Modeling and simulation of longitudinal dynamics for Low Energy Ring–High Energy Ring at the Positron-Electron Project

A time domain dynamic modeling and simulation tool for beam-cavity interactions in the Low Energy Ring (LER) and High Energy Ring (HER) at the Positron-Electron Project (PEP-II) is presented. Dynamic simulation results for PEP-II are compared to measurements of the actual machine. The motivation for this tool is to explore the stability margins and performance limits of PEP-II radio-frequency (RF) systems at future higher currents and upgraded RF configurations. It also serves as a test bed for new control algorithms and can define the ultimate limits of the low-level RF (LLRF) architecture. The time domain program captures the dynamic behavior of the beam-cavity-LLRF interaction based on a reduced model. The ring current is represented by macrobunches. Multiple RF stations in the ring are represented via one or two macrocavities. Each macrocavity captures the overall behavior of all the 2 or 4 cavity RF stations. Station models include nonlinear elements in the klystron and signal processing. This enables modeling the principal longitudinal impedance control loops interacting via the longitudinal beam model. The dynamics of the simulation model are validated by comparing the measured growth rates for the LER with simulation results. The simulated behavior of the LER at increased operation currents is presented via low-mode instability growth rates. Different control strategies are compared and the effects of both the imperfections in the LLRF signal processing and the nonlinear drivers and klystrons are explored.

Figure 1

System block diagram. Fast dynamics (modeled) appear in blue, slow dynamics (fixed parameters in simulation) in green, and not modeled components in red.

Figure 2

Simplified system block diagram. The system transfer function is measured between I (input) and O (output).

Figure 3

Transfer function and parameters of operating station in LER at 1400 mA.

Figure 4

Transfer function and parameters of macrostation from nonlinear simulation in the LER at 1400 mA.

Figure 5

Simulated low frequency beam modes (LER at 2500 mA).

Figure 6

Measured and simulated growth rates for the LER (simulated for each station as well as the macroklystron). The most unstable mode is $-3$.

Figure 7

Plot of ${\Delta }_l$ with current in mA. The $-6{\mathrm{ms}}^{-1}$ limit is from the low-mode longitudinal control path.

Figure 8

Growth rates for a different number of station and gap voltage for the LER.

Figure 9

Modes $-10$ to 10 for the nominal case.

Figure 10

Direct loop phase: $+10°$ rotation in green, $-10°$ rotation in red, and the nominal case for reference in blue.

Figure 11

Comb loop phase: $+10°$ rotation in green, $-10°$ rotation in red, and the nominal case for reference in blue.

Figure 12

Growth rates with direct and comb loop phase rotation for the LER at 1400 mA.

Figure 13

Measured and simulated growth rates vs comb phase rotation for the LER at 1400 mA. Agreement in both the general form and most unstable mode number.

Figure 14

Driver transfer functions. (a) Driver amplifier transfer function driven by carrier (large-signal response). (b) Driver amplifier transfer function driven by carrier and modulation (small-signal response). The modulation is swept across the band at a level $-30\mathrm{dB}$ below the power carrier at 476 MHz.

Figure 15

Estimated growth rates. Values up to 2500 mA validated for LER with the physical system as shown in Fig. 6.

Figure 16

Plot of ${\Delta }_l+2{\sigma }_l$ for LER.