8 Accelerator Engineering and Technology: Accelerator Technology
445
measurement. The advantage of this technique is that in addition to an amplitude
response it also gives phase information. This makes it more sensitive than the single
kick method and so allows smaller excitation amplitudes to be used.
To have a fully continuous measurement of the tune when no signal is visible
due to external sources, a Phase Locked Loop (PLL) needs to be implemented.
A voltage or numerically controlled oscillator (VCO or NCO) is used to put a
sine wave excitation, Asin(ωt), on the beam. The beam response to this signal is
then observed using a pick-up and will be of the form Bsin(ωt+φ), where φ is the
phase difference between the excitation and the observed signal. A phase detector
multiplies the excitation and observed signals together, giving a signal of the form
½ABcos(−φ) − ½ABcos(2ωt+φ) which has a DC component proportional to the
cosine of the phase difference. This will therefore be zero when the phase difference
is 90 ◦ and the amplitude response is a maximum, i.e. at the tune frequency. The
aim of the PLL is to “lock-in” to this 90 ◦ phase difference between excitation
and observed signal by correcting the VCO frequency until the DC component of
the phase detector output is zero. Since the PLL will always try to maintain this
90 ◦ phase difference, the VCO frequency will track any tune changes, so giving a
continuous tune measurement.
In practice things are not quite that simple. Many parameters have to be optimised
in order to for the PLL to find, lock-in and subsequently track the tune peak. The
beam spectra and dynamics also have to be well understood if the PLL is not to
lock or jump to a spurious line, resonance, synchrotron sideband etc. In addition,
for hadron machines, the continuous excitation will lead to some emittance blowup. In order for this to be kept to a minimum the applied excitation has to be small
and therefore the observation pick-up and following electronics very sensitive. This
is less of a problem for lepton machines, where radiation damping takes care of any
emittance blow-up caused by the excitation, making PLL systems much easier to
implement on such machines.
8.6.3.2 Chromaticity Measurement
For any high energy synchrotron, the control of chromaticity is very important for
beam stability and quality. If the chromaticity is of the wrong sign (positive below
transition energy or negative above) then the beam quickly becomes unstable due to
a head-tail instability. If the chromaticity is too large then the tune spread increases
such that particles are inevitably lost as they hit resonance lines in tune space. The
most common method of measuring the chromaticity of a circular machine is to
measure the betatron tune as a function of the beam momentum and then to calculate
the chromaticity from the resulting gradient. This is usually done by varying the RF
frequency, keeping the magnetic field static. The equations of interest are:
ΔQ = (ξQ)
Δp
p
= Q
Δp
p
= Q
γ
2
t
ΔR
R
= Q
−γ 2
t γ 2
γ 2 − γ 2
t
Δf
f
,
(8.68)
445
measurement. The advantage of this technique is that in addition to an amplitude
response it also gives phase information. This makes it more sensitive than the single
kick method and so allows smaller excitation amplitudes to be used.
To have a fully continuous measurement of the tune when no signal is visible
due to external sources, a Phase Locked Loop (PLL) needs to be implemented.
A voltage or numerically controlled oscillator (VCO or NCO) is used to put a
sine wave excitation, Asin(ωt), on the beam. The beam response to this signal is
then observed using a pick-up and will be of the form Bsin(ωt+φ), where φ is the
phase difference between the excitation and the observed signal. A phase detector
multiplies the excitation and observed signals together, giving a signal of the form
½ABcos(−φ) − ½ABcos(2ωt+φ) which has a DC component proportional to the
cosine of the phase difference. This will therefore be zero when the phase difference
is 90 ◦ and the amplitude response is a maximum, i.e. at the tune frequency. The
aim of the PLL is to “lock-in” to this 90 ◦ phase difference between excitation
and observed signal by correcting the VCO frequency until the DC component of
the phase detector output is zero. Since the PLL will always try to maintain this
90 ◦ phase difference, the VCO frequency will track any tune changes, so giving a
continuous tune measurement.
In practice things are not quite that simple. Many parameters have to be optimised
in order to for the PLL to find, lock-in and subsequently track the tune peak. The
beam spectra and dynamics also have to be well understood if the PLL is not to
lock or jump to a spurious line, resonance, synchrotron sideband etc. In addition,
for hadron machines, the continuous excitation will lead to some emittance blowup. In order for this to be kept to a minimum the applied excitation has to be small
and therefore the observation pick-up and following electronics very sensitive. This
is less of a problem for lepton machines, where radiation damping takes care of any
emittance blow-up caused by the excitation, making PLL systems much easier to
implement on such machines.
8.6.3.2 Chromaticity Measurement
For any high energy synchrotron, the control of chromaticity is very important for
beam stability and quality. If the chromaticity is of the wrong sign (positive below
transition energy or negative above) then the beam quickly becomes unstable due to
a head-tail instability. If the chromaticity is too large then the tune spread increases
such that particles are inevitably lost as they hit resonance lines in tune space. The
most common method of measuring the chromaticity of a circular machine is to
measure the betatron tune as a function of the beam momentum and then to calculate
the chromaticity from the resulting gradient. This is usually done by varying the RF
frequency, keeping the magnetic field static. The equations of interest are:
ΔQ = (ξQ)
Δp
p
= Q
Δp
p
= Q
γ
2
t
ΔR
R
= Q
−γ 2
t γ 2
γ 2 − γ 2
t
Δf
f
,
(8.68)