1. Introduction to RF for Accelerators
Dr G Burt
Lancaster University
Engineering

2. Electrostatic Acceleration+ - - +

3. Van-de Graaff s
A standard electrostatic accelerator is a Van de Graaf
These devices are limited to about 30 MV by the voltage hold off across ceramic insulators used to generate the high voltages (dielectric breakdown).

4. RF Acceleration
By switching the charge on the plates in phase with the particle motion we can cause the particles to always see an acceleration
You only need to hold off the voltage between two plates not the full accelerating voltage of the accelerator.

5. RF acceleration
Alternating gradients allow higher energies as moving the charge in the walls allows continuous acceleration of bunched beams.
We cannot use smooth wall waveguide to contain rf in order to accelerate a beam as the phase velocity is faster than the speed of light, hence we cannot keep a bunch in phase with the wave.

6. Early Linear Accelerators
Proposed by Ising (1925)
First built by Wideröe (1928)
Replace static fields by
time-varying periodic fields by only exposing the bunch to the wave at certain selected points.

7. Cavity Linacs
These devices store large amounts of energy at a specific frequency allowing low power sources to reach high fields.

8. Cavity Quality Factor
An important definition is the cavity Q factor, given by
Where U is the stored energy given by,
The Q factor is 2p times the number of rf cycles it takes to dissipate the energy stored in the cavity.
The Q factor determines the maximum energy the cavity can fill to with a given input power.

9. Cavities (w/c)2=(mp/a)2+ (np/b)2+ (pp/L)2
If we place metal walls at each end of the waveguide we create a cavity.
The waves are reflected at both walls creating a standing wave.
If we superimpose a number of plane waves by reflection inside a cavities surface we can get cancellation of E|| and BT at the cavity walls.
The boundary conditions must also be met on these walls. These are met at discrete frequencies only when there is an integer number of half wavelengths in all directions. a L
The resonant frequency of a rectangular cavity can be given by
(w/c)2=(mp/a)2+ (np/b)2+ (pp/L)2
Where a, b and L are the width, height and length of the cavity and m, n and p are integers

10. Pillbox Cavities Transverse Electric (TE) modes
Wave equation in cylindrical co-ordinates
Transverse Electric (TE) modes
Transverse Magnetic (TM) modes
Solution to the wave equation

11. TM010 Accelerating mode Electric Fields
Almost every RF cavity operates using the TM010 accelerating mode.
This mode has a longitudinal electric field in the centre of the cavity which accelerates the electrons.
The magnetic field loops around this and caused ohmic heating.
Magnetic Fields

12. TM010 Dipole Mode H E
Beam

13. A standing wave cavity What is the purpose
The maximum volts per metre determines the acceleration of the electrons

14. Accelerating Voltage Ez, at t=0
Normally voltage is the potential difference between two points but an electron can never “see” this voltage as it has a finite velocity (ie the field varies in the time it takes the electron to cross the cavity
Ez, at t=z/v
Position, z
The voltage now depends on what phase the electron enters the cavity at.
If we calculate the voltage at two phases 90 degrees apart we get real and imaginary components
Position, z

15. Accelerating voltage
An electron travelling close to the speed of light traverses through a cavity. During its transit it sees a time varying electric field. If we use the voltage as complex, the maximum possible energy gain is given by the magnitude,
To receive the maximum kick the particle should traverse the cavity in a half RF period.

16. Transit Time Factor
An electron travelling close to the speed of light traverses through a cavity. During its transit it sees a time varying electric field.
To receive the maximum kick the particle should traverse the cavity in a half RF period.
We can define an accelerating voltage for the cavity by
This is given by the line integral of Ez as seen by the electron. Where T is known as the transit time factor and Ez0 is the peak axial electric field.

17. For TM010 mode Ez, at t=z/v Position, z
Hence voltage is maximised when L=c/2f
This is often approximated as
Where L=c/2f, T=2/p

18. Peak Surface Fields
The accelerating gradient is the average gradient seen by an electron bunch,
The limit to the energy in the cavity is often given by the peak surface electric and magnetic fields. Thus, it is useful to introduce the ratio between the peak surface electric field and the accelerating gradient, and the ratio between the peak surface magnetic field and the accelerating gradient.
For a pillbox
Electric Field Magnitude

19. Surface Resistance
As we have seen when a time varying magnetic field impinges on a conducting surface current flows in the conductor to shield the fields inside the conductor.
Current Density, J.
However if the conductivity is finite the fields will not be completely shielded at the surface due to ohms law (J=sE where s is the conductivity) and the field will penetrate into the surface. x
Skin depth is the distance in the surface that the current has reduced to 1/e of the value at the surface, denoted by .
This causes currents to flow and hence power is absorbed in the surface which is converted to heat.
The surface resistance is defined as

20. Power Dissipation
The power lost in the cavity walls due to ohmic heating is given by,
Rsurface is the surface resistance
This is important as all power lost in the cavity must be replaced by an rf source.
A significant amount of power is dissipated in cavity walls and hence the cavities are heated, this must be water cooled in warm cavities and cooled by liquid helium in superconducting cavities.

21. Shunt Impedance Another useful definition is the shunt impedance,
This quantity is useful for equivalent circuits as it relates the voltage in the circuit (cavity) to the power dissipated in the resistor (cavity walls).
Shunt Impedance is also important as it is related to the power induced in the mode by the beam (important for unwanted cavity modes)

22. TM010 Shunt Impedance

23. Geometric shunt impedance, R/Q
If we divide the shunt impedance by the Q factor we obtain,
This is very useful as it relates the accelerating voltage to the stored energy.
Also like the geometry constant this parameter is independent of frequency and cavity material.

24. TM010 R/Q

25. Geometry Constant It is also useful to use the geometry constant
This allows different cavities to be compared independent of size (frequency) or material, as it depends only on the cavity shape.
The Q factor is frequency dependant as Rs is frequency dependant.

26. Q factor Pillbox

27. The Pendulum
P.E or E
K.E or B
The high resistance of the normal conducting cavity walls is the largest source of power loss
Resistance of the medium (air << Oil)

28. Capacitor
E-Field
The electric field of the TM010 mode is contained between two metal plates –
This is identical to a capacitor.
This means the end plates accumulate charge and a current will flow around the edges
Surface Current

29. Inductor – B-Field Surface Current
The surface current travels round the outside of the cavity giving rise to a magnetic field and the cavity has some inductance. –

30. Resistor
Finally, if the cavity has a finite conductivity, the surface current will flow in the skin depth causing ohmic heating and hence power loss.
Surface Current
This can be accounted for by placing a resistor in the circuit.
In this model we assume the voltage across the resistor is the cavity voltage. Hence R takes the value of the cavity shunt impedance (not Rsurface).

31. Equivalent circuits
To increase the frequency the inductance and capacitance has to be increased.
The stored energy is just the stored energy in the capacitor.
The voltage given by the equivalent circuit does not contain the transit time factor, T. So remember
Vc=V0 T

32. Equivalent circuits
These simple circuit equations can now be used to calculate the cavity parameters such as Q and R/Q.
In fact equivalent circuits have been proven to accurately model couplers, cavity coupling, microphonics, beam loading and field amplitudes in multicell cavities.

33. Beam Loading
In addition to ohmic losses we must also consider the power extracted from the cavity by the beam.
The beam draws a power Pb=Vc Ibeam from the cavity.
Ibeam=q f, where q is the bunch charge and f is the repetition rate
This additional loss can be lumped in with the ohmic heating as an external circuit cannot differentiate between different passive losses.
This means that the cavity requires different powers without beam or with lower/higher beam currents.

34. Average Heating
In normal conducting cavities, the RF deposits large amounts of power as heat in the cavity walls.
This heat is removed by flushing cooling water through special copper cooling channels in the cavity. The faster the water flows (and the cooler), the more heat is removed.
For CW cavities, the cavity temperature reaches steady state when the water cooling removes as much power as is deposited in the RF structure.
This usually is required to be calculated in a Finite Element code to determine temperature rises.
Temperature rises can cause surface deformation, surface cracking, outgassing or even melting.
By pulsing the RF we can reach much higher gradients as the average power flow is much less than the peak power flow.

35. Pulsed Heating
Pulsed RF however has problems due to heat diffusion effects.
Over short timescales (<10ms) the heat doesn’t diffuse far enough into the material to reach the water cooling.
This means that all the heat is deposited in a small volume with no cooling.
Cyclic heating can lead to surface damage.

36. Field Enhancement Elocal=b E0
The surface of an accelerating structure will have a number of imperfections at the surface caused by grain boundaries, scratches, bumps etc.
As the surface is an equipotential the electric fields at these small imperfections can be greatly enhanced.
In some cases the field can be increase by a factor of several hundred.
Elocal=b E0 2b h

37. Field Emission
As we saw in Lecture 3, high electric fields can lead to electrons quantum tunnelling out of the structure creating a field emitted current.
Once emitted this field emitted current can interact with the cavity fields.
Although initially low energy, the electrons can potentially be accelerated to close to the speed of light with the main electron beam, if the fields are high enough.
This is known as dark current trapping.

38. Breakdown
Breakdown occurs when a plasma discharge is generated in the cavity.
This is almost always associated with some of the cavity walls being heated until it vaporises and the gas is then ionised by field emission. The exact mechanisms are still not well understood.
When this occurs all the incoming RF is reflected back up the coupler.
This is the major limitation to gradient in most pulsed RF cavities and can permanently damage the structure.

39. Kilpatrick Limits
A rough empirical formula for the peak surface electric field is
It is not clear why the field strength decreases with frequency.
It is also noted that breakdown is mitigated slightly by going to lower group velocity structures.
The maximum field strength also varies with pulse length as t-0.25 (only true for a limited number of pulse lengths)
As a SCRF cavity would quench long before breakdown, we only see breakdown in normal conducting structures.

40. Typical RF System A typical RF system contains
feedback
Low Level RF
Transmission System RF
Amplifier
Cavity
DC Power Supply or Modulator
A typical RF system contains
A LLRF system for amplitude and phase control
An RF amplifier to boost the LLRF signal
Power supply to provide electrical power to the Amplifier
A transmission system to take power from the Amplifier to the cavity
A cavity to transfer the RF power to the beam
Feedback from the cavity to the LLRF system to correct errors.

41. Transformer Principle
An accelerator is really a large vacuum transformer. It converts a high current, low voltage signal into a low current, high voltage signal.
The RF amplifier converts the energy in the high current beam to RF
The RF cavity converts the RF energy to beam energy.
The CLIC concept is really a three-beam accelerator rather than a two-beam. RF
Cavity
RF Power
Electron
gun RF
Input RF
Output
Collector

42. Vacuum Tube Principle
RF Vacuum Tubes usually have a similar form. They all operate using high current (A - MA) low voltage (50kV-500kV) electron beams. They rely on the RF input to bunch the beam. As the beam has much more power than the RF it can induce a much higher power at an output stage.
These devices act very much like a transistor when small ac voltages can control a much higher dc voltage, converting it to ac.
Electron
gun RF
Input RF
Output
Collector
Bunched Beam
DC beam

43. Basic Amplifier Equations
Input power has two components, the RF input power which is to be amplified and the DC input power to the beam.
Gain=RF Output Power / RF Input Power = Prf / Pin
RF Efficiency= RF Output Power / DC Input Power = Prf / Pdc
If the efficiency is low we need large DC power supplies and have a high electricity bill.
If the gain is low we need a high input power and may require a pre-amplifier.

44. Generation of RF Power
A bunch of electrons approaches a resonant cavity and forces the electrons within the metal to flow away from the bunch. B
The lower energy electrons then pass through the cavity and force the electrons within the metal to flow back to the opposite side A
At a disturbance in the beampipe such as a cavity or iris the negative potential difference causes the electrons to slow down and the energy is absorbed into the cavity C

45. IOT Schematics
Grid voltage
Density Modulation
Time
Electron bunches

46. IOT- Thales 80kW 34kV 2.2Amp 160mm dia, 800mm long, 23Kg weight
72.6% efficiency
25dB gain
160W RF drive
35,000 Hrs Lifetime
4 IOT’s Combined in a combining cavity
RF Output Power 300kW

47. Klystron Schematics Interaction energy Electron energy
Features highlighted in greater detail.
Electron energy
Electron density

48. Klystron RF Output Power 300kW DC, -51kV, 8.48 Amp 2 Meters tall
60% efficiency
30W RF drive
40dB Gain
35,000 Hrs Lifetime

49. Couplers
The couplers can also be represented in equivalent circuits. The RF source is represented by a ideal current source in parallel to an impedance and the coupler is represented as an n:1 turn transformer.

50. External Q factor
Ohmic losses are not the only loss mechanism in cavities. We also have to consider the loss from the couplers. We define this external Q as,
Where Pe is the power lost through the coupler when the RF sources are turned off.
We can then define a loaded Q factor, QL, which is the ‘real’ Q of the cavity

51. Scattering Parameters
When making RF measurements, the most common measurement is the S-parameters.
Black Box
Input signal
S2,1
S1,1
forward transmission coefficient
input reflection coefficient
The S matrix is a m-by-m matrix (where m is the number of available measurement ports). The elements are labelled S parameters of form Sab where a is the measurement port and b is the input port.
S11
S12
S =
S21
S22
The meaning of an S parameter is the ratio of the voltage measured at the measurement port to the voltage at the input port (assuming a CW input).
Sab =Va / Vb

52. Cavity responses
A resonant cavity will reflect all power at frequencies outwith its bandwidth hence S11=1 and S21=0.
The reflections are minimised (and transmission maximised) at the resonant frequency.
If the coupler is matched to the cavity (they have the same impedance) the reflections will go to zero and 100% of the power will get into the cavity when in steady state (ie the cavity is filled).
The reflected power in steady state is given by
where
S11

53. Resonant Bandwidth P ω0 1 D w= = tL QL ω-ω0
SC cavities have much smaller resonant bandwidth and longer time constants. Over the resonant bandwidth the phase of S21 also changes by 180 degrees.

54. Cavity Filling
When filling, the impedance of a resonant cavity varies with time and hence so does the match this means the reflections vary as the cavity fills.
note:
No beam!
As we vary the external Q of a cavity the filling behaves differently. Initially all power is reflected from the cavity, as the cavities fill the reflections reduce.
The cavity is only matched (reflections=0) if the external Q of the cavity is equal to the ohmic Q (you may include beam losses in this).
A conceptual explanation for this as the reflected power from the coupler and the emitted power from the cavity destructively interfere.

55. Coupling Strength Excited by a square pulse critically coupled
under coupled
over coupled

56. Generation of RF Current
The negative potential difference causes the electrons to slow down and the energy is absorbed into the cavity A C
The lower energy electrons then pass through the cavity and force the electrons within the metal to flow back to the opposite side
A bunch of electrons approaches a resonant cavity and forces the electrons to flow away from the bunch. B

57. Bunch Spectrum
A charged bunch can induce wakefields over a wide spectrum given by, fmax=1/T. A Gaussian bunch length has a Gaussian spectrum.
On the short timescale (within the bunch) all the frequencies induced can act on following electrons within the bunch.
On a longer timescale (between bunches) the high frequencies decay and only trapped low frequency (high Q) modes participate in the interaction.

58. Mode Indices

59. Dipole modes
Dipole mode have a transverse magnetic and/or transverse electric fields on axis. They have zero longitudinal field on axis. The longitudinal electric field increases approximately linearly with radius near the axis.
Electric Magnetic
Wakefields are only induced by the longitudinal electric field so dipole wakes are only induced by off-axis bunches.
Once induced the dipole wakes can apply a kick via the transverse fields so on-axis bunches can still experience the effect of the wakes from preceding bunches.

60. Panofsky-Wenzel Theorem
If we rearrange Farday’s Law ( )and integrating along z we can show
Inserting this into the Lorentz (transverse( force equation gives us
for a closed cavity where the 1st term on the RHS is zero at the limits of the integration due to the boundary conditions this can be shown to give
This means the transverse voltage is given by the rate of change of the longitudinal voltage

61. Multibunch Wakefields
For multibunch wakes, each bunch induces the same frequencies at different amplitudes and phases.
These interfere to increase or decrease the fields in the cavity.
As the fields are damped the wakes will tend to a steady state solution.

62. Resonances
As you are summing the contribution to the wake from all previous bunches, resonances can appear. For monopole modes we sum
Hence resonances appear when
It is more complex for dipole modes as the sum is
This leads to two resonances at +/-some Δfreq from the monopole resonant condition.

63. Damping
As the wakes from each bunch add together it is necessary to damp the wakes so that wakes from only a few bunches add together.
The smaller the bunch spacing the stronger the damping is required (NC linacs can require Q factors below 50).
This is normally achieved by adding external HOM couplers to the cavity.
These are normally quite complex as they must work over a wide frequency range while not coupling to the operating mode.
However the do not need to handle as much power as an input coupler.

64. Coaxial HOM couplers
HOM couplers can be represented by equivalent circuits. If the coupler couples to the electric field the current source is the electric field (induced by the beam in the cavity) integrated across the inner conductor surface area. I Cs R
If the coaxial coupler is bent at the tip to produce a loop it can coupler to the magnetic fields of the cavity. Here the voltage source is the induced emf from the time varying magnetic field and the inductor is the loops inductance. L V R

65. Loop HOM couplers
Inductive stubs to probe couplers can be added for impedance matching to the load at a single frequency or capacitive gaps can be added to loop couplers.
Also capacitive gaps can be added to the stub or loop inductance to make resonant filters. I Cs R L Cf
The drawback of stubs and capacitive gaps is that you get increase fields in the coupler (hence field emission and heating) and the complex fields can give rise to an electron discharge know as multipactor (see lecture 6).
As a result these methods are not employed on high current machines.

66. F-probe couplers
F-probe couplers are a type of co-axial coupler, commonly used to damp HOM’s in superconducting cavities.
Their complex shapes are designed to give the coupler additional capacitances and inductances.
These additional capacatances and inductances form resonances which can increase or decrease the coupling at specific frequencies.
Capacative gaps
Inductive stubs
Output antenna
The LRC circuit can be used to reduce coupling to the operating mode (which we do not wish to damp) or to increase coupling at dangerous HOM’s.
Log[S21]
frequency

67. Waveguide Couplers
Waveguide HOM couplers allow higher power flow than co-axial couplers and tend to be used in high current systems. They also have a natural cut-off frequency.
They also tend to be larger than co-axial couplers so are not used for lower current systems.
waveguide 2
waveguide 1
w2/2
w1/2
To avoid taking the waveguides through the cryomodule, ferrite dampers are often placed in the waveguides to absorb all incident power.

68. Choke Damping
load
choke
cavity
For high gradient accelerators, choke mode damping has been proposed. This design uses a ferite damper inside the cavity which is shielded from the operating mode using a ‘choke’. A Choke is a type of resonant filter that excludes certain frequencies from passing.
The advantage of this is simpler (axially-symmetric) manufacturing

69. Beampipe HOM Dampers
For really strong HOM damping we can place ferrite dampers directly in the beampipes. This needs a complicated engineering design to deal with the heating effects.