Analytical Modeling and Optimization of Partitioned Permanent Magnet Consequent Pole Switched Flux Machine With Flux Barrier

Switched Flux Permanent Magnet Machine (SFPMM) encompass unique features of conventional direct current machine, permanent magnet (PM) synchronous machine and switch reluctance machine therefore, applicable for high-speed applications. However, conventional SFPMM exhibits demerits of high PM volume <inline-formula> <tex-math notation="LaTeX">$(V_{PM})$ </tex-math></inline-formula>, high torque ripples <inline-formula> <tex-math notation="LaTeX">$(T_{rip})$ </tex-math></inline-formula>, higher cogging torque <inline-formula> <tex-math notation="LaTeX">$(T_{cog})$ </tex-math></inline-formula>, lower torque density <inline-formula> <tex-math notation="LaTeX">$(T_{den})$ </tex-math></inline-formula> and significant stator flux leakage. In this paper, a new topology of consequent pole (CP) SFPMM (CPSFPMM) is proposed having partitioned PM that improved flux modulation phenomena utilizing flux barriers. Moreover, due to non-linear behaviour of PM and complex stator structure alternate analytical sub-domain model is utilizes for initial design. However, initial design offers lower open-circuit phase flux linkage <inline-formula> <tex-math notation="LaTeX">$(\Phi)$ </tex-math></inline-formula>, average mechanical torque <inline-formula> <tex-math notation="LaTeX">$(T_{avg})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$T_{den}$ </tex-math></inline-formula>. Aforementioned electromagnetic key performance indicator with <inline-formula> <tex-math notation="LaTeX">$T_{cog}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$T_{rip}$ </tex-math></inline-formula>, total harmonics distortion in <inline-formula> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$\Phi _{THD}$ </tex-math></inline-formula>), average power <inline-formula> <tex-math notation="LaTeX">$(P_{avg})$ </tex-math></inline-formula> and power density <inline-formula> <tex-math notation="LaTeX">$(P_{den})$ </tex-math></inline-formula> are refined utilizing Geometric-Based Deterministic Optimization (GBDO). Analysis reveals that proposed new topology of CPSFPMM with flux barriers reduces <inline-formula> <tex-math notation="LaTeX">$T_{cog}$ </tex-math></inline-formula> by 34.90%, <inline-formula> <tex-math notation="LaTeX">$T_{rip}$ </tex-math></inline-formula> by 20.27%, <inline-formula> <tex-math notation="LaTeX">$\Phi _{THD}$ </tex-math></inline-formula> by 28.08% whereas it enhanced <inline-formula> <tex-math notation="LaTeX">$P_{avg}$ </tex-math></inline-formula> by 17.79%, <inline-formula> <tex-math notation="LaTeX">$T_{den}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$P_{den}$ </tex-math></inline-formula> by 34.38%.


I. INTRODUCTION
Switched Flux Machines (SFMs) are forms of stator-active brushless AC machines where all excitation sources, such as field winding, permanent magnets, and armature winding, are located on the stator, maintaining an inactive rotor constructed lamination sheets. SFMs are divided into three main groups based on the sources of their excitation: permanent magnet (PM) excited, field excited, and hybrid excited (PM and field excited). Dominant three topologies of the SFMs becomes Switched Flux Permanent Magnet The associate editor coordinating the review of this manuscript and approving it for publication was Christopher H. T. Lee .

Machines (SFPMM), Field Excited SFMs (FESFMs) and
Hybrid Excited SFMs (HESFMs). Among these three SFMs topologies, SFPMM provides higher torque density therefore considered here for detail investigation regarding structure variations as well torque capability.
SFPMM constitute passive rotor (made of iron only) and stator encompassing Armature Winding (AW) slots and Permanent Magnet (PM). Alternate polarity PMs are sandwiched between AW slots where both PMs and AW are inserted in the stator. Due to the influence of centrifugal force and heat dissipation, such a design configuration prevents PM demagnetization. The passive rotor of SFPMM is developed from stack of steel lamination VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ sheets. The back electromotive force (EMF) that results from SFPMM's double salient design resembles a sine wave and is consequently preferred for high-speed brushless applications. SFPMMs are consider suitable candidate when high power density are primal requirements which make it applicable for domestic and industrial application such as electric vehicle traction [1], wind application [2], industrial servo and electric aircraft [3], electric bicycle and automotive industry [4]. However, SFPMM utilizes high PM volume which increase machines weight, cost and magnet eddy current losses [5]. Moreover, the PMs are extended to the stator yoke which causes serious issue of the flux leakages. In order to overcomes the aforesaid demerits of SFPMM are classified into various categories based on design topologies which are outer rotor, inner rotor, dual rotor and dual stator. With main concern in simple inner rotor topologies, PM-FSMs are further classified in U-shape Module, Flux adjustor, E/C-Core, π-shape, consequent pole, V-shape, square envelope, and multi-tooth as shown in Figure. 1. The above-mentioned SFMs categories reported in literature are results of overcoming the foregoing demerits of SFPMMs, however each design associates some drawbacks.
In order to suppress PM volume, reduce machine cost, diminish machine weight, and flux leakages, an overview is carried out in literature on different topologies of SFPMM [6], [7]. In [8] author proposed E-Core stator structure which reduces PM volume and increases AW slot area however, there is still significant leakages as shown in Figure. 2. The leakage flux is suppressed utilizing mechanical flux adjustor [9], [10] in all pole and alternate pole. Despite the significant utilisation of PM, this approach demands more mechanical peripherals, which unfavourably raises machine weight, volume, and cost.
Different topologies of SFPMM topologies such as π-form core of stator [11], multi-tooth stator core [12], V-type core [13] and square envelope [14] are recently delineate. However, despite of the high PM volume usage, π-shaped stator is complex in manufacturing and assembling, upper apex of modulation pole of multitooth may saturate due to narrow width, V-shape associate mechanical and electrical loading at the base of support between the V legs whereas  square envelop offer low torque density in compare with conventional SFPMM.
In order to suppress leakage flux recently various topologies of Consequent Pole (CP) machines are introduced. In [15], the author examines the existence of unipolar leakage flux with even order harmonics in CP rotor, while [16] CP and hybrid pole machines. Mechanical limitations aside, torque ripple and cogging torque effects predominate, leading to pulsing instantaneous torque. Using different magnetic poles, [17] suppresses torque ripple and cogging torque; nevertheless, in this approach, average torque drops. In order to preserve average torque, the author [18] adds spoke type rotor CP machines, but regrettably, core less density grows, and irreversible demagnetization occurs at the spoke sort PM corners closer to the air gap.
Author [19] effectively suppress the leakage flux by introducing Consequent Pole SFPMM (CPSFPMM) with flux bridges and flux barriers as illustrated in Figure. 3(a). This excellently supresses flux leakage of stator end however, PM utilization is high. Aside of high usage of magnet, the fine flux bridge saturates that worsen performance. Furthermore, flux bridges provide miniature track to the flux that uplifting flux cancellation and circulation as indicated in Figure. 3(b) [20].
This research proposes a new CPSFPMM with reduced PM consumption, as illustrated in Figure. 4, to address the above-mentioned problems of increased PM consumption, flux leakages, magnet cost, machine weight, flux cancellation and circulation linked with SFPMM and CPSFPMM. Figure 4(a) tends to show clearly that CPSFPMM efficiently reduces the utilisation of the PMs by adding a poles alternative to PM and reducing flux cancellation and circulation. Additionally, Figure 4(b) demonstrates how CPSFPMM successfully lowers flux bridge saturation during initial design, when compared to the conventional current state of the art. Moreover, the new topology of the CPSFPMM offers lower torque ripple, cogging torque, and greater torque and power densities.
Numerical based Finite Element Analysis (FEA) method is intensively utilized for accurate modelling of electric machines and performance evaluation before proceeding to manufacturing and fabrication [21]. However, when used for initial sizing, and performance analysis, FEA incorporates geometric features, magnetic saturation, complicated stator structure, and non-linear behaviour of PMs, resulting in computational complexity and time-consuming [22]. Moreover, Finite Element (FE) analysis requires expensive software/ hardware and large drive due to repeated iteration [23]. Therefore, author [20] suggest analytical model for initial design purpose.
To overcome computationally complexity, computational time, computer memory and drive storage this paper also utilizes alternate analytical sub-domain modelling approach in CPSFPMM for accurate prediction of magnetic field distribution in initial design phase machine sizing. The initial design obtained offer lower open-circuit phase flux linkage and low average mechanical torque therefore progressed to Geometric-Based Deterministic Optimization (GBDO) for enhancing aptitudes to attain improved performance and torque density. Analysis shows that as compared to the existing conventional model, the proposed novel CPSFPMM with less PM consumption demonstrates improved electromagnetic performance.
Main contributions in this paper is, author present CPSFPMM which successfully cuts 46.53% of PM usage. Moreover, analytical model (sub-domain approach) is investigated for preliminary sizing and validated by commercially available FEA package JMAG designer v18.1. Moreover, geometric-based deterministic optimization is carried out for enhancing capabilities of electromagnetic performance. Analysis reveals that the developed CPSFPMM at reduced PM usage offer 34.38% higher torque density and effectively curtailed inherent cogging torque and torque ripples by 34.9% and 20.27% respectively. Detailed performance analysis is listed in the proceeding sections.
In the following, section II extant to CPSFPMM construction, section III demonstrates formulation of analytical model. Section IV explains performance calculation, section V studied analytical model validation, section VI examines geometric-based deterministic optimization, section VII overview performance comparison. Finally, section VIII draws conclusions.   Table 1. CPSFPMM model consist of E-core stator slot having alternate h-shaped stator tooth occupied by CM-PMs pole to reduce flux leakages going across the stator yoke and RM-PMs placed above h-shape stator tooth. Flux leakages from PM in poles are reduced with the help of the reverse magnetized radial PMs resulting in higher magnetic flux distribution in the stator yoke and hence improving the flux linkage by passing through alternative flux bridges and flux barriers and link to the rotor.

II. CPSFPMM CONSTRUCTION
The proposed CPSFPMM improve magnetic field distribution where flux links directly through stator yoke with negligible reluctance. Hence magnetic field distribution and modulation effect in new topology of CPSFPMM is better compared to the existing conventional designs.

III. FORMULATION OF ANALYTICAL MODEL
There are two phases in sub-domain model. As illustrated in Figure. 5, the whole field domain is first separated into subregions. The air gap, stator slots, and PMs are subregions of the field domains for proposed CPSFPMM. In this modeling, it is assumed that iron core is infinite permeable and end effect such as leakage flux is negligible.
By resolving the Poisson function enclosing the source in each subdomain, the general vector potential (GVP) formulation in subregions of the sub-domain is denoted by [24] whereas r is the radius or circumference that varies with each boundary and interface conditions. Sub-domain regions i.e., magnets, air gaps, and stator slots are examined independently for its general vector potential solution.
In the PM area, the GVP solution is stated as Air-gap GVP solution is stated as Stator slot GVP solution is stated as where µ r is relative permeability, k is harmonics order, J is current density, A Ik , B Ik , C Ik , D Ik , A IIk , B IIk , C IIk , and D IIk are Fourier coefficient. The foremost source in CPSFPMM is AW and partitioned PMs. Magnetization of magnet distribution are stated as summation of cosine and sine harmonics as [25] whereas M θ and M r is tangential and radial magnetization components.
where M θ sk and M θck are magnitude of k th order sine and cosine tangential magnetization, M rsk and M rck are magnitude of k th order sine and cosine radial magnetization patterns and ω r is rotor rotational speed. Additionally, the current density for non-overlap winding design in the i th armature winding slot may be given as [25].
Typically, the magnetic flux density (MFD) tangential and radial components are given by [27] Air-gap MFD components are stated as 123908 VOLUME 10, 2022 PMs MFD components are stated as Armature slots MFD components i.e., radial and tangential are expressed as Boundary and interface conditions are employed to compute unknown Fourier coefficients where the stator tooth opening angle is g and the slot opening angle is f . The left and right sides of the interface conditions are translated to the equivalent interim for magnetic flux density constituent since they take the form of a Fourier series of various intervals. Over the interval [−π, π] the left half of the formula transforms into a Fourier series, and the right side of the Fourier series is Using a piecewise function and extending Fourier series in the [−π, π] range, the GVP over boundary condition is represented as The left side transformed to Fourier series over interim to make applying the boundary and interface condition Fourier coefficient equation can be used as follows In the following boundary condition, the tangential magnetic field contribution is zero at the yoke of rotor.
Applying condition M θ = 0, The radial magnetization component is stated as Simplifying B Ik and D Ik in term of A Ik and C Ik as The Fourier coefficients A Ik , B Ik , C Ik , D Ik , A IIk , B IIk , C IIk and D IIk is computed utilizing first order multi-variable equations [24], [25], [26], [27] whereas detailed formulation of the sub-domain model can found in [30].

IV. PERFORMANCE CALCULATION
Performance of the proposed CPSFPMM is calculated under no-load and loaded condition.
The flux linkage related to the AW coils set of the phase makes up the open-circuit phase flux linkage for that phase. Open-circuit phase flux linkage is computed using vector potential distribution [27] = Additionally, the no-load torque that a machine exhibits as a result of the attraction of the PMs and AW slots is known as cogging torque. Cogging torque produces acoustic noise and vibration, which is undesirable in applications, particularly in robots. The Maxwell Stress Tensor (MST) approach described in [3] with cosine and sine component of tangential and radial MFD is useful for computation of the cogging torque CPSFPMM where B rck is radial cosine, B θ ck is tangential cosine, B rsk is radial sine, and B θsk is tangential sine components of MFD. Finally, the torque produced under loaded condition is referred as mechanical torque. Using tangential and radial components of MFD and the Maxwell Stress Tensor (MST) approach described in [1], mechanical torque of CPSFPMM is calculated where B Ir and B I θ are radial and tangential component of MFD, respectively.

V. ANALYTICAL MODEL VALIDATION
Based on the parameters mentioned in table 1, the sub-domain analytical model is validated using JMAG FEA package.
To accurately determine the electromagnetic performance of the CPSFPMM, transient analysis is performed at a speed of 1200 r/min and a current density of 15 A/mm 2 . In order to validate analytical sub-domain model some of the key assumptions, boundary conditions and excitation are (i) Rotor and stator are assigned material considering infinite permeability, (ii) Magnetization direction of the PMs are radial and circumferential alignment, (iii) Position of rotor tooth with stator are in aligned position, (iv) All PMs segments have same magnetic properties and dimensions, (v) Developed CPSFPMM is rotary in nature with periodic air-gap permeances, therefore only one pole pitch periodic boundary is considered and (vi) 15 A/mm 2 excitation current density is applied to the armature winding. Furthermore, it is worth mentioning that both in sub-Domain Modelling and FEA validation, magnetic saturation is not considered yet therefore, the peak values of magnetic flux densities reach to 3.75T in on load radial component.
Under no-load condition, Figure. 6 contrasts the opencircuit phase flux linkage results from the related FE analysis and analytical sub-domain results. Open-circuit phase   Figure. 7 and Figure. 8 respectively whereas on-load radial and tangential MFD components predicted by analytical sub-domain and existing FE analysis are shown in Figure. 9 and Figure. 10 respectively. Analysis reveals that predicted analytical sub-domain modelling under no load and on-load conditions fairly match with globally accepted FEA with peak relative error of ∼ 2%. Moreover, magnitude of the MFD components for the proposed design are higher when compare with the conventional design due to improved modulation effect which result better electromagnetic performances.
As shown in Figure. 11, cogging torque in CPSFPMM using MST is computed and compared with FEA using no load radial and tangential MFD components. According to study, the cogging torque predicted analytically for the CPSFPMM's original design exhibits   immensely durable agreement with FE analysis, with ∼ 1.98% error.
Finally, under loaded condition, mechanical torque is calculated in CPSFPMM using MST and compared with FEA using on-load radial and tangential MFD components as illustrated in Figure. 12. According to study, mechanical torque sub-domain modelling projected analytically for the CPSFPMM's original design exhibits extremely strong agreement with FE analysis, with ∼ 1.76% error.

VI. GEOMETRIC-BASED DETERMINISTIC OPTIMIZATION
GBDO is opted for improving performances of CPSFPMM with key performance pointer i.e., peak to peak open-circuit phase flux linkage ( p−p ), peak to peak cogging torque (T cog ), average mechanical torque (T avg ), torque ripples (T rip ), open-circuit phase flux linkage Total Harmonic Distortion (THD) ( THD ), average power (P avg ), torque density (T den ) and power density (P den ).
GBDO is opted in iteration with the overall flow as shown in the flow chart of Fig. 13. First an initial design is developed and to implement GBDO, different design parameters are selected based on their sensitivity such that design parameter with most sensitive should be followed first and then procced to another parameter with the optimum point of previous stage. GBDO consecutively optimize rotor and stator oneafter-another and in sequence. Optimization goals are defined with main constraints as Objective function max T avg , T den , P avg , P den min T cog , T rip , THD and (40) In order to ensure that the GBDO is applied solely to the design parameters and its configuration, it is important to note that the electrical loading (number of turns and applied current), magnetic loading (PM volume), axial length, slot area, and air gap length are all kept constant during implementation. Different design parameters selected for optimization are split ratio, rotor pole ratio, PM width, Flux bridge height, Flux bridge width, Slot depth, PM opening and stator yoke width. Note that the rotor pole optimization is carried out on rotor tip width and rotor base width for optimal rotor pole and it's respective shape. In order to implement GBDO on aforesaid design parameters following optimization coefficients are specified Note that the order of optimization is same as the optimization coefficient is defined and the rotor pole ratio is separately applied to rotor tip (K rpr_tip ) and base (K rpr_base ). Moreover, quantitative electromagnetic performances and it's influence are discussed based on optimization coefficient as follow.

A. INFLUENCE OF SPLIT RATIO
Split ratio is defined as the ratio of stator inner diameter to the stator outer diameter. This step helps to observe effects of stator and rotor back iron height. Due to magnetic saturation of stator yoke, maximum value of split ratio is restricted by h si /2 and minimum is restricted to 2 * h si to confirm uniform magnetic field distribution in stator tooth. Split ratio is very important design parameter which is utilized to achieve higher average mechanical torque. Electromagnetic performance analysis for different split ratio is shown in Figure. 14 which elaborate that split ratio is much sensitive to average mechanical torque. It is notable that during split ratio optimization electric/magnetic loading, rotor pole width and stator pole width remains same. During this step, only rotor back iron height, stator yoke width and height of the flux bridge vary.
Initially, split ratio is 0.61 but electromagnetic performance analysis for split ratio shows that optimal split ratio obtained is 0.56 (shown by encircle) in which average mechanical torque is improved from 2.90 Nm to 2.99 Nm. Moreover, peak to peak cogging torque is reduces from 1.66 Nm to 1.32 Nm, no load open circuit phase flux THD from 1.31% to 0.91 % and torque ripples are reduced from 3.48 Nm to 2.76 Nm whereas torque density and power density are improved to 369.7 kNm m 3 and 48.96 kw kg respectively. Therefore, the design with β s =0.56 is selected for further optimization stages.

B. INFLUENCE OF ROTOR BASE WIDTH
The ratio ''K rpr_base '' is varied for optimal selection of the rotor base width. Initially the base width is 7 mm which is varied in the range of 2.47 mm to 7.74 mm. Variation of electromagnetic performance for different rotor base width is shown in Figure. 15. It can be clearly seen that, at this stage the rotor base width has slight influence on electromagnetic performance. Therefore, the rotor base width will be optimized again with the rotor tip after stator optimization in the second cycle. Based on electromagnetic performance comparison, design with 4.86 mm rotor base width shows comparatively better results and selected for forgoing stator optimization.

C. INFLUENCE OF CM-PMS WIDTH
This section investigates CM-PMs width dedicated optimization coefficient of K PMw . The coefficient is evaluated in such a way that CM-PMs W PM are extended in the range of 1.71to 3.91 mm for optimal width which ensure least possible slotting effects. At this stage of optimization, the overall PM volume is kept constant by changing the RM-PMs spans above the h-shape tooth. The width of the adjoining flux barriers is adjusted together with the width change in the CM-PMs. Figure 16. shows the performance assessment for various CM-PM widths.
Performance evaluation discloses that CM-PMs W PM not only improve average mechanical torque from 3.05 Nm to 3.18 Nm but also strongly enhance open circuit phase flux linkage from 0.076 Wb to 0.087 Wb due to reduction in interaction between slots and PMs. Moreover, the increase in the average torque results higher torque density of 393.68 kNm m 3 , average power of 528.73 watt and power density of 52.14 kw kg. Based on quantitative electromagnetic performance analysis, CPSFPMM having CM-PMs W PM =2.15mm for the foregoing optimization. Note that the initial CM-PMs W PM was 3.03 mm.

D. INFLUENCE OF FLUX BARRIER HEIGHT
The partitioned PMs (RM-PMs and CM-PMs) used in the CPSFPMM are housed in a stator with an h-shaped design that includes a flux barriers and bridge. While flux barriers serve as the PM pole since the employed PMs are uni-polar whereas flux bridges offer an alternative magnetic channel to flux distribution to boost flux modulation.
In this section, electromagnetic performance is investigated with the height of the flux barrier as shown in Figure.17. Since the height of the flux bridge and barrier are same, only one of their heights needs to be developed, cutting down on computing time and further optimization processes.
During initial design, height of the flux barrier was set to be 37 mm which was later changed to 35.2 mm in split ratio optimization. During this optimization stage, the height of flux barrier is varied between 33.2 mm to 37.2 mm with the difference of 1 mm. In varying flux barrier height, overall magnet usage is kept constant by adjusting it through the RM-PMs span. Moreover, width of the flux bridge, width of the flux barrier, stator tooth, stator yoke, rotor pole, electrical loading, air gap and stack length remain unchanged.   Detailed electromagnetic performances analysis as shown in Figure. 17 reveals that flux barrier is sensitive to open circuit phase flux linkage and average mechanical torque which leads to higher torque density. At optimal flux barrier height (R FBH = 36.2 mm) the open circuit phase flux linkage increased from 0.087 Wb to 0.088 Wb and average mechanical torque from 3.18 Nm to 3.25 Nm. Moreover, in this step cogging torque is diminished by 5%, torque ripples are truncated by 4.8%, THD is reduced by 11.8%. In additional, the torque density and power density are enhanced to 401.69 kNm m 3 , and 53.20 kw kg respectively.

E. INFLUENCE OF FLUX BRIDGE WIDTH
In design of CPSFPMM, partitioned PMs such RM/CM-PMs are physically segregated via flux bridge, which offers an additional channel to working harmonics for enhancing flux modulation. But, as seen in Figure 18, there is a major problem with leaky flux and flux cancellation because of the circulation of flux in the h-form stator tooth via stator yoke, flux bridge, and RM-PMs as display in Figure. 18(a). Therefore, flux bridge width is examined to achieve the least amount of flux circulation.
At prevent stress and saturation, W FB was initially fixed to 1 mm throughout the design process. W FB varies between  Figure. 19 detailed electromagnetic examination demonstrates that electromagnetic performance significantly improves at lower W FB but that a narrow flux bridge is put under stress. The flux bridge is therefore removed in order to address the saturation and stress problems on thin flux bridge.
Quantitative electromagnetic analysis at W FB = 0 mm in Figure. 19 shows that open circuit phase flux linkage is improved to 0.09 Wb from 0.088 Wb due to decrease in circulating and cancellation flux at the h-shaped portion as shown in Figure. 18(b). Due to significant compensation of cancellation flux, the average mechanical torque is boost to 3.48 Nm from 3.25 Nm which leads to higher torque density of 430.72 kNm m 3 . Moreover, average power and power density are enhanced to 566.16 watt and 57.07 kw kg respectively. However, cogging torque and torque ripples slightly increases but still within the desired target range.

F. INFLUENCE OF ROTOR POLE
The rotor pole is optimized in two separate steps. First the rotor base width is optimized and then proceed to the rotor tip width. Both these steps effectively optimized the rotor pole to improve open circuit phase flux linkage without interfering average mechanical torque.
Initially, the rotor base width is 4.68 mm and rotor tip width 3.6 mm. The rotor base width is varied between 3.6 mm to 6 mm whereas rotor tip width is varied between 3 mm and 4.8 mm at the difference of 0.2 mm.
Comprehensive performance with changing rotor base width and rotor tip width as display in Figure. Figure. 22(a) demonstrates that the left edge of the original rotor pole is not entirely aligned with the stator tooth (flux barrier) at the same mechanical degree, ensuing lesser phase flux linkage. A larger percentage of rotor poles aligning with the stator teeth at the same mechanical degree would increase the phase flux linkage. This is done with the help of rotor pole optimization where rotor tip is varied at the same mechanical degree. The phase flux linkage is increased with the wider rotor pole and can be clearly seen from Figure. 22(b) that the rotor pole is fully aligned to stator tooth.

G. INFLUENCE OF ROTOR POLE HEIGHT
CPSFPMM is double salient rotor structure made of iron only. The operating principle clarifies that mechanical torque is produced due to partitioned CM/RM PMs. The magnetic flux produced by partitioned PM flow through air gap in the rotor and stator.
The influence of rotor pole height on electromagnetic performance is investigate in the range of 5.6 mm to 8.6 mm. Initially, the rotor pole height was set to be 6.6 mm however, performance analysis utilizing GBDO at different rotor pole as seen in Figure. 23 shows that higher mechanical torque and open-circuit phase flux linkage can be achieved when the rotor pole height is greater or equal to two times of the stator tooth. It is also confirmed from analysis that the optimal rotor pole height is selected to be 7.8 mm as encircled by dotted line. In this step, the average mechanical torque reaches to 3.49 Nm and the phase flux linkage to 0.0935 Wb whereas cogging torque, torque ripple, THD , average power, torque density and power density almost remain the same.

VII. PERFORMANCE COMPARISON
Electromagnetic performance with metric encompassing graphical and tabulated quantitative analysis before and after GBDO is carried out in this section to validate and investigate electromagnetic performance of initial and optimized design as well as compare with conventional design. Key performance indicators of initial optimize design such as 3-phase no-load open-circuit phase flux linkage is shown in Figure. 24 with corresponding harmonics in Figure.25 whereas cogging torque and instantaneous mechanical torque are shown in Figure. 26 and Figure. 27 respectively.
Detail electromagnetic study of initial, optimized, and conventional designs are listed in table 2. Moreover,   a comprehensive comparison with different topologies of SFPMM are carried out in [28] and [29]. Analysis demonstrates that GBDO successfully achieved the targeted torque, higher torque density and open-circuit phase flux linkage without interfering other parameters. It can be clearly seen that, the peak-to-peak flux linkage is increased by 15.65%, peak to peak cogging torque is diminish by 9.25%, enhance average mechanical torque by 20.09%, supress torque ripples by 16.32%, truncate harmonics of phase flux linkage by 12.94%, raise average power by 11.22%, boost torque density and power density by 18.08% and 18.06% respectively.
In addition, proposed topology of CPSFPMM with reduced PM and without flux bridge is compared with the conventional CPSFPMM with flux bridge and more PM usage. It is worth mentioning that developed model utilizes 46.53% of  the total PM volume. Despite of the reduced PM usage, the develop model successfully suppressed flux circulation and cancellation effect. Table. 2 offers a comprehensive performance illustration of proposed and conventional topology. Analysis reveals that proposed CPSFPMM without flux bridge reduces T cog by 34.90%, T rip by 20.27%, THD by 28.02% whereas it enhanced P avg by 17.79%, T den and P den by 34.38% at the cost of 8.67 % and 9.03% p−p and T avg respectively.
The initial and optimized design with Nephogram is shown in Figure. 28 and their respective parameters are listed in table 3. It can be clary seen from Figure. 28(a) that the flux bridge in the initial design saturate which degrade the electromagnetic performance. Therefore, the flux bridge in optimized design is eliminated during optimization process. The maximum magnetic flux density is 2.418 T at the saturation point of the flux bridge. Moreover, optimized design as shown in Figure 28(b) illustrate that no part of the machine saturates. Hence improve magnetic flux distribution and results better electromagnetic performance.
The torque and power vs speed curves in Figure. 29 demonstrate CPSFPMM design, which has been suggested to be used torque higher speed applications. Moreover, from Figure. 30, it can be clearly seen that when operating the proposed CPSFPMM under higher current densities and varying current phase, it is found that higher mechanical torque and lower torque ripples are achieved when d-axis current are set zero.
Furthermore, proposed CPSFPMM is validated by comparing it with various E/C-core SFPMM. Extensive

VIII. CONCLUSION
In this paper, a novel CPSFPMM design with partitioned PM and a flux barrier is developed, which improves the effects of flux modulation and lowers the overall PM volume by 46.53%. According to FE analysis, removing flux bridges, which impeded flux cancellation and circulation and offered a different track via flux barriers, improved modulation. Additionally, analytical model is used in the initial design and verified using the widely used FEA tool JMAG designer v. 18.1. Performance i.e.,T cog , T rip , THD , P avg , T den VOLUME 10, 2022 and P den are refined utilizing GBDO. FE analysis reveals that proposed CPSFPMM topology reduces T cog by 34.90%, T rip by 20.27%, THD by 28.02% whereas it enhanced P avg by 17.79%, T den and P den by 34.38%. As a result, the authors are satisfied in their decision to use an analytical approach for the initial sizing and GBDO for improving performances.