3. MODELING OF PARALLEL THREE-PHASE CURRENT-UNIDIRECTIONAL CONVERTERS 3. MODELING OF PARALLEL THREE-PHASE CURRENT-

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1 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS 3. MOEING OF PARAE THREE-PHASE URRENT- UNIIRETIONA ONERTERS Ths chater eelos the moels of the arallel three-hase current-unrectonal swtch base conerters, whch nclue three-hase A/ buck rectfers an /A current source nerters. 3. TOPOOGIES AN PARAE ARHITETURES A three-hase buck rectfer an a three-hase current source nerter are shown n Fgures 3. an 3., resectely. To smlfy the scusson, the loa for both conerters s resste. ref () A B a b sa sb sc c c san sbn scn R n Fgure 3. Three-hase buck rectfer. 47

2 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS c sa san sb sbn sc scn a b c B A R n ref () Fgure 3. Three-hase current-source nerter. Both buck rectfer an current source nerter are usually classfe as currentunrectonal conerters because they share the same swtchng cells that are current unrectonal. The swtchng cells ether nherently hae seres oes, for examle, GTOs, or hae external seres blockng oes. The symbolc reresentaton an ts oltage an current oeratonal states are shown n Fgure 3.3. s off on (a) Symbolc reresentaton. (b) oltage an current oeratonal states. Fgure 3.3 urrent-unrectonal swtchng cell. In ths chater, the aerage large-sgnal moels an small-sgnal moels of the arallel three-hase current-unrectonal conerters are eeloe. Smlar to the metho use for the current-brectonal conerters, the current-unrectonal swtchng network s aerage on a ral arm bass. onentonally, the aeragng for a three-hase current-unrectonal conerter s base on the aerage fference between the to an bottom swtches n one hase leg [53]. Ths aroach ntentonally neglects commonmoe comonents of the to an bottom rals. The common-moe comonents are generally of no nterest n the control esgn for a sngle conerter. Howeer, they are 48

3 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS crtcal n the analyss an esgn of arallel conerters. The ral-arm aeragng aots a reference ont n the system, then eres the aerage alues of all other onts referrng to the reference ont. As a result, t allows the moel to resere the common-moe comonents. After the ral-arm aeragng, the aerage moel of a three-hase currentunrectonal conerter can be easly obtane by connectng a to an bottom aerage ral arms. For conenence, the reference ont of the conerter s chosen at the neutral ont of the A se of the crcut, as shown n Fgures 3. an 3.. Fgures 3.4 an 3.5 show the arallel buck rectfers an current source nerters. Assumng the arallel conerter systems ouble the ower ratng, then the outut caactance becomes, an the outut resstance becomes R/, where an R are the arameters of a sngle conerter n Fgures 3. an 3.. In contrast to the sngle conerter, the ral nuctor n the arallel conerters s slt nto two nuctors at both to an bottom rals n orer to ao swtchng nteractons between the arallel conerters. A a sa sb s c / c ref () B b c R / san sbn s cn / n a sa sb s c / b c san sbn s cn n / Fgure 3.4 rculatng current n arallel three-hase buck rectfers. 49

4 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / c sa sb s c a b c B A / s s an bn s cn n ref () R / / sa sb s c a b c / san sbn s cn n Fgure 3.5 rculatng current n arallel three-hase current-source nerters. Because of the rect arallelng, a crculatng current exsts, as hghlghte n the Fgures 3.4 an 3.5, for examle. A ero-seuence current s efne, as n (.), n the arallel current-unrectonal conerters. 3. AERAGE MOES 3.. Ral-Arm Aeragng The swtchng cell n Fgure 3.3(a) can be escrbe by a generc swtchng s as n Fgure.6. When s s oen, that s, ether the swtch tself or the seres oe blocks the oltage, referrng to Fgure 3.3(a), then the current s ero. When s s close, that s, both the swtch an the seres oe conuct a current, then the oltage s ero. Therefore, a swtchng functon s an the corresonng current an oltage alues can be escrbe as n (.). 5

5 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS In the current-unrectonal swtch base conerters, a generc swtchng unt, calle a ral arm, can be entfe, as shown n Fgure 3.6. A ral arm s comose of three swtchng cells, an has oltage sources (or caactors) on one se an a current source (or an nuctor) on the other. These features make the ral arm a generc swtchng unt. sa sb sc ref () Fgure 3.6 To ral arm of current-unrectonal conerters. There are swtchng constrans for the three swtchng cels n the ral arm. To reent the nuctor from beng oen-crcute, one of the three swtchng cells, s a, s b or s c, has to be close at any tme. Meanwhle, to reent the nut oltage sources from beng short-crcute, only one of the three swtchng cells can be close at any tme. Ths leas to the followng relatonsh: s s s. (3.) a b c Therefore, the ral arm can be reresente by a sngle-ole, trle-throw swtch, as shown n Fgure 3.7. The nut an outut arables of nterest are also efne n Fgure 3.7. A a sa s b sc ref () B b c Fgure 3.7 To ral arm reresente as a sngle-ole, trle-throw swtch. The PWM of the ral arm s shown n Fgure 3.8, where T s the swtchng ero, an a, b an c are efne as the uty cycles of the swtchng cells s a, s b an s c, 5

6 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS resectely. The corresonng oltage an current waeforms are also shown n Fgure 3.8. a T s a b T s b c T s c T A B t a t b t c t Fgure 3.8 Ral arm PWM an corresonng current an oltage waeforms. Base on the waeforms, one can obtan the oltage an current relatonshs n aerage, assumng the current an the oltages A, B an are contnuous wth small rles: A [ ] a b c B, (3.) 5

7 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS a b c a b c. (3.3). (3.4) a b c The aerage moel of the to ral arm s ecte n Fgure 3.9. Euaton (3.4) always hols ue to the swtchng constrants of the ral arm. A a a a A b B c ref () B b b c c Fgure 3.9 To ral arm s aerage moel. A smlar moel can be eeloe for the bottom ral arm, shown n Fgure 3.. ref () san sbn scn Fgure 3. Bottom ral arm of current-unrectonal conerters. 53

8 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS The aerage moel of the bottom ral arm s shown n Fgure 3.. A a an n ref () B b c bn n n an A bn B cn cn n n Fgure 3. Bottom ral arm s aerage moel. where A [ ] n an bn cn B, (3.5) a b c an bn cn n, (3.6). (3.7) an bn cn Euaton (3.7) always hols ue the swtchng constrants of the ral arm. From Fgures 3. an 3., t s obous that the current n has a negate alue because the actual current can only flow back nto the A se. Howeer, the relatonsh between n an the hase currents stll hols. 3.. Aerage Moel of Parallel Buck Rectfers After aeragng the ral arms, the aerage moel of a three-hase buck rectfer can be realy obtane by connectng the to an bottom ral arms as well as the rest of the crcut comonents. Fgure 3. shows the aerage moel of the buck rectfer, where 54

9 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS a b c a b c an bn cn n. (3.8) A a / c ref () B b R c n / n Fgure 3. Buck rectfer s aerage moel n statonary coornates. The state-sace euatons of the buck rectfer are: A [ a an b bn c cn] B c t, (3.9) c c. (3.) t R It can be seen from Fgure 3. that, the ero-seuence current s always ero because hyscally there s no such current ath. In arallel oeraton, howeer, a crculatng current ath s forme, as shown n Fgure

10 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS A a / c ref () B b R / c n / n a / b c n / Fgure 3.3 Parallel buck rectfers aerage moel n statonary coornates. The state-sace euatons of the arallel buck rectfers are: n n n c, (3.) t t n n c, (3.) t t n, (3.3) t t n n n, (3.4) t t t (3.)-(3.4) can be smlfe as c c. (3.5) R n c, (3.6) t 56

11 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS n c, (3.7) t t where s efne as n (.), an, (3.8) A [ ] n a an b bn c cn B, (3.9) n A [ ] n a an b bn c cn B. (3.) n ( n ) ( n) ( n n) ( n n), (3.) In steay state, the oltage c s controlle as a constant alue, whle the hase currents a, b, c, a, b an c are controlle to be snusoal an n hase wth the corresonng nut hase oltages A, B an, whch are normally gen as: A B m m m cos( ωt) cos( ωt π /3). (3.) cos( ωt π /3) In orer to hae a steay state oeratng ont (so as to lneare the system to esgn controllers), the moel n statonary coornates s usually transforme nto rotatng coornates. The transformaton matrx s chosen as n (.5). The arables n the statonary coornates can then be transforme nto the rotatng coornates usng (.6). Alyng (.6) to (3.9), one can obtan the aerage moel of the buck rectfer n the rotatng coornates as escrbe n (3.) an (3.4): [ ] c t. (3.3) where 57

12 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS 58 cn c bn b an a T, B A T (3.4) ue to (3.4) an (3.7), the thr row ( ) s always ero. Therefore, t s roe n (3.3). Fgure 3.4 shows the eualent crcut of the moel. R c n Fgure 3.4 Buck rectfer s aerage moel n rotatng coornates. Alyng (.6) to (3.9)-(3.), the aerage moel of the arallel buck rectfers n the rotatng coornates can be obtane as escrbe n (3.5), (3-8) an (3-5)-(3-7): [ ] t c, (3.5) [ ] t c, (3.6) [ ] [ ] [ ] ) ( n n. (3.7) where: cn c bn b an a T, cn c bn b an a T, cn cn bn bn an an n n T. (3.8) The eualent crcut of the moel s shown n Fgure 3.5. It s a fourth-orer system wth sx control arables. It can be seen that a ero-seuence current s resent n

13 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS the channel. The current s etermne by the fference between the two arallel conerters common-moe oltages. c n n R / n n ( / ) Fgure 3.5 Parallel buck rectfers aerage moel n rotatng coornates Aerage Moel of Parallel urrent Source Inerters The aerage moel of the current source nerter can be obtane by connectng the to an bottom ral arms as well as the rest of the crcut comonents, as shown n Fgure

14 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / a A c b B / n c n ref () R Fgure 3.6 urrent-source nerter s aerage moel n statonary coornates. The state-sace euatons of the current source nerter are: c t [ ] a an A a an B b bn t R c cn b bn c A B cn A B, (3.9). (3.3) As shown n Fgure 3.6, the ero-seuence current s always ero because there s no such current ath. Therefore, s forme, as shown n Fgure 3.7. n. In arallel oeraton, a crculatng current ath 6

15 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / a A c b B / n c n R / ref () a / b / n n c Fgure 3.7 Parallel current-source nerters aerage moel n statonary coornates. The state-sace euatons of the arallel current source nerters are: n c n, (3.3) t t n c n, (3.3) t t a an a an A A b bn n b bn n B B, (3.33) t R c cn c cn, (3.34) t t n n n n. (3.35) t t Smlfyng these euatons, one can obtan: t c n, (3.36) 6

16 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS t c n, (3.37) t, (3.38) A a an a an an an B ( b bn b bn bn bn ) t R c cn c cn cn cn A B (3.39) In orer to hae a steay state oeratng ont (so as to lneare the system to esgn controllers), the moels are usually transforme from the statonary nto the rotatng coornates. Smlarly, alyng (.6) to (3.9)-(3.3), one can obtan the aerage moel of the current source nerter n the rotatng coornates: t c t [ ] R ω ω R Fgure 3.8 shows the eualent crcut of the moel., (3.4). (3.4) ω c n R ω R Fgure 3.8 urrent-source nerter s aerage moel n rotatng coornates. 6

17 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS Alyng (.6) to (3.36)-(3.37) an (3-39), one can obtan the aerage moel of the arallel current source nerters n the rotatng coornates, as escrbe n (3.38), an (3.4)-(3.44): c [ ] t, (3.4) c [ ] t, (3.43) ω n ( R ). (3.44) t n ω R The eualent crcut of the moel s shown n Fgure 3.9. c n ω R / n ω R / n ( ) / n Fgure 3.9 Parallel current-source nerters aerage moel n rotatng coornates. 63

18 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS SMA-SIGNA MOES 3.3. Small-Sgnal Moel of Parallel Buck Rectfers To obtan the small-sgnal moel of the arallel three-hase buck rectfers, a steaystate oeratng ont s obtane frst: m 3,,, c,, n n, R I I c, I I, I, (3.45) where: R an m (gen as n (3.)) are gen, c, I, I an I are controlle to ther reference alues. I, I,,,,, n, n are calculate base on the gen alues an the control objectes. Assumng that the nut oltage sources are eal, as n (.48), then the small-sgnal moel of the sngle buck rectfer s: c c R t. (3.46) The small-sgnal moel of the arallel buck rectfers can be ere as follows: c c R t (3.47) The eualent crcut s shown n Fgure 3.. Some terms are omtte because I, n an n roe that the ero-seuence current s controlle to be ero.

19 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS I I I n R / c I ( ) / n Fgure 3. Parallel buck rectfers small-sgnal moel. Unlke the moel of the arallel boost rectfers, the small-sgnal moel of the arallel buck rectfers shows that the channel s not neenent an s coule wth the se. The oen-loo transfer functons are smulate usng MATAB base on the arameters gen below: ; ω π 6 ra/s; ; P 5kW ; R c / Po ; m 5µH ; µ F. Be note that the arallel conerter oeraton has an R/. Fgure 3.(a) shows that, ue to the ero-seuence nteracton, the transfer functon of channel control to ral current n the arallel conerters has a ole at the orgn. Wth the nuctor at the se, the transfer functon oes not hae a ouble ole at rotatng freuency as n the arallel boost rectfers. Fgure 3.(b) shows that the - channel control has no small-sgnal effect on the ral current. Fgure 3. shows that the - an -channel controls hae no small-sgnal effect on -channel current. Fgure 3.3 shows that the -channel control has small-sgnal effect on both -channel an outut currents. Fgure 3.4 shows that the two conerters are cross-coule. 65 c o

20 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / 5 Parallel Phase (eg); Magntue (B) 5 5 Sngle Freuency (ra/sec) (a) / (sngle) an / (arallel). / Phase (eg); Magntue (B) Freuency (ra/sec) (b) / (sngle) an / (arallel). Fgure 3. Oen-loo transfer functons of - an -channel control to outut current. 66

21 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / Phase (eg); Magntue (B) Freuency (ra/sec) (a) /. / Phase (eg); Magntue (B) Freuency (ra/sec) (b) /. Fgure 3. Oen-loo transfer functons of - an -channel control to -channel current. 67

22 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS /elta 5 Phase (eg); Magntue (B) Freuency (ra/sec) (a) /. /elta 5 Phase (eg); Magntue (B) Freuency (ra/sec) (b) /. Fgure 3.3 Oen-loo transfer functons of -channel control to outut an -channel currents. 68

23 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / Phase (eg); Magntue (B) Freuency (ra/sec) Fgure 3.4 Oen-loo transfer functon / shows the two conerters are cross-coule Small-Sgnal Moel of Parallel urrent Source Inerters To obtan the small-sgnal moel of the arallel three-hase current source nerter, a steay-state oeratng ont s obtane as follows: 3 m,, I I, I c, ω, n n I, c I, R c I, I I R, I I, (3.48) where: R, an c are gen, an are so controlle that the transforme oltages are as n (3.). I, I an I are controlle to ther reference alues. I, I, I, I,,,,,, n are calculate base on the gen alues an the control n objectes. Assumng that the nut oltage source s eal, as n (.5), then the small-sgnal moel of the sngle current source nerter s: 69

24 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS 7 I I R R t ω ω. (3.49) The small-sgnal moel of the arallel current source nerters can be ere as follows: I I I I R R t ω ω. (3.5) In the moel, s the control arable of channel. The eualent crcut s shown n Fgure 3.5.

25 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS c n I ω R / I I ω R / I n ( ) / Fgure 3.5 Parallel current-source nerters small-sgnal moel. The oen-loo transfer functons are smulate usng MATAB base on the arameters gen below: ; ω π 6 ra/s; ; P 5kW ; R c / Po ; m 5µH ; µ F. Be note that the arallel conerter oeraton has an R/. Fgure 3.6(a) shows that, ue to the ero-seuence nteracton, the transfer functon of -channel control to ral current n the arallel conerters has a ole at the orgn. Fgure 3.6(b) shows that the channel control, howeer, has no ole at the orgn. Fgure 3.7 shows that the - an -channel controls hae no small-sgnal effect on -channel current ue to. Fgure 3.8 shows that the -channel control has small-sgnal effect on both -channel an outut currents. Fgure 3.9 shows that the two conerters are cross-coule. c o 7

26 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / 5 Parallel Phase (eg); Magntue (B) Sngle Freuency (ra/sec) (a) / (sngle) an / (arallel). / Phase (eg); Magntue (B) 5-5 Sngle Parallel Freuency (ra/sec) (b) / (sngle) an / (arallel). Fgure 3.6 Oen-loo transfer functons of - an -channel control to outut current. 7

27 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / Phase (eg); Magntue (B) Freuency (ra/sec) (a) /. / Phase (eg); Magntue (B) Freuency (ra/sec) (b) /. Fgure 3.7 Oen-loo transfer functons of - an -channel control to -channel current. 73

28 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS /elta 5 Phase (eg); Magntue (B) Freuency (ra/sec) (a) /. /elta 5 Phase (eg); Magntue (B) Freuency (ra/sec) (b) /. Fgure 3.8 Oen-loo transfer functons of -channel control to outut an -channel currents. 74

29 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS / 5 Phase (eg); Magntue (B) Freuency (ra/sec) Fgure 3.9 Oen-loo transfer functon / shows the two conerters are cross-coule. 75

30 3. MOEING OF PARAE THREE-PHASE URRENT-UNIIRETIONA ONERTERS 76