**ABSTRACT**

The Super Proton Synchrotron (SPS) is the last link in the chain of accelerators providing protons to the Large Hadron Collider (LHC). The SPS is currently the limiting factor on the maximum number of protons and thus collisions in the LHC. The SPS upgrade is under way to expand the discovery potential of the LHC. The accelerating system — Radio Frequency (RF) — is being improved. Models of the SPS RF feedback systems were developed. These models could assist with design choices, evaluating the upgraded system performance, and anticipate limitations and issues.

**THE SPS **

In an input-output system, where some perturbation or noise has been added to the input and distorted it in some way, a feedback system can be added to make the input and output values match. At its most basic, a feedback system works by finding the error between the output and desired value, i.e. subtracting, and adding that value to the input, making the next output closer to the desired. For example, consider the system in figure 1. A simple sign wave is added to a noise generator causing a distorted wave in the output figure 2.

However, as seen in figures 5 and 6, the introduction of a delay causes instability even for a gain of one. While higher gain reduces the time it takes a system to reach the desired value, too much gain will cause immediate instability. The SPS accomplishes this through an RF control system. An RF feedback system works to stabilize the effect of the beam traveling through the cavity. The beam stability is modeled in the time domain. It corrects the signal by comparing it to a desired value, finding the error and applying a gain and adding the correction back to the input.

**THE FREQUENCY AND TIME DOMAINS **

The ultimate goal of this project is to find the RF feedback system parameters that will lead to the best possible performance of the SPS. For our purposes performance is measured in two ways, the stability of the beam and the stability of the RF system itself. The stability of the proton beam is best measured in the time domain. A stable beam is characterized by a steady phase of the proton bunches.

**RADIO FREQUENCY SYSTEM STABILITY **

Bode plots are used to represent the frequency response of a system and offer two measures of stability. One measure is known as the gain margin. By plotting amplitude against frequency the response of the signal can be seen over all frequencies. The gain margin refers to the part of the plot where the phase is 180°. If the magnitude is greater than or equal to one at this point, then the system is unstable.

**CREATING THE SIMULINK MODEL **

The 200MHz RF system consists of 4 cavities, two short cavities and two long cavities. The figure above is the Simulink model for the short cavity. Each pair of cavities is modeled by a “macrocavity” with the same characteristics but twice the voltage. It is beneficial for verification purposes to build each model (short and long cavities) separately, to be combined later. The Simulink model does not work on its own, it requires a Matlab script to describe the variables.

**MATCHING DATA**

As always the same was carried out for the long cavity. The best fits for the short cavity for high and low pass gains were used from earlier analysis with good agreement. The long cavity had less success in gain analysis which will be remedied when the delay is analyzed. The data, for both cavities, won’t fully match until the delay is dealt with. Using a similar process with the delay, and a few more tweaks of the gain, the simulation will come closer to attaining an accurate replication of SPS.

**SENSITIVITY ANALYSIS **

There is also a delay that acts between the high and low pass gain in the RF feedback system. Like the loop delay, it is best used for matching the initial spikes and end shape of the response.

The low pass gain changes the shape of the response to the beam. Lower gain will create a steeper, more pronounced disturbance. Its best to analyze this in conjunction with the high pass gain to find the correct ratio.

**CONCLUSION **

The Super-Proton Synchrotron was successful modeled in both the frequency and time domains. Using the frequency domain model, significant gain and phase margins were estimated for the upgraded system. A time domain model was also created with Matlab and Simulink and has been validated with SPS data. Once the data is fully replicated with the simulation, these two models can be used in conjunction as a tool for optimizing the SPS facility, providing useful information to the system designers, and estimating the limitations of the upgraded system before it is even operational.

The time domain model was also used to check beam stability as a function of various operational parameters. The initial analysis presented here, seems to indicate that better beam performance can be attained with a different set of parameters, in particular with adjustments to the low pass gain.

Source: Calpoly

Author: Jake Hargrove

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