1. Power Requirements for High beta Elliptical Cavities Rihua Zeng Accelerator Division Lunds Kommun, Lund 2011-6-13

2. Outline Extra power for the cavity Cavity Filling time Modulator droop and Ripple

3. Extra power required We only consider here the extra needed for the cavity, not the waveguide power loss and reflection. Causes: Lorentz detuning, microphonics, synchronous angle, Ql variation, other perturbations(beam loading, klystron droop and ripple, etc) and over shoot due to feedback control

4. Minimum power required How to achieve the minimum power?(assume no detuning, beam on-crest) The key factor to be optimized: coupling factor β, i.e., Q L RL C Ib Icav Transmission Line Z ext ext V for, I for V ref, I ref Generator power: Pg After optimum coupling, the beam induced voltage equal cavity voltage. Optimum coupling only works for one beam current, not for different currents.

5. Minimum power required In the presence of beam loading, the minimum required power could be achieved if all the power is transferred to the beam and cavity wall loss, without any reflection. The way to realize that is to optimize the coupling β, i.e. optimizing the load Q value Q L Now we get the right accelerate voltage simply like that, fantastic! The ideal is fantastic, but the reality is harsh…

6. The Extra power required We have to deal with the Lorentz force detuning, microphonics, synchronous phase, and Q L variation… Can we manage it? Yes! The method is simple, more power! What amount do we need? Don’t worry, someone has figured it out! It looks complicated, but we can make it clear (Assume the Q L is optimized at the design current )! by detuning, Lorentz force, microphonics… by synchronous phase

7. Extra power by detuning by detuning Detuning Δf/Hz f 1/2 / /Hz Required Extra Power 1005180.93% 2005183.73% 3005188.39% 40051814.91% 50051823.29% 60051833.54% 70051845.65% 80051859.63% 90051875.47% 100051893.17% Lorentz force induced detuning is usually several hundred Hz, while microphonics is several ten Hz The way to improve: Try to use piezo tuner to control the detuning to below 100Hz!

8. Extra power by sychronous phase by synchronous phase The high beta cavity synchronous phase is around -15 deg. at current design, but the value is much higher in normal conducting cavities, need pre-detuning… Φb/deg. Required Extra Power -100.78% -151.79% -203.31% -255.44% -308.33% -3512.26% -4017.60% -4525.00% -5035.51% -5550.99% -6075.00%

9. Extra power by QL variations What happens if we can not optimize the QL to the designed value? What if QL is varied by some unexpected conditions? More power is consumed! 0.8% more power for 10% variation, 2% more power for 20% variation, 3% more power for 30% variation, 14% more power for 50% variation!

10. Extra power by other perturbations Klystron droop and ripple beam loading Others… 1% in voltage 2% in power 5% total various in voltage 10% more power needed!

11. Extra power by overshoot Overshoot due to feedback could be very large, depending on the feedback gain. In the bad condition, up to 100% or even higher extra power is needed.

12. Extra power required Tough: 1%+2%+1%+6%=10% (detuning is highly controlled to below 100Hz, synchronous phase 15 deg., QL variation is below 10%, and other perturbations and overshoot are strictly limited to below 3% in voltage ) Relaxed: 25%+2%+3%+20%=50% (without piezo control but detuning is limited to half bandwidth, synchronous phase 15 deg., QL variation is below 30%, and other perturbations and overshoot are limited to 10% below in voltage.) a compromise extra power between these two? 20%? Note: above is a rough estimate, the total effect is not just simply sum up of individual effects

13. Filling time The filling time is defined as the period that the cavity voltage rises from 0 to the desired value. T fill

14. Filling time In the ideal case, the end of the filling is right the time of beam coming, where ‘0’ reflection. And with beam then bring in a reflectless steady state without any control

15. Filling time Then We can calculate out the filling time

16. Filling time The reality is always a bit different Not all the power can be utilized to filling (detuning, QL variation, perturbations ) Need some time for stabilizing feedback

17. Filling time Feedback before filling stage or after filling stage? Before or at filling start, over shoot results in large reflected peak power which probably causes interlock trip. But keep the less filling time(follow the setpoint) After filling stage? Less peak power due to over shoot. But extra time to stabilize(30~50us) Feed forward to track the cavity resonant frequency, an effective way to reduce detuning affect.

18. Filling time A reasonable filling time: 213 +50+87=350us (50us for stabilize feedback, 87us more for prolonged filling time due to power loss at detuning, variations, perturbations) Can we achieve the same value 213 us or less any way? Yes! But more power!

19. Droop and ripple of Modulator The modulator droop and ripple of 1% will induce about ~10° in klystron output phase and ~1% in amplitude The feed back has to be employed. The errors could be suppressed by a factor of loop gain G

20. Droop and ripple of Modulator The loop gain is limited by loop delay and over shoot Also limited by pass-band mode At SNS, the average gain is about 50, normal conducting cavity is about 5 Not enough to suppress large errors, for example, 45 deg. from 3% modulator droop(assume 15 deg./1%) to 0.5 deg. Worse in normal linac.

21. Droop and ripple of Modulator Integral gain of Ki=2πf HBW is then intorduced to eliminate the steady errors and reduce low frequency noises Ki plays much more key role in normal conducting cavity G=50G=20

22. Droop and ripple of Modulator Assuming that 15 degree error is induced by per 1% error from modulator, to control the phase error to 0.5°:

23. Droop and ripple of Modulator Combination with the normal conducting cavity requirement (probably use the same type of modulator) A bit higher requirement but looks concise, Is it reasonable?

24. Issues Summary The Extra Power for the cavity Tough: 1%+2%+1%+6%=10% Relaxed: 25%+2%+3%+20%=50% Is 20% possible scheme? The Cavity Filling time 213 +50+87=350us, is it OK or we need more? Modulator droop and Ripple Is it reasonable?

25. Thank you for the attention, and especially for your advices!