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A Novel Overmodulation Technique for Space-Vector Pwm Inverters

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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 (MI). In mode I, the reference angles are derived from the Fourier series expansion of the reference
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  1144 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998 A Novel Overmodulation Technique forSpace-Vector PWM Inverters Dong-Choon Lee, Member, IEEE, and G-Myoung Lee  Abstract— In this paper, a novel overmodulation techniquefor space-vector pulsewidth modulation (PWM) inverters is pro-posed. The overmodulation range is divided into two modesdepending on the modulation index (MI). In mode I, the referenceangles are derived from the Fourier series expansion of thereference voltage which corresponds to the MI. In mode II, theholding angles are also derived in the same way. The strategy,which is easier to understand graphically, produces a linearrelationship between the output voltage and the MI up to six-step operation. The relationship between those angles and the MIcan be written in lookup tables or, for real-time implementation,can be piecewise linearized. In addition, harmonic componentsand total harmonic distortion (THD) of the output voltage areanalyzed. When the method is applied to the V/f control of theinduction motor, a smooth operation during transition from thelinear control range to the six-step mode is demonstrated throughexperimental results.  Index Terms— Fourier series, inverter utilization, overmodula-tion, space-vector PWM. I. I NTRODUCTION T HREE-PHASE voltage-source pulsewidth modulation(PWM) inverters have been widely used for dc/ac powerconversion since they can produce a variable voltage andvariable frequency power. However, they require a dead timeto avoid the arm-short and snubber circuits to suppress theswitching spike. Apart from these ancillary aspects, the PWMinverters have an essential problem that they cannot producevoltages as large as the six-step inverters can. That is, the dcbus voltage cannot be utilized to the maximum.To increase the voltage utilization of the sinusoidal PWMinverter, a method of the addition of the third harmonics tothe reference voltage was proposed by which the fundamentalcomponent can be increased by 15.5% [1]. In a space-vectorPWM inverter, which is widely used, the voltage utilizationfactor can be increased to 0.906, normalized to that of the six-step operation [2]. On the other hand, different discontinuousPWM strategies were analyzed in [3], where the modulationwaveform of a phase has at least one segment of 60 whichis clamped to the positive and/or negative dc bus for, at most,a total of 120 in a fundamental period during which noswitching in either inverter arm occurs. Recently, it is shownthat 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 tothe srcinal modulation waveform [4]. By injecting the zero-sequence voltage, the modulation index can be increased upto 0.906.On the other hand, a few off-line PWM methods wereproposed to optimize the performance index. With thosestrategies, not only either particular harmonic components canbe eliminated [5] or total harmonics may be minimized [6], butalso the maximum utilization of the inverter can be obtained.However, since their transient responses are slow, it is difficultfor them to be applied to high-performance motor drives.It had not been a great interest to increase the inverterutilization until a few recent overmodulation methods wereproposed [7]–[11]. Kerkman modeled the inverter gain as afunction of the modulation index (MI) using a describingfunction from which a compensated modulation index to givethe desired fundamental voltage component was approximatelyderived for practical implementation [7]. However, the approx-imate inverter model gives a nonlinear inverter gain. In [8]and [9], this nonlinear characteristic was eliminated by usinga simple lookup table. The result is a linear input to outputvoltage transfer function from PWM to six-step operation of the inverter.Holtz proposed a continuous control of PWM invertersin the overmodulation range [10]. In this scheme, there aretwo modes of overmodulation depending on the modulationindex. In mode I, however, the fundamental voltage cannotbe generated as exactly equal to the reference voltage sincethe contribution of the voltage increment around each cornerof the hexagon to the fundamental component differs fromthat 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, thereis also no adequate explanation of the method of controllingthe fundamental component of the output voltage.Another digital continuous control for the space-vectorPWM 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 lineartransfer characteristic of the inverter is lost in theory and muchhigher harmonics are generated.In this paper, a novel overmodulation strategy for the space-vector PWM to produce the exact fundamental voltage versusthe modulation index is proposed, where reference angles andholding angles based on Fourier series expansion of the desiredoutput voltage are derived. The principle is most simple tounderstand graphically. With this scheme, a linear control of the inverter output voltage can be obtained over the wholeovermodulation range. For the dc-link voltage disturbance, theproposed method is shown to be effective as well. In addition,harmonic components of the output voltage and the totalharmonic distortion (THD) are analyzed. When the schemeis applied to the V/f control of induction motor drives, it isdemonstrated that a smooth transient operation can be obtainedin overmodulation range by experimental results.II. A N OVEL O VERMODULATION S TRATEGY In this section, a novel overmodulation strategy for thespace-vector PWM is derived from developing Fourier seriesexpansion of the waveform of the phase voltage referencewhich gives the desired fundamental component. For simpleanalysis, a dead-time effect is neglected. The modulation indexfor PWM inverters is defined here asMI (1)where is the phase voltage reference and is the inverterinput voltage.According to the modulation index, the PWM range isdivided into three regions as follows.  A. Linear Modulation MI  At first, a principle of the space-vector modulation is de-scribed briefly. The space voltage vectors involve six effectivevectors and two zero vectors as shown in Fig. 1. A voltagereference vector is composed of time-average components of two effective vectors adjacent to it and one zero vector. That is,(2)where is the sampling period of the PWM and andare time intervals of applying and vectors, respectively.The time intervals of and for zero-voltage vectorsare calculated as(3)(4)(5)where is a phase angle of the reference voltage vector.Below MI , the space-vector modulation generatessinusoidal output voltages. The trajectory of output voltagesat MI traces a circle inscribed to the hexagon. Aboveit, the voltage waveform of the inverter is distorted, wheremagnitude becomes smaller than that of the reference voltage.  B. Overmodulation Mode I MI  The overmodulation mode I is operated when the magnitudeof a compensated voltage reference vector which is boostedto produce a desired fundamental voltage of is betweentwo radii of an inscribed circle and a circumscribed circleof the hexagon. Fig. 2 shows the trajectory of three voltagevectors rotating in a complex plane (left part) and the phasevoltage 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, thedenotes a reference angle measured from the vertex to theintersection of the compensated voltage vector trajectory withthe side.For a given voltage reference, the phase voltage waveformis divided into four segments. The voltage equations in each  1146 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998 Fig. 3. Reference angle with regard to modulation index (solid line: numer-ical, dashed line: piecewise linearized). segment are expressed asfor (6)for (7)for (8)for(9)where and is an angular velocity of the fundamentalvoltage reference vector.Expanding (6)–(9) in a Fourier series and taking the funda-mental component of it, the resultant equation can be expressedas(10)where and denote integral ranges of each voltagefunction as shown in Fig. 2. Integrating (10) numerically,we can obtain the value of with regard to theSince represents the peak value of the fundamentalcomponent, from the definition of the modulation index of (1)MI (11)Thus, a relationship between the MI and the which gives alinearity of the output voltage is determined, which is plottedin a solid line in Fig. 3.For the voltage reference vector exceeding the side of thehexagon, the inverter cannot generate the output voltage aslarge as the voltage reference since the maximum output islimited up to the side of the hexagon. Then, the switchingintervals through (3)–(5) are corrected as [13](12)(13)(14)As known from Fig. 2, the upper limit in mode I is whenThen, the modulation index is 0.952, which is knownfrom (10) and (11). When the MI is higher than 0.952, anotherovermodulation algorithm is needed. C. Overmodulation Mode II MI  In mode I, the angular velocity of the compensated andactual voltage reference vectors is both the same and constantfor each fundamental period. Under such a condition, outputvoltages higher than MI cannot be generated sincethere exists no more surplus area to compensate for the voltageloss even if the modulation index is increased above that.In modulation ranges higher than 0.952, the actual voltagereference vector is held at a vertex for particular time andthen moves along the side of the hexagon for the rest of the switching period. The holding angle controls the timeinterval the active switching state remains at the vertices,which uniquely controls the fundamental voltage. A basicconcept of the mode II is similar to [10], where it lacks anexplicit 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. FromFig. 4, the voltage equations in four segments are expressed asfor (15)for (16)for (17)for (18)where(19)(20)The and are phase angles of the actual voltage referencevector rotating for and, respectively, as shown in Fig. 5.The two angles of and are derived as follows. Theactual voltage reference vector rotates from to ata little higher speed while the fundamental one is rotating atconstant 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. 5. Angular displacement of reference and actual voltage vectors. is simply derived from a proportional relationship for angulardisplacements of these two vectors as(21)Thereafter, the actual voltage reference vector is held at avertex while the fundamental one is continuously rotatingfrom to For , thesituation is reversed. The actual voltage reference vector isheld at a vertex while the fundamental one is rotating fromto At , the actualvoltage reference vector starts to rotate and is aligned withthe fundamental one at The same analogy as theabove for gives the expressionof (20), which is also applied forSubstituting (15)–(18) into (10) and matching the result of its integral with (11), a relationship between the modulationindex and the holding angle is obtained, which is plotted ina solid line in Fig. 6.III. H ARMONIC A NALYSIS In Section II, the reference angle and the holding anglewere derived which give a linear inverter gain in thecomplete overmodulation range. Here, harmonic componentsof the output voltage are analyzed using the Fourier series Fig. 6. Holding angle with regard to modulation index (solid line: numerical,dashed line: piecewise linearized). expansion as(22)where is given by (6)–(9) in mode I and (15)–(18) inmode II. A numerical integration of (22) shows that even-orderharmonics and triplen harmonics are eliminated in the outputvoltage. The four lowest harmonic components (5th, 7th,11th, and 13th) versus the MI are illustrated in Fig. 7. Someharmonic components are absent at the particular modulationindex. Fig. 8 shows voltage harmonic spectra through fastFourier transform (FFT). The magnitude of each harmoniccomponent coincides well with the result of (22).The THD factor is defined asTHD (23)where and are the rms value and fundamental componentof the phase voltage, respectively. Fig. 9 shows THD factorof the output voltage. As the modulation index increases,especially in mode II, the THD is deteriorated steeply andit culminates to 0.311 at MI The THD for [8] and [10]is similar to that in this method. However, the THD in [11] ismuch higher, as shown in Fig. 9, since the voltage waveformhas jumps.

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