Perturbational Simulation and Experimentation for Calibrated RF … · 2016. 8. 12. · Perturbational Simulation and Experimentation for Calibrated RF Cavity Measurements Alex Pizzuto

Perturbational Simulation and Experimentation for CalibratedRF Cavity Measurements

Alex Pizzuto

Loyola University ChicagoArgonne National Laboratory

[email protected]

August 12, 2016

Abstract

Field strength measurements for determination of the R/Q of various modes in RF cavities rely on per-turbational techniques with analytically approximated perturbing objects. Simulations and analyticalapproximations were compared to measured results on cavities with calculable fields for a repository ofperturbing objects to perform calibrated measurements. Categorized perturbing objects were then used toperform calibrated measurements of the accelerating mode of 86-cell 2π

3 2.856 GHz cavities that had beenstraightened after deformations due to gravitational effects as well as higher order modes (HOMs) of the 352MHz Advanced Photon Source storage ring cavities.

I. Introduction

Throughout their installation at the Ad-vanced Photon Source (APS) at Ar-gonne, various structures underwent un-

predicted gravitational deformations. Duringvarious shutdowns, these cavities are pulledfrom the linear accelerator to undergo straight-ening and tuning. However, this process mayresult in a shifting of electrical center as wellas offset phase advance in certain cells. Theseimperfections can cause kicks from HOMs. Inorder to properly understand the functionalityof the cavities in the accelerator, we seek tocategorize the modes with their correspondingR/Q, a measure of the cavity shunt impedancedivided by the loaded Q of the cavity to pro-vide a performance parameter completely de-pendent on geometrical factors.In addition, various problematic longitudinalmodes have been witnessed in the 352 MHzcavities in the storage ring. Coupling to lon-gitudinal HOMs leads to increased energyspread and eventual beam loss from receiv-ing energy in the non-restoring sections of theRF voltage curve. In order to better quantifythe magnitude of the effects from these HOMs,each mode needs to be associated with a R/Q,

in order to determine what feedback systemsneed to be put into place. These R/Q values arecross-referenced with simulation results fromCST Microwave Studio.In order to make these measurements, classifi-cation of the form factors of perturbing objects(beads) is necessary and allows for calibratedfield measurements on any future occasion.

II. Theoretical Background

Given a resonant mode of a cavity, the resonantfrequency, ω0, will shift when a perturbingobject is introduced to the cavity. The changein frequency was first solved analytically bysolving Maxwell’s First and Second Equationsin the perturbed and unperturbed cavity in [1].If we denote the local absolute permittivity andpermeability of the perturbing object as ∆ε and∆µ, respectively, then it can be shown that theshift in resonant frequency is given by

∆ω

ω0= −

∫V(∆ε~E · ~E∗0 + ∆µ~H · ~H∗0 ) · dV

4U(1)

where U is the stored energy in the cavity. Ifthe numerator is known for a certain geometry,

1

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RF Simulation and Measurement • 2016 Lee Teng Fellowship

then this reduces to

∆ω

ω0=

1U

(F1E2‖ + F2E2

⊥ + F3H2‖ + F4H2

⊥

)(2)

where the Fi are electric and magnetic form fac-tors of the certain perturbing objects, or beads.In principle, if the fields of a cavity are known,then by measurement of the frequency shift ofa mode in the cavity, one can determine theform factors for a certain bead either by in-troducing the bead into regions where certaincomponents of the field are zero, or by per-turbing different fields and solving a system oflinear equations.In practice, in order to compute the magnitudeof the field at enough locations to determinefield patterns with precision, frequency shift istoo imprecise a measurement. Instead, one canindirectly measure the frequency shift by mea-suring the phase shift of the S21 transmissioncoefficient from the two dimensional scatter-ing matrix, so long as we remain on the linearregime of the curve, we can approximate

∆ω

ω0≈ 1

2Qtan (∆φ21).

Then, if we recognize the R/Q is the squareof the accelerating voltage normalized by thestored energy in the cavity divided by the an-gular frequency, then we have

RQ

=1

2πε0

( ∫dz

√tan ∆φ

2QL

1F1

)2

,

thus by measuring the phase shift as a func-tion of longitudinal position in the cavity, ameasured value of R/Q is obtained [2].

III. Analytical Approximations

When the integral in equation 1 is known, theform factors of the beads are equal to the elec-tric or magnetic polarizability of the objecttimes the permittivity or permeability (respec-tively) of free space. In the case of the perturb-ing objects available, this is only exactly solv-able with a metallic or dielectric (DE) sphere.

However, with cylindrical beads, approxima-tions are made by treating the cylinders asprolate or oblate spheroids dependent on theaspect ratio of the bead.In the case of a sphere, where we substitute ξas the absolute permittivity and permeabilityof the object, then we have the form factor forthe magnitude of either the electric or magneticfield as

F = −πa3 ξ − 1ξ + 2

where a is the radius of the sphere [3]. Whilethe sphere is exactly solvable, its isotropic na-ture does not allow one to separate the com-ponents of the fields. However, if we treat acylinder as a prolate spheroid (the solid gener-ated by rotation of an ellipse around its majoraxis) with major axis l/2 and minor axis a, thenwe obtain form factors depending on whetherthe field is transverse or longitudinal. For ex-ample, in the case of a conducting bead in aperturbed field originally parallel or perpen-dicular to the major axis, the form factors aregiven by

F1 = −πL3ε0

(1− 4a2

L2

)3/2

12

log

√

1− 4a2L2 +1

1−√

1− 4a2L2

− 2√

1− 4a2

L2

F2 =πL3ε0

(1− 4a2

L2

)3/2

6

log

√

1− 4a2L2 +1

1−√

1− 4a2L2

− L2√

1− 4a2L2

2a2

.

Much work has been done by [3],[4],[5], and[6] to create better approximations using differ-ent techniques for different beads. All of ourdata were compared to these existing modelsfor comparison, but in all models the behaviorof longitudinal form factor as opposed to thetransverse for cylinders (both hollow needlesand solid rods) highlights the capability of be-ing able to isolate certain components of thefield.

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Figure 1: Approximations for longitudinal and trans-verse electric field form factors

IV. Simulation Methods

While the analytical approximations provide agood starting point for bead behavior, and al-low for relative measurements, in order to mea-sure precisely, more data is needed that trulyfits measured results. In order for the equation1 to hold, the perturbing object must be smallcompared to the size of the wavelength of thesignal. In our case, calibrated measurementswere desired for an S-Band (2.856 GHz) trav-elling wave structure and structures in lowerfrequency parts of the UHF regime (352 MHZfundamental standing wave structure). Bothof our cavities exhibit cylindrical symmetry,allowing for precise simulations in Superfish,which were also compared to simulations per-formed in CST Microwave Studio on a 1.7 GHzpillbox cavity and 2.8 GHz coupled cavity. Sim-

Figure 2: Microwave Studio Simulations (top) and Su-perfish simulation (bottom)

ulations were performed for metallic spheresand needles, as well as for both metallic anddielectric rods. These simulations were used to

determine which analytical solutions could beused if measured calibrated data did not existfor a bead or if simulation was not available inthe future.

Figure 3: Comparison of Needle and Rod simulation re-sults

V. Experimental Results

In order to find values for the form factors ofdifferent beads, calculable fields must be per-turbed. This requires coupling with cavities insuch a way to ensure that the proper modesare excited as well as being sure that one ismeasuring the fields in the proper transverseposition, as it is not always the case that theelectrical center maps directly onto the mechan-ical center of the cavity. For testing purposes,two different cavities, a 1.7 GHz pillbox cavityand a 2.8 GHz coupled cavity, were used.

Figure 4: Cavities used for calibration measurements

The methodology for locating calculablefields in the pillbox cavity first entailed cou-pling with the probes to ensure that some

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modes were excited.Maximum coupling for Transverse Electric (TE)or Transverse Magnetic (TM) modes led to min-imal coupling with the other. Fundamental TEand TM modes were simulated, and bead-pullswere performed with both maximally coupledmodes to determine which probe orientationresulted in the excitation of which modes, bycomparison of field flatness. Once resonances

Figure 5: TE and TM determination

were visible from the log plot of the S11 reflec-tion coefficient on the precision network ana-lyzer (PNA), bead pulls were performed withdielectric rods (only sensitive to the electricfield, and mainly to the longitudinal compo-nent), to have a cursory relative mapping ofthe longitudinal electric field. Though calibra-tion is easiest in modes with on axis longitudi-nal fields, HOMs with zero field on axis weresearched for in order to find electrical center.

Figure 6: Bead-Pulls at different frequencies in Pillboxto locate modes with zero on-axis field

When modes with small magnitude fieldsat mechanical center were located, their fre-quencies were cross referenced with simula-tion results to locate the dipole mode. Oncethe dipole mode was discovered, at a frequencyof 2.747 GHz, relative field strength measure-

ments were taken in 21 transverse locations for2 different longitudinal coordinates, in orderto map the mode transversely and determinethe linear electrical center.

Figure 7: Mapping of relative field strength of DipoleMode (left) and z = 0, x = 0 orientation toshow field patterns at small radii (right)

i. Mode Identification and Exploita-tion

With the location of electrical center and modeidentification from bead-pulls at different res-onances, we turned to specific modes to cali-brate different form factors for the same beadsin calculable fields for the beads listed below.

Table 1: Beads used for measurement in calculable fields

Bead Length (mm) Radius (mm)

Rod 1 4 0.5Rod 2 5 1Rod 3 5 1.25Rod 4 5 2Needle 1 5 0.425Needle 2 5 0.5Needle 3 5 0.75DE Rod 1 5 1.5DE Rod 2 5 2DE Rod 3 5 2.5Sphere N/A 2.35

Each bead has 4 associated form factors forthe longitudinal and transverse electric andmagnetic field components. By exciting modesthat have only one of these components onaxis, it is possible to measure the correspond-ing form factor for each bead. This was donefor all components sans the longitudinal mag-netic field, as we were not investigating effects

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due to a longitudinal magnetic field. DE beadsonly have form factors for the electric fieldcomponents, as they do not affect the magneticfields. However, the permittivity of the dielec-tric beads was unknown, but simulations wereperformed to place a rough estimate on themagnitude of the permittivity of the beads.

i.1 Longitudinal Electric Field Form Factor

In order to isolate the longitudinal electric fieldform factor for the beads, we excited the TM010mode of the cavity. In a pillbox cavity, thefields found by setting Bz = 0 and solvingMaxwell’s Equations result in the followingfield components

Ez(~r) = E0 J0(rr0) cos(kz−ωt)

Bφ(~r) = B0 J′0(rr0) sin(kz−ωt).

Thus, Ez will take a maximum on the longitu-dinal axis, whereas Bφ is identically zero onaxis.

Figure 8: E-field (left) and H-field (right) for F1

Table 2: Longitudinal Electric Field Form Factor. AllForm Factor values given are ·10−19

Bead Measured F1 Simulated F1 Error

Rod 1 1.079 1.20 9.78%Rod 2 3.0123 2.90 3.70%Rod 3 3.567 3.65 2.32%Rod 4 6.316 6.20 1.85%Needle 1 1.441 1.42 1.52%Needle 2 1.768 1.63 8.77%Needle 3 2.095 2.11 0.75%DE 1 1.573 1.59 1.17%DE 2 2.554 2.69 5.16%DE 3 3.733 3.97 5.88%Sphere 3.665 3.540 3.70%

These errors are well within the errors shownin [1] and [5], and the contrast between themeasured results and available approximationshighlight the necessity for calibrated beads tocompute reliable R/Q values. However, it isworth noting that these simulations were runon both L-Band and S-Band cavities, and sim-ulation and measured results tend to divergefrom the analytical approximations in similarfashions when the size of the wavelength rela-tive to the size of the bead is taken into account.

i.2 Transverse Electric Field Form Factor

The mode used for this form factor, TE11 ap-pears in Figure 5. A bead-pull was performedto locate where longitudinally the field tooka maximum, and then the beads were placedin that region to measure the response of theresonance.


The larger errors from the measurementsof these form factors are likely a result of the

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Table 3: Transverse Electric Field Form Factor. All FormFactor values given are ·10−19


Rod 1 0.201 0.156 28.7%Rod 2 0.802 0.888 9.67%Rod 3 1.204 1.468 17.9%Rod 4 3.877 4.425 12.4%Needle 1 0.134 0.141 4.29%Needle 2 0.201 0.188 7.16%Needle 3 0.402 0.459 12.4%DE 1 1.069 1.337 20.0%DE 2 2.006 2.472 18.9%DE 3 3.276 4.026 18.6%Sphere 3.475 3.659 5.02%

limitations of the PNA, as the magnitude of thefrequency and phase shifts are much smallerfor cylinders with axis perpendicular to theelectric field.

i.3 Transverse Magnetic Field Form Factor

In order to find the transverse magnetic formfactors for the beads, the TM11 mode was ex-cited. This is a dipole mode that provides kicksto beams if energy is stored in the mode frombeam travelling with displacement from electri-cal center. Centering was extremely importantin this case, as with the other modes the elec-tric field did not vary greatly near the axis,but with the dipole mode the R/Q, normalizedradially with the first zero of the lowest or-der Bessel function, increases with the square

of the displacement within the radius of thebeampipe (where measurement is possible). Ingeneral, the magnitude of the frequency shiftsfrom the electric fields is much larger than thatdue to a magnetic field (for fields of typicalrespective strength in the cavities), so care wastaken to isolate the magnetic field.


For these measurements, DE beads were notused as they do not respond to magnetic fields.However, they do therefor prove useful for lo-cating where the electric field is zero, as theresonant frequency of the cavity should be un-perturbed.

Table 4: Transverse Magnetic Field Form Factor. AllForm Factor values given are ·10−15


Rod 1 −2.193 −2.601 15.7%Rod 2 −8.655 −9.541 9.28%Rod 3 −12.97 −15.25 14.9%Rod 4 −34.62 −33.97 1.90%Needle 1 −1.454 −1.729 15.9%Needle 2 −2.885 −3.053 5.5%Needle 3 −4.316 −6.241 30.8%Sphere −25.96 −24.27 7.0%

VI. Calibrated R/Q Measurements

i. Travelling Wave Structures

In order to calculate the R/Q of a multi-cellcavity, one must take into account not only thetransit time factor, but also the phase advanceper cell.

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Figure 11: Structure used for travelling wave R/Q mea-surement

In a single cell cavity, the accelerating voltagefor a particle travelling at speed βc is given by

Vacc =∫

Ezej ωβc zdz

However, in a cavity with multiple cells, eachcell has a corresponding cell to cell phase ad-vance. This is taken into account as an offsetin the argument of the exponential that is in-cremented upon entry of a new cell. First, abead-pull was performed that measured thereflection coefficient, and this data was used toobtain the cell to cell phase advance.

Figure 12: Sample data for phase advance

Next, a bead-pull was performed to measurethe phase shift in the transmission coefficientto calculate the strength of the field, and these

two bead-pulls were combined to calculate theR/Q, where

RQ

=1

ωU|∫

Ezej ωβc z−φ(z)dz|2

where φ(z) is a function of phase advanceper cell.

Figure 13: Sample data for phase shift from perturbation

As these structures are constant gradient,each cell should provide roughly the same con-tribution to the R/Q. Taking into account ageneral shift in the phase of the transmissioncoefficient, and plotting with energy scaled toreflect only the stored energy in the cavity upthe longitudinal coordinate that is our upperbound, we obtain the following plot for theevolution of R/Q in the travelling wave struc-ture.

Figure 14: Evolution of R/Q in the travelling wave cav-ity

Using a calibrated bead to measure the R/Qof the travelling wave structure, we obtain avalue of 71.125 from our first run and 58.0from our second, and when compared to theexpected prediction of around 70 per cell, wesee that our calibrated measuring technique isvalid.After the straightening and tuning process ofthe cavities used in operations, the desired in-vestigation would be an analysis of the dipolemodes of the cavities. However, with opera-tions cavities, there is only the ability to probethe cavity before the input coupler or afterthe output coupler. In order to properly exciteHOMs in a cavity as large as this, probes would

7


need to be placed on individual cells to excitethe modes. We attempted to analyze data justfrom the reflection coefficient in hopes of cir-cumventing the problem of too small a signalbeing transmitted, but this would just provethat the fields fell off too quickly to obtain ameaningful R/Q for HOMs in a travelling wavestructure.

ii. HOM Classification

ii.1 Storage Ring Cavities

Though HOMs on travelling wave structureswere not possible to measure with our method-ology, cavities in the Storage Ring (SR) at theAPS have been under investigation due to theirapparent longitudinal HOMs.These HOMs donot provide transverse deflection to beam, butthey cause increased energy spread. This notonly causes larger buckets and eventual beamloss from leaving buckets, but it also causesdispersion which eventually leads to increasedhorizontal emittance. These HOMs can bedamped with proper feedback systems in place,but for those modes with small R/Q, it is notnecessarily obligatory to damp them. To date,only simulations had been done on the longi-tudinal HOMs, the most problematic of whichare pictured below.

Figure 15: Longitudinal HOMs of interest in StorageRing Cavities

In accordance with practice, a bead-pull wasdone at the monopole mode to confirm agre-

ment between simulation and measurement.

Figure 16: Simulated and Experimental TM010 fieldstrength on axis with experimental fieldstrength in blue

As we went to investigation of the HOMs, itwas noticed that the response of some of thesemodes was on the order of -.3dB, so measure-ment was done simultaneously of φ21 as wellas S11. Fits of simulated data were comparedto measured data, and these pulls were usedto analyze which modes truly pose a threat toenergy spread and which resonances are notof interest.

Figure 17: Fits to longitudinal HOMs

Table 5: Longitudinal R/Q values in Storage Ring Cavi-ties

Frequency (MHz) Measured Simulated Error

352 225.3 226.4 0.5%535 100 81.8 22.2%916 8.66 9.1 4.8%938 8.36 6.6 26.7%

While some modes fit the longitudinal sim-ulations well, this methodology was also use-

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ful in determining which resonances that ap-peared were not problematic. For example, itbecame clear from pulls which resonances hadnegligible on axis fields as well as which fit TEmode simulation better.

Figure 18: Resonant mode at 912 MHz (top) showing noon-axis field and mode at 914 MHz (bottom)with corresponding TE mode from simulation

Figure 19: Early analysis of which resonances are prob-lematic HOMs show that though there aremany resonances, not all pose a threat

VII. Acknowledgements

I would like to thank my mentor on this project,Geoff Waldschmidt, for his unwavering sup-port and guidance. I am also extremely gratefulfor the assistance of Tim Jonasson, Roy Agner,and Terry Smith in the lab and for the con-structive video calls with Massimo dal Forno.In addition, I would like to express my grat-itude to the Linda Spentzouris, the Lee TengInternship, and Illinois Accelerator Institute

for making this research possible, and to EricPrebys for his organizing the internship andlecturing at USPAS.

References

[1] L.C. Maier Jr. and J.C. Slater Field StrengthMeasurements in Resonant Cavities. Journalof Applied Physics 23.68. 1952.

[2] C.M. Bhat Measurements of Higher OrderModes in 3rd Harmonic RF Cavity at Fermi-lab IEEE 1993.

[3] A. Labanc Electrical axes of TESLA-type cav-ities. TESLA Report 2008-01.

[4] H. Hahn and H. Halama Perturbation Mea-surement of Transverse R/Q in Iris-LoadedWaveguides. Microwave Theory and Tech-niques 16.1. 1968.

[5] P. Matthews et al. Electromagnetic FieldMeasurements on a mm-wave Linear Acceler-ator. Argonne National Laboratory 1996.

[6] H. Wang and J Guo Bead-pulling Measure-ment Principle and Technique Used for theSRF Cavities at JLab. USPAS 2015.

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Perturbational Simulation and Experimentation for Calibrated RF … · 2016. 8. 12. · Perturbational Simulation and Experimentation for Calibrated RF Cavity Measurements Alex Pizzuto