Cavity Field Control - Feedback Performance and Stability Analysis LLRF Lecture Part3.2 S. Simrock, Z. Geng DESY, Hamburg, Germany
Motivation Understand how the perturbations and noises influence the feedback control performance field stability Identify the most critical parts of the LLRF system concern to field stability Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 2
Outline Overview of the RF feedback control system Sensitivity of the field error to system parameter variations Sensitivity of the field error to noises Feedback stability Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 3
RF Feedback Control System Overview Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 4
RF Control System Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 5
RF Control System Model Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 6
RF Control Model Questions: How well the output will track the reference input in presence of perturbations and noises? Is the feedback system stable? What factors will influence the stability? Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 7
Sensitivity of the Field Error to System Parameter Variations Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 8
Sources of System Parameter Variations Gain and phase errors of the klystron Amplitude and phase errors of cavities due to Lorenz force detuning and microphonics Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 9
LLRF Feedback System Model - Simplified ( s) K K = G H () s () s jϕ = Ge s ωc = s + ω c ω1/ 2 + ω 1/ 2 jδω P controller High power RF system is modeled as a constant gain and phase shift as an approximation around the working point Cavity transfer function of π mode is considered Detector is modeled as a first order low pass filter Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 1
Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 11 Effect of System Parameter Variations with Feedback Closed loop response () () () () () ( ) () () () s H s G s K s G s K s T s R s T s Y + = = 1, ( ) ω ω ω ϕ Δ + Δ Δ + Δ + Δ = Δ j s j j G G G G 2 1/ Assume transfer function of the plant is changed The error of system output due to the system parameter variations in steady state ( ) Δ Δ + Δ + Δ + = Δ 2 1/ _ 1 1 ω ω ϕ j j G G G K Y Y state steady The effect of the parameter variations is suppressed by a factor of the loop gain (1+K*G) >> 1
Sensitivity of the Field Error to Noises Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 12
LLRF Feedback System Model with Noises Transfer function of the input noise ( s) () s Y T A () s = = A 1+ K G( s) () s G() s H () s Transfer function of the detector noise ( s) () Y K T D () s = = D s 1+ K ( s) G( s) H ( s) () s G() s H () s Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 13
Noise Transfer in Frequency Domain Magnitude / db 1-1 -2-3 -4-5 -6-7 -8-9 Cavity bandwidth Closed loop bandwidth Input noise transfer A(s)->Y(s) Detector noise transfer D(s)->Y(s) -1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 Frequency / Hz Parameters for the bode plot: Cavity detuning = Half bandwidth = 216Hz Loop gain = 1 Detector bandwidth = 1MHz Conclusion: Actuator noise is suppressed by feedback gain Low frequency noise of detector is transferred directly to the cavity output; high frequency noise is filtered by closed loop bandwidth and detector bandwidth Reducing the detector noise will be essential to get highly stable cavity field! Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 14
Feedback Stability Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 15
Items Concern to RF Feedback Stability Some major items that concern to the feedback stability: Loop gain Loop delay Loop phase Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 16
Field Stability Concerns to Loop Gain Question: Is the loop gain as higher as better? It seems right. From the discussion before: ΔY Y = steady _ state 1 1 + K G ΔG G + jδϕ + jδ ω ( Δω) 1/ 2 Magnitude / db 1-1 -2-3 -4-5 -6-7 -8-9 Cavity bandwidth Closed loop bandwidth Input noise transfer A(s)->Y(s) Detector noise transfer D(s)->Y(s) -1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 Frequency / Hz But with higher gain: More detector noise goes into the cavity There will be overshot and rings in transient response in presence of loop delay Feedback becomes unstable if the gain exceeds the gain margin So, loop gain is not as higher as better, a compromised gain should be selected! Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 17
Gain Sweep at ACC1 of FLASH Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 18
Cornell RF Control Test at the TJLab FEL relative rms amplitude stability 2 1.8 1.6 1.4 1.2 1..8.6 x1-4 optimal gain o o o o o Loaded Q=1.2.1 8 2 4 6 prop. feedback gain rms phase stability [deg].5.45.4.35.3.25.2.15.1 o optimal gain o o o o Loaded Q=1.2.1 8 2 4 6 prop. feedback gain Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 19
Feedback Stability at ACC4-6 of FLASH with Different Gain Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 2
Gain Margin and Phase Margin with Loop Delay Magnitude / db 4 2-2 Only consider the π mode Loop gain = 1 Gain margin 1 1 1 2 1 3 1 4 1 5 1 6 Loop delay = 1 μs Phase / degree -5-1 -15 Loop delay = 5 μs Phase margin -2 1 1 1 2 1 3 1 4 1 5 1 6 Frequency / Hz Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 21
Field Stability Concerns to Loop Delay Question again: Is the loop delay as smaller as better? It seems right, because lower loop delay will decrease the overshot and rings of the transient response and increase the gain margin. But if there is other pass band modes: Instability happens for certain delays (even zero delay)! Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 22
H cav Cavity Model with Pass Band Modes 9 n () s = ( 1) H () s = ( 1) n= 1 9 1 n 1 2ω1/ 2, nπ 9 nπ /9 n= 1 s + ω1/ 2, nπ 9 jδωnπ 9 8π/9 mode is the most serious one to influence the feedback stability. Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 23
Magnitude / db 4 2 Consider the π mode and 8π/9 mode Loop gain = 1-2 1 1 1 2 1 3 1 4 1 5 1 6 Gain Margin and Phase Margin with Loop Delay and 8π/9 Mode Loop delay = 8π/9 mode Phase / degree -2-4 -6-8 Loop delay = 1 μs Loop delay = 2 μs -1 1 1 1 2 1 3 1 4 1 5 1 6 Frequency / Hz Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 24
Gain Margin and Phase Margin with Loop Delay and 8π/9 Mode (zoom near the 8π/9 mode ) Magnitude / db 4 2 8π/9 mode -2 6 6.5 7 7.5 8 8.5 9 9.5 1 Loop delay = x 1 5 Phase / degree -2-4 -6-8 Loop delay = 1 μs Loop delay = 2 μs -1 6 6.5 7 7.5 8 8.5 9 9.5 1 Frequency / Hz x 1 5 Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 25
Feedback Stability with Different Loop Delay Tested at ACC1 of FLASH Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 26
Summary of the Effect of Loop Gain and Loop Delay Loop delay of the feedback system should be adjusted in order to avoid the instability caused by the pass band modes, and beside that, it should be as small as possible Compromised loop gain should be selected taking into account the disturbance suppression and the noises of the detector Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 27
How Loop Phase Affect Feedback Stability? Vout Qout Stability Range: [ 9 o, 9 o ] Iout Vin Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 28
Test at ACC1 of FLASH With feedback and feed forward on The loop phase is changed in negative way by about 7 degree With feedback and feed forward on The loop phase is changed in positive way by about 8 degree Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 29
Summary In this part, we have learnt: The RF system parameter variations can be suppressed by the loop gain The input noise can be suppressed by the loop gain The detector noise will go into the cavity field within the closed loop bandwidth The loop gain should be selected as a compromise between the perturbation suppression and noise level The loop delay should be selected to avoid the instability caused by other pass band modes The loop phase should be in the range of -9 degree to 9 degree for stability Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 3
Reference [1] E. Vogel. High Gain Proportional RF Control Stability at TESLA Cavities. Physical Review Special Topics Accelerators and Beams, 1, 521 (27) [2] M. Hoffmann. Development of A Multichannel RF Field Detector for the Low-Level RF Control of the Free-Electron Laser at Hamburg. Ph.D. Thesis of DESY, 28 Stefan Simrock, Zheqiao Geng 4th LC School, Huairou, Beijing, China, 29 LLRF & HPRF 31