CERN Accelerator School. RF Cavities. Erk Jensen CERN BE-RF

Transcription

1 CERN Accelerator School RF Cavities Erk Jensen CERN BE-RF CERN Accelerator School, Varna "Introduction to Accelerator Physics"

2 What is a cavity? 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities

3 Lorentz force A charged particle moving with velocity electromagnetic field experiences a force dp dt q v p mγ ( E + v B) through an The total energy of this particle is W ( ) ( ) W kin kinetic energy is mc ( γ 1) mc + pc γmc, the The role of acceleration is to increase the particle energy! Change of W by differentiation: WdW c p dp dw qv Edt qc p Note: Only the electric field can change the particle energy! ( ) E + v B dt qc p Edt 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 3

4 Maxwell s equations The electromagnetic fields inside the hollow place obey these equations: 1 B E 0 c t E + B 0 t B E With the curl of the 3 rd, the time derivative of the 1 st equation and the vector identity E 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 4 E E this set of equations can be brought in the form E 1 c t E which is the Laplace equation in 4 dimensions. With the boundaries of the solid body around it (the cavity walls), there exist eigensolutions of the cavity at certain frequencies (eigenfrequencies)

5 Homogeneous plane wave E B k r u u y x ω c cos cos ( ωt k r ) ( ωt k r ) ( cos ( ϕ) z + sin( ϕ) x) k k Wave vector : the direction of is the direction of propagation, the length of k is the phase shift per unit length. behaves like a vector. k x k ωc c k ω c E y φ 3-Sept-010 z k z ω c ωc 1 ω CAS Varna/Bulgaria 010- RF Cavities 5

6 The components of are related to the wavelength in the direction of that component as etc., to the phase velocity as Wave length, phase velocity 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 6 z x E y k c k c ω c k ω 1 ω ω ω c z c k c k c ω c k ω z z k π λ z z z f k v λ ω ϕ,

7 Superposition of homogeneous plane waves E y x z + Metallic walls may be inserted where without perturbing the fields. E y 0 Note the standing wave in x-direction! This way one gets a hollow rectangular waveguide 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 7

8 Rectangular waveguide Fundamental (TE 10 or H 10 ) mode in a standard rectangular waveguide. Example: S-band :.6 GHz GHz, Waveguide type WR84 (.84 wide), dimensions: 7.14 mm x mm. Operated at f 3 GHz. electric field power flow power flow: 1 Re cross section E H * d A magnetic field power flow 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 8

9 What happens with different waveguide dimensions (different width a)? Waveguide dispersion 1: a 5 mm, f/f c 1.04 f 3 GHz k z k z ω k c 3 π ω λ c g ωc 1 ω : a 7.14 mm, f/f c cutoff c f c a ω ω c 3: a mm, f/f c.88 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 9

10 The phase velocity is the speed with which the crest or a zero-crossing travels in z-direction. Note on the three animations on the right that, at constant f, it is λ g. Note that at f f c, v ϕ,z! With f, v z c! ϕ, Phase velocity 1: a 5 mm, f/f c 1.04 f 3 GHz k z ω k c k 3 z π ω λ c g ωc 1 ω ω v : a 7.14 mm, f/f c 1.44 ϕ, z 1 cutoff k z ω c f c a 1 v ϕ, z ω ω c 3: a mm, f/f c.88 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 10

11 Rectangular waveguide modes TE 10 TE 0 TE 01 TE 11 TM 11 TE 1 TM 1 TE 30 TE 31 TM 31 TE 40 TE 0 TE 1 TM 1 TE 41 TM 41 TE TM TE 50 TE 3 plotted: E-field 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 11

12 Radial waves Also radial waves may be interpreted as superpositions of plane waves. The superposition of an outward and an inward radial wave can result in the field of a round hollow waveguide. 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 1

13 Round waveguide TE 11 fundamental TM 01 axial field f/f c 1.44 TE 01 low loss f c 87.9 f c f c 18.9 GHz a / mm GHz a / mm GHz a / mm 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 13

14 Circular waveguide modes TE 11 TE 11 TM 01 TE 1 TE 1 TE 31 TE 31 TE 01 TM 11 plotted: E-field 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 14

15 General waveguide equations: Transverse wave equation (membrane equation): TE (or H) modes TM (or E) modes boundary condition: longitudinal wave equations (transmission line equations): propagation constant: characteristic impedance: ortho-normal eigenvectors: transverse fields: longitudinal field: 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 15

16 TE (H) modes: TM (E) modes: a b TE (H) modes: TM (E) modes: Ø a where 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 16

17 Waveguide perturbed by notches notches Signal flow chart Reflections from notches lead to a superimposed standing wave pattern. Trapped mode 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 17

18 Short-circuited waveguide TM 010 (no axial dependence) TM 011 TM 01 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 18

19 Single WG mode between two shorts short circuit a e jk z short circuit 1 Signal flow chart 1 e jk z Eigenvalue equation for field amplitude a: a e jk z a k z π m Non-vanishing solutions exist for : With k z ω c ωc 1 ω, this becomes f 0 f c + c m 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 19

20 Simple pillbox (only 1/ shown) TM 010 -mode electric field (purely axial) magnetic field (purely azimuthal) 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 0

21 Pillbox cavity field (w/o beam tube) CAS Varna/Bulgaria 010- RF Cavities 1 The only non-vanishing field components : h Ø a 3-Sept-010 ( ) a a T J J 1, χ χ ρ χ π ϕ ρ... χ a a a B a a a a E z J J 1 J J 1 j 1 χ ρ χ π µ χ ρ χ π χ ωε ϕ a c pillbox 01 0 χ ω Ω ε µ η + h a a Q pillbox 1 01 ησχ ( ) a h a h Q R pillbox ) ( sin J χ χ π χ η

22 Pillbox with beam pipe TM 010 -mode (only 1/4 shown) One needs a hole for the beam pipe circular waveguide below cutoff 3-Sept-010 electric field magnetic field CAS Varna/Bulgaria 010- RF Cavities

23 A more practical pillbox cavity Round of sharp edges (field enhancement!) TM 010 -mode (only 1/4 shown) 3-Sept-010 electric field magnetic field CAS Varna/Bulgaria 010- RF Cavities 3

24 The energy stored in the electric field is Stored energy cavity ε E dv W E E The energy stored in the magnetic field is cavity µ H dv W M H 1.0 Since E and H are 90 out of phase, the stored energy continuously swaps from electric energy to magnetic energy. On average, electric and magnetic energy must be equal. The (imaginary part of the) Poynting vector describes this energy flux. In steady state, the total stored energy constant in time. W cavity ε µ E + H dv is 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 4

25 Stored energy & Poynting vector electric field energy Poynting vector magnetic field energy 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 5

26 P loss Losses & Q factor The losses are proportional to the stored energy. ω The cavity quality factor Q is defined as the ratio Q 0W. In a vacuum cavity, losses are dominated by the ohmic losses due to the finite conductivity of the cavity walls. If the losses are small, one can calculate them with a perturbation method: The tangential magnetic field at the surface leads to a surface current. This current will see a wall resistance { R is related to the skin depth by δ σ 1. } A The cavity losses are given by If other loss mechanisms are present, losses must be added. Consequently, the inverses of the Q s must be added! P δ loss R A wall R A R A H W ωµ σ t P loss da 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 6

27 I define Acceleration voltage & R-upon-Q V acc variation of the field while particles with velocity (see next page). With this definition, c E e dz z V acc ω j z β. The exponential factor accounts for the are traversing the gap is generally complex this becomes important with more than one gap. For the time being we are only interested in. Attention, different definitions are used! β c V acc The square of the acceleration voltage is proportional to the stored energy W. The proportionality constant defines the quantity called R-upon-Q: R Q 0 Vacc ω W Attention, also here different definitions are used! 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 7

28 TT Transit time factor The transit time factor is the ratio of the acceleration voltage to the (non-physical) voltage a particle with infinite velocity would see. V acc E dz sin χ 01 h χ h 01 a a z E z e ω j z β c E dz The transit time factor of an ideal pillbox cavity (no axial field dependence) of height (gap length) h is: Field rotates by 360 during particle passage. TT z dz h/λ 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 8

29 Shunt impedance The square of the acceleration voltage is proportional to the power loss. The proportionality constant defines the quantity shunt impedance Vacc R loss Attention, also here different definitions are used! Traditionally, the shunt impedance is the quantity to optimize in order to minimize the power required for a given gap voltage. P P loss 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 9

30 Equivalent circuit Simplification: single mode I G V acc I B Generator P Beam β : coupling factor R β C L Cavity R LR/(Qω 0 ) CQ/(Rω 0 ) L C R: Shunt impedance : R-upon-Q 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 30

31 Resonance Q100 ZRQ () /(/) ω 10 5 Q10 1 Q ω/ω 0 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 31 Q1

32 Reentrant cavity Nose cones increase transit time factor, round outer shape minimizes losses. nose cone Example: KEK photon factory 500 MHz - R probably as good as it gets - this cavity optimized pillbox R/Q: 111 Ω Ω Q: R: 4.9 MΩ 4.47 MΩ 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 3

33 Loss factor Impedance seen by the beam k loss R ω Q V 0 acc 4 W 1 C V (induced) I B Beam Energy deposited by a single charge q: k loss q R/β C L Cavity R LR/(Qω 0 ) CQ/(Rω 0 ) Voltage induced by a single charge q: Sept-010 t f 0 CAS Varna/Bulgaria 010- RF Cavities 33

34 Summary: relations between V acc, W, P loss R-upon-Q gap voltage k loss R Q 0 R ω 0 Q Vacc ω W Vacc 4 W Shunt impedance Vacc R P loss Energy stored inside the cavity ω Q 0W P loss Power lost in the cavity walls Q factor 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 34

35 Beam loading RF to beam efficiency The beam current loads the generator, in the equivalent circuit this appears as a resistance in parallel to the shunt impedance. If the generator is matched to the unloaded cavity, beam loading will cause the accelerating voltage to decrease. 1 The power absorbed by the beam is Re{ V * } acc I B, Vacc the power loss P. loss R For high efficiency, beam loading should be high. 1 I B The RF to beam efficiency is η. Vacc 1+ IG R I B 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 35

36 Resonance frequency Characterizing cavities Transit time factor field varies while particle is traversing the gap Shunt impedance gap voltage power relation Circuit definition Linac definition Q factor R/Q independent of losses only geometry! loss factor 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 36

37 Higher order modes external dampers... R 1, Q 1,ω 1 R, Q,ω R 3, Q 3,ω 3... n 1 n n 3 I B 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 37

38 Higher order modes (measured spectrum) without dampers with dampers 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 38

39 Pillbox: dipole mode TM 110 -mode (only 1/4 shown) electric field magnetic field 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 39

40 CERN/PS 80 MHz cavity (for LHC) inductive (loop) coupling, low self-inductance 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 40

41 Higher order modes Example shown: 80 MHz cavity PS for LHC. Color-coded: 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 41

42 What do you gain with many gaps? The R/Q of a single gap cavity is limited to some 100 Ω. Now consider to distribute the available power to n identical cavities: each will receive P/n, thus produce an accelerating voltage of. The total accelerating voltage thus increased, equivalent to a total equivalent shunt impedance of. P/n P/n P/n P/n 1 3 n 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 4

43 Standing wave multicell cavity Instead of distributing the power from the amplifier, one might as well couple the cavities, such that the power automatically distributes, or have a cavity with many gaps (e.g. drift tube linac). Coupled cavity accelerating structure (side coupled) The phase relation between gaps is important! 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 43

44 ω L/c π Brillouin diagram Travelling wave structure speed of light line, ω β /c π π/3 π π/ synchronous π/ Sept-010 π/ β L π CAS Varna/Bulgaria 010- RF Cavities 44

45 Examples of cavities PEP II cavity 476 MHz, single cell, 1 MV gap with 150 kw, strong HOM damping, LEP normal-conducting Cu RF cavities, 350 MHz. 5 cell standing wave + spherical cavity for energy storage, 3 MV 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 45

46 CERN PS 00 MHz cavities 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 46

47 PS 19 MHz cavity (prototype, photo: 1966) 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 47

48 CERN PS 80 MHz Cavity (1997) 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 48

49 Ferrite cavity CERN PSB, MHz PS Booster, MHz, < 10 kv gap NiZn ferrites 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 49

50 CERN PS 10 MHz cavity (1 of 10) 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 50

51 Drift-tube linac (JPARC JHF, 34 MHz) 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 51

52 CERN SPS 00 MHz TW cavity 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 5

53 Travelling wave cavities CLIC T18, 1 GHz CLIC HDS, 1 GHz Shintake structure, 5.7 GHz 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 53

54 Side-coupled cavity (JHF, 97 MHz) 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 54

55 Single- and multi-cell SC cavities (1.3 GHz) SC cavity lab KEK, Japan 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 55

56 SC cavities in a cryostat (CERN LHC 400 MHz) 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 56

57 SC deflecting cavity (KEK-B, 508 MHz) Asymmetric shape to split the two polarizations. 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 57

58 Summary RF Cavities The EM fields inside a hollow cavity are superpositions of homogeneous plane waves. When operating near an eigenfrequency, one can profit from a resonance phenomenon (with high Q). R-upon-Q, Shunt impedance and Q factor were are useful parameters, which can also be understood in an equivalent circuit. The perturbation method allows to estimate losses and sensitivity to tolerances. Many gaps can increase the effective impedance. 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities 58