Ferrite Loaded Cavities for RF Accelerating Systems

Transcription

1 Ferrite Loaded Cavities for RF Accelerating Systems Ian Gardner ISIS, RAL FFAG School September 2011

2 CONTENTS Introduction Ferrite relevant parameters Cavities or Co-axial resonators Tuning systems Common designs and equivalent circuits CERN Accelerator School, 1992 RF Engineering for Particle Accelerators Page 349, Ferrite loaded cavities Contains all formula and their derivation RN V1-V2.pdf?version=1

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4 ISIS 1RF Cavity and HPD

5 Schematic of 1RF cavity and High Power Drive (HPD)

6 Cross section of an ISIS MHz cavity

7 APS LPRF Controls: Amplitude, Phase & Cavity Tuning. LVPS From LPRF Driver Amp. HPD Cavity Bias Supply BASIC RF SYSTEM

8 Ferrite Naturally occurring Ferrous Ferrite Fe 3 O 4 Fe 3+ (Fe 2+ Fe 3+ )O 4 structurally Fe 2+ may be replaced by any divalent metal of similar ion size Typical metals being; Nickel, Manganese, Magnesium, Cobalt, Copper, Zinc and Cadmium Wide range of ferrites available. See Philip s Ferroxcube, TDK Epcos, Toshiba and others. Most accelerator systems use NiZn based ferrites

9 Ferrite in RF Cavities Magnetic permeability Dielectric constant Resistivity Thermal conductivity Effects of temperature and electric and magnetic fields on the above parameters

10 Ferrite B-H loop with added ac field Incremental permeability μ r = B ac / μ o H ac varies with operating point

11 Plots of μ r against bias field for various types of Philip s ferrite

12 Ferrite Inductors Ferrite core increases inductance of a coil Hysteresis and eddy currents lead to losses Loss represented by assigning a complex value to the permeability. μ = μ' - j μ" Inductance impedance, Z = jωμl o = jωμ'l o + ωμ"l o = jωl + r L = μ'l o and r = μ"ωl o L o = the value of the air cored inductance

13 At resonance, Z = QωL = Qωμ'L o = (μ'qf)2πl o The factor (μ'qf) is often used as a figure of merit for ferrite materials in a given set up. For High Q; Q = ωl/r = ω μ' L o / ω μ" L o Q = μ' /μ"

14 Plots of μ' and μ" against frequency for different grades of Philip s Ferrite

15 Dielectric Constant of Ferrite Assign complex value to permittivity ε = ε' - jε" Similarly, this gives a resistance in parallel with a ferrite filled capacitor R = 1/ωεC o Where C o is the value of the air spaced capacitor and Q = ε'/ε" When C is resonated with a lossless coil. However, with NiZn ferrites, ε' remains constant at about 10 and does not vary with magnetic or electric fields. Up to 30 MHz ε" not found significant especially with copper cooling plates separating the ferrite.

16 Initial permeability as a function of temperature

17 Complex permeability as a function of frequency

18 80 60 mu value Ibias (A) 4M2 μ' value with bias current on ISIS

19 Tuning Range Frequency range is given by ω 2 = 1 / LC and L α μ μ max / μ min Variable frequency RF accelerating systems use shorted, ferrite loaded, co-axial transmission lines as inductances to resonate with the accelerating gap or drift tube capacitance.

20 Basic co-axial resonator circuit, short circuit at x = 0 Reactive impedance at the points AB without Cg is: Z = jz o tan(ωl/v) l = length of the line, v = signal velocity, Z o = characteristic impedance For ωl/v < π/2 i.e. 2πl/λ < π/2 and l < λ/4 Z is inductive and lets say = jωl Then L = (Z o /ω) tan(ωl/v) For resonance with Cg ω 2 = 1/LCg l = (v/ω) tan -1 (1/Z o ωcg)

21 Typical resonator with ferrite toroids separated by air-cooled or fluid cooled gaps or water cooled metal plates. Electric field runs from inner conductor to outer and across insulating accelerating gap. Suitable gaps are required to prevent voltage breakdown. Vacuum required for beam pipe. Magnetic field circulates in the ferrite toroids.

22 The line capacitance is: C t = 2π ε e ε o / ln (r 3 /r 1 ) The line inductance is: L t = μ e μ o ln (r 3 /r 1 ) / 2π farads / metre henrys / metre Case 1 ε e = ε d 1 / [(k+ε(1- k)) (d 1 +d 2 )] ε e = The effective permittivity for an air filled gap Case 2 ε e = 1 / (1-k) ε e = Effective permittivity using cooled metal plates In both cases: μ e = (1+k(μ' 1)) d 1 / (d 1 +d 2 ) μ e = The effective permeability And: k = ln (r 3 /r 2 ) / ln (r 3 /r 1 ) Derivation see CAS appendix 1

23 The wave velocity in such a line is: v = 1 / (L t C t ) and: Z o = (L t /C t ) = c / (ε e μ e) Where c = the velocity of light = (1/2π) ln (r 3 /r 1 ) (μ e μ o /ε e ε o ) = 60 ln (r 3 /r 1 ) (μ e /ε e ) If the voltage at the capacitor is V g e iωt, the voltage and current are given by V = -V g e iωt sin(ωx/v) / sin(ωl/v) and: i = jv g e iωt cos(ωx/v) / (Z o cos(ωl/v)) The current is a maximum at the short circuit where x = 0 and the ratio of the current at x = - l to the current at x = 0 is cos (ωl/v) : 1 Derivations CAS Appendix 2

24 This difference is usually kept to ~ 10% to keep the RF magnetic field in the toroids at each end of the resonator similar. This means ωl / v < 26 i.e. much shorter than 90 or λ/4

25 RF Magnetic induction in the ferrite H = i / 2πr and B = μ'μ o i / 2πr B rfmax = μ'μ o i -l / (2πr 2 cos(ωl/v)) i -l = the current at x = -l In NiZn ferrites, where the saturation field is Tesla the maximum value of B rf is usually kept to 0.01 Tesla.

26 Q value and B rfmax Plot of μ'q f with peak rf field at 2.0 MHz for two ferrites Resonator length for a required peak gap voltage V g and lowest operating frequency ω, can be estimated from l = V g μ' / (μ e r 2 ln(r 3 /r 1 ) ω B rfmax ) Derivation CAS Appendix 3

27 Power dissipation in the ferrite Power dissipation in the resonator is: P = V g 2 / 2R = V g 2 / (2QωL) Mean Power / unit length P mean = V g 2 / (2lQωL) W / m Q will vary with voltage and frequency. The power dissipation along the resonator P(x) will vary with i 2. P(x) = P max cos 2 (ωx/v) P mean = P max (1 + (sin(2ωl/v) / 2ωl/v)) / 2 W / m W / m

28 Power density and temperature rise in the ferrite P dmax = P max (d 1 + d 2 ) / (π(r 3 2 r 22 )d 1 ) W / m 3 T = P d (d 1 /2) 2 / k Where k = Ferrite thermal conductivity in W / m. C Degrees P d = The power density in the ferrite in W / m 3 T = temperature rise relative to cooling plate or cooling fluid. Typically k = 3.5 W / m. C, d 1 = 25 mm and d 2 = 6 mm and this limits the power density to W/cc for most NiZn ferrites. High Loss Effect

29 Effect of Magnetic Bias Fields μ' is a function of applied field Plots of μ' against H not usually given so must be measured H usually parallel to the rf field but perpendicular bias has been used Dynamic loss of Q with rate of change of bias field Ferrite characteristics must be measured at the anticipated bias field rate of change

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39 Nickel plated mild steel vacuum tube ready for assembly into the cavity. Provides magnetic shielding for the beam

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43 CAVITY TUNE SERVO CONTROL C b L C g L C g Bias current adjusted to keep cavity impedance purely resistive Typical bias currents vary from few amps to thousands of amps Minimum current must be kept high enough to cater for μ' variation

44 TUNING CONTROL Cavity bandwidth to amplitude and phase modulation is f o /2Q The capacitor C b and the two resonators form a tuned circuit for the bias system with a resonant frequency f = 1/(2π (2LC b )) The tuning control loop needs to account for both these time constants For rapid cycling FFAGs this can be a challenge The Q value of the resonators will vary with the rate of change of bias current and the gap voltage Worst case bandwidths must be catered for or dynamic control systems used

45 TUNING ERROR Phase error, due to bandwidth limitations, is given by the response to a ramped frequency input and: Where: φ o = K[t + τ(e (-t/τ) 1)] φ e steady state will be φ e = Kτ 1/(2πτ) = the tuning system bandwidth φ e = the cavity voltage phase error and: K = df/dt (πq/2f o ) f o FFAGs like PAMELA have very high df/dt = frequency at which df/dt is a maximum Methods to reduce these errors use additional digital control

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47 Cavity equivalent circuit

48 Sum of all Cavity Gap Voltages 1RF4 Gap Volts & Anode Current Accel Volts (kv) Gap V (kv) & Ia (A) 10 5 Gap V Ia+beam Time (ms) Time (ms) Frequency Cavity Impedance Frequency (MHz) Cavity Impedance (kohms) Time (ms) Time (ms)

49 Q-Value 100 With R(L) No R(L) TRig Q Time (ms) Dynamic Q

50 Q-Value 100 With R(L) No R(L) TRig Q Frequency (MHz)

51 FNL Booster Synchrotron drift tube system (33-53 MHz) Three resonators in parallel to achieve sufficiently low inductance No decoupling capacitor. No induced RF voltage in bias winding Inner wall of the resonator is resistive and must be driven as a shorted turn by the 10 turn bias field supply

52 FNL Booster RF system equivalent circuit

53 No RF voltage on the bias supply CERN PS Booster Cavity (3 8.4 MHz) Virtual RF ground in the middle of the cavity Resonator RF voltages in anti-phase Balanced RF voltages on the figure of eight bias windings Bias windings give strong coupling between the two resonators Single ended or differential drive can be used

54 CERN PS Booster equivalent circuit

55 Bias field bandwidth set by f o /(2Q) and 1/(2π (2LC b )) RF voltage is carried by the bias windings Need to avoid voltage breakdown from bias conductors Need to avoid RF resonances involving the bias windings. This gets more difficult with higher numbers of turns

56 Single gap design CERN LEAR cavity ( MHz) RF voltage on the bias winding is made zero at the bias supply feed point by splitting the winding at the mid voltage point The two windings are crossed and the parallel bias field is in opposite directions in each half of the cavity

57 CERN LEAR Bias system equivalent circuit Resonance produced by L b and C b

58 The Los Alamos/TRIUMF single gap cavity (46 61 MHz) Perpendicular solenoidal bias of microwave ferrite Elaborate cuts in cavity outer to reduce eddy currents Metallic cooling plates excluded

59 Los Alamos/TRIUMF cavity equivalent circuit High bias field used for low μ and high Q Some stray field on beam axis

60 HIGH ORDER MODES First resonant mode above fundamental occurs for 180 As resonators usually less than 26 next mode at least 5 times the fundamental These modes usually well damped by ferrite losses Additional modes are most likely caused by multi-turn bias windings and co-axial coupling arms (ISIS 11 MHz) Measure, model and damp as necessary

61 SUMMARY Reviewed ferrite properties; magnetic, electric and thermal From μ' & μ" select suitable ferrites to match required frequency range of accelerating system Estimate the size and length of a resonator for a particular gap voltage Determine the cooling required to stay below Curie temperature Select a suitable cavity design Estimate the tuning system bandwidth and tuning errors Measure the ferrite properties at the right rate of change of bias and voltage to obtain Q-value and impedance

62 ISIS Fundamental RF Cavity Parameters 7 Sept 2011 ISKG wl/v (rad) 0.38 wl/v (deg) 21.7 Resonator length (m) 1.12 Mean Ferrite Power (kw) Mean resonator Ferrite Power (kw) Ferrite vol for 1 resonator (cu m) Power density (W/ml) Power mean (W/m) Power max (W/m) Power density max (W/ml) Power density max (W/cu m) Temp rise (deg C) 5.51 Original data lost following laptop disk failure some time ago No of ferrites per resonator 35 k1 4.30E-08 No of cooling disks 36 k2 3.05E+06 Length of resonator (m) 1.12 r1 (m) r2 (m) r3 (m) d1 (mm) d2 (mm) fo (Hz) fo (MHz) µ epsil x µe epsil-e v (m/s) E lmax (m) Brfmax (T) lbmax(m) Zo (ohm) wl/v (deg) L (µh) Vgap (volts pk) Vgap max (kv pk) Vol fer (cu m) C (nf)