Ferrite Loaded Cavities for RF Accelerating Systems Ian Gardner ISIS, RAL FFAG School September 2011
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 http://cdsweb.cern.ch/record/211448/files/ce RN-92-03-V1-V2.pdf?version=1
ISIS 1RF Cavity and HPD
Schematic of 1RF cavity and High Power Drive (HPD)
Cross section of an ISIS 1.3 3.1 MHz cavity
APS LPRF Controls: Amplitude, Phase & Cavity Tuning. LVPS From LPRF Driver Amp. HPD Cavity Bias Supply BASIC RF SYSTEM
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
Ferrite in RF Cavities Magnetic permeability Dielectric constant Resistivity Thermal conductivity Effects of temperature and electric and magnetic fields on the above parameters
Ferrite B-H loop with added ac field Incremental permeability μ r = B ac / μ o H ac varies with operating point
Plots of μ r against bias field for various types of Philip s ferrite
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
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"
Plots of μ' and μ" against frequency for different grades of Philip s Ferrite
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.
Initial permeability as a function of temperature
Complex permeability as a function of frequency
80 60 mu value 40 20 0 0 500 1000 1500 2000 2500 Ibias (A) 4M2 μ' value with bias current on ISIS
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.
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)
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.
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
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
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
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 0.2 0.3 Tesla the maximum value of B rf is usually kept to 0.01 Tesla.
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
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
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 0.1 0.3 W/cc for most NiZn ferrites. High Loss Effect
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
Nickel plated mild steel vacuum tube ready for assembly into the cavity. Provides magnetic shielding for the beam
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
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
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
Cavity equivalent circuit
Sum of all Cavity Gap Voltages 1RF4 Gap Volts & Anode Current 200 20 150 15 Accel Volts (kv) 100 50 Gap V (kv) & Ia (A) 10 5 Gap V Ia+beam 0-5 0 5 10 15 Time (ms) 0-5 0 5 10 15 Time (ms) Frequency Cavity Impedance 3.5 1 3 0.8 Frequency (MHz) 2.5 2 Cavity Impedance (kohms) 0.6 0.4 1.5 0.2 1-5 0 5 10 15 Time (ms) 0-5 0 5 10 15 Time (ms)
200 150 Q-Value 100 With R(L) No R(L) TRig Q 50 0-5 0 5 10 15 Time (ms) Dynamic Q
200 150 Q-Value 100 With R(L) No R(L) TRig Q 50 0 1 1.5 2 2.5 3 3.5 Frequency (MHz)
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
FNL Booster RF system equivalent circuit
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
CERN PS Booster equivalent circuit
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
Single gap design CERN LEAR cavity (0.38 3.5 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
CERN LEAR Bias system equivalent circuit Resonance produced by L b and C b
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
Los Alamos/TRIUMF cavity equivalent circuit High bias field used for low μ and high Q Some stray field on beam axis
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
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
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) 26.13 Mean resonator Ferrite Power (kw) 13.06 Ferrite vol for 1 resonator (cu m) 0.11975 Power density (W/ml) 0.109094 Power mean (W/m) 11664.28 Power max (W/m) 12232.62 Power density max (W/ml) 0.123432 Power density max (W/cu m) 123432.3 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) 0.121 0.121 0.121 0.121 0.121 0.121 0.121 0.121 0.121 r2 (m) 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 r3 (m) 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 d1 (mm) 25 25 25 25 25 25 25 25 25 d2 (mm) 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 fo (Hz) 1300000 1403462 1536347 1715900 1977923 2414121 3379459 9152923 1403462 fo (MHz) 1.30 1.40 1.54 1.72 1.98 2.41 3.38 9.15 1.40 µ 70 60 50 40 30 20 10 1 60 epsil 10 10 10 10 10 10 10 10 10 x 0.703936 0.703936 0.703936 0.703936 0.703936 0.703936 0.703936 0.703936 0.703936 µe 39.09432 33.54277 27.99122 22.43968 16.88813 11.33658 5.785036 0.788644 33.54277 epsil-e 3.37765 3.37765 3.37765 3.37765 3.37765 3.37765 3.37765 3.37765 3.37765 v (m/s) 26107008 28184768 30853397 34459235 39721272 48481142 67867351 1.8E+08 28184768 lmax (m) 5.020579 5.020579 5.020579 5.020579 5.020579 5.020579 5.020579 5.020579 5.020579 Brfmax (T) 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 lbmax(m) 1.208319 1.208319 1.208319 1.208319 1.208319 1.208319 1.208319 1.208319 1.208319 Zo (ohm) 148.129 137.2091 125.3413 112.2255 97.35855 79.76721 56.98182 21.03893 137.2091 wl/v (deg) 21.6606 21.6606 21.6606 21.6606 21.6606 21.6606 21.6606 21.6606 21.6606 L (µh) 7.202341 6.179581 5.15682 4.134059 3.111298 2.088537 1.065776 0.145292 6.179581 Vgap (volts pk) 6000 6483.965 7107.771 7955.011 9201.571 11308.46 16156.43 59653.08 6483.965 Vgap max (kv pk) 6.0 6.5 7.1 8.0 9.2 11.3 16.2 59.7 6.5 Vol fer (cu m) 0.119749 0.119749 0.119749 0.119749 0.119749 0.119749 0.119749 0.119749 0.598746 C (nf) 2.081037 2.081037 2.081037 2.081037 2.081037 2.081037 2.081037 2.081037 2.081037