RF System Stability Analysis by Simulink on Matlab
Huang Gang, Guangwei Wang
Acc. Lab. Department of Engineering Physics ,Tsinghua University, Beijing China
huangg@cenopk.net
Abstract
Matlab is a widely used software in the field of control.
In this paper, we analyze the stability of RF system with it.
A RF system is made up of beams, cavities and low level
control loops. There are complicated feedback relationships
among the phases and amplitudes. In this paper, we present
a mothod to analyze the RF control system by Simulink onMatlab. We got some fundamental results. Compared withother methods, it′s easier to manipulate.
1 Introduction of the RF system
RF system is one of the most important subsystem in anaccelerator. RF system accelerates beam, supplying thepower of beam. The power feed into the RF system is used
to create high electronic field strength and supply theenergy lost while beam goes through undulator wiggler orB-magnet.

Fig. 2 The Pedersen Model with feedback
2 Stability analysis of RF system by simulink onmatlab

Fig.1 Block diagram of RF system
A RF system always includes cavity, power source, andlow level control system. The low level control system isused to keep RF system in a stability area.
The stability of RF system is usually restricted tordgeneration∆ V / V ≤ 1 % a n d ∆ ϕ ≤ 0 .5 ° for a 3
synchrotron. A long and stable running of accelerator
demands high quality of RF system.
To analyse the performance of RF cavity, F. Pedersen
has built a transfer model of cavity including the interactionof beam and cavity, which is widely used in this area.
Considering the feedback loop over Pedersen model, wecan get a pedersen model with feedback (Fig. 2). In thefollowing section, we will analysis the stability of RFsystem based on such a model.
To analyse the system more efficiently, we selectSimulink on Matlab to calculate the system, instead of C orFORTRAN programming. Similarly works have been doneby R. Tighe, SLAC on PEP-II. By learning his idea andmethod, we analyze a supposed system, which is based onSSRF.

To analyse the stability of RF system, we follow thesesteps:i) Give the simple model of RF system as blockdiagramii) Express the system in simulinkiii) Write m-file in matlab to calculate the stability areaand draw stability area
2.1 Single cavity stability
First we calculate the situation of single cavity. Such asituation is a bit simple than real system that is always has
double or more cavity. We can simplify the system to thefollowing case:
2.1.1 No loop system
No loop is added to the system, so the system contains
cavity, RF power source and beam only, which is a basicsystem. The stability of this system can be calculated out
analytically and got Robinson stability limit. Our result canbe compared with theoretical result.
2.1.2 System with phase loop only
By calculating the stability area of such a system, we canfind that the stability limit moved, system can be stabilizedin part of area while ϕ z < 0 .

Master oscillator

BeamAmplifier
CavityKlystronMatchingload

Fig. 3 No loop system

Fig. 7 System with phase loop express on matlab

Fig. 4 No loop system expressed in Matlab

Fig. 8 Stability area of system with phase loop
2.1.3 System with amplitude loop and phase loop
By adding phase loop and amplitude loop to basicsystem, we get the system near to a real system in single
cavity situation.
Master osc.

Beam

Phase cureElectronic phase shift

Amp. cureElectronic damping
cavityKlystronMatching load

Fig. 5 Stability area of no loop system

Fig. 9 Block diagram of system with phase and amplitude
loop

Master oscillator

Beam
Cure

Electronic shifter
CavityKlystronMatching load

Fig. 6 System with phase loop
Fig. 10 Expression of system with phase and amplitude
loop on matlab

2.2.3 Stability areas of different cases

Fig. 11 Stability area of system with phase and amplitude
loop

Fig. 14 Stability area of double cavity system with phaseloop only

2.2 Double cavities stability analysis
Next, we analyse the stability of system with doublecavities based on the single cavity.
Consider two cavities whose power is feeded by two
different klystrons. There is only beam transfer between
two cavities, that is the first cavity act on the beam andcauses the changing beam phase, then the changed beam
becoming the input of second cavity. We also suppose theerror of phase and amplitudes of the power feeding into twocavities are white noise.
2.2.1 Block diagram of double cavity RF system

Fig. 15 Stability area of double cavity system withamplitude loop only

The follow diagrams show the stability area of phaseloop only, amplitude loop only, phase and amplitude loop
without and with white noise.
Beam

Klystron

Cavity withspecial loops

Klystron

C avity withspecial loops

Fig. 16 Stability area of double cavity system withamplitude and phase loop

Fig. 12 Block diagram of double cavity system
2.2.2 Express on matlab
Compose the component of cavity, then the input ofcavity is phase of beam, white noise of klystron ’ samplitude and phase.

Fig. 17 Stability area of double cavity system withamplitude and phase loop and white noise in the output ofklystron on phase and amplitude
2.3 Influence on system stability area by modifying
amplitude loop (no phase loop)
Fig. 13 Expression of double cavity system

In order to analysis how the system stability rely on theintensity of loops, we try to change the feedback intensity
and calculate the stability area.

3 Summary
4

Simulink on Matlab is a useful tool on the analysis forRF control system. Similar works have been done bySLAC, but never seen in China. We learn the method andtry to do some advanced analysis facing real system. Thiswill help for the design and optimization of the system.

Y=Ib/I0

3

2

10
10000

21
5000

0-1

freqa

0-2

faiz

Fig. 18 System stability area with changing amplitude
feedback (no phase loop)
2.4 Influence on system stability area by modifying phaseloop (no amplitude loop)
We also did similar work to phase intensity: that is, selectthe amplifier as 0,2000,4000,6000,...12000, we can get
different stabilities as follow:

Fig. 19 Systemstability area with changing phase feedback(no amplitude loop)

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1996.10[2] CERN ACCELERATOR SCHOOL RF ENGINEERING FOR PARTICLE ACCELERATORS, ExternCollege, Oxford, UK, 3-10 April 1991
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Univ. Publications, 1996
[4] Flemming Pedersen, RF Cavity Feedback, SLAC 400,
1992
[5] R.Tighe, P.Corredoura, RF Feedback Simulation
Results for PEP-II, SLAC-PUB-95-6856, June 1995
[6] R.Tighe, RF Feedback Simulation for the PEP-II B
Factory, SLAC-PUB-6535, June 1994 (A)[7] R.Tighe, BEAM LOADING EFFECTS IN THE CERNPS BOOSTER, IEEE Transaction on Nuclear Science,
vol. NS-22, No. 3, June, 1993