1、IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 5, MAY 1999617Analytical Technique for Calculating the Output Harmonics of an H-Bridge Inverter withDeadTimeC. M. Wu, Wing-hong Lau, Member, IEEE, and Henry Shu-hung Chung, Member, IEEEand its harmonic spec

2、trum is obtained by applying the Fourier analysis. However, these methods cannot show the detailed composition of the harmonic structure. In 7 an analytical technique based on a three-dimensional (3-D) model, together with a double Fourier series, has been developed to calculate the modulation produ

3、cts. Similar approaches have been used in 8 and 9 to analyze the harmonic characteristics of an H-bridge converter. A general expression for the harmonic components of the PWM pulse train has been derived and the information of the position and width of each PWM pulse are implicitly described by the

4、 Bessel functions, nevertheless, their methods have not considered the dead time. Recently, the dead-time effects on the harmonics have been reported in 10 using a modified 3-D model together with a double Fourier series.This paper presents an analytical method for calculating the harmonic character

5、istics of the output voltage of an H-bridge inverter with the consideration of dead time. The 3-D model described in 7 is modified to incorporate the dead time for generating the PWM signal. Double Fourier analysis is then applied to the 3-D model to obtain an elegant function to describe the harmon

6、ic characteristics. This function can be divided into two parts: an ideal part representing the PWM signal without dead time and a correction part representing the effects caused by the dead time. With this method, the dead-time pulse train can easily be computed without using the sophisticated nume

7、rical techniques in 5 and 6. Moreover, the detailed dead-time effects on the output harmonics can easily be studied. Section II shows the derivation of the 3-D model for generating the PWM pulse train without dead time. Section III covers the detailed derivation of the modified 3-D model incorporate

8、d with the dead time and the analytical solution for characterizing the harmonics. Simulations using PSpice have been carried out to verify the analytical solution developed, and the properties of the harmonics have been studied in detail and the results are given in Section IV. Finally, conclusion

9、is given in Section V.AbstractAn analytical technique for calculating the harmonic characteristics of the output voltage of an H-bridge inverter with dead time is presented. The analysis is based on a three- dimensional (3-D) model derived for generating the pulsewidth- modulated (PWM) pulse train.

10、By applying double Fourier anal- ysis, a generalized and elegant mathematical function for describ- ing the harmonic components of the output voltage is formulated. The function can be divided into two parts: an ideal part rep- resenting the PWM signal without dead time and a correction part represe

11、nting the dead-time effect. The function provides detailed composition of the fundamental component, signal har- monics, carrier harmonics, and cross-modulated harmonics. The proposed technique has been verified using examples and the theoretical predictions are confirmed with the results obtained f

12、rom simulations using PSpice.Index TermsDead time, harmonic analysis, pulse width mod- ulated inverters.I. INTRODUCTIONHE pulsewidth-modulated (PWM) H-bridge inverter is widely used in various industrial applications, such asTuninterruptible power supplies, class-D amplifiers, and active power filte

13、rs. Despite the fact that close to an ideal PWM signal can be generated and applied to a gate driver, the output waveform of the inverter will practically deviate from the expected one due to the nonideal switching devices used in the output stage. In fact, the introduction of the dead time1 for avo

14、iding the cross-conduction current through the leg of the inverter during the crossover between the upper and lower switches produces additional distortions. The distortion effects caused by the dead time have caused interest among researchers for many years. It has been shown in 2 and3 that the dea

15、d time may invoke operation instabilities in the AC motor drives. For the application of class Damplifiers, the dead time produces extra distortions at the output 4. Many methods have been developed to evaluate these distortions. Some approaches 5, 6 use sophisticated numerical techniques to find th

16、e position and width of the PWM pulses. The dead-time pulse train can then be computedII. THE 3-D MODEL FORGENERATING THE PWM PULSE TRAINThis section describes the formulation of the 3-D model for generating an ideal PWM pulse train without considering the dead time. The basic configuration of an H-

17、bridge PWMManuscript received January 11, 1997; revised March 19, 1998. This paper was recommended by Associate Editor J. Suykens.C. M. Wu was with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, China. He is now with the Department of Electric Power Engin

18、eering, South China University of Technology, Guangzhou, China.W-h. Lau and H. S-h. Chung are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, China.Publisher Item Identifier S 1057-7122(99)03881-7.inverter with supply voltageis shown in Fig. 1. The ou

19、tputvoltage of one leg is denoted byThe typical method for10577122/99$10.00 1999 IEEE618IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 5, MAY 1999Fig. 1.The basic configuration of an H-bridge inverter.(a)(b)Fig. 3. (a) 3-D Model of the SSNS PWM. (b) Its

20、 2-D representation.Fig. 2.The conventional representation of the PWM signal.Fig. 3(a) shows the 3-D model and how the PWM signal is generated. The projection of the intersections between thegenerating an ideal single-sided and natural-sampled (SSNS)PWM waveform with modulation signalis3-D functiona

21、nd planeon theplaneshown in Fig. 2. The time axis can also be represented by twoforms the PWM signal. The generation process can also bedifferent angular abscissasandwhereandillustrated in a two-dimensional (2-D) plane, as shown in Fig. 3(b). The curves represented by (4) in each carrier cycleare th

22、e angular frequencies of the carrier and modulationsignal respectively. The ratio betweenandis denotedand the line defined by (2) can be plotted on theplane,by, i.e.,as shown in Fig. 3(b). The projection of the intersection pointsbetween these curves on the axis clearly represents the PWM pulse trai

23、n. In fact, this 2-D representation is equivalent to(1)andis also given bypresent Fig. 2 in angular domainsandFor reasons ofsimplicity, this 2-D representation will be used to illustrate(2)the derivation of the harmonic characteristics of the PWM signal in Section III.This method can also be applied

24、 to the double-sided and natural-sampled (DSNS) PWM signal and the 3-D functionThe PWM pulse widthprojected on theaxis for aparticular cycle of the carrier signal can be expressed asis defined in (7) within the regionand(3)is the modulation index. Alternatively, (3) can bewhere written as(7)(4), a 3

25、-DWithin the regionandfunctioncan be defined aswhere(5)(8)The PWM defined bywaveform function and a plane, representing, is governed by (2), i.e.,(6)The 3-D model of the DSNS PWM and its 2-D represen- tation are shown in Fig. 4.WU et al.: CALCULATING OUTPUT HARMONICS OF H-BRIDGE INVERTER WITH DEAD T

26、IME619(a)Fig. 5.The dead-time effects. (a) Modulation signal vs . (b) Ideal PWMwaveform, upper switch drive signal. (c) Upper switch drive signal. (d) Lower switch drive signal. (e) Polarity of the current ia . (f) Output voltage va .found in the literature. In NTEC mode, changes direction twice or

27、more within one cycle of the modulation signal, or the duration is not equally shared for both directions for the case of changing direction once only. In practice, most of the PWM inverters are operated in TEC mode and the harmonic analysis described in this section is based on this(b)Fig. 4. (a) 3

28、-D Model of the PSNS PWM. (b) Its 2-D representation.situation. It should also be noted that the output current not necessarily sinusoidal.isIII. THE HARMONIC CHARACTERISTICS OF ANH-BRIDGE PWM INVERTER WITH DEAD TIMEThe detailed derivation of the harmonic characteristics, based on a 3-D model and do

29、uble-Fourier series approach in the presence of dead-time for both SSNS and DSNS PWM inverters, will be described in this section.B. The 3-D Model with Dead TimeWhen considering the effects of the dead time, the output PWM waveform will deviate from the ideal one. The 3-D model is valid as long as t

30、he inverter operates in TEC mode. During the dead time, both switches in a leg are off. For an inductive load, the output voltage of an H-bridge PWM inverter depends on the output current direction, which causes either the upper or lower body diodes to conduct during the dead time and, consequently,

31、 a voltage gain or loss will occur. For the PWM pulse train, shown in Fig. 5, whenis withinA. Assumptions and ConstraintAll mathematical derivations in following sections are based on the following assumptions.1)All the switching devices in the H-bridge inverter are ideal.The load is inductive.The d

32、ead time is inserted prior to the leading edges of the PWM pulses, which are applied to the gate driver, as shown in Fig. 5(b)(d).the region fromtoand the output current is negative2)3)(i.e., a gain in output voltage is obtained at the pulsetrailing edges. On the other hand, whenis within the region

33、fromtoand, a loss of voltage occursat the pulse leading edges.The gain and loss of voltage caused by the dead time willThe dead time is represented byin time domain,modify the shape of the 3-D model and its corresponding 2-D representation for a SSNS PWM inverter is shown in Fig. 6(a).in the angular

34、 domain in relation to the carrierfrequency, andin the angular domain relating toConsidering the situation of gaining voltage, the curvewillthe signal frequency. The validity of the harmonic analysis isbe replaced by curve and curveCurveis represented byalso constrained by both the direction and its

35、 correspondingis governed by bothand the to trace theisduration of the output currentwithin one cycle of thedead time. We can use a right-angle trianglemodulation signalThe operating mode of the outputcurve, as shown in Fig. 7. The slope ofcurrent can be classified into two modes: two even crossover

36、 (TEC) mode and non-TEC (NTEC) mode. In TEC mode,and the lengthis fixed to maintain a constant dead timeAs pointmoves alongfromto,can bechanges its direction (i.e.,crosses over the zero currenta curve, representing thenew trailing edge,point from positive value to negative value or vice versa) once

37、only, and the direction remains unchanged for the half- cycle of the modulation signal, as shown in Fig. 5(e). In fact,obtained by tracing the trajectory of point, i.e.,this operating mode has been widely used for the analyses(9)620IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND A

38、PPLICATIONS, VOL. 46, NO. 5, MAY 1999For DSNS PWM with dead timein a manner similar to that used above. follows:Forcan be obtained is defined as(12)and for(a)(b)(13)Fig. 6. The 2-D representations of the 3-D model for (a) SSNS and (b) DSNS PWM with dead time.whereFig. 6(b) shows the 2-D representati

39、on of the 3-D model for DSNS PWM with dead time.C. The Harmonic Characteristics ofis a periodic function in bothandwith periodand can be represented by double Fourier series 9(14)Fig. 7. The relationship between 0 ( ) and ( ):whereForwithin the range fromto, curveisreplaced by the straight linewith

40、slopeThis newcurve ofis used to replace the corresponding portion(15)to indicate the gain of output voltage, as shown inFig. 6(a). Sinceis much greater than(i.e.,formost applications, the linecan be considered in parallel is given in (10) withinBy substituting (10)(13) into (15)orcan bewith axis the

41、 regionand, consequently,expressed as the sum of an ideal part which is for the casewithout dead time and a correction part to describe the effects of the dead time. Using the SSNS PWM as an examplecan be written as(10)Whenis within the region fromtoand, a loss of voltage occurs at the pulse leading

42、 edges.The region corresponding to the leading edges is modified,as shown in Fig. 6(a). The lineand the slope of the line same assumption thatthe regionis replaced by the lineisWith theis given in (11) withinThe first term corresponds to the ideal part, and the sum of the second and third terms corr

43、esponds to the correction part. in (6) can also be written as the sum of an ideal part and(11)correction part, i.e.,Similarly,the ideal part corresponds to the situation without dead timeWU et al.: CALCULATING OUTPUT HARMONICS OF H-BRIDGE INVERTER WITH DEAD TIME621and the correction part represents

44、the distortion due to the deadtime. For SSNS PWMis given by (16) and (17)(16)-order Bessel function of the first kind, i.e.,whereis theFor DSNS PWM,is given by (18) and (19)(18)(17)where(19)where(20)622IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 5, M

45、AY 1999TABLE ITHE HARMONIC CHARACTERISTICS OBTAINED BY BOTH SIMULATIONS AND ANALYTICAL SOLUTIONS FOR MR = 80TABLE IITHE HARMONIC CHARACTERISTICS OBTAINED BY BOTH SIMULATIONS AND ANALYTICAL SOLUTIONS FOR MR = 8 If the dead time is zero, it is obvious that (17) and (19)will be zero andbecomes the solu

46、tion for an ideal PWMsituation. With differentand(16)(20) give a completeanalytical description of the harmonic characteristics of the output voltage of a PWM inverter with and without the dead time.IV. SIMULATION RESULTS AND DISCUSSIONSThe operating conditionsofthe second example are asfollow:Hz,Hz

47、,V.s, is ten times lessis smaller in this case, than that of the first example. SinceAs the DSNS PWM is commonly used in industrial ap- plications, and its analytical expression is much simpler as compared with that of the SSNS PWM, the following discus-the lower order harmonic will be the phasor su

48、m of several harmonics generated by cross-modulation. For example, theseventh harmonic is contributed by both the seventh signalsions are based on the DSNS PWM case. In addition, chosen to be an integer for convenience purposes.isharmonic (for harmonic forand andand the cross-modulation From Table I

49、I, it canbe seen that when the magnitude of the resultant harmonics isA. Illustrative ExamplesIn order to verify the analytical solution derivedlarge enough, i.e., larger than 0.08 for this example, the results obtained from both simulations and analytical solutions areinSection III, we use PSpice t

50、o perform simulations for comparison. When deriving the expressions (10)(13) it isconsistent. For a smallthe number of the pulses withinone signal cycle is less than that of a largehence, smallassumed thatTwo examples with differentwillerrors in the pulse position and width can produce significant e

51、rrors in the simulation. On the other hand, the output PWM waveform is dominated by the harmonics with large power, thus the percentage error in those harmonics with less power will be relatively large as shown from simulation results. This explains the large discrepancies between the simulation and

52、 analytical solutions for harmonics with lower magnitude, as shown in Table II.be used to illustrate the effects of this assumption. The first example is an inverter with the following operating conditions:Hz, Hz,s, represents theV whereload. Table I shows the results obtained from both simulationsa

53、nd analytical solution for low-frequency band and frequency band centered around the carrier. It should be noted that the contribution to a particular harmonic will be the phasor sum of several harmonics generated by cross-modulation. ForB. Dead-Time Pulse TrainApart from providing the harmonic char

54、acteristics, the an- alytical solution can also be used to generate the dead-time pulse train using (19) and its validity is verified by simulations.andthe harmonics in the low-frequency bandcorrespond to the signal harmonics. The even harmonics withinthis range are generated for their negligible va

55、lues. Forand they not listed due toandthe72nd to 88th harmonics are the cross-modulation products,As the range forandextends to, we only useexcept for the 80th harmonic, which is the carrier. It can be seen that the results from both methods are consistent.a limited range for calculating the analyti

56、cal solution, i.e.,forandforWU et al.: CALCULATING OUTPUT HARMONICS OF H-BRIDGE INVERTER WITH DEAD TIME623Fig. 8.Dead-time pulse trains obtained from both simulation and analytical solution (Np= 16; MR = 16):(a)(b)(c)Dead-time pulse trains for different Np and MR combinations. (a) An odd Np = 15 for an even MR = 16. (b) An even Np = 16 for anFig. 9.odd MR = 17. (c) An even= 18 for an odd MR = 17.NpFor an inverter operated with the followingdc component from the sixth term of (19). A dc component can only be generated by the fifth term