4 Impedance and Collective Effects
115
where the driving term is used here instead of dipolar and detuning instead of
quadrupolar (or incoherent). In the frequency domain, Eq. (4.21) leads to the
following generalized impedances
Z x [Ω] = x 1 Z
driving
x
− x 2 Z detuning ,
Z y [Ω] = y 1 Z
driving
x
+ y 2 Z detuning .
(4.22)
Note that in the case β = 1, another quadrupolar term is found [29].
From Eqs. (4.21) and (4.22), the procedure to simulate or measure the driving and
detuning contributions can be deduced. In the time domain, using some time-domain
electromagnetic codes like for instance CST Particle Studio [30], the driving and
detuning contributions can be disentangled. A first simulation with x 2 = 0 gives the
dipolar part while a second one with x 1 = 0 provides the quadrupolar part. It should
be noted that if the simulation is done with x 1 = x 2 , only the sum of the dipolar
and quadrupolar parts is obtained. The situation is more involved in the frequency
domain, which is used for instance for impedance measurements on a bench [31].
Two measurement techniques can be used to disentangle the transverse driving and
detuning impedances, which are both important for the beam dynamics (this can also
be simulated with codes like Ansoft-HFSS [32]). The first uses two wires excited in
opposite phase (to simulate a dipole), which yields the transverse driving impedance
only. The second consists in measuring the longitudinal impedance, as a function of
frequency, for different transverse offsets using a single displaced wire. The sum
of the transverse driving and detuning impedances is then deduced applying the
Panofsky-Wenzel theorem in the case of top/bottom and left/right symmetry [33].
Subtracting finally the transverse driving impedance from the sum of the transverse
driving and detuning impedances obtained from the one-wire measurement yields
the detuning impedance only. If there is no top/bottom or left/right symmetry the
situation is more involved [34].
Both longitudinal and transverse resistive-wall impedances were already calculated 40 years ago by Laslett, Neil and Sessler [35]. However, a new physical regime
was revealed by the CERN LHC collimators. A small aperture paired with a large
wall thickness asks for a different physical picture of the transverse resistive-wall
effect from the classical one. The first unstable betatron line in the LHC is around
8 kHz, where the skin depth for graphite (whose measured isotropic DC resistivity
is 10 μm) is 1.8 cm. It is smaller than the collimator thickness of 2.5 cm. Hence
one could think that the resistive thick-wall formula would be about right. In fact
it is not. The resistive impedance is about two orders of magnitude lower at this
frequency, as can be seen on Fig. 4.3. A number of papers have been published on
this subject using the field matching technique starting from the Maxwell equations
and assuming a circular geometry [36–40]. New results have been also obtained for
flat chambers, extending the (constant) Yokoya factors to frequency and material
dependent ones [41], as was already found with some simplified kicker impedance
models [42, 43]. Note that the material resistivity may vary with the magnetic field
115
where the driving term is used here instead of dipolar and detuning instead of
quadrupolar (or incoherent). In the frequency domain, Eq. (4.21) leads to the
following generalized impedances
Z x [Ω] = x 1 Z
driving
x
− x 2 Z detuning ,
Z y [Ω] = y 1 Z
driving
x
+ y 2 Z detuning .
(4.22)
Note that in the case β = 1, another quadrupolar term is found [29].
From Eqs. (4.21) and (4.22), the procedure to simulate or measure the driving and
detuning contributions can be deduced. In the time domain, using some time-domain
electromagnetic codes like for instance CST Particle Studio [30], the driving and
detuning contributions can be disentangled. A first simulation with x 2 = 0 gives the
dipolar part while a second one with x 1 = 0 provides the quadrupolar part. It should
be noted that if the simulation is done with x 1 = x 2 , only the sum of the dipolar
and quadrupolar parts is obtained. The situation is more involved in the frequency
domain, which is used for instance for impedance measurements on a bench [31].
Two measurement techniques can be used to disentangle the transverse driving and
detuning impedances, which are both important for the beam dynamics (this can also
be simulated with codes like Ansoft-HFSS [32]). The first uses two wires excited in
opposite phase (to simulate a dipole), which yields the transverse driving impedance
only. The second consists in measuring the longitudinal impedance, as a function of
frequency, for different transverse offsets using a single displaced wire. The sum
of the transverse driving and detuning impedances is then deduced applying the
Panofsky-Wenzel theorem in the case of top/bottom and left/right symmetry [33].
Subtracting finally the transverse driving impedance from the sum of the transverse
driving and detuning impedances obtained from the one-wire measurement yields
the detuning impedance only. If there is no top/bottom or left/right symmetry the
situation is more involved [34].
Both longitudinal and transverse resistive-wall impedances were already calculated 40 years ago by Laslett, Neil and Sessler [35]. However, a new physical regime
was revealed by the CERN LHC collimators. A small aperture paired with a large
wall thickness asks for a different physical picture of the transverse resistive-wall
effect from the classical one. The first unstable betatron line in the LHC is around
8 kHz, where the skin depth for graphite (whose measured isotropic DC resistivity
is 10 μm) is 1.8 cm. It is smaller than the collimator thickness of 2.5 cm. Hence
one could think that the resistive thick-wall formula would be about right. In fact
it is not. The resistive impedance is about two orders of magnitude lower at this
frequency, as can be seen on Fig. 4.3. A number of papers have been published on
this subject using the field matching technique starting from the Maxwell equations
and assuming a circular geometry [36–40]. New results have been also obtained for
flat chambers, extending the (constant) Yokoya factors to frequency and material
dependent ones [41], as was already found with some simplified kicker impedance
models [42, 43]. Note that the material resistivity may vary with the magnetic field