The main objective of this work is to present a
simple method, based on analytical expressions, for obtaining
a quick approximation of the temperature rise due to the Joule
effect inside the metallic walls of an RF accelerating device. This
proposal relies on solving the 1D heat-transfer equation for a
thick wall, where the heat sources inside the wall are the ohmic
losses produced by the RF electromagnetic fields penetrating
the metal with finite electrical conductivity. Furthermore, it is
discussed how the theoretical expressions of this method can be
applied to obtain an approximation to the temperature increase
in realistic 3D RF accelerating structures, taking as an example
the cavity of an RF electron gun. These theoretical results have
been benchmarked with numerical simulations carried out with
commercial finite-element method codes, finding good agreement
among them

Blanch, C.

Boronat, M.

Esperante, D.

Fuster, J.

Fuster-Martínez, N.

Gimeno, B.

González-Iglesias, D.

Martinez-Reviriego, P.

Martín-Luna, P.

English

UPCommons. Portal del coneixement obert de la UPC

1Study of the RF Pulse Heating Phenomenon inHigh Gradient Accelerating Devices by Means ofAnalytical ApproximationsD. González-Iglesias, D. Esperante, B. Gimeno, Member, IEEE, M. Boronat, C. Blanch, N. Fuster-Martı́nez,P. Martinez-Reviriego, P. Martı́n-Luna, J. FusterAbstract—The main objective of this work is to present asimple method, based on analytical expressions, for obtaininga quick approximation of the temperature rise due to the Jouleeffect inside the metallic walls of an RF accelerating device. Thisproposal relies on solving the 1D heat-transfer equation for athick wall, where the heat sources inside the wall are the ohmiclosses produced by the RF electromagnetic fields penetratingthe metal with finite electrical conductivity. Furthermore, it isdiscussed how the theoretical expressions of this method can beapplied to obtain an approximation to the temperature increasein realistic 3D RF accelerating structures, taking as an examplethe cavity of an RF electron gun. These theoretical results havebeen benchmarked with numerical simulations carried out withcommercial finite-element method codes, finding good agreementamong them.Index Terms—RF pulse heating, thermal analysis, RF acceler-ating structures.I. INTRODUCTIONRF pulse heating is a phenomenon by which metals areheated by the electric induced currents generated by a pulsedhigh power RF electromagnetic field. This heating has twodifferent effects. The first one, which happens during the RFpulse, causes a superficial temperature increase within the RFskin depth range in a short time compared to the expansionmechanical time response of the material inducing stressesthat might damage the surface. According to recent studies ofbreakdown rates in high gradient linear accelerators, there isa direct correlation between these rates and RF pulse heating[1]. Because of that, the reliable operating gradient could bepotentially limited by the thermal stress suffered by thesestructures. The second effect occurs after the RF pulse endsand consists in the heat diffusion along the metallic slab andthe temperature decrease in the surface until the next RFpulse. As a result, there is an overall temperature rise on themetallic walls of the cavity from one pulse until the arrivalof the next. After many RF pulses the device temperatureincreases, the body expands and the resonant frequency shifts.This frequency shift will detune the cavities and might reflectthe RF power.All the Authors are with the Instituto de Fı́sica Corpuscular (IFIC,UV-CSIC), Valencia, Spain. This work was supported by the EuropeanUnion’s Horizon 2020 research and innovation programme under the grantagreement No 777431 (XLS CompactLight). This research was also sup-ported by the Valencian Regional Government VALi+D postdoctoral grant(APOSTD/2019/155).This entails that the RF pulse heating effects must betaken into consideration when designing high gradient RFaccelerating structures. There are several commercial codes(ANSYS [2], SIMULIA [3], etc.), based on Finite Differencesin Time Domain (FDTD) or Finite Elements Method (FEM),which allow a thermal analysis of the accelerating components.However, these software tools usually have a considerablecomputational cost in terms of time and hardware in orderto get accurate results.In this paper we describe a single procedure for analysingthe RF pulse heating within a thick metallic wall exposed to anRF pulsed high power electromagnetic field. First, in SectionII, the theoretical model employed for deriving analytical ex-pressions for the thermal heating induced inside the componentis discussed. Afterwards, in Section III the feasibility of usingthe 1D theoretical expressions for analysing the RF pulseheating phenomenon in 3D structures is explored by meansof performing the thermal analysis of an RF photoinjector.Finally, in Section IV, the main conclusions of this study areoutlined.II. THEORETICAL MODELIn this Section we present the theoretical background re-quired to derive the analytical formulas that allow to describethe temperature increase inside the device wall as a functionof time, caused by an electromagnetic RF pulsed signal. Atypical RF pulsed signal consists in a gated harmonic signal.To analyse this phenomenon, the component wall is modelledas a metallic layer with thickness L (see Fig. 1) that extendsinfinitely in the y and z dimmensions. The left side of thewall corresponds to the inner side of the device, i.e., wherethe RF electromagnetic fields are present. The right side of thewall is in contact with the cooling mechanism of the device,modelled as a convective heat exchange, which is characterizedby its heat transfer coefficient h and the fluid temperature T8.Assuming the metal has a finite electrical conductivity σ, theRF electromagnetic fields inside the cavity (vacuum side inFig. 1) penetrate into the conductor wall and are attenuatedexponentially in a characteristic length δ, known as the skindepth. The time harmonic RF electromagnetic fields insidethe metallic wall will decay exponentially according to theskin depth length δ b2σµ0ω[4], where µ0 is the vacuumpermeability, ω  2πf , and f is the RF frequency. As it iswell known, the RF electric field inside the metal will induce2electric currents which will dissipate power due to the Jouleeffect [4]. This dissipated power constitutes the heat sourcethat increases the temperature of the component walls.Fig. 1. Scheme of the device metallic wall. The inner surface (x  0)corresponds to the vacuum side inside the component, the outer surface (x L) corresponds to the cooling side modelled as a convective heat exchange.The slab extends infinitely in the y and z directions.As a consequence of the symmetry of the slab in the y-zplane, the RF electromagnetic fields only depend on the x-coordinate and the thermal problem is reduced to 1-D. The 1Dheat transfer equation can be solved analytically as an infiniteseries in some cases. In the case we are interested in, theheat sources are time-dependent since the RF electromagneticsignal is composed by a succession of pulses with a repetitionrate fp  1{Tp (Tp being the pulse repetition period). Duringthe pulse (characterized by ton), the electromagnetic fieldsoscillate harmonically in time, with an harmonic amplitudethat also varies with time depending on the characteristic shapeof the pulse. When the pulse ends there is a time lapse inwhich there are no heat sources in the wall until the start ofthe next one. In this work, two pulse shapes will be taken intoconsideration, namely, the square pulse, and the pulse withtransient, which is the transformation of the square pulse in aStanding Wave (SW) structure.A. Temperature increase during the pulseThe typical values of ton in pulsed RF accelerators is usuallywithin the range from several hundred nanoseconds to a fewmicroseconds. These time lapses are shorter than the typicaltimes that takes the diffusion mechanism to transfer the heatacross the wall from the vacuum side to the cooling boundary.Since a negligible amount of heat is transferred across thewall during the RF pulse, the heat exchange between theconvection medium and the right wall boundary will affectthe temperature only in the neighbourhood of such boundary.If we are interested in obtaining the temperature increaseduring the pulse, which only is noticeable in a length of a fewskin depths from the vacuum wall, the convection mechanismof the right wall can be dropped from the calculations byassuming a thermally isolated wall. This assumption simplifiesthe mathematical problem of solving the heat transfer equation,which allows to obtain a simple analytical expression for thetemperature increase during the pulse. Next, the solution of theheat transfer differential equation is provided for the squarepulse and the pulse with transient cases.1) Square pulse: The temperature increase ∆T is given by:∆T px, tq  g0t 8̧n1gnpπDnL q21  epπDnL q2t	cosπnxL	,0 ¤ t ¤ ton(1a)∆T px, tq  g0ton 8̧n1gnpπDnL q21  epπDnL q2ton	epπDnL q2pttonq cosπnxL	, t ¡ ton (1b)withg0 2αL1  e2Lδ	(2a)gn 8αL4L2   π2δ2n21  e2Lδ p1qn	(2b)α Rs|H||,0|22ρCe; D cκρCewhere H||,0 is the amplitude of the parallel RF magneticfield at the interface, Rs  1{pδσq is the surface resistance, κis the thermal conductivity, ρ is the density, Ce is the specificheat.2) Pulse with transient: Let us assume that the RF genera-tor provides a square pulse to the RF circuit. For a SW cavity,such square pulse will be transformed into a transient witha characteristic filling time τ for the cavity. In this case, thetemperature increase is given by:∆T px, tq  u0ptq  8̧n1unptq cosπnxL	, 0 ¤ t ¤ ton(3a)∆T px, tq  u0ptonq  12v0pt tonq 8̧n1vnpt tonq   unptonqepπDnL q2pttonq	cosπnxL	, t ¡ ton (3b)withu0ptq g02t  τ2etτ 12e2tτ 32unptq gn πDnL21  epπDnL q2t	 gn πDnL2 2τe2tτ  epπDnL q2t	 2gn πDnL2 1τepπDnL q2t  etτ	3vnptq gn1  etonτ	2 πDnL2 2τe2tτ  epπDnL q2twhere g0 and gn are the same as in eqs. (2a)-(2b).B. Average temperature increase after many RF pulsesIn order to obtain the average temperature increase in thewall after many RF pulses have been driven through the de-vice, two assumptions are taken to simplify the mathematicalprocedure of solving the 1D heat transfer differential equation.First, the volume heat sources in the wall are replaced by anequivalent superficial heat flux entering through the vacuumwall, which is obtained by integrating the total contribution ofvolume sources along the wall depth. This assumption givesgood results since the skin depth values in RF accelerators ismuch shorter than the wall depth and hence the volume dis-tribution of the heat sources is focused on the neighbourhoodof the vacuum wall. The second assumption considers that thetemporal variation of the heat source can be replaced by atime-constant heat source with the time-average value of theheat generated along the RF pulse sequence. For this case, thepresence of the cooling system cannot be neglected in the cal-culations and it is taken into account. Finally, the temperature,assuming as initial conditions a uniform temperature along thewall T0, is given byT px, tq  T8  qs,0κL κh	qs,0κx (4) 8̧n1Cn cos pλnxqepDλnq2tCn 1λnsin pλnLqrT0  T18s qs,0λ2nκ1  1   hLκcos pλnLqL2  sin 2λnL4λnwhereT18  T8  qs,0κL κh	λn tan pλnLq hκ(5)where qs,0 is the time-averaged total RF power dissipatedin the wall in terms of heat. Eq. (5) has to be solvednumerically to obtain the eigenvalues λn that allow us to getthe coefficients Cn. As it can be noticed in eq. (4), the functiondependence on time is of the form epDλnq2t. Thus, for asufficiently long time, the temperature in the wall will reach asteady state distribution and the summation can be neglected.III. THERMAL ANALYSIS OF AN RF ELECTRON GUNPHOTOINJECTORThe purpose of this section is to analyse the suitability ofapplying the 1D theoretical expressions (derived in SectionII) in order to obtain a first approximation of the temperatureincrease caused by the RF pulse heating phenomenon in 3Dcavities which are part of RF accelerating systems. In SectionII the temperature increase in a 1D metallic wall was studiedby considering separately two effects due to the RF pulseheating.First, the sharp temperature increase occurring during theRF pulse very close to the vacuum side of the wall. The pulseduration in typical RF accelerating devices is too short to alloweffective thermal diffusion across the wall. Therefore, this isa phenomenon depending only on the electromagnetic fieldintensity at the local point of the surface and thus the 3Ddetails of the device (including the cooling system) can beneglected in the analysis. Hence, the 1D theoretical modelpresented in Subsection II-A is expected to give good resultsfor realistic 3D devices.Second, the average temperature increase in the wall aftermany RF pulses are driven into the structure, which finallyevolves to a steady state case with a spatial temperaturedistribution along the wall. In this case, the 3D details ofthe geometry of the device might play an important role inthe final temperature distribution. As a consequence, furtherinvestigation must be carried out for this case in order toevaluate the goodness of using the 1D theoretical expressionsto approximate the RF pulse heating phenomenon in realis-tic 3D devices. To reach this aim, the average temperatureincrease at the steady state is analysed for an RF electrongun accelerating cavity, comparing the results obtained withthe Thermal Transient Ansys (TTA) numerical simulations andthe predictions of the 1D theoretical expressions.The device that we have studied is properly described in[6]. It consists of six accelerating cavities, each of them withlength λ{2 (being λ the wavelength of the RF electromagneticwave in free space), except the first one which is 0.6 times thelength of the others. The present RF gun is designed to operatein SW mode with the π-mode advance per cell operating ata frequency of f  11.994 GHz. The RF gun is connectedto the RF generator external circuit by means of a coaxialcoupler. The filling time of the RF gun is τ  112.5ns. Thegun is intended to operate in pulsed mode with an RF pulseduration of ton  400 ns and fp  400Hz repetition rate.We will focus on the thermal analysis of the full cells, inwhich the magnitude of the RF electric field is almost thesame for all of them. Thanks to this, the thermal analysis canbe reduced to the study of one single full cell. The schemeof the photoinjector cavity in TTA is depicted in Fig. 2. Thecavity radius is Rc  11.029 mm and the cavity length isLcav  12.498mm. The presence of a water cooling channelhas been considered, with square section (a  10mm), heattransfer coefficient h  1.2  104Wm2K1, and fluidtemperature T8  0C. The separation between the topof the cavity and the lower side of the cooling channel isL  15mm. The material for the bulk is copper. The initialtemperature is assumed to be uniform for all the cavity bulk,T0  0C. TTA simulations were carried out for several valuesof the RF electric field at cathode, E0.Now, it remains to estimate the maximum temperatureincrease at the steady state in the RF gun cavity using thecorresponding analytical expression of the 1D model (recallthe eq. (4)). Two aspects must be taken into considerationto explore properly this 1D formula to the case of the 3Dcavity. First, an equivalent heat transfer coefficient heq must4Fig. 2. Scheme of the RF gun cavity with water cooling channel analyzedwith TTA.be chosen for the 1D model in order to compensate thepossible difference between the area size of the pillbox innersurface, Aheat, where the heat sources are located; and thearea of the water cooling channel, Acooling, where the heattransfer due to the convection phenomenon occurs. In the 1Dmodel, it is implicitly assumed that the vacuum surface (wherethe heat sources are) and the cooling system surface havethe same area. As a general rule, for 3D devices, it is notexpected that both areas are equal. To adjust this mismatch,we propose to take in the 1D model an equivalent heat transfercoefficient defined as heq AcoolingAheath, where h representsthe heat transfer coefficient of the water cooling channel ofthe 3D structure. Regarding the area of the water coolingchannel, Acooling, only the area corresponding to the bottomand the lateral faces (neglecting the top face) is taken intoaccount. According to this, the area of cooling is given byAcooling  2πpL Rcqa 2πrpL Rc aq2pL Rcq2s. Thephysical reason that motivates neglecting the top face is thatthis surface has no direct line of sight with the heat sources andhence no significant effects in the cooling are expected. Thischoice might seem a rough approximation but in fact givessatisfactory results as we will see next. For the particular caseof this example, heq  5.087h. The second consideration isrelated to the fact that in the gun cavity the surface lossesare not uniform, whilst the 1D model assumes uniform powerlosses per unit area, qs,0. Consequently, a spatial averagepower losses per unit area coefficient qs,0,avg has to be defined: qs,0,avg  PlossAheat where Ploss is the total RF power lost byJoule effect in the walls of the cavity. The value of Ploss canbe calculated using either HFSS or other RF electromagneticcodes like SUPERFISH [5].In Fig. 3 the maximum temperature increase in the RF guncavity at the steady state is depicted for different values ofthe RF electric field amplitude at the photoinjector cathode.The results of both TTA simulations and the 1D model areincluded for comparison. As it can be observed, there is goodagreement between both data.Fig. 3. Maximum temperature increase in the RF gun at the steady state fordifferent values of electric field amplitude at cathode. Comparison betweenthe TTA simulations and the results of the 1D model.IV. CONCLUSIONSIn this paper, the temperature increase due to the RF pulseheating phenomenon has been analysed in RF acceleratorstructures. For this purpose, the 1D heat transfer differentialequation has been solved in a thick metallic slab for certainspecific cases, allowing to obtain simple analytic mathematicalexpressions. The peak temperature increase during the pulseis obtained for the cases of a square RF pulse and an RF pulsewith transient. Similarly, the average temperature increaseafter many RF pulses is also provided. In the latter case, thetheoretical model assumes that the device exchanges heat withthe environment by means of the convection mechanism. Thisallows us to consider the interesting case of a cooling systembased on a water turbulent flow through a pipe.The 1D analytical expressions for the temperature increasein the slab presented in this paper are aimed at improvingthe design procedure of RF accelerating cavities. This isaccomplished by reducing the computational time required toperform the thermal analysis with regard to using of 3D FEMcodes.Moreover, a 3D RF photoinjector has been analysed bothwith the 1D analytical formulas and the Ansys numerical sim-ulations. The results evidence that the 1D analytical formulasprovide an useful approximation to the temperature increasein the device. However, it must be remarked that the methoddescribed in this manuscript is intended to provide a quickapproximation to the temperature increase in the device, butof course it will not be as accurate as proper 3D numericalsimulations.REFERENCES[1] V. Dolgashev et al., ”Status of high power tests of normal conductingsingle-cell structures”,Proceedings of the 11th European Particle Accel-erator Conference, pp. 742-744, Genoa, 2008.[2] www.ansys.com[3] www.3ds.com/products-services/simulia[4] J. D. Jackson, Classical Electrodynamics, 3rd edition, John Wiley & SonsInc., New York, NY, 1999.[5] K. Halbach and R. F. Holsiger, ”SUPERFISH - A Computer Programfor Evaluation of RF Cavities with Cylindrical Symmetry”, ParticleAccelerators, vol. 7, pp. 213-222, 1976.[6] D. González-Iglesias et al., X-band RF photoinjector design for theCompactLight project, Nucl. Instrum. Meth. A, 1014, 2021.

Study of the RF pulse heating phenomenon in high gradient accelerating devices by means of analytical approximations

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Study of the RF pulse heating phenomenon in high gradient accelerating devices by means of analytical approximations

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