Analytical Model of Slotted Air-Gap Surface Mounted Permanent-Magnet

The Neutral Grounding Resistor Sizing Using an Analytical Method Based on Nonlinear Transformer Model for Inrush Current MitigationGholamabas M.H.Hajivar Shahid Chamran University,Ahvaz, Iranhajivar@S.S.MortazaviShahid Chamran University,Ahvaz, IranMortazavi_s@scu.ac.irMohsen SanieiShahid Chamran University,Ahvaz, IranMohsen.saniei@Abstract-It was found that a neutral resistor together with 'simultaneous' switching didn't have any effect on either the magnitudes or the time constant of inrush currents. The pre-insertion resistors were recommended as the most effective means of controlling inrush currents. Through simulations, it was found that the neutral resistor had little effect on reducing the inrush current peak or even the rate of decay as compared to the cases without a neutral resistor. The use of neutral impedances was concluded to be ineffective compared to the use of pre-insertion resistors. This finding was explained by the low neutral current value as compared to that of high phase currents during inrush. The inrush currents could be mitigated by using a neutral resistor when sequential switching is implemented. From the sequential energizing scheme performance, the neutral resistor size plays the significant role in the scheme effectiveness. Through simulation, it was found that a few ohms neutral grounding resistor can effectively achieve inrush currents reduction. If the neutral resistor is directly selected to minimize the peak of the actual inrush current, a much lower resistor value could be found.This paper presents an analytical method to select optimal neutral grounding resistor for mitigation of inrush current. In this method nonlinearity and core loss of the transformer has been modeled and derived analytical equations.Index Terms--Inrush current, neutral grounding resistor, transformerI.I NTRODUCTIONThe energizing of transformers produces high inrush currents. The nature of inrush currents have rich in harmonics coupled with relatively a long duration, leads to adverse effects on the residual life of the transformer, malfunction of the protection system [1] and power quality [2]. In the power-system industry, two different strategies have been implemented to tackle the problem of transformer inrush currents. The first strategy focuses on adapting to the effects of inrush currents by desensitizing the protection elements. Other approaches go further by 'over-sizing' the magnetic core to achieve higher saturation flux levels. These partial countermeasures impose downgrades on the system's operational reliability, considerable increases unit cost, high mechanical stresses on the transformer and lead to a lower power quality. The second strategy focuses on reducing the inrush current magnitude itself during the energizing process. Minimizing the inrush current will extend the transformer's lifetime and increase the reliability of operation and lower maintenance and down-time costs. Meanwhile, the problem of protection-system malfunction is eliminated during transformer energizing. The available inrush current mitigation consist "closing resistor"[3], "control closing of circuit breaker"[4],[5], "reduction of residual flux"[6], "neutral resistor with sequential switching"[7],[8],[9].The sequential energizing technique presents inrush-reduction scheme due to transformer energizing. This scheme involves the sequential energizing of the three phases transformer together with the insertion of a properly sized resistor at the neutral point of the transformer energizing side [7] ,[8],[9] (Fig. 1).The neutral resistor based scheme acts to minimize the induced voltage across the energized windings during sequential switching of each phase and, hence, minimizes the integral of the applied voltage across the windings.The scheme has the main advantage of being a simpler, more reliable and more cost effective than the synchronous switching and pre-insertion resistor schemes. The scheme has no requirements for the speed of the circuit breaker or the determination of the residual flux. Sequential switching of the three phases can be implemented through either introducing a mechanical delay between each pole in the case of three phase breakers or simply through adjusting the breaker trip-coil time delay for single pole breakers.A further study of the scheme revealed that a much lower resistor size is equally effective. The steady-state theory developed for neutral resistor sizing [8] is unable to explain this phenomenon. This phenomenon must be understood using transient analysis.Fig. 1. The sequential phase energizing schemeUPEC201031st Aug - 3rd Sept 2010The rise of neutral voltage is the main limitation of the scheme. Two methods present to control the neutral voltage rise: the use of surge arrestors and saturated reactors connected to the neutral point. The use of surge arresters was found to be more effective in overcoming the neutral voltage rise limitation [9].The main objective of this paper is to derive an analytical relationship between the peak of the inrush current and the size of the resistor. This paper presents a robust analytical study of the transformer energizing phenomenon. The results reveal a good deal of information on inrush currents and the characteristics of the sequential energizing scheme.II. SCHEME PERFORMANCESince the scheme adopts sequential switching, each switching stage can be investigated separately. For first-phase switching, the scheme's performance is straightforward. The neutral resistor is in series with the energized phase and this resistor's effect is similar to a pre-insertion resistor.The second- phase energizing is one of the most difficult to analyze. Fortunately, from simulation studies, it was found that the inrush current due to second-phase energizing is lower than that due to first-phase energizing for the same value of n R [9]. This result is true for the region where the inrush current of the first-phase is decreasing rapidly as n R increases. As a result, when developing a neutral-resistor-sizing criterion, the focus should be directed towards the analysis of the first-phase energizing.III. A NALYSIS OF F IRST -P HASE E NERGIZING The following analysis focuses on deriving an inrush current waveform expression covering both the unsaturatedand saturated modes of operation respectively. The presented analysis is based on a single saturated core element, but is suitable for analytical modelling of the single-phase transformers and for the single-phase switching of three-phase transformers. As shown in Fig. 2, the transformer's energized phase was modeled as a two segmented saturated magnetizing inductance in series with the transformer's winding resistance, leakage inductance and neutral resistance. The iron core non-l inear inductance as function of the operating flux linkages is represented as a linear inductor inunsaturated ‘‘m l ’’ and saturated ‘‘s l ’’ modes of operation respectively. (a)(b)Fig. 2. (a) Transformer electrical equivalent circuit (per-phase) referred to the primary side. (b) Simplified, two slope saturation curve.For the first-phase switching stage, the equivalent circuit represented in Fig. 2(a) can accurately represent behaviour of the transformer for any connection or core type by using only the positive sequence Flux-Current characteristics. Based on the transformer connection and core structure type, the phases are coupled either through the electrical circuit (3 single phase units in Yg-D connection) or through the Magnetic circuit (Core type transformers with Yg-Y connection) or through both, (the condition of Yg-D connection in an E-Core or a multi limb transformer). The coupling introduced between the windings will result in flux flowing through the limbs or magnetic circuits of un-energized phases. For the sequential switching application, the magnetic coupling will result in an increased reluctance (decreased reactance) for zero sequence flux path if present. The approach presented here is based on deriving an analytical expression relating the amount of inrush current reduction directly to the neutral resistor size. Investigation in this field has been done and some formulas were given to predict the general wave shape or the maximum peak current.A. Expression for magnitude of inrush currentIn Fig. 2(a), p r and p l present the total primary side resistance and leakage reactance. c R shows the total transformer core loss. Secondary side resistance sp r and leakage reactance sp l as referred to primary side are also shown. P V and s V represent the primary and secondary phase to ground terminal voltages, respectively.During first phase energizing, the differential equation describing behaviour of the transformer with saturated ironcore can be written as follows:()())sin((2) (1)φω+⋅⋅=⋅+⋅+⋅+=+⋅+⋅+=t V (t)V dtdi di d λdt di l (t)i R r (t)V dt d λdt di l (t)i R r (t)V m P ll p pp n p P p p p n p PAs the rate of change of the flux linkages with magnetizing current dt d /λcan be represented as an inductance equal to the slope of the i −λcurve, (2) can be re-written as follows;()(3) )()()(dtdi L dt di l t i R r t V lcore p p P n p P ⋅+⋅+⋅+=λ (4) )()(L core l p c l i i R dtdi−⋅=⋅λ⎩⎨⎧==sml core L L di d L λλ)(s s λλλλ>≤The general solution of the differential equations (3),(4) has the following form;⎪⎩⎪⎨⎧>−⋅⋅+−⋅+−−⋅+≤−⋅⋅+−⋅+−⋅=(5) )sin(//)()( )sin(//)(s s 22222221211112121111λλψωττλλψωττt B t e A t t e i A t B t e A t e A t i s s pSubscripts 11,12 and 21,22 denote un-saturated and saturated operation respectively. The parameters given in the equation (5) are given by;() )(/12221σ⋅++⎟⎟⎠⎞⎜⎜⎝⎛⋅−++⋅=m p c p m n p c m m x x R x x R r R x V B()2222)(/1σ⋅++⎟⎟⎠⎞⎜⎜⎝⎛⋅−++⋅=s p c p s n p c s m x x R x x R r R x V B⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅−+++=⋅−−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−c p m n p m p c m R x x R r x x R x σφψ111tan tan ⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅−+++=⋅−−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−c p s n p s p c m R R r x x R x σφψ112tan tan )sin(111211ψ⋅=+B A A )sin(222221s t B A A ⋅−⋅=+ωψ mp n p m p m p m p c xx R r x x x x x x R ⋅⋅+⋅−⋅+−⋅+⋅⋅⋅=)(4)()(21211σστm p n p m p m p m p c xx R r x x x x x x R ⋅⋅+⋅−⋅++⋅+⋅⋅⋅=)(4)()(21212σστ s p n p s p s p s p xx R r x x x x x x c R ⋅⋅+⋅−⋅+−⋅+⋅⋅⋅=)(4)()(21221σστ sp n p s p s p sp c xx R r x x x x x x R ⋅⋅+⋅−⋅++⋅+⋅⋅⋅=)(4)()(21222σστ ⎟⎟⎠⎞⎜⎜⎝⎛−⋅==s rs s ri i λλλ10 cnp R R r ++=1σ21221112 , ττττ>>>>⇒>>c R , 012≈A , 022≈A According to equation (5), the required inrush waveform assuming two-part segmented i −λcurve can be calculated for two separate un-saturated and saturated regions. For thefirst unsaturated mode, the current can be directly calculated from the first equation for all flux linkage values below the saturation level. After saturation is reached, the current waveform will follow the second given expression for fluxlinkage values above the saturation level. The saturation time s t can be found at the time when the current reaches the saturation current level s i .Where m λ,r λ,m V and ωare the nominal peak flux linkage, residual flux linkage, peak supply voltage and angular frequency, respectivelyThe inrush current waveform peak will essentially exist during saturation mode of operation. The focus should be concentrated on the second current waveform equation describing saturated operation mode, equation (5). The expression of inrush current peak could be directly evaluated when both saturation time s t and peak time of the inrush current waveform peak t t =are known [9].(10))( (9) )(2/)(222222121//)()(2B eA t e i A peak peak t s t s n peak n n peak R I R R t +−⋅+−−⋅+=+=ττωψπThe peak time peak t at which the inrush current will reachits peak can be numerically found through setting the derivative of equation (10) with respect to time equal to zero at peak t t =.()(11) )sin(/)(022222221212221/ψωωττττ−⋅⋅⋅−−−⋅+−=+−⋅peak t s t B A t te A i peak s peakeThe inrush waveform consists of exponentially decaying'DC' term and a sinusoidal 'AC' term. Both DC and AC amplitudes are significantly reduced with the increase of the available series impedance. The inrush waveform, neglecting the relatively small saturating current s i ,12A and 22A when extremely high could be normalized with respect to theamplitude of the sinusoidal term as follows; (12) )sin(/)()(2221221⎥⎦⎤⎢⎣⎡−⋅+−−⋅⋅=ψωτt t t e B A B t i s p(13) )sin(/)()sin()( 22221⎥⎦⎤⎢⎣⎡−⋅+−−⋅⋅−⋅=ψωτωψt t t e t B t i s s p ))(sin()( 2s n n t R R K ⋅−=ωψ (14) ωλλλφλφωλλφωmm m r s s t r m s mV t dt t V dtd t V V s=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−+−⋅=+⋅+⋅⋅==+⋅⋅=−∫(8) 1cos 1(7))sin((6))sin(10The factor )(n R K depends on transformer saturation characteristics (s λand r λ) and other parameters during saturation.Typical saturation and residual flux magnitudes for power transformers are in the range[9]; .).(35.1.).(2.1u p u p s <<λ and .).(9.0.).(7.0u p r u p <<λIt can be easily shown that with increased damping 'resistance' in the circuit, where the circuit phase angle 2ψhas lower values than the saturation angle s t ⋅ω, the exponential term is negative resulting in an inrush magnitude that is lowerthan the sinusoidal term amplitude.B. Neutral Grounding Resistor SizingBased on (10), the inrush current peak expression, it is now possible to select a neutral resistor size that can achieve a specific inrush current reduction ratio )(n R α given by:(15) )0(/)()(==n peak n peak n R I R I R α For the maximum inrush current condition (0=n R ), the total energized phase system impedance ratio X/R is high and accordingly, the damping of the exponential term in equation (10) during the first cycle can be neglected; [][](16))0(1)0()0(2212=⋅++⎥⎦⎤⎢⎣⎡⋅−+===⎟⎟⎠⎞⎜⎜⎝⎛+⋅⋅n s p c p s pR x n m n peak R x x R x x r R K V R I c s σ High n R values leading to considerable inrush current reduction will result in low X / R ratios. It is clear from (14) that X / R ratios equal to or less than 1 ensure negative DC component factor ')(n R K ' and hence the exponential term shown in (10) can be conservatively neglected. Accordingly, (10) can be re-written as follows;()[](17) )()(22122n s p c p s n p R x m n n peak R x x R x x R r V R B R I c s σ⋅++⎥⎦⎤⎢⎣⎡⋅−+=≈⎟⎟⎠⎞⎜⎜⎝⎛+⋅Using (16) and (17) to evaluate (15), the neutral resistorsize which corresponds to a specific reduction ratio can be given by;[][][](18) )0()(1)0( 12222=⋅++⋅−⋅++⋅−+⋅+=⎥⎥⎦⎤⎢⎢⎣⎡⎥⎥⎦⎤⎢⎢⎣⎡=n s p c p s p n s p c p s n p n R x x R x x r R x x R x x R r R K σσα Very high c R values leading to low transformer core loss, it can be re-written equation (18) as follows [9]; [][][][](19) 1)0(12222s p p s p n p n x x r x x R r R K +++++⋅+==α Equations (18) and (19) reveal that transformers require higher neutral resistor value to achieve the desired inrush current reduction rate. IV. A NALYSIS OF SECOND-P HASE E NERGIZING It is obvious that the analysis of the electric and magnetic circuit behavior during second phase switching will be sufficiently more complex than that for first phase switching.Transformer behaviour during second phase switching was served to vary with respect to connection and core structure type. However, a general behaviour trend exists within lowneutral resistor values where the scheme can effectively limitinrush current magnitude. For cases with delta winding or multi-limb core structure, the second phase inrush current is lower than that during first phase switching. Single phase units connected in star/star have a different performance as both first and second stage inrush currents has almost the same magnitude until a maximum reduction rate of about80% is achieved. V. NEUTRAL VOLTAGE RISEThe peak neutral voltage will reach values up to peak phasevoltage where the neutral resistor value is increased. Typicalneutral voltage peak profile against neutral resistor size is shown in Fig. 6- Fig. 8, for the 225 KVA transformer during 1st and 2nd phase switching. A del ay of 40 (ms) between each switching stage has been considered. VI. S IMULATION A 225 KVA, 2400V/600V, 50 Hz three phase transformer connected in star-star are used for the simulation study. The number of turns per phase primary (2400V) winding is 128=P N and )(01.0pu R R s P ==, )(05.0pu X X s P ==,active power losses in iron core=4.5 KW, average length and section of core limbs (L1=1.3462(m), A1=0.01155192)(2m ), average length and section of yokes (L2=0.5334(m),A2=0.01155192)(2m ), average length and section of air pathfor zero sequence flux return (L0=0.0127(m),A0=0.01155192)(2m ), three phase voltage for fluxinitialization=1 (pu) and B-H characteristic of iron core is inaccordance with Fig.3. A MATLAB program was prepared for the simulation study. Simulation results are shown in Fig.4-Fig.8.Fig. 3.B-H characteristic iron coreFig.4. Inrush current )(0Ω=n RFig.5. Inrush current )(5Ω=n RFig.6. Inrush current )(50Ω=n RFig.7. Maximum neutral voltage )(50Ω=n RFig.8. Maximum neutral voltage ).(5Ω=n RFig.9. Maximum inrush current in (pu), Maximum neutral voltage in (pu), Duration of the inrush current in (s)VII. ConclusionsIn this paper, Based on the sequential switching, presents an analytical method to select optimal neutral grounding resistor for transformer inrush current mitigation. In this method, complete transformer model, including core loss and nonlinearity core specification, has been used. It was shown that high reduction in inrush currents among the three phases can be achieved by using a neutral resistor .Other work presented in this paper also addressed the scheme's main practical limitation: the permissible rise of neutral voltage.VIII.R EFERENCES[1] Hanli Weng, Xiangning Lin "Studies on the UnusualMaloperation of Transformer Differential Protection During the Nonlinear Load Switch-In",IEEE Transaction on Power Delivery, vol. 24, no.4, october 2009.[2] Westinghouse Electric Corporation, Electric Transmissionand Distribution Reference Book, 4th ed. East Pittsburgh, PA, 1964.[3] K.P.Basu, Stella Morris"Reduction of Magnetizing inrushcurrent in traction transformer", DRPT2008 6-9 April 2008 Nanjing China.[4] J.H.Brunke, K.J.Frohlich “Elimination of TransformerInrush Currents by Controlled Switching-Part I: Theoretical Considerations” IEEE Trans. On Power Delivery, Vol.16,No.2,2001. [5] R. Apolonio,J.C.de Oliveira,H.S.Bronzeado,A.B.deVasconcellos,"Transformer Controlled Switching:a strategy proposal and laboratory validation",IEEE 2004, 11th International Conference on Harmonics and Quality of Power.[6] E. Andersen, S. Bereneryd and S. Lindahl, "SynchronousEnergizing of Shunt Reactors and Shunt Capacitors," OGRE paper 13-12, pp 1-6, September 1988.[7] Y. Cui, S. G. Abdulsalam, S. Chen, and W. Xu, “Asequential phase energizing method for transformer inrush current reduction—part I: Simulation and experimental results,” IEEE Trans. Power Del., vol. 20, no. 2, pt. 1, pp. 943–949, Apr. 2005.[8] W. Xu, S. G. Abdulsalam, Y. Cui, S. Liu, and X. Liu, “Asequential phase energizing method for transformer inrush current reduction—part II: Theoretical analysis and design guide,” IEEE Trans. Power Del., vol. 20, no. 2, pt. 1, pp. 950–957, Apr. 2005.[9] S.G. Abdulsalam and W. Xu "A Sequential PhaseEnergization Method for Transformer Inrush current Reduction-Transient Performance and Practical considerations", IEEE Transactions on Power Delivery,vol. 22, No.1, pp. 208-216,Jan. 2007.。

第38卷第3期计算机仿真2021年3月文章编号：1006 - 9348 (2021)03 - 0190 - 04高频低压平面变压器磁芯气隙的研究王星，程志江，孟德炀，翁雄亮(新疆大学电气工程学院，新疆乌鲁木齐830047)摘要:变压器时常发生磁饱和现象。

为防止磁饱和的发生，通常在磁路中加人一段气隙或减少绕组匝数。

研究在磁回路中 加人气隙来避免磁饱和时，气隙量对高频平面变压器特性参数的影响。

以TDK铁氧体设计的髙频平面变压器为研究对象，通过理论分析计算出饱和电流和气隙量。

在ANSYS中建立3D仿真模型和电路简化模型，改变气隙量，分析高频平面变压 器的涡流场、静电场特性以及原副边电压特性，将仿真数据用MATLAB处理后，得出其电气参数变化规律。

关键词：髙频平面变压器;涡流场;静电场;磁芯气隙;磁饱和中图分类号:TP391.9 文献标识码:BResearch on Air Gap of Magnetic Core of Low Voltage H P TWANG Xing,CHENG Zhi - Jiang,MENG De - Yang, WENG Xiong - Liang(College of Electrical Engineering,Xinjiang University,WulumuqiXinjiang830047,China) ABSTRACT:Magnetic saturation often occurs in transformers.In order to prevent the occurrence of it,an air gap is usually added to the magnetic circuit or the number of winding turns is reduced.This paper studied the effect of air gap on the characteristic parameters of high frequency planar transformer when air gap was added to the magnetic cir­cuit to avoid magnetic saturation.Taking the high frequency planar transformer designed by TDK ferrite as the re­search object,the saturation current and air gap were calculated through theoretical analysis.In ANSYS,a3D simu­lation model and a simplified circuit model were established to change the air gap and analyze the eddy current field, electrostatic field characteristics and the original and secondary voltage characteristics of high frequency planar trans­former.After processing the simulation data with MATLAB,the variation law of electrical parameters was obtained.KEYW ORDS：High frequency planar transformer；Eddy current field；electrostatic field；Magnetic core gap；Magnetic saturationi引言在磁化曲线（B-H曲线）中，当磁场强度（H)达到某一 值时，磁感应强度（B)就不再随磁场强度的增加而增加了，这种现象就叫做磁饱和现象。

冰雹动态本构建模与验证作者：王计真来源：《航空科学技术》2023年第08期摘要：飞机飞行过程中，机身外表容易遭受冰雹撞击，造成机体破坏，危害乘客安全。

为研究结构抗冰雹冲击性能，本文基于拉格朗日（Lagrange）方法、耦合欧拉-拉格朗日方法（CEL）和光滑粒子动力学（SPH）方法，建立冰雹动态本构模型，并根据冰撞刚性靶试验数据，修正冰雹模型参数，验证三种方法的有效性。

数值计算结果表明，三种方法均能较好地描述冰雹动态破碎行为，且在中低速冰撞分析时，基于应变率强化效应和张力失效准则的Lagrange方法与试验有更好的一致性。

关键词：冰雹撞击；动态本构；冰撞刚性靶；应变率强化；张力失效中图分类号：TG146.23 文献标识码：A DOI：10.19452/j.issn1007-5453.2023.08.007基金项目：航空科学基金（2016ZA23005）民机在飞行或起降过程中，容易遭受冰雹的冲击威胁。

冰雹冲击过程属于小质量高速冲击，对飞机结构尤其是复合材料结构安全产生重大影响，严重威胁乘员的生命安全，带来严重的经济损失。

飞机结构抗冰雹撞击研究具有重要理论与工程意义。

冰有20余种结构形式（晶体结构或非晶体状态），因此，构建一种普适材料模型描述其动态行为极为困难，且冰雹撞击分析仅在航空、轨道交通和风电等少数行业存在需求，因而针对冰雹动态本构研究有限。

20世纪70年代，Haynes[1]首次实现了冰的静态及准静态压缩性能测试。

随后，Schulson等[2-3]和Dempsey等[4]研究了不同温度和晶体结构下冰的静态力学性能和裂纹扩展属性。

研究结果表明，冰雹的拉伸强度远小于压缩强度，且静态压缩过程表现出韧性破坏特性。

近年来，冰雹动态本构的研究逐步开展，表明冰为典型的应变率敏感材料。

Botto[5]开展了低应变率下（10？8 ～10？3s？1）冰结构的动态力学性能测试，观察到随应变率增加材料特性由韧性向脆性转变。

CCARPG01MODEL RPGH - HEAVY-DUTY SEALED HOUSINGThese heavy duty units feature a heavy cast aluminum housing with 1/4"thick aluminum cover plates and 0-ring seals. Heavy duty bearings are double-sealed and allow radial shaft loading of 30 lbs (13.6 Kg) and axial loading of 10 lbs (4.5 Kg). Starting Torque is 1 oz-in (7.06 N-mm). Weight of the RPGH is 3.8 lbs (1.7 Kg). Maximum operating speed is 6000 RPM.DIMENSIONS In inches (mm)A 1/2" (12.7 mm) NPT Conduit entry permits signal wiring to be run via flex-conduit to an internal terminal block. Electrical characteristics are identical to those for the Model RPGB. Terminal board markings correspond to the Pin-Out identification of the RPGB.!CURRENT SINK OUTPUTS!HIGH PULSE PER REVOLUTION (PPR) RATESUp to 1200 PPR for fine, high-resolution counting or precision speed measurement from slow shaft speeds.!QUADRATURE OUTPUTFor position measurement, bi-directional counting and in systems with backlash counting requirements.!EASY INSTALLATIONEliminates air-gap, sensing distance, and beam alignment procedures of other types of sensing.!IDEAL FOR DUSTY, DIRTY ENVIRONMENTSWhere “Non Contact” sensing means are impractical.ROTARY PULSE GENERATORS (RPG’S)TRANSFORM SHAFT ROTATION TO COUNT PULSE TRAINS FORCOUNTERS, TACHOMETERS, MOTION MONITORS & CONTROLSBulletin No. RPGB/H-N Drawing No. LP0202Released 4/03RPG’s contain an L.E.D. light source and a photo sensor that scans a shaft-mounted, slotted disc. An internal pulse-shaping amplifier circuit delivers a rectangular pulse signal from the current sinking output in response to the passing slots as it rotates. RPG’s can be direct-coupled to a machine shaft by means of a flexible-bellows, spring, or rubber sleeve type coupling that allowsfor axial and radial misalignment. They can also be coupled with light instrument timing-belts. Timing-belt drives also allow convenient gear-up or gear-down speed ratio changes that can be useful for obtaining non-standard PPR rates.ORDERING INFORMATIONTel +1 (717) 767-6511Fax +1 (717) RPGFC003RPGFC002470120047010004700600----12001000600Flexible Coupling (1" Length) 0.375"-0.375"Flexible Coupling (1" Length) 0.250"-0.375"Single Channel Type “0”RPGFCRPGHPART NUMBERPPR*DESCRIPTIONMODEL NO.* Only Stock PPR’s listed, other PPR’s as well as Quad. (Type “1”) output available on special order.ABOUT QUADRATURE OUTPUTA quadrature output consists of two pulse trains, one of which is 90°(electrical) out of phase with the other. By virtue of this arrangement the direction of motion can be determined. Referring to the following waveform,when the CNT output (Pin C) goes from low to high while the Quadrature output (Pin D) is low, the motion is counter-clockwise.MODEL RPGB - UP TO 1200PPR, WITH SINGLE CHANNEL OR QUADRATURE OUTPUT FOR GENERAL INDUSTRIAL SERVICE DIMENSIONS In Inches (mm)SPECIFICATIONS1. SUPPLY VOLTAGE:+5 to +28 VDC2. OUTPUT:Current SinkingType “0” Single Channel : 250 mA max.Type “1” Quadrature : 250 mA max. per output (Quad. Phase relationship is 90° ±36°)Note: NPN Transistor outputs have 1.5 K Ωload resistors returned to supply for internal feed back purposes. This does not interfere with the ability to use these outputs as conventional “Open-Collector” outputs as long as the supply voltage for the RPGB is supplied by the indicator or control receiving its output signal. The RPGB’s internal load resistor also allows the output to be used as a current source, however, load current must be limited to 1 mA max.3. MAXIMUM SHAFT SPEED:6000 RPM4. MAXIMUM PULSE RATE:Type “0” Single Channel : 20 KHz Type “1” Quadrature : 20 KHzPPR available up to 1270 for both type “1” and “0”.5.HOUSING : Black non-corrosive finished 6063-T6 aluminum.6.BEARINGS :ABEC3 double sealed ball bearings7.RADIAL SHAFT LOAD : 20 lbs. operating (9 kg)8.AXIAL SHAFT LOAD : 10 lbs operating (4.5 kg)9.STARTING TORQUE : 0.38 oz-in (2.68 N-mm)10.MOMENT OF INERTIA : 2.5 x 10-3oz-in-sec 2(1.77 x 10-2N-mm-sec 2)11. CONNECTIONS:Mating 6-pin MS connector #14S-6P-6 or cable/connector assembly (4-wire shielded), 10´ (3.05 m),25´ (7.62 m), or 50´ (15.24 m) long,must be ordered separately. Consult factory for special lengths.12. OPERATING TEMPERATURE:0° to 70°C.13. WEIGHT:10 oz (283.5 g)ORDERING INFORMATION* Other PPR’s available on special order, only stock PPR’s listed.** For quadrature PPR’s above 600, the Gemini Series or a Model BDMD can be used to double or quadruple the effective PPR’s listed, (See the Accesory section of the Catalog.)When the desired output of an RPG and wheel combination is either in feet or inch units, selection of the proper combination is relatively straight forward.For example, with a 1-foot wheel circumference, a 1 PPR Rotary Pulse Generator will deliver 1 pulse/ft, 12 PPR would deliver 12 pulses/ft (1pulse/inch);100 PPR would yield 100 pulses/ft; and 120 PPR would permit measuring to 1/10th of an inch (1/120th of a foot).Measuring in yards or meters, however, is a bit more involved since a 1-yard or 1-meter circumference wheel would be prohibitively large. Instead, 4/10 yard and 4/10 meter wheels can be used in conjunction with RPGB.SELECTING APPROPRIATE WHEEL SIZE & PPR (Pulses Per Rev.) OF ROTARY PULSE GENERATORLENGTH SENSOR CONVERSION BRACKET (P/N LSCB1000)ADAPTS RPGB ROTARY PULSE GENERATOR TO LENGTH MEASUREMENTFOR BI-DIRECTIONAL MOTION APPLICATIONS REQUIRING FOR FINE RESOLUTION, HIGH-PULSE-RATE APPLICATIONSThis conversion bracket allows the customer to assemble a custom length (as shown).Apply thread locking material to The tubular arm length of this bracket, related to the wheel axis center-line of the RPGB is 6.8" similar to the LSQ. The 10´ long, 4-wire, shielded cable (included with conversion bracket)has the same color coding as described for the RPGB cable P/N CCARPG01. Screws for mounting the conversion bracket to the RPGB are included.LENGTH SENSOR MEASUREMENT ACCURACYFactors which affect measurement accuracy include Measuring Wheel accuracy and wear, and material conditions. Ideally, materials which are hard,thin and strong provide good readings, conversely, soft, thick and elastic materials can present problems in obtaining true readings. The great majority of these situations, where this effect is consistant, can be compensated for by applying a multiplier to the quadrature output pulse train so as to obtain aLENGTH SENSOR ACCESSORIESSEPARATE LENGTH MEASURING WHEELS - DIMENSIONS In Inches (mm)MODEL LSAHC - LENGTH SENSOR HINGE CLAMP ASSEMBLYNote: After installation of measuring wheels, ensure guards,shields or other devices are in place to protect personnel from rotating equipment.WHEELS & REPLACEMENT TIRES FOR CODE OR WHEELSRed Lion Controls 20 Willow Springs Circle York PA 17402Tel +1 (717) 767-6511Fax +1 (717) 764-0839Red Lion Controls Asia31, Kaki Bukit Road 3 #06-04/05 TechLinkSingapore 417818Tel +65 6744-6613Fax +65 6743-3360Red Lion Controls BVBasicweg 11b NL - 3821 BR Amersfoort Tel +31 (0) 334 723 225Fax +31 (0) 334 893 793CCARPG01。

飞机机身结构的模态分析与优化设计随着民用航空业的飞速发展，航空器的结构设计也得到了极大的改善。

飞机机身结构作为飞机重要的组成部分，其优化设计与模态分析对于飞机的安全性、舒适度、减少疲劳损伤以及航空器加速度降低等方面都有极为重要的影响。

因此，这篇文章将介绍飞机机身结构的模态分析与优化设计，以促进航空器的发展。

一、机身结构的模态分析在机身结构设计中，模态分析是非常重要的步骤。

模态分析是指对一种结构在一定的边界条件和外荷载作用下，研究其自由振动频率、振型以及对外部激励的响应情况。

模态分析的结果可以用来指导设计工作和预测结构运行和安全。

1、有限元法在模态分析中，有限元法是一种广泛使用的方法。

它可以将结构离散化成各种复杂的形式，如单元板、单元梁、单元壳体等，用矩阵方法求解复杂结构的振动特性。

有限元法具有计算精度高、处理能力强和适用范围广等优点，在机身结构的模态分析中的使用也是十分广泛。

2、振型及频率分析模态分析时，振型及频率是求得的主要指标之一。

振型是指结构在自由振动时的振动状态。

在模态分析中，振型可以描述结构运动的特点，用于确定结构的刚度和几何形状，通过振型的分析可以了解结构的哪些部位较为关键，以便进行后续的优化设计。

频率是指结构在自由振动状态下所具有的振动周期。

在模态分析中，频率越高，表示结构越容易发生共振或者很容易出现破坏，因此，频率的分析为航空器的设计提供了参考和依据。

3、模态优化模态优化是指通过对机身结构进行振动模态分析，找到机身结构的主要振动模态和对应频率，从而进行优化设计。

模态优化设计可以减少机身结构共振的可能性，从而避免机身结构发生破坏，保证飞机安全飞行。

二、机身结构的优化设计机身结构的优化设计是对航空器机身设计的一个重要环节。

通过对机身结构的优化设计，可以提高航空器的性能和安全水平。

具体的优化设计包括如下方面。

1、结构的减重结构的减重是对机身结构的安全性能、效率和可靠性都有极高的要求。

在设计机身结构时，减轻重量可以增加载荷能力、降低阻力、减轻燃料消耗等。

Development of Analytical Equations to Calculate the Cogging Torquein Transverse Flux MachinesM. V. Ferreira da Luz (1), P. Dular (2), N. Sadowski (1), R. Carlson (1) and J. P. A. Bastos (1)(1) GRUCAD, Dept. of Electrical Engineering, Federal University of Santa Catarina, Brazil.(2) Dept. of Electrical Engineering and Computer Science, F.N.R.S., ULG, Belgium.Po. Box 476, 88040-900, Florianópolis, Santa Catarina, Brazil.E-mail of Corresponding Author: mauricio@grucad.ufsc.brAbstract - Cogging torque is produced in a permanent magnetmachine by magnetic attraction between the rotor permanentmagnets and the stator teeth. It is an undesirable effect thatcontributes to torque ripple, vibration and noise of the machine. In this paper, the resultant cogging torque values are computed using a three-dimensional (3D) finite element analysis. For this, the rotor movement is modeled by means of the moving bandtechnique in which a dynamic allocation of periodic or anti-periodic boundary conditions is performed. The 3D finite element method is the most accurate tool to carry out cogging torque. However, it does not easily allow a parametric study. For this reason, an analytical model was developed in order to predict the cogging torque. The tools are intended to be used for the study of transverse flux machines.I.I NTRODUCTIONAlthough permanent magnet (PM) machines are high performance devices, there are torque variations that affect their output performance. These variations during one revolution arise from factors as: commutation of the phase currents; ripple in the current waveform caused by chopping; variations in the reluctance of the magnetic circuit due to slotting as the rotor rotates. This last effect is called cogging [1]. Cogging torque arises from the interaction between permanent magnets and slotted iron structure and occurs in almost all types of PM motors. It manifests itself by the tendency of a rotor to align in a number of stable positions even when the machine is unexcited, and results in a pulsating torque, which does not contribute to the net effective torque. Therefore, one major task in developing PM machines is to minimize the cogging torque. Several methods are known. Some researchers minimize the cogging torque by skewing, an asymmetric distribution of the magnets or pole shifting [2]. Others works consider the relative air-gap permeance by modeling the shape of slots, the tooth width, or using teeth pairing, extra slots or notches in the teeth [3]. Others works control the function of the magnetization manipulating the shape of the magnets, the magnetization of the magnets themselves, the pole arc to pole pitch ratio, and the shape of the iron core [4].To verify the effects of machine geometry on the cogging torque is important to determinate its waveform. The electromagnetic torque can be calculated analytically or numerically in a variety of ways, such as Maxwell Stress and co-energy methods. However, they require very accurate global and local field solutions, particularly for the determination of cogging torque. In other words, a high level of mesh discretisation is required in a finite element method (FEM) calculation, whilst a reliable physical model is essential to an analytical prediction. A lot of work has been done on prediction of cogging torque in PM motors. They are divided into three groups. The first group uses analytical approaches [4, 5]. The second group uses the bi-dimensional (2D) and three-dimensional (3D) FEM simulation [6] and the third one uses a combined numerical and analytical method [7].In the last years we have developed a set of numerical tools for efficiently studying PM machines with FEM. A 3D magnetodynamic formulation, using the magnetic vector potential as the main unknown, discretized with edge finite elements, has been developed with adapted techniques for considering stranded conductors, periodicity and anti-peridodicity boundary conditions, moving band connection conditions and moving parts. The rotor displacement is modeled by means of a layer of finite elements placed in the air gap [8]. This method, named Moving Band Method, uses an automatic relocation of periodicity or anti-periodicity boundary conditions allowing the simulation of any displacement between stationary and moving parts of an electrical machine. The 3D FEM is the most accurate tool to carrying out cogging torque. However, it does not easily allow a parametric study. Moreover, the 3D simulation demands a high computation time. Hence, the purpose of this paper is to develop an analytical model and to compare it with 3D FEM for a transverse flux machine. This comparison allows finding an analytical model fast and precise to study the cogging torque behavior in order to satisfy some industrial design constraints for machines.The contribution of this paper could be divided in two aspects: the first one is the cogging torque calculation using the Moving Band Method for a 3D problem considering two moving bands in the same motor. The second aspect is the development of the analytical model to the transverse flux permanent magnet (TFPM) machine.TFPM machines have been found to be highly viable candidates in electric and hybrid propulsion applications [6]. Of particular interest are the double-sided topologies where high energy permanent magnets are mounted in the rotor rims in a flux concentration arrangement, yielding high air gap flux densities. The topology of such a machine requires 3D finiteelement analyses to accurately predict the machine parameters [6]. II. M AGNETODYNAMIC F ORMULATIONA bounded domain Ω of the two or three-dimensionalEuclidean space is considered. Its boundary is denoted Γ. Theequations characterising the magnetodynamic problem in Ω are[9]:j h = curl , b e t curl ∂−=, 0 div =b , (1a-b-c) r b h b +μ=, e j σ=, (2a-b)where h is the magnetic field, b is the magnetic flux density, b r is the permanent magnet remanent flux density, e is the electric field, j is the electric current density, including source currents j s in Ωs and eddy currents in Ωc (both Ωs and Ωc are included in Ω), μ is the magnetic permeability and σ is the electric conductivity.The boundary conditions are defined on complementary parts Γh and Γe , which can be non-connected, of Γ,0h =×Γh n , 0 . e =Γb n , 0e=×Γe n , (3a-b-c) where n is the unit normal vector exterior to Ω. Furthermore, global conditions on voltages or currents in inductors can be considered [8]. The a -formulation, with a magnetic vector potential a and an electric scalar potential v, is obtained from the weak form of the Ampère equation (1a) and (2a-b) [9], i.e.,0)' ,()' ,grad v ( )' , ( ',)' curl , ()' curl , url c (s h s c c t s r =−σ+∂σ+>×<+ν−νΩΩΩΓΩΩa j a a a a h n a b a a),(F 'a Ω∈∀awhere s h n × is a constraint on the magnetic field associated with boundary Γh of the domain Ω and μ=ν/1 is themagnetic reluctivity.F a (Ω) denotes the function space defined on Ω which contains the basis and test functions for both vector potentials a and a'. (. , .)Ω and <. , .>Γ denote a volume integral in Ω and a surface integral on Γ of products of scalar or vector fields.Using edge finite elements for a , a gauge condition associated with a tree of edges is generally applied.III. P ERIODICITY C ONDITIONS AND M OVING B AND M ETHOD Another important point is the simulation of the rotor movement. The applied technique permits the use of only one mesh for the calculation.Generally, to model electrical machines not presenting fractional windings, the calculation domain can be reduced to one or two poles using anti-periodic or periodic boundary conditions [9]. The discretisation of these boundaries is performed in a similar way, linking all their geometricalentities (nodes, edges and facets) by pairs. These boundaries are denoted ΓA and ΓB , respectively the reference boundary(which contains all the degrees of freedom) and its associatedboundary [8].For the a -formulation, periodicity conditions are split up into a strong relation on the normal component of b and a weakrelation on the tangential component of the magnetic field h .When edge finite elements are used for a , the strongperiodicity (anti-periodicity, with the other sign) relation for apair of equally oriented edges on ΓA and ΓB is a B = ± a A , (5) where a A and a B are the circulations of a along the considered edges on ΓA and ΓB . In 3D, periodicity conditions have to be consistent with gauge conditions (when used) associated with trees of edges [8].The periodicity boundary conditions can be directly applied to the moving band [8] connection (Fig. 1). The connection between the moving and the stationary regions (both being separately meshed), through the moving band, is similar to a periodicity connection (direct identification of the degrees offreedom; Fig. 1, boundaries b-b'). When (anti-) periodicity conditions are considered on both sides of the band (Fig. 1,boundaries a-a'), a complementary part of this band has to be connected through the same conditions to the moving region (Fig. 1, boundaries c-c') [8].Such connection conditions have to be updated for each position during the movement. When the calculation domain angle is exceeded, the moving part must be relocated in front of the stationary part, while inverting the connection conditions (i.e., inverting the rotor field sources) if anti-periodicity conditions are used.The movement is considered using the Lagrangian approach, i.e. with a moving coordinate system [10]. This approach is easily and implicitly considered with the a -formulation because no deformation is done in the domains involving the time derivative, i.e., in the conducting regions.IV. N UMERICAL P REDICTION OF C OGGING T ORQUE The cogging torque is computed at each angular position by means of 3D FEM analysis, integrating the Maxwell stress tensor on a surface containing the rotor, with null stator currents.To the aim of reducing the numerical errors, the cogging torque should be computed as the mean value of the Maxwellstress tensor on the whole airgap volume V g [9], i.e.∫∫∫∧=gV cogging dv )d (T F r , (6)where F is the Maxwell stress and the r is the dummy radius.V. A NALYTICAL P REDICTION OF C OGGING T ORQUE The cogging torque experienced by all estator teeth has the same shape, but are offset from each other in phase by the angular slot pitch [11]. The cogging torque experienced by the k th stator tooth can be written as the Fourier series()∑∞=ϕ+−+=θ1n n s n o ck )θn(θ2 cos T 2 T )(T , (7)where θ is the mechanical angular position of the rotor and ϕn is the phase angle of the k th harmonic component. T n are the Fourier series coefficients and they are determined by the magnetic field distribution around each tooth, the air gap length, and the size of the slot opening between teeth [11]. The method is based on the derivation of the flux density distributions in airgaps as a function of the machine design parameters. θs is the angular slot pitch calculated by sms N N π=θ, (8) where N m is the number of stator slots and N s is the number of magnet poles.Since the cogging torque of each tooth adds to create the net cogging torque of the motor, the motor cogging torque can be written as ∑−==θ1N 0k ck 2n cogging s )(θT S )(T , (9)where S 2n is the skew factor, which is given by ⎟⎟⎠⎞⎜⎜⎝⎛απαπ=s sk m sk m sn 2N N n sin N n N S , (10) where αsk is the slot pitches.In the analytical approach the assumptions used supposed that the end effects and the iron saturation are negligible.VI. R ESULTSThe analyzed TFPM machine as shown in Fig. 2 has 90 poles, a rated power of 10 kW, a rated voltage 220 V, and a rated speed of 200 rpm. This motor was manufactured by WEG Industries - Brazil. Fig. 3 shows a CAD model of the TFPM machine. Fig. 4 and Fig. 5 show the assembly details of the inner and outer stator for one phase of the TFPM machine. In this doubled-sided construction, the rotor is arranged between an inner and an outer stator.Figure 2. The TFPM machine manufactured by WEG Industries - Brazil.Figure 3. A CAD model of the TFPM machine.Figure 4. Assembly details of the inner and outer stator - one phase of theTFPM machine.Fig. 6 shows the ring-shaped windings of the TFPM machine.The Nd-Fe-B permanent magnets in the rotor are magnetized with an alternating polarity in circumferential direction. Therefore, the flux concentrating elements in the rotor increase the magnetic flux density in the airgaps beyond the remanent flux density of the Nd-Fe-B magnets. Fig. 7 shows the magnetic flux distribution due to the Nd-Fe-B magnets to the one-phase of TFPM machine.Figure 5. Assembly details of the inner stator - one phase of the TFPMmachine.Figure 6. Ring-shaped windings of the TFPM machine.Figure 7. Magnetic flux distribution to the one phase of the TFPM machine. The typical feature of TFPM machine is the magnetic flux path which has sections where the flux is transverse to the rotation plane and the ring-shaped winding in the stator in which the direction of the current corresponds to the movement direction of the rotor. This design leads to a structure in which the design of the magnetic circuit becomes almost independent from the design of the electrical circuit. Hence, there is the possibility to achieve higher torque values by increasing the number of pole pairs without affecting the electrical circuit parameters [12]. Also the absence of end-turns in stator winding which results in reduced copper losses is one of the major advantages of this machine structure.Considering the electromagnetic symmetries and using periodic boundary conditions, the smaller domain of study consists of an 8-degree sector of the whole structure. The 3D mesh without the air elements is shown in Fig. 8. In this figure we can see the stator, the coils, the rotor with the permanent magnets and the two moving bands (one inner and another external to the rotor). Each air gap was divided in three equal layers, being the moving band located in the central layer. Hexahedra in the moving band and prisms elsewhere have been used. The mesh of the structure has 40 divisions along the moving band.Figure 8. The studied domain and 3D mesh for TFPM machine. Results are presented for a speed of 200 rpm and when the machine operates at no-load condition, i.e. only the permanent magnet excitation is considered. Fig. 9 shows the cogging torque produced by both outer and inner parts of one phase.Figure 9. The cogging torque (normalized) produced by both outer and inner parts of one phase versus angle for TFPM machine.VII.C ONCLUSIONSIn this paper, the cogging torque was calculated with a 3D magnetodynamic formulation and with adapted techniques for considering stranded conductors, periodicity and anti-peridodicity boundary conditions, moving band connection conditions and moving parts. The Moving Band Method was implemented for 3D problems considering one or more moving bands in the same motor.The 3D FEM is the most accurate tool to carrying out cogging torque. However, it does not easily allow a parametric study. For this reason, an analytical model was developed in order to predict the cogging torque of TFPM machine. The comparison of the results between the analytical model and the 3D FEM simulation was satisfactory.Consequently, the developed analytical model allows fast and precise study of the influence of rotor permanent magnet distribution as well as the opening of stator auxiliary poles on the cogging torque behaviour in order to satisfy some industrial design constraints for machines. The skewing of the stator slots or, alternatively, of the permanent magnets also is taken into account with the analytical model.A CKNOWLEDGMENTThe authors thank the cooperation of the WEG Industries - Brazil. This work was supported by National Council for Scientific and Technological Development (CNPq) of Brazil.R EFERENCES[1] J. R. Hendershot Jr. and T. J. E. Miller, Design of Brushless Permanent-Magnet Motors, Magna Physics Publishing and Clarendon Press - Oxford, 1994.[2] N. Bianchi and S. Bolognani, “Design techniques for reducing thecogging torque in surface-mounted PM motors”, IEEE Transactions on Industry Applications, Vol. 38, No. 5, pp. 1259-1265, 2002.[3] R. Carlson, A. A. Tavares, J. P. A. Bastos and M. Lajoie-Mazenc.“Torque ripple attenuation in permanent magnet synchronous motors”.In: IEEE-IAS Annual Meeting, San Diego. Proceedings of IEEE-IAS, p.57-62, 1989.[4] S. M. Hwang, J. B. Eom, Y. H. Jung, D. W. Lee and B. S. Kang.“Various design techniques to reduce cogging torque by controlling energy variation in permanent magnet motors”. I EEE Transactions on Magnetics, Vol. 37, No. 4, pp. 2806-2809, 2001.[5] J. F. Gieras, “Analytical approach to cogging torque calculation of PMbrushless motors”. IEEE Transactions on Industry Applications, Vol. 40, No. 5, pp. 1310-1316, 2004.[6] E. Schmidt, “3-D Finite element analysis of the cogging torque of atransverse flux machine”. IEEE Transactions on Magnetics, Vol. 41. No.5, pp. 1836-1839, 2005.[7] C. Schlensok, M. H. Gracia and K. Hameyer, “Combined numerical andanalytical method for geometry optimization of a PM motor”. IEEE Transactions on Magnetics, Vol. 42, No. 4, pp. 1211-1214, 2006.[8] M. V. Ferreira da Luz, P. Dular, N. Sadowski, C. Geuzaine, J. P. A.Bastos, “Analysis of a permanent magnet generator with dual formulations using periodicity conditions and moving band”, IEEE Transactions on Magnetics, Vol. 38, No. 2, pp. 961-964, 2002.[9] J. P. A. Bastos and N. Sadowski, Electromagnetic Modeling by FiniteElements. Marcel Dekker, Inc, New York, USA, 2003.[10] K. Muramatsu, T. Nakata, N. Takahashi, and K. Fujiwara, “Comparisonof coordinate systems for eddy current analysis in moving conductors”, IEEE Transactions on Magnetics, Vol. 28, No. 2, pp. 1186-1189, 1992. [11] D. C. Halselman, Brushless Permanent Magnet Motor Design. SecondEdition, Published by The Writers’Collective, 2003.[12] M. Bork, G. Henneberger, “New transverse flux concept for an electricvehicle drive system”, ICEM 96 Proceedings, International Conference on Electrical Machines, Vol. 2, pp. 308-313, 1996.。

阿基米德蜗杆Archimedesworm安全系数safetyfactor factorofsafety安全载荷safeload凹面、凹度concavity扳手wrench板簧flatleafspring半圆键woodruffkey变形deformation摆杆oscillatingbar摆动从动件oscillatingfollower摆动从动件凸轮机构camwithoscillatingfollower 摆动导杆机构oscillatingguide-barmechanis m摆线齿轮cycloidalgear摆线齿形cycloidaltoothprofile摆线运动规律cycloidalmotion摆线针轮cycloidal-pinwheel包角angleofcontact保持架cage背对背安装back-to-backarrangement背锥backcone；normalcone背锥角backangle背锥距backconedistance比例尺scale比热容specificheatcapacity闭式链closedkinematicchain闭链机构closedchainmechanism臂部arm变频器frequencyconverters变频调速frequencycontrolofmotorspeed变速speedchange变速齿轮changegear changewheel变位齿轮modifiedgear变位系数modificationcoefficient标准齿轮standardgear标准直齿轮standardspurgear表面质量系数superficialmassfactor表面传热系数surfacecoefficientofheattransfer 表面粗糙度surfaceroughness并联式组合combinationinparallel并联机构parallelmechanism并联组合机构parallelcombinedmechanis m并行工程concurrentengineering并行设计concurreddesign,CD不平衡相位phaseangleofunbalance不平衡imbalanceorunbalance不平衡量amountofunbalance不完全齿轮机构intermittentgearing波发生器wavegenerator波数numberofwaves补偿compensation参数化设计parameterizationdesign,PD残余应力residualstress操纵及控制装置operationcontroldevice槽轮Genevawheel槽轮机构Genevamechanism；Maltesecross槽数Genevanumerate槽凸轮groovecam侧隙backlash差动轮系differentialgeartrain差动螺旋机构differentialscrewmechanism差速器differential常用机构conventionalmechanism mechanis mincommonuse 车床lathe承载量系数bearingcapacityfactor承载能力bearingcapacity成对安装pairedmounting尺寸系列dimensionseries齿槽toothspace齿槽宽spacewidth齿侧间隙backlash齿顶高addendum齿顶圆addendumcircle齿根高dedendum齿根圆dedendumcircle齿厚tooththickness齿距circularpitch齿宽facewidth齿廓toothprofile齿廓曲线toothcurve齿轮gear齿轮变速箱speed-changinggearboxes齿轮齿条机构pinionandrack齿轮插刀pinioncutter pinion-shapedshapercutter 齿轮滚刀hob,hobbingcutter齿轮机构gear齿轮轮坯blank齿轮传动系pinionunit齿轮联轴器gearcoupling齿条传动rackgear齿数toothnumber齿数比gearratio齿条rack齿条插刀rackcutter rack-shapedshapercutter 齿形链、无声链silentchain齿形系数formfactor齿式棘轮机构toothratchetmechanism插齿机gearshaper重合点coincidentpoints重合度contactratio冲床punch传动比transmissionratio,speedratio传动装置gearing transmissiongear传动系统drivensystem传动角transmissionangle传动轴transmissionshaft串联式组合combinationinseries串联式组合机构seriescombinedmechanism 串级调速cascadespeedcontrol创新innovation creation创新设计creationdesign垂直载荷、法向载荷normalload唇形橡胶密封liprubberseal磁流体轴承magneticfluidbearing从动带轮drivenpulley从动件drivenlink,follower从动件平底宽度widthofflat-face从动件停歇followerdwell从动件运动规律followermotion从动轮drivengear粗线boldline粗牙螺纹coarsethread大齿轮gearwheel打包机packer打滑slipping带传动beltdriving带轮beltpulley带式制动器bandbrake单列轴承singlerowbearing单向推力轴承single-directionthrustbearing 单万向联轴节singleuniversaljoint单位矢量unitvector当量齿轮equivalentspurgear virtualgear当量齿数equivalentteethnumber virtualnumberofteeth当量摩擦系数equivalentcoefficientoffriction当量载荷equivalentload刀具cutter导数derivative倒角chamfer导热性conductionofheat导程lead导程角leadangle等加等减速运动规律parabolicmotion constantaccelerationanddecelerationmotion 等速运动规律uniformmotion constantvelocitymotion等径凸轮conjugateyokeradialcam等宽凸轮constant-breadthcam等效构件equivalentlink等效力equivalentforce等效力矩equivalentmomentofforce等效量equivalent等效质量equivalentmass等效转动惯量equivalentmomentofinertia等效动力学模型dynamicallyequivalentmodel底座chassis低副lowerpair点划线chaindottedline疲劳点蚀pitting垫圈gasket垫片密封gasketseal碟形弹簧bellevillespring顶隙bottomclearance定轴轮系ordinarygeartrain geartrainwithfixedaxes 动力学dynamics动密封kinematicalseal动能dynamicenergy动力粘度dynamicviscosity动力润滑dynamiclubrication动平衡dynamicbalance动平衡机dynamicbalancingmachine动态特性dynamiccharacteristics动态分析设计dynamicanalysisdesign动压力dynamicreaction动载荷dynamicload端面transverseplane端面参数transverseparameters端面齿距transversecircularpitch端面齿廓transversetoothprofile端面重合度transversecontactratio端面模数transversemodule端面压力角transversepressureangle锻造forge对称循环应力symmetrycirculatingstress对心滚子从动件radialorin-linerollerfollower对心直动从动件radialorin-linetranslatingfollower对心移动从动件radialreciprocatingfollower对心曲柄滑块机构in-lineslider-crankorcrank-slidermechanis m 多列轴承multi-rowbearing多楔带polyV-belt多项式运动规律polynomialmotion多质量转子rotorwithseveralmasses惰轮idlegear额定寿命ratinglife额定载荷loadratingII级杆组dyad发生线generatingline发生面generatingplane法面normalplane法面参数normalparameters法面齿距normalcircularpitch法面模数normalmodule法面压力角normalpressureangle法向齿距normalpitch法向齿廓normaltoothprofile法向直廓蜗杆straightsidednormalworm法向力normalforce反馈式组合feedbackcombining反向运动学inverseorbackwardkinematics反转法kinematicinversion反正切Arctan范成法generatingcutting仿形法formcutting方案设计、概念设计conceptdesign,CD防振装置shockproofdevice飞轮flywheel飞轮矩momentofflywheel非标准齿轮nonstandardgear非接触式密封non-contactseal非周期性速度波动aperiodicspeedfluctuation 非圆齿轮non-circulargear粉末合金powdermetallurgy分度线referenceline standardpitchline分度圆referencecircle standardcuttingpitchcircle 分度圆柱导程角leadangleatreferencecylinder 分度圆柱螺旋角helixangleatreferencecylinder 分母denominator分子numerator分度圆锥referencecone standardpitchcone分析法analyticalmethod封闭差动轮系planetarydifferential复合铰链compoundhinge复合式组合compoundcombining复合轮系compoundorcombinedgeartrain复合平带compoundflatbelt复合应力combinedstress复式螺旋机构Compoundscrewmechanis m复杂机构complexmechanism杆组Assurgroup干涉interference刚度系数stiffnesscoefficient刚轮rigidcircularspline钢丝软轴wiresoftshaft刚体导引机构bodyguidancemechanism刚性冲击rigidimpulseshock刚性转子rigidrotor刚性轴承rigidbearing刚性联轴器rigidcoupling高度系列heightseries高速带highspeedbelt高副higherpair格拉晓夫定理Grashoff`slaw根切undercutting公称直径nominaldiameter高度系列heightseries功work工况系数applicationfactor工艺设计technologicaldesign工作循环图workingcyclediagram工作机构operationmechanis m工作载荷externalloads工作空间workingspace工作应力workingstress工作阻力effectiveresistance工作阻力矩effectiveresistancemoment 公法线commonnormalline公共约束generalconstraint公制齿轮metricgears功率power功能分析设计functionanalysesdesign共轭齿廓conjugateprofiles共轭凸轮conjugatecam构件link鼓风机blower固定构件fixedlink frame固体润滑剂solidlubricant关节型操作器jointedmanipulator惯性力inertiaforce惯性力矩momentofinertia,shakingmoment 惯性力平衡balanceofshakingforce惯性力完全平衡fullbalanceofshakingforce 惯性力部分平衡partialbalanceofshakingforce 惯性主矩resultantmomentofinertia惯性主失resultantvectorofinertia冠轮crowngear广义机构generationmechanis m广义坐标generalizedcoordinate轨迹生成pathgeneration轨迹发生器pathgenerator滚刀hob滚道raceway滚动体rollingelement滚动轴承rollingbearing滚动轴承代号rollingbearingidentificationcode 滚针needleroller滚针轴承needlerollerbearing滚子roller滚子轴承rollerbearing滚子半径radiusofroller滚子从动件rollerfollower滚子链rollerchain滚子链联轴器doublerollerchaincoupling滚珠丝杆ballscrew滚柱式单向超越离合器rollerclutch过度切割undercutting函数发生器functiongenerator函数生成functiongeneration含油轴承oilbearing耗油量oilconsumption耗油量系数oilconsumptionfactor赫兹公式H.Hertzequation合成弯矩resultantbendingmoment合力resultantforce合力矩resultantmomentofforce黑箱blackbox横坐标abscissa互换性齿轮interchangeablegears花键spline滑键、导键featherkey滑动轴承slidingbearing滑动率slidingratio滑块slider环面蜗杆toroidhelicoidsworm环形弹簧annularspring缓冲装置shocks shock-absorber灰铸铁greycastiron回程return回转体平衡balanceofrotors混合轮系compoundgeartrain积分integrate机电一体化系统设计mechanical-electricalintegrationsystemdesign 机构mechanis m机构分析analysisofmechanis m机构平衡balanceofmechanism机构学mechanism机构运动设计kinematic内方头紧定螺钉n.square-socket set-screw开槽锥端紧定螺钉Slotted set screws with cone point锥销锁紧挡圈Lock rings with cone pin螺钉锁紧挡圈Lock ring with screw限位钉stop pin限位块stop button热敏电阻器预留空hole for thermistor带隔离板的注模浇口Injection gate plane with isolation plate浇模和注模器的温度隔板T emperature isolation plate at injection and injector 拔模锥度（的定位）销draft pins注模插件和分离（夹层）打磨Moulding inserts and partings grinded定距拉杆length bolt定距拉板puller bolt一模多穴Multi-Cavity流道凝料condensed material in runner利用率utilization ratio自动脱螺纹automatic thread demoulding 、motorized thread unwinding, 生产现场Production scene车间workshop生产线production line模具专业英语入水：gate 进入位：gate location 水口形式：gate type 大水口：edge gate 细水口：pin-point gate 水口大小：gate size 转水口：switching runner/gate唧嘴口径：sprue diameter二、流道: runner热流道：hot runner,hot manifold 热嘴冷流道: hot sprue/cold runner唧嘴直流: direct sprue gate 圆形流道：round(full/half runner流道电脑分析：mold flow analysis 流道平衡:runner balance热嘴：hot sprue 热流道板：hot manifold发热管：cartridge heater 探针: thermocouples插头：connector plug 插座：connector socket密封/封料：seal三、运水：water line 喉塞：line lpug喉管：tube塑胶管：plastic tube 快速接头：jiffy quick connectorplug/socker四、模具零件：mold components三板模：3-plate mold 二板模：2-plate mold边钉/导边：leader pin/guide pin 边司/导套：bushing/guide bushing中托司：shoulder guide bushing 中托边L：guide pin顶针板：ejector retainner plate 托板：support plate螺丝：screw 管钉：dowel pin开模槽：ply bar scot 内模管位：core/cavity inter-lock顶针：ejector pin 司筒：ejector sleeve司筒针：ejector pin 推板：stripper plate缩呵：movable core,return core core puller扣机（尼龙拉勾）：nylon latch lock 斜顶：lifter模胚（架）：mold base 上内模：cavity insert下内模：core insert 行位（滑块）：slide镶件：insert 压座/斜鸡：wedge耐磨板/油板：wedge wear plate 压条：plate撑头: support pillar 唧嘴：sprue bushing挡板：stop plate 定位圈：locating ring锁扣：latch 扣鸡：parting lock set推杆：push bar 栓打螺丝：S.H.S.B顶板：eracuretun 活动臂：lever arm分流锥：spure sperader 水口司:bush垃圾钉：stop pin 隔片：buffle弹弓柱：spring rod 弹弓:die spring中托司：ejector guide bush 中托边：ejector guide pin镶针：pin 销子：dowel pin波子弹弓：ball catch模具成形不良用语英汉对照aberration 色差atomization ?化bank mark ?料纹bite 咬入blacking hole 涂料孔(铸疵) blacking scab 涂料疤blister 起泡blooming 起霜low hole 破孔blushing 泛白body wrinkle 侧壁皱纹breaking-in 冒口带肉bubble 膜泡burn mark 糊斑burr 毛边camber 翘曲cell 气泡center buckle 表面中部波皱check 细裂痕checking 龟裂chipping 修整表面缺陷clamp-off 铸件凹痕collapse 塌陷color mottle 色斑corrosion 腐蚀crack 裂痕crazing 碎裂crazing 龟裂deformation 变形edge 切边碎片edge crack 裂边fading 退色filler speak 填充料斑fissure 裂纹flange wrinkle 凸缘起皱flaw 刮伤flow mark 流痕galling 毛边glazing 光滑gloss 光泽grease pits 污斑grinding defect 磨痕haircrack 发裂haze 雾度incrustation 水锈indentation 压痕internal porosity 内部气孔mismatch 偏模mottle 斑点necking 缩颈nick 割痕orange peel 橘皮状表面缺陷overflow 溢流peeling 剥离pit 坑pitting corrosion 点状腐蚀plate mark 模板印痕pock 麻点pock mark 痘斑resin streak 树脂流纹resin wear 树脂脱落riding 凹陷sagging 松垂saponification 皂化scar 疤痕scrap 废料scrap jam 废料阻塞scratch 刮伤/划痕scuffing 深冲表面划伤seam 裂痕shock line 模口挤痕short shot 充填不足shrinkage pool 凹孔sink mark 凹痕skin inclusion 表皮折叠straightening 矫直streak 条状痕surface check 表面裂痕surface roughening 橘皮状表皮皱折surging 波动sweat out 冒汗torsion 扭曲warpage 翘曲waviness 波痕webbing 熔塌weld mark 焊痕whitening 白化wrinkle 皱纹各式模具分类用语英汉对照landed plunger mold 有肩柱塞式模具burnishing die 挤光模landed positive mold 有肩全压式模具button die 镶入式圆形凹模loading shoe mold 料套式模具center-gated mold 中心浇口式模具loose detail mold 活零件模具chill mold 冷硬用铸模loose mold 活动式模具clod hobbing 冷挤压制模louvering die 百叶窗冲切模composite dies 复合模具manifold die 分歧管模具counter punch 反凸模modular mold 组合式模具double stack mold 双层模具multi-cavity mold 多模穴模具electroformed mold 电铸成形模multi-gate mold 复式浇口模具expander die 扩径模offswt bending die 双折冷弯模具extrusion die 挤出模palletizing die 叠层模family mold 反套制品模具plaster mold 石膏模blank through dies 漏件式落料模porous mold 通气性模具duplicated cavity plate 复板模positive mold 全压式模具fantail die 扇尾形模具pressure die 压紧模fishtail die 鱼尾形模具profile die 轮廓模flash mold 溢料式模具progressive die 顺序模gypsum mold 石膏铸模protable mold 手提式模具hot-runner mold 热流道模具prototype mold 雏形试验模具ingot mold 钢锭模punching die 落料模lancing die 切口模raising(embossing) 压花起伏成形re-entrant mold 倒角式模具sectional die 拼合模runless injection mold 无流道冷料模具sectional die 对合模具segment mold 组合模semi-positive mold 半全压式模具shaper 定型模套single cavity mold 单腔模具solid forging die 整体锻模split forging die 拼合锻模split mold 双并式模具sprueless mold 无注道残料模具squeezing die 挤压模stretch form die 拉伸成形模sweeping mold 平刮铸模swing die 振动模具three plates mold 三片式模具trimming die 切边模unit mold 单元式模具universal mold 通用模具unscrewing mold 退扣式模具yoke type die 轭型模各种模具常用成形方式英汉对照accurate die casting 精密压铸powder forming 粉末成形calendaring molding 压延成形powder metal forging 粉末锻造cold chamber die casting 冷式压铸precision forging 精密锻造cold forging 冷锻press forging 冲锻compacting molding 粉末压出成形rocking die forging 摇动锻造compound molding 复合成形rotary forging 回转锻造compression molding 压缩成形rotational molding 离心成形dip mold 浸渍成形rubber molding 橡胶成形encapsulation molding 注入成形sand mold casting 砂模铸造extrusion molding 挤出成形shell casting 壳模铸造foam forming ?泡成形sinter forging 烧结锻造forging roll 轧锻six sides forging 六面锻造gravity casting 重力铸造slush molding 凝塑成形hollow(blow) molding 中空(吹出)成形squeeze casting 高压铸造hot chamber die casting 热室压铸swaging 挤锻hot forging 热锻transfer molding 转送成形injection molding 射出成形warm forging 温锻investment casting 精密铸造matched die method 对模成形法laminating method 被覆淋膜成形low pressure casting 低压铸造lost wax casting 脱蜡铸造matched mould thermal forming 对模热成形模机械类常用英语：冲压模具-零件类punch冲头insert入块(嵌入件)deburring punch压毛边冲子groove punch压线冲子stamped punch字模冲子round punch圆冲子special shape punch异形冲子bending block折刀roller滚轴baffle plate挡块located block定位块supporting block for location定位支承块air cushion plate气垫板air-cushion eject-rod气垫顶杆trimming punch切边冲子stiffening rib punch = stinger 加强筋冲子ribbon punch压筋冲子reel-stretch punch卷圆压平冲子guide plate定位板sliding block滑块sliding dowel block滑块固定块active plate活动板lower sliding plate下滑块板upper holder block上压块upper mid plate上中间板spring box弹簧箱spring-box eject-rod弹簧箱顶杆spring-box eject-plate弹簧箱顶板bushing bolck衬套cover plate盖板guide pad导料块机械类常用英语：冲压模具-模板类top plate上托板（顶板）top block上垫脚punch set上模座punch pad上垫板punch holder上夹板stripper pad脱料背板up stripper上脱料板male die公模(凸模)feature die公母模female die母模(凹模)upper plate上模板lower plate下模板die pad下垫板die holder下夹板die set下模座bottom block下垫脚bottom plate下托板(底板) stripping plate内外打(脱料板) outer stripper外脱料板inner stripper内脱料板lower stripper下脱料板冲压模具-冲压名称类英汉对照plain die简易模pierce die冲孔模forming die成型模progressive die连续模gang dies复合模shearing die剪边模riveting die铆合模pierce冲孔forming成型(抽凸,冲凸) draw hole抽孔bending折弯trim切边emboss凸点dome凸圆semi-shearing半剪stamp mark冲记号deburr or coin压毛边punch riveting冲压铆合side stretch侧冲压平reel stretch卷圆压平groove压线blanking下料stamp letter冲字(料号) shearing剪断tick-mark nearside正面压印tick-mark farside反面压印extension dwg展开图procedure dwg工程图die structure dwg模具结构图material材质material thickness料片厚度factor系数upward向上downward向下press specification冲床规格die height range适用模高die height闭模高度burr毛边gap间隙weight重量total wt.总重量。

表贴式永磁无刷电机直接解析计算方法李节宝;井立兵;周晓燕;章跃进【摘要】采用直接解析法计算表贴式永磁无刷电机空载气隙主磁场、齿槽转矩和反电动势。

求解场域划分为气隙、永磁体和槽区域，三个子区域的拉普拉斯方程或泊松方程通过交界条件建立联系，然后采用解析法直接求解。

计算中确切计及了实际槽深和槽间距离。

开槽气隙主磁场、齿槽转矩和反电动势解析计算波形分别与二维有限元法计算结果和实验波形作比较，比较结果较为一致，证明了本文方法的正确性和有效性。

%The exact analytical method is applied to compute air-gap main magnetic field distribution, cogging torque and electromotive force (EMF) in surface-mounted permanent-magnet (PM) motors. The solution regions are divided into air-gap domain, permanent magnet domain and slot domains. The Laplace's or Poisson's equations of the sub-domains are solved by the exact analytical method and the solutions are obtained using boundary and interface conditions. The actual depth of slot and distance between slots are taken into account in the computation. Air-gap magnetic field distributions, cogging torque and EMF waveform computed with the proposed analytical method are respectively compared with those issued from two-dimensional finite-element analysis and experimental waveforms, the comparison results are consistent and show the correctness and effectiveness of the proposed analytical method.【期刊名称】《电工技术学报》【年(卷),期】2012(027)011【总页数】6页(P83-88)【关键词】表贴式;永磁无刷电机;气隙磁场;齿槽转矩;反电动势;直接解析法【作者】李节宝;井立兵;周晓燕;章跃进【作者单位】上海大学机电工程与自动化学院,上海200072;上海大学机电工程与自动化学院,上海200072;上海大学机电工程与自动化学院,上海200072;上海大学机电工程与自动化学院,上海200072【正文语种】中文【中图分类】TM3511 引言永磁电机气隙磁场分析是电机设计和性能计算的基础。