Naturally Commutating Converters

Transcription

1 BWW aturally Cmmutatig Certer aturally cmmutatig certer Durig dide cducti the circuit i defied by the Kirchhff ltage euati di + i + i ( ( dt where i the rm ac uly ltage Slig euati ( yield the lad (ad dide curret - /ta φ { } i( i ( - φ + i φ e (A Z 0 β (rad ( The certer circuit cidered i thi chater hae i cmm a ac ltage uly iut ad a dc lad utut The fucti f the certer circuit i t cert the ac urce eergy it ctrlled dc lad wer, maily fr highly iductie lad Tur-ff f certer emicductr deice i brught abut by the ac uly ltage reeral, a rce called lie cmmutati r atural cmmutati Certer circuit emlyig ly dide are termed uctrlled (r rectifier while the icrrati f ly thyritr reult i a (fully ctrlled certer The fuctial differece i that the dide cduct whe frward-biaed wherea the tur- f the frward-biaed thyritr ca be ctrlled frm it gate A uctrlled certer ride a fixed utut ltage fr a gie ac uly ad lad Certer emlyig a cmbiati f bth dide ad thyritr are geerally termed half-ctrlled (r emi-ctrlled Bth fully ctrlled ad half-ctrlled certer allw a adjutable utut ltage by ctrllig the hae agle at which the frward biaed thyritr are tured The larity f the utut (lad ltage f a fully ctrlled certer ca reere (but the curret flw directi i t reerible, allwig wer flw it the uly, a rce called ieri Thu a fully ctrlled certer ca be decribed a a bidirectial certer a it facilitate wer flw i either directi The half-ctrlled certer, a well a the uctrlled certer, ctai dide which reet the utut ltage frm gig egatie Such certer ly allw wer flw frm the ac uly t the dc lad, termed rectificati, ad ca therefre be decribed a uidirectial certer Althugh all thee certer tye ride a dc utut, they differ i characteritic uch a utut rile ad mea ltage a well a efficiecy ad ac uly harmic A imrtat certer characteritic i that f ule umber, which i defied a the reetiti rate i the direct utut ltage durig e cmlete cycle f the iut ac uly A ueful way t judge the uality f the reuired dc utut, i by the ctributi f it uerimed ac harmic The harmic r rile factr F i defied by rm dc rm F FF dc dc where FF i termed the frm factr F i a meaure f the ltage harmic i the utut ltage while if curret are ued i the euati, F i gie a meaure f the curret harmic i the utut curret Bth FF ad F are alicable t the iut ad utut, ad are fully defied i ecti 0 The geeral aalyi i thi chater i ccered with igle ad three hae ac ulie maily feedig iductie dc lad A lad dc back emf i ued i mdellig the dc machie Geerally, uctrlled rectifier euati ca be deried frm the crredig ctrlled certer circuit euati by ettig the ctrlled delay agle t zer Al urely reitie lad euati geerally ca be deried by ettig iductace t zer i the - lad euati ad - lad euati ca be deried frm --E euati by ettig E, the lad back emf, t zer r x r x 0 di/dt curret le 0 i Z +ω Φ X ω β - curret exticti agle Sigle-hae uctrlled certer circuit ac rectifier Half-wae rectifier circuit with a - lad A imle half-wae dide rectifyig circuit i hw i figure a, while ariu circuit electrical waefrm are hw i figure b ad curret cmmece whe the uly ltage ge itie at 0 t will be ee that lad curret flw t ly durig the itie half f the ac uly ltage, 0, but al durig a rti f the egatie uly ltage, β The lad iductr tred eergy maitai the lad curret ad the iductr termial ltage reere ad i able t ercme the egatie uly ad kee the dide frward-biaed ad cductig Thi curret ctiue util all the iductr eergy, i, i releaed (i 0 at the curret exticti agle (r cut-ff agle, β Figure Half-wae rectifier with a - lad: (a circuit diagram ad (b waefrm, illutratig the eual area ad zer curret le criteria where Z ( + ω (hm ta φ ω/ Q ad Z cφ i( 0 (A ( β (rad The curret exticti agle β i determied lely by the lad imedace Z ad ca be led frm euati ( whe the curret, i 0 with β, uch that β >, that i

2 Pwer Electric aturally cmmutatig certer 4 - /ta e β φ i( β - φ + i φ 0 (4 Thi i a tracedetal euati which ca be led by iteratie techiue Figure a ca be ued t determie the exticti agle β, gie ay lad imedace (wer factr agle φ ta ω/ The mea alue f the rectified curret, the utut curret,, i gie by itegrati f euati ( β ( (A i d 0 (5 ( c β (A while the mea utut ltage i gie by β i ω ω ( c β ( t d t 0 (6 Sice the mea ltage acr the lad iductace i zer, (ee the eual area criteri belw Figure b hw the rmalied utut ltage / a a fucti f ω / The rm utut (lad ltage ad curret are gie by rm β ( i ω ω { } 0 β i β t d t irm (7 cφ i βc( β + φ i βc( β + φ β β cφ Z cφ Frm euati (6 ad (7 the harmic ctet i the utut ltage i idicated by the ltage frm factr { β iβ} rm FF (8 c β Fr a reitie lad, whe β, the frm factr reduce t a alue f 57 The rile factr i therefre FF Fr a urely reitie lad the ltage ad curret frm factr are eual The wer deliered t the lad, which i the wer deliered t the lad reitace, i P irm (9 The uly wer factr, uig the rm curret i euati (7, i wer, P f aaret wer (0 i i c rm ( rm i rm β β + φ β cφ µ cφ i cφ rm The characteritic fr a --E lad ca be determied by uig 0 i the cae f the half-wae ctrlled certer i ecti iii Fr a urely iductie lad,, β i ubtituted it the arriate euati The aerage utut ltage ted t zer ad the curret i gie by i( { c } (A ω which ha a mea curret alue f /ω / 045 Figure Sigle-hae half-wae certer characteritic: (a lad imedace agle ø eru curret exticti agle β ad (b ariati i rmalied mea utut ltage / eru ω/ i ductr eual ltage area criteri The aerage utut ltage, gie by euati (6, i baed the fact that the aerage ltage acr the lad iductace, i teady tate, i zer The iductr ltage i gie by di / dt ( which fr the circuit i figure a ca be exreed a β / ω iβ ( t dt di ( i i β 0 0 ( i f the lad curret i i teady tate the i i, which i zer here, ad i geeral β dt 0 ( ( The iductr ltage waefrm fr the circuit i figure a i hw i the lat lt i figure b The iductr eual ltage area criteri imlie that the haded itie area mut eual the haded egatie area, i rder t atify euati ( The et iductr eergy at the ed f the cycle i zer (ecifically, uchaged ice i i β, that i, the eergy i eual the eergy ut Thi i a ueful aid i redictig ad drawig the lad curret waefrm t i ueful t uerime the uly ltage, the lad ltage, ad the reitr ltage waefrm the ame time axi, The lad reitr ltage, i, i directly related t the lad curret, i The iductr ltage will be the differece betwee the lad ltage ad the reitr ltage, ad thi buded et area mut be zer Thu the aerage utut ltage i The eual ltage area aciated with the lad iductace are hw haded i tw lt i figure b ii - ad curret zer le criteri The lad iductace ltage larity chage frm itie t egatie a eergy iitially traferred it the iductr, i releaed The tred eergy i the iductr allw curret t ctiue t flw after the iut ac ltage ha reered At the itat whe the iductr ltage reere, it termial ltage i zer, ad di/ dt 0 ( that i di / dt 0 The curret le chage frm itie t egatie, whece the ltage acr the lad reitace ceae t icreae ad tart t decreae, a hw i figure b That i, the i waefrm cre the uly ltage waefrm with zer le, whece whe the iductr ltage i zer, the curret begi t decreae The fact that the reitr ltage le i zer whe 0, aid redicti ad ketchig f the ariu circuit waefrm i figure b, ad ubeuet waefrm i thi chater

3 5 Pwer Electric aturally cmmutatig certer 6 Examle : Half-wae rectifier ctat curret the dc uly half-wae rectifier circuit f figure Examle, the urce ltage i 0 i( 50t with a iteral reitace i hm, 0 hm, ad the filter caacitr C i ery large Calculate i the mea alue f the lad ltage, ii the dide aerage ad eak curret, D, D iii the caacitr eak chargig ad dichargig curret i the dide reere blckig ltage, D i i D i C i C i D θ β D iii i Alteratiely, a wuld be exected, the aerage dide curret i 0995 D 0A 0Ω The eak dide curret i 0 0 D 5A Ω i The caacitr eak chargig curret i the differece betwee the eak dide curret ad the lad curret, iz, 5A - A 94A, while the eak dichargig curret i the aerage lad curret f A The dide reere ltage i the differece betwee the itataeu uly ltage ad the utut ltage 0 Thi i a maximum at the egatie eak f the ac uly, whe the dide ltage i Durig ay erid whe the lad i diruted, the utut caacitr ca charge t 0, hece the dide ca exeriece, wre cae, Sluti Figure Examle Sigle-hae half-wae rectifier: (a circuit ad (b waefrm i Becaue the lad filter caacitr i large, it i aumed that the dc utut ltage i rile free ad ctat The caacitr ride the lad curret whe the ac uly leel i le that the dc utut The lad curret ad eak dide (hece uly curret are therefre D i The ac uly ride curret, thrugh the rectifyig dide, durig the erid i ( i β i f the caacitr ltage i t be maitaied ctat, the charge it the caacitr mut eual the charge deliered by the caacitr whe the dide i t cductig, that i ii β + ( ω β i i d t i d Al i θ + θ β Maiulati yield i Ω taθ θ 0 0Ω A iteratie luti yield θ 996º, that i, the dide cduct fr a erid f 55m, eery cycle f the ac uly The caacitr ltage i θ i i i 0995 The aerage dide curret i gie by β ( θ D i d c θ i i c A Ω 80 i D Half-wae circuit with a - lad ad freewheel dide The circuit i figure a, which ha a - lad, i characteried by dictiuu curret (i 0 ad high rile curret Ctiuu lad curret ca reult whe a dide D f i added acr the lad a hw i figure a Thi freewheel dide reet the ltage acr the lad frm reerig durig the egatie half-cycle f the ac uly ltage The iductr eergy i t retured t the ac uly, rather i retaied i the lad circuit The tred eergy i the iductr cat reduce t zer itataeuly, the curret i frced t fid a alteratie ath whilt decreaig tward zer Whe the rectifier dide D ceae t cduct at zer lt it blck, ad dide D f ride a alteratie lad curret freewheelig ath, a idicated by the waefrm i figure b The utut ltage i the itie half f the iuidal iut ltage The mea utut ltage (thece mea utut curret i iω ω t d t 0 (4 045 ( The rm alue f the lad circuit ltage 0 i gie by rm ( i d 0 (5 07 ( The utut rile (ac ltage i defied a i rm ( hece the lad ltage frm ad rile factr are defied a FF / rm F / rm i ( 57 FF ¼ (7 After a large umber f ac uly cycle, teady-tate lad curret cditi are etablihed, ad frm Kirchhff ltage law, the lad curret i defied by di + i dt i (A 0 (8 ad whe the freewheel dide cduct di + i 0 dt (A (9

4 7 Pwer Electric aturally cmmutatig certer 8 /taφ ( ( e i φ + D ( + e i φ /taφ Df D ( figure b it will be ee that althugh the lad curret ca be ctiuu, the uly curret i dictiuu ad therefre ha a high harmic ctet r x r x The utut ltage Furier erie ( + +, 4, 6 i ( t + i c (,4,6 ( Diidig each harmic utut ltage cmet by the crredig lad imedace at that freuecy gie the harmic utut curret, whece rm curret That i Z + jω + ω (4 ad ( (5 rm,, 4, 6 + Examle : Half-wae rectifier Figure Half-wae rectifier with a lad freewheel dide ad a - lad: (a circuit diagram ad arameter ad (b circuit waefrm Durig the erid 0, whe the freewheel dide curret i gie by i Df 0, the uly curret, which i the lad curret, are gie by i( i ( t /ta i( ( ω φ φ + + i φ e (A Z Z (0 0 fr /taφ + e i φ /ta φ /taφ Z e e (A where Z + ( ω (hm ta φ ω/ Durig the erid, whe the uly curret i 0, the freewheel dide curret ad hece lad curret are gie by i ( i Df ( ( /taφ e (A ( fr /ta e φ (A Fr dictiuu lad curret (the freewheel dide curret i Df fall t zer befre the rectifyig dide D recmmece cducti, the arriate itegrati gie the aerage dide curret a the circuit f figure, the urce ltage i 40 i( 50t, 0 hm, ad 50 mh Calculate i the mea ad rm alue f the lad ltage, ad rm ii the mea alue f the lad curret, iii the curret budary cditi, amely ad i the aerage freewheel dide curret, hece aerage rectifier dide curret the rm lad curret, hece lad wer ad uly rm curret i the uly wer factr f the freewheel dide i remed frm acr the lad, determie ii a exrei fr the curret hece the curret exticti agle iii the aerage lad ltage hece aerage lad curret ix the rm lad ltage ad curret x the wer deliered t the lad ad uly wer factr Frm the rm ad aerage utut ltage ad curret, determie the lad frm ad rile factr Sluti i Frm euati (4, the mea utut ltage i gie by Frm euati (5 the lad rm ltage i / 40 / 697 rm ii The mea utut curret, euati (5, i 40 0Ω 08A iii The lad imedace i characteried by Z + ( ω 0 + ( 50Hz Ω ta φ ω/ 50Hz 005H /0Ω 57 r φ 575 rad Frm ecti, euati (0

5 9 Pwer Electric aturally cmmutatig certer 0 /taφ + e iφ Z /ta φ /taφ e e 86Ω + e e e Hece, frm euati ( /ta φ /57 e 4 e 5A Sice 4A 0, ctiuu lad curret flw > 40 i(ta 57 /57 4A /57 /57 i tegrati f the dide curret gie i euati ( yield the aerage freewheel dide curret /taφ i ( d e d Df 0 0 Df t /57rad 5A ω 57 5A e d 57rad e 546A 0 The aerage iut curret, which i the rectifyig dide mea curret, i gie by 08A 546A 54A D Df The lad ltage harmic gie by euati (4 ca be ued t ealuate the lad curret at the lad imedace fr that freuecy harmic t ( + i c(,4,6 ( The fllwig table hw the calculati fr each freuecy cmet harmic ( ( ( ω Z + (Ω Z (A 0 ( (67 ( The rm lad curret i A rm,, 4 The wer diiated i the lad reitace i therefre P 0Ω 0 68A 6067W rm The freewheel dide rm curret i ( /taφ ( ω e d t 0 Df ( /57rad ( 5A ω 88A e d t 0 Thu the iut (ad rectifyig dide rm curret i gie by rm D rm rm Df rm A i The iut ac uly wer factr i Put 6067W f A rm rm ii f the freewheel dide D f i remed, the curret i gie by euati (, that i i( - /ta φ { i ( - φ + i φ e } Z 40 - /57 { i ( e } 86Ω - /57 8 i ( e (A 0 β (rad { } The curret exticti agle β i fud by ettig i 0 ad lig iteratiely fr β Figure a gie a iitial etimate f 40 (49 rad whe φ 575 ( rad That i - /57 0 i ( β e β gie β 408 rad r 8, after iterati iii The aerage lad ltage frm euati (6 i 40 ( c β ( c The aerage lad curret i / 860/0 Ω 860A ix The lad rm ltage i 697 with the freewheel dide ad icreae withut the dide t, a gie by euati (7, { β iβ} rm 40 { 408 i 408} 86 The rm lad curret frm euati (7 i decreaed t ( + + i βc β φ i β rm Z cφ 40 i 408c( A 86Ω c57 emal f the freewheel dide decreae the rm lad curret frm 68A t 968A x The lad wer i reduced withut a lad freewheel dide, frm 6067W with a lad freewheel dide, t P i 0Ω rm W The uly wer factr i al reduced, frm 074 t Put 97W f A ad factr rm rm circuit with freewheel dide circuit withut freewheel dide frm factr rile factr frm factr rile factr FF rm / ae F FF - FF rm / ae F FF - ltage factr 697/ /86 86 Curret factr 68A/08A A/860A 057 Sigle-hae, full-wae bridge rectifier circuit Sigle-hae uctrlled full-wae bridge circuit are hw i figure 4a ad 4b Bth circuit aear idetical a far a the lad ad uly are ccered t will be ee i art b that tw fewer dide ca be emlyed but thi circuit reuire a cetre-taed ecdary trafrmer where each ecdary ha ly a 50% cer utiliati factr Fr the ame utut ltage, each f the ecdary widig i figure 4b mut hae the ame rm ltage ratig a the igle ecdary widig f the trafrmer i figure 4a The rectifyig dide i figure 4b exeriece twice the reere ltage, (, a that exerieced by each f the fur dide i the circuit f figure 4a, (

6 Pwer Electric aturally cmmutatig certer Figure 4c hw bridge circuit ltage ad curret waefrm With a iductie aie lad, ( back emf ctiuu lad curret flw, which i gie by i ( iφ /taφ i ( φ + e /taφ Z e 0 (6 Arriate itegrati f the lad curret uared, gie the rm lad (ad ac uly curret: /taφ + 4i φtaφ ( + e Z (7 rm The lad exeriece the trafrmer ecdary rectified ltage which ha a mea ltage (thece mea lad curret f iω ω t d t ( (8 Sice the aerage iductr ltage i zer, the aerage reitr ltage eual the aerage - ltage The rm alue f the lad circuit ltage 0 i i d ( (9 rm r x r x 0 ( Frm the lad ltage defiiti i ecti 0, the lad ltage frm factr i rm FF (0 The lad rile ltage i rm ( ( 045 ( hece the lad ltage rile factr i F / FF F ( ( / 048 which i igificatly le (better tha the half-wae rectified alue f frm euati (7 The utut ltage ad curret (rm ad aerage ca be deried frm the ltage Furier exai frm fr a half-ie wae: ( + c (,4,6 The firt term i the aerage utut ltage, a gie by euati (8 te the harmic magitude decreae raidly with icreaed rder, amely : : : : The utut ltage i therefre dmiated by the dc cmet ad the harmic at ω The utut curret ca be deried by diidig each ltage cmet by the arriate lad imedace at that freuecy That i (4 fr, 4,6 Z + ω ( The lad rm curret whece lad wer, critical lad iductace, ad wer factr, are gie by + rm rm,4 P P rm f ( ee euati 8 critical ω rm (5 Each dide rm curret i / Fr the circuit i figure 4a, the trafrmer ecdary widig rm rm curret i rm, while fr the cetre-taed trafrmer, fr the ame lad ltage, each widig ha a rm curret ratig f rm / The rimary curret ratig i the ame fr bth trafrmer ad i related t the ecdary rm curret ratig by the tur rati i - Sigle-hae full-wae bridge rectifier circuit with a utut -C filter A with a utut -C filter ad ctiuu lad curret Table hw three tyical igle-hae, full-wae rectifier utut tage, where art c i a tyical utut filterig tage ued t btai a ear ctat dc utut ltage f it i aumed that the lad iductace i large ad the lad reitace mall uch that ctiuu lad curret flw, the the bridge aerage utut ltage i the ame a the aerage ltage acr the lad reitr ice the aerage ltage acr the filter iductr i zer Frm euati (, the dmiat lad ltage harmic i due t the ecd harmic therefre the ac curret i redmiately the ecd harmic curret, By eglectig the higher rder harmic, the ariu circuit ac,, curret ad ltage ca be readily btaied a hw i table Frm euati ( the utut ltage i gie by Figure 4 Sigle-hae full-wae rectifier bridge: (a circuit with fur rectifyig dide; (b circuit with tw rectifyig dide; ad (c circuit waefrm

7 Pwer Electric aturally cmmutatig certer 4 ( ω +, t c + c fr + c c (6 With the filter caacitr acr the lad reitr, the aerage iductr curret i eual t the aerage reitr curret, ice the aerage caacitr curret i zer With ctiuu iductr curret, the iductr curret i i ( + c, c + c (7 Z + ω ( Frm euati (7 fr ctiuu iductr curret, the aerage curret mut be larger tha the eak ecd harmic curret magitude, that i > >, Z (8 Table Sigle-hae full-wae uctrlled rectifier circuit ctiuu iductr curret Full-wae rectifier circuit d harmic aerage utut curret curret utut wer, P +P E ad circuit (A (A (W Sice the lad reitr mut be mall eugh t eure ctiuu iductr curret, the ω > uch that Z + ( ω ω Euati (8 therefre gie the fllwig lad idetity fr ctiuu iductr curret > > Z ω that i ω (9 The lad ad uly (eak ac curret are The utut ad uly rm curret are, ac, ac, + + (40,,,, ad the wer deliered t reitace i the lad i P (4, rm B with a utut -C filter ad dictiuu lad curret f the iductr curret reduce t zer, at agle β, all the lad curret i rided by the caacitr t ltage fall t (< ad iductr curret recmmece whe i (4 at a agle i (4 By itegratig di/dt fr i, the iductr curret i f the frm i ( ( ( c c ( ω (44 where β The ltage i fud frm euati (44 by iteratie techiue Examle : Full-wae uctrlled rectifier with a -C filter ad ctiuu lad curret (a - ee ecti ad 0, + ( ω, rm A igle-hae, full-wae, dide rectifier i ulied frm a 0 ac, 50Hz ltage urce ad ue a -C utut filter with a reitr lad, a hw i the lat circuit i Table The aerage iductr curret i 0A with a 4A rm rile curret dmiated by the 00Hz cmet grig dide ltage dr ad iitially aumig the utut ltage i rile free, determie i the dc utut ltage, hece lad reitace ad wer ii the dc filter iductace ad it aerage ltage, whilt eglectig ay caacitr ltage rile iii the dc filter caacitace if it eak-t-eak rile ltage i 5% the aerage ltage i dide aerage, rm, ad eak curret the uly wer factr Sluti (b --E ee ecti 4 0 E +, + ( ω E E E +, rm Sice <, ( 4A < 0A rm, the utut curret i ctiuu i The dc utut ltage i Aumig the 07 i rile free, that i, rm dc, the the lad reitace ad wer diiated are Ω 0A P 07 0A 070W (c -//C ee ecti i,ac C,dc, ω, rm ii The 00Hz ltage cmet i the utut ltage i gie by euati (6, that i c, c c 8 c which ha a rm alue f 8/ 976 The 00Hz rm curret / rduced by thi, ltage i 4A thu

8 5 iii i,, frm ω 976 ω 50Hz 4A,, Pwer Electric 88mH The aerage iductr ltage i zer Frm art i, the dc utut ltage i 07 The eak-t-eak rile ltage i 5% f 07, that i 05 Thi gie a rm alue f 05 / 66 Frm,,, X c, 00Hz ωc, 4A C 7mF ω 50Hz 66 The dide curret are, / + / 0A + 4A / 08A / 764A D, rm, rm, D / 0A/ 5A + 0A + 4A 57A D, aturally cmmutatig certer 6 A the ac uly ltage rie t it extreme each half cycle, a hw i figure 5b, a air f rectifier dide D-D r D-D4, alterately becme frward biaed at time The ac uly ride lad reitr curret ad imultaeuly charge the caacitr, it ltage haig dred whilt ridig the lad curret durig the reiu dide -cducti erid The caacitr chargig curret erid θ c arud the ac uly extreme i hrt, giig a high eak t rm rati f dide ad uly curret Whe all the rectifier dide are reere biaed at β becaue the caacitr ltage i greater tha the itataeu uly ac ltage, the caacitr ulie the lad curret ad it ltage decreae with a -C time ctat util + The utut ltage ad dide ltage, lu lad curret /, ad caacitr curret C d /dt are defied i Table The tart f dide cducti,, the dide curret exticti agle, β, hece dide cducti erid, θ c, are ecified by the fllwig euati Frm i c + i 0 at β : c β + i β 0 X (49 β ta ωc ta ωc β ( ( By euatig the tw exrei fr utut ltage at the budary + gie i( + ( iβ e + β/ta (50 β ad a tracedetal exrei fr reult: ( i i β e + β/ 0 (5 ωc The iut ad utut rm curret i + 0A + 4A 08A, rm, Aumig the iut wer eual the utut wer, the frm art i, P P i 070W The uly wer factr i P P i i 070W f 08 S 0 08A ii - Sigle-hae full-wae bridge circuit with highly iductie lad ctat lad curret D4 D D D i (a id C ic i With a highly iductie lad, which i the uual ractical cae, irtually ctat lad curret flw, a hw dahed i figure 4c The bridge dide curret are the uare wae 80º blck f curret f magitude The dide curret ratig ca w be ecified ad deed the ule umber Fr thi full-wae igle-hae alicati each iut cycle cmrie tw 80º utut curret ule, hece The mea curret i each dide i (A (45 D D D θc D D4 θc D D D D4 cductig dide ad the rm curret i each dide i / (A (46 D whece the dide curret frm factr i FD / (47 D D Sice the lad curret i arximately ctat, wer deliered t the lad i P (48 8 (W Ca β + The uly wer factr i f / / 090, ice rm id iii - Sigle-hae full-wae bridge rectifier circuit with a C-filter ad reitie lad The caacitr mthed igle-hae dide rectifier circuit hw i figure 5a i a cmm wer rectifier circuit ued t btai uregulated dc ltage The circuit i imle ad chea but the iut curret ha high eak ad rm alue, high harmic, ad a r wer factr The caacitr reduce the rile ltage, large ltage-laried caacitace i ued t rduce a almt ctat dc utut ltage lati ad ltage matchig (te-u r te dw are btaied by uig a trafrmer befre the dide rectificati tage a hw i figure 4a ad b The reitr acr the filter caacitr rereet a reitie diiatie lad β + (b Figure 5 Sigle-hae full-wae rectifier bridge: (a circuit with C-filter caacitr ad (b circuit waefrm i

9 7 Pwer Electric aturally cmmutatig certer 8 Table Sigle-hae, full-wae rectifier ltage ad curret ( i Dide cductig β Outut ltage ( i Dide ltage D ( 0 ad - i Caacitr curret eitr curret Dide bridge curret i c ( i ( D( i c(+ i ( cω t X iω t i cφ + ( φ Dide -cductig β + iβ e ( β/taβ ( β/taβ iβ e + i ( β/taβ e Z ( β /taβ e i ω c t Z The dide curret cducti erid θ c i gie by θ β c (5 Whe the dide cduct, ad C are i arallel ad taφ ωc Whe the dide are t cductig, the utut circuit curret flw i a erie -C circuit with a fudametal imedace f: Z + X ad X ω C The reitr aerage ltage ad curret are ( cθc c β (5 The maximum utut ltage ccur at whe, while the miimum utut ltage ccur at the ed f the caacitr dicharge erid whe ad i The utut eak-t-eak rile ltage i therefre the differece: i ( i (54 By aumig, β, ad a erie exai fr the exet ωc f C (55 The ac urce curret i the um f the dide curret, that i i i D id i + i c (56,,4 whe < < β Otherwie i 0 Sice the caacitr ltage i i teady-tate, the aerage caacitr curret i zer, thu fr full-wae rectificati, the aerage dide curret i half the aerage lad curret The eak caacitr curret ccur at, whe the dide firt cduct Frm the caacitr curret euati i table : ωc c (57 c Frm table, the eak dide curret ccur at the ame time a the eak caacitr curret, : i ( + + i ( + c D ωcc + Z (58 i c + i i( + φ X X Similar exrei ca be deried fr the half-wae rectifier cae Fr the -cducti erid, β+ The utut rile ltage i abut twice that gie by euati (55 ad the aerage reitr ltage i euati (5 (after mdificati, i reduced Examle 4: Sigle-hae full-wae bridge rectifier circuit with C-filter ad reitie lad 0 ( i exrei fr the utut ltage ii utut ltage rile ad the % errr i uig the arximati euati (55 iii exrei fr the caacitr curret i dide eak curret the aerage lad ltage ad curret Aumig the utut rile ltage i triagular, etimate i the aerage utut ltage ad rm utut rile ltage ii caacitace C fr % f the maximum utut ltage Sluti The uly ltage i 0 i 50t, which ha a eak alue f 5 ωc 50Hz 00Ω 000µF 46 rad Thu X /ωc 8Ω ad Z Ω Frm euati (49 the dide curret exticti agle β i β ta ωc ta 46rad 606rad 98 ( ( The dide curret tur- agle i le iteratiely frm euati (5, that i ( + β / ωc i i β e 0 ( + 60 / 46 i i60 e 0 gie 6095 rad r 665 The dide cducti erid i θ c β rad r 5 i Frm table, the utut ltage, which i the caacitr ltage, i gie by ( i 57 i ( ( 606rad /46rad ( 0 i606rad e 5 e ii The utut ltage rile i gie by euati (54, that i ( i 0 ( i Frm euati ( f C 50Hz 00Ω 000µF The arximati redict a higher rile: a +% er-etimate iii 606rad / 46rad Frm table, the caacitr curret i i ( c ωcc 0 50Hz 000µF c 0 c i ( c i β ( β / ωc 0 i6 ( 6 / 4 ( 6 / 4 e e 0 e 00Ω i The eak dide curret i gie by euati (58: ωcc + i D Hz 000µF c606 + i606 00Ω 407A + A 47A The eak dide curret i dmiated by the caacitr iitial chargig curret f 407A A igle-hae, full-wae, dide rectifier i ulied frm a 0 ac, 50Hz ltage urce ad ue a caacitr utut filter, 000µF, with a reitr 00Ω lad, a hw i Figure 5a grig dide ltage dr, determie

10 9 Pwer Electric aturally cmmutatig certer 40 The aerage lad ltage ad curret are gie by euati (5 ( cθc c β 0 ( c 0447 c60 A 00Ω i f the rile ltage i aumed triagular the (a The aerage utut ltage i the eak utut ltage miu half the rile ltage, that i which i le tha that gie by the accurate euati (54, (b f the 69 - rile ltage i aumed triagular the it rm alue i 69/ 78 rm Sigle-hae full-wae half-ctrlled certer Whe a certer ctai bth dide ad thyritr, fr examle a hw i figure 7 art a t d, the certer i termed half-ctrlled (r emi-ctrlled Thee fur circuit rduce idetical lad ad uly waefrm, eglectig ay differece i the umber ad tye f emicductr ltage dr The wer t the lad i aried by ctrllig the agle, hw i figure 7e, at which the bridge thyritr are triggered (after firt becmig frward biaed The circuit dide reet the lad ltage frm gig egatie, exted the cducti erid, ad reduce the utut ac rile The articular alicati will determie which e f the fur circuit huld be emlyed Fr examle, circuit figure 7a ctai fie deice f which fur are thyritr, wherea the ther circuit ctai fewer deice, f which ly tw are thyritr The circuit i figure 7b ue the fewet emicductr, but reuire a trafrmer which itrduce extra ct, weight, ad ize Al the thyritr exeriece twice the ltage f the thyritr i the ther circuit, rather tha The trafrmer de ride ilati ad ltage matchig T ii e-arragemet f euati (55, which uder-etimate the caacitace reuiremet fr % rile, gie C f %f % f f 5, 000µF 50Hz 00Ω 00 r x r x (a (b T D i - Other igle-hae bridge rectifier circuit cfigurati Df Figure 6a hw a trafrmer ued t create a tw-hae uly (each hae i 80 aart, which u rectificati rduce eual lit-rail dc utut ltage, + ad - The electrical characteritic ca be aalyed a i the cae f the igle-hae full-wae bridge rectifier circuit with a caacitie C- filter ad reitie lad, i ecti iii the lit rail cae, the rectifier cduct eery 80, alterately feedig each utut ltage rail caacitr Thu the dide aerage ad rm curret are icreaed by ad reectiely, abe the f a cetial igle hae rectifier The ltage dubler i figure 6b ca be ued i euimet that mut be able t erate frm bth 5ac ad 0 ac ltage ulie, withut the aid f a ltage-matchig trafrmer With the witch i the 5 iti, the utut i twice the eak f the iut ac uly The caacitr C charge thrugh dide D, ad whe the uly reere, caacitr C charge thrugh D Sice C ad C are i erie, the utut ltage i the um C + C, where each caacitr i alterately charged (half-wae rectified frm the ac urce The ther, uued, tw dide remai reere biaed, ad are ly eceary if the dual iut ltage fucti i reuired With the witch i the 0 ac iti (e circuit, tadard rectificati ccur, with the tw erie caacitr chargig imultaeuly eery half cycle dual freuecy alicati (0 ac, 60Hz ad 0 ac, 50Hz, the caacitace reuiremet are baed the uly with the lwer freuecy, 50Hz (c (d :: C + + 5/0 ac iut D C C C + D (a (b Figure 6 Bridge rectifier: (a lit rail dc ulie ad (b ltage dubler Circuit a T ad T4 D T ad T D b T D T D c T ad D T ad D T ad D T ad D d T ad D D ad D T ad D D ad D (e Figure 7 Full-wae half-ctrlled certer with freewheel dide: (a, (b, (c, ad (d differet circuit cfigurati rducig the ame utut; ad (e circuit ltage ad curret waefrm ad deice cducti table

11 4 Pwer Electric aturally cmmutatig certer 4 The thyritr triggerig reuiremet f the circuit i figure 7b ad c are imle ice bth thyritr hae a cmm cathde cecti Figure 7c may uffer frm rlged hut-dw time with highly iductie lad The dide i the freewheelig ath will hld the freewheelig thyritr, allwig cducti durig that thyritr ext itie cycle withut ay gate drie reet The extra dide D f i figure 7c byae the bridge thyritr allwig them t dr ut f cducti Thi i achieed at the exee f a extra deice, but the freewheel ath cducti le are decreaed ice that erie circuit w ile ly e emicductr ltage dr Thi ctiued cducti rblem de t ccur i circuit 7a ad d ice freewheelig de t ccur thrugh the circuit thyritr, hece they will dr ut f cducti at certer hut-dw The table i figure 7e hw which emicductr are actie i each circuit durig the ariu erid f the lad cycle Circuit waefrm are hw i figure 7e Sice the lad i a aie - circuit, ideedet f whether the lad curret i ctiuu r dictiuu, the mea utut ltage ad curret (eglectig dide ltage dr are i( ( c ( d + (59 ( + c (A where i the delay agle frm the it at which the aciated thyritr firt becme frward-biaed ad i therefre able t be tured ad cduct curret The maximum mea utut ltage, / (al redicted by euati (8, ccur at 0 The rmalied mea utut ltage i / (+ c (60 The Furier cefficiet f the -ule utut ltage are gie by euati (76 Fr the iglehae, full-wae, half-ctrlled cae,, thu the utut ltage harmic ccur at, 4, 6, Euati (59 hw that the lad ltage i ideedet f the aie lad (becaue the dide clam the lad t zer lt thereby reetig the lad ltage frm gig egatie, ad i a fucti ly f the hae delay agle fr a gie uly ltage The rm alue f the lad circuit ltage i rm ( + i i ω ω t d t ( (6 Frm the lad ltage defiiti i ecti 0, the lad ltage frm factr i ( + i rm FF ( + c (6 The rile ltage i i rm (6 hece the ltage rile factr i K i / FF (64 The lad ad uly waefrm ad euati, fr ctiuu ad dictiuu lad curret, are the ame fr all the circuit i figure 7 The circuit differ i the deice cducti ath a hw i the table i figure 7e After deriig the geeral lad curret euati, the curret euati alicable t the differet circuit deice ca be decded i - Dictiuu lad curret, with < ad β <, the lad curret (ad uly curret i baed euati ( amely + ta ( t ( t φ i ω i ω ( i( φ i ( φ e (A Z (65 ( ω where Z + ad φ ta ω After the lad curret decreae exetially t zer thrugh the freewheel dide accrdig t i( idf ( /taφ e (A 0 (66 0 where fr i euati (65 /taφ i( φ ( e Z The ariu emicductr aerage curret ratig ca be determied frm the aerage half-cycle freewheelig curret,, ad the aerage half-cycle uly curret, F Fr dictiuu lad curret ( ( ( F iφ iφ i φ e (67 /taφ ( ( ( /taφ φ φ φ - F c + c + i i e (68 ii - Ctiuu lad curret, with < φ adβ, the lad curret i gie by euati imilar t euati (0 ad (, ecifically /taφ iφ e i( φ + taφ i( i ( ω t i( φ + e Z /taφ e (69 (A while the lad curret whe the freewheel dide cduct i /taφ i( i ( e (A Df 0 (70 0 where, fr i euati (69 /taφ iφ i( φ e + (A 0 Z /taφ e The ariu emicductr aerage curret ratig ca be determied frm the aerage half cycle freewheelig curret, F, ad the aerage half cycle uly curret, Fr ctiuu lad curret + /taφ iφ i ( φ e /taφ F iφ /ta ( e φ (7 e -F ( + /taφ e (7 taφ cφ taφ /ta ( e iφ i ( φ + cφ + c φ ( φ e Table Semicductr aerage curret ratig Bridge circuit umber Aerage deice curret figure 7 f deice Thyritr Dide a 4T+D F b T+D F c T+D d T+D + F The deice cducti table i figure 7e ca be ued t ecify aerage deice curret, fr bth ctiuu ad dictiuu lad curret fr each f the circuit i figure 7, art a t d Fr a highly iductie lad, ctat lad curret, the uly wer factr i f c Critical lad iductace The critical lad iductace, t reet the curret fallig t zer, i gie by ω crit + i + cθ θ + (7 + c fr θ where c i + θ i (74 The miimum curret ccur at the agle θ, where the mea utut ltage eual the itataeu lad ltage, Whe the hae delay agle i greater tha the critical agle θ, θ i euati (74 yield (ee figure ω crit + i + c + (75 + c t i imrtat t te that certer circuit emlyig dide cat be ued whe ieri i reuired Sice the certer dide reet the utut ltage frm beig egatie, (ad the curret i uidirectial, regeerati frm the lad it the uly i t achieable Figure 7a i a fully ctrlled certer with a - lad ad freewheel dide igle-hae circuit, thi certer eetially behae a a half-ctrlled certer

12 4 Pwer Electric aturally cmmutatig certer 44 Sigle-hae ctrlled thyritr certer circuit Half-wae ctrlled circuit with a - lad The rectifyig dide i the circuit f figure ca be relaced by a thyritr a hw i figure 8a t frm a half-wae ctrlled rectifier circuit with a - lad The utut ltage i w ctrlled by the thyritr trigger agle, The utut ltage rile i at the uly freuecy Circuit waefrm are hw i figure 8b, where the lad iductr ltage eual area are haded The utut curret, hece utut ltage, fr the erie circuit are gie by di + i i ( dt (76 β (rad where hae delay agle ad curret exticti agle β are hw i the waefrm i figure 8b ad are the zer lad (ad uly curret it Slig euati (76 yield the lad ad uly curret ( - /taφ i( {i( - φ - i( - φ e } (A Z where Z + ( ω (hm β (77 ta φ ω/ ad zer elewhere r x r x The curret exticti agle β i deedet the lad imedace ad thyritr trigger agle, ad ca be determied by lig euati (77 with β whe i(β 0, that i ( - /ta i( β - φ i( - φ e β φ (78 Thi i a tracedetal euati A family f cure f curret cducti agle eru delay agle, that i β - eru, i hw i figure 9a The traight lie lt fr φ i fr a urely iductie lad, wherea ø 0 i a traight lie fr a urely reitie lad The mea lad ltage, whece the mea lad curret, i gie by β i t d t ω ω (79 (c c β ( where the agle β ca be extracted frm figure 9a The rm lad ltage i ( β i ω ω rm t d t {( β (iβ i } (80 The rm curret ile itegrati f euati (77, uared, giig i ( β c( + φ + β ( β (8 rm Z cφ teratie luti t euati (78 are hw i figure 9a, where it i ee that tw traight-lie relatihi exit that relate ad β- Exact luti t euati (78 exit fr thee tw cae That i, exact tractable luti exit fr the urely reitie lad, Φ 0, ad the urely iductie lad, Φ (a (a traight lie β - traight lie β ⅓ ⅔ (b i Z E ( T / E (8 ⅓ (c Figure 8 Sigle-hae half-wae ctrlled certer: (a circuit diagram; (b circuit waefrm fr a - lad; ad (c urely iductie lad Delay agle (degree Figure 9 Half-wae, ctrlled certer thyritr trigger delay agle eru: (a thyritr cducti agle, β-, ad (b rmalied mea lad curret

13 45 Pwer Electric aturally cmmutatig certer 46 i - Cae : Purely reitie lad Frm euati (77, Z, φ 0, ad the curret i gie by i( i( (A (8 ad β The aerage lad ltage, hece aerage lad curret, are i t d t ω ω (8 (+ c ( where the maximum utut ltage i 045 fr zer delay agle The rm utut ltage i ( i ω ω rm t d t (84 {( + i } Sice the lad i urely reitie, rm / ad the ltage ad curret factr (frm ad rile are rm eual The wer deliered t the lad i P rm The uly wer factr, fr a reitie lad, i P ut / rm rm, that i f i - 4 (85 ii - Cae : Purely iductie lad Circuit waefrm hwig eual iductr ltage area are hw i figure 8c Frm euati (77, Z ω, φ, ad the utut ltage ad curret are gie by i ( (86 0 elewhere i( ( i( -i( - (A ω (87 ( c - c β ad β ω The aerage lad ltage, baed the eual area criteri, i zer - i d 0 (88 The aerage utut curret i {c - c } d ω (89 ( c + i ω The rm utut curret i deried frm ( c - cω ω ω t d t rm (90 ( ( + c + i X The rm utut ltage i ( i ω ω t d t rm (9 {( + i } Sice the lad i urely iductie, P 0 ad the lad ltage rile factr i udefied ice 0 By ettig 0, the euati (8 t (9 are alid fr the uctrlled rectifier cidered i ecti, fr a urely reitie ad urely iductie lad, reectiely iii - Cae : Back emf E ad - lad With a lad back emf, curret begi t flw whe the uly itataeu ltage exceed the back emf magitude E, that i whe i E (9 Whe curret flw, Kirchhff ltage law gie di iω t i + + E dt (9 which yield i ( E ta E φ taφ i( φ e i( φ e Z Z (94 β ad The exticti agle β i fud frm the budary cditi i( i(β 0, fr β > The lad wer i gie by P + rm (95 while the uly wer factr i gie by P + rm f (96 rm rm The luti fr the uctrlled certer (a half-wae rectifier i fud by ettig, e (9 Examle 5: Half-wae ctrlled rectifier The ac uly f the half-wae ctrlled igle-hae certer i figure 8a i 40 i Fr the fllwig lad ad #: 0Ω, ω 0 Ω ad #: 0 Ω, ω 0Ω ad #: 7Ω, ω 7Ω Determie i each lad cae, fr a firig delay agle /6 the cducti agle γβ -, hece the curret exticti agle β the dc utut ltage ad the aerage utut curret the rm lad curret ad ltage, lad curret ad ltage rile factr, ad wer diiated i the lad the uly wer factr Sluti ad #: Z 0Ω, ω 0 Ω Frm euati (77, Z 0Ω ad φ 0 Frm euati (8, β fr all, thu fr /6, γ β - 5/6 Frm euati (8 ( + c ( + c / The aerage lad curret i / ( + c 009/0 Ω0A The rm lad ltage i gie by euati (84, that i {( + i } rm 40 {( / 6 + i / } 67 Sice the lad i urely reitie, the wer deliered t the lad i P rm rm / 67 /0Ω 7970W / 679 /0Ω 68A rm rm

14 47 Pwer Electric aturally cmmutatig certer 48 Fr a urely reitie lad, the ltage ad curret factr are eual: 67 68A FF FF i A F F FF i The wer factr i 797W f A Alteratiely, ue f euati (85 gie /6 i /6 f ad #: 0 Ω, Z X ω 0Ω Frm euati (77, Z X 0Ω ad φ Frm euati (87, which i baed the eual area criteri, β -, thu fr /6, β /6 whece the cducti erid i γ β 5/ Frm euati (88 the aerage utut ltage i 0 The aerage lad curret i ( c + i ω 40 ( 5 / 6 c / 6 + i / 6 49A 0 Uig euati (90 ad (9, the lad rm ltage ad curret are rm rm { } 40 + i {( ( c i + + } 79A 0Ω 6 Sice the lad i urely iductie, the wer deliered t the lad i zer, a i the wer factr, ad the utut ltage rile factr i udefied The utut curret rile factr i rm 79A FF i 54 49A whece F i 54 4 ad #: 7Ω, ω 7Ω Frm euati (77, Z 0Ω ad φ ¼ Frm figure 9a, fr φ ¼ ad /6, γ β 95º whece β 5º terati f euati (78 gie β 55º 96 rad Frm euati (79 (c c β 40 (c0 c The aerage lad curret i / 850/7Ω 0A Alteratiely, the aerage curret ca be extracted frm figure 9b, which fr φ ¼ ad /6gie the rmalied curret a 05, thu Z A 0Ω Frm euati (8, the rm curret i ( i β c( + φ+ β ( β rm Z cφ 40 i ( 9 c( + ¼ (9 88A 0Ω 6 c¼ The wer deliered t the lad reitr i P 88A 7Ω 46W rm The lad rm ltage, frm euati (80, i {( β (i β i } rm 40 {( 94 6 (i ( 94 i ( 6 } 75 The lad curret ad ltage rile factr are 88A FF i 55 0A F i FFi 8 75 FF F FF 8 The uly wer factr i 46W f A Half-wae half-ctrlled The half-wae ctrlled certer waefrm i figure 8b hw that whe < <, durig the itie half f the uly cycle, eergy i deliered t the lad But whe < <, the uly reere ad me eergy i the lad circuit i retured t the uly Mre eergy ca be retaied by the lad if the lad ltage i reeted frm reerig A lad freewheel dide facilitate thi bjectie The igle-hae half-wae certer ca be ctrlled whe a lad cmmutatig dide i icrrated a hw i figure 0a The dide will reet the itataeu lad ltage 0 frm gig egatie, a with the igle-hae half-ctrlled certer hw i figure 7 The lad curret i defied by euati (8 fr ad euati (9 fr +, amely: di + i i (A dt (97 di + i 0 (A + dt At the thyritr i lie cmmutated ad the lad curret, ad hece freewheel dide curret, i f the frm f euati ( A hw i figure 0b, deedig the delay agle ad - lad time ctat (/, the lad curret may fall t zer, rducig dictiuu lad curret The mea lad ltage (hece mea utut curret fr all cducti cae, with a aie - lad, i i t d t ω ω (98 (+ c ( which i half the mea ltage fr a igle-hae half-ctrlled certer, gie by euati (59 The maximum mea utut ltage, / (euati (4, ccur at 0 The rmalied mea utut ltage i / (+ c (99

15 49 Pwer Electric aturally cmmutatig certer 50 r x r x i i ii - Fr ctiuu cducti the lad curret i defied by /taφ iφ e i( φ + taφ i( i ( ω t i( φ + ( e /taφ Z e /taφ i( i ( e Df 0 (A ( /taφ i i( /ta φ φ e + φ (A /taφ e Z e + The adatage f icrratig a lad freewheel dide are the iut wer factr i imred ad the lad waefrm i imred (le rile giig a better lad erfrmace Full-wae ctrlled rectifier circuit with a - lad fr > Figure 0 Half-wae half-ctrlled certer: (a circuit diagram ad (b circuit waefrm fr a iductie lad The Furier cefficiet f the -ule utut ltage are gie by euati (76 Fr the iglehae, half-wae, half-ctrlled cae,, thu the utut ltage harmic ccur at,,, The rm utut ltage fr bth ctiuu ad dictiuu lad curret i ( i ω ω t d t rm {( + i } (00 i - Fr dictiuu cducti the lad curret i defied by euati (77 durig thyritr cducti + taφ ( ω φ ( φ i( i ( i( Z t i e (A i( idf ( /taφ e 0 /taφ i( ( t /ta φ e ω + φ e Z (A t + The aerage thyritr curret i c φ + c + iφ i ( φ e T while the aerage freewheel dide curret i iφ iφ i ( φ e Df T ω ( /ta φ /taφ ( (0 (0 (0 Full-wae ltage ctrl i ible with the circuit hw i figure a ad b The circuit i figure a ue a cetre-taed trafrmer ad tw thyritr which exeriece a reere bia f twice the uly At high wer where a trafrmer may t be alicable, a fur-thyritr cfigurati a i figure b i uitable The ltage ratig f the thyritr i figure b are half the f the deice i figure a, fr a gie certer iut ltage ad ltage ad curret waefrm are hw i figure art c, d, ad e fr three differet hae ctrl agle cditi The lad curret waefrm becme ctiuu whe the hae ctrl agle i gie by ta ω/ φ (rad (05 at which agle the utut curret i a rectified ie wae Fr > ø, dictiuu lad curret flw a hw i figure c At ø the lad curret becme ctiuu a hw i figure d, whece β + Further decreae i, that i < ø, reult i ctiuu lad curret that i alway greater tha zer ( zer curret erid, a hw i figure e i - > φ, β - <, dictiuu lad curret The lad curret waefrm i the ame a fr the half-wae ituati cidered i ecti, gie by euati (77 That i i( {( [i( - φ - i( - φ e φ} ] (A - /ta Z (06 < β (rad The mea utut ltage fr thi full-wae circuit will be twice that f the half-wae cae i ecti, gie by euati (79 That i i d β (07 (c - c β ( where β ca be extracted frm figure 9 Fr a urely reitie lad, β The aerage utut curret i gie by / ad the aerage ad rm thyritr curret are ad /, reectiely rm The rm lad ltage i β i ω ω t d t rm {( β (i β i } (08 The rm lad curret i i ( β c( + φ+ β ( β c Z φ (09 rm The lad wer i therefre P rm

16 5 Pwer Electric aturally cmmutatig certer 5 iii - < φ, β -, ctiuu lad curret (ad al a urely iductie lad r x r x Figure Full-wae ctrlled certer: (a ad (b circuit diagram; (c dictiuu lad curret; (d erge f ctiuu lad curret, whe ø; ad (e ctiuu lad curret ii - φ, β -, erge f ctiuu lad curret - Whe φ ta ω/, the lad curret gie by euati (06 reduce t i( i( φ (A Z (0 fr φ φ+ (rad ad the mea utut ltage, reducig euati (07 uig β +, i gie by c ( ( - which i deedet the lad uch that φ ta ω/ Frm euati (08, with β, the rm utut ltage i, rm /Z, ad wer cφ rm Uder a ctiuu lad curret cducti cditi, a thyritr i till cductig whe ather i frward-biaed ad i tured The firt deice i itataeuly reere-biaed by the ecd deice which ha bee tured The firt deice i cmmutated ad lad curret i itataeuly traferred t the cmig deice The lad curret i gie by i( - φ {( - /ta φ} i( [i( - φ - e ] ( /taφ Z e Thi euati reduce t euati (0 fr φ ad euati (6 fr 0 The mea utut ltage, whece mea utut curret, are defied by euati ( c ( which i uiuely defied by The maximum mea utut ltage, / (euati (8, ccur at 0 Geerally, fr >, the aerage utut ltage i egatie, reultig i a et eergy trafer frm the lad t the uly The rmalied mea utut ltage i / c ( The rm utut ltage i eual t the rm iut uly ltage ad i gie by + ( i d rm (4 The ac i the utut ltage i 8 + c ac rm (5 The ac cmet harmic magitude i the lad are gie by c + (6 ( ( + ( ( + fr ee, amely, 4, 6 The lad ltage frm factr, (thece rile factr, i FF (7 c The curret harmic are btaied by diii f the ltage harmic by it lad imedace at that freuecy, that i 0,, 4,6, (8 Z + ( ω tegrati f euati (, uared, yield the lad rm curret (r euati (7 fr 0 i( φ i / ta ( φ φ / taφ + taφ /ta ( 4 iiφ /ta ( φ e φ e (9 rm Z e e Thyritr aerage curret i, while thyritr rm curret ratig i / The ame thyritr curret rm ratig exrei are alid fr bth ctiuu ad dictiuu lad curret cditi Fr a highly iductie lad, ctat lad curret, the uly wer factr i f ( c + The harmic factr r ltage rile factr fr the utut ltage i rm F c (0 8 which i a miimum f 048 fr 0 ad a maximum f whe Critical lad iductace (ee figure The critical lad iductace, t reet the lad curret fallig t zer, i gie by ω crit c i θ + c( θ c + + ( fr θ where c i θ i (

17 5 Pwer Electric aturally cmmutatig certer 54 The miimum curret ccur at the agle θ, where the mea utut ltage eual the itataeu lad ltage, Whe the hae delay agle i greater tha the critical agle θ, ubtitutig θ i euati ( gie ω crit ta ( Fr a urely reitie lad ( + c (4 Examle 6: Ctrlled full-wae certer ctiuu ad dictiuu cducti The fully ctrlled full-wae, igle-hae certer i figure a ha a urce f 40 rm, 50Hz, ad a 0Ω 50mH erie lad f the delay agle i 45, determie i the aerage utut ltage ad curret, hece thyritr mea curret ii the rm lad ltage ad curret, hece thyritr rm curret ad lad rile factr iii the wer abrbed by the lad ad the uly wer factr f the delay agle i icreaed t 75 determie i the lad curret i the time dmai umerically le the lad curret euati fr β, the curret exticti agle i the lad aerage curret ad ltage ii the lad rm ltage ad curret hece lad rile factr ad wer diiated iii the uly wer factr Sluti The lad atural wer factr agle i gie by - - φ ta ω/ ta ( 50 50mH /0Ω 575 rad Ctiuu cducti Sice < φ ( 45 < 575, ctiuu lad curret flw, which i gie by euati ( 40 i( - {( - /56} i( [i( - - e ] /56 86Ω e - /56 8 [i( e ] i The aerage utut curret ad ltage are gie by euati ( c c / 58 /0Ω 5A Each thyritr cduct fr 80, hece thyritr mea curret i f 5A 765A ii The rm lad curret i determied by harmic aalyi The ltage harmic (eak magitude are gie by euati (6 c + ( ( + ( ( + fr, 4,6, ad the crredig curret i gie frm euati (8 Z + ω harmic ( ( Z + ω Z 0 ( ( The dc utut ltage cmet i gie by euati ( Frm the calculati i the table, the rm lad curret i A rm Sice each thyritr cduct fr 80, the thyritr rm curret i f 5A 08A The rm lad ltage i gie by euati (4, that i 40 rm 5A FF i 0 5A F i FFi rm 40 FF F FF 57 iii The wer abrbed by the lad i P 5A 0Ω 44W rm The uly wer factr i P 44W f A rm rm Dictiuu cducti i Whe the delay agle i icreaed t 75 ( rad, dictiuu lad curret flw ice the atural wer factr agle φ ta ω/ ta ( 50 50mH /0Ω 575 rad i exceeded The lad - - curret i gie by euati (06 i( 40 {( - /56} [i( - - i( - e ] 86Ω - 8 [i( e ] /56 Slig the euati i art i fr β ad zer curret, that i - /56 0 i( β e β gie β 409 rad r 4 i The aerage lad ltage frm euati (07 i 40 (c 75 - c A 0Ω ii The rm lad ltage i gie by euati (08 rm { } 40 (409 (i88 i The rm curret frm euati (09 i 40 i ( 409 c( ( A 86Ω c rm The lad ltage ad curret frm ad rile factr are rm 55A FF i A F i FFi 49 rm 646 FF F FF 8 6 The wer diiated i the 0Ω lad reitr i P 55 0Ω 86W iii The uly wer factr i P 86W f A rm rm rm

18 55 Pwer Electric aturally cmmutatig certer 56 4 Full-wae, fully-ctrlled circuit with - ad emf lad, E A emf urce ad - lad ca be ecutered i dc machie mdellig The emf rereet the machie eed back emf, defied by E kφω DC machie ca be ctrlled by a fully ctrlled certer cfigurati a hw i figure a, where T -T 4 ad T -T are triggered alterately f i each half ie erid the thyritr firig delay agle ccur after the rectified ie uly ha falle belw the emf leel E, the lad curret flw ice the bridge thyritr will alway be reerebiaed Thu the zer curret firig agle i: i ( E/ (rad fr < < (5 where it ha bee aumed the emf ha the larity hw i figure a With dictiuu utut curret, lad curret cat flw util the uly ltage exceed the back emf E That i i ( E/ (rad fr 0< < (6 ad curret ca alway flw with a firig agle defied by (rad (7 The lad circuit curret ca be ealuated by lig di i ω t + i+ E dt ( (8 The lad ltage ad curret rile are bth at twice the uly freuecy r x r x T T T T4 4i - Dictiuu lad curret The lad curret i gie by E E { } {( - ω ( [c i( c i( - /ta φ ω φ ω t φ + φ φ } ] (9 β < + (rad Fr dictiuu lad curret cducti, the curret exticti agle β, hw figure b, i led by iteratie techiue fr i(β 0 i euati (9 cφi( β - φ - E + { - cφi( - φ } e {( 0 (0 The mea utut ltage ca be btaied frm euati (07, which i alid fr E 0 Fr ay E, icludig E 0 β ( i+ E d E c c β + ( + β ( ( 0 < β < (rad The curret exticti agle β i lad-deedet, beig a fucti f Z ad E, a well a Sice E+, the mea lad curret i gie by E E c c β ( β (A ( 0 < β < (rad The rm utut ltage i gie by rm β β β i + E ( ( The rm ltage acr the - art f the lad i gie by rm rm E (4 The ttal wer deliered t the --E lad i P + E (5 rm where the rm lad curret i fud by itegratig the curret i euati (9, uared, etc Figure A full-wae fully ctrlled certer with a iductie lad which iclude a emf urce: (a circuit diagram; (b ltage waefrm with dictiuu lad curret; (c erge f ctiuu lad curret; ad (d ctiuu lad curret 4ii - Ctiuu lad curret With ctiuu lad curret cducti, the lad rm ltage i The lad curret i gie by E i( - φ {( - /ta φ} i( [i( - φ - + e ] /taφ Z cφ e + (rad (6 The eridic miimum curret i gie by /taφ e + E E i ( φ i ( φ tah /taφ Z e Z taφ (7 Fr ctiuu lad curret cditi, a hw i figure c ad d, the mea utut ltage i gie by euati ( with β

19 57 Pwer Electric ( + i d E+ c ( (8 i c The aerage utut ltage i deedet ly the hae delay agle (ideedet f E, ulike the mea lad curret, which i gie by E E c (A (9 The wer abrbed by the emf urce i the lad i P E, while the ttal wer deliered t the -- E lad i P + E rm The utut curret ad ltage rile i at multile f twice the uly freuecy The utut ltage harmic magitude fr ctiuu cducti, gie by euati (6, are c + ( ( ( ( + + fr, 4,6, (40 The dc cmet acr the - (ad jut the reitr art f the lad i E (4 c E The ac cmet f the utut ltage i ac rm c (4 ad the utut ltage frm factr i FF c (4 Thyritr aerage curret i, while thyritr rm curret ratig i / Thee ame tw thyritr rm exrei are alid fr bth ctiuu ad dictiuu lad curret cditi Critical lad iductace Frm euati (7 et t zer (r i 0 i euati (6, the budary betwee ctiuu ad dictiuu iductr curret mut atify E i ( φ tah > (44 Z ta φ eri f the larity f the back emf E i reered a hw i figure a, waefrm a i art b ad c f figure reult The emf uly ca ride a frward bia acr the bridge thyritr ee after the uly larity ha ge egatie The zer curret agle w atifie < < /, a gie by euati (5 Thu lad ad uly curret ca flw, ee fr > The relatihi betwee the mea utut ltage ad curret i w gie by E+ i c with (45 That i, the emf term E i euati (5 t (44 i arriately chaged t - E The lad curret flw frm the emf urce ad if >, the aerage lad ltage i egatie Pwer i beig deliered t the ac uly frm the emf urce i the lad, which i a eergy trafer rce called wer ieri geeral 0< < 90 > 0 P > 0 i > 0 rectificati 90 < < 80 < 0 P < 0 i > 0 ieri aturally cmmutatig certer 58 T T T T4 r x r x Figure A full-wae ctrlled certer with a iductie lad ad egatie emf urce: (a circuit diagram; (b ltage waefrm fr dictiuu lad curret; ad (c ctiuu lad curret Examle 7: Ctrlled certer ctiuu cducti ad back emf The fully ctrlled full-wae certer i figure a ha a urce f 40 rm, 50Hz, ad a 0Ω, 50mH, 50 emf ig erie lad The delay agle i 45 Determie i the aerage utut ltage ad curret ii the rm lad ltage ad the rm ltage acr the - art f the lad iii the wer abrbed by the 50 lad back emf i the rm lad curret hece wer diiated i the reitie art f the lad the lad efficiecy, that i ercetage f eergy it the back emf ad wer factr i the lad ltage ad curret frm ad rile factr Sluti Frm examle 5, ctiuu cducti i ible ice φ ( i The aerage utut ltage i gie by euati (8 c 40 c45 58 < <