IEEJ Journal of Industry Applications Vol.4 No.4 pp.352–359 DOI: 10.1541/ieejjia.4.352
Paper
Fast Analytical Model for Switched Reluctance Machines Senad Smaka∗a)
Non-member
(Manuscript received July 31, 2014, revised Jan. 16, 2015)
This paper describes a fast analytical model for computing the nonlinear magnetization and static torque characteristics of a switched reluctance machine (SRM). This model is developed using the flux-tube and gage-curve methods. The proposed model is used for computing the magnetization (flux-linkage) and torque characteristics of three and four-phase SRMs. The simulation results obtained using the proposed analytical model are compared to those obtained using magnetostatic finite-element analysis (FEA) for a three-phase 12/8 SRM and for a four-phase 8/6 SRM. Finally, experimental verification of the analytical model is presented for the 12/8 SRM and 8/6 SRM prototypes. Keywords: analytical model, flux-tube, gage-curve, switched reluctance machine
1.
In the proposed model gage-curve method is used to compute ψ(I) characteristics at other rotor positions. Gage-curve method is introduced in (10) where the ψ(I) characteristics at the unaligned and the aligned position are approximated using three precalculated points, first-order, and second-order functions. These characteristics are used to determine flux linkage characteristics at other rotor positions. In this paper ψ(I) characteristics at the unaligned and the aligned position are calculated by flux-tube method instead of using their approximation like in (10). Then, these characteristics are exploited to compute ψ(I, θ) curves at intermediate rotor positions, as described in Section 2.2. By using this approach, the results of FEA or measurements in several characteristic points are not needed, which is the advantage in comparison to classical gage-curve models described in (10) and (12). The comparison of the analytical results to FEA is given in Section 3 for the 12/8 SRM and 8/6 SRM. In Section 4, analytical results are compared with measurements for the 12/8 SRM and 8/6 SRM. Finally, the conclusions are presented in Section 5.
Introduction
Switched reluctance machine (SRM) has highly saturated doubly salient pole structure. The requirement for predicting the steady state and dynamic performance of SRM is to generate its magnetization characteristic ψ(I, θ) and static torque characteristic T (I, θ), where ψ is the flux linkage, I is the excitation current, θ is the rotor mechanical angular position, and T is the torque. The magnetization characteristic can be obtained by magnetostatic FEA (1)–(3) or by measurements done on the existing machine. However, both methods are not particularly suitable to implement during initial stages of the machine’s design process. An alternative approach is to develop analytical model of SRM to compute its magnetization characteristic. This model has to be faster than FEA but still accurate enough. The SRM modeling using analytical methods have been reported in literature (4)–(18) . The most common models are based on the use of: magnetic equivalent circuit (MEC) (4)–(6) ; implementation of basic laws of physics on simplified motor geometry (7)–(9) ; analytical equations that approximate magnetization characteristic (10)–(15); several predefined flux-tubes (16)–(18) . This paper presents fast analytical model of SRM, which is intended to be a part of a sizing-design estimation process of the machine. The analytical model will be discussed in Section 2. Proposed model combines two already known techniques, flux-tube and gage-curve. In this model flux-tube method is used to compute ψ(I) characteristics at the aligned and the unaligned rotor position. Implementation of flux-tube method is not based on the use of iterative computation like in (16)–(18). Instead of using numerical iteration with a prescribed value of error or with the specified number of iterations to compute magnetomotive force, in the proposed model flux linkage contributed by each flux-tube is computed for the assumed values of stator pole magnetic flux density, as described in Section 2.1.
2.
2.1 Implementation of Flux-tube Method The fluxtube based models that enable computation of flux linkage characteristics in arbitrary rotor positions are reported in literature, but the achieved level of accuracy in intermediate rotor positions can be a quite low (19) . In this paper flux-tube model is used only for computing ψ(I) characteristics at the aligned and the unaligned rotor position, where substantial accuracy can be obtained. The level of accuracy depends strongly on the number of flux-tubes that are used to represent the actual flux paths. Two flux-tubes are assumed to be sufficient to represent the actual flux paths at the aligned position and seven flux-tubes are used at the unaligned position (16) . The considered fluxtubes are shown in Fig. 1. Assumptions common to all flux-tube models are also applied here: the air gap flux lines consists of concentric arcs and straight-line parts; the flux lines enter and leave iron
a) Correspondence to: Senad Smaka. E-mail:
[email protected] ∗ Faculty of Electrical Engineering, University of Sarajevo Zmaja od Bosne bb, 71000 Sarajevo, Bosnia and Herzegovina c 2015 The Institute of Electrical Engineers of Japan.
Analytical Model
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(a) Aligned position
(b) Unaligned position
Fig. 1. Assumed flux tubes—aligned and unaligned position
Fig. 2. SRM geometry parameters for flux-tube model
normally; the flux lines in the poles are parallel to the pole axes; the flux lines in the stator and rotor yokes are concentric. Model input data are: B-H curve of the lamination material, arc angles βs and βr , stator and rotor outer radii Rs and Rr , air gap length δ, stator and rotor yokes radii Rsy and Rry , shaft radius Rsh , stack length lstk , number of turns per phase winding Nph , and number of phases m. The SRM geometry parameters for analytical computation of flux linkage characteristics using flux-tube model are emphasized in Fig. 2. The cross-section of 12/8 SRM is given as an example. Flowchart for computing ψ(I) curves at the aligned and the unaligned position is shown in Fig. 3. Usually, only a small number of points k define original B-H curve of the lamination material. In order to achieve an acceptable level of computational accuracy, it was necessary to use cubic spline interpolation to generate interpolated BH curve with more defined points n (n > k). Interpolated magnetization curve is recalculated to generate μint = f(Bint ) curve. Flux linkage characteristics are computed for n values of magnetic flux density. Since the computational time is rising with n, this number should be chosen carefully. The equations for computing flux linkage at two extreme rotor positions are given below. Due to limited available space, only the equations that are derived to compute flux linkage of flux-tubes No.1 and No.6 are given in this paper in order to explain the computation procedure. Flux tube No.1 is one of two flux-tubes used to represent the actual flux paths at the aligned position. Flux-tube No.6 is one of seven fluxtubes considered at the unaligned position. The list of equations used to compute flux linkage of all other defined flux-tubes is given in (20).
Fig. 3. Flowchart for the computation of flux linkage at the aligned and unaligned rotor position
Example: Computing flux linkage of flux-tube No.1 The areas of cross-section penetrated by the flux-tube No.1 are given in Table 1. The lengths of the individual machine segments encountered by flux-tube No.1 are given in Table 2. 353
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Fast Analytical Model for Switched Reluctance Machines（Senad Smaka） Table 1. Areas of Cross-Section for Flux-Tube No.1
Table 2. Lengths of Machine Segments for Flux-Tube No.1
Fig. 4. MEC for one part of flux-tube No.1 Table 3. Areas of Cross-Section for Flux-Tube No.6
The array of magnetic flux density values in the stator pole [Bsp1 ]1×n is assumed to be equal to the magnetic flux density array [Bint ]1×n of interpolated curve μint = f(Bint ) Bsp1 = Bsp11 Bsp12 . . . Bsp1n = [Bint ]· · · · · · · (1)
Rrp1 =
Lrp1 ; μrp1 · Arp1
Rry1 =
Lry1 · · · · · (4) μry1 · Ary1
The reluctance in the air gap is computed as follows Rδ1 =
The array of magnetic flux values in the stator pole [φ1 ]1×n is calculated as φ1 = Bsp1 · Asp1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (2)
Lδ1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·(5) μ0 · Aδ1
The array of currents [I1 ]1×n corresponding to the array of magnetic flux density values [Bsp1 ]1×n is computed using binary addition and arraywise multiplication in Matlab by means of expression ⎧ ⎫ ⎪ ⎪ Rsy1 Rry1 ⎪ ⎪ ⎪ φ1 ⎪ ⎨ ⎬ [I1 ] = + 2 Rsp1 + Rrp1 +Rδ1 + ⎪ ⎪ ⎪ ⎪ 2Npole ⎪ 2 2 ⎪ ⎩ ⎭
The arrays of magnetic flux density values in other parts of SRM are determined as φ1 φ1 ; Brp1 = ; Bsy1 = 2 · Asy1 Arp1 φ1 φ1 Bry1 = ; [Bδ1 ] = · · · · · · · · · · · · · · · · · · · (3) 2 · Ary1 Aδ1
· · · · · · · · · · · · · · · · · · · · · · · · · (6) where Npole is number of turns per pole (Npole = Nph · m/Ns ). The array of flux linkage values of flux-tube No.1 corresponding to the array of currents [I1 ] is ψ1 = φ1 · Nph · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (7)
Then on the basis of μint = f(Bint ) curve, the arrays of magnetic permeability [μsp1 ]1×n , [μsy1 ]1×n , [μrp1 ]1×n , and [μry1 ]1×n , corresponding to the arrays of magnetic flux density values computed by Eqs. (1) and (3), are obtained. The magnetic equivalent circuit created for one quarter of flux-tube No.1 is shown in Fig. 4. Now it is possible to compute the arrays of reluctances related to various laminated parts of the magnetic circuit [Rsp1 ]1×n , [Rsy1 ]1×n , [Rrp1 ]1×n , and [Rry1 ]1×n . These arrays are computed using arraywise right division in Matlab as Lsp1 Lsy1 ; Rsy1 = ; Rsp1 = μsp1 · Asp1 μsy1 · Asy1
Example: Computing flux linkage of flux-tube No.6 In Fig. 5, the detailed sketch of flux-tube No.6 and magnetic equivalent circuit created for this flux-tube are shown. The areas of cross-section penetrated by the flux-tube No.6 are given in Table 3. The lengths of flux-tube No.6 in one stator pole lsp6 and in the stator yoke lsy6 are: lsp6 = lsp1 and lsy6 = lsy1 . The lengths of flux-tube No.6 in one rotor pole lrp6 and in the rotor yoke lry6 are computed by means of expressions 354
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Fig. 6. ψ(θ) curve for a fixed value of current I
(a) Detailed sketch of flux-tube No.6
⎤ ⎡ ⎥⎥⎥ ⎢⎢⎢ π Rr − Rry 4 · βr · Rr ⎢
xC = Rr − ⎥⎥⎥⎦ · sin ⎢⎢⎣ − 8 Nr 8Rr − Rr − Rry
yC = Rr −
Rr − Rry 8
· · · · · · · · · · · · · · · · · · · (16) ⎤ ⎡ ⎥⎥⎥ ⎢⎢⎢ π 4 · βr · Rr ⎢
⎥⎥⎥⎦ · cos ⎢⎢⎣ − Nr 8Rr − Rr − Rry · · · · · · · · · · · · · · · · · · · (17)
The length of flux-tube No.6 in the air gap is lδ6 = (xC − xB )2 + (yC − yB )2 · · · · · · · · · · · · · · · · · (18) where the coordinates xB , yB , xC , and yC are given by Eqs. (10), (11), (16) and (17). The reluctances in various segments of the machine encountered by flux-tube No.6 are calculated in a similar way as in Eqs. (4) and (5). The array of currents [I6 ]1×n corresponding to the array of magnetic flux density values in the stator pole [Bsp6 ]1×n is computed as follows φ6 [I6 ] = 2 Rsp6 + Rrp6 +Rδ6 + Rsy6 + Rry6 Nph · · · · · · · · · · · · · · · · · · · (19)
(b) MEC of flux-tube No.6
Fig. 5. Detailed sketch and MEC of flux-tube No.6
7
· Rr − Rry · · · · · · · · · · · · · · · · · · · · · · · · · · · · (8) 8 Rr − Rry 2π = 2 · Rsh + · · · · · · · · · · · · · · · · · · (9) · 2 Nr
lrp6 = lry6
The length in the air gap lδ6 is given as an arch between points B and C, as shown in Fig. 5. The coordinates of point B are β s xB = (Rr + δ) · sin θ9 = (Rr + δ) · sin · · · · · · · (10) 2 β s yB = (Rr + δ) · cos θ9 = (Rr + δ) · cos · · · · · · · (11) 2
The array of flux linkage values of flux-tube No.6 corresponding to the array of currents [I6 ] is ψ6 = ntube6 · φ6 · Nph · · · · · · · · · · · · · · · · · · · · · · · · ·(20) where ntube6 is the number of flux-tubes No.6 at the unaligned position. For example, ntube6 = 4 for 12/8 SRM (see Fig. 1(b)). 2.2 Implementation of Gage-curve Method In this paper gage-curve method is used for analytical computation of flux linkage characteristics at intermediate rotor positions using flux linkage characteristics computed at the aligned and unaligned rotor position and equations given in (10). These equations are based on the analysis of ψ(θ) curve corresponding to a fixed value of current I (Fig. 6.). The rotor positions are divided into three regions. The equations for modeling ψ(I, θ) characteristics for any chosen value of excitation current in these regions are: - region I (θua < θ < θ1 ) Xi · (θ−θ1 ) ψ (Ii , θ) = kai · (θ1 −θ0i )+Lu · Ii + Zi −(θ−θ1 ) · · · · · · · · · · · · · · · · · · · (21)
The lengths of arcs CF and DE shown in Fig. 5 are Rr − Rry lCF = Rr − · θ10 · · · · · · · · · · · · · · · · · · · · · (12) 8 βr lDE = Rr · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (13) 2 Angles θ10 and θ11 are calculated as follows θ10 θ11
βr 2 = · · · · · · · · · · · · · · · · · · · · · · · · · · · (14) Rr − Rry Rr − 8 π = − θ10 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (15) Nr Rr ·
The coordinates of point C are 355
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- region II (θ1 ≤ θ ≤ θ2 ) ψ (Ii , θ) = ψuai + ki · ψali − ψuai · · · · · · · · · · · · · · · · · (22)
3.
To investigate whether the proposed analytical model is suitable to predict SRM magnetization and static torque characteristics, two experimental machines, 12/8 SRM and 8/6 SRM, are used for analytical and numerical computation. The basic parameters of 8/6 SRM and 12/8 SRM are given in (20) and (21), respectively. Figure 7 and Fig. 8 shows the flux linkage and static torque characteristics of 12/8 SRM and 8/6 SRM, respectively, computed by both analytical model and 2-D FEA. An soft Maxwell software is used for numerical computation. The magnetization characteristics obtained using analytical model matched very well with those obtained by FEA within acceptable accuracy. However, there is a discrepancy between analytical and FEA torque profiles, especially at higher current levels with saturated magnetic material. Actual torque curves tend not to be flat in the region II. The flat part of torque curves corresponding to the linear part of ψ(I, θ) characteristics modeled by Eq. (22). It is necessary to improve modeling of ψ(I, θ) curve in the region II in order to boost the accuracy of proposed model.
- region III (θ2 < θ ≤ θal ) Xi · (θ−θ2 ) Zi −(θ−θ2 ) · · · · · · · · · · · · · · · · · · · (23)
ψ (Ii , θ) = kai · (θ2 −θ0i )+Lu · Ii +
where Ii (i = 1, 2, . . . , ncurrent ) represent actual value of excitation current selected for computation, while ψuai and ψali are the flux linkages at the unaligned and the aligned position, respectively, corresponding to Ii . The boundaries between regions are defined as π βr + βs π |βr − βs | ; θ3 = ; − − Nr 2 Nr 2 θ1 + θ3 π ; θal = θ2 = · · · · · · · · · · · · · · · · · · · · · · · · (24) 2 Nr
θua = 0◦ ; θ1 =
The coefficients in Eqs. (21)–(23) are determined as kai ·(θ1 −θ0i )+Lu ·Ii −ψuai ·(θ1 −θua ) Zi = kai ·(θ1 −θua )− kai ·(θ1 −θ0i )+Lu ·Ii −ψuai ··················· Xi = kai · Zi · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · θ − θ0i ki = ··································· θ3 − θ1 ψali −kai ·(θ2 −θ0i )−Lu ·Ii ·(θal −θ2 ) Zi = kai ·(θal −θ2 )− ψali −kai ·(θ2 −θ0i )−Lu ·Ii ··················· Xi = kai · Zi · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
Comparison of Analytical Model and FEA
(25) (26) (27)
(28) (29)
The unaligned inductance Lu can be found from the ψ(I) characteristic at the unaligned position. The offset angle θ0i is given as θ0i = θ1 −
ψ (Ii−1 , θ) · θal · · · · · · · · · · · · · · · · · · · · · · · · (30) ψm · ξ
where ψm is maximum flux linkage at the aligned position corresponding to maximum value of excitation current, ξ is an empirical coefficient with the value in the range from 8 to 12, and ψ(Ii−1 , θ) is the flux linkage in the position θ obtained for excitation current Ii−1 selected in the previous step of computation. Parameter kai is given as
(a) Flux-linkage characteristics
ψali − ψuai · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (31) θ3 − θ1 2.3 Computation of Static Torque Characteristics T(I, θ) The static torque characteristics can be calculated on the basis of magnetic coenergy Wm , using first central difference approximation method of numerical differentiation, as: ∂Wm (I, θ) T= · · · · · · · · · · · · · · · · · · · · · · · · · · (32) ∂θ I=const. kai =
where the magnetic coenergy can be obtained using numerical integration method based on Simpson’s rule as I ψ (I, θ)dI · · · · · · · · · · · · · · · · · · · · · · · · · · · · (33) Wm =
(b) Static torque characteristics
Fig. 7. Flux-linkage and static torque characteristics computed by analytical model and FEA—12/8 SRM
0
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Fig. 9. Photo of experimental setup
(a) Flux-linkage characteristics
(a) Voltage and current waveforms for 8/6 SRM—aligned position
(b) Static torque characteristics
Fig. 8. Flux-linkage and static torque characteristics computed by analytical model and FEA—8/6 SRM
4.
Experimental Verification of Analytical Model (b) Voltage and current waveforms for 8/6 SRM—unaligned position
The photo of the test setup used for experimental verification of analytically computed magnetization and static torque characteristics is shown in Fig. 9. The three-phase full-wave rectifier is used as power supply during standstill test, while the classic three-phase and fourphase asymmetric bridge converters feed the 12/8 SRM and 8/6 SRM during normal operation, respectively. The tested machine was loaded with eddy current brake WEKA Power LPS 800 LK. An opto-interrupter with slotted disk is employed to determine the rotor position. The torque is obtained from a strain gauge force sensor that is a part of the test bench. Power analyzer LEM NORMA D6133M is used to measure the electrical quantities. Phase current is measured using current channel with triaxial shunt 61I3. Voltage channel 61U1 is used to measure applied voltage. Standstill test is conducted as follows. The rotor of the tested machine is blocked on a chosen position. Step voltage is applied to one of the three or four phases of the motor thus resulting in phase winding direct current flow. Voltage and current waveforms were visualized and recorded on a power analyzer. The produced torque is transmitted to a force sensor using a lever arm and obtained results are also recorded.
Fig. 10. Voltage and current waveforms for 8/6 SRM
Then, all measuring quantities are transferred and stored on a PC. Direct method for magnetization characteristic measurement described in (22) is used. It consists of voltage and phase current waveform storing and off-line flux linkage calculation from the measured voltage and current using a numerical integration ψ (t) = [v (t) − R · i (t)]dt · · · · · · · · · · · · · · · · · · · · · (34) where R is the phase resistance, v(t) and i(t) are stored waveforms of voltage and phase current. Stored voltage and phase current waveforms are postpro cessed in Matlab according to Eq. (34). Figure 10 shows the recorded voltage and phase current waveforms for 8/6 SRM at the aligned and the unaligned position. The slope of the phase current is lower at the aligned position, suggesting higher inductance in comparison to the unaligned rotor position. Computed and measured magnetization and static torque 357
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(a) Flux-linkage characteristics
(a) Flux-linkage characteristics
(b) Static torque characteristics
(b) Static torque characteristics
Fig. 11. Flux-linkage and static torque characteristics computed analytically and measured—12/8 SRM
Fig. 12. Flux-linkage and static torque characteristics computed analytically and measured—8/6 SRM
characteristics for 12/8 SRM and 8/6 SRM are shown in Fig. 11 and Fig. 12, respectively. Only a few measuring points are presented. For example, torque measurements for 12/8 SRM are presented for 24 rotor positions and only 5 excitation currents. Good agreements in Fig. 11(a) and Fig. 12(a) are evident, but the number of flux-tubes can be increased in order to improve accuracy of flux linkage computation at the aligned and unaligned rotor position leading to an increase in the overall accuracy of proposed model. The discrepancies between analytical and experimental results are higher for static torque characteristics especially when there is no overlapping between the poles of excited phase and rotor poles. One of the causes can be inaccuracy in retaining the rotor in these positions when one phase is excited during the experiment that has occurred because of the characteristics of used eddy current brake.
ψ(I, θ) and torque lookup table T (I, θ) for 31 values of current I and 46 values of angle θ lasted approximately nine seconds when the proposed analytical method is used while it lasted more than three hours when FEA software is used on a same PC. Comparison of magnetization characteristics computed using analytical model with both FEA and experiment shows that the proposed analytical model is accurate enough. However, there is a discrepancy between the analytical and FEA and experimental results, especially for a higher torques. The improvement of presented analytical model’s accuracy especially regarding static torque characteristics should be future work. The idea is to model variation of flux-linkage with rotor position in the central region II using second-order function instead of using linear variation as in this paper. Also, the accuracy of analytical computation of flux linkage and static torque characteristics can be improved by defining more flux-tubes at the aligned and unaligned rotor position. Since the high speed PC’s and sophisticated FEA software are available, coupling of FEA and gage-curve method is also one of the possibilities. In this case FEA software will be used to compute flux linkage characteristics at the extreme rotor positions while gage-curve method will be used to compute magnetization characteristics at the arbitrary rotor
5.
Conclusions
Fast analytical model for computing nonlinear magnetization characteristics and static torque characteristics of SRM is presented in this paper. The model is proved as computationally fast. For example, computation of the 12/8 SRM magnetization characteristics 358
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positions like in this paper. Modeling time will increase but it should be acceptable. Computational time also will increase but model’s accuracy can be improved. The rotation interval between the unaligned and aligned position can be divided into higher number of regions instead of using only three regions as in this paper. Investigation of the influence of this parameter on the model’s accuracy is recommended.
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Senad Smaka (Non-member) was born on 1969 in Sarajevo, Bosnia and Herzegovina. He graduated from the Faculty of Electrical Engineering at University of Sarajevo in 1996, received M.S. degree in electrical engineering from the Faculty of Electrical Engineering and Computing at University of Zagreb in 2004 and Ph.D. degree in electrical engineering from the Faculty of Electrical Engineering at University of Sarajevo in 2012. From 2000 he works as teaching assistant and associate professor on Department of Power Engineering of Faculty of Electrical Engineering in Sarajevo. His research interests include HEVs, modeling and numerical analysis of electrical machines and drives. He is member of IEEE.
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