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Introduction to RF for
Accelerators
Dr G Burt
Lancaster University
Engineering
Electrostatic Acceleration
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Van-de Graaff - 1930s
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).
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
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You only need to hold off the voltage between two plates
not the full accelerating voltage of the accelerator.
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.
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.
Cavity Linacs
• These devices store large amounts of
energy at a specific frequency allowing low
power sources to reach high fields.
Circular Acceleration
Synchrotron Radiation from
an electron in a magnetic field:
B
e 2c 2
P 
C E 2 B 2
2
Energy loss per turn of a
machine with an average
bending radius :
E / rev 
C E 4

Energy loss must be replaced by RF system
cost scaling $ Ecm2
Overvoltage
• To provide a stable bunch you often will accelerate off
crest. This means the particles do not experience the
maximum beam energy.
• Vb=Vc cos(fs) = Vc q
• Where Vc is the cavity voltage and Vb is the voltage
experienced by the particle, f is the phase shift and q is
known as the overvoltage.
V
Vp
fs
Stable
region
f
Phase stability is given by
off-crest acceleration
Cavity Quality Factor
• An important definition is the cavity Q factor, given
by
U
Q0 
Pc
Where U is the stored energy given by,
1
2
U  0  H dV
2
The Q factor is 2 times the number of rf cycles it
takes to dissipate the energy stored in the cavity.
 t 
U  U 0 exp   
 Q0 
• The Q factor determines the maximum energy the
cavity can fill to with a given input power.
Cavities
• 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
(/c)2=(m/a)2+ (n/b)2+ (p/L)2
Where a, b and L are the width, height
and length of the cavity and m, n and p
are integers
Pillbox Cavities
Wave equation in cylindrical co-ordinates
1     1 
2
2
r




k
z   0
 r r  r  r 2  2




Solution to the wave equation
  A1 J m (k t r )e  im
• Transverse Electric (TE) modes
  ' m,n r  im
e
H z r ,    A1 J m 
 a 
ik z a 2
H t  2 t H z
 'm , n
ia 2
E t   2  zˆ   t H z 
 ' m,n
• Transverse Magnetic (TM) modes
  m,n r  im
e
E z r ,    A1 J m 
 a 
Et 
ik z a 2

2
m, n
t Ez
Ht 
ia 2

2
m,n
zˆ   t E z 
Cavity Modes
rθ
TE2,1
TE1,1
TE0,1
TM0,1
TEr,θ
Cylindrical (or pillbox) cavities are more common than rectangular cavities.
The indices here are
m = number of full wave variations around theta
n = number of half wave variations along the diameter
P = number of half wave variations along the length
The frequencies of these cavities are given by f = c/(2 * (z/r)
Where z is the nth root of the mth bessel function for TE modes or the nth root of
the derivative of the mth bessel function for TE modes or
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
TM010 Dipole Mode
H
 2.405r   it
Ez  E0 J 0 
e
 R 
Hz  0
Hr  0
i
 2.405r   it
H 
E0 J1 
e
Z0
 R 
E  0
E
Beam
Er  0
Z0=377 Ohms
A standing wave cavity
Accelerating Voltage
Ez, at t=0
Ez, at t=z/v
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
Position, z
The voltage now depends
on what phase the electron
enters the cavity at.
Position, z
If we calculate the voltage at
two phases 90 degrees
apart we get real and
imaginary components
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,
L/ 2
E  eVb  e

Ez  z, t  ei z / c dz
L/ 2
•
To receive the maximum kick the particle should traverse the cavity
in a half RF period.
c
L
2f
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.
V
•
1
maximum possible energy gain during transit
e
To receive the maximum kick the particle should
traverse the cavity in a half RF period. L  c
2f
• We can define an accelerating voltage for the cavity by
 L / 2

Vb     Ez  z, t  ei z / c dz   Ez 0 LT cos t 
 L / 2

• 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.
For TM010 mode
Ez, at t=z/v
 L / 2

i z / c
V     Ez  z , t  e dz 
 L / 2

 E0 cos t 
L/2

cos  z / c  dz
L/2
Position, z
L/2
sin  z / c  

  E0 cos t 


/
c

L/ 2
2sin  L / 2c 
 E0 cos t 
 /c
Hence voltage is maximised when L=c/2f
This is often approximated as
Where L=c/2f, T=2/
V  Ez 0 LT cos t 
V  E0 cos t 
2

L
Peak Surface Fields
• The accelerating gradient is the average gradient seen by an
electron bunch,
Eacc 
Vc
L
• 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.
Emax 

Eacc 2
For a pillbox
H max
A/ m
 2430
Eacc
MV / m
Electric Field Magnitude
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.
However if the conductivity is
finite the fields will not be
completely shielded at the
surface and the field will .
penetrate into the surface.
This causes currents to flow
and hence power is absorbed
in the surface which is
converted to heat.
Skin depth is the distance in
the surface that the current has
reduced to 1/e of the value at
the surface, denoted by
Current
Density, J.
x
1
1

cm
2  f r
The surface resistance is defined as
Rsurf 
1

For copper 1/ = 1.7 x 10-8 Wm
Power Dissipation
• The power lost in the cavity walls due to ohmic heating is given by,
Pc 
1
2
Rsurface  H dS
2
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.
Shunt Impedance
• Another useful definition is the shunt impedance,
2
1 Vc
Rs 
2 Pc
• 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)
TM010 Shunt Impedance
Vc 
2 E0 L
Pc 

i
 2.405r 
H 
E0 J1 

Z0
R


1
2
Rsurface  H dS
2
Pc ,ends
Pc , walls
2
E0
 2.405r 
 2 Rsurface  2 r J1 
 dr
Z0
 R 
2
2
E0 2
  RL 2 Rsurface J1  2.405 
Z0
2
E0 2
Pc   R  R  L  2 Rsurface J1  2.405 
Z0
2Z 0 L 

Rs 
2
3
 R  R  L  Rsurface J1  2.405
2
5 x104

Rsurface
Geometric shunt impedance,
R/Q
• If we divide the shunt impedance by the
Q factor we obtain,
2
R Vc

Q 2U
• 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.
TM010 R/Q
V
2 E0 L

i
 2.405r 
H 
E0 J1 

Z0
R


1
2
U  0  H dV
2
2
E0
 2.405r 
U  2 L  0   r J1 
 dr
Z0
 R 
2
U
 0 E0 2
2
R L J1  2.405 
2
8Z0 2
R
L
L

 150  196Ohms
2
Q  0c  2.405 J1  2.405 R
R
2
Geometry Constant
• It is also useful to use the geometry constant
G  RsurfaceQ0
• 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.
Q factor Pillbox
2
E0 2
Pc   R  R  L  2 Rsurface J1  2.405
Z0
U
 0 E0 2
2
Q0 
R L J1  2.405
2
0 RL
2  R  L  Rsurface
453L / R
G
 260
1  L / R 
2
453L / R

Rsurface 1  L / R 
Frequency Scaling
• Rsurf ~ f0.5 normal conducting
• Rsurf ~ f2 superconducting
• Qo ~ f0.5 normal conducting
• Qo ~ f0.5 superconducting
• Rs ~ f0.5 normal conducting
• Rs ~ f-1 superconducting
• R/Q ~ f0 normal conducting
• R/Q ~ f0 superconducting
Multicell
• It takes x4 power to double the voltage in one cavity but only x2 to
use two cavities/cells to achieve the same voltage (Rs ~number of
cells).
• To make it more efficient we can add either more cavities or more
cells. This unfortunately makes it worse for wakefields (see later
lectures)
• In order to make our accelerator more compact and cheaper we can
add more cells. We have lots of cavities coupled together so that we
only need one coupler.
This however adds
complexity in tuning,
wakefields and the gradient of
all cells is limited by the worst
cell.
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. (Limit is ~ 1 MW but 500
kW is safer)
• 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.
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.
Field Emission
• 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.
Field Enhancement
• 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.
10000
Beta
Elocal=b E0
100000
1000
100
10
2b
1
1
10
100
h/b
h
1000
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.
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.
Maximum Gradient Limits
• All the limiting
factors scale
differently with
frequency.
• They also mostly
vary with pulse
length.
• The limiting
factor tends to
be different from
cavity to cavity.
For a CW machine the gradient is limited by average heating instead. Also
need to think about the electricity bill as 1 MW is £200 per day.
Lecture 2
Capacitor
The electric field of the
TM010 mode is contained
between two metal
plates
E-Field
–
This is identical to a capacitor.
This means the end plates
accumulate charge and a
current will flow around the
edges
Surface
Current
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.
Resistor
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).
Finally, if the cavity has
a finite conductivity, the
surface current will flow
in the skin depth
causing ohmic heating
and hence power loss.
Equivalent circuits

To increase the
frequency
the
inductance
and
capacitance has to
be increased.
1
LC
2
Vc
Pc 
2R
CVc
U
2
2
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
Equivalent circuits
These simple circuit equations
can now be used to calculate
the cavity parameters such as Q
and R/Q.
U
C
Q0 

R
Pc
L
R
V2
1
L



Q0 2U C
C
In fact equivalent circuits have been proven to accurately
model couplers, cavity coupling, microphonics, beam loading
and field amplitudes in multicell cavities.
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.
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,
P Q
U
Qe 
Pe
b
e
Pc

0
Qe
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
1
1
1


QL Qe Q0
U
QL 
Ptot
Scattering Parameters
When making RF measurements, the most common measurement is the Sparameters.
Input signal
S1,1
Black Box
S2,1
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.
S=
S11 S12
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
Resonant Bandwidth
1.00
0.75
P
0.50
ω
  1 = 0
tL QL
0.25
0.00
-10
-5
0
5
10
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.
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).
1.00
The reflected power in steady
state is given by
S11
0.75
1  be
S11 
1  be
0.50
0.25
0.00
-10
-5
0
5
10
where
Q0
be 
Qe
Cavity Coupling
Cavity Behaviour examples
•Steady state
The most important behaviour we must understand is when
the cavity is in steady state (ie when the cavity stored energy
is constant and U=U0). We can use the definitions of beta and
Q to derive,
4bPf Q0
U0 
1  b 2 
We can also get voltage by using R/Q (remember the
overvoltage). From this equation we can see that the
cavity energy is maximum when β=1.
2
 b 1
Pr  
 Pf
 b 1
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.
Pref
Pfor
note:
No beam!
1
b  0.1
0.8
0.6
b 1
b  10
0.4
0.2
0
0
1
2
3
4
5
0t / 2QL
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.
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.
Coupling with Beam Loading
• The rf source will not see any difference between the
power dissipated in the cavity walls and the power
extracted by the beam hence we can calculate a new Q
factor, Qcb.
U
Qcb 
Pc  Pb
• this Qcb will replace Q0 when calculating cavity filling.
This means the match will change as well as needing
more power.
Qcb
b eb 
Qe
U0 
4beb Pf Qcb
1  beb 
2

• Normally we aim for b=1 with beam and have reflections
when filling.
Typical RF System
feedback
Low
Level
RF
RF
Amplifier
Transmission
System
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.
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
RF
Cavity
RF Power
Electron
RF
RF
gun
Input
Output
Collector
• The RF cavity converts the RF energy to beam energy.
• The CLIC concept is really a three-beam accelerator rather than a
two-beam.
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
Gain(dB)  10.log10 Gain 
• 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.
Electron Guns (Diodes)
• When a cathode is heated,
electrons are given sufficient
energy to leave the surface.
• When a high enough voltage
is applied, electrons will travel
across the voltage gap.
• A current is then measured
on the anode.
Triode Guns
• A grid can be inserted into a
diode to control the voltage on
the cathode surface.
Grid voltage
• An RF voltage can be applied
to the grid to produce bunches
of electrons.
Time
Electron bunches
Triodes and Tetrodes
The most basic types of RF amplifiers
are triodes and tetrodes. These
operate by using the grid to bunch the
beam and then the beam is collected
at the anode.
These are
usually low
frequency
tubes.
The anodes potential fluctuates with the
electron beam hence providing an ac voltage.
A tetrode also has a 2nd grid to screen the
control grid from the anode to avoid feedback.
Triode Theory
• The Beam induced from the cathode has a transient
current. The current is given by I=Idc+Iac
• The dc input power is then given by Pdc=VanodeIdc
• The ac input power is given by Pin=VgridIac
• The ac output power is given by Prf=VanodeIac
• In Class A Idc=Iac
Class A
• Efficiency= Prf / Pdc=50%
• Gain =Prf/Pin
Using different ratio of AC
to DC current we can
improve the efficiency at the
expense of Gain
Class B
CERN Tetrode Example
•
•
•
•
•
Frequency=200 MHz
Power= 62 kW
Gain=14 dB
Efficiency = 64%
Cathode Voltage= 10 kV
• Gain is low so needs a SSPA or IOT
driver. This lowers the overall
efficiency and increases the cost.
• A diacrode is a sort of two sided
tetrode that doubles the power.
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.
A
B
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
The lower energy electrons
then pass through the cavity
and force the electrons
within the metal to flow back
to the opposite side
C
Grid voltage
IOT Schematics
Time
Electron bunches
Density Modulation
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
Klystron Schematics
Interaction
energy
Electron
energy
Electron
density
Klystron
• RF Output Power
300kW
• DC, -51kV, 8.48 Amp
• 2 Meters tall
• 60% efficiency (40%
operating)
• 30W RF drive
• 40dB Gain
• 35,000 Hrs Lifetime
Combining Tubes
•IoT’s, tetrodes or SSPA’s are often combined to give a higher power output.
•This reduces efficiency as the combiners are lossy (perhaps 5-10% less).
•It is more reliable as if one amplifier breaks you only loose some of the power.
•Power output limited by heating, normally under 500 kW-1 MW.
Technical Data
Klystron
IOT
Density modulation
Electron Bunches formed by
direct from the cathode
velocity modulation from the cavities.
Several bunching cavities
High Gain
Long Device
Expensive
Considerable velocity spread
Maximum gap voltage determined
by the slower electrons
Rapid reduction in efficiency for
reduced output power
High Gain
Little velocity spread
Higher gap voltage
Increased output power
Higher efficiency
Efficiency
is
approximately
constant for reduced output power
Low Gain
Grid geometry will not permit
IOTs
to
operate
at
high
frequencies like Klystrons.
Solid State Power Amplifier (SSPA)
• We can also make a
high power amplifier by
combining hundreds of
low power solid state
amplifiers
SSPA vs Tubes
Advantages
• No warm-up time
• High reliability
• Low voltage (<100 V)
• Air cooling
• High stability
• Graceful degradation
Disadvantages
• Complexity
• Losses in combiners
• Failed transistors
must be isolated
• Electrically fragile
• High I2R losses
• Low efficiency
• High maintenance
Magnetrons
• For small industrial
accelerators the most
common source is the
magnetron.
• This works by having
an electron cloud
rotate around a
coaxial cathode.
• They are cheap and
fairly efficient and can
reach powers of 5 MW
pulsed or 30 kW CW
at 3 GHz (100 kW at
lower frequencies).
Phase stability is not good enough for large
accelerators.
It may be possible to phase-lock magnetrons to
allow them to be used for larger accelerator.
Magnetrons for medical linacs
Pulse Compression
For pulse linacs it is often
cheaper and easier to produce
longer RF pulses and compress
them to produce higher peak
powers.
Power
This is performed by storing the
RF in a cavity and switching the
external Q of the cavity (or
otherwise increasing the output
power).
Compressed
Pulse
Klystron
Pulse
time
When to use what types?
When to use what types
• In the range of 400 MHz to 1.3 GHz you have a choice.
There is no right answer different accelerators make
different choices.
• IoTs are higher efficiency but limited to <100 kW and
normally need combining.
• SSPA’s are very low down-time but expensive, inefficient
and need a parts replaced a lot.
• Klystrons are high power and difficult to swap so if one
breaks you have trouble.
• Tetrodes are very low gain so need more amplifiers to
drive them.
• Magnetrons are unstable so are not used for large
machines with multiple cavities.
Device frequency
• You can only buy many tubes for accelerators at
discrete frequencies hence most accelerators have to
use common frequencies. The frequencies are:
• 200 MHz, 267 MHz, 352 MHz, 400 MHz, 508 MHz,
650 MHz, 704 MHz
• 1.3 GHz, 2.87 GHz, 3 GHz, 3.7 GHz, 3.9 GHz, 5.6
GHz, 9.3 GHz, 11.424 GHz, 11.994 GHz
• The frequencies tend to correspond to integer
wavelengths in mm and inches and try to avoid
frequencies used in broadcast and comms.
Lecture 3
Generation of RF Current
A
A bunch of electrons
approaches a resonant
cavity and forces the
electrons to flow away
from the bunch.
The negative potential
difference causes the
electrons
to
slow
down and the energy
is absorbed into the
cavity
B
C
The lower energy electrons
then pass through the cavity
and force the electrons
within the metal to flow back
to the opposite side
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.
 2 z 2 
exp  
2 
2
c


• 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.
Mode Indices
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.
Panofsky-Wenzel Theorem
If we rearrange Farday’s Law (   E   dB )and integrating along z we
dt
can show
 E  z, t 

c  dzB  z ,  c   c  dz  dt 
   Ez  z , t  

z


0
0
t0
L
z
L
c
z
Inserting this into the Lorentz (transverse( force equation gives us
 dE  z , t 

z
z
dz
E
z
,

cB
z
,

c
dz
dt


E
z
,
t

 z 
0    c   c   0 t  dz

0
L
L
z
c
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
L
ic mV||
V    dz  Ez  z , c  ~ 
0
 rm
ic
z
This means the transverse voltage is given by the rate of change of the
longitudinal voltage
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.
Resonances
• As you are summing the contribution to the wake
from all previous bunches, resonances can
appear. For monopole modes we sum
 cos(n ) exp( n

2Q
)
• Hence resonances appear when  
n
2
n
• It is more complex for dipole modes as the sum
is

 sin( n ) exp( n 2Q )
n
• This leads to two resonances at +/-some Δfreq
from the monopole resonant condition.
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.
Beampipe cutoff
rθ
TEr,θ
TE1,1
TM0,1
In order to provide heavy damping it
is necessary to have the beampipes
cutoff to the TM01 mode at the
operating frequency but not to the
other modes at HOM frequencies.
In a circular waveguide/beampipes the indices here are
m = number of full wave variations around theta
n = number of half wave variations along the diameter
The cutoff frequencies of these are given by fc = c/(2 * (z/r)
Where z is the nth root of the mth bessel function for TM modes or the nth root of
the derivative of the mth bessel function for TE modes or (=2.4 for TM01 and 1.8
for TE11)
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.
V
L
R
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.
L
L
I
Cs
R
I
Cs
R
Cf
Also capacitive gaps can be added to the stub or loop inductance to make
resonant filters.
1
c 
LCs
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.
F-probe couplers
Capacative
gaps
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.
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]
Inductive
stubs
These additional capacatances and
inductances form resonances which can
increase or decrease the coupling at specific
frequencies.
frequency
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
To avoid taking the waveguides through the
cryomodule, ferrite dampers are often placed in
the waveguides to absorb all incident power.
waveguide 1
w2/2
w1/2
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 (axiallysymmetric) manufacturing
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.
Decay in beampipe
• When a mode is resonant in the cavity but below the
cut-off frequency of the beampipe or waveguide
dampers the fields decay exponentially in the beampipe.
• A=exp(-kz*z), where kz = 1/c*sqrt(c2 - 2)
The TM010 mode will also decay and
some fields will be absorbed in any
absorbers
It is necessary to tailor the beampipe
size and length to make sure the
TM010 mode is sufficiently attenuated
but all the HOMs are damped.
Often the beampipe can have flutes
added to reduce the cutoff of HOMs
without affecting the TM01 mode.
Multicell cavity damping
• Each coupler removes a given power when a field is applied to it.
• The Q factor and hence damping is given by Qe=U/P
• Multicell cavities have more stored energy hence have higher Q
factors.
• In addition HOMs can be trapped in the middle cells and will have
low fields at the couplers.
• Damping requirements must be carefully balanced vs the length and
cost of the RF section.
•
•
•
•
CEBAF = 5 cells, high current but a linac
DLS = 1 cell, high current storage ring
SOLIEL = 2 cell, high current storage ring
ILC =9 cells, high gradient low current
SCRF Cavities
The power required to keep a
cavity on filled to a set voltage is
the power extracted by the beam
plus the ohmic heating in the
walls.
In order to increase the efficiency
of coupling power to the beam we
need to minimise the ohmic
heating in the cavity walls.
The ohmic heating can be reduced by 5-6 orders of magnitude with the
use of a superconducting cavity.
RF superconductivity
The surface resistance has
the following dependence
• Rs increases with frequency squared
• Rs increases exponentially with
temperature
SCRF cavities have higher losses
as they increase in frequency,
for this reason there are few
SCRF cavities above 4GHz.
2
RBCS
1 f 
 17.67 
 2 10
  exp  

T  1.5 
T


4
Residual Resistance
An SRF cavity will decrease its resistance with temperature in theory.
However in practice there is often a minimum resistance due to the effects of
normal conducting impurities in the niobium.
One of the main effects is flux pinning where magnetic fields are frozen into
normal conducting impurities inside the superconductor. This can be avoided by
shielding the cavity from magnetic fields during cooldown.
RF Critical B field
When the electrons condence into
cooper pairs the resulting
superconducting state is more
ordered than the normal-conducting
state.
When a magnetic field is applied to a superconductor, supercurrents flow.
This increases the free energy of the superconducting state. When the free
energy of the superconducting state equals the normal conducting state the
flux enters the material.
For RF fields the flux continues to be excluded in a metastable state unit the
field reaches the critical superheating field (240 mT for Nb)
SRF Couplers
• Also a limited power in SRF couplers.
• WG limited to 500 kW due to multipactor (electron
cloud).
• Coax is limited to a similar amount by limited cooling of
inner coax.
Microphonics
Microphonics: Changes in frequency
caused by connections to the outside
world
•Vibrations
•Pressure Fluctuations
This means the cavity is not always on
resonance and will require more RF
power to fill.
Additionally the constantly varying
frequency will cause phase errors.
To avoid problems we need to artificially
broaden the cavity bandwidth by using
a lower Qe of at least 106
Filling factor
• The cryomodule in
an SRF system also
takes up significant
space.
• The filling factor is
the ratio of the cavity
length to cryomodule
length.
• It varies from a factor
of 5 for single cells to
1.5 for a 9 cell cavity
(typically its 2 extra
cells on each side).
Cryogenic systems
• All refrigerators have a technical efficiency, ηT of 20%-30%
• The Carnot efficiency is given by
T
c 
300  T
• The dynamic heat load, Pc, is the rf power dissipated in the
cavity walls.
• A static heat load, Ps, adds an additional heating
• Liquid helium transfer lines require 1W per metre, so total
loss is length L (More efficient lines can be used)
• It is standard to fill to an overcapacity, O
Total Power for N cryostats= O N (Pc +Ps +L) / (ηT ηc )
RF Cavities for Linacs and
Circular Accelerators
Circular Accelerators
Linac (High energy)
•High HOM damping
•CW operation  high power
couplers required
•High gradient cavities
•Multicell
•Pulsed operation (often)
CESR cavity
TESLA cavity
SCRF vs NCRF
SCRF
• More efficient (even
when including
cryogenic losses)
• Higher CW gradient
• Long pulse or CW only
• Complex system
needing cryostats and
cryogenics
• Only frequencies below
4 GHz.
NCRF
• Less efficient.
• Higher pulsed gradient
• Simpler systems, water
cooled
• More reliable
• Lower capital costs
• Smaller apertures mean
higher wakefields
RF for High Energy Linacs
• Linear accelerators RF requirements are very different to
those of circular acclerators.
Circular Accelerator
•Acceleration over many passes
•Emphasis on beam current
•Need to reduce instabilities
 HOM damping required
•CW operation
•Big SR contribution to RF losses
(lighter particles in particular)
 few high energy storage rings
as SR losses increase with E^4
Linac
•Acceleration in one pass
 High gradients and high
efficiency required
•Beam current limited by source
(no stacking)
•Emphasis on beam energy
•Often pulsed
Putting it all together
• First we need to know the beam current, how much it
needs to be accelerated by, and the overvoltage.
• Can use this to calculate required power and Q factors
for an SRF and/or NCRF system based on pillbox
numbers.
• Investigate possible power sources.
• Single or multicell?
• SCRF or NCRF
• Choose frequency.
• Model real cavity and look at HOM damping.
• Adjust calculations using numbers from RF models.