Analysis, design and control of a hybrid multilevel switching converter for synchrotron ring-magnet power

Switching Converter for Synchrotron Ring-Magnet Power

Supplies

by

PAVAN KUM AR M. R.

B .E n g ., Bangalore University, Bcingalore, 1988 M .Tech, Indian In stitu te of Technology, Madras, 1990 A D issertation Subm itted in Partial Fulfillment of the

Requirem ents for the Degree of D O C T O R OF PH ILO SO PH Y

in the

D ep artm en t o f Electrical and C om puter Engineering W e accept this thesis as conform ing

th e required standard

Dr. J. M. S. K im , Supervisor, D ept, o f E lect. &: Comp. Eng.

Dr. A. K. S. B h a t, D epartm ental Member, D ep t, of Elect. & C om p. Eng.

Dr. W . S. Lu, D ep a rtm en ta ^ Ie m b e r, D ep t, o f E lect. <§2 Com p. Eng.

Dr. D . E. Lobb, O utside M ember, D ept, o f Physics

Dr. G. G. Karady, External Examiner, (A rizona State U niversity)

© PAVAN KUM AR M. R ., 1996 U niversity o f V ictoria

A ll rights reserved. T his disserta tio n m ay n ot be re-produced in whole o r in p art by ph o to co p y o r o th er means, -without the p erm ission o f the author.

ABSTRACT

This thesis presents th e developm ent o f a Hybrid M ultilevel Sw itching Converter (H M SC ) for R ing-M agnet Pow er Supplies (R M PS). T h e thesis includes th e an alysis, design and control o f th e proposed converter. It in­ troduces m u ltilev el sw itching converters to th e field of ring-m agnet power supplies.

RM PS feed th e electrom agnets th at produce th e m agnetic field required to energize and g u id e subatom ic particles in a synchrotron. T h e accuracy and extrem e precision o f this m agnetic field im poses stringent restrictions on th e design and perform ance o f th e RM PS used. S tu dy of conventional power supplies highlights th e need for m odern power supply solutions w hich can m eet the specifications o f RM PS.

T he com p lete frequency-dom ain analysis o f th e conventional resonant- ty p e RM PS along w ith th e Energy Make-up U nit (EM U) is presented. T he resonant frequency drift is identified as th e m ain factor in th e design o f th e EM U. T he analysis o f the input filter netw ork is presented for developm en t o f design criteria for input filter com ponents. T h e principle advantages and disadvantages o f th e resonant-type R M PS are sum m arized before id en tifying m u ltilevel converters as a viable option am on g sw itching converters for a non-resonant ty p e o f R M PS.

T he Hybrid M u ltilevel Switching C onverter (HM SC) is proposed as a non-resonant ty p e R M PS to overcom e th e disadvantages o f th e resonant- ty p e RM PS. T h e op eration al features o f th e HM SC are explained and th e sim plification o f th e general HMSC configuration for positive output currents is identified. T h e stea d y -sta te analysis o f th e HM SC develops com prehensive design criteria for th e d evice ratings and com p on en t stresses, including th e m eth od s for reducing th e sw itching losses in th e HM SC. M ultilevel converters

encounter vo lta g e balancing problem am ong th e D C -link capacitors. It is shown th a t th e HM SC configuration is versatile in m in im izin g this problem . H arm onic sp ectru m o f th e o u tp u t voltage o f th e HM SC is derived and th e effect of num ber o f o u tp u t voltage levels in reducing th e harm onic contents is esta b lish ed .

A d etailed survery o f different current control techniques is presented to form th e background for developing an effective current control algorithm for m u ltilev el converters. A dead-beat current control strategy is chosen as an appropriate control tech n iq u e to suit the needs o f R M PS. T he control schem e is ex ten d ed to th e control o f m ultilevel converters in general. T h e control algorithm is d eveloped to track a given arbitrary current reference signal for b o th single-variable and m ulti-variable sy ste m s. It is also shown th a t th e o u tp u t dead-beat control is a special case o f th e pole placem ent technique. T h e transient behaviour of th e sy ste m has been studied and sta b ility considerations of th e system are exam ined.

E x ten sive com puter sim u lation studies have been perform ed using S A B E R to stu d y th e reference tracking nature of the proposed control schem e. T h e o u tp u t current o f th e HMSC using the m odified dead -b eat control schem e is shown to follow a given arbitrary reference w ith very sm all tracking error. T he reference tracking nature has been sim ulated for a sim ple R L m agnet load and a m agn et load w ith L C R filter. E xperim ental results obtained from a laboratory p rototyp e o f th e HM SC w ith an R L load, have been presented to su b sta n tia te th e an alytical resu lts. Criteria for im provem ent in th e reference tracking properties o f th e proposed system have been identified.

Exam iners:

Dr. J. M. S. K im , Supervisor, D ept, o f Elect. & C om p. Eng.

Dr. A. K. S. B h at, D epartm ental M em ber, Dept, of E lect. & C om p. Eng.

Dr. VV. S. Lu, Depaætrnental M em ber, D ept, of E lect. &: C om p. Eng.

Dr. D. E. Lobb, O utside M ember, Dept, o f Physics

____ __________________

T itle Page

i

A bstract

ii

Table of C onten ts

v

List of Tables

ix

List of F igures

x

A cknow ledgem ent

x v

D edication

x v i

1 In trod uction

1

1.1 R ing-M agnet Power S u p p lie s ... 2

1.1.1 Synchrotrons and R M P S ... 2

1.1.2 R equirem ents of R M P S ... 5

1.2 T y p es o f R ing-M agnet Power Supplies ... 6

1.2.1 R esonan t-T yp e Magnet Power S u p p l ie s ... 7

1.2.2 Power Supplies using Phase-Controlled Rectifiers . . . 10

1.3 M otivation for th e T h e s is ... 12

2 A nalysis o f R eson ant-T ype R ing-M agnet Power Supplies

16

2.1 R esonant Circuit and Pulse Forming Network A nalysis . . . . 17

2.2 A nalysis o f Input Filter o f Energy Make-up U n i t ...33

2.3 S w itch in g c o n v e r t e r s ... 45

2.4 C o n c lu s io n s ... 48

3 H ybrid M u lti-lev el Sw itching C onverter

50

3.1.1 N eutral-P oint-C lam ped I n v e r t e r ... 52

3.1.2 G eneralized M ulti-level C o n v e r t e r ... 54

3.1.3 C ascaded M ulti-level C o n v e r t e r ...57

3.2 Hybrid M ultilevel Sw itching C o n v e r t e r ... 59

3.3 Sim plified Hybrid M ultilevel Sw itching C o n v e r t e r ...64

3.4 Stead y S ta te A nalysis o f the H M S C ... 72

3.4.1 S tea d y State Analysis ... 72

3.4.2 Power Circuit Param eters o f th e HMSC ... 78

3.4.3 S w itch in g Stresses ... 81

3.5 V oltage B alancing Problem in M ulti-level C o n v e r t e r s ... 88

3.6 H arm onic A nalysis o f the HM SC ...94

3.6.1 Effect o f Number of O u tp u t Voltage Levels on the Har­ m o n ic S p e c tr u m 98 3.7 C o n c lu s io n s ...105

4

Current C ontrol of th e H M SC

106

4.2 M odified D ead-B eat Control T e c h n iq u e ...113

4 .2 .2 M odified D ead -B eat Current Control for M u lti-level

C o n v e r te r s ... 123

4 .2 .3 M odified D ead -B eat Control for M ultiple S ta te V ariables 134 4.3 P o le P lacem en t for Feedback C o n t r o l ... 138

4 .3 .1 M odified D ead -B eat Control as an O utput D ead -B eat C ontrol L a w ...142

4 .3 .2 O p tim al P ole P lacem en t T e c h n iq u e ...146

4.4 Effect o f S w itching Frequency on Tracking E r r o r ...147

4.5 T ransient A nalysis o f th e H M S C ...151

4.5.1 S ystem A n a ly sis for Transient Behaviour and S ta b ility 152 4.6 O bservations an d C o n c lu s io n s ... 157

5

C o m p u ter Sim u lation s and Experim ental R esu lts

158

5.1.1 S ystem P a r a m e t e r s ...159

5 .1 .2 C om puter S im u lation o f HMSC using S A B E R 160 5.2 E xp erim en tal R e s u l t s ... 185

5.2.1 Laboratory P rototyp e Power Circuit D e s i g n ... 187

5 .2 .2 Control C ircuit D e s i g n ... 192

5 .2 .3 Hardware and Software Considerations for Im prove­ m en t in R eference T r a c k i n g ... 198

5.3 O bservations and C o n c lu s io n s ... 203

6

C on clu sion s

204

6.2 C ontrib u tion s o f th e T hesis ...207

B ib lio g r a p h y 2 1 0

A S A B E R S im u la t io n a n d I n te r fa c e D e t a ils 2 2 1 A .l External C R outine for Modified D ead-B eat C o n t r o l...223

B A D C & D S P T i m e r I n te r f a c e 2 2 9

C D S P C o n t r o lle r S o f t w a r e 2 3 4

C .l R eduction in M ultiplications in DSP s o f t w a r e ...234 C.2 A ssem bly Level Program L is tin g ... 236

List o f T ables

1.1 T y p ical Specifications for a R M P S ... 7

3.1 S w itch in g S tates of the Sim plified H M S C ... 68

3.2 N um ber o f Switching S t a t e s ... 70

3.3 S w itch in g Transition Table: Higher to Lower State ... 84

3.4 S w itch in g Transition Table: Lower to Higher State ... 85

1.1 T ypical dc-B iased ac Current E xcitation s for Ring-M agnets . . 4

1.2 R esonant-type Ring-M agnet Power S u p p l y ... 8

1.3 Non-resonant RM PS w ith Phase-C ontrolled R e c tifie r s ... 11

2.1 R esonant-type R ing-M agnet Power S u p p l y ... 18

2.2 Circuit Diagram o f Single Resonant C e l l ... 19

2.3 Current Source M odelling o f Pulse Form ing N e t w o r k ... 22

2.4 Variation of Pulse Current and its Fundam ental Com ponent . 27 2.5 Bode Plot o f Current Gain A n / / p ... 31

2.6 Peak Pulse Current in Per U nit (P U ) o f Peak Magnet Current as a function o f Resonant Frequency ...32

2.7 Input Filter w ith Pulsed Power S u p p l y ... 34

2.8 Inductor Current and C apacitor V oltage W aveforms for C yclic O p e r a t i o n ... 34

2.9 R ipple Voltage in Per Unit (P U ) o f P eak C apacitor Voltage as a Function o f PU C a p a c it a n c e ... 40

2.10 Filter Harmonic Current in PU o f Peak M agnet Current as a Function of Filter I n d u c t a n c e ... 44

3.2 (a) Full Bridge N P C Inverter, (b) Sw itching S ta tes. S w itches S2, S3, S6 and S 7 are operated in a com plem entary m anner to sw itch es S4, S i , S8 and S5 r e s p e c t iv e ly ...5-5

3.3 G eneralized M u ltilev el C o n v e r t e r ...56

3.4 (a )C ascad ed Full-B ridge Inverter (b) Sw itching S t a t e s ... 58

60 66 83 87 3.5 Hybrid M u ltilevel S w itching Converter: C ircuit D iagram . . 3.6 Sim plified Hybrid M u ltilevel Switching Converter for R M PS 3.7 E xam p le o f S w itch in g Pattern to Reduce S w itching Losses 3.8 S w itch in g T ransition E xam ple to Reduce Sw itching Losses 3.9 E xam p le o f E q u alization o f Charge am ong Input C apacitors by Prudent C hoice o f Sw itching Pattern: a 1 in d icates charge drawn from a sou rce where as 0 indicates th e source discon­ n ected from th e l o a d ... 93

3.10 Sw itch in g Instants in each Sam pling I n te r v a l... 95

3.11 Variation o f F undam ental and 399th H a r m o n ic... 102

3.12 O u tp u t V oltage H arm onic Spectrum for Varying Load Cur­ rents: Load C urrent is (a) 0.1 pu and (b) 0.25 pu o f peaJc value...103

3.13 O u tp u t V oltage H arm onic Spectrum for Varying Load Cur­ rents: Load C urrent is (a) 0.5 pu and (b) 0.75 pu o f peak value...104

4.1 S tate-S p ace R ep resen tation o f a S y s te m ... 108

4.2 B lock Diagram o f a PID C o n t r o lle r ...108

4.3 Block Diagram o f F in ite T im e Settling C o n tr o l...110

4.4 Block Diagram o f F T S C w ith Model Reference A d ap tive Schem e ( M R A S ) ... 110

4.5 S ta te M odel o f th e M agnet Load ... 115

4.6 Piecew ise C ontinuous Input in any Sam pling I n t e r v a l ... 117

4.7 M agnet Load fed by M ulti-level Input F u n c t i o n ... 124

4.8 M odified D ead -B eat Control S t r a t e g y ...126

4.9 Circuit D iagram o f a M agnet Load with O utput F ilter . . . . 135

4.10 Pole-Zero P lo t for a O pen Loop S y s te m ... 140

4.11 Pole-Zero D iagram o f th e Closed-Loop S y s t e m ...142

4.12 Pole-Zero P lo t o f (a) O pen Loop System (b) Closed-Loop Sys­ tem w ith O u tp u t D ead-B eat C o n t r o l ... 145

4.13 Pole-Zero D iagram for O p tim al Pole A ssignm ent (a) when is sm all (b ) w hen u;„ is l a r g e ...148

4.14 Effect o f S w itch in g Frequency on Tracking E r r o r ... 150

4.15 Control B lock D iagram o f th e S y s t e m ... 153

5.1 Block D iagram o f HM SC w ith Control for Reference Tracking using S A B E R ... 162

5.2 Circuit D iagram of Single Bridge of HMSC used in S A B E R S im ulation. T he sw itch m od el is also s h o w n ...164

5.3 T he Control C ircuit Diagram o f the HMSC used in S A B E R Sim ulation. T h e use o f a N on-linear Block to Interface W ith an E xternal C R outin e is show n... 165

5.4 D C -B iased Sinusoidal Reference Tracking using S A B E R . . . . 169

5.5 D C -B iased Sinusoidal Reference: Tracking E r r o r ...170

5.6 D C -B iased Sinusoidal Reference: O utput V o l t a g e ... 171

5.7 D C -B iased Triangular Reference Tracking using S A B E R . . . 173

5.8 D C -B iased Triangular Reference: Tracking E r ro r ... 174

5.10 D C -B iased Sinusoidal Reference Tracking with, an LCR filter . 178

5.11 Tracking Error w ith .an LCR f il t e r ...179

5.12 O utput V oltage o f th e HMSC with an LCR F i l t e r ...179

5.13 Load (M agn et) V oltage with an LCR f i l t e r ... 180

5.14 O utput Current o f the HMSC with an LCR F i l t e r ...180

5.15 Input C apacitor Voltages for Arbitrary S w itch in g Pattern . . . 182

5.16 Input C apacitor V oltages using Sw itching Pattern to Reduce V oltage U n b a l a n c e ... 182

5.17 H arm onic Sp ectru m o f the Output V o lt a g e ... 184

5.18 Harm onic Spectru m o f the Magnet C u r r e n t...186

5.19 Circuit D iagram for Laboratory P rototype o f HMSC ... 189

5.20 Block D iagram o f D SP Interface to H M S C ... 192

5.21 Circuit D iagram o f the A D C /T im er Interface C a r d ...194

5.22 Flow Chart for th e Im plem entation o f M odified D ead-Beat C ontrol ... 195

5.23 T im in g Diagram for One Cycle Operation o f D SP Controller . 197 5.24 D C -B iased Sinusoidal Output Current: E xperim ental Wave­ form. H orizontal Axis: 5 ms per div, Vertical Axis: 1 Amp per div, Origin is at th e bottom left corner... 199

5.25 O u tp u t V oltage for Sinusoidal Reference: E xperim ental Wave­ form. H orizontal Axis: 5 ms per div, Vertical Axis: 20 V per d iv. Origin is in th e center... 199

5.26 D C -B iased Triangular O utput Current: E xperim ental Wave­ form. H orizontal Axis: 5 ms per div, Vertical Axis: 1 Amp per d iv. Origin is a t the bottom left corner... 200

5.27 O u tp u t V oltage for Triangular Reference: E xperim ental Wave­ form . H orizontal Axis: 5 ms per div, Vertical Axis: 20 V per d iv , Origin is in the cen ter...200

A .l E x a m p le o f a Sim ple E xternal C Routine Interface to S A B E R 222

B .l Pin ou t D etails and Circuit Connections of the ADC ... 230 B .2 8254 T im er Connections to th e D SP ... 231 B.3 PAL C onnections and M em ory Map o f Tim er and A D C . . . . 232 B.4 C lock C ircuit C onnections to T im er/A D C C a r d ...233 B.5 Feedback Signal Buffer D e t a i l s ...233

A ck n o w led g m en t

I am greatly in d eb ted to m y supervisor Dr. J. M. S. Kim for his encour­ agem en t, ad vice and p a tien ce during the course o f this research and for his invaluable su ggestion s for th e preparation o f this m anuscript. He was alw ays ready to discuss a ll sorts o f problem s, work or otherw ise and h elp ed m e to solve th em .

I would also like to thank m y supervisory com m ittee m em bers for their suggestions during th e course o f th is research and preparation o f th is th esis.

Financial assita n ce received from th e U niversity o f V ictoria, in th e form o f U niversity Fellow hsip, and research assistantship from Dr. K im is gratefully acknowledged.

1 gratefully ack now ledge th e help received from the tech n ical support staff o f the D ep artm en t o f E lectrical Engineering during th e course o f th is research. Special m ention n eed s to be m ade o f Kevin Jones, Roger K elly and S teve C am pbell for th eir invaluable support. T he support o f Mr. YongRui Feng during th e ex p erim en ta l phase o f this project is gratefully acknow ledged.

T h e encouragem ent received from m any friends during m y sta y in C anada m erits special m en tio n and thanks. Especially A nand and V ivek were a constant source o f support.

I would like to sp ecia lly thank m y parents who have been a gu id in g light for all m y endeavours. F inally I would like to thank m y dear w ife Par for her unending su p p ort, encouragem ent and sacrifice during th e course o f this thesis.

W ith Love To m y parents

In tr o d u c tio n

T his thesis describes th e developm ent of a Hybrid M ultilevel Sw itching C on­ verter (HM SC) for R ing-M agnet Power Supplies (R M PS) used in synchrotrons. E xtrem ely accurate and precise m agn etic field is required in a synchrotron ring for guiding and energizing subatom ic particles. This m agnetic field is provided by a series o f electrom agnets distributed along the synchrotron ring and dem ands stringent specifications on the design and performance o f power supplies feeding the m agnets.

C onventionally Ring-M agnet Power Supplies are designed using distributed resonant networks w ith dc-bias power supplies [1] or phase-controlled recti­ fiers [2,3]. These power supply configurations satisfy the stead y-state per­ form ance criteria using large reactive com ponents in addition to the m agnet load. T hey have lim ited dynam ic response and often rely on corrector-m agnet power supplies or other auxiliary power supply networks for the fast d ynam ic com pensation required for output regulation and reference tracking. This thesis applies modern sw itching converters to the area o f ring-magnet power supplies to achieve improved perform ance in term s o f fcist dynam ic response, good reference tracking capability and low ou tp ut current ripple contents.

1.) T h e frequency-dom ain analysis of the resonant-type ring-m agnet power supply and th e a ssociated energy make-up u n it.

2.) T h e developm en t o f a non-resonant type Hybrid M ultilevel Sw itch­ ing Converter for R M PS, including th e o p tim ization of th e converter configuration.

3.) T h e analysis and design of th e proposed HM SC, including the steady- s ta te and tran sien t analysis.

4.) T h e form ulation o f a m odified dead-beat control algorithm for current control in m u ltilev el converters in general, and th e HMSC in particular.

5.) T h e im p lem en tation o f the modified dead-beat control technique using a fast D igital Signal Processor (D SP ).

1.1

R in g -M a g n e t Pow er S u p p lies

This section describes th e role o f RMPS in a synchrotron, to identify their im portant requirem ents. It also exam ines the presently used power supply configurations and th eir perform ance, and is concluded w ith the advantages and feasib ility of a sw itch in g power supply configuration for R M PS.

1.1.1

S y n c h r o tro n s and R M P S

Synchrotrons accelerate a beam of subatom ic particles in a m agnetic guiding field. T h e m agn etic gu id in g field strength or pattern should rise gradually

ticle beam is injected into th e synchrotron at the low value of the field and ex tra cted from the synchrotron at the peak value. A fter the extraction , th e field p attern is reduced to its in itial low value and th e cycle is repeated. T his cy cle tim e determ ines the o p eratin g frequency o f th e synchrotron. T h e oper­ atin g frequency and th e variation o f th e m agn etic field pattern are d ep en d en t on th e typ e o f su batom ic p articles accelerated. T h e repetition rates or th e operating frequency varies w id ely from 0.1 Hz to 50 Hz. Synchrotrons w ith operating frequencies less th an 10 Hz are classified under slow -cycling syn ­ chrotrons where as th ose w ith operating frequencies above 10 Hz are grouped under fast- or rapid-cycling synchrotrons.

T h e m agnetic guiding field pattern required in a synchrotron can b e g en ­ erated by excitin g a series o f electrom agn ets, d istrib u ted around th e syn ­ chrotron ring, with a direct current (d c)-b iased alternating current (ac) e x c i­ ta tio n . Ring-M agnet Power S upplies provide the required excitation current to th e electrom agnets. T h e power supplies are called Ring-M agnet Power Supplies since th ey feed th e electrom agn ets arranged in a ring. T he e le ctri­ cal equivalent circuit o f a synchrotron is a group o f inductances (representing th e electrom agn ets) arranged in th e form o f a ring and fed by a power su p p ly unit.

T here are various typ es o f dc-biased ac ex cita tio n s o f the ring-m agnets used in synchrotrons. C om m only used ex cita tio n s are dc-biased sinuosidal ex cita tio n , dc-biased sin u soid al w ith dual frequency [4], triangular a n d /o r trapezoidal w ith flat to p /b o tto m [5]. Som e exam p les o f ring-m agnet e x c i­ tation currents are illu strated in Fig. 1.1. T he B ooster Ring RMPS a t the T R IU M F KAON factory proposes to use a dc-biased sinusoidal ex cita tio n

2850 A

1200 A

lOms

20ms

(a)

5553 A

3110 A

6 67A

75 ms

_{10000 A}

1040 A

(ms)

50

150

25

(c)

Ring dipole m agn ets are to be ex cited w ith a dual-frequency current wave shape w ith a 10 Hz repetition rate as shown in Fig. 1.1(b) [6,7]. In contrast the KAON accelerator at Los Alam os uses a flat-topped trapezoidal current excitation as shown in Fig. 1.1(c) [8,9]. Thus the power supply configuration used in ex c itin g th e ring-m agnets should be able to generate th e required current wave shape.

1.1.2

R e q u ir e m e n ts o f R M P S

RMPS com e under th e high-voltage high-current power supply category. They have to provide a gradual increase of the excitation current from a specified low level to a peak value. A lthough the rise is gradual and could be over a long period o f tim e, the precise nature of the m agnetic guiding field needed im poses strict specifications on th e performance of the RMPS. T he most im portant requirem ents o f RMPS can be listed under:

1.) Low o u tp u t current tracking error

T he m a g n etic guiding field strength has to be uniform and precise. Any d eviation in th e output current from the specified reference leads to variations in th e guide field. Thus the output current should precisely track a given reference signal w ith a very low tracking error. Typically th e tracking error is specified to b e less than 500 parts per m illion (p p m ).

2.) Low o u tp u t current ripple content

Ripple in th e ou tp ut current feeding the m agnets results in variations in the m a g n etic guiding field which is not desirable. Thus an output

m illion (p p m ). T h e gradual increase in the field is necessary since sudden variations in the field leads to eddy current disturbances in the electro m a g n ets and this in turn reflects as variations in the guiding field.

3.) O utput current regulation

T h e term regulation in this case refers to reproducibility. T he particle beam has to be accelerated and guided consistently in successive cycles. T h e variation in th e output current betw een successive cycles should be as low as 200 ppm .

4.) Fast d y n a m ic response

T h e tracking error and the o u tp u t current ripple contents can be m ain­ tain ed at a low level if the sy stem has a fast dynam ic réponse. The supply can respond quickly to changes in the reference and m aintain good tracking capability. In other words th e power supply should have a wide b andw idth.

T he sp ecification s for th e Injector synchrotron at th e Argonne N ational Lab­ oratory (A N L ), Illinois, are listed in Table 1.1 as an exam p le [2]. T his illus­ trates th e strict nature o f the requirem ents involved in RM PS.

1.2

T y p e s o f R in g -M a g n et P ow er S u p p lies

The stea d y -sta te requirem ents of R M PS as described in the previous section are at present served by two major typ es o f power supplies. T h ey can be categorized under:

Param eters R equirem ents

R ipple con ten ts ± 1 X 10~4 ( _ 100 ppm ) R egu lation (R eproducibility) ± 2 X 10"* ( = 200 ppm ) Tracking error ± 5 X 10"* ( = 500 ppm ) Injection current 61 A E x traction current 1044 A Injection voltage 42, 1140 V E xtraction voltage 724, 1822 V R eset voltage -373, -1055 V A cceleration tim e 250 ms R eset tim e 250 ms O p eratin g frequency 2 Hz (T = 500 ms)

1.) R esonan t-T yp e Power Supplies

2.) Power Supplies using Phase-Controlled Rectifiers.

1.2.1

R e s o n a n t-T y p e M a g n et P o w e r S u p p lies

R esonan t-typ e m a g n et power supplies use inductance-capacitance (LC) n et­ works, tuned to th e accelerator operating frequency, to generate th e sinu­ soid al current w aveshape. T h e dc bias current is supplied by a separate dc power supply co n n ected in series w ith th e ring m agnets. A typical resonant- ty p e m agnet power supply is shown in Fig 1.2. It is based on the principle o f d istrib u ted resonance netw ork [I]. T he operating frequency of the network is determ ined by th e m agnet inductance {Lm) and th e capacitance (C ). How­ ever due to th e dc-biased ac nature o f th e ex cita tio n current, a path for the dc-bias current is needed. This is provided by th e bypass choke (Lch)- A

Lm Lm VDC 2C 2C C EMa Bypass cboka Ring Magnats

Rasonant capacitor bank Enargy Maka-up (hit

Figure 1.2: R eson an t-typ e Ring-M agnet Power Supply

d istributed resonant netw ork, instead of a single L C circuit, is used in order to lim it th e peak voltage around th e synchrotron ring.

Under ideal conditions th e resonant network can provide ripple free dc- biased ac excitation once a p u lse o f energy is introduced into th e resonant tank. T h e energy transfers back and forth between th e inductance and the capacitance. However there are ac losses in the resonant network associated w ith non-ideal resistive effects. Thus energy has to be m ade up in order to m aintain a constant a m p litu d e o f ac excitation in the network. This is

losses can b e fed through coupled auxiliary windings on the bypass choke. The pulse o f energy, equal to the cyclic ac power loss, is introduced either during th e ascending or descending portion of the m agnet current waveform through th e auxiliary winding on th e bypass choke. T his pulse o f energy is produced by a pulsed power supply. T he bypeiss choke and the pulsed power su pp ly together from the EM U. T he pulsed power supply consists o f a rectifier, a L C filter and a switch which transfers power to the bypass choke 16].

A variation o f th e distributed resonant RM PS has been proposed by Praeg et al. [4,5,9-11], as th e wave-shaped resonant RM PS. T his is also term ed as the D ual-Frequency resonant RM PS. T he idea is to use L C networks and switches to o b ta in different resonant frequencies w ithin one cycle. This con­ figuration can produce dual-frequency excitation current shown in Fig. 1.1(b). T h e reson an t-type of ring-m agnet power supplies can provide a ripple- free e x cita tio n current to the m agnets and can be used for fast-cycling syn­ chrotrons w ith a typical acceleration frequency of 50 Hz. However, the resonant-type m agnet power supplies require a large resonant capacitor bank, an extra d c power supply for the dc bias current, dc bypass chokes for pro­ viding a p a th for th e dc bias current. T h e dynam ic response of the supply is slow sin ce th e pulsed power supply in the EMU has a large filter at its output. T h e effects o f the resonant frequency drifts on th e operation of the resonant R M PS is very dram atic due to the high Q factor of the network [12,13]. Secondary effects such as tem perature leads to variations in system param eters, w hich in turn results in resonant frequency drifts, altering the operating con d ition s o f the power supply system .

1,2.2

P o w e r S u p p lie s u sin g P h a s e -C o n tr o lle d R e c ti­

fiers

T h e other co m m on ly used power supply configuration in th e area o f ring- m agn et power supplies are the phase-controlled rectifiers. T h ese supplies are th e non-resonant ty p e. T h e supply usually consists o f a sp lit 24-phase ac be­ ing rectified and filtered to obtain the ou tp ut voltage. T h e circuit diagram of a ty p ica l p hase-controlled supply for RM PS is shown in Fig. 1.3. Fathizadeh et al [2,3,14] have show n that a dc-biased triangular waveform can be gener­ ated w ith high accuracy w ith the help of phase-controlled rectifiers feeding th e ring-m agnets. T h e advantage of such supplies is th a t any desired current w aveshape can b e gen erated with a proper reference w aveform . Furthermore a separate dc power su pp ly is not required and the power supply operation is in d ep en d en t of th e load circuit variations. T his results in th e elim ination of th e effects o f load param eter variations, unlike the case o f resonant-type supplies.

T here are certain disadvantages associated w ith the phase-controlled rec­ tifier su pp lies. T heir bandw idth is lim ited and their d y n a m ic response is slow. T h e phase-controlled rectifiers axe usually con n ected to split 24-phase or sp lit 48-phase 60 Hz u tility interface and in turn generate an o u tp u t ripple frequency o f 1440Hz or 2880Hz respectively. T he o u tp u t filter w ill have a ban dw id th w hich is m uch less than this ripple frequency in order to reduce th e o u tp u t ripple con ten ts. Hence th e power supply b andw idth is inherently lim ited by th e filter size. Thus these supplies are m ainly used in slow -cycling synchrotrons w ith operatin g frequency o f less than 10 Hz. M ultiphase con­ trolled rectifiers can b e used for som e m arginal im provem ent in the dynam ic response and reduction in the o u tp u t filter size w ithout increasing th e

out-MAGNET LOAD 3 phase AC

Phase Controlled Rectifier

(a) Lm PCR P C R Lm Lm

(b)

PCR _____ Phase Controlled Rectifier

put ripple con ten t. Due to th e slow dynam ic response, th e m agnetic field provided by th e phase-controlled rectifiers can be easily d istorted by distur­ bances such as u tility fluctuations [15,16].

1.3

M o tiv a tio n for th e T h esis

T he stu d y o f existin g R ing-M agnet Power Supplies highlights th a t the need for dc-biased ac excitation for ring-m agnets is at present b ein g satisfied by reson an t-type power supplies for fast-cycling synchrotrons and by phase- controlled rectifiers for slow -cyclin g synchrotrons. In both cases the power supplies have poor dynam ic characteristics. The following desirable features o f any R M PS can be deduced from th e survey:

1.) A dc-biased arbitrary current excitation output.

2.) Fcist d yn am ic response for disturbance rejection and reference tracking.

3.) Low o u tp u t current ripple con ten ts or provision for ea sy elim ination o f ripple contents with a reduced ou tp u t filter.

T h e m otivation for th is th esis stem s from the fact th at there is a need for a ring-m agnet power supply configuration which caters to b oth slow and rapid-cycling synchrotrons. T h e power supply should have a faist dynam ic response, low o u tp u t ripple con ten ts and it should have a current programma­ b ility feature to generate th e different current waveshapes th a t are required. In ad d ition , any proposed power su pp ly configuration has to overcom e the drawbacks o f th e resonant-type R M P S and phase-controlled R M PS.

T h e application of sw itching converters in the area o f high-perform ance m agn et power supplies, in genered, has so far been lim ited. W ith the advances

in power electronics and power sem iconductor technology, it is now possible to apply sw itching converters to achieve the strict specifications o f th ese m agnet power supplies. Furthermore, th e application o f modern sw itching converters in this area can improve the power supply perform ance such as dynam ic response and o u tp u t ripple content. Much work is needed in term s of analysis, design and control o f modern switching converters, to introduce them to th e area of m agn et power supplies. T his thesis is an a tte m p t to apply m odern sw itching converters to the area o f m agnet power su pp lies to achieve b etter perform ance.

1.4

O rg a n iza tio n o f th e T h esis

The th esis has been divided into six chapters. The ou tlin e of th ese chapters are as follows:

Chapter 2 is devoted to th e analysis o f the R esonant-Type R ing-M agnet Power Supplies. The analysis is categorized under:

1.) Frequency dom ain analysis of the resonant-type RMPS and th e energy m ake-up unit associated w ith it.

2.) T h e effect o f resonant frequency drift on the peak current a n d /o rv o lta g e stress on the pulse form ing network.

3.) T h e effect o f param eter values of the input filter com ponents on th e ripple voltage and harm onic currents on the input side o f th e energy m ake-up network.

Chapter 2 also sum m arizes the advantages and disadvantages of th e resonant- typ e R M PS and introduces switching converters as an alternative solution.

M ultilevel converters are identified as a viable option am ong sw itch in g con­ verters for non-resonant R M PS.

C hapter 3 presen ts th e Hybrid M ultilevel Switching C onverter (HMSC) as a R ing-M agnet Pow er Supply. T he high-voltage high-current nature and other stringent sp ecification s o f RM PS narrow the choice of sw itching con­ verters to one area, n a m ely m ultilevel converters. T h e m ain ob jectives of this chapter are:

1.) To survey different m u ltilevel converter structures to find an appropri­ ate converter stru ctu re to serve as RM PS.

2.) To perform th e general stead y-state analysis of the proposed converter for com p on en t stresses.

3.) To form ulate design principles for the proposed converter.

The sim plified HM SC is derived from the general HMSC configuration. The general stea d y -sta te analysis is performed to determ ine the com p on en t stresses. A general design o f th e converter for a selected reference signal is presented. The voltage balan cin g problem , com m only encountered in m u ltilev el struc­ tures is discussed and m eans to overcom e or m inim ize this problem is ou t­ lined. Harm onic an alysis o f th e ou tp ut quantities of th e HMSC is presented. Chapter 4 delves in to th e control aspects o f the proposed converter. The main ob jectives o f th is chapter are:

1.) To form ulate an effective control technique to acheive th e strict speci­ fications o f R M PS.

2.) To analyze th e power supply system operating w ith th e proposed con­ trol schem e for sta b ility under normal and disturbed conditions.

An ex te n siv e sum m ary o f th e different current control techniques is pre­ sen ted . T h e o u tp u t dead-beat control schem e (also known as the in sta n ta ­ neous p red ictive control schem e) is selected as a suitable control tech n iq u e. T h e proposed control schem e is ex ten d ed to su it m u ltilevel converters in g en ­ eral. T h e con trol algorithm is show n to be valid for b oth single and m u ltip le variable sy ste m s. It is shown th a t th e proposed control algorithm is su ited to a ccu ra tely track a given reference signal. T he m odified dead-beat control a lgorith m is stu d ied as a sp ecial case o f the pole placem ent technique. T h e tran sien t behaviour of th e sy stem under norm al and disturbed conditions is stu d ied . It is shown th at th e sy ste m is stable under disturbed circum stances.

C hapter 5 presents the com puter sim ulation and experim ental results peform ed on th e HMSC working in conjunction with a modified d ead -b eat control algorith m . Com puter sim u lation results obtained using S A B E R axe presented to su bstan tiate th e reference tracking capability of the m odified d ead -b eat control technique. E x ten sive m odelling details o f the power su p p ly sy ste m are presented. T he fast d ig ita l nature o f the com pensation required in th is case has been m od elled effectively using SA B E R . Sim ulation results for a sim p le R L m agnet load and a m agnet load w ith R L C filter are pre­ sen ted . E xperim ental results ob ta in ed on a laboratory prototype m od el are also presented to verify th e reference tracking nature o f th e proposed control tech n iq u e.

Finally, C hapter 6 lists th e im p ortan t contributions o f this thesis work and identifies th e future challenges th a t need to be addressed.

C h a p te r 2

A n a ly s is o f R e so n a n t-T y p e

R in g -M a g n e t P o w er S u p p lies

This chapter presents the analysis o f th e resonant type Ring-M agnet Power Supplies and describes the effect o f param eter variations on th e perform ance o f the pulse form ing network. T h e frequency-dom ain analysis o f the resonant- typ e R ing-M agnet Power Supply w ith th e associated pulse-form ing network is presented in Section 2.1. T he effect o f resonant-frequency variation on the pulse currents o f th e energy maJce-up unit is also discussed. T h e analysis of th e input filter o f th e energy m ake-up unit is presented in Section 2.2, where th e effect of ripple voltage on th e filter capacitor and input harm onic currents through th e filter inductance is studied. T h e foundation for th e developm ent o f non-resonant ty p e ring-m agnet power supplies using sw itching converters is established in S ection 2.3, including th e feasibility and advantages o f using sw itching converters in m agnet power supply area and th e advantages of m ulti-level converters to m eet th e specifications. T he principle conclusions are sum m arized in Section 2.4.

2.1

R eson an t C ircu it and P u lse F orm ing N e t ­

work A n alysis

T his section describes the resonant typ e ring-m agnet power supply and the energy make-up unit associated w ith it. T he principle o f operation o f the system is described and the analysis of th e system is carried out to determ ine the critical factors that affect th e perform ance o f the energy make-up unit.

T h e resonant typ e R ing-M agnet Power Supply (R M P S) is based on the principle of distributed resonance network proposed by J. Fox [1]. T he circuit diagram of a typical resonant ty p e RM PS shown in Fig. 1.2 is repeated here as Fig. 2.1. T he electrom agnets represented by m agnet inductances Lm are arranged along th e periphery o f a circle interspersed w ith a dc choke Lch and a capacitor C to form th e resonant network. The dc choke is provided such th at the dc bias current finds a path and also to introduce a pulse of energy to make up for the losses dissociated w ith th e network. T he distributed resonant network instead o f a single L C circuit is used in order to lim it th e peaJc voltage around the synchrotron ring.

T h e ac losses in the resonant netw ork has to be made up in each cycle in order to maintain the constant am p litu d e of ac excitation required for proper operation. This is achieved by th e Energy MaJce-up Unit (E M U ). T he power to m ake up the cyclic losses is fed through coupled auxiliary windings on the dc choke. A pulse of energy is introduced either during th e ascending or descending portion of the m agnet current waveform. This pulse of energy is produced by a pulsed power supply unit. T h e pulsed power supply unit and the dc choke together form the EM U.

Lm

Lm

Lm

2C

2C

C

Resonémt capacitor bank Energy Make-up Unit

Rf/2

.

chp ^ h s

Figure 2.2: Circuit D iagram of Single Resonant Cell

due to secon d ary effects like tem p eratu re variation, on the perform ance o f the resonant ty p e RM PS have not been analyzed. T he allowable tolerance in the sp ecification s o f th e choke and its effect on th e pulse power su p p ly have b een sim u la ted w ith the help o f SPIC E [12,1.3,17,18] and listed in th e literature. However circuit analysis is not available in the literature to stu d y the effect o f param eter variations on the performance o f the sy stem . T h e changes in param eter values in th e d c choke and resonant capcitor can g iv e rise to resonant frequency drift. T h e effect of resonant frequency drift on th e operation o f th e energy make-up u n it is also not available in th e literature. A d etailed frequency-dom ain an alysis o f th e resonant network along w ith the pulsed power supply is presented for b etter understanding o f the d esign considerations for th e energy m ake-up unit.

T h e circu it diagram of the p ulsed power supply unit, th e dc choke and a single cell equivalent o f the resonant network is shown in Fig. 2.2. T h e non-linear in d u ctan ce o f the dc choke can be m odelled by a dependent cur­ rent source w hich is a function o f th e choke current. T he nom inal design param eters for a single cell are listed from th e Accelerator Design Report [6j;

D ip ole M agnet Current

m axim u m current : 4500 A m inim um current : 1200 A

RMS current (ac) : 1167 A (1650 A peak) dc bias current : 2850 A

R esonant N etw ork

D ipole m a g n ets : Lm = 25.0 m H per cell Copper Loss : = 12.5 mLl per cell Coreloss : Rm = 3325 Çl per cell

C apacitor bank : Cres = S27fiF per cell R esonant Frequency : = 49.5 H z A ccelerator Frequency : fa = 50.0 H z

E nergy M ake-up Network

Input F ilter Inductor : L p = 0.4 H

Input F ilter C apacitor : C p = 400 fiF { f a / f p = 4) Pulse Inductor : Lp = 3.25 m H

Frequency o f P u lse Form ing Network : /p = 150 H z

Energy Storage Choke

Turns ra tio : n = 1:3

C oupling coefficient : kch = 0.99

N onlinear Indu ctan ce = 1% of full scale

Primary W in d in g (energy make-up network side): Self In d u ctan ce : Lchp = 2.778 m H per cell

Copper Loss : = 2.31 m fi per cell Secondary W inding (resonant network side): Self Inductance : Lchs = 25 m H per cell Copper Loss : R^hs = 20.8 m f l per cell Core Loss : R^h = 2800 ÎÎ

T he energy m ake-up network consists o f the input filter stage formed by

Lp and Cf, th e pulse forming stage represented by th e sw itch and the pulse

inductor Lp and th e energy storage choke Lch- T h e energy from the pulse forming network is injected into the energy storage choke when the switch is closed. A discontinuous current pulse transfers th e energy stored in the input filter capacitor to th e choke. T he current pulse rep etition rate is determined by the accelerator operating frequency (w^). T hus the fundam ental frequency o f the pulse current is also Wa- However, this need not necessarily be the resonant frequency (wg) of the resonant network.

Since th e pulse frequency and the pulse w aveshape is fixed by the pulse form ing network, the pulse waveform is well defined and its frequency com­ ponents can be determ ined. In other words the pulse form ing network may be m od elled by a dependent current source as show n in Fig. 2.3. Thus the system can be considered as a current source w hose output changes accord­ ing to the desired level of m agnet current. T he interm ediate network can be considered as a gain stage. The resonant network am plifies the funda­ m ental com ponent o f the pulse current and atten u a tes all other harmonic com ponents. T he m agnet current is an am plified version o f the fundamen­ tal pulse current which is sinusoidally varying at th e required frequency Wg. The system has been analyzed in the frequency dom ain to obtain a transfer characteristic betw een the m agnet current and th e pulse current. The circuit

Rf/2

^chp ^ chs

Figure 2.3: Current Source M odelling o f Pulse Forming Network param eters as listed above have been used.

A ssu m in g th a t th e saturation of the energy storage choke is negligibly sm all, we have;

(2 . 1) (

2

2

ichs — ^C3 — ^cr "h

(2.3)

Also th e laplace-transform ed voltage across th e m agnet load can be expressed as

(2.4)

where L mj is th e equivalent inductance o f th e parallel com bination of Lt

and Lchs given by

f'm/ — ^m\\[^cha — L-m Lchs

T he choke resistan ce Rchs is sm all and is neglected. Therefore I (2,6) L l “ m r S Lmf ) J = / m ( s ) F i ( s ) (2.7) where r- / ^ s m R m + s L m f ( R s m + R m )

’

The o u tp u t voltage is also equal to

K n (i) = f « r ( s ) ^ 4 . + (2.9)

= U s ) { + " } (2.10)

)

Hence

S u bstitu tin g for K n(s) from Eqn. (2.7) we have

T he transform ed expression for th e choke current can be w ritten from Eqn. (2.3) as

which can be rewritten by using Eqn. (2.12) as

= f’2 i s ) l m { s ) (2.16)

where

The prim ary side choke current i'^ is th e reflected secondary current and is given by th e expression

ip = nichs (2.18)

where n is th e turns ratio and therefore

Ip{s) = nichsis) (2.19)

= nF2{s)Im{s) (2.20)

The secondary side choke voltage Vch is given by th e expression

K:a(5) = icsis) R c h s + V m { s ) (2.21) Using Eqns. (2.16) &: (2 .7 ) in Eqn. (2.21) we have

Vch{s) = RchsF2{ s )I m{ s ) + Fi {s)Im{s) (2.22) = [RchsF2{s) + Fi{s)] Im{s) (2.23)

The prim ary side voltage is given by

(2.24)

= ~[RchsF2{s) + F i { s ) ] L i s ) (2.25)

T h e coreloss com p on en t of the choke current Ipr can now b e determ ined as

W = ^ (2.26)

^ [RchsF2{s) + F , { s ) ] U s ) (2.27)

f^Rch

T he pulse current ip is the sum o f the reflected secondary side current and th e coreloss com ponent and hence

fp = I p + fp r (2.28) 4 ( 4 = 4 ( s ) + V ( s ) (2.29) = n F 2 { s ) U { s ) + - 4 - [RchsF2[s) + Fi(s)I U { s ) (2.30) Hitc/ip (2.31) ^Rchp /m (4 (2.32)

Therefore the transfer function between the ou tp ut current and the pulse current is given by th e expression

_ ^Rchp_____ /g

S u b stitu tin g for Fi{ s) and /^ ( s ) from Eqn. (2.8) and Eqn. (2.17) respec­ tively and sim p lifyin g we have

[ m { s ) _ 023^ -f a i S + Co /p(s) &23^ + biS 4- bo (2.34) where 0-2 — ^FchpL/rnf^rcsFcr (2.35) Cl = nRchp{LmJ -\-CresRcxRm) (2.36) Co = nRchpRm (2.37) ^2 — LynfCTcsR-cri^Fm. Rsm) + LmfCrcsi^ + Rchp){Rcr + R-sm + ^m) (2.38) b\ — Cres RcrRsmRrn 4" ijmf{,Rsm 4" Rm') 4- Rchp)[Rmf + CresRm(Rcr + Ram)] (2.39) bo = Rm(Rsm + I + n^Rchp) (2.40)

T h e B o d e plot o f th e transfer function given by Eqn. (2.34) can be ob­ tained for any given resonant frequency /<,. T he resonant frequency can vary due to th e variation o f the resonant capacitor value due to tem perature changes or due to th e change in the choke inductance value due to saturation effects. T h u s th e ac transfer charactersitics provides th e gain o f th e resonant network a t any given resonant frequency.

Since th e pulse ringing frequency (wp = 2irfp) is known, the frequency spectrum o f th e pulse currents can be determ ined. T h e fundam ental com ­ ponent o f th e pulse current can be adjusted to obtain th e desired level of th e m agn et current by using the ac transfer characteristics of the resonant

2 a

_ZL_ “a

Figure 2.4; V ariation o f Pulse Current and its Fundam ental C om ponent

network at th e given resonant frequency. T h e peak value o f the pulse current and its dc com ponent can be d eterm in ed once the fundam ental com p on en t o f the pulse current is fixed. T h e procedure can be repeated for a set o f resonant frequencies to obtain th e effect o f resonant frequency variation on th e pulse current.

The frequency sp ectru m o f th e pulse current has to be determ ined before em barking on using th e transfer characteristics obtained by Eqn. (2 .3 4 ). T h e variation o f th e pulse current and th e in p u t filter current for a general case is shown in Fig. 2.4 to obtain th e frequency spectrum . T he peak value o f th e pulse current is represented as ip and th e peak value o f the fu n dam en tal com ponent is shown as Iip. T he pulse ringing frequency is Wp.

The pulse current can b e defined as

ip s i n { u p t ) 0 < wf < (r/w p ) 0 (7r/wp) < Ljt < (7r/w,)

(2.41)

where

T h e frequency com p on en ts can be found by expressing the pulse current in a fourier series exp an sion as

where

and

ip{t) = Idcp + An s i n {nujat) + B„ cos {nujJ.)

1 r’^/p.

Idcp = r - / ip s i n ujpt d{ujat) 2/k Jo 2 f^lP = I T I s i n {p (jJat) d{uat) Lk Jq 2X wp An =

l p f sin{ p — n)uat ttIp s in {p + n)uat

2w \ ( p - n ) _{0 (P + n)} i r / p ' i ^ ( s i n (p — n)wfp s i n ( p + n ) w /p 27t \ ( p - n )

(2.43)

(2.44)

(2.46)

(2.47)

— / s i n (pu>at) s in (nuat) d {uat ) (2.49) w Jo

i rif/p

— {cos (p - n)(jJat - COS { pn) ( j LJat } d{Uat) (2.50)

(2.51)

ip + n)

(7iÿ6p)

(2.52)

I f'^/p *

I f^iP

= — / s i n {'pUat) cos {nuJat) d{u}at) (2.54)

7T Jo

I /r/p

= ^ { s i n { p + n) uat + s i n {p — n)ujat} d{uat) (2.55)

— ip \ \ c O s { p + n)(jJat \cOs{p — n)Uat

27t

I

+

Zp f

- 2 p

2 r ( ( n 2 - p 2 ) ' ^

( n - p )

(n + p)

j

X _ h f (p - l) f f /p am ( p + l ) r / p ]

- 2 Ï 1 —

(F ^rij

D _ h f 2p c o s ( p - I ) 7 r / p c n s ( p + l) , r / p ) - s l ( ? r T ) - ---( F T Ï ) — /

The peak value o f th e fundam ental com ponent of the pulse current is given by

= Kpip (2.63)

where Kp is a p o sitiv e number.

If th e gain from th e ac transfer characteristics {i m/ i p) defined as Kg is m ade equal to th e fundam ental com ponent o f th e pulse current to obtain th e desired level o f m agn et current for a given resonant frequency, we have;

Y = K g (2.64)

‘p For th e fundam ental com ponent

/lp — (2.65)

l \ g

where lac is the peak value of the ac com ponent o f the m agnet current. Therefore

K , I , = ^ (2.66)

" =

T h e above procedure can be repeated for a set o f resonant frequencies to d eterm ine th e variation o f the peak pulse current as th e resonant frequency varies. T h e B ode p lot of th e transfer function /m //p for a resonant frequency of 49.9 Hz is shown in Fig. 2.5. It is seen th a t as th e resonant frequency approaches th e accelerator frequency the gain o f the network increases and hence th e peak value o f the pulse current required to m aintain a constant am p litu d e o f th e m agn et excitation reduces. T he phase plot shows that as

10" '

Frequency in Hz

Frequency in Hz

Figure 2.5: Bode Plot of Current Gain I m/

th e resonant frequency approaches th e accelerator frequency th e phase shift increases, i.e, the pulse current m oves towards th e zero crossing o f th e ac com p on en t of th e m agn et current. T h is zero crossing of th e m agnet current corresponds to th e peak value o f th e input voltage to th e pulse form ing netw ork. Thus th e input voltage increases.

T h e variation o f th e peak pulse current as a function o f the resonant frequency is shown in Fig. 2.6. T h e plots show th a t as the resonant frequency m oves away from th e accelerator frequency th e current gain decreases and larger peak currents are required to m aintain a constant ac excita tio n o f the m agn et current. T he average input current w hich is the dc com ponent o f the pulse current also increases.

por-0.9

000 ù)p/

= 2.5

XXX cOp/ co^ = 3.0

/

= 3.5

(Dy,/ CD_ =s 4.0

Figure 2.6: Peak Pulse Current in Per Unit (P U ) o f Peak M agnet Current as a fu n ctio n o f Resonant Frequency

tant factor in th e performance o f th e energy m ake-up u n it. T he peak current and v o lta g e stresses on the pu lsed power supply varies dram atically as the resonant frequency drifts away from th e accelerator op eratin g frequency. The changes in th e peak currents and voltages plays an im p ortan t role in the de­ sign o f th e input filter to the pu lsed power supply u n it. T h ese aspects are d iscussed in th e next section.

2.2

A n a ly sis o f In p u t F ilter o f E n erg y M ake­

up U n it

T he effect o f th e resonant frequency drift on the peak pulse currents o f the energy m ake-up units were discussed in th e previous section. T he dram atic variation o f th e peak value o f th e pulse currents and the in p u t voltage deter­ m ine the op eratin g characteristics o f the input L j C j filter. T he eflfect o f the variation in th e L C param eters on th e performance o f the sy stem is discussed in this section.

T he pulsed power supply extracts the energy from the input filter capaci­ tor and transfers th e charge to th e energy storage choke. D uring this interval the voltage across th e filter capacitor ( u f ) reduces to som e value, before be­ ing restored to its originzil value by th e cyclic charging through the input dc source and th e filter inductor ( Lp ) . T he value of the capacitance o f the filter capacitor determ ines the change in th e m agnitude o f the capacitor voltage. Thus it is necessary to determ ine th e variation in the capacitor voltage to be able to design th e filter capacitor. A lso the harm onic currents th at flow through the filter inductor determ ines th e value o f the inductance required. Hence an analysis o f the input filter circuit is essential to design th e filter.

T he input filter network w ith the pulsed power supply m odelled as a current source is shown in Fig. 2.7. T h e equivalent series resistance o f the filter is sm all and is neglected in the analysis. The waveforms o f the inductor current and th e capacitor voltage for cy clic operation o f the sy stem is shown in Fig. 2.8. T h e pulse current (ip) is also shown to depict the different instants at which the changes in the variables occur. The boundary conditions for