TESLA Cavity driving with FPGA controller

1. Tomasz Czarski WUT - Institute of Electronic Systems PERG-ELHEP-DESY Group TESLA Cavity driving with FPGA controller

2. Main topics • Low Level RF system introduction • Superconducting cavity modeling • Cavity features • Recognition of real cavity system • Cavity parameters identification • Control system algorithm • Experimental results

3. CAVITY RESONATORandELECTRICFIELD distribution FIELD SENSOR COUPLER KLYSTRON CAVITY ENVIRONS ANALOG SYSTEM PARTICLE DOWN-CONVERTER 250 kHz MASTER OSCILATOR 1.3 GHz VECTOR MODULATOR FPGA I/Q DETECTOR FPGA CONTROLLER DAC ADC DIGITAL CONTROL SYSTEM CONTROL BLOCK PARAMETERS IDENTIFICATION SYSTEM Functional block diagram of Low Level Radio Frequency Cavity Control System

4. CAVITY transfer function Z(s) = (1/R+ sC+1/sL)-1 RF voltage Z(s)∙I(s–iωg) RF current I(s–iωg) Analytic signal: a(t)·exp[i(ωgt + φ(t))] Demodulation exp(-iωgt) Modulation exp(iωgt) Complex envelope: a(t)·exp[iφ(t)] = [I,Q] Low pass transformation Z(s+iωg) ≈ ω1/2·R/(s +ω1/2–i∆ω) half-bandwidth = ω1/2 detuning = ∆ω Input signal I(s)↔i(t) Output signal Z(s+iωg)∙I(s)↔v(t) State space – continuous and discrete model vn = E·vn-1+ T·ω1/2·R·in-1 E = (1-ω1/2·T) + i∆ω·T dv/dt = A·v + ω1/2·R·i A = -ω1/2+i∆ω Superconductive cavity modeling

5. Particle Field sensor Coupler Cavity features Cavity parameters: resonance circuitparameters: R L C

6. Internal CAVITY model Internal CAVITY model Internal CAVITY model Output distortion factor Input distortion factor vk = Ek-1·vk-1 + uk-1 - (ub)k-1 E = (1-ω1/2·T) + i∆ω·T D C v u + u0 Input offset phasor CAVITY environs FPGA controller area D’ Input calibrator Output calibrator C’ Input offset compensator Output phasor + v’ u’0 External CAVITY model v’k = Ek-1·v’k-1 + C·C’·D·[u0 + D’·(u’k-1 + u’0)] – C·C’·(ub)k-1 Input phasor u’ Algebraic model of cavity environment system

7. Functional black diagram of FPGA controller And control tables determination F P G A C O N T R O L L E R I/Q DETECTOR CALIBRATION & FILTERING – + CALIBRATION + GAIN TABLE SET-POINT TABLE FEED-FORWARD TABLE

8. Cavity parameters identification idea External CAVITY model vk = Ek·vk-1 + F·uk-1Ek= (1-ω1/2·T) + i∆ωk·T Time varying parametersestimation by series of base functions: polynomial or cubic B-spline functions Linear decomposition of time varying parameter x:x = w * y w – matrix of base functions, y – vector of unknown coefficients Over-determined matrix equation for measurement range: V = W*z V– total output vector,W – total structure matrix of the model z– total vector of unknown values Least square (LS) solution:z= (WT*W)-1*WT*V

9. Functional diagram of cavity testing system CAVITY SYSTEM FPGA SYSTEM CONTROLLER MEMORY u v CONTROL DATA PARAMETERS IDENTIFICATION CAVITY MODEL ≈ MATLAB SYSTEM

10. Feed-forward cavity driving CHECHIA cavity and model comparison

11. Feedback cavity driving CHECHIA cavity and model comparison

12. CONCLUSION • Cavity model has been confirmed according to reality • Cavity parameters identification has been verified for control purpose