A study of dynamic Lorentz force detuning of 650 MHz βg¼0.9superconducting radiofrequency cavity

Abhay Kumar a,n, Arup Ratan Jana b, Vinit Kumar b

a Proton Linac and Superconducting Cavities Division, Raja Ramanna Centre for Advanced Technology, Indore 452013, Indiab Materials & Advanced Accelerator Sciences Division, Indore, India

a r t i c l e i n f o

Article history:Received 3 November 2013Received in revised form19 February 2014Accepted 1 March 2014Available online 13 March 2014

Keywords:Dynamic Lorentz force detuningStructural resonanceCavity stiffeningSuperconducting cavity

a b s t r a c t

The small bandwidth of superconducting cavities makes the study of dynamic Lorentz force detuningand its compensation indispensable in case of pulsed mode operation of high gradient accelerators. Inthis paper, we present the study of this detuning and also propose an optimized design for five cell650 MHz βg¼0.9 elliptic superconducting cavities, which will be used in the high energy section of the1 GeV H� linear accelerator for the proposed Indian spallation neutron source project, by suitablyinserting the inter-cell stiffeners. The paper presents a sequential design methodology which starts withstudy of static Lorentz force detuning and tunability; and progresses to find out the structural modes andrelated dynamic detuning values by performing transient structural dynamics calculations. Thedeveloped methodology is general in nature and can be used for a three dimensional model of anygeometry. The work will be useful for optimizing the design against dynamic Lorentz force detuning ofsuperconducting radiofrequency cavities of any shape.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Indian spallation neutron source (ISNS) is envisioned to facilitateneutron based multi-disciplinary research in the fields of condensedmatter physics, material science, chemistry, biology and engineeringsciences. The proposed design of the ISNS consists of a pulsed linearaccelerator (LINAC) that delivers 4 mA pulse current, 1 GeV H� ionbeam at a pulse repetition rate (PRR) of 50 Hz [1] and a duty cycle(percentage ratio of beam bunch length and the time period) of 10%.ISNS LINAC will use superconducting radio frequency (SRF) cavities toavail several advantages such as less power dissipation on the cavitywall and the possibility of operating at larger beam aperture radiusthat allows higher beam current to be accelerated [1,2]. Two sets ofmulti-cell elliptic cavities will be used—one set for medium energythat will accelerate the H� beam from 200MeV to around 500MeV,and the other set for high energy that will accelerate the beam from500MeV to 1 GeV [3–5]. Electromagnetic design of 650MHz, five cellSRF cavity of the high energy section has been recently optimized for aparticle velocity of 0.9 times that of light (βg¼0.9) [1].

In an RF cavity, electromagnetic field induces surface current andsurface charges on the wall of the cavity. This in turn, generates aLorentz pressure on the cavity surface which, depending on the RF

pulse repetition rate, sets up mechanical vibrations in the structure. Asa consequence, the resonating frequency of the cavity starts changingdynamically. This detuning is called dynamic Lorentz force detuning(LFD) [6].

The detuning changes the impedance of the cavity and if it isnot compensated swiftly, then the cavity would reflect a fraction ofthe input power due to loss of impedance matching with thepower source [6]. The half power bandwidth is inversely propor-tional to the quality factor; thus the detuning tolerance of the SRFcavities is extremely limited [7].

In order to reduce the magnitude of the dynamic LFD; the SRFcavities of high gradient pulsed accelerators are stiffened byplacing ring stiffeners [8] between the cells as well as betweenthe end cell and the end cover of a vessel that acts as a 2 K liquidhelium bath for the niobium cavity. This vessel is made of titaniumand is generally known as helium vessel.

We performed a literature survey of the methodologies used indynamic LFD studies of SRF cavities. Mitchel et al. [8] performedthe dynamic LFD studies for SRF cavities of spallation neutronsource (SNS) using the computer codes ABAQUS™ for calculationsof structural deformations, and Poisson Superfish [10] for calcula-tion of electromagnetic modes and fields. Lin et al. showed thatANSYSTM can be used as a tool for the coupled field analysis for thedesign of RF cavities [9]. Later, the same tool was used by them toperform coupled field simulation of 500 MHz SRF cavity tocalculate frequency shift due to cool-down using MSC/PATRANTM

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Nuclear Instruments and Methods inPhysics Research A

http://dx.doi.org/10.1016/j.nima.2014.03.0050168-9002/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: +91 731 2442267; fax: +91 731 2442255.E-mail addresses: abhay@rrcat.gov.in, abhayk99@gmail.com (A. Kumar).

Nuclear Instruments and Methods in Physics Research A 750 (2014) 69–77

as the pre-processor to build the finite element model [10]. Lositoet al. also showed that ANSYSTM can be used to perform coupledfield simulation including Lorentz force detuning studies [11].They applied this analysis tool on a 352 MHz βg¼0.7 four cellSRF cavity and calculated static Lorentz force detuning. Later, SunAn [12] developed a simulation method by combining thestrengths of Poisson Superfish in electromagnetic domain andANSYS™ in the structural domain and showed it to be a precisetool for simulating changes in the cavity frequency and fieldflatness due to the elastic deformations by performing measure-ments on the cavity. This procedure was implemented to performdynamic Lorentz force detuning studies of 700 MHz, βg¼0.42elliptic superconducting RF cavity for the LINAC of Proton Engi-neering Frontier Project (PEFP) [13]. Bandyopadhyay et al. [14]reported the use of ANSYSTM in electromagnetic, thermal and struc-tural analysis of LINAC cavity. Recently, the elliptic SRF cavity of CornellEnergy Recovery LINAC was optimized for stiffening for minimizingthe sensitivity against helium pressure fluctuation using ANSYS™ byperforming elastic static displacement calculations in the structuralmodule and resulting change in the electromagnetic mode in the highfrequency electromagnetic module [15]. Static LFD calculations werealso performed where the Lorentz pressure used in the calculation ofstructural deformation was calculated using the electromagnetic fieldsin ANSYS™ [15]. We have also preferred a single software environ-ment of ANSYS™ with its seamless programming interface calledANSYS™ parametric design language (APDL). We generated a fullycompatible finite element mesh at the interface of structural mesh inniobium and electromagnetic mesh in vacuum for ensuring one-to-one correspondence between the calculated electromagnetic fieldsand applied Lorentz pressure on the structural elements. This alsoensured an accurate data exchange at every time step of analyses inthe high frequency electromagnetic and transient dynamics modules.

The boundary stiffness i.e. the stiffness of the helium vessel atthe extremities of the cavity plays a very important role inreducing the Lorentz force detuning [1]. The choice of stiffenerlocation was optimized by varying the boundary stiffness duringthe design studies of elliptic SRF cavities of SNS [8]. In addition, thechoice of two stiffener rings was considered in the design of SNScavities and also in PEFP cavity [8,12]. We started with the study ofthe time domain and frequency domain variation of the Lorentzpressure on the cavity wall. This knowledge was augmented by theknowledge of mechanical modes of the cavity structure at differ-ent stiffener locations and was very helpful in allowing us to workwith single stiffener design and fixing the helium vessel designprior to the study of the dynamic response.

This paper presents a sequential approach for optimizing thestiffener location for SRF cavities of high gradient pulsed accel-erator while quantitatively discussing the significance of con-straints that lead to the selection. First, the temporal variation ofthe Lorentz pressure during a pulse and its Fourier series expan-sion are obtained. Then, the finite element model of the cavity isdiscussed. After this, we discuss the limit imposed on the choice ofstiffener locations from the considerations of feasibility of tuningthe cavity and the field flatness. Finally, the results are presented,discussed and concluded with a selection of stiffener location.The work pertains to 650 MHz βg¼0.9 elliptic SRF cavity of highenergy section of ISNS. The procedure developed here is general innature and can be applied to the medium beta cavities as well as tothe low beta non-elliptic cavities.

2. Dynamic Lorentz force detuning

The Lorentz pressure P on the cavity's internal surface is givenby [2,6,8],

P ¼ 14ðε0E2�μ0H

2Þ ð1Þ

E and H are electric field amplitude and magnetic field intensityamplitude respectively.

The Lorentz pressure acts outward near the equator andinward near the iris (Fig. 1). It can be seen that outward pressureof lower magnitude acts over a larger area in the equator regionwhereas inward pressure of larger magnitude acts over a smallerarea in the iris region. If gradually applied, this distribution willreduce the length of each cell of the cavity due to larger momentproduced by the forces in the iris region about the equator ofthe cavity.

The frequency change is proportional to the square of electro-magnetic fields. The relation [6,8] between detuning Δf andaccelerating gradient Eacc is given by:

Δf ¼ �KLE2acc ð2Þ

KL used in (2) is known as Lorentz force detuning coefficient andhas the unit of Hz/(MV/m)2 and the accelerating gradient is18.6 MV/m for the cavity.

Eq. (1) shows that the Lorentz pressure on the cavity walls isproportional to the square of the electromagnetic field amplitude.Therefore, Lorentz pressure pulse shape can be readily derived fromcavity voltage pulse shape. The ratio of the dynamic displacement

Fig. 1. Typical Lorentz pressure distribution on cavity wall.

A. Kumar et al. / Nuclear Instruments and Methods in Physics Research A 750 (2014) 69–7770

amplitude and the static displacement, known as dynamic amplifica-tion factor, depends on the frequency ratio of harmonics of excitationforce and the structural mode frequencies. The dynamic amplificationfactor is larger than unity if the frequency ratio lies between 0 and

ffiffiffi2

p

for a lightly damped system [16]. This broad range indicates apossibility of a dynamic amplification of the static Lorentz forcedetuning. The dynamic amplification factor is less than unity if thefrequency ratio is larger than

ffiffiffi2

p[16]. The magnitude of dynamic LFD

depends on many parameters like frequency ratio, damping and pulsetime structure apart from the accelerating gradient; therefore we havenot represented dynamic LFD as a function of the acceleratinggradient. While comparing static and dynamic LFD magnitudes, itappears more appropriate to show their magnitudes rather than thecorresponding value of KL.

3. Lorentz pressure pulse shape

A typical temporal shape of the cavity voltage due to input RFpower during a pulse is shown in Fig. 2. H� ion beam is injected inthe flat top duration of 2 ms. The rising and falling part of the RFpulse shape is determined by the loaded quality factor of thecavity i.e. QL(�5�106) [1].

It is evident that the Lorentz pressure is periodic and willrepeat with the same frequency as that of the input RF pulse. Themechanical vibration of the cavity wall in response to the time-dependent Lorentz pressure P can be analyzed in terms of itsnatural modes of vibration. Let the cavity surface be parameterizedin terms of appropriately defined variables u and v. In thisnotation, the cavity surface has parametric representation givenby x¼x (u,v), y¼y(u,v) and z¼z(u,v) and displacement at differentpoints on the cavity wall surface due to vibration is denoted byχ(u,v,t). Decomposing into Eigen modes of vibration, we can writeχðu; v; tÞ ¼∑mϕmðu; vÞηmðtÞ, where ϕmðu; vÞ denote the mth ortho-normal Eigen mode shape function and ηm denotes the amplitudeof the mth mode. The orthonormality condition for the Eigenmode shape functions is given by

∬ϕmðu; vÞϕnðu; vÞdS¼ S δmn; ð3Þ

where the integration is carried over the surface of the cavity, S isthe surface area, and δmn¼1 if m¼n, and δmn¼0 if man.

As shown in Appendix A, the equation for the evolution of ηm islike the forced simple harmonic damped oscillator equation, and is

given by

€ηmðtÞþ2ξm _ηmðtÞþω20mηmðtÞ ¼

1M∬ Pðu; v; tÞϕmðu; vÞdS; ð4Þ

where the integration is carried over the surface of the cavity, ‘ � ’and ‘ � � ’ denote the first and second derivative respectively, ωom isthemth normal mode frequency of the structure andM is the massof the cavity. Here, ξm is the damping for the mth mode, expressedas a fraction of critical damping [16]. The space dependent andtime dependent parts of the pressure function P(u,v,t) are separ-able, and therefore Eq. (3) can be simplified by expressing P(u,v,t)as

Pðu; v; tÞ ¼ FðtÞ℘ðu; vÞ ¼ FðtÞS

∑nAnϕnðu; vÞ; ð5Þ

where the space dependent part of the pressure function, which isdecided by the particular electromagnetic mode excited in thecavity, has been expanded in terms of the Eigen modes ofmechanical vibration. The temporal dependence is present in thefunction F(t), and is described by the pulse shape of RF powerinside the cavity. Substituting Eq. (5) in Eq. (4), we get

€ηmðtÞþ2ξm _ηmðtÞþω20mηmðtÞ ¼ FðtÞAm ð6Þ

Since F(t) is a periodic function, we can express it as a Fourierseries and write F(t)¼∑nf n sin ðnωtþθnÞ, where ω¼2π times therepetition rate of the pulses. This model ensures that Eq. (6) isconverted into simple homogeneous equation for all other fre-quencies except nω. The forced oscillation will appear only at thestructural mode frequencies. It also indicates that resonance willoccur if ω0m ¼ nω is satisfied for a particular m and n. The discreteparticipating frequencies are the integral multiples of the pulserepetition rate of 50 Hz. Fig. 3 shows the frequency domainrepresentation of the forcing function. It is seen that the structuralmode frequencies higher than 250 Hz have progressively lowercontributions in the total response. Based on this analysis, animportant conclusion can be drawn that there is a radical reduc-tion in amplitudes corresponding to the higher participatingfrequencies if the structural mode frequencies are higher than250 Hz.

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

0.0

0.5

1.0

1.5

2.0

pulse shape- pressure

f(t)| N

orm

F(t)|

Nor

m

t (ms)

pulse shape- cavity voltage

Fig. 2. The normalized amplitude of the Lorentz Pressure Pulse (in black) and theinput RF pulse (in blue) as a function of time are shown by the black line and blueline respectively. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

0 200 400 600 800 1000 1200 1400 16000.0

0.2

0.4

0.6

0.8

1.0

|F(ω

)| Nor

m

ω/2π (Hz))

Fig. 3. Fourier spectrum of the normalized Lorentz Pressure Pulse Shape forcalculation of dynamic LFD; |F(ω)|Norm is the normalized amplitudes correspondto the participating frequencies.

A. Kumar et al. / Nuclear Instruments and Methods in Physics Research A 750 (2014) 69–77 71

4. The finite element model

The internal geometry of the five cell elliptic cavity has beenoptimized to obtain the maximum value of allowed accelerationgradient, keeping the resonant frequency for the TM010-π modefixed at 650 MHz [1] and is presented in Fig. 4 and Table 1. Thewall thickness of the cavity has been kept at 4 mm on the basis ofstudies performed on a similar structure [17].

It is evident from Eq. (4) that the Lorentz pressure can excite onlythose structural modes in the cavity, which have their shape functionsimilar to the Lorentz pressure distribution on the surface. Therefore, a51 sector in azimuthal direction with full cavity length was modeledfor transient analysis to capture all longitudinal modes. The stiffenerswere modeled with 3 mm thickness of niobium. The material proper-ties used in the calculations have been taken from Ref. [21]. Thelongitudinal modes of vibration depend on the boundary stiffness ofthe structure. Mitchell et al. [8] modeled the boundary stiffness bysprings to simulate the helium vessel. This boundary stiffness can besplit into two parts—the bending stiffness of the end cover of theheliumvessel and the membrane stiffness of cylindrical portion. Out ofthis, the former is much smaller than the later and would directlyaffect themechanical mode frequency. In our model, we havemodeledthe full helium vessel without its tuner and joined with the cavity neariris (Fig. 5); thus the bending stiffness of the end cover is correctlyrepresented. The mode shapes of the excited longitudinal modes showthat the cylindrical portion remains stationary and the mode shapes

are generated due to bending of the end cover alone, therefore theabsence of tuner would have least effect on the final results.

The helium vessel is a cylinder with internal diameter of 504 mm.Its end cover is a combination of torus and flat geometries. It wasfound in a previous study that the increase in the stiffness of this SRFcavity along with its helium vessel decreases the static LFD [1].Therefore, the helium vessel stiffness was increased by increasing itswall thickness from 4mm to 5mm [18] and increasing the internaltorus radius of the end cover from 35mm to 120mm. The majorcontribution in the increase of stiffness came from the highercurvature of the end cover; thus this change increased the stiffnessappreciably with some decrease in the mass for the same thickness.The frequency of natural modes of vibration is proportional to thesquare-root of ratio of stiffness and mass [16]; therefore this wouldshift them upwards. This would make a favorable condition forlowering the dynamic response as the relative importance of higherharmonics of the Lorentz pressure pulse is low.

The welding of the titanium end cover of helium vessel withniobium beam pipe requires a transition piece of 55Ti–45Nb inorder to facilitate joining [19]. The 55Ti–45Nb transition piece hasbeen limited to the flat portion of the end cover to avoid metal-forming of 55Ti–45Nb. Therefore, the flat part of the end cover hasto be split into two parts. The one joined with cavity beam pipewill be made of niobium and will be a part of cavity end group. Thesecond flat part will be made of 55Ti–45Nb and this will be weldedto the niobium part at one end and torus part of the helium vesselend cover at the other end (Fig. 5). In order to restrict the loss ofstiffness arising from a lower elastic modulus of 55Ti–45Nb, thewidth of the annular plate was kept smallest minimum necessaryvalue allowed on the basis of manufacturing considerations.

ANSYS™ uses vector finite element method (VFEM) for electro-magnetic computations to enforce continuity of the electric field onthe inter-element boundaries to ensure accuracy [20]. The meshdensity was kept fine near the surfaces and on the axis to ensureaccurate calculations of surface fields and fields on the axis. Secondorder hexahedral elements were used in both the analysis domains.

ANSYS™ parametric design language (APDL) was used tocalculate the Lorentz pressure using Eq. (1) after scaling thenormalized surface electromagnetic fields from an electromag-netic modal analysis. The scaling factor was obtained on the basisof the accelerating gradient. This pressure was subsequentlyapplied in the structural analysis for obtaining the deformedgeometry of the cavity from which the changed frequency ofelectromagnetic mode was obtained.

Rayleigh proportional damping model, in which damping isassumed to be a linear combination of the mass and stiffnessmatrices has been used during the calculations. Following equa-tion gives the damping matrix C in terms of mass matrix ℳ andstiffness matrix K [22,23]:

C ¼ αℳþ K : ð7Þ

here, α is mass damping coefficient and β is stiffness dampingcoefficient and they are given by:

α¼ 2ωmωn

ωmþωn; β¼ 2

ωmþωnð8Þ

Fig. 4. The internal geometry of cavity [1].

Table 1Values of internal geometry parameters [1].

Parameters Value for mid-cells Value for end half-cells

Riris 50.00 mm 50.00 mmReq 199.925 mm 199.925 mmL 103.774 mm 105.8 mmA 83.275 mm 83.275 mmB 84.00 mm 84.00 mma 16.788 mm 16.788 mmb 29.453 mm 29.453 mmα 841 82.751

Fig. 5. The model of cavity with helium vessel.

A. Kumar et al. / Nuclear Instruments and Methods in Physics Research A 750 (2014) 69–7772

The amount of damping present in the system for all the modeshas been kept at 0.3% of critical damping which is an experimen-tally determined value during the development of similar cavities[8]. The critical damping corresponds to the smallest value ofdamping coefficient above which the transient response justattains an exponentially decaying non-oscillatory state [16]. Here,ωm has been taken as the first mode that has maximum contribu-tion in transient displacements and ωn has been taken as the lastcontributing mode [22].

In order to get steady amplitude during the RF pulse, thestructural transient analysis was run for 4 s (200 pulses) by modesuperposition method. The time step size for dynamic analyseswas kept at 50 μs which is less than one twentieth of the timeperiod of the largest frequency of natural modes of vibration thatmay get excited during the pulse [24].The electromagnetic reso-nating frequency and corresponding dynamic LFD values arecalculated at every time step of the 200th pulse by using thedeformed geometry at that instant.

5. Effects of tunability and field flatness

Figure-24 and 25 of Ref. [1] show that the static LFD decreasesas the stiffener position moves radially outward but tuning thecavity becomes impossible if the stiffeners are placed towards theequator. The stiffeners can be positioned without having a sub-stantial effect on the tunability of the cavity (Fig. 6) [1] if they areplaced within the common tangent region of elliptic curvatures ofindividual cells. The stiffener location can be varied in this regionwhile optimizing against the dynamic effects.

The constraints for restricting the radial position of stiffeneralso come from field flatness [25] defined as

η¼ 1� Emax�Emin

ð1=NÞΣN1 ðEiÞ

!; ð9Þ

Here, Ei is the maximum electric field amplitude in ith cell and Emax

and Emin denote the maximum and minimum of these amplitudesrespectively and N represents the number of cells in a cavity. A value ofη close to 100% is desired to maintain a good synchronism of theaccelerating particle with the electromagnetic field.

The slow tuner (mechanical tuner) is operated to tune thecavity after its integration with the helium vessel. The stiffenerlocation should be such that the field flatness remains largelyunaffected from these tuning operations. ηZ98% has been sug-gested as the design goal in the electromagnetic design of the endcell [26]. The requirement for SNS cavity field flatness was set to

be more than 92% [27]. The achieved field flatness in the electro-magnetic design of this cavity is 99.4% [1]. Therefore, the stiffenerlocation that ensures a reasonable axial electrical field flatness of98% when the slow tuner is moved by 1 mm, (i.e. a tuning of�100 kHz) could be a safe and tolerant design choice.

The field flatness considerations favor the stiffeners betweenhelium vessel and end-half-cells and that between the mid-half-cells at the same radial position as the asymmetric placement ofstiffeners produces difference between the change in the opti-mized length of the end cell and that of the mid cells at the time oftuning which may lead to the loss of field flatness. It was alsoobserved that even a uniform radial location for all the stiffenerrings higher than a mean radius of 122.5 mm deteriorates the fieldflatness value to below 98.2% when the slow tuner is displaced by1 mm (Fig. 7). The consequence of asymmetric placement ofstiffeners in the end cell and mid-cell will be discussed later inSection 6.

6. Results

The static LFD without a stiffener ring is �1.3 kHz. This value istoo high for the design of an SRF cavity for a pulsed accelerator andhence, it is evident that the stiffeners are mandatorily required.Table 2 summarizes the results.

The structural modes are well above 250 Hz for the stiffeners'radial position above 116.5 mm and stability is observed indynamic LFD values. This confirms our earlier prediction basedon Fourier expansion of the Lorentz pressure pulse. For thestiffener's radial position at 122.5 mm, we can observe thatalthough the first two structural modes are close to multiples of50 Hz, the dynamic LFD has not shot up. Table 2 shows that thedynamic LFD is not only small but the location is also less sensitiveif the stiffeners' radial location is at a mean radius of 119.5 mm.

Stiffener location at mean radii of 80.5 mm, 101.5 mm and113.5 mm show resonance and the dynamic LFD figures show a jump.Very high resonance is observed for stiffener mean radius at 80.5 mmas the first natural mode of vibration is at 150 Hz which coincides withLorentz pressure harmonics having a high coefficient.

The dynamic LFD during a time period after attainment ofsteady state is shown in Fig. 8.

The important figure of merits for mean stiffener radius of119.5 mm is given in Table 3.

The dynamic LFD is a result of transient displacements of theentire cavity structure which includes shape deformations andlength shortening [15]. We found that it is the second one which ismore important. If we plot normalized change in cavity lengthFig. 6. Tuned LFD when stiffener position is varied for a fixed tuning displacement [1].

95.5

96.0

96.5

97.0

97.5

98.0

98.5

99.0

99.5

100.0

70 80 90 100 110 120 130 140 150

Fiel

d Fl

atne

ss (%

)

Stiffener Radial Position (mm)

Fig. 7. Field flatness reduction with increase in stiffeners' radial position.

A. Kumar et al. / Nuclear Instruments and Methods in Physics Research A 750 (2014) 69–77 73

along with normalized dynamic LFD (Fig. 9), then the associationof change in cavity length and the corresponding dynamic LFDbecomes clear. Therefore, in order to estimate the dynamic LFD, aplot of transient displacement of end of cavity beam pipe isenough. The scale factor between change in the cavity lengthand the dynamic LFD is 17075 Hz/mm for all stiffener locations.This scale factor and the cavity end sensitivity are not identical asthe displacement of piezoelectric transducer (PZT) produces adifferent deformation pattern as compared to that produced bythe Lorentz pressure.

The transient displacements of the cavity extremities indicatethe expected LFD variation during the pulse. Figs. 10–13 show thechange in half of the cavity length for four cases—stiffener at themean radius of 119.5 mm and at mean radii of 113.5, 101.5 and 80.5for showing resonant coupling. Since the time-averaged Lorentzpressure always produces an effective compression in the cavitylength; the displacements shown in the figures are asymmetric.

7. Discussion and conclusion

The core of the ISNS project i.e. the 1 GeV H- linear accelerator willbe a pulsed machine. Therefore, the pulsed Lorentz pressure, repeatingat a 50 Hz frequency, will result in a time varying oscillations in thecavity structure. The consequence is the dynamic LFD of the cavity.Since the SRF cavities have high quality factor and thus an extremelynarrow bandwidth (�120 Hz); this necessitates that this detuning iskept small and manageable by the movement of PZT [6]. It may benoted that the power requirement increases by 6.25% if the uncom-pensated detuning is half of the bandwidth [6].

The static LFD study gave us an idea of the stiffener position fromthe consideration of tunability. Thereafter, a systematic and sequentialapproach towards the estimation of dynamic LFD and stiffenerposition has been developed. First, we performed the Fourier analysisof the periodic pressure pulses to obtain excitation force in terms ofdiscrete participating frequencies and their corresponding relativeamplitudes. The important conclusion of this step is that there is aradical reduction in amplitudes corresponding to the higher partici-pating frequencies if the structural mode frequencies are higher than250 Hz. Therefore, the system should be modified by changing theradial location of the stiffener rings in such away that the participatingstructural mode frequencies are kept above 250 Hz. In addition to this,the resonances should be avoided as the dynamic amplification for alightly damped system could still be high.

The next step is to calculate the participating structural modes.This can be calculated in two ways—first from the knowledge thatthe transient excitations coming from the application of Lorentzpressure can produce longitudinally symmetric modes only or byextracting all the modes in a full model and then performing modesuperposition analysis by selecting only one mode at a time. Therelative magnitude of displacements with respect to the displace-ments calculated with a large number of modes gives an idea ofthe participating modes. The first one is an efficient choice as itallows a smaller model to be used in the analyses.

Then, we calculated the transient displacements for 200RFpulses to get the stabilized transient response using the modesuperposition method. The computed displacements during the

Table 2Summary of transient response of the cavity.

No. Stiffenermean radius(mm)

Frequency oflongitudinally symmetricmodes (Hz)

StaticLFD(Hz)

DynamicLFD (Hz)

Cavitystiffness(kN/mm)

1 122.5 296, 551, 757, 882, 925 �660 �794 17.842 119.5 280, 534, 745, 787, 932 �654 �652 15.673 116.5 266, 515, 733, 877, 939 �649 �656 13.854 113.5 252, 493, 721, 876, 943 �647 �1229 12.275 110.5 239, 471, 711, 874, 945 �648 �648 10.896 107.5 227, 449, 701, 869, 942 �653 �923 9.707 104.5 216, 427, 691, 862, 937 �659 �800 8.688 101.5 205, 407, 680, 853, 929 �669 �1120 7.789 98.5 195, 387, 669, 843, 918 �682 �670 7.01

10 95.5 186, 369, 657, 832, 906 �699 �877 6.3411 92.5 178, 352, 645, 820, 892 �719 �815 5.7512 89.5 170, 336, 633, 807, 878 �742 �916 5.2313 86.5 163, 322, 621, 795, 864 �768 �1098 4.7814 83.5 156, 308, 609, 783, 850 �796 �1574 4.3915 80.5 150, 296, 597, 771, 836 �828 �11480 4.0416 77.5 144, 284, 586, 760, 823 �862 �1177 3.7317 74.5 139, 273, 575, 749, 810 �898 �781 3.45

0

0.2

0.4

0.6

0.8

1

1.2

-100

0

100

200

300

400

500

600

700

3.98 3.985 3.99 3.995 4

Nor

mal

ized

Pul

se S

hape

Dyn

amic

LFD

(Hz)

Time (s)

-1 x Dynamic LFDNormalized Pulse Shape

Fig. 8. Dynamic LFD with mean radial position of stiffener at 119.5 mm.

Table 3Figures of merit of design with mean radial position of stiffener at 119.5 mm.

No. Figures of merit Value

1 Required total tuning range for dynamic LFD compensation 5.3 μm2 Field flatness after 1 mm movement of tuner 98.58%3 Cavity end sensitivity while tuning ( Δf

Δcavity) 208.29 kHz/mm

4 Tuning efficiency ( ΔfΔtuner) 123.83 kHz/mm

Fig. 9. Normalized dynamic LFD and change in cavity length with Stiffener's meanradial position at 119.5 mm during a single pulse.

A. Kumar et al. / Nuclear Instruments and Methods in Physics Research A 750 (2014) 69–7774

last pulse were taken for dynamic LFD computation. Since, theprocess requires several exchanges between the structural andelectromagnetic modules; ANSYS™'s programming interfaceproved to be very useful for the analyses.

The radial position of 119.5 mm is selected for the stiffener ring.The dynamic LFD value is very close to static LFD value for this positionas the first structural mode is well above 250 Hz and is also away fromresonances. Secondly, the dynamic LFD value appears to be stable with

Fig. 10. Longitudinal Displacements with mean radial position of stiffeners at 119.5 mm.

Fig. 11. Resonant conditions with mean radial position of stiffeners at 113.5 mm.

Fig. 12. Resonant conditions with mean radial position of stiffeners at 101.5 mm.

A. Kumar et al. / Nuclear Instruments and Methods in Physics Research A 750 (2014) 69–77 75

respect to slight change in the position of stiffener ring if a downwardposition tolerance is applied during manufacturing. Therefore, theselection has the ability to tolerate the difference between the actualfabricated assembly and the FE model arising from the manufacturingerrors. It also satisfies all other constraints like field flatness andtunability. Operating the cavity at lower pulse repetition rate duringthe trials and commissioning stages will be a favorable condition forthe dynamic LFD as the frequency ratios of excitation force harmonicsand mechanical mode frequencies will become smaller. In addition,the normalized amplitude of harmonics that are near the mechanicalmodes will be very small.

Stiffener placement at a mean radius of 110.5 mm is notselected in spite of a lower magnitude of dynamic LFD as highvalues are observed on both the sides of this location. It can beobserved that the dynamic LFD is 11480 Hz if the stiffeners areplaced at a mean radial position of 80.5 mm. This was the clearsignature of a dynamic resonance with a participating frequency ofthe excitation force that has a high value of normalized amplitude.

An asymmetric stiffener placement i.e. different radial positionof the end-cell stiffener as compared to that of the mid-cellstiffener was also considered. Asymmetric placement of stiffenersproduces large difference between the change in the optimizedlength of the end cell and that of the mid cells at the time oftuning leading to the loss of field flatness. A higher radial locationof end cell stiffener as compared to that of the mid cell stiffenersreduce the required displacement for compensation but alsodeteriorate the field flatness substantially. For example, the fieldflatness deteriorates to 95% when the cavity is extended by 1 mmfor tuning for radial location of end cell stiffener at 121 mm andthat of the mid cell stiffeners at 91 mm.

This should be noted that we need to compensate for thedynamic LFD only during the flat top of the Lorentz pressure pulseshape. Therefore, a pre-elongation of cavity using the slow tunercan be used to bring down the range of PZT. When a limited pre-elongation of slow tuner is applied such that the incident power isnot significantly reflected (�1.6 μm), the partially compensateddynamic LFD, reduces to 454 Hz only. The tuning force requiredwill be less than 60 N for the PZT and the total range of PZT will beless than 4 μm [28]. PZT operated at cryogenic temperatures canprovide several tens of Newton combined with a fast response(o100 ms) [6]; therefore this is not a concern.

On the basis of the design study, a design schematic isproposed in Fig. 14.

The study revealed the design bases of selection of stiffenerposition of elliptic multi-cell SRF cavities of high gradient pulsed

accelerators. The design approach is sequential and can be used toarrive at a design for any SRF cavity. The limitations coming fromelectromagnetic design of the cavity and PZT have been ade-quately addressed. Since the design is an iterative process, changesare inevitable and the approach developed here will be useful forswiftly evaluating the effect of the changes. The Lorentz pressurecalculation subroutine is general in nature and can be used for athree dimensional model of any geometry. The work will bedirectly useful for designing other SRF cavities.

Acknowledgements

We acknowledge the encouragement received from P D Guptaand S B Roy for publishing this work. We thank Vivek Bhatnagarfor discussing the fabrication related constraints. Vivek Bhasin of

Fig. 13. Resonant Conditions with mean radial position of stiffeners at 80.5 mm.

Fig. 14. Proposed design.

A. Kumar et al. / Nuclear Instruments and Methods in Physics Research A 750 (2014) 69–7776

Bhabha Atomic Research Centre, Mumbai participated in discus-sions on the choice of appropriate time history solution methodand structural damping. Vikas Jain provided useful suggestions forimproving the manuscript.

Appendix A

Here we briefly discuss the proof of Eq. (4). We start with theequation for free oscillation of the cavity surface given in Ref. [16]:

€ηmðtÞþ2ξmω0m _ηmðtÞþω20mηmðtÞ ¼ 0: ðA1Þ

Multiplying Eq. (A1) with ϕmðu; vÞ and summing over all mgives

€χ ¼ �2∑mξmω0mϕmðu; vÞ _ηmðtÞ�∑

mω2

omϕmðu; vÞηmðtÞ: ðA2Þ

Physically, the first term of right hand side is damping forceterm, and the second term is restoring force term. Eq. (A2) is validfor an element of area dS at the location (u,v) on the surface in theabsence of Lorentz pressure. Each term on the right side is theforce on element dS divided by its mass.

Now, we add the external force responsible for the forcedvibration. On the right hand side, it should only make one change—i.e. add the term ½Pðu; v; tÞdS�=½mðu; vÞdS�, where mðu; vÞ is massper unit area. This gives us

€χ ¼ �2∑mξmω0mϕmðu; vÞ _ηmðtÞ�∑

mω2

omϕmðu; vÞηmðtÞþPðu; v; tÞmðu; vÞ :

ðA3ÞNow, multiplying the above equation with ϕnðu; vÞ, integrating

over entire area and using the orthonormality condition, we getZ∑mϕmðu; vÞ €ηmðtÞϕnðu; vÞdS¼

�2Z

∑mξmω0mϕmðu; vÞ _ηmðtÞϕnðu; vÞdS

�∑mω2

omϕmðu; vÞηmðtÞϕnðu; vÞdS

þZ

Pðu; v; tÞmðu; vÞ ϕnðu; vÞdS ; ðA4Þ

where S is the total area of the cavity surface. If mass per unit areais constant; and it is assumed that this is not getting affected dueto Lorentz pressure induced deformation of the cavity; then wecan take mðu; vÞ outside the integration and we get Eq. (4).

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