RF system models for the CERN Large Hadron Collider with application to longitudinal dynamics

The Large Hadron Collider rf station-beam interaction strongly influences the longitudinal beam dynamics, both single-bunch and collective effects. Nonlinearities and noise generated within the radio frequency (rf) accelerating system interact with the beam and contribute to beam motion and longitudinal emittance blowup. Thus, the noise power spectrum of the rf accelerating voltage strongly affects the longitudinal beam distribution. Furthermore, the coupled-bunch instabilities are also directly affected by the rf components and the configuration of the low level rf (LLRF) feedback loops. In this work we present a formalism relating the longitudinal beam dynamics with the rf system configurations, an estimation of collective effects stability margins, and an evaluation of longitudinal sensitivity to various LLRF parameters and configurations.

Figure 1

Simplified LHC rf block diagram.

Figure 2

Noise sources. Blocks in red represent the major noise sources, in blue are the equivalent noise sources for simulations purposes, and the other components are shown in green.

Figure 3

Beam transfer function during physics defined by $[10](2\pi ifI-A{)}^{-1}[0{\omega }_s^2{]}^T$. As expected, the resonance is at the synchrotron frequency of about 22 Hz for the physics configuration. The resonance is even sharper during injection due to the longer synchrotron radiation damping time.

Figure 4

The transfer function between the rf accelerating voltage phase and the noise at the $q$ channel at the input of the modulator $\{[{\varphi }_{\mathrm{cav}}(f)]/[n_q^{\mathrm{mod}}(f)]\}(\mathrm{rad}/V)$.

Figure 5

Noise power spectral density at the output of the LHC rf feedback for channel $Q$ (wideband).

Figure 6

Noise power spectral density at the output of the LHC rf feedback for channel $Q$ (narrow band).

Figure 7

Modal growth rates for configuration injection end 0.3 A with 1-Turn feedback (OTFB) on or off.

Figure 8

Tune shift for configuration injection end 0.3 A with 1-Turn feedback (OTFB) on or off.