Novel Electrically Excited Doubly Salient Variable Reluctance Machine With High-Order-Harmonic Winding

The dc excited doubly salient variable reluctance machine (VRM) has received more attention due to its robust rotor structure, low cost, and permanent magnet (PM)-free. In this article, a short-pitch distributed winding connection with improved fifth-order harmonic is newly applied in this kind of machine to enhance torque production under the same current excitation. Its electromagnetic performance is analyzed by finite element analysis (FEA). The simulation results verify that compared with the traditional concentrated winding distribution, the proposed winding arrangement can effectively improve the back-electromotive force (EMF) as well as the motor output torque density.


I. INTRODUCTION
W ITH the rising price of permanent magnets (PMs), more environmentally friendly motors without PMs are gaining more and more attention for their low cost and robustness although motors with PMs have already been used in a large number of industrial applications [1], [2], [3], [4].
Typical PM-free motors include synchronous reluctance motors (SynRMs), switched reluctance motors (SRMs), etc. Synchronous reluctance machine has the same stator design as the conventional PM motor, but its torque density and power factor are lower that limit its application in the industry [5].SRM motors have significant torque ripple and mechanical vibration due to the half-cycle conductivity principle.Since the SRM drive circuit is an asymmetric half-bridge converter, the commercial inverter cannot be used [6].
To overcome these disadvantages, some new motor structures have been proposed, such as dc excited variable reluctance machines (dc-VRMs).It has a simple rotor structure compared to SynRMs which means dc-VRM has better mechanical strength [7].It also achieves a much lower torque ripple than that of SRMs at the same time.However, these dc-VRMs have full dc coil structures resulting in higher dc copper losses in the motor and the dc coil occupies a larger slot area [8], [9], [10], [11], [12], [13], [14].
In this article, a novel electrically excited doubly salient VRM (dc-DSVRM) with a dc coil spanning three stator teeth is investigated.Compared to the existing doubly salient machine, it has a symmetrical magnetic circuit and sinusoidal backelectromotive force (EMF) to reduce torque ripple.It shows comparable torque density but reduced dc copper losses compared to dc-VRMs using full dc coils.To further enhance the torque density of the motor, an armature winding configuration based on the flux modulation theory is proposed in this article.Compared to the conventional winding design, this novel armature winding configuration can fully utilize the higher-order harmonics generated by field winding based on its unique distribution in a doubly salient machine, thus increasing the motor's back-EMF and torque density.The rest of this article is organized as follows.In Section II, the machine structure of the novel electrically excited doubly salient VRM and working principle are introduced with the analysis of the air-gap harmonic.In Section III, the novel armature winding design is given based on the flux modulation theory and the feasibility is verified.In Section IV, the model with the conventional armature winding and the novel winding design are established and the electromagnetic performances are compared.Finally, the conclusion is drawn in Section V.

A. Machine Structure
In this article, an 18-stator slot and 13-rotor pole dc-DSVRM are analyzed.In the stator, both dc field winding and ac armature winding are housed.The dc field coils, which are wound across every three stator teeth, can produce the flux density of fundamental harmonic and other higher-order harmonics.
In the existing doubly salient machines, concentrated armature winding connections are usually utilized in the conventional design.In this article, a distributed winding design utilizing working harmonics modulated from high-orderharmonics is proposed and compared with the conventional design.Two models with different armature windings are shown in Fig. 1.For convenience, the model with conventional concentrated winding is named Model I, and the model with proposed distributed winding is named Model II.
The dimensions of both models are optimized using genetic algorithm (GA) and finite element methods to obtain larger torque and lower torque pulsation while fixing the stator outer radius and rotor inner radius.The main parameters for 0018-9464 © 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
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TABLE I MAIN PARAMETERS OF THE DC-DSVRM
the dc-DSVRMs are denoted in Fig. 2. The values of these parameters are listed in Table I.

B. Working Principle and Harmonic Distribution
In order to prove the feasibility of the novel winding, the field generated by the dc field coils which are wound across three stator teeth is studied with a simple airgap MMF and permeability model.The 1-D model of the dc-DSVRM is established in Fig. 2.
To simplify the calculation, we assume that the permeability of the stator and rotor cores is infinite, and ignore the effect of flux leakage.
Fig. 3 shows the MMF waveform generated by the dc field taking the midpoint of the stator slot where the dc winding is placed as the origin.θ s is the angle of the stator tooth,F dc is the amplitude of the MMF across an effective air gap.Since the dc coils are wound across three stator teeth, the fundamental pole pair number (PPN) of dc excitation is calculated as follows: where N s is the number of stator slots, which equals to 18.According to Fourier transform theory, the MMF of dc field winding can be expressed as follows: where F n is the Fourier coefficient, n is a positive integer number.Furtherly, we can obtain the expression for F n as follows: where n(n = 1, 2, 3, . ..) is the order of harmonics, θ is the air-gap position, and θ s is the angle of slot opening.
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TABLE II SPACE HARMONIC COMPONENTS OF FLUX DENSITY
The MMF of dc field winding is then modulated by the rotor teeth whose waveform is shown in Fig. 4 The rotor permeance waveform can be expressed by the following equation: where P max is the permeance of rotor salient poles, P min is the permeance of rotor slots, N r is the number of rotor salient poles, ω is the mechanical angular speed of the rotor, and θ 2 is the mechanical angle of the rotor salient pole.(P 0 /2) represents the dc component of the MMF.
Multiplying the expression for MMF with the expression for rotor permeance yields the air gap flux density According to the expression of the air gap flux density, the PPN of each harmonic component can be obtained.Due to the magnetic modulation theory, the air-gap flux density consists of many space harmonic components whose PPN can be divided into three groups, n N p , n N p + k N r , n N p − k N r , respectively.The order and the speed of each harmonic component group are shown in Table II.
The first group is not modulated by the rotor and remains stationary.Thus cannot be utilized as working harmonics.The second and third groups are modulated by the rotor poles which means some of them can be chosen as working harmonics.

III. ARMATURE WINDING DESIGN UTILIZING HIGH-ORDER-HARMONICS
Based on the above analysis, the main working harmonics of the traditional winding configuration are modulated from the fundamental harmonic of the dc field, whose PPNs can be expressed as follows: where N r is the number of salient rotor poles.Working harmonics of the novel winding configuration is modulated from a high-order harmonic of the dc field.Different from working harmonics modulated from fundament harmonics of the dc field, the working harmonics modulated from high-order harmonics featured higher gear ratios, whose PPN can be expressed as follows: It is found that modulated by stator teeth, there is abundant fifth-order harmonic in the air-gap flux density, which indicates k = 5 in the proposed motor with an 18/13 slot/pole combination [13].
Fig. 5 depicts the detailed connections of both winding designs.As shown in Fig. 5, the concentrated over-lapped winding is adopted in Model I, and the slot pitch of the proposed distributed winding equals 4 in Model II.
The reasonability of the proposed winding design can be explained by the ratio of flux density amplitudes of the two winding design's working harmonics mentioned above.From (6), the working harmonics consist of harmonics modulated from fundamental and higher harmonics.For the winding distribution of Model I, its main working harmonic selection is the component modulated from the fundamental dc field harmonic (P w = N p − N r = 10).While for Model II, its main working harmonic is selected from the harmonic component modulated from the fifth-order dc current field harmonics (P ′ = 5N p − N r = 2).Then the ratio of the flux density amplitudes of these two working harmonics can be calculated from ( 6) The calculation shows that the amplitude of the flux density of Model II is about 34% lower than the amplitude of Model I, but with a much higher gear ratio.It can be calculated that, the gear ratio of the main working harmonics of Model I with conventional winding configuration is 1.3, while that of the main working harmonics of Model II equals 6.5.
The angle difference of the adjacent angle θ can be calculated as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.According to the PPNs of the main working harmonics of the two windings, θ of the proposed winding design is equal to 40 • , while that of the conventional winding equals 100 • .The winding factor k w of these two models is also calculated as follows:

TABLE III WINDING FACTOR COMPARISON
q sin(n α 2 ) k p = sin y τ π 2 (11) where k p is the pitch factor, k d is the distributed factor, q is the number of the least EMF vector per phase, n is the pole-pair number, and α is the phasor angle between the neighboring coils.
Thus, the armature winding design with a winding pole pitch equal to four slots can be adopted.Thus, the concentrated over-lapped winding can also be adopted but will have a lower winding factor than the proposed armature winding design.
The results for the models in this article are shown in Table III.The result shows that the proposed armature winding design utilizing the fifth-order harmonics in the air-gap exhibits a higher winding factor than the conventional winding design, which implies the improvement of the power density of the

IV. PERFORMANCE COMPARISON A. Air-Gap Flux Density
In order to verify the theory of second part, the element analysis (FEA) model of the motor is built with the parameters in Table I.The air-gap flux density of motor at no load is calculated and analyzed in Fig. 6. result shows that due to the modulation of the stator tooth and rotor poles, there are a lot of space harmonic components in the air-gap, including the 2nd, 10th, 16th, 28th and so on.
The phase of the air gap harmonics is calculated and analyzed in Fig. 7.As can be seen from the graph, the harmonics   whose order equal to n N p remain stationary because these harmonics do not go through the modulation of the rotor.The other four harmonics travel through an electrical cycle over one rotor tooth.
The on-load air-gap flux density is also analyzed as shown in Fig. 8 to verify the working harmonics of both armature winding designs.The figure shows that the harmonics whose order is second in the air-gap magnetic field are utilized using the proposed distributed armature winding with its amplitude much higher than that in the model with the conventional concentrated armature winding design.The result of no-load and on-load air-gap flux density is consistent with the analysis in Section II.

B. Back EMF
From the flux modulation theory, the second harmonic's rotation speed is larger than that of the tenth harmonic.Therefore, Model II with a higher amplitude of the second Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.harmonic will produce higher back-EMF at no load.The two models were analyzed using the FEA, and the results are shown in Fig. 9.As shown in Fig. 9, the amplitude of the back-EMFs of Model II is about 93% higher than that of Model I, which is in accordance with theoretical derivation above.The 3-D simulation result of Model II is also shown in the figure, which agrees well with the 2-D result.
According to the results of the fast Fourier transformation (FFT) of the back-EMF waveform as shown in Fig. 10, the fundamental harmonic amplitude of Model II with the proposed winding is significantly increased due to the utilization of the high-order-harmonics in air-gap.Although motor with the novel winding increases the harmonic amplitude of the specific number of back-EMF waveforms, such as fifth, seventh, and tenth, they are too small to be negligible when compared with the amplitude of the fundamental harmonic.

C. Torque Performance
The cogging torque of the model with concentrated winding and the proposed distributed armature winding design are shown in Fig. 11.The cogging torque's amplitude is much smaller than its corresponding average torque.
The electromagnetic torques at a full load of rated current density are compared in Fig. 12.As shown in Fig. 12. Model II has 54% higher torque.Due to the increased harmonics content in the induced voltage, the torque ripple of the machine with the proposed winding configuration is also boosted.This can be furtherly suppressed by the skew method.The 3-D simulation result of Model II is also shown in the figure, which agrees well with the 2-D result.
In order to further analyze the characteristics of the motor, the overload performances of the motors are also calculated, whose current densities vary from 0 to 12 A/mm 2 and the results are shown in Fig. 13.The simulation results show that   the Model II saturates more easily than the Model I, but has a larger torque output at the same current density even when the current is three times of the rated current.Table IV summarizes the output torque performance of the two models.

V. CONCLUSION
In this article, the flux modulation theory is introduced into an electrically excited doubly salient VRM for armature winding design.The novel armature winding connection method for 18s/13r dc-DSVRM is based on the working harmonics modulated from the high-order harmonic of dc fields.
It was revealed that there is a rich content of fifth-order harmonic in the MMF of dc field excitation.Thus, it is reasonable to choose a working harmonic modulated from this high-order MMF for its higher gear ratio.The proposed armature winding configuration is designed according to the PPN which equals to |5N p − N r | using distributed winding method.Two models with the same stator/rotor slots, external dimensions, field winding configuration, and loading are compared in this article to verify the merits of the proposed armature winding design.The simulation shows that the machine with novel distributed windings exhibits a 93% higher back-EMF as well as higher generated torque compared to an existing machine with a conventional concentrated armature winding design under the same excitation condition.

Fig. 1 .
Fig. 1.Winding connection of 18/13 dc-DSVRM with dc field coils across three stator teeth.(a) Model I concentrated winding design.(b) Model II distributed winding design.

Fig. 7 .
Fig. 7. Phase of harmonics of flux density in the air-gap at no-load.

Fig. 8 .
Fig. 8. Amplitude of harmonics of flux density in the air-gap at on-load.

TABLE IV PERFORMANCE
COMPARISON OF TWO MOTORS