OF PM DRIVESEffects of the speed-controller bandwidth
in permanent magnet synchronous machines
B Y W I E H A N L E R O U X , R O N A L D G . H A R L E Y , & T H O M A S G . H A B E T L E R
MOTOR CURRENT SIGNATURE ANAL-
ysis (MCSA) has been used for quite some
time to detect rotor faults in induction
motors [1]. These faults include static and
dynamic rotor eccentricities, broken rotor bars, and bear-
ing faults. In this kind of analysis, the motor is usually fed
directly from the line and, hence, the voltage is assumed
to be sinusoidal.
Other authors have done work with permanent mag-
net synchronous machines (PMSMs) to detect turn-
to-turn faults of the stator windings [2] and open- and
short-circuit failures of a converter-fed PMSM [3], [4].
Some have also reported on modeling rotor eccentric-
ities either by modulating the air gap (varying
permeance method) [5] or by finite element methods.
Although there has been some investigation into the
MCSA of a PMSM with a rotating fault [6], how the
converter influences this process has not been investi-
gated yet.
A PMSM has to be fed by a converter of some kind.
This drive usually contains an outer speed feedback loop
to set the torque reference (q-axis reference current) and
two current loops to control the q- and d-axis currents.
In line-fed induction motors, it is the stator current (or
air gap flux) that is analyzed in the literature for har-
monic content at specific frequencies. However, in a
PMSM, the fault information is distributed between the
current, voltage, and electromagnetic torque. Further-
more, the value of harmonic fault information and the
degree of division of this information between the cur-
rent, voltage, and torque will vary with the bandwidth of
the speed and current controllers.Digital Object Identifier 10.1109/MIA.2007.91578923
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These effects have to be investi-gated if MCSA is used for rotor faultdetection. Two common phenomenathat are investigated in this articleare dynamic eccentricity and varyingload torque.
Fault Mechanism and SpeedFeedback Effects on FaultHarmonicsIt was shown in [7] that rotor faultsmay be detected by monitoring theamplitudes of certain stator currentharmonic components. In [7], espe-cially those harmonics at the rotorspeed (the inverse of the pole-pairnumber and integer multiples thereof )that were of primary importance indetecting these rotor faults were iden-tified. These harmonics are from thispoint called the fault harmonics.
In short, the presence of these new fault harmonics isexplained as for the dynamic eccentricity example: for afour-pole PMSM with a dynamic eccentricity, plots ofthe flux linkage and inductances against rotor positionhave frequency components with a 0.5 harmonic num-ber and integer multiples thereof. This happens sinceone stator electrical fundamental period produces onlyhalf a rotation for a four-pole machine. Because the
position of minimum air gap rotateswith the rotor in this case, the fluxlinkage and winding inductance willbe different in one half of the rotationcompared with the other half of rota-tion. This produces harmonic com-ponents at the rotating frequency(harmonic number of 0.5). Thus, theposition of the harmonic componentto be monitored is 2/P, where P is thenumber of poles.
These are not new harmonics inthe sense that they have not beendefined before. These are the samesideband harmonics (reported in [8])of an induction machine except thatthe sideband slip-harmonics are notpresent. The harmonics are new in thesense that, for a machine operatingwithout dynamic eccentricity, theyare very small (theoretically zero). In
the case of the PMSM, it is mainly dynamic eccentricitythat influences these harmonics and does not show upwith static eccentricity since the flux linkages and induc-tances of the stator windings are not different in the firsthalf of rotation than the second half of rotation (in a four-pole machine) [7].
In an earlier work [9], these fault harmonics have notbeen shown to detect dynamic eccentricities in synchro-nous machines since the salient-pole synchronousmachine used for the analysis and experiments in theinvestigation was a two-pole machine, and the new har-monics coincide with the fundamental stator electricalfrequency and integer multiples, thus not so easily seen.Other works in detecting dynamic eccentricity insynchronous machines with more than one pole-pairconcentrate on the changes in the existing integer mul-tiples of the stator electrical fundamental frequency[10]. As shown in [7], it is much easier to detect a har-monic that is zero or very small under normal situationsand increases when the fault occurs than to detectchanges in existing harmonics since these changes willbe different for different machines and different operat-ing conditions.
An example of the measured fault harmonics for adynamic eccentricity fault on an actual PMSM is shown inFigure 1. A fast Fourier transform (FFT) of the measuredstator current of a PMSM with a dynamic eccentricityfault is shown in Figure 1.
The motor used for this example is a 2.84-kW, 6.64-A,460-V, 3,000-rev/min (314.2 rad/s), four-pole, surface-mounted PMSM with a 33% eccentricity (rotor is movedoff-center at the driving end by 33% of the air-gaplength). Harmonic 1 is the fundamental current compo-nent and carries on out of scale up to just below 1 per unit(p.u.). In Figure 1, one can clearly see the 0.5, 1.5, and 2.5harmonic current components that are caused by thedynamic eccentricity. These harmonic numbers are fixedsince this is a four-pole machine.
These fault harmonic components increase in ampli-tude for all of the following conditions: rotor faults, exter-nal source vibration, and pulsating load torque. Thus, the
12.0
0.5 1 1.5 2Harmonic Number
2.5
Stator Current for Dynamic Eccentricity Case×10–3
10.5
9.5
7.5
6.0
Cur
rent
p.u
.
4.5
3.0
1.5
0
1FFT of measured abc-frame stator current for the dynamiceccentricity case (speed = 1,280 rev/min or 0.43 p.u.; load =0.97 p.u.).
MOTOR CURRENTSIGNATURE
ANALYSIS HASBEEN USED FOR
QUITE SOME TIMETO DETECT ROTOR
FAULTS ININDUCTIONMOTORS.
2
q qT ∗ = ir
∗qν r∗
di r∗= 0 dν
r∗PI PI
PI–Σ
PWMInverter
PMSM
abc
qd
Encoder
−
−
dir
qir
ω*r
ωr
Σ
Σ
Flow diagram of PMSM drive system.24
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monitoring scheme cannot use the cur-rent harmonics as the sole characteris-tic for fault detection.
PMSMs are mostly controlled inthe dq-axis rotor rotating referenceframe. A flow diagram of this controlmethod is shown in Figure 2. In thiscontrol system, there is an outer speedcontrol loop and two inner currentcontrol loops. It is assumed that this isused in a constant speed commandapplication.
The fault harmonics caused byrotor faults, external vibration, orload torque pulsations can be seen in acombination of the following param-eters: electromagnetic torque, Te; load torque, Tl; statorcurrent; stator voltage; and rotor speed, xr. The relation-ships between these quantities (in the stationary abc-frame and the qd-rotor rotation reference frame) aregiven in (1)–(3) [11]
Te ¼ Jdxr
dtþ Tl (1)
Te ¼3
2
P
2(kr
dsirqs � kr
qsirds)
¼ 1
xr
(eaia þ ebib þ ecic) (2)
�vrqd0s ¼ rs
�irqd0s þWrL
rqd0si
rqd0s þ p(Lr
qd0s)irqd0s
þ Lrqd0s p(ir
qd0s)þWrkrqd0ðpmÞ þ p(kr
qd0ðpmÞ): (3)
In (1)–(3), P is the number of poles; J is the polarmoment of inertia of the motor and load combined; ir
dsis the d-axis rotor rotating reference frame stator flux link-age; ea is the a-phase back-emf; rs is the stator resistance ofone phase; Lr
qd0, vrqd0, ir
qd0, and krqd0ðpmÞ are the qd0-frame
stator inductance vector, stator voltage vector, stator cur-rent vector, and permanent magnet flux linkage vector,respectively, all in the rotor frame; p is the differentialoperator; and Wr is given by
Wr ¼0 xr 0
�xr 0 0
0 0 0
264
375: (4)
For this section, it is assumed that the two current con-trollers are ideal (follow current commands without error).In the case of controlling the d-axis current to be zero, as inFigure 2, the electromagnetic torque equation reduces to
Te ¼3
2
P
2(kr
dsirqs): (5)
According to (5), the electromagnetic torque is propor-tional to the q-axis stator current.
Dynamic EccentricityIn the ideal case with an ideal normalmotor (no rotor unbalances, no vary-ing load torque, and ideal sinusoidalback-emf), all the parameters in (5) areconstant, thus also the electromag-netic torque, Te.
For a dynamic eccentricity, the air-gap length changes dynamically as therotor rotates. For this reason, the statorwinding inductances and the fluxlinkages will have frequency compo-nents at multiples of the rotationalspeed frm; i.e., at k � frm [6] (as shownin Figure 1 for the current), withfrm ¼ 0:5fe, where fe is the electrical
fundamental frequency. These are the fault harmonics.Since the speed reference and thus also the q-axis currentreference are constant, these harmonics will be present inthe electromagnetic torque according to (5). For a con-stant load torque, these harmonics in the electromag-netic torque give rise to variations Dxr in the speedaccording to (1).
These speed variations are fed back by the speed control-ler. Assuming that the bandwidth of the current controllersis infinite (stator currents follow the current commandsexactly), the effect of the speed-controller bandwidth is as fol-lows: when the bandwidth is very low, the controller cannotcompensate for the variations in speed and outputs a torquecommand that leads to sinusoidal currents (in the abc-frame)or a dc current (in the qd-frame). When the speed-controllerbandwidth is very high, variations in the speed are fedthrough to the torque command to compensate for the varia-tions in the speed, and these harmonic variations are seen inthe qd-currents and therefore in the abc-currents as well.
These observations can be summarized as follows: thevariations in the speed because of a dynamic eccentricitywill decrease when the speed-controller bandwidthincreases since the speed controller compensates for thevariations in speed. According to (1), these variations inthe electromagnetic torque have to be decreased as thecontroller bandwidth increases. Therefore, the variationsin ir
qs have to increase as the speed-controller bandwidthincreases to cancel the variations in kr
ds to decrease the var-iations in Te [see (5)].
Thus, increasing the speed-controller bandwidth, withthe presence of a dynamic eccentricity, increases the ampli-tude of the fault harmonics in the current and decreases theamplitude of these harmonics in the electromagnetic torque.
External VibrationExternal vibration modulates the air gap at the frequencyof vibration and therefore changes the inductances. There-fore, the effect on the harmonics in the motor currents,torque, and speed is the same for the rotor faults that affectthe flux linkages and winding inductances at certain fre-quencies and for external vibrations at the frequencies thatproduces the same frequency variations in the inductances.
Load Torque PulsationsWith a variation DTL in the load torque (varying load tor-que) at the frequency of the rotor speed, the speed will
PMSMS AREMOSTLY
CONTROLLED INTHE DQ-AXIS
ROTOR ROTATINGREFERENCE
FRAME.
25
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exhibit similar variations Dxr in the speed according to(1) as in the case of the dynamic eccentricity.
Now, with a speed controller with very low bandwidththat does not measure Dxr, the reference for the q-axiscurrent, ir�
qs, and the electromagnetic torque, Te (assumingperfect current controllers), will be constant. If the speed-controller bandwidth is increased, the controller introdu-ces variations in ir�
qs to compensate for Dxr. Thus, thecurrent fault harmonic amplitude increases with speed-controller bandwidth. In the varying load torque case,compensating for Dxr means that the electromagnetictorque has to contain increased variations DTe as the band-width is increased (1).
Thus, increasing the speed-controller bandwidth,with the presence of a varying load torque, increases theamplitude of the fault harmonics in the current andincreases the amplitudes of these harmonics in the elec-tromagnetic torque.
ConclusionAs discussed in the previous paragraphs, the dynamiceccentricity and a varying load torque at the frequency ofthe rotor speed influence the stator current in the sameway. Any increase in the bandwidth of the speed controllerchanges the harmonics in the current and speed in thesame fashion for both cases. Thus, it is difficult to say
whether a dynamic eccentricity or a varying load torque iscausing the fault harmonic components in the current.
However, the variations in the electromagnetic torqueincrease for increased speed-controller bandwidth in thecase of the varying load torque but decrease for thedynamic eccentricity case. This phenomenon may be usedto distinguish between these two faults. This is investi-gated in the next section.
Table 1 demonstrates how the amplitudes of the faultharmonics change with changes in speed-controller band-width for the dynamic eccentricity and the varying loadtorque cases. For example, assume that a four-pole PMSMis used and that the fault frequency under investigation isthe 0.5 harmonic (see Figure 1). The top half of Table 1shows how the harmonics change for the dynamic eccen-tricity. The first row shows that the 0.5 harmonic will notbe present (or be very small) in the stator current with alow speed-controller bandwidth, but the amplitude of the0.5 harmonic will increase when the speed-controllerbandwidth increases. It can be seen from Table 1 that the0.5 harmonic in the electromagnetic torque decreases withan increase in speed-controller bandwidth in the dynamiceccentricity case but increases in the varying load torquecases. The 0.5 harmonic in the stator voltage is accordingto (3) since this is a current-controlled application.
Effect of Speed-Controller Bandwidthon Fault Harmonics
Distinguishing Between Varying Load Torqueand Dynamic EccentricityIt is postulated in the previous section that the harmonicsin the stator current and voltage that results from a varyingload torque or a dynamic eccentricity increase for increasedspeed-controller bandwidth. However, these harmonics inthe electromagnetic torque will decrease with increasedbandwidth for a dynamic eccentricity while they willincrease for the varying load torque case. It is thus possibleto distinguish between a rotor fault (such as the dynamiceccentricity) and a varying load torque.
Dynamic EccentricityThe simulation of a complete 2-kW, 6-A, 380-V, 3,000-rev/min (314.2 rad/s), four-pole, surface-mounted PMSM drivewith a speed controller supported these hypotheses. ThePMSM is modeled by using parametric finite element analy-sis (FEA) to obtain lookup tables of stator winding induc-tances and permanent magnet flux linking the statorwindings as functions of the rotor position. This is first donefor the normal motor by rotating the rotor in the center ofthe stator while calculating these parameters with FEA forone full rotation. This procedure is repeated, but now therotor is displaced from the center of the stator (by 30%) andstill rotates around the center of the stator while calculatingthese parameters at different rotor positions for one full rota-tion. This latter procedure yields lookup tables for thedynamic eccentricity case. These lookup tables were thenused in a transient MATLAB simulation of the drivedescribed in Figure 2 and (1)–(3). Details of this procedurecan be found in [6]. Simulation of the system in Figure 2 isdone with a constant speed command and yields speed, volt-age, current, and electromagnetic torque simulation data
TABLE 1. CHANGES IN HARMONIC COMPONENTSAS SPEED-CONTROLLER BANDWIDTH CHANGESFOR DYNAMIC ECCENTRICITY AND VARYINGLOAD TORQUE.
Dynamic Eccentricity
ControllerBandwidth Low
$ ControllerBandwidth High
NoHarmonic
HarmonicPresent
NoHarmonic
HarmonicPresent
irqds $ irqds
vrqds vr
qds
Lr, kr Lr, kr
Te $ Te
Tl Tl
xr $ xr
Varying Load Torque
ControllerBandwidth Low
$ ControllerBandwidth High
NoHarmonic
HarmonicPresent
NoHarmonic
HarmonicPresent
irqds $ irqds
vrqds $ vr
qds
Lr, kr Lr, kr
Te $ Te
Tl Tl
xr $ xr26
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against time. Fourier transforms of the current is done in thedq-rotor rotating reference frame and the harmonics arepresented and analyzed.
The PMSM is simulated under two conditions: 1) witha dynamic eccentricity of 30% and a constant load torqueand 2) with a normal rotor but with an added load torquecomponent varying purely sinusoidally (0.16 p.u. peak topeak) at the frequency of the rotor speed. Thus, the loadtorque in the latter case contained a varying component
DTl ¼ 0:5 coshr
2
� �, (6)
where hr is the rotor angle in electrical radians.All simulations are done at 0.7 p.u. speed and 0.55 p.u.
load. As a result of the simulation, the amplitude of the0.5 harmonic stator current components is shown inFigure 3 as a function of the speed-controller bandwidth.
The results shown in Figure 3 are for the dynamic eccen-tricity case. Figure 3(a) is the case where the current con-trollers have infinite bandwidth. The infinite bandwidth isobtained by actually setting the line q-d currents equal tothe current commands in the simulation and not solvingthe electrical transients. It can be seen that the d-axis cur-rent stays zero all the time as can be expected whereas theq-axis current increases for an increase in speed-controllerbandwidth. The same is shown in Figure 3(b) but this timeusing a finite bandwidth for the current controllers. A pro-portional (p) current controller is used with a bandwidthstill much higher than the bandwidth of the speed control-ler. The q-axis current shows the same trend as a function ofspeed-controller bandwidth. The d-axis current compo-nent, however, is not zero any more because of the nonidealcurrent controllers.
The amplitudes of the 0.5 harmonic in Figure 3 are verysmall, and one can argue that they might be too small to bemeasured in an actual machine. However, these amplitudesare calculated from the simulation result with a particularsimulated machine and machine constants only to showthe trend. The experimental work was done on a largermachine under heavier load conditions, and the fault har-monics (0.5, 1.5, etc.) can clearly be seen in Figure 1 andcan easily be measured. This can also be seen in themeasured results later in this article (Figure 9). The meas-urability of these components for this type of experimentalmachine has been established in [6], [7], and [12].
Varying Load TorqueThe 0.5 harmonic components of the stator currentobtained from the simulation of the varying load torquecase is shown in Figure 4. Again, the current componentsare plotted against speed-controller bandwidth for currentcontrollers with infinite and finite bandwidth. The cur-rent components follow the same trends, with increases inspeed-controller bandwidth, as in this case of the dynamiceccentricity (Figure 3).
Motor Torque Dependencyon Speed-Controller BandwidthThe relationships between the speed-controller bandwidthand the 0.5 harmonic variations in the current, electro-magnetic torque, and speed are compared in Figure 5.
Figure 5(a) and (c) again shows the 0.5 harmonicstator current amplitude against speed-controller band-width for the dynamic eccentricity and varying load tor-que cases. This time, the total current, ir
3Simulated q- and d-axis currents: (a) for an infinite currentbandwidth and (b) a finite current bandwidth for thedynamic eccentricity case.
4
25.0
16.7
8.3
0
0.5 Harmonic—Infinite Current Bandwidth
Speed-Controller Bandwidth [rad/s]
(a)
101 102
×10–3
Cur
rent
p.u
.
25.0
16.7
8.3
0
0.5 Harmonic—Infinite Current Bandwidth
Speed-Controller Bandwidth [rad/s](b)
101 102
×10–3
iqid
Cur
rent
p.u
.
iqid
Simulated q- and d-axis current for the varying load case:(a) for an infinite current bandwidth and (b) for a finitecurrent bandwidth.
27
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In Figure 5(b), the 0.5 harmoniccomponent in the speed and electro-magnetic torque is shown against anincrease in speed-controller band-width. First, it should be explainedwhat is meant by the 0.5 harmoniccomponent in the speed. The speedsignal is sampled (or saved in thesimulated case). Then, an FFT is doneon this speed signal the same way it isdone on the current signal. An exam-ple of an FFT on a speed signal isshown in Figure 6 for the varying loadtorque case, with harmonic 1 the sameas the fundamental frequency of thestator current signal. The very largezero component (or dc component)that carries on out of scale in Figure 6(up to 0.7 p.u.) corresponds to the average speed. The 0.5harmonic component in Figure 6 corresponds to speedoscillations at the frequency of rotation speed (for the four-pole machine). The same procedure is followed for theelectromagnetic torque signal. In the simulation, the tor-que is calculated and in the experimental setup, the torqueis estimated.
Now in Figure 5(b), the 0.5 harmonic components inthe speed and the electromagnetic torque decrease for an
increase in speed-controller band-width. This is the dynamic eccentric-ity case. In the varying load torquecase [Figure 5(d)], the 0.5 harmoniccomponent in the speed decreaseswith an increase in speed-controllerbandwidth but the 0.5 harmoniccomponent in the electromagnetictorque increases.
These simulation results confirmwhat is stated in ‘‘Fault Mechanismand Speed Feedback Effects on FaultHarmonics’’ section: the fault fre-quency components in the electro-magnetic torque caused by a dynamiceccentricity and a varying load tor-que react differently to changes inspeed-controller bandwidth. This
property might therefore be used to distinguish be-tween the two phenomena.
This simulation result was repeated on an experimentalsetup of a 2.85-kW, four-pole, surface-mounted PMSMwith a dc machine as load. However, the measured resultscannot be presented because a technical problem prohib-ited obtaining good results as the PMSM uses a resolverfor position and speed feedback with the resolver fitted onthe shaft of the PMSM. The dynamic eccentricity was
implemented on thePMSM [7] by firstremoving the bearingfrom the shaft. The shaftwas then machinedsmaller at the drivingend of the motor overthat length where thebearing has to fit. Thenthe bearing was placedback onto the shaft, butthis time with shimsbetween the bearing andthe shaft on the oppositeside of the machining tomove the rotor from thecenter of the stator. Thisis shown in Figure 7.
With this, the rotorwas moved 15 thou-sandths of an inch fromthe center of the statorwhile the nominal air-gaplength is 50 thousandthsof an inch. When therotor rotates, the shims(and thus the position ofmaximum air-gap length)rotate with the rotor, cre-ating the dynamic eccen-tricity. However, movingthe rotor off-center meansthat the rotor of theresolver is also movedoff-center. Now the
5
2.5
Dynamic Eccentricity
1.7
Cur
rent
p.u
0.8
0
×10–5
Speed-Controller Bandwidth [rad/s]
(c) (d)
(a) (b)
101 102
2.5
Dynamic Eccentricity
4
3
2
1
Spe
ed p
.u. a
nd T
orqu
e p.
u.
0
×10–5
Speed-Controller Bandwidth [rad/s]101 102
2.5
Varying Load Torque
1.7
Cur
rent
p.u
0.8
0
×10–3
Speed-Controller Bandwidth [rad/s]101 102
2.5
Varying Load Torque
4
3
2
1
Spe
ed p
.u. a
nd T
orqu
e p.
u.
0
×10–5
Speed-Controller Bandwidth [rad/s]101 102
Te
�r
Te
�r
(a)–(d) A 0.5 harmonic component of stator current, speed, and electromagnetic torqueagainst speed-controller bandwidth for the dynamic eccentricity and varying load torquecases.
WITH A VERYLOW SPEED-
CONTROLLERBANDWIDTH,THERE IS VERY
LITTLEINFORMATION
IN THE CURRENT.
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(previously normal working) resolver also experiences adynamic eccentricity and the stator inductances containthe same fault frequencies since the resolver is also a four-pole machine. The position and speed feedback wouldthus be erroneous. Thus, the results of this experimentcould not be trusted.
Monitoring Stator Voltage or Currentto Detect a Rotor FaultSince the fault information is contained in both the statorvoltage and current, it is necessary to investigate whetheror not there are situations when either the current or thevoltage should not be monitored. When the speed-controller bandwidth is very low, the variation in the speedthat is caused by the dynamic eccentricity (rotor fault) isvirtually zero at the output of the speed controller and doesnot influence the current command. Thus, for a low speed-controller bandwidth, the fault information will not bevisible in the current and has then to be in the voltage. Asthe speed-controller bandwidth increases, the fault infor-mation will increase in the current and will decrease in thevoltage. This is first shown in the simulated results in Fig-ure 8 and later on by measurement in Figure 9.
Figure 8(a) shows how the 0.5 harmonic voltage in-creases from the normal motor to the dynamic eccentricity(labeled D Eccen in the graph) case. Two cases are shown,one with a speed-controller bandwidth of 5.4 rad/s andone of 54 rad/s. It can be seen that there is a bigger increasein voltage fault harmonic amplitude with the lower band-width. A much bigger effect can be seen in the current[Figure 8(b)]. The increase in current fault harmonic withthe higher speed-controller bandwidth is much bigger
than for a smaller speed-controller bandwidth for the samedynamic eccentricity fault.
This can also be seen in the measured results (Figure 9)of the experimental 2.8-kW, four-pole PMSM drive.
Figure 9(a) shows how the amplitudes of the fault harmon-ics in the voltage (0.5, 1.5, and 2.5) changes from the normalmotor case to a dynamic eccentricity case for low (harmonic1.5 increases from point N to point A in Figure 3) and high(harmonic 1.5 increases from point N to point B) speed-con-troller bandwidths. The low bandwidth is 3.5 rad/s and thehigh bandwidth is 35.3 rad/s. Figure 9(b) shows the same har-monics in the current (the 0.5 current component increasesfrom point N to point F for a low bandwidth, but to the
6
1.9
×10–3 FFT of Speed
Harmonic Number
1.61.3
Spe
ed p
.u.
1.00.60.3
00 0.5 1 1.5 2 2.5
FFT of speed signal showing 0.5 harmonic component forthe varying load torque case.
7
Stator
Bearing
ShimsShaftMachined
Shaft
Sketch showing implementation of dynamic eccentricity.
8
6.8
Stator Voltage
Normal Rotor Fault
(a)
(b)
D Eccen
×10–5
4.6
Vol
tage
p.u
.
2.3
0
16.7
13.3
Stator Voltage
Normal Rotor Fault D Eccen
×10–6
10.0
Vol
tage
p.u
.
6.7
3.3
0
Bandwidth = 5.4 rad/s
Bandwidth = 54 rad/s
Bandwidth = 5.4 rad/s
Bandwidth = 54 rad/s
Simulated results: Increase in 0.5 harmonic levels fordifferent speed-controller bandwidths: (a) for stator voltageand (b) for stator current for the dynamic eccentricity rotorfault.
9
5.6×10–3 Stator Voltage (abc-Frame)
Normal D EccenRotor Fault
(a)
(b)
AB
CD
N
N
3.8
Vol
tage
p.u
.
1.9
0
9.07.5
×10–3 Stator Current (abc-Frame)
Normal D EccenRotor Fault
E
F6.04.5
Cur
rent
p.u
.
3.01.5
0
1.5 Harm; Low BW1.5 Harm; High BW0.5 Harm; Low BW0.5 Harm; High BW2.5 Harm; Low BW2.5 Harm; High BW
1.5 Harm; Low BW1.5 Harm; High BW0.5 Harm; Low BW0.5 Harm; High BW2.5 Harm; Low BW2.5 Harm; High BW
Measured results: Increase in harmonic levels for differentspeed-controller bandwidths: (a) for stator voltage and(b) for stator current for the dynamic eccentricity rotor fault.
29
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higher point E for a high bandwidth). It can be seen that thevoltage harmonics increase more with a low speed-controllerbandwidth for a rotor fault while the current harmonicsincrease more with a high speed-controller bandwidth.
This should be taken into account when implementinga condition-monitoring scheme and is discussed in theconclusions.
Separating Varying Load Torqueand Dynamic EccentricityIn the previous section, it was discussed in detail that esti-mating the electromagnetic torque while changing thespeed-controller bandwidth could be used to investigatewhether current fault harmonics are a result of a rotor faultor an external varying load torque. However, if both ofthese phenomena exist together, this method will not dis-tinguish between these effects. Often, the current harmon-ics caused by a varying load torque are much larger thanthe current harmonics caused by the dynamic eccentricityor other rotor faults. This means that a dynamic eccentric-ity cannot be detected in the presence of a varying load tor-que. Some investigation [13] has been conducted intodetecting the rotor fault harmonics in the presence of loadtorque variations at the same frequency for using an induc-tion motor. This requires the separation of the rotor faultand varying load torque effects.
In the PMSM, the command for the q-axis current isdirectly related to the torque command. Changing the band-width of the speed controller changes the amplitude of theharmonic components of the q-axis current, as discussed inthe previous section. The command for the d-axis currentcommand stays constant at zero and is not affected by varia-tions in the speed and the speed-controller bandwidth. Thelogical step is thus to monitor the q- and d-axis currents indi-vidually and use this information to distinguish between therotor fault and pulsating load torque cases [13]. However,these two currents are highly coupled, as can be seen whenstudying the realistic (not the ideal) motor equations. Thesecurrents can be decoupled only if the flux linkages andinductance terms are known exactly. In the literature, averagevalues of the flux linkages and inductances are calculatedfrom measurements on the machine. However, with rotorfaults, the flux linkage and inductance components have var-iations and the decoupling will be incorrect. Moreover,decoupling does not normally take place in inverter drives.
Another attempt at separating these effects in induc-tion motor drives was done in [14]. As discussed in theprevious paragraph, changes in the q-axis current causedby variations in the load torque have an impact on thed-axis current. To remove this effect, the actual d-axis sta-tor current is compared with an estimated current thatdoes not contain any rotor fault-induced harmonic compo-nents. This is accomplished by using the inductionmachine equations to predict what the d-axis currentshould be for an ideal machine operating under the sameload conditions. Subtracting this estimated current fromthe actual current gives an error that contains the varia-tions caused by the rotor fault. In the investigation in[14], the voltage equations are transformed to the syn-chronous reference frame with the d-axis aligned with therotor flux linkage. This allows the estimation of the d-axiscurrent without measuring the rotor speed, xr.
Following the same procedure for the PMSM drive, thevoltage and current equations in the rotor rotating refer-ence frame are given by
vrqs ¼ rsi
rqs þ pkr
qs þ xrkrds,
vrds ¼ rsi
rds þ pkr
ds � xrkrqs, (8)
where the flux linkages are given by
krqs ¼ Lr
qsirqs þ kr
qðpmÞ,
krds ¼ Lr
dsirds þ kr
dðpmÞ, (9)
where the inductances are assumed to be constant.The derivative of the d-axis current can be extracted
from (8) and (9) as
d
dtirds ¼
1
Lrds
� ��vr
ds � rsirds þ xrL
rdsi
rqs
� pkrdðpmÞ þ xrk
rqðpmÞ
�: (10)
In (10), pkrdðpmÞ and xrk
rqðpmÞ are both zero when still
assuming constant inductances and ideal magnet fluxlinkage. Thus, (10) reduces to
d
dtirds ¼
1
Lrds
� �vr
ds � rsirds þ xrL
rdsi
rqs
h i, (11)
where irqs, irds, and vr
ds are actual currents and voltages.The change in d-axis current in (11) is still influenced
by the load torque variations (irqs is part of equation) but itassumes constant inductances, i.e., no rotor fault. Theideal d-axis current (for a machine with or without anyrotor faults) is then predicted for the next step by
irds(kþ 1) ¼ ir
ds(k)þ H � d
dtirds(k), (12)
where the derivative is given by (11), and H is the timestep of integration. At the next time step, the actual d-axiscurrent is subtracted from the estimated current, whichyields the error defined by
ei(kþ 1) ¼ irds(kþ 1)� ir
ds(kþ 1): (13)
The error between predicted and actual current wave-forms given in (13) now represents the variations in currentcaused by the rotor fault but not the load torque variations.
A disadvantage of working in the rotor rotating qd-reference frame and using (11)–(13) is that the rotor angle isneeded for the transformation from the stationary abc-reference frame. Another disadvantage is that the rotor speedmust be known. Changing the analysis to using the synchro-nous reference frame might offer a solution. Rewriting (10)in the synchronously rotating reference frame yields
d
dtieds ¼
1
Leds
� ��ve
ds � rsieds þ xeL
edsi
eqs
�pkedðpmÞ þ xek
eqðpmÞ
�: (14)30
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NE�
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In (14), neither the rotor speed nor position is required.However, the q- and d-axis magnet flux linkages now con-tain harmonics because the transformation from the rotorreference frame (where the magnet flux linkages are inde-pendent of load torque variations) to the synchronousreference frame uses the rotor position that contains thesame frequency variations as the speed due to the varyingload torque. Thus, pke
dðpmÞ and xekeqðpmÞ cannot be ignored
as before, and they are in fact unknown. An analysis in thestationary reference frame suffers from the same problems.
Both (10) and (14) were used to estimate the d-axis cur-rent in the varying load torque and dynamic eccentricitycases. The magnet flux linkage terms were neglected whilefollowing this estimation procedure in both the rotor andsynchronous rotating reference frames. Neither of thesemethods was successful in separating the harmonics in thed-axis current caused by the varying load torque fromthose caused by dynamic eccentricity. In the synchronousreference frame, pke
dðpmÞ and xekeqðpmÞ could not be ignored
because they contain harmonic components caused by thevarying load torque as explained before. The rotor referenceframe used in (10) and (11) requires the use of the rotor speedand position, and there is another reason that this methodcannot be used. Because of nonidealities, the magnet fluxlinkages are not constant in time, and thus the harmoniccomponents in xrk
rqðpmÞ cannot be ignored. Thus, none of
these methods can be used to distinguish between the vary-ing load torque and rotor fault effects.
ConclusionsThis article investigates the effects of the speed-controllerbandwidth on the detection of rotor faults by monitoringthe stator electrical quantities. The main contributions arelisted in the following:
n The mechanism by which fault harmonics showup in the stator current and speed was discussed.
n It was shown that, by the action of the speed con-troller, increasing the speed-controller bandwidthincreases these fault harmonics in the current andvoltage.
n Also, for a load torque varying at the frequency ofthe rotor speed, the same fault harmonics willshow up in the current and voltage as in the rotorfault case. Again, these harmonics increase withincreased speed-controller bandwidth.
n It was shown that the fault harmonics in the electro-magnetic torque can be used to distinguish betweenthe rotor fault and a varying load torque by increas-ing the speed-controller bandwidth online. If thefault harmonic components in the electromagnetictorque decrease with an increase in speed-controllerbandwidth, then the fault frequencies are caused bya rotor fault such as a dynamic eccentricity and notby a varying load torque.
n It was shown that the rotor fault and varying loadtorque cannot be separated by looking at the faultfrequencies in the same way as an inductionmachine. Thus, with the technique used in this arti-cle, it is very difficult to detect a dynamic eccentric-ity in the presence of a varying load torque at thesame frequency. It might be worthwhile to investi-gate other methods to separate these two effects.
n Last, it was shown that for a very low speed-controller bandwidth and this type of drive, thevoltage should be monitored to detect the rotorfaults and not the stator currents. With a very lowspeed-controller bandwidth, there is very littleinformation in the current. This was verifiedexperimentally.
The results of this investigation is specific to the typeof drive used and should be repeated for different drivesand controllers.
References[1] G. B. Kliman and J. Stein, ‘‘Methods of motor current signature
analysis,’’ Elec. Mach. Power Syst., vol. 20, no. 5, pp. 463–474, Sept.
1992.
[2] J. A. Haylock, B. C. Mecrow, A. G. Jack, and D. J. Atkinson, ‘‘Oper-
ation of fault tolerant machines with winding failures,’’ IEEE Trans.Energy Conversion, vol. 14, no. 4, pp. 1490–1495, Dec. 1999.
[3] N. Bianchi, S. Bolognani, M. Zigliotto, and M. Zordan, ‘‘Influence
of the current control strategy on the pmsm drive performance dur-
ing failures,’’ in Proc. 7th European Conf. Power Electronics and Applica-tions, Trondheim, Norway, 1997, pp. 1.330–1.335.
[4] J.-P. Martin, F. Meibody-Tabar, and B. Davat, ‘‘Multiple-phase
permanent magnet synchronous machine supplied by VSIs, working
under fault conditions,’’ in Proc. World Congress Industrial Applicationsof Electrical Energy and 35th IEEE-IAS Annual Meeting, Rome, Italy,
2000, pp. 1710–1717.
[5] C. M. Riley and T. G. Habetler, ‘‘Current-based sensorless vibra-
tion monitoring of small synchronous machines,’’ in Proc. PowerElectronics Specialist Conference—PESC’98, Fukuoka, Japan, 1998,
pp. 108–112.
[6] W. le Roux, R. G. Harley, and T. G. Habetler, ‘‘Rotor fault analysis
of a permanent magnet synchronous machine,’’ in Proc. 15th Int.Conf. Electrical Machine on CD, Brugge, Belgium, 2002.
[7] W. le Roux, R. G. Harley, and T. G. Habetler, ‘‘Detecting rotor faults
in permanent magnet synchronous machines,’’ in Symp. for Diagnosticsof Electric Machines, Power Electronics and Drives, Atlanta, GA, 2003.
[8] D. G. Dorrell, W. T. Thomson, and S. Roach, ‘‘Analysis of airgap
flux, current, and vibration signals as a function of the combination
of static and dynamic airgap eccentricity in 3-phase induction
[11] D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of acDrives. New York: Oxford Univ. Press, 1996.
[12] W. le Roux, R. G. Harley, and T. G. Habetler, ‘‘Detecting rotor
faults in low power permanent magnet synchronous machines,’’ IEEETrans. Power Electron., to be published.
[13] R. R. Schoen and T. G. Habetler, ‘‘Effects of time-varying loads on
rotor fault detection in induction machines,’’ IEEE Trans. Ind. Applicat.,vol. 31, no. 4, pp. 900–906, July/Aug. 1995.
[14] R. R. Schoen and T. G. Habetler, ‘‘Evaluation and implementation of
a system to eliminate arbitrary load effects in current-based monitor-
ing of induction machines,’’ IEEE Trans. Ind. Applicat., vol. 33,
no. 6, pp. 1571–1577, Nov./Dec. 1997.
Wiehan Le Roux ([email protected]) is with ConditionAssessment Technologies, Spoornet, Johannesburg, South Africa.Ronald G. Harley and Thomas G. Habetler are with Electri-cal and Computer Engineering, Georgia Institute of Technol-ogy, Atlanta. Harley and Habetler are Fellows of the IEEE.This article first appeared as ‘‘Converter Control Effects onCondition Monitoring of Rotor Faults in Permanent MagnetSynchronous Machines’’ at the 2003 IEEE Industry Applica-tions Society Annual Meeting.
