1

Abstract— In the past couple of years, the

permanent magnet synchronous generator has been a widely used generator. In spite of its many advantages

it also presents some disadvantages. This papers aims to identify them and propose a topology that nullifies

this disadvantages. A single phase low voltage generator belonging to an

isolated system and providing 20kW to a load modeling

a residence is proposed. The presented generator is aimed to work through the kinetic energy extracted

from a river current.

This research will study and analyze the

electromagnetic waveforms and the thermal distribution of the generator through a finite element

model. The equations best suited to calculate the power losses in the low speed permanent magnet

generator will be identified. Particular attention will be allocated to the identification of the main constraints in the generator sizing.

Finally, the dielectric insulation lifetime will be estimated.

Index Terms— PM synchronous generator, power

losses, finite-element analysis (FEA), thermal analysis , electromagnetic analysis.

I. INTRODUCTION

he permanent magnet synchronous generator was

developed around 1950 [1], and since then, has

been the main electrical machine topology in both low

velocities applications, such as energy production

through alternative sources [2,3], as well as high

velocities applications, such as aeronautical industry

and flywheels [4,5]. The permanent magnet generator

has great benefits such as the absence of brushes,

smaller volume and higher efficiency.

In this type of generator, the permanent magnets are

usually located in the inner part of the machine.

However, this topology has some drawbacks, the main

one being the occasional ungluing of the magnets due

to centrifugal forces originated by the rotational motion

of the generator. Another drawback is the fact that this

configuration makes the magnetization in loco of the

magnets difficult.

The study of a new topology in which the magnets

are located in the outer part of the machine, that part

being the rotor, is proposed in this paper. With this

approach the centrifugal force tends to compress the

magnets instead of ungluing them, while making this

component more easily accessible allowing for an easy

maintenance.

The proposed generator was designed to operate

through the kinetic energy extracted from river

currents. The name of the used system is RiverSails,

[6], and was developed by the company Tidal Sails. The

system consists in a series of extruded aluminum sails,

attached to wire ropes strung across the tidal stream.

It forms a geometric figure similar to that represented

in Fig 1. The generator studied is aimed to be placed in

the corners of that system.

Fig.1. RiverSails system

Based on the river speed the generator was targeted

to rotate at 100 rpm at nominal operation.

II. MATERIALS THAT FORM THE GENERATOR

In this section the materials that form the generator

are presented.

A. Permanent magnets

Permanent magnets were chosen to be made of

NdFeB. This choice was based on the fact that this type

of magnets presents the higher value of residual

magnetic flux density at satisfactory working

temperatures [7]. The demagnetization curves of the

selected magnets are represented in Fig. 2.

Electric Rotating Stator Generator with

Permanent Magnets and Fixed Rotor with Concentrated Windings: Analysis and Study

on its Magnetic Circuit

Gonçalo Miguéis, Student, DEEC/AC Energia and P.J. Costa Branco, LAETA/IDMEC

T

2

Fig.2. Demagnetization curves of the permanent magnets

B. Soft magnetic material

Due to its low cost, a non-oriented type of magnetic

material was chosen. Its magnetization curve is

illustrated in Fig. 3. This material is constituted by

laminated and dielectrically insulated sheets.

Fig.3. BH curve of the soft magnetic material

C. Shaft

Steel was selected to form the shaft of the generator .

This material was chosen due to its strong mechanical

characteristics associated with the fact that it is

nonmagnetic and so does not influence the magnetic

circuit of the generator.

D. Conductors

The conductors are made of copper. This is a common

choice in the construction of electric machines due to

relation between its price and electrical conductivity.

III. POWER LOSSES CALCULATION

In addition to reducing the efficiency, the power

losses also have the adverse effect of the heating the

electrical machines. The methods used to calculate

these losses are presented in this section.

A. Copper losses

The copper losses in the generator windings, 𝑃𝑐𝑢 ,

were calculated through equation (1). This equation is

only valid for sinusoidal systems. As it is not the case,

the total copper losses are the some of the equation (1)

applied to all the harmonics of the generator current.

The parameter 𝑟𝑐𝑢 is the winding resistance and 𝐼𝐼𝑛 is

the rms value of the generator current. 𝑃𝑐𝑢 = 𝑟𝑐𝑢 𝐼𝐼𝑛

2 (1)

B. Ferromagnetic materials losses

Due to low velocity of the generator and consequent

low electric frequency of its currents, it is sometimes

difficult to estimate the Steinmetz coefficients from the

manufacturer data. Therefore, the Steinmetz equation,

regardless of being one of the most used in that matter,

is not used in this paper to estimate losses in the

ferromagnetic materials.

The selected equation is (2). The symbol 𝐽𝐹

represents the rms value of the current density of the

Foucault currents and 𝜎 is the electrical conductivity of

the material where the losses are calculated.

𝑃𝐹 = ∫𝐽𝐹

2

𝜎𝑉

𝑑𝑉 (2)

The ferromagnetic materials include the permanent

magnets and the soft ferromagnetic material. Since the

soft magnetic material is formed by a series of

insulated electrical sheets, its electrical conductivity

differs from that of a solid block. The equation (4)

allows the estimation of the electrical conductivity of the laminated material, [8]. The symbol 𝜎𝑒𝑞 is the

equivalent electrical conductivity of the laminated

material, 𝜎𝑀 is the electrical conductivity of the non-

laminated material, 𝑥 𝑙𝑎𝑚 is the number of laminations

that the material has and can be calculated through

the equation (3). In this equation 𝐷 is the depth of the

generator and 𝜀𝑙𝑎𝑚 is the thickness of one sheet.

𝑥𝑙𝑎𝑚 =𝐷

𝜀𝑙𝑎𝑚

(3)

𝜎𝑒𝑞 =𝜎𝑀

𝑥𝑙𝑎𝑚2 (4)

IV. ELECTRICAL GENERATOR, POWER CONVERTER AND

LOAD

The electrical generator geometry as well as the

power converter and load, which forms the isolated

system in study, is presented in this section.

3

A. PM synchronous generator

Fig.4. Magnetization direction of the permanent magnets and

winding direction of the electrical conductors

Fig. 5 illustrates a 3D image of the permanent

magnet generator. Its dimensions are synthetized in

Tab. I.

Fig. 5 Electrical generator proposed geometry

TABLE I GEOMETRIC CHARACTERISTICS OF THE ELECTRICAL GENERATOR

Parameter Value

Stator diameter [mm] 486

Air gap [mm] 1

Permanent magnets

height [mm]

6

Rotor diameter [mm] 540

Depth [mm] 480

Number of pole pairs 10

A single phase low voltage generator is proposed in

this paper. This implies that the maximum value of its

rms voltage has to be lower than 1kV, [9].

In Fig. 4 the directions of magnetization of the

permanent magnets as well as the winding direction of

the conductors are presented. Due to difficulties in

representation it is only pictured a winding per pole in

Fig. 4. However, the generator has 𝑁𝑠 windings in

series per slot. In order to reduce copper losses there

are also 𝑁𝑃 of this circuits in parallel.

B. Power converter

The use of a power converter is necessary due to the

voltage amplitude and electric frequency difference

between the generator and load. An AC/DC/AC

converter is proposed and can be seen in Fig. 6. This

converter consists of a single-phase rectifier followed

by a capacitor (DC link) and a three-phase inverter.

Fig. 6. Isolated electrical system with focus on power converter

C. Electric load

Fig. 7. Isolated electrical system with focus on electrical load

The electrical load is intended to model an average

residence. For this reason, it is constituted by a three

phase resistance in series with an inductor with a

power factor of 0.86. It is also intended that the

generator provides 20 kW of active power to the load.

The electrical load is represented Fig. 7.

𝐿𝑜

𝑎𝑑

Load Rectifier Inverter

𝐺𝑀𝑃

4

V. REQUIREMENTS IN SIZING PM GENERATOR

The sizing of the proposed generator implies the

fulfilment of certain conditions, both electromagnetic

and thermal. This section lists these constraints.

A. Electromagnetic constraints

The first of these constraints are expressed in

equations (5) and (6) and require that the magnetic flux

density in soft magnetic material in the rotor, 𝐵𝐼𝑑𝑡 , and

the stator, 𝐵𝐼𝑑𝑧, of the generator are less than the value

that forms a “knee” in the magnetization curve of this material, 𝐵𝐽 . Its conditions assure the non-saturation

of the material. 𝐵𝐼𝑑𝑡 ≤ 𝐵𝐽 (5)

𝐵𝐼𝑑𝑧 ≤ 𝐵𝐽 (6)

The following of these constraints are depicted in

equation (7) and determines the maximum number of

windings in a slot, 𝑁𝑀𝑎𝑥, assuring that the area of the

simulated number of windings is less than the area of

the slot in which they will be allocated. The variable

𝐴𝑟𝑒 𝑎𝐶𝑜𝑛𝑑 is the cross section of a conductor, 2.09 𝜇 𝑚2,

𝐴𝑟𝑒 𝑎𝑆𝑙𝑜𝑡 is the area of the slot, and 𝐾𝑐𝑢 is the window

the utilization factor and represents the fraction of the

core window area that is filled by copper, its usual

values vary between 0.3 and 0.7 [10]. The selected value

was 0.5.

𝑁𝑆 ∙ 𝑁𝑃 ≤𝐾𝑐𝑢 𝐴𝑟𝑒 𝑎𝑆𝑙𝑜𝑡

𝐴𝑟𝑒𝑎𝐶𝑜𝑛𝑑

= 𝑁𝑀𝑎𝑥 (7)

The following constraints, eq. (8) and eq. (9), assure

that the generator doesn’t exceed a maximum voltage

limit, eq. (8), and that the active power delivered to the load, 𝑃𝐶 , is the desired, eq. (9). 𝑉𝐺𝑒 𝑟𝑀𝑎𝑥

is the maximum

value of the voltage at the generator terminals. This

voltage value was imposed in order to avoid the use of

𝑑𝑉/𝑑𝑡 filters, [11]. 𝑉𝐺𝑒 𝑟𝑀𝑎𝑥

< 1 kV (8)

𝑃𝐶 = 20 kW (9)

The last constraint is related to the ability of the

permanent magnets to keep its magnetization

competence. For this it is required that the flux density in the magnet, 𝐵𝑀𝑎𝑔 , is more than that forming the

“knee” in Fig. 2, [12], in this case 0. 35 T. 𝐵𝑀𝑎𝑔 > 0.35 T (10)

If the constraints (7) to (10) are not met, the

parameter 𝑁𝑆 which represents the number of windings

in series per slot, should be decreased. If the

constraints (5) and (6) are not met the geometry of the

generator should be rearranged.

B. Thermal requirements

Once again the first thermal requirement is related

to the competence of the permanent magnets to keep

its magnetization ability. For this it is required that

the temperature in the magnets, 𝑇𝑀𝑎𝑔 , is less than his

maximum operation temperature value, [12]. For the

chosen permanent magnets this temperature is 150℃ ,

but in order to assure a safety margin it was decided

that the temperature in the magnets must be less than

135℃ . 𝑇𝑀𝑎𝑔 < 135℃ (11)

Another temperature limit that has to be respected

concerns the conductor’s insulation, 𝑇𝐶𝑜𝑛𝑑 . For the

present machine, an insulator with a maximum

operating temperature of 180℃ was chosen. 𝑇𝐶𝑜𝑛𝑑 < 180℃ (12)

If the constraints (11) to (12) are not met, the

parameter 𝑁𝑃 which represents the number of circuits

in parallel per slot, should be increased.

VI. ELECTROMAGNETIC WAVEFORMS IN NOMINAL

OPERATION MODE

The generator topology presented in chapter IV was

simulated in an electromagnetic finite element model

from a 2D geometry. The variables 𝑁𝑆, 𝑁𝑃 and 𝐷 were

changed in order to fulfil all the constraints mentioned

in the previous chapter.

A. Electromagnetic waveforms in nominal operation mode

Fig. 8 shows the magnetic flux density distribution.

It can be seen in this figure that the constraints (5) and

(6) are met.

Fig. 8. magnetic flux density distribution

Fig. 9 illustrates the isolated system in study. In this

figure, the voltage and currents presented in the course

of this paper are schematized.

Fig. 10 shows the voltage at the generator terminals

and Fig. 11 shows its harmonics. In Fig. 11 can be seen

that the generator voltage has a 3rd harmonic of high

amplitude, however it is not a requirement that the

generator waveform be close to sinusoidal. Only the

load voltage and current need to meet this criterion. We

can see in Fig. 10 that the constraint (8) is met.

𝐵 [T]

5

Fig. 9. Electrical system in study

Fig. 10. Voltage at generator terminals

Fig. 11. Harmonic content of voltage at generator terminals

Fig. 12 shows the current in the generator

conductors and Fig. 13 shows its harmonics. These

results show once again that the generator current has

some harmonics of considerable amplitude.

Fig. 12. Current in the generator windings

The Fig. 14 and Fig. 15 show us the load voltage and

current respectively. Based on this results It can be

concluded that these waveforms constitute a balanced

alternate sinusoidal system.

Fig. 13. Harmonic content of current in generator windings

Fig. 14. Load voltage

Fig. 15. Load current

Fig. 16 shows the load voltage waveform harmonics.

It can be observed that this waveform it is not purely

sinusoidal due to the existence of a harmonic of 7th

order. However, the amplitude of this harmonic is less

than 2% than the amplitude of the fundamental.

Therefore, it can be considered negligible.

Fig. 16. Harmonic content of load voltage

Fig. 17. Harmonic content of load current

𝐿𝑜

𝑎𝑑

𝐼𝐿𝑜𝑎𝑑𝐴

𝐼𝐿𝑜𝑎𝑑𝐵

𝐼𝐿𝑜𝑎𝑑𝐶

𝑉𝐿𝑜𝑎 𝑑𝐴 𝑉𝐿𝑜𝑎 𝑑𝐶

𝑉𝐿𝑜𝑎 𝑑𝐵

𝐼𝐼𝑛

𝑉𝐼𝑛

6

On the other hand, we can see in Fig. 17 that the load

current waveform is purely sinusoidal.

Fig. 18 shows the magnetic flux density in the

permanent magnets during an electrical cycle. This

waveform is not purely sinusoidal due to effects of the

magnetic flux density produced by the current that

circulates in the generators conductors, according to

Lenz law. Analyzing Fig. 18 it can be concluded that

constraint (10) is also met.

Fig. 18. Magnetic flux density in the permanent magnets

VII. THERMAL FINITE ELEMENT ANALYSIS

With the aim of checking whether the generator

materials have a temperature value above its

maximum operating temperature, the thermal

behavior of the generator was simulated with the aid

of a finite element program.

A. 2D finite element thermal model

The temperatures in the materials of the electrical

generator are dependent on the surrounding external

temperature and on the heat dissipation created by the

power losses. The permanent magnet generator

operates at low electric frequency, 𝑓, of 50

3 Hz according

to equation (13) where 𝑛 is the velocity, 100 rpm and 𝑝

is the number of pole pairs, 10. Since the electric

frequency has a low value and since the power losses in

the soft magnetic material are proportional to its

frequency, we conclude that these losses can be

considered negligible.

𝑓 =𝑛 𝑝

60 (13)

On the other hand, the permanent magnet, 𝑃𝑀𝑃 ,

and copper losses, 𝑃𝐶𝑢 , have a high impact on the

generator temperature and are presented in Tab. II.

The copper losses are the most significant ones,

accounting for approximately 80% of the generator

losses

TABLE II LOSSES IN THE PERMANENT MAGNET GENERATOR

Parameter Value 𝑃𝐶𝑢 [W] 497

𝑃𝑀𝑃 [W] 132

Based on these power losses values, the generator

temperature distribution illustrated in Fig. 19 was

obtained.

Fig. 19. Thermal distribution of generator simulated in 2D finite

element model

Fig. 19 shows that the constraints (11) and (12) are

met since the temperature on the magnets is 134℃ and

the temperature on the copper conductors is 155℃ .

Since Fig. 19 was obtained through a 2D thermal finite

element model the heat distribution was only

considered radially due to software limitations.

B. 3D finite element thermal model

In spite of Fig. 19 presenting the worst possible

scenario of the machines operation conditions, this is

not the thermal distribution that it is obtained in

nominal operation conditions. In order to accurately

obtain this distribution, the same geometry with the

same power losses values was simulated, on a 3D

thermal finite element model. The thermal distribution

obtained by this model is illustrated in Fig. 20.

Fig. 20. Thermal distribution of generator simulated in 3D finite

element model

It can be seen in Fig. 19 that the heat distribution

now occurs radially and through the generator shaft.

In this model the temperature on the magnets is 8℃

𝑇 [℃]

𝑇 [℃]

7

less and the temperature on the copper conductors is

12℃ less than the temperature present in Fig. 19.

VIII. UNBALANCED OPERATION LOAD

The performance of the generator in the event of loss

of one phase of the three-phase load will be analyzed in

this section. In this event, the load voltage and the load

current have the waveforms represented in Fig. 21 and

Fig. 22 respectively. It can be verified that the phase

differences between currents or voltages becomes 90º

instead of the 120º shown in Fig. 14 and Fig. 15.

Fig. 21. Load voltage with loss of one of the load phases

Fig. 22. Load current with loss of one of the load phases

It can be concluded that the load current waveform

is almost the same as the one in balanced load

operation. On the other hand, the voltage waveform

has a slight distortion, in addition to having a rms

value of 221 V which corresponds to a reduction of 4%

comparing with the balanced load operation.

Regarding the generator’s voltage waveform, it can

be concluded that this waveform, Fig. 23, is similar to

that verified in balanced load operation.

Fig. 23. Generator’s voltage with loss of one of the load phases

On the other hand, due to the loss of one of the load

phases the equivalent load impedance has a lower

value than that in the balanced load operation, leading

to a 40% lower rms generator’s current, Fig. 24.

Fig. 24. Generator’s current with loss of one of the load phases

Due to this low rms generator’s current, the

generator’s copper losses drop 64% . The permanent

magnet losses also suffer a reduction of 38% comparing

to the balanced load operation.

Since the generator’s power losses in unbalanced

load operation has lower values that the ones in

balanced load operation, it can be implied that the

generator material’s temperatures will be also lower in

these operation conditions.

Therefore, it can be concluded that the generator

supports the loss of one of the load phase without

damaging its lifetime expectancy.

IX. DIELECTRIC INSULATION LIFETIME ESTIMATION

The electrical insulation material used to isolate

between wires has a certain lifetime duration. This

lifetime can be defined as the period of time since the

completion of its fabrication until the point where its

required performance can no longer be achieved [13].

To determine the insulation lifetime, 𝐿𝑖𝑠 equation

(14) [14], was used. In this equation 𝑇𝑖𝑠 is the

temperature in the dielectric insulation, 𝑘𝑖𝑠 is the

Stefan-Boltzmann constant, 𝜗𝑖𝑠 is the activation energy

and 𝐵𝑖𝑠 is a constant associated with the material.

𝐿𝑖𝑠 = 𝐵𝑖𝑠 𝑒𝜗𝑖𝑠

𝑘𝑖𝑠 𝑇𝑖𝑠 (14)

The material chosen to perform dielectric insulation

in the copper conductors was polyester epoxy. This

material has a maximum operation temperature of

180℃ and can withstand the 155℃ verified in the worst

case scenario. The values of the parameters present in

Eq.(14) are in Tab. III, [13,15], and the insulation

lifetime in function of the temperature is illustrated in

Fig.25.

TABLE III

VALUES OF THE PARAMETERS IN EQ. (14)

Parameter Value

𝐵𝑖𝑠 [h] 8.97

𝜗𝑖𝑠 [eV] 1.38

𝑘𝑖𝑠 [ μ eV/K] 86.17

If the permanent magnet generator is always

operating at nominal condition, the insulation lifetime

is 144 340 ℎ (approximately 16 years).

8

Fig. 25. Insulation lifetime of dielectric insulator

X. CONCLUSIONS

During this paper, a permanent magnet topology

that prevents the ungluing of the permanent magnets

was studied. For this, a topology where the stator is

located in the inner part of the machine and the rotor

is located in his outer part was proposed.

The permanent magnet generator was inserted in an

isolated system composed by the generator, a power

converter and an electrical load modeling a residential

load. The generator was dimensioned, to have 10 pole

pairs and deliver 20 kW of active power to the load.

In this paper it was concluded that the losses in the

soft magnetic material are negligible at low frequency,

and that the Steinmetz equation is not a good approach

to calculate power losses due to lack of data from the

material manufactures regarding losses at low electric

frequencies. An alternative approach to calculate these

same power losses is presented, as well as an

alternative approach to calculate electrical

conductivity of the soft magnetic material due to its

constitution of laminated sheets.

In section V the electromagnetic and thermal

constraints related to the sizing of the generator were

presented.

It was concluded in this paper, that although the

voltage and current waveforms at the generator

terminals have some harmonic content, with the

application of the power converter and its filters, the

voltage and current waveforms delivered to the load

have almost no harmonic content.

Regarding the thermal analysis, it was concluded

that the thermal finite element 2D model is not as

accurate as the thermal finite element 3D model since

the first only considers heat dissipation radially.

In the following section it was concluded that the

generator supports the loss of one of the load phases

without damaging its lifetime expectancy.

Finally, in section VIII, the lifetime of the insulating

material between the wires of the winding was

estimated. This lifetime was estimated assuming the

generator operation at nominal conditions. It was

concluded that the insulation lifetime is approximately

16 years.

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