State-Space Model for Induction Motors with
Static Eccentricity Faults
Zifeng Gong1
Philip Desenfans1
Davy Pissoort1
Hans Hallez2
Dries Vanoost1
1
Department of Electrical Engineering, KU Leuven, Bruges, Belgium
{zifeng.gong, philip.desenfans, davy.pissoort, dries.vanoost}@kuleuven.be
2
Department of Computer Science, KU Leuven, Bruges, Belgium
hans.hallez@kuleuven.be
Abstract
This work proposes a novel state-space model for eccentric induction motors, allowing the use of
modern control theory to diagnose the eccentricity faults and to mitigate their effects. The proposed
model is derived from the inductance-based multiple coupled circuit model. Instead of actual stator
windings and rotor bars placed in slots, equivalent ideally distributed stator and rotor windings are
considered in the motor modelling. The self and mutual inductances that incorporate the effects of
static eccentricity in motors are evaluated using the modified winding function approach. Moreover, the
rotor quantities are referred to the stator side to reduce the number of characteristic inductances. An
eccentricity-oriented coordinate transformation is implemented to decouple the flux linkages in the new
coordinate system and to reduce the system complexity further. Hereafter, stator currents and rotor
fluxes are selected as system states, forming a fourth-order state-space model for the electromagnetic
system of an eccentric motor. The proposed model is validated through comparisons with the reference
data sourced from both the experiments and the finite element method. The potential of the proposed
model for static eccentricity fault diagnosis purposes is explored.
Key words
Induction motor, static eccentricity fault, multiple coupled circuit approach, modified winding function
approach, state-space model
1
Draft
Correspondence
Zifeng Gong
Department of Electrical Engineering, KU Leuven
Spoorwegstraat 12, 8200 Bruges, Belgium
Tel: +32 497844547
Email: zifeng.gong@kuleuven.be
2
Draft
Nomenclature
Ni
Winding function of the winding i
λ
Flux vector
npp
Machine pole pair number
I
Identity matrix
O
Geometric centre
i
Current vector
r
Resistance
L
Inductance matrix
rstin
Stator inner radius
R
Resistance matrix
Te
Electromagnetic torque
T
Transformation matrix
Tl
Load torque
v
Voltage vector
W
Number of turns
µ0
Permeability of vacuum
Wco
Magnetic co-energy
ωr
Rotor speed
Superscripts
ϕ
Arbitrary angular position
′
ϕs
Static eccentricity position
Subscripts
ρs
Static eccentricity degree
θr
Rotor mechanical angle
G
Coefficient in series
g
Air gap length
g −1
Inverse air gap length
g0
Normal air gap length
J
Rotor inertia
L
Inductance
Lm
Magnetising inductance
Llr
Leakage inductance per rotor phase
Lls
Leakage inductance per stator phase
LM
Converted magnetising inductance
s, r
Rotor quantities referred to stator side
Staor and rotor
a, b, c Phases a, b and c
α, β
3
Draft
Axes α and β
1
Introduction
Induction motors (IMs) are widely used in modern industry for energy conversion due to their simple
structure and high efficiency. However, as a consequence of worn bearings, manufacturing tolerance, and
misalignment, eccentricity faults (EFs) may occur that lead to motor performance degradation and
give rise to an unbalanced magnetic pull which may cause catastrophic structural damage [1]. Therefore,
models that can predict the behaviour of an eccentric IM are of great significance to investigate since they
facilitate the understanding of EFs and provide the theoretical foundation for EF impact mitigation [2].
Existing models that integrate EFs into an IM are mainly numerical models. For instance, the
finite element method (FEM), the magnetic equivalent circuit (MEC), and the multiple coupled circuit
(MCC) have been reported to characterise the behaviour of eccentric IMs [3–5]. FEM is considered the
most accurate one among these approaches as it comprehensively analyses the magnetic flux distribution
in an eccentric motor according to the motor’s full geometry [6, 7]. However, FEM’s accuracy comes
at a cost of considerable computation. As a result, FEM is commonly used as a validation for other
modelling methods. MEC is a lumped-parameter model, which also takes geometrical nuances into
account, although with less precision [8]. MEC allows IM modelling with nonlinearities such as slotting
effect and saturation, which makes MEC an attractive approach for simulating eccentric motors [9, 10].
In the case of an eccentric IM, the air gap permeances connecting the stator and rotor teeth are evaluated
based on the effective air gap length [11]. MCC was initially developed for simulating a conventional IM
with a centered rotor and stator, which utilised the winding function approach (MFA) to calculate the
self and mutual inductances between any two circuits [12]. Later, a more general theory, the modified
winding function approach (MWFA), was proposed, which extends MCC models to cover the case of
eccentric IMs [13, 14].
Despite of numerical models’ excellent accuracy, their complexity restricts their applicability in developing active EF mitigation. This constraint arises from the fact that (fault-tolerant) controllers for IMs
are predominantly designed within the modern control theory framework which requires a state-space
representation of IMs. For control and fault diagnosis purposes, the DQ circuit-based state-space model
and its variants are the most commonly used models. For instance, the conventional DQ model was
derived and its observability was thoroughly examined in [15]. Its variants have been used in the design
of IM speed sensorless control [16], where the rotational speed and the rotor resistance are assumed
unknown and reconstructed by measured signals. Moreover, based on the DQ model, a current sensor
fault-tolerant scheme has been proposed for IM drives [17], which compensates the sensor fault using
the estimated current. In [18], an open-circuit fault-tolerant controller was developed for IMs driven by
four-switch three-phase inverters. The dynamics of IMs with potential faults were described using a DQ
model. Across these approaches, the fundamental concept is to reconstruct the quantities of interest (e.g.
4
Draft
rotor resistances and stator currents) using observers, and to further compensates any adverse effects
resulting from faults. It is evident that the quantities to be reconstructed must present in the model.
However, unlike quantities such as currents, eccentricity faults are not naturally accounted for in the conventional state-space model and remain unmodelled. Therefore, a state-space model that encompasses
the effects of static eccentricity needs to be developed prior to the design of model-based fault diagnosis
and fault-tolerant controllers. This motivates us to address the analytical modelling of EFs in IMs in
this paper.
In light of the previous discussion, we extend the conventional state-space model to incorporate static
eccentricity. Our aim is to spark research interest regarding EF diagnosis and its active mitigation.
Similar to the construction of a conventional model, the proposed approach first begins with a simplified
MCC model, replacing both stator windings and rotor bars with equivalent three-phase ideally distributed
windings. The inductances under static eccentricity faults between any two windings are calculated using
the MWFA. The model is transformed into a stationary frame before being formulated in the state-space
format. The remainder of the paper is organised as follows. Section 2 formulates the governing equations
and evaluates the inductances. Section 3 reformulates the model in a state-space representation. Section
4 validates the proposed model and discusses the potential for its application in fault diagnosis. Section
5 draws the conclusion remarks.
2
Eccentric Motor Modelling
This section formulates the governing equations for an IM with static eccentricity and subsequently
evaluates the self and mutual inductances. As shown in Fig. 1, both the stator and rotor have three
magnetic axes, each separated by an angle of 120 degrees. In a healthy IM, the air gap is uniformly
distributed along the stator inner surface with a fixed length g0 . However, due to the static eccentricity,
the rotor centre Or is displaced from the stator centre Os by a distance of ρs g0 at the angular position
ϕs .
2.1
Governing Equations
Based on the MCC theory, the governing equations of the IM with three stator phases (as, bs, cs)
and three equivalent rotor phases (ar, br, cr) shown in Fig. 1 can be written in the vector-matrix form
as:
5
Draft

d


v abcs = Rs iabcs + λabcs


dt




d


0 = Rr iabcr + λabcr
dt
,



λabcs = Lss iabcs + Lsr iabcr






λ
= LT i
+L i
(1)
v abcs = [vas , vbs , vcs ]T ,
(2)
iabcs = [ias , ibs , ics ]T ,
(3)
iabcr = [iar , ibr , icr ]T ,
(4)
λabcs = [λas , λbs , λcs ]T ,
(5)
λabcr = [λar , λbr , λcr ]T .
(6)
abcr
sr abcs
rr abcr
where
Moreover, Rs = rs I 3 is the stator resistance matrix and Rr = rr I 3 is the rotor resistance matrix. Lss ,
Lsr , Lrs , and Lrr are the matrices containing the self/mutual inductances that will be evaluated in
Section 2.2.
The mechanical equation of the IM is given by:
Te − TL = J
ωr =
dωr
,
dt
dθr
,
dt
(7)
(8)
where TL is the load torque, J the rotor inertia, ωr the rotor rotational speed, θr the rotor mechanical
angle. The electromagnetic torque Te produced in the IM can be calculated as the partial derivative of
the magnetic co-energy Wco with respect to the rotor mechanical angle θr :
∂Wco
,
∂θr
(9)
1 T
T
T
T
iabcs Lss iabcs + iT
abcs Lsr iabcr + iabcr Lsr iabcs + iabcr Lrr iabcr .
2
(10)
Te =
where
Wco =
2.2
Inductance Calculation
According to the MWFA [13], the mutual inductance between any two windings i and j in an
6
Draft
eccentric IM can be expressed as:
< g −1 Ni >< g −1 Nj >
Lij = 2πµ0 rstin l < g −1 Ni Nj > −
,
< g −1 >
(11)
where Ni and Nj are the winding functions of the winding i and j, r and l are the stator inner radius
and the axial length, µ0 is the permeability of vacuum, g −1 is the inverse air gap function (17), and
< · > is defined as:
< f (ϕ) >=
1
2π
Z 2π
f (ϕ)dϕ.
(12)
0
Note that the self inductance of the winding i can be obtained by replacing Nj with Ni in (11).
The winding functions of the stator A phase and the equivalent rotor A phase of a three-phase IM
are given by:
∞
Nas =
Ws X
sk cos(knpp ϕ)
2
(13)
k=1
and
∞
Nar =
Wr X
rk cos(knpp (ϕ − θr )),
2
(14)
k=1
where Ws and Wr are the effective turns per stator phase and per rotor phase. Coefficients sk and rk
depends on the winding arrangements of the machine. θr is the rotor angle. The winding functions of the
other two phases can be obtained by shifting the cosines of (13) and (14) by − 23 π and 23 π, respectively.
In a healthy IM, the air gap is uniformly distributed along the stator inner surface with a fixed length
g0 . However, the presence of static eccentricity will lead to an asymmetrical air gap distribution, as
shown in Fig. 1. The effective air gap length at any arbitrary angular location ϕ is given by:
g(ϕ) = g0 (1 − ρs cos(ϕ − ϕs )),
(15)
where ρs is the static eccentricity degree and ϕs is the position where the minimum air gap length is
located.
From (11), it is clear that the inverse of (15) is involved in the inductance calculation:
g −1 (ϕ) =
1
.
g0 (1 − ρs cos(ϕ − ϕs ))
(16)
However, the integration of (16) is not analytically solvable. In order to obtain analytical expressions
for the inductances, we perform a series expansion on (16):
1
g −1 (ϕ) =
g0
G0 +
Nk
X
!
Gk cos (k (ϕ − ϕs )) ,
k=1
7
Draft
(17)
where
G0 = q
1
1 − (ρs )
,
(18)
2
k
q
2
2 1 − 1 − (ρs )
q
.
Gk =
k
2
(ρs )
1 − (ρs )
(19)
Note that equation (17) is equivalent to (16) if Nk = ∞. Fig. 2 shows the inverse air gap lengths
calculated by (16) and (17) for Nk = 1, Nk = 2, and Nk = 3. These illustrative results correspond to a
motor with a 0.3 mm air gap and an eccentricity of ρs = 0.4 at ϕs = 0. The average percentage errors
of the approximations (17) for Nk = 1, Nk = 2, and Nk = 3 relative to (16) are computed to be 5.88%,
1.23%, and 0.25%. Meanwhile, the maximum percentage errors are 11%, 2.30%, and 0.48%, respectively.
Since our objective is to establish a practical and applicable state-space representation used for modelbased EF diagnosis and mitigation, strong nonlinearities should be avoided. Therefore, we opt to neglect
higher-order spacial harmonics in the winding functions (for k > 1) given in (13) and (14). Additionally,
we adopt Nk = npp in the series expansion (17) for modelling a npp pole pair machine. Subsequently,
by substituting (13), (14), and (17) into (11), all the inductances in (1) can be calculated. For instance,
the mutual inductance Lasbs is given by:
Lm G0 2 + G2npp cos (npp ϕs ) cos npp ϕs − 2π
3
Lasbs = −
2G0
,
(20)
where Gnpp is the nth
pp term of Gk and
Ls =
Ws2 πµ0 rl
.
4g0
(21)
Additionally, the mutual inductance Larbr is given by:
Lr G0 2 + G2npp cos (npp ϕsr ) cos npp ϕsr − 2π
3
Larbr = −
2G0
,
(22)
where
Lr =
Wr2 πµ0 rl
W2
= r2 Ls
4g0
Ws
(23)
ϕsr = ϕs − θr .
(24)
and
3
Model Transition
In the previous section, we evaluated all the inductances necessary for the model construction.
However, as indicated by (20) and (22), the inductance expressions are complex, rendering the state8
Draft
space model derived in the conventional frame impractical. Therefore, in this section, we begin by
reducing the system complexity before proceeding with the conversion to a state-space model.
3.1
Reduction of Characteristic Inductances
In general, the numbers of turns per stator phase are not equal to those per rotor phase, i.e., Ws ̸= Wr .
This will result in that more than one characteristic inductances (e.g. Ls (21) and Lr (23)) are needed
to describe the motor electromagnetic behaviour, increasing the complexity of the model. However, in
real applications, rotor quantities such as currents and fluxes are not measured and their magnitudes are
not particularly interesting to investigate. Therefore, all the quantities on the rotor’s side are referred
to the stator to reduce the number of involved parameters by letting:
Wr
i′r =
ir ,
Ws
Ws
λ′r =
λr ,
Wr
R′r =
Ws
Wr
2
Ws
Rr , L′sr =
Lsr ,
Wr
L′rr =
Ws
Wr
2
Lrr ,
where the superscript ′ denotes the rotor quantities referred to the stator. As a result, we have:

dλs


v s = R s is +


dt


′


dλ

r
 0 = R′r i′r +
dt
.


′ ′

λ = Lss is + Lsr ir


 s



λ′ = L′ T i + L′ i′
r
s
sr
(25)
rr r
In the remainder of this paper, the superscript ′ is omitted for a simpler notation.
3.2
Coordinate Transformation
It is widely recognised that linear dependence is inherent in a three-phase electrical system. For
example, the sum of three-phase stator currents is zero according to the Kirchhoff’s current law, implying
one phase-current can be represented by the linear combination of the other two. This redundancy can
be eliminated by performing a coordinate transformation. The transformation matrix T is given by:

θ − 2π
3
cos
r  cos(θ)
2
 − sin(θ) − sin θ − 2π T (θ) =
3
3
 q
q
1
2
1
2
cos
θ + 2π
3
− sin
θ + 2π
3
q
1
2



.


(26)
Applying the transformation T (npp θ) to the stator quantities and T (npp (θ−θr )) to the rotor quantities
9
Draft
gives rise to:

d


T v abcs = T Rs T −1 T iabcs + T T −1 T λabcs


dt




d


0 = T Rs T −1 T iabcr + T T −1 T λabcr
dt
.


−1
−1

T λabcs = T Lss T T iabcs + T Lsr T T iabcr






T λ
= T LT T −1 T i
+ T L T −1 T i
abcr
abcs
sr
rr
(27)
abcr
In (27), the quantities in the new frame are identified as xαβ0 = T xabc . For instance,
T v abcs = v αβ0s .
(28)
The resistance and inductance matrices in the new αβ0 frame are:


T Rs T −1 = Rs








T Rr T −1 = Rr




T Lss T −1 = Lmag + Lls I 3 + Ladd ,







T Lsr T −1 = Lmag + Ladd






T Lrr T −1 = Lmag + Llr I 3 + Ladd
(29)
where Lls and Llr denote the leakage inductance per stator phase and per rotor phase. I 3 represents
the third-order identity matrix. Moreover, Lmag and Ladd are given by:



Lmag = 


3
2 G0 Lm
0
0
3
2 G0 Lm
0
0

3G2npp Lm cos2 (npp (θ−ϕs ))
−
4G0

 3G2 Lm sin(2npp (θ−ϕs ))
npp
Ladd = 

8G0

0

0 

0 
,

0
3G2npp Lm sin(2npp (θ−ϕs ))
8G0
3G2npp Lm sin2 (npp (θ−ϕs ))
−
4G0
0
(30)

0

0
.

0
(31)
From (29), it is obvious that the complexity of the system equations mainly comes from the inductance
matrix Ladd . Since Ladd is non-diagonal, the α- and β-axis flux linkages are coupled. However, this
complexity can be significantly reduced by a proper selection of the angle θ used for coordinate transformation. As shown in Fig. 3, an eccentric IM still holds symmetry about the direction of the eccentricity.
10
Draft
If we align the α-axis with the eccentricity i.e., let θ = ϕs , Ladd becomes:

 −

Ladd = 


3.3
3G2npp Lm
4G0
0
0

0 

0 0 
.

0 0
0
(32)
State-Space Formulation
Assume a balanced three-phase voltage supply, the 0-axis equation can be neglected since the voltage
excitation v0s is 0. Thus, (27) with θ = ϕs becomes:

dλαs


vαs = rs iαs +


dt




dλβs


vβs = rs iβs +


dt




dλαr


vαr = rr iαr + npp ωr λβr +



dt



dλ

vβr = rr iβr − npp ωr λαr + βr
dt
,




λαs = (k2 LM + Lls ) iαs + k1 LM iαr







λβs = (k1 LM + Lls ) iβs + k1 LM iβr







λαr = k1 LM iαs + (k2 LM + Llr ) iαr




λ = k L i + (k L + L ) i
βr
1 M βs
1 M
lr βr
(33)
where
LM =
3
Lm ,
2
(34)
k1 = G0 ,
(35)
1
k2 = G0 − G2npp .
2
(36)
Further, considering stator currents are typically available from sensors and rotor fluxes are commonly
used for control purposes, we construct the fourth-order state-space model as:
dx
= f (x, u),
dt
(37)
where

iαs





 iβs 


x=
,


λ
 αr 


λβr
11
Draft
(38)

γ1 iαs − δ1 λαr − npp ωr β1 ξ1 λβr − ξ1 vαs





 γ2 iβs − δ2 λβr + npp ωr β2 ξ2 λαr − ξ2 vβs 


f =
,


rr β1 iαs − rr α1 λαr − npp ωr λβr




rr β2 iβs − rr α2 λβr + npp ωr λαr

(39)

 uαs 
u=
,
uβs
(40)
α1 = (Llr + k2 LM )−1 ,
(41)
α2 = (Llr + k1 LM )−1 ,
(42)
β1 = k1 LM α1 ,
(43)
β2 = k1 LM α2 ,
(44)
γ1 = (rs + rr β12 )ξ1 ,
(45)
γ2 = (rs + rr β22 )ξ2 ,
(46)
δ1 = rr α1 β1 ξ1 ,
(47)
δ2 = rr α2 β2 ξ2 ,
(48)
and
ξ1 = −Lls + LM −k2 +
ξ2 =
k12 LM
Llr + k2 LM
−1
−1
k1 LM Llr
.
−Lls −
Llr + k1 LM
,
(49)
(50)
Note that the rotor speed ωr is treated as a known constant in the fourth-order model (37), implying
the rotor speed is measured. One can also extend the model to a fifth-order system to incorporate the
mechanical equation (7), with the speed modelled as an additional state.
4
Results and Discussion
This section first focuses on the validation of the proposed model using the results from both the
FEM and the experiment. Hereafter, the potential applications of the state-space model in fault diagnosis
are briefly discussed.
12
Draft
4.1
Model Validation: FEM
In this subsection, the proposed motor model reproduces a two-pole, 400 V/50 Hz, 1.1 kW motor whose
characteristic parameters are given in Table 1. The reference method is the time-stepping FEM, in
which the motor model is built and solved using the Finite Element Method Magnetics software [19].
The ferromagnetic material in the FEM model has a linear permeability of 4000.
To begin with, the dynamics of the motor are simulated for 1.5 s with a time step of 1 ms. The
machine is powered by the nominal supply and the nominal load is applied after t = 0.8 s. Figures
4 and 5 show the comparisons regarding the output electromagnetic torque Te and the stator α-axis
currents iαs obtained from both the FEM and the proposed model. The transient behaviour during the
motor start-up can be observed at the first 0.3 seconds of the simulation. After the motor is loaded, the
produced electromagnetic torque Te increases to balance the load torque. The root-mean-square errors
(RMSEs) of Te and iαs in the steady state are 0.1 Nm and 0.25 A, showing a good alignment between
the two approaches. Moreover, the percentage errors of Te and iαs are calculated to be 2.13% and 6.08%,
respectively.
Next, the stator α-axis self inductances Lαsαs calculated by both the proposed model and the FEM
are presented in Fig. 6. The calculation of Lαsαs using the proposed method is straightforward, as given
in (33). In the FEM, this calculation is done in two steps. First, the stator self and mutual indutances
in the conventional abc frame are obtained by calculating the flux linkages under a 1 A current supply to
each phase. Hereafter, the inductances are transformed to the αβ0 frame to obtain Lαsαs . It is observed
from Fig. 6 that Lαsαs calculated through the FEM varies with respect to the shaft position, while
Lαsαs calculated by the proposed model is constant. This is because the state-space model 1) assumes a
ideal geometry and therefore neglects the slotting effects and 2) discards the higher-order magnetomotive
force harmonics in the winding functions. Nevertheless, it is noteworthy that the Lαsαs calculated by the
state-space model (672 mH) closely approximates the mean self inductance (672.7 mH) obtained from
the FEM.
Furthermore, the stator self and multual inductances Lαsαs and Lαsβs calculated by the proposed
model with their mean values calculated by the FEM under various fault severity conditions are compared.
In all cases, the static eccentricity is set aligned with the stator A-phase winding magnetic axis. The
results presented in Table 2 show a good alignment between the two methods, particularly at low fault
severity. Additionally, it is noticed that the error in Lαsαs increases with the escalating fault severity.
The observed discrepancy can be primarily attributed to the simplifications made during the derivation
of the inductance expressions. Additionally, it can be seen that the mutual inductances between the
stator α- and β-axes, as calculated by both methods under various fault conditions, are 0. This confirms
the α- and β-axis fluxes can be decoupled by performing the proposed eccentricity-oriented coordinate
13
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transformation.
4.2
Model Validation: Experiment
The experimental setup is shown in Fig. 7, where the test machine is driven by an inverter under
a scalar control and is coupled with a servo machine as a load. The test machine under investigation is
a two-pole, 400 V/50 Hz, 1.1 kW double-end squirrel cage induction machine. The motor parameters
necessary for reproducing the results are given in Table 3. The stator winding currents and line voltages
are displayed and recorded using an oscilloscope. The double-end feature of the test machine allows
us to modify the non-load end for introducing different levels of static eccentricity, as shown in Fig. 8.
Specifically, the original internal bearing at this end is removed and an external bearing is used to support
the rotor. The offsets of the rotor centre can be adjusted by changing the bearing housing position. The
0.24 mm and 0.12 mm offsets correspond to roughly 50% and 25% static eccentricities at the non-load
end. Assuming no eccentricities present at the load-end, these two offsets introduce 25% and 12.5%
static eccentricities in the machine on average.
Firstly, the motor’s torque-speed characteristics at the nominal voltage supply and frequency are
simulated. The results are compared to the measured torque-speed characteristics, as shown in Fig. 9.
The proposed model closely approximates the test machine near the nominal speed (2860 rpm). Deviations are observed at a lower speed, which can be partially attributed to the change in the rotor circuit
resistance due to the skin effect that is not taken into account in the proposed method.
Secondly, Table 4 shows a thorough comparison of the machine performance under different load
conditions. This comparison confirms that the proposed method produces a good estimation of the
machine behaviours at the typical working points. Despite the accuracy achieved, the proposed method
tends to underestimate the consumed power and overestimate the efficiency, especially at a low load.
This stems from the negligence of losses such as the core and friction losses. Additionally, the waveforms
of the simulated and measured stator phase winding currents collected in the steady state at half-load
are presented in Fig. 10 for two electrical periods. The RMSE of the presented currents is 0.47 A and
the percentage error of the current root-mean-square values is 6.14%.
The recorded currents are further transformed into the frequency domain for a more comprehensive
comparison. As shown in Fig. 11, the proposed model accurately predicts the fundamental frequency
component with a small error. However, higher-order harmonics are not accounted for. This is mainly
because 1) the higher-order spacial magnetomotive force harmonics in the winding functions (13) and
(14) are discarded and 2) the MWFA method neglects the slotting effects and the material nonlinearity. The measured current shows much more abundant harmonic components, with the fundamental
frequency component dominating the spectrum. In addition to the 50 Hz component, it is observed that
the measured current spectrum contains multiple peaks that correspond to the odd-order, even-order
14
Draft
harmonics of the fundamental frequency, and the rotor slot harmonics. These three groups of harmonics
mainly stem from the saturation, unbalance, and slotting in the machine.
Lastly, the amplitudes of the fundamental component of the machine stator α-axis current, as obtained
from the simulated and measured data, are compared across various levels of static eccentricity and under
different load conditions. The results are summarised in Table 5. The average percentage errors of iαs
and iβs are 4.84% and 3.12%, respectively. Notably, the measured currents show discrepancies in iαs and
iβs in the absence of eccentricities, implying the existence of unbalance in the motor. This is backed up
by the presence of the even-order frequency components in the current spectrum, as shown in Fig. 11.
Moreover, it is observed that the simulated α- and β-axis currents decrease with the increased fault
severity. This trend is anticipated since the faults lead to changes in the characteristic inductances, as
shown in Table 2. Most of the experimental results align with this trend. Given the small magnitude
of differences caused by the static eccentricities, they are highly sensitive to subtle alterations in the
experiment. Within the presented experimental framework, exceptions found in the measured values
may be attributed to the manufacturing tolerances of the fault injection setup, the speed oscillation of
the servo machine, and measurement noises.
4.3
Prospects for Further Development
The capability of the proposed model does not only lie in predicting the eccentric motor performance,
but also in serving as a foundational model for fault diagnosis and fault tolerant controller designs. For
simulating an eccentric motor, one calculates currents and fluxes for given supply voltages with known
eccentricity degrees and positions. The perspective shifts when considering the fault diagnosis. In this
context, the goal is to determine the eccentricity degree and angular position using measured voltages
and currents. Within the modern control theory framework, this can be typically resolved by 1) system
identification techniques [20], where eccentricity information is achieved using optimisation techniques
or 2) fault estimation approaches [21], where eccentricity is reconstructed by designing an observer for
the augmented system:
d
x̄ = f̄ (x, u),
dt
with


 x 



x̄ = 
 ρs  ,


ϕs
and
(51)

f
(x,
u)





.
f̄ (x, u) = 
0



0
(52)

15
Draft
(53)
5
Conclusion
In this paper, a state-space representation has been developed for IMs with static eccentricity. The
proposed model is derived from an inductance-based multiple coupled circuit model, in which actual
stator windings and rotor bars are replaced with ideally distributed windings. Based on the MWFA, the
inductances are evaluated using the winding functions and the inverse air gap function. This study finds
that the α- and β- axes can be decoupled by performing an eccentricity-oriented coordinate transformation, reducing the system complexity significantly. Depending on whether the mechanical equation is
integrated, IMs with static eccentricity can be described as either a fourth-order or a fifth-order system.
The alignment of the results with the FEM and experiments illustrates the effectiveness of the proposed model. Since state-space representations seamlessly integrate with modern control frameworks,
the proposed model holds significant promise for fault diagnosis applications employing techniques such
as system identifications and observer designs.
16
Draft
Acknowledgement
This work was supported by the China Scholarship Council (No. 202007000007).
17
Draft
Table 1: Motor 1 Parameters
Parameter
Symbol
Value
Stator phase winding resistance
rs
5.35 Ω
Rotor phase winding resistance
rr
4.38 Ω
Magnetising indutance
LM
0.662 H
Stator leakage per phase
Lls
0.01 H
Rotor leakage per phase
Llr
0.01 H
Rotor inertia
J
0.01 kg-m2
18
Draft
Table 2: Inductance Comparison (Eccentric Motor)
Data Lαsαs Lαsβs
Data Lαsαs Lαsβs
ρs
ρs
source [mH]
[mH]
source [mH]
[mH]
SS
672.0
0
SS
703.9
0
0
0.3
FEM
672.7
0
FEM
687.0
0
SS
672.8
0
SS
716.6
0
0.05
0.35
FEM
673.3
0
FEM
692.2
0
SS
675.3
0
SS
732.3
0
0.1
0.4
FEM
674.4
0
FEM
698.6
0
SS
679.6
0
SS
751.2
0
0.15
0.45
FEM
676.3
0
FEM
706.1
0
SS
685.5
0
SS
774.4
0
0.2
0.5
FEM
679.0
0
FEM
714.7
0
SS
693.7
0
0.25
FEM
682.5
0
Remark: SS denotes the state-space model.
19
Draft
Table 3: Motor 2 Parameters
Parameter
Symbol
Value
Stator phase winding resistance
rs
6.2 Ω
Rotor phase winding resistance
rr
5.3 Ω
Magnetising indutance
LM
0.3984 H
Stator leakage per phase
Lls
0.0124 H
Rotor leakage per phase
Llr
0.0124 H
Rotor inertia
J
0.01 kg-m2
20
Draft
Data
Source
Table 4: Machine Static Performance at Different Loads
Load I
P1
P2
Speed
η
cos φ
[%]
[A]
[kW] [kW] [rpm]
[%]
State-space model
Experiment
Error [%]
125
125
/
2.92
2.96
1.35
1.62
1.70
4.70
1.36
1.38
1.45
2822
2818
0.14
0.802
0.829
3.26
84.18
80.94
4.00
State-space model
Experiment
Error [%]
100
100
/
2.53
2.51
0.79
1.29
1.34
3.73
1.11
1.10
0.91
2862
2861
0.04
0.734
0.770
4.68
85.85
82.07
4.61
State-space model
Experiment
Error [%]
75
75
/
2.21
2.15
2.79
0.97
1.00
3.00
0.84
0.83
1.20
2899
2900
0.03
0.633
0.677
6.50
86.87
81.92
5.70
State-space model
Experiment
Error [%]
50
50
/
1.96
1.88
4.26
0.66
0.70
5.71
0.57
0.55
3.64
2935
2934
0.03
0.483
0.534
9.48
86.45
79.14
9.24
State-space model 25
1.81 0.35
0.29
2968
0.282 81.19
Experiment
25
1.73 0.40
0.28
2966
0.335 68.41
Error [%]
/
4.62 12.50 3.57
0.07
15.82 18.68
Remark: I, P1, P2, cos φ, and η denote the stator current RMS (abc frame),
input power, output mechanical power, power factor, and efficiency.
21
Draft
Table 5: Amplitude Comparison of Fundamental Frequency Component
Data
Eccentricity
Amplitude [A]
0%
load
25%
load
50%
load
75%
load
100%
load
iαs
(State-space model)
0%
12.5%
25%
3.0938
3.0868
3.0666
3.1611
3.1546
3.1357
3.4176
3.4121
3.3963
3.9998
3.9959
3.9853
4.4495
4.4466
4.4388
iαs
(Experiment)
0%
12.5%
25%
3.1765
3.1367
3.1135
3.2406
3.2898
3.2313
3.5985
3.6421
3.5841
4.1532
4.1462
4.1403
4.8805
4.9090
4.9222
iβs
(State-space model)
0%
12.5%
25%
3.0934
3.0770
3.0271
3.1607
3.1448
3.0966
3.4172
3.4025
3.3577
3.9994
3.9865
3.9472
4.4491
4.4373
4.4012
iβs
(Experiment)
0%
12.5%
25%
2.9995
2.9641
2.9569
3.1459
3.1231
3.1097
3.3884
3.4166
3.3943
3.9550
4.0162
3.9464
4.7009
4.7233
4.7090
22
Draft
#"-axis
!%-axis
#"
(
'(()
#%-axis
!%
#%
*" '0
&"
&%
&%
&"
)%
!"
("
$%
!"-axis
!"-axis
$"
$%-axis
$"-axis
Figure 1: Equivalent windings of an IM.
23
Draft
Inverse airgap
length [1/mm]
6
5
4
3
2
0
1
2
3
4
5
6
Angular position [rad]
Figure 2: Inverse air gap length at 40% eccentricity.
24
Draft
)-axis
#"-axis
!%-axis
#"
'-axis
#%-axis
!%
#%
&%
("
&"
!"
$%
!"-axis
$"
$%-axis
$"-axis
Figure 3: Eccentricity-oriented coordinate transformation.
25
Draft
Output torque [Nm]
40
40
30
20
20
10
0
0
0.1
0.2
0
0
0.5
1
1.5
Time [s]
Figure 4: Electromagnetic torque comparison between the proposed model (labelled with SS) and the
FEM.
26
Draft
Currents [A]
40
20
0
50
0
-50
0
0.1
0.2
-20
-40
0
0.5
1
1.5
Time [s]
Figure 5: Stator current comparison between the proposed model (labelled with SS) and the FEM.
27
Draft
Inductance [mH]
676
674
672
670
668
0
1
2
3
4
5
6
Augular position [rad]
Figure 6: Stator α-axis self inductances in one shaft revolution obtained from the state-space model
(labelled with SS) and the FEM.
28
Draft
Servo machine
drive
Test machine dri
drive
Oscilloscope
Current and voltage probes
Figure 7: Experimental setup.
29
Draft
No offset
0.24mm offset
0.12mm offset
Injection of eccentricity
External bearing housing
Figure 8: Eccentricity fault injection. The 0.24 mm and 0.12 mm offsets correspond to the 25% and
12.5% static eccentricities.
30
Draft
Torque [N-m]
15
10
5
State-space model
Experiment
0
0
500
1000
1500
2000
Speed [rpm]
2500
3000
Figure 9: Comparison of the simulated and measured torque-speed characteristics.
31
Draft
Current [A]
5
Experiment
State-space model
0
-5
0
0.01
0.02
0.03
0.04
Time [s]
Figure 10: Time domain comparison between the predicted and measured stator α-axis currents.
32
Draft
Figure 11: Frequency domain comparison between the predicted and measured stator α-axis currents.
33
Draft
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