Fine-Tuning theRFQ End Region

“…The Devil is in the Detail”• RFQ bulk design very close to completion

• But before drafting need to check:• Repeatability & agreement of codes/meshes• Frequency of full 4m RFQ• If asymmetry is caused if pumps only on top• Tunability of cavity using slug tuners (Saad)• Field flatness• If small machining details have an effect

ANSYS Mesh Quality

• In ANSYS, results converge for vane-tip mesh < 2mm and quadrant mesh < 15mm.• Full 4m RFQ solution needs careful allocation of mesh due to size of problem!• Now Compare with other codes…

ANSYS Maximum Resolution: 324.131 MHz

Superfish Maximum Resolution:324.137 MHz

CST Maximum Resolution:324.129 MHz

Vacuum Port Placement

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, but confirmed real when solving one entire 4m long quadrant

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

50% Field drop at ends is unacceptable and cannot be tuned out!

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

[Ref: Thomas Wangler, Michigan State UniversityLinac Seminar Series – “RFQ Basics”]

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.

[Ref: Thomas Wangler, Michigan State UniversityLinac Seminar Series – “RFQ Basics”]

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

[Ref: Thomas Wangler, Michigan State UniversityLinac Seminar Series – “RFQ Basics”]

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.

[Ref: Thomas Wangler, Michigan State UniversityLinac Seminar Series – “RFQ Basics”]

Having matcher on/off does indeed drastically affect field flatness due to its local frequency error

Matcher On

Matcher Off

Matcher Off/On

Adding the matcher hugely affects the capacitance (inductance to a lesser extent)Want 324MHz (almost midway between these two) which suggests:1)Increase capacitance as much as possible by reducing material removed for matcher2)Increase inductance by reducing vacuum volume removed to ensure 7mm end gap

This region removed lowers inductance

Addition of matcher lowers capacitance

LC

1

Initial design of radial matcher was a circle, tangent to vane at two points.

Quadrupole frequency of this end region = 375.125 MHz

No radial matcher on cold model: vane had square ends.

Quadrupole frequency of this end region = 291.078 MHz

H-field

Modified Radial Matcher Design7mm

R 5

R 21.8

E-field

334.708 MHz

To fine-tune toward 324 MHz, modify cutback radius

H-field

Fine-tune Inductance by Varying Cut-back radius

Original cutback radius = 15mm

Rounding Off Corners

High magnetic field region, so will affect inductance & frequency

To reduce sparking and hot-spots, and to ease machining, the corners will be radiused

Fine-tune Inductance by Varying Cut-back radius

ANSYS Maximum Resolution: 324.131 MHz

Superfish Maximum Resolution:324.137 MHz

CST Maximum Resolution:324.129 MHz44.0mm Quadrant

radius is uncomfortably close to 324 MHz

∴ Relax it slightly by increasing to 44.1mm

44.0mm

Varying Quadrant Radius in CST

Fine-tune Inductance by Varying Cut-back radius

As a bonus, these changes also improve the Q by ~20%

Final Design

CST and ANSYS results for final geometry, 4m RFQ with flush tuners.

Taking into account meshing accuracy for these large models (see slides 3 & 4), both agree with the frequency being (323.5 ± 0.5) MHz

Original Matcher

No Matcher

Optimised Matcher

New matcher design achieves considerably flatter field and ensures (323.5 ± 0.5) MHz along the entire RFQ

Questions?