Precise calibration of cavity forward and reflected signals using low-level radio-frequency system

Jin-Ying Ma • Feng Qiu, • Long-Bo Shi • Zheng-Long Zhu • Tian-Cai Jiang • Zong-Heng Xue • Ke-An Jin • Qi Chen • Cheng-Ye Xu • Xing-Hao Ding • Zheng Gao • Lie-Peng Sun, • Gui-Rong Huang, • Yuan He,

Abstract Precise measurements of the cavity forward(Vf)and reflected signals (Vr) are essential for characterizing other key parameters such as the cavity detuning and forward power.In practice,it is challenging to measure Vf and Vr precisely because of cross talk between the forward and reflected channels (e.g.,coupling between the cavity reflected and forward signals in a directional coupler with limited directivity).For DESY,a method based on the cavity differential equation was proposed to precisely calibrate the actual Vf and Vr.In this study,we verified the validity and practicability of this approach for the Chinese ADS front-end demo superconducting linac(CAFe)facility at the Institute of Modern Physics and a compact energy recovery linac (cERL) test machine at KEK.At the CAFe facility,we successfully calibrated the actual Vf signal using this method.The result demonstrated that the directivity of directional couplers might seriously affect the accuracy of Vf measurement.At the cERL facility,we calibrated the Lorentz force detuning (LFD) using the actual Vf.Our study confirmed that the precise calibration of Vf significantly improves the accuracy of the cavity LFD measurement.

Keywords Forward and reflected signals · Measurement ·Calibration

1 Introduction

The Chinese ADS front-end demo superconducting linac (CAFe) was constructed at the Institute of Modern Physics to develop the key technologies for a superconducting(SC)front-end linac.It was used to demonstrate the possibility of a 10-mA high-power continuous-wave (CW)proton beam for the China Initiative Accelerator-Driven System project [1-3].The CAFe facility is a 162.5-MHz SC radio-frequency (RF) facility operating in CW mode and consists of a normal conducting section and an SC section.As shown in Fig.1,the normal conducting section is composed of an ion source,low-energy beam transport line (LEBT),RF quadrupole accelerator,and medium-energy beam transport line (MEBT).The SC section is composed of SC accelerating units,which consist of 23 SC half-wave resonator cavities and are assembled in four cryomodules (CM1-CM4).

Fig.1 (Color online) Layout of CAFe facility.The half-wave SC cavities are mounted in four cryomodules (CM1-CM4).Cavity CM3-5 is marked by a red triangle

The cavity forward and reflected signals(VfandVr)are typically used to control other key parameters such as the cavity voltage signal (Vc),cavity detuning (Δf),and forward power.For example,at the SC test facility (STF) at KEK[4],because there is no pickup antenna in the STF RF gun cavity,Vcis reconstructed by computing the superposition ofVfandVr(Vc=Vf+Vr) [5].Note that in this paper,Vc,Vf,andVrrefer to baseband signals,which can be represented by their amplitude and phase (e.g.,Vc=|Vc|ej∠Vc).The cavity pickup signal at the Paul Scherrer Institute was found to be corrupted by noise [6].Therefore,Vcwas recalculated using the actualVfandVr,and the noise was successfully removed.In addition,Vfmeasurement can affect the accuracy of the cavity dynamical detuning.For instance,at the compact energy recovery linac (cERL) test facility at KEK,we compared the cavity Lorentz force detuning (LFD,ΔfLFD) calculated using the cavity differential equation with that calibrated using the square of the accelerating gradient.We found that the results of these two methods do not agree at the beginning of cavity filling.This discrepancy was attributed to the inaccuracy ofVfmeasurement,which was caused by the limited directivity of the RF directional coupler [7].

At CAFe,we measured theVc,Vf,andVrsignals and the low-level RF (LLRF) output in cavity CM3-5(see Fig.2).The location of the LLRF output signal is marked in Fig.3.The cavity amplitude loop was operated in closed-loop mode,whereas the phase loop was operated in open-loop mode.As shown in Fig.2,the cavity amplitude error was well compensated by the feedback loop.By contrast,because the phase loop was open,the cavity phase fluctuated greatly owing to perturbations such as microphonics and ΔfLFD.Moreover,Fig.2 shows that there is a significant difference between theVfphase and LLRF output phase.One possible reason for the phase difference is the nonlinearity of the solid-state amplifier (SSA).Another possible reason involves the inaccurate measurement ofVfowing to cross talk between the forward and reflected channels,for example,coupling between the cavity reflected and forward signals in the directional coupler.Later sections of this article demonstrate that the nonlinearity of the SSA is not the primary reason for this phase difference.

Fig.2 (Color online)Comparison of amplitude and phase of cavity voltage signal(Vc),cavity forward signal (Vf),cavity reflected signal (Vr),and LLRF output signal.The cavity amplitude loop was operated in closed-loop mode,whereas the phase loop was operated in open-loop mode.Figure shows that the phase of Vf differs from that of the LLRF output

Fig.3 Schematic of digital LLRF system at CAFe.The location of the LLRF output signal is marked

Because of cross talk between the measurement channels,the measuredVfsignal is mixed with theVrsignal,resulting in inaccurateVfmeasurement.To address this problem,Brandt proposed a method of eliminating the effect ofVron theVfsignal [8].We calibrated the actual value ofVfin CM3-5at CAFe and in an SC cavity at KEKcERL using this method.The measurement results confirmed that this method is effective.

The remainder of this paper is organized as follows:Section II briefly describes the CAFe LLRF system.Section 3 reviews the algorithms used to calibrate the actual values ofVfandVr.In addition,the actualVfandVrvalues are estimated with higher precision.Section 4 demonstrates the validity of the method using the experimental results for CAFe and cERL.Section 5 presents a discussion of our future work.Finally,we summarize our study in Sect.6.

2 LLRF system

At CAFe,the LLRF data acquisition system is used to measure the amplitude,phase,and frequency and then use various signal processing methods to monitor and control the RF fields.Figure 3 shows a simplified illustration of the layout of the digital LLRF system at CAFe.The frequencies of the RF signals and the intermediate frequency (IF)signal are 162.5 and 25 MHz,respectively.A field-programmable gate array(FPGA)module is used for real-time signal processing and high-speed data acquisition.RF signals such asVc,Vf,andVrare first converted to IF signals.The IF signals are sampled at 100 MHz via a 16-bit analog-to-digital converter (ADC) and are fed to the FPGA.Then,the amplitude and phase ofVcare extracted from the IF signals.Subsequently,the amplitude and phase signals are compared with their setpoints,and their errors are calculated.These error signals are controlled by a proportional and integral (PI) controller.The controlled amplitude and phase signals are used to rebuild the IF signal.Next,the IF signal is up-converted,and the upconverted signal is used to drive the SSA and ultimately power the cavity.An ARM chip is integrated into the FPGA.An experimental physics and industrial control system (EPICS) is installed on the ARM chip as an embedded Linux system for data acquisition.The amplitude and phase waveforms ofVc,Vf,andVrare saved to the hard drive of the CPU controller during measurement,and the measurement data are analyzed offline.

3 Calibration of real cavity forward signal

To calibrate the realVfandVrsignals (see Fig.2),we review the method of Brandt and present an example for cavity CM3-5at CAFe.CM3-5is mounted in the third cryomodule (CM3).It is the fifth cavity in CM3,and the loaded quality factor is 3·8×105,as illustrated in Fig.1.

Figure 4 shows theVc,Vf,andVrsignals measured by the LLRF system during the pulse conditioning of cavity CM3-5.According to these results,the cavity signal cannot be obtained by computing the measured signal between theandvectors if the two signals are uncalibrated.The actual cavity signalVcis a superposition of the actualVfandVrsignals,and it can be reconstructed using the formula [8-10]

The actualVfandVrsignals can be calibrated using the measured cavity forward and reflected signals according to Omet et al.[5] and Alexanderc [8]

whereandare the measured values.In this equation,a,b,c,anddare complex constants related to the actual and measured values.Then,Eq.(1) can be written as

whereX=a+candY=b+d.The complex constantsXandYcan be estimated by the multiple linear regression method.The specific calculation methods are as follows.

The complex signalsV,X,andYcan be written asV=R（V）+jI（V）,X=C1+jC2,andY=C3+jC4,respectively.R(V),C1,andC3represent the real parts,whereasI(V),C2,andC4represent the imaginary parts.Then,Eq.(3) can be written as

For a single pulse with sampled data pointsVck,,and,wherekruns from 1 ton,this method can be used to obtain a linear,overdetermined system of equations.LetVcbe the response variable;then

In addition,let the measured valuesandform the regressor matrix

Then the relation

is approximately satisfied for some parameter vector

where C represents the values of interest.The best estimates ofC1,C2,C3,andC4can be determined using multiple linear regression.The complex constantsXandYcan be obtained using these C values.The crosstalk elementsbcan be taken into account using the information from the RF decay time (after 3.5 ms in Fig.4).During decay,the actualVfshould be 0,and the factorsaandbare then limited by another complex factorZas

Fig.4 (Color online)Measured cavity voltage,forward,and reflected signals in CM3-5 during RF commissioning

Then the complex factors can be expressed in terms ofX,Y,andZaccording to

The final unknown complex factoracan be extracted numerically from the cavity differential equation assuming a constant cavity half-bandwidthω0·5,which can be estimated in terms of the field decay curve (t＞3.5 ms).

The cavity equation can be written in polar coordinates as

Consequently,the differential of |Vc| can be computed using the formula

whereθandφrepresent the phases ofVc andVf,respectively.The variabley(a) represents the calibratedas a function of the factoraaccording to Eq.(13).By contrast,can be directly calibrated from the amplitude ofVc,which is independent ofa.The difference between these two calculations should be minimized by optimizinga.Therefore,we denote the difference asλ2（a）.

Note thatais a complex number with two components.A two-dimensional map can be obtained by sweeping the real and imaginary components ofa,as shown in Fig.5.Here,the color axis represents the value ofλ2（a）.TheXandYaxes represent the real and imaginary components of(a/X),respectively.The optimal value of (a/X) is clearly 0.99+0.07i.To better illustrate this result,Figs.6 and 7 compare the waveforms ofVfandy(a),respectively,fora/X=1 anda/X=0·99+0·07i.The difference betweenandyis minimized when the optimal value ofais used.

Fig.5 (Color online) Scanning result of the parameter λ2.The optimized factor a/X is 0·99+0·07i.The X and Y axes represent the real and imaginary parts of a/X,respectively

Fig.6 (Color online) Comparison of cavity forward voltage V f for a/X=1 and a/X=0·99+0·07i.The waveforms of XV*f and V c are also presented for comparison

Fig.7 (Color online)Comparison of parameter y for a/X=1 and a/X=0·99+0·07i.The quantity is also plotted as a reference

4 Experimental verif ication

To demonstrate the method described in the previous section,theVfmeasurement data were processed and analyzed at the CAFe and cERL facilities.

4.1 Experimental result at CAFe

During the RF commissioning at CAFe,the LLRF output,SSA output,andandsignals of CM3-5were down-converted and sent to the ADCs of the LLRF system.The positions of the four signals (points A,B,C,and D,respectively)are shown in Fig.8.As mentioned in Sect.1,the phase ofdiffers from the LLRF output phase according to the measurement (see Figs.2,9).In theory,the LLRF output phase andVfphase should be in agreement,whereas the phases at points A and C differ greatly,according to Fig.9.As shown in Fig.8,in the ideal case,the phases of the LLRF output (point A) andVf(point C)should be in agreement.The discrepancy has two possible causes:the nonlinearity of the SSA and cross talk between the measurement channels (that is,the unwanted coupling of some components ofVrwithVf).We analyze the effects of these two causes as follows.

Fig.8 Diagram of signal measurement.Points A and B represent the LLRF and SSA output signals,respectively.Points C and D represent the measured values of the cavity forward and reflected signals ( and ),respectively

To eliminate the impact of the nonlinearity of the SSA,we measured the output characteristics of the SSA,as shown in Fig.10.The amplitude and phase characteristics of the SSA output were measured using a network analyzer.The operating point for Fig.2(and Fig.9)is marked in Fig.10 asP(x,y).The amplitude nonlinearity curve can be f itted using the polynomial formula.[11]

Fig.9 (Color online) Amplitude and phase of LLRF output signal,which corresponds to point A in Fig.8 (green, VLLRF),SSA output signal considering the effect of SSA nonlinearity,which corresponds to point B in Fig.8(red),and forward signal of measured corresponds to (aqua).When the nonlinearity of the SSA is taken into account,the phases at point B (fθ（|VLLRF|）) and point A (∠) still differ greatly

Fig.10 (Color online) Schematic of amplitude and phase characteristics of SSA.The SSA and LLRF output amplitudes are normalized to the saturation point of the SSA

The phase characteristic curves can be fitted by

The amplitude (fA（|VLLRF|）) and phase (fθ（|VLLRF|）) of the SSA output can be calibrated using Eqs.(14) and (15),as shown in Fig.9.

Furthermore,fA（|VLLRF|）andfθ（|VLLRF|）can also be estimated using the differential near the operating point.If the SSA is linear,its output amplitude increases linearly as the input amplitude increases,and the linear gain isK=y/x.In fact,the output amplitude of the SSA is often nonlinear with respect to the input amplitude;thus,the differential at pointPis approximatelyG=Δy/Δx,as illustrated in Fig.9[11-13].The amplitude variation of the SSA output signal depends on the value ofG/K.ForG/K＞1,the amplitude fluctuation of the LLRF output will beG/Ktimes larger than that of the SSA.By contrast,forG/K＜1,the amplitude fluctuation of the LLRF output will beG/Ktimes smaller than that of the SSA.We can see from Fig.10 that the value ofG/Kis approximately 1.8;thus,a 1% fluctuation in the LLRF output amplitude near the operating point corresponds to a 1·8%fluctuation in the SSA output amplitude.It is easy to see that the ratio of the LLRF output amplitude andfAis in good agreement with the value ofG/Kin Fig.9.In addition,a 1% fluctuation in the LLRF output amplitude near the operating point will result in an error of 0·015°in the SSA output phase,according to the phase curves in Fig.9.Noise from various sources also contributes to the error,which is difficult to determine from the LLRF output phase,as shown in Fig.9.Thus far,we have estimated the amplitude and phase at point B(fAandfθ)successfully.Figure 9 shows thatfAandfθdiffer greatly in amplitude and phase (especially in phase) from.Thus,we deduce that the nonlinearity of the SSA is not the main reason for the disagreement in phase by comparing the phase curves offθ（|VLLRF|）andfA.

To determine whether theVrsignal is coupled with theVfsignal,it is necessary to calculate the actualVfvalue of CM3-5.The values of the complex constantsaandbmust be known before this calculation can be performed.The exact value ofafor CM3-5was obtained by theoretical analysis and derivation in Sect.3.The exact value ofbcan then be calculated using Eq.(9).From the knownaandbvalues,the actual cavity forward signal can be calibrated using the measured signalsand.Figure 11 shows the actualVfsignal,measuredVfsignal,and SSA output signal considering the effects of the nonlinearity of the SSA.Panels (a)-(c) represent the measurement results using the same calibration factors (i.e.,aandb) but for different dates.The red lines represent the amplitude and phase of the SSA output signal considering the effect of the nonlinearity of the SSA.The blue lines show the calibrated actual forward signal,which representsVf2in the figure.The aqua lines represent the measured forward signal,as in the previous section,which representsVf1.Figure 11a shows good phase agreement between the blue and red lines.The LLRF output phase is in agreement with the phase of theVfsignal in theory.This result confirms that theVrsignal is coupled with theVfsignal in the signal measurement.It is demonstrated that cross talk in the measurement channels (e.g.,cross talk caused by the limited directivity of the directional coupler) is the major reason for the inaccurate measurement ofVfand thus the phase discrepancy.

To further confirm the results ofVfsignal calibration,we used the same calibration array(that is,the same values ofa,b,c,andd) to calibrate the measurement results for other cases.In these cases,the working point of the SSA is the same as the working point in Fig.10.The blue and red lines show good agreement (see Fig.11b,c) in all cases,indicating that the calibration array has high universality.

Fig.11 (Color online) Amplitude and phase of SSA output signal considering the effects of the nonlinearity of the SSA (red),Vf2 (the actual forward signal,blue),Vf1 (the measured forward signal,aqua).a-c The measurement results obtained using the same calibration factors a and b but for different dates.These cases show good phase consistency between the blue and red lines

4.2 Experimental result at KEK-cERL

The cERL facility was constructed at KEK to accommodate industrial applications of SC technology.It is a 1.3-GHz SC machine operated in CW mode[14-16].At cERL,two nine-cell cavities(ML1 and ML2)with highQLvalues of up to 1×107were installed in the main linac.

At cERL,we observed that the LFD calculated usingVfdiffers from the theoretical value at the beginning of cavity filling in ML2.The discrepancy was assumed to be related to the inaccuracy of theVfmeasurement due to signal cross talk between the measurement channels.To verify the correctness of this assumption,the experimental data were re-analyzed.The cavity detuning(Δω)during the RF pulse without the beam can be estimated using the cavity differential equation [7,8,12]

whereφandθrepresent the phases ofVcandVf,respectively.The quantityω0·5represents the cavity half-bandwidth.The detuning calculated using Eq.(16) includes other sources of detuning in addition to the LFD,although the LFD plays a major role in the overall detuning when the SC cavity is operated in pulse mode.

Moreover,for a given pulse pattern,the LFD can be calibrated using the LFD transfer function modelK(s)[7,17-19] as

whereK(s) is the LFD transfer function,andEacc is the accelerating gradient.The detuning at the beginning of the filling time can be estimated using the above equation(see Δfmodelin Fig.12b).

Fig.12 (Color online)a Two methods of calibrating the actual cavity forward signal.Green line,Cali1:calibration of Vf without considering signal cross talk.Blue line,Cali2:calibration of Vf considering signal cross talk. b Comparison of LFD calculated using Eq.(16)without considering signal cross talk (green),using Eq.(16) considering signal cross talk (blue),and using Eq.(17) (red)

According to Eq.(16),the amplitude and phase (|Vf|andθ,respectively) ofVfare required to calculate the cavity detuning Δω.When the crosstalk componentsbandcare not considered,the calibrated cavity forward signalVfis equal to(Cali1 in Fig.12a).Under this assumption,the detuning ΔfCail1is calibrated as shown in Fig.12b.Note that in Fig.12a,the amplitude of the Cali1 curve does not decrease to zero immediately when the RF power is switched off.As shown in Fig.12b,the waveforms of ΔfCail1and Δfmodel(red curve) are slightly different at the beginning of the filling time.

Next,the crosstalk coefficientsbandcin Eq.(2) are considered,and the coefficientsa,b,c,anddare recalculated using the method described in Sect.3.Then,the actual cavity forward signalVfis calibrated byVf=+(Cali2 in Fig.12a).As shown in Fig.12a,the amplitude of the Cali2 curve decreases to zero immediately when the RF power is turned off.Using the waveform ofVfin the cali2 curve,the LFD (ΔfCail2) is obtained,as shown in Fig.12b.As expected,the trends of ΔfCail2and Δfmodelare in good agreement at the beginning of the filling time.

From the analysis above,we conclude that the inaccurate measurement ofVfcan result in the miscalculation of the LFD.In addition,the analysis demonstrates the validity of the method of Brandt.

5 Future work

In the future,after estimating the values ofa,b,c,anddin the offline state,we will implement Eq.(2)in the FPGA.The actual cavity forward signalVfand the actual reflected signalVrare expected to be calibrated in real time.In the next stage,the calibratedVfcan be used to calibrate other parameters such as the cavity detuning and beam current.

In addition,the directivity of the directional coupler is related to its input power (the output of the RF source);therefore,the value ofa/Xmay also depend on the location of the nonlinear operating point of the SSA.In the next step,we will attempt to determine the dependence of the parametera/Xon the operation point of the RF source.This research will be based on many measurements and calculations at CAFe at various SSA powers according to the research purpose.

Finally,this paper assumes that cross talk exists only between the measured forward and reflected waves.However,RF components could also contribute to the crosstalk signal.We will study these crosstalk relationships further in future work.

6 Summary

In this study,the actualVfwas successfully calibrated using the method proposed by Brandt.We verified the validity of this method for the CAFe and cERL facilities.The study at CAFe indicated that inaccurateVfmeasurement is caused mainly by the directivity of the couplers.We confirmed that at cERL,the calibration accuracy of the cavity detuning can be significantly improved by considering the crosstalk components resulting from,for example,the limited directivity of the directional coupler.

AcknowledgementsWe thank all members of the CAFe and cERL commissioning teams for providing stable beam operation.We also thank all the operation staff for their cooperation and help during the machine study.

Author ContributionsAll authors contributed to the study conception and design.Material preparation,data collection and analysis were performed by Feng Qiu,Jin-Ying Ma and Gui-Rong Huang.The first draft of the manuscript was written by Jin-Ying Ma and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.