Low-frequency quadrupole impedance of undulators and wigglers

An analytical expression of the low-frequency quadrupole impedance for undulators and wigglers is derived and benchmarked against beam-based impedance measurements done at the 3 GeV NSLS-II storage ring. The adopted theoretical model, valid for an arbitrary number of electromagnetic layers with parallel geometry, allows to calculate the quadrupole impedance for arbitrary values of the magnetic permeability ${\mu }_r$. In the comparison of the analytical results with the measurements for variable magnet gaps, two limit cases of the permeability have been studied: the case of perfect magnets (${\mu }_r\to \infty$), and the case in which the magnets are fully saturated (${\mu }_r=1$).

Figure 1

Damping wiggler chamber cross section with magnetic gap closed, $d_{dw}>t_{dw}+b_{dw}$ and $t_{dw}\ll b_{dw}$.

Figure 2

Damping wiggler at open position and the aluminum vacuum chamber (side view).

Figure 3

(a) Image of IVU without the side cover. (b) Schematic cross section of IVU in the middle of the device. The full gap $2b_i$ between the magnets can be varied from 6 to 40 mm for ID10 and from 5.7 to 40 mm for ID16.

Figure 4

Schematic layout of two magnet arrays (top and bottom) with magnetic field lines (courtesy of Clarke [6]).

Figure 5

Geometry of the vacuum chamber in planar approximation.

Figure 6

Horizontal (a) and vertical (b) betatron tune shifts versus average current for lattices with local ${\beta }_x=21\mathrm{m}$.

Figure 7

Summary results of the quadrupole impedance vs magnetic gap $d$ for IVU and DW. Blue dots and green solid line are measured data vs analytically obtained by Eq. (29) for the IVU case. Dark red dot and red solid line are for one DW.

Figure 8

Horizontal (a) and vertical (b) betatron tune shifts versus average current for lattices with local ${\beta }_x$-bumps in IDs location, where the local beta functions ${\beta }_x=31\mathrm{m}$ and ${\beta }_y=3.6\mathrm{m}$ (IVUs).