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Abstract--Department of Electrical Engineering at Tallinn

University of Technology has been involved in the development of novel electrical machines for wind applications. This paper presents the analysis of the electrical parameters of a novel slow-speed slotless permanent magnet synchronous generator. Firstly, some of the design parameters are calculated and a simplified analytical mathematical model of the generator is constructed to calculate the output variables under different operational conditions. Secondly, some FEM calculations are carried out to check the validity of the analytical model. Thirdly, test results are analyzed and compared to the calculated values. Finally, the analytical model is evaluated and final output parameters of the generator are determined.

Index Terms--Generators, magnetic circuits, magnetic flux, permanent magnet machines, variable speed drives, wind power generation.

I. INTRODUCTION N recent years two trends in the wind energy industry have been observed. An increasing number of wind turbines have

been developed to use directly coupled slow-speed generators. [1]-[5] This has been done in order to improve the reliability and cost-effectiveness of wind turbines by leaving out the gearbox. At the same time, more and more specific wind turbine power converters have been developed in order to construct wind turbines with a full power converter system. [2], [6]-[9] This enables the use of direct drive variable speed generators and more efficient control of wind turbines.

Fig. 1. An illustration of the studied generator.

K. Tuttelberg is with with the Department of Electrical Power Engineering, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia (e-mail:kaur.tuttelberg@ttu.ee).

A. Kallaste and T. Vaimann are with the Department of Electrical Engineering, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia (e-mail:ants.kallaste@ttu.ee; toomas.vaimann@ttu.ee).

Generators with a high pole number and small rotational speed have been designed to be implemented on wind turbines, for which a 5 kW generator has been designed. [10] In this paper the mentioned slow-speed slotless permanent magnet synchronous generator is studied. The machine is a radial flux generator characterized by a relatively large diameter and small axial length. In order to achieve minimal cogging torque the stator has a slotless design. The stator has concentrated windings housed in resin. On the rotor there are permanent magnet poles. An illustration of the generator can be seen in Fig. 1.

II. MATHEMATICAL MODEL To evaluate the electrical parameters of the generator a

mathematical model is composed. In order to do that, some initial parameters are set. From that the dimensions of the generator and its components can be derived. Based on this a magnetic circuit is composed and analyzed. Continuing the electromagnetic calculation leads to finding the output parameters.

A. Initial parameters Firstly, some of the design parameters are set. The power

rating Pn and a suitable rotational speed nn are determined by requirements from the design of the turbine. A suitable expected output voltage level can be chosen to fit the input voltage Uin of power electronics. In order to compose an analytical model of the generator the following parameters are set beforehand: number of phases m, number of pole pairs p, number of coils nc, pole pitch τ, dimensions of magnets hm, wm, lm, remanence Br and relative permeability μr of magnets, current density jt and fill factor kf in windings. f

Fig. 2. Top view of a stator winding.

A conceptual layout with the relevant dimensions of a stator coil can be seen in Fig. 2. To continue the analysis it is assumed that the length of the linear part of the coil wcl is as

Analysis of a Slow-Speed Slotless Permanent Magnet Synchronous Generator

Kaur Tuttelberg, Toomas Vaimann, and Ants Kallaste

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978-1-4673-5556-8/13/$31.00 ©2013 IEEE

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long as the magnet so that wcl is equal to lm. Next the height of the coil hcw, insulation thickness hins, circumferential separation between coils wcs, and clearance between the stator and magnet surfaces hcl are set. As it has proven beneficial to use parallel windings in the coil, the number of these parallel paths is set as np. This concludes the list of necessary parameters to calculate the output parameters of the generator. As always the design of a generator involves several iterations and these parameters have to be adjusted if the output characteristics do not meet the requirements. This paper, however, focuses mostly on the analysis of a particular machine and not on the methods of designing it.

B. Dimensions The circumferential distance between the magnets ws can

be found as follows:

(1)

Where wp is the width of one pole and wm the width of the magnet. Based on this the diameter of the rotor dr can be found from the equation:

(2)

Next the air-gap for the magnetic field model lg can be calculated as:

(3) If now the height of the magnet is taken into account the inner diameter of the stator ds is calculated as:

(4) Following this leads to dimensioning the windings. The width of the coil wco is as follows:

(5)

As a simplification the width of the core of the coil wci is regarded as ¼ of the outer width wco. Next the cross sectional area of the winding Sc can be found as:

(6)

Where wcw corresponds to the width of the wound part of the coil (see Fig. 2 for clarification). A suitable diameter for wires in the winding dt can be evaluated as:

(7)

Consequently the number of turns in the winding nt can be calculated as:

(8)

C. Magnetic flux calculation With all the relevant dimensions found a magnetic circuit

can be studied. The physical layout of the magnets, windings, and stator and rotor laminations is depicted on Fig. 3.

Fig. 3. Layout of magnets and windings in the air-gap.

The magnetic flux is described as four components: flux in the air-gap Φg, leakage flux on the sides of the magnet Φl, leakage flux to the neighboring pole Φs, and remanence flux Φr which is equal to the sum of the other three fluxes. The distribution of the fluxes can be seen in Fig. 4.

Fig. 4. Magnetic fluxes in the generator.

An equivalent magnetic circuit is composed which can be seen in Fig. 5. The non-linear permeances of the steel laminations are several orders of magnitude larger than the other permeances and are disregarded as the model is simplified. As the circuit is symmetrical it can be further simplified by splitting it along the axis of symmetry. Further analysis of the circuit is trivial and is not described here.

Fig. 5. Magnetic circuit used to calculate air-gap flux.

Next the parameters of the circuit are determined and the magnetic flux in the air-gap derived. The permeance of the magnet itself Ppm can be calculated as:

(9)

Where Apm is the surface area of the magnet. Leakage flux on the sides of the magnets can be described by the following permeance Pl:

(10)

The permeance for the air-gap leakage Ps can be derived as:

(11)

And the permeance for the air-gap flux Pg is as follows:

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(12)

After relating the magnetic field to the geometry of the machine the air-gap flux itself Φg can be calculated as:

(13)

Where PΣ is the resulting permeance of the split circuit.

D. Electrical parameters Based on the value of the magnetic flux in the air-gap the

EMF of a single phase E at a certain frequency f can be found based on the following equation:

(14)

Where kw is the winding factor. To calculate the output voltages and load currents and analyze the operation of the generator a simple equivalent circuit is used as shown in Fig. 6. [11]

Fig. 6. Equivalent circuit of the generator for one phase.

The resistance per one phase R is calculated as:

(15)

Where ρ is the resistivity of the winding material. Next the reactance per phase X at a certain frequency f can be found as:

(16)

Where Lc is the inductance of a single coil. As a simplification the inductance can be calculated using an empirical formula for a multilayer air-core coil which can be found in different handbooks. A more specific calculation for the inductance could be carried out but is assumed to give insignificant improvements to the already simplified model.

(17)

Where req is the equivalent radius of the coil and lc the total length of wire in the winding both of which can easily be calculated based on the dimensions of the coil. For the equivalent circuit it can be written:

(18)The calculation of the voltage and current is carried out in iterations. The phase current I at a certain power level for iteration i is:

(19)

For the first iteration the expected voltage level Uin can be used. To evaluate the output parameters at different power

levels and frequencies (18) is transformed into the following function:

(20) The rated frequency of the generator fn can be calculated as:

(21)

E. Output characteristics When (20) is solved for different power levels and both

U(P,fn) and U(P,fn)·I(P) are plotted against I a characteristic graph is obtained as can be seen in Fig. 7.

Fig. 7. Characteristic output curves of the generator at rated speed.

III. FEM ANALYSIS In order to verify the integrity of the model a 2D FEM

analysis was conducted. In this analysis the air-gap flux derived from the magnetic circuit was compared to the air-gap flux calculated from the FEM model. The results of FEM modeling are shown in Fig. 8.

Fig. 8. Graphical results from the FEM calculations showing flux lines and flux density. Range is from 0 to 1.9 T where cyan corresponds to the lowest and magenta to the highest values.

The air-gap flux from the magnetic circuit at given parameters was calculated as 1.26 mWb and the same value from the FEM calculation was 1.20 mWb which means an error of less than 5%. Even though the entire modeling process was rather simplified the resulting equations can be used. The rest of the model which describes the output parameters can be verified by testing the machine.

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IV. TEST RESULTS In addition to analytical modelling some tests were carried

out in the lab of electrical machines of the Department of Electrical Engineering at Tallinn University of Technology. As a part of this paper the no-load and load test results are examined. Also the phase resistance measured during the tests is used to verify calculations. The load test was carried out by applying various loads on the generator at rated speed. The no-load test was performed at different rotational speeds. In order to compare the results to the mathematical model corresponding data points were calculated using the analytical model described earlier. The results of the no-load test can be seen in Fig. 9 and the results of the load test at rated speed can be seen in Fig. 10.

Fig. 9. No-load test results and calculated values.

Fig. 10. Load test results and calculated values (“m” denotes measured and “a” denotes calculated).

As can be seen from Fig. 9, calculated and measured no-load values vary very little. A linear characteristic was obtained and this was as expected. No-load test results exceed calculated values. When examining load test results as in Fig. 10, it can be seen that the difference of calculated and measured results increase with load. Measured values of the load test are lower than calculated.

V. RESULTS AND CONCLUSIONS

The main results of interest were the electrical parameters of the generator. These results are shown in Table I.

TABLE I ELECTRICAL PARAMETERS OF THE STUDIED GENERATOR Parameter Value Phase EMF at rated speed 225 V Voltage at rated load 180 V Rated load current 9.3 A Phase resistance 3.1 Ω

The first part of the analytical model was verified by comparing the calculated air-gap flux to the value obtained from FEM analysis and the resulting error was less than 5%. The analysis of electrical parameters was evaluated by testing the generator. The difference in the calculated and measured phase resistance was negligible. However, from the load test it can be seen that at higher loads the mathematical model deviates increasingly from test results. The most probable cause to this is considered to be the inaccuracy of the calculation of inductances.

In general the proposed method can be considered usable as a first approximation. It can be used as a simplified model to evaluate similar multi-pole slotless PM generators.

VI. REFERENCES [1] Polinder, H.; van der Pijl, F.F.A.; de Vilder, G.-J.; Tavner, P.; ,

"Comparison of direct-drive and geared generator concepts for wind turbines," Electric Machines and Drives, 2005 IEEE International Conference on , vol., no., pp.543-550, 15-15 May 2005

[2] Banham-Hall, D.D.; Taylor, G.A.; Smith, C.A.; Irving, M.R.; , "Towards large-scale direct drive wind turbines with permanent magnet generators and full converters," Power and Energy Society General Meeting, 2010 IEEE , vol., no., pp.1-8, 25-29 July 2010

[3] Polinder, H.; Bang, D.; van Rooij, R.P.J.O.M.; McDonald, A.S.; Mueller, M.A.; , "10 MW Wind Turbine Direct-Drive Generator Design with Pitch or Active Speed Stall Control," Electric Machines & Drives Conference, 2007. IEMDC '07. IEEE International , vol.2, no., pp.1390-1395, 3-5 May 2007

[4] Hui Li; Zhe Chen; , "Design optimization and comparison of large direct-drive permanent magnet wind generator systems," Electrical Machines and Systems, 2007. ICEMS. International Conference on , vol., no., pp.685-690, 8-11 Oct. 2007

[5] Karmaker, H.; Chen, E.; Chen, W.; Gao, G.; , "Stator design concepts for an 8 MW direct drive superconducting wind generator," Electrical Machines (ICEM), 2012 XXth International Conference on , vol., no., pp.769-774, 2-5 Sept. 2012

[6] Zheng Chen; Xiangning Xiao; Haitao Wang; Mengwei Liu; , "Analysis of converter topological structure for direct-drive wind power system with PMSG," Power System Technology (POWERCON), 2010 International Conference on , vol., no., pp.1-5, 24-28 Oct. 2010

[7] Di Gerlando, A.; Foglia, G.M.; Perini, R.; Ubaldini, M.; , "Operation and sizing aspects of converters for wind energy systems equipped with direct-drive, permanent magnet generators," Electrical Machines, 2008. ICEM 2008. 18th International Conference on , vol., no., pp.1-6, 6-9 Sept. 2008

[8] Jianlin Li; Ying Zhu; Hongyan Xu; Honghua Xu; , "CPS-SPWM flying capacitor three-level back-to-back converter applicative direct-drive wind power generator system," Sustainable Power Generation and Supply, 2009. SUPERGEN '09. International Conference on , vol., no., pp.1-6, 6-7 April 2009

[9] Jun Li; Huang, A.Q.; Bhattacharya, S.; Wei Jing; , "Application of active NPC converter on generator side for MW direct-driven wind turbine," Applied Power Electronics Conference and Exposition (APEC), 2010 Twenty-Fifth Annual IEEE , vol., no., pp.1010-1017, 21-25 Feb. 2010

[10] A. Kallaste, T. Vaimann, O. Pabut, “Slow-Speed Ring-Shaped Permanent Magnet Generator for Wind Applications,” in 11th International Symposium “Topical Problems in the Field of Electrical and Power Engineering” and “Doctoral School of Energy and Geotechnology II”: Elektriajam, 2012, pp. 66-69.

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[11] V. Ostovic, Computer-aided Analysis of Electric Machines: A Mathematica approach. Hemel Hempstead : Prentice Hall International Ltd, 1994, p. 191

VII. BIOGRAPHIES Kaur Tuttelberg was born in Rakvere, Estonia in 1990 and received his BSc degree in electrical power engineering from Tallinn University of Technology in 2012. He is currently a master’s student in Tallinn University of Technology.

He is currently working for Goliath Wind as an electrical engineer. His main research interest is power generation.

Toomas Vaimann was born in Pärnu, Estonia, in 1984 and received his BSc and MSc degrees in electrical engineering from Tallinn University of Technology, Estonia, in 2007 and 2009 respectively. He is currently a PhD student in Tallinn University of Technology Department of Electrical Engineering.

He has been working in several companies as an electrical engineer. Presently he is working at the Tallinn University of Technology Department of Electrical Engineering as an engineer.

His main research interests include diagnostics of electrical machines.

Ants Kallaste was born in Pärnu, Estonia in 1980 and received his BSc and MSc degrees in electrical engineering from Tallinn University of Technology, Estonia, in 2004 and 2006 respectively. He is currently a PhD student in Tallinn University of Technology Department of Electrical Engineering.

He has been working in several companies as an electrical engineer. Presently he is working at the Tallinn University of Technology Department of Electrical Engineering on a researcher’s position.

His main research interests include PM machine design and wind turbines.