Low Level RF feedback loop design CSEM-EIC/EOn/Sep02 1 Main - PowerPoint PPT Presentation

Low Level RF feedback loop design CSEM-EIC/EOn/Sep02 1

Main variables ϕ the instantaneous phase deviation of the bunch from the synchronous phase. δ R the variations of the beam radius ω rf the RF frequency δω b the variations of the beam frequency ϕ b the phase of the beam with respect to the RF CSEM-EIC/EOn/Sep02 3

Transfer functions ϕ s = = B ( s ) B scaling factor, ϕ δω 2 + ω 2 s rf s ω s the synchronousfrequency. R b = = B ( s ) R δω 2 + ω 2 s rf s tt δω s = = synchronization system rb B ( s ) ω δω 2 + ω 2 s rf s Phase and radial system CSEM-EIC/EOn/Sep02 4

Phase and radial loop         x 0 1 x 0 d  1     1    = + U         − ω 2 dt  x   0   x   k  s 0 2 2 � � � � � � B A s s ϕ         0 1 x 0      1    = = + y U          R   b 0   x   0  2 � � � � D C s s CSEM-EIC/EOn/Sep02 5

Discrete representation, pole placement Add an integral action   A 0 sdiscr = F   −  C 1  sdiscr [ ] K K K ϕ R int Set of gains ( ) = − ϕ + − Feedback: U K K R K Z ϕ R int CSEM-EIC/EOn/Sep02 6

Pole placement Poles of a 3rd order Bessel filter Non overshooting behaviour 200 Hz (20 ms) CSEM-EIC/EOn/Sep02 7

Feedback structure Cascaded implementation plus antiwindup CSEM-EIC/EOn/Sep02 8

Closed loop response 1 mm step -3 Phase and radial loop response Phase and radial loop response 1.2 x 10 12 150 100 10 1 mag, [dB] 50 8 Radius and its reference [m] 0.8 0 6 Phase [deg] -50 0 1 2 3 4 0.6 10 10 10 10 10 4 freq, [Hz] -50 0.4 2 -100 phase [deg] 0 -150 0.2 -200 -2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 -250 time [s] 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 time [s] -300 0 1 2 3 4 10 10 10 10 10 freq, [Hz] CSEM-EIC/EOn/Sep02 9

Synchronization Beam rigidly phase with its reference ϕ b - ϕ ref = ϕ set Phase reference can be incremented at each clock cycle Output of the phase detector: sawtooth corresponding to the freq difference CSEM-EIC/EOn/Sep02 10

Synchronization Force the output of the phase detector to a ct value during acceleration using an offset (Moving reference) CSEM-EIC/EOn/Sep02 11

Synchronization Moving reference Error: diff between an extrapolated rf phase and the ref. Allows the closing of the synchron CSEM-EIC/EOn/Sep02 12

State space model  1 = ϕ = ω x  1 b b s  ω 2  = ω = ω s x  2 b rf + ω 2 2 s  s  s = ϕ = ω x  3 rf + ω 2 2  s s CSEM-EIC/EOn/Sep02 13

Loop gains After going to discrete, set of gains: ( ) = − ϕ + ω + ϕ − ϕ − ϕ ∫ U K K K K ( ) ϕ ω ϕ b b int ref b b b Using pole placement CSEM-EIC/EOn/Sep02 14

Closed loop behaviour Synchronization loop response 1.4 150 100 1.2 mag, [dB] 50 Beam phase and its reference [rad] 1 0 -50 0.8 0 1 2 3 4 10 10 10 10 10 freq, [H z] 0.6 -50 -100 0.4 phase [deg] -150 -200 0.2 -250 0 -300 0 1 2 3 4 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 10 10 10 10 tim e [s] freq, [H z] CSEM-EIC/EOn/Sep02 15