3.5 Conclusions
3.5.2 Test Box Conclusions
To characterise the transmission response of the couplers, a ‘test-box’ was designed and manufactured and the response of each HOM coupler built for the SPS DQW was measured.
The test box measurements were split into two main measurement campaigns. The first measured the frequency of the stop-band and the second characterised the high frequency transmission response.
Difficulties with accurately measuring the stop-band frequency were observed as a result of errors imposed by the test-box geometry. This result suggests the same might be true with installation on the crab cavity and that the effect of the 3 MHz difference in stop-band frequency should be evaluated. Regardless of the error, the test-box allowed the intra-coupler stop-band frequency spread to be quantified.
The high frequency transmission analysis is not perturbed with any signifi- cance by the test-box or pick-up geometry because it is a function of the coupler’s transmission line lengths and as such insertion or rotational errors have a negligi- ble effect on the transmission response. The deviation of the manufactured HOM coupler’s high frequency response from that of the simulated was quantified (∆S21)
showing two frequency bands of decreased damping:
2) 1070 MHz→ narrowband.
The use of these measurements to assess the ability to predict deviations in HOM damping from simulations to measurements should be verified. To do this, the simulated dressed cavity mode parameters should be compared to the measured and the resulting ratios in quality factor compared to the test-box measurements.
Chapter 4
Impedance and Power
Calculations
Following the analysis of the SPS DQW HOM coupler’s equivalent circuit opera- tion, the impedance spectra of the cavity with three HOM couplers mounted was evaluated, assessing the ability of the HOM couplers to damp the HOMs within given thresholds.
In Chapter 3, the HOMs in the DQW crab cavity were characterised as ‘ver- tical’, ‘horizontal’ and ‘longitudinal’ depending on there r/Q values. The same nomenclature will be used hereafter, but with reference to the HOM impedances, i.e. taking into account their quality factors.
As a result of the HL-LHC’s high beam current, high HOM powers as a result of the interaction with longitudinal HOMs is feasible. As such, after evaluating the longitudinal impedance, the HOM power was calculated and evaluated statistically, using measured deviation values, to assess the worst-case.
4.1
Impedance Calculations
As discussed in Chapter 2, an important parameter used to quantify the effect of the higher order modes on the charged particle beam is the impedance. Impedance simulations are generally carried out using either wakefield or Eigenmode solvers. Wakefield simulations return the impedance as a function of frequency, whilst the
Eigenmode simulations generate the mode parameters and field topology from the solutions of the eigenvalue problem [106]. In Eigenmode simulations, the impedance is calculated for the respective resonant modes.
The wakefield simulations are limited by the computationally intensive cal- culations that are needed for high-Q cavities. As such, using the mathematics detailed in Chapter 3, the Eigenmode solver in CST MWS [104] was used for the impedance simulations. The ‘non-linear Eigenmode solver’ with tetrahedral meshing was used for the calculations. The non-linear solver avoided problems with ‘Q-switching’ (where the quality factors are mistakenly assigned to adjacent modes) often seen with the standard Eigenmode solver for broadband simulations. For this reason and for general accuracy, this solver was recommended by the software developers [107]. Figure 4.1 shows the dressed SPS DQW crab cavity, annotated with the coupler nomenclature used throughout this thesis, alongside the vacuum model used for simulations.
(a) HOM coupler nomenclature. (b) Vacuum model in CST MWS [104].
Figure 4.1: Dressed SPS DQW crab cavity.
The longitudinal and transverse impedance was calculated for the dressed SPS DQW crab cavity1. The transverse impedance was calculated in the vertical and
horizontal planes where a linear voltage relationship with radial position was as- sumed, modelling each mode as if it were a pure dipole. To account for the band- width of each mode, and since it was the resistive part of the losses (i.e. power
1The design for preliminary tests in the Super Proton Synchrotron (SPS) at CERN before
loss) for evaluation, the real part of the resonator model
Z(f) = Zn
1 +Q2
e(x/f −f /x)2
(4.1) was applied over a frequency range up to the beam-pipe cut-off frequency of 2.1 GHz. Here,Znis the impedance at the resonant frequencyf,Qe is the external quality factor and xis the frequency array.
For an ‘exotic’ cavity geometry such as that of the DQW crab cavity, some HOMs have significant r/Qk and r/Q⊥ values. As such, the longitudinal and
transverse impedance was calculated for every HOM up to the beam-pipe cut-off frequency. The resulting impedance spectra are shown in Fig. 4.2.
)UHTXHQF\>0+]@ =l > 2 K P V FD Y LW \ @ (a) Longitudinal. )UHTXHQF\>0+]@ =v > 2 K P V P F DY LW \ @ (b) Vertical. )UHTXHQF\>0+]@ =h > 2 K P V P F DY LW \ @ (c) Horizontal.
Figure 4.2: Impedance spectra in each plane for the dressed SPS DQW crab cavity. The design limits for the LHC DQW1, shown as red lines in Fig. 4.2, are
200 kΩ/cavity and 1 MΩ/m/cavity for the longitudinal and transverse impedance respectively [108]. For the SPS DQW, this criteria was not met for all modes.
This was because of a space constraint associated with the cryomodule in the SPS which limited the size of the couplers.
Modes above the impedance threshold as well as low frequency1 (i.e. <1 GHz),
high impedance modes are detailed in Tab. 4.1.
Frequency [MHz] Qe Rv [kΩ/m]a Rh [kΩ/m] Rl [kΩ]b 570.36 3080 3 0 77 590.14 1920 0 0 61 681.62 1160 0 175 0 746.67 6160 1889 0 0 926.80 12600 0 4020 0 958.87 10300 15 0 100 1620.25 187670 1 2187 1 1659.75 106680 0 2911 0 1662.25 3377420 2 4542 2 1746.23 35440 3132 0 3 1754.42c 23520 0 2047 0 1789.31 104020 1339 7 413 1840.93 16630 1108 0 1 1856.09 126610 1 4013 0 1953.89 57330 530 4 227
Table 4.1: Modes which are higher in impedance than the design limit (boxed) and low frequency, high impedance modes.
aTransverse threshold: 1 MΩ/m/cavity bLongitudinal threshold: 200 kΩ/cavity
cDamped by the pick-up probe on the cavity beam-pipe.
Referring to the modes detailed in Tab. 4.1, three points of interest were noted from the impedance simulations:
• Lowest frequency mode ‘split’: With the addition of the HOM couplers to the cavity, the lowest frequency HOM was ‘split’ into two modes. This is a result of the HOM coupler’s transmission line resonance from the hook to the band-stop filter geometry. The resonance has a high amplitude within
1Due to the profile of the proton bunch, the HOM excitation is inversely proportional to the
the bandwidth of the first HOM, but has a much lower bandwidth than the HOM.
• Many horizontal modes are over the impedance threshold: There are six modes for which the horizontal impedance is over the impedance threshold. Furthermore, they are all more than a factor of two higher than the design threshold and one is low frequency. This is a result of the general field topology of the horizontal modes. Whereas the highest impedance verti- cal modes generally have a high magnetic flux around the central resonators at the cavity base, which couples well to the hook type HOM couplers po- sitioned at 90◦ to this flux, the high impedance horizontal modes do not
necessarily have the highest magnetic field at this location.
• The mode which is damped by the pick-up probe is over the
impedance threshold: The mode (1754 MHz) has a high field area on the upper surface of the cavity, but it is concentrated in a plane which the HOM couplers cannot efficiently couple to. There is also a high field area on the beam-pipe blend. It is hence damped by the Pick-Up (PU) probe, which uses a ‘mushroom-like’ geometry to coupler to the electric field at the beam-pipe. The mode’s field topology is shown in Fig. 4.3 alongside a 3D model of the pick-up probe.
(a) Electric (left) and magnetic (right) field topology of the 1.75 GHz mode (arbitrary units).
(b) Pick-up probe used for sam- pling the fundamental mode and damping the 1754 MHz mode.