With the proposed technique, those two problems will be overcome. reduction is to eliminate the specific harmonics, which are the lowest orders.

Transcription

1 CHAPTER 3 OPTIMIZED HARMONIC TEPPED-WAVEFORM TECHNIQUE (OHW The obective o the proposed optimized harmonic stepped-waveorm technique is to reduce, as much as possible, the harmonic distortion in the load voltage, working with a reduced switching requency. The quarter-wave symmetric waveorm concept used in this thesis is proposed by teanovic et al. [3]. The paper also presents the total harmonic distortion minimization, which is very diicult to solve the switching angles because nonlinear equations tend to be much more complicated with the higher number o voltage levels. In [], the authors present the multilevel waveorm, which has very high THD with low modulation index. With the proposed technique, those two problems will be overcome. Basically, the concept o the proposed technique is to combine the idea o the selective harmonic eliminated PWM presented by Patel et al [5] with the quarter-wave symmetric idea concept presented by teanovic et al [3]. The concept o the harmonic reduction is to eliminate the speciic harmonics, which are the lowest orders. In this Chapter, the method used to solve the appropriated switching angles o the Optimized Harmonic tepped-waveorm (OHW technique will be presented. Equal step waveorm with optimized switching angles will be ocused in this thesis. The presented technique is to synthesize waveorm by the multilevel inverter using cascadedinverter with separated dc sources (DCs, which is presented in Chapter. 37

2 3. Introduction Vdc Full Bridge Cell V - Vdc Full Bridge Cell V - V out Vdc Full Bridge Cell V - Figure 3. chematic diagram o an s H-bridge series-connected multilevel inverter (single phase. As shown in Fig. 3., s H-bridge cells are connected in series. An output voltage waveorm can be generated by summation o the output voltage o each cell, i.e. V out = V V K V (3. Fig. 3. illustrates a generalized waveorm o s H-bridge inverters in seriesconnection. As discussed in Chapter, s output levels can be synthesized with s H- bridge inverters and s separate dc sources (DCs. From the voltage waveorm in Fig. 3., it consists o s switching angles,,,, (s-, and s, in each cycle. The voltage o the irst level equals V ; the voltage o the second level equals V and so on. These voltage levels are supplied by DCs, whose amplitudes may be dierent. By 38

3 considering the waveorm in Fig. 3., there are three possible optimization techniques to reduce the voltage THD: step heights are optimized with equally spaced steps; step spaces are optimized with the steps o equal height; and 3 optimizing both heights and spaces. Based on the circuit complexities and control possibilities, this thesis will ocus on the second method, which uses equal voltage amplitude and optimizes the switching angles. To achieve these optimized angles, the numerical calculation will be applied and will be presented later. Vout V V(- V V -V -V 3... (s- s (s π/ (s... (s- (s- (s π 3π/ (s π ωt -V(- -V Figure 3. Output voltage waveorm o the s H-bridge cell series-connected multilevel inverter. 39

4 In general, the modulation index o PWM is the ratio o the modulating signal amplitude to the carrier signal amplitude. For the speciied multilevel case, the modulation index is deined as ollows: V out M = (3. sv dc where V out is the amplitude o the output voltage at the undamental requency. s is the number o dc sources or H-bridge cells per phase. and V dc is the amplitude o dc sources. 3. Optimized Harmonic tepped Waveorm 3.. Quarter-Wave ymmetric Multilevel Waveorm The optimized harmonic stepped waveorm is assumed to be the quarter-wave symmetric. The relationship among the switching angles o the waveorm shown in Fig. 3. can be ound as ollows: In the second quarter; = π s s M s = π = π s

5 In the third quarter, s = π M 3s = π s = π 3s s In the orth quarter, 3s = π s M s = π s = π (3.3 The irst hal cycle o the proposed quarter-wave symmetric waveorm is depicted in Fig The output voltage level is zero rom ωt = to ωt =. At ωt =, the output voltage level is changed rom zero to V, and rom V to (V V at ωt =. The process will be repeated until ωt = π/, and the output voltage level becomes V V V (- V. Then, in the second quarter, the level o output voltage will be decreased to V V V (- at ωt = π-. The process will be repeated until ωt = π- and output voltage becomes zero again. In the second hal o the waveorm, the process will be repeated all o previous steps except the amplitude o the dc sources change rom positive to negative. The next period will then repeat the same cycle.

6 Vout V V (- V V ωt 3... (s- π s π/ s π (s-... π π π 3 π Figure 3.3 the quarter-wave symmetric s H-bridge cell multilevel waveorm. 3.. Fourier eries o the Proposed Waveorm Because o odd quarter-wave symmetric characteristic, which is illustrated in Fig. 3., the Fourier series coeicient are given by and where a n π = ( ωtsin( nωt d( ωt,or odd n (3. π a =,or even n (3.5 n b =,or all n (3.6 n ( ω t = Vout ( ωt

7 Vout E E (-... E E 3... (s- s π/ ωt Figure 3. The irst quarter o the quarter-wave symmetric waveorm. For all n, rom equations (3. to (3.6, the Fourier series is given as From equation (3., let Hence, = ωt ( ωt = an sin( nωt (3.7 = n a n π = ( sin( n d( (3.8 π From equation (3.8 and Fig. 3., a = π 3 E sin( n d E sin( n d K E π sin( n d n 3

8 / [ E cos( n E cos( n 3 E cos( n ] π = K nπ = [ E cos( n E cos( n E cos( n 3 E cos( n K nπ π E cos( n E cos( n [ E cos( n ( E E cos( n ( E E cos( n ] = K ( (3.9 nπ From Fig. 3.3 and 3., the ollowing relationship can be ound. V = E V = E E M = E E( V (3. ubstitute equation (3. into equation (3.9, we get a n = [ V cos( n V cos( n K V cos( n ] (3. nπ uppose the steps o equal heights, as discussed in 3., let Thereore, or any s and odd n, a n is given by V = V = K = V = E (3. a n E = K nπ [ cos( n cos( n cos( n ] or a n E = cos( n nπ k = k (3.3

9 Finally, the Fourier series o the quarter-wave symmetric s H-bridge cell multilevel waveorm is written as ollows: v out E ( ωt = cos( n k sin( nωt n= nπ (3. k= where k is the switching angles, which must satisy the ollowing condition π,, K, < s is the number o H-bridge cells. n is odd harmonic order. and E is the amplitude o dc voltages Optimized Harmonic witching Angles In this section, the method to solve the optimized harmonic switching angles will be explained. From equation (3.3 and (3., the harmonic components in the waveorm can be describes as ollows: the amplitude o dc component equals zero the amplitude o the undamental component, n =, and odd harmonic component are given by E h = cos( π k = k and h n E = cos( n nπ k = k (3.5 where s is the number o ull-bridge cells. 5

10 E is the amplitude o dc voltage. k are the switching angles. n is the odd harmonic order. 3 the amplitude o all even harmonics equals zero Thus, only the odd harmonics in the quarter-wave symmetric multilevel waveorm need to be eliminated. The switching angles o the waveorm will be adusted to get the lowest output voltage THD. Form the waveorm shown in Fig. 3., s switching angles, namely,,, and, need to be known. Mathematically, s equations obtained rom (3.5 are set up. Unortunately, these equations are nonlinear as well as transcendental in nature, which suggests a possibility o multiple solutions. There are no general methods that can be applied to solve such equations. The practical method o solving these equations is a trial and error process. A numerical technique is the best approach in solving the equations. Usually, the Newton-Raphson method is used to solve such nonlinear equation systems The Newton-Raphson Method The Newton-Raphson method or computing the root o an equation is a successive-approximation procedure, which is suitable or implementation in a computer program. Generally, the system o nonlinear equation in s variables can be represented as ( = k,, K, ( K = k,,, 6

11 M K = k (3.6 (,,, Then, (3.6 can be written in vector notation as where F ( = K (3.7 F =,,, ] [ K T = T [,, K, ] K = k, k,, ] [ K k T and F,, K are s matrices Nonlinear equation system can be solved by using a linearization technique, which the nonlinear equations are linearized about an approximate solution. As shown in Fig. 3.5, an equation in a single independent variable y = ( one starts with an estimate o near the root o ( = k. One then computes the intersection o the tangent line to the graph at this estimate with the axis and uses the intersection as the abscissa o the new estimate. The process is then repeated until a desirable solution is achieved. 7

12 ( ( k ( ( - 3 Figure 3.5 Linearization technique. In a multivariable nonlinear system, a set o independent variables is ormed in matrix ormat and the statement o the algorithm o Newton s method can be shown as ollows: Guess a set o initial values or with = uppose T = [,, K, ] (3.8 Calculate the value o ( (3.9 F = F 3 Linearize equation (3.7 about F d = K (3. 8

13 9 where = L M O M M L L and [ ] T d d d d L = olve d rom (3. by ( F K INV d = (3. where INV is the inverse matrix o 5 As updated the initial values, d = (3. 6 Repeat the process, equations (3.9 to (3., until d is satisied to the desired degree o accuracy Apply Newton s Method to olve the Optimized Harmonic witching Angles From 3..3, the amplitude o odd harmonic component in quarter-wave symmetric multilevel waveorm is written again here. = = k k n n n E h cos( π, or n = 3, 5,7, (3.3

14 A set o nonlinear equation corresponding to (3.3 can be given as ollows: The nonlinear equation o the undamental component: E π [ cos( cos( cos( ] = h The nonlinear equations o the odd harmonic component: E 3π E 5π E nπ K (3.a [ cos(3 cos(3 cos(3 ] = h3 [ cos(5 cos(5 cos(5 ] = h5 M K (3.b [ cos( n cos( n cos( n ] = hn K (3.c K (3.n For an output voltage waveorm o s ull-bridge cell circuit, s switching angles, namely,,, and, need to be known. It is implied that s equations are required to solve such switching angles. Importantly, the expected solutions must be less than π /. From nonlinear equations system (3., equation (3. is the nonlinear equation o the undamental component o an output voltage waveorm, whereas (3.b, (3.c, and (3.n are the nonlinear equation o the 3 rd, the 5 th, and the n th odd harmonic component o the output voltage, respectively. On the right hand side o these equations, they are the amplitudes o the components, which can be controlled. Basically, the lowest odd harmonic components should be eliminated rom a single-phase system. In a threephase system, because o electrical degree phase shit, the lowest non-triplen harmonic components need to be removed rom phase voltage. All even harmonics are not existed because o the symmetric characteristic o the waveorm. 5

15 To control the amplitude o the undamental component, (3.a will be considered. As introduced in 3., the modulation index or the multilevel waveorm is given as M h = (3.5 se where h is the amplitude o the undamental component. s is the number o dc sources or H-bridge cells. and E is the voltages o the dc sources. From (3.a and (3.5, the ollowing equation can be obtained. smπ cos( cos( K cos( = (3.6 From equation (3.6, varying the modulation index value can control the amplitude o the undamental component. The other s- nonlinear equations, which are the undesirable harmonic components, can be eliminated. Usually, they are set to be zero. Thereore, in a singlephase multilevel inverter, the lowest s- odd harmonics can be removed, and in a threephase multilevel inverter, the lowest s- non-triplen harmonics can be eliminated. Table 3. shows the number o harmonic contents which can be eliminated as unction o the number o dc sources, s, in both single-phase and three-phase system. 5

16 Table 3. The number o eliminated harmonic components o OHW technique. ingle-phase system Three-phase system The number o dc sources s s (per phase The number o output voltage levels s phase voltage levels s line to line voltage levels The harmonics in the output phase voltage, which need to be eliminated The lowest s- odd harmonics The lowest s- non-triplen odd harmonics Example 3. To explain how to apply Newton-Raphson method to the presented waveorm, the output waveorm o a nine-level inverter using cascaded-inverter with separated dc sources (DCs shown in Fig. 3.6 will be used as an example. In this example, singlephase system is assumed, and modulation index o output voltage, M, is.85. From the waveorm shown in Fig. 3.6, our unknowns,,, 3, and, need to be known. Because o a single-phase system, the lowest three odd harmonics, i.e., the 3 rd, 5 th, and 7 th, should be eliminated. 5

17 E Vout v 3 π/ π π ωt E Figure 3.6 Output voltage waveorm o a nine-level inverter. To control the undamental amplitude and eliminate the harmonics, our nonlinear equations can be set up as ollows: 3π cos( cos( cos( 3 cos( =.85 (3.7a cos( 3 cos(3 cos(3 3 cos(3 = (3.7b cos( 5 cos(5 cos(5 3 cos(5 = (3.7c cos( 7 cos(7 cos(7 3 cos(7 = (3.7d To solve the switching angles o nonlinear equations (3.7a to (3.7d, the Newton s method explained in is applied, and the ollowing matrices are implemented: The switching angle matrix, T = [,, 3, ] (3.8a 53

18 5 The nonlinear system matrix, = cos(7 cos(7 cos(7 cos(7 cos(5 cos(5 cos(5 cos(5 cos(3 cos(3 cos(3 cos(3 cos( cos( cos( cos( F (3.8b and = 7 sin(7 7sin(7 7sin(7 7sin(7 5sin(5 5sin(5 5sin(5 5sin(5 3sin(3 3sin(3 3sin(3 3sin(3 sin( sin( sin( sin( (3.8c 3 The corresponding harmonic amplitude matrix T T =.85(3 ( π (3.8d Then, equations (3.7a to (3.7d can be rewritten in the ollowing matrix ormat: F = T ( (3.9 By using matrices (3.8a to (3.8d and the Newton s method, the statement o algorithm, which is suitable or implement in a computer program, is shown as ollows: Guess a set o initial values or with = Assume T ],,, [ 3 = (3.3 Calculate the value o ( F F = (3.3 3 Linearize equation (3.9 about T d F = (3.3 and [ ] T d d d d d 3 =

19 olve d rom equation (3.3, i.e. d = INV ( T F (3.33 where INV is the inverse matrix o 5 As updated the initial values, = d (3.3 6 Repeat the process, equations (3.3 to (3.3, until d is satisied to the desired degree o accuracy, and the solutions must satisy the ollowing condition: π,, 3, < (3.35 To implement this algorithm in a computer, MATLAB programming is used. Ater running the program, each iteration result is tabulated in table 3. In this example, the process is repeated until all elements in d matrix are less than. The inal solutions, in degree, are: = = 8. 3 = = (3.36 Because these switching angles satisy the condition (3.35, they are the correct solutions. By using these switching angles in the waveorm, the 3 rd, 5 th, and 7 th harmonics will be eliminated with the modulation index equal to.85. As the results, the irst harmonics content, which shows in the waveorm, is the 9 th harmonic. 55

20 Table 3. Content o matrix, F, and Newton s method d rom MATLAB program by using the Iteration index (degree F d = = = = = = The Total Harmonic Distortion Calculation As introduced in the irst chapter, the total harmonics distortion (THD is mathematically given by n= ( n H THD = H (

21 where H is the amplitudes o the undamental component, whose requency isω and H (n is the amplitudes o the n th harmonics at requency nω From (3.5, the amplitude o the undamental and harmonic components o the quarter-wave symmetric multilevel waveorm can be express as: h n E = nπ k = cos( n k (3.38 From (3.37, let H = h ( n n and H = h n= h n THD = (3.39 h ubstitute equation (3.38 to (3.39, THD = n k = n= k = cos( n cos( k k (3. By using (3., thereore, output voltage THD o the presented waveorm can be calculated. Theoretically, to get exact THD, ininite harmonics need to be calculated. However, it is not possible in practice. Thereore, certain number o harmonics will be given. It relies on how precise THD is needed. Usually, n = 63 is reasonably excepted. The THD calculation or example 3. will be presented in the ollowing example. 57

22 Example 3. Replacing with 63 or (3., the equation becomes 63 ( cos( n cos( n cos( n 3 cos( n n= n THD = (3. [ cos( cos( cos( cos( ] By substituting the switching angle values rom example 3. into equation (3., the output voltage THD o nine-level cascaded inverter waveorm, which is calculated up to the 63 rd harmonic, is.73%. 3 Veriy the results rom example 3. and 3. with PPICE The calculated results rom example 3. and 3. will be veriied by PPICE. The nine-level inverter using our V dc sources is simulated. The operating requency is Hz and the modulation index is.85. Ater simulated, the output voltage waveorm o the nine-level inverter, its spectrum, and the output ile o PPICE are shown in Fig. 3.7, 3.8, and 3.9, respectively. As a consequence, the 3 rd, 5 th, and 7 th is eliminated rom the waveorm as shown in Fig The undamental component o the waveorm is 3V, which is equal to 85% o V, or the modulation index is.85. The most signiicant harmonic, which is existed in the waveorm, is the 9 th and has the amplitude o.v, which is 7.% o the undamental component. From Fig. 3.9, the output voltage THD is shown as.73%, which is equal to the calculated result. 58

23 Figure 3.7 The output voltage waveorm o a nine-level inverter. Figure 3.8 The requency spectrum o the waveorm in Fig

24 **** 7/3/99 :7:5 ********* NT Evaluation Ppice(July 997************ * C:\My Documents\Thesis\multi_5_7 test.sch **** FOURIER ANALYI TEMPERATURE = 7. DEG C **************************************************************************** FOURIER COMPONENT OF TRANIENT REPONE V(Vout DC COMPONENT =.E HARMONIC FREQUENCY FOURIER NORMALIZED PHAE NORMALIZED NO (HZ COMPONENT COMPONENT (DEG PHAE (DEG.E 3.E.E -7.E-.E 8.E 6.6E-3.95E E 5.557E 3.E3.775E-3.E-5.669E.67E.6E3.367E-3.8E E 5.378E 5.E3 6.8E-3.88E E -.77E 6.E3.79E-3 5.5E-6.37E.3E 7.8E3 5.7E-3.68E E -.738E 8 3.E3 9.8E-3.7E E E 9 3.6E3.E 7.78E E E-.E3.339E-3.76E E 7.E.E3 7.7E.67E-.79E.793E.8E3 6.9E-3.9E E 9.6E 3 5.E3.85E 5.7E E E- M 6.E 5.398E-3.587E E -.55E 6.E 3.37E 9.7E-3 -.E -.3E 6.8E 3.58E-3.3E-5 -.6E -.6E 63.5E.6E.36E- -.56E -.7E TOTAL HARMONIC DITORTION =.739E PERCENT Figure 3.9 The output ile o PPICE. 3.3 Conclusions To optimized the output multilevel voltage THD, the optimized harmonic stepped-waveorm technique is presented. By applying this technique, the s- lowest odd harmonics can be eliminated rom a single-phase output voltage. In three-phase system, the triplen harmonics in line voltage do not exist; thereore, the s- non-triplen harmonics need to be eliminated rom a phase voltage. By using the Newton-Raphson method, the switching angles can be easily solved. This method is very useul and eective technique, which is easy to apply in a computer program. In this thesis, MATLAB program is a main mathematical tool to do the ob. An m-ile used to solve the switching angles is shown in Appendix B. 6

25 PPICE is used to veriy the MATLAB solutions. As an example, the simulation results o the 9-level output voltage are very consistent with the calculated results. In chapter 5, a set o switching angles o the OHW technique, which will be considered rom ive levels up to iteen levels, will be solved by MATLAB program using the Newton-Raphson method. The MATLAB results will be then veriied by PPICE to support the theory. In the ollowing chapter, the elective Harmonic Eliminated Pulse Width Modulation (HE PWM technique will be discussed. The purpose to present this technique is because it will be used to compare with the OHW technique in chapter 5 and 6. The comparison will be presented in several aspects. 6