Fundamental cavity impedance and longitudinal coupled-bunch instabilities at the High Luminosity Large Hadron Collider

The interaction between beam dynamics and the radio frequency (rf) station in circular colliders is complex and can lead to longitudinal coupled-bunch instabilities at high beam currents. The excitation of the cavity higher order modes is traditionally damped using passive devices. But the wakefield developed at the cavity fundamental frequency falls in the frequency range of the rf power system and can, in theory, be compensated by modulating the generator drive. Such a regulation is the responsibility of the low-level rf (llrf) system that measures the cavity field (or beam current) and generates the rf power drive. The Large Hadron Collider (LHC) rf was designed for the nominal LHC parameter of 0.55 A DC beam current. At 7 TeV the synchrotron radiation damping time is 13 hours. Damping of the instability growth rates due to the cavity fundamental (400.789 MHz) can only come from the synchrotron tune spread (Landau damping) and will be very small (time constant in the order of 0.1 s). In this work, the ability of the present llrf compensation to prevent coupled-bunch instabilities with the planned high luminosity LHC (HiLumi LHC) doubling of the beam current to 1.1 A DC is investigated. The paper conclusions are based on the measured performances of the present llrf system. Models of the rf and llrf systems were developed at the LHC start-up. Following comparisons with measurements, the system was parametrized using these models. The parametric model then provides a more realistic estimation of the instability growth rates than an ideal model of the rf blocks. With this modeling approach, the key rf settings can be varied around their set value allowing for a sensitivity analysis (growth rate sensitivity to rf and llrf parameters). Finally, preliminary measurements from the LHC at 0.44 A DC are presented to support the conclusions of this work.

Figure 1

Rf loop model block diagram.

Figure 2

Rf loop model block diagram with detailed llrf.

Figure 3

Modulus of cavity impedance with and without rf feedback. Injection settings: $Q_L=20000$, $V_c=1\mathrm{MV}$, cavity on tune.

Figure 4

Modulus of cavity impedance with OTFB. Injection settings: $Q_L=20000$, $V_c=1\mathrm{MV}$, cavity on tune.

Figure 5

Modulus of cavity impedance with and without rf feedback (OTFB off). Physics settings: $Q_L=60000$, $V_c=2\mathrm{MV}$, cavity on tune.

Figure 6

LHC Cavity 1B1 measured and modeled open loop transfer function ($Q_L=20000$).

Figure 7

LHC Cavity 1B1 measured and modeled closed loop transfer function ($Q_L=20000$).

Figure 8

Plot of $H_m(\omega )$ with $(\omega )$, for $p$ from 1 to 20 for ${\sigma }_{\tau }=250\mathrm{p}\mathrm{s}$.

Figure 9

Growth rates with mode number. Rf feedback off. $Q_L=20000$, Detuning $\mathrm{\Delta }f=-10.4\mathrm{kHz}$.

Figure 10

Modulus of $Z_{cl}$. Physics settings: $Q_L=60000$, $V_c=2\mathrm{MV}$, detuning of $-8.2\mathrm{kHz}$. Rf station optimized with zero detuning.

Figure 11

$Z_{\text{cl}}$ phase for detuning of zero and $-8.2\mathrm{kHz}$. Physics settings: $Q_L=60000$, $V_c=2\mathrm{MV}$. Rf station optimized with zero detuning.

Figure 12

${\mathrm{\Lambda }}_n={\sigma }_n+j\mathrm{\Delta }{\omega }_n$ and stability threshold. Configuration ($\beta$), OTFB Off.

Figure 13

Growth rates with mode number. Configuration ($\alpha$), OTFB Off.

Figure 14

Tune shift with mode number. Configuration ($\alpha$), OTFB Off.

Figure 15

Highest $|{\mathrm{\Lambda }}_n|$ with llrf gain change. Configuration ($\beta$), OTFB Off.

Figure 16

Highest $|{\mathrm{\Lambda }}_n|$ with llrf phase change. Configuration ($\beta$), OTFB Off.

Figure 17

Measured and threshold $4{\sigma }_{\tau }$ bunch length (ns).