Introduction to RF for Accelerators. Dr G Burt Lancaster University Engineering. Electrostatic Acceleration. +. .      . + + + + + +. Vande Graaff  1930s. A standard electrostatic accelerator is a Van de Graaf.
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Introduction to RF for Accelerators
Dr G Burt
Lancaster University
Engineering
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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).
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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.
Replace static fields by
timevarying periodic fields by only exposing the bunch to the wave at certain selected points.
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.
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
Wave equation in cylindrical coordinates
Solution to the wave equation
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
H
E
Beam
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
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
For a pillbox
Electric Field Magnitude
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
Rsurface is the surface resistance
P.E or
E
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)
EField
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
BField
Surface Current
The surface current travels round the outside of the cavity giving rise to a magnetic field and the cavity has some inductance.
–
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).
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
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.
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.
Elocal=b E0
2b
h
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.
feedback
Low Level RF
Transmission System
RF
Amplifier
Cavity
DC Power Supply or Modulator
RF
Cavity
RF Power
RF
Input
RF
Output
Collector
Electron
gun
RF Vacuum Tubes usually have a similar form. They all operate using high current (A  MA) low voltage (50kV500kV) 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.
RF
Input
RF
Output
Collector
Electron
gun
Bunched Beam
DC beam
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
Grid voltage
Density Modulation
Time
Electron bunches
4 IOT’s Combined in a combining cavity
Interaction energy
Electron energy
Electron density
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.
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
When making RF measurements, the most common measurement is the Sparameters.
Black Box
Input signal
S2,1
S1,1
forward transmission coefficient
input reflection coefficient
The S matrix is a mbym 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
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
ω0
QL
P
1
D w=
=
tL
ωω0
SC cavities have much smaller resonant bandwidth and longer time constants. Over the resonant bandwidth the phase of S21 also changes by 180 degrees.
note:
No beam!
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.
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.
critically coupled
under coupled
over coupled
The negative potential difference causes the electrons to slow down and the energy is absorbed into the cavity
C
A
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
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 offaxis bunches.
Once induced the dipole wakes can apply a kick via the transverse fields so onaxis bunches can still experience the effect of the wakes from preceding bunches.
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
I
Cs
R
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.
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
L
L
I
Cs
R
I
Cs
R
Cf
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.
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.
Fprobe couplers are a type of coaxial 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
Log[S21]
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.
frequency
waveguide 2
waveguide 1
w2/2
w1/2
Waveguide HOM couplers allow higher power flow than coaxial couplers and tend to be used in high current systems. They also have a natural cutoff frequency.
They also tend to be larger than coaxial couplers so are not used for lower current systems.
To avoid taking the waveguides through the cryomodule, ferrite dampers are often placed in the waveguides to absorb all incident power.
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
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.