amplitude of torque pulsation. Two machine models with different rotor geometry designs and similar stators were investigated using FEM software tool. Rotor geometry of the reference model is done by using non-optimized predefined geometry primitives. Both designs were compared in terms of air-gap flux density and quality of the produced torque.

Slight difference in the radial component of the air-gap flux density for two rotor designs and the air-gap flux density harmonic content are represented in Fig. 12(b), (c). The amplitude of the fundamental wave is increased by 3.6%. Also space harmonics with even ordinal numbers are present due to the asymmetry of the designed rotor. Furthermore, 5^{th} and 7^{th} harmonics are increased by 9.4% and 22.7%, respectively. However, amplitudes of 11^{th}, 13^{th}, 17^{th} and 19^{th} harmonics are reduced. Great torque ripple reduction (from 42% to 6.9%) can be observed in Fig. 12(d), however the average torque is slightly reduced by 0.6%. Smoother produced torque with low harmonic distortion (see Fig. 12(e)) was achieved after the optimization process.

Unequal number of flux barriers inside rotor poles (see Fig. 11) has positively influenced the machine behavior in terms of torque pulsation. Additional fifth barriers were added inside one pair of poles. In this way amplitudes of the peaks in the produced torque were significantly reduced. However, the amount of iron loss in the model with the optimized rotor geometry is increased by 8.5% due to the presence of the additional harmonics with even ordinal numbers in the flux density.

Fig. 12. Model with concentrated winding. Simulation results.

## IV. Conclusion

This work represents the know-how in terms of the new time-saving approach for designing the flux-barriers of SynRM without utilization of additional Multiphysics tools. The CAD development phase for designing the flux barriers is not present anymore. The geometry design of the flux barriers made with FBDM can be easily adjusted within the FEM software without reexport and reimport of the barriers with other tools. Time-saving geometry optimization process is achieved using FBDM. Furthermore, this method is advantageous in the manufacturing process, due to the possibility of defining the flux barriers form with the conversion of mathematical description based on simple mathematical expressions.

The new designed flux barriers were implemented within the FEM models. The simulation results show the reduced values for the torque ripples. The applicability of the method to different winding types was shown. The effect of the rotor design with asymmetrical flux barriers on the iron loss is mentioned.

Although the accent in the utilization of the FBDM, described in this publication, was made on reduction of the torque ripples, further analysis, investigations, calculations and optimizations can be made to achieve other design objectives, for example increasing the power factor.

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