Side Coupling Cavity

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EU contract number RII3-CT-2003-506395 CARE-Report-2008-011-HIPPI

Development of Side Coupled Cavities

J.-M. De Conto, J.-M. Carretta, Y. Gomez-Martinez, R. Micoud

Laboratoire de Physique Subatomique et de Cosmologie (LPSC)(Universit Joseph Fourier, CNRS/IN2P3, Institut National Polytechnique Grenoble)

53, avenue des Martyrs38026 Grenoble Cdex - France

Abstract

Side coupled Cavities are good candidates for proton accelerations in the 90-180MeV range, as it has been first proposed for the CERN LINAC4 project. This is not anew technology used, for example, for the Spallation Neutron Source (SNS). Thegoal for HIPPI was the development of technical knowledge about it. We summarizehere the theoretical and experimental studies. This work is not completed and tuningprocedures are under study and must be proven on the prototype.

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1 Side Coupled Cavities

We present here a brief summary of side coupled cavities. A Side Coupled Linac ismade of a lump chain of resonant cavities, alternatively accelerating and coupling, asshown on the drawing given Figure 1.1. This set of cavities is equivalent to a lumpedchain of RLC circuits (figure 1.2).

Figure 1.1: Accelerator tank (figure CERN) as initially proposed for the high energypart of LINAC4

Figure 1.2: Equivalent circuit

100MeV

120MeV

140MeV

160MeV

0.428 0.462 0.492 0.52D (cm) 28.83 28.99 28.78 28.86L (cm) 9.108 9.838 10.48111.06g (cm) 2.6 3.1 3.4 3.8g/L 0.285 0.315 0.324 0.343Q 20795.122120.823003 23884.4ZT2(M /m)

34.863 37.623 39.77141.486

T 0.893 0.894 0.897 0.896Ep/Eo 5.62 5.35 5.374 5.249Ep (Kil.,3.5MV/m)

0.799 0.761 0.764 0.746

ZT2 (3%Coupling)

25.56 27.61 29.19 30.45

Table 1.1: SCL parameters for linac 4 (courtesy CERN, Eva Benedico-Mora)

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If each cavity has the same resonnant frequency 0, and considering the classical couplingfactor k, the eigenmodes i X

r

and the eigenfrequencies i of such a structure are obtainedfrom the following relations:

N i

k

k k k

k

M

ii ..0,

100

2/002/12/

001

2

20 ==

=

LL

( )

pq

i

iin

X X

N i

k

cavity Nthit j N

in X

rr

+=

=

=

cos1

..0 expcos

0

The i values are the eigenvalues of M. The eigenvectors are orthogonal and we supposenow that 1=q X

r

(orthonormal basis). For a perfect structure, all the modes are on a coslike

curve (figure 1.3 left). For a real structure (ie: different frequencies for each cell), the centralmode (the /2 mode) is not defined in an unique way (depending on the boundaryconditions), and this causes a discontinuity in the curve (the so-called stop-band, figure 1.3right).

685

690

695

700

705

710

715

0 1 2 3 4 5 6

Dispersion curve for CERN/SCL cavitiesf0=700 MHz, k=3%

MHz

Mode number

Figure 1.3: Mode frequency for CERN SCL cavities versus phase shift from cell to cell(radians). The accelerating mode is the central one ( /2 mode ). Left: perfect structure

(identical cells). Right: imperfect structure (different frequency between accelerating andcoupling cells).

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2 Design

The Microwave Studio (MWS) software has been used at CERN, as well as HFSS at LPSCto define the cavity structure. The so-called M3 solution has been chosen. Its geometry is230 mm between cavity axes (accelerating/coupling) and 20mm gap in accelerating cells.

0 mode /2 mode mode

Figure 2.1: Magnetic field for the basic pattern of SCL (2 half accelerating cells + 1 couplingcell), calculated with HFSS

M1 Q( /2) k

1/2 MHz

2/2 MHz

3/2 MHz

a/2 MHz

c/2 MHz

MWStudio 10155 0,4% 738,6 739,67 741,4 739,67740,32HFSS 19956 0,5% 705,8 706,5 711.3 706,5 710.59M2 MWStudio 10902 3% 721,68 731,33 744.07 731.33 733.92HFSS 18586 3,4% 697,32 701.13 738.38 701,13 734.05M3 HFSS 19128 3% 688,71 702,5 710,33 702,5 696,08

Table 2.1: comparison MWS/HFSS simulations. The M3 solution has been chosen.

Figure 2.2: SCL assembly. The module is made of elementary parts including half anaccelerating cell and half a coupling cell. Here, the cooling system for a 15% duty cuycleoperation is represented.

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3 E 3.1 Sensitivity do errors

A frequency error (due for example to machining) an a cell error gives a modification of the M

matrix, who becomes (I+ )M, as follows:

The goal of the study is how to get the mechanical tolerances of the structure. Theknowledge of the dependence frequency versus mechanical error is obtained by Superfishsimulations.

3.2 Single cell perturbation - First order

The cavity number k est perturbed by an amount . The accelerating mode is written versusthe up to second order.

The X symbols are the normalized eigenvectors. The star notation is for the perturbedvalues. The /2 mode has the q index.

We write also :

rror studies

i

i

i

i

i

N

f f

f f

M I

+

=

=+

2

1

11

00

0

0

00

)(

2

2

1

L

LLL

LL

L

where

22

12/*

2/

2/22

12/*

2/

++=+++= E X A A X X

***

)(

X AMX

A I

==+

A is the identity matrix except for the k th term, equal to 1+ . At first order, writing the development at =0, we get :

The prime symbols are for the derivative versus Ais zero, except for the k th term, equal to 1.

For the /2 mode, (eigenvalue 1), we have :

Then

111*

******

A X MA MX A

X X X AM MX A

+=++=+

111

0

0

A X MA X qqk +=+

L

L

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0

L

inally

=

0L

qk q X Y

F

111 A X MAY qq +=+ The A 1 vector is decomposed over the orthonormal eigenvectors X i, with i components. We

et:g

qi X X

X X

X X

i

ik qk iiiiik qk

qq qk qk

==+

=+=+

pour

1

211

2

The q coefficient is got by writing the conservation of the norm. So, still at first order :021 22* =+== qqqk qq X X X We observe immediatly that i is zero (it is proportional to an alternate sum of cosine terms,

The accelerating mode perturbation is a second order perturbation (at least)

3.3 Single cell pertur The calculation is done

A A +

=0.

111

111111

0

' Z X A X MA A

Y A X MA A X MAY

qqk k qk

qqqq

+=

+=+=+

L

he A vector is decomposed again among the eigenvectors, with the i cfficients, leading

over all the modes)

bation - Second order

now to second order

211221

******** 22

X AMA MA A

X X X X AM X M A MX A

+=+++=++

A vaut is the identity matrix for

0

L

21122 A A X MA Z qq ++=+ T 2to:

i

iik qk iiiiiik qk

qk qk qqqk qk

X Z X Z

X Z X Z

=++=+

=++=+

10

0

11

22

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The q cfficient is obtained from the norm:

( ) ( )

( )

==

+++=+++=

=++

2

222

22212

21

2

22

21

1221

201221

1)(

i

ik qk iq

iqqqq

q

X X

A X A X A X

A A X

The calculation is the same, to second order. We suppose (for an easier understanding) thattwo cavities (1 and 2) are perturbed.The second order terms have been already calculated (previous paragraph), except for thecross terms between cavities 1 and 2.

e rewrite this quation (q is the accelerating mode) :

12

3.4 Several cavities error

If the symbols are used to write the derivation versus the cavity number, we get:*

12***

1**

12*

12*

12*

21*

12 X X X X AM X AM X AM AMX ++=+++

W

qqqqq X V X X M U 1212 ++=+

o, for i q :S

( )1

12

1

0

=

++=+

iqqi

iiqiiiq

ii

X V U

X V X U

X

The main parameters are the number of cells (N), the coupling factor (k) and the standarddeviation of the cell frequency error.The perturbed mode is now known and written:

iq X

And something similar for the q term

3.5 Global sensitivity to errors

+++= ipi ji ijp ji piqp C B A X 2

1* 1

he mean value of the electric field can be deduced:

Where C if a function of the C ip coefficients only.

The RMS value is, hence :

T2* 1 C X q +>=<

C C B A X ipijpipq22424222* 21 ++++>=

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Side Coupling Cavity

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