energies
Article Analytical Calculation of D- and Q-axis Inductance for Interior Permanent Magnet Motors Based on Winding Function Theory
Peixin Liang 1,2, Yulong Pei 2, Feng Chai 1,2,* and Kui Zhao 2
1 State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China; [email protected] 2 Department of Electrical Engineering, Harbin Institute of Technology, Harbin 150001, China; [email protected] (Y.P.); [email protected] (K.Z.) * Correspondence: [email protected]; Tel.: +86-451-8640-3480
Academic Editor: K. T. Chau Received: 30 April 2016; Accepted: 21 July 2016; Published: 25 July 2016
Abstract: Interior permanent magnet (IPM) motors are widely used in electric vehicles (EVs), benefiting from the excellent advantages of a more rational use of energy. For further improvement of energy utilization, this paper presents an analytical method of d- and q-axis inductance calculation for IPM motors with V-shaped rotor in no-load condition. A lumped parameter magnetic circuit model (LPMCM) is adopted to investigate the saturation and nonlinearity of the bridge. Taking into account the influence of magnetic field distribution on inductance, the winding function theory (WFT) is employed to accurately calculate the armature reaction airgap magnetic field and d- and q-axis inductances. The validity of the analytical technique is verified by the finite element method (FEM).
Keywords: interior permanent magnet motor; d- and q-axis inductances; lumped parameter magnetic circuit model; winding function theory; armature reaction magnetic field; saturation; nonlinearity
1. Introduction In recent decades, permanent magnet (PM) motors cover a wide range of industrial applications, especially for electric vehicles (EVs), attributing to the compact structure, high efficiency, high power/torque density and excellent dynamic performances [1–6]. As a typical topology of PM motors, the interior permanent magnet (IPM) motors with a V-shape provide a more superior performance of the flux-weakening capability and wider constant-power speed range [7–10], achieving considerable attention. Along with these excellent advantages, IPM motors offer a great alternative opportunity to contribute for a more rational use of energy, which makes EVs have the potential to provide a perfect solution to energy security and environmental impacts in the future [11,12]. The d- and q-axis inductances are key parameters for the performance design including flux-weakening capability, power factor and control strategy [13–15]. For the first stage of the IPM motor design, the calculation of d- and q-axis inductances in no-load condition is crucial. For IPM motors, on one hand, the magnetic bridges protect the permanent magnets from flying away from the rotor. On the other hand, they provide a path for the permanent-magnet leakage flux and cause saturation. The influence of magnetic bridge saturation is the biggest obstacle for the calculation of d- and q-axis inductances. In addition, the accurate paths of d- and q-axis fluxes are another crucial point. Various researches on inductance with finite element method (FEM) have been successfully investigated in the past [16], e.g., a 3D FEM model is used to calculate the inductances considering the end effect and magnetic flux leakage [17]. In [18], taking into account saturation and nonlinearity, the
Energies 2016, 9, 580; doi:10.3390/en9080580 www.mdpi.com/journal/energies Energies 2016, 9, 580 2 of 11 cross-coupling inductance characteristics are investigated. FEM can effectively analyze the inductances accountingEnergies 2016 for, 9 the, 580 saturation, nonlinearity, complex geometry and cross-coupling influence. However,2 of 11 FEM is too theory-lacking and time-consuming to provide a physical insight to the motor design, especiallythe cross for‐coupling the influence inductance of parameter characteristics variation are at investigated. the first stages FEM of can the designeffectively process. analyze the inductancesComparatively accounting speaking, for analyticalthe saturation, methods nonlinearity, are often complex preferred. geometry Lumped and parameter cross‐coupling magnetic influence. However, FEM is too theory‐lacking and time‐consuming to provide a physical insight to the circuit model (LPMCM) is a common method. In [19,20], the d- and q-axis inductances are calculated motor design, especially for the influence of parameter variation at the first stages of the design process. with LPMCMs, which are established based on the d- and q-axis flux paths matching the practical Comparatively speaking, analytical methods are often preferred. Lumped parameter magnetic operatingcircuit model point of(LPMCM) the control is a issue.common The method. influence In [19,20], of saturation the d‐ and and q‐axis nonlinearity inductances is analyzed.are calculated In [21], the motorwith LPMCMs, is divided which into severalare established mechanical based phase on the regions d‐ and accordingq‐axis flux topaths the windingmatching distribution.the practical On the basisoperating of the point mechanical of the control phase issue. region The and influence equivalent of saturation magnetic and circuit, nonlinearity the d- andis analyzed.q-axis inductances In [21], are predicted.the motor is Althoughdivided into LPMCM several mechanical has advantages phase regions in saturation according and to the nonlinearity, winding distribution. the drawback On of the methodthe basis is of that the the mechanical calculation phase accuracy region is greatlyand equivalent affected magnetic by the way circuit, flux pathsthe d‐ areand divided q‐axis and analyticalinductances models are arepredicted. built. OnAlthough the other LPMCM hand, has though advantages the LPMCM in saturation expresses and nonlinearity, the relationship the of permeance,drawback it of is the complicated method is tothat classify the calculation the permeance accuracy according is greatly toaffected their contributions.by the way flux Especially paths are for the motordivided with and distributed analytical models winding, are the built. amount On the of teethother ishand, too much, though which the LPMCM makes it expresses more difficult the to classifyrelationship the flux of paths permeance, and calculate it is complicated the inductance to classify based the permeance on permeance. according Furthermore, to their contributions. the accurate Especially for the motor with distributed winding, the amount of teeth is too much, which makes it flux used for the inductance calculation is the summation of the main and leakage magnetic field. more difficult to classify the flux paths and calculate the inductance based on permeance. However, the LPMCM calculates the main permeance, neglecting the influence of leakage permeance, Furthermore, the accurate flux used for the inductance calculation is the summation of the main and whichleakage is what magnetic causes calculationfield. However, error. the LPMCM calculates the main permeance, neglecting the influenceThis paper of leakage presents permeance, an analytical which methodis what causes for d- calculation and q-axis error. inductance calculation in no-load conditionThis based paper on LPMCMpresents an and analytical winding method function for theory d‐ and (WFT). q‐axis LPMCMinductance is adaptedcalculation to thein no calculation‐load of magneticcondition bridge based saturation,on LPMCM and and WFT winding is employed function to theory divide (WFT). accurate LPMCM paths ofis dadapted- and q-axis to the fluxes andcalculation consider the of magnetic influence bridge of magnetic saturation, field and distribution. WFT is employed The d- to and divideq-axis accurate flux linkages paths of produced d‐ and q‐ by a veryaxis small fluxes current and consider are calculated, the influence based of magnetic on which fieldd- anddistribution.q-axis inductances The d‐ and q can‐axis be flux obtained. linkages The FEMproduced results verify by a very the validitysmall current of the are proposed calculated, method. based on which d‐ and q‐axis inductances can be obtained. The FEM results verify the validity of the proposed method. 2. Analytical Method 2. Analytical Method A 48-slot/8-pole IPM motor with a V-shaped rotor is instanced to verify the proposed method. Figure1 showsA 48‐slot/8 the‐ modelpole IPM with motor magnetic with a bridges, V‐shaped where rotor isα instancedis the magnet to verify pole-arc the proposed to pole-pitch method. ratio, β Figure 1 shows the model with magnetic bridges, where α is the magnet pole‐arc to pole‐pitch ratio, is the barrier arc to pole-pitch ratio, Rs is the stator bore radius, Rr is the rotor outer radius, tb is the β is the barrier arc to pole‐pitch ratio, Rs is the stator bore radius, Rr is the rotor outer radius, tb is the bridge thickness, wbar is the barrier width, lbar1 and lbar2 are the barrier length, wm is the PM width, lm bridge thickness, wbar is the barrier width, lbar1 and lbar2 are the barrier length, wm is the PM width, lm is is the PM length in magnetization direction, wr is the rib width, lr is the rib length. For IPM motors, the PM length in magnetization direction, wr is the rib width, lr is the rib length. For IPM motors, the the saturationsaturation mainly mainly focuses focuses on on the the bridge. bridge. Here, Here, we we assume assume that that the the permeability permeability of stator of stator and androtor rotor (except(except the the bridge) bridge) core core is is infinite. infinite.
FigureFigure 1. Model1. Model of of a a 48-slot/8-pole 48‐slot/8‐pole Interior Interior permanent permanent magnet magnet (IPM) (IPM) motor. motor.
Energies 2016, 9, 580 3 of 11 Energies 2016, 9, 580 3 of 11 Energies 2016, 9, 580 3 of 11 2.1. Saturation Calculation Energies2.1. Saturation 2016, 9, 580 Calculation 3 of 11 2.1.Energies Saturation 2016, 9, 580 Calculation 3 of 11 TheThe leakage leakage fluxes fluxes pass pass the bridgesthe bridges causing causing different differe saturationnt saturation characteristics characteristics depending depending on theon the 2.1. SaturationThe leakage Calculation fluxes pass the bridges causing different saturation characteristics depending on the rotor2.1.rotor shape. Saturation shape. For For theCalculation sakethe sake of simplicity, of simplicity, the rotorthe rotor is divided is divided into into two two pieces: pieces: saturation saturation regions regions with with a a constantrotor shape. permeability For the sake and of non-saturation simplicity, the regionrotor is withdivided infinite into two permeability pieces: saturation shown regions in Figure with2. a The constantThe leakage permeability fluxes passand thenon-saturation bridges causing region different with saturation infinite permeabilitycharacteristics showndepending in Figureon the 2. The constant permeability and non‐saturation region with infinite permeability shown in Figure 2. The saturationrotorsaturation shape. regions regions For ofthe bridges ofsake bridges of aresimplicity, are widened, widened, the whererotor where is the divided th equivalente equivalent into two bridge pieces:bridge width saturationwidth can can be regions be expressed expressed with asa as saturationconstant permeability regions of bridges and non are‐ saturationwidened, whereregion the with equivalent infinite bridgepermeability width canshown be expressedin Figure as2. The constant permeability and non‐saturation region with=+ infinite permeability shown in Figure 2. The saturation regions of bridges are widened, wwhereb “ www thebbar equivalent` bar2tb 2 t b bridge width can be expressed as (1)(1) saturation regions of bridges are widened, wherewwbbarb the equivalent 2 t bridge width can be expressed as (1)
wwbbarb 2 t (1) wwbbarb 2 t (1)
Figure 2. Simplified model of rotor. FigureFigure 2. Simplified 2. Simplified model model of rotor. of rotor. Figure 2. Simplified model of rotor. As shown in Figure 3, the fluxes,Figure produced 2. Simplified by magnets, model of pass rotor. the magnet, magnetic bridges, and airAs‐gap. shownAs Considering shown in Figure in Figure the3, theflux 3, fluxes,the paths fluxes, producedaround produced the by magnets, magnets, by magnets, an pass LPMCM pass the magnet,the is employedmagnet, magnetic magnetic to analyze bridges, bridges, the and and As shown in Figure 3, the fluxes, produced by magnets, pass the magnet, magnetic bridges, and air-gap.saturationair-gap. Considering Considering and nonlinearity the the flux offlux paths bridges. paths around Figure around the 4 depicts the magnets, magnets, the LPMCM an LPMCMan LPMCMof one is‐half employed is of employed two magnets to analyzeto withanalyze the the air‐gap. Considering the flux paths around the magnets, an LPMCM is employed to analyze the saturationoppositesaturation and poles and nonlinearity and nonlinearity the associated of bridges. of bridges. bridges Figure Figurebehind4 depicts 4 the depicts magnet the LPMCMthe halves, LPMCM of where one-half of one-half Rm is of the two of reluctance two magnets magnets of with with saturation and nonlinearity of bridges. Figure 4 depicts the LPMCM of one‐half of two magnets with oppositeopposite poles gpoles and and the associatedthe associated bridges bridges behind behind the magnetthe magnet halves,b halves, where whereR isRm the is the reluctance reluctancer 0 of of magnet,opposite R poles is the and reluctance the associated of one ‐bridgeshalf of the behind per pole the magnetair gap, halves,R is the where reluctance Rm ism ofthe bridge, reluctance Φ , Φ of, opposite poles and the associated bridges behind the magnet halves, where Rm is the reluctance of g b r 0 magnet,Φmagnet,magnet,g andR Φg R bisR gare isthe is thethe the reluctance reluctanceremanent reluctance ofmagnetof ofone one-half one-ha‐half flux, of of lfmagnet the of the theper per leakageperpole pole pole air flux, airgap, air gap, airgap,Rb ‐isgapR theRb is fluxis reluctance thethe and reluctancereluctance bridge of bridge, leakage ofof bridge, Φ rflux,, Φ0, Φr,, Φ , magnet, Rg is the reluctance of one‐half of the per pole air gap, Rb is the reluctance of bridge, Φr, Φ0, 0,respectively.ΦΦgg andand Φ Φb bare areare the the the remanent remanent remanent magnet magnet magnet flux, flux, flux, magnet magnet magnet leakage leakage leakage flux, flux,air flux,‐gap air-gap air-gap flux and flux flux bridge and and bridge leakage bridge leakage leakageflux, flux, Φg and Φb are the remanent magnet flux, magnet leakage flux, air‐gap flux and bridge leakage flux, flux,respectively.respectively. respectively.
Figure 3. Flux line distribution in no‐load condition.
FigureFigure 3. 3.Flux Flux line line distributiondistribution in in no no-load‐load condition. condition. Figure 3. Flux line distribution in no-load condition. Φg Rg Rg
Φg Φg Rg Rg Rg Rg Φb Φg Rb Rg Rb Rg Φb Φb Rb Rb Φ0 Rb Rb Φ0 Φb R Rb Φ0 b Φr RΦ0 Φ0 m Φ0 Rm Φr Φ0 Φr Rm ΦR0m Φr Φr Rm Rm Φr
Φr Rm Rm Φr
Figure 4. Magnetic circuit model. Figure 4. Magnetic circuit model. FigureFigure 4. 4. MagneticMagnetic circuit model. model.
Figure 4. Magnetic circuit model.
Energies 2016, 9, 580 4 of 11
Energies 2016, 9, 580 4 of 11 In the LPMCM, the permeability of bridge is a nonlinear uncertain parameter, which is the basic of the solution for the all fluxes. The iterative method is effective for this problem. Figure5 shows the In the LPMCM, the permeability of bridge is a nonlinear uncertain parameter, which is the basic iterative process: of the solution for the all fluxes. The iterative method is effective for this problem. Figure 5 shows the iterative process: 1. Firstly, the initial flux density of bridge is supposed as Bb. 2.1. Firstly,Secondly, the calculate initial flux all density the reluctances of bridge and is supposed fluxes according as Bb. to the material B–H profile and the 2. Secondly,law of magnetic calculate flux all continuity. the reluctances and fluxes according to the material B–H profile and the 3. lawThirdly, of magnetic compare flux the continuity. obtained flux density of bridge Bb1 with Bb. 4.3. Thirdly,Finally, adjustcompareBb andthe obtained repeat this flux process density until of bridge the error Bb1 is with satisfied. Bb. 4. Finally, adjust Bb and repeat this process until the error is satisfied.
Initial flux density Bb
Based on material B-H profile, calculate all reluctances
Calculate all fluxes Modify Bb
Calculate flux density Bb1
N Check error Y Obtain the permeability
Figure 5. Flow chart of the iterative process.
All the parameters can be expressed as: All the parameters can be expressed as:
lm Rm (2) lmwL Rm “ 0 rm (2) µ0µrwm L RR R Rssr´ Rr Rgg “ (3) RRsr 2 (3) µ pRs`Rrq 2π LL 0 0 242 4pp wb Rb “ w (4) µ pBbqb ¨ tb L Rb (4) BtLbb φr “ Brwm L (5)
rrmRBwmRg L (5) φb “ φr (6) RmRb ` RbRg ` RgRm RRmg br (6) RR RRφb RR mbBb1 “ bg gm (7) tb L where L is effective length of the rotor and stator, Brb is the remanence of magnet, p is the number Bb1 (7) tL of pole pairs, µr is the relative permeability of magnet,b µ0 and µ (Bb) are the permeability of air and bridge, respectively. where L is effective length of the rotor and stator, Br is the remanence of magnet, p is the number of 2.2.pole Inductance pairs, μr is Calculation the relative permeability of magnet, μ0 and μ (Bb) are the permeability of air and bridge, respectively. When the current is very small, the influence of armature-reaction magnetic field on saturation can2.2. Inductance be neglected, Calculation and the d- and q-axis inductances in no-load condition are obtained by:
When the current is very small, the influenceL “ φofd `armatureL ‐reaction magnetic field on saturation d Id 0 can be neglected, and the d‐ and q‐axis inductancesφ in no‐load condition are obtained by: (8) L “ q ` L # q Iq 0
Energies 2016, 9, 580 5 of 11
L d L d 0 Id
(8) L q L q 0 Iq Energies 2016, 9, 580 5 of 11 where φd and φq are the main flux linkages in d‐ and q‐axis, Id and Iq are the currents in d‐ and q‐axis, respectively. L0 is the leakage inductance. whereTakingφd and intoφq areaccount the main the influence flux linkages of magnetic in d- and fieldq-axis, distribution,Id and Iq are the the WFT currents is used in tod- accurately and q-axis, calculaterespectively. the dL‐ 0andis the q‐axis leakage inductances. inductance. For the clear expression of magnetic motive force (MMF) and rotorTaking magnetic into potential, account thethe influence initial condition of magnetic is that field all distribution, the angular thepositions WFT is are used expressed to accurately with electricalcalculate degree. the d- and q-axis inductances. For the clear expression of magnetic motive force (MMF) and rotorBased magnetic on the winding potential, function, the initial the conditionarmature‐ isreaction that all MMF the angular of each positions phase can are be expressed expanded withinto aelectrical Fourier degree.series as [22,23]: Based on the winding function, the armature-reaction MMF of each phase can be expanded into a Fourier series as [22,23]: 2Nkwv I Ftas cos cos vp 2Nkwv I Fa “ vpπ cos pνθsq cos pωt ` ϕq ν 2Nkwv I 22 Ftbscos cos (9) $ 2Nkwv I 2π 2π Fb “ ř vp cos νθs ´ 33cos ωt ` ϕ ´ ’ vpπ 3 3 (9) ’ ν ’ 2Nk2Nkwv I I 444π 4 π & Fc “ ř wvcos `ν` θs ´ ˘˘ cos` ωt ` ϕ ´ ˘ Ftcs vpπ cos3 cos 3 ν vp 33 ’ ´´ ¯¯ ´ ¯ ’ ř where N is the amount%’ of winding turns in series of stator phase, k is the winding factor, I is the where N is the amount of winding turns in series of stator phase, kwv is the winding factor, I is the amplitude of phase current, ϕϕ is the current phase measured from the a-axis,a‐axis, ω ω isis the the rated electrical angular velocity, and θ is the angular position in the stator reference frame measured from the axis of angular velocity, and θss is the angular position in the stator reference frame measured from the axis ofphase phase a shown a shown in in Figure Figure6. 6.
(a) (b)
Figure 6. Model with different rotor position: (a) d‐axis overlaps with a‐axis; (b) q‐axis overlaps Figure 6. Model with different rotor position: (a) d-axis overlaps with a-axis; (b) q-axis overlaps with a‐axis. with a-axis.
The synthetic armature‐reaction MMF is obtained by: The synthetic armature-reaction MMF is obtained by: FtFFFss(,) a b c
Fspθs,Fvkttqcos( “ Fa` Fb ` (Fc )) (10) sv (10) “ v Fcospvθs ` kvpωt ` ϕqq v where ř where Nkwv F“ 3 NkIwv FI 3vpπ k “ ´1,vpf or v “ 6pm ´ 1q ` 1 $ v (11) kforvm1, 6( 1) 1 ’ kv “v 1,f or v “ 6m ´ 1 (11) &’ mkforvm“ 1,1, 2, 3... 6 1 v ’ ’ m 1,2,3... 2.2.1. D-Axis Magnetic Potential %
When ϕ = 0, t = 0 and the d-axis overlaps with the a-axis, the flux passes the d-axis path shown in Figure7. As the permeability of the non-saturation region in rotor is much larger than that of the magnet and barrier, the magnet and barrier can be regarded as equal-potential area. The magnetic potential Energies 2016, 9, 580 6 of 11 Energies 2016, 9, 580 6 of 11 2.2.1. D‐Axis Magnetic Potential 2.2.1. D‐Axis Magnetic Potential When ϕ = 0, t = 0 and the d‐axis overlaps with the a‐axis, the flux passes the d‐axis path shown When ϕ = 0, t = 0 and the d‐axis overlaps with the a‐axis, the flux passes the d‐axis path shown Energiesin Figure2016 7., 9 , 580 6 of 11 in FigureAs the 7. permeability of the non‐saturation region in rotor is much larger than that of the magnet and barrier,As the permeability the magnet and of the barrier non‐ saturationcan be regarded region as in equalrotor‐ ispotential much larger area. thanThe magneticthat of the potential magnet distributionand barrier, ofofthe thethe magnet rotor in inand the barrier d-axis‐axis iscanis shown be regarded in Figure as8 8. .equal In In one-halfone‐potential‐half of of the the area. electricalelectrical The magnetic period, period, the potentialthe rotor rotor magneticdistribution potential of the rotor cancan bebe in expressedexpressedthe d‐axis is asas shown in Figure 8. In one‐half of the electrical period, the rotor magnetic potential can be expressed as βπ απ απ 4 sinsinjjjj 4 sinsin j 2 ´ sinsin j 2 442222 (12) FFdp()θsq “UjUjUd1 sinjjjcos cospjθsq ` Ud2 sin sin cos cospjθsq (12) ds d1244πj 222 s dπ ´ ¯ j ` ˘ s jjj `jj˘ j (12) Fds() UjUj d12ÿ cos s dÿ cos s jjjj wherewhere UUdd1 andand UUdd22 areare the the magnetic magnetic potentials potentials shown shown in in Figure Figure 7.7. where Ud1 and Ud2 are the magnetic potentials shown in Figure 7.
Figure 7. Flux line distribution in the d-axis.‐axis. Figure 7. Flux line distribution in the d‐axis. a-axis a-axis
Ud1
Ud1 Ud2 απ a pole pitch
Ud2 απ a pole pitch θs θ βπ s βπ
Figure 8. Magnetic potential distribution in the d‐axis. Figure 8. Magnetic potential distribution in the d-axis.‐axis. According to the continuity of magnetic flux, the flux through the air‐gap is equal to that passing According to the continuity of magnetic flux, the flux through the air‐gap is equal to that passing the rotor.According The path to the of continuitythe d‐axis flux of magnetic in the rotor flux, consists the flux of through the magnet, the air-gap barrier is and equal bridge. to that Based passing on the rotor. The path of the d‐axis flux in the rotor consists of the magnet, barrier and bridge. Based on thethe rotor.path, the The fluxes path ofin thethe drotor-axis can flux be in calculated the rotor consists by of the magnet, barrier and bridge. Based on the path, the fluxes in the rotor can be calculated by the path, the fluxes in the rotor can be calculated by 22llL BtL 2 00RRsr 2 rm wL 01bar bar 2 0 b b ((,0)())FFss ds LdU s d1 RR 2 wL2201llLbar bar 2 0 BtL b b 2απ2 pRs00 R rsr2 l m rm w bar w b ((,0)())FFss dsµ R ` LdUR s d1 2µ µrwm L 2µ pl `l qL 2µ µpB qt L 2 0 s r 0 0 bar1 bar2 0 b b (13) 2 απ pFspθs, 0q ´ FdppRθsqqs R r2 Ldθs “ Ud1 l m` w bar` w b ´ ppRs´RRRrq 2 BtLlm wbar wb 2 2 0 sr 0 bb (13) $ βπ((,0)())FFss ds LdU s d2 (13) 2 0 µ0 RRsrRs`Rr 0 µ0´BtLµbbpBbqtb L ¯ ş22 pRsr R2 w b ’ απ((,0)())FFpssFspθs, 0q ´ dsFdpθsqq LdULd sθs “ d2Ud2 w & 2 ppRs´Rrq 2 b 2 pRsr R2 w b şSubstituting (10) and (12) into (13), the magnetic potentials can be obtained as: %’ Substituting (10) and (12) into (13), thethe magneticmagnetic potentialspotentials cancan bebe obtainedobtained as:as: v vαπ 2sinFP21FP sin 1 vp2 q Ud1 “ 2 U 2sinFPvpP1 2`P3`P4`P5q d1 v 2 $ v vP2345 P Pvβπ P vαπ (14) Ud1 FP sin ´sinp q ’ vPř 1 P P2 P 2 (14) ’ Ud2 v“ 2345 & ´ vvv´p0.5P¯5`P6q ¯ FPv sin sin (14) 1 vv ’ ř 22 ’U FP1 sin sin where % d 2 22 v vPP0.5 56 Ud 2 µ0 Rs ` Rr P “v vPP0.5 L (15) 1 p pR ´ R q 562 where s r where µ0 Rs ` Rr P2 “ Lαπ (16) p pRs ´ Rrq 2
2µ0µrwm L P3 “ (17) lm Energies 2016, 9, 580 7 of 11 Energies 2016, 9, 580 7 of 11
0 RRsr P1 RR L (15) pR0 R sr2 P1 sr L (15) pRsr R 2
0 RRsr PL2 RR (16) pR0 R sr2 PL2 sr (16) pRsr R 2
20 rmwL P3 2wL (17) 0 l rm P3 m (17) Energies 2016, 9, 580 lm 7 of 11 201llLbar bar 2 P4 2 llL (18) 01 barw bar 2 P4 bar (18) 2µ0 plbarw1bar` lbar2q L P4 “ (18) 20 wbarBtLbb P5 2 BtL (19) 0 w bb P5 2µ0µ pBb q t L (19) P “ w b b (19) 5 wb RRb PL 0 sr 6 RR (20) pRµ00 R Rsrs 22` Rr pβ ´ αq π P6PL“6 sr L (20)(20) ppRpRssr´ R rq 222 2
2.2.2. Q Q-Axis‐Axis Magnetic Potential 2.2.2. Q‐Axis Magnetic Potential When φϕ == 0,0, tt == 00 andand thetheq q-axis‐axis overlaps overlaps with with the thea -axis,a‐axis, the the flux flux passes passes the theq-axis q‐axis path path shown shown in inFigure FigureWhen9. Similarly, 9. ϕSimilarly, = 0, t the= 0 the magneticand magnetic the q potential‐axis potential overlaps distribution distribution with the of a rotor‐ axis,of rotor in the the in flux qthe-axis passes q‐axis is shown theis shown q‐ inaxis Figure inpath Figure 10shown, and 10, andcanin Figure becan expressed be 9. expressed Similarly, in a Fourier inthe a magneticFourier series series potential as as distribution of rotor in the q‐axis is shown in Figure 10, and can be expressed in a Fourier series as 11p1´αqπ p1´ βqπ sin j ´ sin j 4 sinjj112 sin 2 Fqpθsq “ Uq4 sinjj22 sin cos pjθsq (21)(21) F () Ujπ ´ ¯ j ´ ¯ cos qs q4 j 22s (21) j j Fqs() Uj q ÿ coss j j where Uq isis the magnetic potential shown in Figure9 9.. where Uq is the magnetic potential shown in Figure 9.
Figure 9. Flux line distribution in the q‐axis. Figure 9. Flux line distribution in the q-axis.‐axis. a-axis a-axis
Uq Uq βπ a pole pitch βπ a pole pitch θs θs απ απ
Figure 10. Magnetic potential distribution in the q‐axis. Figure 10. Magnetic potential distribution in the q-axis.‐axis.
As shown in Figure9, the flux is divided into two parts in bridge region. The two parts operate in parallel, and the length of each path is 0.5 wb. The flux in the q-axis can be calculated by
p1´αqπ 2 µ0 Rs ` Rr µ0µ pBbq tb L µ0µ pBbq tb L Fs pθs, 0q ´ Fq pθsq Ldθs “ Uq ` (22) p1´βqπ p pRs ´ Rrq 2 0.5wb 0.5wb ż 2 ˆ ˙ ` ˘ Energies 2016, 9, 580 8 of 11
From Equation (17), we can get
vp1´αqπ vp1´βqπ FP1 sin 2 ´ sin 2 U “ (23) q ´ ´ ¯ ´ ¯¯ v v p2P5 ` P6q ÿ 2.2.3. D-and Q-Axis Inductances The armature reaction airgap magnetic fields in d-axis can be obtained by
µ0 B “ pFs pθs, 0q ´ F pθsqq d d pRs´Rrq βπ απ sin j ´sinpj απ q (24) 4 sinpj 2 q 4 2 2 µ0 “ Fcos pvθsq ´ U cospjθsq ´ U cospjθsq d1 π j d2 π ´ ¯ j pRs´Rrq ˜ v j j ¸ ř ř ř The d-axis inductance can be calculated by
Ld “ Lmd ` L0 2π 2Nkwv Rs`Rr 0 vpπ cospνθsqBd 2 Ldθr (25) $ L “ ν & md ş ř I where Lmd is the main inductance% in the d-axis. L0 consists of end-winding leakage inductance, slot leakage inductance and tooth-tip leakage inductance. The detailed computation processes of the leakage inductance are expressed in [24]. The end-winding leakage inductance can be calculated by
4m L “ µ N2q p2l λ ` w λ q (26) ew 0 w h ew w Q where q is the number of stator slots per pole and phase, lw is the average length of the end winding, m is the number of phases, Q is the number of slots, wew is the coil span, λh is the empirical axial permeance factor and λw is the empirical span permeance factor of the end winding. The slot leakage inductance is 4m L “ µ N2L λ (27) u 0 Q u where λw is the slot leakage permeance factor, which is relative to slot dimensions [13,24]. The tooth-tip leakage inductance is
4m L “ µ N2L k λ (28) tt 0 Q tt tt where λtt is the tooth-tip leakage permeance factor, and ktt is a factor that takes into account the presence of different phases in a slot [24]. The armature reaction airgap magnetic fields in the q-axis can be obtained by
µ0 Bq “ Fs pθs, 0q ´ Fq pθsq pRs´Rrq p ´ q sin j p1´αqπ ´sin j 1 β π ` 4 ˘ 2 2 µ0 (29) “ Fcos pvθsq ´ Uq cos pjθsq π ´ ¯ j ´ ¯ pRs´Rrq ˜ v j ¸ ř ř The q-axis inductance can be calculated by
Lq “ Lmq ` L0 2π 2Nkwv Rs`Rr 0 vpπ cospνθsqBq 2 Ldθr (30) $ L “ ν & mq ş ř I where Lmq is the main inductance% in the q-axis. Energies 2016, 9, 580 9 of 11
3. Results and Analysis The armature reaction airgap magnetic fields and inductances in the d- and q-axis are calculated by the proposed method, and the validity is verified by FEM. The FEM results are obtained by the commercial software ANSYS (version 16.0, ANSYS, Pittsburgh, PA, USA). The main parameters of the motors are listed in Table1. Based on these parameters, the saturation of bridge is firstly investigated. Table2 depicts the results of the analytical method and FEM. The comparison shows that the proposed method is very effective for bridge saturation.
Table 1. Main parameters of the 8-pole/48-slot motor.
Symbol Parameter Results and Unit p Number of pole pairs 4 Q Number of slots 48 I Amplitude of phase current 1A N Amount of winding turns in series of stator phase 32 Br Remanence of magnet 1.25 µr Relative permeability of magnet 1.024 α Magnet pole-arc to pole-pitch ratio 0.742 β Barrier arc to pole-pitch ratio 0.886 Rs Stator bore radius 91 mm Rr Rotor outer radius 90 mm tb Bridge thickness 1 mm wbar Barrier width 5 mm lbar1 Barrier 1 length 5 mm lbar2 Barrier 2 length 5.9 mm wm PM width 20 mm lm PM length in magnetization direction 6 mm wr Rib width 8 mm lr Rib length 8.8 mm
Table 2. Permeability of bridge.
Method Value Unit Proposed method 4.40 ˆ 10´5 H/m Finite element method (FEM) 4.52 ˆ 10´5 H/m
Taking into account the influence of the armature-reaction magnetic field on saturation, a very small current is employed to calculate no-load d- and q-axis inductances. Freeze permeability is adopted to calculate the armature-reaction magnetic field. Figure 11 shows the waveforms of airgap flux density in the d- and q-axes. It is observed that the analytical results match well with that of FEM. Based on Equations (25) and (30), the d- and q-axis inductances in the no-load condition can be obtained shown in Table3. The analytical data and simulation data tally well, which indicates that the proposed method is quite accurate for d- and q-axis inductance calculations.
Table 3. The inductances of the two methods.
Item Proposed Method FEM Unit D-axis inductance 0.524 0.513 mH Q-axis inductance 1.267 1.271 mH Energies 2016, 9, 580 10 of 11
Taking into account the influence of the armature‐reaction magnetic field on saturation, a very small current is employed to calculate no‐load d‐ and q‐axis inductances. Freeze permeability is adopted to calculate the armature‐reaction magnetic field. Figure 11 shows the waveforms of airgap flux density in the d‐ and q‐axes. It is observed that the analytical results match well with that of FEM. Based on Equations (25) and (30), the d‐ and q‐axis inductances in the no‐load condition can be obtainedEnergies 2016 shown, 9, 580 in Table 3. The analytical data and simulation data tally well, which indicates10 that of 11 the proposed method is quite accurate for d‐ and q‐axis inductance calculations.
0.004 FEM 0.010 FEM 0.003 0.008 Analytical 0.006 Analytical 0.002 0.004 0.001 0.002 0.000 0.000 -0.001 -0.002 -0.002 -0.004 -0.006 Flux Density (T) Flux -0.003 (T) Density Flux -0.008 -0.004 -0.010 0 60 120 180 240 300 360 0 60 120 180 240 300 360 Electrical Angle (Deg) Electrical Angle (Deg) (a) (b)
Figure 11. Comparison of airgap flux density: (a) in the d‐axis; (b) in the q‐axis. Figure 11. Comparison of airgap flux density: (a) in the d-axis; (b) in the q-axis.
Table 3. The inductances of the two methods. 4. Conclusions Item Proposed Method FEM Unit In this paper, a novel analytical method of d- and q-axis inductance calculation in a no-load D‐axis inductance 0.524 0.513 mH condition is proposed and applied to IPM motors with V-shaped rotors. Taking bridge saturation into Q‐axis inductance 1.267 1.271 mH consideration, the rotor is divided into two pieces: saturation regions with a constant permeability and non-saturation regions with infinite permeability. The LPMCM and iterative method effectively solve 4. Conclusions the saturation and nonlinearity of the bridge. The rotor magnetic potentials in the d- and q-axis are investigatedIn this paper, according a novel to the analytical flux paths. method Based onof thed‐ and WFT, q‐ theaxis influence inductance of magnetic calculation field in distribution a no‐load conditionon inductance is proposed is considered. and applied The analytical to IPM resultsmotors of with bridge V‐shaped permeability, rotors. armature-reaction Taking bridge saturation magnetic intofield andconsideration,d- and q-axis the inductances rotor is divided match wellinto with two the pieces: ones obtainedsaturation by regions FEM, which with confirms a constant the permeabilityvalidity of the and proposed non‐saturation model. regions with infinite permeability. The LPMCM and iterative method effectively solve the saturation and nonlinearity of the bridge. The rotor magnetic potentials inAcknowledgments: the d‐ and q‐axisThis are investigated study was carried according out as to a partthe flux of the paths. industrial Based application on the WFT, technology the influence in electric of vehicles and supported by the National Natural Science Foundation of China (51477032) and Self-Planned Task magnetic(NO. SKLRS201608B) field distribution of the State on inductance Key Laboratory is considered. of Robotics andThe System analytical (HIT). results of bridge permeability, armature‐reaction magnetic field and d‐ and q‐axis inductances match well with the ones obtained by Author Contributions: This paper is the results of the hard work of all authors. Peixin Liang, Yulong Pei and FEM,Feng Chai which conceived confirms and the designed validity the of proposed the proposed method. model. Kui Zhao built the FEM models and analyze the data. Peixin Liang wrote the paper. All authors gave advices for the manuscripts. Acknowledgments:Conflicts of Interest: ThisThe authorsstudy was declare carried no conflictout as a of part interest. of the industrial application technology in electric vehicles and supported by the National Natural Science Foundation of China (51477032) and Self‐Planned Task (NO.References SKLRS201608B) of the State Key Laboratory of Robotics and System (HIT). Author Contributions: This paper is the results of the hard work of all authors. Peixin Liang, Yulong Pei and 1. Wang, Y.; Niu, S.; Fu, W. Electromagnetic Performance Analysis of Novel Flux-Regulatable Permanent Feng Chai conceived and designed the proposed method. Kui Zhao built the FEM models and analyze the data. Magnet Machines for Wide Constant-Power Speed Range Operation. Energies 2015, 8, 13971–13984. Peixin Liang wrote the paper. All authors gave advices for the manuscripts. 2. Wang, K.; Zhu, Z.Q.; Ombach, G. Torque Enhancement of Surface-Mounted Permanent Magnet Machine ConflictsUsing of Third-Order Interest: The Harmonic. authors declareIEEE Trans.no conflict Magn. of2014 interest., 50, 104–113. [CrossRef] 3. Li, Y.; Zhao, J.; Chen, Z.; Liu, X. Investigation of a Five-Phase Dual-Rotor Permanent Magnet Synchronous ReferencesMotor Used for Electric Vehicles. Energies 2014, 7, 3955–3984. 4. Zheng, P.; Wu, F.; Lei, Y.; Sui, Y.; Yu, B. Investigation of a Novel 24-Slot/14-Pole Six-Phase Fault-Tolerant 1. Wang, Y.; Niu, S.; Fu, W. Electromagnetic Performance Analysis of Novel Flux‐Regulatable Permanent Modular Permanent-Magnet In-Wheel Motor for Electric Vehicles. Energies 2013, 6, 4980–5002. Magnet Machines for Wide Constant‐Power Speed Range Operation. Energies 2015, 8, 13971–13984. 5. Zhao, J.; Li, B.; Gu, Z. Research on an Axial Flux PMSM with Radially Sliding Permanent Magnets. Energies 2. Wang, K.; Zhu, Z.Q.; Ombach, G. Torque Enhancement of Surface‐Mounted Permanent Magnet Machine 2015, 8, 1663–1684. Using Third‐Order Harmonic. IEEE Trans. Magn. 2014, 50, 104–113. 6. Wu, X.; Wang, H.; Huang, S.; Huang, K.; Wang, L. Sensorless Speed Control with Initial Rotor Position 3. Li, Y.; Zhao, J.; Chen, Z.; Liu, X. Investigation of a Five‐Phase Dual‐Rotor Permanent Magnet Synchronous Estimation for Surface Mounted Permanent Magnet Synchronous Motor Drive in Electric Vehicles. Energies Motor Used for Electric Vehicles. Energies 2014, 7, 3955–3984. 2015, 8, 11030–11046. 7. Chai, F.; Liang, P.; Pei, Y.; Cheng, S. Analytical Method for Iron Losses Reduction in Interior Permanent
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