Transcript Introduction to RF Cavities for Accelerators

Introduction to RF Cavities
for Accelerators
Dr G Burt
Lancaster University
Engineering
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
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  eV  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

V     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.
Peak Surface Fields
• The accelerating gradient is the average gradient seen by an
electron bunch,
Eacc 
V
d
• 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.
Eacc
E max
Eacc
Bmax
Electric Field Magnitude
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.
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 2p 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.
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.
Shunt Impedance
• Another useful definition is the shunt impedance,
2
1V
R
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)
Geometric shunt impedance,
R/Q
• If we divide the shunt impedance by the
Q factor we obtain,
2
V
R

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.
Higher Order Modes
• There are a number of modes other than the TM010
mode. They have the same notation as waveguide
modes with the addition indice, p, notating the
longitudinatl variation.
• TE/Mmnp
• Modes are often classified by their m, indice. In circular
cavities m is the azimuthal variation.
• Monopole modes, m=0, accelerate and decelerate the
beam
• Dipole modes, m=1, kick the beam transverely
• Quadropole modes, m=2 can also kick the beam but are
weak near the axis.
Higher Order Modes
• Monopole modes, include the accelerating mode and
have m=0 (no azimuthal variation)
• There are TM and TE monopole modes, TM monopoles
decelerate the beam and are a problem.
• TE monopole mode are low loss and are useful for
energy storage, they have little interaction with beams.
TM011
Beam
Dipole Modes
E
E
Beam
TM110 Dipole Mode
TE111 Dipole Mode
H
H
Beam
Multi-cell structures
In a multi-cell
structure the
coupling between
the cells causes
each mode to split
into a number of
modes equal to the
number of cells.
The Pendulum
f0 
1
2p LC
The high resistance of the normal conducting
cavity walls is the largest source of power loss
P.E or
E
P.E or
E
K.E or B
Resistance of the
medium (air << Oil)
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.
Cavity Coupling
Probe coupling to E-field
Capacitive coupling
Higher penetration
higher coupling
Loop couples to the
B-field
Inductive coupling
Higher penetration
lower coupling
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

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
Beam Loading
• In addition to ohmic and external 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.