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### RF System models for the Decay Ring

G. Burt, A. Dexter (Lancaster Uni)

With thanks to E. Jensen (CERN)

Issues

- Huge beam current 50-250 Amps.
- Huge RF power is required.
- Beam Current in quadrature with the RF (cavity will be detuned when the beam arrives).
- Very transient, ring partially filled with 20 bunches (500 ns).
- A tuner could not react that fast.

Solution (suggested by E. Jensen)

- If we split the RF into real and imaginary parts, the beam loading adds qwR/Q = Ib R/Q to the real voltage at 40 MHz.
- Detune the cavity so that the cavity phase is advanced between bunches (real part becomes finite and negative).
- This causes a phase (and frequency) shift as the imaginary part remains the same. Phase shift = atan [(Ib R/Q)/Vg]
- Dw ~ g ~ w0(Ib R/Q)Vg
- If correct cavity frequency is chosen beam loading is reduced as the real parts cancel. However imaginary part also changes.

Beam V

V before beam arrives

Example System:

- Ib = 224 Amps
- Vg = 300 kV
- R/Q = 25 Ohms
- Q = 20, 000
- f1 = 40.0 MHz
- f2 = 39.2 MHz

A simple code has been written to understand the behaviour of such a system.

It includes a simple LLRF system that responds instantly (unrealistic) and can look at the effect of a varying current or frequency.

No detuning

If we do not detune the cavity and we only have a small RF power available the gap voltage quickly rises to 750 kV and the phase tends towards 180 degrees.

To keep the cavity on amplitude and phase with the cavity tuned to 40 MHz takes ~9 MW.

How much RF is required?

- To get the phase and amplitude correct with a detuned cavity requires 200 kW in this case with no charge or frequency errors.
- Power not linear with charge (P ~ Q4)
- This is significantly less than the 9 MW required for a non-detuned system.

What about errors in charge?

If the charge in the beam varies we need more power to keep the cavity on phase as the capacitive loading of the cavity by the beam varies (real parts of the voltage no longer cancel).

This doesn’t include the generator mismatch which will mean even more power is required.

During filling and other slow charge variations this can be corrected with a tuner in the cavity to keep the cavity frequency correct for minimum beam loading.

Filling

- As the decay ring fills the bunch charge will vary.
- This means the beam-loading/detuning will also vary.
- We will have to change the cavity frequency. 23 ms is very fast and probably not possible.
- Will be difficult to keep phase correct during a frequency sweep.

Options

- Ferrite based cavity – Maximum voltage is around 20 kV. Would require 1000 cavities.
- Broadband Cavity – To cover the full frequency range would require a Q of 40. This needs 45 MW of RF to fill without ferrites.
- Use a cavity just broadband enough to cover the frequency jump of one injection and slowly tune the cavity between injections.
- Otherwise use brute force- no detuning SRF cavity

Brute force approach

- If we design a cavity to have a low R/Q we can minimise the impact of the beam.
- If we use an SRF cavity we can reduce the power overhead to 170 kW.
- This does require a very low R/Q of only 0.5 Ohms (PS buncher cavity is 33 Ohms by comparison).
- As there is no detuning it operates very stably during filling.

Broadband detuned cavity

- In the earlier calculations (assuming the PS buncher cavity) the cavity frequency changed by 50 kHz everytime a new bunch is injected and merged.
- This would require a cavity with a Q of around 400.
- This requires 270 kW of RF power which is more than the brute force approach.

Combing detuning with brute force

- If we half the R/Q of the cavity the detuning doesn’t have to be as much and the beam induces less voltage.
- This means a Q of 800 is sufficient and we only need 80 kW of RF power (half that of the brute force method).
- However this method requires a very complex LLRF system to cope with the slow frequency variation between injections.

Proposed Cavity

- The cavity will be a variant of the PS cavity with a modified gap/aperture to reduce the R/Q.
- Diameter is 1.6 m and length is 1 m.
- The cavity can be plated with metals (other than copper) to reduce the Q.

Tuning

- The PS cavity uses a variable capacitance servo tuner with sufficient range for our needs.

Coarse and fine tuners for the CERN PS 40 MHz buncher cavity, A. Mitra

Whats next

- Need to finish full LLRF system model to understand how bad transients are during filling and merging. (A. Dexter)
- Need to decide on detuned or brute force method.
- Need to design appropriate cavity.
- HOM studies will be very important.

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