Dec 17, 1983 - while the motor is a modified standard three-phase squirrel-cage motor. .... squirrel-cage induction motor excited by a full-bridge voltage.
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Six-Phase Voltage Source Inverter Driven Induction Motor MOHAMED A. ABBAS,
MEMBER, IEEE,
ROLAND CHRISTEN,
Abstract-A six-phase six-step voltage-fed induction motor is presented. The inverter is a transistorized six-step voltage source inverter, while the motor is a modified standard three-phase squirrel-cage motor. The stator is rewound with two three-phase winding sets displaced from each other by 30 electrical degrees. A model for the system is developed to simulate the drive and predict its performance. The simulation results for steady-state conditions and experimental measurements show very good correlation. It is shown that this winding configuration results in the elimination of all air-gap flux time harmonics of the order (6v ± 1, v = Consequently, all rotor copper losses produced by these 13,5,* harmonics as well as all torque harmonics of the order (6v, v = 1,3,5, ) are eliminated. A comparison between-the measured instantaneous torque of both three-phase and six-phase six-step voltage-fed induction machines shows the advantage of the six-phase system over the three-phase system in eliminating the sixth harmonic dominant torque ripple.
I. INTRODUCTION DURING THE PAST decade, interest in multiphase systems with more than three phases has increased for various reasons. For power system applications, the high phase order (HPO) transmission system has been investigated as a means of increasing the capacity of overhead electric power transmission rights of way [1], [2]. For motor drive applications, multiphase system could potentially meet the demand for high-power electric drive systems which are both rugged and energy-efficient. Key advantages of multiphase induction motor drive systems over conventional three-phase systems are summarized as follows. 1) Reduction of the required inverter phase current permits the use of a single power device for each inverter switch instead of a group of devices connected in parallel. Problems of static and dynamic current sharing among parallel devices, such as bipolar transistors, are therby eliminated in large drive systems. 2) The sixth harmonic pulsating torque associated wih conventional three-phase six-step voltage-fed induction motor drives is eliminated by the appropriate choice of a multiphase motor winding configuration.
Paper IPCSD 83-63, approved by the Industrial Drive's Committee of the IEEE Industry Applications Society for presentation at the 1983 Industry Applications Society Annual Meeting, Mexico City, Mexico, October 3-7. A version of this paper was first published in the Conference Record IEEE-LAS1983 Annual Meeting. Manuscript released for present publication December
17, 1983. M. A. Abbas and R. Christen are with Gould Inc., Gould Research Center, 40 Gould Center, Rolling Meadows, IL 60008. T. M. Jahns is with the General Electric Company, Research and Development Center, Building 37-325, P.O. Box 43, Schenectady, NY 12301.
AND
THOMAS M. JAHNS
3) Rotor harmonic losses are reduced from the level produced in three-phase six-step systems. 4) The total system reliability is improved by providing for continued system operation with degraded performance following loss of excitation to one of the machine stator phases. Previous multiphase investigations have been conducted to explore the merits of multiphase motor drive systems and to develop mathematical models for analyzing the performance of these systems. For the case of voltage excitation, Ward and Harer [3] investigated the performance of a five-phase ten-step voltage-fed induction motor drive system. The dominant torque ripple in this system was the tenth harmonic with amplitude of one-third of the sixth harmonic produced in similar three-phase system. However, the line current was rich in third and higher order harmonics which resulted in higher motor losses. Analog computer simulation of a six-step voltage-fed six-phase motor with two three-phase stator winding sets in [4] predicted that 30 electrical degrees displacement between the two sets eliminates the sixth harmonic pulsating torque component. Simulation results for this case predicted torque ripple amplitudes of only 20 percent of the comparable three-phase system values along with substantial reductions of the rotor current harmonic components.
The steady-state torque-speed characteristics with (n) and (n - 1)-phase sinusoidal excitation for three-, six-, and ninephase motors were presented by Klingshirn [5], [6]. The effects of the coil pitch on the harmonic content of the stator current were pointed out. A six-phase induction generator has been reported [7]. For the current excitation case, Jaschke [8] analyzed the effect of the number of stator phases on the torque pulsation in current source inverter (CSI) induction motor drive systems. Lipo [9] developed a d-q model for the six-phase induction machine including the effects of slot leakage coupling. The resulting model was simulated on an analog computer to investigate the influence of this coupling on the performance of a CSI-fed machine. Andersen and Bieniek [10], [11] described the performance of a six-phase CSI-fed induction motor and its advantages over three-phase machines fed by either single or double current source inverters. An extensive investigation of the steady-state and transient characteristics of n-phase induction motor drive systems with (n - 1)-phase excitation for both voltage and current source invetters was presented in [12]. This paper presents the steady-state characteristics of a six-
0093-9994/84/0900-1 2511$1 .00 © 1984 IEEE
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Fig. 1. Six-phase VSI induction motor drive system. A
E
B
C
Fig. 2. Six-phase stator winding configuration.
phase squirrel-cage induction motor excited by a voltage source inverter. The transistorized inverter delivers six-step voltage excitation to the motor, which is a modified industrial three-phase machine rewound with a six-phase stator winding. The stator windings consist of two three-phase winding sets displaced by 30 electrical degrees. A model is presented for the six-phase induction machine based on the generalized twophase real component tranformation. This model has been used to simulate the drive performance during steady-state operation and provide results to compare with experimental measurements. Measured instantaneous torque waveforms for both six-phase and three-phase systems are presented and compared. II. SYSTEM DESCRIPTION The system investigated in this work consists of a six-phase squirrel-cage induction motor excited by a full-bridge voltage source inverter as shown in Fig. 1. A phase-controlled rectifier provides a variable dc voltage for the dc link inverter input. At the heart of this system is an industrial-grade 20-hp squirrel-cage induction motor modified for this program. The stator of this machine has been rewound in such a way that the windings can be connected in either a three-phase or six-phase configuration. As illustrated in Fig. 2, -the six-phase configuration consists of two three-phase;sets separated by 30 electrical degrees. Note that the neutrals of these two sets are not connected. By setting the separation angle at 30 degrees, all of the air-gap flux components of orders (6v + 1, v = 1,3,5,***) contributed by the six stator phases cancel each other during balanced excitation. As a result, the air-gap flux .is inherently free of several potentially troublesome time harmonic components, such as the fifth and seventh harmonics, regardless of the harmonic contents of the stator phase excitation waveforms. Note also that if the separation angle had been set at 60 electrical degrees, the air-gap flux
D
distribution during balanced excitation would be the same as for a single three-phase winding; in;this case, motor performance would be indistinguishable for the three- and six-phase configurations. In the laboratory system, the six-phase inverter consists of 12 power transistor switches arranged in two three-phase fullbridge modules. During balanced six-phase excitation, these inverter switches are controlled to deliver standard six-step voltage waveforms to each of the three-phase sets of stator windings. The two three-phase sets of excitation waveforms are separated in time phase by 30 electrical degrees, consistent with the space separation of the stator winding sets for balanced excitation. Motor terminal volts/hertz is held nearly constant by deriving the excitation frequency from measurements of the dc link voltage. Power transistor base drive circuits and all other low power control circuits are of conventional design and require no additional discussion to be devoted to these parts of the experimental system. SYSTEM MODEL In order to gain a fuller understanding of system behavior, an analytical model of the multiphase motor-inverter system has been developed for system simulation on a digital computer. Considering the limited amount of technical literature describing the detailed operation of such systems, availability of this type of model is invaluable for investigating the effects of individual motor and excitation parameters on system performance. The following assumptions and approximations have been adopted in the process of developing this model. III.
1) The machine air gap is uniform. 2) Motor stator and rotor windings are sinusoidally distributed. 3) Magnetic saturation of the machine iron is neglected.
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4) Eddy current, friction, and windage losses are neglected. 5) Inverter switching devices are ideal (zero impedance when on, infinite impedance when off). 6) The dc link power supply is an ideal voltage source. The first four assumptions are often adopted in this type of induction machine analysis in order to neglect air-gap space harmonics and magnetic nonlinearities which complicate the use of linear analysis techniques. The last two assumptions eliminate the effects of power supply and inverter impedance from the analysis, resulting in ideal voltage excitation waveforms at the motor stator terminals. Taken together, these assumptions simplify the analysis and are consistent with the objectives of the present investigation. At the center of this system model is a set of decoupled motor equivalent circuits resulting from application of the generalized two-phase real component transformation as developed by White and Woodson [13]. As the name suggests, this transformation represents a generalization of the familiar three-to-two-phase d-q transformation to multiphase motor configurations with any number of stator phases. Equations defining this transformation are summarized in Appendix I; additional information concerning its development can be found in [12], [13]. This transformation is attractive for this application since the resulting motor equivalent circuits require only real variables rather than the complex variables required by the instantaneous symmetrical component transformation [13]. In addition, these equivalent circuits for the six-phase motor hold much in common with familiar d-q model circuits, as will be apparent shortly. The generalized two-phase real component transformation is appropriate only for symmetrical n-phase machines in which each stator phase is separated from each adjacent stator phase by 3601n electrical degrees. As a result, this transformation cannot be applied directly to the six-phase motor configuration illustrated in Fig. 2 since the six stator phases are not separated by equal 60 (360/6) electrical degree angles. Instead, the generalized two-phase transformation appropriate for a symmetrical 12-phase machine is applied since the six magnetic axes established in the 12-phase motor are identical to those of the six-phase configuration in Fig. 2. This relationship is illustrated in Fig. 3, with lower case letters identifying phase variables of the 12-phase machine, and upper-case for the sixphase configuration. The next step in applying this tranformation is to make use of the special relationships among the phase variables of the symmetrical 12-phase motor during balanced excitation, listed
A
a
F
i
D
9
Fig. 3. Winding axes for 12-phase and six-phase windings.
as follows: Va =
-Vg
Ub =
-
.a =
g
lb
h
Vh
.Vf =
=
=-ui
1'I
(The variables are defined in the Nomenclature at the end of the paper.) With these relationships, the 12-phase configuration can be reduced to a form equivalent to the six-phase configuration of Fig.. 2 by connecting each of the six pairs of stator phases sharing the same magnetic axes in antiparallel, and relabeling the combined phase variables as follows: iA = ia-ig = 2ia
Vg VB= Vb= -Vh
iB = ib - ih = 2ib
-Vd
IF= Ij -Id= 21j.
UA= Va
VF=Vj
Initial application of the generalized two-phase transformation to the symmetrical 12-phase machine result in 12 independent transformed stator (and rotor) differential equations in a form summarized in Appendix II. Connecting the stator winding pairs in antiparallel, as described earlier, has the effect of cutting the number of differential equations in half. This number can be reduced still further by noting that the neutrals of the two three-phase sets comprising the sixphase configuration of Fig. 2 are separated. This 'is iB + iD + iF= 0.
iA + iC + iE = 0
This neutral configuration has the effect of eliminating two more of the transformed stator equations. These constraints, together with the standard current and voltage constraints imposed by a squirrel-cage rotor, have the net effect of reducing the total number of transformed machine differential equations to six. These six equations, four for the remaining stator components and two for the rotor components, are given in matrix form as follows:
-Vs;
RRs+LsP
0
0
0
VS3
0
R+LsP
0
0
VS-v
0
0
Rs+Ls5P
0
0
0
VS6
0
0
0
Rs+LLsP
0
0
0
MP
wrM
0
0
Rr+LrP
crLr
MP
0
O
-corLr
Rr+LrP
-
crM
1B
MP
0
MP
is-v
ia
'
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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. IA-20, NO. 5, SEPTEMBER/OCTOBER 1984
The electromechanical balance equation in terms of the transformed machine variables becomes
Rs
Ls
Lr +
JPW r= Te-TL
WrMIs
°
M1,
5 rLr Ir
Rr
where Te = qM(i,sjicr -i.,isr)
RS
LS
Lr
M= (6\l/k/2)Lsr.
The transformed motor electrical differential equations listed can be usefully depicted by the equivalent circuits shown in Fig. 4. The first two circuits represent the torque-producing circuits in which the rotor and stator are magnetically coupled. These two circuits have exactly the same form as the equivalent circuits resulting from a standard d-q transformation of a three-phase induction motor in the stationary frame [13]. The remaining two equivalent circuits of Fig. 4 represent transformed stator variables which do not interact with the rotor. As a result, these circuits consist only of stator resistance and stator leakage inductance, similar to the zerosequence circuits associated with standard three-phase transformations. If the stator phase excitation voltages of the six-phase motor are expressed in Fourier component form and then transformed by the generalized two-phase component transformation as described earlier, some very interesting facts emerge. First, this algebraic manipulation indicates that the tranformed voltage excitation variables Vs and Vos contain only time harmonic components of the order (12v ± 1, v = 1,2, * * *) in addition to the fundamental. As a result, the voltages and currents in the upper two transformed equivalent circuits of Fig. 4 are inherently free of all time harmonic components which could contribute torque harmonics of the order (6v, v = 1,3,5,* Further, this algebraic manipulation shows that the voltages Ves and V6S exciting the remaining two equivalent circuits of Fig. 4 consist solely of time harmonics of the orders (6v ± 1, u = 1,3,5, ). Thus excitation voltage time harmonics such as the fifth and seventh are prevented from contributing to the air-gap flux and pulsating torque by the six-phase configuration but do appear as excitation to the uncoupled 'y and 6 component equivalent circuits. As a result, these voltage harmonics can contribute significantly to the stator current since the impedance of the -y and 6 component circuits is low, consisting entirely of stator resistance and leakage inductance. In particular, peak stator phase currents flowing in the six-phase motor are very sensitive to -y and 6 component current amplitudes. Careful attention during the motor design and excitation waveform selection' processes must be devoted to insuring that the current amplitudes in the 'y and 6 components circuits do not reach unacceptably high levels.
+ M
Rs
RrrI
i
Ls
vs-Y Rs
rMIsa
-
LS
=
LS-M
Lr
=
Lr_ M
La
Fig. 4. Generalized two-phase equivalent circuit for six-phase (30' displacement) induction motor.
is), the rotor currents (ijr, ir)% and the rotor angular velocity (co'). However, simulation results presented in this paper were generated by holding the rotor speed constant (i.e., infinite rotor inertia), reducing the system order to six. A fourth-order Runge-Kutta numerical integration algorithm with errordetection and adjustable time step was used in predicting system response to nonsinusoidal excitation waveforms. Since results presented in this work are for steady-state operating conditions, care was taken to allow all transient modes to disappear before steady-state waveforms were recorded. Motor parameters for the 20-hp prototype six-phase machine studied in this simulation are listed in Appendix III. These parameters have been obtained directly from machine measurements supplemented by manufactuer-supplied design data. Both no-load and locked-rotor tests were performed on the prototype motor, providing data for determination of all motor equivalent circuit parameters except Lz and Lb. These last two important stator leakage inductances were measured directly by exciting their corresponding stator component voltages Vy and Va. All parameter measurements were performed at 60 Hz with sinusoidal excitation. All machine parameters and variables have been subsequently normalized based on rated stator phase current and voltage. Following these initial measurements, the prototype sixphase induction motor was coupled to a laboratory dynamometer through an intervening rotating torque transducer and speed reduction gear. After attaching additional instrumentation, the inverter-excited motor was operated over a range of speeds and loading conditions. Data including motor terminal IV. SIMULATION AND EXPERIMENTAL RESULTS voltages, phase currents, shaft speed, and shaft torque were The system model equations presented in the preceding gathered at a variety of steady-state operating points for section were programmed on a digital computer for numerical subsequent analysis. A view of the experimental equipment simulation. As described, the system is seventh-order with during this testing is provided in Fig. 5. state variables contributed by the stator currents (ij,S, jis, yS, Computer simulation of the prototype system proceeded in
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ABBAS et al.: SIX-PHASE VOLTAGE SOURCE INVERTER DRIVEN INDUCTION MOTOR 20.00-
-
0
0
i
0.00-
0
110.00
- 20.00
114.00 NORMALIZED TIME
118.00
(rad)
(a) setup for six-phase induction motor-drive system.
parallel with the dynamometer testing, providing predictions of drive steady-state performance at operating points matching those chosen in the empirical testing. Correlation between empirical and calculated drive performance has been generally quite good over a range of test conditions, confirming the basic accuracy of the adopted machine model. Qualitative agreement between measured and predicted system variable waveshapes, such as the motor phase currents, has been uniformly good. Quantitative agreement for key performance variables, including peak stator currents and average torque, has also been good in many cases. Nevertheless, discrepancies in certain specific cases have exceeded 15 percent, likely due to a combination of experimental measurement errors and limitations imposed by the model assumptions. Several of these issues are addressed more completely in the following paragraphs in the context of a comparison of specific measured and predicted system performance results. In order to provide some insight into performance of the prototype system and analytical model, examples of measured and calculated machine data are provided first for typical test conditions at 60-Hz excitation frequency. For reference, Fig. 6 shows the six-step phase voltage excitation waveform developed by the simulation and by the actual inverter. Predicted and measured stator phase current waveforms for light load and 70-percent rated machine torque are provided, respectively, in Fig. 7 and 8. Correlation between simulated and empirical current waveshapes is very good in both cases. Quantitative agreement for the peak stator current amplitudes in the light-load case (Fig. 7) is very good, with the model predicting 21.8 A against a measured peak current of 21.0 A. However, agreement in the loaded case (Fig. 8) is not as good, with the predicted peak current of 28.8 A falling below the measured value of 33.5 A by 14 percent. Although the exact cause of this discrepancy is still under investigation, suspected contributing factors include the neglect of magnetic saturation in the motor model. Calculated waveforms for the developed electromagnetic torque for the same operating conditions as in Figs. 7 and 8 are provided in Fig. 9. As noted earlier, constant rotor speed (i.e., infinite rotor inertia) has been assumed in making these
(b) Fig. 6. Six-step voltage waveform. (a) Simulated. (b) Measured, 50 V/div, 2 ms/div.
calculations. These predicted torque waveforms show no trace of the sixth harmonic pulsating torque component which dominates in three-phase six-step inverter drive systems. Instead, the major pulsating torque component visible in the Fig. 9 torque waveforms is the twelth harmonic, produced principally by the interaction of the eleventh and thriteenth components of the rotor current with the fundamental air-gap flux. The amplitude of the twelth harmonic pulsating torque component is small, calculated to be only two percent of the average torque for the 0.7 pu load torque case. Unfortunately, mechanical vibration generated by the speed reduction gear made it impossible to record the six-phase motor torque waveforms at 60-Hz excitation frequency, thereby precluding a direct comparison between simulated and measured torque waveforms. In order to circumvent this vibration problem, a prony brake was used to allow low-noise measurements of the shaft instantaneous torque waveform under low-speed conditions. By reconnecting the motor stator windings, torque waveforms were recorded for both three-phase and six-phase operation with inverter excitation. Fig. 10 provides examples of these three-phase and six-phase torque waveforms at two different excitation frequencies, 4.8 Hz and 10 Hz. At both
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'a
E
E
Ul
-
Z
Lu
0.00
z wr
-
cc,
C).l
-40.00 _ 110.00
114.00
118.00
NORMALIZED TIME (rad)
(a)
110.00
114.00 NORMALIZED TIME (rad)
118.00
(a)
(b) Fig. 7. Phase current waveform at light load. (a) Simulated. (b) Measured, 10 A/div, 2 ms/div.
(b) Fig. 8. Phase current waveform at 0.7 pu rated torque. (a) Simulated. (b) Measured, 10 A/div, 2 ms/div.
frequencies the presence of sixth harmonic pulsating torque component is clearly apparent only in the torque waveforms recorded with three-phase excitation. The amplitude of the torque ripple in Fig. 9 for the three-phase cases is roughly 11 percent of the average torque, while the detectable torque ripple for the six-phase torque waveforms is only 3.5 percent. Even though such quantitative comparisons must be treated carefully, the qualitative advantage of the six-phase configuration in reducing torque ripple amplitude is clearly evident in the empirical results. During the course of this empirical testing, motor phase current waveforms in the three-phase and six-phase stator winding configurations were observed and compared. Similarities between the current waveshapes for three-phase and sixphase excitation are quite apparent in the waveforms provided in Fig. 11 for similar lightly loaded test conditions at 10-Hz excitation frequency. It has been observed that the peak normalized phase currents measured under similar test conditions during three-phase and six-phase excitation are not significantly different in the prototype system, although no far-reaching conclusion on this point is attempted here. Nevertheless, these comparisons suggest that it is possible to limit peak stator phase current in the six-phase system to
tolerable levels, provided that the multiphase induction motor is properly designed and excited. V. CONCLUSION This paper presents a summary of several key analytical and experimental results gathered during a recent investigation of multiphase (in excess of three) induction motors excited by voltage source inverters. Specifically, a six-phase squirrelcage induction motor consisting of two three-phase stator winding sets separated by 30 electrical degrees is studied in this work. In the laboratory prototype system, the motor is excited by a transistorized voltage-controlled inverter. An analytical model for the multiphase motor system has been presented based on the generalized two-phase real component transformation of motor equations. It has been shown that this transformation leads to. the generation of four independent machine equivalent circuits, only two of which involve coupling between the stator and rotor quantities and hence, torque production. Similarities between these equivalent circuits for the six-phase motor and the familiar d-q-o circuits associated with a three-phase machine have been pointed out. These transformed machine equivalent circuits also indicate
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ABBAS et al.: SIX-PHASE VOLTAGE SOURCE INVERTER DRIVEN INDUCTION MOTOR
40.00
-
E uJ 3 20.00-
= 20.00-
cc00
rs
0 I-
I
110.00
114.00 NORMALIZED TIME (rad)
110.00
118.00
114.00 NORMALIZED TIME (rad)
(a)
(b)
Fig. 9. Developed torque waveforms. (a) Light load. (b) 0.7
pu
rated
torque.
4.8 Hz. 20 ms /div, 3 Nm/div
(b)
(a)
10 Hz. 10 ms
(d) (c) Instantaneous shaft torque for both three-phase and six-phase sixstep voltage-fed induction motor. (a) 3 o, 250 r/min. (b) 6 X, 248 r/min. (c) 3 o, 586 r/min. (d) 6 X, 589 r/min.
Fig.
10.
(a) Fig.
11.
(b)
Phase current waveform at 10-Hz fundamental frequency. (a) Three-phase. (b) Six-phase.
118.00
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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. IA-20, NO. 5,
that current harmonics in the six-phase motor of orders (6v + do not contribute to the air-gap flux and 1, V = 1,3,5, thus do not participate in pulsating torque or rotor copper loss production. However, the amplitudes of these current harmonics are a matter of concern in the voltage source system, being limited only by the stator resistance and leakage inductances between the stator phases. Additional investigation into techniques for controlling the amplitudes of these undesirable current components is presently underway. A digital computer has been used to integrate numerically the transformed motor differential equations, thereby producing system performance predictions under a variety of test conditions. System parameters used in these simulations have been derived directly from prototype equipment measurements. Good agreement between predicted and measured system performance under a variety of test conditions has served to verify key features of the analytical model. In particular, the model has demonstrated a capability for predicting important system variables such as peak stator phase currents and average torque with reasonable accuracy. Qualitative waveshape agreement is consistently very good, and suspected sources for residual quantitative discrepancies are under investigation. A key goal of this research program has been to investigate the desirability of multiphase induction motor configurations of high power density electric motor drive systems. By a combination of analytical and empirical results, several of the anticipated advantages of the multiphase system have been demonstrated during the course of this study. For example, the inherent capability of the six-phase configuration to eliminate the sixth harmonic pulsating torque component, which is the dominant pulsating torque component in a three-phase system,
1/4
1/1
1/4
A -i
~1
1 cos
cos a
cos 2cr
2c
cos 4(x
Number of pole pairs. Moment of inertia. Torque. Differential operator. Number of stator, rotor phases.
q J T P
n,k
Subscripts Six-phase variable in alphabetical order. A,B,Twelve-phase variable in alphabetical order. a,b, x,~,, * * Generalized two-phase transformation component variables. Electromagnetic. e Load. L -
-
Superscripts Stator and rotor variable. s,r APPENDIX I
GENERALIZED TWO-PHASE REAL COMPONENT
TRANSFORMATION
This power invariant transformation represents a generalization of the faimiliar two-phase stationary reference frame tranformation to symmetrical multiphase systems with an arbitrary number of phases n. The transformation equation for the n-phase system is
XT=A-lX where XT=[XT0,
Xn 9
..
*
..
XTl(n
01)
9
two phase component variables vector. The transformation matrix is
1/12 ...
SEPTEMBER/OCTOBER 1984
...
1/U2 1/U2
...
0 sin 2a sin 4cr
0
sin sin
a a
This column exists
n
i/1.2
cos (n-l)ct
cos2(n-1)a
.
only if n is even
1/12
*. **
sin (n - l)cr
sin 2(n - I)a
.X=-- [Xa, Xb,9 , Xz], has been empirically verified. Results from the six-phase prototype system also demonstrate that it is possible to design phase variables vector, the system such that the peak normalized stator phase currents 2r= are not significantly higher than in an equivalent three-phase asystem with basic six-step voltage excitation waveforms. n Taken together, these results encourage further development A -1 -At. of the multiphase induction motor concept for specialized high APPENDIX II power drive applications. =
VI. NONENCLATURE
Variables Instantaneous current, voltage. i.v Instantaneous angular velocity. X inductance. Resistance, R,L M Mutual inductance.
PHASE VARIABLE MACHINE MODEL The phase variable mathematical model for n-phase stator and k-phase rotor induction machine in matrix form is Lv J
L
PL'
Rrr+ PLrr]
[i]
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ABBAS et al.: SIX-PHASE VOLTAGE SOURCE INVERTER DRIVEN INDUCTION MOTOR
where
"Der Asynchronmotor mit drei und sechs Wicklungs-strangen Stromeinpragenden Wechselrichter," Archiv Elektrotech., vol. 63, pp. 153-167, Mar. 1981. [12] T. M. Jahns, "Improved reliability in solid-state drives for large asynchronous ac machines by means of multiple-independent phasedrive units," Ph.D. thesis, Mass Inst. Technol., Cambridge, Apr. 1978. [13] D. White and H. Woodson, Electromechanical Energy Conversion. New York: Wiley, 1959.
[11]
Vs, Is
stator phase voltage and current vectors,
vr, Ir Rss
rotor phase voltage and current vectors, =Rs In} stator n x n diagonal resistance matrix, = R Ik, rotor k x k diagonal resistance matrix, stator n x n inductance matrix, rotor k x k inductance matrix, = (Lrs)t, stator to rotor n x k mutual inductance
-R"r
r
Ls
L"r Lsr
,
am
matrix.
APPENDIX III
Mohamed A. Abbas (S'77-M'80) received the
MOTOR PARAMETERS The tested prototype 20-hp two-pole 60-Hz standard squirrel-cage induction motor rewound as a six-phase machine has the following parameters: stator winding resistance/phase stator leakage inductance/phase referred to stator rotor resistance/phase referred to stator rotor leakage
0.317 0 2.947 mH 0.2836 0
inductance/phase 1.929 mH mutual inductance between stator and rotor/phase 99.97 mH inductance phase leakage 2.61 mH y-component o-component leakage inductance phase 2.61 mH. REFERENCES [1] J. R. Stewart and D. D. Wilson, "High phase order transmission-a feasibility analysis, Part I-Steady state considerations," IEEE2 Trans. Power App. Syst., vol. PAS-97, pp. 2300-2307, Nov./Dec. 1978. 12] W. C. Guyker et aL., " 138-kV six-phase transmission system feasibility," in Conf. Rec. 1978 American Power Conf., pp. 1293-1305. [3] E. E. Ward and H. Harer, "Preliminary investigation of an inverter fed 5-phase induction motor," Proc. Inst. Elec. Eng., vol. 116 (B), pp. 980-984, June 1969. [4] R. H. Nelson and P. C. Krause, "Induction machine analysis for arbitrary displacement between multiple winding sets," IEEE Trans. Power App. Syst., vol. PAS-93, pp. 841-848, May/June 1974. [5] E. A. Klingshirn, "High phase order induction motors, Part IDescription and theoretical considerations," IEEE Trans. Power App. Syst., vol. PAS-102, pp. 47-53, Jan. 1983. [6] "High phase order induction motors, Part 11-Experimental results," IEEE Trans. Power App. Syst., vol. PAS-102, pp. 54-59, Jan. 1983. [7] M. L. Kostyrev, "Equations anid parameters of a multi-winding asynchronous valve-type generator with shorted rotor," Elec. Technol. (USSR), pp. 19-23, 1980. [8] R. Jaschke, "Aligemeine Theorie des umrichtergespeisten Kafiglaufermotors mit beliebiger Strangzahl der Standerwicklung unter Berucksichtigung der Oberfelder," Archiv far Elektrotech., vol. 62, pp. 91101, 1980. [9] T. A. Lipo, "A d-q model for six-phase induction machines," in Proc. Int. Conf. Electrical Machines, vol. 2, 1980, pp. 860-867. [101 E. Andersen and K. Bieniek, "6-Phase induction motors for currentsource inverter drives," in Conf. Rec. 16th Annu. Meet. IEEE Ind. Appl. Soc., 1981, pp. 607-618. ,
B.S.
degree from the Higher Industrial Institute of
Cairo, Egypt, in 1964, the M.S. degree from the Polytechnic Institute of New York, Brooklyn, NY, in 1976 and the Ph.D.
degree from the University
of
Wisconsin-Madison in 1980, all in electrical engineermg.
Since joining the power electronics group of the Gould Research Center in 1980 he has been
involved in the design, development, analysis and simulation of multiphase (more than three-phases) high power density induction motor drive systems.
Roland Christen recieved the B.S.S.E. degree from Rochester Institute of Technology, Troy, NY,
After graduation he joined Lear Siegler Power Equipment Division to participate in the design of cycloconverter static power systems for aircraft and ground power applications. In 1972 he began work for Borg Warner in the field of industrial motor drives. In 1974 he joined the power electronics group at Sundstrand Corporation to develop controls and power circuitry for advanced aircraft generating systems. Since 1980 he has been working at the Gould Research Center as Project Manager of the electric propulsion group. His duties include the development of high power transistor ac drives for torpedo applications.
Thomas M. Jahns (S'72-M'78) received the S.B. and S. M. degrees in 1974 and the Ph.D. degree in 1978 from the Massachusetts Institute of Technology, Cambridge, all in electrical engineering. He joined Alexander Kusko Inc., Needham Heights, MA, in 1978 as a consulting engineer where he worked on power engineering projects in several areas including mass transportation and industrial drives. From 1979 to 1983 he worked at Gould Laboratories, Gould, Inc., in Rolling Meadows, IL. At Gould he participated in a variety of programs developing new ac drive systems for both land and marine propulsion applications, as well as leading laboratory investigations of highperformance ac drives for industrial applications. In September 1983 he joined the General Electric Company at the Corporate Research and Development Center, Schenectady, NY, where he is pursuing research interests in variablespeed drive systems and power electronics. Dr. Jahns is an active member of the IEEE-IAS Industrial Drives Committee and the recipient of two IEEE-IAS prize paper awards.
