Vtusolution.in AC VOLTAGE CONTROLLER CIRCUITS (RMS VOLTAGE CONTROLLERS) Voltage. Controller

Transcription

1 AC TAGE CNTRER CRCUT (RM TAGE CNTRER) AC volage conrollers (ac line volage conrollers) are eployed o vary he RM value of he alernaing volage applied o a load circui by inroducing Thyrisors beween he load and a consan volage ac source. The RM value of alernaing volage applied o a load circui is conrolled by conrolling he riggering angle of he Thyrisors in he ac volage conroller circuis. n brief, an ac volage conroller is a ype of hyrisor power converer which is used o conver a fixed volage, fixed frequency ac inpu supply o obain a variable volage ac oupu. The RM value of he ac oupu volage and he ac power flow o he load is conrolled by varying (adjusing) he rigger angle AC npu olage f s s f s AC olage Conroller (RM) ariable AC RM /P olage There are wo differen ypes of hyrisor conrol used in pracice o conrol he ac power flow n-ff conrol Phase conrol These are he wo ac oupu volage conrol echniques. n n-ff conrol echnique Thyrisors are used as swiches o connec he load circui o he ac supply (source) for a few cycles of he inpu ac supply and hen o disconnec i for few inpu cycles. The Thyrisors hus ac as a high speed conacor (or high speed ac swich). PHAE CNTR n phase conrol he Thyrisors are used as swiches o connec he load circui o he inpu ac supply, for a par of every inpu cycle. Tha is he ac supply volage is chopped using Thyrisors during a par of each inpu cycle. The hyrisor swich is urned on for a par of every half cycle, so ha inpu supply volage appears across he load and hen urned off during he reaining par of inpu half cycle o disconnec he ac supply fro he load. By conrolling he phase angle or he rigger angle (delay angle), he oupu RM volage across he load can be conrolled. The rigger delay angle is defined as he phase angle (he value of ω) a which he hyrisor urns on and he load curren begins o flow. Thyrisor ac volage conrollers use ac line couaion or ac phase couaion. Thyrisors in ac volage conrollers are line couaed (phase couaed) since he inpu supply is ac. When he inpu ac volage reverses and becoes negaive during he negaive half cycle he curren flowing hrough he conducing hyrisor decreases and f

2 falls o zero. Thus he N hyrisor naurally urns off, when he device curren falls o zero. Phase conrol Thyrisors which are relaively inexpensive, converer grade Thyrisors which are slower han fas swiching inverer grade Thyrisors are norally used. For applicaions upo 4Hz, if Triacs are available o ee he volage and curren raings of a paricular applicaion, Triacs are ore coonly used. Due o ac line couaion or naural couaion, here is no need of exra couaion circuiry or coponens and he circuis for ac volage conrollers are very siple. Due o he naure of he oupu wavefors, he analysis, derivaions of expressions for perforance paraeers are no siple, especially for he phase conrolled ac volage conrollers wih R load. Bu however os of he pracical loads are of he R ype and hence R load should be considered in he analysis and design of ac volage conroller circuis. TYPE F AC TAGE CNTRER The ac volage conrollers are classified ino wo ypes based on he ype of inpu ac supply applied o he circui. ingle Phase AC Conrollers. Three Phase AC Conrollers. ingle phase ac conrollers operae wih single phase ac supply volage of 3 RM a 5Hz in our counry. Three phase ac conrollers operae wih 3 phase ac supply of 4 RM a 5Hz supply frequency. Each ype of conroller ay be sub divided ino Uni-direcional or half wave ac conroller. Bi-direcional or full wave ac conroller. n brief differen ypes of ac volage conrollers are ingle phase half wave ac volage conroller (uni-direcional conroller). ingle phase full wave ac volage conroller (bi-direcional conroller). Three phase half wave ac volage conroller (uni-direcional conroller). Three phase full wave ac volage conroller (bi-direcional conroller). APPCATN F AC TAGE CNTRER ighing / lluinaion conrol in ac power circuis. nducion heaing. ndusrial heaing & Doesic heaing. Transforer ap changing (on load ransforer ap changing). peed conrol of inducion oors (single phase and poly phase ac inducion oor conrol). AC agne conrols. PRNCPE F N-FF CNTR TECHNQUE (NTEGRA CYCE CNTR) The basic principle of on-off conrol echnique is explained wih reference o a single phase full wave ac volage conroller circui shown below. The hyrisor swiches T and T are urned on by applying appropriae gae rigger pulses o connec he inpu ac supply o he load for n nuber of inpu cycles during he ie inerval N. The

3 hyrisor swiches T and T are urned off by blocking he gae rigger pulses for nuber of inpu cycles during he ie inerval FF. The ac conroller N ie N usually consiss of an inegral nuber of inpu cycles. R R oad Resisance Fig.: ingle phase full wave AC volage conroller circui s i g o i o i g n Gae pulse of T Gae pulse of T w w w w Fig.: Wavefors Exaple Referring o he wavefors of N-FF conrol echnique in he above diagra, n Two inpu cycles. Thyrisors are urned N during N for wo inpu cycles.

4 ne inpu cycle. Thyrisors are urned FF during FF for one inpu cycle Fig.: Power Facor Thyrisors are urned N precisely a he zero volage crossings of he inpu supply. The hyrisor T is urned on a he beginning of each posiive half cycle by applying he gae rigger pulses o T as shown, during he N ie N. The load curren flows in he posiive direcion, which is he downward direcion as shown in he circui diagra when T conducs. The hyrisor T is urned on a he beginning of each negaive half cycle, by applying gaing signal o he gae of T, during N. The load curren flows in he reverse direcion, which is he upward direcion when T conducs. Thus we obain a bi-direcional load curren flow (alernaing load curren flow) in a ac volage conroller circui, by riggering he hyrisors alernaely. This ype of conrol is used in applicaions which have high echanical ineria and high heral ie consan (ndusrial heaing and speed conrol of ac oors). Due o zero volage and zero curren swiching of Thyrisors, he haronics generaed by swiching acions are reduced. For a sine wave inpu supply volage, vs sinω sinω RM value of inpu ac supply RM phase supply volage. f he inpu ac supply is conneced o load for n nuber of inpu cycles and disconneced for nuber of inpu cycles, hen n T, T N FF Where T inpu cycle ie (ie period) and f f inpu supply frequency. conroller on ie n T. N conroller off ie T. FF T upu ie period ( ) ( nt T ) N + +. FF

5 We can show ha, N upu RM volage ( RM ) i ( RM ) T Where i( RM ) is he RM inpu supply volage. T N T DERE AN EXPREN FR THE RM AUE F UTPUT TAGE, FR N-FF CNTR METHD. Now upu RM volage in ω. d ( ω) ubsiuing for RM ω N ω T ω N in d RM ωt ω. ( ω ) ω Cosθ in θ ( ω ) ω N Cos ω d RM ωt ωn ωn. RM d ω Cos ωd( ω) ωt ωt ωn in ( ω) RM ( ωn ) RM ω ω N sin ωn sin ωt N An inegral nuber of inpu cycles; Hence T, T,3 T, 4 T,5 T,... & ω,4,6,8,,... N N Where T is he inpu supply ie period (T inpu cycle ie period). Thus we noe ha sin ω N ω ω T T N N ( RM )

6 N T N RM i( RM ) T Where i ( RM ) RM value of inpu supply volage; N nt n T + nt + T n + N N FF n k ( + n) RM k duy cycle (d). PERFRMANCE PARAMETER F AC TAGE CNTRER RM upu (oad) olage n sin. RM ωd( ω) ( n+ ) n RM i( RM ) k k ( + n) k k i( RM ) RM Where i ( RM ) RM value of inpu supply volage. Duy Cycle N N nt k T + + nt N FF Where, k n ( + n) RM oad Curren duy cycle (d). RM upu AC (oad) Power ( RM ) ( RM ) Z R ; for a resisive load Z R. P R RM

7 npu Power Facor P oupu load power P PF A inpu supply vol aperes R RM PF i( RM ) in( RM ) ; in( RM ) RM inpu supply curren. The inpu supply curren is sae as he load curren in Hence, RM supply curren RM load curren; in RM ( RM ). R RM ( RM ) i ( RM ) k PF PF k i RM in RM i RM i RM n + n The Average Curren of Thyrisor T ( Avg ) i T Wavefor of Thyrisor Curren n 3 ω n sin ωd. T Avg + ( n) ( ω ) n sin ωd. T Avg + ( n) ( ω ) k n cosω + ( n) T Avg n + n [ cos + cos ] T Avg

8 n ( ) + ( + n) T Avg n + n [ ] T Avg Where n k. ( + n) T Avg N k duy cycle ( + n) T Avg ( + ) ( n+ ) N FF n k., axiu or peak hyrisor curren. R RM Curren of Thyrisor T ( RM ) n sin ω. ( ω ) d T RM ( n+ ) n sin. T RM ωd( ω) ( n+ ) n ( cos ω ) n T RM d( ω) ( n+ ) n T ( RM ) d d 4 ( n+ ) 4 ( n ) T RM ( ω ) cos ω. ( ω ) n sin ω ( ω) + n sin sin ( ) 4 + ( n ) T RM

9 n { } T RM 4 n+ n 4 ( n+ ) 4( n+ ) T RM n n ( + n) T RM T RM k k PRBEM. A single phase full wave ac volage conroller working on N-FF conrol echnique has supply volage of 3, RM 5Hz, load 5Ω. The conroller is N for 3 cycles and off for 4 cycles. Calculae N & FF ie inervals. RM oupu volage. npu P.F. Average and RM hyrisor currens. 3, in RM, 35.69, T.sec f 5Hz, T s. n nuber of inpu cycles during which conroller is N; n 3. nuber of inpu cycles during which conroller is FF; 4. n T 3 s 6s.6sec N N n T.6sec conroller N ie. FF T 4 s 8s.8sec T.8sec conroller FF ie. FF Duy cycle n 3 k.485 ( + n) ( 4 + 3) RM oupu volage i( RM ) ( + n) RM n

10 ( + ) RM RM 5.57 RM RM npu Power Facor PF. RM ( RM ) A Z R 5Ω RM P R W k n 3 PF 7 ( + n) PF Average Thyrisor Curren Raing n k T ( Avg ) + n where T Avg R A Peak (axiu) hyrisor curren A T Avg RM Curren Raing of Thyrisor n k T RM + n A T RM

11 PRNCPE F AC PHAE CNTR The basic principle of ac phase conrol echnique is explained wih reference o a single phase half wave ac volage conroller (unidirecional conroller) circui shown in he below figure. The half wave ac conroller uses one hyrisor and one diode conneced in parallel across each oher in opposie direcion ha is anode of hyrisor T is conneced o he cahode of diode D and he cahode of T is conneced o he anode of D. The oupu volage across he load resisor R and hence he ac power flow o he load is conrolled by varying he rigger angle. The rigger angle or he delay angle refers o he value of ω or he insan a which he hyrisor T is riggered o urn i N, by applying a suiable gae rigger pulse beween he gae and cahode lead. The hyrisor T is forward biased during he posiive half cycle of inpu ac supply. can be riggered and ade o conduc by applying a suiable gae rigger pulse only during he posiive half cycle of inpu supply. When T is riggered i conducs and he load curren flows hrough he hyrisor T, he load and hrough he ransforer secondary winding. By assuing T as an ideal hyrisor swich i can be considered as a closed swich when i is N during he period ω o radians. The oupu volage across he load follows he inpu supply volage when he hyrisor T is urned-on and when i conducs fro ω o radians. When he inpu supply volage decreases o zero a ω, for a resisive load he load curren also falls o zero a ω and hence he hyrisor T urns off a ω. Beween he ie period ω o, when he supply volage reverses and becoes negaive he diode D becoes forward biased and hence urns N and conducs. The load curren flows in he opposie direcion during ω o radians when D is N and he oupu volage follows he negaive half cycle of inpu supply. Fig.: Halfwave AC phase conroller (Unidirecional Conroller)

12 Equaions npu AC upply olage across he Transforer econdary Winding. v sinω s in ( RM ) RM value of secondary supply volage. upu oad olage v o v ; for ω o v v sinω ; for ω o. o upu oad Curren i o vo sinω i ; for ω o. R R i o i ; for ω o. T DERE AN EXPREN FR RM UTPUT TAGE ( RM ) sin d. RM ω ( ω) cos ω. d RM ( ω)

13 ( cos ). d RM ω ( ω) 4 d( ω) cos ωd. ω RM sin ω ( ω) RM ( ) RM sin ω sin 4 sin ;sin 4 RM sin + ( ) RM sin + ( ) RM sin + ( ) RM sin + ( i RM ) RM sin + ( ) RM Where, i ( RM ) RM value of inpu supply volage (across he ransforer secondary winding). Noe: upu RM volage across he load is conrolled by changing ' ' as indicaed by he expression for ( RM )

14 PT F ( RM ) ERU TRGGER ANGE FR A NGE PHAE HAF- WAE AC TAGE CNTRER (UNDRECTNA CNTRER) sin + ( ) RM sin + ( ) RM By using he expression for ( RM ) we can obain he conrol characerisics, which is he plo of RM oupu volage ( RM ) versus he rigger angle. A ypical conrol characerisic of single phase half-wave phase conrolled ac volage conroller is as shown below Trigger angle Trigger angle ( RM ) in degrees in radians ; 6 ( 6) 6 ; 3 ( 6) 9 ; ( 3 6) ; 3 ( 4 6) 5 5 ; 6 ( 5 6) 8 ;( 6 ) (RM) % % 7.7% % 6 8 Trigger angle in degrees

15 Fig.: Conrol characerisics of single phase half-wave phase conrolled ac volage conroller Noe: We can observe fro he conrol characerisics and he able given above ha he range of RM oupu volage conrol is fro % of o 7.7% of when we vary he rigger angle fro zero o 8 degrees. Thus he half wave ac conroller has he draw back of liied range RM oupu volage conrol. T CACUATE THE AERAGE AUE (DC AUE) F UTPUT TAGE sin ωd. ( ω) d sin ω. ( ω ) cosω [ cos + cos ] ; cos [ cos ] ; Hence ( cos ) When ' ' is varied fro o. varies fro o DADANTAGE F NGE PHAE HAF WAE AC TAGE CNTRER. The oupu load volage has a DC coponen because he wo halves of he oupu volage wavefor are no syerical wih respec o level. The inpu supply curren wavefor also has a DC coponen (average value) which can resul in he proble of core sauraion of he inpu supply ransforer. The half wave ac volage conroller using a single hyrisor and a single diode provides conrol on he hyrisor only in one half cycle of he inpu supply. Hence ac power flow o he load can be conrolled only in one half cycle. Half wave ac volage conroller gives liied range of RM oupu volage conrol. Because he RM value of ac oupu volage can be varied fro a axiu of % of a a rigger angle o a low of 7.7% of a Radians. These drawbacks of single phase half wave ac volage conroller can be over coe by using a single phase full wave ac volage conroller.

16 APPCATN F RM TAGE CNTRER peed conrol of inducion oor (polyphase ac inducion oor). Heaer conrol circuis (indusrial heaing). Welding power conrol. nducion heaing. n load ransforer ap changing. ighing conrol in ac circuis. Ac agne conrols. Proble. A single phase half-wave ac volage conroller has a load resisance R 5Ω, inpu ac supply volage is 3 RM a 5Hz. The inpu supply ransforer has a urns raio of :. f he hyrisor T is riggered a 6. Calculae RM oupu volage. upu power. RM load curren and average load curren. npu power facor. Average and RM hyrisor curren. Given, f R p 3, RM priary supply volage. npu supply frequency 5Hz. 5Ω 6 radians. 3 RM secondary volage. p Np N Therefore 3 Where, p N p Nuber of urns in he priary winding. N Nuber of urns in he secondary winding.

17 RM alue of upu (oad) olage ( RM ) d RM sin ω. ( ω ) We have obained he expression for ( RM ) as sin + ( ) RM RM RM sin [ ] RM RM oad Curren ( RM ) RM upu oad Power P ( RM ) Aps R 5 P R Was P RM KW npu Power Facor P PF RM secondary supply volage 3. RM secondary supply curren RM load curren Aps RM W PF W ( )

18 Average upu (oad) olage sin d. ω ( ω) We have obained he expression for he average / DC oupu volage as, [ cos ] cos( 6 ) [.5 ] ols [ ] Average DC oad Curren ( ) Aps R 5 Average & RM Thyrisor Currens i T ( + ) Fig.: Thyrisor Curren Wavefor Referring o he hyrisor curren wavefor of a single phase half-wave ac volage conroller circui, we can calculae he average hyrisor curren T ( Avg ) as sin d. T Avg ω ( ω) 3 ω sin d. T Avg ω ( ω)

19 ( cosω) T Avg cos( ) cos T Avg + Where, [ + cos ] T Avg Peak hyrisor curren Peak load curren. R Aps R [ + cos ] T Avg + T Avg T Avg 3 cos [ ].553 Aps T Avg RM hyrisor curren T ( RM ) can be calculaed by using he expression sin d. T RM ω ( ω) ( ω) cos. d T RM ( ω) d( ) cos d. T RM ω ω ( ω) 4 sin ω T ( RM ) ( ω) 4

20 sin sin 4 ( ) T RM sin 4 + ( ) T RM sin + ( ) T RM T RM T RM sin A T RM.9746 Aps T RM NGE PHAE FU WAE AC TAGE CNTRER (AC REGUATR) R RM TAGE CNTRER WTH RETE AD ingle phase full wave ac volage conroller circui using wo CRs or a single riac is generally used in os of he ac conrol applicaions. The ac power flow o he load can be conrolled in boh he half cycles by varying he rigger angle ' '. The RM value of load volage can be varied by varying he rigger angle ' '. The inpu supply curren is alernaing in he case of a full wave ac volage conroller and due o he syerical naure of he inpu supply curren wavefor here is no coponen of inpu supply curren i.e., he average value of he inpu supply curren is zero. A single phase full wave ac volage conroller wih a resisive load is shown in he figure below. is possible o conrol he ac power flow o he load in boh he half cycles by adjusing he rigger angle ' '. Hence he full wave ac volage conroller is also referred o as o a bi-direcional conroller.

21 Fig.: ingle phase full wave ac volage conroller (Bi-direcional Conroller) using CRs The hyrisor T is forward biased during he posiive half cycle of he inpu supply volage. The hyrisor T is riggered a a delay angle of ' ' ( radians). Considering he N hyrisor T as an ideal closed swich he inpu supply volage appears across he load resisor R and he oupu volage v v during ω o radians. The load curren flows hrough he N hyrisor T and hrough he load resisor R in he downward direcion during he conducion ie of T fro ω o radians. A ω, when he inpu volage falls o zero he hyrisor curren (which is flowing hrough he load resisor R ) falls o zero and hence T naurally urns off. No curren flows in he circui during ω +. o The hyrisor T is forward biased during he negaive cycle of inpu supply and when hyrisor T is riggered a a delay angle ( + ), he oupu volage follows he negaive halfcycle of inpu fro ω ( ) + o. When T is N, he load curren flows in he reverse direcion (upward direcion) hrough + o radians. The ie inerval (spacing) beween he gae rigger pulses of T and T is kep a radians or 8. A ω he inpu supply volage falls o zero and hence he load curren also falls o zero and hyrisor T urn off naurally. T during ω ( ) nsead of using wo CR s in parallel, a Triac can be used for full wave ac volage conrol. Fig.: ingle phase full wave ac volage conroller (Bi-direcional Conroller) using TRAC

22 Fig: Wavefors of single phase full wave ac volage conroller EQUATN npu supply volage v sinω sinω; upu volage across he load resisor v v sinω ; for ω o R ; and ( + ) o ω upu load curren v sinω i sinω ; R R for ω o and ( + ) o ω T DERE AN EXPREN FR THE RM AUE F UTPUT (AD) TAGE The RM value of oupu volage (load volage) can be found using he expression RM v d ; ( ω RM )

23 For a full wave ac volage conroller, we can see ha he wo half cycles of oupu volage wavefors are syerical and he oupu pulse ie period (or oupu pulse repeiion ie) is radians. Hence we can also calculae he RM oupu volage by using he expression given below. RM sin ωd. ω RM. ( ω ) v d ; v v sinω; For ω o and ( + ) o ω Hence, ( sin ) d( ) ( sin RM ω ω ω ) d( ω) + + sin ωd. ( ω) sin ωd. ( ω) + + cos ω cos ω d( ω) d( ω) + + d( ω) cos ωd. ( ω) + d( ω) cos ωd. ( ω) + + sin ω sin ω + 4 ( ω) ( ω) ( ) ( ) ( sin sin ) sin 4 sin ( ) sin sin 4 + ( ) ( ) ( ) sin sin ( + ) ( ) ( + ) sin sin ( ) + + 4

24 sin sin.cos cos.sin ( ) ( ) sin & cos Therefore, 4 sin sin + + RM ( ) sin 4 + ( ) sin RM 4 + Taking he square roo, we ge ( ) sin RM + ( ) sin RM + ( ) sin RM + sin + ( ) RM sin + ( ) RM sin + i ( RM ) ( ) RM sin + ( ) RM Maxiu RM volage will be applied o he load when, in ha case he full sine wave appears across he load. RM load volage will be he sae as he RM supply volage. When is increased he RM load volage decreases.

25 ( ) RM sin + ( ) RM + i( RM ) RM The oupu conrol characerisic for a single phase full wave ac volage conroller wih resisive load can be obained by ploing he equaion for ( RM ) CNTR CHARACTERTC F NGE PHAE FU-WAE AC TAGE CNTRER WTH RETE AD The conrol characerisic is he plo of RM oupu volage ( RM ) versus he rigger angle ; which can be obained by using he expression for he RM oupu volage of a full-wave ac conroller wih resisive load. sin + ( ) RM Where RM value of inpu supply volage Trigger angle Trigger angle ( RM ) % in degrees in radians % 3 ; 6 ( ) 98.54% 6 ; 3 ( ) 89.69% 9 ; ( ) 7.7% ; 3 ( ) 44.% 5 5 ; 6 ( ) 6.98% 8 ;( 6 6) ;

26 (RM) Trigger angle in degrees We can noice fro he figure, ha we obain a uch beer oupu conrol characerisic by using a single phase full wave ac volage conroller. The RM oupu volage can be varied fro a axiu of % a o a iniu of a 8. Thus we ge a full range oupu volage conrol by using a single phase full wave ac volage conroller. Need For solaion n he single phase full wave ac volage conroller circui using wo CRs or Thyrisors T and T in parallel, he gaing circuis (gae rigger pulse generaing circuis) of Thyrisors T and T us be isolaed. Figure shows a pulse ransforer wih wo separae windings o provide isolaion beween he gaing signals of T and T. Gae Trigger Pulse Generaor Fig.: Pulse Transforer G K G K NGE PHAE FU-WAE AC TAGE CNTRER WTH CMMN CATHDE is possible o design a single phase full wave ac conroller wih a coon cahode configuraion by having a coon cahode poin for T and T & by adding wo diodes in a full wave ac conroller circui as shown in he figure below

27 Fig.: ingle phase full wave ac conroller wih coon cahode (Bidirecional conroller in coon cahode configuraion) Thyrisor T and diode D are forward biased during he posiive half cycle of inpu supply. When hyrisor T is riggered a a delay angle, Thyrisor T and diode D conduc ogeher fro ω o during he posiive half cycle. The hyrisor T and diode D are forward biased during he negaive half cycle of inpu supply, when rigged a a delay angle, hyrisor T and diode D conduc ogeher during he negaive half cycle fro ω ( ) + o. n his circui as here is one single coon cahode poin, rouing of he gae rigger pulses o he hyrisor gaes of T and T is sipler and only one isolaion circui is required. Bu due o he need of wo power diodes he coss of he devices increase. As here are wo power devices conducing a he sae ie he volage drop across he N devices increases and he N sae conducing losses of devices increase and hence he efficiency decreases. NGE PHAE FU WAE AC TAGE CNTRER UNG A NGE THYRTR + D T D 3 AC upply D 4 D R -

28 A single phase full wave ac conroller can also be ipleened wih one hyrisor and four diodes conneced in a full wave bridge configuraion as shown in he above figure. The four diodes ac as a bridge full wave recifier. The volage across he hyrisor T and curren hrough hyrisor T are always unidirecional. When T is riggered a ω, during he posiive half cycle ( ), he load curren flows hrough D, T, diode D and hrough he load. Wih a resisive load, he hyrisor curren (flowing hrough he N hyrisor T ), he load curren falls o zero a ω, when he inpu supply volage decreases o zero a ω, he hyrisor naurally urns FF. n he negaive half cycle, diodes D3& D 4 are forward biased during ω o radians. When T is riggered a ω ( + ), he load curren flows in he opposie direcion (upward direcion) hrough he load, hrough D 3, T and D 4. Thus D 3, D4 and T conduc ogeher during he negaive half cycle o supply he load power. When he inpu supply volage becoes zero a ω, he hyrisor curren (load curren) falls o zero a ω and he hyrisor T naurally urns FF. The wavefors and he expression for he RM oupu volage are he sae as discussed earlier for he single phase full wave ac conroller. Bu however if here is a large inducance in he load circui, hyrisor T ay no be urned FF a he zero crossing poins, in every half cycle of inpu volage and his ay resul in a loss of oupu conrol. This would require deecion of he zero crossing of he load curren wavefor in order o ensure guaraneed urn off of he conducing hyrisor before riggering he hyrisor in he nex half cycle, so ha we gain conrol on he oupu volage. n his full wave ac conroller circui using a single hyrisor, as here are hree power devices conducing ogeher a he sae ie here is ore conducion volage drop and an increase in he N sae conducion losses and hence efficiency is also reduced. The diode bridge recifier and hyrisor (or a power ransisor) ac ogeher as a bidirecional swich which is coercially available as a single device odule and i has relaively low N sae conducion loss. can be used for bidirecional load curren conrol and for conrolling he RM oupu volage. NGE PHAE FU WAE AC TAGE CNTRER (BDRECTNA CNTRER) WTH R AD n his secion we will discuss he operaion and perforance of a single phase full wave ac volage conroller wih R load. n pracice os of he loads are of R ype. For exaple if we consider a single phase full wave ac volage conroller conrolling he speed of a single phase ac inducion oor, he load which is he inducion oor winding is an R ype of load, where R represens he oor winding resisance and represens he oor winding inducance.

29 A single phase full wave ac volage conroller circui (bidirecional conroller) wih an R load using wo hyrisors T and T ( T and T are wo CRs) conneced in parallel is shown in he figure below. n place of wo hyrisors a single Triac can be used o ipleen a full wave ac conroller, if a suiable Traic is available for he desired RM load curren and he RM oupu volage raings. Fig: ingle phase full wave ac volage conroller wih R load The hyrisor T is forward biased during he posiive half cycle of inpu supply. e us assue ha T is riggered a ω, by applying a suiable gae rigger pulse o T during he posiive half cycle of inpu supply. The oupu volage across he load follows he inpu supply volage when T is N. The load curren i flows hrough he hyrisor T and hrough he load in he downward direcion. This load curren pulse flowing hrough T can be considered as he posiive curren pulse. Due o he inducance in he load, he load curren i flowing hrough T would no fall o zero a ω, when he inpu supply volage sars o becoe negaive. The hyrisor T will coninue o conduc he load curren unil all he inducive energy sored in he load inducor is copleely uilized and he load curren hrough T falls o zero a ω β, where β is referred o as he Exincion angle, (he value of ω ) a which he load curren falls o zero. The exincion angle β is easured fro he poin of he beginning of he posiive half cycle of inpu supply o he poin where he load curren falls o zero.

30 The hyrisor T conducs fro ω o β. The conducion angle of T is δ ( β ), which depends on he delay angle and he load ipedance angle φ. The wavefors of he inpu supply volage, he gae rigger pulses of T and T, he hyrisor curren, he load curren and he load volage wavefors appear as shown in he figure below. Fig.: npu supply volage & Thyrisor curren wavefors β is he exincion angle which depends upon he load inducance value. Fig.: Gaing ignals

31 Wavefors of single phase full wave ac volage conroller wih R load for > φ. Disconinuous load curren operaion occurs for > φ and β < ( + ) ; i.e., ( β ) <, conducion angle <. Fig.: Wavefors of npu supply volage, oad Curren, oad olage and Thyrisor olage across T Noe The RM value of he oupu volage and he load curren ay be varied by varying he rigger angle. This circui, AC RM volage conroller can be used o regulae he RM volage across he erinals of an ac oor (inducion oor). can be used o conrol he eperaure of a furnace by varying he RM oupu volage.

32 For very large load inducance he CR ay fail o couae, afer i is riggered and he load volage will be a full sine wave (siilar o he applied inpu supply volage and he oupu conrol will be los) as long as he gaing signals are applied o he hyrisors T and T. The load curren wavefor will appear as a full coninuous sine wave and he load curren wavefor lags behind he oupu sine wave by he load power facor angle φ. T DERE AN EXPREN FR THE UTPUT (NDUCTE AD) CURRENT, DURNG ω o β WHEN THYRTR T CNDUCT Considering sinusoidal inpu supply volage we can wrie he expression for he supply volage as v sinω insananeous value of he inpu supply volage. e us assue ha he hyrisor T is riggered by applying he gaing signal o T a ω. The load curren which flows hrough he hyrisor T during ω o β can be found fro he equaion di + Ri sinω d ; The soluion of he above differenial equaion gives he general expression for he oupu load curren which is of he for sin ( ω φ) + ; Z i τ Ae Where axiu or peak value of inpu supply volage. ( ω ) Z R + oad ipedance. φ ω oad ipedance angle (power facor angle of load). R an τ oad circui ie consan. R Therefore he general expression for he oupu load curren is given by he equaion R i sin ( ω φ) + Ae ; Z

33 The value of he consan A can be deerined fro he iniial condiion. i.e. iniial value of load curren i, a ω. Hence fro he equaion for i equaing i o zero and subsiuing ω, we ge i sin ( φ) + Ae Z R R sin Z Therefore Ae ( φ) A sin R Z e + R Z ( φ) e sin ( φ) A R ( ω) ω e sin ( φ) A Z By subsiuing ω, we ge he value of consan A as R ( ) ω e sin ( φ) A Z ubsiuing he value of consan A fro he above equaion ino he expression for i, we obain R R( ) ω i sin ( ω φ) + e e sin ( φ) Z Z ; ( ω ) ( ) R R i e e + Z Z ω ω sin ( ω φ) sin ( φ) + Z Z R i e ω ω sin ( ω φ) sin ( φ) Therefore we obain he final expression for he inducive load curren of a single phase full wave ac volage conroller wih R load as i sin ( ω φ) sin ( φ) e Z R ( ω ) ω ; Where ω β.

34 The above expression also represens he hyrisor curren i T, during he conducion ie inerval of hyrisor T fro ω o β. To Calculae Exincion Angle β The exincion angle β, which is he value of ω a which he load curren i falls o zero and T is urned off can be esiaed by using he condiion ha i, a ω β By using he above expression for he oupu load curren, we can wrie As i sin sin Z Z we can wrie ( β φ) ( φ) Therefore we obain he expression ( β φ) ( φ) R ( β ) ω sin sin e sin ( β φ) sin ( φ) e R ( β ) ω ( β ) ω The exincion angle β can be deerined fro his ranscendenal equaion by using he ieraive ehod of soluion (rial and error ehod). Afer β is calculaed, we can deerine he hyrisor conducion angle δ ( β ). β is he exincion angle which depends upon he load inducance value. Conducion angle δ increases as is decreased for a known value of β. For δ < radians, i.e., for ( β ) < radians, for ( + ) > β he load curren wavefor appears as a disconinuous curren wavefor as shown in he figure. The oupu load curren reains a zero during ω β +. This is referred o as o disconinuous load curren operaion which occurs for β ( ) < +. When he rigger angle is decreased and ade equal o he load ipedance angle φ i.e., when φ sin β φ, e R we obain fro he expression for ( β φ) ; Therefore sin Exincion angle β ( φ) ( ) β φ radians. + + ; for he case when φ δ β radians 8 ; for he case when φ Conducion angle Each hyrisor conducs for 8 ( radians ). T conducs fro + φ and provides a posiive load curren. T conducs fro ( φ) ω + o ( φ) φ o + and provides a negaive load curren. Hence we obain a coninuous load curren and he

35 oupu volage wavefor appears as a coninuous sine wave idenical o he inpu supply volage wavefor for rigger angle φ and he conrol on he oupu is los. v v v i φ φ φ φ Fig.: upu volage and oupu curren wavefors for a single phase full wave ac volage conroller wih R load for φ Thus we observe ha for rigger angle φ, he load curren ends o flow coninuously and we have coninuous load curren operaion, wihou any break in he load curren wavefor and we obain oupu volage wavefor which is a coninuous sinusoidal wavefor idenical o he inpu supply volage wavefor. We loose he conrol on he oupu volage for φ as he oupu volage becoes equal o he inpu supply volage and hus we obain ( RM ) ; for φ Hence, RM oupu volage RM inpu supply volage for φ T DERE AN EXPREN FR RM UTPUT TAGE ( RM ) F A NGE PHAE FU-WAE AC TAGE CNTRER WTH R AD. φ 3 ω ω

36 When >, he load curren and load volage wavefors becoe disconinuous as shown in he figure above. β sin d. RM ω ( ω) upu vo sinω, for ω o β, when T is N. ( ω) β cos ( RM ) d( ω) β β d d RM ( ω ) cos ω. ( ω ) β sin ω ( ω) RM ( β ) RM β sin β sin + sin sin β + ( β ) RM sin sin β + ( β ) RM The RM oupu volage across he load can be varied by changing he rigger angle. For a purely resisive load, herefore load power facor angle φ. ω φ an ; R β radians 8 Exincion angle

37 PERFRMANCE PARAMETER F A NGE PHAE FU WAE AC TAGE CNTRER WTH RETE AD sin + RM upu olage ( ) inpu supply volage. RM ; RM ( RM ) ( RM ) ( RM ) RM value of load curren. R RM value of inpu supply curren. upu load power npu Power Facor Average Thyrisor Curren, i T P R RM R R P ( RM ) ( RM ) PF RM ( RM ) sin PF ( ) + ( + ) Fig.: Thyrisor Curren Wavefor 3 ω i d d T Avg T ( ω ) sin ω. ( ω ) sin ωd. ( ω) cosω T Avg T Avg [ cos + cos ] [ + cos ]

38 Maxiu Average Thyrisor Curren, for, T ( Avg ) RM Thyrisor Curren sin d. T RM ω ( ω) sin + ( ) T RM Maxiu RM Thyrisor Curren, for, T ( RM ) n he case of a single phase full wave ac volage conroller circui using a Triac wih resisive load, he average hyrisor curren. Because he Triac conducs in T Avg boh he half cycles and he hyrisor curren is alernaing and we obain a syerical hyrisor curren wavefor which gives an average value of zero on inegraion. PERFRMANCE PARAMETER F A NGE PHAE FU WAE AC TAGE CNTRER WTH R- AD The Expression for he upu (oad) Curren The expression for he oupu (load) curren which flows hrough he hyrisor, during ω o β is given by R ( ω ) ω i it sin ( ω φ) sin ( φ) e Z ; for ω β Where, Maxiu or peak value of inpu ac supply volage. ( ω ) Z R + oad ipedance. ω φ an oad ipedance angle (load power facor angle). R Thyrisor rigger angle Delay angle. β Exincion angle of hyrisor, (value of curren falls o zero. β is calculaed by solving he equaion sin ( β φ) sin ( φ) e R ( β ) ω ω ) a which he hyrisor (load)

39 Thyrisor Conducion Angle δ ( β ) Maxiu hyrisor conducion angle δ ( β ) radians 8 for φ. RM upu olage sin sin β + ( β ) RM The Average Thyrisor Curren β T Avg id T ( ω) β R ( ω ) ω sin ( ) sin T Avg ω φ ( φ) e d( ω) Z β β R ( ω ) ω sin ( ). d( ) sin T Avg ω φ ω ( φ) e d( ω) Z Maxiu value of T ( Avg ) occur a. The hyrisors should be raed for axiu T ( Avg ), where. Z RM Thyrisor Curren T ( RM ) β T RM id T ( ω) Maxiu value of T ( RM ) occurs a. Thyrisors should be raed for axiu T ( RM ) When a Triac is used in a single phase full wave ac volage conroller wih R ype of load, hen and axiu T Avg T ( RM )

40 PRBEM. A single phase full wave ac volage conroller supplies an R load. The inpu supply volage is 3, RM a 5Hz. The load has H, R Ω, he delay angle of hyrisors T and T are equal, where. Deerine 3 a. Conducion angle of he hyrisor T. b. RM oupu volage. c. The inpu power facor. Coen on he ype of operaion. Given s 3, f 5Hz, H, R Ω, 6, radians, ( ω ) ( ω ) oad pedance Z R ( f) ω Ω Z Ω A Z.488 ω oad pedance Angle φ an R φ an an (.3459) Trigger Angle > φ. Hence he ype of operaion will be disconinuous load curren operaion, we ge β < + β < ( 8 + 6) ; β < 4 Therefore he range of β is fro 8 degrees o 4 degrees. 8 < β < 4

41 Exincion Angle β is calculaed by using he equaion sin ( β φ) sin ( φ) e R ( β ) ω n he exponenial er he value of and β should be subsiued in radians. Hence R ( βrad Rad ) ω sin ( β φ) sin ( φ) e ; Rad 3 Assuing β 9 ; ( φ) ( ) ( β ) sin 7.44 sin e ( β ) ( β ) 3.83( β ) sin e 8 radians, β Rad β Rad β β H.: R.H.: Assuing β 83 ; sin sin e 4.94 β Rad β ( β ) sin β φ sin sin h.: R.H.: e Assuing β 8 β Rad β β 3 3

42 .H.: ( β φ) R.H.: sin sin e Assuing β 96 β Rad β H.: ( β φ) R.H.: Assuing β 97 sin sin e β Rad β sin β φ sin H.: R.H.: Assuing β e β Rad β H.: R.H.: 4 sin β φ sin e 3.79 Conducion Angle δ ( β ) RM upu olage sin sin β ( RM ) ( β ) + RM RM sin 6 sin RM

43 npu Power Facor P PF RM ( RM ) A Z.488 P R W RM 3, RM P PF A single phase full wave conroller has an inpu volage of (RM) and a load resisance of 6 oh. The firing angle of hyrisor is. Find a. RM oupu volage b. Power oupu c. npu power facor d. Average and RM hyrisor curren. oluion 9,, R 6Ω RM alue of upu olage sin + RM upu Curren sin ols A R 6 oad Power P R P was

44 npu Curren is sae as oad Curren Therefore 4.4 Aps npu upply ol-ap A Therefore oad Power npu ol-ap npu Power Facor.77( lag ) Each Thyrisor Conducs only for half a cycle Average hyrisor curren T ( Avg ) RM hyrisor curren T ( RM ) d T Avg R sin ω. ( ω ) ( + cos ) ; R [ + cos9 ] 4.5 A 6 sin ω d T RM R ( ω ) ( cos ω ) R sin R + ( ω) d sin R + sin8 + Aps 6

45 3. A single phase half wave ac regulaor using one CR in ani-parallel wih a diode feeds kw, 3 heaer. Find load power for a firing angle of 45. power oluion 45, 3 ; P KW W 4 A sandard rs supply volage of 3, he heaer dissipaes KW of oupu Therefore Resisance of heaer P R R 3 R 5.9Ω P RM value of oupu volage sin + ; for firing angle 45 sin ols 4 RM value of oupu curren Aps R 5.9 oad Power P R Was 4. Find he RM and average curren flowing hrough he heaer shown in figure. The delay angle of boh he CRs is 45. CR + i o -φ ac CR kw, heaer

46 oluion 4 45, Resisance of heaer R 48.4Ω R Resisance value of oupu volage sin + sin ols 4 RM curren flowing hrough heaer Aps R 48.4 Average curren flowing hrough he heaer 5. A single phase volage conroller is eployed for conrolling he power flow fro, 5 Hz source ino a load circui consising of R 4 Ω and ω 6 Ω. Calculae he following a. Conrol range of firing angle b. Maxiu value of RM load curren c. Maxiu power and power facor d. Maxiu value of average and RM hyrisor curren. oluion For conrol of oupu power, iniu angle of firing angle is equal o he load ipedance angle θ θ, load angle Avg ω 6 θ an an 56.3 R 4 Maxiu possible value of is 8 Therefore conrol range of firing angle is 56.3 < < 8

47 Maxiu value of RM load curren occurs when θ A his value of he Maxiu value of RM load curren Aps Z Maxiu Power P R W npu ol-ap W Power Facor P npu A Average hyrisor curren will be axiu when θ and conducion angleγ 8. Therefore axiu value of average hyrisor curren + sin T Avg ( ω θ) d( ω) Z Noe: sin ( ω θ) sin ( θ) A, Bu θ, i it e Z it i sin Z ( ω θ) Z cos( ω θ) T Avg Z ( ω ) ω + R cos( + θ) + cos( θ) T Avg Z + Z Z cos( ) cos( ) [ ] T Avg Aps T Avg Z iilarly, axiu RM value occurs when and γ. Therefore axiu value of RM hyrisor curren + TM sin d Z ( ω θ) ( ω )

48 ( ω θ) + cos TM d Z ( ω ) TM ( ω θ) sin ω 4 Z + TM TM 4 Z + [ ].5777 Aps Z 4 + 6

49 CNTRED RECTFER (ine Couaed AC o DC converers) NTRDUCTN T CNTRED RECTFER Conrolled recifiers are line couaed ac o power converers which are used o conver a fixed volage, fixed frequency ac power supply ino variable oupu volage. AC npu olage ine Couaed Converer Type of inpu: Fixed volage, fixed frequency ac power supply. Type of oupu: ariable oupu volage + DC upu (d c ) The inpu supply fed o a conrolled recifier is ac supply a a fixed rs volage and a a fixed frequency. We can obain variable oupu volage by using conrolled recifiers. By eploying phase conrolled hyrisors in he conrolled recifier circuis we can obain variable oupu volage and variable (average) oupu curren by varying he rigger angle (phase angle) a which he hyrisors are riggered. We obain a unidirecional and pulsaing load curren wavefor, which has a specific average value. The hyrisors are forward biased during he posiive half cycle of inpu supply and can be urned N by applying suiable gae rigger pulses a he hyrisor gae leads. The hyrisor curren and he load curren begin o flow once he hyrisors are riggered (urned N) say a ω. The load curren flows when he hyrisors conduc fro ω o β. The oupu volage across he load follows he inpu supply volage hrough he conducing hyrisor. A ω β, when he load curren falls o zero, he hyrisors urn off due o AC line (naural) couaion. n soe bridge conrolled recifier circuis he conducing hyrisor urns off, when he oher hyrisor is (oher group of hyrisors are) urned N. The hyrisor reains reverse biased during he negaive half cycle of inpu supply. The ype of couaion used in conrolled recifier circuis is referred o AC line couaion or Naural couaion or AC phase couaion. When he inpu ac supply volage reverses and becoes negaive during he negaive half cycle, he hyrisor becoes reverse biased and hence urns off. There are several ypes of power converers which use ac line couaion. These are referred o as line couaed converers. - Differen ypes of line couaed converers are Phase conrolled recifiers which are AC o DC converers. AC o AC converers AC volage conrollers, which conver inpu ac volage ino variable ac oupu volage a he sae frequency. Cyclo converers, which give low oupu frequencies.

50 All hese power converers operae fro ac power supply a a fixed rs inpu supply volage and a a fixed inpu supply frequency. Hence hey use ac line couaion for urning off he hyrisors afer hey have been riggered N by he gaing signals. DFFERENCE BETWEEN DDE RECTFER AND PHAE CNTRED RECTFER The diode recifiers are referred o as unconrolled recifiers which ake use of power seiconducor diodes o carry he load curren. The diode recifiers give a fixed oupu volage (fixed average oupu volage) and each diode recifying eleen conducs for one half cycle duraion (T/ seconds), ha is he diode conducion angle 8 or radians. A single phase half wave diode recifier gives (under ideal condiions) an average oupu volage ( ) and single phase full wave diode recifier gives (under ideal condiions) an average oupu volage ( ), where is axiu value of he available ac supply volage. Thus we noe ha we can no conrol (we can no vary) he oupu volage or he average load curren in a diode recifier circui. n a phase conrolled recifier circui we use a high curren and a high power hyrisor device (silicon conrolled recifier; CR) for conversion of ac inpu power ino oupu power. Phase conrolled recifier circuis are used o provide a variable volage oupu and a variable (average) load curren. We can conrol (we can vary) he average value ( value) of he oupu load volage (and hence he average load curren) by varying he hyrisor rigger angle. We can conrol he hyrisor conducion angle δ fro 8 o by varying he δ rigger angle fro o 8, where hyrisor conducion angle APPCATN F PHAE CNTRED RECTFER DC oor conrol in seel ills, paper and exile ills eploying oor drives. AC fed racion syse using racion oor. Elecro-cheical and elecro-eallurgical processes. Magne power supplies. Reacor conrols. Porable hand ool drives. ariable speed indusrial drives. Baery charges. High volage DC ransission. Uninerrupible power supply syses (UP). oe years back ac o power conversion was achieved using oor generaor ses, ercury arc recifiers, and hyraorn ubes. The odern ac o power converers are designed using high power, high curren hyrisors and presenly os of he ac- power converers are hyrisorised power converers. The hyrisor devices are phase conrolled o obain a variable oupu volage across he oupu load erinals. The

51 phase conrolled hyrisor converer uses ac line couaion (naural couaion) for couaing (urning off) he hyrisors ha have been urned N. The phase conrolled converers are siple and less expensive and are widely used in indusrial applicaions for indusrial drives. These converers are classified as wo quadran converers if he oupu volage can be ade eiher posiive or negaive for a given polariy of oupu load curren. There are also single quadran ac- converers where he oupu volage is only posiive and canno be ade negaive for a given polariy of oupu curren. f course single quadran converers can also be designed o provide only negaive oupu volage. The wo quadran converer operaion can be achieved by using fully conrolled bridge converer circui and for single quadran operaion we use a half conrolled bridge converer. CAFCATN F PHAE CNTRED RECTFER The phase conrolled recifiers can be classified based on he ype of inpu power supply as ingle Phase Conrolled Recifiers which operae fro single phase ac inpu power supply. Three Phase Conrolled Recifiers which operae fro hree phase ac inpu power supply. DFFERENT TYPE F NGE PHAE CNTRED RECTFER ingle Phase Conrolled Recifiers are furher subdivided ino differen ypes Half wave conrolled recifier which uses a single hyrisor device (which provides oupu conrol only in one half cycle of inpu ac supply, and i provides low oupu). Full wave conrolled recifiers (which provide higher oupu) o Full wave conrolled recifier using a cener apped ransforer (which requires wo hyrisors). o Full wave bridge conrolled recifiers (which do no require a cener apped ransforer) ingle phase sei-converer (half conrolled bridge converer, using wo CR s and wo diodes, o provide single quadran operaion). ingle phase full converer (fully conrolled bridge converer which requires four CR s, o provide wo quadran operaion). Three Phase Conrolled Recifiers are of differen ypes Three phase half wave conrolled recifiers. Three phase full wave conrolled reciriers. o ei converer (half conrolled bridge converer). o Full converer (fully conrolled bridge converer). PRNCPE F PHAE CNTRED RECTFER PERATN The basic principle of operaion of a phase conrolled recifier circui is explained wih reference o a single phase half wave phase conrolled recifier circui wih a resisive load shown in he figure.

52 R R oad Resisance Fig.: ingle Phase Half-Wave Thyrisor Converer wih a Resisive oad A single phase half wave hyrisor converer which is used for ac- power conversion is shown in he above figure. The inpu ac supply is obained fro a ain supply ransforer o provide he desired ac supply volage o he hyrisor converer depending on he oupu volage required. v P represens he priary inpu ac supply volage. v represens he secondary ac supply volage which is he oupu of he ransforer secondary. During he posiive half cycle of inpu supply when he upper end of he ransforer secondary is a a posiive poenial wih respec o he lower end, he hyrisor anode is posiive wih respec o is cahode and he hyrisor is in a forward biased sae. The hyrisor is riggered a a delay angle of ω, by applying a suiable gae rigger pulse o he gae lead of hyrisor. When he hyrisor is riggered a a delay angle of ω, he hyrisor conducs and assuing an ideal hyrisor, he hyrisor behaves as a closed swich and he inpu supply volage appears across he load when he hyrisor conducs fro ω o radians. upu volage v v, when he hyrisor conducs fro ω o. For a purely resisive load, he load curren i (oupu curren) ha flows when he hyrisor T is on, is given by he expression v i, for ω R The oupu load curren wavefor is siilar o he oupu load volage wavefor during he hyrisor conducion ie fro o. The oupu curren and he oupu volage wavefor are in phase for a resisive load. The load curren increases as he inpu supply volage increases and he axiu load curren flows a ω, when he inpu supply volage is a is axiu value. The axiu value (peak value) of he load curren is calculaed as i( ax). R

53 Noe ha when he hyrisor conducs ( T is on) during ω o, he hyrisor curren i T, he load curren i hrough R and he source curren i flowing hrough he ransforer secondary winding are all one and he sae. Hence we can wrie v sinω i it i ; for ω R R is he axiu (peak) value of he load curren ha flows hrough he ransforer secondary winding, hrough T and hrough he load resisor R a he insan ω, when he inpu supply volage reaches is axiu value. When he inpu supply volage decreases he load curren decreases. When he supply volage falls o zero a ω, he hyrisor and he load curren also falls o zero a ω. Thus he hyrisor naurally urns off when he curren flowing hrough i falls o zero a ω. During he negaive half cycle of inpu supply when he supply volage reverses and becoes negaive during ω o radians, he anode of hyrisor is a a negaive poenial wih respec o is cahode and as a resul he hyrisor is reverse biased and hence i reains cu-off (in he reverse blocking ode). The hyrisor canno conduc during is reverse biased sae beween ω o. An ideal hyrisor under reverse biased condiion behaves as an open swich and hence he load curren and load volage are zero during ω o. The axiu or peak reverse volage ha appears across he hyrisor anode and cahode erinals is. The rigger angle (delay angle or he phase angle ) is easured fro he beginning of each posiive half cycle o he ie insan when he gae rigger pulse is applied. The hyrisor conducion angle is fro o, hence he conducion angle δ ( ). The axiu conducion angle is radians (8 ) when he rigger angle. Fig: Quadran Diagra The wavefors shows he inpu ac supply volage across he secondary winding of he ransforer which is represened as v, he oupu volage across he load, he oupu (load) curren, and he hyrisor volage wavefor ha appears across he anode and cahode erinals.

54 Fig: Wavefors of single phase half-wave conrolled recifier wih resisive load EQUATN v sinω he ac supply volage across he ransforer secondary. s ax. (peak) value of inpu ac supply volage across ransforer secondary. RM value of inpu ac supply volage across ransforer secondary. v v he oupu volage across he load ; i i oupu (load) curren.

55 When he hyrisor is riggered a ω (an ideal hyrisor behaves as a closed swich) and hence he oupu volage follows he inpu supply volage. v v sinω; for ω o, when he hyrisor is on. i v R i oad curren for o ω, when he hyrisor is on. T DERE AN EXPREN FR THE AERAGE (DC) UTPUT TAGE ACR THE AD f is he peak inpu supply volage, he average oupu volage can be found fro v. d ( ω) d sin ω. ( ω ) d sin ω. ( ω ) d sin ω. ( ω ) cosω [ cos + cos ] ; cos + ; [ cos ] The axiu average () oupu volage is obained when and he axiu oupu volage ( ax) d. The average oupu volage can be varied by varying he rigger angle fro o a axiu of 8 ( radians). We can plo he conrol characerisic, which is a plo of oupu volage versus he rigger angle by using he equaion for ( ).

56 CNTR CHARACTERTC F NGE PHAE HAF WAE PHAE CNTRED RECTFER WTH RETE AD The average oupu volage is given by he expression [ + cos ] We can obain he conrol characerisic by ploing he expression for he oupu volage as a funcion of rigger angle Trigger angle ( ) in degrees % d % d d 93.3 % d 6.75 d 75 % d 9.5 d 5 % d.5 d 5 % d d 6.69 % d 8 () d.6 d. d d ( ax) 6 8 Trigger angle in degrees Fig.: Conrol characerisic Noralizing he oupu volage wih respec o volage ( ) n ( ax) d d, he noralized oupu

57 n n d ( + cos ) n ( + cos ) d n T DERE AN EXPREN FR THE RM AUE F UTPUT TAGE F A NGE PHAE HAF WAE CNTRED RECTFER WTH RETE AD The rs oupu volage is given by v. d RM ( ω) upu volage v sin ω ; for ω o sin d. RM ω ( ω) cos ω By subsiuing sin ω, we ge ( ω) cos. d RM ( ω) ( cos ). d RM ω ( ω) 4 d d RM 4 ( ω ) cos ω. ( ω ) sin ω ( ω) RM ( ) RM ( sin sin ) ; sin

58 Hence we ge, sin + ( ) RM + ( ) RM sin PERFRMANCE PARAMETER F PHAE CNTRED RECTFER upu power (average or oupu power delivered o he load) Where P ; i.e., P upu ac power average or value of oupu (load) volage. average or value of oupu (load) curren. P ac RM RM Efficiency of Recificaion (Recificaion Raio) P ( ) Efficiency η ; P ac P ( ) % Efficiency η P The oupu volage can be considered as being coposed of wo coponens The coponen ( ) DC or average value of oupu volage. The ac coponen or he ripple coponen ac r ( rs) RM value of all he ac ripple coponens. ac The oal RM value of oupu volage is given by + RM r rs Therefore ac r rs RM

59 For Facor (FF) which is a easure of he shape of he oupu volage is given by ( RM ) RM oupu load volage FF DC oupu load volage The Ripple Facor (RF) which is a easure of he ac ripple conen in he oupu volage wavefor. The oupu volage ripple facor defined for he oupu volage wavefor is given by Therefore r rs ac rv RF RM ( ) ( RM ) rv ( ) ( ) r v FF Curren Ripple Facor defined for he oupu (load) curren wavefor is given by r( rs) ri Where ac r rs ( RM ) ( ) ac oe ies he peak o peak oupu ripple volage is also considered o express he peak o peak oupu ripple volage as peak o peak ac ripple oupu volage r pp The peak o peak ac ripple load curren is he difference beween he axiu and he iniu values of he oupu load curren. ( ax) ( in) r pp Transforer Uilizaion Facor (TUF) Where TUF P ( ) RM value of ransforer secondary oupu volage (RM supply volage a he secondary)

60 RM value of ransforer secondary curren (RM line or supply curren). φ v upply volage a he ransforer secondary side. i npu supply curren (ransforer secondary winding curren). i Fundaenal coponen of he inpu supply curren. P Peak value of he inpu supply curren. φ Phase angle difference beween (sine wave coponens) he fundaenal coponens of inpu supply curren and he inpu supply volage. φ Displaceen angle (phase angle) For an R load φ Displaceen angle oad ipedance angle ω φ R an for an R load Displaceen Facor (DF) or Fundaenal Power Facor DF Cosφ Haronic Facor (HF) or Toal Haronic Disorion Facor (THD) The haronic facor is a easure of he disorion in he oupu wavefor and is also referred o as he oal haronic disorion (THD) Where HF RM value of inpu supply curren. RM value of fundaenal coponen of he inpu supply curren. npu Power Facor (PF)

61 PF cosφ cosφ The Cres Facor (CF) ( peak ) Peak inpu supply curren CF RM inpu supply curren For an deal Conrolled Recifier FF ; which eans ha RM ( ). Efficiency η % ; which eans ha P P ( ac). ac ; so ha RF r v ; Ripple facor (ripple free converer). r rs TUF ; which eans ha P HF THD ; which eans ha PF DPF ; which eans ha φ NGE PHAE HAF WAE CNTRED RECTFER WTH AN R AD n his secion we will discuss he operaion and perforance of a single phase half wave conrolled recifier wih R load. n pracice os of he loads are of R ype. For exaple if we consider a single phase conrolled recifier conrolling he speed of a oor, he load which is he oor winding is an R ype of load, where R represens he oor winding resisance and represens he oor winding inducance. A single phase half wave conrolled recifier circui wih an R load using a hyrisor T ( T is an CR) is shown in he figure below.

62 The hyrisor T is forward biased during he posiive half cycle of inpu supply. e us assue ha T is riggered a ω, by applying a suiable gae rigger pulse o T during he posiive half cycle of inpu supply. The oupu volage across he load follows he inpu supply volage when T is N. The load curren i flows hrough he hyrisor T and hrough he load in he downward direcion. This load curren pulse flowing hrough T can be considered as he posiive curren pulse. Due o he inducance in he load, he load curren i flowing hrough T would no fall o zero a ω, when he inpu supply volage sars o becoe negaive. A phase shif appears beween he load volage and he load curren wavefors, due o he load inducance. The hyrisor T will coninue o conduc he load curren unil all he inducive energy sored in he load inducor is copleely uilized and he load curren hrough T falls o zero a ω β, where β is referred o as he Exincion angle, (he value of ω ) a which he load curren falls o zero. The exincion angle β is easured fro he poin of he beginning of he posiive half cycle of inpu supply o he poin where he load curren falls o zero. The hyrisor T conducs fro ω o β. The conducion angle of T is δ β, which depends on he delay angle and he load ipedance angle φ. The wavefors of he inpu supply volage, he gae rigger pulse of T, he hyrisor curren, he load curren and he load volage wavefors appear as shown in he figure below. i i i Fig.: npu supply volage & Thyrisor curren wavefors

63 β is he exincion angle which depends upon he load inducance value. Fig.: upu (load) volage wavefor of a single phase half wave conrolled recifier wih R load Fro β o, he hyrisor reains cu-off as i is reverse biased and behaves as an open swich. The hyrisor curren and he load curren are zero and he oupu volage also reains a zero during he non conducion ie inerval beween β o. n he nex cycle he hyrisor is riggered again a a phase angle of ( + ), and he sae operaion repeas. T DERE AN EXPREN FR THE UTPUT (NDUCTE AD) CURRENT, DURNG ω o β WHEN THYRTR T CNDUCT Considering sinusoidal inpu supply volage we can wrie he expression for he supply volage as v sinω insananeous value of he inpu supply volage. e us assue ha he hyrisor T is riggered by applying he gaing signal o T a ω. The load curren which flows hrough he hyrisor T during ω o β can be found fro he equaion di + Ri sinω d ; The soluion of he above differenial equaion gives he general expression for he oupu load curren which is of he for sin ( ω φ) + ; Z i τ Ae Where axiu or peak value of inpu supply volage. ( ω ) Z R + oad ipedance.

64 φ ω oad ipedance angle (power facor angle of load). R an τ oad circui ie consan. R Therefore he general expression for he oupu load curren is given by he equaion R i sin ( ω φ) + Ae ; Z The value of he consan A can be deerined fro he iniial condiion. i.e. iniial value of load curren i, a ω. Hence fro he equaion for i equaing i o zero and subsiuing ω, we ge i sin ( φ) + Ae Z R sin Z Therefore Ae ( φ) A sin R Z e + R Z ( φ) e sin ( φ) A R ( ω) ω e sin ( φ) A Z R By subsiuing ω, we ge he value of consan A as A R ( ) Z ω e sin ( φ) ubsiuing he value of consan A fro he above equaion ino he expression for i, we obain R R( ) ω i sin ( ω φ) + e e sin ( φ) Z Z ; ( ω ) ( ) R R i e e + Z Z ω ω sin ( ω φ) sin ( φ)

65 R + Z Z ω ω sin ( ω φ) sin ( φ) i e Therefore we obain he final expression for he inducive load curren of a single phase half wave conrolled recifier wih R load as i sin ( ω φ) sin ( φ) e Z R ( ω ) ω ; Where ω β. The above expression also represens he hyrisor curren i T, during he conducion ie inerval of hyrisor T fro ω o β. T CACUATE EXTNCTN ANGE β The exincion angle β, which is he value of ω a which he load curren i falls o zero and T is urned off can be esiaed by using he condiion ha i, a ω β By using he above expression for he oupu load curren, we can wrie As i sin sin Z Z, we can wrie ( β φ) ( φ) Therefore we obain he expression ( β φ) ( φ) R ( β ) ω sin sin e sin ( β φ) sin ( φ) e R ( β ) ω e R ( β ) ω The exincion angle β can be deerined fro his ranscendenal equaion by using he ieraive ehod of soluion (rial and error ehod). Afer β is calculaed, we δ β. can deerine he hyrisor conducion angle β is he exincion angle which depends upon he load inducance value. Conducion angle δ increases as is decreased for a specific value of β. Conducion angle δ ( β ) ; for a purely resisive load or for an R load when he load inducance is negligible he exincion angle β and he conducion angle δ ( )

66 Equaions v sin ω npu supply volage s v v sin ω upu load volage for ω o β, when he hyrisor T conducs ( T is on). Expression for he load curren (hyrisor curren): for ω o β i sin ( ω φ) sin ( φ) e Z R ( ω ) ω Exincion angle β can be calculaed using he equaion sin ( β φ) sin ( φ) e R ( β ) ω ; Where ω β. T DERE AN EXPREN FR AERAGE (DC) AD TAGE v d. ( ω ) β v. d( ) v. d( ) v. d ω + ω + ( ω) β ; v for ω o & for ω β o ; β v. d( ω) ; v sin ω for ω o β β sin d. ω ( ω) β cosω ( cos cos β) β ( cos cos ) Noe: During he period ω o β, we can see fro he oupu load volage wavefor ha he insananeous oupu volage is negaive and his reduces he average or he oupu volage when copared o a purely resisive load.

67 Average DC oad Curren ( ) ( ) ( Avg ) R R ( cos cos β) NGE PHAE HAF WAE CNTRED RECTFER WTH R AD AND FREE WHEENG DDE T + s ~ Fig. : ingle Phase Half Wave Conrolled Recifier wih R oad and Free Wheeling Diode (FWD) Wih a R load i was observed ha he average oupu volage reduces. This disadvanage can be overcoe by connecing a diode across he load as shown in figure. The diode is called as a Free Wheeling Diode (FWD). The wavefors are shown below. s i G i FWD Gae pulses β upply volage oad curren β oad volage - ωβ + i R 3 + ω ω ω 3 ω

68 A ω, he source volage v falls o zero and as v becoes negaive, he free wheeling diode is forward biased. The sored energy in he inducance ainains he load curren flow hrough R,, and he FWD. Also, as soon as he FWD is forward biased, a ω, he CR becoes reverse biased, he curren hrough i becoes zero and he CR urns off. During he period ω o β, he load curren flows hrough FWD (free wheeling load curren) and decreases exponenially owards zero a ω β. Also during his free wheeling ie period he load is shored by he conducing FWD and he load volage is alos zero, if he forward volage drop across he conducing FWD is negleced. Thus here is no negaive region in he load volage wave for. This iproves he average oupu volage. The average oupu volage [ + cos ], which is he sae as ha of a purely resisive load. The oupu volage across he load appears siilar o he oupu volage of a purely resisive load. The following poins are o be noed. f he inducance value is no very large, he energy sored in he inducance is able o ainain he load curren only upo ω β, where < β <, well before he nex gae pulse and he load curren ends o becoe disconinuous. During he conducion period o, he load curren is carried by he CR and during he free wheeling period o β, he load curren is carried by he free wheeling diode. The value of β depends on he value of R and and he forward resisance of he FWD. Generally < β <. f he value of he inducance is very large, he load curren does no decrease o zero during he free wheeling ie inerval and he load curren wavefor appears as shown in he figure. i 4 CR FWD CR FWD + 3 ω 3 Fig. : Wavefor of oad Curren in ingle Phase Half Wave Conrolled Recifier wih a arge nducance and FWD

69 During he periods, 3,... he CR carries he load curren and during he periods, 4,... he FWD carries he load curren. is o be noed ha The load curren becoes coninuous and he load curren does no fall o zero for large value of load inducance. The ripple in he load curren wavefor (he aoun of variaion in he oupu load curren) decreases. NGE PHAE HAF WAE CNTRED RECTFER WTH A GENERA AD A general load consiss of R, and a DC source E in he load circui + ~ v n he half wave conrolled recifier circui shown in he figure, he load circui consiss of a source E in addiion o resisance and inducance. When he hyrisor is in he cu-off sae, he curren in he circui is zero and he cahode will be a a volage equal o he volage in he load circui i.e. he cahode poenial will be equal o E. The hyrisor will be forward biased for anode supply volage greaer han he load volage. When he supply volage is less han he volage E in he circui he hyrisor is reverse biased and hence he hyrisor canno conduc for supply volage less han he load circui volage. The value of ω a which he supply volage increases and becoes equal o he load circui volage can be calculaed by using he equaion sinω E. f we assue he value of ω is equal o γ hen we can wrie sinγ E. Therefore γ is E calculaed as γ sin. For rigger angle < γ, he hyrisor conducs only fro ω γ o β. For rigger angle > γ, he hyrisor conducs fro ω o β. i + R E v The wavefors appear as shown in he figure

70 v oad volage E γ + ω i δ oad curren β + +β β Equaions v sin ω npu supply volage. v sin ω upu load volage for ω o β v E for ω o & for ω β o Expression for he oad Curren When he hyrisor is riggered a a delay angle of, he equaion for he circui can be wrien as di sin ω i R+ +E ; ω β d The general expression for he oupu load curren can be wrien as Z E R τ i sin ( ω φ) + Ae Where ( ω ) Z R + ω φ R oad pedance an oad ipedance angle ω τ R oad circui ie consan The general expression for he oupu load curren can be wrien as

71 Z E R R i sin ( ω φ) + Ae To find he value of he consan A apply he iniial condiion a ω, load curren i. Equaing he general expression for he load curren o zero a ω, we ge R E ω i sin ( φ) + Ae Z R We obain he value of consan A as R ω E A R Z sin ( φ) e ubsiuing he value of he consan A in he expression for he load curren, we ge he coplee expression for he oupu load curren as E E i sin ( ω φ) + sin ( φ) e Z R R Z R ( ω ) ω The Exincion angle β can be calculaed fro he final condiion ha he oupu curren i a ω β. By using he above expression we ge, i E E sin + sin Z R R Z R ( β φ) ( φ) To derive an expression for he average or load volage v d. ( ω ) ( β ) ω e β v. d( ) v. d( ) v. d ω ω ( ω) + + β v sin ω upu load volage for ω o β v E for ω o & for ω β o β Ed. ( ) sin Ed. ω ω ( ω) + + β β E E ( ω ) + ( cosω ) + ( ω ) β

72 E E ( ) ( cos β cos) + ( β) E β β ( cos cos ) + ( + ) ( cos cos β) ( ) β + E Conducion angle of hyrisor δ ( β ) RM upu olage can be calculaed by using he expression v. d RM ( ω) DADANTAGE F NGE PHAE HAF WAE CNTRED RECTFER ingle phase half wave conrolled recifier gives ow oupu volage. ow oupu power and lower efficiency. Higher ripple volage & ripple curren. Higher ripple facor. ow ransforer uilizaion facor. The inpu supply curren wavefor has a coponen which can resul in sauraion of he ransforer core. ingle phase half wave conrolled recifiers are rarely used in pracice as hey give low oupu and low oupu power. They are only of heoreical ineres. The above disadvanages of a single phase half wave conrolled recifier can be over coe by using a full wave conrolled recifier circui. Mos of he pracical converer circuis use full wave conrolled recifiers. NGE PHAE FU WAE CNTRED RECTFER ingle phase full wave conrolled recifier circui cobines wo half wave conrolled recifiers in one single circui so as o provide wo pulse oupu across he load. Boh he half cycles of he inpu supply are uilized and convered ino a uni-direcional oupu curren hrough he load so as o produce a wo pulse oupu wavefor. Hence a full wave conrolled recifier circui is also referred o as a wo pulse converer. ingle phase full wave conrolled recifiers are of various ypes ingle phase full wave conrolled recifier using a cener apped ransforer (wo pulse converer wih id poin configuraion). ingle phase full wave bridge conrolled recifier Half conrolled bridge converer (sei converer). Fully conrolled bridge converer (full converer).

73 NGE PHAE FU WAE CNTRED RECTFER UNG A CENTER TAPPED TRANFRMER i T A + AC upply B v T v upply olage across he upper half of he ransforer secondary winding v v sinω A R v FWD vb va sinω supply volage across he lower half of he ransforer secondary winding. This ype of full wave conrolled recifier requires a cener apped ransforer and wo hyrisors T and T. The inpu supply is fed hrough he ains supply ransforer, he priary side of he ransforer is conneced o he ac line volage which is available (norally he priary supply volage is 3 RM ac supply volage a 5Hz supply frequency in ndia). The secondary side of he ransforer has hree lines and he cener poin of he ransforer (cener line) is used as he reference poin o easure he inpu and oupu volages. The upper half of he secondary winding and he hyrisor T along wih he load ac as a half wave conrolled recifier, he lower half of he secondary winding and he hyrisor T wih he coon load ac as he second half wave conrolled recifier so as o produce a full wave load volage wavefor. There are wo ypes of operaions possible. Disconinuous load curren operaion, which occurs for a purely resisive load or an R load wih low inducance value. Coninuous load curren operaion which occurs for an R ype of load wih large load inducance. i Disconinuous oad Curren peraion (for low value of load inducance) Generally he load curren is disconinuous when he load is purely resisive or when he R load has a low value of inducance. During he posiive half cycle of inpu supply, when he upper line of he secondary winding is a a posiive poenial wih respec o he cener poin he hyrisor T is forward biased and i is riggered a a delay angle of. The load curren

74 flows hrough he hyrisor T, hrough he load and hrough he upper par of he secondary winding, during he period o β, when he hyrisor T conducs. The oupu volage across he load follows he inpu supply volage ha appears across he upper par of he secondary winding fro ω o β. The load curren hrough he hyrisor T decreases and drops o zero a of load and he hyrisor T naurally urns off a ω ω β. β, where β > for R ype v i β β ( + ) ( +β) Fig.: Wavefor for Disconinuous oad Curren peraion wihou FWD During he negaive half cycle of he inpu supply he volage a he supply line A becoes negaive whereas he volage a line B (a he lower side of he secondary winding) becoes posiive wih respec o he cener poin. The hyrisor T is forward biased during he negaive half cycle and i is riggered a a delay angle of ( + ). The curren flows hrough he hyrisor T, hrough he load, and hrough he lower par of he secondary winding when T conducs during he negaive half cycle he load is conneced o he lower half of he secondary winding when T conducs. For purely resisive loads when, he exincion angle β. The load curren falls o zero a ω β, when he inpu supply volage falls o zero a ω. The load curren and he load volage wavefors are in phase and here is no phase shif beween he load volage and he load curren wavefor in he case of a purely resisive load. For low values of load inducance he load curren would be disconinuous and he exincion angle β > bu β < ( + ). For large values of load inducance he load curren would be coninuous and does no fall o zero. The hyrisor T conducs fro o ( + ), unil he nex hyrisor T is riggered. When T is riggered a ω ( + ), he hyrisor T will be reverse biased and hence T urns off. 3 ω ω

75 T DERE AN EXPREN FR THE DC UTPUT TAGE F A NGE PHAE FU WAE CNTRED RECTFER WTH R AD (WTHUT FREE WHEENG DDE (FWD)) The average or oupu volage of a full-wave conrolled recifier can be calculaed by finding he average value of he oupu volage wavefor over one oupu cycle (i.e., radians) and noe ha he oupu pulse repeiion ie is T seconds where T represens he inpu supply ie period and T ; where f inpu supply frequency. f Assuing he load inducance o be sall so ha β >, β < ( + ) we obain disconinuous load curren operaion. The load curren flows hrough T for ω o β, where is he rigger angle of hyrisor T and β is he exincion angle where he load curren hrough T falls o zero a ω β. Therefore he average or oupu volage can be obained by using he expression v d. ( ω ) β ω v d. ( ω ) β ω β sin ωd. ( ω) cosω β β ( cos cos ) Therefore ( cos cos β ) < β < ( + )., for disconinuous load curren operaion, When he load inducance is sall and negligible ha is, he exincion angle β radians. Hence he average or oupu volage for resisive load is obained as ( cos cos ) ; cos ( cos ( ) )

76 ( cos ) + ; for resisive load, when THE EFFECT F AD NDUCTANCE Due o he presence of load inducance he oupu volage reverses and becoes negaive during he ie period ω o β. This reduces he oupu volage. To preven his reducion of oupu volage due o he negaive region in he oupu load volage wavefor, we can connec a free wheeling diode across he load. The oupu volage wavefor and he oupu volage obained would be he sae as ha for a full wave conrolled recifier wih resisive load. When he Free wheeling diode (FWD) is conneced across he load When T is riggered a ω, during he posiive half cycle of he inpu supply he FWD is reverse biased during he ie period ω o. FWD reains reverse biased and cu-off fro ω o. The load curren flows hrough he conducing hyrisor T, hrough he R load and hrough upper half of he ransforer secondary winding during he ie period o. A ω, when he inpu supply volage across he upper half of he secondary winding reverses and becoes negaive he FWD urns-on. The load curren coninues o flow hrough he FWD fro ω o β. v i β β ( + ) ( +β) Fig.: Wavefor for Disconinuous oad Curren peraion wih FWD 3 ω ω EXPREN FR THE DC UTPUT TAGE F A NGE PHAE FU WAE CNTRED RECTFER WTH R AD AND FWD v d. ( ω ) ω Thyrisor T is riggered a ω. T conducs fro ω o

77 upu volage v sin ω ; for ω o FWD conducs fro ω o β and v during disconinuous load curren Therefore sin ωd. ( ω) cosω [ cos + cos ] ; cos Therefore ( + cos ) The DC oupu volage is sae as he DC oupu volage of a single phase full wave conrolled recifier wih resisive load. Noe ha he oupu volage of a single phase full wave conrolled recifier is wo ies he oupu volage of a half wave conrolled recifier. CNTR CHARACTERTC F A NGE PHAE FU WAE CNTRED RECTFER WTH R AD R R AD WTH FWD The conrol characerisic can be obained by ploing he oupu volage versus he rigger angle. The average or oupu volage of a single phase full wave conrolled recifier circui wih R load or R load wih FWD is calculaed by using he equaion ( + cos ) can be varied by varying he rigger angle fro of rigger angle is fro o radians). Maxiu oupu volage is obained when + ax ( cos ) o 8. (i.e., he range Therefore ( ax) for a single phase full wave conrolled recifier. Noralizing he oupu volage wih respec o is axiu value, we can wrie he noralized oupu volage as

78 n n d ( ax) n ( + cos ) n + cos ( ) n + cos Therefore ( ) n + cos ( ) d d Trigger angle Noralized ( ) in degrees oupu volage n d () d.6 d. d 6 8 Trigger angle in degrees Fig.: Conrol characerisic of a single phase full wave conrolled recifier wih R load or R load wih FWD

79 CNTNUU AD CURRENT PERATN (WTHUT FWD) For large values of load inducance he load curren flows coninuously wihou decreasing and falling o zero and here is always a load curren flowing a any poin of ie. This ype of operaion is referred o as coninuous curren operaion. Generally he load curren is coninuous for large load inducance and for low rigger angles. The load curren is disconinuous for low values of load inducance and for large values of rigger angles. The wavefors for coninuous curren operaion are as shown. v i T N T N T N ( +) ( +) Fig.: oad volage and load curren wavefor of a single phase full wave conrolled recifier wih R load & wihou FWD for coninuous load curren operaion n he case of coninuous curren operaion he hyrisor T which is riggered a a delay angle of, conducs fro ω o ( + ). upu volage follows he inpu supply volage across he upper half of he ransforer secondary winding v va sinω. The nex hyrisor T is riggered a ω ( + ), during he negaive half cycle inpu supply. As soon as T is riggered a ω ( + ), he hyrisor T will be reverse biased and T urns off due o naural couaion (ac line couaion). The load curren flows hrough he hyrisor T fro ω ( + ) o ( + ). upu volage across he load follows he inpu supply volage across he lower half of he ransforer secondary winding v v sinω. operaion. B Each hyrisor conducs for radians ( 8 ) 3 ω ω in he case of coninuous curren

80 T DERE AN EXPREN FR THE AERAGE R DC UTPUT TAGE F NGE PHAE FU WAE CNTRED RECTFER WTH ARGE AD NDUCTANCE AUMNG CNTNUU AD CURRENT PERATN. ( + ) v d. ( ω ) ω ( + ) sin ωd. ( ω) cosω ( + ) cos cos( + ) ; cos( ) [ cos + cos ] cos + cos The above equaion can be ploed o obain he conrol characerisic of a single phase full wave conrolled recifier wih R load assuing coninuous load curren operaion. Noralizing he oupu volage wih respec o is axiu value, he noralized oupu volage is given by Therefore n ( cos ) n cos ( ax) cos n n

81 Trigger angle in degrees d Rearks d 6.5 d 9 d -.5 d d 8 d () d.6 d. d -. d -.6 d - d Maxiu oupu volage ( ax) d 5 8 Trigger angle in degrees Fig.: Conrol Characerisic We noice fro he conrol characerisic ha by varying he rigger angle we can vary he oupu volage across he load. Thus i is possible o conrol he oupu volage by changing he rigger angle. For rigger angle in he range of o 9 degrees ( ie.., 9 ), is posiive and he circui operaes as a conrolled recifier o conver ac supply volage ino oupu power which is fed o he load. For rigger angle > 9,cos becoes negaive and as a resul he average oupu volage becoes negaive, bu he load curren flows in he sae posiive direcion. Hence he oupu power becoes negaive. This eans ha he power flows fro he load circui o he inpu ac source. This is referred o as line couaed inverer operaion. During he inverer ode operaion for > 9 he load energy can be fed back fro he load circui o he inpu ac source.

82 T DERE AN EXPREN FR RM UTPUT TAGE The rs value of he oupu volage is calculaed by using he equaion ( + ) v. d RM ( ω) ( + ) sin d. RM ω ( ω) ( + ) sin d. RM ω ( ω) ( + ) ( ω) cos. d RM ( ω) ( + ) ( + ) d d RM ( ω ) cos ω. ( ω ) sin ω ( RM ) ( ω) ( ) RM ( + ) ( + ) sin + sin + sin cos + cos sin sin ( ) RM + sin sin ( ) RM RM ( ) Therefore ( RM ) ; The rs oupu volage is sae as he inpu rs supply volage.

83 NGE PHAE EMCNERTER Erraa: Consider diode D as D in he figure and diode D as D ingle phase sei-converer circui is a full wave half conrolled bridge converer which uses wo hyrisors and wo diodes conneced in he for of a full wave bridge configuraion. The wo hyrisors are conrolled power swiches which are urned on one afer he oher by applying suiable gaing signals (gae rigger pulses). The wo diodes are unconrolled power swiches which urn-on and conduc one afer he oher as and when hey are forward biased. The circui diagra of a single phase sei-converer (half conrolled bridge converer) is shown in he above figure wih highly inducive load and a source in he load circui. When he load inducance is large he load curren flows coninuously and we can consider he coninuous load curren operaion assuing consan load curren, wih negligible curren ripple (i.e., consan and ripple free load curren operaion). The ac supply o he seiconverer is norally fed hrough a ains supply ransforer having suiable urns raio. The ransforer is suiably designed o supply he required ac supply volage (secondary oupu volage) o he converer. During he posiive half cycle of inpu ac supply volage, when he ransforer secondary oupu line A is posiive wih respec o he line B he hyrisor T and he diode D are boh forward biased. The hyrisor T is riggered a ω ; ( ) by applying an appropriae gae rigger signal o he gae of T. The curren in he circui flows hrough he secondary line A, hrough T, hrough he load in he downward direcion, hrough diode D back o he secondary line B. T and D conduc ogeher fro ω o and he load is conneced o he inpu ac supply. The oupu load volage follows he inpu supply volage (he secondary oupu volage of he ransforer) during he period ω o. A ω, he inpu supply volage decreases o zero and becoes negaive during he period ω o ( + ). The free wheeling diode D across he load ω o +. becoes forward biased and conducs during he period

84 Fig:. Wavefors of single phase sei-converer for RE load and consan load curren for > 9

85 The load curren is ransferred fro T and D o he FWD D. T and D are urned off. The load curren coninues o flow hrough he FWD D. The load curren free wheels (flows coninuously) hrough he FWD during he free wheeling ie period o +. During he negaive half cycle of inpu supply volage he secondary line A becoes negaive wih respec o line B. The hyrisor T and he diode D are boh forward biased. T is riggered a ω ( + ), during he negaive half cycle. The FWD is reverse biased and urns-off as soon as T is riggered. The load curren coninues o flow hrough T and D during he period ( + ) o ω T DERE AN EXPREN FR THE AERAGE R DC UTPUT TAGE F A NGE PHAE EM-CNERTER The average oupu volage can be found fro sin ωd. [ cosω ] ( ω ) [ cos + cos ] ; cos Therefore [ + cos ] can be varied fro The axiu average oupu volage is o by varying fro o. ( ax) d Noralizing he average oupu volage wih respec o is axiu value n n.5 + cos d ( ) The oupu conrol characerisic can be ploed by using he expression for

86 T DERE AN EXPREN FR THE RM UTPUT TAGE F A NGE PHAE EM-CNERTER The rs oupu volage is found fro sin d. RM ω ( ω) ( cos ). d RM ω ( ω) RM sin + NGE PHAE FU CNERTER (FUY CNTRED BRDGE CNERTER) The circui diagra of a single phase fully conrolled bridge converer is shown in he figure wih a highly inducive load and a source in he load circui so ha he load curren is coninuous and ripple free (consan load curren operaion). The fully conrolled bridge converer consiss of four hyrisors T, T, T 3 and T 4 conneced in he for of full wave bridge configuraion as shown in he figure. Each hyrisor is conrolled and urned on by is gaing signal and naurally urns off when a reverse volage appears across i. During he posiive half cycle when he upper line of he ransforer secondary winding is a a posiive poenial wih respec o he lower end he hyrisors T and T are forward biased during he ie inerval ω o. The hyrisors T and T are riggered siulaneously ω ; ( ), he load is conneced o he inpu supply hrough he conducing hyrisors T and T. The oupu volage across he load follows he inpu supply volage and hence oupu volage v sinω. Due o he inducive load T and T will coninue o conduc beyond ω, even hough he inpu volage becoes negaive. T and T conduc ogeher

87 during he ie period o ( ) +, for a ie duraion of radians (conducion angle of each hyrisor 8 ) During he negaive half cycle of inpu supply volage for ω o he hyrisors T 3 and T 4 are forward biased. T 3 and T 4 are riggered a ω ( + ). As soon as he hyrisors T 3 and T 4 are riggered a reverse volage appears across he hyrisors T and T and hey naurally urn-off and he load curren is ransferred fro T and T o he hyrisors T 3 and T 4. The oupu volage across he load follows he supply volage and v sinω ω + o +. n during he ie period he nex posiive half cycle when T and T are riggered, T 3 and T 4 are reverse biased and hey urn-off. The figure shows he wavefors of he inpu supply volage, he oupu load volage, he consan load curren wih negligible ripple and he inpu supply curren.

88 During he ie period ω o, he inpu supply volage v and he inpu supply curren i are boh posiive and he power flows fro he supply o he load. The converer operaes in he recificaion ode during ω o. During he ie period ω o ( + ), he inpu supply volage v is negaive and he inpu supply curren i is posiive and here will be reverse power flow fro he load circui o he inpu supply. The converer operaes in he inversion ode during he ie period ω o ( + ) and he load energy is fed back o he inpu source. The single phase full converer is exensively used in indusrial applicaions up o abou 5kW of oupu power. Depending on he value of rigger angle, he average oupu volage ay be eiher posiive or negaive and wo quadran operaion is possible. T DERE AN EXPREN FR THE AERAGE (DC) UTPUT TAGE The average () oupu volage can be deerined by using he expression v. d( ω) ; The oupu volage wavefor consiss of wo oupu pulses during he inpu supply ie period beween & radians. n he coninuous load curren operaion of a single phase full converer (assuing consan load curren) each hyrisor conduc for radians (8 ) afer i is riggered. When hyrisors T and T are riggered a ω T and T conduc fro o ( + ) and he oupu volage follows he inpu supply volage. Therefore oupu volage v sinω ; for ω o ( + ) Hence he average or oupu volage can be calculaed as + sin ωd. ( ω) + sin ωd. ( ω) + sin ωd. ( ω) [ cos ] + ω cos( + ) + cos ; cos( ) Therefore cos + cos

89 The oupu volage can be varied fro a axiu value of for o a iniu value of for radians 8 The axiu average oupu volage is calculaed for a rigger angle and is obained as ( ax) d cos( ) Therefore ( ax) d The noralized average oupu volage is given by ( ) n n n ( ax) d cos n cos Therefore n n cos ; for a single phase full converer assuing coninuous and consan load curren operaion. CNTR CHARACTERTC F NGE PHAE FU CNERTER The oupu conrol characerisic can be obained by ploing he average or oupu volage versus he rigger angle For a single phase full converer he average oupu volage is given by he equaion cos Trigger angle in degrees Rearks d d 6.5 d 9 d -.5 d d 8 d Maxiu oupu volage ( ax) d

90 () d.6 d. d -. d -.6 d - d Trigger angle in degrees Fig.: Conrol Characerisic We noice fro he conrol characerisic ha by varying he rigger angle we can vary he oupu volage across he load. Thus i is possible o conrol he oupu volage by changing he rigger angle. For rigger angle in he range of o 9 degrees ( ie.., 9 ), is posiive and he average load curren is also posiive. The average or oupu power P is posiive, hence he circui operaes as a conrolled recifier o conver ac supply volage ino oupu power which is fed o he load. For rigger angle > 9,cos becoes negaive and as a resul he average oupu volage becoes negaive, bu he load curren flows in he sae posiive direcion i.e., is posiive. Hence he oupu power becoes negaive. This eans ha he power flows fro he load circui o he inpu ac source. This is referred o as line couaed inverer operaion. During he inverer ode operaion for > 9 he load energy can be fed back fro he load circui o he inpu ac source TW QUADRANT PERATN F A NGE PHAE FU CNERTER

91 The above figure shows he wo regions of single phase full converer operaion in he versus plane. n he firs quadran when he rigger angle is less han 9, and are boh posiive and he converer operaes as a conrolled recifier and convers he ac inpu power ino oupu power. The power flows fro he inpu source o he load circui. This is he noral conrolled recifier operaion where P is posiive. When he rigger angle is increased above 9, becoes negaive bu is posiive and he average oupu power ( oupu power) P becoes negaive and he power flows fro he load circui o he inpu source. The operaion occurs in he fourh quadran where is negaive and is posiive. The converer operaes as a line couaed inverer. T DERE AN EXPREN FR THE RM AUE F THE UTPUT TAGE The rs value of he oupu volage is calculaed as v. d RM ( ω) The single phase full converer gives wo oupu volage pulses during he inpu supply ie period and hence he single phase full converer is referred o as a wo pulse converer. The rs oupu volage can be calculaed as + v. d RM ( ω) + sin ωd. RM ( ω) + sin d. ω RM ( ω) ( ω) + cos. d RM ( ω) + + d( ) cos d. RM ω ω ( ω) sin ω ( ω) RM + +

92 ( ) RM sin + sin + RM sin + sin ; sin ( + ) sin sin sin ( ) RM ( ) RM Therefore ( RM ) Hence he rs oupu volage is sae as he rs inpu supply volage The rs hyrisor curren can be calculaed as Each hyrisor conducs for radians or operaing a coninuous and consan load curren. Therefore rs value of he hyrisor curren is calculaed as T RM ( RM ) RM T RM RM 8 in a single phase full converer The average hyrisor curren can be calculaed as T ( Avg ) ( ) ( ) T Avg ( )

93 NGE PHAE DUA CNERTER

94 We have seen in he case of a single phase full converer wih inducive loads he converer can operae in wo differen quadrans in he versus operaing diagra. f wo single phase full converers are conneced in parallel and in opposie direcion (conneced in back o back) across a coon load four quadran operaion is possible. uch a converer is called as a dual converer which is shown in he figure. The dual converer syse will provide four quadran operaion and is norally used in high power indusrial variable speed drives. The converer nuber provides a posiive oupu volage and a posiive load curren, when operaed in he recificaion ode. The converer nuber provides a negaive oupu volage and a negaive load curren when operaed in he recificaion ode. We can hus have bi-direcional load curren and bi-direcional oupu volage. The agniude of oupu load volage and he load curren can be conrolled by varying he rigger angles & of he converers and respecively. Fig.: Four quadran operaion of a dual converer There are wo odes of operaions possible for a dual converer syse. Non circulaing curren ode of operaion (circulaing curren free ode of operaion). Circulaing curren ode of operaion. NN CRCUATNG CURRENT MDE F PERATN (CRCUATNG CURRENT FREE MDE F PERATN) n his ode of operaion only one converer is swiched on a a ie while he second converer is swiched off. When he converer is swiched on and he gae rigger signals are released o he gaes of hyrisors in converer, we ge an average oupu volage across he load, which can be varied by adjusing he rigger angle of he converer. f is less han 9, he converer operaes as a conrolled recifier and convers he inpu ac power ino oupu power o feed he load. and are boh posiive and he operaion occurs in he firs quadran. The average oupu power P is posiive. The power flows fro he inpu ac supply o he load. When is increased above 9 converer operaes as a line couaed inverer and becoes negaive while is posiive and he oupu power P becoes negaive. The power is fed back fro he load circui o he inpu ac source hrough he converer. The load curren falls o zero when he load energy is uilized copleely. The second converer is swiched on afer a sall delay of abou o ill seconds o allow all he hyrisors of converer o urn off copleely. The gae signals