Phase modulation in storage-ring RF systems

TL;DR

This paper introduces a novel phase modulation technique for storage-ring RF systems, demonstrating its implementation on Dimtel controllers and validating it through tests on a cavity simulator and the KARA electron storage ring.

Contribution

It presents a new conceptual approach to phase modulation in storage-ring RF systems and its practical implementation on existing low-level RF controllers.

Findings

Successful implementation on Dimtel controllers

Effective phase modulation demonstrated on cavity simulator

Validated approach with tests on KARA storage ring

Abstract

This paper presents a conceptual approach to phase modulation of the cavity field in storage ring RF systems. An implementation of the concept on Dimtel low-level RF controllers is also presented. The method is illustrated with the test results from a cavity simulator, as well as an electron storage ring KARA.

Tables

Table 1. Table 1: KARA parameters

Parameter	Value
Injection energy, GeV	0.5
Operating energy, GeV	2.5
Beam current, $\mathrm{mA}$	180
Circumference, $m$	110
Harmonic number $h$	184
Synchrotron tune, $Q_s$	0.013

Equations

$$V_c(t)=V_{C0}\cos ({\omega }_{\mathrm{rf}}t+\beta \sin ({\omega }_{m}t))$$

$$V_c(t)$$

$$V_c(\omega )=H_{\mathrm{cl}}(\omega )V_s(\omega )$$

$$H_L$$

$$H_U$$

$$V_s(t)$$

$$a_U$$

$$a_L$$

$${\varphi }_U$$

$${\varphi }_L$$

$${\omega }_U$$

$${\omega }_L$$

$$V_s(t)$$

$$V_s(t)=V_0(1+a_a\cos ({\omega }_{m}t+{\varphi }_a))\cos ({\omega }_{\mathrm{rf}}t+{\beta }_p\sin ({\omega }_{m}t+{\varphi }_p))$$

$${\varphi }_a$$

$${\varphi }_p$$

$$a_a$$

$$\frac{J_1({\beta }_p)}{J_0({\beta }_p)}$$

$$V_0=\frac{V_{C0}{J}_0(\beta )}{J_0({\beta }_p)}$$

$$a_a=-0.039$$

$${\varphi }_a=-0.89$$

$${\beta }_p=-0.065$$

$${\varphi }_p=-1.46$$

$$V_0=1.001V_{C0}$$

Full text

I Introduction

Phase modulation of RF cavity field is commonly used in electron storage rings to excite quadrupole oscillation of the beam. This is typically done to increase the effective bunch length and to improve the Touschek lifetime Sakanaka et al. (2000); Mitsuhashi et al. (2001); Sommer et al. (2018). Such modulation is normally generated by the low-level RF (LLRF) system. When the desired modulation frequency is within the bandwidth of the cavity field control loop or loops, modulation is easily achieved by applying the necessary signal to the feedback loop setpoint. At higher modulation frequencies, comparable or exceeding LLRF loop bandwidth, the task is more complicated. Firstly, the cavity field no longer precisely follows the setpoint. Secondly, due to cavity detuning from the resonance, upper and lower sidebands can be strongly asymmetric, converting pure phase modulation at the setpoint to a mix of amplitude and phase modulations of the cavity field.

This paper will describe a concept for creating precise phase modulation of the cavity field and a practical implementation of this concept on Dimtel LLRF controller (LLRF9). The derivation of the phase modulation method is presented in Sec. II. Section III contains the results of both laboratory and accelerator tests of this approach.

II Phase Modulation Concept

The goal is to produce phase modulation of the cavity field at a given modulation frequency. The approach, presented in this paper, works as follows:

Define the desired cavity field signal; 2. 2.

Measure the closed-loop transfer function from the station setpoint to the cavity probe; 3. 3.

Calculate the setpoint signal that will produce the desired cavity field under closed-loop operation; 4. 4.

Compute the relevant configuration parameters to generate the setpoint waveform.

II.1 Desired Cavity Field Signal

For modulation amplitude $\beta$ (in radians) and frequency $\omega_{m}$ we have:

[TABLE]

where $V_{0}$ is the cavity field amplitude setpoint. For small modulation amplitudes, we can drop the sidebands at the harmonics of the modulation frequency, leaving the following expansion:

[TABLE]

where $J_{0}$ and $J_{1}$ are Bessel functions of the first kind.

II.2 Transfer Function

Let $H_{\mathrm{cl}}(\omega)$ be the closed loop transfer function between the station setpoint and the cavity field probe:

[TABLE]

where $V_{s}$ is the setpoint voltage. Using the network analyzer, integrated in LLRF9, we measure $H_{\mathrm{cl}}(\omega)$ at three frequencies: $\omega_{\mathrm{rf}}-\omega_{m}$, $\omega_{\mathrm{rf}}$111Transfer function measurement at the RF frequency is difficult due to the large interfering signal. In practice, this measurement is approximated by an average of the transfer function measurements made at small (3 $\mathrm{Hz}$) symmetric offsets around the RF frequency, and $\omega_{\mathrm{rf}}+\omega_{m}$. Let $H_{L}$ and $H_{U}$ be normalized complex gains at lower and upper sidebands respectively:

[TABLE]

In theory, closed-loop response at RF frequency should be unity for the loop topology where feedback error signal is the difference of the setpoint and cavity field probe signals. Normalization allows us to avoid shifting cavity setpoint at the RF in response to transfer function measurement errors.

II.3 Setpoint Waveform

Using Equations 1–2, we calculate the desired setpoint voltage:

[TABLE]

If one had access to an arbitrary waveform generator as the source of the setpoint signal, our task would be complete — just applying the waveform described by Eq. 3 to the system setpoint is sufficient. In practice, one more step is needed to map this waveform to the modulation capabilities of a physical low-level RF system.

II.4 Mapping to LLRF9

Setpoint signal generation in LLRF9 supports simultaneous amplitude and phase modulation profiles dim . To generate the waveform in Eq. 3, we need to express the setpoint profile in terms of four parameters: voltage setpoint $V_{0}$, amplitude modulation magnitude $a_{a}$, phase modulation magnitude $\beta_{p}$, and relative phase between amplitude and phase modulation waveforms $\Delta\phi_{\mathrm{ap}}$.

Equation 3 defines the setpoint waveform as a sum of three terms — RF carrier and two sidebands. Let’s use the following definitions:

[TABLE]

to shorten that equation to:

[TABLE]

Setpoint signal with amplitude and phase modulation is given by

[TABLE]

Expanding Eq. 5 and discarding sidebands at the harmonics of $\omega_{m}$ as well as the terms proportional to $a_{a}J_{1}(\beta_{p})$, then matching the coefficients of sine and cosine terms of three spectral lines to those in Eq. 4, we get:

[TABLE]

Numerical solver is then used to compute modulation argument $\beta_{p}$ from the ratio of Bessel functions in Eq. 9. Finally,

[TABLE]

III Measurements

III.1 Bench Measurements

Bench measurements, presented in this section, were collected with a cavity filter ($Q=7600$). The filter was tuned 80 \kilo below the RF frequency, simulating typical situation in a storage ring and both proportional and integral feedback loops were closed. From the closed-loop transfer function measurement shown in Figure 1, setpoint parameters to get 1$\mathrm{\SIUnitSymbolDegree}$ cavity field phase modulation at 120 $\mathrm{kHz}$ were computed:

[TABLE]

Cavity output was then measured under three conditions: no modulation, full modulation, as computed above, and phase modulation only. Results are presented in Fig. 2. Standard deviation of the amplitude rises from $3.4\times 10^{-4}$ without modulation to $1.1\times 10^{-3}$ with combined amplitude and phase modulation of the setpoint. Modulation leakage into field amplitude is due to the approximations in the parameter derivation as well as the errors in the transfer function measurement. When the amplitude modulation term is removed, amplitude standard deviation rises to $8\times 10^{-3}$.

Setpoint phase modulation computed in this configuration is 3.7$\mathrm{\SIUnitSymbolDegree}$, reflecting the magnitude of the transfer function. Cavity phase modulation is 1.04$\mathrm{\SIUnitSymbolDegree}$ — close to the design target.

III.2 Accelerator Measurements

Phase modulation technique was tested at KARA storage ring in Karlsruhe, Germany. Main parameters of KARA are summarized in Table 1.

KARA RF system consists of two stations, each with one klystron driving two cavities. Phase modulation of 2$\mathrm{\SIUnitSymbolDegree}$ at 67 $\mathrm{kHz}$ was applied to one of two stations (Sector 4). Machine was operating at 2.5 $\mathrm{GeV}$ with insertion device gaps closed and beam current of 30.9 $\mathrm{mA}$.

Amplitude and phase signals in cavities 1 and 2 are shown in Fig. 3. The difference in average amplitudes and phases between the two cavities is due to the station configuration imperfections. Phase and amplitude modulation levels are 2.2$\mathrm{\SIUnitSymbolDegree}$/0.18% and 2.1$\mathrm{\SIUnitSymbolDegree}$/0.15% in cavities 1 and 2 respectively. Most likely reason for slightly elevated phase modulation levels in the cavity signals is the imperfect tuning of the vector combiner, with cavity signals being added at an angle. In this loop configuration, there is no independent control of modulation levels in individual cavities.

IV Conclusions

This works presents a method of generating phase modulation in an RF cavity under beam loading. The method seamlessly accounts for closed-loop response of the RF station, including amplitude roll-off and asymmetry. The method was demonstrated on the bench as well as in a real accelerator with beam. In all cases, low level of amplitude modulation was seen. The levels of phase modulation in operation were in good agreement with the desired setting.