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And Q-Axis Inductance for Interior Permanent Magnet Motors Based on Winding Function Theory

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And Q-Axis Inductance for Interior Permanent Magnet Motors Based on Winding Function Theory 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 isa instancedis the magnet to verify pole-arc the proposed to pole-pitch method. ratio, b 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.
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