Cavity Field Control - Cavity Theory. LLRF Lecture Part3.1 S. Simrock, Z. Geng DESY, Hamburg, Germany

Transcription

1 Cavity Field Contol - Cavity Theoy LLRF Lectue Pat3.1 S. Simock, Z. Geng DESY, Hambug, Gemany

2 Outline Intoduction to the Cavity Model Baseband Equations fo Cavity Model Diving Tem in Cavity Equations RF Powe Dissipation and Reflection of a Cavity RF Powe Dissipation and Reflection at Filling Stage RF Powe Dissipation and Reflection at Flattop Stage Mechanical Model of the Cavity Pass Band Modes of the Cavity Cavity Simulato Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF

3 Intoduction to the Cavity Model Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 3

4 Context of the Cavity in the Contol Loop The cavities ae the plant to be contolled by the LLRF system. The cavities ae diven by the RF powe amplifies such as klyston. Nomally, the cavities ae equipped with pobes fo picking up the RF signal to be measued by the field detecto. Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 4

5 Motivation fo Cavity Model Study Undestand the popety and behavio of the cavity Model the cavity and the LLRF contol system fo algoithm study and contolle design Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 5

6 9-Cell Cavity Woking Mode: TM1 Paametes fo TESLA cavity Effective length Apetue diamete Cell to cell coupling /Q Unloaded Q 136 mm 7 mm 1.98 % 136 Ω Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 6

7 Equivalent RLC Cicuit Model The 9-cell cavity is modeled with nine magnetically coupled esonatos. Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 7

8 Pass Band Modes of the 9-Cell Cavity π mode is selected fo acceleating the beam 8π/9 is close (8 khz away) to the opeation mode, which may influence the stability of the acc. field Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 8

9 Pass Band Modes of the 9-Cell Cavity Aows show the diection and amplitude of the electic field component along the axis π mode is used fo beam acceleation Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 9

10 Pinciple of Beam Acceleation with π Mode Maximum acceleation voltage V acc L / E L / z e j ( z / c) dz Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 1

11 Cavity Model fo the π Mode with RF Diving 1:n The whole 9-cell cavity is modeled as an single RLC cicuit fo the π mode Klyston is modeled as a constant-cuent souce Powe couple of the cavity is modeled as a lossless tansfome Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 11

12 View fom Diffeent Refeence Plane View fom cavity side of the tansfome. Used to study the cavity behavio as a diffeential equation. View fom the powe tansmission line side. Used to study the powe tansmission. Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 1

13 Baseband Equations fo Cavity Model Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 13

14 Diffeential Equation of the Cicuit Model dvc 1 dvc 1 1 di + + V c dt RLC dt LC C dt The equation is descibed with cicuit paametes (R, L, C), which need to be mapped to the measuable cavity chaacteistics (quality facto, bandwidth, shunt impedance ) R L R // Z I I g I b ext 1+ R R n Z Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 14

15 Relationship between Cicuit Paametes and Cavity Chaacteistics Resonance fequency πf 1 LC (f₀ 1.3GHz) Numbes in Backet: Typical Values fo TESLA Cavity Quality facto and input coupling facto Q Q W P diss, cav RC W n P Z ext diss, ext R L C (~1e1) β Q L P P diss, ext diss, cav Q 1 + β Q Q ext (~3e6) R n Z (>3) Shunt impedance and nomalized shunt impedance (/Q) 1 1 R ( / Q ) Q L ( ) L n R β Z R R // n Z (~1554MΩ) R 1+ β 1 Q Q Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 15

16 Diffeential Equation of the Cavity dvc 1 dvc 1 1 di + + V c dt RLC dt LC C dt d V dt c + dvc 1/ + Vc 1/ dt R L di dt The half bandwidth of the cavity is defined as 1 1/, τ is the time constant of the cavity τ Q L The cavity voltage Vc and diving cuent I ae always sine signals with phase and amplitude modulation Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 16

17 Diffeential Equation of the Envelope Define the phaso fo sine signals: jt Vce, When studying the cavity behavio with klyston powe and beam cuent, the caie fequency tem is not inteested. The base band (envelope) equation will be used. V c I Ie jt d V dt c + dvc 1/ + Vc 1/ dt R L di dt Cavity baseband equation: dv dt c + ( jδ) V R I 1/ c 1/ L Detuning is defined as: Assumptions: Δ << 1/ << Valid fo high Q cavities Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 17

18 Vaiations of the Cavity Equation Voltage diven cavity equation dv dt c + ( 1/ jδ) Vc 1/ V fo Fo supeconducting cavity β >>1 So the cavity equation can be simplified as dv dt c + β β + 1 ( 1/ jδ) Vc 1/ V fo Valid fo both nomalconducting and supeconducting cavity Valid fo cavities with lage coupling facto, such as the supeconducting cavity Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 18

19 State Diffeential Equation of the Cavity Sepaate the eal and imaginay pats of the complex cavity equation, we will get the state diffeential equation dv dt c + ( jδ) V R I 1/ c 1/ L V V + jv, I I + c c ci ji i d dt V V c ci 1/ Δ Δ V 1/ V c ci + 1/ R L I I i State equation is suitable fo digital simulation and implementation, and can fit to the famewok of moden signal pocessing and contol theoy Complex equation is suitable fo analysis Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 19

20 Steady State Behavio of the Cavity Resonance Cuves Steady State: no tansient, the item with time deivative equals to zeo V c 1/ RLI jδ 1/ Vc RL I cosψ, 1 wheeψ tan V ( Δ ) 1/ c I is ψ the detuning angle Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF

21 Steady State Behavio of the Cavity Resonance Cicle V R I cosψ, V I ψ c L c Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 1

22 Tansient Behavio of the Cavity Tansient behavio: step esponse of the cavity R ( ) () ( j t ) LI Vc t 1/ 1/ Δ 1 e jδ 1/ 1 Δ/ 1/ 1 Amplitude Δ/ 1/.5 Δ/ 1/ 1 Δ/ 1/ 1.5 Δ/ 1/ Δ/ 1/ 3 Δ/ 1/ Time / μs Phase / deg Δ/ 1/ 5 Δ/ 1/ 3 Δ/ 1/ Δ/ 1/ 1.5 Δ/ 1/ 1 Δ/ 1/.5 Δ/ 1/ Time / μs Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF

23 Tansient Behavio of the Cavity Imaginay pat of V c Δ/ 1/ Real pat of V c Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 3

24 Diving Tem in Cavity Equations Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 4

25 Typical Paametes in Pulsed System Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 5

26 Relationship of Diving Tem in Cavity Equation to Klyston Powe and Beam Cuent Remind the cavity baseband equation: dv c + 1/ jδ Vc 1/ R dt I I I g b v I g + I b f g ( P Z, Z, β ) fo, cav ( Q, f ) bunch bunch ( ) I The diving tem is the supeposition of the geneato cuent and beam cuent: L The geneato cuent is a function of the klyston powe, cavity and tansmission line impedance, and input coupling facto The beam cuent is a function of bunch chage and bunch epetition ate Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 6

27 Relationship of the Geneato Cuent and Klyston Powe Cavity equation diven by klyston powe and beam: dv dt ( jδ) V R ( I + I ) Z cav Z Γ Z cav + Z V ef ΓV fo V fo Z I Z cavz V c V fo + V ef Z + Z Tanslate to the cavity side of the tansfome, we will get Duing steady state, when thee is no beam and detuning (Ohm s law), so I fo βp P fo fo P I g I fo, I fo n R R Z c + 1/ c 1/ L L V V fo c c g cav R R b L L fo I I I g fo fo Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 7

28 Relationship of the Beam Cuent and Aveage DC Beam Cuent Cavity equation diven by klyston powe and beam: dv dt c ( jδ) V R ( I + I ) + 1/ c 1/ L g b A single bunch is descibed by a Gaussian cuve: I () t Q πσ t e t σ t I peak e t σ t Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 8

29 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 9 Relationship of the Beam Cuent and Aveage DC Beam Cuent (cont d) Fouie decomposition of bunch tain: () ( ) ( ) [ ] + + L L 1,,,1,, sin cos / 1 n b n e T I a T t n b t n a a t I n n b t peak n b b n b n b n b σ t σ π π Aveage DC beam cuent: b b t peak b T Q T I I σ π Beam cuent in cavity equation: ( ) / 13MHz b n b n b n I a I e I a t b σ Bunch length σt ~ 3 ps

30 Beam Phase Respect to RF ϕb, V c Cavity voltage changing with time ϕ b dv c With items in cavity equation: / jδ Vc dt ϕb 18 Ib Vc Enegy gain of single paticle: ΔE cosϕ ( ) R ( I + I ) + 1 1/ ( ) o o Beam Phase Definition: V c b L g mod b 36 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 3

31 Beam Loading in the Cavity RF geneato induced voltage Beam induced voltage On-cest acceleation Cavity voltage / MV Beam loading is significant in supeconducting cavities Beam induced voltage cancels the exponential incease of the geneato induced voltage, esulting in a flattop Ig / ma Ib / ma Time / μs Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 31

32 Beam Injection Time fo Flattop If the beam is acceleated on-cest and thee is no detuning, the beam injection time fo flattop is t inj 1 1/ ln I I g b τ ln I I g b Fo TESLA cavity, tinj 734μs * ln(16ma/8ma) 51 μs Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 3

33 RF Powe Dissipation and Reflection of a Cavity Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 33

34 RF Powe Pofile fo Cavity Opeation Filling Stage: cavity voltage inceases fom zeo with the cavity diving powe Flattop Stage: cavity voltage keeps constant fo beam acceleation, which is a nealy steadystate condition Cavity voltage / MV Diving powe / kw No Beam Reflected powe / kw Time / μs Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 34

35 RF Powe Dissipation and Reflection at Filling Stage Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 35

36 RF Powe Dissipation at Filling Stage Filling of cavity: fill the cavity to a desied voltage V fom within a peiod of Tfill Factos influence the equied filling powe Desied cavity flattop voltage (V) Filling time (Tfill) Loaded Q of cavity (QL) Detuning of cavity (Δ) Fom the cavity tansient behavio, the RF powe equied fo filling stage is P fill 4 Q Q L V 1 + e ( + 4Q Δ ) T Q L fill L e T Q fill L cos ( ΔT ) fill Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 36

37 Filling Powe fo Diffeent Loaded Q and Filling Time 14 T fill 4μs Filling Powe / kw Cavity Voltage 35MV Detuning Hz T fill 5μs T fill 6μs T fill 7μs T fill 8μs x 1 6 Q L Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 37

38 Filling Powe fo Diffeent Loaded Q and Detuning Filling Powe / kw Cavity Voltage 35MV Filling time 5μs When QL 3e6 3% moe filling powe is equied fo Hz detuning Detuning 1Hz Detuning 8Hz Detuning 6Hz Detuning 4Hz Detuning Hz Detuning Hz x 1 6 Q L Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 38

39 RF Powe Reflection at Filling Stage Reflection powe can be calculated via the elationship of V c V fo + V ef Fowad Voltage Cavity Voltage At the beginning of the RF pulse, the cavity voltage is zeo, so the eflection powe equals to the fowad powe, and when the cavity voltage inceases, the eflection powe decease Voltage / MV Reflected Voltage Time / μs Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 39

40 RF Powe Dissipation and Reflection at Flattop Stage Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 4

41 RF Powe Dissipation at Flattop Stage Flattop of cavity: keep the cavity voltage to a desied value V in pesence of beam cuent Ib Factos influence the equied flattop powe Desied cavity flattop voltage (V) Beam cuent and beam phase (Ib) Loaded Q of cavity (QL) Detuning of cavity (Δ) Duing flattop, the cavity is appoximately in steady state, so steady state equations can be used Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 41

42 Induced Cavity Voltage by the Klyston Powe Recall: cavity esonance cicle fo steady state behavio Vc RL I cosψ V I ψ c If thee is no beam, the cavity is diven by the fowad voltage concen to klyston powe R V c L ψ I R V tan fo 1 fo cosψ e ( Δ ) (detuning angle) L I V 1/ jψ fo Cavity voltage on esonance: V fo Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 4

43 Vecto Diagam fo Cavity Diving Resonance cicle fo beam cuent induced voltage Resonance cicle fo klyston powe induced voltage Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 43

44 Effect of Detuning Detuning will decease the cavity voltage and shift the cavity phase Moe input powe will be needed to maintain the cavity voltage Input phase should be changed to compensate the phase shift Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 44

45 Klyston Powe in Pesence of Beam and Detuning Optimization fo minimizing the klyston powe equied: Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 45

46 Powe Requied as Function of Detuning Example: Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 46

47 Flattop Powe fo Diffeent Loaded Q and Detuning Flattop Powe / kw Cavity Voltage 35MV Beam Cuent 9mA (On-cest acceleation) When QL 3e6 5% moe flattop powe is equied fo Hz detuning Detuning 1Hz Detuning 8Hz Detuning 6Hz Detuning 4Hz Detuning Hz Detuning Hz Q L x 1 6 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 47

48 RF Powe Reflection at Flattop Stage Enegy consevation yields Pfo Pdiss + Pef + Pbeam + dw dt Fo supeconducting cavity, duing flattop, the eflected powe can be appoximated as P ef P fo P beam P fo V c I b cosϕ b 35 3 Fowad powe Reflected powe 35 3 Fowad powe Reflected powe RF powe / kw No beam RF powe / kw With beam Time / μs Time / μs Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 48

49 Ideas fo Minimizing the Requied Klyston Powe Reduce the detuning effect At filling stage, tack the fequency of the input RF with the esonance fequency of the cavity The pe-detuning of the cavity should be adjusted to minimize the aveage detuning duing flattop Piezo tune can be used to compensate the Loenz foce detuning duing flattop If thee is beam, optimize the loaded Q, detuning and filling time When the beam is lage, optimize the loaded Q and detuning duing the flattop with the equations in this section When the beam is small, matching of the beam is not feasible, compomise should be made fo the selection of the loaded Q and filling time Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 49

50 Mechanical Model of the Cavity Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 5

51 Cavity Defomation by Electomagnetic Field Pessue Radiation pessue μ P H ε E Resonance fequency shift 4 Δf K E acc Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 51

52 Static Loenz Foce Detuning Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 5

53 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 53 Dynamic Loenz Foce Detuning ( ) ( ) + Δ Δ Δ Δ 1 m m m m m m m m m f K V Q f f dt d π π π π & & Δ + + Δ Δ Δ N m m t 1 ' ) ( State space equation of the mth mechanical mode

54 Pass Band Modes of the Cavity Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 54

55 Cavity Model with Pass Band Modes Goal: Model the cavity including the pass band modes by extending the π mode cavity equation discussed befoe dv c + ( 1/ jδ) Vc 1/ V fo dt Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 55

56 Cicuit Model of the 9-Cell Cavity Electical coupled seies esonance cicuits diven by a voltage souce Assume all cells ae identical Cells ae coupled via Ck Beam tube effect at the 1 st and 9 th cell is modeled with Cb Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 56

57 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 57 Steady State Equation of the Cavity M M L M O M L CV g j I I I I a a a b κ κ κ κ κ κ κ κ ( ) κ β κ Q j b Q j a the input powe couple to the1st cell the coupling facto of is the single cell the unloaded quality facto of is the single cell the esonance feqency of is, the diving fequency of is coupling facto the cell- to - cell is β κ Q V g

58 Fequency Response of the 9-Cell Cavity Keep the diving voltage amplitude constant, change the fequency: 15 Cell 1 15 Cell 15 Cell 3 Amplitude Cell 4 15 Cell 5 15 Cell 6 Amplitude Cell 7 15 Cell 8 15 Cell 9 Amplitude Fequency / MHz Fequency / MHz Fequency / MHz Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 58

59 Pass Band Modes of the Cavity Pass band modes can be calculated by solving the eigenvalue poblem by emoving the diving tem in the cavity equation, as esults Fequency of nπ/9 mode: nπ, n π 9 1+ κ 1 cos 9 Nomalized field distibution in diffeent cell of nπ/9 mode : I nπ / 9, m B nπ / 9 B nπ / m 1 sin[ nπ ], 18 when n 1to 8 when n 9 m 1,,...,9 is the cell numbe Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 59

60 Amplitude Amplitude Amplitude 1.5 Relative Field Distibution of Pass Band Modes π/9 Mode 5 1 4π/9 Mode π/9 Mode Cell No. Amplitude Amplitude Amplitude.5 π/9 Mode π/9 Mode π/9 Mode Cell No Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 6 Amplitude Amplitude Amplitude.5.5 3π/9 Mode 6π/9 Mode π Mode Cell No.

61 Quality Facto of Diffeent Pass Band Modes Fo supeconducting cavity, the intenal powe loss of the cavity can be neglected, so the loaded quality facto can be appoximated to be Q L, nπ / 9 Qext, nπ / 9 P, nπ / 9 U ext, nπ / 9 nπ / 9 Fom the nomalized field distibution of the pass band modes, the effective stoed enegy in the cavity fo each pass band mode is the same. 9 m 1 I nπ / 9, m 1 U nπ / 9 is identical If we ignoe the diffeence of the esonance fequency, the loaded quality facto of each mode is invesed popotional to the stoed enegy in the fist cell, use the loaded quality facto in π mode as efeence, we get Q L, nπ / 9 Q L, π Iπ,1 1, n I / 9,1 nπ nπ sin 18 1,,...,8 Fo example, the QL fo the π/9 mode is 16 times lage than the QL fo π mode Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 61

62 Amplitude Field in the 9 th Cell Field in the 9 th cell is impotant because the pobe is installed thee The field fo diffeent pass band modes in the 9 th cell has the popety of Have the same amplitude to the same input powe The phase diffeence of the neaest pass band modes is 18 degee The loaded Qs ae diffeent C ell 9 3 π/9 π/9 3 π/9 4 π/9 5 π/9 6 π/9 7 π/9 8 π/9 π Phase / deg Cell π/9 3 π/9 5 π/9 7 π/9 π π/9 4 π/9 6 π/9 8 π/ Fequency / MHz Fequency / MHz Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 6

63 Cavity Model with Pass Band Modes Based on the π mode model discussed befoe, the cavity model with pass band modes can be made as follows Each pass band mode is modeled with a base band equation simila with the π mode The diving tem of all pass band modes ae the same fo the same input powe Diffeent pass band mode has diffeent bandwidth The oveall cavity voltage is the supeposition of the voltage of all the pass band modes, use the equations below dv dt V c Δ c, nπ 9 9 ( 1) n 1 nπ 9 1/, nπ 9 + is ( jδ ) 1/, nπ 9 n 1 V, nπ 9 c, nπ 9 the half is nπ 9 V c, nπ 9 bandwidth of 1/, nπ 9 V fo, n 1,,..,9 Hee we have consideed the phase diffeence of pass band modes in the 9th cell the detuning of the nπ/9 mode the nπ/9 mode Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 63

64 Tansfe Function of the Cavity Fom the cavity equations, the tansfe function of the cavity is H cav 9 n () s ( 1) H () s ( 1) n n 1 1/, nπ 9 nπ /9 n 1 s + 1/, nπ 9 jδnπ 9 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 64

65 Bode Plot of the Cavity Tansfe Function Magnitude / db Phase / degee Fequency / Hz Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 65

66 Cavity Simulato Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 66

67 Cavity Simulato Idea: build a hadwae cavity simulato to simulate the cavity behavio including the Loenz foce detuning Use cases: Test the LLRF hadwae such as down convete, contolle and actuato befoe the eal cavity is eady Contol algoithm study Opeato taining LLRF system on-line calibation if integated with woking system RF o base band input Cavity Simulato (DSP o FPGA) RF o base band output Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 67

68 Possible Cavity Simulato Integation with LLRF Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 68

69 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 69 Discete Cavity Equation ( ) I R V j dt dv L c c 1/ / 1 Δ + i i c i L i i ji I I jv V V I I R V V V V dt d Δ Δ,, 1/ 1/ 1/ + Δ Δ 1, 1, 1/ 1, 1, 1/ 1/,, 1 1 n i n L n i n n i n I I R T V V T T T T V V T is the sampling peiod. The discete equation can be ealized in digital pocessos, such as FPGA, fo simulation (state space equation) (discete)

70 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 7 Discete Mechanical Equation fo Loenz Foce Detuning ( ) ( ) + Δ Δ Δ Δ 1 m m m m m m m m m f K V Q f f dt d π π π π & & Loenz foce detuning by the mth mechanical mode: Δ + + Δ Δ Δ N m m t 1 ' ) ( ( ) ( ) Δ Δ Δ Δ ' 1, 1, ' ' ',, 1 1 V f K T Q f T f T T m m m n m n m m m m n m n π π π π & & (discete) (sum up)

71 Cavity Simulato Block Diagam I g I b d dt V n ( V I, I ) f,δ n 1, g b Δ ( Δ, Δ & V ) m, n g m, n 1 m, n 1, V Δ m Seveal mechanical modes ae included V Δ Δm + Δ Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 71

72 Cavity Simulato at DESY Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 7

73 Cavity Simulato at KEK Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 73

74 Softwae Real Time Cavity Simulato at Femilab Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 74

75 Summay In this pat, we have leant: Model the cavity with esonance cicuits Baseband equations fo cavity model RF powe dissipation and eflection of a cavity Mechanical model of the cavity Pass band modes model of the cavity Concept fo cavity simulato Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 75

76 Refeence [1] E. Vogel. High Gain Popotional RF Contol Stability at TESLA Cavities. Physical Review Special Topics Acceleatos and Beams, 1, 51 (7) [] T. Schilche. Vecto Sum Contol of Pulsed Acceleating Fields in Loentz Foce Detuned Supeconducting Cavities. Ph. D. Thesis of DESY, 1998 [3] A. Bandt. Development of a Finite State Machine fo the Automated Opeation of the LLRF Contol at FLASH. Ph.D. Thesis of DESY, 7 [4] H. Padamsee, J. Knobloch, et.al. RF Supeconductivity fo Acceleatos. John Wiley & Sons, Inc [5] W.M. Zabolotny et al. Design and Simulation of FPGA Implementation of RF Contol System fo TESLA Test Facility. TESLA Repot 3-5, 3 Stefan Simock, Zheqiao Geng 4th LC School, Huaiou, Beijing, China, 9 LLRF & HPRF 76