THREE-PHASE voltage-source pulsewidth modulation

1144 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998 A Novel Overmodulation Technique for Space-Vector PWM Inverters Dong-Choon Lee, Member, IEEE, and G-Myoung Lee Abstract In this paper, a novel overmodulation technique for space-vector pulsewidth modulation (PWM) inverters is proposed. The overmodulation range is divided into two modes depending on the modulation index (). In mode I, the reference angles are derived from the Fourier series expansion of the reference voltage which corresponds to the. In mode II, the holding angles are also derived in the same way. The strategy, which is easier to understand graphically, produces a linear relationship between the output voltage and the up to sixstep operation. The relationship between those angles and the can be written in lookup tables or, for real-time implementation, can be piecewise linearized. In addition, harmonic components and total harmonic distortion (THD) of the output voltage are analyzed. When the method is applied to the V/f control of the induction motor, a smooth operation during transition from the linear control range to the six-step mode is demonstrated through experimental results. Index Terms Fourier series, inverter utilization, overmodulation, space-vector PWM. I. INTRODUCTION THREE-PHASE voltage-source pulsewidth modulation (PWM) inverters have been widely used for dc/ac power conversion since they can produce a variable voltage and variable frequency power. However, they require a dead time to avoid the arm-short and snubber circuits to suppress the switching spike. Apart from these ancillary aspects, the PWM inverters have an essential problem that they cannot produce voltages as large as the six-step inverters can. That is, the dc bus voltage cannot be utilized to the maximum. To increase the voltage utilization of the sinusoidal PWM inverter, a method of the addition of the third harmonics to the reference voltage was proposed by which the fundamental component can be increased by 15.5% [1]. In a space-vector PWM inverter, which is widely used, the voltage utilization factor can be increased to 0.906, normalized to that of the sixstep operation [2]. On the other hand, different discontinuous PWM strategies were analyzed in [3], where the modulation waveform of a phase has at least one segment of 60 which is clamped to the positive and/or negative dc bus for, at most, a total of 120 in a fundamental period during which no switching in either inverter arm occurs. Recently, it is shown that discontinuous PWM schemes and the space-vector PWM Manuscript received August 20, 1997; revised January 28, 1998. This work was supported by the Electrical Engineering and Science Research Institute (EESRI), Korea, under Project 95-67. Recommended by Associate Editor, O. Ojo. The authors are with the School of Electrical and Electronic Engineering, Yeungnam University, Kyungbuk 712-749, Korea. Publisher Item Identifier S 0885-8993(98)08236-2. Fig. 1. Diagram of space voltage vectors. can be obtained by properly adding a zero-sequence voltage to the original modulation waveform [4]. By injecting the zerosequence voltage, the modulation index can be increased up to 0.906. On the other hand, a few off-line PWM methods were proposed to optimize the performance index. With those strategies, not only either particular harmonic components can be eliminated [5] or total harmonics may be minimized [6], but also the maximum utilization of the inverter can be obtained. However, since their transient responses are slow, it is difficult for them to be applied to high-performance motor drives. It had not been a great interest to increase the inverter utilization until a few recent overmodulation methods were proposed [7] [11]. Kerkman modeled the inverter gain as a function of the modulation index () using a describing function from which a compensated modulation index to give the desired fundamental voltage component was approximately derived for practical implementation [7]. However, the approximate inverter model gives a nonlinear inverter gain. In [8] and [9], this nonlinear characteristic was eliminated by using a simple lookup table. The result is a linear input to output voltage transfer function from PWM to six-step operation of the inverter. Holtz proposed a continuous control of PWM inverters in the overmodulation range [10]. In this scheme, there are two modes of overmodulation depending on the modulation index. In mode I, however, the fundamental voltage cannot be generated as exactly equal to the reference voltage since the contribution of the voltage increment around each corner of the hexagon to the fundamental component differs from that of the voltage decrement around the center of each side 0885 8993/98$10.00 1998 IEEE

LEE AND LEE: NOVEL OVERMODULATION TECHNIQUE FOR SPACE-VECTOR PWM INVERTERS 1145 Fig. 2. Trajectory of reference voltage vector and phase voltage waveform in mode I. of the hexagon since it is dealt with in an average meaning. So, it gives somewhat nonlinear transfer characteristics of the inverter in overmodulation mode I. For mode II, there is also no adequate explanation of the method of controlling the fundamental component of the output voltage. Another digital continuous control for the space-vector PWM inverter was proposed in [11], where two modes of the overmodulation in [10] are incorporated in single mode, by which the implementation becomes simpler, but the linear transfer characteristic of the inverter is lost in theory and much higher harmonics are generated. In this paper, a novel overmodulation strategy for the spacevector PWM to produce the exact fundamental voltage versus the modulation index is proposed, where reference angles and holding angles based on Fourier series expansion of the desired output voltage are derived. The principle is most simple to understand graphically. With this scheme, a linear control of the inverter output voltage can be obtained over the whole overmodulation range. For the dc-link voltage disturbance, the proposed method is shown to be effective as well. In addition, harmonic components of the output voltage and the total harmonic distortion (THD) are analyzed. When the scheme is applied to the V/f control of induction motor drives, it is demonstrated that a smooth transient operation can be obtained in overmodulation range by experimental results. II. A NOVEL OVERMODULATION STRATEGY In this section, a novel overmodulation strategy for the space-vector PWM is derived from developing Fourier series expansion of the waveform of the phase voltage reference which gives the desired fundamental component. For simple analysis, a dead-time effect is neglected. The modulation index for PWM inverters is defined here as (1) where is the phase voltage reference and is the inverter input voltage. According to the modulation index, the PWM range is divided into three regions as follows. A. Linear Modulation At first, a principle of the space-vector modulation is described briefly. The space voltage vectors involve six effective vectors and two zero vectors as shown in Fig. 1. A voltage reference vector is composed of time-average components of two effective vectors adjacent to it and one zero vector. That is, where is the sampling period of the PWM and and are time intervals of applying and vectors, respectively. The time intervals of and for zero-voltage vectors are calculated as where is a phase angle of the reference voltage vector. Below, the space-vector modulation generates sinusoidal output voltages. The trajectory of output voltages at traces a circle inscribed to the hexagon. Above it, the voltage waveform of the inverter is distorted, where magnitude becomes smaller than that of the reference voltage. B. Overmodulation Mode I The overmodulation mode I is operated when the magnitude of a compensated voltage reference vector which is boosted to produce a desired fundamental voltage of is between two radii of an inscribed circle and a circumscribed circle of the hexagon. Fig. 2 shows the trajectory of three voltage vectors rotating in a complex plane (left part) and the phase voltage waveform of an actual voltage reference vector (bold line) transformed in a time domain (right part) [12], which is modulated actually by the inverter. Here, the denotes a reference angle measured from the vertex to the intersection of the compensated voltage vector trajectory with the side. For a given voltage reference, the phase voltage waveform is divided into four segments. The voltage equations in each (2) (3) (4) (5)

1146 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998 limited up to the side of the hexagon. Then, the switching intervals through (3) (5) are corrected as [13] (12) (13) (14) As known from Fig. 2, the upper limit in mode I is when Then, the modulation index is 0.952, which is known from (10) and (11). When the is higher than 0.952, another overmodulation algorithm is needed. Fig. 3. Reference angle with regard to modulation index (solid line: numerical, dashed line: piecewise linearized). segment are expressed as for (6) for (7) for (8) where and is an angular velocity of the fundamental voltage reference vector. Expanding (6) (9) in a Fourier series and taking the fundamental component of it, the resultant equation can be expressed as for (9) C. Overmodulation Mode II In mode I, the angular velocity of the compensated and actual voltage reference vectors is both the same and constant for each fundamental period. Under such a condition, output voltages higher than cannot be generated since there exists no more surplus area to compensate for the voltage loss even if the modulation index is increased above that. In modulation ranges higher than 0.952, the actual voltage reference vector is held at a vertex for particular time and then moves along the side of the hexagon for the rest of the switching period. The holding angle controls the time interval the active switching state remains at the vertices, which uniquely controls the fundamental voltage. A basic concept of the mode II is similar to [10], where it lacks an explicit explanation about how to derive the algorithm. Here, detailed expressions based on Fourier series expansion just in the same way as in mode I will be developed. From Fig. 4, the voltage equations in four segments are expressed as for (15) for (16) for (17) (10) where and denote integral ranges of each voltage function as shown in Fig. 2. Integrating (10) numerically, we can obtain the value of with regard to the Since represents the peak value of the fundamental component, from the definition of the modulation index of (1) where for (18) (19) (20) (11) Thus, a relationship between the and the which gives a linearity of the output voltage is determined, which is plotted in a solid line in Fig. 3. For the voltage reference vector exceeding the side of the hexagon, the inverter cannot generate the output voltage as large as the voltage reference since the maximum output is The and are phase angles of the actual voltage reference vector rotating for and, respectively, as shown in Fig. 5. The two angles of and are derived as follows. The actual voltage reference vector rotates from to at a little higher speed while the fundamental one is rotating at constant speed from to Equation (19)

LEE AND LEE: NOVEL OVERMODULATION TECHNIQUE FOR SPACE-VECTOR PWM INVERTERS 1147 Fig. 4. Trajectory of reference voltage vector and phase voltage waveform in mode II. Fig. 6. Holding angle with regard to modulation index (solid line: numerical, dashed line: piecewise linearized). Fig. 5. Angular displacement of reference and actual voltage vectors. is simply derived from a proportional relationship for angular displacements of these two vectors as expansion as (22) (21) Thereafter, the actual voltage reference vector is held at a vertex while the fundamental one is continuously rotating from to For, the situation is reversed. The actual voltage reference vector is held at a vertex while the fundamental one is rotating from to At, the actual voltage reference vector starts to rotate and is aligned with the fundamental one at The same analogy as the above for gives the expression of (20), which is also applied for Substituting (15) (18) into (10) and matching the result of its integral with (11), a relationship between the modulation index and the holding angle is obtained, which is plotted in a solid line in Fig. 6. III. HARMONIC ANALYSIS In Section II, the reference angle and the holding angle were derived which give a linear inverter gain in the complete overmodulation range. Here, harmonic components of the output voltage are analyzed using the Fourier series where is given by (6) (9) in mode I and (15) (18) in mode II. A numerical integration of (22) shows that even-order harmonics and triplen harmonics are eliminated in the output voltage. The four lowest harmonic components (5th, 7th, 11th, and 13th) versus the are illustrated in Fig. 7. Some harmonic components are absent at the particular modulation index. Fig. 8 shows voltage harmonic spectra through fast Fourier transform (FFT). The magnitude of each harmonic component coincides well with the result of (22). The THD factor is defined as THD (23) where and are the rms value and fundamental component of the phase voltage, respectively. Fig. 9 shows THD factor of the output voltage. As the modulation index increases, especially in mode II, the THD is deteriorated steeply and it culminates to 0.311 at The THD for [8] and [10] is similar to that in this method. However, the THD in [11] is much higher, as shown in Fig. 9, since the voltage waveform has jumps.

1148 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998 Fig. 7. Harmonic voltage components. Fig. 10. Experiment system. (a) (b) (c) Fig. 8. Harmonic spectra by FFT, normalized to fundamental component at = 1:0 (simulation). (d) Fig. 11. Phase voltage waveforms. (a) =0:78; (b) =0:938; (c) =0:968; and (d) =1:0: Fig. 9. Total harmonic distortion. IV. EXPERIMENTS AND DISCUSSIONS To confirm the validity of the proposed scheme, experiments were performed for the V/f control of induction motor drive fed by an insulated gate bipolar transistor (IGBT) PWM inverter. Fig. 10 shows the experiment system with a digital signal processor (DSP) board. In practice, such a highperformance DSP is not required for implementation of the V/f control of the induction motor. Also, the dc-link voltage is usually measured for the space-vector modulation and overvoltage protection, and the current is measured only for monitoring. The inverter switching frequency is 3.5 khz, and the dc-link voltage is 287 V, which is set a little lower than at nominal operation in order to show distinctly the effect of the overmodulation algorithm. The induction motor used for experiments is rated at 3 Hp, 220 V, and 60 Hz. Let us consider a case using lookup tables for data angle. First, the reference and holding angles are calculated off line and stored in the memory with regard to the increment of 0.001 from to. If a desired reference voltage is given, a modulation index is calculated by (1) and the reference angle or holding angle corresponding to it is read out from the lookup table. In the case of the mode I, the magnitude of a compensated reference voltage vector is easily calculated using the reference angle as from which the switching interval is calculated. In mode II, the holding angle according to the is first determined, and then the phase angle of the actual voltage reference vector is determined by considering (19) and (20) with regard to the Then, of course, its magnitude reaches the side of the hexagon. Fig. 11 shows the waveform of the output phase voltage at different modulation indices, which represents the voltage

LEE AND LEE: NOVEL OVERMODULATION TECHNIQUE FOR SPACE-VECTOR PWM INVERTERS 1149 (a) (b) (c) (d) Fig. 14. Transient responses for dc-link voltage disturbance (experimental). Fig. 12. Phase current waveforms. (a) = 0:78; (b) =0:938; (c) =0:968; (d) =1:0: (a) Fig. 15. Transient responses for dc-link voltage disturbance (simulation). (b) Fig. 13. Transient responses for motor frequency change. (a) Linear! mode I and (b) mode I! mode II. value averaged over each switching period for easy monitoring. The phase currents corresponding to each voltage in Fig. 11 are illustrated in Fig. 12. According to the increase of the modulation index, the phase currents are more distorted. Fig. 13 shows transient responses of the voltage and current for the change of the motor frequency. Since a linearity of the voltage modulation is guaranteed, the motor current is not changed abruptly, but smoothly. When a disturbance in the dc-link voltage occurs, the inverter is often operated in overmodulation range. Fig. 14 shows the transient responses in case of the decrease of 10% from the operating dc-link voltage. Since the inverter input voltage is decreased, the modulation index is boosted so that the fundamental component of the output voltage can be kept the same. In Fig. 15, at a similar condition to that in Fig. 14, torque ripples due to current harmonics are generated, but the average torque is kept constant. Since the torque ripples are filtered by the motor inertia, the motor speed is little changed. Fig. 16 shows FFT spectra of the phase voltage analyzed by a digital oscilloscope of which the results are the same as those in Fig. 8. If the hardware memory cannot allow lookup tables for the reference angle and the holding angle, they can be calculated in real time by piecewise-linear approximation as shown in dashed lines in Figs. 3 and 5. Then, a transfer characteristic of the output to the modulation index is shown in Fig. 17, from which it is known that the nonlinearity is sufficiently tolerable

1150 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998 It is expected that the overmodulation algorithm is very effective to the PWM inverter controls in a frequent variation of the utility voltage or battery-fed inverter system. APPENDIX The equations of the reference and holding angle piecewise linearized as a function of the are as follows. A. Mode I Fig. 16. Harmonic spectra by FFT, normalized to fundamental component at = 1:0 (experimental). B. Mode II REFERENCES Fig. 17. Output voltage versus modulation index in piecewise linearization. for practical purposes. The piecewise-linearized equations for two overmodulation modes are given in the Appendix. V. CONCLUSIONS In the space-vector modulation, a linear control of the inverter output voltage was obtained up to by a novel overmodulation strategy. The method is based on Fourier series representation of the reference voltage, where a graphical transformation between complex voltage vectors and the phase voltage in time domain is used implicitly. The reference angles in mode I and the holding angles in mode II were derived as a function of the modulation index by numerical analysis. These data can be written in lookup tables or, for real-time implementation, piecewise linearized. In addition, each harmonic component and the THD of the output voltage were analyzed. The THD factor in this scheme was shown to be lower than that of other method. In spite of the dc-link voltage disturbance, the fundamental component of the inverter output voltage can be kept constant by boosting the modulation index. When the method is applied to the V/f control of the induction motor, a smooth operation during transition from the linear control range to the six-step mode was demonstrated through experimental results. [1] G. Buja and G. Indri, Improvement of pulse width modulation techniques, Arch. fr Elektrotech., vol. 57, pp. 281 289, 1975. [2] H. W. van der Broek, H. C. Skudelny, and G. V. Stanke, Analysis and realization of PWM based on voltage space vectors, IEEE Trans. Ind. Applicat., vol. 24, no. 1, pp. 142 150, 1988. [3] J. W. Kolar, H. Ertl, and F. C. Zach, Influence of the modulation method on the conduction and switching losses of a PWM converter system, IEEE Trans. Ind. Applicat., vol. 27, no. 6, pp. 1064 1075, 1991. [4] V. Blasko, A hybrid PWM strategy combining modified space vector and triangle comparison methods, in IEEE PESC Conf. Rec., 1996, pp. 1872 1878. [5] H. S. Patel and R. G. Hoft, Generalized techniques of harmonic elimination and voltage control in thyristor inverters: Part I Harmonic elimination, IEEE Trans. Ind. Applicat., vol. 9, pp. 310 317, May/June 1973. [6] G. S. Buja, Optimum output waveforms in PWM inverters, IEEE Trans. Ind. Applicat., vol. 16, no. 6, pp. 830 836, 1980. [7] R. J. Kerkman, B. J. Seibel, D. M. Brod, T. M. Rowan, and D. Leggate, A simplified inverter model for on-line control and simulation, IEEE Trans. Ind. Applicat., vol. 27, no. 3, pp. 567 573, 1991. [8] R. J. Kerkman, T. M. Rowan, D. Leggate, and B. J. Seibel, Control of PWM voltage inverters in the pulse dropping region, IEEE Trans. Power Electron., vol. 10, no. 5, pp. 559 565, 1995. [9] R. J. Kerkman, D. Leggate, B. J. Seibel, and T. M. Rowan, Operation of PWM voltage source-inverters in the overmodulation region, IEEE Trans. Ind. Electron., vol. 43, no. 1, pp. 132 141, 1996. [10] J. Holtz, W. Lotzkat, and A. M. Khambadkone, On continuous control of PWM inverters in the overmodulation range including the six-step mode, IEEE Trans. Power Electron., vol. 8, no. 4, pp. 546 553, 1993. [11] S. Bolognani and M. Zigliotto, Novel digital continuous control of SVM inverters in the overmodulation range, IEEE Trans. Ind. Applicat., vol. 33, no. 2, pp. 525 530, 1997. [12] P. Vas, Electrical Machines and Drives: A Space-Vector Theory Approach. New-York: Oxford Science, 1992. [13] T. G. Habetler, A space vector-based rectifier regulator for ac/dc/ac converters, in Proc. EPE, vol. 2, 1991, pp. 101 107.

LEE AND LEE: NOVEL OVERMODULATION TECHNIQUE FOR SPACE-VECTOR PWM INVERTERS 1151 Dong-Choon Lee (S 90 M 95) was born in Korea in 1963. He received the B.S., M.S., and Ph.D. degrees in electrical engineering, all from Seoul National University, Seoul, Korea, in 1985, 1987, and 1993, respectively. He was a Research Engineer at Daewoo Heavy Industry from 1987 to 1988. He also was at the Research Institute of Science Engineering of Seoul National University under a Post-Doctoral Fellowship for one year. He has been a Faculty Member of the School of Electrical and Electronic Engineering, Yeungnam University, Kyungbuk, Korea, since 1994. Also, he is currently a Visiting Scholar at the Department of Electrical Engineering, Texas A&M University, College Station. His research interests include ac machine drives, static power converters, and DSP applications. G-Myoung Lee was born in Korea in 1970. He received the B.S. degree from Kyungil University, Korea, in 1995 and the M.S. degree from Yeungnam University, Kyungbuk, Korea, in 1997, both in electrical engineering. He is currently working toward the Ph.D. degree at Yeungnam University. His research interests are motor drives and controls and PWM converters and inverters.