Comparison of Different FE Calculation Methods for the Electromagnetic Torque of PM Machines

1NAFEMS Seminar: Numerical Simulation of Electromechanical Systems

October 26 - 27, 2005 Wiesbaden, Germany

Comparison of Different FE Calculation Methods for the Electromagnetic Torque of PM Machines

Dieter Gerling

Institute for Electrical Drives, University of Federal Defense Munich Neubiberg, Germany

Summary: Generally, the torque of electrical machines can be calculated applying different methods. In thispaper, the Maxwells stress tensor method, the magnetic co-energy method, and the lumped-parameter method are investigated. As an example, a PM machine with surface mounted magnets inthe rotor is analyzed by means of the FE-software package ANSYS. For the lumped-parametermethod, the dq-parameters of the electric machine are derived with the Fixed Permeability Method(FPM). With this method the parameters of the PM machine can be calculated with high precision. Theobtained results for the electromagnetic torque applying the different calculation methods arecompared concerning accuracy, ease of use and computing time. Keywords: Torque Calculation, Finite Element Method, ANSYS, PM Machine, Fixed Permeability Method (FPM)



1. Introduction The basic task of any electric machine is to generate torque to accelerate and drive a load over a specific range of speeds. Thus, torque is a very important consideration for both analysis and design of electrical machines. The finite element method provides an accurate approach to torque evaluation from the derivation of electromagnetic field distribution. This calculation method allows precise determination of machine parameters through the magnetic field solutions as it takes into account the actual distribution of windings, details of geometry, and non-linearity of magnetic materials of an electrical machine. In this paper, a PM machine with surface mounted magnets in the rotor is analysed with ANSYS. Figure 1 shows the geometry of the studied PM machine. The main geometrical data are presented in the Table 1. Fig. 1: Geometry of the studied PM machine Different methods based on finite element solutions have been used for the calculation of the electromagnetic torque of this machine. The methods analysed here are Maxwells stress tensor-, dq-model of PM machine-, and magnetic co-energy method. The use of Maxwells stress tensor is probably the simplest method, since it requires only the local flux density distribution along a specific contour around the air-gap of the machine. The accuracy of torque calculation with this method relies on model discretization and on contour selection. According to the virtual work principle, the electromagnetic torque equals the derivative of the magnetic co-energy with respect to angular position at constant current. With this method at least two finite element solutions are required to obtain the co-energy change due to an incremental displacement, and this inevitably increases the computing time. Calculation of the electromagnetic torque with the third method is based on the dq-mathematical model of the PM machine. The dq-parameters of the machine are derived accurately using fixed permeability method (FPM) [2-4]. This method ensures that the saturation effect which occurs in the load model is not ignored during calculation of the

dq-axes inductances (single current excitation of the stator windings), back emf (single magnet excitation).

The calculation algorithm for the dq-parameters of the PM machine with the fixed permeability method is presented in the fourth section of this paper.

TABLE 1- GEOMETRY DATA Rotor radius [mm] 21.6 Air gap length [mm] 0.9 Thickness of magnet [mm] 4 Magnet pole arc [deg] 127.8 Number of poles [ -- ] 4 Stator outer radius [mm] 40 Stack length [mm] 35



2. Electromagnetic Torque

2.1 Maxwells Stress Tensor Method The Maxwells stress tensor method is probably the most commonly used since it is uncomplicated to apply and needs a relatively small calculation time. But, on the other hand, the accuracy of this method is obviously dependent on the type of the problem to be solved, on the model discretization and on the selection of the integration line or contour. If a Maxwells stress tensor is used, it is important to force at least three layers of elements in the small air-gap and to calculate the torque in the middle element layer. The electromagnetic torque eT in the motor air-gap, on a closed surface of radius r, can be calculated by integrating the Maxwells stress tensor. For two-dimensional electromagnetic field models:

2

2

0 0e r

LT r B B d

= (1) where rB and B are radial and tangential components of the flux density, and L is the active length of the machine. Taking advantage of the specific periodicity of electrical machines, the integral can be performed from 0 to 2 electrical radians and the result multiplied by the number of pole pairs p.

2.2 Co-Energy Method The second most frequently used method is based on the stored magnetic co-energy change or the virtual work with a small displacement. According to the virtual work principle, the electromagnetic torque eT equals the derivative of the magnetic co-energy co-engW with respect to angular position at constant current,

co-enge

i const

WT =

= (2) In co-energy method the calculation time is doubled, because with this method the field is solved twice to get an energy difference.

2.3 Electromagnetic Torque Based on the dq-Mathematical Model of PM Machine The PM motor is usually modeled and analyzed by means of the space-vector theory, which is well known and effective. The motor equations, valid also when the iron is saturated, are reported in the following. The electromagnetic torque based on the dq -formulation is

( )32e d q q d

T p i i = (3) while the voltage dynamic equations are

d d d q

q q q d

du R idtdu R idt

= +

= + + (4)

where, in general, the flux linkages depend on both current components, according to the following magnetic model

( )( )

,

,

d d d q

q q d q

i i

i i

== (5)



If the motor works in the magnetic linear region (low current), the d- and q-axis flux linkages vary linearly with the corresponding d- and q-axis current components, i. e.

m

d d d d

q q q

L iL i

= += (6)

where, md is the d-axis flux-linkage due to magnets. For the non-saturation case the total PM flux-linkage flows through the d-axis. When the motor operates with high torque, hence high current, saturation effects are not negligible and the dq flux linkages become involved functions of both d- and q-axis current components. The complete magnetic model for a saturated motor can be thus expressed as

( ) ( ) ( )( ) ( ) ( )

, , ,

, , ,

md d q d d q d d d q

mq d q q d q q q d q

i i L i i i i i

i i L i i i i i

= += + (7)

Compared with the equ. (6), an additional q-axis PM flux-linkage component is present in the expression of the total flux-linkage. Like dL and qL , when the saturation occurs the

md and mq also

depend on the d- and q-axis current components. In an analysis made in [1] for a PM machine with inset magnets in the rotor (IPM machine), it is shown that the md increases with di , while it decreases with qi components. In the same way, when the saturation occurs, the magnitude of

mq (which is

negative for the motor operation condition of the PM machine) increases with qi , while it decreases

with di ; its variation reaches up to 24% of the md . This is the main effect of the saturation.

Using the equ. (3) and (7), the following expression for the electromagnetic torque is obtained

{ }3 ( , ) ( , ) ( , ) ( , )2 m me d d q q d q d q d d q q q d q dT p L i i L i i i i i i i i i i = + (8) The above expression is used to calculate the electromagnetic torque with the third FE method. The dq-parameters of the PM machine are calculated using the fixed permeability method (FPM) [2, 3, 4]. The main point of this method is that it transforms non-linear problems into linear ones by storing the permeabilities from the non-linear analysis. For each operation point the magnetic permeability of each element obtained from the previous FE simulation under the double excitation is fixed and stored for the further analysis. With this fixed permeability, a second calculation is carried out either with current or magnet single excitation. This ensures that the saturation effect which occurs in the load model is not ignored during calculation of the dq-parameters of the PM machine.

3. Finite Element Analysis Results Based on the above calculation FE methods, in the following the electromagnetic torque is calculated for 2.56 AI = (peak value) and 0 = operation condition. is the electrical angle between total flux linkage due to the stator currents and the rotor q-axis (in the literature this angle is known as torque angle or load angle). The flux line distribution and the flux density along the surface of a PM pole pair for this operation condition are shown in the figure 2. It is shown that the rotor surface is subjected to a relatively large flux density fluctuation due to the varying reluctance as teeth are passed. All following finite element calculations have been performed using the software package ANSYS.



I=2.559 A; Torque-load angle

-1,2-1

-0,8-0,6-0,4-0,2

00,20,40,60,8

11,2

0 20 40 60 80 100 120 140 160 180

delta [elec. degree]

T [N

m]

T_maxw T_Coen

Fig. 2: Flux line distribution and the flux density along the surface of a PM pole pair. Firstly, the electromagnetic torque is evaluated using the Maxwells stress tensor and the magnetic co-energy method for a fixed current excitation in the stator and for different incremental rotor positions. Figure 3 compares the results obtained with these methods. It is shown that the results obtained from the Maxwells stress tensor method are in good agreement with the results obtained with the magnetic co-energy method. The torque curves vary sinusoidally with the load angle (for a fixed current in the stator, the load angle is equivalent with the rotor position). As is known the PM machine with surface mounted magnets in the rotor gives the maximum torque for the case when the load angle is zero ( 0 = , qI I= ). Due to the slotting effect some harmonics (torque pulsations) are added in the toque curve of this type of the machine. This torque component is known as cogging torque. The cogging torque is often the largest component of torque pulsation in permanent magnet motors. It is caused by the interaction between the magnets and the stator teeth. This effect is normally not depending on the stator current. Only if the teeth are saturated due to stator current, usually the cogging torque increases because of the wider effective slot openings.

Fig. 3: Electromagnetic torque versus load angle obtained with Maxwells stress tensor- and the magnetic co-energy method.

B [T

]



For the same operation condition as before in the following the electromagnetic torque versus rotor position is evaluated using Maxwells stress tensor method and the dq mathematical model of the PM machine. The fixed permeability method is used to derive the motor parameters. Using this method the saturation condition of the machine under the double excitation is fixed and stored for the further analysis. The dq-inductances and flux linkages due to magnets are derived under fixed permeability from the double excitation condition. The dq-parameters of the motor are: 2.559 A, 0 A,q di i= = L 6.483 mH,q = 1 4L 6.483 mH, 1.531 10 Vs, 7.663 10 Vsm md d q = = = . Figure 4 shows the saturation condition of the iron parts of the PM machine for the given operation condition. Three regions of the machine can be seen to have low relative permeability values: the stator teeth (in the main flux path), the stator yoke (in the main flux path), and the rotor yoke (in the main flux path).

Fig. 4: The saturation condition (relative permeability) of the PM machine at 2.56 AI = and 0 = . The electromagnetic torque results versus rotor position obtained with the Maxwells stress tensor and from the dq mathematical model of the PM machine are presented and compared in the figure 5. Also here the obtained results show a good agreement between these methods.

Fig. 5: Electromagnetic torque versus rotor position obtained with Maxwells stress tensor- method and the dq-mathematical model of PM machine.

delta=0, I=2.559 A: Torque-rotor position

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 12 24 36 48 60 72 84 96 108

120

132

144

156

168

180

192

204

216

228

240

theta [elec. degree]

T [N

m]

T_maxw T_dq



As the calculation of the electromagnetic torque with the Maxwells stress tensor method gives the total torque components in the air-gap, the dq model gives only the average components of the electromagnetic torque. The dq-theory is established based on the assumption that both the winding flux-linkage and the currents are sinusoidal. The air-gap harmonics are ignored using this model.

4. Computing time As is discussed in the previous sections, the Maxwells stress tensor method is probably the most commonly used method for the torque calculation since it is uncomplicated to apply and needs a relatively small calculation time. Only one simulation for each operation point is needed with this method. In co-energy method the calculation time is doubled, because with this method the field is solved twice to get an energy difference. Also, the torque calculation with dq-mathematical model of the PM machine parameters of which are derived with fixed permeability method needs two simulations for each operation point. The first simulation should be done under the double excitation, from which the permeability of each element is fixed and stored for the further analysis. With this fixed permeability, a second calculation is carried out either with current or magnet single excitation. The calculation time for the second simulation with the frozen permeability condition is faster because the electromagnetic problem is linearized. Figure 6 shows the calculation procedure of the dq-parameters with fixed permeability method. If the saturation effect occurs it is required that the calculation procedure to be repeated for different rotor positions, therefore to derive the average values for these parameters. For this case this method is very time consuming.

Fig. 6: Calculation flow-chart of the dq-parameters of the PM machine.

start

Total flux linkage: T

, I

Fixed Permeability method

Current flux-linkage: i

Magnet flux-linkage: m T i =

dq-transformation theory

, , , m mq d q dL L

Torque, T

Double excitation condition

0 : : P =

n P

Documents

Comparison of Different FE Calculation Methods for the Electromagnetic Torque of PM Machines