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Theoretical Prediction and Experimental Verificationof Light-Load Instability in a 11-kW Open-loop
Induction Motor DriveAnirudh Guha ∗, Abhishek Chetty †, C Kumaresan †, G Narayanan ∗, R Krishnamoorthy †
Indian Institute of Science∗, L&T Technology Services †Email:[email protected];[email protected];[email protected];
[email protected];[email protected]
Abstract—This paper presents the small-signal stability anal-ysis of a 11-kW open loop inverter-fed induction motor drive,including the effect of inverter dead-time. The analysis is carriedout using an improved small-signal model of the drive that hasbeen reported in literature recently, and was used to demonstratesmall-signal instability on a higher power level motor. Throughsmall-signal stability analysis, the region of oscillatory behaviouris identified on the voltage vs frequency plane (V-f plane),considering no-load. These predictions using the improved modelare also compared against predictions of a standard model ofan inverter-fed induction motor including dead-time effect. Theanalytically predicted region of oscillatory behaviour is verifiedextensively through time-domain simulations and experimentalresults. The experimental measurements with sine-triangle andconventional space vector PWM schemes validate the applicabil-ity of the modelling and analysis for both these schemes.
I. INTRODUCTION
Induction machines are widely used in the industry. Open-loop volts-per-hertz control of induction motors is a simplecontrol scheme which is commonly used in applications suchas fans and pumps. Energy efficiency considerations havepaved the way for variable speed operation of fans and pumpsas against fixed speed operation. V/f control of inductionmotors is also widely employed in the industry during thetesting stage of inverters and induction motors.
A few open-loop V/f induction motor drives have beenreported to exhibit oscillatory behaviour at low and mediumspeeds, and particularly under light-load conditions [1]–[10].This light-load instability phenomenon is impacted by motorparameters such as stator and rotor resistances, leakage andmagnetising reactances, and inertia [1], [3], [10]. Increasedstator resistance of the motor is an important factor whichresults in this light-load instability [2]–[5]. Further, the in-verter dead-time, which is introduced for safe commutation ofswitches in an inverter is reported to result in or accentuatethese sub-harmonic oscillations in the induction motor [4]–[9].
Considering the importance of V/f drives in the industry, thispaper attempts to study the prevalence of such oscillations incommercial motor drives. Specifically, a case study on a 11-kW industrial motor drive is reported in this paper. The studypresented here is a collaborative effort between an industryand an academic institution. This study comprises small-signal stability analysis, time-domain numerical simulationsand actual measurements on a 11-kW motor.
To predict and explain oscillatory behaviour of an inductionmotor, the standard classical approach models the effect ofinverter dead-time as an equivalent series resistance (Req0).The standard small-signal model of the induction motor, butwith the effective stator resistance taken as the sum of thedead-time equivalent resistance and actual stator resistance isused for stability analysis [5]. The stability analysis predicts aregion of oscillatory behaviour on the voltage versus frequency(V-f) plane, considering no-load [5]. Recently, inverter dead-time was shown to result in significant sustained sub-harmonicoscillations in a 100-kW induction motor [7]–[9]. An improvedsmall-signal model of an induction motor was derived in [8],[9]. The improved small-signal model was shown to predictthe region of oscillatory behaviour of the 100-kW drive moreaccurately than the standard model [8], [9]. This was demon-strated for different values of dead-time in [8], [9]. Analysisis presented here for the 11-kW motor drive considering boththe improved and standard small-signal models.
The modelling of the induction motor drive including dead-time is reviewed, and stability analysis of a 11-kW motor driveis presented in section II. Simulation results that validate theanalysis are presented in section III. Experimental validationconsidering both sine-triangle and conventional space vectormodulation schemes is presented in section IV.
II. MODELLING AND STABILITY ANALYSIS OF ANOPEN-LOOP INVERTER-FED INDUCTION MOTOR DRIVE
The dynamic model of the induction motor drive [8], [9],[11] and the linearised small-signal models of the drive [5],[8], [9] are reviewed in the following subsections.
A. Modified stator voltage equations
The modified stator voltage equations incorporating theeffect of inverter dead-time are expressed as follows [8], [9]:
vqs,id =
rs + |−→V s,err,f |√i2qs + i2ds
+ Lsd
dt
iqs + ωsLsids
+ Lmdi
′
qr
dt+ ωsLmi
′
dr
(1)
vds,id =− ωsLsiqs +
rs + |−→V s,err,f |√i2qs + i2ds
+ Lsd
dt
ids
− ωsLmi′
qr + Lmdi
′
dr
dt
(2)
The stator q-axis and d-axis currents are given by iqs and ids,respectively. |
−→V s,err,f | = 4
πVdctdTsw
is the magnitude of thedead-time error voltage vector, where Vdc is the dc bus voltage,td is the inverter dead-time, and Tsw is the switching cycleduration [12]. vqs,id is the ideal q-axis stator voltages appliedby an ideal inverter without dead-time. The ideal voltagevector is aligned along the q-axis, and hence its componentvds,id along the d-axis is zero. The rotor voltage equations andthe mechanical dynamic equations are the standard equationsas given in [8], [9], [11].
The improved small-signal model is derived by linearisingthe modified stator dynamic equations (1) and (2), and bylinearising the standard rotor voltage equations and mechanicaldynamic equations, about a steady operating point.
B. Small-signal model
The linearised small-signal model is expressed in the stan-dard form [8]:
∆x = A ∆x + B ∆u (3)
where
A =
[−L−1R −L−1λ10(
32J
) (poles
2
)2λT20 −BJ
](4)
The various constituent matrices are given in [8]. Importantly,the matrix R may be expressed as
R =
[Rss Rsr
Rrs Rrr
](5)
1) Improved small-signal model: In the improved small-signal model obtained by linearising the modified statordynamic equations, the sub-matrix Rss that is derived isexpressed as shown below in (6) [8].
Rss,impr =
[rs +Rq,eq0 ωs0Ls −Xqd,eq0
−ωs0Ls −Xqd,eq0 rs +Rd,eq0
](6)
where
Rq,eq0 =Req0
(i2ds0
i2qs0 + i2ds0
);Rd,eq0 =Req0
(i2qs0
i2qs0 + i2ds0
)
Xqd,eq0 = Req0
(iqs0ids0i2qs0 + i2ds0
);Req0 =
|−→V s,err,f |√i2qs0 + i2ds0
2) Standard small-signal model: In steady-state, the effectof dead-time is seen as an equivalent resistance (Req0) thatadds to the stator resistance of the motor [5]–[7]. In the stan-dard model [5], the dead-time equivalent resistance (Req0) wasjust added to the stator resistance; there was no linearisation ofdead-time error voltage terms considered as in the improved
TABLE IINDUCTION MOTOR PARAMETERS (STAR EQUIVALENT)
rs (stator resistance) 0.333Ω
r′r (rotor resistance referred to stator side) 0.359Ω
Ls (stator inductance) 82.46 mH
L′r (rotor inductance referred to stator side) 84.94 mH
Lm (mutual inductance) 79.82 mH
J (inertia) 0.0685 kgm2
model. The Rss sub-matrix used in the standard model is givenbelow.
Rss,std =
[rs +Req0 ωs0Ls−ωs0Ls rs +Req0
](7)
C. Small-signal stability analysis of a 11-kW motor drive
Stability analysis is performed on a 11-kW inverter-fedinduction motor, considering no-load. The motor is consideredto be fed from an inverter with a dc bus voltage Vdc = 600V,switching frequency fsw = 5kHz, and dead-time td = 3µs.The steady-state solution of the drive for a particular appliedvoltage and frequency is obtained. The matrix A (see eqn 4)is evaluated about this steady operating point using the motorparameters (see Table I) and steady-state solution. The eigenvalues of the matrix A are then evaluated. An operating pointis identified as an unstable operating point if any eigen valuehas a positive real part. The eigen values of the matrix A aresimilarly evaluated at various operating points, correspondingto different V/f ratios and different fundamental frequencies.
Considering the ideal case, with no dead-time effect, theeigen values (or root loci) are shown plotted in Fig. 1 for aV/f ratio of 0.89 pu. The loci of eigen values obtained using thestandard small-signal model [5] are shown in Fig. 2, while theroot loci obtained using the improved small-signal model [8]are shown in Fig. 3. Considering the ideal case with no dead-time (Fig. 1), and the case of the standard model includingdead-time (Fig. 2), a complex conjugate pair of eigen valuesare seen close to the jω axis. But these eigen values haveonly negative real parts. However, when the improved small-signal model is considered for analysis, it is observed in Fig.3 that the root-loci branch close to the jω axis traverse to theright of the jω axis for the speeds ranging between 10Hz and30Hz. This indicates small-signal instability and oscillatorybehaviour in the afore-mentioned speed range.
The stability analysis is performed considering no-load, forfundamental frequencies ranging between 5Hz and 50 Hz, andfor V/f ratios ranging between 0.5 pu and 1 pu. Consideringthe ideal case without dead-time, and considering the standardmodel where dead-time is just an equivalent resistance, nounstable operating points are seen on the V-f plane in Fig. 4.However, when the stability analysis is performed using theimproved small-signal model, a significant region of instabilityis identified on the V-f plane, as shown in Fig. 5.
It is interesting to note that for this 11 kW motor, the small-signal instability is predicted only by the improved small-signal model, and not by the standard model. Whereas, in
−100 −80 −60 −40 −20 0−500
0
500
Realdpartdofdeigendvalued(sec−1)
Imag
inar
ydpa
rtof
deig
endv
alue
d(ra
d/se
c)
−25 −20 −15 −10 −5 0 5 10−150
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150
Realdpartdofdeigendvalued(sec−1)
Imag
inar
ydpa
rtof
deig
endv
alue
d(ra
d/se
c)
5Hz
5Hz
50Hz
50Hz
Fig. 1. Locus of the eigen values of a 11-kW inverter-fed induction motorunder no-load for a speed range of 5Hz to 50Hz. V/f = 0.89 pu, dc bus voltageVdc = 600V, switching frequency fsw = 5kHz, dead-time td = 0.
−300 −250 −200 −150 −100 −50 0−500
0
500
Real part of eigen value (sec−1)
Imag
inar
y pa
rtof
eig
en v
alue
(ra
d/se
c)
−15 −10 −5 0 5 10−150
−100
−50
0
50
100
150
Real part of eigen value (sec−1)
Imag
inar
y pa
rtof
eig
en v
alue
(ra
d/se
c)
5Hz
5Hz
50Hz
50Hz
Fig. 2. Locus of the eigen values of a 11-kW inverter-fed induction motorunder no-load for a speed range of 5Hz to 50Hz, as predicted by the standardmodel [5]. V/f = 0.89 pu, dc bus voltage Vdc = 600V, switching frequencyfsw = 5kHz, dead-time td = 3µs.
the stability analysis of a 100 kW induction motor presentedin [9], the standard model predicts a wider range of instabilitycompared to the improved model.
The analytically predicted region of instability of the 11kWmotor drive is verified through time-domain simulations in thefollowing section.
III. SIMULATION RESULTS
Simulations are performed considering two models.
−15 −10 −5 0 5 10−150
−100
−50
0
50
100
150
Real part of eigen value (sec−1)
Imag
inar
y pa
rtof
eig
en v
alue
(ra
d/se
c)
−200 −150 −100 −50 0 50−500
0
500
Real part of eigen value (sec−1)
Imag
inar
y pa
rtof
eig
en v
alue
(ra
d/se
c)
5Hz
5Hz
50Hz
50Hz
10Hz
10Hz
30Hz
30Hz
Fig. 3. Locus of the eigen values of a 11-kW inverter-fed induction motorunder no-load for a speed range of 5Hz to 50Hz, as predicted by the improvedmodel [8]. V/f = 0.89 pu, dc bus voltage Vdc = 600V, switching frequencyfsw = 5kHz, dead-time td = 3µs.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
StatorIfrequencyI(inIp.u)
Idea
lISta
torI
volt
ageI
(inI
p.u)
A(0.4,I0.356)
B(0.7,I0.623)
x
x
Fig. 4. No unstable operating points are observed on the V-f plane for the11-kW induction motor drive under no-load, as predicted by the standardsmall-signal model [5]. Vdc = 600V; fsw = 5kHz; td = 3 µs. The stabilityanalysis is limited to frequencies between 0.1 pu and 1 pu and for V/f ratiosbetween 0.5 pu and 1 pu. The frequency base is 50Hz, and the line to linevoltage base is 415V.
a) Simulation model-1: The standard dynamic equationsof the induction motor in the synchronous dq-reference frameare considered [11]. The ideal sinusoidal inverter outputvoltages are modified by adding the switching-cycle-averagedsquare-wave error voltages on account of inverter dead-time[7], [12]. The resultant inverter output voltages are transformedto voltages in the synchronous dq reference frame [12] and fedto the standard induction motor dynamic model.
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
StatorIfrequencyI(inIp.u)
Idea
lISta
torI
volt
ageI
(inI
p.u)
unstableIoperatingIpoints
A(0.4,I0.356)
B(0.7,I0.623)
x
x
Fig. 5. Stable and unstable equilibrium points on the V-f plane for a 11-kWinduction motor drive under no-load, as predicted by the improved small-signal model [8]. Vdc = 600V; fsw = 5kHz; td = 3 µs The stability analysisis limited to frequencies between 0.1 pu and 1 pu and for V/f ratios between0.5 pu and 1 pu. The frequency base is 50Hz, and the line to line voltagebase is 415V.
b) Simulation model-2: The modified stator voltageequations (equations (1) and (2)) that are obtained consider-ing only the fundamental component of the dead-time errorvoltage are considered along with the standard rotor andmechanical dynamic equations of the induction motor.
The parameters of the induction motor considered are givenin Table I. An inverter dc bus voltage of 600V, switchingfrequency of 5kHz, and a dead-time of 3 µs are considered.
Using model-1, the simulated waveforms at operating pointA (shown in Fig. 5), corresponding to a V/f ratio of 0.89 puand a fundamental frequency of 20Hz, are shown in Fig. 6.The simulated waveforms using model-2 are shown in Fig. 7.Sub-harmonic oscillations of about 10 Hz are observed inthe q-axis and d-axis currents simulated using both model-1and model-2, indicating the operating point A to be unstable.These q-axis and d-axis currents are expected to be constantat steady-state for a stable operating point. The amplitude ofsub-harmonic oscillation observed using model-2 (see Fig. 7)is slightly higher than that observed with model 1 (see Fig. 6).Either of these models could be used to simulate the oscillatorybehaviour of the drive.
The simulated R-phase current at the unstable operatingpoint A is shown in Fig. 8 for comparison with experimentalwaveforms presented in the subsequent section.
Simulations are repeated at various operating points on theV-f plane, and the set of unstable operating points are identifiedas shown in Fig. 9. This region where oscillatory behaviour ofthe motor is observed, is in close coherence with the regionpredicted by the improved model in Fig. 5.
IV. EXPERIMENTAL RESULTS
The experimental setup consists of a 3-phase, 415V, 11-kW, squirrel cage induction motor, fed from a two-level IGBT
9 9.1 9.2 9.3 9.4 9.5−10
−5
0
5
10
15
Time (s)
Sta
tor
q−
axis
cu
rren
ti q
s (A
)
(a)
9 9.1 9.2 9.3 9.4 9.5−10
0
10
20
30
40
Time (s)
Sta
tor
d−
ax
is c
urr
en
ti d
s (A
)
(b)
Fig. 6. Sub-harmonic oscillations observed in the simulated (a) q-axis (b)d-axis motor currents at the unstable equilibrium point A (shown in Fig. 5)using simulation model-1.
9 9.1 9.2 9.3 9.4 9.5−10
−5
0
5
10
15
Time (s)
Sta
tor
q−
axis
curr
ent
i qs (
A)
(a)
9 9.1 9.2 9.3 9.4 9.5−10
0
10
20
30
40
Time (s)
Sta
tor
d−
ax
is c
urr
en
ti d
s (A
)
(b)
Fig. 7. Sub-harmonic oscillations observed in the simulated (a) q-axis (b)d-axis motor currents at the unstable equilibrium point A (shown in Fig. 5)using simulation model-2.
based voltage source inverter. The voltage source inverter isswitched at a frequency of 5kHz, and the inverter dead-timeis 3 µs. The dc bus is about 600V, and is supplied froma controlled voltage source. The V/f control is implementedin a TMS320F28035 based controller, and the gating pulsesgenerated are fed to the inverter switches through the gate-drive circuitry.
The measured R-phase stator current (trace 1), Y-phasestator current (trace 2) and dc bus voltage (trace 3), at
9 9.1 9.2 9.3 9.4 9.5−20
0
20
40
Time (s)
Sta
tor
R−
phas
e cu
rren
t i R
(A
)
Fig. 8. Sub-harmonic oscillations in the simulated R-phase stator current atunstable operating point A (shown in Fig.5) using simulation model-1.
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Statorvfrequencyv(invp.u)
Idea
lvSta
torv
volt
agev
(inv
p.u)
unstablevoperatingvpoints
Fig. 9. Stable and unstable equilibrium points on the V-f plane for a 11-kW induction motor drive under no-load, as observed from the time-domainsimulation model (model-1). Vdc = 600V; fsw = 5kHz; td = 3 µs.
operating point A (see Fig. 5) are shown in Fig. 10. Themeasurement with sine-triangle PWM scheme (SPWM) areshown in Fig. 10(a), and the measurements with conventionalspace vector modulation (CSVPWM) are shown in Fig. 10(b).Sub-harmonic oscillations are observed in the current and DCbus voltage waveforms in Fig. 10, indicating operating point Ato be unstable. It is to be noted that the controlled DC voltagesource is unable to regulate and maintain the dc bus voltage at600V when these oscillations occur. The nature of oscillatorywaveforms with both SPWM and CSVPWM are quite similar.
The measured stator currents and DC bus voltage at operat-ing point B (see Fig. 5) are shown in Fig. 11. The waveforms inFig. 11(a) and Fig. 11(b) correspond to SPWM and CSVPWMschemes, respectively. No sub-harmonic oscillations are ob-served in the measured currents and DC bus voltage, indicatingthe operating point B to be stable.
Considering the SPWM scheme, the set of unstable operat-ing points are identified experimentally, and are indicated onthe V-f plane in Fig. 12(a). Similarly, the region of oscillatorybehaviour with CSVPWM is shown in Fig. 12(b). The regionof oscillatory behaviour remains nearly the same with SPWMand CSVPWM as seen in Fig. 12(a) and Fig.12(b), respec-tively. This is expected, since with both switching schemes,
(a)
(b)
Fig. 10. Measured stator currents and DC bus voltage at unstable operatingpoint A (see Fig. 5). (a) Sine-triangle PWM (b) Conventional Space VectorPWM. Trace 1: R-phase stator current (10A/div), Trace 2: Y-phase statorcurrent (10A/div), Trace 3: DC bus voltage (25V/div), Time scale: 25ms/div.
the dead-time error voltage remains the same theoretically(neglecting any effect of switching ripple on dead-time effect).
The experimentally determined region of oscillatory be-haviour (Fig. 12(a) and Fig. 12(b)) match reasonably wellwith the simulated (Fig. 9) and analytically predicted (Fig. 5)regions. Any deviations between the experimental and sim-ulated results could be on account of mismatch in motorparameters, impact of friction damping in the experimentalsetup, and possible impact of unaccounted effects such asimpact of device switching characteristics and current rippleon the effective dead-time error voltage.
V. CONCLUSION
Oscillatory behaviour has been observed in a 11-kW open-loop induction motor drive, on account of the inverter dead-time. Considering no-load, the region on the V-f plane exhibit-ing oscillatory behaviour has been determined analytically andis verified through simulations and experiments. The analyt-ical predictions are confirmed to be of reasonable accuracy.Further, the region of oscillatory behaviour has been shown tobe almost the same with sine-triangle and conventional spacevector modulation schemes.
Identifying the region of oscillatory behaviour could poten-tially help in handling the problem. Dead-time compensation
(a)
(b)
Fig. 11. Measured stator currents and DC bus voltage at stable operatingpoint B (see Fig. 5). (a) Sine-triangle PWM (b) Conventional Space VectorPWM. Trace 1: R-phase stator current (10A/div), Trace 2: Y-phase statorcurrent (10A/div), Trace 3: DC bus voltage (25V/div), Time scale: 10ms/div.
schemes [7] and oscillation mitigation schemes [10] could mit-igate these oscillations. Various mitigation schemes/algorithmsto solve the problem are currently being explored by theauthors.
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0 0.2 0.4 0.6 0.8 10
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Stator)frequency)(in)p.u)
Idea
l)Sta
tor)
volt
age)
(in)
p.u)
unstable)operating)points
Linear)modulation)limit
(a)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
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0.5
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1
Statorvfrequencyv(invp.u)
Idea
lvSta
torv
volt
agev
(inv
p.u)
unstablevoperatingvpoints
(b)
Fig. 12. Experimentally observed set of stable and unstable equilibriumpoints on the V-f plane for a 11kW induction motor drive under no load,considering (a) Sine-triangle PWM (SPWM) (b) Conventional Space VectorPWM (CSVPWM). Vdc = 600V; fsw = 5kHz; td = 3 µs.
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