Download Automatic magnetic cycle editor for the CERN PSB

BE/Note/2010/NNN (sub ref)
2017-05-07
[email protected]
Automatic magnetic cycle editor for the CERN PSB
Alfred BLAS / BE-RF
Magnetic cycle, synchrotron.
Summary
The main horizontal bending field in the CERN PSB synchrotron is programmed as a function
of time called magnetic cycle. The creation of such a cycle takes into account enough
parameters to unease its empirical elaboration. In the present context with Linac 2 as the
injecting machine, there is typically only one operational cycle used for all proton beams.
The foreseen operation of the PSB with an extraction kinetic energy of 2 GeV (instead of 1.4
GeV as at present), together with a possible doubling of the injected beam charge with Linac
4, pushes the rf and the magnetic setups towards new limits. Using the existing hardware to
achieve higher performances means an optimisation of the magnetic cycle.
This note describes an application creating the synchrotron magnetic cycle from different
input parameters, either equipment or beam related.
The excel file used for the simulations can be found here:
https://blas.web.cern.ch/blas/Publications/PSB_Cycle.xlsm
Table of Contents
1.
Introduction .................................................................................................................................. 3
2. Magnetic cycles obtained with the automatic editor and simulation results .......................... 4
2.1 Fastest cycle achievable in the present PSB context (50 MeV at injection up to 1.4 GeV
at extraction) ..................................................................................................................................... 4
2.2 Maximum intensity possibly accelerated from 160 MeV to 2 GeV with the present
bendings and rf setup, but with a new 5 kV MPS ....................................................................... 7
2.3 Maximum LHC intensity (ultimate beam) accelerated on h=2, with the present
bendings and rf setup, but with a new 5 kV MPS ..................................................................... 10
2.4 Rf power required to accelerate 2.5 E13 protons per ring to 2 GeV. ............................... 14
3.
4.
Global approach used for the cycle editor.............................................................................. 17
3.1
Definition of the system ..................................................................................................... 17
3.2
Main constraints and core processing ............................................................................. 18
Description of the different ingredients or sub-systems that need to be considered....... 21
4.1.
Power supply feeding the PSB bending magnets .......................................................... 21
4.1.1
4.2
MPS model.................................................................................................................. 22
Magnetic circuit driven by the MPS................................................................................. 22
4.2.1
MPS load model ......................................................................................................... 25
4.3
RF system ............................................................................................................................. 26
4.4.
Beam dynamics ................................................................................................................... 29
4.4.1 Basic formulas ................................................................................................................... 29
4.4.2 Incoherent tune shift [10, 12] ........................................................................................... 31
4.4.3 Momentum p [eV/c] ........................................................................................................ 35
4.4.4 Beta...................................................................................................................................... 35
4.4.5 Adiabaticity........................................................................................................................ 36
4.4.6 Beam signal spectrum (cosine bunch shape) ................................................................ 36
4.4.7 Beam signal spectrum (rectangular bunch shape) ....................................................... 41
4.4.8 Space charge voltage ........................................................................................................ 42
4.4.9 MPS model ......................................................................................................................... 43
5. Required precision of the bending field ...................................................................................... 46
6. Tune shift and losses ...................................................................................................................... 48
Conclusion ........................................................................................................................................... 49
Acknowledgements ............................................................................................................................ 49
References ............................................................................................................................................. 50
Annex – Visual basic functions used in the Excel cycle editor..................................................... 51
-2-
1.
Introduction
The future Linac 4 beam will be injected into full height rf buckets in the PSB, using
optionally a process allowing for a shaping of its surface in the longitudinal phase space
(longitudinal painting [1]).
The created bunched beam current, with various possible intensities, will induce space
charge effects superimposed to the cavities rf driving voltage, thus impacting the rf acceptance.
The acceleration needs to fulfil adiabatic conditions and be such as to maintain the
acceptance above the beam emittance despite its reduction due to the change of stable phase.
The acceleration rate should also be limited in order not to require an rf current above the
capabilities of the rf system.
A contrario, the acceleration needs to be as fast as possible, within the limits imposed by
the magnetic circuit hardware, in order to limit the power dissipation within the dipoles in the
ring and also to reduce the time spent at low energy where high space charge forces dominate.
An automatic cycle editor that takes into account the different requirements and
constraints is described in this note.
-3-
2. Magnetic cycles obtained with the automatic editor and simulation
results
Results are presented here first without explanations on how they are obtained. This will allow
those only interested in the PSB 2 GeV upgrade feasibility study, to have the result they are
looking for, without being overwhelmed by details. How the simulations have been carried
out, is presented in the following chapters.
2.1 Fastest cycle achievable in the present PSB context (50 MeV at injection
up to 1.4 GeV at extraction)
This first example depicts a cycle accelerating particles injected the from Linac 2 output
kinetic energy of 50 MeV up to the present extraction energy of 1.4 GeV. The beam intensity
is chosen to be equal to the typical maximum intensity obtained in the PSB (1.1 E13 particles
accelerated on harmonic 1). This example is proposed in order to allow making a direct
comparison with the cycles being presently used.
The assumptions being made here are: single h=1 accelerating voltage (no dual harmonic),
maximum C02 cavity current available for acceleration = 2.5A, maximum bucket filling
during acceleration = 80%.
Figure 2.1.1: 50 MeV to 1.4 GeV magnetic cycle or current in the main dipoles [A] (in red or
top trace) with 1.1E13 protons accelerated with a single h=1cavity; 20 ms flat-top.
The bottom trace is the RMS average current [A] remaining well below the
present accepted limits.
Injection at C275, extraction at C595 (C805 with the actual 1.4 GeV cycles)
-4-
Figure 2.1.2: Part of the cavity (C02 for a single harmonic h=1 beam) rf current dedicated to
acceleration. The available limit is 3 A and the cycle is designed in order to
maintain a 0.5 A margin that allows for some tuning errors. Do not take into
account the line corresponding to the C04 limit, which is not involved here.
Figure 2.1.2 shows only the current required from the cavity amplifier in order to accelerate
the beam. Note that the cycle editor has limited the ramp to limit the current to 2.5 A as
specified. The value required to create the nominal 8 kV without beam is about 3A but does
not appear here although it is supplied by the amplifier.
Figure 2.1.3: Effective rf voltage when space charge effects have been deduced (bottom
trace); The top trace is the cavity voltage (8 kV peak)
-5-
Figure 2.1.4: h=1 bunch emittance [eV.s] bottom red trace compared to the h=1 rf bucket
acceptance (top trace). The part of the curve after extraction should not be considered. The
emittance remains below 80% of the acceptance as required by a programmable input
parameter
Figure 2.1.5: Vertical incoherent tune spread. A weighting coefficient has been applied to the
results of the formulae that provide the tune shift, in order to obtain the tune spread value
actually encountered in the machine. This weighting coefficient (0.4) take into account the
percentage of the tune shift that is actually a tune spread, the effect of the real transverse
distribution and the beneficial effect of the dual harmonic acceleration. This curve will be
used as a reference to evaluate losses due to tune spread in the following simulations.
-6-
2.2 Maximum intensity possibly accelerated from 160 MeV to 2 GeV with
the present bendings and rf setup, but with a new 5 kV MPS
This simulation describes a cycle accelerating, from 160 MeV to 2 GeV, the maximum
possible intensity (1.1E13 p) on h=1 and achievable with the new foreseen MPS when
respecting the other present hardware limits (power dissipation in the bendings, maximum rf
current and voltage).
At injection, the bucket is filled up to 80% and the filling ratio remains below this value
during the entire cycle.
Figure 2.2.1: 160 MeV to 2 GeV magnetic cycle or current in the main dipoles [A] (in red or
top trace) with 1.1E13 protons accelerated with a single h=1cavity; 20 ms flat-top.
The bottom trace is the RMS average current [A] remaining just below the present
accepted limits.
Injection at C275, extraction at C675 (C805 with the actual 1.4 GeV cycles)
-7-
Figure 2.2.2: Part of the cavity (C02 for a single harmonic h=1 beam) rf current dedicated to
acceleration. The available limit is 3 A and the cycle is designed in order to
maintain a 0.5 A margin that allows for some tuning errors. Do not take into
account the line corresponding to the C04 limit, which is not involved here.
Figure 2.2.2 shows only the current required from the cavity amplifier in order to accelerate
the beam. The value required to create the nominal 8 kV without beam, about 3A, does not
appear here although it is supplied by the amplifier.
-8-
Figure 2.2.3: Effective rf voltage when space charge effects have been deduced (bottom
trace); The top trace is the cavity voltage (8 kV peak)
Figure 2.2.4: h=1 bunch emittance [eV.s] bottom red trace compared to the h=1 rf bucket
acceptance (top trace). The part of the curve after extraction should not be considered. The
emittance remains below 80% of the acceptance as required by a programmable input
parameter
Figure 2.2.5: Vertical incoherent tune spread. A weighting coefficient has been applied to the
results of the formulae that provide the tune shift, in order to obtain the tune spread value
actually encountered in the machine. This weighting coefficient (0.4) take into account the
percentage of the tune shift that is actually a tune spread, the effect of the real transverse
distribution and the beneficial effect of the dual harmonic acceleration. The maximum tune
spread of -0.14 at 160 MeV compares favourably with the – 0.38 obtained at 50 MeV (see
figure 2.1.5 )
-9-
2.3 Maximum LHC intensity (ultimate beam) accelerated on h=2, with the
present bendings and rf setup, but with a new 5 kV MPS
This second example depicts a cycle accelerating the maximum LHC foreseen intensity
(3.25E12 p per ring on LHC 25 Ultimate), from 160 MeV to 2 GeV, on h=2. At Injection the
h=2 buckets are filled up to 80 %. This intensity value corresponds to about 13 injected turns
from Linac 4.
Figure 2.3.1: 160 MeV to 2 GeV magnetic cycle or current in the main dipoles [A] (in red or
top trace) with 3.25 E12 protons accelerated with a single h=2 cavity; 20 ms flattop. The bottom trace is the RMS average current [A] remaining just below the
present accepted limits.
Injection at C275, extraction at C645 (C805 with the actual 1.4 GeV cycles)
- 10 -
Figure 2.3.2: Part of the cavity (C02 for a single harmonic h=1 beam) rf current dedicated to
acceleration. The available limit of 2.5 A is not reached for this beam intensity.
Figure 2.3.3: Effective rf voltage when space charge effects have been deduced (bottom
trace); The top trace is the cavity voltage (8 kV peak)
- 11 -
Figure 2.3.4: h=2 bunch emittance [eV.s] bottom red trace compared to the h=2 rf bucket
acceptance (top trace). The part of the curve after extraction should not be considered. The
emittance remains below 80% of the acceptance as required by a programmable input
parameter
Figure 2.3.5: Vertical incoherent tune spread. A weighting coefficient has been applied to the
results of the formulae that provide the tune shift, in order to obtain the tune spread value
actually encountered in the machine. This weighting coefficient (0.5) take into account the
percentage of the tune shift that is actually a tune spread and the effect of the real transverse
- 12 -
distribution. The maximum tune spread of -0.31 at 160 MeV compares favourably with the –
0.38 obtained at 50 MeV (see figure 2.1.5 )
- 13 -
2.4 Rf power required to accelerate 2.5 E13 protons per ring to 2 GeV.
In order to achieve the acceleration of the maximum intensity foreseen in the Linac 4 era with
the present dipole cooling setup (2267 A RMS max), the C02 rf power needs to be increased
such that the cavity amplifier supplies 6.3 amperes for the acceleration only (this value
corresponds to a strict minimum without margin). The results with such improved rf
characteristics are shown below:
Figure 2.4.1: 160 MeV to 2 GeV magnetic cycle or current in the main dipoles [A] (in red or
top trace) with 2.5E13 protons accelerated with a single h=1cavity; 20 ms flat-top.
The bottom trace is the RMS average current [A] remaining just below the present
accepted limits.
Injection at C275, extraction at C675 (C805 with the actual 1.4 GeV cycles)
- 14 -
Figure 2.4.2: Part of the cavity (C02 for a single harmonic h=1 beam) rf current dedicated to
acceleration. The required 6.3 A are well above the present C02 limits.
Figure 2.4.3: Effective rf voltage when space charge effects have been deduced (bottom
trace); The top trace is the cavity voltage (8 kV peak)
- 15 -
Figure 2.4.4: h=1 bunch emittance [eV.s] bottom red trace compared to the h=1 rf bucket
acceptance (top trace). The part of the curve after extraction should not be considered. The
emittance remains below 80% of the acceptance as required by a programmable input
parameter
Figure 2.4.5: Vertical incoherent tune spread. A weighting coefficient has been applied to the
results of the formulae that provide the tune shift, in order to obtain the tune spread value
actually encountered in the machine. This weighting coefficient (0.4) take into account the
percentage of the tune shift that is actually a tune spread, the effect of the real transverse
distribution and the beneficial effect of the dual harmonic acceleration. The maximum tune
spread of -0.34 at 160 MeV compares well with the – 0.38 obtained at 50 MeV (see figure
2.1.5 ), meaning that low energy transverse losses are likely to be of the same order.
- 16 -
3.
Global approach used for the cycle editor
3.1
Definition of the system
The required PSB incoming beam parameters
Injection energy
Ion type with mass and charge
Longitudinal and transverse emittance at injection with a possible increase law along the cycle
The rf system (only the narrow band ferrite tuned cavity case is treated here)
Maximum cavity voltage
Maximum available current for beam acceleration
Harmonic of the accelerating signal (w.r.t. the revolution
The main power supply (MPS) with its bending magnets circuit
Maximum voltage
Maximum and minimum currents
R, L, C impedance values
Maximum power dissipation
Maximum Vdot (voltage variation rate)
- 17 -
3.2
Main constraints and core processing
The main goal is to increase the magnetic field up to the extraction value as fast as possible in
order to limit the power dissipation within the bending magnets and also to keep the PSB cycle
length within the 1.2 s present value.
At each time step, when increasing the field, the following parameters will have to be checked
for:
Bucket filling ratio below the maximum allowed value Bucket_Fill_max
Cavity rf current requirement below I_RF_max
Current requirements in the bendings below the MPS limits
Bendings power dissipation below limits
Rf bucket changes to be “adiabatic”
List of the effects encountered when the bending field ramp is increased (faster cycle):
The power dissipation within the bendings is lowered
The rf bucket and the bunch length shrink meaning more peak current and more space
charge effects (longitudinally and transversally), and the risk that the acceptance goes below
the beam emittance with induced losses
The rf cavity current required for acceleration is increased
The variation d2IMPS/dt2 causes a bucket shape change with induced non-adiabatic effects
- 18 -
Requirements for the flat-top duration.
On the extraction plateau, the beam needs to be synchronized and beforehand optionally
splitted.
The required time for the synchronisation is in theory null as there are (not yet tested) means to
prepare this process during the end of the acceleration with the new digital low-level rf
hardware. For information, the synchrotron frequency at 2 GeV, h=1 with 8kV rf voltage is 256
Hz.
The splitting process takes about 10 ms in the present 1.4 GeV context. Although this might not
be the shortest possible value (no one has yet tried to make it shorter), we can estimate that by
multiplying these 10 ms by the ratio of synchrotron frequencies (1.74 = 447 Hz/256 Hz), we get
a safe estimation of the required time (17.4 ms).
Another way of approaching the problem would be to estimate the splitting process time by
comparing it to the time required for bunching at injection. At 50 MeV, with a maximum
synchrotron frequency of 2 kHz at the final voltage of 8 kV, the bunching takes 1 ms (this is an
optimized time). Multiplying this value by the ratio of synchrotron frequencies (7.81 =
2kHz/256 Hz) one gets 7.81 ms. This value could be multiplied by two for a even more
conservative approach by considering that the splitting could be compared to a de-bunching
followed by a re-bunching. With this comparison, the splitting would last 16 ms.
In the simulations that are presented at the end of the document, the value of 20 ms for the flattop has been used.
The above constraints have been included in an ExcelTM simulation program to provide results
precise enough to elaborate strategies for a future upgrade of the PSB to 2 GeV.
The global processing is described in figure 3.2.1.
The 1.2 s cycle is divided into time intervals of an arbitrary duration along which the different
calculated values are averaged. At each time sample corresponds a value of all the required
parameters needed for a good overview of the acceleration process.
Note that the adiabaticity does not appear as a constraint within the processing loop depicted in
figure 3.2.1. This is just due to the fact that it didn’t appear as a critical parameter (the value is
nevertheless observed in the Excel spreadsheet); it could be easily added in the script.
The MPS power also doesn’t appear. The integrated energy is plotted and observed sideways. If
it exceeds the limits, the user needs to lower the constraints by either lowering the beam current
or by increasing the maximum allowed bunching factor. He can also decrease the injected beam
longitudinal emittance.
The core script depicted in figure 3.2.1 allows defining the maximum field increase in order to
fulfil the basic rf current and Bucket filling constraints. The demanded B value is then loaded
into an MPS model that determines the value that can actually be reached. The obtained field is
then used to calculate all the beam and rf parameters for the actual time sample and for the
next.
- 19 -
Step = 1E-4 * Bdot_max / (t0 – t -1)
i=0
B = B -1
Bdot = Bdot -1
Bunch Length = Bunch_Length -1
B = B + i * Step
Bdot = B - B -1 / (t0 – t -1)
Ion charge
Ion Mass
rho
Radius
Beta
Gamma
eta
Frev
P
ΔVrf / turn
Obtained
B0
i=i+1
Bunch_Q
Beam pipe size
Beam transverse size
VRF set-point
IBEAM RF
VSpace charge
Sin φs
VMAX, VdotMAX
MPS
Model
NewBunch
Length
IRF_MAX
BF
>=
BFMAX
YES
IRF_MAX
BFMAX
YES
EXIT
use B as
required value
BF = Bucket Filling
NO
Bunch Emittance
Figure 3.2.1
Bucket Area
Bucket filling
Core process used to calculate the accelerating cycle
- 20 -
B_over_I
R, L
I RF
I RF
>=
IMAX, IMIN
4.
Description of the different ingredients or sub-systems that need
to be considered
Each sub-system will be depicted with a summary of the constraints related to cycle
edition. The resulting Visual basic code employed in Excel is available in the annex.
4.1.
Power supply feeding the PSB bending magnets
In order to edit a magnetic cycle, the limitations of the power source need to be known.
The main power supply (MPS) is presently constructed [2] using 5 sets of thyristors in
charge of rectifying a high voltage 3-phase source in order to supply a maximum output voltage
of 1 kV. 4 of these sets are combined so as to add-up their voltage for a maximum of 4 kV. The
fifth and last thyristor’s cell is used as a spare.
The system block diagram is available using
https://edms.cern.ch/file/1070963/1/1070963_V1_TEEPC_A2.pdf
the
following
link:
Using a thyristor means changing its angle of conduction by means of changing its firing
time. This allows transforming duration into the integral of a sine wave voltage along the same
time duration. The thyristor setup being followed by a low-pass filter, only remains the DC
value of the voltage. In summary, the MPS is basically a time to voltage converter.
As the MPS finally allows creating a magnetic field (to a certain extent proportional to
the electric current), it was found convenient to use a current probe within a feedback loop so
as to make the system controlled as a current source.
In the near future (within the Linac 4 era), the dipolar field measurement will be used into
the loop so as to make the MPS a horizontal bending field source, which is finally the most
convenient option. This will allow cancelling the non linear effects of magnet saturation
together with the eddy current induced field ripple.
For the cycle editing, parameters in table 3.1.1 need to be taken into account [3] and [4].
MPS parameters
Present limits
Future (Linac 4) limits
Maximum voltage
+ 3800 / -3400 V
+/- 5000V
Maximum current
+ 4200 A
+ 5500 A
Minimum current
100 A
100 A
Maximum Voltage ramp
0.14 kV/ms
1 kV/ms
Table 4.1.1
List of PSB MPS limiting parameters
- 21 -
Note in table 4.1.1 that a minimum current value is required for the MPS to maintain the
current (or field) control loop into a stable state. The given empirical value [2] needs to be
confirmed after a deeper analysis.
4.1.1 MPS model
The 4 parameters in table 4.1.1 were selected as they represent a restriction to what can
be achieved for the cycle. Table 4.1.1.1 summarizes how these values will be used in a context
were they will limit the value of the field or its rate of change.
MPS limiting parameters
Employment within the cycle editor
Max voltage VMPS MAX
Limits the rate of field
Max current
IMPS MAX
Limits the maximum field achievable
Min current
IMPS MIN
Gives the minimum field value
Max voltage ramp (dVMPS/dt)MAX
Table 4.1.1.1
4.2
Limits the rate of field change
Impact of the MPS limiting factors on a cycle edition
Magnetic circuit driven by the MPS
The power source having been described with its limitations, now comes the turn of the
load that includes the bending magnet. These two ingredients will finally allow estimating the
total field dynamics.
The main power supply (MPS) is feeding a circuit [5] including different magnetic
elements: the horizontal dipoles (BHZ and reference magnet), the focusing quadripoles (QFO)
and the defocusing quadripoles (QDE).
The “trim” power supplies have been represented; their aim is to provide adjustments to
change the tune of the synchrotron (QFO, QDE) or to compensate the saturation effect of the
bendings.
- 22 -
TRIM
A
BHZ (64x)
Ring 1+4
TRIM
QFO
QFO (128x)
4 Rings
Reference
magnet
MPS
TRIM
QDE
5 kΩ
QDE (64x)
4 Rings
BHZ (64x)
Ring 2+3
Figure 4.2.1
Magnetic circuit driven by the PSB main power supply [2]
From the MPS viewpoint the circuit can be represented as in figure 4.2.2.
Rm
Ie
Ip
Im
MPS
Figure 4.2.2
Eddy
current
model
Lm
Ie1
Ie2
Re1
Re2
Le1
Le2
Cp
Equivalent electric circuit driven by the PSB main power supply [5]
- 23 -
The electrical components represented in figure 4.2.2 have values that can be found in
table 4.2.1. Below 600 kHz, this model gives less than +/- 5% error on the estimation of Eddy
current effects.
Table 4.2.1
Rm
400 mΩ
Re1
110 Ω
Re2
1420 Ω
Lm
191 mH
Le1
0.957 mH
Le2
0.583 mH
CP
280 pF
List of values to be used in figure 3.2.2 [4] [5]
From table 4.2.1 it can be noted that the circuit elements used to model the eddy current
effect (Lm, Le1, Le2, CP) can be neglected during most of the cycle where the current transitions
are slow. Things are different during transients, like when approaching the flat-top with a high
current slope and then abruptly changing this slope to zero, where resonant effects have been
observed [6].
Figure 4.2.3
current ramps [6]
Current fluctuations observed on the PSB flat top for different initial
- 24 -
Figure 4.2.3 shows how the field can fluctuate at the flat top for durations up to 100 ms
depending on the initial dI/dt slope. This effect has not yet been analysed in terms of interaction
with the beam, but is likely to be deleterious for the splitting process.
Knowing the equivalent circuit (figure 4.2.2), these parasitic transients can be avoided by
adapting the shape of the cycle close to the magnetic plateau, at the price of a slower cycling
together with an associated increase of the dissipated power in the magnets. This latter effect is
not wanted as the thermal situation is already critical presently at 1.4 GeV.
Hopefully, in the foreseen PSB MPS setup, the feedback control loop will be using the
field measurement as the observed variable. This means that these parasitic current/field
oscillations will be removed. In terms of power, figure 4.2.3 suggests that the MPS will be
asked to lower its voltage to compensate the Eddy current effect.
Simplifying the equivalent load circuit of the MPS described in figure 4.2 by removing
the components corresponding to the Eddy current thus doesn’t imply an underestimation of the
power requirements. The resulting circuit (figure 4.2.4) is finally more precise in a context with
a feedback loop using the value of the actual magnetic field.
Rm
Im
Lm
MPS
Figure 4.2.4
The MPS with its load equivalent circuit in the context of the measured
field feedback control
A major characteristic of the magnetic circuit is the B = f(H) curve. This function allows
to estimate the field as a function of the current. As the saturation effects in the PSB are delt
with the trim power supplies the magnetization curve can be approximated by a straight line.
4.2.1 MPS load model
The simple R/L load obtained after simplification means only 2 parameters need to be
taken into account for the edition of the magnetic cycle, as summarized in table 4.2.1.1.
- 25 -
MPS load parameters
Employment within the cycle editor
Rm [Ω]
Lm/Rm time constant, required MPS voltage, actual
bending field and dissipated energy are a function of Rm.
Lm [H]
Lm/Rm time constant, required MPS voltage and actual
bending field are a function of Rm.
B_over_I [T/A]
Allows estimating the field as a function of the current
= 2.14 * 10-4 T/A in the PSB
Table 4.2.1.1
4.3
List of the MPS load values to be used for the cycle editor
RF system
This paragraph depicts how the cavity power setup limits the rate of acceleration due to
its limit in available current from the final amplifier. Only the narrow band ferrite loaded cavity
case is treated here.
The energy required to accelerate the beam is supplied by an rf cavity. The present
cavities are narrow-band ferrite loaded cavities tuned to the required frequency by a current
loop that applies a magnetic field to the ferrite. This slow-changing field, varying with the
accelerating frequency, changes the permeability of the ferrite and thus the cavity resonant
frequency.
Figure 4.3.1 gives a simplified overview of the cavity setup with the different involved
currents.
Fast
Feedback
RF
TETRODE
CAVITY
BEAM
Power
Amplifier
Set point
Low Level
RF
RPD
Tuner
Filter
Reactive
Power
Detector
Figure 4.3.1
Block diagram of the PSB ferrite cavity setup
- 26 -
The bunched beam in the ring corresponds to an rf current in quadrature phase with the
cavity rf signal. In a below transition case as encountered in the PSB, the beam current is
actually in quadrature advance (= lag due to the sign of the proton charge) and is thus perceived
by the power amplifier (equivalent current source using a tetrode) as an extra capacitive load.
The effect is depicted in figure 4.3.2 where the different signals are represented as vectors in
the context of a non-accelerated beam.
TETRODE
INDUCED
VOLTAGE
CAVITY
DETUNING
SET-POINT
VOLTAGE
CAVITY
TETRODE
TOTAL
VOLTAGE
CURRENT
BEAM
CURRENT
Figure 4.3.2
BEAM
INDUCED
VOLTAGE
BEAM
INDUCED
VOLTAGE
Vector diagram of rf signals within a cavity (non-accelerating case)
Figure 4.3.2 shows how by detuning the cavity, the tetrode current can be in phase with
the cavity voltage despite the presence of beam. The reactive (capacitive) effect of the beam is
compensated by an increase of the cavity inductance. The cathode current is multiplied by the
out of tune cavity impedance to create the tetrode induced voltage and the beam current
undergoes the same effect. The closed-loop system is stabilized when the voltage sum of these
two vectors is in phase with the driving signal (set-point voltage). One interesting effect of such
a technique is that, in a non-accelerating case, the tetrode current isn’t required to increase to
compensate for the presence of beam. Actually, when the cavity is detuned to compensate for
the beam presence, its impedance modulus is lower, implying more current for the same
voltage, but this effect is counteracted by the beam induced voltage, which due to the cavity
detuning, has a beneficial effect on the required rf voltage.
When the beam is accelerated, its rf vector component turns toward the real axis in
opposite phase with respect to the cavity rf voltage. This effect is depicted in figure 3.3.3.
CAVITY
DETUNING
SET-POINT
VOLTAGE
BEAM
CURRENT
Figure 4.3.3
TETRODE
INDUCED
VOLTAGE
BEAM
INDUCED
VOLTAGE
CAVITY
TETRODE
TOTAL
VOLTAGE
CURRENT
BEAM
INDUCED
VOLTAGE
Vector diagram of rf signals within a cavity (accelerating case)
- 27 -
As the beam current is less reactive in the accelerating case, the tuning loop needs to
compensate less, up to a theoretical point where the stable phase is 90o and the rf beam current
is real and in opposite phase with respect to the cavity voltage. In such an extreme case, no
tuning compensation is required and the tetrode current needs to be increased by an amount
equal to the beam current in order to maintain the expected rf cavity voltage.
The required tetrode current can be expressed from a geometrical description, like in
figures 4.3.2 and 4.3.3, as a function of the number of charges, particle velocity, stable phase
value and cavity Q value. It can also be estimated with enough precision (in the context of
power requirements estimation) as the value IB*sin(φS); IB being the amplitude of the rf
component of the beam current and φS the stable phase. Note that the stable phase is also an
approximation of the beam vector angle, valid in the case of short bunches. Longer bunches
have an asymmetric shape and thus the stable phase does not represent precisely the beam
phase. This long bunch effect translates into a beam phase that is larger than the actual stable
phase.
The approximation ITETRODE = ITETRODE NO BEAM + IB*sin(φS) remains nevertheless valid in
our context.
Table 4.3.1 summarizes the information required to establish the magnetic cycle
Cavity parameters
Employment within the cycle editor
RC
Cavity resistive value, required to evaluate the tetrode current
necessary to create the expected rf voltage without beam.
ITETRODE 0 peak = VRF peak / RC
ITETRODE MAX peak
Maximum tetrode current value (measured at the cavity gap)
which allows to check for the following requirement:
ITETRODE MAX peak - ITETRODE 0 peak > IB peak*sin(φS)
Table 4.3.1
band cavity only)
List of the cavity related values to be used for the cycle editor (narrow
- 28 -
4.4.
Beam dynamics
This paragraph depicts how different beam dynamics related parameters are evaluated in
the Excel application
4.4.1 Basic formulas
R [m] = synchrotron radius = 25 m for the PSB
c [m/s] = velocity of light = 2.99792458 * 108
α(ΓS) [ ] = moving bucket function with ΓS = |sin(φS)| and φS [rad] the stable phase. Using
α(ΓS) = 1 gives the stationary bucket area.
α(ΓS) is approximated with less than 0.5% error by the polynomial:
α(ΓS ) = −4.869298085 ∙ ΓS7 + 21.563437656 ∙ ΓS6 − 38.304144731 ∙ ΓS5 +
35.560953904 ∙ ΓS4 − 18.839475275 ∙ ΓS3 + 6.323558981 ∙ ΓS2 − 2.435032449 ∙ ΓS +
1
e [ ] = elementary charge = 1 when eV is used as the energy unit
VRF [V] = peak rf voltage at the cavity gap
ES [eV/u] = 𝛾 ∙ 𝐸0𝑢 = total energy per atomic mass unit (amu) of the synchronous particle
ζ [ ] = total charge
Ar [ ] = ion mass relative to the amu
η [ ] = 1⁄ 2 − 1⁄𝛾 2 = frequency slip factor
𝛾𝑇
h [ ] = harmonic number (number of rf period per revolution)
𝛾=
1
√1 − 𝛽 2
𝐹𝑅𝐸𝑉 =
p=
𝛽∙𝑐
2𝜋 ∙ 𝑅
√ES2 − E02
eV
𝑜𝑟 = √ES2 − E02 when p expressed in [ ]
c
c
E0 [eV] is the energy of the rest particle
- 29 -
Moving_Bucket_Area [eV.s/nucleon] = 16 ∙
𝑅
∙ 𝛼 (ΓS )√
𝑐
𝑒∙𝑉𝑅𝐹 ∙𝐸𝑆 ∙𝜁
2𝜋∙|𝜂|∙ℎ3 ∙𝐴𝑟
[7]
α(Γs) the moving bucket coefficient, can be approximated with less than 0.5% by a power
series:
𝛼(ΓS ) ≈ −4.869298085 ∙ Γ𝑆7 + 21.563437656 ∙ Γ𝑆6 − 38.304144731 ∙ Γ𝑆5 + 35.560953904
∙ Γ𝑆4 − 18.839475275 ∙ Γ𝑆3 + 6.323558981 ∙ Γ𝑆2 − 2.435032449 ∙ Γ𝑆 + 1
Bunch length in a stationary bucket (approximation with less than 0.5% error) [7]
𝜋 3 ⁄8 − √(𝜋 3 ⁄8) − 16 ∙ 𝐵𝑓 ∙ {(𝜋 3 ⁄16) − 1}
√
(𝐵𝑓𝑟)
[𝑟𝑓
𝜑𝑆𝑡𝑎𝑡 𝐵𝑢𝑛𝑐ℎ
𝑟𝑎𝑑] ≈ 𝜋 ∙
(𝜋 3 ⁄16) − 1
Bfr [ ] is the bucket filling ratio = beam emittance / bucket area
Bunch length in a moving bucket [9]
𝜑𝑀𝑜𝑣 𝐵𝑢𝑛𝑐ℎ ∝ {
𝜑𝑀𝑜𝑣 𝐵𝑢𝑛𝑐ℎ
𝜂
𝐸𝑆 ∙ 𝑅𝑆2 ∙ 𝑉̂𝑅𝐹 ∙ √1 − 𝑠𝑖𝑛2 (𝜑𝑆 )
𝜂
∝ {
}
𝐸𝑆 ∙ 𝑅𝑆2 ∙ 𝑉̂𝑅𝐹
1⁄4
}
1⁄4
∙ {1 − 𝑠𝑖𝑛2 (𝜑𝑆 )}−1⁄8
At a given energy, the bunch length is thus proportional to {1 – sin2(φS)}-1/8, value equal to one
when φS =0.
=> φMov Bunch = φStat Bunch . {1 – sin2(φS)}-1/8
⇒ 𝜑𝑀𝑜𝑣 𝐵𝑢𝑛𝑐ℎ (𝐵𝑓) [𝑟𝑓 𝑟𝑎𝑑]
≈𝜋∙√
𝜋 3 ⁄8 − √(𝜋 3 ⁄8) − 16 ∙ 𝐵𝑓 ∙ {(𝜋 3 ⁄16) − 1}
∙ {1 − 𝑠𝑖𝑛2 (𝜑𝑆 )}−1⁄8
(𝜋 3 ⁄16) − 1
Bfr = bucket filling ratio
- 30 -
Beam Transverse dimensions from emittance value
x’, y’


x,y
Area = π.ε
x, y
  x, y
Figure 4.4.1.1
Beam emittance ε and associated physical dimension
The RMS normalized emittance ε* = β.γ.ε is the value obtained in the PSB when using the wire
scanners application. From this value, the RMS physical beam radius can be obtained (see
figure 4.4.1.1).
𝑟𝑥,𝑦
𝜀 ∗ ∙ 𝛽𝑥,𝑦
√
=
𝛽∙𝛾
4.4.2 Incoherent tune shift [10, 12]
Δ𝑄𝑥,𝑦 𝑖𝑛𝑐 = −
𝐵𝑢𝑛𝑐ℎ_𝑄_𝑒 ∙ 𝑟0 ∙ 〈𝛽𝑥,𝑦 〉
𝜋 ∙ 𝛽2 ∙ 𝛾
∙{
(1 − 𝛽 2 ) 𝒙,𝒚
1
𝛽2
1
𝐷𝑓 ∙ 𝛽 2 𝒙,𝒚
𝒙,𝒚
𝒙,𝒚
∙
𝜺
+
∙
𝜺
−
∙
(
−
1)
∙
𝜺
+
∙ 𝜺𝟐 }
𝟎
𝟏
𝐵𝑓. 𝑏 2
𝐵𝑓 ∙ ℎ2 𝟏
ℎ2 𝐵𝑓
𝑔2
Expression of the tune shift when the beam current is at its local maximum
The first term corresponds to space charge self-forces, the second to electric image forces in
vacuum chamber, the third to magnetic image forces from the vacuum chamber (high
frequency magnetic beam components only) and the last to the magnetic image in magnet poles
(Low frequency magnetic beam components only).
r0 [m] =
𝑒2
4∙𝜋∙𝜀0 ∙𝑚0 ∙𝑐 2
= classical particle radius = 1.5347 . 10-18 for protons
〈𝛽𝑥,𝑦 〉 = average beta function amplitude
- 31 -
b,g, h [m] : see figure 3.4.1.1 and table 3.4.1.1. For the beam dimensions, one should use the
RMS value  b = √𝟐 ∙ 𝝈𝒚 and a = √𝟐 ∙ 𝝈𝒙 .
Df [ ]: fraction of the ring circumference where the beam is sandwiched between bending
magnetic poles. This is how Df is defined in [12] for a coasting beam with a uniform line
density. As we treat separately the section with dipoles (elliptic chambers) and the remaining
circular beam pipe sections, we will use f=0 in the circular part and f = 1 for the elliptic
chamber.
Bf [ ]: Bunching factor = average particle line density / peak line density
Figure 4.4.2.1 Beam and vacuum pipe dimensions as to be used in table 4.4.2.1 (copy from [10])
In the PSB 1/3 of the ring is composed of elliptic beam pipe sections (where the dipoles stand)
and 2/3 of circular beam pipes. The tune shift will thus be calculated for both parts and the final
result will be obtained by making a weighted sum of the two ingredients.
- 32 -
Circular beam pipe
Parameters
(for high intensity 1e13
per ring)
beam radius
= (a+b)/2
Elliptic beam pipe
1/3 of the PSB circumference
2/3 of the PSB
circumference
(see figure below for the meaning of a, b and
h)
〈𝛽𝑥 〉 [ ]
6.15107
6.54203
〈𝛽𝑦 〉 [ ]
6.60863
5.19698
25 e-6
25 e-6
of the vertical normalized
beam emittance
7 e-6
7 e-6
h [m]
8 e-2
3.2 e-2
w [m]
8 e-2
8 e-2
g [m]
10 e-2 (estimation)
∞
𝜀0𝑥
0.5 ∙ 𝑓𝑜𝑟𝑚
𝑓𝑜𝑟𝑚 ∙ 𝑏 2
𝑎 ∙ (𝑎 + 𝑏)
𝑦
0.5 ∙ 𝑓𝑜𝑟𝑚
𝑓𝑜𝑟𝑚 ∙ 𝑏
𝑎+𝑏
𝜀1𝑥
0
ℎ2
2∙𝐾 2
′2 )
∙ [(1 + 𝑘 ∙ (
) − 2]
12 ∙ (𝑤 2 − ℎ2 )
𝜋
𝑦
0
= −𝜀1𝑥
0
-π2/24
εH [m] 1σ value
of the horizontal
normalized beam
emittance
εV [m] 1σ value
𝜀0
𝜀1
𝜀2𝑥
(no bendings in the cannot be calculated, thus assumed to be the
circular sections)
same as in the parallel plates case
π2/24
0
𝑦
𝜀2
Df
(no bendings in the cannot be calculated, thus assumed to be the
circular sections)
same as in the parallel plates case
0
1
Table 4.4.2.1 Laslett coefficients to be used to evaluate the transverse incoherent tune shift.
- 33 -
form = Beam form factor = 1 for a uniform transverse distribution; =3 for a bi-Gaussian
distribution (the tune shift is 3 times as large in the centre of the distribution as the tune shift in
a uniform distribution).
Note that the tune spread rather than the tune shift is a relevant value in a context where losses
are being evaluated. An hypothetic beam where all particles encounter the same tune shift is not
critical as a simple change of the optics set-point is enough to compensate for this effect.
In a bi-Gaussian distribution beam with an infinite transverse radius, the tune spread is equal to
the maximum tune shift as the particle on the periphery undergoes no beam-beam effect. For
practical beams, the Gaussian distribution is truncated and the space charge tune spread can be
evaluated using figure 3.4.1.2.
k’ =
[
𝑛 𝑛
1+2∙∑∞
𝑛=0 (−1) ∙𝑞
 k’ =
1+2∙∑∞
𝑛=0
[
𝑞
𝑛2
2
2
]
with
1+2∙(1−𝑞+𝑞4 −𝑞9 +⋯ )
1+2∙(1+𝑞+𝑞4 +𝑞9 +⋯ )
]
𝑞=
𝑤−ℎ
𝑤+ℎ
= 0.428 in the PSB elliptic chamber
2
with q = 0.428
𝑘 = √1 − 𝑘 ′2
∞
2
(2 ∙ 𝑛)!
𝜋
𝜋
1
9
25
𝐾 = 𝐾(𝑘) = ∙ ∑ [ 2∙𝑛
] ∙ 𝑘 2∙𝑛 = ∙ [1 + ∙ 𝑘 2 +
∙ 𝑘4 +
∙ 𝑘6 + ⋯ ]
2
2
2 ∙ (𝑛!)
2
4
64
256
𝑛=0
Figure 4.4.2.2 Space charge tune shift of a particle with betatron amplitude r as a fraction of the
maximum space charge tune shift of a bi-Gaussian distributed round beam with rms beam size σr
(copy from [12])
- 34 -
4.4.3 Momentum p [eV/c]
Equilibrium of forces within a dipole:
(_i stands for ion)
𝑚_𝑖 ∙ 𝑣 2
𝑞_𝑖 ∙ 𝑣 ∙ 𝐵 =
𝜌
⇒ 𝑝_𝑖 = 𝑚_𝑖 ∙ 𝑣 = 𝑞_𝑖 ∙ 𝐵 ∙ 𝜌
⇒ 𝑝_𝑝𝑒𝑟_𝑎𝑚𝑢 =
𝑚_𝑖 ∙ 𝑣 𝜁 ∙ 𝑒 ∙ 𝐵 ∙ 𝜌
=
𝑚_𝑖_𝑎
𝑚_𝑖_𝑎
⇒ 𝑝_𝑝𝑒𝑟_𝑎𝑚𝑢 [ 𝐽 ∙ 𝑠/𝑚] =
𝜁∙𝑒∙𝐵∙𝜌
𝑚_𝑖_𝑎
amu [kg] = atomic mass unit: 1.660538782E-27 kg
ζ [ ] = number of elementary charges in the ion
1 eV = e*J
Expression of the momentum in eV/c (1 [J.s/m] = c/e [eV/c]):
⇒ 𝑝_𝑝𝑒𝑟_𝑎𝑚𝑢 [ 𝑒𝑉/𝑐] = 𝐵 ∙ 𝜌 ∙ 𝑐 ∙
𝜁
𝑚_𝑖_𝑎
4.4.4 Beta
Equilibrium of forces within a dipole:
( _i stands for ion)
𝑚_𝑖 ∙ 𝑣 2
𝑞_𝑖 ∙ 𝑣 ∙ 𝐵 =
𝜌
⇒ 𝑝_𝑖 = 𝑚_𝑖 ∙ 𝑣 = 𝑞_𝑖 ∙ 𝐵 ∙ 𝜌 (1.1)
We also have:
𝑝_𝑖 = 𝑚_𝑖 ∙ 𝑣 =
𝐸_𝑖
𝐸_𝑖 ∙ 𝛽
∙𝑣 =
(1.2)
2
𝑐
𝑐
(1.1)+ (1.2) =>
𝑞_𝑖 ∙ 𝑣 ∙ 𝐵 =
⇔𝛽 =
𝐸_𝑖 ∙ 𝛽
𝑐
𝑞_𝑖 ∙ 𝐵 ∙ 𝜌 ∙ 𝑐
𝛾 ∙ 𝐸_𝑖0
- 35 -
using:
𝛾=
⇒ 𝛽2 =
⇒ 𝛽 2 ∙ {1 +
1
√1 − 𝛽 2
(𝑞_𝑖 ∙ 𝐵 ∙ 𝜌 ∙ 𝑐)2
𝐸_𝑖0 2
(𝑞_𝑖 ∙ 𝐵 ∙ 𝜌 ∙ 𝑐)2
𝐸_𝑖0 2
∙ (1 − 𝛽 2 )
}=
(𝑞_𝑖 ∙ 𝐵 ∙ 𝜌 ∙ 𝑐)2
𝐸_𝑖0 2
let’s define:
𝐵0 =
𝐸_𝑖0
𝑚_𝑖0 ∙ 𝑐
=
𝑞_𝑖 ∙ 𝜌 ∙ 𝑐 𝜁 ∙ 𝑒 ∙ 𝜌
2
⇒ 𝛽 ∙ {1 +
𝐵2
𝐵0 2
}=
𝐵2
𝐵0 2
𝐵⁄
𝐵0
⇒𝛽 =
√1 + (𝐵⁄ )
𝐵0
2
4.4.5 Adiabaticity
A longitudinal process is called adiabatic when the synchrotron motion parameters are
changed slowly enough to preserve the beam emittance in a process that is reversible. When
editing a cycle, one needs to check that the variations of the rf bucket fulfil this constraint.
One adiabaticity parameter often used for the rf capture of unbunched beams is:
ka =
𝑑𝐴 𝑇𝑆
𝑑𝑡
∙
𝐴
A [eV.s] = Bucket area
TS [s] = synchrotron period
A process is considered as adiabatic when |ka| < 0.5, which corresponds to:
𝑑𝐴
𝐴
| | < 0.5 ∙
𝑑𝑡
𝑇𝑆
4.4.6 Beam signal spectrum (cosine bunch shape)
- 36 -
λ
λPP
t, φ
TRF
φRF
TBunch
φBunch
Figure 4.4.6.1 : Beam line density along the synchrotron circumference
The beam line density as a function of time represents the function dQ/dt (Q being the charge
of the beam), which is thus equivalent to the corresponding electrical current. The integral of
this function along a bunch period correspond to the number of particle in the bunch.
In the context of this note, the interesting value is the beam current at the rf harmonic, which is
the value being sensed by a narrow band cavity (the actual ferrite types installed in the PSB).
The shape of the beam can be obtained analytically, taking into account the rf voltage, the
stable phase, the emittance and the particle distribution. A Fourier analysis for each obtained
shapes would then give the beam current at each harmonic.
In the present context, it was found convenient to make a simplification, assuming a
cosinusoidal shape, visually very similar to the one actually observed in a single harmonic
acceleration context.
Due to the possible dual-harmonic acceleration, bunches may have different shapes, but in the
practical case where the second harmonic is used to flatten bunches, the second harmonic has
typically its zero phase (=0 voltage) corresponding to the centre of the bunch (more precisely
the location of the synchronous particle). In this case the second harmonic cavity is not
providing any average longitudinal energy, meaning that the energy increase is all provided by
the first harmonic cavity. As the energy increase is a function of the stable or synchronous
phase (energy per turn =VRF*sinφS*IBEAM*TREV) and not of the distribution of the particles, the
beam current at the accelerating harmonic needs to be equal to that one obtained in a single
harmonic case. This means that in the case of a rectangular shaped bunch, the bunch will get
wider in order to have the same current at the main accelerating rf as compared to the single
harmonic case.
Treating the case of single harmonic type of bunch shape is thus a good approach as long as the
conditions above are fulfilled, which is the case for high intensity beams requiring the most
from the rf power equipment (requirements that we are trying to establish).
The rectangular shaped beam will also be analysed, as its harmonics will have to be dealt with
by the highest harmonics cavities.
In figure 4.4.6.1 the beam line density can be expressed between –TBunch/2 and +TBunch/2 as
𝜆(𝑡) = 𝜆𝑃𝑃 ∙ 𝑐𝑜𝑠 (𝜋 ∙
- 37 -
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
)
with:
∫
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
−𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
⇒ 𝑄𝐵𝑢𝑛𝑐ℎ = 𝜆𝑃𝑃 ∙ ∫
𝜆(𝑡) 𝑑𝑡 = 𝑄𝐵𝑢𝑛𝑐ℎ
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
−𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
𝑐𝑜𝑠 (𝜋 ∙
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
) 𝑑𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
⇒ 𝑄𝐵𝑢𝑛𝑐ℎ
⁄2
𝑇𝐵𝑢𝑛𝑐ℎ
𝑡
= 𝜆𝑃𝑃 ∙
∙ [𝑠𝑖𝑛 (𝜋 ∙
)]
𝜋
𝑇𝐵𝑢𝑛𝑐ℎ −𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
⇒ 𝑄𝐵𝑢𝑛𝑐ℎ = 2 ∙ 𝜆𝑃𝑃 ∙
⇔ 𝜆𝑃𝑃 [𝐶 ⁄𝑠 𝑜𝑟 𝐴] =
𝑇𝐵𝑢𝑛𝑐ℎ
𝜋
𝜋 𝑄𝐵𝑢𝑛𝑐ℎ 𝜋
2 ∙ 𝜋 ∙ ℎ𝐵𝑢𝑛𝑐ℎ ∙ 𝐹𝑅𝐸𝑉
∙
= ∙ 𝑄𝐵𝑢𝑛𝑐ℎ ∙
2 𝑇𝐵𝑢𝑛𝑐ℎ 2
φBunch
= 𝑄𝐵𝑢𝑛𝑐ℎ ∙
𝜋 2 ∙ ℎ𝐵𝑢𝑛𝑐ℎ ∙ 𝐹𝑅𝐸𝑉
φBunch
Peak beam current with bunch length phase φBunch related to the main rf at harmonic
hBunch*FREV
Beam signal in Fourier series:
∞
1
𝜆(𝑡) = ∙ 𝑎0 + ∑ 𝑎𝑛 ∙ 𝑐𝑜𝑠(𝑛 ∙ 𝜔𝑅𝐹 ∙ 𝑡) + 𝑏𝑛 ∙ 𝑠𝑖𝑛(𝑛 ∙ 𝜔𝑅𝐹 ∙ 𝑡)
2
𝑛=1
bn = 0 as λ(t) is even in the representation of figure 3.4.5.1
∞
1
⇒ 𝜆(𝑡) = ∙ 𝑎0 + ∑ 𝑎𝑛 ∙ 𝑐𝑜𝑠(𝑛 ∙ 𝜔𝑅𝐹 ∙ 𝑡)
2
𝑛=1
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
4 ∙ 𝜆𝑃𝑃
𝑎𝑛 =
∙∫
𝑇𝑅𝐹
0
_____________________________
cos (𝜋 ∙
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
) ∙ cos(n ∙ ωRF ∙ t) 𝑑𝑡
∫ 𝑢′ ∙ 𝑣 = 𝑢 ∙ 𝑣 − ∫ 𝑢 ∙ 𝑣′
𝑢′ = 𝑐𝑜𝑠 (𝜋 ∙
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
)
𝑣 = cos (n ∙ 2π ∙
𝑇𝐵𝑢𝑛𝑐ℎ
𝑡
∙ 𝑠𝑖𝑛 (𝜋 ∙
)
𝜋
𝑇𝐵𝑢𝑛𝑐ℎ
_____________________________
𝑢=
- 38 -
𝑣 ′ = −n ∙
t
)
TRF
2π
t
∙ sin (n ∙ 2π ∙
)
TRF
TRF
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
4 ∙ 𝜆𝑃𝑃 𝑇𝐵𝑢𝑛𝑐ℎ
𝑡
t
⇒ 𝑎𝑛 =
∙
∙ [𝑠𝑖𝑛 (𝜋 ∙
) ∙ cos (n ∙ 2π ∙
)]
𝑇𝑅𝐹
𝜋
𝑇𝐵𝑢𝑛𝑐ℎ
TRF 0
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
2π ∙ n
∙
∙∫
TRF
0
⇒ 𝑎𝑛 =
𝑠𝑖𝑛 (𝜋 ∙
𝑇𝐵𝑢𝑛𝑐ℎ
4 ∙ 𝜆𝑃𝑃 𝑇𝐵𝑢𝑛𝑐ℎ
∙
𝑇𝑅𝐹
𝜋
t
) 𝑑𝑡
TRF
) ∙ sin (n ∙ 2π ∙
4 ∙ 𝜆𝑃𝑃 𝑇𝐵𝑢𝑛𝑐ℎ
TBunch
4 ∙ 𝜆𝑃𝑃 𝑇𝐵𝑢𝑛𝑐ℎ 2π ∙ n
∙
∙ cos (n ∙ π ∙
)+
∙
∙
𝑇𝑅𝐹
𝜋
TRF
𝑇𝑅𝐹
𝜋
TRF
∙∫
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
𝑠𝑖𝑛 (𝜋 ∙
0
⇔
𝑡
+
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
) ∙ sin (n ∙ 2π ∙
t
) 𝑑𝑡
Trf
𝜋 ∙ 𝑇𝑅𝐹
∙𝑎
4 ∙ 𝜆𝑃𝑃 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ 𝑛
= cos (n ∙ π ∙
∙∫
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
TBunch
2π ∙ n
)+
TRF
TRF
𝑠𝑖𝑛 (𝜋 ∙
0
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
) ∙ sin (n ∙ 2π ∙
t
) 𝑑𝑡
TRF
_____________________________
∫ 𝑢′ ∙ 𝑣 = 𝑢 ∙ 𝑣 − ∫ 𝑢 ∙ 𝑣′
𝑢′ = 𝑠𝑖𝑛 (𝜋 ∙
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
)
𝑣 = sin (n ∙ 2π ∙
−𝑇𝐵𝑢𝑛𝑐ℎ
𝑡
∙ 𝑐𝑜𝑠 (𝜋 ∙
)
𝜋
𝑇𝐵𝑢𝑛𝑐ℎ
_____________________________
𝜋 ∙ 𝑇𝑅𝐹
⇔
∙𝑎
4 ∙ 𝜆𝑃𝑃 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ 𝑛
𝑢=
= cos (n ∙ π ∙
𝑣′ = n ∙
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
t
∙ [𝑐𝑜𝑠 (𝜋 ∙
) ∙ sin (n ∙ 2π ∙
)]
𝑇𝐵𝑢𝑛𝑐ℎ
TRF 0
∙∫
0
2π
t
∙ cos (n ∙ 2π ∙
)
TRF
TRF
TBunch
2π ∙ n −𝑇𝐵𝑢𝑛𝑐ℎ
)+
∙
TRF
TRF
𝜋
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
t
)
TRF
𝑐𝑜𝑠 (𝜋 ∙
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
) ∙ cos (n ∙ 2π ∙
- 39 -
+
2π ∙ n 𝑇𝐵𝑢𝑛𝑐ℎ 2π ∙ n
∙
∙
TRF
𝜋
TRF
t
) 𝑑𝑡
TRF
⇔
𝜋 ∙ 𝑇𝑅𝐹
∙𝑎
4 ∙ 𝜆𝑃𝑃 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ 𝑛
= cos (n ∙ π ∙
∙∫
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
0
⇔
TBunch
2π ∙ n 𝑇𝐵𝑢𝑛𝑐ℎ 2π ∙ n
)+0+
∙
∙
TRF
TRF
𝜋
TRF
𝑐𝑜𝑠 (𝜋 ∙
𝑡
𝑇𝐵𝑢𝑛𝑐ℎ
) ∙ cos(n ∙ ωrf ∙ t) 𝑑𝑡
𝜋 ∙ 𝑇𝑅𝐹
TBunch
2π ∙ n 𝑇𝐵𝑢𝑛𝑐ℎ 2π ∙ n 𝑇𝑅𝐹
∙ 𝑎𝑛 = cos (n ∙ π ∙
)+
∙
∙
∙
∙𝑎
4 ∙ 𝜆𝑃𝑃 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
TRF
TRF
𝜋
TRF 4 ∙ 𝜆𝑃𝑃 𝑛
𝜋 ∙ 𝑇𝑅𝐹
π ∙ n2 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
TBunch
⇔(
−
) ∙ 𝑎𝑛 = cos (n ∙ π ∙
)
4 ∙ 𝜆𝑃𝑃 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
𝜆𝑃𝑃 ∙ TRF
TRF
2
2
𝜋 𝑇𝑅𝐹
− 4 ∙ 𝑛2 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
TBunch
⇔( ∙
) ∙ 𝑎𝑛 = cos (n ∙ π ∙
)
4 𝜆𝑃𝑃 ∙ TRF ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
TRF
⇔ 𝑎𝑛 [𝐶 ⁄𝑠] =
4 𝜆𝑃𝑃 ∙ TRF ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
TBunch
∙ 2
∙
cos
(n
∙
π
∙
)
2
𝜋 𝑇𝑅𝐹 − 4 ∙ 𝑛2 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
TRF
Replacing λ by its charge equivalent:
𝜋 𝑄𝐵𝑢𝑛𝑐ℎ
4 2 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ ∙ TRF ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
TBunch
⇔ 𝑎𝑛 = ∙
∙ cos (n ∙ π ∙
)
2
2
2
𝜋 𝑇𝑅𝐹 − 4 ∙ 𝑛 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
TRF
⇔ 𝑎𝑛 =
2
𝑇𝑅𝐹
2 ∙ 𝑄𝐵𝑢𝑛𝑐ℎ ∙ TRF
TBunch
∙ cos (n ∙ π ∙
)
2
TRF
− (2 ∙ 𝑛 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ )
T
cos (n ∙ π ∙ Bunch
2 ∙ 𝑄𝐵𝑢𝑛𝑐ℎ
TRF )
⇔ 𝑎𝑛 =
∙(
)
TRF
𝑇𝐵𝑢𝑛𝑐ℎ 2
1 − (2 ∙ 𝑛 ∙
)
TRF
This expression can be written in terms of angles; n is the harmonic of bunches repetition rate:
φBunch
cos (n ∙ π ∙ φ
)
RF
⇒ 𝑎𝑛 [𝐶 ⁄𝑠] = 2 ∙ ℎ ∙ 𝐹𝑅𝐸𝑉 ∙ 𝑄𝐵𝑢𝑛𝑐ℎ ∙ (
)
φBunch 2
1 − (2 ∙ 𝑛 ∙ φ
)
RF
In the PSB context, where all the rf buckets are filled and the bunches are considered equal in
shape and amplitude, the beam signal is only composed of rf harmonics. As we are here
interested in beam loading issues within cavities, we will retain the values of the beam
amplitudes at these rf harmonics where the cavities sit.
- 40 -
To avoid confusions, we will replace n by hRF_Meas and h by hBunches ; the latter being the
harmonic of the dominating rf.
⇒ 𝑎𝑛
φBunch
cos (hRF_Meas ∙ π ∙ φ
)
RF
[𝐶 ⁄𝑠] = 2 ∙ ℎ𝐵𝑢𝑛𝑐ℎ𝑒𝑠 ∙ 𝐹𝑅𝐸𝑉 ∙ 𝑄𝐵𝑢𝑛𝑐ℎ ∙ (
)
φBunch 2
1 − (2 ∙ hRF_Meas ∙ φ
)
RF
The expression of the cosine-shaped beam signal is thus:
𝜆(𝑡)𝑟𝑓
φBunch
cos (hRF_Meas ∙ π ∙ φ
)
RF
= 2 ∙ ℎ𝐵𝑢𝑛𝑐ℎ𝑒𝑠 ∙ 𝐹𝑅𝐸𝑉 ∙ 𝑄𝐵𝑢𝑛𝑐ℎ ∙ (
) ∙ 𝑐𝑜𝑠(𝜔𝑅𝐹 ∙ 𝑡)
φBunch 2
1 − (2 ∙ hRF_Meas ∙ φ
)
RF
and the amplitude of the hRF_Meas harmonic is:
‖𝐼𝐶𝑜𝑠 𝐵𝑒𝑎𝑚 ‖𝑎𝑡 ℎ𝑅𝐹_𝑀𝑒𝑎𝑠 = ‖𝜆(𝑡)‖𝑎𝑡 ℎ𝑅𝐹_𝑀𝑒𝑎𝑠
φBunch
cos (hRF_Meas ∙ π ∙ φ
)
RF
= 2 ∙ ℎ𝐵𝑢𝑛𝑐ℎ𝑒𝑠 ∙ 𝐹𝑅𝐸𝑉 ∙ 𝑄𝐵𝑢𝑛𝑐ℎ ∙ |
|
φBunch 2
1 − (2 ∙ hRF_Meas ∙ φ
)
RF
and when the bunch length [rad] is defined in terms of the dominating rf phase angle, where
φRF represents 2π by definition:
φ
cos (hRF_Meas ∙ Bunch
2 )|
‖𝐼𝐶𝑜𝑠 𝐵𝑒𝑎𝑚 ‖𝑎𝑡 ℎ𝑅𝐹_𝑀𝑒𝑎𝑠 = 2 ∙ ℎ𝐵𝑢𝑛𝑐ℎ𝑒𝑠 ∙ 𝐹𝑅𝐸𝑉 ∙ 𝑄𝐵𝑢𝑛𝑐ℎ ∙ |
2
φ
1 − (hRF_Meas ∙ Bunch
)
π
4.4.7 Beam signal spectrum (rectangular bunch shape)
λ
λPP
t, φ
TBunch
φBunch
TRF
φRF
Figure 4.4.7.1 Beam line density along the synchrotron circumference
- 41 -
The rectangular bunch shape is a convenient simplification of the actual bunch shape
encountered in a dual harmonic acceleration where a low bunching factor is required. The value
of the beam current at the two first rf harmonics will be used in our context to evaluate the
beam current sensed by the h1 and h2 cavities.
In figure 4.4.7.1 the beam line density can be expressed between –TBunch/2 and +TBunch/2 as
𝜆(𝑡) = 𝜆𝑃𝑃
with:
∫
𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
−𝑇𝐵𝑢𝑛𝑐ℎ⁄
2
𝜆(𝑡) = 𝑄𝐵𝑢𝑛𝑐ℎ
⇒ 𝑄𝐵𝑢𝑛𝑐ℎ = 𝜆𝑃𝑃 ∙ 𝑇𝐵𝑢𝑛𝑐ℎ
𝑄𝐵𝑢𝑛𝑐ℎ
2 ∙ 𝜋 ∙ ℎ𝐵𝑢𝑛𝑐ℎ
= 𝑄𝐵𝑢𝑛𝑐ℎ ∙ 𝐹𝑅𝐸𝑉 ∙
𝑇𝐵𝑢𝑛𝑐ℎ
φBunch
Peak beam current with bunch length phase φBunch related to the main rf at harmonic
hBunch*FREV
⇔ 𝜆𝑃𝑃 [𝐶 ⁄𝑠] =
The same mathematical approach as for cosine shaped bunches leads to the following formula:
φ
sin (hRFMeas ∙ Bunch
2 )|
‖𝐼𝑅𝑒𝑐𝑡 𝐵𝑒𝑎𝑚 ‖𝑎𝑡 ℎ𝑅𝐹𝑀𝑒𝑎𝑠 = 2 ∙ ℎ𝐵𝑢𝑛𝑐ℎ𝑒𝑠 ∙ 𝐹𝑅𝐸𝑉 ∙ 𝑄𝐵𝑢𝑛𝑐ℎ ∙ |
φ
hRFMeas ∙ Bunch
2
Expression of the beam amplitude for rectangular bunches at harmonic hRF_Meas when the bunch
length is expressed as an angle of the dominating rf in radians
4.4.8 Space charge voltage
The total coupling impedance can be expressed as [8]:
𝑍𝑖
𝑔0 ∙ 𝑍0
= 𝑅 + 𝑗 ∙ (𝜔𝑅𝐸𝑉 ∙ 𝐿 −
)
ℎ
2 ∙ 𝛽 ∙ 𝛾2
R [Ω] = resistive wall impedance; can be neglected in the PSB
L [H] = total wall inductance over one revolution; can be neglected in the PSB
𝑍0 [𝛺] = free space impedance = √
𝜇0
= 377Ω
𝜀0
- 42 -
𝑝𝑖𝑝𝑒 𝑟𝑎𝑑𝑖𝑢𝑠
g0 [ ] = Longitudinal space charge coupling coefficient = 0.5 + 𝐿𝑛 (2𝜎 𝑏𝑒𝑎𝑚 𝑟𝑎𝑑𝑖𝑢𝑠) [13]
formulae valid within 15% precision over all binomial distributions from
Kapchinsky-Vladimirsky to Gaussian for a circular beam into a circular beam pipe.
This coefficient will be calculated for beams with an elliptic cross section, taking the
average of the H and V dimensions as the beam radius; same for the pipe radius
where the pipe is elliptic (within magnets).
h [ ] = mode number = ω/ωREV
Neglecting R and L:
𝑍𝑖
𝑔0 ∙ 𝑍0
= −𝑗 ∙
ℎ
2 ∙ 𝛽 ∙ 𝛾2
Space charge force has therefore the same effect as a negative wall inductance, implying that
the induced voltage at any beam longitudinal phase is proportional to the derivative of the beam
current at this given phase.
The beam line density or current having the shape of the potential and the potential being the
integral of the voltage, the space charge induced voltage is proportional to the derivative of the
integral of the rf voltage. It is therefore proportional to the rf voltage itself.
The space charge induced voltage can be written:
𝑍𝑖
𝑉𝑆𝐶 (ℎ) = −𝐼𝑏𝑒𝑎𝑚 (ℎ) ∙ ℎ ∙ | |
ℎ
The minus sign is valid below transition only
𝑉𝑆𝐶 (ℎ) = −𝐼𝑏𝑒𝑎𝑚 (ℎ) ∙
ℎ ∙ 𝑔0 ∙ 𝑍0
2 ∙ 𝛽 ∙ 𝛾2
4.4.9 MPS model
The main power supply with its magnetic circuit has been modelled in figure 3.2.4. It consists
of a current generator feeding a serial L-R network.
In our context where the program is asking for a field value at a given time sample, the exercise
will be to evaluate the required MPS source voltage to check that it remains elow the maximum
value.
𝑉𝑀𝑃𝑆 = 𝑅𝑀𝑃𝑆 ∙ 𝐼𝑀𝑃𝑆 + 𝐿𝑀𝑃𝑆 ∙
𝑑𝐼𝑀𝑃𝑆
𝑑𝑡
Special solution: IMPS_S = constant
⟹
𝑑𝐼𝑀𝑃𝑆_𝑆
=0
𝑑𝑡
𝑅𝑀𝑃𝑆 ∙ 𝐼𝑀𝑃𝑆_𝑆 = 𝑉𝑀𝑃𝑆 ⟹ 𝐼𝑀𝑃𝑆_𝑆 =
- 43 -
𝑉𝑀𝑃𝑆
𝑅𝑀𝑃𝑆
General solution:
𝑅𝑀𝑃𝑆 ∙ 𝐼𝑀𝑃𝑆_𝐺 + 𝐿𝑀𝑃𝑆 ∙
⟹
𝑑𝐼𝑀𝑃𝑆_𝐺
=0
𝑑𝑡
𝑑𝐼𝑀𝑃𝑆_𝐺
𝑅𝑀𝑃𝑆
=−
∙ 𝑑𝑡
𝐼𝑀𝑃𝑆_𝐺
𝐿𝑀𝑃𝑆
⟹ ∫
𝑑𝐼𝑀𝑃𝑆_𝐺
𝑅𝑀𝑃𝑆
=−
∙𝑡
𝑑𝐼𝑀𝑃𝑆_𝐺
𝐿𝑀𝑃𝑆
𝐼𝑀𝑃𝑆_𝐺
𝑅𝑀𝑃𝑆
⟹ 𝐿𝑛 (
)=−
∙𝑡
𝑘
𝐿𝑀𝑃𝑆
𝐼𝑀𝑃𝑆_𝐺 = 𝑘 ∙
−𝑅𝑀𝑃𝑆
∙𝑡
𝑒 𝐿𝑀𝑃𝑆
−𝑡
=𝑘∙𝑒 𝜏∙
Complete solution with τ = LMPS / RMPS:
𝐼𝑀𝑃𝑆 =
−𝑡
𝑉𝑀𝑃𝑆
+ 𝑘∙𝑒 𝜏∙
𝑅𝑀𝑃𝑆
If at t= 0, IMPS = IMPS_0
𝑘 = 𝐼𝑀𝑃𝑆_0 −
⇒ 𝐼𝑀𝑃𝑆 =
𝑉𝑀𝑃𝑆
𝑅𝑀𝑃𝑆
−𝑡
−𝑡
𝑉𝑀𝑃𝑆
⋅ (1 − 𝑒 𝜏 ∙ ) + 𝐼𝑀𝑃𝑆_0 ∙ 𝑒 𝜏 ∙
𝑅𝑀𝑃𝑆
−𝑡
(𝐼𝑀𝑃𝑆 − 𝐼𝑀𝑃𝑆_0 ∙ 𝑒 𝜏 ∙ ) ∙ 𝑅𝑀𝑃𝑆
𝑉𝑀𝑃𝑆 =
𝑉𝑀𝑃𝑆 =
−𝑡
(1 − 𝑒 𝜏 ∙ )
−𝑡
𝐼
(𝐼 𝑀𝑃𝑆 − 1 + 1 − 𝑒 𝜏 ∙ ) ∙ 𝑅𝑀𝑃𝑆 ∙ 𝐼𝑀𝑃𝑆_0
𝑀𝑃𝑆_0
𝑉𝑀𝑃𝑆 = 𝑅𝑀𝑃𝑆 ∙ 𝐼𝑀𝑃𝑆_0 +
−𝑡
(1 − 𝑒 𝜏 ∙ )
𝐼
𝐼
(𝐼 𝑀𝑃𝑆 − 𝐼𝑀𝑃𝑆_0 ) ∙ 𝑅𝑀𝑃𝑆 ∙ 𝐼𝑀𝑃𝑆_0
𝑀𝑃𝑆_0
𝑀𝑃𝑆_0
−𝑡
(1 − 𝑒 𝜏 ∙ )
𝑉𝑀𝑃𝑆 = 𝑅𝑀𝑃𝑆 ∙ {𝐼𝑀𝑃𝑆_0 +
(𝐼𝑀𝑃𝑆 − 𝐼𝑀𝑃𝑆_0 )
(1 −
−𝑡
𝑒 𝜏 ∙)
}
As the magnets have supposedly a linear magnetization curve (if not, the trims will correct
this), with a conversion factor “B_over_I”:
- 44 -
𝑉𝑀𝑃𝑆 =
(𝐵 − 𝐵0 )
𝑅𝑀𝑃𝑆
∙ {𝐵0 +
−𝑡 }
𝐵_𝑜𝑣𝑒𝑟_𝐼
(1 − 𝑒 𝜏 ∙ )
(𝐼𝑀𝑃𝑆 − 𝐼𝑀𝑃𝑆_0 )
(1 −
−𝑡
𝑒 𝜏 ∙)
=
𝑉𝑀𝑃𝑆
− 𝐼𝑀𝑃𝑆_0
𝑅𝑀𝑃𝑆
−𝑡
𝐼𝑀𝑃𝑆 − 𝐼𝑀𝑃𝑆_0 + 𝐼𝑀𝑃𝑆_0 ∙ (1 − 𝑒 𝜏 ∙ ) =
−𝑡
𝐼𝑀𝑃𝑆 − 𝐼𝑀𝑃𝑆_0 ∙ 𝑒 𝜏 ∙ =
−𝑡
𝑉𝑀𝑃𝑆
∙ (1 − 𝑒 𝜏 ∙ )
𝑅𝑀𝑃𝑆
−𝑡
𝑉𝑀𝑃𝑆
∙ (1 − 𝑒 𝜏 ∙ )
𝑅𝑀𝑃𝑆
as B = IMPS . B_over_I:
𝐵=
−𝑡
−𝑡
𝑉𝑀𝑃𝑆
∙ (1 − 𝑒 𝜏 ∙ ) + 𝐵0 ∙ 𝑒 𝜏 ∙
𝑅𝑀𝑃𝑆 ∙ 𝐵_𝑜𝑣𝑒𝑟_𝐼
Maximum Bdot = dB/dt :
𝑉𝑀𝑃𝑆 = 𝑅𝑀𝑃𝑆 ∙ 𝐼𝑀𝑃𝑆 + 𝐿𝑀𝑃𝑆 ∙
𝑑𝐼𝑀𝑃𝑆
𝑑𝑡
𝑑𝐼𝑀𝑃𝑆
𝑉𝑀𝑃𝑆_𝑀𝐴𝑋 − 𝑅𝑀𝑃𝑆 ∙ 𝐼𝑀𝑃𝑆
|
=
𝑑𝑡 𝑀𝐴𝑋
𝐿𝑀𝑃𝑆
𝑑𝐵
𝐵_𝑜𝑣𝑒𝑟_𝐼
𝑅𝑀𝑃𝑆
|
=
∙ 𝑉𝑀𝑃𝑆_𝑀𝐴𝑋 −
∙𝐵
𝑑𝑡 𝑀𝐴𝑋
𝐿𝑀𝑃𝑆
𝐿𝑀𝑃𝑆
Maximum Bdouble_dot = d2B/Δt2 :
𝑑2𝐵
𝐵_𝑜𝑣𝑒𝑟_𝐼 𝑑𝑉𝑀𝑃𝑆
𝑅𝑀𝑃𝑆 𝑑𝐵
|
=
∙
|
−
∙
2
𝑑𝑡 𝑀𝐴𝑋
𝐿𝑀𝑃𝑆
𝑑𝑡 𝑀𝐴𝑋 𝐿𝑀𝑃𝑆 𝑑𝑡
- 45 -
5. Required precision of the bending field
The cycle editor provides set point values for the magnetic field without specifying the
required precision. These specifications are detailed in this chapter.
The precision at injection (160 MeV), with a nominal Linac 4 beam momentum and a nominal
fixed injection rf frequency, a field variation will translate into a bucket shift in energy.
With a h=1, 8kV bucket filled-up to 80 %, there is a 50 keV margin at the top and at the
bottom area of the rf bucket (0.674 eV.s beam emittance).
Dividing arbitrarily this gap by 2, one gets +/- 25 keV of possible fluctuation of the rf bucket.
This leads [11] to a requirement in term of precision for the dipolar field such as:
𝑑𝐵 𝑑𝑝
𝑑𝑅
2
=
− 𝛾𝑡𝑟
∙
𝐵
𝑝
𝑅
dR/R is equal to zero in this context where we are comparing different energy (or momentum)
equilibrium states.
𝑑𝐵
𝑑𝑝
⇒
=∙
𝐵
𝑝
with
𝑣
𝑝=𝐸∙ 2
𝑐 ∙𝑞∙𝜌
ρ [m] = bending radius
𝑑𝐵
𝑑𝐸
⇒
=∙
𝐵
𝐸
At injection E = 938 + 160 = 1098 MeV
𝑑𝐵
25 ∙ 103
⇒ | | ≤∙
𝐵
1098 ∙ 106
𝑑𝐵
⇒ | | ≤∙ 23 ∙ 10−6
𝐵
Required relative precision of the magnetic field at injection (160 MeV) for a maximum 5%
energy error in the longitudinal painting process.
This requirement corresponds to the required real value of the bending field with respect to
the set value. Its measurement, resulting in the B-train, is not involved here.
At extraction (2GeV), where the rf frequency is determined by a fixed reference, any variation
of the B field will translate into momentum and trajectory deviation.
In the transverse plane of the receiving machine (PS), the effect of radial offsets (injection
errors) on the transverse emittance should be cancelled by the transverse feedback acting
bunch by bunch. At a first order, the requirements for the PSB B field precision at extraction
will only come from trajectory issues (septum, transfer line) not covered here. The increase of
energy with its associated transverse beam size shrinking should lower the constraints, but the
possible doubling of the intensity may ask for more to keep radio-activation effects due to
losses, low.
- 46 -
In the longitudinal plane a field error means a shift in energy with respect to the downstream
machine (PS). If all bunches in the PS suffer the same energy offset, this will be dealt with by
the PS phase loop. In case we are in a double batch filling process, where bunches from a
previous shot circulate in the PS, the energy error will translate into an emittance increase, and
as the filamentation process is not perfect, to unwanted tails.
In the longitudinal phase space, the coordinates of the outer particle trajectory on both axes
are proportional to the square root of the emittance. Increasing the amplitude on the energy
axis by a factor (1+x) means thus increasing the emittance by a factor (1+x)2.
1% emittance increase means thus an excursion on the energy axis multiplied by (1+x) such
that (1+x)2 = 1.01, meaning x = 0.005.
Note that the blow-up effect is averaged over the previously injected bunches. To avoid taking
into account all the possible cases, the computation will be made as if this averaging process
didn’t occur.
We have:
2
𝑑𝐵
𝑑𝑓 𝛾 2 − 𝛾𝑡𝑟
𝑑𝑝
2
= 𝛾𝑡𝑟
+
∙
2
𝐵
𝑓
𝛾
𝑝
with df = 0, as the frequency is fixed during synchronisation =>
2
𝑑𝐵 𝛾 2 − 𝛾𝑡𝑟
𝑑𝑝
=
∙
2
𝐵
𝛾
𝑝
γ = 3.13 at 2 GeV
γtr = 4.07 in the PSB
𝛽
𝑝=𝐸∙
𝑐∙𝑞∙𝜌
Beam energy spread at 2 GeV = 9.04 MeV (with 0.674 eV.s beam emittance as at injection)
Energy offset for 1% emittance increase = 9.04 MeV * 0.005 = 45 keV
Total beam energy at 2 GeV: E = 2.92 GeV
𝑑𝐵
𝑑𝐸
| | ≤ −0.69 ∙
𝐵
𝐸
𝑑𝐵
45 ∙ 103
| ≤ −0.69 ∙
= 15.4 ∙ 10−6
𝐵
2.92 ∙ 109
Required relative precision the magnetic field at extraction (2 GeV) for a 1% maximum blowup in the receiving machine
|
This requirement corresponds to the required real value of the bending field with respect to
the set value. Its measurement, resulting in the B-train, is not involved here.
- 47 -
6. Tune shift and losses
The cycle editor allows tracking the value of the incoherent tune shift without specifying the
actual effects translating into blow-up or losses.
Measurements made by B. Mikulec and G. Rumolo in the PSB seem to show that losses
during the high space charge phase are somehow proportional to the intensity (for a given
acceleration rate).
The best linear fit of the results is:
Losses [%] = 2.02 *(nb of E12 protons) – 1.05
Best quadratique fit:
Losses [%] = 0.12 *(nb of E12 protons)^2 – 0.2 *(nb of E12 protons) + 3.82
The tune shift being proportional to the intensity (see chapter 4.4.2), suggests that losses are
somehow proportional to the tune shift.
There is an average 18% loss with a 1E13 p beam injected at 50 MeV, with ΔQV = -0.38
(simulated value in chapter 2.1)
2.5E13 p injected at 160 MeV corresponds to a ΔQV = -0.35.
This factor 0.92 = 0.35/0.38 in tune spread suggests a possible 18 * 0.92 = 16.5% loss within
the high space charge time window or a maximum of 2.08 E13 protons accelerated from the
2.5E13 injected.
This result remains to be verified.
- 48 -
Conclusion
An automatic cycle editor has been proposed for the CERN PS Booster. It takes into account
the main characteristics of the synchrotron hardware together with beam parameters. It
outputs the fastest possible cycle, allowing minimum losses at low energy where space charge
effects dominate and allowing for a minimization of the energy dissipated in the ring dipoles.
The developed software indicates that the PSB, with the present hardware, but a new foreseen
Main Power Supply, is capable of accelerating up to 1.1E13 protons per ring from 160 MeV
to 2 GeV. This intensity corresponds to three times the maximum LHC requirements.
Upgrading the C02 cavity amplifier in order to increase its output current capability by an
extra 3.3 Amperes (+ some margin) would allow accelerating the most intense - 2.5 E13
protons per ring - injected beam foreseen with Linac 4, while keeping the present power
dissipation limits of the bendings. This 2.5 E13 protons per ring value is theoretical as losses
are expected at low energy, mainly due to incoherent tune spread. The actual estimated
maximum accelerated intensity is around 2.1 E13 p per ring. The simulations should be
conservative as they take into account a single harmonic acceleration with higher space charge
effects as actually encountered in the foreseen dual harmonic acceleration.
Acknowledgements
Fulvio Boattini, Marco Buzio, Max Chamiot-Clerc, Antony Newborough, and Serge Pittet in
charge of the PSB main power supply and magnet systems are warmly thanked for having
shared their knowledge of the present and future systems. Mauro Paoluzzi has been very
helpful providing information about the PSB power rf equipments and Giovanni Rumolo
about the PSB beam parameters and machine optics. Elias Metral has been very helpful for
discussing the longitudinal space charge impedance. Many thanks to Bettina Mikulec who
carefully read this document before publication and provided valuables suggestions. Klaus
Hanke has supported this work in the frame of the PSB energy upgrade study.
- 49 -
References
[1] Active Longitudinal painting for the H-charge exchange injection of the Linac 4
beam into the PS Booster, Carli C., Garoby R., CERN-AB-Note-2008-011, 2008.
[2] Discussions with Max Chamiot-Clerc and Serge Pittet from CERN TE-EPC group,
https://edms.cern.ch/file/1070963/1/1070963_V1_TEEPC_A2.pdf
[3] Power converters implications for Booster Energy Upgrade, Burnet, J-P (CERN) ;
Pittet, S, (CERN) ;04.05.2010, presentation,
https://dfs.cern.ch/dfs/Departments/TE/Groups/EPC/Projects/BOOSTER%20Upgrade/TETM%20Booster%20upgrade.pdf
[4] Operation of the CERN PS-Booster above 1 GeV; Saturation effects in the main
bending magnets and required modifications, M. Benedikt, C. Carli, CERN/PS 98-059 (OP)
[5] Eddy Current Modeling and Measuring in Fast-Pulsed Resistive Magnets /
Arpaia, P (Sannio U. ; CERN) ; Buzio, M (CERN) ; Gollucio, G (Sannio U. ; CERN) ;
Montenero, G (Sannio U. ; CERN), CERN-sLHC-PROJECT-Report-0047.
[6] Presentation by M. Buzio to the PSB energy upgrade working group,
https://twiki.cern.ch/twiki/pub/PSBUpgrade/MinutesMeeting17June2010/PSBMainDipoleEddy
CurrentTestJune2010.pdf
[7] Some useful formulae for longitudinal phase space, Flemming Pedersen, document
not published
[8] Bunches with local elliptic energy distribution, A. Hofmann, F. Pedersen, IEEE
trans. Nucl. Sci. NS 26 No 3 page 3526, 1979.
[9] Longitudinal beam dynamics in circular accelerators, J. Le Duff, IEEE trans.
Nucl. Cern accelerator school proceedings, 1992 Jyväskylä, p307.
[10] Space Charge, K. Schindl, CAS - CERN Accelerator School: Intermediate Course
on Accelerator Physics, Zeuthen, Germany, 15 - 26 Sep 2003, pp.305-320.
[11] A selection of formulae and data useful for the design of A. G. synchrotrons, C.
Bovet, R. Gouiran, I. Gumowski, K.H. Reich, CERN/MPS-SI/int, DL/70/4, 1970
[12] Physics of Intensity Dependent Instabilities, Lecture Notes by K. Y. Ng, USPAS,
Los Angeles, January 2002
[13] Form Factor g in Longitudinal Space Charge Impedance, R. Baartman, June
1992
- 50 -
Annex – Visual basic functions used in the Excel cycle editor
' Global constants
Public Const gC = 299792458
Public Const ge = 1.6021773349E-19
Public Const gPi = 3.14159265358979
' MPSvoltage_for_B [V] evaluates the MPS voltage required from t_now [s] to t_aft [s] to
obtain a field change from B_now [T] to B_aft [T]
' the parameters are:
' MPS_R [Ohm] = resistance of the magnetic circuit;
' MPS_L [H]= Inductance of the circuit;
' B_over_I [T/A] = courant to field conversion factor
' MPS_Vmax [V] = maximum voltage (absolute value as the MPS is supposed to provide both
positive and negative voltages) available from the MPS ;
' t_now [s] = initial time value (present time);
' t_aft [s] = final time;
' B_now [T] = field obtained at time t_now;
' B_aft [ T] field required at time t_aft;
Function MPSvoltage_for_B(MPS_R, MPS_L, B_over_I, MPS_Vmax, t_now, t_aft, B_now,
B_aft)
If MPS_R < 0 Or MPS_L < 0 Or B_over_I < 0 Or MPS_Vmax < 0 Or t_aft < t_now Then
MPSvoltage_for_B = "Input error"
Exit Function
Else
MPSvoltage_for_B = (MPS_R / B_over_I) * (B_now + (B_aft - B_now) / (1 - Exp(-(t_aft
- t_now) / (MPS_L / MPS_R))))
If MPSvoltage_for_B > MPS_Vmax Then MPSvoltage_for_B = "+Saturated"
If MPSvoltage_for_B < -MPS_Vmax Then MPSvoltage_for_B = "-Saturated"
End If
End Function
' Bfield_from_V [T] evaluates the obtained bending field B_now [T] obtained at t_now [s]
when a voltage MPS_V is applied by the MPS from t_bef to t_now
' the parameters are:
' MPS_R [Ohm] = resistance of the magnetic circuit;
' MPS_L [H] = Inductance of the circuit;
' B_over_I [T/A] = courant to field conversion factor
' MPS_Vmax [V] = maximum voltage (absolute value as the MPS is supposed to provide both
positive and negative voltages) available from the MPS
' MPS_V [V] = voltage created by the MPS from t_bef to t_now;
' t_bef [s] = initial time value;
' t_now [s] = final time value = present time;
' B_bef [T] = field at time t_bef;
Function Bfield_from_V(MPS_R, MPS_L, B_over_I, MPS_Vmax, MPS_V_bef, t_bef, t_now,
B_bef)
- 51 -
If MPS_R < 0 Or MPS_L < 0 Or B_over_I < 0 Or t_now < t_bef Then
Bfield_from_V = "Input error"
Exit Function
Else
If MPS_V_bef = "+Saturated" Then MPS_V_bef = MPS_Vmax
If MPS_V_bef = "-Saturated" Then MPS_V_bef = -MPS_Vmax
Bfield_from_V = (B_over_I * MPS_V_bef / MPS_R) * (1 - Exp(-(t_now - t_bef) /
(MPS_L / MPS_R))) + B_bef * Exp(-(t_now - t_bef) / (MPS_L / MPS_R))
End If
End Function
' Max_Bdot_from_B [T/s] evaluates the maximum possible dipolar field increase during the
next time step as a function of the actual magnetic field
' the parameters are:
' MPS_R [Ohm] = resistance of the magnetic circuit;
' MPS_L [H] = Inductance of the circuit;
' B_over_I [T/A] = courant to field conversion factor
' MPS_Vmax [V] = maximum voltage (absolute value as the MPS is supposed to provide both
positive and negative voltages) available from the MPS
' t_now [s] = initial time value = present time step value;
' t_aft [s] = final time value = next time step value;
' B_now [T] = field at present time t_now;
' Acc [ ] = status bit indicating acceleration (if = 1) or deceleration (if = 0);
Function Max_Bdot_from_B(MPS_R, MPS_L, B_over_I, MPS_Vmax, t_now, t_aft, B_now,
Acc)
If MPS_R < 0 Or MPS_L < 0 Or B_over_I < 0 Or t_aft < t_now Then
Max_Bdot_from_B = "Input error"
Exit Function
Else
If Acc = 1 Then MPS_V = MPS_Vmax
If Acc = 0 Then MPS_V = -MPS_Vmax
Max_B_t_aft = (B_over_I * MPS_V / MPS_R) * (1 - Exp(-(t_aft - t_now) / (MPS_L /
MPS_R))) + B_now * Exp(-(t_aft - t_now) / (MPS_L / MPS_R))
Max_Bdot_from_B = (Max_B_t_aft - B_now) / (t_aft - t_now)
End If
End Function
' Beta [ ] evaluates the value v/c as a function of the magnetic field
' the parameters are:
' m_i_0 [kg] = rest mass of the ion;
' zeta [ ] = total charge of the ion in terms of number of elementary charges e;
' rho [m] = magnetic radius of the main dipoles;
' B [T] = main dipolar field
Function Beta(m_i_0, zeta, rho, B)
If m_i_0 < 0 Or zeta < 1 Or rho < 0 Or B < 0 Then
Beta = "Input error"
Exit Function
Else
B0 = m_i_0 * gC / (zeta * ge * rho)
- 52 -
Beta = (B / B0) / Sqr(1 + (B / B0) ^ 2)
End If
End Function
' RevFreq [Hz] evaluates the value of the revolution frequency
' as a function of:
' Beta [ ] = relative velocity (v/c);
' Radius [m] = Synchrotron mean radius
Function RevFreq(Beta, Radius)
If Beta < 0 Or Radius < 0 Then
RevFreq = "Input error"
Exit Function
Else
RevFreq = Beta * gC / (2 * gPi * Radius)
End If
End Function
' Momentum_per_amu [eV/c/u] evaluates the value of the momentum per atomic mass unit
' as a function of:
' Beta [ ] = relative velocity (v/c);
' rho [m] = magnetic radius of the main dipoles;
' B [T] = main dipolar field
' zeta [ ] = total charge of the ion in terms of number of elementary charges e;
' m_i_a [ ] = rest mass of the ion in terms of amu (atomic mass unit)
Function Momentum_per_amu(B, rho, zeta, m_i_a)
If B < 0 Or rho <= 0 Or zeta <= 0 Or m_i_a < 0 Then
Momentum_per_amu = "Input error"
Exit Function
Else
Momentum_per_amu = B * rho * gC * zeta / m_i_a
End If
End Function
' Tot_Energy_per_amu [eV/u] evaluates the value of the total energy (rest + kinetic) per
atomic mass unit
' as a function of:
' Beta [ ] = relative velocity (v/c);
' rho [m] = magnetic radius of the main dipoles, B [T] = main dipolar field;
' zeta [ ] = total charge of the ion in terms of number of elementary charges e;
' m_i_a [ ] = rest mass of the ion in terms of amu (atomic mass unit);
' amu_e [eV] = atomic mass unit in eV;
Function Tot_Energy_per_amu(B, rho, zeta, m_i_a, amu_e)
If B < 0 Or rho <= 0 Or zeta <= 0 Or m_i_a < 0 Or amu_e < 0 Then
Tot_Energy_per_amu = "Input error"
Exit Function
Else
Tot_Energy_per_amu = Sqr(Momentum_per_amu(B, rho, zeta, m_i_a) ^ 2 + amu_e ^ 2)
End If
- 53 -
End Function
' Kin_Energy_per_amu [eV/u] evaluates the value of the total energy (rest + kinetic) per
atomic mass unit
' as a function of:
' Beta [ ] = relative velocity (v/c);
' rho [m] = magnetic radius of the main dipoles;
' B [T] = main dipolar field;
' zeta [ ] = total charge of the ion in terms of number of elementary charges e;
' m_i_a [ ] = rest mass of the ion in terms of amu (atomic mass unit);
' amu_e [eV] = atomic mass unit in eV;
Function Kin_Energy_per_amu(B, rho, zeta, m_i_a, amu_e)
If B < 0 Or rho <= 0 Or zeta <= 0 Or m_i_a < 0 Or amu_e < 0 Then
Kin_Energy_per_amu = "Input error"
Exit Function
Else
Kin_Energy_per_amu = Tot_Energy_per_amu(B, rho, zeta, m_i_a, amu_e) - amu_e
End If
End Function
' IcosBeam [A] evaluates the amplitude of the beam current at the harmonic h_rf_meas [ ] for a
bunch with a cosine shape
' as a function of:
' h_bunches [ ] = bunches repetition rate = harmonic of the dominating cavity (in terms of
voltage applied),
' Bunch_Q_e [ ] = number of elementary charges within the bunch;
' Bunch_Length [rf rad] = bunch length expressed in angle of the dominating rf period;
' Frev [Hz] = revolution frequency;
' h_bunches [ ] = dominating rf harmonic of the revolution;
' h_rf_meas [ ] = harmonic of the dominating rf at which the beam current IcosBeam is
evaluated;
Function IcosBeam(Bunch_Q_e, Bunch_Length, Frev, h_bunches, h_rf_meas)
If Bunch_Q_e < 0 Or Bunch_Length < 0 Or Frev < 0 Or h_bunches < 1 Or h_rf_meas < 1
Then
IcosBeam = "Input error"
Exit Function
Else
IcosBeam = 2 * h_bunches * Frev * Bunch_Q_e * ge * Abs(Cos(h_rf_meas *
Bunch_Length / 2) / (1 - (h_rf_meas * Bunch_Length / gPi) ^ 2))
End If
End Function
' IrectBeam [A] evaluates the amplitude of the beam current at the harmonic h_rf_meas [ ] for
a bunch with a rectangular shape
' as a function of:
' h_bunches [ ] = bunches repetition rate = harmonic of the dominating cavity (in terms of
voltage applied),
' Bunch_Q_e [ ] = number of elementary charges within the bunch;
' Bunch_Length [rf rad] = bunch length expressed in angle of the dominating rf period;
- 54 -
' Frev [Hz] = revolution frequency;
' h_bunches [ ] = dominating rf harmonic of the revolution;
' h_rf_meas [ ] = harmonic of the dominating rf at which the beam current IcosBeam is
evaluated;
Function IrectBeam(Bunch_Q_e, Bunch_Length, Frev, h_bunches, h_rf_meas)
If Bunch_Q_e < 0 Or Bunch_Length < 0 Or Frev < 0 Or h_bunches < 1 Or h_rf_meas < 1
Then
IrectBeam = "Input error"
Exit Function
Else
IrectBeam = 2 * h_bunches * Frev * Bunch_Q_e * ge * Abs(Sin(h_rf_meas *
Bunch_Length / 2) / (h_rf_meas * Bunch_Length / 2))
End If
End Function
' g0 [ ] evaluates the beam form factor used in the formula evaluating the longitudinal space
charge impedance
' this evaluation is valid for the PSB only where it is taken into account that the beam pipe is
circular for 2/3 of the circumference and elliptic in the remaining 1/3
' as a function of:
' Vacuum_chamber_half_height_ellip [m] = Beam pipe half height in the elliptic chamber ;
' Vacuum_chamber_half_width_ellip [m] = Beam pipe half width in the elliptic chamber;
' Vacuum_chamber_radius_circ [m] = Beam pipe radius in the circular chamber;
' Beam_norm_emit_h_1sig [m] = Beam horizontal normalized emittance at 1 sigma;
' Beam_norm_emit_v_1sig [m] = Beam vertical normalized emittance at 1 sigma;
' Beta_h_circ [m] = average beta function amplitude in the horizontal plane of the circilar pipe
sections;
' Beta_v_circ [m] = average beta function amplitude in the vertical plane of the circular pipe
sections;
' Beta_h_ellip [m] = average beta function amplitude in the horizontal plane of the elliptic pipe
sections;
' Beta_v_ellip [m] = average beta function amplitude in the vertical plane of the elliptic pipe
sections;
' Bet [ ] = beta value = normalized velocity = v/c;
Function g0(Vacuum_chamber_half_height_ellip, Vacuum_chamber_half_width_ellip,
Vacuum_chamber_radius_circ, Beam_norm_emit_h_1sig, Beam_norm_emit_v_1sig,
Beta_h_circ, Beta_v_circ, Beta_h_ellip, Beta_v_ellip, Bet)
If Bet < 0 Or Bet > 1 Then
g0 = "Input error"
Exit Function
Else
Gamma = 1 / Sqr(1 - Bet ^ 2)
B = (1 / 3) * (Vacuum_chamber_half_height_ellip + Vacuum_chamber_half_width_ellip)
/ 2 + (2 / 3) * Vacuum_chamber_radius_circ
A_circ = (Sqr(Beam_norm_emit_v_1sig * Beta_v_circ) + Sqr(Beam_norm_emit_h_1sig *
Beta_h_circ)) / Sqr(Bet * Gamma)
A_ellip = (Sqr(Beam_norm_emit_v_1sig * Beta_v_ellip) + Sqr(Beam_norm_emit_h_1sig
* Beta_h_ellip)) / Sqr(Bet * Gamma)
A = (2 / 3) * A_circ + (1 / 3) * A_ellip
- 55 -
If (2 * A) > B Then
g0 = "beam too wide"
Exit Function
Else
g0 = 0.5 + 2 * Log(B / (2 * A))
End If
End If
End Function
' VspaceChargeCos [V] evaluates the peak longitudinal space charge induced voltage, at
h_rf_meas, for a cosinusoidal bunch shape.
' as a function of:
' Bunch_Q_e [ ] = number of elementary charges within the bunch;
' Bunch_Length [rf rad] = bunch length expressed in angle of the dominating rf period;
' Frev [Hz] = revolution frequency;
' h_bunches [ ] = dominating rf harmonic of the revolution;
' h_rf_meas [ ] = harmonic of the rf at which the space charge voltage is evaluated;
' g0 [ ] = space charge coupling coefficient;
' Bet [ ] = v/c = relative velocity
Function VspaceChargeCos(Bunch_Q_e, Bunch_Length, Frev, h_bunches, h_rf_meas, g0,
Bet)
If Bunch_Q_e < 0 Or Bunch_Length < 0 Or Frev < 0 Or h_bunches < 1 Or h_rf_meas < 1 Or
Bet > 1 Or Bet < 0 Then
VspaceChargeCos = "Input error"
Exit Function
Else
VspaceChargeCos = IcosBeam(Bunch_Q_e, Bunch_Length, Frev, h_bunches, h_rf_meas)
* h_rf_meas * g0 * 377 * (1 - Bet ^ 2) / (2 * Bet)
End If
End Function
' AlphaS [ ] evaluates the moving bucket area factor (value to multiply to the stationanry
bucket area to obtain the moving bucket area) by polonomial approximation
' as a function of:
' sin_phiS [ ] = sinus of accelerating stable phase
Function AlphaS(Sin_PhiS)
Dim Coeff(8)
Coeff(8) = -4.86929808544067
Coeff(7) = 21.5634376561353
Coeff(6) = -38.3041447316052
Coeff(5) = 35.5609539048259
Coeff(4) = -18.8394752753217
Coeff(3) = 6.32355898129261
Coeff(2) = -2.43503244988618
Coeff(1) = 1
If Abs(Sin_PhiS) > 1 Then
AlphaS = "Out of range"
Exit Function
- 56 -
Else
AlphaS = 0
For i = 0 To 7
AlphaS = AlphaS + Coeff(i + 1) * Abs(Sin_PhiS) ^ i
Next i
End If
End Function
' Moving_Bucket_Area_per_amu [eV.s/u]evaluates the area of a single harmonic (h_bunches)
rf bucket
' as a function of:
' Radius [m] = synchrotron radius;
' sin_phiS [ ] = sinus of stable phase;
' V_rf [V]= peak value of the rf signal;
' E_tot_u [eV/u] = total energy per mass unit (gamma*rest energy of atomic mass unit);
' h_bunches [ ] = harmonic of the dominating rf signal with respect to the revolution;
' eta [ ] = frequency slip factor (1/gammaTransition^2) - (1/gamma^2) ;
' zeta [ ]= value of the charge state (number of protons - number of electrons);
' m_i_a [amu]= ion mass in atomic mass units
Function Moving_Bucket_Area_per_amu(Radius, m_i_a, zeta, h_bunches, E_tot_u, eta,
V_rf, Sin_PhiS)
If Radius < 0 Or Sin_PhiS < -1 Or Sin_PhiS >= 1 Or E_tot_u <= 0 Or h_bunches < 1 Or
V_rf < 0 Then
Moving_Bucket_Area_per_amu = "Input error"
Exit Function
Else
Moving_Bucket_Area_per_amu = 16 * (Radius / gC) * AlphaS(Sin_PhiS) * Sqr(V_rf *
E_tot_u * zeta / (2 * gPi * Abs(eta) * h_bunches ^ 3 * m_i_a))
End If
End Function
' BunchLength_from_stat_bucket_fill [rf rad] evaluates, for stationary buckets, the full bunch
length as an angle of the dominating rf at harmonic h_bunches
' as a function of:
' Bucket_filling [ ] = bunch emittance [eV.s] / bucket area [eV.s]
Function BunchLength_from_stat_bucket_fill(Bucket_filling)
If Bucket_filling < 0 Or Bucket_filling > 1 Then
BunchLength_from_stat_bucket_fill = "Input error"
Exit Function
Else
BunchLength_from_stat_bucket_fill = gPi * Sqr((gPi ^ 3 / 8 - Sqr(gPi ^ 6 / 64 - 16 *
Bucket_filling * (gPi ^ 3 / 16 - 1))) / (gPi ^ 3 / 16 - 1))
End If
End Function
' Max_possible_B [T] evaluates the required Bending field B for the fastest accelerating cycle
' in order to have a Bucket_Filling ratio as specifified by the parameter Bucket_Fill_max [ ]
' without exceeding the rf power capabilities (I_rf_max [A])
' The parameters are:
- 57 -
' m_i_0 [kg] = rest mass of the ion;
' m_i_a [ ] = rest mass of the ion in terms of atomic mass units (amu);
' m_i_e [eV/c^2] = rest mass of the ion in eV/c^2;
' amu_e [eV/c^2] = rest mass of the atomic mass unit in eV/c^2;
' zeta [ ] = net number of elementary charges of the ion;
' Radius [m] = mean radius of the synchrotron;
' rho [m] = magnetic radius of the main dipoles;
' Gamma_tr [ ] = normalized energy at transition
' h_bunches [ ] = revolution harmonic of the bunches repetition rate = harmonic of the
dominating rf signal;
' Bunch_Q_e [ ] = number of elementary charges within the bunch;
' Bunch_emit [eV.s] = bunch emittance;
' Vrf0_now [V] = peak programmed rf voltage at present time (space charge effect not
deduced)
' B_bef [T] = Dipolar magnetic field obtained at previous time step;
' B_ej [T] = required extraction dipolar field;
' B_min [T] = mimimum allowed main bending field
' t_bef [s] = cycle time corresponding to the previous time step;
' t_now [s] = cycle time corresponding to the present time step;
' Bunch_Length_bef [rf rad] = bunch length as an angle of the dominating rf harmonic (=
h_bunches) at the previous time step;
' Bucket_Fill_max [ ] = maximum allowed bucket filling = bunch emittance [eV.s] / bucket area
[eV.s];
' I_rf_max [A] = Maximum current available from the rf power amplifiers to accelerate the
beam;
' Acc [ ] = status bit indicating acceleration (if = 1) or deceleration (if = 0);
' g0_bef [ ] = space charge coupling coefficient;
' T_FT_bef [s] = Extraction flat-top duration measured at the previous time step;
' Ej_FT_duration [s] = required value for the extraction flat-top duration
Function Max_possible_B(m_i_0, m_i_a, amu_e, zeta, Radius, rho, Gamma_tr, h_bunches,
Bunch_Q_e, Bunch_emit, Vrf0_now, B_bef, B_ej, Bdot_bef, B_min, Bdot_max, t_bef, t_now,
Bunch_Length_bef, Bucket_Fill_max, I_rf_max, Acc, g0_bef, T_FT_bef, Ej_FT_duration)
If m_i_0 <= 0 Or m_i_a <= 0 Or zeta <= 0 Or Radius <= 0 Or rho <= 0 Or Gamma_tr <= 0
Or h_bunches <= 0 Or Bunch_Q_e <= 0 Or Vrf0_now < 0 Or B_bef < 0 Or Bdot_max <= 0 Or
t_now - t_bef < 0 Or Bucket_Fill_max < 0 Or Bucket_Fill_max > 1 Or (Acc <> 1 And Acc <>
0) Then
Max_possible_B = "Input error"
Exit Function
Else
If T_FT_bef >= Ej_FT_duration Then
Max_possible_B = B_min
Exit Function
Else
B = B_bef
Bunch_Length = Bunch_Length_bef
If Acc = 1 Then step = Bdot_max * (t_now - t_bef) / 100000 Else step = -Bdot_max *
(t_now - t_bef) / 100000
For i = 0 To 100000
B = B + i * step
- 58 -
Max_possible_B = B
Bdot = (B - B_bef) / (t_now - t_bef)
Bet = Beta(m_i_0, zeta, rho, B)
Frev = RevFreq(Bet, Radius)
p = Momentum_per_amu(B, rho, zeta, m_i_a)
Gamma = 1 / Sqr(1 - Bet ^ 2)
eta = (1 / Gamma_tr ^ 2) - (1 / Gamma ^ 2)
Ib = IcosBeam(Bunch_Q_e, Bunch_Length, Frev, h_bunches, h_bunches)
Vsc = VspaceChargeCos(Bunch_Q_e, Bunch_Length, Frev, h_bunches, h_bunches,
g0_bef, Bet)
Vrf = Vrf0_now - Vsc
Vrf_turn = 2 * gPi * rho * Radius * Bdot
Sin_PhiS = Vrf_turn / Vrf
Irf = Ib * Sin_PhiS
If B >= B_ej Then B = B_ej
If Irf >= I_rf_max Or B = B_ej Then Exit Function
Energy = Tot_Energy_per_amu(B, rho, zeta, m_i_a, amu_e)
BucketArea = Moving_Bucket_Area_per_amu(Radius, m_i_a, zeta, h_bunches, Energy,
eta, Vrf, Sin_PhiS)
Bucket_filling = Bunch_emit / BucketArea
If Bucket_filling >= Bucket_Fill_max Then Exit Function
Stat_BucketArea = Moving_Bucket_Area_per_amu(Radius, m_i_a, zeta, h_bunches,
Energy, eta, Vrf, 0)
Stat_Bucket_filling = Bunch_emit / Stat_BucketArea
Bunch_Length = BunchLength_from_stat_bucket_fill(Stat_Bucket_filling) / (1 Sin_PhiS ^ 2) ^ (1 / 8)
Next i
Max_possible_B = "not converging"
End If
End If
End Function
' Effective_Vrf_from_B [V] evaluates from the new value of the bending field, the effective
peak rf voltage experienced by the beam when space charge effects are deduced.
' As I didn't find how to impose a sequence of calculation to excel, I couldn't just take the value
of B resolved from the function Max_possible_B
' this is why the script here is typically a copy of the function Max_possible_B.
' The program will exit at the same B value as the function Max_possible_B, allowing to take
the value to calculate the effective rf voltage.
' The parameters are:
' m_i_0 [kg] = rest mass of the ion;
' m_i_a [ ] = rest mass of the ion in terms of atomic mass units (amu);
' m_i_e [eV/c^2] = rest mass of the ion in eV/c^2;
' amu_e [eV/c^2] = rest mass of the atomic mass unit in eV/c^2;
' zeta [ ] = net number of elementary charges of the ion;
' Radius [m] = mean radius of the synchrotron;
' rho [m] = magnetic radius of the main dipoles;
' Gamma_tr [ ] = normalized energy at transition
' h_bunches [ ] = revolution harmonic of the bunches repetition rate = harmonic of the
dominating rf signal;
' Bunch_Q_e [ ] = number of elementary charges within the bunch;
- 59 -
' Bunch_emit [eV.s] = bunch emittance;
' Vrf0_now [V] = peak programmed rf voltage at present time (space charge effect not
deduced)
' B_bef [T] = Dipolar magnetic field obtained at previous time step;
' B_ej [T] = required extraction dipolar field;
' B_min [T] = mimimum allowed main bending field
' t_bef [s] = cycle time corresponding to the previous time step;
' t_now [s] = cycle time corresponding to the present time step;
' Bunch_Length_bef [rf rad] = bunch length as an angle of the dominating rf harmonic (=
h_bunches) at the previous time step;
' Bucket_Fill_max [ ] = maximum allowed bucket filling = bunch emittance [eV.s] / bucket area
[eV.s];
' I_rf_max [A] = Maximum current available from the rf power amplifiers to accelerate the
beam;
' Acc [ ] = status bit indicating acceleration (if = 1) or deceleration (if = 0);
' g0_bef [ ] = space charge coupling coefficient;
' T_FT_bef [s] = Extraction flat-top duration measured at the previous time step;
' Ej_FT_duration [s] = required value for the extraction flat-top duration
Function Effective_Vrf_from_B(m_i_0, m_i_a, amu_e, zeta, Radius, rho, Gamma_tr,
h_bunches, Bunch_Q_e, Bunch_emit, Vrf0_now, B_bef, B_ej, B_min, Bdot_bef, Bdot_max,
t_bef, t_now, Bunch_Length_bef, Bucket_Fill_max, I_rf_max, Acc, g0_bef, T_FT_bef,
Ej_FT_duration)
If m_i_0 <= 0 Or m_i_a <= 0 Or zeta <= 0 Or Radius <= 0 Or rho <= 0 Or Gamma_tr <= 0
Or h_bunches <= 0 Or Bunch_Q_e <= 0 Or Vrf0_now < 0 Or B_bef < 0 Or Bdot_max <= 0 Or
t_now - t_bef < 0 Or Bucket_Fill_max < 0 Or Bucket_Fill_max > 1 Or (Acc <> 1 And Acc <>
0) Then
Effective_Vrf_from_B = "Input error"
Exit Function
Else
B = B_bef
If T_FT_bef >= Ej_FT_duration Then
Effective_Vrf_from_B = Vrf0_now
Exit Function
Else
If Acc = 1 Then step = Bdot_max * (t_now - t_bef) / 100000 Else step = -Bdot_max *
(t_now - t_bef) / 100000
Bunch_Length = Bunch_Length_bef
For i = 0 To 100000
B = B + i * step
Bdot = (B - B_bef) / (t_now - t_bef)
Bet = Beta(m_i_0, zeta, rho, B)
Frev = RevFreq(Bet, Radius)
p = Momentum_per_amu(B, rho, zeta, m_i_a)
Gamma = 1 / Sqr(1 - Bet ^ 2)
eta = (1 / Gamma_tr ^ 2) - (1 / Gamma ^ 2)
Ib = IcosBeam(Bunch_Q_e, Bunch_Length, Frev, h_bunches, h_bunches)
Vsc = VspaceChargeCos(Bunch_Q_e, Bunch_Length, Frev, h_bunches, h_bunches,
g0_bef, Bet)
Vrf = Vrf0_now - Vsc
Effective_Vrf_from_B = Vrf
- 60 -
Vrf_turn = 2 * gPi * rho * Radius * Bdot
Sin_PhiS = Vrf_turn / Vrf
Irf = Ib * Sin_PhiS
If B >= B_ej Then B = B_ej
If Irf >= I_rf_max Or B = B_ej Then Exit Function
Energy = Tot_Energy_per_amu(B, rho, zeta, m_i_a, amu_e)
BucketArea = Moving_Bucket_Area_per_amu(Radius, m_i_a, zeta, h_bunches,
Energy, eta, Vrf, Sin_PhiS)
Bucket_filling = Bunch_emit / BucketArea
If Bucket_filling >= Bucket_Fill_max Then Exit Function
Stat_BucketArea = Moving_Bucket_Area_per_amu(Radius, m_i_a, zeta, h_bunches,
Energy, eta, Vrf, 0)
Stat_Bucket_filling = Bunch_emit / Stat_BucketArea
Bunch_Length = BunchLength_from_stat_bucket_fill(Stat_Bucket_filling) / (1 Sin_PhiS ^ 2) ^ (1 / 8)
Next i
Effective_Vrf_from_B = "not converging"
End If
End If
End Function
' Effective_Vrf_Inj [V] evaluates the effective peak RF voltage after space charge effects have
been deduced
' in order to have a Bucket_Filling ratio as specifified by the parameter Bucket_Fill [ ]
' Acc is a bit indicating if we are accelerating or decelerating; 1 means accelerate, 0 means
decelerate)
Function Effective_Vrf_Inj(m_i_0, m_i_a, m_i_e, zeta, Radius, rho, Gamma_tr, h_bunches,
Bunch_Q, Bucket_Fill_Inj, B_Inj, Bdot_Inj, Vrf0_now)
If m_i_0 <= 0 Or m_i_a <= 0 Or m_i_e <= 0 Or zeta <= 0 Or Radius <= 0 Or rho <= 0 Or
Gamma_tr <= 0 Or h_bunches <= 0 Or Bunch_Q <= 0 Or Bucket_Fill_Inj < 0 Or B_Inj < 0 Or
Vrf0_now < 0 Then
Effective_Vrf_Inj = "Input error"
Exit Function
Else
Bet = Beta(m_i_0, zeta, rho, B_Inj)
Frev = RevFreq(Bet, Radius)
p = Momentum_per_amu(B_Inj, rho, zeta, m_i_a)
Gamma = 1 / Sqr(1 - Bet ^ 2)
eta = (1 / Gamma_tr ^ 2) - (1 / Gamma ^ 2)
Energy = Sqr(p ^ 2 + m_i_e ^ 2)
Vrf_turn = 2 * gPi * rho * Radius * Bdot_Inj
Vsc_mem = 0
Vsc = 0
For i = 0 To 1000
Vrf_eff = Vrf0_now - Vsc
Sin_PhiS = Vrf_turn / Vrf_eff
Moving_Bucket_Area = Moving_Bucket_Area_per_amu(Radius, m_i_a, zeta,
h_bunches, Energy, eta, Vrf_eff, Sin_PhiS)
- 61 -
Bunch_emit = Moving_Bucket_Area * Bucket_Fill_Inj
Stat_Bucket_Area = Moving_Bucket_Area_per_amu(Radius, m_i_a, zeta, h_bunches,
Energy, eta, Vrf_eff, 0)
Stat_Bucket_filling = Bunch_emit / Stat_Bucket_Area
Bunch_Length = BunchLength_from_stat_bucket_fill(Stat_Bucket_filling) / (1 Sin_PhiS ^ 2) ^ (1 / 8)
Ib = 2 * Bunch_Q * Frev * h_bunches * Abs(Cos(Bunch_Length / 2) / (1 (Bunch_Length / gPi) ^ 2))
Vsc = 1.69 * 377 * Ib / (2 * Bet * Gamma ^ 2)
Effective_Vrf_Inj = Vrf_eff
If Abs(Vsc_mem - Vsc) < 1 Then Exit Function
Vsc_mem = Vsc
Next i
Effective_Vrf_Inj = "not converging"
End If
End Function
' Flat_Top_duration [s] evaluates the duration of the extraction flat-top
' as a function of:
' Ej_FT_duration [s] = required value for the extraction flat-top duration
' B_ej [T] = value of the required extration bending field
' B_bef [T] = value of the field at the previous time step;
' T_FT_bef [s] = previous value of the flat-top duration;
' t_now [s] = present time step value;
' t_aft [s] = next time step value;
Function Flat_Top_duration(Ej_FT_duration, B_ej, B_bef, T_FT_bef, t_now, t_aft)
If Ej_FT_duration < 0 Or B_ej <= 0 Or B_bef < 0 Or T_FT_bef > 2 * Ej_FT_duration Or
t_aft < t_now Then
Flat_Top_duration = "Input error"
Exit Function
Else
Flat_Top_duration = T_FT_bef
If T_FT_bef >= Ej_FT_duration Then
Exit Function
Else
If B_bef < B_ej - 0.001 Then ' 0.001 is equivalent to a 10 Gauss error margin
Flat_Top_duration = 0
Exit Function
Else
Flat_Top_duration = T_FT_bef + (t_aft - t_now)
End If
End If
End If
End Function
' SyncFrequency [Hz] evaluates the synchrotron frequency
' as a function of:
' Radius [m] = mean synchrotron radius;
' sin_phiS [ ] = sinus of stable phase;
- 62 -
' V_rf [V]= peak value of the rf signal;
' E_tot_u [eV/u] = total energy per mass unit (gamma*rest energy of atomic mass unit);
' h_bunches [ ] = harmonic of the dominating rf signal with respect to the revolution;
' eta [ ] = frequency slip factor (1/gammaTransition^2) - (1/gamma^2) ;
' zeta [ ]= value of the charge state (number of protons - number of electrons);
' m_i_a [amu]= ion mass in atomic mass units
Function SyncFrequency(Radius, Sin_PhiS, V_rf, E_tot_u, h_bunches, eta, zeta, m_i_a)
If Radius < 0 Or Sin_PhiS < -1 Or Sin_PhiS > 1 Or E_tot_u <= 0 Or h_bunches < 1 Then
SyncFrequency = "Input error"
Exit Function
Else
SyncFrequency = (gC / (2 * gPi * Radius)) * Sqr(h_bunches * Abs(eta) * V_rf * Sqr(1 Sin_PhiS ^ 2) * zeta / (2 * gPi * E_tot_u * m_i_a))
End If
End Function
' Bucket_Height [eV] evaluates the full height of the moving rf bucket
Function Bucket_Height(Sin_PhiS, Beta, Vrf_now, h_bunches, eta, tot_Energy, m_i_a, zeta)
If Beta < 0 Or Beta > 1 Or Sin_PhiS < 0 Or Sin_PhiS >= 1 Or h_bunches < 1 Then
Bucket_Height = "Out of Range"
Exit Function
Else
Y = Sqr(2 * Sqr(1 - Sin_PhiS ^ 2) + (2 * Atn(Sin_PhiS / Sqr(1 - Sin_PhiS ^ 2)) - gPi) *
Sin_PhiS)
Bucket_Height = 2 * Y * Beta * Sqr(Vrf_now * tot_Energy * zeta / (gPi * h_bunches *
Abs(eta) * m_i_a))
End If
End Function
' Bunch_Height [eV] evaluates the full height of the bunch
' in a stationary bucket
Function Bunch_Height(Beta, Bunch_Length, Vrf_now, tot_Energy, zeta, h_bunches, eta,
m_i_a)
If Beta < 0 Or Beta > 1 Or h_bunches < 1 Or Bunch_Length < 0 Or Bunch_Length > 2 * gPi
Then
Bunch_Height = "Out of Range"
Exit Function
Else
Bunch_Height = 2 * Beta * Sin(Bunch_Length / 4) * Sqr(2 * Vrf_now * tot_Energy * zeta
/ (gPi * h_bunches * Abs(eta) * m_i_a))
End If
End Function
' B_from_kin_energy [T] evaluates the main dipolar field required for a specified kinetic
energy per amu [eV/u]
' as a function of
' Radius [m] = synchrotron mean radius;
' sin_phiS [ ] = sinus of stable phase;
- 63 -
' V_rf [V]= peak value of the rf signal;
' E_kin [eV/u] = total energy per mass unit (gamma*rest energy of atomic mass unit);
' h_bunches [ ] = harmonic of the dominating rf frequency with respect to the revolution;
' eta [ ] = frequency slip factor (1/gammaTransition^2) - (1/gamma^2);
' zeta [ ]= value of the charge state (number of protons - number of electrons);
' m_i_a [amu]= ion mass in atomic mass units
Function B_from_kin_energy(Radius, Sin_PhiS, V_rf, E_kin, h_bunches, eta, zeta, m_i_a,
m_i_e)
If Radius < 0 Or Sin_PhiS < -1 Or Sin_PhiS > 1 Or E_kin <= 0 Or h_bunches < 1 Then
B_from_kin_energy = "Input error"
Exit Function
Else
B_from_kin_energy = (gC / (2 * gPi * Radius)) * Sqr(h_bunches * Abs(eta) * V_rf *
Sqr(1 - Sin_PhiS ^ 2) * zeta / (2 * gPi * (E_kin + m_i_e) * m_i_a))
End If
End Function
' AlphaPhil [rf rad] evaluates relative bucket filling
' as a function of full bunch length phi_l
Function AlphaPhil(Phi_l)
If Phil < 0 Or Phi_l > 2 * gPi Then
AlphaPhil = "Out of Range"
Exit Function
Else
AlphaPhil = gPi * Phi_l ^ 2 + (1 - gPi ^ 3 / 16) * Phi_l ^ 4 / (16 * gPi ^ 4)
End If
End Function
' Sin_Stable_Phase [ ] evaluates the sinus of the stable phase in a single harmonic acceleration
case
' as a function of:
' V_rf [V] = Amplitude of the dominating Rf signal
' V_rf_per_turn [V] = voltage per turn required on the cavity gap to provide the requested
energy for the actual Bdot
Function Sin_Stable_Phase(V_rf, V_rf_per_turn)
If Abs(V_rf_per_turn) <= V_rf Then
Sin_Stable_Phase = V_rf_per_turn / V_rf
Else
Sin_Stable_Phase = "Out of Range"
End If
End Function
' Delta_Q_inc_circ [ ] evaluates the incoherent transverse tune shift in a circular beam pipe, in
the chosen plane (H or V)
' as a function of:
' Bunch_Q_e [] = the number of electric charge per bunch
' Beta_h_circ [m] = average beta function amplitude in the horizontal plane of the circular pipe
sections
- 64 -
' Beta_v_circ [m] = average beta function amplitude in the vertical plane of the circular pipe
sections
' Beam_norm_emit_h_1sig [m] = beam transverse half height measured at 1 sigma
' Beam_norm_emit_v_1sig [m] = beam transverse half width measured at 1 sigma
' pole_piece_dist [m] = distance from the middle of the vacuum chamber to the bending
magnetic pole.
' Bet [ ] = beta value = normalized velocity = v/c
' Bunching_factor [ ] = average particle line density / peak line density
' H_plane [ ] = index indicating the chosen transverse plane, write 1 for the horizontal plane, 0
for the vertical plane
' Form [ ] = empirical factor taking into account the type of transverse distribution (Gaussian,
uniform etc..)
'
and translating tune shift into tune spread by removing the min tune shift value, and
taking into account the longitudinal distribution (with or without second harmonic)
Function Delta_Q_inc_circ(Bunch_Q_e, Beta_h_circ, Beta_v_circ, Beam_norm_emit_h_1sig,
Beam_norm_emit_v_1sig, Pole_piece_dist, Bet, Bunching_factor, H_plane, Form)
If Bunch_Q_e < 0 Or Beta_h_circ <= 0 Or Beta_v_circ <= 0 Or (H_plane <> 0 And H_plane
<> 1) Then
Delta_Q_inc_circ = "Out of Range"
Exit Function
Else
r0 = 1.5347E-18
Gamma = (1 - Bet ^ 2) ^ -0.5
B = (Sqr(2) / 2) * (Sqr(Beam_norm_emit_h_1sig * Beta_h_circ) +
Sqr(Beam_norm_emit_v_1sig * Beta_v_circ)) / Sqr(Bet * Gamma) ' B = radius of the beam as
the average of Height and width
BF = Bunching_factor
g = Pole_piece_dist
eps_0_x = 0.5 * Form
eps_0_y = 0.5 * Form
If H_plane = 1 Then
Delta_Q_inc_circ = -(Bunch_Q_e * r0 * Beta_h_circ / (gPi * Bet ^ 2 * Gamma)) *
(eps_0_x / (B ^ 2 * Gamma ^ 2 * BF))
Else
Delta_Q_inc_circ = -(Bunch_Q_e * r0 * Beta_v_circ / (gPi * Bet ^ 2 * Gamma)) * ((1 Bet ^ 2) * eps_0_y / (B ^ 2 * Gamma ^ 2 * BF))
End If
End If
End Function
' Delta_Q_inc_ellip [ ] evaluates the incoherent transverse tune shift in an elliptic beam pipe, in
the chosen plane (H or V)
' as a function of:
' Bunch_Q_e [] = the number of electric charge per bunch
' Beta_h_ellip [m] = average beta function amplitude in the horizontal plane of the elliptic pipe
sections
- 65 -
' Beta_v_ellip [m] = average beta function amplitude in the vertical plane of the elliptic pipe
sections
' Beam_norm_emit_h_1sig [m] = beam transverse half height measured at 1 sigma
' Beam_norm_emit_v_1sig [m] = beam transverse half width measured at 1 sigma
' Vacuum_chamber_half_height_ellip [m] = vacuum chamber half height (= radius in case of a
circular tube)
' Vacuum_chamber_half_width_ellip [m] = vacuum chamber half height (= radius in case of a
circular tube)
' pole_piece_dist [m] = distance from the middle of the vacuum chamber to the bending
magnetic pole.
' Bet [ ] = beta value = normalized velocity = v/c
' Bunching_factor [ ] = average particle line density / peak line density
' H_plane [ ] = index indicating the chosen transverse plane, write 1 for the horizontal plane, 0
for the vertical plane
' Form [ ] = empirical factor taking into account the type of transverse distribution (Gaussian,
uniform etc..)
'
and translating tune shift into tune spread by removing the min tune shift value, and
taking into account the longitudinal distribution (with or without second harmonic)
Function Delta_Q_inc_ellip(Bunch_Q_e, Beta_h_ellip, Beta_v_ellip,
Beam_norm_emit_h_1sig, Beam_norm_emit_v_1sig, Vacuum_chamber_half_height_ellip,
Vacuum_chamber_half_width_ellip, Pole_piece_dist, Bet, Bunching_factor, H_plane, Form)
If Bunch_Q_e < 0 Or Beta_h_ellip <= 0 Or Beta_v_ellip <= 0 Or (H_plane <> 0 And
H_plane <> 1) Then
Delta_Q_inc_ellip = "Out of Range"
Exit Function
Else
r0 = 1.5347E-18
Gamma = (1 - Bet ^ 2) ^ -0.5
A = Sqr(2) * Sqr(Beam_norm_emit_h_1sig * Beta_h_ellip) / Sqr(Bet * Gamma)
B = Sqr(2) * Sqr(Beam_norm_emit_v_1sig * Beta_v_ellip) / Sqr(Bet * Gamma)
BF = Bunching_factor
h = Vacuum_chamber_half_height_ellip
w = Vacuum_chamber_half_width_ellip
g = Pole_piece_dist
eps_0_x = Form * B ^ 2 / (A * (A + B))
eps_0_y = Form * B / (A + B)
eps_2_x = -(gPi ^ 2) / 24
eps_2_y = gPi ^ 2 / 24
q = (w - h) / (w + h)
k_prime = ((1 + 2 * (1 - q + q ^ 4 - q ^ 9)) / (1 + 2 * (1 + q + q ^ 4 + q ^ 9))) ^ 2
SK = Sqr(1 - k_prime ^ 2)
K = (gPi / 2) * (1 + (1 / 4) * SK ^ 2 + (9 / 64) * SK ^ 4 + (25 / 256) * SK ^ 6)
eps_1_x = (h ^ 2 / (12 * (w ^ 2 - h ^ 2))) * ((1 + k_prime ^ 2) * (2 * K / gPi) ^ 2 - 2)
eps_1_y = -eps_1_x
If H_plane = 1 Then
Delta_Q_inc_ellip = -(Bunch_Q_e * r0 * Beta_h_ellip / (gPi * Bet ^ 2 * Gamma)) *
(eps_0_x / (B ^ 2 * Gamma ^ 2 * BF) + (eps_1_x / h ^ 2) / BF - (eps_1_x / h ^ 2) * Bet ^ 2 *
((1 / BF) - 1) + (eps_2_x / g ^ 2) * Bet ^ 2)
Else
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Delta_Q_inc_ellip = -(Bunch_Q_e * r0 * Beta_v_ellip / (gPi * Bet ^ 2 * Gamma)) *
(eps_0_y / (B ^ 2 * Gamma ^ 2 * BF) + (eps_1_y / h ^ 2) / BF - (eps_1_y / h ^ 2) * Bet ^ 2 *
((1 / BF) - 1) + (eps_2_y / g ^ 2) * Bet ^ 2)
End If
End If
End Function
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