Impact of Rotor Flux-Barriers on Coupling Coefficient in a Brushless Doubly-Fed Reluctance Machine

A brushless doubly-fed reluctance machine offers several advantages over a single-port machine since it requires a partially-rated power converter, provides reduced maintenance, and operates without permanent magnets. The complexity of the air-gap magnetic fields due to differing winding-pole numbers interacting with a reluctance rotor have precluded the use of an analytical framework to optimize the rotor. Instead, researchers have relied upon time-consuming finite element analysis (FEA). This paper presents an analytical method to compute closed-form solutions for the mean torque, captured by the coupling coefficient, using a practical circular ducted rotor. The method also accounts for the potential drop across the flux-barriers, leading to a better estimation of mean torque. The model is validated using both FEA and experimental tests.


I. INTRODUCTION
A STANDARD approach for constructing the megawattscale electromechanical drivetrains, such as in turboelectric aircraft and offshore wind turbines, has been to use single electrical port machines. A power-electronic converter is required between the electrical machine and the ac source to enable complete torque-speed control of the drivetrain. The converters along with the associated filters process the full-rated machine power, leading to a bulky, expensive and less efficient system. In contrast, a brushless doubly-fed reluctance machine (BDFRM) has two stationary electrical ports. The torque is produced due to electromagnetic coupling between two stator windings with different pole numbers using a variable-reluctance rotor that modulates the air-gap field [1], [2], [3]. A BDFRM when used in a limited speed range applications such as in a wind turbine requires a fractionally-rated power converter allowing a significant reduction in the drive size [4], [5]. Using a switched-drive Manuscript  architecture can extend the use of BDFRM to other wide-speed range applications, such as turbo-electric aircraft propulsion system, while preserving the benefit of fractionally-rated power electronics [6], [7]. In addition, the absence of brushes/slip-rings and permanent magnets result in a highly reliable and faulttolerant drive system. While a BDFRM-based drive exhibits several advantages, as mentioned, achieving a high-torque-dense BDFRM remains as a major challenge [8]. The stator-design optimization has been thoroughly investigated using pole numbers, stator geometry, and current excitations as design variables to enhance the torque density [9], [10], [11]. However, the rotor design has still primarily relied on finite-element-anlysis (FEA) based tools, making it challenging to expand the design space for optimization. Understanding the impact of rotor-design parameters on torque production plays a crucial role in enhancing the torque density. Coupling between the two stationary windings, defined and quantified by a coupling coefficient, strongly affects the torque density and can differ greatly among various rotor designs [12]. A simple-yet-elegant approach to calculate the coupling coefficient assumes the rotor to have an infinite number of iron ducts of zero width [13]. However, because the only parameters are the stators and rotor pole-pair combinations, it is ineffective for detailed rotor design optimization. A recent analytical approach computes the coupling coefficient by considering a practical rotor with a finite number of iron ducts, dividing the flux-guide layer into several equal segments, and then superimposing Fourier expansions [14]. Uniform flux-barriers and flux guide thicknesses are assumed. While this is more accurate in computing the coupling coefficient than using the ideal ducted-rotor assumption, as it introduces three more design parameters-flux-barrier thickness, number of flux-barriers, and iron-duct thickness-the method assumes the flux remains confined within a duct. In practice, a BDFRM has a non-zero flux crossing the barriers when a finite number of ducts is considered, leading to relaxed flux paths, as shown in Fig. 1. Reference [15] proposes an alternative framework to capture the relaxed flux paths to compute instantaneous torque. A finite number of flux barriers in the rotor with non-uniform thickness and stator winding harmonics are considered. This framework is used to maximize torque density while minimizing torque ripple. An iterative procedure is developed to solve this multi-objective optimization making it computationally intensive. The key contribution of this paper is an approach that an derives explicit function of the coupling coefficient, like the one derived using the ideal-ducts approximation, but with an added dependence on the rotor design parameters. The framework computes the magnetic potential in different ducts, thereby capturing the effect of the flux crossing the barriers-an approach that has been explored to model rotors in synchronous reluctance machines [16]. The proposed method is verified using FEA showing that the coupling coefficient and mean torque are strongly dependent on flux-barrier location and thickness. A BDFRM experimental prototype validates the proposed method. It is 50% more accurate in estimating the coupling coefficient compared to the ideal ducted-rotor assumption at the rated condition. The proposed approach opens up opportunities to use the analytical closed-form solutions to find optimum flux-barrier half-span angles and thickness to maximize torque density.
The remainder of the paper is organized as follows. Section II presents the analytical electromagnetic model of a BDFRM in which the air-gap flux density and mean torque are derived. Section III illustrates different approaches to calculate the magnetic potential of rotor ducts and coupling coefficient. For a practical rotor, the coupling coefficients with one and two fluxbarriers per rotor pole are also derived. The impact of the fluxbarriers location on the coupling coefficient is observed using the proposed analytical model and FEA. Experimental results and conclusions are presented in Section IV and V, respectively.

II. AIR-GAP FLUX DENSITY AND MEAN TORQUE IN BDFRM
An analytical model to compute air-gap flux-density and mean torque is derived in this section. The resulting torque expression is used in Section III to compute the coupling coefficient. The following assumptions are made: 1) The primary and secondary stator winding harmonics are ignored since the spatial winding harmonics have negligible impact on mean torque.
2) The core material is assumed to have infinite permeability. This assumption implies that the magnetic potential drop in the stator core and across each individual rotor duct is negligible.
3) The tangential and radial iron ribs in the rotor, necessary for mechanical integrity, are also neglected. At rated conditions the iron ribs saturate due to their small dimensions, and thus ribs have a negligible impact on machine performance. 4) Effects of both stator and rotor slots on the air-gap flux density distribution are ignored. The net electrical loading A in stationary frame as a result of exciting primary stator having p p poles with current amplitude I p and frequency ω p and secondary stator having p s poles with current amplitude I s and frequency ω s , is given by where A p = 6k wp N p I p πD and A s = 6k ws N s I s πD , k wp and k ws are the fundamental winding factors, D is the air-gap diameter, δ is the initial phase offset between the two stator currents, φ is the stator angular coordinate, and N p and N s are the total series turns per phase for the primary and secondary stators, respectively. A rotor reference transformation of the stator electrical loading using a rotor angular coordinate φ r gives where ω rm is the mechanical speed of the rotor with p r poles and is given by ω rm = ω p + ω s p r . The MMF due to this electrical loading along the air-gap is described as The radial flux density in the air-gap of thickness g is given by where F r is the rotor magnetic potential. Instantaneous torque τ e is the product of air-gap radius and Lorentz force along the air-gap surface and is calculated by Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where L stk is the stack length. Using (4), the torque is simplified as: where The terms I and II are zero because of orthogonality between (F prim , A prim ) and (F sec , A sec ). The terms III and IV are zero because of the absence of common spatial harmonics between (F prim , A sec ) and (F sec , A prim ).
Hence the instantaneous torque is given by While this result is similar to that of a SynRm [16], the integration in (7) cannot be further simplified over only one rotor pole for a BDFRM. The complexity arises due to asymmetry in rotor magnetic potential F r (t, φ r ) as a result of different number of stator and rotor poles. Using (7), torque is computed from the integral product of net electrical loading and rotor magnetic potential. The net electrical loading A(t, φ r ) contains spatial harmonics of n 1 = p p /2 due to primary stator and n 2 = p s /2 due to secondary stator. Thus, the average torque τ e is only produced by the n 1 and n 2 spatial harmonic components of F r . The fourier series for rotor magnetic potential F r is given by where the coefficients a n and b n are given by Using the useful spatial harmonic components of F r in torque production, mean torque τ e in (7) is expressed as: which can be further simplified to give Flux paths in ducted rotor [13].
The fourier series coefficients a n 1 , a n 2 , b n 1 and b n 2 require the computation of rotor magnetic potential F r . Different frameworks have been used to model the rotor structure and will be discussed in Section III.

III. COMPUTATION OF ROTOR MAGNETIC POTENTIAL AND COUPLING COEFFICIENT
As shown in Section II, both the air-gap flux density in (4) and mean torque in (10) require the computation of rotor magnetic potential F r . This section describes solving the rotor magnetic potential using two approaches: (a) ideal-ducts approximation with infinite number of iron ducts having infinitesimal thickness, and (b) a practical ducted rotor having n b number of flux-barriers per rotor pole.
The ideal-ducts approximation is used extensively because of its simplicity to model the rotor and BDFRM initial sizing. The complete model is described in [13]; however, for the sake of completeness, a brief description is hereafter presented.

A. Ideal-Ducts Approximation
In this model, the rotor consists of a infinitely permeable material with infinitesimal thickness that are laid out in a concentric way to allow a low-reluctance magnetic flux path, as depicted in Fig. 2. The key assumption made in this model is that the flux ϕ entering a rotor duct at an air-gap angle of φ r in exits from the same duct at φ r out , as shown in Fig. 2.
Using Ampere's law, the air-gap flux density B ideal with an ideal ducted rotor in the rotor reference frame is Equation (11) describes the modulation of the original MMF F s by the ducted rotor. Using (4) and (11), F r is Using (11) and (12), the mean torque τ e in (10) simplifies to [10] The coupling coefficient C ps denotes coupling between the primary and secondary stator achieved through rotor modulation. According to this model, C ps depends solely upon the choice of pole combination without taking any rotor dimensions into account.

B. Practical Rotor With Finite Number of Rotor Ducts
Next, a rotor modeling framework is discussed that solves the rotor magnetic potential for a generic ducted rotor having n b number of flux-barriers per rotor pole. Unlike the idealducts approximation, the proposed approach also considers the magnetic potential drop across the flux barriers to give a more accurate estimation of mean torque. Since the iron is assumed to have infinite permeability, the rotor magnetic potential F r is considered to be constant in an individual iron duct and zero in the inner-most duct, as shown in Fig. 3(a). Magnetic flux crossing the barriers causes the magnetic potential to be different between various ducts of the same rotor pole. Assuming a uniform air-gap field H ij in the flux-barrier, the difference in the magnetic potentials between two successive rotor-iron ducts i and j is defined as: (14) where parameters t b ij and l b ij represent the thickness and length, respectively, of the barrier spanning an angle 2φ ij between the iron ducts i and j and ϕ ij denotes the flux crossing the flux-barrier. The same flux passes through the rotor surface and therefore, (14) can be rewritten as (15) Using (4) and (15) at the outermost barrier: where R b 1 is the reluctance of the outermost flux barrier placed between ducts 1 and 2 and having a barrier span-angle of 2φ b 1 .
Since the rotor magnetic potential F r 1 is constant throughout the integral limits in (16), rearranging the terms in (16) gives the rotor potential F r 2 as: where the constants a b 1 and ρ b 1 given by depend upon the circular flux-barrier geometry and stator excitation, respectively. However, F r 2 can not be computed directly from (17) since F r 1 is unknown. The innermost rotor duct n b + 1, assumed to be at zero potential, must be set as a reference for potential of other rotor ducts. A recursive procedure is hence derived to compute F r i (i = 1, 2, . . .n b ) with F r n b +1 = 0. Using (4) and (15) at the second flux barrier: Since the rotor potential F r is constant at F r 1 for π/p r − φ b 1 ≤ φ r ≤ π/p r + φ b 1 and F r 2 elsewhere, the above equation is simplified to give   is declared and added in (17) and (20) as that leads to the following recursive process of computing F r i+1 (i = 2, 3, . . .n b ) in terms of F r 1 : Equation (22) is used n b times along with F r n b +1 = 0 to compute the rotor magnetic potential F r . Two example rotor designs with one and two flux-barriers per rotor pole are next presented to compute the fourier series coefficients a n 1 , a n 2 , b n 1 and b n 2 in (10) and calculate the coupling coefficient C ps as a function of rotor flux-barrier parameters.
1) One-Barrier Per Rotor Pole: Considering a flux-barrier span angle of 2φ b , the rotor magnetic potential is given by Figs. 4 and 5, where m = 1, 2, . . .p r are the p r rotor segments. Using (17) and F r0 = 0, the magnetic potential F r m is computed by The fourier coefficient a n of F r is calculated as: Similarly the coefficient b n is given by Using (25), the first component τ e 1 in (10) is computed as: As seen in (24), the rotor pole magnetic potential F r m comprises of two terms -one generated due to primary stator excitation and another due to secondary stator excitation. Using (24), both the terms are simplified in (27) to give Since n 1 + n 2 = p r , the summation terms in (28) are further simplified to which gives the first component of mean torque τ e 1 as The other components of mean torque τ e 2 , τ e 3 and τ e 4 are derived similarly to give The net mean torque for a rotor with one flux-barrier per pole is given by where C ps represents the coupling coefficient between the primary and secondary windings as defined in [13] including the rotor barrier effects. Fig. 6(a) compares the effect of flux-barrier half-span angle on the coupling coefficient C ps between the ideal-duct approximation given by (13) and the proposed analytical model given by (33). Unlike the ideal-duct approximation, the coupling coefficient and hence the mean torque changes with the location of flux-barrier. Four rotor structures with same flux-barrier thickness and half-span angles of 10 • , 15 • , 20 • and 25 • are analyzed in FEA using ANSYS Maxwell and the resulting coupling coefficient is compared with the proposed analytical model, as shown in Fig. 6(a). The machine parameters are given in Table I. As observed from FEA simulations, the coupling coefficient is dependent on the flux-barrier position. The differences between the FEA and proposed analytical model are attributed to ignoring the effect of stator and rotor slots, presence of iron ribs and saturation of core in the analytical model. The impact of flux-barrier thickness on C ps is shown in Fig. 6(b). The analytical model trend matches with the one  Table. predicted by FEA until it reaches a barrier thickness of 5 mm. At higher barrier thicknesses, the slotting effect due to the barrier opening becomes significant and leads to a wider mismatch between the FEA results and the computed coupling coefficient. The model assumes that the rotor magnetic potential F r changes from 0 to F r m at the center of flux-barrier, as shown in Fig. 7(a). At a higher barrier thickness, approximating that F r changes from 0 to F r m at the start of iron duct, as shown in Fig. 7(b), leads to a better match with the FEA results. The rotor slotting effect can be included in the analytical model by replacing φ b with φ b − φ t /2 in (33) to yield the following expression for the coupling coefficient: Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. Fig. 7. The rotor slotting can be approximated by assuming F r changes from 0 to F rm at the start of iron duct. where φ t = 2t b /D. The analytical results using these approximations are also included in Fig. 6(b). Two additional examples of 6-2-4 and 10-2-6 pole combinations are shown in Fig. 8 to show the applicability of the proposed approach for different slot-pole/duct combinations. The proposed analytical model follows the trend obtained using FEA. Total number of 18 and 30 stator slots are used, respectively, for these two cases to allow an integral number of slots per-pole per-phase for both the stator windings.

2) Two Flux-Barriers Per Rotor Pole:
The magnetic potential of a m th rotor pole segment having two flux-barriers per pole is denoted by F r 1 m , F r 2 m and F r 3 m as shown in Figs. 9 and 10. Using the recursive relations in (22), the magnetic potential of the three ducts are given by: where the barrier constants C r 11 , C r 12 , C r 21 and C r 22 are given by: Using the principle of superposition as shown in Fig. 10, the fourier series coefficient a n of F r is a n = 4 nπ sin(nφ b 1 ) where the variable F r 1 m is introduced to use the superposition principle and is computed by The fourier series sine component b n of F r is computed similarly. The first component of mean torque τ e 1 in (10) is computed as Using F r 1 m and F r 2 m as defined in (38) and (35), respectively, and the summation terms derived earlier in (30), τ e 1 is simplified to The other components of mean torque τ e 2 , τ e 3 and τ e 4 are derived similarly to give Using the definition of n 1 , n 2 and k τ , the net mean torque for a rotor with two flux-barriers per rotor pole is given by A similar comparison between the ideal-duct approximation and the proposed analytical rotor models is also carried out for a motor with two flux-barriers per rotor pole. Fig. 11 shows the change in coupling coefficient with different flux-barriers span angles φ b1 and φ b2 according to (42).

IV. EXPERIMENTAL RESULTS
Experimental tests are performed on a 220 V (primary and secondary), 2 A (primary), 1.8 A (secondary), 1200 rpm BD-FRM designed in-house using a 145 T stator frame size. Both the stator and rotor were laser-cut from M-19 C5 steel laminations and bonded using Remisol EB 548. Fig. 12 shows the machine geometry and dimensions. The rotor parameters are shown in Table II. Tangential iron ribs of thickness 0.35 mm-the minimum thickness required for laser cutting-are used in the prototype. The hardware setup shown in Fig. 13 comprises the prototype BDFRM, a dynamometer, voltage and current sensors, a dc supply and a three-phase inverter. The coupling coefficient is first estimated using no-load tests and then verified under loaded conditions, as explained below.

A. Estimation of C ps Using No-Load Tests
In the first set of tests, the BDFRM is operated at no-load with the primary winding kept open and the secondary winding excited with a dc current I dc . The A-phase of the secondary stator is connected to the positive terminal and B-and C-phases are shorted together to the negative terminal of the dc source. A dynamometer is used to spin the rotor at a constant speed ω rm . In this operation condition, the peak of induced line voltage V p and frequency ω p measured in the primary stator are given by  Tables I and II is used for experimental validation. where L ps is the mutual inductance between the primary and secondary stator windings. The coupling coefficient is related to the mutual inductance between the two stator windings as given by [10]: Using (43), (44) and the machine parameters from Table I, the coupling coefficient C ps is given by Fig. 14(a) shows the primary-stator induced voltage with the secondary stator excited by a dc current of 2.5 A at a rotor speed of 600 rpm. Fig. 14(b) shows the Fourier specturm of this waveform. The fundamental component (60 Hz) is used to compute C ps , as given by (45) and listed in Table III. The  proposed method predicts the coupling coefficient within 10%   TABLE III COMPARISON OF COUPLING COEFFICIENT AT RATED I s = 2.5 A AND ω rm = 600 RPM error, which is significantly better compared to the ideal-ducts approximation. The procedure is repeated for different secondary stator dc excitations and the results are shown in Fig. 15. Obtained C ps from experimental tests shows that the coupling coefficient varies with the current excitation. This variation is primarily due to the non-linear behavior of the magnetic material. At lower current excitation, the iron ribs are highly permeable and acts as a magnetic short circuit thereby degrading the rotor modulation capability [17], [18]. At rated current of 2.5 A, the iron ribs saturate due to their small dimensions and they have a negligible impact on the coupling coefficient. Increasing excitation current beyond this magnitude, results in bulk saturation, which again reduces the C ps . FEA results corroborate this experimental trend.
Similar tests are performed at varying excitation patterns and rotor speeds to evaluate the coupling coefficient at different operating conditions as shown in Fig. 16. The rotor speed does not have any impact on C ps as expected from the analytical model.

B. Verification of C ps Under Loaded Conditions With Closed-Loop Speed Control
The BDFRM is next operated under different loading conditions with the secondary stator connected to a three-phase drive operating in closed-loop speed-control mode while the primary stator windings are shorted. The secondary stator current controllers, speed controller, and the primary stator flux estimator are designed based on [19]. The estimated coupling coefficient and mutual inductance from the no-load test are used for the d-q reference frame transformation in the primary stator flux reference frame. The reference speed and d-axis secondary stator current are set as 300 rpm and 2 A, respectively. The load torque  is increased in steps to a maximum torque of 0.85 Nm using a dynamometer. Fig. 17(a) shows the estimated electromagnetic torque and speed response of the BDFRM. A sample primary and secondary stator-phase currents are shown in Fig. 17(b) when the estimated electromagnetic torque changes from 0.35 Nm to 0.85 Nm. The primary stator windings being shorted operates at a slip frequency, which increases with load. An increase in the demanded torque leads to an increase in the q-axis secondary stator current i sq , as shown in Fig. 17(c). The q-axis primary stator current i pq responds to this increase proportionately given by [20] i pq = L ps L p i sq where L p is the primary stator self-inductance. This equation allows us to estimate L ps and the corresponding C ps using (44) and the measured primary and secondary q-axis stator current. The experimentally obtained coupling coefficient matches closely with the values predicted by the proposed analytical model, as Fig. 17. While the secondary stator is connected to a 3-phase drive operating in speed-control mode and primary stator is shorted, the load torque is changed and the currents are measured.
shown in Fig. 18. The test is repeated for a d-axis secondary stator current of 1.5 A.

V. CONCLUSION AND FUTURE WORK
This paper proposes an analytical approach to model the effect of rotor flux-barriers on the coupling coefficient, which strongly impacts torque density of a BDFRM. The proposed framework allows closed-form solutions that can be used for optimizing rotor design in place of computationally time intensive FEA-based approaches. A rotor with one and two flux-barriers per rotor pole are used as examples to illustrate the proposed approach. Taking the flux crossing the barriers into consideration leads to a more accurate coupling coefficient estimation compared to the widely-used ideal-ducts approximation. The FEA and experimental results are used to show the influence of fluxbarrier parameters on the coupling coefficient and validate the proposed analytical model. Experimental results also show the influence of the core material's non-linear permeability on the coupling coefficient. As a future work, the proposed model can be further augmented to capture this non-linearity and saturation, as described in [21], to have a better machine performance prediction over the entire operating range. Further, loss models for a synchronous reluctance motor such as in [22] can also be used to compute rotor iron loss for a high-power BDFRM where a high frequency operation is needed. The prototype drive will also be tested in sub-synchronous, synchronous, and super-synchronous speed regions as in [23] using the vector control methods in [24].