circuital model for post coupler stabilization in a drift tube...
TRANSCRIPT
sLH
C-P
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This is an internal CERN publication and does not necessarily reflect the views of the CERN management.
sLHC Project Note 0014
2010-05-10
Circuital Model for Post Coupler Stabilization in a Drift Tube Linac
F. Grespan, G. De Michele, S. Ramberger, M. Vretenar / BE-RF
Keywords: Linac4, DTL, Post Couplers, Equivalent Circuit.
Summary
Linac4 Drift Tube Linac (DTL) cavities will be equipped with Post Couplers (PCs) for field
stabilization. The study presented in this paper starts with the analysis of 2D and 3D
simulations of post couplers in order to develop an equivalent circuit model which can explain
the post coupler stabilization working principle and define a tuning strategy for DTL cavities.
Simulations and equivalent circuit results have been verified by measurements on the Linac4
DTL prototypes at CERN.
1. Introduction
DTLs are accelerating cavities operating in the TM010 mode. The fields in all the cells
have the same phase, so that the overall cell array operates in a zero mode. The group velocity
of a resonating mode is proportional to the slope of the dispersion curve at that point,
zg kv / . Since the group velocity of the TM010 mode is zero, there is no power flowing
along the structure and the field distribution is very sensitive to frequency perturbations of the
cells. This problem is particularly serious for long DTLs, where the difference in frequency
between modes is lower. The field stabilization in accelerators operating in a zero mode can be
obtained with the resonant coupling approach [1].
Post couplers (PCs) are used in DTL cavities to create a secondary coupled resonator
system, which is then coupled to the main resonator system, formed by the accelerating cells
(DT cells) resonating in the TM010 mode. The purpose of the PC resonator system is to stabilize
the accelerating field in case of local frequency errors in high power operation. At the point of
confluence PCs increase the slope (group velocity) of the dispersion curve for the accelerating
mode and consequently they increase the power flow as required in transient conditions. The
system, composed by two chains of coupled resonators, has two bands of frequencies: the TM
band and the PC band (Fig. 1). Since the PC mode field distributions correspond to capacitively
loaded TE modes (Fig. 1), the PC0 mode cannot be excited in a real tank with conducting end
walls.
- 2 -
Figure 1 Dispersion curves for TM and PC modes at 3 different PC lengths. Below, the
electric field pattern of PC1 mode.
The Linac4 DTL structure [2, 3] consists of 3 cavities accelerating the beam from 3 MeV
up to 50 MeV. In their final version, the lengths of the 3 DTL tanks are 3.90 m, 7.34 m and
7.25 m respectively, with 39, 42 and 30 cells per cavity. The operating frequency is 352.2 MHz,
the tank diameter is 520 mm and the drift tube diameter 90 mm.
A DTL hot prototype and a DTL cold model have been built and are available at CERN
for the study of field flatness and stabilization.
A 1034 mm long prototype (Fig. 2) [3] for high power tests has been machined by
CINEL, and copper-plated and assembled at CERN. The operating frequency is 352.2 MHz and
the average field is 3.3 MV/m. The prototype consists of 13 cells with a tank diameter of
520 mm and a DT diameter of 90 mm. The cell length increases along the prototype
corresponding to beam energy going from 3 MeV to 5.4 MeV. Field stabilization is achieved by
PCs, with one PC every three DTs at the longitudinal positions of the 2nd
, 5th
, 8th
and 11th
DT.
The PC diameter is 20 mm.
A DTL cold model with scaled dimensions (Fig. 2) [4] for tuning studies has been built
by KASCT in Saudi Arabia. The operating frequency is 538.7 MHz, the tank diameter 340 mm
- 3 -
and the DT diameter 58.8 mm. It consists of a 16 cells cavity equipped with holes for PC
positioning at each DT, in order to test the effectiveness of stabilization for different PC
configurations.
Figure 2 Linac4 DTL hot prototype (left) and DTL cold model (right) at CERN.
2. Simulations and measurements on the DTL hot prototype
2D and 3D simulations and low-level RF measurements were performed on the DTL hot
prototype. The study of the influence of PCs in this structure has the objective of better
understanding the PC stabilization mechanism and defining a PC tuning strategy for tank
stabilization.
2.1 Slater perturbation theorem calculations versus measurements
Since PCs only have a negligible effect on the nominal accelerating field, the TM010
frequency shift due to PC insertion can be estimated with the Slater perturbation theorem, using
the field distribution computed by Superfish [5]. The formula used for the calculation of the
frequency shift is [6]:
2 2
2 2
2
4 4
cyl cylV
H E dVk H k E
df f R dr fU U
where R is the PC radius, dr is the step length between 2 Superfish data, f is the cavity
frequency, kcyl = 2 is the shape factor due to the field distortion close to the PC surface, µ and ε
are the material permeability and permittivity, H and E are the unperturbed magnetic and
electric fields and U is the stored energy.
The frequency shift calculated as function of the PC length is shown in Fig. 3, together
with measurements on the DTL hot prototype equipped with 4 PCs. The deviation from the
simulated curve [7] has been investigated with a more accurate measurement over a range of
PC lengths from 155 mm to 180 mm (Fig. 3), indicating that in this range the 3 highest modes
of the PC band cross the TM010, and couple with it. The Slater perturbation theorem describes
the TM010 frequency shift due to the PC insertion, but cannot give information on the coupling
between the TM01 band and the PC band and consequently on the optimum length of the PCs
for stabilization of the tank.
- 4 -
0.00
0.05
0.10
0.15
0.20
0.25
0 50 100 150 200 250
PCs length [mm]
TM
01
0 f
req
ue
nc
y s
hif
t [M
Hz]
.
Measured Freq. Shift
Simulated Freq. Shift
351.0
351.4
351.8
352.2
352.6
353.0
158 162 166 170 174 178
PCs length [mm]
Fre
qu
en
cy
[M
Hz]
.
TM010_mode
PC3 mode
PC2 mode
PC1 mode
Figure 3 Simulated (Superfish) and measured TM010 detuning as function of PC length (left)
and measurements showing PC modes crossing TM010 mode (right).
2.2 3D simulations and bead-pull measurements on post coupler modes
3D HFSS [8] simulations and bead-pull measurements have been undertaken on the four
PC modes. PC modes are easily recognized in simulations by a characteristic field pattern with
E field between PCs and drift tubes and H field around PCs. The simulated axial field
corresponds well with the bead-pull measurements performed on the PC modes close to
confluence. Fig. 4 shows the highest PC mode (PC1 mode), which presents the same axial field
pattern as the TM011 mode. For this reason the PC1 mode has a stabilizing effect with respect to
perturbations induced by the TM011 mode on the accelerating field.
Using the Perturbation Theory formalism limited to PC1 and TM011 modes, the electric
field in a generic point (x,y,z) inside the cavity is given by
0 0 0 1 011
1 011( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , )pert PC TM
PC TME x y z E x y z E x y z E x y z E x y z E x y z .
- 5 -
The quantity measured with the bead-pull technique is the electric field z-component on
the beam axis
),0,0(),0,0(),0,0(),0,0(),0,0(),0,0( 011
011
1
1
000 zEzEzEzEzEzE TM
zTM
PC
zPCzzz
pert
z .
Coefficients
1 01,0
1 2 2 2 2
1 1
( , , ) ( , , ) ( , , )PCPCi i j j
i jcavity
PC
o PC o PC
E x y z P x y z E x y z dxdydzE P E
and 011TM
depend on the local geometry perturbation described by the matrix ),,( zyxPP and on the
field pattern of modes. In this formula index i and j represent vector components. If mode
resonant frequencies 1PCand
011TM are tuned such that coefficients
1PC and
011TMare equal
and opposite, the accelerating field perturbation can be cancels out.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 0.2 0.4 0.6 0.8 1
z axis [m]
Ma
gE
Simulation
Bead-pull Meas.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
z axis [m]
Ma
gE
.
TM011 bead pull meas
Figure 4 Simulated and measured field on axis for the highest PC mode (PC1 mode) (left)
and measurement for TM011 mode in a cavity without PCs (right).
2.3 Study of geometric post coupler parameters
3D HFSS simulations have been used in order to obtain values of the PC1 mode frequency
as function of the number of PCs per unit length of the tank (Fig. 5). Taking a very small gap
between PC and DT (gap PC-DT << /4) the PC mode electric field is concentrated in the gap
area, so it is possible to apply the quasi-static approximation [9] in order to calculate values of
the capacitance Cp associated to the gap PC-DT using the formula2
2p
UC
V, where U and V are
calculated from the simulation. For a gap PC-DT = 3 mm, the results show a value of Cp
distributed with respect to an average value of (2.8 0.2)pC pF , while the frequency
increases with the number of PCs per unit length (Fig. 6).
From this analysis we can conclude that, for small variations of the distance between PC
and DT, the inductance Lp associated to the PC is a function of the distance between PCs.
Figure 7 shows the values calculated from simulations as 2
1
(2 )p
p p
Lf C
compared with the
curve obtained by the formula 72 10 lneq
p PC
PC
DL l
dof a coaxial inductor where the PC is
the inner conductor and the outer conductor has an equivalent diameter of
- 6 -
2
tankeq tank o
PCs PC
LD D
N l where Ltank and Dtank are the tank length and diameter, lpc is the PC
length and N°PCs is the number of PCs inside the tank. This formula takes into account the
distance between Post Couplers.
Figure 5 Two cavity configurations equivalent in the number of PCs
per unit length.
y = 3.7735x + 249.01R² = 0.9333
2.1
2.4
2.7
3
3.3
3.6
3.9
4.2
0
50
100
150
200
250
300
350
0 2 4 6 8 10 12 14 16
pF
MH
z
number of PCs per meter
Simulation settings: Gap PC-DT=3mm, DT_radius=45mm, Tank_radius=260mm; PC_radius=10mm
PC freq
PC capacitance
Linear (PC freq)
Linear (PC capacitance)
Figure 6 PC mode frequency and Cp capacitance from 3D simulations of different
distributions of PCs inside the tank.
- 7 -
90
100
110
120
130
140
150
160
0 2 4 6 8 10 12 14 16
PC
in
du
cta
nce L
p [
nF
]
number of PCs per meter
Figure 7 PC inductance Lp as function of the number of PCs per meter. The solid line is
given by the Lp formula, crosses are obtained from 3D simulations in quasi-static
approximation. An example of an LHC drawing
In quasi-static conditions it is possible to estimate the Cp value starting form electrostatic
considerations. The electrostatic definition for the capacitance is
0 0
A A
d d
dA E d AQ
CV E ds E ds
. We want to get a simple empirical formula 0 Q
p
av
AC
g,
where AQ is the area where the surface charge density induced by the electric field is
distributed, and gav is the average integral path of the electric field. Fig. 8 shows in detail the
electric field distribution between PC and DT in a PC mode. The surface charge on the DT is
distributed over an ellipsis of area AQ = ( ∙ xtg ∙ stg) and the average distance from the PC is
gav = ½ (dDT + 2g - ytg), where Ptg = (xtg, ytg) is the tangent point to the drift tube cylinder of a
line starting from the border tip point of the PC, stg is the arc length from Ptg to the top of the
drift tube, dDT is the DT diameter and g is the gap PC-DT. The capacitance is independent from
the length of the DT and the formula is:
0 0
arccos(2 / )
22
Q tg DT tg DT
pDTav
tg
A x d y dC
dgg y
Fig. 9 shows the curve given by the formula with the values calculated from simulations
as 2
2p
UC
V. The equation overestimates the capacitance by about 10% with respect to
simulation results.
- 8 -
Figure 8 electric field pattern between PC and DT in a PC mode.
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
0 2 4 6 8 10 12 14 16
PC
cap
acit
an
ce C
p [
pF
]
gap PC-DT [mm]
Figure 9 PC capacitance Cp as function of the gap PC-DT. The solid line is given by the Cp
formula, crosses are obtained from 3D simulations in quasi-static approximation.
The interaction of PCs and stems has also been studied with 3D simulations, in order to
properly insert stems in the DTL equivalent circuit.
Stem modes can be distinguished in RF measurements because of the much lower
frequencies with respect to the operating mode and because of the low sensitivity to gap
displacements. 3D simulations show that the presence of DT stems weakly affects the field
pattern of the PC modes (Fig. 10): the electromagnetic energy is concentrated around the PCs,
with a slight deviation of the H field around the stems, and the change in mode frequency can
be estimated using the Slater perturbation theorem. The same behavior can be noticed for the
stem modes in relation to the presence of PCs (Fig. 11).
From this we conclude that DT stems and PCs can be considered separately in the
equivalent circuit.
- 9 -
Figure 10 E and H field magnitude of the PC1 mode showing only a small influence of DT
stems on mode pattern.
Figure 11 A 3D view of the magnetic field of the stem mode, showing weak coupling between
stems and PCs.
3. Equivalent circuit
3.1 Circuit equations and matrix form
A complete equivalent circuit for a DTL cell equipped with PCs is shown in Figure 12. In
this circuit C0 represents the gap capacitance, L0/2 represents the inductance of half a drift tube
(DT), and C is the capacitance between a DT and the tank wall, represented by the ground
conductor. Stems are represented by inductors Ls in parallel with the shunt capacitance Cs. Cp is
the capacitance between PC and DT, and Lp is the inductance of a PC.
C0 can be estimated with the parallel plate capacitance formula, L0 and Cs with coaxial
inductance and capacitance formulas and Ls with straight wire inductance formula [10]. Cp and
Lp estimation formulas are described in section 2.3. Geometrical formulas, DTL hot prototype
dimensions and calculated circuit element values are listed in Table 1 and resulting frequency
values are compared to measured frequencies. These approximations are obtained by
- 10 -
electrostatic considerations, but they can give an idea of the orders of magnitude of the circuit
elements.
There are 3 main resonator chains in a DTL: drift tube resonators, stem resonators and PC
resonators. The TM010 mode frequency is given by 000 /1 LC and we define a PC
frequency ppp LC/1 and a stem frequency 1/s s sC L . Frequency values estimated
from geometrical parameters give s (≈ 200 MHz) lower than 0 and p (≈ 350 - 370 MHz).
This is confirmed by measurements on the DTL prototype, where the stem dispersion curve is
lower than TM and PC dispersion curves (Figure 13). Since for frequencies close to the
operating mode the stem impedance 2
21 s
stem sY j C is capacitive and there is no coupling
between stem and PC, as discussed in Paragraph 2, the circuit model reduces to the circuit in
Figure 14, where the capacitance C is defined 2
21 s
sC C .
Figure 12 Equivalent circuit for a DTL including DT, stem and PC resonators.
0
100
200
300
400
500
600
0 1 2 3 4 5 6
MH
z
mode number
TM modes
Stem modes
PC modes
Figure 13 Frequency bands in the DTL prototype.
- 11 -
Table 1
Formulas, geometric dimensions and circuit element values for DTL hot prototype geometry.
Formula Geom. Parameters Calc.
Values
Frequency
[MHz]
2
0 04
DTdC
gap dDT=9 cm, gap=1.5 cm 3.76 pF
370
7
0 2 10 ln tank
DT
DL
d
Dtank=52 cm, dDT=9 cm,
=14 cm 49 nH
0
2
ln
s
tank
DT
CD
d
Dtank=52 cm, dDT=9 cm,
=14 cm 4.44 pF
200
7 42 10 2.303log st
s st
st
lL l
d
Lst=21.5 cm,
Dst=2.9 cm 146 nH
0
Q
p
av
AC
g dpc=2 cm, gappc=10 mm 1.96 pF
330
72 10 lneq
p pc
pc
DL l
d
lpc=20.5 cm,
Ltank=56 cm, NPCs=3 117 nH
Figure 14 DTL tank top view and equivalent circuit.
Solving the mesh equations for the drift tube and the PC currents, we obtain the system
- 12 -
1
0 1 1 1
0
1 1 1
1 10
1 12 0
1 10
n p n n n
p
n n n n n n
n p n n n
p
ip j L i i ipj C j C
i j L i i i ip ipj C j C
ip j L i i ipj C j C
This system is equivalent to the following, with some substitutions and simplifications:
21
2
21 1 1
2
0 0 0
21 11 2
1 1
2 1
1 1
n n nn
p p p
n n n n nn
n n nn
p p p
i i ipip
C C C C C
i ip ip i ii
C C C C C C C
i i ipip
C C C C C
This system of equations can be rewritten as an eigenvalue problem in matrix
form IIM 2 . For a cavity of 3 cells + 2 PCs the 5 x 5 matrix is:
2
00
2
2
2
2
00
1
2
1
2
00
0
2
2
00
2
2
2
2
00
1
2
1
2
00
0
0
2
00
22
00
2
0022
00
2
00
2
0
2
00
22
00
2
0022
00
2
00
2
0
2
00
1100
1100
12
0011
0011
C
i
C
ip
C
i
C
ip
C
i
C
i
C
ip
C
i
C
ip
C
i
CCC
C
C
C
C
C
C
CCC
C
C
C
C
C
C
CCC
C
C
C
C
C
C
CCC
C
C
C
C
C
C
CCC
PP
PP
PP
PP
pp
p
pp
pppp
p
pp
pp
The eigenvector elements are currents that are scaled with capacitances and element
frequencies squared. The eigenvalues are the mode frequencies squared.
The voltage through a gap is proportional to the current divided by the capacitance C0 of
the gap iii CIV ,0,0,0 / , and the average field E0 is defined by the voltage divided by cell length
iii LcellVE /,0,0 . This distinction is negligible in case of all identical gaps, but is important in
case of increasing gap lengths or gap perturbations.
3.2 Circuit derivation of the stabilizing PC condition
In the following, the circuit in figure 14 is used to derive a straightforward criterion for
post-coupler stabilizationon in periodic structures.
Let us use a transport matrix notation for voltages Vn and currents In along the circuit
chain [11]:
- 13 -
n
n
nnn
n
n
n
I
V
ZYY
Z
I
V
1
1
1
1where
2
0
2
0
11
CjZ n and
22
2
p
p
pn CCjY .
If the 1st cell capacitance has a perturbation C0, and there are no PCs (Fig. 15a), the field
flatness of the whole structure is perturbed (the quantity proportional to the gap field is the
current)
2
0 02
0 0 01 0 0
21 0 0 0
00200 0
1 1( 1)
(1 )1 ( 1)1
d I d Ij C j CV Z I
I Z Y I Cj Cd Id I
Cj C
where 0
0 0
Cd
C C and the driving frequency is 0 . For the next unperturbed n cells the
impedance Zn = 0, and we obtain 2 1 1 1 1 1 1 0 0 1 1 0
0
(1 ) ( 1)C
I YV Y Z I Y Z I I I d IC
.
The difference 1 0
0
( 1)n n
CI I d I
C is constant along the cell gaps.
As the cell lengths scale with , the perturbation effect increases where the ratio C/C0.
Note that the effect is therefore stronger at the high energy end of the DTL!
Let us now suppose to have one PC every other cell (Fig. 15b).
2
2 1 1 1 1 1 1 0 0 1 1 02 2
0 0 0
(1 ) ( 1) ( 1)p p
p
CCI YV Y Z I Y Z I I I d d I
C C
The formula shows that the presence of the PCs can modify the propagation of the
perturbation from one cell to the next.
The PC effect is stabilizing if I2 = I0:
2
2 0 0 02 2
0 0 0 0
1 ( 1) ( 1) ( 1)p p
p
CC CI d I d d I I
C C C
and from this equation a condition for the optimum value of the PC frequency is obtained
p
pCC
C
2
2 2
02.
Assuming that the variation of Lp is negligible with respect to the variation of Cp when
changing the PC to DT gap (gappc), it is useful to solve the previous relation for CP:
- 14 -
12
22
0p
stab
pCL
CC .
In general the conditions are 2
2 0
1p
pn C
C
and 2
0
stab
p
p
CC
CL n where
number post couplersn
number cells. Let us define the coupling coefficient
p
p
Ck
C and the previous
relation becomes 2
2 0
1p
pn k where both p and kp depend on Cp.
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
0 2 4 6 8 10 12 14 16 18
cell number
E0
(a)
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
0 2 4 6 8 10 12 14 16 18
cell number
E0
(a)
0.992
0.994
0.996
0.998
1
1.002
1.004
1.006
1.008
1.01
1.012
0 2 4 6 8 10 12 14 16 18
cell number
E0
(b)
0.992
0.994
0.996
0.998
1
1.002
1.004
1.006
1.008
1.01
1.012
0 2 4 6 8 10 12 14 16 18
cell number
E0
(b)
Figure 15 perturbed E0 without PCs (a) and with PCs at the stabilizing value of Cp
(b) from a circuit simulation.
The stabilizing Cp value decreases as function of the capacitance C (Fig. 16). Where DTs
are longer, gappc must be larger; where DTs are shorter, gappc must be smaller.
- 15 -
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 5 10 15 20
Cp
(p
F)
C (pF)
Cp (n=1)
Cp (n=1/2)
Figure 16 Cp as function of C (case of n=1 and n=2).
Some observations can be made about the distance between PC and DT in order to
understand the stabilizing setting of gappc (Fig. 17):
1. when gappc is too small (stab
pp CC ), PCs can compensate the perturbation, but not
completely
2. when gappc is slightly larger than the stabilizing condition
( stab
pp CC with stab
pp0 ) there is a kind of overcompensation that changes the slope
with respect to the perturbation
3. when gappc is much larger than the stabilizing condition ( stab
pp CC and
0p ), the PC effect is de-stabilizing.
0.94
0.96
0.98
1
1.02
1.04
1.06
0 2 4 6 8 10 12 14 16 18
E0
cell number
under-compensation
over-compensated
de-stabilized
Figure 17 The PC effect in a perturbed field with gappc at 3 different configurations.
- 16 -
3.3 1st and 2
nd magnetic coupling between post couplers
A comparison of the measured PC dispersion curve with the curve obtained by the
equivalent circuit (Fig. 20a) shows that one needs to take a next nearest coupling between PCs
into account, which couples PCs placed in opposite sides of the DTL tank. This nearest
neighbour coupling is equivalent to a magnetic coupling between PC inductors Lp (Fig. 18)
with 2
1 0/p p pk M L at the operating frequency.
A next nearest neighbour coupling kp2 between PCs is included as well in the matrix. It is
equivalent to a magnetic coupling Mp2 connecting PCs placed at the same side of the DTL tank.
Mp2 has an opposite sign with respect to Mp. Fig. 18 shows the circuit including PC couplings,
Fig. 19 shows an example of matrix and Fig. 20 shows how the dispersion curves change taking
into account the PC couplings.
The effect of a second nearest neighbor magnetic coupling between PCs has been clearly
observed in 3D HFSS simulations. A structure of 4 cells, each 7 cm in length, equipped with 3
PCs, shows that the 2nd
PC-mode, characterized by the un-exited central PC (Fig. 21), is higher
in frequency with respect to the 1st PC-mode, which is usually the highest in frequency. This
effect is caused by the stronger negative coupling and it becomes less important with 10 cm
long cells. Finally, with 14 cm long cells, the 2nd
PC-mode frequency is lower than the 1st
mode.
Mp Mp
Mp2
Mp Mp
Mp2
Figure 18 equivalent circuit including nearest and next nearest coupling between PCs. 2 2
2 0 00 0
0
2 220 0 0 0
1 2
2 22 220 0 0 0
0 0
0
2 220 0 0 0
1 1
2 2220 0
0 0
0
1 10 0 0 0
1 10 0
2 10 0
1 10 0
2 10 0
p p
p p p p
p
p p p p
p p p p
p
p p p p
C CC
C C C C
C CC k k
C C C C
C CC CC
C C C C C C
C Ck C k
C C C C
C CCC
C C C C
2
0 0
2 220 0 0 0
2 1
2220 0
0 0
0
1 10 0
1 10 0 0 0
p p p p
p
p p
C
C C
C Ck k C
C C C C
CCC
C C C C
Figure 19 circuit matrix including nearest and next nearest coupling between PCs.
- 17 -
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6 7 8
mode number
fre
qu
en
cy
(M
Hz)
.
TM01n modes
PC modes (circuit)
PC modes (measured)
(a)
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6 7 8
mode number
fre
qu
en
cy
(M
Hz)
.
TM01n modes
PC mdoes (circuit)
PC modes (measured)
(b)
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6 7 8
mode number
fre
qu
en
cy
(M
Hz)
ù
.
TM01n modes
PC mdoes (circuit)
PC modes (measured)
(c)
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6 7 8
fre
qu
en
cy (
MH
z)
mode number
TM01n modes
PC modes (circuit)
PC modes (measured)
(a)
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6 7 8
fre
qu
en
cy (
MH
z)
mode number
TM01n modes
PC modes (circuit)
PC modes (measured)
(b)
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6 7 8
fre
qu
en
cy (
MH
z)
mode number
TM01n modes
PC mdoes (circuit)
PC modes (measured)
(c)
Figure 20 PC dispersion curve computed by equivalent circuit without PC coupling (a), with
nearest PC coupling (b) and next nearest PC coupling (c).
- 18 -
Figure 21 electric field pattern of PC modes in a DTL cavity equipped with 3 PCs.
Fig. 22 illustrates a lattice model equivalent to a DTL tank equipped with PCs. The matrix
in Fig. 19 can be written showing couplings between resonant elements (Fig. 22). Graphically it
is clear that the coupling k0 between DT cells can be increased by shortening the cell length
(capacitance C decreases), the coupling kp between PCs and DTs increases when gappc is
smaller (capacitance Cp increases), couplings kp1 and kp2 between PCs increase if the spacing
between PCs is smaller (increase the ratio Mp/Lp). Table 2 shows coupling coefficient orders of
magnitude extrapolated by fitting frequency measurements on the DTL cold model with 7 PCs.
The main coupling k0 is the strongest, the second nearest neighbor magnetic coupling kp2
between PCs is the weakest.
2
0 0 0
2
0 0 1 2
2
0 0 0 0
2
1 0 0 1
2
0 0 0 0
2
2 1 0 0
2
0 0 0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0 0 0
p
p p p p
p p
p p p p
p p
p p p p
p
k k k
k k k k k
k k k k k
k k k k k
k k k k k
k k k k k
k k k
Figure 22 Lattice model with couplings between resonant elements. The matrix shows all
couplings between elements.
- 19 -
Table 2
Coupling strengths for the DTL cold model with 7 PCs at gappc = 19 mm.
k0/ 02 kp/ 0
2 kp1/ 0
2 kp2/ 0
2
2.3 0.19 0.096 -0.044
4. Circuit extrapolation of stabilizing PC setting
The following results refer to the 16 cell DTL cold model at CERN, equipped with 5 and
7 PCs. Because PCs are installed every three or two drift tubes respectively, we minimize the
average value of the tilt sensitivity, resulting in a saw-tooth pattern. Tilt Sensitivity is defined as
fE
EETS
unpert
i
unpert
i
pert
i
i
1
0
00 and Tilt Sensitivity Slope as
CellsofN
TSTSSlopeTS
firstlast
__
)()(_ .
A procedure for finding an optimum average PC length has been defined:
1. The gap capacitance C0 is calculated from a SUPERFISH simulation, using the formula
UVC 2
02
1
2. Measurement of the TM010 and TM011 (without PCs) to calculate the coupling
capacitance C between tank and drift tubes (Note: here we make the assumption that all the
cells have the same average length, we don’t take into account the increasing cell length)
3. Measurement of the PC frequency band (at least 3 modes, for example PChighest,
PClowest, PCcentral modes) and of the TM010 and TM011 modes at different length of the PCs (at
least 3 lengths)
4. Fitting of the measured frequencies with dispersion curves computed by the circuit, by
adjusting the circuit parameters 0 p, Cp, kp1, kp2
5. Insertion of a perturbation C0 in the end capacitances of the circuit
6. Looking to the fields given by the previously fitted circuits: the goal is to minimize the
Tilt Sensitivity (Fig. 23,24,25)
7. Extrapolation of the stabilizing PC length from the parameter curves (Table 3), finding
at the zero of TS_Slope curve (Fig. 26).
- 20 -
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6
mode number
fre
qu
en
cy
(M
Hz)
.
Fitted_freq
Meas_Freq
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 14 16 18
cell number
TS
Figure 23 frequency fit and E0 perturbed field in an equivalent circuit without PCs.
- 21 -
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6
mode number
fre
qu
en
cy
(M
Hz)
.
fitted freq
meas freq
fitted freq
meas freq
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 14 16 18
cell number
TS
Figure 24 Frequency fit and E0 perturbed field in an equivalent circuit with 5 PC at 10 mm
gappc.
- 22 -
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6
mode number
fre
qu
en
cy
(M
Hz)
.
fitted freq
meas freq
fitted freq
meas freq
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 14 16 18
cell number
TS
Figure 25 Frequency fit and perturbed accelerating field E0 in an equivalent circuit with 5 PC
with gappc = 25 mm.
- 23 -
7 PCs settings
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0 5 10 15 20 25
PC-DT space [mm]
kp
1, kp
2, T
S_S
lop
e .
0
100
200
300
400
500
600
700
800
900
1000
freq
_P
C, C
p .
kp1
kp2
TS_Slope
freq_PC [MHz]
Cp [pF*1000]
7 PCs settings
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0 5 10 15 20 25
PC-DT space [mm]
kp
1, kp
2, T
S_S
lop
e .
0
100
200
300
400
500
600
700
800
900
1000
freq
_P
C, C
p .
kp1
kp2
TS_Slope
freq_PC [MHz]
Cp [pF*1000]
Figure 26 Circuit parameters and tilt sensitivity Slope as function of the gappc for 5 PCs and
7 PCs settings. The point where the tilt sensitivity crosses zero is marked.
Table 3
Stabilizing PC setting for DTL cold model equipped with 5 PCs and 7 PCs.
Stabilizing Parameters (5 post–couplers)
gappc
[mm] p [MHz] Cp
[pF] k0/ 0
2 kp/ 0
2 kp1/ 0
2 kp2/ 0
2
22.5 485 0.62 2.3 0.20 0.087 -0.033
Stabilizing Parameters (7 post–couplers)
gappc
[mm] p [MHz] Cp
[pF] k0/ 0
2 kp/ 0
2 kp1/ 0
2 kp2/ 0
2
19.0 477 0.61 2.3 0.19 0.096 -0.044
Let us consider the ratio pcgap
SlopeTSr
)_( as sensitivity criterion to the PC length
adjustment. This sensitivity is larger in case of 5 PCs, where r = 9 around the optimum
gappc = 22.5 mm, than in the case of 7 PCs, where r = 2 around the optimum gappc = 19 mm.
An increased sensitivity to the PC length adjustment is related to a larger value of the gappc
which makes it more difficult to fix the optimum PC length for stabilization.
5. Measurements on the DTL cold model
In order to verify the equivalent circuit results, we use the tilt sensitivity measure to find
the stabilization length on the DTL cold model equipped with 5 and 7 PCs.
- 24 -
First, we need a measurement of the reference field (Fig. 27) to be stabilized, which is
called the “natural field distribution” of the tank [12]. In order to verify the effectiveness of PCs
in stabilizing the accelerating field, it is not necessary to refer to a perfectly flat field.
Then the first and the last gap are perturbed, to create a tilt in the field (Fig. 27).
Now starting at a too small value gappc (15 mm), the PCs are not able to stabilize the tilt
of the field as expected from the equivalent circuit (Fig. 28).
When gappc becomes too large (25 mm for 5 PCs, 22 mm for 7 PCs), the field reverses its
slope, and we are in over-compensation case (Fig. 29).
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
TS
.
E0 field
Tilt Sensitivity
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
TS
.
E0 field
Tilt Sensitivity
Figure 27 Reference field and perturbed field.
- 25 -
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
TS
.
E0 field
Tilt Sensitivity
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
TS
.
E0 field
Tilt Sensitivity
Figure 28 gappc = 15 mm is too small for both 5 PCs (left) and 7 PCs (right) case.
- 26 -
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
TS
.
E0 field
Tilt Sensitivity
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
TS
.
E0 field
Tilt Sensitivity
Figure 29 gappc = 25 mm is too large for 5 PCs case (left) and gappc = 22 mm is too large for 7
PCs case (right).
Finally we set the PCs at gappc calculated with the equivalent circuit: 22.5 mm gap for 5
PCs, 19 mm gap for 7 PCs. The tilt sensitivity curve approaches zero, except for cells close to
the tank ends (Fig. 30).
This effect can be mitigated by adjusting gappc in order to take the increasing cell length
in this model into account. An optimum PC configuration thus sets the central PC at the
computed gappc, and varies gappc towards lower values at the low energy end, abd larger values
at the high energy end (Fig. 31). Table 4 shows the final PC settings.
- 27 -
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
TS
.
E0 field
Tilt Sensitivity
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
TS
.
E0 field
Tilt Sensitivity
Figure 30 PCs are set all at the gappc calculated by the equivalent circuit simulation (left: 5
PCs, right: 7PCs).
- 28 -
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
TS
.
E0 field
Tilt Sensitivity
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10 12 14 16 18
cell number
E0
.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
TS
.
E0 field
Tilt Sensitivity
Figure 31 PCs are adjusted taking the increasing cell length along the cavity into account (left:
5 PCs, right: 7 PCs).
Table 4
Optimum PC settings for the DTL cold model at CERN.
PC
number 1 2 3 4 5
- -
Gap [mm] 21.5 22 22.5 23 23.5 - -
PC
number 1 2 3 4 5 6 7
Gap [mm] 17.5 18 18.5 19 19.5 20 20.5
6. Conclusions
A circuit model for drift tube linacs has been developed in order to study RF stabilization
by post-couplers. 2D and 3D simulations and measurements have been performed, a circuit
topology has been deduced and values for circuit elements have been found. The resulting
equivalent circuit has been studied in various conditions and compared with measurements. In
- 29 -
particular the stabilization effect of post-couplers has been modeled and the optimum post-
coupler length could be deduced without lengthy bead-pull measurements.
2D simulations, performed with Superfish code, allow for a good estimation of frequency
detuning and power dissipation due to post coupler insertion, as has been shown by the
comparison with measurements taken on the DTL “hot” prototype. 2D simulations however
cannot analyze the coupling between post couplers (PCs) and accelerating cells (DT cells)
resulting in strong variations of mode frequencies. 3D simulations, performed with HFSS code,
have been demonstrated to be a means for computing such coupled cavity fields and the
resulting mode frequencies in agreement with measurements taken on the hot DTL prototype.
In an analytic approach, the stabilization of the accelerating mode by PCs can be
interpreted on the basis of a mutual cancelation of the perturbations induced by the TM011 and
PC1 mode. While the perturbations are equal in absolute value (with opposite sign), the
differences in frequency towards the accelerating mode TM010 are not. Further 3D simulations
have been undertaken to study PC mode frequencies as a function of the number of PCs per
meter and of the gap size PC-DT. In a quasi-static approach, approximate formulas for PC
circuit parameters as a function of the PC geometry have been deduced and verified to be
consistent with simulation results.
It has been shown that the equivalent circuit for a DTL equipped with PCs represents a
complete description of the cavity behavior around the operating frequency. It gives an
explanation of stabilizing PC settings in terms of circuit parameters and shows clearly that PC
frequency tuning is directly related to the PC-DT coupling. The matrix form of circuit equations
allows the introduction of nearest neighbor coupling and next nearest neighbor coupling
between PCs. With the insertion of couplings, experimental dispersion curves are precisely
reproduced by the equivalent circuit.
A tuning procedure based on the equivalent circuit and on frequency measurements has
been defined, tested and validated with measurements on the DTL aluminum model in 2
different PC configurations. The particular advantage of the new procedure is that optimum
post-coupler length for stabilization is deduced from spectral measurements and that the
number of lengthy bead-pull measurements thus can be considerably reduced.
References
[1] D. E. Nagle, E. A. Knapp, B. C. Knapp, “ Coupled Resonator Model for Standing Wave Accelerator Tanks”, Rev. Sci. Instrum., 1967, Vol. 38, Number 11.
[2] F. Gerigk et al., “RF structures for Linac4”, Proc. PAC07.
[3] S. Ramberger et al., “Drift Tube Linac Design and Prototyping for the CERN Linac4”, Proc. LINAC08.
[4] N. Alharbi, F. Gerigk and M. Vretenar, “Field Stabilization with Post Couplers for DTL Tank1 of Linac4”, Tech. Rep. CARE-Note-2006-012-HIPPI, CARE, 2006.
[5] Los Alamos Accelerator Code Group, Poisson Superfish.
[6] Los Alamos Accelerator Code Group, Poisson Superfish User’s Manual.
[7] S. Machida, T. Kato, S. Fukumoto, “Stabilizing characteristics of post-couplers”, Transaction on Nuclear Science, 1985, Vol. 32.
[8] Ansoft Corporation, HFSS 10.1.
[9] T. Wangler, “RF Linear Accelerators”, John Wiley & Sons, Inc. 1998, par.5.14.
- 30 -
[10] F.E. Terman, “Radio engineer's handbook”, McGraw-Hill 1943.
[11] T. Wangler, “RF Linear Accelerators”, John Wiley & Sons, Inc. 1998, par.3.4.
[12] J. Billen, private communication.