Chapter 4 DC to AC Conversion (INVERTER)

Transcription

1 Chapter 4 DC t AC Cnversin (INERTER) General cncept Single-phase inverter Harmnics Mdulatin Three-phase inverter Drives (ersin 3-003): 1

2 DC t AC Cnverter (Inverter) DEFINITION: Cnverts DC t AC pwer by switching the DC input vltage (r current) in a pre-determined sequence s as t generate AC vltage (r current) utput. General blc diagram I DC I ac DC ac TYPICAL APPLICATIONS: Un-interruptible pwer supply (UPS), Industrial (inductin mtr) drives, Tractin, HDC Drives (ersin 3-003):

3 Simple square-wave inverter (1) T illustrate the cncept f AC wavefrm generatin S1 S3 S4 S Drives (ersin 3-003): 3

4 AC Wavefrm Generatin S1,S ON; S3,S4 OFF fr t 1 < t < t v O S1 S3 DC DC + v O t 1 t t S4 S S3,S4 ON ; S1,S OFF fr t < t < t 3 v O S1 S3 DC + v O t t 3 t S4 S - DC Drives (ersin 3-003): 4

5 AC Wavefrms INERTER OUTPUT OLTAGE - FUNDAMENTAL COMPONENT 1 4DC RD HARMONIC 5 RD HARMONIC 1 5 Drives (ersin 3-003): 5

6 Harmnics Filtering DC SUPPLY INERTER (LOW PASS) FILTER LOAD L + v O 1 C + v O BEFORE FILTERING AFTER FILTERING v O 1 v O Output f the inverter is chpped AC vltage with zer DC cmpnent. It cntain harmnics. An LC sectin lw-pass filter is nrmally fitted at the inverter utput t reduce the high frequency harmnics. In sme applicatins such as UPS, high purity sine wave utput is required. Gd filtering is a must. In sme applicatins such as AC mtr drive, filtering is nt required. Drives (ersin 3-003): 6

7 ariable ltage ariable Frequency Capability Higher input vltage Higher frequency 1 Lwer input vltage Lwer frequency t Output vltage frequency can be varied by perid f the square-wave pulse. Output vltage amplitude can be varied by varying the magnitude f the DC input vltage. ery useful: e.g. variable speed inductin mtr drive Drives (ersin 3-003): 7

8 Output vltage harmnics/ distrtin Harmnics cause distrtin n the utput vltage. Lwer rder harmnics (3 rd, 5 th etc) are very difficult t filter, due t the filter size and high filter rder. They can cause serius vltage distrtin. Why need t cnsider harmnics? Sinusidal wavefrm quality must match TNB supply. Pwer Quality issue. Harmnics may cause degradatin f equipment. Equipment need t be de-rated. Ttal Harmnic Distrtin (THD) is a measure t determine the quality f a given wavefrm. Drives (ersin 3-003): 8

9 Ttal Harmnics Distrtin (THD) ltage THD : If THDv n, RMS Current THD : ( ) If the rms vltage fr the vavefrm is nwn, I Z n n THDv THDi Z n n n n n, RMS 1, RMS + 3, RMS ( ) ( ) RMS ( I ) I n, RMS 1, RMS n 1, RMS is the nth harmnic vltage, 1, RMS , RMS, RMS is the impedance at harmnic frequency. Drives (ersin 3-003): 9

10 Furier Series Study f harmnics requires understanding f wave shapes. Furier Series is a tl t analyse wave shapes. Furier Series a a b n n f ( v) where f ( v) dθ f ( v)cs f ( v)sin Inverse Furier a + θ ωt ( nθ ) ( nθ ) dθ dθ ( a csnθ + b sin nθ ) n 1 ("DC" term) n ("cs" term) ("sin" term) n Drives (ersin 3-003): 10

11 Drives (ersin 3-003): 11 Harmnics f square-wave (1) ( ) ( ) ( ) ( ) + θ θ θ θ θ θ θ θ θ θ sin sin 0 cs cs 0 1 d n d n b d n d n a d d a n n - θω

12 Harmnics f square wave () Slving, b n n n n n [ ( ) ( ) ] cs nθ + cs nθ [(cs0 csn ) + (csn csn )] [(1 cs n ) + (1 csn )] [(1 csn )] When n is even, csn 1 b n 0 (i.e. even harmnics d nt exist) 0 When n is b n 4 n dd, csn 1 Drives (ersin 3-003): 1

13 Spectra f square wave Nrmalised Fundamental 1st 3rd (0.33) 5th (0.) 7th (0.14) 9th (0.11) 11th (0.09) n Spectra (harmnics) characteristics: Harmnic decreases with a factr f (1/n). Even harmnics are absent Nearest harmnics is the 3rd. If fundamental is 50Hz, then nearest harmnic is 150Hz. Due t the small separatin between the fundamental an harmnics, utput lw-pass filter design can be very difficult. Drives (ersin 3-003): 13

14 Quasi-square wave (QSW) α α α - Nte that α 1 bn α n Expanding : csn a n ( ) [ α ] nθ dθ csnθ [ cs( nα ) csn( α )] ( α ) cs( n nα ) csn csnα + sin n sin nα b n n cs n 0. (due t half - wave symmetry) sin [ cs( nα ) csn csnα ] ( nα )[ 1 csn ] n α csn csnα Drives (ersin 3-003): 14

15 If n is even, b Nte : Harmnics cntrl b 4 If n is dd, b n cs( nα ) n In particular, amplitude f the fundamental is : 1 4 cs n ( α ) The fundamental, b 0, 1, is cntrlled by varying Harmnics can als be cntrlled by adjusting α, Harmnics Eliminatin : Fr example if α 30 general, harmnic n, then b harmnic is eliminated frm the wavefrm.in 90 α n 3 will be eliminated if Drives (ersin 3-003): 0,r the third : 15

16 Example A full - bridge single phase inverter is fed by square wave signals. The DC lin vltage is100. The lad is R 10R and L 10mH in series. Calculate: a) the THDv using the "exact" frmula. b) the THDv by using the first three nn - zer harmnics c) the THDi by using the first three nn - zer harmnics Repeat (b) and (c) fr quasi -square wave case with α degrees 30 Drives (ersin 3-003): 16

17 Half-bridge inverter (1) + C1 - G S S1 ON S OFF t + C - R L S S1 OFF S ON Als nwn as the inverter leg. Basic building blc fr full bridge, three phase and higher rder inverters. G is the centre pint. Bth capacitrs have the same value. Thus the DC lin is equally spilt int tw. The tp and bttm switch has t be cmplementary, i.e. If the tp switch is clsed (n), the bttm must be ff, and vice-versa. Drives (ersin 3-003): 17

18 Sht thrugh fault and Dead-time In practical, a dead time as shwn belw is required t avid sht-thrugh faults, i.e. shrt circuit acrss the DC rail. Dead time creates lw frequency envelpe. Lw frequency harmnics emerged. This is the main surce f distrtin fr high-quality sine wave inverter. + S1 I shrt S 1 signal (gate) G S RL S signal (gate) "Sht thrugh fault". I shrt is very large t d "Dead time' t d t d Drives (ersin 3-003): 18

19 Single-phase, full-bridge (1) Full bridge (single phase) is built frm tw halfbridge leg. The switching in the secnd leg is delayed by 180 degrees frm the first leg. RG + LEG R LEG R' ωt G S1 R + - S3 R' R ' G ωt + - S4 S RG R ' G G is "virtual grumd" ωt Drives (ersin 3-003): 19

20 Three-phase inverter Each leg (Red, Yellw, Blue) is delayed by 10 degrees. A three-phase inverter with star cnnected lad is shwn belw + + / S1 S3 S5 G R Y B + / S4 i R i Y i B S6 S i a i b Z R Z Y Z B N Drives (ersin 3-003): 0

21 Three phase inverter wavefrms Inverter Phase ltage (r ple switching wavefrm) RG DC / - DC / YG DC / DC / DC / 40 0 BG - DC / line-t -ine ltage RY DC - DC Six-step Wavefrm RN DC /3 DC /3 - DC /3 - DC /3 Interval Psitive device(s) n Negative device(s) n 1 3,4 3, ,6 4 1, ,6 6 1,3 Quasi-square wave peratin vltage wavefrms Drives (ersin 3-003): 1

22 Pulse Width Mdulatin (PWM) +1 M 1 Mdulating Wavefrm Carrier wavefrm t t 0 1 t t 3 t t 4 5 Triangulatin methd (Natural sampling) Amplitudes f the triangular wave (carrier) and sine wave (mdulating) are cmpared t btain PWM wavefrm. Simple analgue cmparatr can be used. Basically an analgue methd. Its digital versin, nwn as REGULAR sampling is widely used in industry. Drives (ersin 3-003):

23 PWM types Natural (sinusidal) sampling (as shwn n previus slide) Prblems with analgue circuitry, e.g. Drift, sensitivity etc. Regular sampling simplified versin f natural sampling that results in simple digital implementatin Optimised PWM PWM wavefrm are cnstructed based n certain perfrmance criteria, e.g. THD. Harmnic eliminatin/minimisatin PWM PWM wavefrms are cnstructed t eliminate sme undesirable harmnics frm the utput wavefrm spectra. Highly mathematical in nature Space-vectr mdulatin (SM) A simple technique based n vlt-secnd that is nrmally used with three-phase inverter mtrdrive Drives (ersin 3-003): 3

24 Mdulatin Index, Rati +1 M 1 Mdulating Wavefrm Carrier wavefrm t0 t1 t t3 t t 4 5 Mdulatin Index (Mdulatin Depth) M I M Amplitude f the mdulating wavefrm Amplitude f the carrier wavefrm I : Mdulatin Rati (Frequency Rati) M R p M R ( p) Frequency f the carrier wavefrm Frequency f the mdulating wavefrm Drives (ersin 3-003): 4

25 Mdulatin Index, Rati Mdulatin Index deterrmines the utput vltage fundamental cmpnent If 0 < 1 M where M I I 1 < 1, in, in are fundamental f the utput vltage and input (DC) vltage, respectively. Mdulatin rati determines the incident (lcatin) f harmnics in the spectra. The harmnics are nrmally lcated at : f M where and f R m ( f ) is m the frequency f the mdulating signal is an integer (1,,3...) Drives (ersin 3-003): 5

26 Regular sampling t 1 t Sinusidal mdulating wavefrm, v m (t) Carrier, v c (t) t Regular sampling wavefrm, v s (t) t' 1 t' v pwm t Regular sampling PWM Drives (ersin 3-003): 6

27 Asymmetric and symmetric regular sampling T +1 sample pint M1sinωmt T 4 3T 4 5T 4 4 t 1 t0 t1 t t3 asymmetric sampling t symmetric sampling Generating f PWM wavefrm regular sampling Drives (ersin 3-003): 7

28 Biplar Switching +1 M 1 Mdulating Wavefrm Carrier wavefrm t t 0 1 t t3 t t 4 5 Drives (ersin 3-003): 8

29 Uniplar switching A Carrier wavefrm B (a) S1 (b) S3 (c) pwm (d) Uniplar switching scheme Drives (ersin 3-003): 9

30 Biplar PWM switching: Pulsewidth characterizatin δ 4 mdulating wavefrm carrier wavefrm th pulse δ 1 δ α Drives (ersin 3-003): 30

31 The th Pulse δ0 δ0 δ0 δ0 + δ 1 δ + α The th PWM pulse Drives (ersin 3-003): 31

32 Determinatin f switching angles fr th PWM pulse (1) A S1 A S v msin( θ) + A p1 A p Equating the vlt -secnd, A A s1 s A A p1 p Drives (ersin 3-003): 3

33 Drives (ersin 3-003): 33 PWM Switching angles () ( ) ( ) ( )( ) ( ) ( ) ( )( ) [ ] ) sin( Similarly, ) sin( sin cs ) cs( sin - secnd supplied by the sinusid, vlt The Similarly fr the secnd half, given as : the PWM pulse is f cycle the first half - secnd during The lt m s m m m s p p A d A A A δ α δ δ α δ α δ α θ θ δ δ δ δ δ δ δ δ δ δ α δ α +

34 Fr small angle δ sinδ ( )( δ δ ) f Switching angles (3) δ δ T derive the mdulatin strategy, A A p1 p s1 s ;, sin( α sin( α Hence, fr the the first half cycle f PWM pulse, δ δ ) δ ) sin( α ( δ δ ) m ( δ sin( α δ ) 1 By definitin, the Mdulatin Rati, M I A A is nwn as mdulatin ) Thus, the pulse width fr the first half the PWM wavefrm is given by : δ 1 δ A A s1 1 s ( δ m m m [ 1+ M sin( α δ )] I m δ ) cycle Drives (ersin 3-003): 34

35 PWM switching angles (4) Thus the leading edge switching angle f the th pulse is : α δ 1 Using similar methd, pulse width f the secnd half δ α δ + δ δ δ Mdulatin, i.eδ δ Hence δ [ 1+ M sin( α + δ )] And the trailing edge angle: cycle f δ I The abve equatin is valid fr Asymmetric [ 1+ M sinα ] I and δ Fr Symmetric Mdulatin, 1 1 PWM wavefrm: are different. Drives (ersin 3-003): 35

36 Example Fr the PWM shwn belw, calculate the switching angles pulses n carrier wavefrm mdulating wavefrm t1 t t3 t4 t5 t6 t7 t8 t9 t10 t11 t1 t13 t14 t15 t16 t17 t18 α1 Drives (ersin 3-003): 36

37 Harmnics f biplar PWM Assuming the PWM wavefrm is half wave symmetry, harmnic cntent f b n α δ α + δ α δ α δ α + δ α + δ T each (th) PWM pulse can be cmputed as : f ( v)sin nθdθ sin nθdθ sin nθdθ + + sin nθdθ δ0 δ0 δ0 δ0 δ 1 α δ Drives (ersin 3-003): 37

38 This Harmnics f Biplar PWM Which can be reduced t : { csn( α δ ) csn( α δ1 ) n + csn( α + δ ) csn( α δ ) + csn( α Yeilding, b n n [ csn( α + csnα csnδ equatin cannt be simplified prductively.the Furier cefficent fr the PWM wavefrm isthe sum f pulses ver ne perid, i.e.: b b n n p 1 b n + δ δ 1 ) csn( α ) csn( α Next slide shws the cmputatin f this equatin. b n ] 1 fr the p 1 + δ ) ) } Drives (ersin 3-003): 38

39 PWM Spectra M I 0. Amplitude 1.0 M I M I 0.8 M I 0.6 Mdulatin Index 0 p p 3p 4 p M I 1.0 Fundamental NORMALISED HARMONIC AMPLITUDES FOR SINUSOIDAL PULSE-WITDH MODULATION Drives (ersin 3-003): 39

40 PWM spectra bservatins The harmnics appear in clusters at multiple f the carrier frequencies. Main harmnics lcated at : f p (f m ); 1,,3... where f m is the frequency f the mdulatin (sine) wavefrm. There als exist side-bands arund the main harmnic frequencies. Amplitude f the fundamental is prprtinal t the mdulatin index. The relatin ship is given as: 1 M I in The amplitude f the harmnic changes with M I. Its incidence (lcatin n spectra) is nt. When p>10, r s, the harmnics can be nrmalised. Fr lwer values f p, the side-bands clusters verlap-nrmalised results n lnger apply. Drives (ersin 3-003): 40

41 Tabulated Biplar PWM Harmnics n M I M R M R M R M R M R M R M R M R M R M R M R M R M R M R Drives (ersin 3-003): 41

42 Three-phase harmnics Fr three-phase inverters, there is significant advantage if M R is chsen t be: Odd: All even harmnic will be eliminated frm the ple-switching wavefrm. triplens (multiple f three (e.g. 3,9,15,1, 7..): All triplens harmnics will be eliminated frm the line-t-line utput vltage. By bserving the wavefrm, it can be seen that with dd M R, the line-t-line vltage shape ls mre sinusidal. As can be nted frm the spectra, the phase vltage amplitude is 0.8 (nrmalised). This is because the mdulatin index is 0.8. The line vltage amplitude is square rt three f phase vltage due t the three-phase relatinship Drives (ersin 3-003): 4

43 Effect f dd and triplens RG YG RY p 8, M 0.6 RG YG RY p 9, M 0.6 ILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIO THAT IS A MULTIPLE OF THREE IN A THREE PHASE INERTER Drives (ersin 3-003): 43

44 Spectra: effect f triplens Amplitude (Line t line vltage) B A Fundamental Harmnic Order COMPARISON OF INERTER PHASE OLTAGE (A) & INERTER LINE OLTAGE (B) HARMONIC (P1, M0.8) Drives (ersin 3-003): 44

45 Cmments n PWM scheme It is desirable t have M R as large as pssible. This will push the harmnic at higher frequencies n the spectrum. Thus filtering requirement is reduced. Althugh the vltage THD imprvement is nt significant, but the current THD will imprve greatly because the lad nrmally has sme current filtering effect. Hwever, higher M R has side effects: Higher switching frequency: Mre lsses. Pulse width may be t small t be cnstructed. Pulse drpping may be required. Drives (ersin 3-003): 45

46 Example The amplitudes f the ple switching wavefrm harmnics f the red phase f a three-phase inverter is shwn in Table belw. The inverter uses a symmetric regular sampling PWM scheme. The carrier frequency is 1050Hz and the mdulating frequency is 50Hz. The mdulatin index is 0.8. Calculate the harmnic amplitudes f the line-t-vltage (i.e. red t blue phase) and cmplete the table. Harmnic number Amplitude (ple switching wavefrm) Amplitude (line-t line vltage) Drives (ersin 3-003): 46