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1372 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 35, NO. 3, SEPTEMBER 2020

Vibration Analysis of Brushless Doubly Fed

Machines in the Presence of Rotor Eccentricity

Salman Abdi , Ehsan Abdi , Senior Member, IEEE, Hamid Toshani , and Richard McMahon

Abstract—In this work, an analytical study has been performed

on the Brushless Doubly Fed Machine’s (BDFM) vibration due

to the interaction of its fundamental magnetic ﬁelds, exerting

bending forces in the back iron. The effects of rotor eccentricity

on exacerbating the machine’s vibration have been considered by

assessing the stator back iron displacement function in the presence

of rotor eccentricity. Finite element analysis is carried out for a

250 kW BDFM built in frame size D400 to validate the analytical

methods. The stator back iron displacement is determined for an

ideally-constructed machine as well as when the rotor has static

and dynamic eccentricity. In addition, the prototype BDFM was

tested at different operating conditions in order to examine its

noise and vibration levels. A set of measurements was conducted

to assess the main vibration component frequencies developed by

the machine at different rotor speeds. It is shown that the main

vibration components are created by bending set-up in the back

iron, rotor eccentricity, and the components with time and space

harmonic natures. The results obtained from ﬁnite element analysis

and experimentally agree with the analytical theory of BDFM

vibration.

Index Terms—Brushless Doubly Fed Machine (BDFM),

Vibration analysis, Rotor eccentricity, Beam theory, Time

harmonics, Space harmonics, Rotor speed ripples, Finite element

analysis.

I. INTRODUCTION

THE Brushless Doubly Fed Machine (BDFM) is a variable

speed generator or motor, which in recent years has been

investigated as a possible replacement for the Doubly-Fed In-

duction Generator (DFIG) [1], currently used in majority of

large wind turbines. Similar to the DFIG concept, a BDFM

allows variable speed operation using a variable voltage, variable

frequency (VVVF) converter rated at only a fraction (30–50%)

of the generator rating [2], [3], but it also beneﬁts from the fact

that it does not require any brush gear, eliminating this source

Manuscript received August 16, 2019; revised February 19, 2020; accepted

March 29, 2020. Date of publication April 15, 2020; date of current version

August 20, 2020. Paper no. TEC-00846-2019. (Corresponding author: Salman

Abdi.)

Salman Abdi is with the School of Engineering, University of East Anglia

(UEA), Norwich NR4 7TJ, U.K. (e-mail: s.abdi-jalebi@uea.ac.uk).

Ehsan Abdi is with the Wind Technologies Ltd, St. John’s Innovation Centre,

Cambridge CB4 0WS, U.K. (e-mail: ehsan.abdi@windtechnologies.com).

Hamid Toshani is with the Iran University of Science and Technology, Tehran

1684613114, Iran (e-mail: h_toshani@elec.iust.ac.ir).

Richard McMahon is with the Warwick Manufacturing Group (WMG),

University of Warwick, Coventry CV4 7AL, U.K. (e-mail: r.mcmahon.1@

warwick.ac.uk).

Color versions of one or more of the ﬁgures in this article are available online

at https://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TEC.2020.2987100

Fig. 1. A schematic of the wind turbine drive train with a BDFM as generator.

of failure and reducing machine maintenance, which is the key

advantage of the machine [4]. The BDFM also shares with the

DFIG the ability to control the reactive power ﬂow through the

machine.

To date, several large BDFMs have been manufactured, for

instance in China with a 200 kW machine [5], in Brazil with

the design of a 75 kW machine [6], and in the UK with the

largest BDFM ever reported. The later was designed and built

in a frame size D400 and tested by the authors and some aspects

of the machine’s performance were reported in [7] and [8].

However, to achieve successful large scale BDFMs for wind

generation application with competitive design and performance

speciﬁcations over its counterparts, it is essential to improve

the machine design including the vibration and acoustic noise

characteristics.

The BDFM is operated with one of its windings, called the

power winding (PW), connected directly to the 3-phase grid.

The other winding, called the control winding (CW), may be

either open circuited or short-circuited, thus operating similar

to induction machines, but at two different speeds [9]. However,

the desirable mode of operation for a BDFM is the ‘synchronous

mode’ in which the control winding is connected to a variable

voltage variable frequency converter as shown in Fig. 1 [7]. The

machine’s operating point is set as with a standard synchronous

machine, but by adjusting the control winding frequency this

operation can be at any rotor speed, leading to variable speed

generator or drive operation [10].

The magnetic ﬁelds in an induction machine exert signiﬁcant

forces on the machine’s stator and rotor iron parts. Such forces

can result in displacements and can be observed in the form

of vibration and noise in the machine. In addition, because the

BDFM has two main ﬁeld components with different frequencies

and pole numbers resulted from two stator windings, the vibra-

tion pattern is more complex in the BDFM than in the induction

machine.

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ABDI et al.: VIBRATION ANALYSIS OF BDFM IN THE PRESENCE OF ROTOR ECCENTRICITY 1373

The presence of rotor eccentricity further increases the

iron displacements and hence vibration levels by essentially

modulating the magnetic ﬁelds produced by the stator wind-

ings and thus introducing further harmonic components [11].

Similar problem has been reported for other two-ﬁeld electri-

cal machines, such as dual stator induction machine [12]. In

this work, an analytical study is conducted on the BDFM’s

modes of vibration caused by the interaction of its fundamental

magnetic ﬁelds exerting bending forces in the back iron. The

effects of rotor eccentricity, including both static and dynamic

eccentricities, on exacerbating the machine’s vibration are taken

into account. Finite element (FE) analysis was carried out for

a large-scale BDFM built in frame size D400 to validate the

analytical methods. The FE models were validated by the experi-

mental data obtained from ﬂux measurements of stator teeth and

back iron. The stator back iron displacement is then determined

for an ideally-constructed machine as well as when the rotor

has static and dynamic eccentricity. The prototype BDFM was

also tested at different operating conditions in order to examine

its noise and vibration levels. A set of measurements was con-

ducted to assess the main vibration frequencies at different rotor

speeds.

II. PREVIOUS WORKS ON BDFM VIBRATION

There are only few studies carried out on the vibration analysis

of BDFMs. Logan et al. [13] derived equations for vibration

components magnitudes generated by the PW and CW mag-

netic ﬁelds as functions of air-gap ﬂux density, pole numbers

and machine dimensions. In that study however, the effects of

ﬁeld harmonics and eccentricity were neglected. Abdi et al.

in [14] proposed a new parallel winding design for the stator

PW and CW to mitigate the vibration level in the presence of

rotor eccentricity. Analytical and experimental analysis of the

BDFM vibration were performed in [15] noting that the vibration

amplitude may be decreased if special attention is given to the

choice of loops number in the rotor cage, number of power and

control windings poles and careful design of the stator core.

Dorrell et al. in [16] proposed a number of rotor designs with

suitable stator pole pair combinations in order to improve the

BDFM design by reducing the unbalanced magnetic pull (UMP).

In order to minimise the magnetising currents in induction

type machines, it is essential to construct the machine with

shortest practical air gap. However, these induction machines

experience strong magnetic ﬁelds across the air gap, which

can cause considerable forces exerted on the iron parts of the

machine. These can ultimately lead to time-varying displace-

ments on the machine’s surface and hence transmit noise to the

surrounding air.

The UMP is an undesirable characteristic in dual stator ma-

chines by which vibrations and acoustic noise can be produced.

It was shown in [16] that UMP occurs when the stator windings

pole-pair numbers differ by one, so this vibration source can be

avoided by careful stator winding design. However, it was shown

in [11], [14] that another source of vibration known as ‘bending

set-up in the back iron’ occurs in all BDFMs irrespective of

pole-pair number combinations causing displacements of stator

back iron and leading to bending mode vibration in the machine.

It was shown in [13] that the dominant term in the calculation

of displacements in a BDFM is related to the difference of stator

pole pair numbers p1–p2and angular frequencies ω1-ω2.This

term essentially represents the existence of two fundamental

ﬁeld components in the machine generated by stator windings

with different frequencies and pole numbers. It was also shown

that the vibration component with angular frequency ω1-ω2

has the greatest contribution in the vibration of an ideally

constructed BDFM.

As with the induction machine, the eccentricity of the rotor

further modulates the ﬁeld patterns, and therefore increases

vibration and noise levels in the machine [17]; however, these

effects are not straightforward to precisely determine through

analytical calculations. Nevertheless, it will be shown in this

paper that in BDFMs with static and dynamic eccentricities,

vibration components at angular frequencies of ωr,ω1-ω2and

ω1-ω2±ωrare developed. The effects of eccentricity in the

vibration spectrum are also discussed in section III.

There are other important factors that can contribute to the

vibration and noise in the BDFM, such as the presence of time

harmonics in stator and rotor winding currents, air gap magnetic

ﬁeld space harmonics, and the dynamics of the mechanical

system (including natural frequencies of various components).

These effects are present in the experimental results shown in

section V, but their analytical investigation is outside the scope

of this paper.

III. MAGNETIC FORCES AND RESULTING DISPLACEMENTS

A. Perfectly Constructed BDFM:

As mentioned in Section II, vibration caused by bending set-

up in the back iron is occurred in BDFMs irrespective of pole

pair number combinations causing stator back iron displacement

and leading to bending mode vibration in the BDFM.

The magnetic ﬁeld in a BDFM air gap comprises two fun-

damental ﬁeld components, one with 2p1poles and the mean

absolute ﬂux density of ¯

B1rotating at ω1rad/s, and another

with 2p2poles and ¯

B2ﬂux density rotating at ω2. The net ﬂux

density is essentially the superposition of the above two ﬁeld

components and can be expressed as a function of time and

space angle [18]:

B(θ, t)=π

2¯

B1cos (p1θ+ω1t+φ1)

+¯

B2cos (p2θ+ω2t+φ2)(1)

where ω1and ω2are the frequencies of the two stator supplies,

and φ1and φ2are phase angles. In (1) any harmonic ﬁeld

components generated by saturation, rotor structure, slotting,

and rotor eccentricity are ignored.

In [13] a theoretical analysis of vibration patterns in the

BDFM was proposed using beam theory as described for the

induction machine by, for example, Alger [19]. Generally, this

assumes that transverse deﬂections dominate shear deﬂections

and that the underlying longitudinal strains are small, so that

changes in curvature can be directly related to the bending mo-

ment on a given cross-section in a linear fashion [13]. Using this

method, the forces exerted by the magnetic ﬁeld are calculated

and the resulting displacement of the iron is determined from

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1374 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 35, NO. 3, SEPTEMBER 2020

Fig. 2. A block diagram showing how stator back iron deﬂection is obtained

using air gap ﬂux density waveform [13].

the procedure illustrated in Fig. 2. The magnitudes of the main

vibration components as function of the machine dimensions

and pole numbers can then be obtained. The algebraic expression

of vibration components gives an insight on how they may be

changed to reduce vibration during the design of BDFMs and

therefore having an analytical expression of the vibration pattern

is important for the BDFM design optimisation.

Based on the above approach, the resulting displacement in

the stator back iron from the exerted magnetic force produced

by the air gap magnetic ﬁeld of (1) can be calculated as [14]:

v(θ, t)=−Kv

×⎡

⎢

⎣

1

(2p1)2−12¯

B2

1cos (2p1θ+2ω1t+2φ1)

+1

(2p2)2−12¯

B2

2cos (2p2θ+2ω2t+2φ2))

+2

(p1−p2)2−12¯

B1¯

B2cos ((p1−p2)θ+(ω1−ω2)t

+(φ1−φ2)) + 2

(p1+p2)2−12¯

B1¯

B2

cos ((p1+p2)θ+(ω1+ω2)t+(φ1+φ2))⎤

⎥

⎦(2)

Kvis dependent on material properties and machine geometry

and expressed as [13]:

Kv=3DaD3

cπ2

64Eymμ0y3

c

(3)

where Dais the air gap diameter, Dcis the median diameter of

the stator back iron, ycis the back iron depth and Eym is the

Young’s modulus of the material. Hence, Kvis constant for a

particular BDFM. The full derivation of (2) can be found in [13].

Equation (2) comprises four terms, the ﬁrst two of which are sin-

gle ﬁeld components, as would be expected in an induction ma-

chine. The latter two terms are dependent on the difference and

sum of the pole pairs, respectively. Since each term is inversely

proportional to the fourth power of the pole pairs, the term that

includes the difference of pole pairs, i.e. the third term with angu-

lar frequency of ω1−ω2, can become relatively large in BDFMs

with a small difference between power and control winding pole

pairs. This will be investigated experimentally in Section V.

B. BDFM With Rotor Eccentricity:

The previous section showed that a perfectly constructed

BDFM experiences vibration modes in addition to those that

would appear in an equivalent induction machine because of the

interaction between two stator ﬁelds. In this section, the effect

of rotor eccentricity is studied as a source of noise and vibration.

Rotor and stator eccentricity can essentially introduce addi-

tional components of ﬂux with different pole numbers from those

of the windings. Two types of eccentricity are considered in this

analysis: static eccentricity where the central axes of the stator

bore and the rotor shaft are offset, and dynamic eccentricity

where the centre axis of rotor lamination stack is offset from

the centre axis of rotor shaft. The air gap length as a result of

eccentricity is:

g(θ, t)=g0−gscos (θ+φs)−gdcos (θ+ωrt+φd)(4)

where g0is the average air gap length, gsand gdare the amplitude

of the static and dynamic eccentricity components respectively,

and ωris the rotor angular velocity. φsand φdare arbitrary angles

relative to the stator reference axis. The magnetic ﬂux density in

the air gap as a result of the surface magnetizing current density

Jm(θ, t)may be derived from the Ampere’s law as:

B(θ, t)=μ0g−1(θ, t)Da

2Jm(θ, t)dθ (5)

where the MMF drops across the iron parts of the magnetic

path are neglected. The inverse of air gap function in (4) can be

expressed using Fourier series as:

g−1(θ, t)=g−1

0+g−1

s1cos (θ+φs1)

+g−1

d1cos (θ+ωrt+φd1)+...

=g−1

0[1 + ds1cos (θ+φs1)

+dd1cos (θ+ωrt+φd1)+...](6)

where ds1and dd1are the static and dynamic eccentricity coef-

ﬁcients, respectively. Given these coefﬁcients are generally very

small, ds1and dd1may be approximated to gs/g0and gd/g0,

respectively [20].

The BDFM magnetizing current density contains p1pole pair

PW and p2pole pair CW components:

Jm(θ, t)= ˆ

Jm1cos (p1θ+ω1t+φ1)

+ˆ

Jm2cos (p2θ+ω2t+φ2)(7)

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ABDI et al.: VIBRATION ANALYSIS OF BDFM IN THE PRESENCE OF ROTOR ECCENTRICITY 1375

TAB L E I

AIR GAP MAGNETIC FLUX DENSITY COMPONENTS IN THE

PRESENCE OF ROTOR ECCENTRICITY

After substituting (6) and (7) into (5), the components of the

ﬂux are shown in Table I, where

¯

Bi=Da

π

g−1

0μ0

pi

ˆ

Jmi (8)

Consequently, the displacement function of the stator back

iron in the presence of rotor eccentricity can be derived from the

procedure described in Fig. 2. Essentially, the exerted magnetic

force produced by the air gap magnetic ﬁeld components of (8)

can be obtained from a series of integrations in order to derive

the resultant back iron displacement. The displacement com-

ponents that arise are summarized in Table II, neglecting those

containing the product of two or more eccentricity coefﬁcients

(e.g. ds1.ds2), which are relatively smaller than other terms.

The magnitudes of displacement components given in Table II

include the constant Kvgiven in (3). The ﬁrst four components

given in rows 1–4 in Table II are also present in the analysis

of displacements for a perfectly constructed BDFM given in

(2). The remaining components (5–24 in Table II) are created

from eccentricities. As previously noted, the displacement terms

have magnitudes inversely proportional to the fourth power of

the pole-pairs and thus it is those with low number of pole pairs

that make signiﬁcant contribution in the machine vibration.

In a 2/4 pole pair machine, as for the D400 BDFM, there are

six components with a single pole pair waveform (i.e. comps. 5,

6, 12, 15, 16 and 22 in Table II) in addition to the main displace-

ment component present in the ideal machine analysis (comp. 3

in Table II), that together give the following approximation for

the back iron displacement:

v(θ, t)=2KV¯

B1¯

B2cos (θ+(ω1−ω2)t+(φ1−φ2))

+ds1¯

B2

1+¯

B2

2cos (θ+φs1)

+ds1

16 ¯

B1¯

B2cos(2θ+(ω1−ω2)t+(φ1−φ2+φs1))

+dd1¯

B2

1+¯

B2

2cos (θ+ωrt+φd1)

+dd1

16 ¯

B1¯

B2cos (2θ+(ω1−ω2+ωr)t

+(φ1−φ2+φd1))] (9)

TAB L E I I

DISPLACEMENTS COMPONENTS IN THE PRESENCE OF ROTOR ECCENTRICITY

From (9) it can be realised that the most important vibration

frequency components expected in the BDFM vibration spec-

trum are:ωr,ω1−ω2and ω1−ω2±ωr. It should be noted that

the vibration component with frequency signature of ω1−ω2

rad/s is created due to both bending set-up in the back iron (as

shown in (2)) and eccentricity (as shown in (9)). Hence, the

two mechanisms superimpose but not necessarily with the same

phase over the operating speed range.

IV. FINITE ELEMENT ANALYSIS OF BDFM MAGNETIC FIELDS

Table III gives details of the 250 kW BDFM used in this study.

Both stator PW and CW are connected in delta. The rotor is a

nested-loop design comprising six nests, each with ﬁve loops

[21]. All rotor loops are terminated with a common end-ring at

one end only [22].

Finite element (FE) analysis has been widely used, for exam-

ple in [23]–[26] to study UMP and its resulting vibration in elec-

trical machines. Finite element analysis of the 250 kW BDFM

was performed using the commercial software application EFFE

1376 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 35, NO. 3, SEPTEMBER 2020

TABLE III

SPECIFICATIONS OF THE 250 KW D400 BDFM

[26]. The model was solved as a voltage-fed problem so that

simulation results can be compared directly to experimental

measurements. A 2-D analysis was performed by assuming that

the machine is inﬁnitely long in the direction parallel to the

shaft to reduce the computational time. The end region leakage

effects were computed using the method presented in [27] and

incorporated into the analysis using lumped parameters. The

modelling was performed using the time-stepping method for

accurate analysis.

In the synchronous mode of operation, the PW is connected

directly to the grid and the CW is supplied with variable voltage

at variable frequency from a converter. The implementation of

BDFM synchronous operation in FE is particularly challenging

because the CW excitation voltage required to set a speciﬁc load

condition cannot be predetermined as the machine is not stable in

open loop. Therefore, a closed-loop controller was implemented

with details described in [28]. The PW and CW voltage sources

used in simulations only included the fundamental frequencies

and did not contain time harmonics.

In order to validate the numerical computation of magnetic

ﬁelds in FE models, the stator back iron and teeth ﬂux densi-

ties obtained by FE analysis have been compared to the same

parameters measured using ﬂux search coils. The search coils

were ﬁtted into the BDFM as shown in Fig. 3 and the results are

shown in Fig. 4. The close agreement validates the use of FE

modelling in obtaining the air gap magnetic ﬁelds in the BDFM,

which may be subsequently used for the analysis of stator back

iron displacement.

The 250 kW BDFM is modelled in the synchronous operating

mode for the two cases of ideally constructed rotor as well as

when the rotor has different degrees of static and dynamic ec-

centricity. The ﬂux density in the air gap is then obtained by post

processing of the FE models. The stator back iron displacement

can be subsequently determined using the procedure described in

Fig. 2. Different levels and types of rotor eccentricity are studied.

The back iron displacement for an ideally-constructed rotor as

well as when the rotor has static and dynamic eccentricity are

shown in Figs. 5 and 6.

Fig. 3. Stator ﬂux search coils ﬁtted on to (a) stator tooth (b) stator back iron

Fig. 4. Measured and predicted ﬂux density in (a) stator tooth (b) stator back

iron

From Figs. 5 and 6 the following can be concluded:

rIn both static and dynamic eccentricity cases, the back iron

displacement is substantially larger than when the rotor is

ideally constructed.

ABDI et al.: VIBRATION ANALYSIS OF BDFM IN THE PRESENCE OF ROTOR ECCENTRICITY 1377

Fig. 5. Displacement in the stator back iron for ideally-constructed machine

and different levels of static eccentricity.

Fig. 6. Displacement in the stator back iron for ideally-constructed machine

and different levels of dynamic eccentricity.

rFor the same levels of eccentricity, the severity of back

iron displacement caused by rotor dynamic eccentricity is

considerably higher than that of static eccentricity.

rFor the case of ideally constructed rotor, the back iron

displacement (green lines in Figs. 5 and 6) have 4-pole

(p1–p2pole-pair) space distribution. This corresponds to

the 3rd component of displacement function with angular

frequency of ω1−ω2, as predicted in Table II. It conﬁrms

the analysis of section III-A that the dominant component

of back iron displacement for the ideally-constructed ma-

chine has ω1−ω2angular frequency and p1–p2pole pair

number.

rThe displacement for all other cases with static and dy-

namic eccentricity, have 2-pole (p1–p

2+1pole-pair)

space distribution. This conﬁrms the ﬁndings of section III-

B that in a BDFM with non-ideal rotor construction, the

dominant displacement components have angular veloci-

ties of ωr,ω1−ω2and ω1−ω2±ωr,which correspond

to the 11th, 12th, 21st and 22nd components predicted in

Table II, all with pole pair number equal to |p1–p

2±1|.

Fig. 7. 250 kW BDFM and the load machine on test rig.

Fig. 8. Location of accelerometers on the BDFM test bed for vibration

measurements.

V. E XPERIMENTAL ASSESSMENT OF VIBRATION IN THE BDFM

A. Vibration Test Set-Up

The 250 kW BDFM is shown in Fig. 7 on the experimental

rig. The machine’s control system includes grid-side inverter

(GSI) and machine-side inverter (MSI). The GSI was developed

with an embedded control system to stabilize the DC-link and

synchronize to the 690 V grid voltage. The MSI was developed

to control the PW real and reactive power using a Speedgoat

controller [7].

A set of measurements was conducted on the machine to

validate the vibration analysis carried out in the previous section.

The machine was instrumented with a number of accelerometers

positioned on the bedplate, frame and terminal box of the ma-

chine, as shown in Fig. 8, and was operated in the synchronous

mode of operation at a speed range of 320–620 rpm. Vibration

was measured using Bruel & Kjaer 2260 instruments from

the drive end of the machine with the speciﬁcations shown in

Table IV. The accelerometers’ signals were digitised using a

1378 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 35, NO. 3, SEPTEMBER 2020

TAB L E I V

SPECIFICATIONS OF THE VIBRATION MEASUREMENT INSTRUMENTS

Fig. 9. Vibration velocity over the BDFM operating speed range.

data logger controlled by LabVIEW software. The fast Fourier

transform of the signals were also computed using a built-in

algorithm. The measured averaged root mean square (rms)vi-

bration velocity over a range of operating speeds is shown in

Fig. 9.

As can be seen from Fig. 9, for the rotor speed between

540-680 rpm, the rms values of vibration velocity remain within

the range of 2.87–4.01 mm/s. This is close to the standard limit,

based on the ISO 10816 standard for vibration assessment of

a 250 kW class M electrical machine. Nevertheless, for the

operating speed range of 320–520 rpm, the measured vibration

rms values are between 5.15 and 12.70 mm/s, which is above

the tolerable limit.

B. Measured Vibration Spectra

The spectra of vibration amplitudes in dB have been obtained

for the 250 kW BDFM at the operating conditions where the

machine’s vibration velocity is at its highest level. Fig. 10(a)

shows the vibration spectrum at the rotor speed of 440 rpm

and the PW and CW supplied frequencies of 50 Hz and 6 Hz,

respectively. The supply voltages were set to give magnetic

loadings of 0.32 and 0.38 T for the PW and CW, respectively,

in order to achieve the nominal ﬁeld strength in the machine air

gap.

Air gap non-uniformity was present in the prototype 250 kW

BDFM, partly due to the rotor shape not being purely circular

and partly due to the presence of rotor eccentricity. Experimental

measurements showed that the rotor axis was off centre by

approximately 15% of the air gap length. In addition, rotor

whirling was observed when experimental tests were carried

Fig. 10. Vibration amplitude spectrum at the machine’s drive end at 0 kW

(no-load), 100 kVAR, (a) 440 rpm, (b) 320 rpm.

out at rated operating conditions, showing the presence of both

static and dynamic eccentricities at normal operation.

Fig. 10(a) shows three distinct peaks at 56, 48.7 and 63.3

Hz frequencies. The 56 Hz vibration component is consistent

with ω1−ω2component predicted in section III-A as the main

vibration component caused by the bending set-up in the back

iron of the ideally constructed BDFM. It is also consistent with

ω1−ω2component predicted in section III-B as a result of

rotor eccentricity. The other two frequencies, i.e. 48.7 Hz and

63.3 Hz, however, are consistent with ω1−ω2±ωrvibration

components created solely by the presence of rotor eccentricity.

It is important to note that there is always a degree of rotor

eccentricity present in a real machine due to manufacturing

imperfection, which may cause signiﬁcant vibration and noise

issues in the BDFM as discussed in Section II.

Fig. 10(b) shows the vibration spectrum at the rotor speed of

320 rpm and the PW and CW supplied frequencies of 50 Hz and

18 Hz, respectively. The main peak vibration frequencies are 68,

63 and 73.3 Hz, which correspond to ω1−ω2and ω1−ω2±

ωr, similar to what was found at 440 rpm.

The time and space harmonic contents in the BDFM torque,

PW and CW currents, and active powers and their contribution

in the machine vibration were studied in [29], [30]. Fig. 10 also

demonstrates these harmonic-driven sources of vibration, which

are denoted with different colours and numbers. Table V shows

different vibration mechanisms and their associated numbers

and colours. These additional vibration components generated

from harmonic effects can exacerbate the vibration and noise

levels and therefore, special design considerations such as rotor

design optimisation and use of magnetic wedges in rotor slot

openings need to be employed to reduce the harmonic effects.

The vibration spectrum for a range of synchronous speeds was

also obtained where the output active and reactive power were

ABDI et al.: VIBRATION ANALYSIS OF BDFM IN THE PRESENCE OF ROTOR ECCENTRICITY 1379

TAB L E V

THE ASSOCIATED COLOR TO EAC H SOURCE OF VIBRATION

FREQUENCY IN FIG.10

Fig. 11. Dominant frequencies in the vibration spectrum. For each speed the

diamond, triangle and rectangle are the frequencies of largest, second largest

and third largest components, respectively. The solid, dashed and dotted lines

correspond to ω1−ω2,ω1−ω2+ωr,andω1−ω2−ωr, respectively.

kept constant at 230 kW and 98 kVAR, respectively. For each

speed, the three dominant frequencies with largest magnitudes

were determined and are shown in Fig. 11. The solid, dashed and

dotted lines in Fig. 11 correspond to ω1−ω2,ω1−ω2+ωr,

and ω1−ω2−ωr,respectively.

As can be seen, the largest displacement component at most

speeds has the frequency of ω1−ω2, which corresponds to the

source of vibration that includes the effects of both back iron

bending set-up and rotor eccentricity. There are also reasonable

correlations between the frequencies of second and third largest

displacement components and the predicted rotor eccentricity

vibration sources with ω1−ω2+ωr,and ω1−ω2−ωr,re-

spectively. However, the variation in the frequency of second

and third largest components seen in Fig. 11 suggests that other

vibration mechanisms are taking part in the BDFM’s vibration

and noise patterns. These frequencies correspond mostly to the

vibration mechanisms noted in Table V.

VI. CONCLUSION

In this paper, various vibration mechanisms present in the

BDFM’s main mode of operation, the synchronous mode, were

studied. It was shown that one of the main sources of vibration is

created by bending set-up in the back iron, which occurs in any

BDFM irrespective of pole pair combinations, causing a stator

back iron displacement. It was also shown analytically that the

displacement component with an angular frequency and pole-

pair number of ω1−ω2and p1–p2(e.g. 4-pole displacement

component for a 4/8 pole prototype BDFM) can become very

large in BDFMs with a small difference between power and

control winding pole pairs and hence is the dominant vibration

component in an ideally-constructed BDFM.

In addition, it was shown that the rotor eccentricity introduces

additional components of back iron displacement and results

in BDFM vibration with the dominant frequencies of ω1−ω2

and ω1−ω2±ωrand pole-pair number |p1–p

2±1| (e.g.

2-pole displacement component for a 4/8 pole BDFM). The

analytical theory was validated through deriving the back iron

displacement in a 250 kW BDFM from the air gap magnetic

ﬁelds obtained from the ﬁnite element analysis.

A set of vibration tests were carried out on the prototype

BDFM in order to assess the machine’s vibration pattern. The

experimental results conﬁrmed the dominant vibration frequen-

cies of ω1−ω2and ω1−ω2±ωras predicted analytically for

an eccentric BDFM. A degree of air gap non-uniformity was

present in the prototype machine due to the uneven air gap length

caused by manufacturing tolerance as well as rotor eccentricity.

Therefore, it is essential to implement appropriate design

considerations in order to mitigate the vibration and noise levels

before a large-scale BDFM is constructed for wind turbine

drive trains. These may include increasing the air gap length,

introduction of damping in the rotor winding, introduction of

damping in the stator winding with parallel paths, and isolation

of the stator frame from the stator core.

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Salman Abdi received the B.Sc. degree from Fer-

dowsi University, Mashhad, Iran, in 2009, and the

M.Sc. degree from the Sharif University of Technol-

ogy, Tehran, Iran, in 2011, both in electrical engineer-

ing. He then completed the Ph.D. degree in electrical

machines design and modeling from Cambridge Uni-

versity, Cambridge, U.K, in 2015. He is currently an

Assistant Professor in Electrical Engineering at the

University of East Anglia (UEA), Norwich, UK. His

main research interests include electrical machines

and drives for renewable power generation and auto-

motive applications.

Ehsan Abdi (Senior Member, IEEE) received the

B.Sc. degree from the Sharif University of Technol-

ogy, Tehran, Iran, in 2002, and the M.Phil. and Ph.D.

degrees, from Cambridge University, Cambridge,

U.K., in 2003 and 2006, respectively, all in electrical

engineering. He is currently the Managing Director

of Wind Technologies Ltd., Cambridge, where he has

been involved with commercial exploitation of the

brushless doubly fed induction generator technology

for wind power applications. He became a Senior

Member of the IEEE in 2012. His main research

interests include electrical machines and drives, renewable power generation,

and electrical measurements and instrumentation.

Hamid Toshani received the B.S. degree in electrical

engineering from the Ferdowsi University of Mash-

had, Mashhad, Iran, in 2009 and the M.Sc. degree in

control engineering from Iran University of Science

and Technology, Tehran, Iran, in 2012. He is currently

a Ph.D. Student majoring in Control Engineering in

the Department of Electrical Engineering, Iran Uni-

versity of Science and Technology, Tehran, Iran. His

research interests include robust control, optimiza-

tion techniques and intelligent approaches in control

systems.

Richard McMahon received the B.A. degree in

electrical sciences and the Ph.D. degree from the

University of Cambridge, Cambridge, U.K., in 1976

and 1980, respectively. Following post-doctoral work

on semiconductor device processing, he became a

University Lecturer in electrical engineering with the

Engineering Department, University of Cambridge,

in 1989, where he was a Senior Lecturer in 2000. In

2016, he joined the Warwick Manufacturing Group

(WMG), University of Warwick, Coventry, U.K., as

a Professor of power electronics. His current re-

search includes electrical machines, power electronics and the electriﬁcation of

transport.