rf measurements and tuning of the 1-meter-long 750 MHz radio-frequency quadrupolefor artwork analysis

Hermann W. Pommerenke,1, 2, ∗ Ursula van Rienen,2, 3 and Alexej Grudiev1

1European Organization for Nuclear Research (CERN), CH-1211 Geneva 23, Switzerland2Institute of General Electrical Engineering, University of Rostock, D-18051 Rostock, Germany

3Department Life, Light & Matter, University of Rostock, D-18051 Rostock, Germany(Dated: November 20, 2020)

The 750 MHz PIXE-RFQ (radio-frequency quadrupole), developed and built at CERN, provides2 MeV protons over a length of one meter for proton-induced X-ray emission analysis (PIXE)of cultural heritage artwork. In this paper, we report low-power rf measurements and tuning ofthe PIXE-RFQ, which have been completed mid-2020. Using a novel algorithm based on directmeasurements of the response matrix, field and frequency could tuned at the same time in onlytwo steps to satisfying accuracy. Additionally, we report measurements of single modules, qualityfactors, and coupling strength. In all cases, very good agreement between rf measurement anddesign values was observed.

I. INTRODUCTION

After successful design, construction, and commission-ing of the HF-RFQ, a compact 750 MHz radio-frequencyquadrupole (RFQ) for medical applications [1–8], CERNinitiated the development of a new RFQ operating atthis high frequency in mid-2017. The so-called PIXE-RFQ [3, 8–11] will accelerate protons to 2 MeV over alength of one meter only. A CAD model is shown inFig. 1 and the RFQ key parameters are listed in Table I.

TABLE I. Design parameters of the PIXE-RFQ [2, 3].

Species proton (H+)Input energy 20 keVOutput energy 2 MeVrf frequency 749.480 MHzInter-vane voltage 35 kVRFQ length 1072.938 mmVane tip transverse radius 1.439 mmMid-cell aperture 1.439 mmMinimum aperture 0.706 mmFinal synchronous phase -15 degOutput beam diameter 0.5 mmBeam transmission 30 %Peak beam current 200 nARepetition rate 200 HzPulse length 125 µsDuty factor 2.5 %Unloaded quality factor (Q0) 6000Peak rf power loss 65 kWrf wall plug power ≤ 6 kVA

The PIXE-RFQ has been developed in the context ofthe MACHINA collaboration (Movable Accelerator forCultural Heritage In-situ Non-destructive Analysis) be-tween CERN and INFN [11, 12]. The aim of the project isto build the first transportable system for proton-induced

∗ hermann.winrich.pommerenke@cern.ch

X-ray emission analysis (PIXE) of cultural heritage art-work, allowing employment in museums, restoration cen-ters, or even in the field. Low-power rf measurementsand tuning of the 750 MHz four-vane RFQ have beencompleted in mid-2020.

In any four-vane RFQ, field tuning plays an impor-tant role to achieve the desired transverse and longitu-dinal field distribution. Vane modulation, certain designchoices (e.g. a piecewise-constant cross section, as im-plemented here [10]), as well as manufacturing imperfec-tions and misalignments lead to local variations of thecapacitance and inductance distribution. Consequently,the ideal quadrupole field of the TE210 operating mode isperturbed [13, 14]. The perturbation must be correctedby means of bead-pull measurements and movable pistontuners that allow for locally modifying the inductance.Many rf cavities and RFQs in particular are tuned usingtransmission line models (see e.g. Refs. [15–19]). Con-trarily, we adopted the tuning algorithm developed forthe HF-RFQ by Koubek et al. [5, 6] based on Ref. [14],where the effects of idividual tuner movements on thefield are directly measured and recorded in a responsematrix. Corrective tuner movements are than obtainedby matrix inversion. We augmented this algorithm suchthat both frequency and field could be tuned at the sametime.

This paper covers low-power rf measurements and tun-ing procedure of the PIXE-RFQ. In the following, af-ter reviewing preliminary considerations to the measure-ments (Section II), we report the single module measure-ments carried out on the individual mechanical modulesin Section III. Section IV discusses both algorithm andexecution of the tuning procedure. Lastly, quality factormeasurements are reported in Section V.

II. PRELIMINARY CONSIDERATIONS

This section briefly describes the bead-pull measure-ment, lists the tuning goals, and discusses corrections

arX

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011.

0992

0v1

[ph

ysic

s.ac

c-ph

] 1

9 N

ov 2

020

mailto:hermann.winrich.pommerenke@cern.ch

2

FIG. 1. CAD model of the PIXE-RFQ (without end plates)and labeling of the quadrants and slug tuners. The arrowsindicate the labels of hidden tuners in quadrant 3.

of frequency measurements with respect to temperatureand medium.

A. Bead-pull field measurement

In bead-pull measurement, a small object (the bead) isintroduced into the rf cavity, effectively removing a smallvolume from the resonator. Following Slater’s perturba-tion theorem [20, 21] this can be observed as a changein resonance frequency of the cavity, proportional to thesquared field amplitudes at the location of the bead. InRFQs the bead is typically introduced into the four quad-rants, i.e. the regions occupied by the magnetic field.

In the present case, the bead-pull measurements wereperformed using the phase ∆φ = argS11 of the reflectionS11 = Γ measured through the single input power coupler(upper part of Fig. 1). The same bead-pull bench andpulley system as for the HF-RFQ was used [5, 6]. Thebead was threaded on a 0.3 mm fishing wire. The sizeof the perturbing bead must be small enough such thatthe frequency shift stays within the linear regime of the∆φ(∆f) curve, but large enough such that an acceptablesignal-to-noise ratio (SNR) is achieved. Following themeasurements of the HF-RFQ, a 7 mm×4 mm aluminumbead was used. While the PIXE-RFQ features the samefrequency and a comparable quality factor, its length andthus volume are approximately halved. Therefore, thesame bead introduces roughly double the frequency shiftwhen inserted into the PIXE-RFQ. With ∆f u 5 kHz, or∆φ u 17°, the shift induced by the bead still remainedwell within the linear regime.

From the raw phase measurements of the RFQ quad-rants the field components were extracted and aligned bymeans of a few data processing steps detailed in Ref. [6]and a smoothing Savitzky-Golay filter [22, 23]. With ∆φproportional to the squared magnetic field, the relativequadrant amplitudes a1, . . . , a4 arise by taking the squareroot and assigning the proper sign to account for the the

alternating field orientation of the TE210 mode. The fieldflatness is then quantified by one quadrupole two dipolecomponents of orthogonal polarizations [14]:

Q =1

4(a1 − a2 + a3 − a4) ,

DS =1

2(a1 − a3) ,

DT =1

2(a2 − a4) .

(1)

B. Tuning goals

By design, the inter-vane voltage of the PIXE-RFQis constant at V0 = 35 kV, corresponding to a magneticfield that is constant along the RFQ and equal in allfour quadrants. The goal of the tuning process can thusbe defined as Q = 100 % = const. and DS = DT = 0 atall sampling points. Errors of ±2 % with respect to theaverage quadrupole component are acceptable in each ofthe three field components from a beam dynamics pointof view. These tolerances have been established withrespect to the 2 m long 750 MHz HF-RFQ [5].

The PIXE-RFQ represents a stand-alone machine; norf structures requiring frequency and phase stability areinstalled downstream of the RFQ. Therefore, frequencyaccuracy does not represent a critical tuning goal. Afrequency shift is in fact foreseen by design during nom-inal operation in order to maintain a constant coolingwater temperature [10]. Deviations from the design res-onant frequency as much as few MHz are acceptable forthe beam dynamics [10]. Nevertheless, the frequency wastuned to match the design value: 749.480 MHz under vac-uum at 22 °C with an error smaller than ±60 kHz. Thistolerance emerges from the water temperature range of atypical cavity cooling system: ±5 K around the nominalvalue. The corresponding sensitivity of the PIXE-RFQresonant frequency amounts to −13.3 kHz/K, as deter-mined in Ref. [10].

C. Frequency correction

Most of the measurements were conducted under airsince the bead-pull setup denies evacuating the RFQ cav-ity. Furthermore, the ambient temperature was not con-trolled and deviated significantly from the RFQ designreference temperature of 22 °C. The measured frequencywas therefore affected by two main aspects: (i) thermalexpansion of the RFQ cavity, and (ii) the relative per-mittivity εr of air.

The thermal behavior of the cavity during low-level rfmeasurements was dominated by the ambient tempera-ture that changed during the day. Because of thermalexpansion, the resonant frequency is anti-proportionalto the bulk copper temperature: ∆f/f = −α∆T , whereα = 1.66× 10−5/K is the secant thermal expansion coef-ficient of the copper cavity and ∆T the difference between

3

measured and reference temperature. The ±0.2 K preci-sion of the used thermometers translates to a frequencyuncertainty of ±2.6 kHz. This error is by an order ofmagnitude smaller than the error introduced by humid-ity uncertainty (see the following).

The speed of light and thus the resonant frequencyin a medium are by a factor of

√εrµr lower than the

vacuum values [24]. For air we assume µr = 1 andεr = 1.00058986 [25] at standard temperature and pres-sure (STP, 0 °C, 1 atm, 900 kHz). The measured fre-quency fmeas is thus corrected as follows:

f = fmeas ·√εr

1− α∆T . (2)

A major source of uncertainty is the dependence of εron the ambient humidity. Analytical expressions for thisrelationship have been formulated based on experimentalstudies and can be found in Refs. [26–28]. The influencecan be mitigated by measuring the humidity and calcu-lating the corresponding εr. Alternatively, the cavity canbe flooded with dry nitrogen (N2). The latter representsa standard procedure for rf cavity measurements and hasbeen done during the final tuning steps of the medicalHF-RFQ [5].

However, there are no strict accuracy requirements forthe PIXE-RFQ frequency. Hence, we accepted the hu-midity uncertainty and worked with the constant STPvalue of εr. A humidity uncertainty of ±30 % translatesto an error in frequency of approximately ±30 kHz. Ad-ditional, but much smaller errors originate in pressureand temperature dependence of εr [29]. Thus, we ex-pected that the PIXE-RFQ could be tuned under air toan accuracy of approximately ±30 kHz when measuringonly the cavity temperature. If frequency stability wasrequired, this error would still lie within the ±5 K tuningrange of a typical water cooling system.

In the following, the frequency is exclusively reportedin terms of the corrected value (the value under vacuumat 22 °C).

III. SINGLE MODULE MEASUREMENTS

The PIXE-RFQ consists of two mechanical modules, inthe following denoted as module 1 and module 2, whichwere brazed individually and then clamped together toform the full assembly. Both modules were measured in-dividually to ascertain the manufacturing quality, and todetermine if it would be necessary to use the vacuumpumping ports as “emergency” tuning features in addi-tion to the piston tuners. By comparing both resonantfrequency and field distribution to the simulation (eigen-mode solver of CST Microwave Studio® 2018 [30]) it wasfound that no special measures were required, as veryclose agreement was observed between measurement andsimulation.

Figure III shows module 1 mounted to the measure-ment bench for the single module measurement. No

auxiliaries were installed and all ports were closed byaluminum flange covers. In the absence of coupler ordiagnostic pickup antennas, a small makeshift antennacrafted from simple copper wire was mounted to one ofthe ports. Although the antenna was strongly under-coupled, a phase shift of 4° was observed upon introduc-ing the bead into the cavity. With an SNR of roughly45 dB this was considered as sufficient. As no end plateswere present, the upstream and downstream ends of themodule were terminated by aluminum extension tubesin order to obtain well-defined boundary conditions thatcould be reproduced in a 3D eigenmode simulation.

FIG. 2. Photograph of the first module of the PIXE-RFQ withextension tubes mounted on the bead-pull support frame forsingle module measurements.

Figures 3a and (b) show the measured spectra of theindividual modules with attached extension tubes. Be-cause of the absence of any auxiliaries and the metallicextension tubes, the TE210 frequencies are roughly 6 MHzlower than the RFQ design frequency of 749.48 MHz.The measured frequencies deviate from the simulationvalue by 150 kHz for module 1 and 600 kHz for module 2.The deviations in the dipole-mode frequencies are smallerthan 1 MHz.

A near-perfect agreement between measured and sim-ulated quadrupole component Q [see Eq. (1)] was ob-served with an error less than 1 % of the average Q am-plitude (Fig. 4). The measured dipole components DS,DT, which vanish in the simulation, reach amplitudes ofup to ±4 %. Larger deviations were expected for the fullassembly, as the clamped connection of the two modulesintroduces additional alignment errors.

IV. TUNING

After the assembly of the full structure—two modules,two end plates, pumping ports, power coupler, and diag-nostic antennas—the PIXE-RFQ was tuned by means ofsixteen movable piston tuners. In this section, the tunertooling is described, reliability measurements are shown,the augmented tuning algorithm is derived, and the tun-ing steps are reported. Field and frequency after tuningand final tuner installation are compared to the initial

4

700 705 710 715 720 725 730 735 740 745 750 755 760

−4

−2

0

2×TE110 TE210

Frequency f (MHz)

|S11|(

dB)

meas.sim.

(a) module 1

700 705 710 715 720 725 730 735 740 745 750 755 760

−4

−2

0

Frequency f (MHz)

|S11|(

dB)

(b) module 2

FIG. 3. Measured reflection coefficients of the individual RFQmodules using a small makeshift antenna. Because of thisasymmetric arrangement, one of the dipole modes is excitedmuch stronger than the other.

0 200 400 600 800 1,000 1,200 1,400 1,600

0

0.2

0.4

0.6

0.8

1

Longitudinal position z (arb. unit)

Am

plit

ude

(arb

.uni

t)

sim.Q

DS

DT

(a) module 1

0 200 400 600 800 1,000 1,200 1,400 1,600

0

0.2

0.4

0.6

0.8

1

Longitudinal position z (arb. unit)

Am

plit

ude

(arb

.uni

t)

(b) module 2

FIG. 4. Measured quadrupole component Q and dipole com-ponentsDS, DT of the individual RFQ modules in comparisonto the simulation (dotted lines).

values.

A. Tuner setup and tooling

The PIXE-RFQ features sixteen piston tuners, copperslugs with a conical tip [Fig. 5d]. Four tuners each aremounted at four positions along the RFQ. Their desig-nations are shown in Fig. 1.

(a) (b)

(c) (d)

FIG. 5. PIXE-RFQ on bead-pull bench with tuners mountedin guidance tubes (a). Closeup of tuner tooling is shownin (b). CAD model of the tuner tooling is shown in (c); takenfrom Ref. [5] with courtesy of B. Koubek. Tip of tuner insideRFQ cavity seen from one of the bead-pull holes is shownin (d).

The nominal tuner length is given by the nominal tunerinsertion into a perfect cavity while mounted flush in theso-called flange-to-flange position. Initially the tunerswere machined with an over-length of 11 mm, allowingfor a mechanical tuning range of ±11 mm. A final lengthfor each tuner was determined by iterative tuner adjust-ments and bead-pull measurements. The tooling wasoriginally developed for the HF-RFQ [5, 6] and is shownin Figs. 5b and (c). During the tuning, the slugs weremounted on a threaded, spring-loaded piston, allowingfor adjustment of the tuner penetration into the RFQ.The piston was mounted in a guidance tube fixed to theRFQ flange, which offered accurate transverse position-ing. A scale with 10 µm graduation was used to adjustthe tuner position. The penetration was confirmed bymeans of a caliper before the tooling was removed afterthe final tuning step. Then, the tuners were remachinedto their final lengths and installed flange-to-flange withcopper gaskets.

During the tuning procedure, the field quality was as-sessed by measuring the quadrupole and dipole compo-nents of the TE210 eigenmode, Q, D

S, and DT [Eq. (1)]at discrete points. Each sampling point corresponds to

5

an interval of the continuous field profile over which thedata were averaged to reduce noise (Fig. 6).

200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300 1,40095

100

105

110

1 2 3 4 5 6 7 8

Position along RFQ z (arb. unit)

Am

plit

udeQ

(%)

FIG. 6. Example measurement ofQ component before tuning.Brackets visualize the windows over which the measurementis averaged to obtain the discrete sampling points.

B. Reliability measurements

Before the tuning process, the measurement error wasestimated by means of reliability measurements. The au-thors differentiate between two distinct concepts: (i) re-peatability, the error observed between multiple bead-pull measurements taken without any changes to theRFQ itself, in particular no tuner movements, and (ii) re-producibility, the error introduced by tuner movementsand mechanical hysteresis effects. The errors arising fromthese two aspects pose the ultimate limit for the tuningaccuracy.

1. Repeatability of field measurement

To assess repeatability, thirteen bead-pull measure-ments were carried out over several days without movingany tuners. The deviations between the repeated mea-surements were larger than 0.5 % for one quadrant. Suchan error would limit the dipole-component tuning accu-racy to more than 1 % (Fig. 7). The deviations comprisesystematic errors caused by change of ambient parame-ters such as temperature and humidity. However, theywere expected to be negligible in this case, being sev-eral orders of magnitude smaller than what could be re-solved by bead-pull measurement. A significantly largererror source is random noise introduced by vibration andslippage of the wire and by the vector network analyzer(VNA) itself.

Therefore, we studied to which extent the error couldbe reduced by averaging over several repetitions. Basedon the data from the thirteen runs, all possible com-binations of three, six, or ten measurements whereformed. The average of each combination was calcu-lated at each sampling point, generating new artificialsets of measurements. The resulting error was calculatedas δX = max |X − 〈X〉|, where X ∈ {Q,DS, DT} and 〈·〉indicates averaging over all thirteen available measure-ments. The results are summarized in Fig. 7, given as a

percentage of the quadrupole component (〈Q〉 = 100 %).Averaging over three measurements guarantees a mea-surement error < 0.5 % for DS, DT and < 0.2 % for Q,which lies well within the tuning requirements. Espe-cially for sampling points with large spread, the Q er-ror could be reduced by nearly a factor of two. Withrespect to the considerable effort for carrying out thesemeasurements, it was decided to repeat all tuning-relatedbead-pull measurements three times.

0

0.1

0.2

0.3

0.4

δQ

(%)

Averaging over 1, 3, 6, 10 measurements

0

0.5

1

δD

S(%

)1 2 3 4 5 6 7 8

0

0.5

1

Sampling points along RFQ

δD

T(%

)

FIG. 7. Reduction of error in quadrupole component (δQ) anddipole components (δDS, δDT) when averaging over three,six, and ten measurements, compared to no averaging (onemeasurement). The source data was obtained from thirteenmeasurements carried out over several days.

2. Reproducibility with respect to tuner position

A second study was performed to assess errors thatoriginate from tuner movements, i.e. the measurementreproducibility. Several tuners were moved from 0 mm(nominal position in perfect RFQ) to 3 mm, 6 mm, andback to nominal, after which the sequence was repeated.Results for tuner 4 are shown in Fig. 8 (without anyaveraging as discussed above). The error observed in thisstudy is virtually the same as the one observed during therepeatability study (Fig. 7), indicating that the overallmeasurement error was dominated by the bead-pull setupitself, while mechanical hysteresis played a negligible role.

From Fig. 8, an estimate of the influence of a singletuner on the field distribution could be obtained. An in-sertion by 3 mm reduced the Q component by roughly3 % in the vicinity of the tuner, and modifies DT byroughly 6 %. Contrarily, it has almost no influence onDS—as one would expect because of the dipole compo-nent definitions, each incorporating two opposite quad-rants [Eq. (1)]. In the bottom plot at z ≈ 700 it becomesclear that the relationship is nonlinear; the influence of atuner increases with its insertion depth.

6

90

100

110

Q(%

)Tuner 04 inserted by 0 mm, 3 mm, 6 mm w.r.t. nominal

−10

0

10

DS

(%)

200 400 600 800 1,000 1,200 1,400−10

0

10

Position along RFQ z (arb. unit)

DT

(%)

FIG. 8. Reproducibility of the field components after move-ment of tuner 4 (exemplary). The tuner was repeatedlyinserted by 0 mm (nominal position), 3 mm, 6 mm, and re-tracted again.

C. Augmented tuning algorithm

The PIXE-RFQ was tuned with an improved version ofthe algorithm developed for the HF-RFQ by Koubek [5,6], which is based on the tuning scheme for four-vaneRFQs described in Ref. [14]. For the PIXE-RFQ, theresponse matrix was augmented by the authors to alsoinclude the frequency, such that it could be tuned at thesame time as the field.

1. Previous tuning algorithm used for HF-RFQ

Koubek [5] defined (in a slightly different notation)

xcur =[Q1 · · · Qn DS1 · · · DSn DT1 · · · DTn

]T ∈ R3n(3)

as the vector of quadrupole and dipole field amplitudesQi, D

Si , D

Ti currently measured in the RFQ at the dis-

crete field sampling points i = 1, . . . , n. The correspond-ing target values are summarized in the vector

xtrg =[Qtrg,1 · · · DTtrg,n

]T ∈ R3n. (4)Similarly, the current tuner positions tj and target tunerpositions ttrg,j , j = 1, . . . ,m are collected in

tcur =

t1...tm

∈ Rm, ttrg = ttrg,1...ttrg,m

∈ Rm. (5)By moving each tuner individually by some distance (allother tuners remain in nominal position), the responsematrix R = ∂xcur/∂tcur ∈ R3n×m is obtained. Each ma-trix entry quantifies the effect of one tuner on Q, DS, DT

at one sampling point in first-order approximation. Ina perfectly tuned constant-voltage RFQ the quadrupole

component equals unity (Qtrg,i = 1 ∀i) whereas thedipole components vanish (DStrg,i = D

Ttrg,i = 0 ∀i) at all

sampling points. Thus, the arising system of equationsreads

xtrg − xcur︸ ︷︷ ︸∆x

= (∂xcur/∂tcur)︸ ︷︷ ︸R

· (ttrg − tcur)︸ ︷︷ ︸∆t

. (6)

The new tuner positions required to correct the field dis-tortion are obtained as

ttrg = tcur + ∆t = tcur + R†∆x, (7)

where R† identifies the pseudo-inverse of R.

2. Including the frequency

In this work, we augmented the presented systemof equations to include also the frequency. The mea-sured frequency is introduced as a dimensionless quan-tity f̄ = wff such that it can be combined with the like-wise dimensionless measured field amplitudes Qi, D

Si , D

Ti

by appending it to xcur. Analogously, the target fre-quency f̄trg = wfftrg is appended to xtrg. The normal-izing weight wf (in units of 1/Hz) can be used to con-trol the influence of the frequency compared to the fieldamplitudes. More precisely, wf determines the contribu-tion of the frequency deviation ftrg − f to the residual||∆x−R∆t||, which is minimized when solving the over-determined system by means of the least-squares method.Larger wf translate to higher importance. The frequencyis incorporated by appending a new row to R:

[R]3n+1 =∂f̄

∂tcur=

[∂f̄

∂t1· · · ∂f̄

∂tm

]. (8)

The algorithm presented in Ref. [5] forces the normal-ized quadrupole component to equal unity, Qtrg,i = 1∀i,which might not lead to an optimum solution when in-cluding the frequency. Instead, we use the relaxed condi-tion Qtrg,i = Q̂trg∀i, which only requires all quadrupolecomponent samples to be equal to some value Q̂trg, notnecessarily unity, but close. As an unknown quantity,Q̂trg must be brought to the right-hand side. This isaccomplished by normalizing the tuner positions by aweight wt, i.e. replacing the ti by t̄i = wtti, andttrg,i by t̄trg,i = wtttrg,i. R is augmented with a cor-

7

responding new column, and the system

−Q1...

−Qn−DS1

...

−DSn−DT1

...

−DTn∆f̄

︸ ︷︷ ︸

∆x

=

∂Q1∂t̄1

∂Q1∂t̄2

· · · ∂Q1∂t̄m −1...

. . .. . .

......

∂Qn∂t̄1

∂Qn∂t̄2

· · · ∂Qn∂t̄m −1∂DS1∂t̄1

∂DS1∂t̄2

· · · ∂DS1

∂t̄m0

.... . .

. . ....

...∂DSn∂t̄1

∂DSn∂t̄2

· · · ∂DSn

∂t̄m0

∂DT1∂t̄1

∂DT1∂t̄2

· · · ∂DT1

∂t̄m0

.... . .

. . ....

...∂DTn∂t̄1

∂DTn∂t̄2

· · · ∂DTn

∂t̄m0

∂f̄∂t̄1

∂f̄∂t̄2

· · · ∂f̄∂t̄m 0

︸ ︷︷ ︸

R

t̄trg,1 − t̄1

...

t̄trg,m − t̄mQ̂trg

︸ ︷︷ ︸

∆t

,

(9)emerges, where ∆f̄ = f̄trg−f̄ = wf∆f , and all quantitiesare dimensionless.

3. Matrix inversion by SVD

Eq. (9) must be solved to obtain the tuner corrections.One possible solution is given by

∆t = R†∆x, (10)

where R† = (RTR)−1R denotes the Moore-Penrose in-verse [31] (pseudo-inverse) of the generally non-square R.Koubek [5, 6] pointed out that R is potentially ill-conditioned and proposed to use a special method basedon singular value decomposition (SVD) [31] to computethe inverse.

To simplify the notation, we identify N = 3n + 1and M = m + 1 corresponding to a setup with n lon-gitudinal field sampling points and m tuners. The SVDof R ∈ RN×M is given as

R = UΣVT, (11)

where U ∈ RN×N , V ∈ RM×M are orthonormal ma-trices, and Σ ∈ RN×M is a rectangular diagonal ma-trix whose diagonal entries σ1, . . . , σM are the singularvalues of R in descending order. Note that M ≤ N isan essential requirement of the algorithm, which can al-ways be achieved by increasing the number of samplingpoints. The pseudo-inverse can then be constructed asR† = VΣ†UT, where Σ† is obtained by inverting eachdiagonal entry σi of Σ and transposing the result.

For ill-conditioned, almost singular R, the recipro-cals 1/σi of the smaller singular values (larger i) approachinfinity. This can lead to invalid solutions, i.e. tuner ad-justments that lie outside of the physical tuner movementrange. Koubek [5, 6] proposed to circumvent this prob-

lem by consecutively setting the largest 1/σi of Σ† to

zero. We define

Σ†k = diag[

1σ1· · · 1σM−k 0 · · · 0

]∈ RM×N (12)

as the matrix Σ† where the k largest entries have been setto zero, with k = 0, . . . , (M − 1). Note that Σ†0 = Σ† isthe initial Moore-Penrose inverse, whereas k = M leadsto R† = 0 and no tuner movements at all. The matricesgive M possibly useful solutions

∆tk = VΣ†kU

T∆x. (13)

Naturally, solutions with tuner adjustments outside thephysical movement range must be discarded. The re-maining solutions are checked by computing the predic-tion x̃trg,k for corrected field and frequency using theoriginal response matrix:

x̃trg,k = xcur + R∆tk. (14)

The tuner movement for the current tuning step is chosenas that ∆tk whose corresponding x̃trg,k best fulfills therequirements: a quadrupole component as flat as possi-ble, dipole components and frequency deviation as closeto zero as possible.

After applying one tuner adjustment, field and fre-quency are measured again and a new vector xcuremerges. The process is repeated until the measuredfield components match the requirements to a desiredaccuracy [5, 6, 14]. Koubek showed that it is sufficientto use only the initial R and choose a different solu-tion [Eq. (13)], i.e. a different value for k, for eachtuning iteration [5, 6]. This way, time-consuming re-measurement of R for each tuning iteration is avoided.

D. Tuning steps

The tuning procedure can be structured into twophases: at first, the response matrix R was measuredby means of individual tuner movements. Then, two cor-rective tuner movements were carried out.

1. Measurement of response matrix

The entries of R from Eq. (9) were determined bymeans of spectrum and bead-pull measurements. Eachmatrix column was obtained by moving the correspond-ing tuner a certain length while all other tuners re-mained in nominal position. The probing tuner move-ments ∆tj should resemble the anticipated correctivemovement as close as possible as R contains only first-order approximations of the de facto nonlinear responses:∂Xi/∂tj u ∆Xi/∆tj = const., where X ∈ {Q,DS, DT}.From Fig. 8 it can be seen that a movement of 3 mmaffects DT to an amount similar to the detuning in DT.Therefore, the derivatives in R were approximated as dif-ference quotients where ∆tj = 3 mm, i.e. each tuner was

8

retracted by 3 mm. Naturally, the tuner normalizationconstant was chosen as wt = (3 mm)

−1.

In Fig. 9a, a visual representation of the responsematrix is shown. The last row and column are omit-ted here in order to show only the field sensitivities.All four tuners at the same longitudinal position (e.g.tuners 1 to 4) have approximately the same effect onQ, where tuners located at the extremities of the RFQinduce slightly stronger field tilts than those near thecenter. Each tuner has a strong effect on the dipole com-ponent comprising the quadrant in which the tuner islocated, while the other dipole component is influencedonly marginally [Eq. (1)]. Two tuners located oppositeof each other (e.g. tuners 1 and 3, or 2 and 4, see Fig. 1)have opposite effects on the respective dipole component.The frequency—an integral quantity proportional to thetotal field energy—is affected by each tuner with roughlythe same sensitivity of −50 kHz/mm [Fig. 9b]. Individualdeviations are attributed to the inhomogeneous capaci-tance and inductance distributions caused by vane mod-ulation, misalignments, and design choices, which in factmotivate the RFQ tuning.

1 5 9 13 16

Q1...

DS1...

DT1...

Tuner index

Sam

plin

gpo

ints

−1

0

1

Sens

itiv

ity

(%/

mm

)

(a)

1 5 9 13 16

−60

−40

−20

0

Tuner index

Sens

itiv

ity

(kH

z/m

m)

(b)

FIG. 9. Visual representation of response matrix R exceptfor the last row and column (a). Each entry represents thesensitivity of the Q (upper third), DS (middle third), or DT

component (lower third) at one sampling point with respectto the tuner movement given in units of %/mm (without nor-malization). The responses of the TE210 eigenfrequency tothe movement of each tuner, i.e. the last row of R, is shownin (b). A negative derivative means that the frequency islowered when the tuner is retracted, as the total cavity in-ductance is increased.

In each tuning step, the frequency was enforced toequal the nominal frequency of 749.480 MHz to an accu-racy ensuring that no significant frequency error wouldbe introduced by the tuning algorithm itself. We se-lected wf = (10

4 Hz)−1 such that the predicted frequencyobtained from x̃trg,k [Eq. (14)] matched the desired fre-quency with an error smaller than 0.1 kHz, while stillkeeping the weight as small as possible.

2. Corrective tuner movements

With Eq. (14), M possible solutions ∆tk [Eq. (13)]yield M predictions x̃trg,k of the field distribution af-ter the tuning step, shown in Fig. 10a for the first tun-ing step. Corresponding amplitude errors are reportedin Fig. 10b. Solutions with lower k (less singular val-ues eliminated) generally deliver better corrections. InRefs. [5, 6], Koubek reported that—with the original ver-sion of the algorithm—many of the possible solutionswith small k yielded tuner adjustments that were out-side the physical tuner movement range. In the presentcase this could not be confirmed; all k = 1, . . . ,M − 1solutions provide physically possible tuner movements.This is attributed to the fact that the augmented ver-sion of the algorithm [Eq. (9)] includes the frequency,which would strongly disagree with the desired value forunphysically large tuner movements and thus acts in adampening manner. Furthermore, the response matrixof the HF-RFQ was more affected by measurement noise,as no averaging was performed [5, 6]. Nevertheless, it isstill advantageous to make use of the truncated SVD inthe improved algorithm and select a solution “by eye,”as solution k does not strictly provide a better correctionthan solution k + 1.

The PIXE-RFQ was tuned in only two steps after themeasurement of R. For the first step, k = 1 was chosenas the option that provides very small errors for all threecomponents while equally correcting DS, DT. The corre-sponding field prediction is shown in Fig. 11 (red dashedline). After applying the tuner adjustments, the fre-quency error improved from 624 kHz to 46 kHz (Fig. 12).A significant deviation between field prediction and mea-surement (red solid line) was observed, in particular forDS, where an error of nearly ±3 % remained. The reasonis found in the nonlinear relation between field and tunerposition, that is represented in the response matrix onlyby linear approximation.

A second tuning step with the same response matrix Rwas carried out, however, this time we chose k = 3. Pre-dicted and measured fields after the second tuning stepare reported in Fig. 11 (yellow lines). DS could be sup-pressed to an error of 1.3 %, and the frequency deviationwas improved to 1.5 kHz. This error is of the same orderof magnitude as the error arising from the thermome-ter precision (±2.6 kHz). The second tuning step alreadysaw a slight worsening in DT. This indicates that anaccuracy limit given either by the overall noise level or

9

98

99

100

101Q

(%)

−5

0

5

DS

(%)

1 2 3 4 5 6 7 8−5

0

5

Sampling points along RFQ

DT

(%)

0 5 10 15

Solution index k

(a)

0 2 4 6 8 10 12 14

0

2

4

6

8

Solution index k

Am

plit

ude

erro

r(%

)

δQ

δDS

δDT

0 2 4 60

0.2

0.4

0.6

(b)

FIG. 10. Predicted field components (a) after applying thetuner adjustments yielded during tuning step 1 by discard-ing the smallest k singular values from the response matrix.Corresponding field component errors are reported in (b).

the linear approximation was reached. The errors in fre-quency and field after the second tuning step fulfilledthe requirements listed in Section II B: δQ = ±0.6 %,δDS = ±1.3 %, and δDT = ±0.6 %. As a third iterationdelivered no satisfying predictions of improvement, it wasdecided to stop the tuning procedure after two steps.

3. Tuner recutting

Final tuner positions are reported in Fig. 13. Notethat on average the tuners were retracted, as the initialmeasured frequency was too high and the total cavity in-ductance had to be increased. All adjustments are con-siderably smaller than the maximum foreseen movementrange of ±11 mm. The final tuner lengths were deter-mined both from the scale on the tuner tooling [Fig. 5b],which was used during the tuning process itself, and froman additional measurement using a caliper. The errorsbetween the two measurements read up to 60 µm, likelyoriginating in small inclinations of the tuners within theguidance tubes. The average of both values was used forremachining.

Following remachining, a bead-pull measurement was

98

99

100

101

rr

rr r

r r r

Qua

drup

oleQ

(%)

initial step 1 step 2predicted ∅measured r

−6

−3

0

3

6

9

r rr r r r r r

Dip

oleD

S(%

)

1 2 3 4 5 6 7 8−6

−3

0

3

6

9

r r r r r r rr

Sampling points along RFQ

Dip

oleD

T(%

)

FIG. 11. Field components for nominal tuner position (ini-tial) and after the first and second tuning step. Dashed linesindicate the field predicted by Eq. (14), while solid lines reportthe actually measured field.

initi

al

tuni

ngste

p 1

tuni

ngste

p 2

tune

rsrem

achi

ned

fixed

with

gask

ets

unde

r vac

uum

0

200

400

600Fr

eq.e

rr.∆f

(kH

z)

FIG. 12. Frequency of the TE210 eigenmode for nominal tunerposition (initial), after the first and second tuning step, andafter remachining measured under air and vacuum. The finalfrequency error amounts to ∆f = 23.5 kHz.

carried out with the tuners fixed to the correspondingflanges without seals as a mean of validation, after whichthey were installed with vacuum gaskets and a final mea-surement was performed. The results are reported inFig. 14 in comparison to the initial field and the fieldafter the second tuning step. Corresponding errors aresummarized in Table II. A small deviation, larger thanmeasurement noise, was observed between the field af-ter the second tuning step and the field after they wereremachined and fixed. This deviation likely originates inthe finite accuracy of length measurement and materialcutting.

The final frequency measured under vacuum was23.5 kHz above the target value. This error was achieved

10

1 5 9 13 16

−2

0

2

Tuner index j

Tune

rpo

s.t j

(mm

)

FIG. 13. Tuner positions after the final tuning step. A pos-itive (negative) value indicates that the tuner was retracted(inserted) with respect to the nominal position.

measuring under air, solely correcting for temperatureand constant air permittivity (STP value). The deviationcan be explained by the fact that response matrix and ini-tial frequency were measured on a rainy day (≈ 100 % rel.humidity), while the tuning steps were performed undersunny weather (≈ 50 % rel. humidity). It is likely thatthe remaining error could have been significantly reducedby correcting for the measured humidity or using a dryflooding gas. Nevertheless, the final frequency error is bymore than a factor of two smaller than the tuning rangegiven by a typical water cooling system (±60 kHz).

TABLE II. Errors in field components and frequency duringtuning and subsequent measurements in chronological order.

Step δQ (%) δDS (%) δDT (%) ∆f (kHz)Initial 2.5 7.9 8.4 623.6Tuning step 1 0.6 2.6 0.5 45.5Tuning step 2 0.6 1.3 0.6 1.5Tuners recut 0.3 1.0 0.9 65.7Fixed with gaskets 0.3 0.9 0.9 24.5Under vacuum — — — 23.5

V. QUALITY FACTOR MEASUREMENT

After tuning, the quality factors of the PIXE-RFQwere measured under vacuum to validate the design andprovide information on the input power reflection origi-nating in potential over- or under-coupling.

The raw measurement data were the complex reflectioncoefficient S11 = Γ measured in a bandwidth of 1 MHzsymmetrically around the TE210 eigenfrequency. The re-flection coefficient describes a circle in the complex planein very good approximation, from which unloaded (Q0),external (Qex), and loaded quality factor (Q`) can beobtained, where

1

Q`=

1

Q0+

1

Qex. (15)

Additionally, the coupling coefficient

βc = Q0/Qex (16)

98

99

100

101

Qua

drup

oleQ

(%)

initial last tuning steptuners remachined fixed with gaskets

−6−3

0369

Dip

oleD

S(%

)

1 2 3 4 5 6 7 8−6−3

0369

Sampling points along RFQ

Dip

oleD

T(%

)

FIG. 14. Measured field components before tuning, after tun-ing, with remachined tuners, and for the final installation ofthe tuners with copper gaskets.

emerges. The parameters were extracted by fittingan ideal circle to Γ following the method describedin Refs. [32–35]. Contrarily to e.g. the three-pointmethod [36, 37], the circle-fitting method performs im-plicit averaging by solving a heavily over-determined sys-tem of equations, and is thus significantly less susceptibleto measurement noise.

The technique is based on the Möbius transformation

Γ =b1ω̄ + b2b3ω̄ + 1

, (17)

with ω̄ = 2(ω/ω0−1), which describes a circle in the com-plex plane characterized by the parameters b1, b2, b3. Forω̄ → ±∞, Γ → b1/b3 approaches the detuned reflection,whereas ω̄ = 0 delivers Γ(∆ω = 0) = b2 as the reflec-tion coefficient of the loaded resonator. Their distanceequals the circle diameter D = |b1/b3 − b2|, and the cou-pling coefficient can be determined as βc = (2/D− 1)−1.Furthermore, Q` = ={b3}, from which the two remain-ing quality factors can be calculated using Eqs. (15)and (16) [33]. The VNA measures Γ(ωi) at N discretesampling points ωi. Thus, Eq. (17) can be written inmatrix form:Γ(ω1)...

Γ(ωN )

︸ ︷︷ ︸Γ ∈ CN

=

ω̄1 1 −ω̄1Γ(ω1)... ... ...ω̄N 1 −ω̄NΓ(ωN )

︸ ︷︷ ︸

K ∈ CN×3

b1b2b3

︸ ︷︷ ︸

b ∈ C3

, (18)

a heavily overdetermined system of equations with eachrow representing one measurement point. The curvedescribed by Γ is not a perfect circle—deviations be-come stronger with increasing distance from ω0 (or|ω̄| � 0). Furthermore, equidistant frequency sampling

11

implies lower density of measurement points around thecritical ω0. Therefore, a weighting matrix W is intro-duced, which reduces the significance of measurementpoints further away from ω0:

wi = 1−|Γ(ωi)|2 , W = diag[w1 · · · wN

]∈ RN×N .

(19)Then, the three fitting parameters are obtained as

b = (WK)†WΓ. (20)

0◦

30◦

60◦90◦

120◦

150◦

180◦

210◦

240◦270◦

300◦

330◦

0 0.5 1

Reflection coefficient Γ

measurementcircle fit

FIG. 15. Smith chart of the reflection coefficient Γ measuredover a span of 1 MHz and the fitted circle. Only every tenthmeasurement point is shown for clarity.

In Table III, measured quality factors and couplingcoefficient are compared to the design values [10]. Veryclose agreement was observed: the measured Q0 agreeswith the design value with an error smaller than 0.5 %.This indicates not only the reliability of the power losscalculation carried out at the design stage [10], but alsothe manufacturing quality of the PIXE-RFQ.

TABLE III. Quality factors measured using the circle-fittingmethod compared to the design values.

Measured Design Rel. err.Unloaded quality factor Q0 6018 5995 0.4 %External quality factor Qex 5099 4796 6.3 %Loaded quality factor Q` 2756 2664 3.5 %Coupling coefficient βc 1.18 1.25 5.6 %

A larger deviation of 6 % was observed in Qex. Thistranslates to a comparable error in βc and an error of3 % in Q`. The errors could be introduced by imper-fect machining, brazing, and alignment of the physicalcoupler. A 6 % error in βc could correspond to a me-chanical error of 300 µm in the coupling loop size. Asthe coupler represents a complex geometry composed ofmultipole components assembled by means of bolts, wecannot exclude such an error. (In contrast, different sim-ulation models generally yield errors in Qex which are byan order of magnitude smaller.)

All measurements reported in this section were con-ducted under vacuum (approximately 10−6 mbar). Aninteresting observation was made in that the measuredcoupling increased by approximately 2 % (from 1.16to 1.18) when the RFQ cavity was evacuated. A possibleexplanation is found in the mechanical structure of thepower coupler: The inner conductor is only supported bythe coupling loop and a polyether ether ketone (PEEK)rf window [10]. Subjected to the pressure difference be-tween vacuum and exterior atmosphere, the PEEK win-dow could deform and push inwards the inner conductor.This would slightly increase the effective area of the cou-pling loop, resulting in a larger coupling coefficient.

The discrepancy highlights that it is important to de-sign the coupler with an over-coupling margin of few tenpercent. Although the PIXE-RFQ coupler features a ro-tatable flange that allows for fine-tuning the couplingcoefficient by means of measurements, we decided notto take advantage and rather accepted an over-couplingof 18 % (βc = 1.18). This translates to an input reflectionof −22 dB. If the RFQ cavity consumes 65.0 kW of peakpower in terms of surface losses [10], an input power of65.5 kW is required, and 0.5 kW, i.e. roughly 0.7 % ofthe input power, are reflected back towards the genera-tor. These values are well acceptable for operation.

VI. SUMMARY

In this paper, the low-power rf measurements and tun-ing procedure carried out at CERN on the one-meter-long750 MHz PIXE-RFQ are reported.

Before full assembly, bead-pull measurements wereperformed on mechanical modules to confirm that no se-vere errors occured during machining and brazing. Thefield showed near-perfect agreement with the simulation,whereas the frequency error was equal or smaller than600 kHz in both cases. The small observed deviationsare well within the capabilities of the tuning system.

The main part of this paper is dedicated to the RFQtuning procedure. Longitudinal and transverse fieldtilts were compensated by means of sixteen movablepiston tuners. The aim of the tuning was to achievethe most flat possible quadrupole component and van-ishing dipole components. Furthermore, the operatingmode frequency was tuned to the design frequency of749.480 MHz under vacuum at 22 °C.

Reliability measurements were carried out before theactual tuning procedure in order to assess the accuracylimits given by bead-pull measurement and tuner tooling.The bead-pull measurement error could be improved to0.5 % by performing each measurement thrice. No signif-icant mechanical hysteresis effects were observed.

The tuning algorithm used for HF-RFQ [5] was aug-mented for the PIXE-RFQ such that frequency and fieldcould be tuned at the same time. The results suggest thatincluding frequency and corresponding weights also leadsto a more stable convergence of the tuning iterations. Af-

12

ter measuring the response matrix, the PIXE-RFQ wastuned in only two steps. Subsequently, the tuners wererecut to their final lengths. The measurement under vac-uum revealed a remaining frequency error of 23.5 kHz,which can be explained with the measurement uncer-tainty with respect to air humidity. The error lies wellwithin the ±60 kHz tuning range of a typical water cool-ing system (±5 K) and is in any case acceptable since norf accelerating structures are present downstream of theRFQ.

The three quality factors and the coupling coefficientwere measured using a circle-fitting method. Excellentagreement with an error smaller than 0.5 % was observedinQ0. Qex was measured to be 6 % higher than the designvalue. The discrepancy could be attributed to mechani-cal manufacturing error in the coupler of roughly 300 µm.The larger Qex corresponds to a measured coupling co-efficient of 1.18 compared to a design value of 1.25.The measured over-coupling of 18 % requires an extra

0.5 kW (0.7 %) of rf peak input power. It was decided tonot make use of the possibility of rotating the coupler tocloser approach critical coupling.

With the low-power rf measurements completed inmid-2020, the PIXE-RFQ has been transported to INFN,Florence, Italy for high-power rf conditioning and beammeasurements. The world’s smallest RFQ is expected tocommence operation towards the end of 2020.

ACKNOWLEDGMENTS

We wish to thank Sebastien Calvo, Yves Cuvet, SergeMathot, and the staff of the CERN vacuum brazing work-shop for their help in successfully tuning the PIXE-RFQ.

This work has been supported by the CERN Knowl-edge Transfer Group and the Wolfgang Gentner Pro-gramme of the German Federal Ministry for Educationand Research (BMBF, grant no. 05E15CHA).

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rf measurements and tuning of the 1-meter-long 750 MHz radio-frequency quadrupole for artwork analysisAbstractI IntroductionII Preliminary considerationsA Bead-pull field measurementB Tuning goalsC Frequency correction

III Single module measurementsIV TuningA Tuner setup and toolingB Reliability measurements1 Repeatability of field measurement2 Reproducibility with respect to tuner position

C Augmented tuning algorithm1 Previous tuning algorithm used for HF-RFQ2 Including the frequency3 Matrix inversion by SVD

D Tuning steps1 Measurement of response matrix2 Corrective tuner movements3 Tuner recutting

V Quality factor measurementVI Summary Acknowledgments References