ANSYS Eigenmode Results of Full 4m RFQ with Vacuum

Ports and 43.8mm Quadrant Radius

Study Performed

• Use Pete’s latest 4m RFQ internal models• Vacuum ports only on major top vane• Vacuum ports on top and bottom

• Find all eigenmodes and Q between 300 to 400 MHz (quadrupole and both dipoles)

• Use symmetry planes where possible

Symmetry Planes• Different vacuum port configurations imply

vertical asymmetry∴ Use two quadrants of half (2m) of RFQ

• Once it’s confirmed how vacuum ports affect fields, need to confirm longitudinal modes∴ Use just one quadrant but full 4m length

With One or Two Vac Ports

Vacuum ports top & bottom Vacuum ports top only

Removing the bottom vacuum port increases frequencies by 25 kHz

Open squares indicate theoretical modes, missing due to symmetry

Confirming Missing Odd Modes

Full 4m-long RFQ model confirms missing modes, but slightly poorer mesh quality shifts whole spectra up ~0.5MHz

Two quadrants, longitudinal symmetry One quadrant, full 4m long

Mesh Quality

Results converge for vanetip mesh size < 2mm and quadrant mesh size < 15mm.

Results in previous slides for one quadrant of full 4m RFQ are on limit of memory, using 2mm vanetip and 12mm bulk quadrant mesh sizes, but are sufficiently accurate.

Electric Field in Vane Gap for Different Longitudinal

Modes

TE210: 324.5MHz

Electric Field in Vane Gap for Different Longitudinal

Modes

TE211: 327.7MHz

Electric Field in Vane Gap for Different Longitudinal

Modes

TE212: 334.6MHz

Electric Field in Vane Gap for Different Longitudinal

Modes

TE213: 345.3MHz

Electric Field in Vane Gap for Different Longitudinal

Modes

TE214: 359.9MHz

Electric Field in Vane Gap for Different Longitudinal

Modes

TE215: 378.0MHz

Electric Field in Vane Gap for Different Longitudinal

Modes

TE216: 397.2MHz

Absolute Electric Field of First Four Longitudinal Modes

Conclusions• Mesh density sufficient to solve 4m RFQ

• Longitudinal modes well resolved

• …however the fundamental is not flat!

• Fundamental TE210 mode is ~0.5MHz too high so adjust quadrant radius to 44mm

• Next nearest modes ~3MHz away. All have high Q so should not cross-talk

• Vacuum port asymmetry 25kHz shift

New stuff hot off the press!

18

Example of a frequency error at a single point x0

Suppose the local error is a delta function at some point x0. Local error magnitude is defined as

02 (x) (x x0 )

1 (x x0 )dx

0

V

(1)02 0

2 2V

(x x0 )dx0

V

02 2V

This is the new resonant frequencyof the cavity in terms of local frequency error

0

V0

This relates the cavity frequency change to .

(1)V 0(x) 2

V12V

(x x0 )cos(kmx)dx0

V

02 m

2 cos(kmx)m1

is the new wavefunction

(Talk I’ve found possibly explaining non-flat field)

19

Fractional vane-voltage error

V0 (x)V0

800

V

2 cos(mx0 / V )m2

cos(mx / V )m1

V0 (x)V 0

4 200

V

213

xV

12

xV

2

12x0V

2

, x x0

1

3 x0

V 12

x

V

2

12

x0V

2

, x0 x

An analytic solution exists for this summation. It is.

Each of the higher modes m contributes a term proportional to the voltage value of each mode at the point of the perturbing error, divided by the mode index m squared so nearest modes in frequency contribute most.

(Talk I’ve found possibly explaining non-flat field)

20

Dependence of the fractional voltage error at each point x on the

parameters.

The fractional voltage error at each point increases with the fractional cavity frequency error and as the square of the vane length to wavelength ratio.

This next graph shows that if the local error at some point x0 causes the local resonant frequency to increase, the local voltage decreases, and vice versa.

V0 (x)V0

4 200

V

213

xV

12

xV

2

12x0V

2

, x x0

1

3 x0

V 12

x

V

2

12

x0V

2

, x0 x

(Talk I’ve found possibly explaining non-flat field)

21

x/Vx/lV

V0(x)

m=0

m=0 and 1

m=1 to 20

Perturbed voltage distribution for problem with a -function error at the vane end, where

x0/lV = 0, lV/ = 2 and 0/0 = 0.01.

(Talk I’ve found possibly explaining non-flat field)

As-Designed End RegionNo Radial Matcher

22

As-Designed End RegionWith Radial Matcher

23

On-Tune End RegionNo Radial Matcher

24

On-Tune End RegionWith Radial Matcher

25