A Comprehensive Analytical Sizing Methodology for Transverse and Radial Flux Machines

Transverse flux machines have the potential to offer high torque density in direct-drive vehicle traction applications. Besides, sizing equations are a wide-spread technique for transverse flux machines design, as their computational cost is much lower than the finite element method. In this paper a novel analytical sizing methodology for transverse and radial flux machines is presented, focusing on the current load and the pole length factor as the main design parameters. The motor specifications are intended for a light-duty electric vehicle application. As transverse flux machines have a single, hoop-shaped coil per phase that embraces the flux of all the pole pairs, their principle of operation and therefore their sizing equations differ from radial flux machines. The proposed analytical method allows to compare transverse and radial flux machines easily through a similarity analysis and a parametric study. Furthermore, the discrepancies between the analytical model and the finite element method are quantified and then included in previous equations. Then the analytical model is optimized with a multiobjective genetic algorithm in the final stage. According to the sizing methodology presented here, transverse flux machines have a superior performance than radial flux machines in terms of torque density and efficiency.


A s
Stator current load, RMS. a s Number of parallel paths in the stator.

B0
No-load flux density, peak value.B 0 No-load flux density, mean value.

B01
No-load flux density, peak value of the first harmonic.Baq q-axis armature flux density, peak value.B aq q-axis armature flux density, mean value.Baq1 q-axis armature flux density, peak value of the first harmonic.

I. INTRODUCTION
Transverse flux machines (TFMs) have the potential to offer high torque density in direct-drive applications such as electric vehicles, wind power and fault-tolerant aircraft generators [1].TFMs benefit from magnetic and electrical loadings being decoupled, resulting in high torque density [2].However, their design and operation are still challenging regarding torque ripple and power factor [3], [4].Moreover, TFMs have been seen traditionally as difficult to manufacture due to their complex structure, but in the last few years additive manufacturing is opening up new posssibilies regarding the housing design, in order to accommodate a large number of pieces and facilitate cooling, and improvements in electromagnetic performance with the use of soft magnetic materials [5], [6].
As TFMs design methods based on the finite element method (FEM) require 3D, time-consuming models, sizing equations or magnetic equivalent circuits are generally used for preliminary analysis, at the cost of offering lower accuracy than FEM.In [2] the authors present some sizing equations based on the gap-to-outer diameter ratio, with a remarkable effort to integrate different voltage, current and power waveforms in sizing equations.The analysis is limited to the optimal power density output for different speeds using both rare-earth and ferrite magnets, with the rareearth TFM showing a better performance than the induction machine in terns of power density.This method has been expanded to other TFMs in [7], [8], and [9].The sizing equations presented in [10] and [11] are more accurate, but not straightforward, being based on the complex permeance method to obtain the flux density distribution in the air gap, then the torque is calculated introducing a flux factor that takes into account the effective air gap flux that produces torque.Furthermore, TFMs sizing is addressed using a magnetic equivalent circuit (MEC) in [12], [13], [14], [15], and [16] TFMs modeling is based on a position-dependent air gap flux density and MEC.A TFM with a single-row of magnets is proposed in [13]: validation of MEC with no-load tests shows a 9 % discrepancy in the back-EMF, while FEM shows a great agreement with tests.However, FEM versus experimental error becomes higher under load, and the same conclusion is drawn in [17].The authors of [18] propose a two-phase TFM with skewed PMs and shared-border pole stator, showing a 2.6 % error between FEM and experimental effective values of the back-EMF.Besides, magnetic charge method has been also proved as feasible for TFMs design [19] even though it has not been widely used in the literature.
TFMs have been explored for electric vehicle applications such as e-scooters [20], [21], [22], hybrid cars [23], hybrid buses [24] and ships [25].Recently a claw-pole TFM has been integrated in a commercial motorcycle [26], [27].TFMs efficiency has been proved similar to RFMs in the low-speed, low-torque range, being superior for TFMs in high-torque points and lower in high-speed points [28].However, the overload capability, power factor and efficiency of TFMs must still be improved to compete with radial flux machines performance [29].In [30] three transverse, radial and axial flux machines with the same dimensions and permanent magnet (PM) mass are compared, with the TFM showing a greater torque for low current load, but poor flux-weakening capabilities.
In this work a comprehensive analytical sizing methodology for transverse and radial flux machines is presented.The motor specifications are based on a light-duty electric vehicle, without loss of generality.The model is based on a first-harmonic analysis as a trade-off between accuracy and complexity, and then the underlying assumptions are verified via FEM.All the main dimensions and parameters have been considered in the analysis, not just the outer dimensions as usually done in the literature.Furthermore, the equations are easily adaptable to different transverse and radial flux machine topologies, thus allowing a more straightforward and general analysis than previous gap-toouter diameter analytical models or magnetic equivalent circuits.The proposed equations have a similar form for transverse and radial flux machines, thus allowing to deduce the similarity equations presented here in order to compare both topologies.It means that, given some specifications and design parameters, most of the TFM dimensions can be calculated from a base radial machine applying the similarity equations.
This document is structured as follows.In Section II the main dimensions and sizing equations of TFMs and RFMs are deduced for each of the stator and rotor components.Section III deals with the similarity equations, stating that the TFM has an extra degree of freedom for design, and the parametric study, with the pole length ratio and current load as the main design parameters.In Section IV the discrepancies between the analytical method and FEM are quantified by defining different FEM factors.In Section V the analytical model is optimized, including surface-fitted FEM factors, and Section VI concludes the paper with a brief summary.

II. SIZING EQUATIONS FOR TRANSVERSE AND RADIAL FLUX MACHINES
Conventional radial flux machines (RFMs) and axial flux machines (AFMs) start from a similar concept, as the flux of each pole goes through the coils, distributed in slots, and then the flux linkage of all the series-connected coils in a phase is added up.However, in TFMs there is a unique, concentrated coil per phase that links the flux of all the poles at the same time.In all cases, if the gap diameter is kept constant and the pole pairs are duplicated, the pole section is halved (1).As RFMs and AFMs are generally formed by the series connection of the coils, the number of turns per phase has to be duplicated in order to keep constant the no-load flux linkage (2) and so the back-EMF and torque.However, in TFMs the hoop-shaped coil embraces the flux of all the pole pairs in a phase, so the halving of the pole section is compensated by the duplication of the pole pairs, then an increase in the number of turns results in a multiplication of the back-EMF and torque.Therefore, TFMs take advantage of a high number of pole pairs to deliver high torque.Besides, hybrid machines that combine some features from TFMs and RFMs have been proved recently as a good alternative in order to increase further torque and torque density with respect to conventional induction machines and permanent magnet synchronous machines [31], [32].
Ψ0 ∝ N ms S p p for TFMs N ms S p for RFMs (2) The main parts of the magnetic circuit of TFMs and RFMs and their dimensions are shown in Figure 1.In TFMs the flux path is essentially perpendicular (i.e.''transverse'') to the rotation plane, so the magnetic and electrical loadings are decoupled.Thereby the stator span α s is chosen in a similar way as the rotor span, according to (3), in order to prevent leakage flux.Both rotors in Figure 1 correspond to a surface permanent magnet synchronous motor (S-PMSM), in which the equivalent d-axis gap is the sum of the magnets radial height g m and the air gap g a corrected with the PMs relative permeability µ rm and the Carter factor k C , respectively.As the PMs permeability is close to that of air, both the dand q-axis equivalent gaps are assumed to be equal.Here the magnets-to-air gap ratio (5) has been introduced, as it is practical for sizing purposes: in Section II-B it will be shown that the no-load flux density in the air gap depends only on the PMs remanence and the magnets-to-air gap ratio. ) A. SPECIFICATIONS Table 1 shows the main specifications of the electrical motors that have been designed in this paper.This values correspond to a light-duty electric vehicle application, according to literature.As the aim is to compare TFMs and RFMs for direct-drive vehicle traction applications, unit transmission ratio has been considered.Figure 2 shows the phasor diagram of a S-PMSM in its maximum torque operating point, that is, according to (6), when the stator peak current is fully aligned with the q-axis.At other operating points, the stator current phasor forms an angle δ i with the q-axis (7).The electrical angular frequency is given by (8) as a function of the rated speed and the number of pole pairs.The RMS value of the stator voltage is related to the DC-bus voltage, as in electric vehicles the motor is fed by a power converter: in this work the peak voltage of the extended sine-triangle or space vector modulation techniques at their maximum modulation index has been considered for calculations (9) [33].Once the stator RMS voltage is known, the back-EMF and the power factor angle can be calculated (10) and finally the no-load flux linkage (11) and the inductances ( 12), (13). ) L q = L d (13) In conventional RFMs both the stator and rotor cores are made of electrical steel laminations, whereas in TFMs the use of soft magnetic composites is progressively gaining wider attention [34], [35].Table 2 shows some representative values of the flux density in different parts of the machines that have been taken in this work for calculations, based on the materials properties.
The no-load flux linkage and the inductances are key parameters for electrical machines sizing, as they are related to some important dimensions.The no-load flux linkage is proportional to the number of turns and the pole surface (1), (2).In classical electrical machines theory, the number of turns is related to the stator current and the gap diameter via the current load, that in this paper is defined as (14).Therefore, the torque (6) can be expressed as a function of the gap diameter, the pole length and the current load.If a geometrical ratio that relates the gap diameter and the pole length is defined, both magnitudes can be obtained.Besides, the d-axis inductance is inversely proportional to the d-axis air gap (15), so the d-axis air gap can be calculated once the number of turns is known.Therefore, in this first stage of the analytical design process it is shown that three outer parameters of the motor (torque, no-load flux linkage, inductance) can been correlated with three main dimensions (air gap diameter, pole length, air gap) using an intermediate parameter (the current load) to obtain the number of turns.In Section II-B the definitions of the no-load flux linkage for TFMs and RFMs will be discussed.Then the rest of the machine parameters can be obtained, according to the flowchart in Figure 3.

B. NO-LOAD FLUX LINKAGE AND D-AXIS ARMATURE FLUX LINKAGE
In PMSMs the energy conversion is produced by the interaction of the no-load flux linkage -created by the rotor PMsand the armature flux linkage -created by the stator current flowing through the stator winding.Ideally, both fluxes cross entirely the air gap, thus being named ''magnetizing'' flux linkages.Actually, the stator currents produce an extra flux, i.e. the ''leakage'' flux, that does not cross the air gap but increases the inductance and contributes to magnetic saturation.
If the PMs or the stator coil are shortened β electrical degrees and the air gap is constant along the pole pitch, the flux density waveform in the air gap is rectangular [36], as shown in Figure 4.Only the radial component of the flux density is assumed to contribute to flux, being constant along the axial direction.Assuming that the only reluctance of the magnetic circuit corresponds to the air gap (i.e. the machine is not saturated), the peak value of the no-load flux density (that is, the magnetic field created by PMs) is given by ( 16).On the other hand, the stator current in the q-axis produces a magnetomotive force (MMF) per pole k N N ms Îq along the q-axis air gap g q , leading to a peak armature flux density in the q-axis named Baq (17).N ms is the number of turns per phase, so the equivalent turns factor k N has been introduced as the ratio between the MMF per pole and the MMF per phase N ms Îq .Furthermore, similar magnitudes can be defined for the d-axis, based on the d-axis current Îd .

B0 = Bm
k gma 1 + k gma (16) The equivalent turns factor is defined in a different way for TFMs and RFMs, as stated in (18).In TFMs the armature field is created by a single-phase winding, as all the phases are axially stacked and so independent, and according to the Ampère's law (Figure 5a), the phase MMF N ms Îq acts on each pole pair, the field crossing the air gap four times, that is why k N turns out to be 1/4.On the other hand, in RFMs the field is created by the entire three-phase winding being fed by three-phase, balanced currents (m = 3), so the total MMF is m/2 times the MMF per phase.Moreover, as the number of turns per phase N ms is 2p times the number of turns per pole, the MMF per phase has to be divided by 2p to obtain the MMF per pole, that is why k N equals m/ (4p) in RFMs.
for TFMs m/ (4p) for RFMs (18) Once the peak values of the flux density are known, the amplitude of the first harmonics and their mean values are straightforward according to Fourier analysis ( 19), ( 20), (21), (22).ξ r1 is the first-harmonic ''winding'' factor of the rotor ( 23) and ξ s1 is the stator first-harmonic winding factor, the latter defined as the product of the short-pitching factor and the distribution factor.
In TFMs the magnetic flux circulates mainly through a plane that is perpendicular to the rotational plane, so the magnetic and electrical loadings are decoupled.The phase coil is embraced by the flux of all the pole pairs at the same time, in a different way than RFMs, where each coil embraces the flux of a single pole pair and then the flux linkage per phase is obtained as the sum of all the pole pairs -if the coils are series-connected.In this case, the no-load and q-axis armature flux linkages follow ( 24) and ( 25) respectively for both TFM and RFMs.As the flux density has been reduced to its first harmonic, the flux linkage is assumed to be sinusoidal.Fringing, leakage or saturation effects can be included in the model using some factors, named no-load factor k 0 , armature factor k a , leakage factor k σ and saturation factor k sat .The leakage factor is further explained in Section II-I.Initially all these effects are neglected and in Section IV they will be analyzed via FEM.
In RFMs D 2 g l is the prismatic volume of the air gap, but not in TFMs as the pole length l is much lower than the outer length.Therefore, it is convenient to define a representative aspect ratio of each machine called ''pole length factor'' k l as in (27).In this way, the number of pole pairs p that multiplies the noload linkage of TFMs is canceled, so a common D 3  g sizing equation can be deduced for TFMs and RFMs, as suggested in (28).If a ''pole surface factor'' ( 29) is defined to introduce the factor π/2 of TFMs, a common equation to calculate the gap diameter of TFMs and RFMs is deduced (30).Table 3 summarizes the main design parameters of the rotor and the stator that are needed to obtain the gap diameter.

E. STATOR TEETH, STATOR YOKE, ROTOR YOKE AND TFM BRIDGES
In traditional electrical machines, copper or aluminum conductors are allocated in slots along the stator perimeter.TFMs do not have slots in the classical way, instead they have U-shaped cores that surround the winding -and sometimes I-shaped bridges-that guide the magnetic flux from one pole to another.Therefore, in TFMs the number of U-cores may resemble the number of slots in RFMs (32), as both serve to allocate the stator winding.Anyway, the number of slots and the slot pitch (33) are more relevant in RFMs, as these magnitudes are related to the winding factor and the stator tooth width (34), respectively.In TFMs it is more practical to impose the stator tooth width as a fraction k s of the pole pitch, as the magnetic and electrical loadings are decoupled, and then calculate the apparent flux density in the stator tooth (35).
τ Qs for RFMs (34) B sta = B0 τ p w st for TFMs (35) The dimensions of the stator yoke, the rotor yoke and the TFM bridges are determined from the no-load flux (36) applying the flux conservation law in the corresponding sections of the machines (37), (38), (39).
As in TFMs the U-cores embrace the hoop coil to guide the magnetic flux, they can be treated similarly to slots of conventional RFMs.RFMs have a distributed winding along the pole pairs in q s slots per pole and phase, where q s is not necessarily an integer number.Being N ms the number of turns connected in series to produce flux linkages ( 24) and ( 25), obtained from current load (14), if the winding has a s paths in parallel then the number of turns per slot (40) has to be increased proportionally.
 a s N ms for TFMs a s q s p N ms for RFMs (40) If the RMS current density of each wire is J s , the slot fill factor is k Cu and the slot section is S ss , the sum of currents through the slot section gives (41).Therefore, substituting (40) in (41) gives (42).The current density is a design parameter that is related to the thermal performance of the machine and will be explained in greater detail in Section II-H.Table 4 shows the main design parameters of the stator winding.
for RFMs (42) Equation (42) states that the cross-section of slots in RFMs is q s p times smaller than in TFMs for the same phase MMF, fill factor and current density.However, in RFMs the slot width is limited by the slot pitch and the stator tooth width, whereas in TFMs the conductors can be allocated freely along the axial direction (43).That is why it can be stated that in TFMs the magnetic and electrical loads (no-load flux linkage and current, respectively) do not compete for the same space.Moreover, as the slot height follows (44) and the slot section is imposed by previous design parameters, it is shown that in TFMs the slot width factor k wss adjusts the slot aspect ratio k ss (45), whereas in RFMs it is fixed by other parameters as the slot width cannot be adjusted freely.The slot width factor of TFMs is shown in Table 4 and it Section III-A it will be deduced as a function of the TFMs relative slot aspect ratio.
The slot opening width of the RFM (Figure 6) is calculated according to the Carter factor equation (46) given in [37].The slot opening and the slot angle are h sso = 3 mm and α sso = 20 • , and thereby the slot wedge height follows (47).

G. WINDING LENGTH AND RESISTANCE
In electrical machines the winding is usually divided into active winding and end-winding.The active winding is located into the stator slots, whereas the end-winding serves as a connection between active coil sides.In conventional RFMs, the end-winding has a straight part along the axial direction (length l we ) and a curved part (length w we ).Furthermore, the end-winding length can be reduced using short-pitched coils or a fractional number of slots per pole and phase q s .However, in TFMs the end-winding length is fixed Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.by the machine dimensions as the phase coil is not distributed in slots.Figure 7 illustrates active and end-winding lengths of TFMs and RFMs, given by ( 48), ( 49) and ( 50).The sign ± should be taken as positive if the stator is external (as in this work), and negative otherwise.Then the total length of one turn can be calculated (51), and thereby the phase resistance (52).41) and (52).Therefore, a comprehensive equation of the temperature rise (55) can be written if the current load definition is used (14).It is stated that the thermal constant A s J s is of great importance for the machine thermal performance, then if the current load is increased, the current density must be decreased in order to not overheat the windings.As the heat transfer coefficient cannot be determined accurately without specific FEM or experimental validation, provided that typical values are not available in the literature, only the thermal constant values of literature [38] can be considered in the design process.The current load values were given in Table 3, so Table 4 gives the thermal constant and current density values that have been taken in this work.The thermal constant is the same for all the machines analyzed here as all of them are intended for the same cooling requirements.
P Cu = mR ms I 2 s = mρ Cu l Ns J s N ms I s (54) As PMSMs have magnets in the rotor, only the stator self-inductances are defined.The flux linkage produced by the stator current is usually divided into magnetizing flux and leakage flux, and so the inductance (56), (57).
As d-and q-axis inductances are equal in S-PMSMs, they are interchangeable here.The leakage factor is defined as (58), so the highest the leakage in an electrical machine, the highest the leakage factor.Initially leakage was neglected in the machine sizing (Table 3).As the total inductance is given by the motor requirements (Table 1), an increase in the leakage factor would increase the d-axis gap in order to reduce the magnetizing inductance (31), thus increasing the air gap and the PMs height (4), (5).
The leakage inductance L σ is given by (59).Factors p a s and 2q s p a s are the number of series-connected slots in TFMs and RFMs, respectively.λ is the specific permeance of each component of the flux linkage (Table 5), referred to the pole pitch in case of TFMs and to the pole length in RFMs, as they are the ''slot depths'' along which the conductors are located.
As in TFMs the ''slot depth'' (i.e. the tooth width) depends on the radius, from w st1 in the stator bridge to w st2 in the stator yoke, the proposed formulae for the slot leakage permeances (60), (61) include this variation.A negative sign in the expression of w st (h) must be taken in case of an external stator.Expressions of the end-winding and tooth tip leakage have been taken from [38] and [39], respectively.

N 2
Qs τ p λ ss = N Hitherto, there is no big difference when it comes to designing an electrical machine with an external or internal stator.As long as there is enough space to allocate the stator in the inner part of the machine, only a few signs must be changed when calculating the winding lengths and the slot specific permeance.The major differences when choosing an external or internal stator lay on the outer diameter, as the rotor radial height is usually lower than the stator height.The expressions for the inner diameter, outer diameter, phase length and outer length in the external stator case are given in (62), ( 63), (64), (65).Furthermore, it should be noted that the outer length of TFMs is proportional to the number of phases, as they are generally stacked in the axial direction.The outerlength-to-outer-diameter ratio k loDo (66) is of importance as is determines the true aspect ratio of the machine.Once the outer dimensions have been determined, any volumes and masses can be calculated.In this work the mass of active materials m a , that is, the sum of copper, iron and PMs (67), has been taken as the reference to determine the torque density.

IRON LOSSES AND EFFICIENCY
As the volumes of each part of the machine can be determined, the Steinmetz formula gives iron losses (68).
In this work the specific, rated losses of M250-35A electrical steel have been used in the Steinmetz formula, as the iron losses coefficients of the material were not available.Finally, the efficiency of the machine can be calculated with (69), having neglected mechanical and stray losses.

III. SIMILARITY EQUATIONS AND PARAMETRIC STUDY
The analytical sizing method presented in this work allows to compare TFMs and RFMs, given the specifications and the same design parameters for both topologies, using the similarity equations.Furthermore, the main dimensions depend on the current load in a similar way both in TFMs and RFMs.These analyses are presented in Section III-A and Section III-B, respectively.

A. SIMILARITY EQUATIONS
According with the proposed analytical sizing method and the design parameters given in Table 1, Table 2, Table 3 and Table 4, the main dimensions of TFMs and RFMs can be compared defining some dimensionless numbers that represent the TFM dimensions with respect to the RFM dimensions (70), ( 71), ( 72), ( 73), ( 74), (75).In the definition of the relative PMs volume, 2p * m (T ) is the relative number of PMs: TFMs have 4 magnets per pole pair and m (T ) stacks (one per phase), whereas RFMs have 2 PMs per pole pair and a single-stack.Results are shown in Table 6.For the same torque, no-load flux density, current load and pole length factor, the diameter of a TFM is 0.86 times the diameter of a RFM.Moreover, it should be noted that the PMs volume of the TFM is less than half when compared with the RFM.An increase in the TFM pole pairs would lead to a further reduction in the relative pole pitch and the relative pole length, and so the PMs volume.
106072 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
It should be noted that the relative slot aspect ratio k * ss can be chosen freely, as an additional degree of freedom in TFMs design that results from the decoupling of the magnetic and electric loadings.In this work, unit relative slot aspect ratio has been chosen for the parametric study, and then in the optimization stage is it changed to a design variable.Thereby, the expression of the stator slot width factor k wss in (43) as a function of the design parameters can be deduced from the similarity equations presented here, as shown in (76), (77).
Hitherto the relative dimensions do not depend on the current load or the pole length ratio.The relative outer length (78) and the relative rotor volume (79) depend both on TFM and RFM parameters, but their comparison is quite straightforward.For the values presented in the parametric study (unit relative aspect ratio, RFM pole length factor from 0.5 to 2), the relative outer length and the relative rotor volume are presented in Table 7.Although the TFMs diameter is lower according to Table 6, in some cases they have a bulky rotor regarding the axial length when compared with RFMs.However, their rotor and stator yokes are much smaller, so anyway they can take advantage in terms of the active mass.Figure 8 and Figure 9 show some TFMs and RFMs that result from the analytical sizing method presented here.It is shown that the current load has a great impact on the gap diameter and specially on the slot height.Therefore Section III-B analyzes the dependance of the main dimensions with the current load.

B. PARAMETRIC STUDY
The D 3  g equation ( 30) states that the current load and the pole length ratio are key parameters in electrical machines sizing, as the rest of the parameters are given by the specifications or minor design constraints.Therefore, in this section a parametric study will be conducted with respect to the current load for various pole length ratios, both for TFMs and RFMs.
The proposed analytical model allows to write the main dimensions of the machines as power functions of the current load.The gap diameter, the pole pitch and the pole length are a decreasing function with the current load (80), see Figure 10.The tooth width and the yokes' heigths have the same dependance as they are proportional to the pole pitch.The air gap and the magnets height increase with current load (81), Figure 11.Furthermore, according to (82) the stator slot section increases with the current load at the same that the slot width decreases, thus resulting in a steep increase of the slot height and the slot aspect ratio.Although the outer length of TFMs depends on the additional degree of freedom related to the slot aspect ratio, with the proposed analytical model it can be written as proportional to the pole pitch, so  it follows the same dependance as the diameter (Figure 12).Moreover, neglecting the air gap and the slot opening against the outer diameter, it is shown that the outer diameter and the outer volume have a global minimum (83), as shown in Figure 13 and Figure 14 (here the torque density has been depicted, so it has a global maximum).The torque density with respect to the active mass follows a similar dependance, according to Figure 15. (80) An insight into Table 8 and Table 9 shows that leakage flux of TFMs seems to be much higher than RFMs.Furthermore, the leakage flux through the stator bridges is by far the biggest contribution to leakage in TFMs, and both in the TFM and RFM the slot leakage flux increases a lot with the current load: according to Table 5 it is proportional to h ss /w ss and therefore to A 7/3 (82).

IV. FINITE ELEMENT ANALYSIS
In order to validate the analytical model, finite element simulations have been conducted using Altair Flux3D software.In TFMs, 3D geometries become mandatory for finite element modeling as their flux path is inherently threedimensional, whereas in RFMs 3D models are not essential,    but convenient if 3D effects such as the end-winding leakage flux and the fringing flux at the machine end have to be 106074 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.included in computations.Figure 16 shows the 3D meshed models, together with the boundary conditions implemented in Flux3D software.A second-order, extruded mesh has been used to improve the accuracy and reduce the computation time.It should be noted that only half of the axial length has been modeled both in the TFM and RFM cases, but using different symmetry and periodicity boundary conditions.In the TFM-U (but not in other TFMs) the magnetic field is normal to the middle cross-section plane, whereas in RFMs the magnetic field is tangent and so confined in the middle cross-section plane.Besides, as TFMs geometry is different in the north and south poles, cyclic boundary conditions must be used instead of the anticyclic boundary condition typically applied in RFMs.Furthermore, the magnetic field is assumed to be zero in infinity by use of an infinite box technique.Initially, constant magnetic permeability of iron has been assumed in order to compare the FEM results with the analytical model -based on a linear material.In Subsection IV-B3 the influence of magnetic saturation has been quantified separately.
According to Figure 4, in the analytical sizing equations the radial air gap flux density is reduced to its first harmonic, and so the flux linkages and voltages, neglecting any 3D or higher-order-harmonic effects.Furthermore, in ( 16), ( 17) it is assumed that the only reluctance in the magnetic circuit corresponds to the equivalent air gap (PMs + air gap), with no leakage or fringing effects.Therefore, in this work analytical and FEM models have been compared in order to validate previous assumptions.For this purpose, two factors have been defined, both for the no-load flux and the q-axis armature flux, named ''FEM no-load factor'' and ''FEM armature factor''.At the same time, each factor is the product of the ''FEM peak factor'', the ''FEM form factor'' and the ''FEM flux factor'', and each of these three factors can be calculated dividing the FEM quantities by the analytical magnitudes.For instance, in case of the no-load flux the peak, form  86) and (87), respectively, their denominators being given by ( 16), (19), (20), (24).The definitions in case of the q-axis armature flux are analogous, but using the armature field ( 17), ( 21), (22) and flux (25).It should be noted that the proposed peak and form factors are based exclusively on the air gap flux density waveform, whereas the flux factor takes into account other effects related to flux (such as leakage and fringing) that affect its first harmonic amplitude even though the mean value of the air gap flux density is perfectly known.Ultimately both the no-load flux factor and the armature flux factor can be correlated with the machine design parameters via surface fitting (88) and later re-used to improve the accuracy of the machine sizing method.In this work, the 'poly23' algorithm from MATLAB has been to determine coefficients a ii in (88).Firstly the no-load flux density waveform has been obtained for each TFM and RFM with a static FEM simulation (active PMs but no stator current in a fixed rotor position), and then the first harmonic of the flux linkage has been determined with a multistatic simulation and subsequent Fourier data analysis.Figure 17 illustrates the no-load radial flux density in the air gap for those machines with k l = 1, where the zero electrical angle corresponds to the rotor d-axis being aligned with the stator A-phase.It is shown that the peak values obtained via FEM match the analytical results, however, the no-load flux density waveforms of the TFM seem to be smoother than its RFM counterparts -thus altering the form factor-specially when the current load increases (so the gap diameter decreases).In Figure 18 two field maps have been depicted in the middle cross-section of the rotor PMs.Regarding the arrows density, can be stated qualitatively that the TFM under analysis exhibits higher PMs leakage flux and higher air gap fringing flux than the RFM.This effect could be explained by the fact that in the TFM the rotor inter-pole air space lays entirely under the stator inter-pole air space (Figure 18a), so the air gap reluctance starts to increase at the same time that the PMs flux density starts to decrease near the contact surface between PMs and air.However, in the RFM the rotor inter-pole air space lays mostly under the stator tooth tip (Figure 18b), so the reluctance does not change on the stator side, leading to a steeper decrease in the air gap flux density.In this work the stator tooth width equals the PMs span in all the TFMs, but changing this ratio would probably make the noload flux density waveform to be more rectangular or more sinusoidal.Figure 19 shows the no-load factor or the TFMs and RFMs that have been designed in this work.The no-load factor is close to one in most cases, thus proving the accuracy of the analytical method, but it seems to decrease with the current load both in the TFMs and RFMs. Figure 20 depicts the no-load factor decomposition for those machines with k l = 1.According to the results, the contributions of the no-load factor do not change significantly with the current load, so the decrease in the no-load factor can be attributed equally to the peak factor, the form factor and the flux factor.

B. Q-AXIS ARMATURE FLUX
The armature flux is usually divided into magnetizing flux and leakage flux when designing electrical machinesas the inductance, according to (56)-, so it has been analyzed separately via the armature factor and the leakage factor (linear material) in Subsections IV-B1 and IV-B2,  and then the effect of saturation has been analyzed in Subsection IV-B3.

1) ARMATURE FACTOR
Figure 21 depicts the armature flux waveforms in the q-axis for those machines with k l = 1.It is clearly shown that the FEM waveforms differ from the analytical assumptions.In the TFMs the air gap armature flux density seems to concentrate around the stator tooth corners and then it decreases to zero with a certain slope, and the same occurs in the RFMs near the slot openings, but in case of the RFM the peak values of the flux density are also higher than the expected by the analytical calculations.An insight into the FEM field maps (Figure 22) shows that the TFM exhibits some leakage flux coming from the stator bridges and fringing flux in the rotor inter-pole space that explains the increase of the armature flux density near the stator tooth corners.Besides, the RFM seems to suffer from flux concentration in the top part of the stator tooth tip, i.e. the smallest cross-section are in the magnetic circuit, and this could be behind the higher peak value of the flux density.
According to Figure 23a, in the TFMs the armature factor increases with the current load, and this increase is mainly determined by the form factor according to    23b), the changes in the armature factor are determined by the peak factor, as the form and flux factors do not change with the current load (Figure 24b).

2) LEAKAGE FACTOR
The leakage inductance and so the leakage factor k σ can be deduced from finite element simulations using (56), as the magnetizing fluxes that cross the air gap have been calculated in Subsection IV-B1 and the total flux is computed  by FEM software.Figure 25 illustrates the comparison between analytical and FEM leakage factors, showing a better agreement in case of the RFM.The calculation of the leakage permeances is based on a simplified geometrical description, thus introducing some error, and the endwinding leakage permeance has a great uncertainty, so the proposed analytical model results can be considered as acceptable.Furthermore, in case of TFMs stator bridges performance (that is, the biggest contribution to leakage according to Table 8) a flux distribution other than constant could be considered according to FEM (Figure 26a), as the leakage flux tends to close mostly through the inner boundary producing a non-uniform flux density in the crosssection.In RFMs a flux distribution depending on the radius is considered when calculating the slot permeance, and this flux distribution is in accordance with FEM results of Figure 26b.flux, so a saturation factor k sat (Figure 27) has been calculated -dividing the nonlinear results by the linear results-to correct the sizing equations later.

C. BACK-EMF AND BASE TORQUE, DYNAMIC SIMULATIONS
As the machine torque follows (6), the analytical and FEM discrepancies in the first harmonic of the back-EMF and synchronous torque are given by the no-load flux factor.According to Figure 28 and Figure 17, the back-EMF and torque obtained via FEM agree with the analytical firstharmonic analysis, so it has been proved to be accurate enough to size TFMs and RFMs as a first approach.The rest of effects related to harmonics (such as the cogging torque) should be optimized via FEM at a later stage in the design process.

V. OPTIMIZATION
Once the no-load, armature, leakage and saturation factors have been obtained via FEM, their surface equations (88) are introduced in the analytical model and then it is optimized.The optimization problem is presented in (89), the objective functions are the torque density with respect to the active mass T /m a and the efficiency η (90).The constraints are related to minimum dimensions for manufacturing, performance and the aspect ratio (91), but any other constraints could be included depending on the specific application, such as the overall cost or PMs volume.The design variables are (92), their ranges shown in Table 10.A MATLAB-based multiobjective genetic algorithm (which is a controlled elitist genetic algorithm, a variant of NSGA-II) has been used to solve the problem.The population size, the number of generations, and the mutation function have been left by  Figure 30 illustrates the Pareto front as a result of solving (89) both for the TFM and the RFM.It is clearly shown that for the proposed optimization problem the TFM is the most favorable option both in terms of torque density and efficiency.From the Pareto sets of optimal solutions, the optimal solution has been chosen for each machine topolgy as the one that minimizes the distance to the -theoreticalpoint in with both the torque density and the efficiency are maximum, according to (93).The optimum designs taken from the Pareto fronts are shown in Figure 31, and Table 11 collects their main parameters.It is shown that both the torque density and the efficiency have been improved with respect to the machines given by the parametric study.The introduction a priori of the no-load, armature, saturation and leakage factors from surface fitting has allowed to consider these effects in the analytical model, and now the results given by FEM seem to be more optimistic than the analytical predictions terms of the leakage flux.inally, Table 12 compares optimum designs of this work with other PMSMs in the literature, showing good agreement.The range of active torque densities from previous publications is quite wide, and it depends strongly on specific topologies.As an example, the use of inner permanent magnets in RFMs increases active torque density up to about 10 Nm/kg [40], it is four times higher than the surface-mounted PMSM in [41].Transverse flux machines (TFMs) benefit from higher torque density when a high number of pole pairs is used.This is due to their single-phase, hoop-shaped coils that embrace simultaneously the flux of all the pole pairs, unlike radial flux machines (RFMs) where each coil embraces the flux of a single pole pair.Therefore, sizing equations for TFMs differ from those proposed for radial flux machines in the literature.In this work is has been shown that the main dimensions of TFMs can be written as functions of the current load, as in classical theory for radial flux machines.Moreover, the similarity equations introduced here allow a straightforward comparison of TFMs and RFMs, given the specifications and design parameters.
The finite element method (FEM) has been used to quantify the discrepancies between computational models and the analytical assumptions, defining several FEM factors.The analytical model and FEM show a good agreement, thereby a surface-fitting model has been applied to introduce the FEM factors in a second stage of the design process.Finally the analytical model has been optimized using a multiobjective genetic algorithm, proving that TFMs have a superior performance than radial flux machines in terms of torque density and efficiency.

FIGURE 3 .
FIGURE 3. Flowchart of the sizing method and optimization.

FIGURE 4 .
FIGURE 4. Rectangular flux density waveform assumed by the analytical model (solid blue line), its first harmonic (solid red line) and the mean value of the first harmonic (dashed red line).

FIGURE 5 .
FIGURE 5. Ferromagnetic cores and coil in a TFM-U, single-phase view: (a) Illustration of the Ampère's Law in a single pole-pair, (b) Phase coil linking the flux of all pole pairs.

FIGURE 6 .
FIGURE 6. Slot dimensions in a RFM.

TABLE 8 .TABLE 9 .
Specific leakage permeances and leakage factor in TFMs with k l = 1.Specific leakage permeances and leakage factor in RFMs with k l = 1.

FIGURE
FIGURE 3D FEM meshed models and boundary conditions (BC), having removed the surrounding air: (a) TFM-U with k l = 0.5, A s = 10 kA/m, (b) RFM with k l = 0.5, A s = 10 kA/m.

FIGURE 22 .
FIGURE 22. FEM armature flux density: (a) leakage and fringing fluxes in the TFM-U with A s = 20 kA/m, k l = 1, (b) non-uniform field in the RFM with A s = 20 kA/m, k l = 1.

Figure 24a .
Figure 24a.In case of RFMs (Figure23b), the changes in the armature factor are determined by the peak factor, as the form and flux factors do not change with the current load (Figure24b).
Even though the electrical machines in this work have been designed for unsaturated condition regarding the magnetizing fluxes, leakage flux increases the total flux at the risk of saturating the magnetic circuit.All the previous simulations have been repeated using a nonlinear material (Cogent M250-35A electrical steel), having similar results for linear and nonlinear simulations.The only exception is the armature

FIGURE 30 .
FIGURE 30.Pareto fronts of the multiobjective genetic algorithm optimization.
The associate editor coordinating the review of this manuscript and approving it for publication was Xiaodong Liang .B syStator yoke height.ÎdStator current in the d-axis, peak value.ÎqStator current in the q-axis, peak value.fr1Form factor of the rotor, 1st harmonic.kfs1Form factor of the stator, 1st harmonic.k gma Magnets-to air gap ratio.k l Pole length factor.k loDo Outer-length-to-outer-diameter ratio.k N Equivalent turns factor.s Number of stator slots per pole and phase.Q s Number of stator slots.R ms Stator resistance per phase.R th Thermal resistance.S ss Stator slot cross-section.T 1 Peak motor torque at the base speed.U DC DC-voltage.i Current phasor angle with respect to the q-axis.η Efficiency.λ ew Specific permeance of the end-winding.λ sb Specific permeance of the stator bridge.λ ss Specific permeance of the stator slot.λ sso Specific permeance of the stator slot opening.λ ssw Specific permeance of the stator slot wedge.λ tt Specific permeance of the tooth tip.µ 0 Vacuum permeabiliy.µ rm Relative permeabiliy of the magnets.ω 1 Base angular frequency.ϕ Power factor angle.Φ0 No-load flux, peak value.Ψ 0 No-load flux linkage, peak value.Ψ aq Magnetizing armature q-axis flux linkage, peak value.ρ Cu Copper resistivity.τ p Pole pitch.τ Qs Stator slot pitch.ξ r1

FIGURE 2. Phasor
diagram of a S-PMSM for maximum torque.

TABLE 2 .
Design parameters related to material properties.

TABLE 3 .
Rotor and stator design parameters.
g a /2 + h sb + h ss /2) for TFMs If iron, mechanical and stray losses are neglected against copper losses P Cu , the temperature rise follows (53), being R th the thermal resistance of the stator housing.D o and l o are the outer dimensions of the machine and α th is the heat transfer coefficient.Copper losses are given by (54) substituting ( g a /2 + h sb + h ss /2) for TFMs π D g /2 ± (g a /2+h sso +h ssw +h ss /2) sin

TABLE 6 .
Relative dimensions of TFMs with respect to RFMs in this work.

TABLE 7 .
Relative outer length and rotor volume for different RFMs pole length factor.

TABLE 12 .
Optimum designs and other PMSMs in the literature.