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880 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 3, MAY/JUNE 2001 New Techniques for Inverter Flux Control Mukul Chandorkar, Member, IEEE Abstract—Inverter flux control methods directly control the integral of the inverter output voltage vector, by means of the inverter switching. These methods have found applications in lowand medium-voltage industrial drives, and in inverters with outputs connected to the utility mains. This paper describes a new method for the direct control of the inverter flux vector, with the viewpoint of using this vector as the main forcing quantity in a closed-loop control system. It illustrates the use of the inverter flux in the control of a utility-connected inverter. Index Terms—Pulsewidth modulation, three-phase inverters, utility connection. I. INTRODUCTION I T IS TYPICAL to consider inverter pulsewidth modulation (PWM) methods under two headings—open loop and closed loop. The typical open-loop control method is the well-known sine-triangle intersection method, along with its numerous variations [1]. As the name implies, open-loop methods determine the switching pattern of the inverter independently of the inverter output voltage or current. Closed-loop control methods, on the other hand, use feedback of some output quantity to directly determine the next switching state of the inverter. The next switching state is chosen so as to correct the error between the desired value of the output quantity, and its actual value. For example, the output current of an inverter can be controlled by choosing the next switching state of the inverter to correct the error between a reference current value, and the measured inverter current [2]. Also under the heading of closed-loop modulation methods are the sigma–delta modulation strategies [3], often used to control resonant link converters. Another well-known example of closed-loop inverter control is the so-called direct torque control (DTC) of induction machines [4]. Typically, closed-loop inverter control methods for three-phase inverters control two independent output quantities directly by means of inverter switching. In DTC, the two independent output quantities that are so controlled are the stator flux magnitude and the torque. However, the torque and the flux magnitude are not the only independent quantities that can be controlled directly by the inverter switching. Paper IPCSD 01–001, presented at the 1999 Industry Applications Society Annual Meeting, Phoenix, AZ, October 3–7, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial Power Converter Committee of the IEEE Industry Applications Society. Manuscript submitted for review October 15, 1999 and released for publication February 27, 2001. The author is with the Electrical Engineering Department, Indian Institute of Technology, Mumbai 400076, India (e-mail: [email protected]). Publisher Item Identifier S 0093-9994(01)03933-0. Fig. 1. Flux control by inverter switching. In this paper, the concern is on the independent closed-loop control of the two components of the inverter flux vector. The inverter flux vector is defined as (1) is the instantaneous inverter In this definition, output voltage vector in the stationary – reference frame. The task of the flux vector control method is to guide the end of along a specified path in the – the inverter flux vector plane, by independently controlling the two components of the flux vector, using the inverter switching action. This is shown in and are two inverter voltage vectors, and Fig. 1. In Fig. 1, is the inverter flux vector. The tip of is guided along the desired path by selecting the times and sequence of the voltage vectors. An early and comprehensive description of a method to control the flux vector in the stationary – reference frame, for induction machine control, was given in [5]. The effects of constraints on the maximum inverter switching frequency and minimum switch commutation time—the time between consecutive switchings in an inverter leg—were analyzed in [5]. A flux-based modulation strategy was developed, which resulted in minimum machine losses, while meeting the constraints on maximum switching frequency and minimum commutation time. Subsequently, the “fish” method was proposed to analyze and synthesize inverter switching patterns based on the flux vector observed in a rotating – reference frame [6]. The flux “fish” is 0093–9994/01$10.00 © 2001 IEEE Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply. CHANDORKAR: NEW TECHNIQUES FOR INVERTER FLUX CONTROL defined as the locus of the flux vector observed in the synchronously rotating reference frame. The PWM pattern is generated by choosing the inverter states so as to confine the error between the reference and actual flux vectors within limits along the synchronously rotating and axes. A space-vector-based flux-vector control technique suitable for digital processor implementation has been discussed in [7]. As in [5] and [6], this technique confines the error between a reference flux vector and the actual flux vector to be within a specified tolerance zone. This is done by appropriately choosing zero and active inverter output voltage vectors. A zero voltage vector is chosen when the error magnitude becomes less than a specified tolerance. An active vector is chosen if the error magnitude exceeds the tolerance. The strategy for choosing the active vector is reported to reduce the number of switchings per output cycle compared to the “fish” method of [6], for the same total harmonic distortion (THD) in the inverter output. This paper describes a new method to control the inverter flux vector in rectangular coordinates. The described method is related to the PWM method of [5], in that the control is performed in the stationary rectangular – reference frame. The aim is to develop the flux modulator so that it matches with the outer loops of the inverter control system, which produce the reference flux vector for the modulator. This paper also discusses the switching frequency characteristics and linearity of the resulting modulator. It then describes the design and implementation of a closed-loop control system that uses the flux-vector modulator to effectively control a three-phase inverter connected to the main power system through a sine-wave filter. II. INVERTER FLUX-VECTOR CONTROL Essentially, inverter flux-vector control involves appropriately choosing the inverter voltage vectors, so as to make the flux follow a reference flux vector within a specified tolerance band. Because the modulator does not directly control the load, the response of the load quantities is not as fast as, for example, the motor torque with DTC. However, the advantage here is that it does lead to a general-purpose modulator whose task is to control the inverter flux vector to whatever value is specified by the outer control loop (within the limits of the inverter rating and dc-bus voltage). The outer control loop can then be designed independently to generate the inverter flux-vector set point for achieving the desired final result. An example of the use of flux control for an active filter application can be found in [8]. The well-known vector diagram of a two-level voltage-source inverter is shown in Fig. 2. In Fig. 2, the length of each vector is , where is the inverter dc-bus voltage value. Six sectors—I … VI—are also defined as shown. Each sector spans 30 on either side of each inverter vector. The sector of the inverter flux vector decides the possible voltage vectors that can be used to correct it. , In the following, the reference flux vector is . The actual flux vector is similarly where . The components of both these vectors are observed in the stationary reference frame. In order to control the inverter 881 Fig. 2. Voltage-source-inverter vectors. TABLE I FLUX ERROR TOLERANCE flux vector in rectangular coordinates, the two rectangular comand , are controlled. ponents of the inverter flux vector, The quantity is the permitted error tolerance in both, the and -axes flux components. The flux controller is expected to within the permitted tolerance. keep the error vector and are the results of the error comThe binary variables parison in, respectively, the - and the -axes components of the flux vector. Table I, in which the subscript represents either or , shows the specification of the error tolerance. The rules for determining the output voltage vector are as follows. • If there is an active voltage vector for the present sector, that can correct the errors in both the - and the -axes components simultaneously, then that vector is used. • Otherwise, a zero vector is used. (Of the two possible zero vectors, the one selected should result in the smaller number of switchings.) These rules result in Table II for the inverter voltage vector . The flux control method as described in Table II renders itself very well for a digital signal processor (DSP) implementation. On the DSP, the inverter flux components are updated at con, using rectangular integration stant time intervals of (2) In this equation, the subscript refers to the sample number, and and are, respectively, the - and -axes components of the inverter voltage vector in the previous sampling in- Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply. 882 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 3, MAY/JUNE 2001 TABLE II VOLTAGE-VECTOR SELECTION Fig. 4. Inverter output voltages. Fig. 5. Linear range calculation. tant for the use of the inverter flux modulator in a closed-loop control system. In the linear region, this relationship is Fig. 3. Flux-vector locus. (3) terval. These components are themselves calculated by the DSP on the basis of the information of the previous switching state of the inverter and the dc-bus voltage feedback. The reference flux-vector components are also made available to the modulator from the outer control loop in the time . On the basis of this, the error status information interval and are calculated. These bits, along with the sector bits information, are then used to determine the new inverter voltage . vector Fig. 3 shows one cycle of the inverter flux vector , for a V. The angular velocity of the referdc-bus voltage rad/s , and its magnitude is 0.918 ence flux vector is V s. The sampling V s. The flux tolerance value is s. interval Fig. 4 shows the - and -axes voltages that give rise to the flux-vector locus of Fig. 3. in which is the magnitude of the fundamental inverter output is the output angular frequency. phase voltage, and is the magnitude of the reference flux vector. The linear region can be calculated on the basis of Fig. 5. In Sector I (Fig. 2), the smallest tangential velocity with which the tip of the inverter flux vector can be moved occurs when the inverter flux vector is aligned with the axis. In and have the this situation, both the voltage vectors same tangential component, which is in the direction of the axis. Since the length of each inverter voltage vector is , the maximum peak fundamental output phase voltage possible in the linear range is III. LINEARITY AND FLUX REFERENCE LIMITS The linearity of the relationship between the fundamental output voltage magnitude and the output frequency is impor- (4) Note that this is also the peak phase voltage which, when . If the full-load voltage rectified, results in the dc voltage Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply. CHANDORKAR: NEW TECHNIQUES FOR INVERTER FLUX CONTROL 883 Fig. 7. Switching frequency dependence on h. Fig. 6. Fundamental output voltage magnitude linearity. is to be delivered at the nominal output frequency the nominal value of the reference flux should be , then This section presents simulations of the switching frequency behavior for the flux controller described above. For this purpose, the nominal output frequency and nominal angular velocity are Hz (5) comThis puts a limit on the maximum possible value of mand that can be given by the outer control loop to the flux controller, if the latter is to operate in its linear range. This limit should also account for variations of the dc-bus voltage from its nominal value. The limit is given by (6) is the nominal dc-bus voltage, which gives rise In this, to the nominal inverter output voltage at the nominal frequency . The limit on the and nominal flux reference magnitude reference flux magnitude can be calculated in the DSP on the and the desired output basis of the dc-bus voltage feedback frequency . In practice, this limit on the flux reference magnitude can be best implemented if the outer control loop provides the flux reference vector command in the synchronous reference frame, rotating at an angular frequency . The reference is limited in the synchronous reference frame, and then transformed to the stationary reference frame and provided to the flux controller. Fig. 6 shows the normalized fundamental component output voltage, as a function of the normalized output frequency. This was obtained by a simulation of the flux modulator at various output frequency points, followed by an extraction of the fundamental component magnitude from the resulting switching pulse pattern. The three curves shown in Fig. 6 represent three different settings of the flux magnitude reference, as a multiple [see (5)]. The flux magniof the nominal flux reference tude reference is further limited according to (6). IV. SWITCHING FREQUENCY Predominantly, the output switching frequency is influenced by the flux error tolerance , the fundamental output angular , and the flux reference magnitude . frequency Further, the ratio One switching cycle is considered to be one on and one off event for any switch. In the simulation, the number of switching cycles is counted within five fundamental output cycles. The average value of the total number of switchings for the three phase legs is calculated over the five fundamental output cycles, . and divided by five to give the average switching frequency as a funcFig. 7 shows the averaged switching frequency tion of the output frequency , with the flux error tolerance as the parameter. For Fig. 7, the integration time step [refer to (2)] is set as s. The flux reference magnitude is set as if if (7) and are calculated by (5) and (6), respecwhere tively. Equation (7) ensures that the modulator remains in the linear region, over the entire range of the fundamental frequency . In Fig. 7, the tolerance is represented as a fraction of the . It is important to note that when is specinominal flux fied in this manner, the curves of Fig. 7 are then independent of . the dc-bus voltage as a function of Fig. 8 shows the switching frequency the fundamental output frequency, with the integration time as the parameter. The tolerance value is fixed at step , and the flux magnitude reference is determined from (7). , as a function of the Fig. 9 shows the switching frequency fundamental output frequency , with the reference flux magnitude as the parameter. The tolerance value is fixed at Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply. 884 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 3, MAY/JUNE 2001 Fig. 8. Switching frequency dependence on 1T . Fig. 9. Switching frequency dependence on . , and the integration time step is s. The parameter for the plots in Fig. 9 is the flux reference magnitude, [see (5)]. Specifically, the flux refgiven as a multiple of erence magnitude for Fig. 9 is set as if if (8) , as shown in Fig. 9. With the where is the multiplier of flux reference magnitude set from (8), the modulator is linear over the entire fundamental frequency range of Fig. 9. Some comments regarding the switching frequency characteristics of Figs. 7–9 are in order here. All these figures are in. Furthermore, if both the dependent of the dc-bus voltage and -axes coordinates for these figures are normalized by the , these characteristics are indenominal output frequency as well. pendent of A further point to be noted is that the curves in Figs. 7 and 8 show a maximum switching frequency at a fundamental output . This is a consefrequency of approximately quence of the fact that, for such tolerance band controllers, the characteristic of the switching frequency normalized by the output frequency, as a function of the normalized output frequency, is approximately a straight line with a negative slope . This is shown in Fig. 10, with the in the range tolerance as the parameter for the curves, and the integration s. time step In Fig. 10, the normalized switching frequency characteristics can be approximated by equations of straight lines of the form (9) The values of the constants and are determined by in Fig. 8. Fig. 7, and by From (9), it is easily seen that the maximum value of occurs for Fig. 10. Normalized switching frequency. This maximum value of is given by The fact that the normalized switching frequency curves are approximated by straight lines, as in Fig. 10, can be very useful in developing a switching frequency limiting method for the modulator and can be of importance to high-power converters which have strong average switching frequency constraints. Specifically, the constant , which determines the tolerance value as a fraction of (Fig. 7), can be determined on the basis of , as a (9). This enables the easy determination of function of the desired inverter switching frequency. The following relationships can be determined empirically from Fig. 10: in When these relationships are used with (9), the following equation results: (10) Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply. CHANDORKAR: NEW TECHNIQUES FOR INVERTER FLUX CONTROL Fig. 11. Utility-connected three-phase inverter. 885 either by the use of a phase-locked loop (PLL) or by the use of a power-frequency droop characteristic as in [10], [11]. In either case, in the steady state, the – frame angular frequency equals the mains frequency, and its position relative to the mains voltage determines the sharing of the total load power between the inverter and the utility mains. The inverter controller is required to align the filtered output voltage vector with the rotating axis of the reference frame. Inherent to this requirement is the need to actively damp the and . As shown natural oscillations of the filter formed by below, the inverter flux vector can be used as a very effective forcing quantity to achieve this. Further, being a continuous quantity, it is very convenient to use the flux vector to define the power angle, which essentially determines the flow of real power from the inverter to the load bus [10]. In Fig. 12, this is the angle between the vectors and , where is the flux vector associated with the load bus voltage . For the purpose of the flux-based control of the inverter interface, one of the state variables used to model the interface is the output voltage vector on the filter capacitor. The other state , instead of the variable is chosen to be the filter flux vector conventionally used inductor current in . In the reference frame rotating with the angular velocity , the inverter and filter are modeled, in terms of the flux vectors, by the following vector differential equation: (11) Fig. 12. Flux-vector diagram of inverter interface. Similarly to the curves of Fig. 7, (10) is independent of the , and of the nominal frequency . However, dc voltage s. Using it is valid only for an integration time step (10), it is possible to approximately calculate the value of the tolerance , which results in a specified switching frequency . is the filter natural freIn this equation, is the current fed by the inverter and filter into quency, and the load bus. The inverter flux vector is the forcing quantity, used to align the filtered voltage vector with the rotating axis, and ensure that it has the desired magnitude. It is readily seen that the two eigenvalues of the complex system matrix of (11) are V. UTILITY-CONNECTED INVERTER CONTROL Inverter flux-vector control can be used very effectively for the control of inverters which have their outputs connected to the main utility system [9]. This section describes the flux control of a three-phase inverter connected to the utility system through a sine-wave output filter. Fig. 11 shows the schematic diagram of a three-phase inverter connected to the mains, and together with the mains, feeding a load, in the manner of a line-interactive uninterruptible power supply (UPS) system. Fig. 12 shows the vector diagram of the inverter interface, in a reference frame rotating with the angular velocity . The axes – are stationary, and the axes – rotate as shown, counterclockwise with an angular velocity . The instantaneous position of the rotating – reference frame axes is determined relative to the utility mains voltage, To force the filtered voltage vector to assume the desired can be generated value , the inverter flux-vector reference by a proportional–integral (PI) controller acting on the voltage , in the rotating reference frame vector error (12) and In this equation, the complex constants are the gains of the rotating frame vector PI controller. For the purpose of controller design, it is assumed that the inverter, as controlled by the flux modulator, is capable of producing the commanded flux vector at its output with is negligible delay. That is, the assumption that always valid. With this assumption, the controller action can be Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply. 886 Fig. 13. IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 3, MAY/JUNE 2001 Closed-oop system based on flux-vector control. easily included in the system description of (11). The resulting closed-loop system is as follows: Fig. 14. (13) Fig. 13 shows the schematic diagram of the closed-loop voltage control system, based on inverter flux control. Equation , in which the (13) is in the classical form definitions of the matrices , , and are readily apparent. The complex system of (13) can be expanded into a system of six real first-order differential equations, by expanding all into their components, and setting the vectors and . complex gains to be Three of the six eigenvalues of the real system matrix are the of (13), and the three eigenvalues of the complex matrix other three are their complex conjugates. and can be selected to ensure the The complex gains desired steady-state value for the filtered voltage vector , and to shape its transient response. The characteristic polynomial of the system matrix [see (13)] is found to be (14) and affects the roots of , which The choice of are the eigenvalues of the closed-loop system matrix of (13). and of Conversely, the specification of two distinct roots results in two complex linear simultaneous equations of the type These two equations can be used to solve for unique values and . These values are then used to calculate the third of of , which must be checked for acceptability. If it root is not acceptable, the process of calculating the complex gains must be repeated. are determined as described above, When the roots of the active damping of the filter natural oscillations is guaranteed, in addition to the regulation of the voltage vector . No-load startup of inverter and filter. It is important to note that only a simple PI regulator is sufficient in this case to stabilize the filter, and there is no need for a derivative term or high-pass filtering. This is a consequence of the use of the inverter and filter flux to model the system. If, instead, the traditional model with voltages and currents is used, a PI regulator can never stabilize the filter, even in the rotating reference frame. A. Experimental Results The closed-loop flux-based voltage control method described above was implemented for a 2.25-kW 110-V three-phase MOSFET inverter feeding a load as shown in Fig. 11, together with the 110-V three-phase mains. The system component values were: mH F H The complex controller gains were fixed at The frequency of the rotating – reference frame of the inverter was determined according to a power-frequency droop characteristic as in [11]. The control system implementation including the flux modulator was on a Motorola 56000 fixed-point DSP system operating on a 27-MHz clock. Fig. 14 shows the startup on no load of the isolated inverter. The upper trace of the oscillogram shows the line–neutral voltage across the filter capacitor. The lower trace shows the inverter output line current. The inverter and filter rapidly reach steady state, and the filter natural oscillations are quickly damped. Fig. 15 shows the inverter and filter performance immediately following the connection of the utility mains to the load bus. Prior to the mains connection, the inverter was supplying the load by itself. After mains connection, the utility picks up the load, as is indicated by the increase in the utility current (upper trace). The lower trace shows the line–neutral voltage across the filter capacitor. The middle trace shows the line–neutral voltage, expanded tenfold in time, around the instant of the utility connection. There are no disturbances on the filter output voltage before or after the point of utility connection. This is achieved Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply. CHANDORKAR: NEW TECHNIQUES FOR INVERTER FLUX CONTROL 887 Electronics Consortium (WEMPEC), University of Wisconsin, Madison. The author wishes to thank ABB Corporate Research Ltd., Switzerland, for extending support in carrying out this work. REFERENCES Fig. 15. Utility connection to load bus. by the tight control of the voltage by inverter switching due to flux-vector control. VI. CONCLUSION This paper has presented the direct control of the inverter output flux vector by means of the inverter switching. The two components of the flux vector are independently controlled so as to guide the vector along a desired trajectory. The method lends itself very well to a digital implementation on a DSP. The linearity between the fundamental output voltage magnitude and the output frequency is examined. The limits on the inverter flux reference magnitude, needed to ensure operation in the linear region, are presented. The dependence of the switching frequency on various modulator parameters is examined. Of particular practical importance is the dependence of the switching frequency on the error tolerance band value . An approximate relationship between and the switching frequency is presented, which is independent of the inverter dc voltage, and of the nominal output frequency. This would prove useful for designing a switching frequency limiting scheme, especially for high-power inverters having strong switching frequency limitations. The design and implementation of a flux-based controller for a three-phase inverter connected to the utility mains was presented. This illustrates the advantages of using the inverter flux vector as a forcing variable in the control. ACKNOWLEDGMENT The work reported on utility-connected inverter control was carried out in the Wisconsin Electric Machines and Power [1] A. M. Hava, R. J. Kerkman, and T. A. Lipo, “A high performance generalized discontinuous PWM algorithm,” in Proc. IEEE APEC’97, 1997, pp. 886–894. [2] D. M. Brod and D. W. Novotny, “Current control of VSI-PWM inverters,” IEEE Trans. Ind. Applicat., vol. 21, pp. 562–570, May/June 1985. [3] A. Mertens, “Performance analysis of three-phase inverters controlled by synchronous delta-modulation systems,” IEEE Trans. Ind. Applicat., vol. 30, pp. 1016–1027, July/Aug. 1994. [4] I. Takahashi and T. Noguchi, “A new quick-response and high-efficiency control strategy of an induction motor,” IEEE Trans. Ind. Applicat., vol. 22, pp. 820–827, Sept./Oct. 1986. [5] V. G. Török, “Near-optimum on-line modulation of PWM inverters,” in Proc. IFAC Conf. Control in Power Electronics and Electric Drives, Lausanne, Switzerland, 1983, pp. 247–254. [6] A. Veltman, P. P. J. van den Bosch, and R. J. A. Gorter, “On-line optimal switching for 2-level and 3-level inverters using the fish method,” in Proc. IEEE PESC’93, 1993, pp. 1061–1067. [7] A. M. Trzynadlowski, M. M. Bech, F. Blaabjerg, and J. K. Pedersen, “An integral space-vector PWM technique for DSP-controlled voltagesource inverters,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1998, pp. 1215–1221. [8] S. Bhattacharya, A. Veltman, D. Divan, and R. Lorenz, “Flux based controller for active filter,” IEEE Trans. Ind. Applicat., vol. 32, pp. 491–502, May/June 1996. [9] L. Ängquist and L. Lindberg, “Inner phase angle control of voltage source converter in high power applications,” in Proc. IEEE PESC’91, 1991, pp. 293–298. [10] M. C. Chandorkar, D. M. Divan, and R. Adapa, “Control of parallel connected inverters in stand-alone AC supply systems,” IEEE Trans. Ind. Applicat., vol. 29, pp. 136–143, Jan./Feb. 1993. [11] M. C. Chandorkar, D. M. Divan, and B. Banerjee, “Control of distributed UPS systems,” in Proc. IEEE PESC’94, 1994, pp. 197–204. Mukul Chandorkar (M’84) received the B.Tech. degree from Indian Institute of Technology, Bombay, India, the M. Tech. degree from Indian Institute of Technology, Madras, India, and the Ph.D. degree from the University of Wisconsin, Madison, in 1984, 1987, and 1995, respectively, all in electrical engineering. He has several years of experience in the power electronics industry in India, Europe, and the U.S. During 1996–1999, he was with ABB Corporate Research Ltd., Baden-Dättwil, Switzerland. He is currently an Associate Professor in the Electrical Engineering Department, Indian Institute of Technology, Mumbai, India. His technical interests include uninterruptible power supplies, drives, real-time simulation of power electronic systems, and the measurement and analysis of power quality. Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.