Three-Dimensional Flux Vector Modulation Of

8/4/2019 Three-Dimensional Flux Vector Modulation Of

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1

Three-Dimensional Flux Vector Modulation of

Four-Leg Sinewave Output InvertersDhaval C. Patel, Rajendra R. Sawant, Member, IEEE, and Mukul C. Chandorkar, Member, IEEE

AbstractThe time-integral of the output voltage vector ofa three-phase inverter is often termed as the inverter fluxvector. This paper addresses the control of a three-phase four-leg sinewave output inverter having an LC filter at its output,by controlling the flux vector in three dimensions. Flux vectorcontrol has the property that the output filter resonance isactively damped by the output voltage control loop alone. Further,the inverter switching action inherently regulates the outputvoltage rapidly against dc bus voltage variations. Flux vectorcontrol of sinewave output inverters finds several applicationsin three-phase four-wire systems. This paper presents the fluxmodulation method for three-phase four-leg inverters feeding

unbalanced and nonlinear loads. All the necessary steps for thedigital implementation of the flux modulator are presented. Theswitching behavior of the modulator has been evaluated, whichis useful for variable fundamental frequency applications of theinverters. To provide experimental validation, the modulator isimplemented as a part of the control system for a stand-alonethree-phase four-leg inverter with an LC filter at its output.Control system details are also provided. Experimental resultsindicate the effectiveness of the modulator and the control systemin providing balanced voltages at the output of the LC filtereven under highly unbalanced conditions with nonlinear loads.The resonance damping and voltage regulation properties of themodulator are also apparent from the experimental results.

Index TermsIntegral space vector modulation, Flux modula-tion, Four-leg inverter.

I. INTRODUCTION

CONVENTIONAL three-phase three-wire inverters are

suitable for supplying three-phase balanced loads such

as induction motors. For unbalanced three-phase loads such

as those formed by unequal single-phase loads connected to

the three-phase system, inverters should be able to provide

a path for the neutral current. There are two main ways for

doing this with three-phase inverters.

Inverters with split dc link capacitors [1] Inverters with fourth (neutral) leg [2][6] (see Fig. 1)

The higher dc link utilization, requirement of smaller dc link

capacitors and flexibility in control are inherent advantages of

four-leg inverters over split dc link capacitor inverters. Four-

leg inverters can be used for applications such as stand-alone

sinewave output inverters for non-linear unbalanced loads,

Manuscript submitted on February 10, 2009; revised May 22, 2009.Accepted for publication on July 14, 2009.

Copyright c2009 IEEE. Personal use of this material is permitted. How-ever, permission to use this material for any other purposes must be obtainedfrom the IEEE by sending a request to [email protected]

Dhaval C. Patel, Rajendra R. Sawant, and Mukul C. Chandorkar arewith Department of Electrical Engineering, Indian Institute of Technol-ogy Bombay, Mumbai 400076 INDIA e-mail: ([email protected]; [email protected]; [email protected])

Snp Sap Sbp

Snn San Sbn

Scp

Scn

AB

C

N

abc

n

Vdc

Ln

+

-

Linear/NonlinearBalanced/Unbalanced

Load

Four-Leg VSI

LC Filter

Fig. 1. Four-leg sinewave output voltage inverter

distributed generation interfaces, microgrids, neutral currentcompensators and active filters [6].

Space vector modulation methods for four-leg inverters have

been presented in [2][6]. Space vector modulation for four-leg

inverters is complex [2], [6]. However, it has advantages such

as low output distortion, suitability to digital implementation,

constant switching frequency and good dc bus utilization [7].

The inverter flux vector is the time-integral of the inverter

switching voltage vector. Inverter switching based on the

control of the flux vector has several advantages in the control

of sinewave output inverters having LC output filters. In

contrast to voltage modulation control methods, the output

voltage control loop alone with a flux modulator is sufficient

to actively damp the output filter resonance [8]. Further,the inverter switching inherently regulates the output voltage

against dc bus voltage variations. The method also lends itself

to easy digital implementation on a processor or a field-

programmable gate array (FPGA).

Two-dimensional flux vector modulation of three-leg invert-

ers was presented in [8], [9]. Grid connected applications of

three-leg sinewave output inverters using flux vector modula-

tion were discussed in [8]. A flux vector modulator for a fuel

cell inverter was presented in [10]. An application for active

filter was discussed in [11].

An undesirable feature of flux modulators is the variable

inverter switching frequency that results from the tracking of

the flux reference vector using inverter switching within ahysteresis band. The switching frequency characteristics of the

flux modulator for a three-leg inverter were discussed in [8].

A solution to the problem of variable switching frequency was

presented in [12], [13], which resulted in constant switching

frequency.

Charge modulator for current source inverter, an analogy of

the flux modulator, was presented in [14]. A flux modulator

designed in the synchronous reference frame was employed in

[14][16]. A comparison of different modulators for inverter

control was presented in [14]. An analysis of a flux modulator

was presented in [17]. A voltage modulation index in the

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3

V1

V4

V5

V6

V7

V14

q

d0Reference Flux Vector

Actual Flux Vector

Fig. 3. Graphical representation of reference flux tracking

III. FLU X MODULATION FOR FOU R-L EG INVERTER

A. Principle of the Flux Modulator

The inverter flux vector is defined as

(t) =

t0

V d + (0) (2)

In this, V is the inverter output voltage vector (Fig. 2.)In three-dimensional flux vector modulation, the vector

is made to track a reference vector

by choosing anappropriate sequence of inverter output voltage vectors. An

inverter voltage vector is selected on the basis of the error

between and , so that moves towards . Fig. 3 showsthe actual flux vector tracking the reference flux vector

in the three-dimensional qd0 space. The flux vector erroris sampled at regular intervals T, and the the inverter outputvoltage vector is chosen so as to keep the vector error within

a tolerance band. This is detailed in the next section.

B. Implementation of the Flux Modulator

The flux modulator is implemented in discrete-time on a

digital signal processor (DSP). The sampling time step for

the discrete-time implementation is T. This is the time stepat which the error between the reference and the actual flux

vector, , is sampled for corrective action. In order torealize flux modulator for a four-leg inverter it is necessary to

1) identify the sector on the q d plane in which qd, theq d plane projection of the reference flux vector ,is located, as shown in Table II and Fig. 4

2) generate the error bits for the q, d and 0axiscomponent errors as shown in Table III

3) select the inverter voltage vector that reduces the errors

in the q, d and 0axis components as shown inTables IV and V.

1) Sector identification: The location of qd identifies oneof six sectors (I. . .VI) on the q d plane. This is shownin Table II and Fig. 4. The sector is identified by limits

to the slope of the tangent to the trajectory of qd. Theselimits are given in Table II. It is important to note that,

depending on the application, the trajectory of qd may ormay not be a circle. In applications with balanced loads, the

trajectory would typically be a circle. However, if the inverter

has to produce balanced output voltages when unbalanced and

nonlinear loads are present, the trajectory of qd will not bea circle. Both situations are shown in Fig. 4. In Fig. 4, the

tangents are denoted as T1. . .T6 and the sectors as I. . .VI.

TABLE IISEC TOR WITH C OR RESPONDING LIMITS OF TANGENT SLOPE

Limits of tangent slope Sectors

/3 T < 2/3 I2/3 T < II

T < 4/3 III4/3 T < 5/3 IV5/3 T < 2 V

0 T < /3 VI

T1

T1

T2T2

T3T3

T4

T4

T5T5

T6 T6

II

IIII

III

III

IV

IV

VVVI

VI

qd

(a) (b)

q q

ddFlux Vector Trajectory

Fig. 4. Sector identifi cation for (a) balanced and (b) unbalanced fluxtrajectories

2) Error bits generation: The errors in the q, d and0axis flux vector components are q q, d d and0 0. These errors are used to determine three bits Sq,Sd and S0 as shown in Table III. In this table, the subscript xstands for one of q, d and 0. The error tolerance band is h.

3) Inverter voltage vector selection: The sector information

and error bits determined above are used to select an appropri-

ate inverter voltage vector for output during the current time

step. The selected vector reduces the error during thetime step.

There are eight possible inverter voltage vectors which can

be selected for any given sector. These are given in Table IV.

Further, there are eight possible combinations of the three error

bits Sq, Sd and S0. Each possible vector can correct for a

TABLE IIIER R OR B ITS GENER ATION

Comparison of Vectors Error Bit Next Action

x x h Sx = 1 Increase xx

x h Sx = 0 Decrease

x

h < x x < h Sx = Sx No Change

TABLE IVPO S S I B L E S W I T C H I N G V E C TO R S F O R E A C H S E C T OR

Sector Possible Vectors

I V0 V1 V4 V5 V12 V13 V14 V15II V0 V1 V4 V5 V6 V7 V14 V15III V0 V1 V2 V3 V6 V7 V14 V15IV V0 V1 V2 V3 V10 V11 V14 V15V V0 V1 V8 V9 V10 V11 V14 V15VI V0 V1 V8 V9 V12 V13 V14 V15



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5

Fundamental frequency f (Hz)

Switchingfrequencyfsw

(Hz) 1.50nom

1.25nom

1.00nom

0.75nom

0.50nom

0.25nom

00

200

400

600

800

1000

1200

1400

1600

1800

2000

10 20 30 40 50 60 70 80 90 100

Fig. 7. Switching frequency characteristics for different

Normalized output frequency f/fnom

Normalizedswitchingfrequencyfsw

/f

h = 0.01nom

0.015nom

0.02nom

0.025nom

0.03nom

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10

20

30

40

50

60

70

80

90

100

Fig. 8. Normalized switching characteristics

The maximum possible value of reference flux to getoperation in the linear region is given by

max = nomnom

VdcVdc,nom

(7)

The flux reference vector must be contained in a sphere

of radius max centered at the origin. Here, Vdc,nom isthe nominal dc-bus voltage, which can produce the nominal

inverter output voltage at the nominal frequency and nominal

flux reference magnitude nom. The maximum flux referencecan be calculated using (7) on the basis of the dc-bus voltage

feedback Vdc and the desired output frequency [8].Simulations of switching frequency characteristics are pre-

sented here for a nominal frequency fnom = 50 Hz andnominal angular velocity nom = 2fnom. The dc bus voltageis assumed to be at its nominal value.

Change of the switching state from ON to OFF and OFF

to ON are considered as separate events in the counting of

switching in all four legs. The number of switching events is

averaged over five fundamental cycles.

Fig. 6 shows the averaged switching frequency fsw as afunction of the output frequency f, with the hysteresis bandh as the parameter. For Fig. 6, the integration time step is set

as T = 20 s. The flux reference magnitude is set as

=

nom, if nom maxmax, if nom >

max

(8)

where nom and max are calculated by (6) and (7), respec-

tively. (8) ensures that the modulator remains in the linear

region, over entire range of the fundamental frequency f. Thecurves of Fig. 6 are independent of the dc-bus voltage Vdcbecause the tolerance h is represented as a fraction of thenominal flux nom.

Fig. 7 shows the averaged switching frequency fsw as afunction of the output frequency f, with the reference fluxmagnitude as the parameter. The hysteresis band value isfixed at h = 0.01nom and integration time step T = 20s.

Fig. 8 shows normalized switching characteristics derived

from the variable hysteresis band characteristics shown in

Fig. 6. These plots are useful during variable fundamental

frequency applications of the inverter, as they permit the

implementation of a variable hysteresis band to keep the

switching frequency constant.

V. FOU R-L EG SINEWAVE OUTPUT INVERTER CONTROL

Fig. 9 shows the closed loop control system for a stand-

alone four-leg sinewave output inverter with an LC filter. The

inverter and filter are required to supply regulated and balanced

sinusoidal voltages to unbalanced and nonlinear loads. As

shown in Fig. 9, the reference voltage vector components are

denoted by Eeqref, Eedref and E0ref. The superscript e denotes

quantities in the synchronously rotating qe de referenceframe. These are derived through transformation of phase

reference voltages from the a b c frame.

The q d plane component vector of the reference voltagevector is Eqd. This vector component is controlled by the two-dimensional flux control method detailed in [8]. The gains of

the q and daxis synchronous frame PI controllers shownin Fig. 9 are computed accordingly. The control of the 0axisvoltage is given below.

The model of an LC filter in q d 0 coordinate isVqd0 = Lqd0

ddt

iqd0 + Eqd0 (9)

Lqd0 =

Lf 0 00 Lf 0

0 0 Lf + 3Ln

(10)

Vqd0, iqd0 and Eqd0 are inverter voltage, inverter output currentand filter terminal voltage respectively. The state equations of

the inverter and filter for the 0axis components areE0

e0

=

0 2f01 0

E0

e0

+

2f0 1Cf

0 0

v0

i0

(11)

Equation (11) remains unchanged in synchronous reference

frame. The frequency f0 =1

Cf (Lf+3Ln). The flux compo-

nent e0 is associated with the 0axis voltage component



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6

+

+

+

+

Eeqref

Eedref

E0ref

PI

PI

PI

controller

controller

controller

eqref

edref

0ref

qref

dref

Eeq

Eed

Eq

Ed

E0

ejt

ejt

Ln

n

Lf

Cf

Vabc iabc

Eabc

Eabc

Switching Pulses

FLUX

MODULATOR

FOUR LEGINVERTER

abcto

qd0

Load

Fig. 9. Stand-alone four-leg sinewave output inverter system

across the filter capacitor, and v0 is associated with the

0axis voltage component at the inverter terminals.With a PI regulator, the close loop state equations for the

0axis component are E0e0

v0

=

0

2f0

2f0

1 0 0ki0 kp02f0 kp02f0

E0e0

v0

+

0 0

1Cf

0 0 0ki0 kp0 kp0

1Cf

E0refE0ref

i0

(12)

The characteristic polynomial for this system is

Fs = s3 + kp0

2f0s

2 + 2f0 (1 + ki0) s (13)

This can be used to determine the PI regulator gains for a

specified dynamic response.

V I . EXPERIMENTAL RESULTS

To provide experimental validation, the flux modulator and

voltage control system described above was implemented

to control a stand-alone four-leg sinewave output inverter.

The power circuit was built with four insulated gate bipolar

transistor (IGBT) legs. The IGBT assembly was rated for 35 Arms current and 1200 V dc bus with 850 F/1200 V dc linkcapacitors. The LC filter components values were Lf = 3mH, Ln = 3 mH and Cf = 200 F. The entire control

system including the flux modulator was implemented on aplatform with a Texas Instruments 32-bit floating point DSP

TMS320VC33 with a 13.3 ns instruction cycle. The samplingtime for the control system was T = 20 s.

The synchronous reference frame PI controller gains for

controlling Eqd were set at Kpq = 0.0023, Kiq = 0.175,Kpd = 0.000346, and Kid = 0.538. The gains for the 0-axiscontroller were Kp0 = 0.1 and Ki0 = 0.08. The suffixes pand i stand for proportional and integral gains respectively.The suffixes q, d and 0 stand for the q d 0 coordinates.

Fig. 10 shows the phase voltage VCN and the line voltageVBC at the inverter terminals (Fig. 1.) These were obtained

VCN

VCN

VBC

VBC200 V/div

200 V/div

2 ms/divTime

Fig. 10. Experimental phase and line voltage at inverter terminals

with only the flux modulator, for a 50 Hz output frequency,with balanced flux references. The q and daxis flux refer-ences each had a magnitude of 0.5 Vs, and the 0axis fluxreference was 0 Vs. The value of the tolerance band h = 0.01Vs. The inverter dc bus voltage was 320 V.

Experiments were performed to test different load condi-

tions, such as balanced/unbalanced and linear/nonlinear three-

phase loads. Here the waveforms for two different load con-

ditions are shown.Fig. 11 shows waveforms for a three-phase diode bridge

rectifier load on the inverter. The dc side of the rectifier has

a filter capacitor and a resistive load. The upper three traces

show phase voltages and the corresponding phase currents.

The lowest trace shows the inverter dc link voltage. Initially

the rectifier was not connected to the inverter. It was switched

on to the inverter at a certain time. Fig. 11 shows the no-

load, transient, and loaded steady state performance of the

system. The high quality sinewave voltage output under no

load shows the effectiveness of the active damping of the LC

filter resonance.



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7

VdcVdc

iaia

ibib

icic

EaEa

EbEb

Ec

Ec

Time

40 V/div

40 V/div

40 V/div

50 V/div

8 A/div

8 A/div

8 A/div

20 ms/div

Fig. 11. Experimental waveforms with three-phase rectifi er load

VdcVdc

inin

iaia

ibib

icic

EaEa

EbEb

EcEc

Time

40 V/div

40 V/div

40 V/div

50 V/div

8 A/div

8 A/div

8 A/div

3.3 A/div

20 ms/div

Fig. 12. Experimental waveforms with unbalanced linear load and three-phase rectifi er

VphVph

Vdc

Vdc

50 V/div

50 V/div

200 ms/divTime

Fig. 13. Experimental result of voltage regulation of flux modulator

Load

currentsia

,ib

,ic(A)

Time (s)

ia ib ic

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

20

10

10

20

Fig. 14. Experimental load currents for unbalanced linear and nonlinear load

Magnitude(%

offundamental)

Frequency (Hz)

Fundamental (50 Hz) = 112.4 V peakTotal Harmonic Distortion (THD) = 4.20 %

100

80

60

40

20

00

200 400 600 800 1000 1200 1400 1600 1800 2000

Fig. 15. Harmonic spectrum of the load voltage Ea

Fig. 12 shows waveforms with unbalanced linear load and

balanced nonlinear loads connected to the inverter. The upper

three traces show phase voltages and the corresponding phase

currents. The next trace shows the neutral current supplied bythe inverter to the unbalanced load. The lowest trace shows

the inverter dc link voltage. As in the previous experiment,

the inverter was operated on no load initially. The load was

switched on to the inverter subsequently.

Voltage regulation of the flux modulator is shown in Fig. 13.

Here the flux modulator was operated without the output

voltage control loop. The reference flux magnitudes were set as

a = 0.4, b = 0.4 and c = 0.4. The upper trace shows theoutput phase voltage across the LC filter capacitor. The lower

trace shows the inverter dc link voltage. In the experiment, the

dc link voltage was reduced from 330 V to 250 V. It is apparentthat there is no change in the output voltage amplitude even

after the dc link voltage is reduced.The performance of the four-leg inverter controlled by the

control system shown in Fig. 9 is shown here by means of

oscillograms and harmonic spectrum plots. Combinations of

balanced and unbalanced, linear and nonlinear loads were

connected to the inverter. The load on the inverter consisted of

three single-phase diode bridge rectifiers with unequal dc-side

resistances, in parallel with unbalanced linear (resistance in

series with inductance) load. This was a case of severe load

unbalance, both for the linear and the nonlinear load. The peak

of the distorted load current was about 20 A. Fig. 14 showsthe three-phase load current oscillogram. The highly distorted



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8

VoltagesEe q

,Ee d

,E0

(V)

Time (s)

Eeq

EedE0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

120

100

80

60

40

20

0

020

Fig. 16. Experimental load voltages in the synchronous reference frame

q (Vs)d (Vs)

0

(V

s)

0.1

0.2

0.20.2

0.4

0.4

0

00

0.1

0.2

0.2 0.20.40.4

Fig. 17. Experimental inverter flux vector locus for unbalanced linear andnonlinear load

and unbalanced nature of the load is apparent from this.

Fig. 15 shows the load voltage harmonic spectrum. It is

apparent that all harmonics are negligibly small compared to

the fundamental. Fig. 16 shows the load voltages Eeq , Eed

and E0 in the synchronous reference frame (refer Fig. 9.)The references were set to be Eeqref = 110 V, E

edref = 0

V and E0ref = 0 V. It is apparent that E0 is controlledclose to zero, indicating that the zero sequence load voltage

component is negligibly small. The control system orients the

load voltage vector so that the daxis voltage component Eedis zero on average, and the qaxis component Eeq has thedesired magnitude.

In order to achieve balanced sinusoidal load voltages in thepresence of such severe load nonlinearity and unbalance, the

inverter flux vector locus needs to deviate substantially from

a circle lying in the qd plane. The inverter flux vector locusis shown in Fig. 17.

VII. CONCLUSION

A flux vector modulation method has been proposed for

the control of a sinewave output four-leg inverter. Digital

processor implementation of the flux modulator for a four-leg

inverter is simple. This paper has described the implementation

details of the modulator. The switching behavior described

here is useful for a variable hysteresis band operation of the

modulator. This paper has also described the implementation

of a voltage control system to regulate the output sinewave

voltages feeding unbalanced and nonlinear loads using the flux

modulator. The paper has presented experimental validation

of the modulator and control system. The results show that

the flux modulator proposed here works satisfactorily under

balanced and unbalanced, linear and nonlinear load conditions

on the inverter.

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[6] R. R. Sawant and M. C. Chandorkar, A multifunctional four-leg grid-connected compensator, in IEEE Trans. Ind. Appl., vol. 45, no. 1, Jan/Feb2009, pp. 249-259.

[7] D. Shen, P. W. Lehn, Fixed-frequency space-vector-modulation controlfor three-phase four-leg active power fi lters, in IEE Proc.- Electric Power

Appl., vol. 149, no. 4, July 2002, pp. 268-274.[8] M. Chandorkar, New techniques for inverter flux control, in IEEE Trans.

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integral space-vector PWM technique for DSP-controlled voltage-sourceinverters, in IEEE Trans. Ind. Appl., vol. 35, no. 5, Sept./Oct. 1999, pp.1091-1097.

[10] F. Jurado, Novel Fuzzy Flux Control for Fuel-Cell Inverters, in IEEETrans. Ind. Electron., vol. 52, no. 6, Dec. 2005, pp. 1707-1710.

[11] S. Bhattacharya, A. Veltman, D. M. Divan and R. D. Lorenz, Flux-based active fi lter controller, in IEEE Trans. Ind. Appl., vol. 32, no. 3,May/June 1996, pp. 491-502.

[12] B. Shi, M. Chandorkar and G. Venkataramanan, Modeling and Designof a Regulator for Three Phase PWM Inverters with Constant SwitchingFrequency, in European Power Electron. Conf., Toulouse, France, 2003.

[13] P. C. Loh and D. G. Holmes, A multidimensional variable band fluxmodulator for four-phase-leg voltage source inverters, in IEEE Trans.Power Electron., vol. 18, no. 2, March 2003, pp. 628-635.

[14] P. C. Loh and D. G. Holmes, A variable band universal flux/chargemodulator for VSI and CSI modulation, in IEEE Trans. Ind. Appl., vol.38, no. 3, May/June 2002, pp. 695-705.

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Dhaval C. Patel received the B.E. degree in PowerElectronics Engineering from the Saurashtra Univer-sity, India in 2002 and the M.E. degree in ElectricalEngineering with specialization in Power Electronicsand Drives from the Gujarat University, India in2004. Currently he is working toward the Ph.D.degree at the Department of Electrical Engineering,Indian Institute of Technology - Bombay, India. Hiscurrent research interests are in the area of diagnosisand analysis of electical machines, power converters

and modulation techniques for inverters.

Rajendra R. Sawant (M2000) was born in Ma-harashtra State, India on February 19, 1968. Hereceived B. E. degree in Electrical Engineering fromMarathwada University, Aurangabad, MaharashtraState in 1988, M. Tech. and Ph. D. degree from

Power Electronics and Power Systems group, de-partment of Electrical Engineering, Indian Instituteof Technology-Bombay, India, in 1996 and 2009,respectively.

He was involved in the development of ResonantConverter based Induction Heating Systems as a

Power Electronics Consultant and Researcher from 1996 to 2002 with differentsmall scale industries in Mumbai, India. He is involved in teaching PowerElectronics and different basic subjects in Electrical Engineering for the last18 Years in Mumbai University at the undergraduate and graduate level.Presently, he is working as a Professor and Head with the Dept. of Electronicsand Telecom. at Rajiv Gandhi Institute of Technology, University of Mumbai,India. His research interest are active power fi lters and power conditioners,grid connected converter control, converters for distributed generations andmicro-grid, resonant converters for induction heating systems, simulation ofelectric circuits and systems, etc.

Mukul C. Chandorkar (M84) received the B.Tech. degree from the Indian Institute of Technology- Bombay, the M. Tech. degree from the IndianInstitute of Technology - Madras, and the Ph.D.degree from the University of Wisconsin-Madison,in 1984, 1987 and 1995 respectively, all in electricalengineering.

He has several years of experience in the powerelectronics industry in India, Europe and the USA.During 1996-1999, he was with ABB CorporateResearch Ltd., Baden-Daettwil, Switzerland. He is

currently a professor in the electrical engineering department at the IndianInstitute of Technology - Bombay. His technical interests include electricpower quality compensation, drives, and the real-time simulation of electricalsystems.

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Three-Dimensional Flux Vector Modulation Of