Transient beam loading at injection

Ivan Karpov and Philippe Baudrenghien

Power requirements at injection

The full-detuning scheme has no advantage during machine filling (previous meeting)

→ The half-detuning scheme needs to be used

Required peak power in steady-state situation 𝑃HD =𝑉cav መ𝐼b,rf

8, but is it

the same during injection process?

→ Evaluation of power including details of LLRF system is necessary

Peak beam rf current

2

Cavity-beam-generator model developed for FCC

3

rf cavity

Load

Circulator

Generator

LLRF Σ

–

+

𝐼b,rf, rf component of

the beam current

𝑉ref, reference voltage

𝑉, cavity voltage

𝐼g, generator current

𝐼r, Reflected current

𝑉 𝑡 , 𝐼b,rf 𝑡 , 𝐼g 𝑡 , 𝐼r 𝑡 are time-varying complex phasors rotating with angular rf frequency 𝜔rf

𝑑𝑉 𝑡

𝑑𝑡= −𝑉 𝑡

1

𝜏− 𝑖Δ𝜔 + 𝜔rf 𝑅/𝑄 𝐼𝑔 𝑡 −

𝐼b,rf 𝑡

2

*J. Tückmantel, Cavity-Beam-Transmitter Interaction Formula Collection with Derivation, CERN-ATS-Note-2011-002, 2011

For given 𝐼g 𝑡 and 𝐼b,rf 𝑡 the cavity voltage can be found from*

Cavity filling time 𝜏 = 2𝑄L/𝜔rf, cavity detuning Δ𝜔 = 𝜔r − 𝜔rf, 𝑅/𝑄 = 45 Ω

→ How do we get 𝐼b,rf 𝑡 and 𝐼g 𝑡 ?

𝜖 = 𝑉ref − 𝑉, error signal

rf component of the beam current

4

The rf power chain (amplifier, circulator, etc.) has limited bandwidth

For power transient calculations, we are interested dynamics of the system for the first few turns after injection

→ 𝐼b,rf 𝑡 can be replaced by a stepwise function 𝑓(𝑡) with sampling rate 1/𝑡bb = 40 MHz (𝑡bb - bunch spacing), so

𝐼b,rf 𝑡 = −𝑖 መ𝐼b,rf 𝑓(𝑡)

→ Synchrotron motion can be neglected

Peak rf current መ𝐼b,rf =𝑒𝑁p𝐹b

𝑡bbBunch form factor 𝐹𝑏 = 2𝑒−

𝜔rf2 𝜎2

2 𝑁p - number of particles per bunch

Fourier transform

Generator current as output of LLRF module

5

Delay, 𝜏delay Gain, G

OTFB

AC coupling AC coupling

𝐼g 𝑡 𝜖 𝑡Σ+

+

First simplified model (analog direct rf feedback): 𝐼g 𝑡 = 𝐺 𝜅 𝑡 − 𝜏delay = 𝐺𝜖(𝑡 − 𝜏delay)

Correction signal Error signal

𝜅 𝑡

The direct feedback gain is defined by the loop stability 𝐺 = 2 𝑅/𝑄 𝜔rf𝜏delay−1

for 𝜏delay = 650 ns

For the finite gain cavity voltage will be lower than 𝑉ref

It improves longitudinal multi-bunch stability

Generator current as output of LLRF module

6

Delay, 𝜏delay Gain, G

OTFB

AC coupling AC coupling

𝐼g 𝑡 𝜖 𝑡Σ+

+

Model for analog and digital direct rf feedback:𝑑𝐼g 𝑡

𝑑𝑡=𝐼g 𝑡

𝑎d𝜏d+𝐺

𝜏d𝜅 𝑡 − 𝜏delay + 𝐺

𝑑𝜅 𝑡 − 𝜏delay

𝑑𝑡

Correction signal Error signal

𝜅 𝑡

In the LHC 𝑎d = 10, 𝜏d ≈2

𝜔rev=

𝑡rev

𝜋, for the revolution period 𝑡rev ≈ 88.9 μs

Frequency dependent gain

𝜔𝜔rev

1

𝑎d

1/𝜏d1/𝑎d𝜏d

With digital rf feedback error in cavity voltage can be reduced

Generator current as output of LLRF module

7

Delay, 𝜏delay Gain, G

OTFB

AC coupling AC coupling

𝐼g 𝑡 𝜖 𝑡Σ+

+

Model for one-turn delay feedback:

Correction signal Error signal

𝜅 𝑡

In the LHC 𝑎OTFB =15

16, 𝐾 = 10, 𝜏AC = 100 μs.

OTFB reduces transient beam loading and improves longitudinal multi-bunch stability

Frequency dependent gain

𝜔𝜔rev

1

𝑎d

1/𝜏d1/𝑎d𝜏d

𝑦 𝑡 = 𝑎OTFB𝑦 𝑡 − 𝑡rev + 𝐾 1 − 𝑎OTFB 𝑥(𝑡 − 𝑡rev + 𝜏delay)

Removes DC offset

from the signal

Model AC coupling: 𝑦 𝑡

𝑑𝑡= −

𝑦 𝑡

𝜏AC+

𝑑𝑥 𝑡

𝑑𝑡

𝑥

𝑦

𝑥𝑦

Results: analog DFB only (1/2)

8

Injection of 3 × 48 bunches with 𝐹𝑏 = 1 and 𝑁𝑝 = 2.3 × 1011; rf cavities are pre-detuned with Δ𝜔 = 2𝜋Δ𝑓

𝑉cav መ𝐼b,rf8

→ The requested power is below steady-state limit, but what happens with cavity voltage?

𝑃 𝑡 =1

2𝑅/𝑄 𝑄L 𝐼g 𝑡

2

*J. Tückmantel, Cavity-Beam-Transmitter Interaction Formula Collection with Derivation, CERN-ATS-Note-2011-002, 2011

Generator power*

Results: analog DFB only (2/2)

9

Injection of 3 × 48 bunches with 𝐹𝑏 = 1 and 𝑁𝑝 = 2.3 × 1011; rf cavities are pre-detuned with Δ𝜔 = 2𝜋Δ𝑓

𝑉cav

As expected for the finite gain, the voltage is lower than it is requested

→ This explains lower power consumption

Results: analog + digital DFB (1/2)

10

Injection of 3 × 48 bunches with 𝐹𝑏 = 1 and 𝑁𝑝 = 2.3 × 1011; rf cavities are pre-detuned with Δ𝜔 = 2𝜋Δ𝑓

There is a small overshoot in power after injection

Results: analog + digital DFB (2/2)

11

Injection of 3 × 48 bunches with 𝐹𝑏 = 1 and 𝑁𝑝 = 2.3 × 1011; rf cavities are pre-detuned with Δ𝜔 = 2𝜋Δ𝑓

Some modulation of the cavity voltage amplitude and more significant modulation of the cavity voltage phase

Results: analog + digital DFB + OTFB (1/2)

12

Injection of 3 × 48 bunches with 𝐹𝑏 = 1 and 𝑁𝑝 = 2.3 × 1011; rf cavities are pre-detuned with Δ𝜔 = 2𝜋Δ𝑓

There is a difference between first and the second turn after injection

Significant overshoot due to action of OTFB

First turn

Second turn

Results: analog + digital DFB + OTFB (2/2)

13

Injection of 3 × 48 bunches with 𝐹𝑏 = 1 and 𝑁𝑝 = 2.3 × 1011; rf cavities are pre-detuned with Δ𝜔 = 2𝜋Δ𝑓

First turn

Second turn

Better compensation of the cavity voltage at the second turn by OTFB costs significantly more power

Conclusions

• Detailed model of LLRF in the LHC was implemented in the time-domain beam-cavity-generator interaction equations.

• Preliminary results show that one turn delay feedback can cause problems during injection process resulting in large power transients. Possible solution would be reduction of OTFB gain during machine filling.

• Next steps:

• Comparison with MD data and BLonD model

14

Benchmarks

15

Expected impulse response constant of OTFB

𝜏OTFB =𝑡rev

1 − 𝑎OTFB≈ 1.5 ms

Long term evolution

16

17

18

19