Power Overhead Calculation For
Lorentz Detuning Force
Rihua Zeng
2012-Mar-20
Power over head calculat ion for non-opt imal QL
For thefor
prot on linac,
Q is not opt imaloverhead
for each cavity but usually set t o a fixed design value fo
Pre-detuning
reducing
L
same type of cavit ies. In t he case of non-opt imal QL and non-opt imal beam velocity for t he cavity
required generat or power Pg and power overhead ρmax for LFD can be calculat ed as follows (assum
t hat proper pre-detuning has been applied t o cancel t he effect of synchronous phase operat ion)[8]:
2
1 Vcav
Pg =
8 RL
ρmax (β) =
RL
1+
(β) I bcosϕ b
Vcav
2
+
∆ ωL (t)
ω1/ 2
2
|∆ ωL (t)|max
1
·
ω1/ 2
1 + 0.5(R/ Q) β QL I bcosϕ b/ Vcav
2
As ment ioned in last section, t he det uning range ∆ f = f 1/ 2 and ∆ f = 2f 1/ 2 will be assumed res
t ively t o estimat e power overhead for LFDs in different levels. Figure 3 shows t he correspoding over
in half bandwidt h detuning for all t he superconduct ing cavit ies, by employing equat ion 5 and assum
2
opt imal pre-det uning for LFD is applied. It should be not ed t hat the t erm VRcaLv is dependent no
beam velocity β but on t he accelerat ing field, which means t hat for t he same det uning fact or ∆ f /
in t he same type of cavity operat ing at same accelerat ing field, t he extra power for LFDs are t he s
In contrast , t he generat or power Pg required under no LFD does depend on β and Pg goes up
increases. As a result , the rat io of t he ext ra power for LFD to generat or power wit hout LFD decre
as β increases, which is t he case in Figure 3 for spoke and most high bet a cavit ies operat ing near
t he same accelerat ing fields. In t he med beta cavities, power overhead t rend behaviours different
t hey are designed t o operat e at much different accelerat ing fields.
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Overhead estimation for cavity with optimal Ql
Overhead calculation for cavity with non-optimal Ql
•
It makes calculation easier to discuss detuning according to the rate Δf/f1/2
•
Below f1/2 , ( K~1.5 for high beta, K~2 for med beta), most cavity overhead is <7%.
•
25% or more are required for detuning > 2 f1/2 (K~3 for high beta, K~4 for med beta)
•
Appropriate pre-detuning for both sync. phase and LFD is assumed
Overhead for Lorentz force detuning under Ql variations
Overhead for Lorentz force detuning in Spoke cavity
Some measured data about LFD of spokes is
4~8 Hz/(MV/m)2.
Worse data in some individual measurement.
Overhead is relatively higher in the beginning
section of spokes
Generator power (W)
1000000
900000
800000
700000
600000
500000
400000
300000
200000
100000
0
0
50
100
150
200
fraction of cavities. The detuning coefficients are about
3~4 Hz/(MV/m)2 and 1-2 Hz/(MV/m)2 in medium and
high beta cavities respectively. Some cavities show higher
Lorentz
detuning
under
mechanical
resonance
detuning force
coefficients
at higher
repetition
rates as shown
in
Fig. 7, implying dynamic resonance condition.
hape
could
(Fig.
800
60Hz
Dynamic Detuning (Hz)
high
ts in
ating
V/m).
safe
rates
many
pends
s and
Fig.
ature
s are
1000
600
30Hz
400
200
15Hz
0
flattop
filling
-200
0
500
1000
1500
Time (us)
Figure 7. Lorentz force detuning at various repetition rates
Typical mechanical modes in superconducting cavity
1st and 2nd order equation simulation for LFD
Some measurement data by piezo sensor at FLASH
Conclusion
•
The overhead could be reduced to a small value if the detuning is limited into
half cavity bandwidth with lower LFD coefficient, optimal pre-detuning for
LFD, and avoiding mechanical resonance.
•
Overhead increases by the square of rate Δf/f1/2 . 25% or more are required for
detuning Δf/f1/2 > 2 (K~3 for high beta, K~4 for med beta) , under optimal predetuning for LFD
•
It seems non-significant overhead is required for Ql variations ±5%
•
More overhead is required for some spoke cavities at the beginning of spoke
section.
•
Need to investigate more details for mechanical modes of cavities at ESS.