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:
[email protected]
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.
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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,
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Determination
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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”, IPEC-Yokohama, April 3-7, 1995, Vol. 1, pp. 58-63. [15] W. C. Y. Chan and C. K. Tse, “Studies PESC’96,
DC/DC
of Routes to-
Converters”,
IEEE -
Braveno, Italy, June 23-27, 1996, pp. 784-795.
[16] Tamotsu Ninomiya, and Akira Takeuchi, “Analysis
and T. A. Lipo,
Discontinuous GA, February
PWM
“A
Phenomena
Chaotic Oscillation IECON’91, oct. 28-
of
in Resonant NOV. 1, 1991,
[17] Seiichi Ozaki, “Nonlinear Effects and Chaos in a Inverter-Induction Machine Drive System”, Transactions of the IEEJ, Vol. 115-D, May 1995, pp. 585-590. Pierre
Liberty,
Lemieux,
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Complexity,
and Anarchy”,
Vol. 7, No. 3, March 1994, pp. 21-29.
[19] J. Schemer,
“Bezugsspannung
Zurunrichtersteuerung”,
in ETZ-b, Bd. 27, 1975, pp. 151-152. “Pulse width control of a 3-phase [20] M. Depenbrock, inverter with nonsinusoidal phase voltages”, IEEE-ISPC Conf. Rec., 1977, pp. 399-403.
High
Algorithm”,
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[18]
23, 1987, pp. 330-337.
[8]
of an
Industry
J. D. Thunes, T. M. Rowan, and D. W.
Frequency-Based
[12] R. J. Kerkman,
Beat
[5] J. B. Scott, R. F. Jackson, P. R. Stubis, and I. T. Howard, “Parallel Inverters for AC Motor Speed Control”, Conference
[7] R. J. Kerkman,
“A
With Split Phase Stator
2-6, 1994, pp. 569-574.
on Industrial
on
No. 4, July/August
Chaos for Current-Programmed
927-932.
“Vector
Transactions
“Operation Dropping Electronics,
[2] Y. Zhao and T. A. Lipo, “Space Vector PWM Control IEEE
“A
Applications,
UK, 1993, pp. 39-44.
Decomposition”,
for Indirect Field Orientation IEEE
Vol. IA-27,
and D. Leggate,
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r,= 0.855 Q
European
T. M.
Simple On-Line
Motor Parameters
L.=
r, = 2.238 Q
[10]
A
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07803-4070-1/97/$10.00 (c) 1997 IEEE
