Analytical Model for Sizing the Magnets of Permanent Magnet Synchronous Machines
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1 Journal of Electrical Engineering 3 (2015) doi: / / D DAVID PUBLISHING Analytical Model for Sizing Magnets of Permanent Magnet Synchronous Machines George Todorov and Bozhidar Stoev Department of Electrical Machines, Faculty of Electrical Engineering, Technical University of Sofia, Sofia 1000, Bulgaria Abstract: The paper presents a procedure for determining dimensions of magnets that confirm required characteristics of a PMSM (permanent magnet synchronous motors). The magnet sizes influence on saliency of motor and consequently on its parameters and characteristics. An analytical model that accounts salience, stator resistance and saturation has been developed. The model uses FEA (finite element analysis) to obtain stator windings flux linkage, direct-axis and quadrature-axis inductances and phase shift between quantities. The distribution of magnetic field produced by permanent magnets and magnetic field at operation with rated load has been obtained using this model. An iterative procedure changes dimensions of magnets at each step and performs analysis of model until an operating point with required characteristics is achieved. Key words: Permanent magnets, PMSM, sizing, volume. 1. Introduction The increased efficiency requirements of international standards for alternating current motors make designers to look for improved technologies and constructions. PMSM (permanent magnet synchronous motors) give an attractive alternative for a wide range of applications from line-start constant speed to inverter-fed variable speed drives. Due to ir high efficiency, power factor, power and torque density, PMSM are nowadays an important component of high-performance electric drives for electric and hybrid electric vehicles, aircraft, marine, mill, paper etc. applications. Equations for calculating size and volume of PM (permanent magnets) have been developed by Balagurov [1] and Gieras [2]. These equations have been originally developed for ferrite and alnico magnets, which possess a non-linear demagnetization characteristic. Nowadays most commonly used permanent magnets are rare-earth magnets. They characterize with high energy and linear demagnetization characteristic, but a high temperature Corresponding author: George Todorov, Ph.D., associate professor, research field: electrical machines. sensitivity. Their high energy production allows designer to use as little permanent magnet material as possible, without sacrificing performance of designed motor. Sekerák [3] presents influence of magnet material on volume and size of magnets. Yoshioka and Morimoto [4] analyze and show influence of PM arrangement on parameters of PMSM for automotive applications and Sorgdrager and Grobler [5] show influence of magnets size on air-gap flux density distribution, but none of m emphasize on sizing of magnets. Mi [6] presents a sizing procedure for rare-earth magnets (NdFeB and SmCo) based on equations of Balagurov and Gieras. Although procedure developed in Ref. [6] describes a full estimation of magnets volume and size, some assumptions were made, such as neglecting pole salience, stator resistance and saturation. These assumptions can affect accuracy of calculations and proper size estimation in design process. This paper proposes a procedure for sizing rare-earth permanent magnets for PMSM with both salient and non-salient rotor configurations. Using FEA (finite element analysis) and space vector diagram of
2 Analytical Model for Sizing Magnets of Permanent Magnet Synchronous Machines 135 PMSM, saturation and stator resistance are accounted for in process of determining a proper volume and size. 2. Sizing Procedure Before starting procedure permanent magnets arrangement of rotor configuration have to be defined. The available choices for PM arrangements are: surface mounted magnets, inset surface mounted magnets, interior radial flux magnets and interior circumferential flux magnets Fig. 1. The sizing procedure is subdivided to seven main sections and its structure diagram is shown in Fig Initial Design Parameters After arrangement of rotor configuration is chosen initial design parameters have to be defined. The initial design parameters are divided into two groups. The first group includes required values of output power, efficiency and power factor, phase voltage and frequency, number of pole pairs, characteristics of magnets, stator winding arrangement and its parameters. The second group of initial parameters includes stator cross-section, i.e., outer and inner diameter, air-gap length, stator slot dimensions and rotor shaft diameter. The laminations stack length is also included in second group. Based on cross-section dimensions, maximum possible magnet width is determined. The width of magnet per pole, calculated in following sections must not be larger than one determined here. 2.2 Initial Electric Values Electromagnetic Power and Rated Current At this stage of sizing procedure, initial values of electromagnetic power and rated current have to be defined. The rated current is: (a) (b) (c) (d) Fig. 1 Available permanent magnet arrangements: (a) Surface mounted PM; (b) Inset surface mounted PM; (c) Interior radial flux PM; (d) Interior circumferential flux PM.
3 136 Analytical Model for Sizing Magnets of Permanent Magnet Synchronous Machines Fig. 2 Procedure structure....cos where, m is number of phases,, cos are required values of efficiency and power factor of motor. The electromagnetic power expressed by input power, efficiency, iron and copper losses is: 1 2 where, denotes copper losses and iron losses. 2.3 Initial Permanent Magnet Volume The relationship between magnets volume and electromagnetic power is necessary for defining initial PM volume. The electromagnetic power is:...cos 3
4 Analytical Model for Sizing Magnets of Permanent Magnet Synchronous Machines 137 where, denotes back e.m.f. induced from permanent magnets flux and β is current phase angle angle between vectors of stator current and back e.m.f (see space vector diagram of synchronous motor in Fig. 4). The back e.m.f. is expressed as: 2. π In above equation, is number of turns per phase, is winding factor, is frequency of input voltage and is air-gap flux per pole. The rated armature current expressed in terms of magnetomotive force (m.m.f.) of direct-axis armature reaction is:. 5 0,45....sin where is direct-axis armature reaction factor and p is number of pole pairs. By substituting Eqs. (4) and (5) into Eq. (3), following expression for electromagnetic power is derived:.. 2.π , sin To calculate initial volume of permanent magnets anor parameter is needed. It is defined as magnet usage ratio in Ref. [6], or coefficient of utilization on permanent magnets in Ref. [2]:. 7. Here is m.m.f. of magnetic circuit at maximum demagnetizingg current. ; is total flux per pole with demagnetizing current zero; is total residual flux per pole and is coercivee m.m.f. per pole. The definition of magnet usage ratio is illustrated in Fig. 3. For surface mounted and interior radial flux magnets, total flux and coercive m.m.f per pole are: and for interior circumferential flux magnets: where, is permanent magnet width, is permanent magnet thicknesss and is length of permanent magnet along shaft direction. Usually magnet length is set to be equal to laminations stack length. and are residual flux density and coercivity of magnets. By expressing armature reaction from Eq. (7) and substituting in Eq. (6), following expression for total volume of magnets is derivedd Ref. [6]: Fig. 3 Illustration of magnet usage ratio. Fig. 4 Space vector diagram of PMSM.
5 138 Analytical Model for Sizing Magnets of Permanent Magnet Synchronous Machines. 0,45..sin π..cos... where is maximum armature current per unit and is flux leakage factor. At this step of procedure values of current angle β, leakage factor, maximum armature current per unit and direct-axis armature reaction factor are unknown. To calculate an initial volume and size of magnets, a value has to be assigned to each of above parameters. The initial values for those parameters are chosen as recommended in literature [1, 2, 6]. The magnet pole coverage α should also be assigned for calculating armature reaction factor. After volume is determined, sizes of magnet per pole are calculated. The thickness of magnet is:. 13 where τ is pole pitch. Since magnet length is chosen to be equal to stack length and magnet thickness is determined from Eq. (13), width of magnet can be calculated: The width from Eq. (14) should be compared to maximum possible width, defined by cross-section geometry of motor in initial design parameters section. If width is larger than maximum possible one, magnet thickness must be readjusted. 2.4 FEM Model and with Zero Demagnetizing Current The object of this section is to calculate e.m.f., induced by permanent magnet flux. The flux corresponds to operating point with demagnetizing current zero point M in Fig. 3. This is joint point of demagnetizing line of PM (line 1) and working line that characterizes permeance of magnetic circuit with demagnetizing current zero (line 2). The operating point M is determined via FEA of motor. Once initial volume and sizes have been determined, a Finite Element Model of PMSM is built and analyzed. From results of FEA characteristic of magnetic circuit and m.m.f F M is determined. The e.m.f. is n calculated by substituting magnetic flux into Eq. (4). 2.5 Calculation at Rated Load After m.m.f. and magnetic flux with zero armature current are defined, operating point of motor at rated current, for given permanent magnet volume and dimensions, is determined. The three phase currents are taken for a moment of time where current in phase A is at maximum and are assigned to corresponding windings (with assumption of sinusoidal distribution): 2.. cos cos 2π 3 2..cos 4π The phase winding current creates an armature reaction, opposing field of permanent magnets. The m.m.f. of direct axis armature reaction per pole:. 2 π This armature reaction tends to magnetize circuit in a negative direction and moves operating point of PM down in demagnetizing line. Again, operating point of magnet is found from intersection between demagnetizing line (Fig. 3, line 1) and working line (line 2). In case of a rated current and a corresponding armature reaction, working line 2 is shifted from origin of coordinate system with field strength equal to demagnetizing m.m.f. of armature reaction. The operating point P determines value of magnetic flux at rated current, which is used to calculate back e.m.f. from Eq. (4).
6 Analytical Model for Sizing Magnets of Permanent Magnet Synchronous Machines 139 After performing an FEA with phase currents assigned to phase windings, flux linkages of each phase are found:.. 19 А where, is number of turns per coil, is lamination stack length, is cross section of stator slot and A is value of magnetic vector potential for positive and negative sides of coils respectively. To determine parameters of motor for rated operating point, a coordinate system fixed to rotor, i.e., d-q coordinate system and Park transformations are used. The direct-axis and quadrature-axis components of current space vector are calculated by: 2 3.cos θ.cos θ 2π 3.cos θ 4π sin θ.sin θ 2π 3.sin θ 4π 3 23 The d-axis and q-axis components of stator flux linkage are n determined by: 2 3..cos θ.cos θ 2π 3.cos θ 4π where is stator winding flux linkage obtained for operating point M. The d-axis and q-axis synchronous reactances are: where is angular frequency. The determination of operating point of motor at rated load is done by an iterative procedure for defining values of,,,, current phase angle β and load angle θ. The procedure stops when such values are determined, that balance voltage equation that is graphically depicted with space vector diagram in Fig. 4. If current values of quantities do not balance Eq. (30) current phase angle must be readjusted. The correct phase angle is determined by an iterative loop. For each new value of angle, phase currents, magnet operating point, flux linkages and inductances are recalculated using FEA. The loop stops when voltage drops balance out input voltage. 2.6 Torque Calculation This section checks wher obtained magnet volume and sizes will provide enough energy to produce required output torque and power. The electromagnetic torque is computed using values of,,, obtained for rated point: 2 3..sin θ.sin θ 2π and electromagnetic power is: 31.sin θ 4π 3 25 where θ r is rotor position angle ( angle between d-axis and magnetic axis of phase A).The d-axis and q-axis inductances of motor are:. 2. π where is synchronous speed of motor in turn per minute. If power calculated from Eq. (32) is lower than
7 140 Analytical Model for Sizing Magnets of Permanent Magnet Synchronous Machines one calculated from Eq. (2) in section 2, width of magnets must be adjusted. Based on difference between previously defined power and one computed here a new width of magnets can be found as:. 33 where is magnet width calculated from Eq. (14), is required electromagnetic power from Eq. (2) and is power calculated from Eq. (32). When a new magnet width is found, model has to be rebuild and calculations have to be repeated starting from section 4. The iterative procedure for new magnet width stops when such dimensions are found, that satisfy required electromagnetic power and provide proper work of motor. 2.7 Final Parameter Calculation Having dimensions of PM, values of all quantities, current phase angle β and load angle θ at rated operating point, efficiency and power factor are calculated. At end of sizing procedure final values for initially assigned leakage factor, maximum per unit armature current and magnet usage ratio ξ are calculated. The leakage factor is calculated as: 34 where is air-gap flux at rated operating point of motor. The maximum armature current occurs, when voltage is out of phase with back e.m.f., i.e.: 35 The per unit maximum current is defined as:. 2 π 3. Design Example To illustrate sizing procedure described above, a 1.5 kw, 1,500 rpm, 4-pole, three phase interior PM synchronous motor is chosen and its magnets are sized according to required characteristics. Efficiency of 92% and power factor of 0.96 are required at rated power and speed. Magnets used for excitation are circumferentially magnetized NdFeB permanent magnets, with a residual flux density B 1.05 T and coercivity H C ka/m. The main initial design parameters are: stator outer diameter 136 mm, stator inner diameter 80 mm, air-gap length 0.55 mm, lamination stack length 80 mm and turns per phase 336. By using proposed sizing procedure, following magnet volume and sizes are determined: PM volume 42, m ; magnet width 21 mm; magnet thickness 6 mm; magnet length 80 mm; Fig. 6 shows flux distribution from final FEA solution. The values of initial parameters, used to define an initial magnet volume, and final values calculated at end of sizing procedure are shown in Table The final value of magnet usage ratio ξ is calculated from Eq. (7) for operating point K with armature reaction m.m.f. at maximum demagnetizing current: Fig. 5 Permanent magnet dimensions.
8 Analytical Model for Sizing Magnets of Permanent Magnet Synchronous Machines 141 (a) (b) Fig. 6 Flux distribution: (a) with armature current zero; (b) at rated armature current. Table 1 Parameter values for design example. cos η Initial values % Final values % 4. Conclusions This paper presents an analytical procedure for determining volume and size of permanent magnets of a PMSM. As PMSM characteristics are strongly influenced by characteristics and size of magnets, it is important to know application of motor. The core idea is to determine such volume and size that will satisfy initial requirements, provide proper work of motor in whole range of operating modes and use minimum permanent magnet material. The model accounts saliency of motor, stator resistance and uses FEA which has made it possible to automatically include saturation of all parts of magnetic circuit. The procedure requires number of iterations to achieve required solution. Performing design using model of motor that apply FEA combined with vector diagrams and Park transformations allows sizing of permanent magnets with good accuracy, verified by obtained results. References [1] Balagurov, V. A., Galtieev, F. F., and Larionov, A. N Permanent Magnet Electrical Machines (in Russian, and translation in English). Moscow, Russia: Energia. [2] Gieras, J. F., and Wing, M Permanent Magnet Motor Technology: Design and Applications. 2nd ed. New York: Marcel Dekker. [3] Sekerák, P., and Hrabovcová, V Synchronous Motors with Different PM Materials. ELEKTRO 2012, [4] Yoshioka, S., and Morimoto, S Influence of Magnet Arrangement on Performance of IPMSMs for Automotive Applications. Energy Conversion Congress and Exposition (ECCE) 2014 IEEE, [5] Sorgdrager, A., and Grobler, A Influence of Magnet Size and Rotor Topology on Air-Gap Flux Density of a Radial Flux PMSM. IEEE International Conference on Industrial Technology (ICIT) 2013, [6] Mi, C. C Analytical Design of Permanent Magnet Traction-Drive Motors. IEEE Transactions on Magnetics, 42 (7):
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