CN117828244B - Linear induction motor chain type equivalent circuit and characteristic analysis method - Google Patents
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Abstract
Description
技术领域Technical Field
本发明属于直线感应电机领域，更具体地，涉及一种直线感应电机链式等效电路及特性分析方法。The present invention belongs to the field of linear induction motors, and more specifically, relates to a linear induction motor chain equivalent circuit and a characteristic analysis method.
背景技术Background technique
直线感应电机无需齿轮箱等传动机构可以直接产生直线运动，具有结构简单、维护量小、成本低等优点，在城轨交通、工业驱动等场合广泛应用。然而，因初级铁芯两侧开断、半填充槽双层叠绕组等特殊结构，直线感应电机面临着严重的纵向端部效应及三相不对称性，导致气隙磁场成分及作用机理十分复杂，且随着运行速度的增加纵向端部效应的影响将逐渐加剧。以上因素使得直线感应电机等效参数具有高阶、非线性、强耦合的变化特性。因此，推导精确的直线感应电机分析方法对于直线感应电机电磁和驱动特性的研究十分重要。Linear induction motors can directly generate linear motion without the need for transmission mechanisms such as gearboxes. They have the advantages of simple structure, low maintenance, and low cost. They are widely used in urban rail transit, industrial drives, and other occasions. However, due to the special structure of the primary core with two sides disconnected, half-filled slots and double-layer windings, linear induction motors face serious longitudinal end effects and three-phase asymmetry, resulting in very complex air gap magnetic field components and action mechanisms, and the influence of longitudinal end effects will gradually increase with the increase in operating speed. The above factors make the equivalent parameters of linear induction motors have high-order, nonlinear, and strongly coupled variation characteristics. Therefore, it is very important to derive an accurate linear induction motor analysis method for the study of the electromagnetic and drive characteristics of linear induction motors.
部分学者将直线感应电机初级运动产生次级涡流作为切入点，假设气隙磁密从初级运行方向的入端到出端呈指数形式衰减，认为气隙磁通的畸变对电机励磁电感产生影响。以一个与速度及电机结构参数相关的函数对励磁电感和铁损电阻进行修正，由此得到电机等效模型。但因前提假设过于简单，该模型不能合理描述高速、大电流等工况下的电机特性，且没有考虑绕组三相不对称性。还有学者将初级载流绕组等效为行波电流层，基于许-克变换法求解端部外侧磁场分布并建立求解域边界条件，进而通过偏微分方程解得气隙磁场分布，然后通过坡印廷矢量积分求得初级通过气隙向次级传递的功率，最终得到电机等效模型。但这种未考虑绕组自身丰富的空间谐波磁场。Some scholars take the secondary eddy current generated by the primary motion of the linear induction motor as the starting point, assuming that the air gap magnetic flux decays exponentially from the input end to the output end in the primary running direction, and believe that the distortion of the air gap magnetic flux affects the motor excitation inductance. The excitation inductance and iron loss resistance are corrected by a function related to the speed and motor structural parameters, thereby obtaining an equivalent model of the motor. However, due to the overly simple premise assumptions, this model cannot reasonably describe the motor characteristics under high-speed, high-current and other working conditions, and does not consider the three-phase asymmetry of the winding. Other scholars have equated the primary current-carrying winding to a traveling wave current layer, and based on the Xu-Ke transform method, the magnetic field distribution outside the end is solved and the boundary conditions of the solution domain are established. Then, the air gap magnetic field distribution is solved by the partial differential equation, and then the power transmitted from the primary to the secondary through the air gap is obtained by the Poynting vector integral, and finally the motor equivalent model is obtained. However, this does not take into account the rich spatial harmonic magnetic field of the winding itself.
因此，有必要提供一种直线感应电机链式等效电路及特性分析方法以准确分析直线感应电机气隙磁场的组成成分及不同工况下的运行特性。Therefore, it is necessary to provide a linear induction motor chain equivalent circuit and characteristic analysis method to accurately analyze the components of the linear induction motor air gap magnetic field and the operating characteristics under different working conditions.
发明内容Summary of the invention
针对现有技术的缺陷，本发明的目的在于提供一种直线感应电机链式等效电路及特性分析方法，旨在解决现有的分析直线感应电机磁场成分不充分、运行特性不精确的问题。In view of the defects of the prior art, the purpose of the present invention is to provide a linear induction motor chain equivalent circuit and characteristic analysis method, aiming to solve the problems of insufficient analysis of the magnetic field components and inaccurate operating characteristics of the prior art linear induction motor.
为实现上述目的，本发明提供一种直线感应电机链式等效电路及特性分析方法，包括以下步骤：To achieve the above object, the present invention provides a linear induction motor chain equivalent circuit and characteristic analysis method, comprising the following steps:
S1：根据短初级直线电机绕组磁动势分析方法求解出的初级各对极谐波磁动势，对其进行位置函数的偏导，得到初级各对极谐波电流层密度表达式；S1: Based on the short primary linear motor winding magnetomotive force analysis method, the harmonic magnetomotive force of each primary pole pair is solved, and the partial derivative of the position function is performed to obtain the expression of the harmonic current layer density of each primary pole pair;
S2：基于许-克变换法建立的求解域边界条件及Maxwell方程组求解计及纵向端部效应的各对极谐波气隙磁场表达式；S2: Based on the Xu-Ke transformation method, the boundary conditions of the solution domain and the Maxwell equations are established to solve the harmonic air gap magnetic field expressions of each pair of poles taking into account the longitudinal end effect;
S3：基于各对极谐波气隙磁场与电流层密度表达式，分析直线感应电机稳态推力特性；S3: Analyze the steady-state thrust characteristics of the linear induction motor based on the expressions of the harmonic air gap magnetic field and current layer density of each pole pair;
S4：根据各对极谐波气隙磁场表达式及绕组排布方式，求解三相绕组各对极谐波感应电动势表达式；S4: According to the expression of the harmonic air gap magnetic field of each pair of poles and the winding arrangement, solve the expression of the harmonic induced electromotive force of each pair of poles of the three-phase winding;
S5：根据励磁电感定义求解各对极谐波励磁电感；设定三相绕组电流表达式，结合各对极谐波励磁电感及感应电动势，求解各对极谐波次级电流表达式；S5: Solve the harmonic excitation inductance of each pole pair according to the definition of excitation inductance; set the three-phase winding current expression, combine the harmonic excitation inductance of each pole pair and the induced electromotive force, and solve the harmonic secondary current expression of each pole pair;
S6：基于各对极谐波感应电动势及次级电流，求解次级阻抗表达式，并结合初级阻抗，得到直线感应电机链式等效电路及特性分析方法。S6: Based on the harmonic induced electromotive force and secondary current of each pole pair, the secondary impedance expression is solved, and combined with the primary impedance, the linear induction motor chain equivalent circuit and characteristic analysis method are obtained.
进一步地，在所述步骤S2中各对极谐波电流层密度表达式为：Furthermore, in step S2, the expression of the harmonic current layer density of each pair of poles is:
其中，/>，/>。in ,/> ,/> .
式中，v为谐波极对数，L为初级铁芯纵向长度，ω e为初级角频率，J 1v+、J 1v-为正、反转各对极谐波行波电流层密度幅值，F v+、F v-、φ +、φ -分别为短初级直线电机绕组磁动势分析方法求解出的正转磁动势与反转磁动势的幅值、相位。In the formula, v is the number of harmonic pole pairs, L is the longitudinal length of the primary iron core, ωe is the primary angular frequency, J1v + and J1v- are the amplitudes of the harmonic traveling wave current layer density of each pair of positive and reverse poles, Fv + , Fv- , φ + , and φ- are the amplitudes and phases of the forward and reverse magnetomotive force solved by the magnetomotive force analysis method of the short primary linear motor winding.
进一步地，在所述步骤S2计及纵向端部效应的各对极谐波气隙磁场表达式为：Furthermore, in step S2, the expression of the harmonic air gap magnetic field of each pole pair taking into account the longitudinal end effect is:
； ;
式中，B 1v+、B 2v+、B 3v+、B 1v-、B 2v-、B 3v-分别为正转行波、反转行波各对极谐波正常行波、入端行波、出端行波磁密的幅值，α 1、α 2为入端行波、出端行波磁密的衰减系数，τ e为入端行波、出端行波磁密的极距。In the formula, B1v + , B2v + , B3v + , B1v- , B2v- , and B3v- are the amplitudes of the normal traveling wave, input-end traveling wave, and output-end traveling wave magnetic density of each pair of harmonics of the forward traveling wave and reverse traveling wave respectively, α1 and α2 are the attenuation coefficients of the input-end traveling wave and output - end traveling wave magnetic density, and τe is the pole pitch of the input-end traveling wave and output-end traveling wave magnetic density.
进一步地，在所述步骤S3直线感应电机稳态推力特性为电机稳态运行工况下推力随时间变化的特点，其由各对极谐波电流层密度及气隙磁场进行计算，其表达式为：Furthermore, in step S3, the steady-state thrust characteristic of the linear induction motor is the characteristic of thrust variation over time under the steady-state operation condition of the motor, which is calculated by the harmonic current layer density of each pair of poles and the air gap magnetic field, and its expression is:
； ;
式中，W为初级铁芯的横向长度。Where W is the transverse length of the primary core.
进一步地，在所述步骤S4的各对极谐波感应电动势考虑了纵向端部效应对气隙磁密的畸变作用。Furthermore, the harmonic induced electromotive force of each pole pair in step S4 takes into account the distortion effect of the longitudinal end effect on the air gap magnetic flux density.
进一步地，所述方法应用于不考虑铁耗、集肤效应及横向边缘效应的直线感应电机。Furthermore, the method is applied to a linear induction motor without considering iron loss, skin effect and lateral edge effect.
在所述步骤S6的直线感应电机链式等效电路可以分析各对极谐波对电机性能的影响规律。The linear induction motor chain equivalent circuit in step S6 can analyze the influence of each pole pair harmonic on the motor performance.
总体而言，通过本发明所构思的以上技术方案与现有技术相比，具有以下有益效果：In general, the above technical solution conceived by the present invention has the following beneficial effects compared with the prior art:
相比于现有方法，本发明充分考虑了绕组自身方法的谐波，以各极对数谐波电流层密度作为激励源，结合基于许-克变换法建立的求解域边界条件及Maxwell方程组求解气隙磁场的组成成分及变化规律；以各极对数谐波电流层密度及气隙磁场求解直线感应电机稳态推力特性，分析直线感应电机不同工况下稳态推力大小及时变特性，揭示直线感应二倍频推力波动产生的原因，计算各极对数谐波磁场对推力特性的影响规律；根据绕组排布方式及各极对数谐波磁场，求解各相绕组各极对数谐波感应电动势，再根据定义求解各谐波极对数励磁电感，进一步，求解出各谐波极对数次级电流及次级阻抗，结合初级阻抗得到直线感应电机链式等效电路。Compared with the existing methods, the present invention fully considers the harmonics of the winding method itself, takes the harmonic current layer density of each pole pair as the excitation source, and combines the boundary conditions of the solution domain established based on the Xu-Ke transform method and the Maxwell equations to solve the composition and change law of the air gap magnetic field; the steady-state thrust characteristics of the linear induction motor are solved by the harmonic current layer density of each pole pair and the air gap magnetic field, and the steady-state thrust size and time-varying characteristics of the linear induction motor under different working conditions are analyzed, the cause of the linear induction double frequency thrust fluctuation is revealed, and the influence of the harmonic magnetic field of each pole pair on the thrust characteristics is calculated; according to the winding arrangement and the harmonic magnetic field of each pole pair, the harmonic induced electromotive force of each pole pair of each phase winding is solved, and then the excitation inductance of each harmonic pole pair is solved according to the definition, and further, the secondary current and secondary impedance of each harmonic pole pair are solved, and the chain equivalent circuit of the linear induction motor is obtained in combination with the primary impedance.
本发明提供的方法能够更全面、准确地分析直线感应电机气隙磁场的组成成分及运行特性。The method provided by the present invention can more comprehensively and accurately analyze the components and operating characteristics of the air gap magnetic field of the linear induction motor.
附图说明BRIEF DESCRIPTION OF THE DRAWINGS
图1是本发明提供的直线感应电机链式等效电路及特性分析方法流程图；FIG1 is a flow chart of a linear induction motor chain equivalent circuit and a characteristic analysis method provided by the present invention;
图2（a）为本发明应用的直线感应电机结构示意图；FIG2 (a) is a schematic diagram of the structure of a linear induction motor used in the present invention;
图2（b）为本发明应用的直线感应电机分析模型；FIG2( b ) is an analysis model of a linear induction motor used in the present invention;
图3是本发明实施例提供的额定速下气隙磁密对比结果图；FIG3 is a graph showing a comparison of air gap flux density at rated speed provided by an embodiment of the present invention;
图4是本发明提供的直线感应电机链式等效电路；FIG4 is a chain equivalent circuit of a linear induction motor provided by the present invention;
图5是实施例提供的电机推力特性分析图。FIG. 5 is a diagram showing the thrust characteristics of a motor provided in an embodiment.
具体实施方式Detailed ways
为了使本发明的目的、技术方案及优点更加清楚明白，以下结合附图及实施例，对本发明进行进一步详细说明。应当理解，此处所描述的具体实施例仅用以解释本发明，并不用于限定本发明。此外，下面所描述的本发明各个实施方式中所涉及到的技术特征只要彼此之间未构成冲突就可以相互组合。In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.
本发明提供的一种直线感应电机链式等效电路及特性分析方法流程图，如图1所示，具体包括以下步骤：A linear induction motor chain equivalent circuit and a flow chart of a characteristic analysis method provided by the present invention, as shown in FIG1 , specifically comprises the following steps:
S1：根据短初级直线电机绕组磁动势分析方法求解出的各对极谐波磁动势，对其进行位置函数的偏导，得到各对极谐波电流层密度表达式；S1: Based on the short primary linear motor winding magnetomotive force analysis method, the harmonic magnetomotive force of each pole pair is solved, and the partial derivative of the position function is performed to obtain the expression of the harmonic current layer density of each pole pair;
S2：结合基于许-克变换法建立的求解域边界条件及Maxwell方程组求解计及纵向端部效应的各对极谐波气隙磁场表达式；S2: Combine the boundary conditions of the solution domain established based on the Xu-Ke transformation method and the Maxwell equations to solve the harmonic air gap magnetic field expression of each pole pair taking into account the longitudinal end effect;
S3：基于各对极谐波气隙磁场与电流层密度表达式，分析直线感应电机稳态推力特性；S3: Analyze the steady-state thrust characteristics of the linear induction motor based on the expressions of the harmonic air gap magnetic field and current layer density of each pole pair;
S4：根据各对极谐波气隙磁场表达式及绕组排布方式，求解三相绕组各对极谐波感应电动势表达式；S4: According to the expression of the harmonic air gap magnetic field of each pair of poles and the winding arrangement, solve the expression of the harmonic induced electromotive force of each pair of poles of the three-phase winding;
S5：根据励磁电感定义求解各对极谐波励磁电感；设定三相绕组电流表达式，结合各对极谐波励磁电感及感应电动势，求解各对极谐波次级电流表达式；S5: Solve the harmonic excitation inductance of each pole pair according to the definition of excitation inductance; set the three-phase winding current expression, combine the harmonic excitation inductance of each pole pair and the induced electromotive force, and solve the harmonic secondary current expression of each pole pair;
S6：基于各对极谐波感应电动势及次级电流，求解次级阻抗表达式，并结合初级阻抗，得到直线感应电机链式等效电路。S6: Based on the harmonic induced electromotive force of each pole pair and the secondary current, the secondary impedance expression is solved, and combined with the primary impedance, the chain equivalent circuit of the linear induction motor is obtained.
下面具体说明直线感应电机链式等效电路及特性分析方法。The following is a detailed description of the linear induction motor chain equivalent circuit and characteristic analysis method.
在忽略铁芯损耗、集肤效应以及横向边缘效应的前提条件下，推导直线感应电机链式等效电路及特性分析方法。Under the premise of ignoring core loss, skin effect and lateral edge effect, the chain equivalent circuit and characteristic analysis method of linear induction motor are derived.
（1）建立直线感应电机一维场分析模型(1) Establishing a one-dimensional field analysis model of a linear induction motor
如图2（a）所示，初级铁芯运动方向为x轴（纵向），初级绕组电流的流通方向为z轴（横向），气隙磁通密度的有效方向为y轴。而绕组的排布方式为当前轨道交通直线感应电机最常用的半填充槽双层叠绕组。该一维场分析模型满足以下条件：As shown in Figure 2 (a), the primary core movement direction is the x-axis (longitudinal), the primary winding current flow direction is the z-axis (transverse), and the effective direction of the air gap flux density is the y-axis. The winding arrangement is the most commonly used half-filled slot double-layer winding for current rail transit linear induction motors. The one-dimensional field analysis model meets the following conditions:
（a）初级铁芯和次级铁轭磁导率为无穷大，不考虑铁芯损耗；(a) The magnetic permeability of the primary core and secondary yoke is infinite, and the core loss is not considered;
（b）气隙磁场值包含y轴分量，且与y轴坐标y无关，行波磁场及电机运行方向沿x轴，所有电磁场参量为x轴坐标x和时间t的正弦函数；(b) The air gap magnetic field value contains the y-axis component and is independent of the y-axis coordinate y. The traveling magnetic field and the motor running direction are along the x-axis. All electromagnetic field parameters are sinusoidal functions of the x-axis coordinate x and time t.
（c）不考虑次级集肤效应，且次级端部电气参数全部归算到初级。(c) The secondary skin effect is not considered, and all electrical parameters at the secondary end are attributed to the primary.
（2）根据直线绕组磁动势分析方法，求解各极对数谐波磁动势：(2) According to the linear winding magnetomotive force analysis method, the harmonic magnetomotive force of each pole pair is solved:
； ;
其中。in .
式中，v为谐波极对数，L为初级铁芯纵向长度，ω e为初级角频率，F v+、F v-、φ +、φ -分别为初级正转磁动势与反转磁动势的幅值、相位。In the formula, v is the number of harmonic pole pairs, L is the longitudinal length of the primary iron core, ωe is the primary angular frequency, Fv + , Fv- , φ + , φ- are the amplitude and phase of the primary forward magnetomotive force and reverse magnetomotive force respectively .
（3）对各极对数谐波磁动势进行位置函数x的偏导，求解初级各极对数谐波电流层密度：(3) Take the partial derivative of the position function x for the harmonic magnetomotive force of each pole logarithm, and solve the primary harmonic current layer density of each pole logarithm:
； ;
式中，，/>；In the formula, ,/> ;
（4）基于图2（a）所示直线感应电机磁场分析模型及基于许-克变换法建立的求解域边界条件，求解气隙磁场：(4) Based on the linear induction motor magnetic field analysis model shown in Figure 2 (a) and the solution domain boundary conditions established based on the Xu-Ke transformation method, the air gap magnetic field is solved:
； ;
式中，B 1v+、B 2v+、B 3v+、B 1v-、B 2v-、B 3v-分别为正转行波、反转行波各对极谐波正常行波、入端行波、出端行波磁密的幅值，α 1、α 2为入端行波、出端行波磁密的衰减系数，τ e为入端行波、出端行波磁密的极距。In the formula, B1v + , B2v + , B3v + , B1v- , B2v- , and B3v- are the amplitudes of the normal traveling wave, input-end traveling wave, and output-end traveling wave magnetic density of each pair of harmonics of the forward traveling wave and reverse traveling wave respectively, α1 and α2 are the attenuation coefficients of the input-end traveling wave and output - end traveling wave magnetic density, and τe is the pole pitch of the input-end traveling wave and output-end traveling wave magnetic density.
具体的步骤包括如下：The specific steps include the following:
由于正转分量和反转分量的表达式几乎相同所以就已正转分量作为分析例子进行说明。Since the expressions of the positive rotation component and the negative rotation component are almost the same, the positive rotation component is used as an analysis example for explanation.
（4-1）根据本发明应用的直线感应电机一维场分析模型图2（b），通过安培环路定律，可得：(4-1) According to the one-dimensional field analysis model of the linear induction motor applied in the present invention, FIG2(b), through Ampere's loop law, it can be obtained that:
； ;
其中g e 为等效电磁气隙，μ 0为空气磁导率，j 1v+为正转初级v对极谐波面电流密度，j 2v+为正转次级导体的v对极谐波等效行波电流层密度。Where g e is the equivalent electromagnetic air gap, μ 0 is the air magnetic permeability, j 1 v + is the positive primary v- pole harmonic surface current density, and j 2 v + is the v -pole harmonic equivalent traveling wave current layer density of the positive secondary conductor.
（4-2）引入矢量磁位，并根据电流密度、电导率以及气隙电场的关系得到正转次级导体各极对数谐波等效电流层密度与矢量磁位的关系式：(4-2) The vector magnetic potential is introduced, and the relationship between the harmonic equivalent current layer density of each pole of the positive secondary conductor and the vector magnetic potential is obtained based on the relationship between current density, conductivity and air gap electric field:
； ;
其中矢量磁位与气隙磁通密度、电场强度的关系为：The relationship between the vector magnetic potential and the air gap magnetic flux density and electric field strength is:
， ,
； ;
式中，σ s 为次级导体的表面电导率，σ s =σd，σ为次级导体体积电导率；d为次级导体板厚度；V 2为次级移动速度。In the formula, σs is the surface conductivity of the secondary conductor, σs = σd , σ is the volume conductivity of the secondary conductor; d is the thickness of the secondary conductor plate; V2 is the secondary movement speed.
（4-3）根据（4-1）、（4-2）中的关系式可得矢量磁位的全解为：(4-3) According to the relationship in (4-1) and (4-2), the complete solution of the vector magnetic potential is:
； ;
式中c 1、c 2为待定系数，其它系数表达式为方程特解，/>，，/>。Where c 1 and c 2 are unknown coefficients, and the other coefficients are expressed as is a particular solution of the equation, /> , ,/> .
其中，，/>，/>，/>，/>，/>，，/>。in, ,/> ,/> ,/> ,/> ,/> , ,/> .
式中τ v 为v对极谐波极距，s v+ 为正转v对极谐波滑差，V sv+ 为正转v对极谐波同步速。Where τ v is the harmonic pole pitch of v pairs of poles, s v+ is the harmonic slip of v pairs of poles in the forward rotation, and V sv+ is the harmonic synchronous speed of v pairs of poles in the forward rotation.
（4-4）根据矢量磁位的全解及其与气隙磁通密度的关系式求解气隙磁场：(4-4) The air gap magnetic field is solved based on the full solution of the vector magnetic potential and its relationship with the air gap magnetic flux density:
； ;
进一步，可以变形为：Further, it can be transformed into:
； ;
式中，B 1v+、B 2v+、B 3v+分别为正转行波各对极谐波正常行波、入端行波、出端行波磁密的幅值。In the formula, B1v + , B2v + , and B3v + are the amplitudes of the normal traveling wave, the input end traveling wave, and the output end traveling wave magnetic density of each pair of harmonics of the forward traveling wave, respectively.
（4-5）同理，各极对数谐波气隙磁密的反转分量求解过程与其相同，最终可以得到其表达式：(4-5) Similarly, the solution process of the reverse component of the harmonic air gap magnetic flux of each pole pair is the same, and finally its expression can be obtained:
； ;
（4-6）根据上述表达式分析日本12000型电机在各个工况气隙磁密分布情况，并与有限元计算结果做对比，这里给出额定工况下的气隙磁密对比图，如图3所示，由图可看出解析结果与仿真结果高度吻合，验证本方法的有效性。(4-6) Based on the above expression, the air gap flux distribution of the Japanese 12000 motor under various operating conditions is analyzed and compared with the finite element calculation results. Here is a comparison chart of the air gap flux under rated conditions, as shown in Figure 3. It can be seen from the figure that the analytical results are highly consistent with the simulation results, verifying the effectiveness of this method.
（5）基于各对极谐波气隙磁场与电流层密度表达式，分析直线感应电机稳态推力特性：(5) Based on the expressions of harmonic air gap magnetic field and current layer density of each pole pair, the steady-state thrust characteristics of the linear induction motor are analyzed:
； ;
式中，W为初级铁芯的横向长度。Where W is the transverse length of the primary core.
该式可分析直线感应电机稳态推力特性，准确反映了稳态推力二倍频波动特性。This formula can analyze the steady-state thrust characteristics of the linear induction motor and accurately reflects the steady-state thrust double frequency fluctuation characteristics.
（6）根据各对极谐波气隙磁场表达式及绕组排布方式，求解三相绕组各对极谐波感应电动势表达式：(6) According to the expression of the harmonic air gap magnetic field of each pair of poles and the winding arrangement, the expression of the harmonic induced electromotive force of each pair of poles of the three-phase winding is solved:
由于正转分量和反转分量的表达式几乎相同所以就已正转分量作为分析例子进行说明。Since the expressions of the positive rotation component and the negative rotation component are almost the same, the positive rotation component is used as an analysis example for explanation.
对气隙磁密进行位置函数x的积分，进一步对其进行时间t求导可得：Integrate the position function x of the air gap flux density and further differentiate it with respect to time t to obtain:
； ;
（7）根据励磁电感定义求解各对极谐波励磁电感；设定三相绕组电流表达式，结合各对极谐波励磁电感及感应电动势，求解各对极谐波次级电流表达式；(7) Solve the harmonic excitation inductance of each pole pair according to the definition of excitation inductance; set the three-phase winding current expression, combine the harmonic excitation inductance of each pole pair and the induced electromotive force, and solve the harmonic secondary current expression of each pole pair;
，/>，/>； ,/> ,/> ;
式中，i mv+为正转每对极谐波励磁电流，i 2v+为每对极谐波次级电流，i s为定子电流，N ϕ 为每相串联匝数，k Np 为每对极谐波绕组系数Where, i mv+ is the harmonic excitation current of each pair of poles in the forward direction, i 2v+ is the harmonic secondary current of each pair of poles, i s is the stator current, N φ is the number of series turns per phase, k Np is the harmonic winding coefficient of each pair of poles
（8）基于各对极谐波感应电动势及次级电流，求解次级阻抗表达式：(8) Based on the harmonic induced electromotive force of each pole pair and the secondary current, solve the secondary impedance expression:
； ;
（9）并结合初级阻抗，得到直线感应电机链式等效电路，如图4所示。(9) and combined with the primary impedance, the linear induction motor chain equivalent circuit is obtained, as shown in Figure 4.
（10）基于直线感应电机链式等效电路可分析12000直线感应电机恒流恒频（210A，22Hz）特性曲线，如图5所示。(10) Based on the linear induction motor chain equivalent circuit, the constant current and constant frequency (210A, 22Hz) characteristic curve of the 12000 linear induction motor can be analyzed, as shown in Figure 5.
应用本发明的分析方法，并结合计算机辅助计算，即可较为全面、准确地计算直线感应电机分析直线感应电机气隙磁场的组成成分及运行特性。By applying the analysis method of the present invention and combining it with computer-aided calculation, the components and operating characteristics of the air gap magnetic field of the linear induction motor can be calculated more comprehensively and accurately.
本领域的技术人员容易理解，以上所述仅为本发明的较佳实施例而已，并不用以限制本发明，凡在本发明的精神和原则之内所作的任何修改、等同替换和改进等，均应包含在本发明的保护范围之内。It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.
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