Transcript Cavity BPM Plans

Cavity BPM
A. Liapine, UCL
A Black Box View
Beam
z
out
BPM signal is a mixture of decaying
harmonic signals with different
amplitudes and decay times.
Some of the amplitudes depend mostly
on the bunch charge, some have a
strong offset dependence
nanoBPM Meeting, KEK, March 2005
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Excursion Into Waveguides (1)
The electromagnetic field is known to
propagate through a waveguide as a
wave (or a mixture of a few waves)
with a fixed configuration. This
configuration depends on the
frequency of oscillations, waveguide
type and excitation type.
electric field is shown with red lines,
magnetic with blue ones
► wave is Transverse Electric - the electric
field has no longitudinal component (in
some literature it is marked as H-wave)
► the direction of the propagation is given
by E x H
► nodes and antinodes of transversal
components of E and H coincide in case
of vacuum filling
►
nanoBPM Meeting, KEK, March 2005
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Excursion Into Waveguides (2)
indexes show the numbers of
antinodes of the field for both axes
- x, y for a rectangular, φ, r for a
circular waveguide
► the number of antinodes for the φ
direction is a doubled index (the
field must be continuous among φ)
►
magnetic coupling uses a loop acting to the magnetic field. The coupling strength
depends on the magnetic flux through the loop i.e. inductivity of the loop
► electric coupling uses an antenna , the coupling depends on its capacitance
► electromagnetic coupling is a sum of two – electric and magnetic, they may
sometimes even cancel each other
►
nanoBPM Meeting, KEK, March 2005
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Circular Waveguide
We solve the wave equation
in the cylindrical coordinate system
Look for the solutions in a form
Transversal components
follow from the Maxwell’s
equations
integrating by parts. Solutions are:
The boundary condition gives the critical k:
nanoBPM Meeting, KEK, March 2005
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Cylindrical Cavity Resonator
A cylindrical cavity is a piece of a circular waveguide cut
transversally with conductive planes at z=0 and z=L.
At these planes the sum of the transversal components
of the electric field has to be 0:
This boundary condition
says us that
In that way we get the equations describing all the
standing waves possible in the cavity…
…called eigenmodes and
coinciding frequencies called
eigenfrequencies
nanoBPM Meeting, KEK, March 2005
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Dipole Mode
A bunch propagating through the cavity interacts
with its eigenmodes exciting electromagnetic
oscillations in the cavity.
The excitation of the modes, which have a node at
r=0, is very sensitive to the beam offset, what is
used for the beam position detection.
The first dipole mode TM110 is used because it is the
strongest one among the others.
The phase of the excited field depends on the
direction of the offset.
nanoBPM Meeting, KEK, March 2005
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Useful Definitions
It is convenient to represent a
cavity as an RLC circuit, usually
loaded to external load by means
of an ideal transformator.
The impedance R is called the
shunt impedance.
The voltage in the cavity is
calculated among a certain path,
beam trajectory in our case.
n
L
R
C
Z
The internal quality factor is
introduced to indicate the decay
of the oscillations due to the
losses in the cavity walls.
The external quality factor
indicates the decay due to the
power coupled out of the cavity.
nanoBPM Meeting, KEK, March 2005
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Single Bunch Excitation (1)
The excitation is proportional to the
voltage seen by the bunch
We use the definition of the normalized
shunt impedance
The energy given by the bunch to the
mode n is
and get the excited voltage and stored
energy as
The voltage excited in the cavity is two
times higher
Using the definition of the external Q we
get the output power
With
nanoBPM Meeting, KEK, March 2005
we get the output voltage
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Single Bunch Excitation (2)
The dipole mode electric field in the cavity
Extension to the beam pipe region
Fit both fields in order to get constants at r=a
Using an integral
we get
And the field in the beampipe is
We need the voltage, so integrating
and using * again we get
The voltage is linear vs. offset!
nanoBPM Meeting, KEK, March 2005
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Excitation Summary
►
►
►
►
The bunch excites the eigenmodes of the cavity
passing through it
The dipole mode excitation has a significant
dependence on the beam offset, the phase depends on
the offset direction
The excited signal decays exponentially, depending on
how much power is lost in the walls and coupled out
The excitation is almost linear in the beam pipe range
nanoBPM Meeting, KEK, March 2005
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Multibunch Excitation
Exponential decay of the energy stored in
the cavity is given by
V
Were the loaded Q value is used. It takes
into account walls losses and output power
V
If the mode frequency is a harmonic of the
bunch repetition rate, an infinite bunch
train produces a voltage
t
t
The error can be calculated as
The sum of this series is
A fixed error gives a high limit for the
loaded Q value
nanoBPM Meeting, KEK, March 2005
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Beam Incline Impact
The ratio of the voltages does not
depend on the bunch charge
Incline component of the dipole mode is
excited if the beam trajectory is inclined
with respect to the z axis of the cavity.
We compare the excitation calculating the
voltages for the both cases.
Equivalent offset for a 5.5 GHz cavity
(x’ = 0.5 mrad)
Approximating the Bessel function we get
nanoBPM Meeting, KEK, March 2005
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Monopole Modes Impact
Monopole modes have the
highest excitation among all
other modes. The difference to
the dipole mode excitation
maybe 100 dB and more. The
first two monopole modes
surround the dipole mode
resonance.
Due to the finite Q values these modes
have components at the dipole mode
frequency. These components can not
be filtered out and need a mode
selective solution. A mode selective
coupling realizing the difference in the
field structure of dipole and monopole
modes is used in all the latest designs.
nanoBPM Meeting, KEK, March 2005
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Polarization and Cross-Talk

The excited dipole mode field can be represented as a combination of two polarizations.

Need to align the polarizations to x, y
and separate them in frequency.

nanoBPM Meeting, KEK, March 2005
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Thermal Noise
The spectral noise power density integrated over the bandwidth of the narrowest filter
in the electronics gives us the level of the noise component:
Following the path of the signal in the electronics and taking into account the losses
and the internal noise of the electronics we can estimate the resolution limit:
The final estimation has to take into account also the discretization noise.
nanoBPM Meeting, KEK, March 2005
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Impacts Summary
►
►
►
►
The energy stored in the cavity decays exponentially. If
the decay is not fast enough, the previous bunch signal
contributes to the next bunch signal.
An inclined beam excites the dipole mode even if it
passes through the centre. The phase difference between
position and incline components is 900.
Monopole modes are strongly excited and therefore
generate large backgrounds.
Asymmetries cause a coupling between x and y signals.
nanoBPM Meeting, KEK, March 2005
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Analog Signal Processing
The readings are waveforms in GHz range, so we need a
downconversion electronics. Basically, two methods are
available:
►homodyne receiver
►heterodyne receiver.
An accurate direct conversion is not possible because of
the high frequency.
nanoBPM Meeting, KEK, March 2005
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Homodyne Receiver
The signal is downconverted to the “direct
current” in one stage. Just a few components
are needed, the losses are low.
HR is very sensitive to the isolations
between LO and RF ports of the mixer.
I/Q mixer is usually used.
nanoBPM Meeting, KEK, March 2005
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Heterodyne Receiver
Downconversion is realized in several stages. That gives a better possibility for the
filtering and amplification of the signal. The mirror frequency issue does not seem to
be really dangerous in our case.
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I’m afraid that’s all I can say…
► Check
also
http://www.hep.ucl.ac.uk/~liapine/
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