Calculations of the Resonant Frequency and Thermal Stress of the Front End Test Stand Radio
Frequency Quadrupole
INTRODUCTION
The Front End Test Stand (FETS) [1] being constructed at the Rutherford Appleton Laboratory is entering
the next stage of development, with the three-solenoid magnetic Low Energy Beam Transport (LEBT)
being installed. The next major component of FETS will be the vane-type Radio Frequency Quadrupole
(RFQ). Drawing up to 2 MW of power from an already-commisioned Toshiba Klystron tuned to 324
MHz at a duty cycle of up to 10%, the RFQ will accelerate H- ions from 65 keV to 3 MeV over its four
metre length. The manufacturing process, resonant frequency calculations and tuner stepper motor
controls have been thoroughly tested at Imperial College, UK, with a 40 cm cold model of the RFQ [2].
Since the construction of the cold model, the design of the real four metre RFQ has been modified such
that it will now be fully bolted together, with milled water baths for cooling. This design has benefits such
as ease of construction, no risk of the copper warping under welding and the ability to take the RFQ apart
for cleaning. However, the drawback is that each flange must be approximately 1.5 times the size to allow
for vacuum seals; meaning space for tuners, RF couplers and vacuum feedthroughs is limited.
SIMULATION PROCESS
As such, a comprehensive finite element model has been constructed in ANSYS [3] to calculate the
resonant frequency of the new design and the associated electromagnetic fields in the cavity vacuum. A
macro is then used to convert the magnetic fields into wall current heat loads in order to calculate the
temperature distribution in the copper walls. The heat transfer coefficients of the water baths are then
adjusted in order to reduce the overall temperature of the RFQ and the temperature gradients of the vanes
to acceptable levels. In this manner, the coolant network can be analysed to judge its ability to take away
the RF heat loads.
The calculated temperature distribution can then be input as a structural boundary condition in order to
calculate displacements and stresses as the copper suffers thermal expansion. Subsequently, the calculated
displacements and electromagnetic fields can be exported to Matlab, where a routine has been written to
determine the resonant frequency shift using the Slater perturbation method. The entire model, therefore,
is used to determine whether the frequency shift due to thermal expansion is within the tuning range of
the movable slugs.
PREVIOUS WORK
The RFQ cold model was originally simulated [4] using CST Microwave Studio [5]. In the model, the
copper walls were moved, whilst the vane tips were held fixed, until the resonant frequency of the cavity
became the required 324 MHz. Subsequently, the position of the slug tuners was varied to investigate
how they affected the resonant frequency and Q-value. The simulations were found to be very accurate, as
the manufactured cold model performed exactly as expected [6].
The full four-metre RFQ design [7] needed to be tested for its thermal performance, so a CAD model was
imported into ANSYS workbench to study temperature distributions and structural displacements.
Initially, rough estimates were used for the heat loads applied to the walls. Because of the good thermal
conductivity of copper, the overall steady state temperature distribution was quite insensitive to the
positions of the applied heat flux; however it soon became clear that the magnitude of hot spots made a
large difference, and these could not be estimated. Therefore, the real heat flux needed to be calculated
using the magnetic fields in the vacuum cavity. ANSYS Workbench is presently unable to calculate
resonant frequencies, so the model had to be imported into ANSYS Classic to use its high frequency
eigenmode solver.
PREPARING THE MODEL
ANSYS Classic proves cumbersome and not entirely reliable at importing and manipulating complex
model geometries, so a method was devised to transfer the model from ANSYS Workbench. An IGES
file of the copper wall geometry of a one-meter section of the RFQ was imported into Workbench from
Autodesk Inventor [8]. Figure 1 shows how, using the ANSYS Design Modeller module, the model was
simplified by removing parts either irrelevant to the simulations or which would require high mesh
densities to accurately represent, such as bolt holes. The model was then sliced along all three axes due to
the symmetry of the problem, leaving one eighth of an RFQ section. The space contained by the copper
walls and vanes was then “filled” with a vacuum body in order to simulate the cavity RF eigenmodes.
FIG. 1. Imported IGES model of a 1 m RFQ section (left) and the remaining model after simplification
and slicing by three symmetry planes (right). The ends of the vanes and the cut-backs can be seen in the
left-hand image. The right-hand image shows the internal structure of the cooling channels and how an
end wall was added in order to close the vacuum near the vane ends.
MESHING
There is a facility to communicate between ANSYS Workbench and Classic via ‘Input Files’. By making
one of these files, the mesh and thermal boundary conditions set in Workbench is written in a way that
Classic can interpret and solve by itself. The drawback to this method is that no geometry data is
transferred – only the mesh. Therefore, the usual CAD commands such as picking a face to apply a
boundary condition fails as there are no edges, faces or bodies defined in the input file. All commands
such as applying boundary conditions must be done on the mesh elements or nodes themselves. Clearly, a
typical mesh has many thousands of nodes so one cannot just pick a particular set to apply boundaries to.
Therefore, “Named Selections” must be made in the Workbench simulation and written to the input file
along with the mesh. These named selections record which nodes are attached to the faces you may pick
in Workbench. It is these lists of nodes to which you can assign boundary conditions in Classic. It is
critical to create named selections of all important boundary nodes for all the simulations you wish to
perform in Classic. The entire Eigenmode-Thermal-Structural process was carried out in ANSYS Classic,
so the mesh and named sections had to be prepared to ensure all three steps could be carried out
consecutively. The final mesh of both the copper walls and internal vacuum is shown in Figure 2.
FIG. 2. The copper and vacuum meshes shown together. Note the high mesh density between the vane
tips and at the vane cut-backs, whereas the bulk body meshes can be quite course.
The ANSYS thermal and structural solvers are very accurate even with fairly course meshes. Similarly,
the high frequency eigenmode solutions show very little change with different mesh densities. However,
the boundary between the copper and the vacuum is of critical importance as this is where the magnetic
fields and subsequent heat flux are. The boundary was set to have a much finer mesh than the bulk
vacuum and copper bodies. Preliminary tests showed that the magnetic fields bunched together, as
expected, around corners of the copper, such as at the tuners and the cut-backs in the vane ends. It is at
these positions where the maximum heat flux is. These areas therefore required the highest mesh densities
of the entire model to ensure accurate solutions. The mesh was also fairly fine at the vane tips to ensure a
reasonable representation of the electric quadrupole field; but properties such as field flatness and
quadrupole uniformity were not studied in this simulation so the mesh was not excessively dense there.
RESONANT FREQUENCY CALCULATIONS
The resonant frequency of the one-metre RFQ section was determined using second-order, ten-node
HF119 tetrahedron mesh elements. Second order elements require much more computer memory and
processor time, but provide far greater accuracy to the solutions and subsequent post-processing, so are
highly recommended. “Electric Wall” boundaries were applied to the nodes which coincided with the
copper walls to ensure the magnetic field lines flow parallel to the walls. “Electric Shields” were also
applied there in order to calculate the Q-factor. The vacuum faces created by the symmetry planes were
left open, allowing the magnetic field to enter and exit through them, and for the electric quadrupole field
at the vane tips to be properly formed. Plots of the electromagnetic fields are shown in Figures 3 and 4.
FIG. 3. Electric quadrupole field at the vane tips (left) and the magnetic field flowing past the vane cutback (right).
FIG.4. Contour plot of the magnetic field (left), where light colours show the strongest fields occurs at the
vane cut-backs, as well as at the tuner and vacuum pumping slots. The pumping slots work well in
preventing RF from reaching the vacuum pump itself (right).
The resonant frequency of the one-metre RFQ section was found to be 333 MHz. This is nine megahertz
above the required frequency of the RFQ, which is not tolerable. To ensure the accuracy of the BlockLanczos eigenmode solver used by ANSYS Classic, the RFQ cold model was simulated to benchmark it
against the previous CST solution. The 40 cm cold model was again found to have the desired 324 MHz
resonant frequency. Therefore, it was neither the ANSYS solver nor the meshing to blame for the 9 MHz
discrepancy: it had to be due to the different RFQ geometry.
Two factors are different between the 40 cm cold model and one-metre RFQ section. The length is the
most obvious change, but the tuner, coupler and vacuum pump ports are also positioned differently. The
RF couplers on the cold model were right next to the vane cut-backs which, as has already been
mentioned, is where the maximum magnetic flux density is. The ports on the present RFQ design are far
away from these cutbacks. Therefore, this difference could also affect the flow of the magnetic field and
hence the resonant frequency of the cavity.
The model and mesh were re-designed in ANSYS Workbench to facilitate different parts of the vacuum
to be ‘enabled’ in the eigenmode solver, shown in Figure 5. Each part was separated off into its own
vacuum body, which in ANSYS Classic is held as a separate mesh element type. Therefore, there can be
any number of HF119 element types defined, one for each part. If a part should not be included in the
solution, it can be changed to be made up of ten-node MESH200 elements instead, which are excluded
from the solution. In this way, the active tuner/coupler/pump ports could be varied, as well as the overall
length.
FIG. 5. Each colour denotes a different block of vacuum which can be enabled in the eigenmode solver.
Note that this model is for the entire four-metre RFQ, compared to the one-metre section in Figure 2.
Hence there are considerably more mesh elements, but it can be seen that some areas have had their mesh
density reduced in order to circumvent memory limitations. This does not affect the eigenmode solver,
but would affect subsequent heat load calculations and the thermal solver.
Figure 6 shows how, for a particular RFQ length, the resonant frequency varies as much as 1 MHz
depending on the port positions. This reflects the fact that the frequency does indeed change when the
slug tuners are moved, which is encouraging, but it does highlight the need to consider the placement of
the RF couplers.
Varying the length showed an even more marked difference in the resonant frequency, as Figure 7 shows.
Reducing the length to 40 cm again gave approximately 324 MHz, which is where the cold model was
tuned for. The frequency increased asymptotically toward 335 MHz as the length was increased.
Intuitively, this goes against expectations, as one would ordinarily think that a larger volume would
decrease the frequency of a resonating cavity. As an example, think of how a glass of water with the
water gradually removed (i.e. more air put in) makes a lower note sound when struck. However the subtle
difference in the RFQ is that the resonant frequency, f0 is caused by a balance between the electric and
magnetic fields, or more specifically their associated capacitance, C and inductance, L by the relationship
f0 
1
2 LC
(1)
Using Equation 1 it is clear that increasing either the inductance or capacitance reduces the resonant
frequency. In the relatively short cold model, the capacitance between the vane ends and the end wall is
large relative to the capacitance between the vane tips. Over the entire four-metre RFQ, though, the vane
ends have only a tiny contribution to the total capacitance, so the relative capacitance decreases, leading
to an increased resonant frequency.
FIG. 6. Change in resonant frequency with
active port positions. Note how the vacuum slots
have little effect on the resonant frequency.
FIG. 7. Change in resonant frequency with the
total RFQ length. The lines are arctangent
functions used to give an asymptotic fit.
To counteract this effect, the transverse area of the RFQ must be increased (i.e. the walls moved further
away from the vane tips) in order to increase the inductance of the cavity and hence reduce its resonant
frequency back down. The size should be chosen such that the movable tuners should not have to impinge
on the vacuum too much, in order to tuner either side of the 324 MHz resonance. Additionally, with the
tuners set ideally as close to the walls as possible, this increases the Q-factor of the cavity and reduces the
heat load to the tuners.
TEMPERATURE DISTRIBUTIONS
When the resonant frequency and electromagnetic fields have been calculated, an ANSYS macro donated
by Neal Hartman (LBNL) and Robert Rimmer (JLAB) [9] is used to calculate the surface loss currents
created by the magnetic fields. These loss currents are then converted into a heat load for each node on
the surface and applied as heat flux on MESH152 triangular 3D thermal surface effect elements. The heat
flux is scaled to the expected heat loss of the whole RFQ. So, for example, if the full four-metre RFQ uses
2 MW of power on a 10% duty cycle, then one quarter of a 50 cm section would have
2,000,000 W * 10% / 8 / 4 = 6,250 W
(2)
6.25 kW in total applied to its walls, distributed in accordance with the magnetic surface losses. In reality,
only about a quarter of this power will be used on the FETS RFQ, so this applied heat load includes a
large safety margin. The copper walls (which, during the eigenmode solution, had been left as un-solved
MESH200 elements) are converted to SOLID87 thermal elements whilst the vacuum is converted to
MESH200. As well as the calculated surface heat loads, convective heat transfer coefficients (HTCs) are
applied to the walls exposed to air, as well as to the tuner, coupler and vacuum pump coolant channels.
Ambient air is assumed to have an HTC of 20 Wm-2K-1 and temperature of 22 °C. The cooling pipes for
the tuner, coupler and vacuum flanges are fairly typical shapes which will have a predictable flow pattern
so it is safe to assume that they can easily achieve HTCs in excess of 3000 Wm-2K-1 and water
temperatures of 15 °C. The novel water bath cooling for the vanes, however, is not so straightforward, so
the HTC to give optimum cooling strategy was decided using the simulation, and then it is up to pipe flow
calculations to determine whether the required HTC is achievable.
FIG. 8. Temperature distribution within the copper for 6,250 W of input power and with the water baths
having heat transfer coefficients of 3000 Wm-2K-1 and water temperatures of 15 °C.
As shown in Figure 8, the outer walls and central parts of the vanes reach a temperature of about 40 °C.
The majority of the input heat load is at the vane cut-backs, so the water baths have been placed in order
to maximise the cooling there. However the baths do not penetrate far into the vanes, so there is no direct
cooling applied to the vane tips, especially the very ends. Therefore, it is the vane ends which reach the
highest temperature in the RFQ, being at approximately 100 °C. This means there is a 60 °C temperature
gradient from the major part of the vane to its end tip.
STRUCTURAL DISPLACEMENT
Using the standard equation for the change in length, ΔL due to thermal expansion
L   L L T
(3)
where αL is the thermal expansion coefficient of copper (= 1.7x10-5), one can estimate that the vane ends,
which have a length L of about 10 cm, with an applied temperature gradient ΔT of about 60 °C will grow
in length by something of the order 100 µm. In the ANSYS model, the SOLID87 thermal mesh elements
are converted into SOLID187 structural elements, symmetry planes are defined and the calculated
temperature distribution is input as a structural boundary condition. Upon solving the problem, the vane
ends are found to move by about 230 µm, as Figure 9 shows, which agrees well with the crude estimate
using Equation 3. The vane tips need to be machined to an accuracy of about 50 µm to ensure the RFQ
stays on tune, so at first glance this displacement sounds far too large.
FIG. 9. Longitudinal displacement within the
RFQ. The colour scaling ranges from 0 (blue) to
233 µm toward the left-hand end wall (red).
FIG. 10. Vertical displacement within the RFQ.
The colour scaling ranges from -1.23 mm
upwards (blue) to 37 µm downwards (red).
However, Figure 10 shows that the walls of the RFQ move outward, counteracting any inward movement
of the vanes toward each other, so the only appreciable movement of the vanes is at the ends,
longitudinally. The result is that the vane ends are the only major contributor to a small frequency shift.
Exporting the nodal displacements into a routine written in Matlab to calculate the frequency perturbation
using the Slater algorithm gives a frequency shift of approximately 312 kHz, which is well within the
several-MHz range of the movable slug tuners. Previous calculations of the temperature distribution and
associated frequency shift of the cold model with 60 W of input RF power have been in excellent
agreement with those measured on the real cold model, so the predictions made of the four-metre RFQ
and 2 MW power are expected to be good estimates of the real performance.
Plotting the calculated von Mises stress in Figure 11, it is apparent that the longitudinal expansion of the
vane ends puts a large stress of approximately 200 MPa on the base of the vanes, where the cut-backs
meet the water baths. This stress is significantly higher than the yield stress of copper so more detailed
calculations are needed to check this value. This will be performed by increasing the mesh density in that
region of copper to ensure the structural solver is reliable. If the stress is still found to be large,
modifications will need to be made to the copper body in order to increase the strength at the vane cutbacks without affecting cooling efficiency.
FIG. 11. Total von Mises stress, where yellow-red colours show high stress at the vane cut-backs.
SUMMARY
The ANSYS model of the FETS RFQ has evolved to be a detailed, multi-purpose tool, using the
eigenmode electromagnetic fields to create heat flux on the copper walls. This heat load is then used to
calculate temperature distributions and structural displacements, which can be exported to calculate the
frequency perturbation caused by the heating. The full four-metre RFQ is found to have a larger resonant
frequency than the cold model, so will have to be made larger to bring the frequency down to be in tune
with the klystron. The cooling channel design is able to take away the large heat loads, leaving the copper
a reasonable temperature. The movement of the vane ends is sufficient to change the resonant frequency
by approximately 300 kHz, which is within the range of the tuners. Large stresses are caused in the
copper at the vane cut-backs, however, which need to be addressed. Overall, the simulations show that the
present mechanical design of the RFQ is physically feasible and can be carried forward to manufacture.
REFERENCES
[1] A. Letchford et. al, THPP029, EPAC’08.
[2] A. Letchford and P. Savage, MOPCH117, EPAC’06.
[3] ANSYS, Canonsburg, USA, www.ansys.com.
[4] A. Kurup and A. Letchford, MOPCH116, EPAC’06.
[5] Computer Simulation Technology, Darmstadt, Germany, www.cst.com.
[6] S. Jolly et. al, THPP024, EPAC’08.
[7] Pete’s upcoming IPAC’10 Paper.
[8] Autodesk Inventor, Farnborough, UK, www.autodesk.co.uk.
[9] N. Hartman and R. Rimmer, MPPH060, PAC’01.