Reduced Order Model of Developed Magnetic ... - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 7, JULY 2010

2649

Reduced Order Model of Developed Magnetic Equivalent Circuit in Electrical Machine Modeling Seyed Amin Saied, Karim Abbaszadeh, and Mehdi Fadaie Electrical Engineering Department Seyed-Khandan, K. N. Toosi University of Technology, Tehran 16315-1355, Iran In this paper, we use a detailed magnetic equivalent circuit (MEC) to improve the analysis accuracy in modeling of electrical machines. We do this for those areas in which the flux routes and consequently the so-called flux tubes are not easily definable (e.g., the air gap or saturated parts). In deriving the MEC, we represent these parts by a detailed reluctance network. Moreover, for those parts in which the saturation usually happens, we implement a nonlinear model. The resultant network is complex and contains a large number of nodes. In order to reduce the processing time, we use a novel reduced order model to simplify the network. This reduced model makes the analysis process faster without missing the accuracy of information in the predefined parts of the machine. Finally, we apply the methods to an axial-flux permanent-magnet motor. The experimental results verify the compatibility between the simulation and laboratory results. Index Terms—Axial flux PM motor, brushless rotating machines, finite-element methods, permanent-magnet motors.

I. INTRODUCTION IGH torque density of permanent-magnet (PM) motors makes them attractive for industrial applications (Fig. 1 shows an axial flux PM motor). In practice, these kinds of motors are fed by inverters. This combination forms a complex system to analysis. A proper model of electrical machine plays a very important role in study validation. This model needs to be accurate and fast in simulations. Although there are a lot of papers in this area, the electrical machine modeling is still a challenging and attractive research topic. Numerical methods such as finite-element methods [1] are well-accepted tools for modeling of electrical machines. Although these methods simulate the electrical machines accurately [2], [3], their processing time is very long, especially during a dynamic or three-dimensional simulation [4]. Another method in magnetic system analysis is magnetic equivalent circuit (MEC) [5]–[10]. This method uses the magneto-electric analogy in which reluctances replace resistances. The MEC of an electrical machine is usually a simple network and easy to process. Moreover, this method is capable of taking into account the saturation and rotor movement [11]. Since the accuracy in determination of flux tubes in a magnetic system defines the model precision, a simple MEC is not accurate enough for some purposes. This problem is usually addressed by implementing a more detailed reluctance network in which the magnetic system is subdivided into tiny pieces [11]. In this approach, besides improving the model accuracy, the processing time also increases. Cogging torque is one of the important drawbacks of PM motors [12], which results in shaft vibration and noise. This torque is one of the key factors in pulsating in the motor torque which prevents smooth rotation of the motor. This phenomenon especially appears at low speeds and light loads [13]. The periodical

H

Manuscript received November 04, 2009; revised January 23, 2010; accepted February 12, 2010. First published March 08, 2010; current version published June 23, 2010. Corresponding author: K. Abbaszadeh (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2044511

Fig. 1. Axial flux PM motor structure.

torque ripples pulsate with rotor position [14] and limit the performance of position servo control in tracking applications [15]. With regards to motor-drive analysis, the ability of the model in torque ripple simulation is an important factor. There are several approaches for cogging torque modeling. In case of estimating or calculating, there is a lot of research that approximates the cogging torque by mathematical formulas [15] or calculate and model it by defining some assumptions [16]–[24]. In these research works, although it is possible to state cogging torque analytically, the effect of saturation and motor loading are not taken into account. In addition, in some cases to calculate the air gap reluctances, the flux routes are approximated to simple shapes. Since the flux routes are not so easy to predict, especially in air gap or saturated teeth, the results can be affected by those approximations. This paper introduces a fast processing model that is able to simulate cogging torque and stator saliency. This method is based on the reluctance network. In MEC, which is a well-documented method [25], the magnetic system is modeled by a reluctance network. In conventional MEC, these reluctances are constant (linear elements), flux-dependent (saturated parts), or position-dependent (air gap reluctances). Those reluctances which model the air gap are the most delicate [11]. In this case, the function for the reluctance variation is derived by off-line calculation [25]. This has the disadvantage of not taking into account the effect of winding currents and saturation. Teeth saturation directly affects on air gap flux and consequently on the air gap reluctances. Moreover, to

0018-9464/$26.00 © 2010 IEEE

2650


calculate the position-dependent reluctances, the flux tubes are usually approximated which affects the results. This paper tries to improve the model accuracy by considering surface mounted PM machine characteristics. This procedure that is explained in next sections, besides improving the model accuracy, increases the processing time. In this case the reduced order model (the main contribution of this paper) is proposed to simplify the network. In the following sections, first the application of the detailed reluctance network for modeling an electrical machine is described (Section II), then the saturation modeling (Section III), and the reduced order model (Sections IV and V). In Section VI, the proposed method is verified by experimental results. II. DETAILED RELUCTANCE NETWORK The essence of MEC analysis is to model a flux tube with its reluctance [25]. In conventional MEC, there are three types of reluctance: constant, flux-dependent, and position-dependent. The constant reluctances can be modeled by linear elements, while variable reluctances must be modeled by nonlinear ones. The modeling procedure for flux-dependent reluctances (saturated iron) is presented in Section III. In case of position-dependent variables (air gap reluctances), this paper proposes a method which is described in this section. While reluctances are defined according to flux routes, those in air gap are not easy to be determined. The saturation and rotor movement cause the flux routes to be affected. Here, the approach is considering the surface-mounted PM machine characteristics. While the magnets’ relative permeability is assumed to be unity; the magnet reluctances can be modeled similar to the air gap. In this regard, it is obvious that during the rotor/ magnet rotation, it is not necessary to alter the reluctances. In other words, since the magnets’ magnetomotive force (MMF) are modeled by voltage sources (as depicted in Fig. 2), to simulate the rotor/magnet movement, it is only necessary to shift the value of these voltage sources. This approach removes the position-dependency of the reluctances in a surface-mounted PM machine. In order to increase the model accuracy, a detailed reluctance network is used for air gap and teeth modeling. This is done by subdividing these areas into tiny pieces in such a way that the flux routes in MEC get similar to those in reality. Fig. 2 shows a portion of an AFPM motor which is modeled by the detailed reluctance network. While a detailed reluctance network is used to model the electrical machine, the number of network nodes, and consequently the order of relevant system of equations get very large. This complexity directly affects the processing time. In Section IV, a method for reducing the system order is presented. III. NONLINEAR MODULE, CONSIDERING THE SATURATION The saturated iron cannot be modeled by a linear element because its flux and MMF have a nonlinear relation ship. Fig. 3 shows this relation in SIMULINK. The look-up table in this figure contains the iron – curve data and defines the element flux according to its MMF. This nonlinear module is used in the teeth corners where the saturation usually happens (these areas are indicted by a circle

Fig. 2. Detailed model of machine portion which lies along with the line depicted in Fig. 1.

Fig. 3. Saturation-modeling module.

in Fig. 2). It is obvious that the nonlinearity of this module refers to the nonlinearity of the look-up table. This feature is further discussed and used in network simplifying procedure later. IV. REDUCED ORDER MODEL FOR A LINEAR CIRCUIT The procedures in the previous parts not only increase the model accuracy, but also lead to exceeding the network size. This has a negative effect on simulation time. On the other hand the growth in network size increases the need for memory capacity. It refers to the dimension of matrix used for network analysis. These drawbacks make serious problems in transient or three-dimensional analysis. As mentioned earlier, The MEC related to a surface mounted PM motor contains: MMF sources as well as constant and fluxdependent reluctances. A glance at this network reveals that the linear elements are used in majority of the network. Therefore, a proper simplification in the linear part has a large influence on reduction of the network size. In the proposed method, first the network elements are divided into two groups: linear and nonlinear elements. Then, the linear parts are modeled by a simplified model. Finally, the nonlinear parts are included to complete the model. In the following part, the method is described for the simple electrical circuit depicted in Fig. 4. Imagine that voltage of node “A” and current of branch “B” are to be determined, while the current and voltage source are known. In other words, in this

SAIED et al.: REDUCED ORDER MODEL OF DEVELOPED MAGNETIC EQUIVALENT CIRCUIT

2651

Fig. 4. Sample network for explaining the simplifying method. Fig. 7. Determination of the arrays related to the independent voltage source.

Fig. 5. Sample network after removing the nonlinear part.

Fig. 8. Simplified model for the network in Fig. 4.

matrix is directly defined (in this case the effect of every input on the network outputs is straightly measured). In this paper, the second method is used. For example, in Fig. 7, when the voltage of the internal voltage source is set to one volt and the other independent sources are deactivated, the meters show the relevant matrix arrays. This procedure is repeated for the other independent sources (inputs) to define the H-matrix completely. According to (1), the network system of equations corresponding to Fig. 7 is in the form of

Fig. 6. Figure depicting the network inputs/outputs.

circuit voltage of node “A” and current of branch “B” are circuit outputs and the current and voltage sources are circuit inputs. Here node “0” is the reference node. Removing the Nonlinear Parts: In this stage the nonlinear part (look-up table) is removed from the network (as shown in Fig. 5). The new network contains the linear elements only. In the next step, the voltage and current of the module are added to the circuit outputs and inputs respectively. Finding the Reduced Order Model: From mathematical point of view, the circuit inputs and outputs are considered as independent and dependent variables. Since these variables are related through a linear circuit, according to the super-position theorem, a linear relation can be stated between them. In other words, it is possible to substitute the circuit in Fig. 5 with a linear MIMO system. The linear MIMO system is stated in (1) where the H-matrix is the coefficient matrix

(1) There are two ways to define the H-matrix for a linear network. The first one is a mathematical method which uses the network system of equations and omits the unnecessary variable from it [26]; the second one considers the linear circuit characteristics and uses the super-position theorem. In the latter, the

(2) Main Model: The above system of equations simply describes the circuit shown in Fig. 7. Here, the order of the simplified model is reduced to three. It is clear that to complete the model, it is necessary to include the nonlinear part (the look-up table). This is done by relating the module voltage to its current via the look-up table. Fig. 8 depicts the final configuration. V. REDUCED ORDER MODEL OF A COMPLEX NETWORK For a complicated network like MEC of an electrical machine (e.g., the one described in Part II), simplifying the entire network at once is a time-consuming process. To solve this problem, the original network is broken into several sub-networks, then for each of them; the reduced order model is derived (using the procedure explained in Part IV). In the end, the models are properly reconnected in order to form the reduced order model of the machine (the main model). For the sake of simplicity, a network

2652


Fig. 9. A sample network for explaining the splitting procedure.

Fig. 11. Modified graph of disconnected sub-networks.

references is set to zero, and considered as the main network reference (here is chosen). In other words, it is the global reference and the other references are considered as local references. It is clear that in each sub-network the voltages are defined in respect to the relevant local reference. 2) Injecting Currents: The KCL theorem states that the injecting currents have a linear relation ship, therefore (4) Fig. 10. Disconnected sub-networks.

with a small number of nodes is used to describe the mentioned procedure. The network is shown in Fig. 9. Here, the current of branches “D” and “B” as well as voltage of nodes “A” and “C” are assumed to be the model outputs. The node “ ” is considered as the network reference. In this paper, the currents flowing from one sub-network toward another are called injecting current. In Fig. 9 these currents are shown with arrows. According to the explained procedure the mentioned network is broken into two sub-networks in the shown location (Breaking line). Then, the connecting equations are derived. These equations relate the variables of each sub-network to the variables of the other. In the next step, the proposed method is used to simplify each sub-network. Finally, the main model is defined using the connecting equations and simplified model of the sub-networks. Connecting Equations: Imagine that both parts of the network are disconnected from each other (Fig. 10 shows the separated sub-networks). In (3), the connecting equations which relate the sub-networks are stated. This can be easily concluded by comparing Figs. 9 and 10:

3) Sub-Networks Reduced Order Model: While (3) shows one set of equations for voltages and currents of the nodes on the breaking line, it is possible to determine another set of equations for those variables using the sub-networks equations. These sets of equations describe the original network completely (in these sets of equations, number of variables are equal to the number of equations). In order to derive the sub-networks equations, it is necessary to consider voltages or currents as independent variables. In this case, the injecting currents and the node voltages are assumed to be independent and dependent variables respectively. According to Part IV, the systems of equations related to the sub-networks are as follows:

(5-a)

(3)

Finding the Reduced Order Mode: 1) Voltage References: To derive the equations of the subnetworks, it is necessary to consider one node in each sub-netare work as the reference node. In this case, the nodes chosen as the reference nodes (as shown in Fig. 11). One of the

(5-b)

Main Model: There are different mathematical methods for solving the main model equations including: (3), (5-a), and


2653

Fig. 13. Back-EMF of the AFPM motor (measured).

Fig. 12. Simplified model of the network in Fig. 9.

(5-b). Here, a method suitable for simulating in SIMULINK is used. In this method, the currents are considered as independent variables in one sub-network, and at the same time as dependent and are variables in the other sub-network. In this case, determined as the dependent variables in (5-b) while I1 and I2 are considered as independent variables in (5-a). In this regard, the equations in (5-b) are modified as Fig. 14. Comparison of the simulated and measured back-EMF.

VI. APPLICATION OF THE METHODS FOR AN AXIAL FLUX MOTOR

(6)

From connecting (3), it is clear that

(7) Substituting (7) for the same in (6)

(8)

Now it is possible to connect (5-a) and (8). Fig. 12 shows the diagram of the main model in SIMULINK. The matrix H and in (5-a) and (8) are determined directly by the procedure explained in Part IV.

In this paper, a three-phase, 4-pole, 15-slot AFPM1 is considered for analysis. Since the slot/pole number is a fractional one, the machine structure is not symmetric. Therefore, the motor is modeled completely. In this case, the proposed methods properly decline the order of the model equations and provide a suitable model for study. In the following subsection, the reduced order model is used for simulations, and the results are compared with experimental results to verify the model. Fig. 13 shows the back-EMF of the machine measured in laboratory test. In Fig. 14, this experimental result is compared with the simulated signal. The signals are normalized in the maximum value of the experimental signals. In order to provide a better vision about the motor magnetic system, it is equipped with a search coil. This coil is mounted on one of the stator teeth. The coil voltage which is derivative of tooth flux provides a base for comparing the simulation and experimental results more accurately. Fig. 16 shows the measured search coil voltage while in Fig. 17, the measured voltage and the simulated one are compared. As Figs. 14 and 17 show, there is a close compatibility between experimental and the proposed method results. In comparison with FEM, the processing of the proposed method is significantly fast. In Fig. 18, the motor flux calculated by the proposed model is compared with the one calculated by FEM. The figure shows that the signals are very similar to each other as expected before. Another important feature of the model is its ability to consider cogging torque. In order to verify the model cogging

= 158 = 118 = 18

1Motor specifications are: motor outer diameter mm, motor : mm, magnet thickness mm, pole step Elec. inner diameter Deg. Air gap : mm, Slot width mm and slot depth mm.

= 88 5 = 1 45

= 10

=5

2654


Fig. 19. Cogging torque calculated by FEM and proposed method. Fig. 15. Motor test bed and the scope showing the measured back-EMF.

VII. CONCLUSION

Fig. 16. Search coil voltage (measured).

Fig. 17. Comparison of simulation and measurement results for search coil voltage.

Fig. 18. Flux density calculated by FEM and proposed method.

torque, the machine is modeled by FEM as well. Fig. 19 depicts cogging torques calculated by the reduced order model and FEM. As the figure shows, the reduced order model simulates the machine behavior precisely, while the processing time is considerably reduced from 55 min in FEM simulation to 4 min in reduced order model simulations.

In this paper, to model the PM machine more precisely, the detailed MEC is used. This complicated model is able to consider the saturation and simulates the flux routes in the air gap more accurately. As a drawback, the increase in the number of nodes makes the processing time long. To solve this problem without missing the model accuracy, the reduced order model is introduced. In the new model, the number of equations, and consequently the processing time significantly declines. The experimental results verify the recommended procedure. In conventional MEC, the air gap reluctances are modeled by variable reluctances which differ according to the rotor position. These variable reluctances must be calculated analytically or numerically in advance, and it is considered as a limitation for the method. On the other hand, the approximation used during these calculations is another disadvantage. However, in the reduced order model, the machine reluctances are determined by on-line calculation which has the advantage of taking into account the saturation and machine loading. The simulation results of the proposed method that are fairly similar to the experimental and FEM verify the proposed procedure. REFERENCES [1] R. J. Hill-Cottingham, P. C. Coles, D. Rodger, and H. C. Lai, “Numerical models of an induction machine,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1551–1553, May 2003. [2] Y. Zhang, K. T. Chau, J. Z. Jiang, D. Zhang, and C. Liu, “A finite element-analytical method for electromagnetic field analysis of electric machines with free rotation,” IEEE Trans. Magn., vol. 42, no. 10, pp. 3392–3394, Oct. 2006. [3] J. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 2002. [4] M. Yilmaz and P. Kerin, “Capabilities of finite element analysis and magnetic equivalent circuit for electrical machine analysis and design,” in IEEE 39th Power Electronics Specialists Conf., Rhodes, Greece, Jun. 15–19, 2008. [5] V. Ostovic, “A method for evaluation of transient and steady state performance in saturated squirrel cage induction machines,” IEEE Trans. Energy Convers., vol. 1, no. 3, pp. 190–197, Sep. 1986. [6] V. Ostovic, “A simplified approach to magnetic equivalent-circuit modeling of induction machines,” IEEE Trans. Ind. Appl., vol. 24, no. 2, pp. 308–316, Mar.–Apr. 1988. [7] A. K. Wallace and A. Wright, “Novel simulation of cage windings based on mesh circuit model,” IEEE Trans. Power App. Syst., vol. PAS-93, pp. 377–382, Jan.–Feb. 1974. [8] C. Delforge and B. Lemaire-Semail, “Induction machine modeling using finite element and permeance network methods,” IEEE Trans. Magn., vol. 31, no. 3, pp. 2092–2095, May 1995. [9] G. R. Slemon, “An equivalent circuit approach to analysis of synchronous machines with saliency and saturation,” IEEE Trans. Energy Convers., vol. 5, no. 3, pp. 538–545, Sep. 1990.


[10] M. Moallem and G. E. Dawson, “An improved magnetic equivalent circuit method for prediction the characteristics of highly saturated electromagnetic devices,” IEEE Trans. Magn., vol. 34, no. 5, pt. 1, pp. 3632–3635, Sep. 1998. [11] T. Raminosoa, J. A. Farooq, A. Djerdir, and A. Miraoui, “Reluctance network modeling of surface permanent magnet motor considering iron nonlinearities,” Energy Convers. Manage., vol. 50, pp. 1356–1361, 2009. [12] S. A. Saied and K. Abbaszadeh, “Cogging torque reduction in brushless dc motors using slot-opening shift,” Adv. Elect. Comput. Eng., vol. 9, no. 1, pp. 28–33, 2009. [13] B. Stumberger et al., “Torque reduction in exterior-rotor permanent magnet synchronous motor,” J. Magn. Magn. Mater., vol. 304, pp. e826–e828, 2006. [14] T. M. Jahns and W. L. Soong, “Pulsating torque minimization techniques for permanent magnet ac motor derives—A review,” IEEE Trans. Ind. Electron., vol. 43, pp. 321–330, 1996. [15] Y. Luo, Y. Chen, and Y. Pi, “Authentic simulation studies of periodic adaptive learning compensation of cogging effect in PMSM position servo system,” in 2008 Chinese Control and Decision Conf., 2008, pp. 4760–4765, CCDC. [16] X. Wang, Y. Yang, and D. Fu, “Study of cogging torque in surfacemounted permanent magnet motors with energy method,” J. Magn. Magn. Mater., vol. 267, pp. 80–85, 2003. [17] L. Zhu, S. Z. Jiang, Z. Q. Zhu, and C. C. Chan, “Analytical method for minimizing cogging torque in permanent magnet machines,” IEEE Trans. Magn., vol. 45, no. 4, pp. 2023–2031, Apr. 2009. [18] D. Zarko, D. Ban, and T. A. Lipo, “Analytical solution for cogging torque in surface permanent magnet motors using conformal mapping,” IEEE Trans. Magn., vol. 44, no. 1, pp. 52–65, Jan. 2008. [19] L. Zhu, S. Z. Jiang, Z. Q. Zhu, and C. C. Chan, “Comparison of alternate analytical model for predicting cogging torque in surface-mounted permanent magnet machines,” in IEEE Vehicle Power and Propulsion Conf. (VPPC), Harbin, China, Sep. 2008. [20] A. B. Proca et al., “Analytical model for permanent magnet motors with surface mounted magnets,” IEEE Trans. Energy Convers., vol. 18, no. 3, Sep. 2003. [21] Z. Q. Zhu, S. Ruangsinchaiwanich, Y. Chen, and D. Howe, “Evaluation of superposition technique for calculating cogging torque in permanent-magnet brushless machines,” IEEE Trans. Magn., vol. 42, no. 5, pp. 1597–1603, May 2006.

2655

[22] Z. Q. Zhu, S. Ruangsinchaiwanich, and D. Howe, “Synthesis of cogging torque waveform from analysis of a single slot,” IEEE Trans. Ind. Appl., vol. 42, no. 3, pp. 650–657, May/Jun. 2006. [23] L. Zhu, S. Z. Jiang, Z. Q. Zhu, and C. C. Chan, “Analytical method for minimizing cogging torque in permanent magnet machines,” IEEE Trans. Magn., vol. 45, no. 4, pp. 2023–2031, Apr. 2009. [24] J. F. Gieras, “Analytical approach to cogging torque calculation of PM brushless motors,” IEEE Trans. Ind. Appl., vol. 40, no. 5, pp. 1310–1316, Sep./Oct. 2004. [25] V. Ostovic, Dynamics of Saturated Electric Machines. New York: Springer-Verlag, 1989. [26] C. Boccaletti, G. Duni, and E. Santini, “Extended thevenin equivalent circuits,” in Int. Symp. Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), 2008.

Seyed Amin Saied is working toward the Ph.D. degree at K. N. Toosi University of Technology, Tehran, Iran. He received the M.Sc. degree in electrical engineering from K. N. Toosi University of Technology in 2006. His main research interest includes design and modeling of electrical machines, and electrical parameter estimation of electrical machines.

Karim Abbaszadeh received the B.S. degree in communication engineering from Khajeh Nasir University of Technology, Tehran, Iran, in 1994, and the M.S. and Ph.D. degrees in electrical engineering from Amir Kabir University of Technology, Tehran, Iran, in 1997 and 2000, respectively. From 2001 to 2003, he was a visiting scholar at Texas A&M University, College Station. In July 20003, he joined the Department of Electrical Engineering, K. N. Toosi University, Tehran, Iran. His main research interests and experience include fault diagnosis of electric machinery, analysis and design of electrical machines, and sensorless variable-speed drives, multiphase variable-speed drives for traction and propulsion applications.

Mehdi Fadaie is working toward the M.Sc. degree at K. N. Toosi University of Technology, Tehran, Iran. His main research interest includes modeling of electrical machines, and electrical machines with inverter supply.