Study of absorber Study of absorber effectiveness in ILC cavitieseffectiveness in ILC cavities

K. Bane, C. Nantista, C. Adolphsen

12 October 2010

Outline of Talk • Introduction

• S-matrix formulation

• The basic rf unit

• Results

• Conclusion

See K. Bane, C. Nantista, C. Adolphsen, “Higher order mode heating analysis for the ILC superconducting linacs,” presented at Linac10, SLAC-PUB-14257, Sep. 2010.

Introduction • The superconducting cavities and interconnects in the ILC linac are designed to operate at 2 K, where cooling costs are very high.

• Ceramic dampers, connected to 70 K, are placed between the cryomodules to absorb HOM power. How effective are they?

• Brute force calculation of the modes in a string of cryomodules is limited by computing capacity. E.g. M. Liepe has calculated ~400 modes in 3 cavities plus absorbers, up to 8 GHz.

• We use an S-matrix approach, applied to an rf unit comprising 26 cavities and 3 absorbers, to address this problem. We perform calculations at sample frequencies (4, 8, 12, 16, 20 GHz) to assess the effectiveness of the absorbers.

S-matrix Formulation

Equations: • In the case of a single mode, at Junction n

rn= (Sn-1)21rn-1+ (Sn-1)22ln

ln= (Sn)11rn+ (Sn)12ln+1

with n= 1,…N, where N is number of objects; S-matrix at freq. f

• With M modes

(ln,1,ln,2,…,ln,M,rn+1,1,rn+1,2,…,rn+1,M)T= S (rn,1,rn,2,…,rn,M,ln+1,1,ln+1,2,…,ln+1,M)T

Fig 1: A string of beamline objects coupling through two beam pipe modes.

• Boundary conditions: for all modes m

r1,m= rN+1,me2πifL/c, lN+1,m= l1,me-2πifL/c

where the repeating unit length is L

• Driving terms moving to the right (in mode 1)

dn’,1= exp(-2πifzn’/c)

with n’ limited to downstream junctions of the cavities

Note: in reality beam excites different modes at specific amplitudes, but

(1)we are only interested in relative power deposition in absorbers vs other objects(2) we are counting on mode mixing to make us insensitive to which mode we excite (to be tested)

• Problem can be written as matrix equation

Ay= d

which can be solved for y, the right and left going voltages in all the junctions in all the modes

with N objects and M modes, the order of the problem is 2M(N+1)

• The power deposited in object n is

pn= m(|rn,m|2+|ln+1,m|2-|ln,m|2-|rn+1,m|2) + (|rn+1,1|2-|rn+1,1-dn+1,1|2)

since our drive moves to the right in mode 1

The Basic RF Unit

Fig. 2: Arrangement of beamline objects in an ILC basic rf unit. Also included, but not shown, are bellows connecting all objects

• The basic rf unit is 40 m long. All objects are modeled as cylindrically symmetric with beam pipes of radius a= 39 mm

• HFSS was used to find S-matrices of the different types of objects. Note that there are 1, 2, 3, 4, 5 TM0 modes for f= 4, 8, 12, 16, 20 GHz.

• The cavities were taken to be perfectly conducting with Cu plated beam pipes (of length 264 mm), the drift is Cu plated (length 1.2 m), the bellows are Cu plated. Conductivity of Cu at 2 K: 1010 Ω-1m-1.

• The absorbers at 70 K are estimated to have tan = 0.23 – 7.5e-3 f [GHz]

Mildner et al., SRF05For lack of a better idea, we approximate tanl for 10 mm ring @ 70K as linearly dropping from 0.2 to 0.08 between 4 GHz and 20 GHz, based on the plots at right.

tanl = 0.2, @ 4 GHz0.17, @ 8 GHz0.14, @ 12 GHz0.11, @ 16 GHz0.08, @ 20 GHz

Ceramic HOM Absorber

142.3 mm

60 mm

142.3 mm

r = 39 mm

8.85% absorbed

4 GHz: = 15, tanl = 0.2

@ 8 GHz: 42% and 73% absorbed

@ 12 GHz: 30%, 54%, and 64% absorbed

(Chris Nantista)

Fig. 3: Output of sample HFSS calculations of: (a) an ILC cavity, (b) an absorber, (c) a bellows. In all cases the beam pipe radius is a= 39 mm. • The basic rf unit has N= 66 objects. At f= 20 GHz, where there are M= 5 modes, the order of the matrix is 670.

• As indicator of effectiveness of the absorbers we use

p2K/ptot= 1 – pabs/ptot

with p2K and pabs, respectively, the power deposited at 2K (into the Cu) and at 70K (into the absorbers).

Results • We perform the calculations for the 5 sample frequencies (see Table 1, nominal)

• We allow for slight variations in the cavity pipe lengths:

Sij CiSijCj , where Ci= exp([i-ii]z/2) ,

and i= 2πfRs/(caZ0i), i= [(2πf/c)2-(j01/a)2]1/2, Rs= (πZ0f/c)1/2

• We added random pipe lengths, with z= 2 mm, to cavities (see Table 1 for statistical results) Table 1: Nominal results and statistics for p2K/ptot when varying

cavity beam pipe length randomly, with z= 2 mm (in [%]).

Fig 4: For f= 4 GHz, |y| for right going waves (blue) and left going waves (brown); |rn,1| for absorbers is shown in red. For this case p2K/ptot= 0.17.

Fig 4b: For f= 4 GHz, power deposition along the rf unit; the absorber losses are shown in red.

(Finer vertical scale)

Fig. 5: For f= 16 GHz, p2K/ptot vs. systematic change in cavity beam pipe length z.

• To find power deposition in frequency gaps, ideally we should repeat the process for more frequencies

• Instead, for our 5 sample frequencies, we systematically lengthen the cavity beam pipes, and take the statistics of the results as indicators of absorber efficiency over the frequency intervals

• The results of 2000 steps, for f= 16 GHz, are shown in Fig. 5.

Fig 6: For f= 16 GHz and z= 0.8 mm, |y| for right (blue) and left (brown) going waves in the first mode (m= 1, for clarity); |rn,1| for absorbers is in red; p2K/ptot= 0.95.

• Some of the high-loss points represent very localized loss patterns (see Fig. 6)

• For f= 16 GHz, we repeated the calculation, allowing the driving term to be in a higher beam pipe mode, m (see Table 3). We see that the statistical results are relatively insensitive (~25%) to which mode is driven.

Table 2: Statistics for p2K/ptot when varying cavity beam pipe length systematically (in [%])

Table 3: For f= 16 GHz, when the driving mode is m. Statistics for p2K/ptot when varying cavity beam pipe length systematically (in [%])

9 cav 4 cav

abs

9 cav

abs abs

4 cav

drift

• A thought experiment: can we improve the absorber effectiveness by moving an absorber close to the drift?

• There seems to be a significant improvement in absorber effectiveness

Table 2b: Statistics for p2K/ptot when varying cavity beam pipe length systematically (in [%]).

Actual configuration Modified configuration

Conclusions • We have presented an S-matrix formulation of power flow in a string of three ILC cryomodules, to study the effectiveness of the ceramic dampers in absorbing the HOM power generated by the beam

• From detailed calculations at 5 sample frequencies, we have obtained estimates of the damper effectiveness in the frequency range up to 20 GHz

• In the frequency range up to 20 GHz, we find that the expectation value of the fraction of HOM power that is not absorbed by the dampers is 1.5%--8.0%

• According to our calculations a highly localized field pattern within a cryomodule, one that avoids the absorbers and renders them ineffective, can occur

• Note that in this simplified, cylindrically-symmetric model, the effect of the fundamental mode and higher mode couplers is not included