July 14

1. July 14

2. Error induced mixing of higher frequency longitudinal modes

3. Mode spectrum of a four-vane RFQ showing: >the desired TE210 operating mode at 425 MHz, >higher longitudinal TE21p modes (p=1, 2, …), >and the family of TE11p dipole modes. A four-rod structure will have the longitudinal modes but not the dipole modes.

4. Why we need tuners. The voltage distribution along the rods or vanes are sensitive to external perturbations. This sensitivity is driven by the other modes • Perturbations can arise from a variety of effects including:> construction errors> RF drive ports> RF pickup ports • How to determine the sensitivity of the voltage distribution to perturbations? • Given an observed voltage distribution that deviates from the design profile, how can you determine where the errors are and how you can fix them? • We can use perturbation theory to help answer these questions.

5. The L0C0 resonators represents the 4 quadrants of the four vane, Dx is an incremental distance along the vanes. L(x) is the vanetip inductance per unit length, I(x) is the longitudinal vanetip current, and V(x) is the intervane voltage. Perturbation-induced mixing of the higher frequency longitudinal modes distorts the intervane voltage distribution along the vanes.This mode mixing is most important for long RFQs (long compared with the wavelength). We can use a transmission line model of the RFQ structure to understand effects of perturbations on the longitudinal field distribution

6. Wave equation describes intervane voltage Distribution. Dispersion relation relates mode frequency to cutoff frequency and wave number k. The operating mode frequency, which is assumed constant along the RFQ. End boundary conditions specifies zero longitudinal current at both ends of the vanes Transmission line model relates longitudinal vane current to voltage gradient. because currents are zero at ends of vanes Wave equation and boundary conditions for transmission line model

7. Solutions to the wave equation that satisfy the boundary conditions give an infinite number of discrete unperturbed modes with voltage distributions given by cosine wavefunctions. , lV is the vane length. The zero superscript means this is the unperturbed solution. lV is the vane length. This gives the normalization of the wave functions Resonant frequency of the nth mode

8. Voltage distribution along the vanes of the three lowest unperturbed modes of the 4-Vane RFQ The boundary conditions are satisfied by an integer number of half wavelengths along the vanes. The operating mode (uniform) is n=0.

9. Suppose we have perturbations along the RFQ that cause local resonant frequency errors. Apply standard perturbation theory. Define this as the squared local resonant frequency error as a function of position along the vanes. The corrections all depend on this function. Results: is the new or perturbed frequency of the nth mode, n=0, 1, 2, … is the new or perturbed voltage function for the nth mode.

10. New perturbed wavefunction for the n=0 RFQ operating mode The perturbed voltage error is a sum over all the other unperturbed wavefunctions. We are not directly exciting these other modes but the perturbations mix them in. Because of the m2 in the denominator the modes nearest the n=0 operating mode contribute most to the voltage error correction. The contribution of each of the other voltage cosine functions also increases in proportion to the overlap between that cosine function and the local squared frequency error distribution.

11. Fixing the unwanted perturbation • To correct such a local frequency error or perturbation you may have to introduce a tuner to compensate for it. • First you may have to find where it is. That is where perturbation theory may help you.

12. 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 G. This is the new resonant frequency of the cavity in terms of local frequency error G This relates the cavity frequency change to G. is the new wavefunction

13. Fractional vane-voltage error Each ofthe 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. An analytic solution exists for this summation. It is.

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

15. m=0 and 1 m=0 m=1 to 20 Perturbed voltage distribution for problem with a d-function error at the vane end, where x0/lV = 0, lV/l = 2 and dw0/w0 = 0.01. V (x) 0 x/lV / l x V

16. Perturbed voltage distributions for problem with a d-function error at the middle of the vane, where x0/lV = 0.5, lV/l = 2 and dw0/w0 = 0.01. V0(x) x/lV

17. Consider delta-function frequency errors at each end that are equal and opposite Resonant frequency error distribution So cavity resonant frequency is unchanged. New voltage* Fractional change in voltage distribution 17 *(Here, dw0 is the cavity frequency change for a local error at one end only.)

18. Voltage distribution for delta-function frequency errors at each end that are equal and opposite is a linear tilt. • The voltage error term is proportional to (lV/l)2. • The perturbed voltage distribution is a linear tilt. • So if you want to introduce a tilt in the voltage distribution at fixed cavity frequency, introducing equal and opposite local errors at the two ends will do it.

19. Perturbed voltage distribution for problem with equal and opposite d-function errors at the ends of the vane.x0/lV = 0,and 1, lV/l = 2 and dw0/w0 = 0.01.Result is a linear voltage tilt. V0(x) x/lV 19

20. Four rod RFQ

21. Four-Rod RFQ Four conducting rods are charged by currents flowing on the “stems”.Each stem connects to an opposite pair ofrods. The next stem connects to the otherpair of rods. Thus, opposite rods are shorted together.

22. Example: ISIS Four-Rod RFQ at Rutherford Appleton Laboratory35 keV to 665 keV H- beam, 202.5 MHz, V=90 kV

23. The charging currents for the rods flow from each stem plate to the adjacent one. The resonant circuit includes the inductance associated with the magnetic flux between stems and the capacitance between pairs of rods.

24. More about 4-Rod RFQ • Now days the rods are really short vanes cut on a milling machine, not circular rods cut on a lathe as was done in the early days. • Outer cylindrical walls carry little current and 4-rod RFQ can be reduced to compact transverse size with not much change in the resonant frequency. • The 4-rod structure is generally chosen for low RF frequencies below 200 MHz for compactness and excellent dipole mode suppression which simplifies the tuning. • The “rods” at high frequency are smaller than full vanes and can be more difficult to cool than the 4-vane.

25. End effects in the four rod RFQ Unlike for the four vane structure there is a lack of symmetry at the ends so that the two opposite rod pairs have different voltages at the ends. This produces a time-varying on-axis potential at the ends and an undesirable beam-energy change, which may perturb the longitudinal matching. This effect is believed to be small for the MSU four-rod, but is under study here and if necessary, can be fixed by proper tuning of the end cells.

26. Four Rod RFQ Lumped-Circuit Model

27. rod pair B rod pair A rod pair A rod pair B Stem connected To rod pair A Stem connected to’ rod pair B Stem connected to rod pair B One period (two cells)of a lumped circuit model of four-rod RFQ 2d

28. V(z) d z rods rods stems Inter-rod voltage is minimum where stems are located and maximum midway between stems. This is because each stem is a l/2 transmission line terminated by the capacitances • The voltage distribution varies approximately as cos(2pz/ l) from • z=0 at capacitances to z=-d/2 and +d/2 at stems, and l is rf wavelength.

29. C C C C Rod capacitance per unit length • Let C be capacitance betweenadjacent rods of a cell of length d • Capacitances in series add. Total capacitance of a cell of length d is 4C. • Capacitance per unit length is Cd=4C/d, which istotal cell capacitance divided by cell length d.

30. 4 rods I h d w Inductance per cell • Neglect inductance of the rods. • Inductance of stems per cell is the flux per unit current that charges the rods. • Flux: F=BA=m0H(hd)=m0(I/w)(hd) • Inductance L=F/I=m0hd/w. I I

31. 4 rods I h d w Lumped circuit model allows us to derive formulas for four-rod RFQ

32. 4 rods I h d w RF power dissipation per cell

33. 4 rods I h d w Stored energy per cell

34. 4 rods I h d w Quality Factor

35. 4 rods I h d w Summary of lumped circuit model of four-rod resonator • In first approximation, the capacitance per unit length Cd can be estimated as ~120 pF/m, same as for the four-vane cavity. • If capacitance and inductance are correctly estimated the model could be used to optimize the efficiency of the four-rod cavity. • If we know the geometry h, d, and w for the MSU four-rod RFQ we could plug in the numbers and compare with expected properties. How well does the model work?

36. Other RFQ structures than the four vane and four rod

37. Courtesy of P. Ostroumov Another RFQ structure is the Four Vane with Windows RFQ. The stronger intervane coupling removes the dipole mode problem.

38. Four Vane with Windows RFQ • Because of the strong magnetic coupling of the quadrants through the windows, the frequency of the quadrupole mode is lowered well below the dipole modes, removing the undesirable degeneracy with the dipole modes. • May be considered as a four-vane with windows in the vanes, or a 4-rod with individual stems supporting each of the four rods, where the stems are oriented 90 degrees apart. • This structure can be considered as an intermediate resonator between the 4-Rod and the 4-Vane RFQ. • Can be configured either with antisymmetric or fourfold symmetric windows. The antisymmetric has a larger mode separation, but has a nonzero on-axis field at entrance or exit, which might perturb the longitudinal matching of the beam at the entrance unless the end cells can be properly tuned.

39. Another version that is topologically equivalent to the four-vane with windows RFQ has four symmetric arrays of posts (also can be called stems) that are 90 degrees apart. Each array of posts supports an individual rod (short vane).