Cavity control system essential modeling for TESLA linear accelerator

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1 control system essential modeling for TESLA linear accelerator Tomasz Czarski, Krzysztof Pozniak, Ryszard Romaniuk Institute of Electronic Systems, Warsaw Uniersity of Technology, Poland Stefan Simrock DESY, TESLA, Hamburg, Germany ABSTRACT The pioneering TESLA linear accelerator and free electron laser project is initially introduced. Elementary analysis of caity resonator with signal and power considerations is presented. Two alternatie simulation models of caity control system are proposed. Keywords: TESLA, free electron laser, accelerator, high power microwae caity, ector and phasor simulation models 1. INTRODUCTION TESLA stands for TeV Energy Superconducting Linear Accelerator a 33 kilometer-long electron-positron collider. This new impressie project is being deeloped and planned at DESY research center in Hamburg. In order to demonstrate the feasibility of such an immense accelerator, the TESLA Test Facility as a prototype was constructed and is still under intensie improement during its operation. Accelerator section uses superconducting niobium structures of so-called caity resonators. The ery low energy spread during acceleration process is necessary to obtain an extremely small particles interaction point and high beam luminosity. The complex control system for the relatiistic beam has been deeloped to cope with signal disturbances, non-linearity and with time-arying parameters, in order to stabilize accelerating fields of resonators. A single accelerating module consists of 32 caities powered by one klystron. The control feedback system regulates the ector sum of pulsed accelerating fields in multiple caities. The superconducting caities, due to their narrow bandwidth, are ery sensitie to mechanical perturbation caused by microphonics and Lorentz force detuning. In addition to the feedback control loop, which suppresses stochastic errors, the adaptie feed forward is applied to compensate repetitie perturbations induced by the beam loading and by the dynamic Lorentz force detuning. The TESLA control is a drien feedback system stabilizing the detected real and imaginary parts of the incident wae and thus affecting the amplitude and the phase of a constant frequency signal. In order to analyze and optimally design the system a sophisticated modeling is necessary. 2. PHYSICAL DESCRIPTION OF THE SYSTEM The TESLA caity is a 9-cell standing wae structure which is about 1m long and whose fundamental TM mode has a frequency of 1300 MHz and the bandwidth of about 430 Hz (FWHM). A multi-cell structure is applied to maximize the actie acceleration space and it is limited by the field s homogeneity requirements and parasitic pass-band modes. The resonator is operated in the π-mode with 180 phase difference between adjacent cells. The RF (radio frequency) oscillating field is synchronized with the motion of a particle moing with the elocity of light across the caity.

2 The equialent representations of the chain of nine cells are resonant LCR circuits which are magnetically coupled. The following considerations are limited to the π-mode of one caity represented as a single LCR circuit. The effectie accelerating oltage V acc is defined by the energy gain of the unity charge. The caity oltage V c is the maximum accelerating oltage acting on the particle. The accelerating oltage of a bunch passing the caity with a time delay t b is V acc (t b ) = V c cos(ωt b ) = V c cos ϕ where ω=2πf is the wae pulsation, ωt b = ϕ is the angle phase of the caity oltage related to the beam current. According to the energy definition of the oltage, the beam loading can be modeled as a current sink feed by additional power through the caity electromagnetic field. Bunched beam current has typically ~2 ps pulsed structure, 1 MHz rate and an aerage alue of 8 ma. One 10 MW klystron, through coupled wae-guide with circulator, supplies RF power to 32 caities which are operated in ~1 ms pulsed mode, 10 Hz rate, with an aerage accelerating gradients of 25 MV/m (Fig. 1). Klystron Klystron coupler Waeguide Circulator Waeguide coupler CAVITY Field sensor Figure 1. Compact block diagram of the caity enironment. The fast amplitude and phase control of the caity field is accomplished by modulation of the signal driing the klystron. The digital controller closes the feedback loop, actuates the ector modulator stabilizing the real (in-phase) and imaginary (quadrature) components of the signal according to set-point input (Fig. 2). enironment Modulator Demodulator Re Im Im Re Controller Set-point input Figure 2. control system schematic block diagram.

3 3. CAVITY SIGNAL CONSIDERATIONS The objectie of the control system is to stabilize the caity oltage which is related to the beam loading current, and to minimize the power consumption supplied by the klystron. Physical model for signal calculations is presented in Figure 3. All the current (J) and oltage (U) quantities are represented in Laplace tranform space. U f J f Coupler J f U c J c J b ~ Z 0 U r J r Wae-guide with Circulator J 1 U 1 1:N J 2 U 2 Z Figure 3. Physical model of caity enironment circuit. The RF generator is modeled as a current source J f driing the wae-guide coupled to the caity. The circulator load matches itself to the wae-guide impedance Z 0 and isolates the generator from the reflected power. Transmission line signals are as follows forward current = J f forward oltage = U f = Z 0 J f reflected current = J r reflected oltage = U r = Z 0 J r Superposition of the forward and reflected waes actuates the coupler, and therefore coupler input current = J 1 = J f - J r coupler input oltage = U 1 = U f U r = Z 0 (J f J r ) The coupler conerts signals according to the transformation ratio 1:N, and therefore coupler output current = J 2 = J 1 /N = (J f - J r )/N coupler output oltage = U 2 = N U 1 = NZ 0 (J f J r ) The beam loading is represented as a current sink J b with its repetitie pulse-function structure. Superposition of the output coupler current and the beam loading current actuates the caity, therefore caity current = J c = J 2 - J b = (J f - J r )/N - J b caity oltage = U c = Z J c = Z ((J f - J r )/N - J b ). where Z = (1/RsC1/sL) -1 is the impedance of the resonant LCR caity circuit representation. The output coupler oltage U 2 equals caity oltage U c, therefore U 2 = NZ 0 (J f J r ) = U c = Z ((J f - J r )/N - J b ), Soling upper equations yields U c = N 2 Z 0 Z /( N 2 Z 0 Z) (2J f /N - J b ) Applying an equialent parallel resistance connection R L = N 2 Z 0 R and the caity trans-impedance Z L = (1/R L sc 1/sL) -1 and substituting the current generator J g = J f /N, results in the caity oltage U c = Z L (2J g - J b ) = Z L J. J(s) = 2J g (s) - J b (s) Z L (s) U c (s) Figure 4. representation with transfer function Z L (s).

4 4. ANALYTICAL SIGNAL MODELING According to the RF generator constant pulsation ω g, and due to a narrow resonator bandwidth, the caity oltage can be modeled in time domain as a generalized oscillation u r = A(t) cos(ω g t ϕ(t)) with its Hilbert transform u i = A(t) sin(ω g t ϕ(t)). Both mutually π/2 shifted oscillations with frequency ω g are applied for further analysis. This ordered pair of complementary signals called analytical signal can be represented as a u r u i ector u = or phasor in complex domain u = (u r, u i ) = u r iu i = A(t) exp(i(ω g t ϕ(t))). Because of disturbances, the amplitude A and the phase ϕ are time -arying components with a relatiely narrow spectral range. The caity control system proceeds with a low leel of frequency watching the real signal component (in-phase) = I = r = Acosϕ and the imaginary signal component (quadrature) = Q = i = Asinϕ. I, O components can be detected by complex demodulation, as a phasor = ( r, i ) = (Acosϕ, Asin ϕ) = Ae iϕ = u exp(-iω g t) = (u r, u i ) exp(-iω g t) where exp(-iω g t) stands for complex demodulation operator, or else ( r, i ) = (u r iu i ) (cos(ω g t) - isin(ω g t)) = u r cos(ω g t) u i sin(ω g t) i(-u r sin(ω g t) u i cos(ω g t)), therefore demodulation representation for a ector is as follows r = cos(ω g t) sin(ω g t). u r = Dem. u r i - sin(ω g t) cos(ω g t) u i u i where Dem is a demodulation matrix or a ector rotational transformation. u u r u i Dem ω g r i u (u r, u i ) exp(-iω g t) ( r, i ) Figure 5. Graphical representation of ector and phasor demodulation. The complex modulation is the reciprocal operation as follows for a phasor u = (u r, u i ) = ( r, i ) exp(iω g t)= exp(iω g t) for a ector u r = cos(ω gt) - sin(ω g t). r r = Mod. u i sin(ω g t) cos(ω g t) i i where modulation matrix = Mod = Dem -1. r i Mod ω g u u r u i ( r, i ) exp(i ω g t) Figure 6. Graphical representation of ector and phasor modulation. u (u r, u i )

5 According to the caity transfer function modeling, the relation between current and oltage analytical signal (both mutually π/2 shifted oscillations) as a phasor or ector representation is gien in Laplace space: U(s) = Z L (s) (2J g (s) - J b (s)) = Z L (s) J(s) where the caity trans-impedance Z L (s) = R L /(1 R L sc R L /sl) = R L /(1 Q L (s/ω 0 ω 0 /s) with parameters: loaded quality factor = Q L, resonance frequency = ω 0, generator frequency = ω g. The doubled caity transfer function representation and the signal conersion modules can be assembled into the complex caity model according to figure 7. I(ω)«i(t) Phasor Modulator ω g current j(t) «J(s) current Voltage U(s)«u(t) oltage Phasor Demodulator ω g (t) V(ω) Figure 7. Functional diagram of complex caity representation. The complex caity representation can be effectiely simplified for signals with a narrow spectral range relatie to the generator frequency ω g. Modulator input phasor and demodulator output phasor are time dependent functions with adequate Fourier transform: i(t) «I(ω) and (t) «V(ω) The complex modulation shifts forward the signal spectrum according to Fourier transform relation i(t) exp(iω g t) = j(t) «J(ω) = I(ω ω g ) output oltage is related to input current according to caity Fourier transfer function U(ω) = Z L (ω) J(ω) = Z L (ω) I(ω ω g ) The complex demodulation shifts back the signal spectrum according to Fourier transform relation u(t) exp( iω g t) = (t) «V(ω) = U(ωω g ) = Z L (ωω g ) I(ω) The caity Fourier transfer function Z L can be presented for frequency (ωω g ) Z L (ωω g ) = R L /(1iQ L ((ωω g )/ ω 0 ω 0 /(ωω g ))) = R L /(1i(ω ω)/ ω 1/2 (ωω 0 ω g )/2(ωω g )) where: detuning = ω = ω 0 ω g, caity half-bandwidth (HWHM) = ω 1/2 = ω 0 /2Q L. Resultant caity Fourier transfer function can be approximated for frequencies ω << ω g ω 0 Z L (ωω g ) R L /(1(iω i ω)/ ω 1/2 ) = Z L (iω). Then signals relation in Fourier space is as follows Moing to Laplace space with the operator s = iω yields V(iω) = Z L (iω) I(iω). V(s) = Z L (s) I(s).

6 where Z L (s) = R L /(1(s i ω)/ ω 1/2 ). Therefore signals relation in Laplace space can be written for phasor representation or for ector representation (ω 1/2 s i ω) (V r,v i ) = ω 1/2 R L (I r, I i ) ω 1/2 s ω. V r I r - ω ω 1/2 s V i = ω 1/2 R L I i Moing to time domain yields state space relation where state matrix A = d /dt = A ω 1/2 R L i -ω 1/2 - ω ω -ω 1/2 5. CAVITY POWER CONSIDERATION According to the analytical signal model, the caity oltage and the current can be represented, after complex demodulation, as a relatiely stable phasor or ector (t) = V and i(t) = I, each with its I, Q components. The real power is gien: in terms of complex oltage and conjugate current for phasor representation or as a scalar product for ector representation P = ½ Re{V I*} P = ½ V I The forward power, which is proided by the wae-guide, reflects itself partly due to the mismatched input coupler and dissipates in the circulator load. The residual transmitted power supplies caity and feeds the beam loading. The objectie of the accelerator system is to delier power to the beam with the best efficiency. Energy conseration yields P f = P r P d dw/dt P b where - forward power P f = ½ Re{V f I f *} = ½ I f 2 Z 0 ½ I g 2 R L (Q L << Q 0 ) - reflected power P r = ½ Re{V r I r *} = ½ I r 2 Z 0 V c R L I g 2 /2R L (Q L << Q 0 ) - dissipated power P d = ½ Re{V c I c *} = V c 2 /2R 0 - electromagnetic energy stored in caity W = V c 2 /2? ω 0 = 73J for 25 MV caity oltage - electromagnetic power transferred to caity in transient states dw/dt - beam loading power P b = ½ Re{V c I b *} The main caity parameters and the typical alue for power considerations are as follows - resonance pulsation ω 0 = (LC) -½ = 2π 1300 [rad/s] - characteristic resistance? = (L/C) ½ = 520 Ω - resonant impedance = R = Z(ω 0 ) = ~5 TΩ - quality factor (unloaded) Q 0? 0 W/P d = R/ρ = ~ fictitious loading as a parallel connection R L N 2 Z 0 R = ~1.5 GΩ - loaded quality factor Q L = R L /ρ = half- bandwidth (HWHM) ω 1/2 = ω 0 /2Q L = 2π 215 [rad/s] In a steady state (dw/dt = 0) without beam loading (I b = 0) on resonance condition (ω = ω 0 ) calculations for forward, dissipated and reflected power yields - P f V c 2 /8R L 50 kw for 25 MV caity oltage (Q L << Q 0 )

7 - P d = 4Q L /Q 0 (1 Q L /Q 0 ) P f 0 - P r = (1 2Q L /Q 0 ) 2 P f 50 kw Effectie quality factor Q f ω 0 W/P f = 4Q L (1 Q L /Q 0 ) 4Q L (Q L << Q 0 ). In a steady state with the beam loading and the detuned caity (ω ω 0) the main signals and parameters are as follows - beam loading aerage current = I b0 = 8mA typical alue - RF component beam loading current I b = I b = 2I b0 - determined as a reference phasor - caity required oltage V c = V c e iϕ - V c, ϕ determined, stabilized amplitude and phase - generator current I g = I g e iθ - I g, θ amplitude and phase actuated by controller - complex trans-impedance Z L = Z L (iω) = R L /(1- iq L ( ω 0 /ω ω/ω 0 ))= R L cosψ e iψ - tuning angle ψ, tanψ = Q L ( ω 0 /ω ω/ω 0 ) ω/ω 1/2 (detuning = ω=ω 0 -ω) - caity actual oltage V c = Z L (iω) (2I g I b ) 2R L /(1- i ω/ω 1/2 ) (I g I b0 ). Equating the required and the actual caity oltage V c, yields the stabilization equation. Soling this equation, the required forward power P f is obtained, which is dependent on the beam loading and the caity detuning (Q 0 >> Q L ): P f = ((1 2R L I b0 cosϕ/ V c ) 2 (( ω/ω 1/2 ) 2 2R L I b0 sinϕ/ V c ) 2 ) V c 2 /8R L The beam loading required power = P b = ½ Re{V c I b *} = V c I b0 cosϕ. 6. COMPLEX CAVITY SYSTEM MODELING The caity control system using complex caity modeling is composed according to figure 8. Beam loading Summing junctions Im j b (t) Re S -_ S Im 2j g (t) Re Current j(t)«j(s) current Voltage U(s)«u(t) oltage Transducer Transducer Phasor Modulator exp(iω g t) exp(-iω g t) Phasor Demodulator Complex GAIN Function generators D - error signal phasor S (t) - oltage phasor Summing junction 0 - reference phasor Figure 8. Functional diagram of complex caity control system.

8 The complex caity plant is represented as a transfer function by the double caity trans-impedance Z L (s) = R L /(1 R L sc R L /sl) = R L /(1 Q L (s/ω 0 ω 0 /s) The input current ector J(s) is Laplace transform of superposition 2j g (t) j b (t), each one with its mutually π/2 shifted components (Re, Im). The caity transfer function Z L (s) transforms the input current ector J(s) to the output oltage ector U(s) = Z L (s) J(s). The output oltage ector U(s), retransformed to the original u(t) with its mutually π/2 shifted components, actuates the transducer and the phasor Demodulator and it is conerted to oltage phasor (t). The summing junction compares the oltage phasor (t) with required oltage 0 as a reference phasor and generates the error signal phasor D. The controller, as a Complex Gain, amplifies the error D and closes the feedback system loop. The phasor Modulator and the transducer conerts signal to the current ector 2j g (t) with its mutually π/2 shifted components oscillating with the frequency ω g. The Beam Loading current sink subtracts its two mutually π/2 shifted pulsed structures components and the resultant current ector superposition j(t) actuates the caity. The complex caity simulation with its doubled π/2 shifted signal trace performs the features of the fundamental circuit model. Neertheless due to practical reasons it should be implemented in a reasonably lower leel of frequency. 7. CONTROL SYSTEM MODELING FOR STATE SPACE CAVITY REPRESENTATION The state space caity representation allows for modeling on a low leel frequency range. The dynamics of the closed loop system are dominated by the low frequency poles of the caity which can be described by the state space relation: d/dt = A ω 1/2 R L (2i g - i b ) where state matrix A = -ω 1/2 - ω ω -ω 1/2 Beam Loading _ i b Summing junctions Σ _ Σ i CAVITY 2i g Transducer Transducer _ Σ _ CONTROLLER Σ Summing junctions Set-point input Figure 9. Functional diagram of control system with caity state space model.

9 The set-point input deliers the required oltage alue which is compared to the actual caity oltage. The controller amplifies the error signal and closes the feedback system loop. The transducers conert signals to match its alue to the caity enironment. The beam loading current is extracted outside the complex caity model showed in fig. 8. It is represented as a RF component which equals double alue of the aerage current (i b = 2i b0 ). 8. SUMMARY The fundamental knowledge about modeling of the caity resonator for TESLA linear accelerator is presented in this paper. Continuous ectored description of the system applies the linear, time-inariant caity model. It is useful for the initial analysis and simulations of the caity behaiors. The complex caity representation simulates the circuit approximation but has rather academic importance. State space equation constitutes the base for further and adanced design of the control feedback. The challenging task for the deelopment of the caity control system is to compensate for the dynamic Lorentz force detuning and to suppress the exciting of parasitic pass-band modes. To obtain the fast and precise stabilization of the caity field, the digital controller with the efficient algorithm should be carefully designed and implemented. REFERENCES 1. TESLA Technical Design Report, Part II The Accelerator, DESY March T. Schilcher, Vector Sum Control of Pulsed Accelerating Fields in Lorentz Force Detuned Superconducting Caities, Ph.D thesis, DESY, Hamburg, August W.M.Zabolotny, K.T.Pozniak, R.S.Romaniuk, I.M.Kudla, K.Kierzkowski, T.Jezynski, A.Burghardt, S.Simrock, Design and Simulation of FPGA Implementation of RF Control System for Tesla Test Facility. This olume.