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Improved equivalent magnetic network modeling for analyzing working points of PMs in interior permanent magnet machine
Accepted Manuscript Research articles Improved Equivalent Magnetic Network Modeling for Analyzing Working Points of PMs in Interior Permanent Magnet Machine Liyan Guo, Changliang Xia, Huimin Wang, Zhiqiang Wang, Tingna Shi PII: DOI: Reference:
S0304-8853(17)32278-3 https://doi.org/10.1016/j.jmmm.2018.01.018 MAGMA 63596
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
19 July 2017 20 December 2017 8 January 2018
Please cite this article as: L. Guo, C. Xia, H. Wang, Z. Wang, T. Shi, Improved Equivalent Magnetic Network Modeling for Analyzing Working Points of PMs in Interior Permanent Magnet Machine, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/j.jmmm.2018.01.018
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Improved Equivalent Magnetic Network Modeling for Analyzing Working Points of PMs in Interior Permanent Magnet Machine Liyan Guo1, Changliang Xia1,2, Huimin Wang2, Zhiqiang Wang2, Tingna Shi1 1
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China 2 Tianjin Engineering Center of Electric Machine System Design and Control, Tianjin Polytechnic University, Tianjin 300387, China As is well known, the armature current will be ahead of the back electromotive force (back-EMF) under load condition of the interior permanent magnet (PM) machine. This kind of advanced armature current will produce a demagnetizing field, which may make irreversible demagnetization appeared in PMs easily. To estimate the working points of PMs more accurately and take demagnetization under consideration in the early design stage of a machine, an improved equivalent magnetic network model is established in this paper. Each PM under each magnetic pole is segmented, and the networks in the rotor pole shoe are refined, which makes a more precise model of the flux path in the rotor pole shoe possible. The working point of each PM under each magnetic pole can be calculated accurately by the established improved equivalent magnetic network model. Meanwhile, the calculated results are compared with those calculated by FEM. And the effects of d-axis component and q-axis component of armature current, air-gap length and flux barrier size on working points of PMs are analyzed by the improved equivalent magnetic network model. Index Terms—interior permanent magnet machine, improved equivalent magnetic network model, working point of PM. I. Introduction The interior permanent magnet (PM) machine has many advantages, such as high efficiency, high power density, and wide speed range, which makes it widely used in automobile, locomotive traction, industrial robot, and aerospace etc.[1]-[2]. The PMs in the interior PM machine are protected by the rotor pole shoe, which makes the machine more suitable for the situations that require high machine speed[3]. When the interior PM machine operates under load condition, the armature current is ahead of back electromotive force (back-EMF). Especially when the machine operates under high speed by the fluxweakening control, the angle between the armature current and the back-EMF is larger, which makes irreversible demagnetization appeared in PMs more easily. Accurately estimating working points of PMs in the early design stage of machine has an important significance in determining the operation range of the machine.
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The Finite Element Method (FEM) is often adopted to analyze demagnetization in PM. By utilizing the FEM, the flux density component in PM along the magnetization direction of PM can be calculated, and the demagnetization in PM can be analyzed [4]-[6]. However, the parameters of the machine need to be changed frequently in the early design stage and the optimization design stage of the machine. The FEM will cost a lot of time and computer resources in the modeling and computation process. While the equivalent magnetic circuit method that is suitable for the interior PM machine will cost less[7]. However, using the equivalent magnetic circuit method, only the average value of working points of PMs under one magnetic pole can be achieved[3]. Many kinds of interior PM machine adopt the “V” type, “U” type, “W” type and multilayer PM structure today. In these cases, each magnetic pole contains several pieces of PMs[8]. When the machine operates under load condition, the magnitudes and directions of armature reaction field acting on the left and the right PM under one magnetic pole are different. That is, the working points of the left and right PM are different[9]. The problem is that the difference of working point in the left and the right PM cannot be calculated by the equivalent magnetic circuit method. Therefore, the demagnetization in PM cannot be predicted timely by utilizing the equivalent magnetic circuit method. Compared with the equivalent magnetic circuit method, the equivalent magnetic network model (EMNM) can divide networks more finely, and it also costs less computation time. The EMNM is built on the basis of the flux tube theory. Certain part in the machine having the same material, the uniform flux distribution and regular shape is equivalent to one permeance unit. Permeance units are connected through nodes. The magnetic potential at each node can be achieved by solving the magnetic potential equation at each node, and then performance parameters can be achieved further[10]. In the EMNM, the saturation of iron in the machine can be modeled by the nonlinear permeance [11]-[13]. The leakage flux in the flux barrier can be modeled by the flux flowing through the nonlinear permeance unit in the flux barrier[14]-[15]. Moreover, effect of the stator slot opening can also be considered in the EMNM. The flux lines distribution in the stator slot opening can be equivalent to the quarter circle[16], and then effect of the stator slot opening can be considered by setting permeance units in the stator slot opening. The published literatures have established EMNMs aiming at the interior PM machines with different rotor magnetic circuit structures, and calculated the magnetic field and electromagnetic performance of machine by the established model further. Reference [15] has established the EMNM aiming at the interior PM machine with V-type PM structure. Meanwhile, it calculates the magnetic field distribution in each point of iron, and calculates the iron loss of machine further. References [14] and [17] have established the EMNMs aiming at the interior PM machines with double layer U-type PM structure, which was used to calculate the iron loss and magnet eddy current loss of machine further. References [18] and [19] have established the EMNM separately aiming at the interior PM machine with multi-layer U-type PM structure and “―”type PM structure to calculate the magnetic field in two machines. Corresponding author: Changliang Xia (e-mail: [email protected]).
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The rotor pole shoe is a part of the flux path in the interior PM machine. If the rotor pole shoe can be modeled accurately, the flowing path of magnetic field will be modeled accurately. However, the rotor pole shoe often has irregular shape and large volume in the interior PM machine, which makes it difficult to model the rotor pole shoe accurately. In order to model the flux path in the rotor pole shoe and calculate the working points of PMs accurately, an improved EMNM is established aiming at the interior PM machine in this paper. In the improved EMNM, each PM under each magnetic pole is segmented, which makes the networks in the rotor pole shoe refined. Meanwhile, both a radial permeance unit and a tangential permeance unit are set for each refined network in the rotor pole shoe. The working points of PMs under no-load and load conditions can be calculated accurately by the established improved equivalent magnetic network model. Further, the demagnetization in PMs can be estimated accurately in the early design stage of the machine. Finally, effects of the armature current, air-gap length and flux barrier size on the working points of PMs are analyzed by utilizing the established improved EMNM. II. Improved Equivalent Magnetic Network Modeling An improved EMNM is established aiming at the interior PM machine in this paper. The structure of interior PM machine is shown in Fig.1, and related parameters of the machine are listed in Table I[1]. The EMNM of one-unit machine is established in this paper to reduce the calculation quantity. One-unit machine has one pole pair, twelve stator slots and twelve stator teeth.
Fig. 1. Structure of the machine. 1-stator; 2-stator winding; 3-rotor; 4-air-gap; 5-PM; 6, 7flux barrier; 8, 9, 10-air flux barrier TABLE I Related parameters of the machine Symbol Quantity p Pole pair number Q Slot number n Rated speed
Value 4 48 1800
Unit --r/min
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IN α Rra δ Rsy Br μr hPM wPM_side wPM_middle l
RMS of rated current Current angle of rated current Outer radius of rotor Air-gap length Outer radius of stator Remeance Relative permeance of PM Thickness of PM Width of the side PM Width of the middle PM Axial length of PM
166 13.68 148.2 1.8 232.5 1.19 1.121 16 30 23 210
A ° mm mm mm T -mm mm mm mm
A. Permeance distribution in each part of machine Certain part in the machine having the same material, the uniform flux distribution and regular shape is equivalent to one permeance unit according to the flux path in the machine. Further, the permeance units distribution in the whole machine can be achieved, as shown in Fig.2.
Fig. 2. Permeance distribution in each part of the machine. (a) stator; (b) rotor; (c) air-gap The permeance units distribution in the stator is shown in Fig.2(a). The magnetomotive force (MMF) source in each tooth represents the equivalent MMF source produced by the armature current under load condition[9]. Because the tooth width along the radial direction is different, each tooth is equivalent to three permeance units. The stator yoke between two stator teeth is equivalent to one permeance unit. The flux lines distribution in the stator slot
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opening can be equivalent to the quarter circle. So the permeance unit distribution in the stator slot opening can be achieved accordingly. Meanwhile, the slot leakage permeance unit is added between adjacent teeth to calculate leakage flux flowing through the stator slot. Rotor pole shoe in the interior PM machine is a part of flux path. However, the rotor pole shoe has irregular shape and large volume in the interior PM machine, which makes it difficult to model the rotor pole shoe accurately. In order to establish an accurate model for rotor pole shoe, each PM under each magnetic pole is segmented, which also makes the networks in the rotor pole shoe refined. Meanwhile, both a radial permeance unit and a tangential permeance unit are set aiming at each network in the rotor pole shoe to model the flux path more accurately, as shown in Fig.2(b). The equivalent flux source and permeance of the segmented PM are calculated using formula (1) and formula (2). seg
GMseg
Br wPM l m
0 r wPM l mhPM
(1) (2)
where m is the number of PM segmentations. Φseg is the equivalent flux source of the segmented PM. GMseg is the permeance of the segmented PM. Br is the remeance of the PM. wPM is the width of the PM. l is the axial length of the PM. hPM is the thickness of the PM. μ0 is the permeability of the vacuum. μr is the relative permeability of the PM. In order to calculate the tangential component of air-gap flux density, two-layer air-gap permeance model is adopted in this paper, as shown in Fig.2(c). Because the rotor rotates, the permeance distribution in the air-gap will change at every moment. At different moments, the permeance of the radial permeance units and the tangential permeance units in the air-gap can be achieved by calculating the arc length of the overlapping regions between the units in the rotor and stator that are next to the air-gap, as shown in Fig.3. In Fig.3, “Teeth1” represents certain teeth in stator. “1”, “2”, and “3” represent three regions in the rotor which are adjacent to the air gap. Fig.3 only takes the air gap between “Teeth1” and three regions in rotor as an example to explain the permeance units distribution in the air gap. The analysis results can be applied to calculate the permeance units distribution in the whole air gap.
Fig.3 Sketch map of calculation for permeance of permeance units in air gap
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It can be seen from Fig.3 that at certain moment t=t1, “Teeth 1” has overlapping regions with rotor region “2” and rotor region “3”. The arc length of the overlapping region between “Teeth 1” and rotor region “2” is l2. The arc length of the overlapping region between “Teeth 1” and rotor region “3” is l3. At next moment t=t2, “Teeth 1” only has overlapping region with rotor region “2”. The arc length of the overlapping region between “Teeth 1” and rotor region “2” is l2. At moment t=t3, “Teeth 1” has overlapping regions with rotor region “1” and “2”. The arc length of the overlapping region between “Teeth 1” and rotor region “1” is l1, and the arc length of the overlapping region between “Teeth 1” and rotor region “2” is l2. And so on, if the electrical angle in radian corresponding to “Teeth1” is [θ1,θ2], and the electrical angle in radian corresponding to rotor region“i” is [αi,αi+1], i=1,2,3, the arc length of the overlapping region between “Teeth1” and rotor region “i” li can be calculated accordingly at certain moment. At certain moment, (1) If αi+1<θ1 or αi>θ2, li=0. (2) If αi>θ1 and αi+1<θ2, li=Rg(αi+1-αi)/p, where p represents pole pair number of machine, and Rg represents the radius in the middle of air gap. (3) If αi<θ1 and θ1≤αi+1≤θ2, li=Rg(αi+1-θ1)/p. (4) If θ1≤αi≤θ2 and αi+1>θ2, li=Rg(θ2-αi)/p. (5) If αi<θ1 and αi+1>θ2, li=Rg(θ2-θ1)/p. At different moments, according to the arc length of overlapping regions, the permeance of radial permeance units and tangential permeance units can be calculated. The permeance of radial permeance units in the air gap are calculated by Gri=2μ0lil/g, and the permeance of tangential permeance units in the air gap are calculated by Gti,i+1=2μ0gl/(li+li+1). Where, g is the air gap length; μ0 is the permeability of vacuum; l is the axial length of machine. And so on, the permeance units distribution in the overall air gap region can be achieved at different moments. In Fig.2, the permeance units with dark color represent the nonlinear permeance units, and those with light color represent the linear permeance units. In order to calculate the permeance of each permeance unit in Fig.2, several typical geometric structures of the permeance units and the corresponding calculation formula of the permeance are shown in Table II. The permeance of each permeace unit in Fig.2 can be calculated further. TABLE II Permeance formula of several typical structure Geometry Permeance formula Ga a l1l2 h
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Gb b a1 a2 b 2h
Gc 4cl3 π
The width of stator tooth changes along the radial direction, so the geometric structure of the permeance unit in stator tooth is similar to the trapezoidal permeance unit in Table II. The geometric structures of the permeance units in stator yoke and stator slot are similar to the rectangular permeance unit in Table II. The flux lines distribution in the stator slot opening can be equivalent to quarter circle. So the geometric structures of the permeance units in the stator slot opening are similar to the fan-shaped permeance unit in Table II. The permeance units in the rotor pole shoe are refined by segmenting PMs, and the geometric structure of each refined permeance unit is similar to the rectangular permeance unit or trapezoidal permeance unit in Table II. The geometric structures of the permeance units in the iron between adjacent poles and the permeance units in the flux barriers are similar to the rectangular permeance unit in Table II. The geometric structure of each permeance unit in the air-gap is similar to the rectangular permeance unit in Table II. Therefore, the permeance of permeance units in the machine can be calculated according to the geometric parameters of the machine and the corresponding formula in Table II. B. Solving Improved Equivalent Magnetic Network Model 1) Solving magnetic field According to the permeance unit distributions in each part of the machine that are given in Fig.2, the corresponding structure diagram of the improved EMNM is given in Fig.4. As can be seen from Fig.4, the EMNM is divided into ten layers. The number of nodes in the first layer, the second layer and the third layer are twelve respectively. The number of nodes in the fourth layer is twenty-four. The number of nodes in the fifth layer is decided by the number of PM segmentations and the position of rotor. The number of nodes in the 6th layer, the 7th layer and the 8th layer are 6m+2 respectively. The number of nodes in the 9th layer is 8m. The number of nodes in the 10th layer is one.
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Fig.4. Structure diagram of the EMNM of the interior PM machine having single-layer Utype PM structure When the rotor rotates to certain position, the MMF equation of the whole EMNM can be described as follow GF Φ
(3)
Where G is the permeance matrix of the whole EMNM; F is the column vector that is composed of the magnetic potential at each node of the whole EMNM; Φ is the column vector that is composed of the equivalent flux source at each node of the whole EMNM. The resolving flow chart of magnetic field utilizing EMNM is shown in Fig.5. In order to consider the saturation of the iron in the solving process, the iterative calculation is needed to get the correct permeability of each nonlinear permeance unit. The permeance matrix G of the whole model can be listed when the iterative calculation is convergent. Further, the magnetic potential at each node can be calculated by solving formula (3). Then the flux density distribution at each permeance unit can be calculated.
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Fig.5. Resolving flow chart of the EMNM. 2) Solving working points of PMs Fig.6 shows the relationship diagram between the demagnetization curve of PM and the magnetization curve of external magnetic circuit.
Fig.6. The relationship diagram between the demagnetization curve of PM and the magnetization curve of external magnetic circuit. In Fig.6, Λn=f(Фm) is the magnetization curve of external magnetic circuit, and the point of intersection between it and the demagnetization curve of PM is the working point of PM (ao is the working point of PM under no load condition, and al is the working point of PM under load condition). The working point of PM can be represented by the coordinate (Фm, Fm). Under no load condition, it is (Фmo, Fmo), and under load condition, it is (Фml, Fml). In order to simplify the calculation, the per-unit value can be used to represent the coordinate of working point of PM. That is, the working point of PM can be represented by (fm, bm), and fm=Fm/Fc, bm=Фm/Фr. The demagnetization curve of PM can be determined uniquely as long as the type and temperature of PM is certain. Therefore, the working point of PM can be determined uniquely as long as the bm or fm is achieved. In this paper, the bm is calculated to determine the working point of PM. Meanwhile, the larger the value of bm is, the farther
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the working point of PM from the knee point of demagnetization curve of PM will be, and the more difficult the irreversible demagnetization in PM will be. The relationship diagram shown in Fig.6 can be converted to the equivalent magnetic circuit diagram correspondingly, as shown in Fig.7. In Fig.7, Λ0 represents the permeance of PM. Λσ represents the leakage permeance of external magnetic circuit. Λδ represents the main permeance of external magnetic circuit. The part in the dashed box represents the external magnetic circuit of the machine. Фmo and Фml represent the flux provided to the external magnetic circuit by PM under no load and load condition, and they are corresponding to the Фmo and Фml in Fig.6. Фr is the flux of the equivalent magnetic flux source of PM, and is corresponding to the Фr in Fig.6. Meanwhile, Фr=BrAm, where Br is the remeance of PM; Am is the sectional area of flux provided by PM.
Fig.7. The equivalent magnetic circuit under no load and load condition. After achieving the magnetic potential at each node, the working points of PMs can be obtained, and the demagnetization in PMs can be estimated in the early design stage of the machine. By calculating the ratio between the flux provided to the external magnetic circuit by PM and the flux of the equivalent flux source of PM, the working point of PM can be obtained uniquely [3][20-22], as shown in formula (4) and (5). In addition, the larger the ratio is, the farther the working point of PM from the knee point in the demagnetization curve of PM will be, that is, the difficult the irreversible demagnetization in PM will be. m (i ) bm (i ) seg m m (i) bm i 1 m seg
(4)
0 (i) FGM (i)GMseg m (i) seg 0 (i)
(5)
Where i=1,2,3,…,m. Φseg is the flux value of the equivalent flux source of the segmented PM, and it is corresponding to the Φr in Fig.6 and Fig.7. △FGM(i) is the magnetic potential difference between two nodes of the permeance unit of the ith segmented PM, and it is corresponding to the magnetic potential difference between two nodes of Λ0 in Fig.7. GMseg is the permeance of the segmented PM, and it is corresponding to the Λ0 in Fig.7. Φ0(i) is the
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flux flowing through the permeance unit of the ith segmented PM, and it is corresponding to the Ф0o and Ф0l in Fig.7. Φm(i) is the flux provided to the external magnetic circuit by the ith segmented PM, and it is corresponding to the Фmo and Фml in Fig.7. bm(i) is the ratio between the flux provided to the external magnetic circuit by the ith segmented PM and the flux of the equivalent flux source of the ith segmented PM, which determined working point of the ith segmented PM. bm is the average value of bm(i) under certain PM, which determines the average value of the working points of certain PM. It can be seen from Fig.1 that each magnetic pole has three PMs, so the bm of three PMs under one magnetic pole are calculated respectively. The bm of the left PM, the middle PM, and the right PM under one magnetic pole are represented by bm_L , bm_M , and bm_R respectively. C. Validating by FEM The calculated results by improved EMNM are compared with those calculated by FEM to validate the improved EMNM. Fig.8 shows the fundamental magnitudes of the radial components of air-gap flux density under no-load and rated load conditions calculated by the improved EMNM when the number of PM segmentations takes different values. It can be seen from Fig.8 that when the number of PM segmentations m is equal or greater than 4, the fundamental magnitudes of air-gap flux density almost change no more, that is, the calculation results converge. Therefore, the number of PM segmentations m needs to be equal or greater than 4 when the established improved EMNM is utilized to analyze the machine. In order to make computation time not too long, the number of PM segmentations is set to be 4, that is, m=4.
Fig.8. Fundamental magnitudes of radial components of air-gap flux density under different numbers of PM segmentations. (a) no-load; (b) rated load. 1) Solving and validating working points of PMs The FEA software used in the manuscript to determine the working point of PM is the Ansys Maxwell. In order to determine the working point of PM, the flux density component along the magnetizing direction of PM at every node of the PM needs to be calculated at first. The diagram of the flux density decomposition in PM is shown in Fig.9.
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Fig.9. Diagram of flux density decomposition in PM. ‘1’represents PM, ‘2’, ‘3’ and ‘4’ represent air flux barrier, ‘5’ and ‘6’ represent flux barrier. The flux density component along the corresponding magnetizing direction of PM at each node of the PM can be achieved according to Fig.9. The flux density component along the corresponding magnetizing direction of PM at certain node of N1 (the left PM), N2 (the middle PM) and N3 (the right PM) under certain N magnetic pole is derived as follows BN1 BN1x cos(0 rNh ) BN1y sin(0 rNh ) BN2 BN2x cos rNh BN2y sin rNh B B cos( ) B sin( ) N3x 0 rNh N3y 0 rNh N3
(6)
Where BNa is the flux density component along the magnetizing direction of certain PM at certain node of certain PM under N magnetic pole, and a=1,2,3, represents the left PM, the middle PM and the right PM under one magnetic pole respectively. BNax and BNay represent the flux density components along the x-axis and y-axis directions at certain node of certain PM under N magnetic pole respectively. θ0 is the angle between the center line of corresponding magnetic pole and the line that is parallel to the magnetizing direction of the side PM in each magnetic pole. θrNh is the angle between the center line of the hst N magnetic pole and the positive direction of the x-axis respectively, and h=1,2,…,p, p represents the number of pole pairs. The flux density component along the corresponding magnetizing direction of PM at certain node of S1 (the left PM), S2 (the middle PM) and S3 (the right PM) under certain S magnetic pole is derived as follows
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BS1 BS1x cos(0 rSh +π) BS1y sin(0 rSh +π) BS2 BS2x cos( rSh +π) BS2y sin( rSh +π) B B cos( +π) B sin( +π) S3x 0 rSh S3y 0 rSh S3
(7)
Where BSa is the flux density component along the magnetizing direction of certain PM at certain node of certain PM under S magnetic pole, and a=1,2,3, represents the left PM, the middle PM and the right PM under one magnetic pole respectively. BSax and BSay represent the flux density components along the x-axis and y-axis directions at certain node of certain PM under S magnetic pole respectively. θrSh is the angle between the center line of the hst S magnetic pole and the positive direction of the x-axis respectively, and h=1,2,…,p, p represents the number of pole pairs. By importing the derived expressions of the flux density component along the corresponding magnetizing direction of PM at each node of the PM to the FEA software, the flux density distribution in the PM along the corresponding magnetizing direction of PM can be calculated by FEA. The flux density distributions in PMs along the corresponding magnetizing directions of PMs are calculated by FEM under no load and rated load conditions, as shown in Fig.10.
Fig.10. Distributions of flux density in PMs along the corresponding magnetizing direction by FEM. (a) no load; (b) rated load In the FEA method, the PM is divided into many meshes. In the jth mesh, by calculating its flux density component Bj along the corresponding magnetizing direction of PM, the flux provided to the external magnetic circuit by the mesh is Фmj=BjAj. Aj is the sectional area of flux provided by the jth mesh in PM. According to the definition, the bmj of the jth mesh in PM can be calculated as follows bmj
mj B j Aj B j rj Br Aj Br
(8)
The average value bm of the bmj in certain PM can be calculated further as follows l
l
bm
B
l
j bmj B B j j 1
l
j 1
l
r
j 1
lBr
(9)
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Where l is the number of mesh in the PM. The bm(i) and bm under no-load and rated-load conditions by improved EMNM are shown in Table III. Further, the bm_L , bm_M , and bm_R achieved by FEM, improved EMNM and traditional EMNM (PMs under each magnetic pole are not segmented, and the permeance networks distribution in the rotor pole shoe is simplified) are listed in Table IV respectively. The traditional EMNM refers to that PMs under each magnetic pole are not segmented, and the permeance networks model in the rotor pole shoe is simplified. The magnetic potential at each node of traditional EMNM is calculated by solving traditional EMNM, and then bm of PM can be calculated by formula (4) and (5). Similarly, the calculation method for working points of PMs by improved EMNM is to calculate magnetic potential at each node by the improved EMNM established in the manuscript. Then the bm of PM can be calculated by formula (4) and (5). Unlike the traditional EMNM, the bm(i)of segmented PM under certain PM can be calculated by improved EMNM. TABLE III Working points of PMs by improved EMNM No load Rated load b b bm bm 0.8717 0.7705 Left PM 0.8717 0.8717 0.7968 0.7992 0.8717 0.8122 0.8718 0.8173 0.8716 0.8175 Middle PM 0.8714 0.8715 0.8172 0.8173 0.8714 0.8172 0.8716 0.8174 0.8721 0.8490 Right PM 0.8718 0.8719 0.8317 0.8292 0.8718 0.8184 0.8718 0.8175 m
m
TABLE IV The calculated working points of PMs by different methods bm_L
No-load
bm_M bm_R
FEM 0.859 0.867 0.859
Improved EMNM 0.8717 0.8715 0.8719
Traditional EMNM 0.8716 0.8716 0.8717
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Rated load
bm_L bm_M bm_R
0.773 0.7992 0.821 0.8173 0.833 0.8292
0.8145 0.8173 0.8146
It can be seen from Fig.10 that the flux density distributions in the left PM and the right PM under one magnetic pole along the corresponding magnetizing directions of PMs are the same under no-load condition. Therefore, the working points of the left PM and the right PM under each magnetic pole are the same under no-load condition. The armature current is ahead of the back-EMF under rated-load condition, which makes the function of the demagnetization field on the left PM under each magnetic pole stronger. Therefore, the working point of the left PM is lower than that of the right PM under each magnetic pole under load condition. Meanwhile, under load condition, the working point of the segmented PM in the left PM shifts up gradually with the increase of the distance between the segmented PM and rotor surface. At the same time, the working point of the segmented PM in the right PM shifts down gradually with the increase of the distance between the segmented PM and rotor surface. It can be seen from Table IV that under no-load condition, the calculated bm of PMs by the traditional EMNM have little difference from those by FEM. However, under load condition, the calculated bm_L , bm_M , and bm_R of PMs by the traditional EMNM are 0.8145, 0.8173, and 0.8146 respectively. The working points of the left PM and the right PM under one magnetic pole are almost the same under load condition, which is not in accordance with the actual situation. Therefore, the demagnetization in PM cannot be estimated accurately by utilizing the traditional EMNM. It can be seen from Table IV that the errors of the calculated bm of PMs between FEM and improved EMNM are little under no load and load conditions. The improved EMNM established in this paper refines the networks in the rotor pole shoe by segmenting the PM, which makes the flux path in the rotor pole shoe modelled more accurately. It can be seen from the calculation results in Table III that the working points of the left PM and the right PM under one magnetic pole by improved EMNM are almost the same under no-load condition. Meanwhile, by the improved EMNM, the working point of the right PM is larger than that of the left PM under load condition. In addition, it can also be seen from Table III that under load condition, the working point of the segmented PM in the left PM by the improved EMNM shifts up gradually with the increase of the distance between the segmented PM and rotor surface. At the same time, the working point of the segmented PM in the right PM by the improved EMNM shifts down gradually with the increase of the distance between the segmented PM and rotor surface. These change trends about the working points of PMs under each magnetic pole achieved by improved EMNM are consistent with those achieved by FEM. Therefore, the working points of PMs can be determined more accurately by the improved EMNM proposed in this paper. Further, the demagnetization in PMs can be estimated more accurately in the early design stage of
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machine. 2) Solving and validating air-gap flux density and electromagnetic performances When the rotor rotates to certain position, the air-gap flux density waveforms by the improved EMNM and FEM are shown in Fig.11 (no-load) and Fig.12 (rated load) respectively. Meanwhile, the fundamental magnitudes of the radial component of the airgap flux density Bgmr_O(no-load) and Bgmr_RL(rated load) by the improved EMNM and FEM are also listed in Table V respectively. It can be seen from Fig.11 and Fig.12 that the air-gap flux density distributions by two methods have a good consistency. Moreover, it can be seen from Table V that the fundamental amplitudes of the radial component of the air-gap flux density by two methods have differences, but the errors are acceptable.
Fig.11. Air-gap flux density distribution by improved EMNM and FEM under no load condition. (a) radial component; (b) tangential component
Fig.12. Air-gap flux density distribution by improved EMNM and FEM under rated load condition. (a) radial component; (b) tangential component TABLE V COMPARISON OF THE CALCULATION RESULTS BETWEEN IMPROVED EMNM AND FEM
Bgmr_O E0 Bgmr_RL Tem Psl_RL
FEM 0.8022 T 494V 0.9792T 951.4Nm 1549W
Improved EMNM 0.8399 T 491.8V 0.9452T 1003Nm 1550.1W
Relative error 4.7% -0.45% -3.06% 5.42% 0.071%
After achieving the magnetic field in each part of the machine by EMNM, the no-load back-EMF E0 under rated speed, the electromagnetic torque Tem and the stator iron loss
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Psl_RL under rated load condition can be calculated further. Moreover, the calculation results by the improved EMNM and FEM are all shown in Table V. It can be seen from Table V that the errors of the calculation results between two methods are acceptable. III. Effect of Machine Parameters When the interior PM machine operates under load condition, the armature current is ahead of back-EMF. The negative d-axis component of armature current makes demagnetization appeared in PMs easily. Effects of d-axis component and q-axis component of armature current on working points of PMs are analyzed in this paper. Airgap length and flux barrier size are key parameters in the machine. Their sizes influence the permeance of the magnetic circuit and leakage flux of magnetic field, and influence the working points of PMs further. Therefore, their effects on working points of PMs are also researched. A. Effect of armature current This section researches effect of the armature current on working points of PMs. Fig.13 shows effects of d-axis component and q-axis component of armature current on working points of PMs. It can be seen from Fig.13 that when the q-axis component of armature current is constant, the working point of each PM under each magnetic pole shifts down with the increase of the negative d-axis component of armature current. This is because that the increase of negative d-axis component of armature current strengthens the demagnetizing field produced by armature current. Meanwhile, it also makes that the directions of armature reaction field acting on PMs are closer to the opposite direction of the magnetizing directions of PMs. At the same time, when the negative d-axis component of armature current is smaller, the working point of each PM under each magnetic pole shifts down with the increase of q-axis component of armature current. However, when the negative d-axis component of armature current reaches certain value, the working point of each PM under each magnetic pole shifts up with the increase of q-axis component of armature current. On the whole, the q-axis component of armature current has little effect on working points of PMs. The d-axis component of armature current has main effect on the working points of PMs.
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Fig.13. Effects of armature current on working points of PMs. (a) right PM; (b) middle PM; (c) left PM B. Effect of air-gap length We usually think that the larger the air-gap length is, the more difficult the irreversible demagnetization in PMs will be. However, above viewpoint proves to be not true by analysis. Fig.14 shows the variation of working points of PMs with the variation of air-gap length. It can be seen from Fig.14 that whether the machine operates under no-load or rated load condition, the working points of PMs shift down with the increase of air-gap length. In addition, with the change of air-gap length, the variations of working points of PMs under load condition are smaller than that under no-load condition. On one hand, increase of airgap length makes the leakage flux of armature reaction field increased, and makes the function of demagnetizing field on the PMs weakened further. On the other hand, increase of air-gap length also makes permeance of external magnetic circuit decreased, which makes the no-load working points of PMs shifted down. Therefore, although increase of airgap length makes the function of demagnetizing field on the PMs weakened, shifting down of no-load working points of PMs consequently may increase the risk that the irreversible demagnetization appears in PMs under the demagnetizing current.
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Fig.14. Effects of air-gap length on working points of PMs. (a)no-load; (b) rated load. C. Effect of flux barrier size Flux barrier is an important part of the interior PM machine. Its size influences the permeance of magnetic circuit and the leakage flux of magnetic field, and influences the working points of PMs further. Fig.15 and Fig.16 show effects of the flux barrier size on working points of PMs under no-load and rated load conditions respectively. It can be seen from two figures that the working points of PMs under no-load and rated load conditions shift up with the increase of flux barrier thickness or the decrease of flux barrier width. This is because that increasing flux barrier thickness or decreasing flux barrier width makes the permeance of the external magnetic circuit increased. Meanwhile, increasing flux barrier thickness or decreasing flux barrier width also makes the leakage flux of armature reaction field increased, which makes the function of armature reaction field on the PMs weakened. Moreover, with the change of flux barrier size, the variations of working points of PMs under load condition are larger than that under no-load condition.
Fig.15. Effects of flux barrier size on no-load working points of PMs. (a) right PM; (b) middle PM; (c) left PM.
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Fig.16. Effects of flux barrier size on working points of PMs under rated load condition. (a) right PM; (b) middle PM; (c) left PM. Moreover, it can be seen from Fig.15-Fig.16 that whether the machine operates under noload or rated load condition, the variations of working points of PMs with the variation of flux barrier width are all smaller than that with the variation of flux barrier thickness. That is, the flux barrier thickness has a larger effect on the working points of PMs. At the same time, it can also be seen from figures that the larger the flux barrier thickness is, the smaller the variations of working points of PMs with the variation of flux barrier width will be. In addition, the smaller the flux barrier width is, the smaller the variations of working points of PMs with the variation of flux barrier thickness will be. IV. CONCLUSION An improved EMNM is established in this paper aiming at the interior PM machine. By the improved EMNM, working points of PMs are determined accurately under no-load and load condition, which makes it possible to analyze demagnetization in PM in the early design stage of machine. Meanwhile, effects of d-axis component and q-axis component of armature current, air-gap length and flux barrier size on the working points of PMs are analyzed by utilizing the improved EMNM, and some conclusions are drawn: (1) d-axis component of armature current has main effect on the working points of PMs, and the q-axis component has little effect; (2) Increase of air-gap length may increase the risk that irreversible demagnetization appear in PMs; (3) Increasing flux barrier thickness or decreasing its width strengthens the antidemagnetization abilities of PMs. Moreover, the flux barrier thickness has larger effect on working points of PMs than flux barrier width.
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ACKNOWLEDGMENT This work was supported in part by the project supported by Major Program of National Natural Science Foundation of China under Grant 51690180, in part by the National Natural Science Foundation of China under Grant 51507111. REFERENCES [1] Changliang Xia，Liyan Guo， Zhen Zhang，Tingna Shi，Huimin Wang．Optimal Designing of Permanent Magnet Cavity to Reduce Iron Loss of Interior Permanent Magnet Machine． IEEE Transactions on Magnetics，2015，51(12)：8115409． [2] Peixin Liang, Feng Chai, Yunlong Bi, Yulong Pei, Shukang Cheng, Analytical Model and Design of Spoke-Type Permanent-Magnet Machines Accounting for Saturation and Nonlinearity of Magnetic Bidges, Journal of Magnetism and Magnetic Materials, 2016, 417:389-396. [3] Tang Renyuan, "Modern Permanent Magnet Machines-Theory and Design," Beijing: Machinery Industry press, 1997, pp. 45-48, 164-174. [4] Vipulkumar I. Patel, Jiabin Wang, Sreeju S. Nair, Demagnetization Assessment of Fractional-Slot and Distributed Wound 6-Phase Permanent Magnet Machines, IEEE Transactions on Magnetics, 2015, 51(6):8105511. [5] Ki-Doek Lee, Won-Ho Kim, Chang-Sung Jin, Ju Lee, Local demagnetisation analysis of a permanent magnet motor, IET Electric Power Applications, 2015, 9(3):280-286. [6] Hong Chen, Ronghai Qu, Jian Li, and Dawei Li, Demagnetization Performance of a 7 MW Interior Permanent Magnet Wind Generator With Fractional-Slot Concentrated Windings, IEEE Transactions on Magnetics, 2015, 51(11):8205804. [7] Zhen Zhang, Changliang Xia, Yan Yan, Qiang Gen, Tingna Shi, A Hybrid Analytical Model for Open-Circuit Field Calculation of Multilayer Interior Permanent Magnet Machines, Journal of Magnetism and Magnetic Materials, 2017, 435:136-145. [8] Ki-Chan Kim, Kwangsoo Kim, Hee Jun Kim, Ju Lee, Demagnetization Analysis of Permanent Magnets According to Rotor Types of Interior Permanent Magnet Synchronous Motor, IEEE Transactions on Magnetics, 2009, 45(6):2799-2802. [9] Jian-Xin Shen, Peng Li, Meng-Jia Jin, Guang Yang, Investigation and Countermeasures for Demagnetization in Line Start Permanent Magnet Synchronous Motors, IEEE Transactions on Magnetics, 2013, 49(7):4068-4071. [10] Shi Wei, "The Research of Anti-Demagnetization Technology of Permanent Magnet in High Density Permanent Magnet Motor," Shanghai: Shanghai University, 2012. [11] Ming Cheng, K. T. Chau, C. C. Chan, E. Zhou, X. Huang, Nonlinear VaryingNetwork Magnetic Circuit Analysis for Doubly Salient Permanent-Magnet Motors, IEEE Transactions on Magnetics, 2000, 36(1):339-348. [12] Yoshiaki Kano, Takashi Kosaka, Nobuyuki Matsui, Simple Nonlinear Magnetic Analysis for Permanent-Magnet Motors, IEEE Transactions on Industry Applications, 2005, 41(5):1205-1214. [13] Z. Q. Zhu, Y. Pang, D. Howe, S. Iwasaki, R. Deodhar, A. Pride, Analysis of Electromagnetic Performance of Flux-Switching Permanent-Magnet Machines by
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Nonlinear Adaptive Lumped Parameter Magnetic Circuit Model, IEEE Transactions on Magnetics, 2005, 41(11):4277-4287. [14] Abdul Rehman Tariq, Carlos E. Nino-Baron, Elias G. Strangas, Iron and Magnet Losses and Torque Calculation of Interior Permanent Magnet Synchronous Machines Using Magnetic Equivalent Circuit, IEEE Transactions on Magnetics, 2010, 46(12):4073-4080. [15] Jagadeesh K. Tangudu, T. M. Jahns, Ayman EL-Refaie, Core Loss Prediction Using Magnetic Circuit Model for Fractional-Slot Concentrated-Winding Interior Permanent Magnet Machines, 2010 IEEE Energy Conversion Congress and Exposition, 2010, 10041011. [16] B. Sheikh-Ghalavand, S. Vaez-Zadeh, A. Hassanpour Isfahani, An Improved Magnetic Equivalent Circuit Model for Iron-Core Linear Permanent-Magnet Synchronous Motors, IEEE Transactions on Magnetics, 2010, 46(1):112-120. [17] Abdul Rehman Tariq, Carlos E. Nino, Elias G. Strangas, A novel numerical method for the calculation of iron and magnet losses of IPMSMs, IEEE International Electric Machines and Drives Conference, 2009, 1605-1611. [18] Seok-Hee Han, Thomas M. Jahns, Wen L. Soong, A Magnetic Circuit Model for an IPM Synchronous Machine Incorporating Moving Airgap and Cross-Coupled Saturation Effects, 2007 IEEE International Electric Machines & Drives Conference, 2007, 21-26. [19] D. J. Gómez; A. L. Rodríguez; I. Villar; A. López-de-Heredia; I. Etxeberria-Otadui; Z. Q. Zhu, Improved permeance network model for embedded magnet synchronous machines, 2014 International Conference on Electrical Machines (ICEM), 2014, 12311237. [20] Jun-Young Son, Jin Hwan Lee, Dae-Woo Kim, Yong-Jae Kim, Sang-Yong Jung, Analysis and Modeling of Concentrated Winding Variable Flux Memory Motor Using Magnetic Equivalent Circuit Method, IEEE Transactions on Magnetics, 2017, 53(6): 8102704. [21] Jun-Young Song, Jin Hwan Lee, Dae-Woo Kim, Yong-Jae Kim, Sang-Yong Jung, Analysis and Modeling of Permanent Magnet Variable Flux Memory Motors Using Magnetic Equivalent Circuit Method, IEEE Transactions on Magnetics, 2017, 53(11), in press. [22] Hui Yang, Heyun Lin, Jianning Dong, Jiahun Yan, Yunkai Huang, Shuhua Fang, Analysis of a Novel Switched-Flux Memory Motor Employing a Time-Divisional Magnetization Strategy, IEEE Transaction on Magnetics, 2014, 50(2): 7021004.
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Highlights 1. Proposing improved equivalent magnetic network model for interior PM machine 2. Calculating working points of PMs accurately 3. Analyzing effects of machine parameters on working points of PMs
S0304-8853(17)32278-3 https://doi.org/10.1016/j.jmmm.2018.01.018 MAGMA 63596
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
19 July 2017 20 December 2017 8 January 2018
Please cite this article as: L. Guo, C. Xia, H. Wang, Z. Wang, T. Shi, Improved Equivalent Magnetic Network Modeling for Analyzing Working Points of PMs in Interior Permanent Magnet Machine, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/j.jmmm.2018.01.018
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Improved Equivalent Magnetic Network Modeling for Analyzing Working Points of PMs in Interior Permanent Magnet Machine Liyan Guo1, Changliang Xia1,2, Huimin Wang2, Zhiqiang Wang2, Tingna Shi1 1
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China 2 Tianjin Engineering Center of Electric Machine System Design and Control, Tianjin Polytechnic University, Tianjin 300387, China As is well known, the armature current will be ahead of the back electromotive force (back-EMF) under load condition of the interior permanent magnet (PM) machine. This kind of advanced armature current will produce a demagnetizing field, which may make irreversible demagnetization appeared in PMs easily. To estimate the working points of PMs more accurately and take demagnetization under consideration in the early design stage of a machine, an improved equivalent magnetic network model is established in this paper. Each PM under each magnetic pole is segmented, and the networks in the rotor pole shoe are refined, which makes a more precise model of the flux path in the rotor pole shoe possible. The working point of each PM under each magnetic pole can be calculated accurately by the established improved equivalent magnetic network model. Meanwhile, the calculated results are compared with those calculated by FEM. And the effects of d-axis component and q-axis component of armature current, air-gap length and flux barrier size on working points of PMs are analyzed by the improved equivalent magnetic network model. Index Terms—interior permanent magnet machine, improved equivalent magnetic network model, working point of PM. I. Introduction The interior permanent magnet (PM) machine has many advantages, such as high efficiency, high power density, and wide speed range, which makes it widely used in automobile, locomotive traction, industrial robot, and aerospace etc.[1]-[2]. The PMs in the interior PM machine are protected by the rotor pole shoe, which makes the machine more suitable for the situations that require high machine speed[3]. When the interior PM machine operates under load condition, the armature current is ahead of back electromotive force (back-EMF). Especially when the machine operates under high speed by the fluxweakening control, the angle between the armature current and the back-EMF is larger, which makes irreversible demagnetization appeared in PMs more easily. Accurately estimating working points of PMs in the early design stage of machine has an important significance in determining the operation range of the machine.
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The Finite Element Method (FEM) is often adopted to analyze demagnetization in PM. By utilizing the FEM, the flux density component in PM along the magnetization direction of PM can be calculated, and the demagnetization in PM can be analyzed [4]-[6]. However, the parameters of the machine need to be changed frequently in the early design stage and the optimization design stage of the machine. The FEM will cost a lot of time and computer resources in the modeling and computation process. While the equivalent magnetic circuit method that is suitable for the interior PM machine will cost less[7]. However, using the equivalent magnetic circuit method, only the average value of working points of PMs under one magnetic pole can be achieved[3]. Many kinds of interior PM machine adopt the “V” type, “U” type, “W” type and multilayer PM structure today. In these cases, each magnetic pole contains several pieces of PMs[8]. When the machine operates under load condition, the magnitudes and directions of armature reaction field acting on the left and the right PM under one magnetic pole are different. That is, the working points of the left and right PM are different[9]. The problem is that the difference of working point in the left and the right PM cannot be calculated by the equivalent magnetic circuit method. Therefore, the demagnetization in PM cannot be predicted timely by utilizing the equivalent magnetic circuit method. Compared with the equivalent magnetic circuit method, the equivalent magnetic network model (EMNM) can divide networks more finely, and it also costs less computation time. The EMNM is built on the basis of the flux tube theory. Certain part in the machine having the same material, the uniform flux distribution and regular shape is equivalent to one permeance unit. Permeance units are connected through nodes. The magnetic potential at each node can be achieved by solving the magnetic potential equation at each node, and then performance parameters can be achieved further[10]. In the EMNM, the saturation of iron in the machine can be modeled by the nonlinear permeance [11]-[13]. The leakage flux in the flux barrier can be modeled by the flux flowing through the nonlinear permeance unit in the flux barrier[14]-[15]. Moreover, effect of the stator slot opening can also be considered in the EMNM. The flux lines distribution in the stator slot opening can be equivalent to the quarter circle[16], and then effect of the stator slot opening can be considered by setting permeance units in the stator slot opening. The published literatures have established EMNMs aiming at the interior PM machines with different rotor magnetic circuit structures, and calculated the magnetic field and electromagnetic performance of machine by the established model further. Reference [15] has established the EMNM aiming at the interior PM machine with V-type PM structure. Meanwhile, it calculates the magnetic field distribution in each point of iron, and calculates the iron loss of machine further. References [14] and [17] have established the EMNMs aiming at the interior PM machines with double layer U-type PM structure, which was used to calculate the iron loss and magnet eddy current loss of machine further. References [18] and [19] have established the EMNM separately aiming at the interior PM machine with multi-layer U-type PM structure and “―”type PM structure to calculate the magnetic field in two machines. Corresponding author: Changliang Xia (e-mail: [email protected]).
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The rotor pole shoe is a part of the flux path in the interior PM machine. If the rotor pole shoe can be modeled accurately, the flowing path of magnetic field will be modeled accurately. However, the rotor pole shoe often has irregular shape and large volume in the interior PM machine, which makes it difficult to model the rotor pole shoe accurately. In order to model the flux path in the rotor pole shoe and calculate the working points of PMs accurately, an improved EMNM is established aiming at the interior PM machine in this paper. In the improved EMNM, each PM under each magnetic pole is segmented, which makes the networks in the rotor pole shoe refined. Meanwhile, both a radial permeance unit and a tangential permeance unit are set for each refined network in the rotor pole shoe. The working points of PMs under no-load and load conditions can be calculated accurately by the established improved equivalent magnetic network model. Further, the demagnetization in PMs can be estimated accurately in the early design stage of the machine. Finally, effects of the armature current, air-gap length and flux barrier size on the working points of PMs are analyzed by utilizing the established improved EMNM. II. Improved Equivalent Magnetic Network Modeling An improved EMNM is established aiming at the interior PM machine in this paper. The structure of interior PM machine is shown in Fig.1, and related parameters of the machine are listed in Table I[1]. The EMNM of one-unit machine is established in this paper to reduce the calculation quantity. One-unit machine has one pole pair, twelve stator slots and twelve stator teeth.
Fig. 1. Structure of the machine. 1-stator; 2-stator winding; 3-rotor; 4-air-gap; 5-PM; 6, 7flux barrier; 8, 9, 10-air flux barrier TABLE I Related parameters of the machine Symbol Quantity p Pole pair number Q Slot number n Rated speed
Value 4 48 1800
Unit --r/min
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IN α Rra δ Rsy Br μr hPM wPM_side wPM_middle l
RMS of rated current Current angle of rated current Outer radius of rotor Air-gap length Outer radius of stator Remeance Relative permeance of PM Thickness of PM Width of the side PM Width of the middle PM Axial length of PM
166 13.68 148.2 1.8 232.5 1.19 1.121 16 30 23 210
A ° mm mm mm T -mm mm mm mm
A. Permeance distribution in each part of machine Certain part in the machine having the same material, the uniform flux distribution and regular shape is equivalent to one permeance unit according to the flux path in the machine. Further, the permeance units distribution in the whole machine can be achieved, as shown in Fig.2.
Fig. 2. Permeance distribution in each part of the machine. (a) stator; (b) rotor; (c) air-gap The permeance units distribution in the stator is shown in Fig.2(a). The magnetomotive force (MMF) source in each tooth represents the equivalent MMF source produced by the armature current under load condition[9]. Because the tooth width along the radial direction is different, each tooth is equivalent to three permeance units. The stator yoke between two stator teeth is equivalent to one permeance unit. The flux lines distribution in the stator slot
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opening can be equivalent to the quarter circle. So the permeance unit distribution in the stator slot opening can be achieved accordingly. Meanwhile, the slot leakage permeance unit is added between adjacent teeth to calculate leakage flux flowing through the stator slot. Rotor pole shoe in the interior PM machine is a part of flux path. However, the rotor pole shoe has irregular shape and large volume in the interior PM machine, which makes it difficult to model the rotor pole shoe accurately. In order to establish an accurate model for rotor pole shoe, each PM under each magnetic pole is segmented, which also makes the networks in the rotor pole shoe refined. Meanwhile, both a radial permeance unit and a tangential permeance unit are set aiming at each network in the rotor pole shoe to model the flux path more accurately, as shown in Fig.2(b). The equivalent flux source and permeance of the segmented PM are calculated using formula (1) and formula (2). seg
GMseg
Br wPM l m
0 r wPM l mhPM
(1) (2)
where m is the number of PM segmentations. Φseg is the equivalent flux source of the segmented PM. GMseg is the permeance of the segmented PM. Br is the remeance of the PM. wPM is the width of the PM. l is the axial length of the PM. hPM is the thickness of the PM. μ0 is the permeability of the vacuum. μr is the relative permeability of the PM. In order to calculate the tangential component of air-gap flux density, two-layer air-gap permeance model is adopted in this paper, as shown in Fig.2(c). Because the rotor rotates, the permeance distribution in the air-gap will change at every moment. At different moments, the permeance of the radial permeance units and the tangential permeance units in the air-gap can be achieved by calculating the arc length of the overlapping regions between the units in the rotor and stator that are next to the air-gap, as shown in Fig.3. In Fig.3, “Teeth1” represents certain teeth in stator. “1”, “2”, and “3” represent three regions in the rotor which are adjacent to the air gap. Fig.3 only takes the air gap between “Teeth1” and three regions in rotor as an example to explain the permeance units distribution in the air gap. The analysis results can be applied to calculate the permeance units distribution in the whole air gap.
Fig.3 Sketch map of calculation for permeance of permeance units in air gap
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It can be seen from Fig.3 that at certain moment t=t1, “Teeth 1” has overlapping regions with rotor region “2” and rotor region “3”. The arc length of the overlapping region between “Teeth 1” and rotor region “2” is l2. The arc length of the overlapping region between “Teeth 1” and rotor region “3” is l3. At next moment t=t2, “Teeth 1” only has overlapping region with rotor region “2”. The arc length of the overlapping region between “Teeth 1” and rotor region “2” is l2. At moment t=t3, “Teeth 1” has overlapping regions with rotor region “1” and “2”. The arc length of the overlapping region between “Teeth 1” and rotor region “1” is l1, and the arc length of the overlapping region between “Teeth 1” and rotor region “2” is l2. And so on, if the electrical angle in radian corresponding to “Teeth1” is [θ1,θ2], and the electrical angle in radian corresponding to rotor region“i” is [αi,αi+1], i=1,2,3, the arc length of the overlapping region between “Teeth1” and rotor region “i” li can be calculated accordingly at certain moment. At certain moment, (1) If αi+1<θ1 or αi>θ2, li=0. (2) If αi>θ1 and αi+1<θ2, li=Rg(αi+1-αi)/p, where p represents pole pair number of machine, and Rg represents the radius in the middle of air gap. (3) If αi<θ1 and θ1≤αi+1≤θ2, li=Rg(αi+1-θ1)/p. (4) If θ1≤αi≤θ2 and αi+1>θ2, li=Rg(θ2-αi)/p. (5) If αi<θ1 and αi+1>θ2, li=Rg(θ2-θ1)/p. At different moments, according to the arc length of overlapping regions, the permeance of radial permeance units and tangential permeance units can be calculated. The permeance of radial permeance units in the air gap are calculated by Gri=2μ0lil/g, and the permeance of tangential permeance units in the air gap are calculated by Gti,i+1=2μ0gl/(li+li+1). Where, g is the air gap length; μ0 is the permeability of vacuum; l is the axial length of machine. And so on, the permeance units distribution in the overall air gap region can be achieved at different moments. In Fig.2, the permeance units with dark color represent the nonlinear permeance units, and those with light color represent the linear permeance units. In order to calculate the permeance of each permeance unit in Fig.2, several typical geometric structures of the permeance units and the corresponding calculation formula of the permeance are shown in Table II. The permeance of each permeace unit in Fig.2 can be calculated further. TABLE II Permeance formula of several typical structure Geometry Permeance formula Ga a l1l2 h
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Gb b a1 a2 b 2h
Gc 4cl3 π
The width of stator tooth changes along the radial direction, so the geometric structure of the permeance unit in stator tooth is similar to the trapezoidal permeance unit in Table II. The geometric structures of the permeance units in stator yoke and stator slot are similar to the rectangular permeance unit in Table II. The flux lines distribution in the stator slot opening can be equivalent to quarter circle. So the geometric structures of the permeance units in the stator slot opening are similar to the fan-shaped permeance unit in Table II. The permeance units in the rotor pole shoe are refined by segmenting PMs, and the geometric structure of each refined permeance unit is similar to the rectangular permeance unit or trapezoidal permeance unit in Table II. The geometric structures of the permeance units in the iron between adjacent poles and the permeance units in the flux barriers are similar to the rectangular permeance unit in Table II. The geometric structure of each permeance unit in the air-gap is similar to the rectangular permeance unit in Table II. Therefore, the permeance of permeance units in the machine can be calculated according to the geometric parameters of the machine and the corresponding formula in Table II. B. Solving Improved Equivalent Magnetic Network Model 1) Solving magnetic field According to the permeance unit distributions in each part of the machine that are given in Fig.2, the corresponding structure diagram of the improved EMNM is given in Fig.4. As can be seen from Fig.4, the EMNM is divided into ten layers. The number of nodes in the first layer, the second layer and the third layer are twelve respectively. The number of nodes in the fourth layer is twenty-four. The number of nodes in the fifth layer is decided by the number of PM segmentations and the position of rotor. The number of nodes in the 6th layer, the 7th layer and the 8th layer are 6m+2 respectively. The number of nodes in the 9th layer is 8m. The number of nodes in the 10th layer is one.
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Fig.4. Structure diagram of the EMNM of the interior PM machine having single-layer Utype PM structure When the rotor rotates to certain position, the MMF equation of the whole EMNM can be described as follow GF Φ
(3)
Where G is the permeance matrix of the whole EMNM; F is the column vector that is composed of the magnetic potential at each node of the whole EMNM; Φ is the column vector that is composed of the equivalent flux source at each node of the whole EMNM. The resolving flow chart of magnetic field utilizing EMNM is shown in Fig.5. In order to consider the saturation of the iron in the solving process, the iterative calculation is needed to get the correct permeability of each nonlinear permeance unit. The permeance matrix G of the whole model can be listed when the iterative calculation is convergent. Further, the magnetic potential at each node can be calculated by solving formula (3). Then the flux density distribution at each permeance unit can be calculated.
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Fig.5. Resolving flow chart of the EMNM. 2) Solving working points of PMs Fig.6 shows the relationship diagram between the demagnetization curve of PM and the magnetization curve of external magnetic circuit.
Fig.6. The relationship diagram between the demagnetization curve of PM and the magnetization curve of external magnetic circuit. In Fig.6, Λn=f(Фm) is the magnetization curve of external magnetic circuit, and the point of intersection between it and the demagnetization curve of PM is the working point of PM (ao is the working point of PM under no load condition, and al is the working point of PM under load condition). The working point of PM can be represented by the coordinate (Фm, Fm). Under no load condition, it is (Фmo, Fmo), and under load condition, it is (Фml, Fml). In order to simplify the calculation, the per-unit value can be used to represent the coordinate of working point of PM. That is, the working point of PM can be represented by (fm, bm), and fm=Fm/Fc, bm=Фm/Фr. The demagnetization curve of PM can be determined uniquely as long as the type and temperature of PM is certain. Therefore, the working point of PM can be determined uniquely as long as the bm or fm is achieved. In this paper, the bm is calculated to determine the working point of PM. Meanwhile, the larger the value of bm is, the farther
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the working point of PM from the knee point of demagnetization curve of PM will be, and the more difficult the irreversible demagnetization in PM will be. The relationship diagram shown in Fig.6 can be converted to the equivalent magnetic circuit diagram correspondingly, as shown in Fig.7. In Fig.7, Λ0 represents the permeance of PM. Λσ represents the leakage permeance of external magnetic circuit. Λδ represents the main permeance of external magnetic circuit. The part in the dashed box represents the external magnetic circuit of the machine. Фmo and Фml represent the flux provided to the external magnetic circuit by PM under no load and load condition, and they are corresponding to the Фmo and Фml in Fig.6. Фr is the flux of the equivalent magnetic flux source of PM, and is corresponding to the Фr in Fig.6. Meanwhile, Фr=BrAm, where Br is the remeance of PM; Am is the sectional area of flux provided by PM.
Fig.7. The equivalent magnetic circuit under no load and load condition. After achieving the magnetic potential at each node, the working points of PMs can be obtained, and the demagnetization in PMs can be estimated in the early design stage of the machine. By calculating the ratio between the flux provided to the external magnetic circuit by PM and the flux of the equivalent flux source of PM, the working point of PM can be obtained uniquely [3][20-22], as shown in formula (4) and (5). In addition, the larger the ratio is, the farther the working point of PM from the knee point in the demagnetization curve of PM will be, that is, the difficult the irreversible demagnetization in PM will be. m (i ) bm (i ) seg m m (i) bm i 1 m seg
(4)
0 (i) FGM (i)GMseg m (i) seg 0 (i)
(5)
Where i=1,2,3,…,m. Φseg is the flux value of the equivalent flux source of the segmented PM, and it is corresponding to the Φr in Fig.6 and Fig.7. △FGM(i) is the magnetic potential difference between two nodes of the permeance unit of the ith segmented PM, and it is corresponding to the magnetic potential difference between two nodes of Λ0 in Fig.7. GMseg is the permeance of the segmented PM, and it is corresponding to the Λ0 in Fig.7. Φ0(i) is the
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flux flowing through the permeance unit of the ith segmented PM, and it is corresponding to the Ф0o and Ф0l in Fig.7. Φm(i) is the flux provided to the external magnetic circuit by the ith segmented PM, and it is corresponding to the Фmo and Фml in Fig.7. bm(i) is the ratio between the flux provided to the external magnetic circuit by the ith segmented PM and the flux of the equivalent flux source of the ith segmented PM, which determined working point of the ith segmented PM. bm is the average value of bm(i) under certain PM, which determines the average value of the working points of certain PM. It can be seen from Fig.1 that each magnetic pole has three PMs, so the bm of three PMs under one magnetic pole are calculated respectively. The bm of the left PM, the middle PM, and the right PM under one magnetic pole are represented by bm_L , bm_M , and bm_R respectively. C. Validating by FEM The calculated results by improved EMNM are compared with those calculated by FEM to validate the improved EMNM. Fig.8 shows the fundamental magnitudes of the radial components of air-gap flux density under no-load and rated load conditions calculated by the improved EMNM when the number of PM segmentations takes different values. It can be seen from Fig.8 that when the number of PM segmentations m is equal or greater than 4, the fundamental magnitudes of air-gap flux density almost change no more, that is, the calculation results converge. Therefore, the number of PM segmentations m needs to be equal or greater than 4 when the established improved EMNM is utilized to analyze the machine. In order to make computation time not too long, the number of PM segmentations is set to be 4, that is, m=4.
Fig.8. Fundamental magnitudes of radial components of air-gap flux density under different numbers of PM segmentations. (a) no-load; (b) rated load. 1) Solving and validating working points of PMs The FEA software used in the manuscript to determine the working point of PM is the Ansys Maxwell. In order to determine the working point of PM, the flux density component along the magnetizing direction of PM at every node of the PM needs to be calculated at first. The diagram of the flux density decomposition in PM is shown in Fig.9.
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Fig.9. Diagram of flux density decomposition in PM. ‘1’represents PM, ‘2’, ‘3’ and ‘4’ represent air flux barrier, ‘5’ and ‘6’ represent flux barrier. The flux density component along the corresponding magnetizing direction of PM at each node of the PM can be achieved according to Fig.9. The flux density component along the corresponding magnetizing direction of PM at certain node of N1 (the left PM), N2 (the middle PM) and N3 (the right PM) under certain N magnetic pole is derived as follows BN1 BN1x cos(0 rNh ) BN1y sin(0 rNh ) BN2 BN2x cos rNh BN2y sin rNh B B cos( ) B sin( ) N3x 0 rNh N3y 0 rNh N3
(6)
Where BNa is the flux density component along the magnetizing direction of certain PM at certain node of certain PM under N magnetic pole, and a=1,2,3, represents the left PM, the middle PM and the right PM under one magnetic pole respectively. BNax and BNay represent the flux density components along the x-axis and y-axis directions at certain node of certain PM under N magnetic pole respectively. θ0 is the angle between the center line of corresponding magnetic pole and the line that is parallel to the magnetizing direction of the side PM in each magnetic pole. θrNh is the angle between the center line of the hst N magnetic pole and the positive direction of the x-axis respectively, and h=1,2,…,p, p represents the number of pole pairs. The flux density component along the corresponding magnetizing direction of PM at certain node of S1 (the left PM), S2 (the middle PM) and S3 (the right PM) under certain S magnetic pole is derived as follows
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BS1 BS1x cos(0 rSh +π) BS1y sin(0 rSh +π) BS2 BS2x cos( rSh +π) BS2y sin( rSh +π) B B cos( +π) B sin( +π) S3x 0 rSh S3y 0 rSh S3
(7)
Where BSa is the flux density component along the magnetizing direction of certain PM at certain node of certain PM under S magnetic pole, and a=1,2,3, represents the left PM, the middle PM and the right PM under one magnetic pole respectively. BSax and BSay represent the flux density components along the x-axis and y-axis directions at certain node of certain PM under S magnetic pole respectively. θrSh is the angle between the center line of the hst S magnetic pole and the positive direction of the x-axis respectively, and h=1,2,…,p, p represents the number of pole pairs. By importing the derived expressions of the flux density component along the corresponding magnetizing direction of PM at each node of the PM to the FEA software, the flux density distribution in the PM along the corresponding magnetizing direction of PM can be calculated by FEA. The flux density distributions in PMs along the corresponding magnetizing directions of PMs are calculated by FEM under no load and rated load conditions, as shown in Fig.10.
Fig.10. Distributions of flux density in PMs along the corresponding magnetizing direction by FEM. (a) no load; (b) rated load In the FEA method, the PM is divided into many meshes. In the jth mesh, by calculating its flux density component Bj along the corresponding magnetizing direction of PM, the flux provided to the external magnetic circuit by the mesh is Фmj=BjAj. Aj is the sectional area of flux provided by the jth mesh in PM. According to the definition, the bmj of the jth mesh in PM can be calculated as follows bmj
mj B j Aj B j rj Br Aj Br
(8)
The average value bm of the bmj in certain PM can be calculated further as follows l
l
bm
B
l
j bmj B B j j 1
l
j 1
l
r
j 1
lBr
(9)
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Where l is the number of mesh in the PM. The bm(i) and bm under no-load and rated-load conditions by improved EMNM are shown in Table III. Further, the bm_L , bm_M , and bm_R achieved by FEM, improved EMNM and traditional EMNM (PMs under each magnetic pole are not segmented, and the permeance networks distribution in the rotor pole shoe is simplified) are listed in Table IV respectively. The traditional EMNM refers to that PMs under each magnetic pole are not segmented, and the permeance networks model in the rotor pole shoe is simplified. The magnetic potential at each node of traditional EMNM is calculated by solving traditional EMNM, and then bm of PM can be calculated by formula (4) and (5). Similarly, the calculation method for working points of PMs by improved EMNM is to calculate magnetic potential at each node by the improved EMNM established in the manuscript. Then the bm of PM can be calculated by formula (4) and (5). Unlike the traditional EMNM, the bm(i)of segmented PM under certain PM can be calculated by improved EMNM. TABLE III Working points of PMs by improved EMNM No load Rated load b b bm bm 0.8717 0.7705 Left PM 0.8717 0.8717 0.7968 0.7992 0.8717 0.8122 0.8718 0.8173 0.8716 0.8175 Middle PM 0.8714 0.8715 0.8172 0.8173 0.8714 0.8172 0.8716 0.8174 0.8721 0.8490 Right PM 0.8718 0.8719 0.8317 0.8292 0.8718 0.8184 0.8718 0.8175 m
m
TABLE IV The calculated working points of PMs by different methods bm_L
No-load
bm_M bm_R
FEM 0.859 0.867 0.859
Improved EMNM 0.8717 0.8715 0.8719
Traditional EMNM 0.8716 0.8716 0.8717
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Rated load
bm_L bm_M bm_R
0.773 0.7992 0.821 0.8173 0.833 0.8292
0.8145 0.8173 0.8146
It can be seen from Fig.10 that the flux density distributions in the left PM and the right PM under one magnetic pole along the corresponding magnetizing directions of PMs are the same under no-load condition. Therefore, the working points of the left PM and the right PM under each magnetic pole are the same under no-load condition. The armature current is ahead of the back-EMF under rated-load condition, which makes the function of the demagnetization field on the left PM under each magnetic pole stronger. Therefore, the working point of the left PM is lower than that of the right PM under each magnetic pole under load condition. Meanwhile, under load condition, the working point of the segmented PM in the left PM shifts up gradually with the increase of the distance between the segmented PM and rotor surface. At the same time, the working point of the segmented PM in the right PM shifts down gradually with the increase of the distance between the segmented PM and rotor surface. It can be seen from Table IV that under no-load condition, the calculated bm of PMs by the traditional EMNM have little difference from those by FEM. However, under load condition, the calculated bm_L , bm_M , and bm_R of PMs by the traditional EMNM are 0.8145, 0.8173, and 0.8146 respectively. The working points of the left PM and the right PM under one magnetic pole are almost the same under load condition, which is not in accordance with the actual situation. Therefore, the demagnetization in PM cannot be estimated accurately by utilizing the traditional EMNM. It can be seen from Table IV that the errors of the calculated bm of PMs between FEM and improved EMNM are little under no load and load conditions. The improved EMNM established in this paper refines the networks in the rotor pole shoe by segmenting the PM, which makes the flux path in the rotor pole shoe modelled more accurately. It can be seen from the calculation results in Table III that the working points of the left PM and the right PM under one magnetic pole by improved EMNM are almost the same under no-load condition. Meanwhile, by the improved EMNM, the working point of the right PM is larger than that of the left PM under load condition. In addition, it can also be seen from Table III that under load condition, the working point of the segmented PM in the left PM by the improved EMNM shifts up gradually with the increase of the distance between the segmented PM and rotor surface. At the same time, the working point of the segmented PM in the right PM by the improved EMNM shifts down gradually with the increase of the distance between the segmented PM and rotor surface. These change trends about the working points of PMs under each magnetic pole achieved by improved EMNM are consistent with those achieved by FEM. Therefore, the working points of PMs can be determined more accurately by the improved EMNM proposed in this paper. Further, the demagnetization in PMs can be estimated more accurately in the early design stage of
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machine. 2) Solving and validating air-gap flux density and electromagnetic performances When the rotor rotates to certain position, the air-gap flux density waveforms by the improved EMNM and FEM are shown in Fig.11 (no-load) and Fig.12 (rated load) respectively. Meanwhile, the fundamental magnitudes of the radial component of the airgap flux density Bgmr_O(no-load) and Bgmr_RL(rated load) by the improved EMNM and FEM are also listed in Table V respectively. It can be seen from Fig.11 and Fig.12 that the air-gap flux density distributions by two methods have a good consistency. Moreover, it can be seen from Table V that the fundamental amplitudes of the radial component of the air-gap flux density by two methods have differences, but the errors are acceptable.
Fig.11. Air-gap flux density distribution by improved EMNM and FEM under no load condition. (a) radial component; (b) tangential component
Fig.12. Air-gap flux density distribution by improved EMNM and FEM under rated load condition. (a) radial component; (b) tangential component TABLE V COMPARISON OF THE CALCULATION RESULTS BETWEEN IMPROVED EMNM AND FEM
Bgmr_O E0 Bgmr_RL Tem Psl_RL
FEM 0.8022 T 494V 0.9792T 951.4Nm 1549W
Improved EMNM 0.8399 T 491.8V 0.9452T 1003Nm 1550.1W
Relative error 4.7% -0.45% -3.06% 5.42% 0.071%
After achieving the magnetic field in each part of the machine by EMNM, the no-load back-EMF E0 under rated speed, the electromagnetic torque Tem and the stator iron loss
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Psl_RL under rated load condition can be calculated further. Moreover, the calculation results by the improved EMNM and FEM are all shown in Table V. It can be seen from Table V that the errors of the calculation results between two methods are acceptable. III. Effect of Machine Parameters When the interior PM machine operates under load condition, the armature current is ahead of back-EMF. The negative d-axis component of armature current makes demagnetization appeared in PMs easily. Effects of d-axis component and q-axis component of armature current on working points of PMs are analyzed in this paper. Airgap length and flux barrier size are key parameters in the machine. Their sizes influence the permeance of the magnetic circuit and leakage flux of magnetic field, and influence the working points of PMs further. Therefore, their effects on working points of PMs are also researched. A. Effect of armature current This section researches effect of the armature current on working points of PMs. Fig.13 shows effects of d-axis component and q-axis component of armature current on working points of PMs. It can be seen from Fig.13 that when the q-axis component of armature current is constant, the working point of each PM under each magnetic pole shifts down with the increase of the negative d-axis component of armature current. This is because that the increase of negative d-axis component of armature current strengthens the demagnetizing field produced by armature current. Meanwhile, it also makes that the directions of armature reaction field acting on PMs are closer to the opposite direction of the magnetizing directions of PMs. At the same time, when the negative d-axis component of armature current is smaller, the working point of each PM under each magnetic pole shifts down with the increase of q-axis component of armature current. However, when the negative d-axis component of armature current reaches certain value, the working point of each PM under each magnetic pole shifts up with the increase of q-axis component of armature current. On the whole, the q-axis component of armature current has little effect on working points of PMs. The d-axis component of armature current has main effect on the working points of PMs.
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Fig.13. Effects of armature current on working points of PMs. (a) right PM; (b) middle PM; (c) left PM B. Effect of air-gap length We usually think that the larger the air-gap length is, the more difficult the irreversible demagnetization in PMs will be. However, above viewpoint proves to be not true by analysis. Fig.14 shows the variation of working points of PMs with the variation of air-gap length. It can be seen from Fig.14 that whether the machine operates under no-load or rated load condition, the working points of PMs shift down with the increase of air-gap length. In addition, with the change of air-gap length, the variations of working points of PMs under load condition are smaller than that under no-load condition. On one hand, increase of airgap length makes the leakage flux of armature reaction field increased, and makes the function of demagnetizing field on the PMs weakened further. On the other hand, increase of air-gap length also makes permeance of external magnetic circuit decreased, which makes the no-load working points of PMs shifted down. Therefore, although increase of airgap length makes the function of demagnetizing field on the PMs weakened, shifting down of no-load working points of PMs consequently may increase the risk that the irreversible demagnetization appears in PMs under the demagnetizing current.
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Fig.14. Effects of air-gap length on working points of PMs. (a)no-load; (b) rated load. C. Effect of flux barrier size Flux barrier is an important part of the interior PM machine. Its size influences the permeance of magnetic circuit and the leakage flux of magnetic field, and influences the working points of PMs further. Fig.15 and Fig.16 show effects of the flux barrier size on working points of PMs under no-load and rated load conditions respectively. It can be seen from two figures that the working points of PMs under no-load and rated load conditions shift up with the increase of flux barrier thickness or the decrease of flux barrier width. This is because that increasing flux barrier thickness or decreasing flux barrier width makes the permeance of the external magnetic circuit increased. Meanwhile, increasing flux barrier thickness or decreasing flux barrier width also makes the leakage flux of armature reaction field increased, which makes the function of armature reaction field on the PMs weakened. Moreover, with the change of flux barrier size, the variations of working points of PMs under load condition are larger than that under no-load condition.
Fig.15. Effects of flux barrier size on no-load working points of PMs. (a) right PM; (b) middle PM; (c) left PM.
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Fig.16. Effects of flux barrier size on working points of PMs under rated load condition. (a) right PM; (b) middle PM; (c) left PM. Moreover, it can be seen from Fig.15-Fig.16 that whether the machine operates under noload or rated load condition, the variations of working points of PMs with the variation of flux barrier width are all smaller than that with the variation of flux barrier thickness. That is, the flux barrier thickness has a larger effect on the working points of PMs. At the same time, it can also be seen from figures that the larger the flux barrier thickness is, the smaller the variations of working points of PMs with the variation of flux barrier width will be. In addition, the smaller the flux barrier width is, the smaller the variations of working points of PMs with the variation of flux barrier thickness will be. IV. CONCLUSION An improved EMNM is established in this paper aiming at the interior PM machine. By the improved EMNM, working points of PMs are determined accurately under no-load and load condition, which makes it possible to analyze demagnetization in PM in the early design stage of machine. Meanwhile, effects of d-axis component and q-axis component of armature current, air-gap length and flux barrier size on the working points of PMs are analyzed by utilizing the improved EMNM, and some conclusions are drawn: (1) d-axis component of armature current has main effect on the working points of PMs, and the q-axis component has little effect; (2) Increase of air-gap length may increase the risk that irreversible demagnetization appear in PMs; (3) Increasing flux barrier thickness or decreasing its width strengthens the antidemagnetization abilities of PMs. Moreover, the flux barrier thickness has larger effect on working points of PMs than flux barrier width.
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ACKNOWLEDGMENT This work was supported in part by the project supported by Major Program of National Natural Science Foundation of China under Grant 51690180, in part by the National Natural Science Foundation of China under Grant 51507111. REFERENCES [1] Changliang Xia，Liyan Guo， Zhen Zhang，Tingna Shi，Huimin Wang．Optimal Designing of Permanent Magnet Cavity to Reduce Iron Loss of Interior Permanent Magnet Machine． IEEE Transactions on Magnetics，2015，51(12)：8115409． [2] Peixin Liang, Feng Chai, Yunlong Bi, Yulong Pei, Shukang Cheng, Analytical Model and Design of Spoke-Type Permanent-Magnet Machines Accounting for Saturation and Nonlinearity of Magnetic Bidges, Journal of Magnetism and Magnetic Materials, 2016, 417:389-396. [3] Tang Renyuan, "Modern Permanent Magnet Machines-Theory and Design," Beijing: Machinery Industry press, 1997, pp. 45-48, 164-174. [4] Vipulkumar I. Patel, Jiabin Wang, Sreeju S. Nair, Demagnetization Assessment of Fractional-Slot and Distributed Wound 6-Phase Permanent Magnet Machines, IEEE Transactions on Magnetics, 2015, 51(6):8105511. [5] Ki-Doek Lee, Won-Ho Kim, Chang-Sung Jin, Ju Lee, Local demagnetisation analysis of a permanent magnet motor, IET Electric Power Applications, 2015, 9(3):280-286. [6] Hong Chen, Ronghai Qu, Jian Li, and Dawei Li, Demagnetization Performance of a 7 MW Interior Permanent Magnet Wind Generator With Fractional-Slot Concentrated Windings, IEEE Transactions on Magnetics, 2015, 51(11):8205804. [7] Zhen Zhang, Changliang Xia, Yan Yan, Qiang Gen, Tingna Shi, A Hybrid Analytical Model for Open-Circuit Field Calculation of Multilayer Interior Permanent Magnet Machines, Journal of Magnetism and Magnetic Materials, 2017, 435:136-145. [8] Ki-Chan Kim, Kwangsoo Kim, Hee Jun Kim, Ju Lee, Demagnetization Analysis of Permanent Magnets According to Rotor Types of Interior Permanent Magnet Synchronous Motor, IEEE Transactions on Magnetics, 2009, 45(6):2799-2802. [9] Jian-Xin Shen, Peng Li, Meng-Jia Jin, Guang Yang, Investigation and Countermeasures for Demagnetization in Line Start Permanent Magnet Synchronous Motors, IEEE Transactions on Magnetics, 2013, 49(7):4068-4071. [10] Shi Wei, "The Research of Anti-Demagnetization Technology of Permanent Magnet in High Density Permanent Magnet Motor," Shanghai: Shanghai University, 2012. [11] Ming Cheng, K. T. Chau, C. C. Chan, E. Zhou, X. Huang, Nonlinear VaryingNetwork Magnetic Circuit Analysis for Doubly Salient Permanent-Magnet Motors, IEEE Transactions on Magnetics, 2000, 36(1):339-348. [12] Yoshiaki Kano, Takashi Kosaka, Nobuyuki Matsui, Simple Nonlinear Magnetic Analysis for Permanent-Magnet Motors, IEEE Transactions on Industry Applications, 2005, 41(5):1205-1214. [13] Z. Q. Zhu, Y. Pang, D. Howe, S. Iwasaki, R. Deodhar, A. Pride, Analysis of Electromagnetic Performance of Flux-Switching Permanent-Magnet Machines by
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Highlights 1. Proposing improved equivalent magnetic network model for interior PM machine 2. Calculating working points of PMs accurately 3. Analyzing effects of machine parameters on working points of PMs