IEEE Industry Applications Society
Annual Meeting
New Orleans, Louisiana, October 5-9, 1997
Current Regulator Instabilities on Parallel Voltage Source Inverters
J. Thunes, R. Kerkman, D. Schlegel, T. Rowan
Rockwell Automation - Allen Bradley Company
6400 W. Enterprise Dr.
Mequon, WI 53092 USA
Phone (414) 242-8288 Fax (414) 242-8300 e-mail: jdthunes@meq1.ra.rockwell.com
ABSTRACT - Parallel inverters are often used to meet
system power requirements beyond the capacity of the
largest single structure. They have also been used to
reduce harmonics, reduce PWM switching frequency and
increase available output voltage or frequency. The type of
parallel structure depends on the construction of the load
motor, the most prevalent are dual three phase machines,
split-phase machines, six phase machines, and a standard
three phase machine with interphase reactors. Operation of
parallel structures presents areas for investigation
encompassing analysis, simulation, control, and design.
The paper reports on the commissioning of a 775 hp dual
winding three phase motor with parallel inverters. A simple
method of paralleling structures with carrier based PWM
current regulators (CRPWM) to independently regulate
each inverter’s current is employed. Experimental results
show a loss of current control that is similar to a random
event. The instability between the parallel inverters and the
common motor can result in large uncontrolled currents.
Simulations established the reduction in controller gain, as
the regulator enters the PWM pulse dropping or
overmodulation region, results in a loss of current control.
Experimental results show the loss of current control is the
result of an interaction between the parallel inverters
through the dual wound three phase motor. Modifications
were made to the modulator and a two phase discontinuous
controller employed; the gain characteristic of the two
phase modulator in the overmodulation region extends the
dynamic range of the motor drive.
I. INTRODUCTION
Parallel inverters have been used to address a variety of
system problems including reduction of harmonic torque
pulsation's, extending high frequency operation, reducing dc
link current harmonics, reducing PWM switching losses and
extending the range of available power structures to larger
motor sizes. Each approach provides the designer with system,
interface, commissioning, control, and motor design problems.
The solution to each of these system problems depends on the
application.
The methods used to parallel inverter structures generally fall
into three main categories: dual three phase machines [1] and
split-phase machines [2,3,4] with individual inverters, and a
standard machine with interphase reactors between inverters
[5,6]. Dual three phase implementations break apart a
standard three phase induction motor into two sets of balanced
windings. The winding break may be local (side-by-side),
axial or centrosymmetric [1]. The windings may have common
or separate neutrals. Split-phase implementations break the
phase belt into two equal halves with an angular separation of
30 degrees. Interphase reactors are used to balance the
currents between two inverter structures wired to standard
motor terminals. In this case, the inverter di/dt is limited by
the interphase reactor’s impedance when the inverter switches
are not exactly matched. An additional control can then be
used to balance the current sharing between the two structures.
In the dual three phase and split-phase motors, the windings
are separate for each inverter and depending on control
implementation may be neutral connected (if available).
Major problems encountered when paralleling inverters are
current form factor, current imbalance, and instability due to
the interaction of the inverters and circulating currents through
the motor. Zhao and Lipo [2] proposed a space vector PWM
control for a 30 degree phase shifted dual three phase motor
drive system to reduce the current distortion observed by
Gopakumar, Ranganathan, and Bhat [3]. Current imbalance and
inverter instability were addressed by Scott, Jackson, Stubis,
and Howard [5] through a complex feedback control to
minimize the effects of dead time, differences between master
and slave on-state voltages and recovery diode characteristics,
and storage time errors. Gopakumar, Ranganathan, and Bhat
[4] proposed a hysteretic current regulated vector controller
for a low power split phase machine. However, the motor
drive was configured as a three phase inverter drive, ignoring
operation in the dual winding configuration. Ogasawara,
Takagaki, Akagi, and Nabae [6] proposed a current controller
for an interphase reactor configuration. The control was
comprised of a basic switch selection component with
additional structures to control current ripple and current
amplitude.
07803-4070-1/97/$10.00 (c) 1997 IEEE
In this paper, a dual (physically separated) winding three
phase motor is used with parallel inverters on a common dc
bus as demonstrated in Fig. 1. Each inverter has its own
carrier based PWM current regulator with a common command
provided by the master inverter. Advantages of this
implementation include: independent current regulators,
incorporating low frequency references, and low cost
implementation because of the absence of a complex
coordinated control of the master and slave inverter switches.
A benefit of this implementation is the ability to parallel more
than two inverters without changing the structure of the control.
The complete drive includes a Field Oriented Controller
(FOC) with field-weakening and model reference adaptive
control (MRAC) for on-line adaptation resident in the master
inverter [7].
However, with a sufficiently high PWM carrier frequency, the
di/dt is acceptably limited. Using the PWM reference from the
master for both current regulators further reduces the
occurrences of higher di/dt.
The disturbance rejection
between the two current regulators tend to result in similar
switching patterns in both inverters.
The commissioning, FOC, and MRAC are implemented only in
the master drive as shown in Fig. 2. Both drives maintain their
own system level communication, speed feedback (ωr) and
PWM modulator.
V_ffwd
V_ffwd
Iqs*
Ids*
Iqs*
ωr
Ids*
Convertor
&
Inverter
Current
Regulator
Induction
Motor
Convertor
&
Inverter
PWM
ωr
II. DRIVE SYSTEM AND MODELS
Master
Field
Oriented
Control
A. System Configuration
The master and slave inverters are connected to a common ac
to dc converter (Fig. 1). This limits any dc bus voltage
unbalance between the inverters. The master drive provides
the control of the motor torque, speed, current, and adaptation.
The slave drive(s) employs a carrier based current regulator
that receives the stationary current commands from the master.
System level control information (enables and faults) is
coordinated between the two drives, but all of the motor
control is from the master drive.
In this case, each inverter regulates its own current relying on
the current regulator to reject the disturbance presented by the
other inverter(s). Intervals exist where the voltages at the
motor are not completely matched, resulting in a larger di/dt.
INDUCTION
MOTOR
MASTER
SLAVE
GATE DRIVER /
INVERTER
COMMOM DC
BUS CONVERTER
GATE DRIVER /
INVERTER
ANALOG
CURRENT
REGULATOR
3 PHASE AC LINE
ANALOG
CURRENT
REGULATOR
MASTER
CP & VP
CONTROL
SLAVE
CP & VP
COMMUNICATION
PLC
COMMUNICATION
PLC
COMMUNICATION
Lσ
3
2
Slave
ωr
rr
Vds
Ks
Low
Pass
Filter
MRAC
Vqs
Fig. 2. Parallel Inverter Controller Implementation.
B. Motor Model
The motor model proposed by Lipo [8] in Fig. 3, is used to
analyze the motor's operation. Separate stator windings are
implemented for each of the parallel drives. Although the
stator windings are physically separated, they are modeled as
connected together through the stator leakage inductance at the
rotor circuit. The motors' magnetic coupling will result in a
disturbance between the two current regulators depending on
each inverters’ switching pattern at any given instant. The
motor used in Fig. 1 is modeled with a split winding designed
for 0 degrees of phase shift between the inverters. This
simplifies the system by allowing the same control signals to
be used for each of the parallel inverter's. The Voltage
Equations for the motor model [8] are:
vqs1 = rs1iqs1 + dλ qs1/dt
vds1 = rs1ids1 + dλ ds1/dt
vqs2 = rs2iqs2 + dλ qs2/dt
(1)
vds2 = rs2ids2 + dλ ds2/dt
vqr = r'ri'qr - ω rλ 'dr + dλ 'qr/dt
vdr = r'ri'dr + ω rλ 'qr + dλ 'dr/dt
PLC
Current
Regulator
PWM
Fig. 1. Parallel Inverter System Diagram.
07803-4070-1/97/$10.00 (c) 1997 IEEE
=0
=0
III. INSTABILITY INVESTIGATION: TEST RESULTS
ωλ
Tests were run on the following motors; 775 hp (575 vac, 8
pole), 1250 hp (575 vac, 6 pole), 1000 hp (460 vac, 4 pole),
200 hp (400 vac, 4 pole), and 20 hp (400 vac, 4 pole). All
motors were parallel stator lap wound, designed for a
minimum phase shift between the master and slave coil groups.
Testing has verified a minimum phase shift in all of the above
cases.
-+
ωλ
+
Fig. 3. d-q Axis Motor Model with Parallel Stator Windings:
top) q-axis bottom) d-axis.
C. Modulation Model
Several different modulation schemes were considered for this
system including sinusoidal, sinusoidal with third harmonic,
space vector and two phase [9]. Each modulation scheme has
its own characteristic gain (Fig. 4) and harmonic spectrum.
The gain of the modulator is determined by its operating point
or bus utilization. Note that only the two phase modulation has
full voltage output with finite gain. Percent Bus Utilization is
defined as:
2
3
* 100
2
*
π
Vll rms *
% BusUtilization =
V
bus
Fig. 4. Inverter Gain vs. Modulation Index.
( 2)
The commissioning procedure estimates Lσ, Rs and field
current [10,11]. For the above 775 hp motor, Rs=0.7% and
Lσ=18.85% of base motor impedance, and field
current=34.7% of rated motor current.
The MRAC
maintained field orientation throughout the speed, load, and
temperature range, while in the linear PWM range [11].
Operation in the overmodulation range, however, produced
instabilities with a loss of current control. No loss of control
was ever encountered in the linear PWM region (loaded or
unloaded). At no load, while operating in the overmodulation
region, loss of control would ALWAYS occur, but the onset
was indeterminate. The instability and its' frequency of
occurrence were operating point dependent. The frequency of
occurrence increased as operation moved further into the pulse
dropping region, but the time and repeatability for the
instability varied greatly. The loss of control could not be
predicted, and appeared to be random with more occurrences
as bus utilization increased. Fig. 5 shows the master and slave
d-axis feedback currents and slave q-axis current regulator
integrator output for a typical transition into the unstable state.
Notice the gradual rise in the slave integrator until finally the
current is completely uncontrolled. Although the response
resembles a limit cycle, tests showed the control
Fig. 5. Experimental Results of a Parallel Inverter System
Transitioning into Unstable Operation.
07803-4070-1/97/$10.00 (c) 1997 IEEE
trajectory may or may not repeat. The 775 hp motor was
running at 670 rpm, no load with a PWM carrier of 2 kHz.
Initial attempts to determine the nature of the "disturbance"
leading to the current regulator instability investigated the
FOC, adaptive control, and PWM carrier frequency as well as
looking for velocity, torque and bus disturbances. Disabling
the control (fixed slip gain and no adaptation) still resulted in
loss of current control. No external disturbances were seen in
the velocity, torque (or current) commands or bus voltage
associated with the loss of control. Although similar operating
points were used, the loss of current control was indeterminate
with varying trajectories to the unstable state.
IV. INSTABILITY INVESTIGATION
The instabilities and their indeterminacy proved a difficult
problem requiring simulation and analysis. The extended run
times necessary to duplicate the instabilities make simulations
tedious and problematical. To replicate the observed
oscillations and instabilities accurate system component
models are necessary; however, complex models often prevent
insight into the cause of the observed phenomena.
A. Simulation Results
Initial studies incorporated an inverter model represented by a
sinusoidal balanced three phase voltage amplifier with the
motor model shown in Fig. 3 [8]. A velocity regulator
controlled the speed of the machine. Simulations were
performed with motor parameters provided by the
manufacturer, and at operating points corresponding to test
results exhibiting the instabilities. With this model, the
simulations failed to predict the observed instabilities.
Machine parameters were altered, including the mutual
leakage inductance; however, the simulations still failed to
duplicate the observed instabilities.
Because experimental results revealed the instability to occur
only at high voltage utilization, more detailed inverter models
were investigated. First, a reduced order model, which
incorporates the nonlinear gain characteristics of the inverter
but retains the sinusoidal excitation [12], was interfaced to the
motor model. Finally, a detailed inverter model, which
included power device switching, dead time compensation,
and device rise and fall time was developed. Both models
successfully reproduced the experimental results with
transitions to unstable operation.
The first simulations to show the instabilities employed a 5 hp
induction machine. Fig. 6 shows the a-phase modulating signal;
the master q-axis current; and the q-axis integrator of the
master current regulator. The time origin has been adjusted to
remove the first 1.5 seconds of simulation results. Parameters
for the system are provided in the appendix. A synchronous
current regulator without feed forward voltage provided a high
bandwidth current regulated inverter. A sine wave PWM
modulator with a PWM carrier frequency of 4 kHz and a 5 µs
dead time was employed. The commanded speed was 1530
rpm, the machine unloaded, and a bus utilization of 95%.
The outbreak of the instability becomes evident at
approximately 1.7 seconds (Fig. 6). Although the instability is
severe, the command voltage does not saturate. The master
current (iqs1) exhibits a more complex characteristic than the
q-axis integrator (q-int1), a consequence of the parallel
inverters and inter-winding magnetic coupling in the motor.
Initially the period of oscillation appears to be three times the
fundamental period, however, observation over an extended
time shows this not to be true. Fig. 7 shows a phase plane plot
of the q-axis and d-axis integrators of the master inverter. This
represents six seconds of simulation time and clearly shows an
unpredictability in the response of the system. Although not
periodic, Fig. 7 indicates a bounded response.
10
Vas *
0
-10
1.5
1 PU =
2.5V
2
5
iqs1
0
-5
1.5
1 PU =
5V
2
10
q-int1
0
-10
1.5
2.5
2.5
1 PU =
2.5V
2
Time (sec)
2.5
Fig. 6. Simulation Results of a Parallel Inverter System
Transitioning into Unstable Operation.
Results of the more detailed models showed the most
important contributors to the observed instabilities were the
inclusion of the nonlinear effects of the inverter gain and signal
level saturation. Adding rise and fall times of the devices,
dead time, and switching delays between the master and slave
inverters did not improve the model of the system sufficiently
to predict the observed instability.
07803-4070-1/97/$10.00 (c) 1997 IEEE
population and r represents the natural rate of growth of the
population. For the parallel inverter system, p(kT) is
analogous to the modulating signal from the current regulator.
The gain of the current regulator decreases nonlinearly as the
modulating signal exceeds Vbus/2 [12] and the modulating
gain curve (Fig. 4) is analogous to r in (3). The parallel
inverter current regulator gain is determined by the degree of
over modulation and the type of modulator used (sine, third
harmonic injection, space vector, or two phase).
8
6
4
2
p(k+1) = p(k) + rp(k)(1-p(k))
dint1 0
1 PU=
2.5 V -2
Fig. 8 displays the response p(kT) for three rates of growth.
The top trace shows a well-damped system, the second and
third demonstrating the existence of bifurcation points; a
doubling and quadrupling of the period of oscillation. This
doubling of the oscillation period is evident in Fig. 9. All three
are equilibrium attractors, and produce predictable
trajectories.
-4
-6
-8
-8
(3)
-6
-4
-2
0
2
qint1 1 PU= 2.5 V
4
6
8
Fig. 7. Phase Plane Simulation Results for the Master
d-q axis Integrators
B. Chaos Theory:
Oscillations
An Explanation for
Indeterminate
The indeterminacy of the instability, and the difficulty
experienced in predicting the instabilities by simulation made
conclusions as to the cause difficult. Review of the literature
together with the results of Fig. 7 suggests one possible
explanation for the instabilities is chaos theory. Chaos theory
provides an alternative to randomness as a source for
unpredictability. Furthermore, all the modulators except the
discontinuous two phase showed characteristics similar to
systems with behavior identified to be chaotic.
In higher order systems, like power inverters, there exist a
locus of points leading to a given attractor and there exists
more than one attractor; thus the system initial conditions will
establish its trajectory, whether a periodic attractor or a
strange attractor. For example, a strange attractor (chaotic
system) occurs for (3) regardless of the initial condition when
r=3.0. Fig. 10 depicts this for p(0) equal to 0.1 and 0.106. The
traces show a response that is not repeated and totally
unpredictable, characteristic of a chaotic system.
2
Per
Unit
r=1.9,p(0)=0.1
1
p(KT )
0
0
20
40
60
80
100
2
Recently, a number of technical papers have appeared
addressing unpredictable dynamic responses resulting from
nonlinearities within power electronic systems [13-17]. Under
certain conditions, a nonlinear system may transition from well
behaved to total unpredictability. The first condition - a
periodic attractor - will result in identical steady state
trajectories regardless of the initial condition. The second
condition - a strange or vague attractor - results in trajectories
that depend on the initial condition of the system. A simple
first order example will clarify the terms and lead to an
explanation for the cause of the parallel inverter instabilities.
Per
Unit
r=2.4,p(0)=0.1
1
p(KT )
0
0
20
2
Per
Unit
40
60
80
100
r=2.55,p(0)=0.1
1
p(KT )
0
0
20
40
60
Time (seconds)
80
Fig. 8. Simulation of Three Growth Rates.
Equation (3) represents a nonlinear first order model of the
earth’s population [18]. In the case of [18], p(kT) represents
the ratio of the actual population to the maximum sustainable
07803-4070-1/97/$10.00 (c) 1997 IEEE
100
been observed
in simulations
or testing when using the two
phase modulator.
1.2
WI
m
V. STABILIZING
08
SYSTEM:
0.6
Ixm 0,4
INVERTER
RESULTS
Studies were conducted to develop a modulation strategy to
improve the dynamic characteristics of the parallel voltage
0.2
source
inverters.
comparison
PWM
Results Demonstrating
the Oscillation
Doubling
of
Many
method
command.
triangle,
Fig. 9, Simulation
algorithms
employ
a
triangle
to convert the voltage command
When the voltage
the overmodulation
command
into a
exceeds the
region is &ntered. As long as the
voltage command is less than or equal to the triangle, the
inverter gain is constant (Fig. 4). For sine wave modulation,
Period.
the modulation
index (or bus utilization)
is approximately
78V0 when
the nonlinear
region
is entered.
Several
modulation
strategies - space vector, two phase, and third
harmonic - extend the linear gain of the inverter to 91?4. of
the six-step limit.
1.5
Per
Unit
THE PARALLEL
EXPERIMENTAL
I
P(KT)
05
0
0
20
40
Time
60
80
100
Schemer’s
(seconds)
Per
Unit
1
modulation
voltage
0.5
P(KT)
o
0
20
Fig. 10. Simulation
C.
was chosen for the
40
Time
60
(seconds)
80
Results for an Unpredictable
System.
100
Chaotic
Selection of a Modulator
simulation
modulator,
studies
indicating
the importance
of
served as motivation
for an investigation
alternative modulators for the parallel inverter system.
harmonic
injection
and space vector modulators
investigated
and simulation
results demonstrated
modulator produced similar results to those of the sine
modulator.
strategy
[9, 12].
that has finite
The two
gain
for
phase modulator
fill
[20],
inverter
Fig.
switching
sequence
for
switching
sequence
is created
the two
phase modulation.
by
summing
the
the
of
Third
were
each
wave
As Fig. 4 shows, all of these modulators exhibit
similar
to that of the sine wave
modulator.
exhibits
a unique characteristic
in
the over modulation region; the gain does not decrease to
zero with increasing modulation index [9]. Consequently, the
dynamic interaction of the modulator and parallel inverter
system is quite different. Although a prediction of the global
stability appears impossible, simulations indicated two phase
modulation
provided
superior
to the other modulators.
performance
Voltage
Reference
Synchronized
2 Phase
carrier
Enable
when compared
As yet, chaotic behavior
has not
Fig. 11. Two Phase Modulator
07803-4070-1/97/$10.00 (c) 1997 IEEE
The
selected
characteristics
The two phase modulator
11,
contains a voltage reference set just above the PWM triangle,
the voltage required to enter the overmodulation
region . This
reference voltage is summed with the most positive (Vmax)
and most negative (Vmin) voltage commands and has a
ripple fi-equency of three times the fundamental. The Vmax
and inverted Vmin voltages are compared to obtain the
maximum
value (y) which in turn selects the desired
The instabilities
of the parallel inverter system with sine
wave modulation,
their similarity
to chaotic systems, and
gain
[19] two phase modulation
parallel drives due to the instabilities observed or predicted
with other modulators.
Two phase modulation
is the only
!.U
Block Diagram.
..-
reference voltage with the voltage commands just before the
The microcontroller
can enable or
PWM comparators.
disable
the two phase modulation
by controlling
2MeEMie
1FU=5V
a digital
switch that sums either the two phase voltage reference (y ‘)
5
or zero with the PWM voltage commands as shown in
Fig. 12. Due to limitations with the analog implementation,
the operation of the two phase modulation
was limited to
above 65°/0 bus utilization.
“A!’RR%
Vd@@nl
4
1IU=25V
1%3
‘14”k
U-it
clnErtF&k
2
1FU=25V
1
‘QAxis
0
11TJ=05V
-1 J
I
i
oa51
I
I
1
I
,.
1
I
1.522533.544.5555
6
Tm?(Secm$)
v= *
Fig. 13. Experimental
Results Demonstrating
a Transition
into Two Phase Operation
E:
o——=f——
I
6T---FF~
Ilr
Fig. 12. Two Phase Control Implementation
Block Diagram.
s V.h”e
—,%,
1 Pu = 650V
5
4
—V,
Crnd
1PU=25V
3
Simulations and experimental testing (on all of the motors)
has shown that using two phase modulation maintains current
control well into the pulse dropping region. Experimental
results based on a 1250 hp, 575 vac induction
dual stator windings
utilization
motor
show good current regulation
with
with bus
of 90% and higher. Lab testing has demonstrated
good regulator operation above 957. bus utilization
Examples of the two phase modulator
current decreases showing
operation
in the field
motor. Fig.
As the bus
two phase
reached, the
weakening
region. The two phase modulator’s dynamic performance is
demonstrated in Fig. 14 by its ability to regulate current with
a 1L1.TO/O transient in the bus voltage. As the bus utilization
changes, a smooth transition into and out of two phase
modulation
0
1PU=025V
-1
14
145
15
155
1,6
4.65
Time
17
175
18
185
19
(S=onds)
levels.
operation are shown in
Figures 13 and 14, on a 500 hp, 460 vat, 4 pole
13 shows an acceleration from O to 2000 rpm.
into
utilization
reaches 650/’, a transition
modulation occurs. As the commanded speed is
I
1
Fig. 14. Experimental Results Demonstrating Two Phase
Modulator Operating with a Bus Disturbance.
nonlinearities,
including
the modulator,
inverter and machine,
result in a chaotic transition
into instability
that is not
predictable or repeatable. Through analysis and simulation, a
solution is proposed. Experimental
results demonstrate that
the two phase modulator
tremendous
has superior
performance
and a
impact on the of the drive system operation.
ACKNOWLEDGMENT
is seen.
The authors wish to thank Kevin
Stachowiak
assistance in this development and implementation.
V1. CONCLUSION
The commissioning,
operation, and simulation
of parallel
inverter motor drive systems are presented. A review of the
major system configurations is included. A system employing
independent current regulators is proposed as a solution to
the operation
of a high performance
FOC for parallel
inverters. Instabilities observed match the characteristics of a
system.
The
system’s
operating
point
and
chaotic
07803-4070-1/97/$10.00 (c) 1997 IEEE
for
his
APPENDIX
5 hp, 460 vac Induction
Induction
0.297 H
Lli = L’l, = 0.012 H
J = 0.0326 kg-m’
Ll~ = 0< 50% of Lll
[11] R. J. Kerkman,
Conference
Record,
Applications,
of
Machine Using Vector Space
Transactions
on
Industry
Vol. 31, No. 5, September/October
1995, pp.
742-749.
[3] K. Gopakumar, V. T. Ranganathan, and S. R. Bhat,
“Split-Phase Induction Motor Operation from PWM Voltage
Source
Inverter”,
IEEE
Transactions
on
Industry
Applications,
Vol. 29, No. 5, September/October
1993, pp.
[4]
K. Gopakumar,
Control
Windings”,
V. T. Ranganathan,
of Induction
Motor
IEEE IAS Annual Meeting,
and S. R. Bhat,
Denver, CO, October
Drives, September 18-20, 1991, pp. 31-37.
[6] S. Ogasawara, J. Takagaki, H. Akagi, and A. Nabae, “A
Novel Control Scheme of Duplex Current-Controlled
PWM
Inserters”, IEEE IAS Annual Meeting, Atlanta, GA, Oct. 18-
T. M. Rowan, and D. Leggate, “Indirect
Field-Oriented
Control of an Induction Motor in the Field.
on Industry
Weakening
Region
“, IEEE Transactions
Applications, Vol. 28, No. 4, July/August 1992, pp. 850-857.
T. A.
Machines”,
Lipo,
“A
d-q Model
International
for
Conference
Six Phase Induction
on Electric
Machines,
Athens, Greece, September 15-17, 1980, pp. 860-867.
[9] A. Hava,
Performance
R. J. Kerkman,
Generalized
IEEE - APEC’97,
Atlanta,
1991, pp. 720-
Determination
Vol. 32, No. 3, May/June
D. Leggate,
Voltage
of PWM
Region”,
of
the
[13]
1996, pp. 577-584.
B. Seibel, and T. Rowan,
Source’’ Inserters
in the Pulse
IEEE - Transactions
on Industrial
Vol. 43, No. 1, February 1996, pp. 132-141.
J R. Wood,
Electronics”,
“Chaos:
A Real Phenomenon
IEEE APEC’89,
Baltimore,
in Power
MD, March
13-17,
1989, pp. 115-124.
[14] I. Nagay, L. Matakas, E. Masada, “Application
of the
Theory of Chaos in PWM Technique of Induction Motors”,
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