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Available online at www.sciencedirect.com Mathematics and Computers in Simulation 81 (2010) 239–251 Optimal design of a high-speed slotless permanent magnet synchronous generator with soft magnetic composite stator yoke and rectifier load Ahmed Chebak , Philippe Viarouge, Jérôme Cros Department of Electrical and Computer Engineering, Laval University, Quebec, QC, G1K 7P4, Canada Received 16 October 2008; received in revised form 6 April 2010; accepted 9 May 2010 Available online 24 May 2010 Abstract This paper presents a specific design methodology of a DC generation system using a high-speed slotless generator with surface- mounted magnets and soft magnetic composite (SMC) stator yoke connected to a rectifier. The method is based on an analytical design model of the machine, an electrical model of the machine–rectifier system and a non-linear optimization procedure. The coupling between both models is achieved by a specific correction mechanism during the iterative process that performs an efficient convergence of the optimization procedure. The machine design model is derived from an analytical computation of the two- dimensional magnetic field distribution created by the magnets, the armature currents and the stator eddy currents that circulate in the SMC material. It has been cross-validated by 2D finite element analysis. The design approach is applied to the specifications of a 1.5 MW, 18,000 rpm slotless permanent magnet generator with a rated DC output voltage of 1500 V. © 2010 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: High-speed generation system; Permanent magnet machine; Optimal design; Soft magnetic composite material; Eddy currents 1. Introduction High-speed slotless permanent magnet (PM) generators can be interesting for gas turbine driven generation systems due to their high power density, high efficiency and small size. These systems can be embedded in many applications such as aircraft, hybrid vehicles, ships, and total energy units [8,7]. With a high-speed slotless generator, a direct coupling to the gas turbine can be performed without gearbox. The stator and rotor magnetic losses due to the slotting effects are reduced and the cogging torque is also eliminated [11,1]. In this paper, the DC generation system presented in Fig. 1 is using a high-speed slotless machine with a stator yoke made of soft magnetic composite (SMC) material and connected to a controlled bridge rectifier. The rotor of the synchronous machine is equipped with surface-mounted magnets (Fig. 2). The rectifier is delivering the active power to the load through a LC filter. The integrated design of such a generation system is a complex problem because there is a strong coupling between the machine and the converter performances that are influenced by the high frequency current commutation in the rectifier. The armature current and Corresponding author. Permanent address: Département de Maths-Info et Génie, Université du Québec à Rimouski, 300 allée des Ursulines, Rimouski, QC, G5L3A1, Canada. Tel.: +1 418 723 1986x1876/656 2131x7139; fax: +1 418 724 1879/656 3159. E-mail addresses: ahmed [email protected], [email protected] (A. Chebak), [email protected] (P. Viarouge), [email protected] (J. Cros). 0378-4754/$36.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2010.05.002

Optimal Design of a High-speed Pmsg

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Page 1: Optimal Design of a High-speed Pmsg

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 81 (2010) 239–251

Optimal design of a high-speed slotless permanent magnetsynchronous generator with soft magnetic composite

stator yoke and rectifier load

Ahmed Chebak ∗, Philippe Viarouge, Jérôme CrosDepartment of Electrical and Computer Engineering, Laval University, Quebec, QC, G1K 7P4, Canada

Received 16 October 2008; received in revised form 6 April 2010; accepted 9 May 2010Available online 24 May 2010

Abstract

This paper presents a specific design methodology of a DC generation system using a high-speed slotless generator with surface-mounted magnets and soft magnetic composite (SMC) stator yoke connected to a rectifier. The method is based on an analyticaldesign model of the machine, an electrical model of the machine–rectifier system and a non-linear optimization procedure. Thecoupling between both models is achieved by a specific correction mechanism during the iterative process that performs an efficientconvergence of the optimization procedure. The machine design model is derived from an analytical computation of the two-dimensional magnetic field distribution created by the magnets, the armature currents and the stator eddy currents that circulate inthe SMC material. It has been cross-validated by 2D finite element analysis. The design approach is applied to the specificationsof a 1.5 MW, 18,000 rpm slotless permanent magnet generator with a rated DC output voltage of 1500 V.© 2010 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: High-speed generation system; Permanent magnet machine; Optimal design; Soft magnetic composite material; Eddy currents

1. Introduction

High-speed slotless permanent magnet (PM) generators can be interesting for gas turbine driven generation systemsdue to their high power density, high efficiency and small size. These systems can be embedded in many applicationssuch as aircraft, hybrid vehicles, ships, and total energy units [8,7]. With a high-speed slotless generator, a directcoupling to the gas turbine can be performed without gearbox. The stator and rotor magnetic losses due to the slottingeffects are reduced and the cogging torque is also eliminated [11,1]. In this paper, the DC generation system presentedin Fig. 1 is using a high-speed slotless machine with a stator yoke made of soft magnetic composite (SMC) materialand connected to a controlled bridge rectifier. The rotor of the synchronous machine is equipped with surface-mountedmagnets (Fig. 2). The rectifier is delivering the active power to the load through a LC filter. The integrated design of sucha generation system is a complex problem because there is a strong coupling between the machine and the converterperformances that are influenced by the high frequency current commutation in the rectifier. The armature current and

∗ Corresponding author. Permanent address: Département de Maths-Info et Génie, Université du Québec à Rimouski, 300 allée des Ursulines,Rimouski, QC, G5L3A1, Canada. Tel.: +1 418 723 1986x1876/656 2131x7139; fax: +1 418 724 1879/656 3159.

E-mail addresses: ahmed [email protected], [email protected] (A. Chebak), [email protected] (P. Viarouge),[email protected] (J. Cros).

0378-4754/$36.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2010.05.002

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240 A. Chebak et al. / Mathematics and Computers in Simulation 81 (2010) 239–251

Fig. 1. High-speed DC generation unit using a controlled rectifier.

the DC output voltage waveforms highly depend on the transient impedance of the machine. This impedance is complexin SMC machines because the eddy currents induced in this kind of magnetic material are not independent of the statoryoke geometry like in laminated yokes [2]. On one hand, the machine transient impedance has a great influence on theaverage output DC voltage of the generation unit. On the other hand, the generator performances in terms of torqueand losses highly depend on the harmonic content of the armature currents. Consequently, the design process of sucha generation unit must use a specific methodology to take the strong machine–converter coupling into account.

In this paper, the authors present a design methodology of this kind of generation unit. It is based on an analyt-ical design model of the machine, an electrical model of the machine–rectifier system and a non-linear constrainedoptimization procedure. The coupling between both models is achieved by a specific correction mechanism duringthe iterative process that performs an efficient convergence of the optimization procedure towards an optimal designsolution. The machine design model is derived from an analytical computation of the two-dimensional magnetic fielddistribution created by the magnets, the armature currents and the stator eddy currents that circulate in the SMC mate-rial. It has been cross-validated by 2D finite element (FE) analysis. The design approach is applied to the specificationsof a 1.5 MW, 18,000 rpm slotless PM generator with a rated DC output voltage of 1500 V.

2. Generator analytical design model

The generator design model is using the analytical modeling method of high-speed slotless PM machines that isdetailed in [2]. It has been adapted in this paper for the generator operation and several post-processors have been addedto compute specific machine performances. The model is based on the computation of the magnetic field distributionthat is derived from the 2D analytical solution of the Maxwell’s equations in the magnets/air-gap/windings/stator coreregions of the slotless machine structure presented in Fig. 2. The model is formulated in polar coordinates and it takesinto account the stator eddy currents and the time and space harmonics of the magnetic field. In this study, the followingassumptions are made:

• The motor axial length is infinite, i.e., the end-effects are negligible and the induced eddy currents are axiallydirected.

• The magnetic saturation and the hysteresis phenomena are absent.• The stator material permeability μs and conductivity σs are constant, isotropic and homogeneous.

Fig. 2. Structure of a 4 poles slotless PM generator with SMC stator yoke.

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A. Chebak et al. / Mathematics and Computers in Simulation 81 (2010) 239–251 241

• The rotor iron core is infinitely permeable.• The eddy currents in the magnets and in the rotor yoke are neglected.• The magnet retaining sleeve is non-conductive and non-magnetic.

2.1. Calculation of the magnetic field distribution

The magnetic field distribution in the generator structure is calculated separately as a superposition of the no-loadmagnetic field produced by the magnets and the armature reaction field produced by the windings currents in termsof potential vectors Am and As, respectively. A general representation of the magnetization vector is used that isapplicable to radial or parallel magnetized magnets and discrete Halbach arrays, where the magnet permeability canbe different from μ0. The stator windings distribution is modeled for generalized balanced three-phase windings withfinite thickness [6]. In order to compute the magnetic fields, the generalized form of the diffusion equation that takesinto account the eddy current effects is applied [5]:

−∇2A = μJs + μσ

(−∂A

∂t+ V × (∇ × A)

)+ μ0∇ × M (1)

where Js is the winding currents density, V is the circumferential speed of the conductive region, M is the magnetizationvector, μ and σ are, respectively, the permeability and the conductivity of each material.

The no-load field is derived by applying (1) in each region i of the generator structure (Fig. 2). By using the variableseparation resolution method, the general solutions in the rotor coordinates and in the four regions can be expressed interms of complex form of Fourier series:

A(I)m (r, θr) =

+∞∑k=−∞

(A

(I)m,kr

α + B(I)m,kr

−α + Sk (r))

expjkpθr (2)

A(i=II,III)m (r, θr) =

+∞∑k=−∞

(A

(i)m,kr

α + B(i)m,kr

−α)

expjkpθr (3)

A(IV )m (r, θr) =

+∞∑k=−∞

(A

(IV )m,k Iα

(τm,kr

) + B(IV )m,k Kα

(τm,kr

))expjkpθr (4)

with:

Sk (r) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

μ0jkpMr,k − Mθ,k

1 − α2 r if α = |k|p /= 1

μ0jkpMr,k − Mθ,k

2r ln (r) if α = |k|p = 1

(5)

where Iα and Kα are modified Bessel functions of the first and second kind of order α, with τ2m,k = −jkpΩμ0μrsσs.

p is the number of pole-pairs, Ω is the rotor speed, and μrs and σs are the relative permeability and the conductivityof the SMC material, respectively. Mr,k and Mθ,k are the complex Fourier coefficients of the radial and tangentialcomponents of the magnetization vector distribution. A

(i)m,k and B

(i)m,k are constant coefficients that can be determined

by the boundary conditions, given by:

H(I)θ (r, θr) |r=Rro = 0

H(i)θ (r, θr) |r=Ri+1 = H

(i+1)θ (r, θr) |r=Ri+1

B(i)r (r, θr) |r=Ri+1 = B(i+1)

r (r, θr) |r=Ri+1

⎫⎬⎭ for i = I, II, III

A(I)m (r, θr) |r=Rso = 0

(6)

where B and H are the flux density and the magnetic field strength, respectively.

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The same computation method is used to predict the armature reaction field produced by the stator currents. Thederivation of the total currents density for the three-phase stator windings under steady-state operation is described in[2]. It can be generalized in the following complex form in the stator reference frame:

Js (θs, t) =+∞∑

k=−∞

+∞∑h=−∞

− 3

4δkIhexpj(kpθs+hpΩt−ϕh) if k + h = 3l, l ∈Z (7)

where δk are the Fourier coefficients of the winding density distribution, k is the order of the odd space-harmonics. Ih

and ϕh are the amplitude and the phase angle of the stator current nontriplen odd time-harmonics of order h.By applying (1) in each region of the generator, the following solutions are obtained:

A(i=I,II)s (r, θs, t) =

+∞∑k=−∞

+∞∑h=−∞

(A

(i)s,k,hr

α + B(i)s,k,hr

−α)

Jk,hexpj(kpθs+hpΩt−ϕh) (8)

A(III)s (r, θs, t) =

+∞∑k=−∞

+∞∑h=−∞

(A

(i)s,k,hr

α + B(i)s,k,hr

−α + Rp,k,h (r))

Jk,hexpj(kpθs+hpΩt−ϕh) (9)

A(IV )s (r, θs, t) =

+∞∑k=−∞

+∞∑h=−∞

(A

(IV )s,k,hIα

(τs,k,hr

) + B(IV )s,k,hKα

(τs,k,hr

))Jk,hexpj(kpθs+hpΩt−ϕh) (10)

where τ2s,k,h = jkpΩμ0μrsσs and Jk,h = −(3/4)δkIh. Rp,k,h (r) is a particular solution of the field problem in the

winding region that can be expressed as:

Rp,k,h (r) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−μ0r2

4 − α2 if α = |k|p /= 2

−μ0r2 ln (r)

4if α = |k|p = 2

(11)

The constant coefficients A(i)s,k,h and B

(i)s,k,h can be derived by using the same boundary conditions given in (6).

From the knowledge of the resultant magnetic field in terms of total potential vector Atot = Am + As, all elec-tromagnetic characteristics required for the design model such as magnetic field density, torque, losses, e.m.f. andinductances can be derived. As an example, the calculated radial and tangential components of the flux density dueto the magnets and the stator currents are reported in Fig. 3 for a typical 4 poles, 1.5 MW, 18,000 rpm slotless PMgenerator equipped with radial magnetized magnets. The results have been also validated by 2D time-stepping FEanalysis by assuming that the magnetic saturation of materials is negligible. One can notice an excellent agreementbetween the two calculation methods.

2.2. Generator losses computation

2.2.1. Stator magnetic lossesThe stator magnetic losses in the SMC material can be derived from the distribution of the resultant electromagnetic

field calculated under generator full-load operation [3]. One assumes that they can be separated into eddy current lossescomponent and hysteresis losses component. The eddy current losses are computed by using the Poynting’s vectormethod in the stator reference system [10]:

PEC = 1

∫ 2π

0

[∫∫(S)

Real( �E × �H∗)

d�S]

dωt (12)

where E and H are the resultant electric and magnetic fields, ω = pΩ is the angular frequency and S is the integrationsurface located at the inner radius of the stator core. The resultant electric field is axially directed and can be determined

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A. Chebak et al. / Mathematics and Computers in Simulation 81 (2010) 239–251 243

Fig. 3. Flux density components due to magnets (a) and armature currents (b) at inner radius of the winding (analytical model vs. FE analysisresults).

form the total potential vector Atot in the SMC stator yoke region:

E = −∂A(IV )tot (r, θs, t)

∂t= −∂A(IV )

m (r, θs, t)

∂t+ −∂A

(IV )s (r, θs, t)

∂t(13)

The eddy current loss computation has been also verified by using a second method based on the integration of theresultant eddy current density over the stator core volume.

The hysteresis losses are calculated by a specific postprocessor based on an analytical expression of the SMChysteresis loss density [4]:

PH

[W/kg

] = CmBxmaxf (14)

where f is the electric frequency, Cm and x are specific SMC material coefficients and Bmax is the amplitude of thefull-load stator core flux density over one steady-state period. The stator core is split in small rings. For each ringaverage radius, Bmax is derived from the resolution of the magneto-dynamic Eq. (1). The total hysteresis losses arethen computed by integration on the volume of the stator core.

Fig. 4 presents the variations of the stator magnetic losses components vs. rotor speed for the optimized slotless PMgenerator (1.5 MW, 18,000 rpm) described in Table A.1 of Appendix A, where the 51 first time and space harmonicsare taken into account (k = h = 51). The analytical computation of the eddy current losses is also validated by FEsimulations. One can notice that the eddy current losses remain more significant than the hysteresis losses in this highpower machine despite the use of a less conductive SMC material. An opposite result has been obtained for low powerslotless PM machines [2,3].

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244 A. Chebak et al. / Mathematics and Computers in Simulation 81 (2010) 239–251

Fig. 4. Variation of stator magnetic losses with rotor speed.

2.2.2. Copper lossesA conventional approach is used to compute the copper losses in the windings as a sum of two components: the

RI2 losses corresponding to the winding DC resistance R and the eddy current losses due to proximity effect resultingfrom the magnet movement. The skin effect is neglected since Litz wire will be used, with a suitable strand radius rLZ

lower than the skin depth corresponding to the operation frequency. The average of the extra eddy current losses pervolume for round wire can be written as the summation over the space frequency components [1]:

PE =+∞∑k=1

(B2

r,k + B2θ,k

) k2ω2r2LZ

8ρc

(15)

where Br,k and Bθ,k are the peak values of the flux density of the k th space-harmonic radial and tangential components,respectively, and ρc is the copper resistivity.

2.2.3. Mechanical lossesThe windage losses of the rotating rotor are derived from the drag torques of a rotating cylinder and a rotating disk

corresponding to the rotor ends, respectively. The corresponding losses are:

PF = CFπρΩ3R4oL (16)

PF,end = 1

2CF,endρΩ3

(R5

o − R5i

)(17)

where CF and CF,end are specific friction coefficients [9]. ρ is the density of the fluid. Ro and Ri are the outer and theinner radius of the rotor, and L is the length of the cylinder.

2.3. Torque computation

The generator instantaneous electromagnetic torque is determined by integrating the Maxwell Stress Tensor alongthe air-gap:

Tem (t) = roμ0

∫∫(S′)

HrHθdS′ (18)

where S′ is an integration surface that can be located at any radius ro in the air-gap region. Hr and Hθ are the radialand tangential components of the resultant magnetic field. This torque takes into account the interaction between themagnets, the armature currents and the eddy currents induced in the stator yoke. The average electromagnetic torque canbe derived from the combinations of the time and space harmonics that are synchronous with the rotor, i.e. k + h = 0.

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A. Chebak et al. / Mathematics and Computers in Simulation 81 (2010) 239–251 245

Fig. 5. Instantaneous electromagnetic torque and its average value.

The torques corresponding to the hysteresis losses and the mechanical losses are added to the average electromagnetictorque to get the input generator torque delivered by the gas turbine.

Fig. 5 depicts a comparison between the instantaneous electromagnetic torque calculated by the analytical methodand that computed by the time-stepping FE analysis for the current waveform and magnets of the optimized slotless PMgenerator. The average value of the electromagnetic torque is also presented. One can notice a good correspondencebetween the results obtained by the two calculation methods.

2.4. Generator equivalent electrical circuit

The electrical parameters of the generator single phase equivalent circuit must be derived from the machine analyticalmodel and transferred to the electrical model of the machine–rectifier system to perform the simulations of the wholegeneration system. The e.m.f. is computed from the analytical resolution of the magnetic field by taking the effects ofharmonics into account. The inductance is calculated for each space and time-harmonic. Fig. 6 presents the variationsof the machine inductance vs. frequency for the optimal generator characteristics (Table A.1). One can notice that theinductance corresponding to each current time-harmonic is nearly constant in the frequency domain. According to thisresult, we assume that the inductance of the equivalent circuit is constant. The stator phase resistance is derived fromthe stator dimensions and it also takes into account the effects of the extra eddy current losses. The generator equivalent

Fig. 6. Frequency response of the generator inductance.

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246 A. Chebak et al. / Mathematics and Computers in Simulation 81 (2010) 239–251

Fig. 7. Inputs and outputs of the generator analytical design model.

circuit is assumed to be composed of a non-sinusoidal voltage source corresponding to the e.m.f. connected in serieswith the stator inductance and resistance.

The input and output variables of the generator analytical design model are presented in Fig. 7. The geometricaldimensions, the structural parameters (i.e. number of poles, number of slots, ...), the speed, a corrected value Kc(i)Vdo ofthe specified rated DC output voltage Vdo and the harmonic content of the armature currents (amplitude and phase angle)are the main input variables. Kc(i) is a variable input correction factor that is used to implement the correction mechanismof the coupling between the machine analytical design and the electrical model of the machine–rectifier system describedin the next paragraph. The performances in terms of torque and losses, the equivalent circuit parameters, the output DCvoltage Vdi computed by the model are the main output variables. It must be noticed that any stator current waveformcan be imposed in the generator analytical design model in terms of fundamental and time-harmonics.

3. Electrical model of the generation system

The electrical model of the whole generation system is presented in Fig. 8: it is composed of the generator electricalequivalent circuit linked to the rectifier circuit. For simplification purpose, the rectifier output circuit is modeled witha current source Ido (specified rated DC current of the generation unit). One assumes that the output current rippleis negligible according to a suitable choice of the LC filter inductance. The electrical circuit of Fig. 8 is simulatedin Matlab/Simulink and an FFT analysis of the steady-state current and voltage waveforms is performed. The outputvariables of this electrical model are the harmonic content of the armature currents and the rectifier output DC voltageVdsim that takes the influence of the current commutation into account. These variables are used to perform the correctionmechanism of the coupling between both design models.

4. Optimization design process

The generator analytical design model and the electrical model of the generation system have been implemented ina same design environment with a non-linear constrained optimization procedure. The flowchart of the optimal designprocess including the iterative correction mechanism of the coupling between both design models is presented in Fig. 9.In a first step, the main requirements of the generation system are derived from the specifications of the application:unit output power Pout , generator speed Ω, specified rated DC output voltage Vdo, overall dimensional constraintsof the generator, magnet characteristics, winding configuration, number of slots and poles, filling factor and materialdata. The initial set of amplitude and phase angle of the armature current harmonics corresponds to a square armaturecurrent waveform (instantaneous current commutation) and the initial correction factor is set to Kc(1) = 1.

Fig. 8. Inputs and outputs of the electrical model of the generation system.

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A. Chebak et al. / Mathematics and Computers in Simulation 81 (2010) 239–251 247

Fig. 9. Design optimization method with correction mechanism.

In the next step, the generator design is performed by solving a nonlinear constrained optimization problem for thegiven set of amplitude and phase of the armature current harmonics and correction factor value. The optimization statevariables are the main dimensions defined in Fig. 2 and some structural and winding parameters:

• The rotor yoke inner radius Rri.• The rotor yoke thickness hr.• The magnet thickness hm.• The mechanical air-gap e.• The windings thickness hw.• The stator yoke thickness hs.• The stator axial length l.• The magnet width to pole pitch ratio β.• The number of turns per phase Nsp.

The objective function to be minimized is related to the materials cost:

Obj = 10Pmag + 7Pcopp + Piron (19)

where Pmag, Pcopp and Piron are the magnet, copper and SMC weights, respectively.Several constraints are imposed:

• Magnet demagnetization constraints: the PM demagnetization is calculated at the inner and outer magnet radius forthe worst-case of armature demagnetizing reaction. The PM demagnetization limit is 0.3 T at 125 ◦C for the selectedmagnet material.

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Table 1DC voltages and correction factors.

i = 1 i = 2 i = 3 i = 4 i = 5

Vdo [V] 1500 1500 1500 1500 1500Vdi [V] 1515.8 1754.4 1813.6 1827 1826.8Vdsim [V] 1296 1451 1489 1501 1500Kc(i) 1 1.1696 1.2091 1.218 1.2172Kc(i+1) 1.1696 1.2091 1.218 1.2172 1.2172

• Loss constraint: a maximal value of generator losses is imposed according to the specified cooling system perfor-mance (water cooling system).

• Mechanical constraints: the mechanical stress due to high-speed operation defines the maximal rotor radius and theminimal retaining sleeve thickness.

• Geometrical constraints for the overall dimensions.• Saturation constraints in the stator and rotor cores.• Specified rectifier output power constraint.• DC voltage constraint: the DC voltage Vdi computed by the generator design model must respect Vdi ≥ Kc(i)Vdo.

The electrical parameters of the equivalent circuit are transferred to the electrical model of the generation systemand a simulation of the circuit is carried out. New values of the amplitude and phase of the armature current harmonicsand of the output DC voltage Vdsim are then computed.

A specific correction mechanism is then applied after this step: the new current harmonic content is reinjected inthe generator analytical design model and the DC voltage constraint Kc(i)Vdo is actualized by injection of a new valueof the correction factor Kc(i+1) that is computed from (Fig. 8):

Kc(i+1) = Vdi

Vdsim

(20)

A new execution of the nonlinear constrained optimization problem of the generator is then performed. The processis repeated until a stable optimal solution is found in terms of output DC voltage (i.e. Vdsim = Vdo) and all generatorcharacteristics are identical to those of the previous optimal solution. The convergence of the preceding design processis reached with a limited number of correction steps.

Fig. 10. Generator optimal structure.

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Fig. 11. e.m.f., current and phase voltage waveforms.

5. Optimization results

The proposed design procedure has been implemented in Matlab and applied to design a 3 phase, 4 poles, 36 slots,high-speed PM generator using radial sintered NdFeB magnets. The slotless stator yoke is made of SMC material witha relative permeability of 200 and a conductivity of 2500 S/m. The rated speed of the generator is 18,000 rpm. Thespecified output power of the generation system is 1.5 MW at 1500 V DC voltage and full rectifier operation (zerofiring angle).

The final optimal design solution is obtained in 3 iterations of the preceding correction mechanism. Table 1 presentsthe values of the calculated and simulated DC voltages for each intermediate optimal solution and the iterations of thecorrection factor Kc(i). One can notice that the convergence of the mechanism is obtained after 4 executions of thenonlinear constrained optimization procedure only.

Fig. 10 shows the final optimal structure of the slotless PM generator. Its main characteristics are listed in Table A.1 ofAppendix A. One can notice that the PM demagnetization is the most important constraint, since the slotless generatortopology has a high electrical loading and low magnetic loading. Therefore, the copper losses are more significant thanother losses and represent 67% of the total losses with 31% for the magnetic losses.

Fig. 11 presents the waveforms of the e.m.f., current and terminal voltage in a phase coil of the stator winding.One can notice that the current commutation has a great influence on the stator voltage. Since the slotless generatorinductance is low, the use of a conductive retaining sleeve is not necessary to improve the current commutation like in

Fig. 12. Calculated and simulated phase voltage waveforms.

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250 A. Chebak et al. / Mathematics and Computers in Simulation 81 (2010) 239–251

slotted structures [11]. When the firing angle increases, the commutation time decreases. The output DC voltage andcurrent also decrease.

Fig. 12 compares the waveform of the stator line voltage computed with the generator analytical model by summationof the different time-harmonics and waveform of the stator line voltage obtained by simulating the electrical model ofthe whole generation system. One can notice a good agreement between these two waveforms. This result validatesthe assumption made on the inductance and on the structure of the generator equivalent circuit.

6. Conclusion

A specific design method of high-speed DC generation system using a slotless PM machine with a SMC statoryoke connected to a controlled rectifier has been developed. This method is based on a generator analytical designmodel, an electrical model of the machine–rectifier system, a simulation tool and a non-linear constrained optimizationprocedure. The coupling between both models is achieved by a specific correction mechanism that performs an efficientconvergence of the optimization procedure. The efficiency of the method has been validated and the overall convergenceis reached with a limited number of circuit simulations.

Appendix A.

Table A.1.

Table A.1Characteristics of the generator optimal structure.

Parameter Value Unit

Number of poles 4 [–]Number of slots 36 [–]Short pitch factor 1 [–]NdFeB magnet remanent flux density 1.2 [T]NdFeB magnet relative permeability 1.05 [–]Copper filling factor 0.37 [–]Number of turns per phase 24 [–]Stator axial length 300 [mm]Outer stator diameter 291.4 [mm]Stator yoke thickness 34.5 [mm]Inner winding diameter 183.7 [mm]Windings thickness 19.4 [mm]Retaining sleeve thickness 3 [mm]Mechanical air-gap 1.5 [mm]Magnet thickness 28.5 [mm]Rotor yoke thickness 27 [mm]Magnet width to pole pitch ratio 0.748 [–]Specific loading 187680 [A/m]Current density 24.96 [A/mm2]RMS value of e.m.f. 993 [V]Cyclic inductance 0.2086 [mH]Effective stator phase resistance 0.0237 [�]No-load maximal air-gap flux density 0.55 [T]Magnet weight 21.71 [kg]Copper weight 20.34 [kg]Iron weight 77.71 [kg]Total generator weight 119.76 [kg]Copper losses 40.22 [kW]Stator eddy current losses 14.73 [kW]Stator hysteresis losses 4.07 [kW]Windage losses 0.97 [kW]Total generator losses 60 [kW]Efficiency 96.16 [%]

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