The dream of the authors was .... (4-13) where k is the Boltzmann constant and Î is the temperature. Inserting (4-11), (4-12) and (4-13) into (4-10) gives. 2. 2. 2. 2.
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH ORGANISATION EUROPEENNE POUR LA RECHERCHE NUCLEAIRE CERN – AB DIVISION
LHC-Project-Report 853
THE PRINCIPLE AND FIRST RESULTS OF BETATRON TUNE MEASUREMENT BY DIRECT DIODE DETECTION
M. Gasior, R. Jones
Abstract The fractional part of the tune value of a circular accelerator can be measured by observing the betatron oscillations of the beam on a position sensitive pick-up. In the frequency domain the betatron signal is seen as sidebands on the revolution harmonics. The bunches in the beam often have a very short length with respect to the revolution period, resulting in a wideband pick-up signal spectrum, containing many betatron lines. Classical tune measurement systems filter out just one or a few of these betatron sidebands. As a consequence, most of the betatron energy is lost and only a very small fraction remains for further processing. This paper describes a new method, referred to as Direct Diode Detection (3D), which overcomes this and a few other problems. The basic idea is to time stretch the beam pulses from the pick-up in order to increase the betatron frequency content in the baseband. This can be accomplished by a simple diode detector followed by an RC low pass filter, as used in the common envelope detection technique for demodulating AM signals. It will be shown that such a circuit can increase the betatron signal level by orders of magnitude compared to classical systems. The 3D method was recently tested in the CERN SPS and PS machines with prototype base band tune (BBQ) measurement systems. Results will be presented showing that the prototypes were sensitive enough to see betatron oscillations in both machines with no added external excitation. The SPS system is also shown to be capable of measuring tunes with a resolution of 10-5 with no explicit excitation. Copies of the BBQ prototype are currently being tested on BNL-RHIC and FNAL-Tevatron as part of the US-LARP collaboration, with a view to installing a similar system for tune measurement at the LHC. Geneva, Switzerland August 2005
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– II –
Table of contents
1.
Introduction ....................................................................................................................................1
2.
Classical approach..........................................................................................................................1
3.
Direct Diode Detection ...................................................................................................................4 3.1. Simulating Direct Diode Detection ..........................................................................................4 3.2. Analytical approach to Direct Diode Detection .......................................................................5 3.3. Suppression of the revolution frequency..................................................................................9
4.
3D vs classical approach ..............................................................................................................12 4.1. Signal to noise ratio................................................................................................................12 4.2. Further 3D signal to noise improvement ................................................................................17 4.3. Assumed simplifications of the 3D circuitry..........................................................................17 4.4. 3D method with many bunches ..............................................................................................19 4.5. Other 3D method advantages .................................................................................................19 4.6. 3D method disadvantages.......................................................................................................20
5.
BBQ Prototypes ............................................................................................................................21
6.
SPS results.....................................................................................................................................23 6.1. Mains ripple in the SPS beam spectrum.................................................................................23 6.2. Observation of transverse damper noise.................................................................................23 6.3. Tune changes induced by a collimator ...................................................................................23
7.
PS results.......................................................................................................................................27 7.1. Measurements without kicking ..............................................................................................27 7.2. Measurements with small kicks .............................................................................................28 7.3. Mains ripple in the PS beam spectrum ...................................................................................28 7.4. DC detector voltages ..............................................................................................................29
8.
Experience with beam sounds .....................................................................................................31
9.
Summary and conclusions ...........................................................................................................31
10.
Acknowledgements...................................................................................................................31
11.
References .................................................................................................................................31
– III –
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– IV –
Re
se1(t)
1. Introduction The fractional part of the tune value of a circular accelerator can be measured by observing the betatron oscillations of the beam on a position sensitive pick-up. In the frequency domain the betatron signal is seen as sidebands on the revolution harmonics. The bunches in the beam often have a very short length with respect to the revolution period, resulting in a wideband pick-up signal spectrum, containing many betatron lines. Classical tune measurement systems filter out just one or a few of these betatron sidebands. As a consequence, most of the betatron energy is lost and only a very small fraction remains for further processing. This paper describes a new method, referred to as Direct Diode Detection (3D), which overcomes this and a few other problems. The basic idea is to time stretch the beam pulses from the pick-up in order to increase the betatron frequency content in the baseband. This can be accomplished by a simple diode detector followed by an RC low pass filter, as used in the common envelope detection technique for demodulating AM signals. It will be shown that such a circuit can increase the betatron signal level by orders of magnitude compared to classical systems. The 3D method was recently tested in the CERN SPS and PS machines with prototype base band tune (BBQ) measurement systems. Results will be presented showing that the prototypes were sensitive enough to see betatron oscillations in both machines with no added external excitation. The SPS system is also shown to be capable of measuring tunes with a resolution of 10-5 with no explicit excitation. Copies of the BBQ prototype are currently being tested on BNL-RHIC and FNAL-Tevatron as part of the US-LARP collaboration, with a view to installing a similar system for tune measurement at the LHC.
Classical tune measurement systems are based on the linear processing of signals from a beam position pick-up (PU). A functional diagram of such an approach is shown in Fig. 2-1. The PU electrodes of one plane are loaded with resistances Re, representing the whole load seen by each electrode. The PU position signal (2-1)
being a difference of opposite electrode signals, contains a component related to the betatron motion of the beam. For the following analysis it is assumed that there is only one bunch circulating in the machine, since this can be considered as the most difficult case to deal with. It is also assumed that the bunch undergoes a simple betatron motion of a small, constant amplitude (no chromaticity, no momentum spread, etc. included). Under these circumstances, the position signal from a PU with impulse response hPU(t) can be modelled as ∞ ⎛ ⎞ ~ sc (t ) = cos(2π f b t )⎜⎜ hPU (t ) ∗ sb1 (t ) ∗ ∑ δ (t − nT ) ⎟⎟ + n = −∞ ⎝ ⎠
+ hPU (t ) ∗ so1 (t ) ∗
∞
∑ δ (t − nT )
n = −∞
sc(t) Re se2(t) Fig. 2-1. The principle of classical tune measurements.
where T is the machine revolution period, sb1(t) is the bunch intensity response during this time, and fb is the betatron frequency. The resulting signal can be represented as a convolution (* operator) of the PU response hPU(t) with the bunch signal, with a periodicity resulting from a further convolution with a comb of Dirac delta functions δ(t), spaced by intervals of length of T. This periodic signal is then subject to a modulation at the betatron frequency. The common mode signal so1(t) represents the beam offset signal during one revolution period. This originates from any beam offset from the centre of the PU and from imperfections in circuits subtracting the signals from opposite PU electrodes (usually a 180° hybrid or a differential amplifier), and is independent of the betatron motion. The aim of a tune measurement system is to determine the betatron frequency fb. Usually this is done by performing a spectral analysis of the PU position signal ~sc (t ) , yielding the signal magnitude spectrum ~ S c ( f ) . Employing two Fourier transform (FT) properties
FT{a (t ) * b(t )} = A( f ) ⋅ B( f )
(2-3)
and
2. Classical approach
sc (t ) = se1 (t ) − se 2 (t )
PU
(2-2)
FT{cos(2π f m )a (t )} =
A( f − f m ) + A( f + f m ) 2
(2-4)
where A(f) and B(f) are the FTs of a(t) and b(t) respectively, the magnitude spectrum of ~sc (t ) (2-2) can be calculated as ∞ n⎞ ~ 1 1 ⎛ Sc ( f ) = H PU ( f − f b ) Sb1 ( f − f b ) ∑ δ ⎜ f − f b − ⎟ + T 2 T⎠ n = −∞ ⎝
+
∞ n⎞ 1 ⎛ H PU ( f + f b ) Sb1 ( f + f b ) ∑ δ ⎜ f + f b − ⎟ + T ⎠ (2-5) 2 n = −∞ ⎝
+ H PU ( f ) So1 ( f )
∞
∑
⎛ ⎝
δ ⎜ f−
n = −∞
n⎞ ⎟ T⎠
where HPU(f), Sb1(f) and So1(f) are the FTs of hPU(t), sb1(t) and so1(t), respectively. The normalization by T arises from the fact that ⎧ ∞ ⎫ 1 FT ⎨ ∑ δ (t − nT ) ⎬ = ⎩n = −∞ ⎭ T
∞
⎛ ⎝
∑ δ⎜ f
n = −∞
−
n⎞ ⎟ T⎠
(2-6)
Analysis of (2-5) can be simplified by assuming a perfect PU with infinite bandwidth, i.e. HPU(f) = 1. In this case, the influence of the PU on the signals can be
–1–
neglected, and the response (2-2) becomes
+ so1 (t ) ∗
∞
(2-7)
∑δ (t − nT )
n = −∞
whose magnitude spectrum is
0
∞
n⎞ 1 ⎛ S b1 ( f − f b ) ∑ δ ⎜ f − f b − ⎟ + T 2T ⎝ ⎠ n = −∞ + S b1 ( f + f b ) +
∞
⎛ ⎝
n⎞
∑ δ ⎜ f + fb − T ⎟
n = −∞
⎠
+
Time Fig. 2-2. An example of time domain signals, illustrating (2-7).
(2-8)
∞ n⎞ 1 ⎛ So1 ( f ) ∑ δ ⎜ f − ⎟ T T ⎝ ⎠ n = −∞
The pulse shape spectra Sb1(f) and So1(f) give envelopes to the betatron and offset spectrum components. The time domain periodicity causes the spectrum to have energy grouped at harmonics of the revolution frequency for the offset component, and on their sidebands for the betatron component. An example of the response denoted in (2-7) is shown in Fig. 2-2. Assuming that beam signals se1(t) and se2(t) from the PU electrodes are of Gaussian shape, signals sb1(t) and so1(t) are also Gaussians. For illustration purposes the bunch length on the plot is made comparable to the revolution period and the modulated component is made unrealistically big. In real cases the bunch length and the modulated component of the PU signal are orders of magnitude smaller than in the sketch. In this example the signals also contain frequency components at DC, which is a consequence of the assumed infinite PU bandwidth. In reality this is not the case, with the average signal value being zero and the signal shapes modified by the real PU response (e.g. the signals are differentiated according to the low cut-off frequency of the PU). The magnitude spectrum of a unit amplitude Gaussian pulse of standard deviation σ ⎛ t2 ⎞ ⎟ s g (t ) = exp⎜⎜ − 2 ⎟ ⎝ 2σ ⎠
(2-9)
is given by
(
S g ( f ) = 2πσ exp − 2 π 2σ 2 f 2
)
ln(2) 0.133 ≅ 2πσ σ
(2-11)
Since the beam spectrum for short bunches extends to very high frequency, the content of signal at any one
Sb( f )
fb
So( f ) fr
Frequency
0
Fig. 2-3. An example of a beam magnitude spectrum, illustrating (2-8).
betatron sideband is very small. Assuming an infinite bandwidth PU and a beam offset resulting in a unit amplitude Gaussian pulse (2-9) seen by the PU once every revolution, the average value of the signal, i.e. the DC signal content, can be written as s p (t ) =
1 T
t0 +T
⎛ t2 ⎞ ⎟dt exp⎜⎜ − 2 ⎟ ⎝ 2σ ⎠ t =t0
∫
(2-12)
Assuming that σ ∆b
(3-11)
where ∆c is the relative discharge of the storage capacitor, and ∆b is the maximal decrease of the betatron modulation envelope. Assuming one bunch in the machine, the reference period for these changes is one revolution period, T. For this analysis it is convenient to look on the unipolar voltage on one of the peak detectors. ∆c is the difference in the hold function amplitude (3-5) for t = 0 and t = T ⎛ T⎞ ⎛ 1⎞ ∆c = 1 − exp⎜ − ⎟ = 1 − exp⎜ − ⎟ ⎝ τ ⎠ ⎝ n⎠
(3-12)
Sb( f ) So( f )
fb 0
T
=
Cf Rf T
(3-13)
For large n (τ >> T ) ∆c (3-12) can be simplified to ∆c =
1 n
(3-15)
for an interval of T. Substituting fb = q/T and using (3-11) and (3-14) yields the threshold relative time constant nd nd =
1 2 β sin (π q )
1.5
1
2
f / fr
-1 -2
t=¥ t = 30 T t = 10 T t=3T
-3 -4 -5
0.1
0.2
0.3
0.4
0.5
| f / fr | Fig. 3-21. A comparison of spectrum (3-7) for a few values of the decay time constant τ and the rectangular window spectrum (3-8) – the curve for τ = ∞.
(3-14)
As sketched in Fig. 3-23, the change in the modulation envelope is fastest around the zero crossing of the betatron motion and is given by T⎞ ⎛ ∆b = 2 β sin⎜ 2π f b ⎟ 2⎠ ⎝
0.5
0
(3-16)
fr attenuation [dB]
τ
fb
0
where n is the relative time constant in terms of machine turns n=
fb
Fig. 3-20. An example of a beam magnitude spectrum, illustrating (3-5) for tune values larger than ½. Sb(f) and So(f) are spectrum contents related to beam betatron oscillations and a beam offset, respectively..
Magnitude spectrum [dB]
ξ =4
Magnitude spectrum
Now for small arguments coth(x) ≅ 1/x, hence for large time constants, where τ > T, (3-9) can be simplified to
60
~ x
40
4 t/T
20 0 0.1
1
10
100
Normalized time constant t / T
Fig. 3-22. fr attenuation curves according to (3-9) and (3-10).
– 10 –
1000
Detector signal (a)
fb = 0.1 fr detector signal fastest change envelope b cos(2p fb t) 0
Detector signal
For time constants larger than nd the dragging effect occurs and the betatron signal from the detector is attenuated. This is seen in Fig. 3-10, showing simulation results for q = 0.1 and β = 0.05. For such conditions nd (3-16) is 32 and, therefore, the curve corresponding to the case with τ = 30 T is close to the dragging limit. Values of threshold time constant nd (3-16) as a function of β and q are plotted in Fig. 3-24 and 3-25 respectively. For a relative time constant of n = 100, no dragging is observed even for betatron oscillation amplitudes corresponding to 0.5 % of the total bunch intensity signal amplitude. If for some reason the amplitude becomes larger than this (e.g. a strong beam kick), then the amplitude of the detector output signal will remain approximately the same, providing that betatron frequency remains similar. This can be considered as an advantage, as the detector output signal amplitude can be kept below a certain limit, to avoid saturation of the first stage differential amplifier. When dragging occurs, a 3D system still gives a proper signal, but with a reduced amplitude.
(b)
1
2
3
4
5
t/T
fb = 0.5 fr detector signal fastest change envelope b cos(2p fb t) 0
1
t/T
Fig. 3-23. A sketch illustrating the calculation of the largest filter time constant with no dragging, nd (3-16). Two cases with the tunes of 0.1 (a) and 0.5 (b) are shown.
Fig. 3-24. The largest relative filter time constant nd (3-16) for no dragging as a function of the betatron modulation depth β.
Fig. 3-25. The largest relative filter time constant nd (3-16) for no dragging as a function of the tune value q.
– 11 –
dv =
4. 3D vs classical approach 4.1.
Signal to noise ratio One of the most important parameters of a tune measurements system is the signal to noise ratio (SNR). This defines the smallest signals, and hence the smallest beam oscillations that can be detected. Minimizing beam oscillations is particularly important for large hadron machines such as the LHC in order to conserve the transverse beam emittance. The dream of the authors was to construct a system so sensitive, that it would detect any residual coherent beam oscillations present due to machine imperfections, without the need for explicit excitation. As it will be shown in the next chapters, the 3D method makes this possible. The comparison of the 3D method with the classical approach based on filtering a single betatron harmonic should be made in such a way as to be independent of the measurement conditions. The simplest way of doing this seemed to be to compare the spectral densities of both signal and noise for the two approaches. In this way the comparison is independent of tune spread and betatron oscillation amplitude. It is assumed that there is a single bunch in the machine, which undergoes simple betatron oscillation of a small, constant amplitude. During this analysis some real numbers will be plugged into equations, to give them a quantitative meaning. The numbers used and therefore results obtained should be treated as indicative. The aim of the noise analysis is to reveal the principal limits and allow for the optimization of the 3D circuitry. If Sd and Nd are magnitude spectrum signal and noise densities of a system using the 3D method and Sc and Nc are the corresponding quantities of the reference system using the classical one betatron harmonic processing, then the improvement factor, or the gain, of the 3D method can be defined as GD =
Sd / N d S /S G = d c = S Sc / N c N d / N c GN
S GS = d Sc
(4-1a)
dv =
χ=
are the ratios of the signal and noise levels respectively, for the 3D and reference classical systems. To calculate GS one can assume that both systems use the same PU, with its electrodes terminated in the same manner. Furthermore, the 3D circuitry is considered to be ideal, using perfect diodes and with the detectors not representing a significant load for the electrodes. For the reference system a beam charge change dρb observed on the PU electrode due to betatron motion gives the corresponding voltage change. If an electrode has a capacitance Cpu, then the voltage change is
(4-3)
C pu
(4-4)
C pu + C f
with respect to the reference system, which does not have any capacitive load on the PU electrodes. Here it is assumed that the instantaneous voltage on the storage capacitor is independent of Rf , as the bunch pulses are considered to be very short, making capacitor reactance dominant. Assuming that the reference system is operating at a frequency well below the bunch energy spectrum cut-off fcg (2-11) and within the PU bandwidth, then the spectral content of a unit amplitude betatron signal is defined by (2-16) and gives 2πσ 2T
Sc =
(4-5)
The corresponding spectral content for the 3D system is described by (3-5). At the betatron frequency it is just the hold function spectrum (3-7) divided by 2T, giving Sd =
1 τ (1 − exp(− j 2π q − T / τ )) 2T 1 + j 2π q
(4-6)
The signal increase GS (4-1a) is then the quotient of (4-6) and (4-5) with the attenuation factor (4-4) GS =
1
⋅
C pu
2πσ C pu + C f
⋅
τ (1 − exp(− j 2π q − T / τ )) 1 + j 2π q
(4-7)
As discussed before, the observed fractional tune, q, can be assumed to be less than 0.5, and with τ >> T (4-7) simplifies to GS =
(4-1b)
1 dρ b C pu + C f
The voltage at the detector output is then attenuated by a factor
and N GN = d Nc
(4-2)
In the 3D system, when the diode is conducting, the PU capacitance is connected to the storage capacitance Cf. In this case, and assuming an ideal diode (in particular no voltage drop), the voltage change is
(4-1)
where
1 dρ b C pu
T
⋅
C pu
2π σ C pu + C f
⋅
sin(π q) πq
(4-8)
This can also be linked to the duty cycle η (2-14) to give GS =
C pu 4 sin( π q) ⋅ ⋅ πq 2π η C pu + C f
(4-9)
The increase of the 3D signal with respect to the classical measurement method is seen to be inversely dependent on the bunch length. This is a consequence of the fact that the 3D system makes use of the whole beam spectrum (within the PU and detector bandwidths). The smaller the bunch length, the wider the spectrum and the less signal is seen by the classical system looking at a single betatron harmonic, while the signal of the 3D system remains unchanged, as long as the bunch signal
– 12 –
amplitude does not change. GS (4-9) is listed in Tab. 4-1 for Cf = Cpu and q = 0.3. It can be seen that for a single short bunch in the machine the signal improvement factor GS grows with T, i.e. the machine size, increasing from 360 (≈51 dB) for the PS to 61000 (≈96 dB) for the LHC. Unfortunately, the signal increase GS is not the final gain of the SNR obtained by the 3D method. This is because the 3D circuitry is of high impedance, and as such introduces significantly more noise than a good 50 Ω input amplifier of a classical system. The noise performance of one branch of the 3D circuitry as in Fig. 3-1 with a high impedance amplifier, can be modelled by the circuit shown in Fig. 4-1. The model contains: • the noise current source InD, representing the shot noise associated with the diode leakage current with reverse polarity; • the thermal noise of the discharge resistor Rf, represented by InR; • the noise current of the high impedance amplifier, InA; • the noise voltage of the amplifier, VnA. The model assumes that the noise performance is defined by the circuit behaviour when the diodes are switched off, i.e. the state in which they reside most of the time. All noise contributions when the diodes conduct are neglected (e.g. noise of the diodes and of terminating resistors), as they are considered much smaller than the contributions of the model of Fig. 4-1. It is also assumed that the PU does not introduce any noise (no losses, which is not far from reality for a stripline or a capacitive PU), nor does the beam itself. The total noise voltage squared per unit bandwidth (the squared voltage noise spectral density) Vn2 seen on the amplifier input is
(
Vn 2 = 2 VnA 2 + Z RC
2
(I
nD
2
+ I nR 2 + I nA 2
))
(4-10)
where |ZRC| is the magnitude of the RC filter impedance Z RC =
Rf 1 + (2π f R f C f ) 2
(4-11)
The shot noise current squared of the diode leakage current is I nD 2 = 2 e I RD
(4-12)
where e is the elementary electron charge and IRD is the DC diode leakage current with reverse polarity. The thermal noise current squared of the resistor is I nR 2 =
4k Θ Rf
(4-13)
where k is the Boltzmann constant and Θ is the temperature. Inserting (4-11), (4-12) and (4-13) into (4-10) gives ⎞ ⎛ 4k Θ T 2 R f 2 ⎜ 2 e I RD + + I nA 2 ⎟ ⎟ ⎜ Rf ⎠ ⎝ Vn = 2 VnA 2 + 2 2 T + ( 2π q R f C f )
(4-14)
Now assuming that the classical system has the noise
InA Cf
Rf
VnA
InD
InR
1 2
2
Vn
Fig. 4-1. Noise model of the 3D circuitry in Fig. 3-1 (one detector shown). Noise contributions: InD – diode leakage current shot noise, InR – Rf thermal noise, InA, Vna – current and voltage noise of the amplifier.
voltage Vnc, then GN as defined in (4-1b) can be written as ⎞ ⎛ 4k Θ T 2 R f 2 ⎜ 2 e I RD + + I nA 2 ⎟ ⎟ ⎜ Rf 2 ⎠ ⎝ GN = VnA 2 + 2 2 VnC T + (2π q R f C f )
(4-15)
Taking (4-7) and (4-15) yields the general formula for the overall gain in signal to noise, GD (4-1)
(
GD =
−2 VnC T R f C f C pu 1 − exp − j 2π q − T ( R f C f ) ⋅ 1 + j 2π q πσ C pu + C f
⎛ ⎞ 4k Θ + I nA 2 ⎟ T 2 R f 2 ⎜ 2 e I RD + ⎜ ⎟ R f ⎝ ⎠ VnA 2 + 2 2 T + ( 2π q R f C f )
) (4-16)
This expression is plotted in Fig. 4-2 as a function of Rf and Cf for q = 0.3. Machine parameters were taken as those listed in Tab. 2-1. The parameters related to 3D circuitry, namely Cpu, IRD, InA, VnA, VnC, were taken from Tab. 4-2, and can be considered as approximating the actual parameters of the built BBQ prototypes described in the next chapter. The noise spectral density Vnc of a classical reference system was assumed as 1 nV/ Hz . Although the thermal noise spectral density corresponding to a 50 Ω electrode termination resistor is 0.9 nV/ Hz , this can be neglected at higher frequencies since it is shunted by the small reactance of the PU electrode capacitance. This assumed noise spectral density is therefore the noise contribution of a 180° hybrid and an amplifier or of a differential amplifier. Since in the classical case the first amplifier has to have a huge dynamic range, it is difficult to make it low noise. The maximum gain in signal to noise of the 3D system with respect to a classical system is shown in Fig. 4-2 to be some 35 dB for the PS and up to some 60 dB for the LHC. It can be seen that there is an optimum value for Cf, whereas increasing Rf always improves GD. In reality, as was shown in Chapter 3, the product Rf Cf, i.e. the filter time constant τ, cannot be made too large in order to avoid the dragging effect. For further discussions and quantitative estimates a time constant of 100 T will be systematically assumed. Time constants below T do not give significant fr suppression and above 1000 T are likely to be difficult to realize, due to the inconveniently large resistor values required,
– 13 –
(a)
(b)
(c)
(d) Fig. 4-2. 3D signal to noise gain factor GD (4-16) as a function of the storage capacitor Cf and discharge resistor Rf, with respect to a classical reference system. Machine parameters were taken as those in Tab 2-1 and circuit parameters as those in Tab. 4-2.
(e)
especially for large machines (i.e. of large T). Rewriting the filter impedance magnitude (4-11) as Z RC =
Z RC =
Rf Rf Cf ⎛ 1 + ⎜⎜ 2π q T ⎝
⎞ ⎟ ⎟ ⎠
2
(4-17)
T 2π qC f
(4-18)
i.e. the resistance Rf is much larger than the reactance of Cf, and can be neglected. With this assumption, the noise voltage (4-14) becomes
and considering τ = Rf Cf as much larger than T, one obtains – 14 –
Vn = 2 VnA 2 +
⎛ ⎞ ⎜ 2 e I RD + 4 k Θ + I nA 2 ⎟ ⎜ ⎟ R ( 2π qC f ) ⎝ f ⎠ T2
2
(4-19)
Tab. 4-1. A quantitative comparison of classical and 3D tune measurement approaches.
PS (*)
Parameter (#)
Signal gain GS (4-7) with Cf = Cpu and q = 0.3 SNR gain GD (4-20) with Rf, Cf optimized for q = 0.3 (Tab. 4-4) Ratio GS /GD, i.e. noise increase GN (4-15) (*) (#)
dB dB dB
RHIC Tevatron SPS (*)
51.1 35.8 15.3
69.3 48.0 21.3
69.2 45.5 23.7
LHC
77.9 53.8 24.1
95.7 63.4 32.3
with LHC beam. GS is 1.2 dB higher for q = 0.1 and 2.6 dB lower for q = 0.5.
Tab. 4-2. Values of parameters related to 3D circuitry, used for quantitative estimates. The values approximate parameters of the BBQ prototypes.
Pick-up electrode capacitance Diode leakage current (with reverse polarity) (*) Voltage spectral noise density of the high impedance detector amplifier Current noise spectral density of the high impedance detector amplifier Voltage noise spectral density of the classical reference system (#) (*) (#)
Cpu IRD VnA InA Vnc
50 pF 5 nA 5 nV/ Hz 3 pA/ Hz 1 nV/ Hz
Θ k e
300 K 1.38⋅10-23 J/K 1.6⋅10-19 C
The corresponding spectral density of shot current noise is 40 fA/ Hz . It has to include losses of the hybrid and of the first amplifier.
Tab. 4-3. Constants used for quantitative estimates.
Temperature Boltzmann constant Elementary electron charge
Tab. 4-4. Examples of optimal Rf and Cf values, calculated from (4-26) and (4-24) for a few operating frequencies. The filter time constant was 100 T, machine parameters as in Tab. 2-1 and other noise model parameters as in Tab. 4-2.
Parameter
PS
Optimal Rf value calculated at f = 0.1 fr Optimal Rf value calculated at f = 0.2 fr Optimal Rf value calculated at f = 0.3 fr Optimal Rf value calculated at f = 0.5 fr Optimal Cf value calculated at f = 0.1 fr Optimal Cf value calculated at f = 0.2 fr Optimal Cf value calculated at f = 0.3 fr Optimal Cf value calculated at f = 0.5 fr
MΩ MΩ MΩ MΩ pF pF pF pF
The signal to noise gain of the 3D system, assuming a rectangular hold function spectrum (3-8) can then be written as VnC GD = VnA 2 +
T
⋅
C pu
2 π σ C pu + C f
⋅
sin(π q) πq
⎛ ⎞ T2 ⎜ 2 e I RD + 4k Θ + I nA 2 ⎟ 2 ⎜ ⎟ Rf (2π q C f ) ⎝ ⎠
(4-20)
The noise performance of the 3D circuitry is described by the denominator. For small machines T is small, and the reactance of Cf is small, so the total current noise I nT = 2 e I RD
4k Θ + + I nA 2 Rf
RHIC Tevatron
4.2 7.6 11 16 50 28 20 13
GD =
9.0 16 21 31 142 82 60 41
LHC
11 19 26 38 202 118 87 60
19 32 43 62 463 277 207 144
C pu VnC T sin(π q) ⋅ ⋅ ⋅ VnA 2 π σ C pu + C f πq
(4-22)
In this case, GD does not change significantly (< 4 dB) with the observed tune q, due to the hold function spectrum shape. For larger machines T increases, and the denominator of (4-20) becomes dominated by the total current noise InT. This is due to the fact that as the Cf reactance gets smaller, the voltage developed by the noise current becomes more important. In this case VnA can be neglected and VnC GD =
(4-21)
⋅
C pu C f
2 π σ C pu + C f 2 e I RD +
does not develop significant voltage on Cf (i.e. output of the 3D circuitry). If this voltage is small with respect to VnA, the noise current total can be neglected, and
11 19 25 37 191 112 82 57
SPS
⋅ sin(π q)
4k Θ + I nA 2 Rf
(4-23)
and the 3D circuitry noise performance improves with q. Signal to noise optimization relies on finding the
– 15 –
34
46
32 30
Optimized at 0.1 fr 0.2 fr 0.3 fr 0.5 fr
PS
28 0.1
0.2
0.3
0.4
0.5
0.1
52
Optimized at 0.1 fr 0.2 fr 0.3 fr 0.5 fr
38 0.1
0.2
0.3
50 48
SPS
46 0.4
0.1
0.5
f / fr
(c)
0.3
(b)
44
Tevatron
0.2
Optimized at 0.1 fr 0.2 fr 0.3 fr 0.5 fr 0.4
0.5
0.4
0.5
f / fr
54
42
RHIC
40
46
40
0.2
Optimized at 0.1 fr 0.2 fr 0.3 fr 0.5 fr 0.3
f / fr
(d)
Fig. 4-3. 3D signal to noise gain factor GD (4-20) as a function of the observed signal frequency. GD was calculated with Rf and Cf optimal values as those in Tab. 4-4, calculated for the four operating frequencies specified on the plots. Machine parameters were taken as those in Tab. 2-1 and other model parameters as those in Tab. 4-2.
64 62
GD [dB]
44 42
f / fr
(a)
GD [dB]
GD [dB]
48
GD [dB]
GD [dB]
36
60 58
Optimized at 0.1 fr 0.2 fr 0.3 fr 0.5 fr
LHC
56 0.1
0.2
0.3
0.4
0.5
f / fr
(e)
maximum gain GD (4-20), for given noise parameters of the 3D circuitry and a given time constant. Large time constant are favourable for decreasing the noise contribution from Rf. However, if one asks for a given time constant, this has to be done at the expense of decreasing Cf which in turn makes all noise current contributions more important. In addition, increasing Cf decreases the signal, due to the divider Cpu / Cf. For these reasons, the surfaces in Fig. 4-2 show that Cf has an optimum value which is Rf dependent. The optimal value of Cf can be found by adequate analysis of (4-20) and shown to be
Inserting in (4-24) and solving for Rfo, yields R fo =
C fo
1 ⎛ ⎞⎞ ⎜ C pu 4 k Θ ⎟ ⎟ ⎞3 ⎟ + 4 k Θ⎜ − 1⎟ ⎟ 1 ⎟ ⎜⎜ ⎟⎟ ⎟ 2 3 ⎠ ⎟ ⎝ C pu A ⎠⎠
(
(4-24)
(4-25)
)
A = 2 B⎛⎜ 3B + 3T 3B 2 − C pu ( 4k Θ) 3 ⎞⎟ − C pu (4 k Θ) 3 ⎝ ⎠
(4-26)
(4-26a)
and B is defined as 3
Cfo is, as expected, Rf dependent. For a given time constant, the optimal Rf value, denoted as Rfo, can be found by requiring C fo R fo = nT
3I nC 2
⎛ ⎜⎛ A ⎜⎜ ⎜ ⎜⎝ C pu ⎜ ⎝
where A is defined as
1
2 ⎛ ⎞⎞ 3 ⎞ ⎛ ⎛ T 4k Θ ⎟⎟ ⎜ 2 e I RD + + I nA 2 ⎟ ⎟ = ⎜ C pu ⎜⎜ ⎟⎟ ⎜ ⎜ Rf ⎝ 2π q VnA ⎠ ⎝ ⎠⎠ ⎝
1
B = 3πVnA q I nC 2 n 2
(4-26b)
I nC = 2 e I RD + I nA 2
(4-26c)
where
Examples of optimal Rf values, calculated according to (4-26), and of optimal Cf values, calculated according to (4-24), are listed in Tab. 4-4. These are given for
– 16 –
Optimized at 0.1 fr
60
80
Rf IRD VnA
Noise contribution [%]
Noise contribution [%]
80
40 20
60
20 0
Noise contribution [%]
RHIC
Tevatron
Optimized at 0.3 fr
60
SPS
LHC
PS
(b) 80
Rf IRD VnA
Noise contribution [%]
PS
80
40 20
RHIC
Tevatron
Optimized at 0.5 fr
60
SPS
LHC
Rf IRD VnA
40 20
0
(c)
Rf IRD VnA
40
0
(a)
Optimized at 0.2 fr
0
PS
RHIC
Tevatron
SPS
LHC
(d)
PS
RHIC
Tevatron
SPS
LHC
Fig. 4-4. Noise contributions resulting from analysis of (4-19). Optimal values of Rf and Cf were taken as those in Tab. 4-4, calculated for the four operating frequencies (0.1fr, 0.2fr, 0.3fr, 0.5fr). Machine parameters were taken as those in Tab. 2-1 and other model parameters as those in Tab. 4-2.
several tune values, with the filter time constant set to 100 T. Figure 4-3 shows the GD curves as a function of the observation frequency for several machines using Rf and Cf values optimized for a given tune. It can be seen that the change in gain over the whole tune range is only slightly affected by the choice of tune used for Rf and Cf optimization. Figure 4-4 shows the relative noise contributions from Rf, IRD and VnA for various machines and for various tune values, with Rf and Cf optimized for a filter time constant τ of 100 T. InA is not shown, as it contributes much less than a percent for all shown cases. It can be seen that for each machine as the relative Rf contribution decreases for higher tunes, that of VnA increases, while IRD remains practically constant. This can be explained by the fact that for increasing frequency Cf represents a smaller reactance, and in consequence, noise currents from Rf and IRD develop less noise voltage on the amplifier input. Since VnA is constant for all tunes, its relative contribution therefore increases. For the PS VnA dominates for all tune values, while for the LHC IRD contributes the most. For other three machines the dominant contribution depends on the q value for which Rf, Cf were optimized Figure 4-5 shows the influence of changes of Rf, Cf, IRD and VnA on GD (4-20) for a fixed tune of 0.3 fr. Each variable is changed independently within two orders of magnitude. The corresponding changes in GD are plotted with respect to the standard GD value, GD0 (i.e. GD value as in Tab. 4-1). Note that changes in GD are not very large, despite the big parameter sweeps, indicating that the choice of component values for the 3D circuit is not critical.
4.2.
Further 3D signal to noise improvement So far it has been assumed that the diode leakage current IRD and the amplifier noise voltage VnA are 5 nA and 5 nV/ Hz respectively, corresponding to the approximate values achieved in the first prototypes. To check how much the noise performance could be improved, the signal to noise gain of the 3D system, GD (4-20), was calculated for the same filter time constant τ of 100 T and an order of magnitude improvement in IRD and VnA, i.e. 0.5 nA and 0.5 nV/ Hz respectively. These values would be extremely difficult to achieve in practice and should be considered as lower limits. Table 4-5 shows the results of these calculations and the improvement that could be gained over the original SNR figures listed in Tab. 4-1. The improvement is seen to be very small, with the largest increase in signal to noise of 7 dB for the PS. Figure 4-6a shows the relative noise contributions using the improved values of IRD and VnA. The noise performance is now determined nearly entirely by Rf. Further improvement can therefore only be obtained by increased the filter time constant, to lower the thermal noise caused by Rf. The noise contribution for such a situation is shown in Fig. 4-6b, for which the time constant was increased to 1000 T. The corresponding GD values are listed in Tab. 4-5, with the largest increase in signal to noise of 15 dB for the PS. 4.3.
Assumed simplifications of the 3D circuitry The noise analysis in the previous sections was performed assuming ideal diodes. In particular the following effects were ignored:
– 17 –
4
GD0 = 35.8 dB
2 0 -2 -4 -6
Rf Cf IRD VnA
PS
-8
GD0 = 48.0 dB
2
GD / GD0 [dB]
GD / GD0 [dB]
4
0 -2 -4 -6
RHIC
-8 10
- 10 0.1
0.2
0.5
1
2
5
0.1
10
0.2
1
2
5
10
(b)
(a) 4
4
GD0 = 45.5 dB
2 0 -2 -4 -6
Rf Cf IRD VnA
Tevatron
-8
GD0 = 53.8 dB
2
GD / GD0 [dB]
GD / GD0 [dB]
0.5
Parameter relative change
Parameter relative change
0 -2 -4 -6
Rf Cf IRD VnA
SPS
-8 - 10
- 10 0.1
0.2
0.5
1
2
5
0.1
10
0.2
0.5
1
2
5
10
Parameter relative change
Parameter relative change
(c)
(d) 4
GD / GD0 [dB]
Rf Cf IRD VnA
Fig. 4-5. 3D signal to noise gain factor GD (4-20) as a function of the circuit parameters with respect to the reference gain GD0, calculated for standard parameter values. All curves are plotted for frequency of 0.3 fr, at which Rf and Cf values were optimized for each machine; their values are listed in Tab. 4-4. Values of machine specific parameters were taken as those in Tab. 2-1 and noise model parameters as those in Tab. 4-2. Coordinates (0 dB,1) correspond to the standard parameter values.
GD0 = 63.4 dB
2 0 -2 -4 -6
Rf Cf IRD VnA
LHC
-8 - 10 0.1
0.2
0.5
1
2
5
10
Parameter relative change
(e) • Diode forward voltage. In practice a small-signal silicon Schottky diode can be considered as a switch for voltages from a fraction of a volt onwards. In the BBQ prototypes the diodes were operated with a small DC bias current, to enlarge the linear region of their operation. Nevertheless, for PU electrode voltages below a volt, the detector signals should be considered as significantly smaller than the peak values. • Diode series resistance. It was assumed that the diodes do not have any resistance and the storage capacitors are charged with no time constant. Small-signal silicon Schottky diodes have considerable series resistances, in the tens of Ω range, resulting in significant charging
time constants, especially for the very short beam pulses coming from the PU. This effect is nevertheless small for large time constants, where the storage capacitor voltage does not decay significantly between two consecutive bunch passages, such that the charging current remains relatively small. • Diode parasitic capacitance. This capacitance, in the order of a pF, forms a capacitive divider with the storage capacitor. This causes a fraction of the beam pulses to appear at the storage capacitor in addition to the desired signal. The parasitic signal increases the dynamic range of the signal at the input of the differential amplifier.
Tab. 4-5. Noise performance of the 3D circuit with the diode leakage current IRD and the amplifier voltage noise VnA improved by a factor of 10.
Parameter
PS
GD (4-20) with IRD and VnA improved 10 times and τ = 100 T Difference with respect to the standard GD values in Tab. 4-1 GD (4-20) with IRD and VnA improved 10 times and τ = 1000 T Improvement with respect to the standard GD values in Tab. 4-1 – 18 –
dB dB dB dB
35.8 6.9 50.8 15.0
RHIC Tevatron 48.0 3.2 60.3 12.3
45.5 2.6 57.2 11.7
SPS
LHC
53.8 2.5 65.4 11.6
63.4 2.3 73.8 10.4
Rf IRD VnA
60
Noise contribution [%]
Noise contribution [%]
80
40 20
80 60 40 20
0
(a)
Rf IRD VnA
0
PS
RHIC
Tevatron
SPS
LHC
(b)
PS
RHIC
Tevatron
SPS
LHC
Fig. 4-6. Noise contributions for IRD and VnA improved one order of magnitude. (a) Standard filter time constant of 100 T. (b) Increased filter time constant of 1000 T.
All these details which were not included in the present analysis are unlikely to degrade the calculated improvement by more than a few dB, providing that the diodes work in the linear regime. On the other hand, the measurements presented in Section 6.2 will show that the 3D method can give unprecedented sensitivity even for low intensity beams, with the diodes operated with signals much below the linear regime. 4.4.
3D method with many bunches All the previous noise analyses were performed assuming a single bunch in the machine. In theory, for a large time constant, adding more bunches does not lead to a significant increase in the signal level, and only reduces the revolution frequency content (see the simulation results in Fig. 3-8 and 3-9). In practice, however, the signal does increase, due to the fact that the storage capacitor requires less charging current per bunch, so making the series resistances of the diodes less important. This leads to a reduction in the diode forward voltage, increasing the detector voltages, especially for small beam signals. For many bunches in the machine one can operate the 3D system in a completely different regime, namely with much smaller time constants. This makes it possible to use low impedance, low noise amplifiers. When the BBQ prototypes were operated with many bunches (e.g. a full 200 MHz fixed target beam in the SPS), the observed spectral noise floor was defined by the beam. It was observed that once the beam was injected into the machine, the noise floor of the BBQ signals rose by a few tens of dB. The most likely explanation for this is that the variation in bunch amplitudes gets downmixed in the same way as the betatron signal, appearing as a baseband noise floor. The spectral distribution of this noise was seen to have a negative slope, i.e. dominated by low frequency noise, which changed with beam conditions. This phenomenon is still not fully understood. However, it means that for operation with many bunches, the noise performance of the BBQ prototypes is defined by the beam, and any improvement in the electronic noise would not result in significantly better SNR for the observed signals.
4.5.
Other 3D method advantages The main advantage of the 3D technique is the signal to noise improvement discussed in the previous sections. There are, however, several other advantages, which are discussed below. Simplicity and cost The 3D method can work with virtually any position PU as the signal sensor, and does not require the complication of a resonant or moveable PU. In addition it does not require the use of ultra low noise amplifiers nor mixers, to down-convert the detected frequency. Revolution frequency suppression The other important feature of the 3D method is the fact that the revolution frequency is already considerably suppressed before the first amplifier stage. This removes both the need to centre the beam by using electronic or mechanical means and all the electronics required for feedback on such systems to maintain the centring during orbit variations. Robustness against saturation The harmonics of the revolution frequency are converted by the 3D system to a DC voltage. This is easily eliminated using a series capacitor between the storage capacitor and the differential amplifier input. In addition, the frequency information is kept even if the input signal is clamped by the amplifier. This makes the 3D method very robust against saturation and in consequence, gain control of consecutive amplifier stages can, to a large extent, be relaxed. Flattening out the beam dynamic range The 3D circuitry gives a similar response for a single bunch as it does for many bunches with the same amplitude. This means that the amplitude of the BBQ signals depends mainly on the bunch intensity and very little on the number of bunches. This considerably reduces the required dynamic range of the processing chain. In the LHC case, for example, the overall dynamic range is some 100 dB, of which only ≈30 dB comes from bunch intensity changes. Note that this is the complete opposite for a system using a resonant PU. In this case the more bunches in the machine, the more efficient the resonator and the more
– 19 –
signal is obtained. For few bunches the resonance cannot be efficiently maintained for large machines, considerably reducing the available signal. As a result, the dynamic range of signals from such a system is considerably larger than that needed to take into account only bunch intensity variation. Independence from the beam filling pattern Since the 3D signal observed in the baseband originates from a wideband beam signal spectrum, it is practically immune to changes in the bunch filling pattern. This is not the case for classical or resonant systems, where the amount of signal available at a given detection frequency (usually in the MHz range) is filling pattern dependent.
bunch contributes to the detector output voltage (this is equivalent to extending the detector output bandwidth to much higher frequencies). Then the detector differential voltage is digitized with the clock synchronized to the bunch repetition, yielding samples of the detector voltage corresponding to each individual bunch. Taking the LHC as an example, for bunches spaced by 25 ns the filter time constant would be only a few times larger than the bunch spacing and the ADC clock would be of 40 MHz, synchronized to the beam.
4.6. 3D method disadvantages Operation in the low frequency range The fact that 3D-based systems are operated in the baseband means working at relatively low frequencies, where cable shielding is not very efficient. Furthermore, the working band is close to the mains frequency and its low harmonics. Such interference is difficult to filter in a passive way. On the other hand, for these low frequencies, the performance of active components, in particular operational amplifiers, is very good. Hence it is possible to achieve effective active filtering of mains ripple and associated harmonics. In addition, such operational amplifiers have high power supply rejection ratios. Practice showed that with a careful design of the power supplies, PCBs and grounding scheme, it is possible to keep the signal spectrum free from interference in the range of interest. Response dominated by largest bunches When the 3D circuitry is operated with large time constants and with many bunches in the machine, the slow decay of the storage capacitor voltage causes the diodes to switch on only for the bunches with highest intensity. The PU pulses corresponding to lower intensity bunches are smaller than the voltage on the storage capacitor, and hence the diodes do not switch on. These bunches therefore do not contribute to the detector output voltage. A response to such lower intensity bunches may be enabled at the expense of reducing the time constant and decreasing the revolution harmonic suppression. In this case, the differential amplifier after the detectors has to cope with a larger dynamic range at the input. No bunch to bunch tune measurement The BBQ system as described is only sensitive to the bunch majority and will not give bunch to bunch tune information, since it does not have sufficient time resolution after the diodes. It might, however, be possible to gate the signal to allow the possibility of measuring the tune of a specific bunch or group of bunches. In such a set-up a fast switch could be used to pass the PU signals to the diode detectors only during the period of interest. Alternatively, gating may be used at the detector output. In such a scheme, a moderate filter time constant of the detector filters is used, to make sure that each – 20 –
5. BBQ Prototypes A block diagram of the Base Band Q Measurement System is depicted in Fig. 5-1. It consists of diode detectors, a front-end with amplifiers and filters, and an observation system. The diode detectors were built as separate boxes, which could be connected directly to the pick-up outputs. A photograph of one detector is shown in Fig. 5-2. Two pairs of such detectors for horizontal (H) and vertical (V) planes, were connected through ~1.5 m of cable each to the front-end. This allows the front-end box to be located further from the beam so that it is less affected by radiation. The capacitance of the cables contributed to the value of the storage capacitor Cf of the peak detectors. A photograph of the front-end box is shown in Fig. 5-3. It is seen that the power supply is placed in a separate compartment from the two processing channels. Since the maximum gain of each channel could be as high as 100 dB and that analogue processing was carried out at low frequency, the PCB layout, power supply and grounding had to be done very carefully. A few front-end parameters could be changed remotely over a parallel logic bus. This included the value of the time constant, turning on and off the DC bias current for the diodes, connecting the output stage directly to the DC voltage on the detectors, and changing the front-end gain. The gain could be varied from 30 dB to 100 dB in 5 dB steps. The detectors used in the PS, SPS, RHIC and
Re PU
Input Select
Cs D
Cf
Rf
+
DA
Re
Tevatron were all identical, while the front-ends for these four machines were built using identical PCBs. Only the machine specific parts, in particular the filters, used different component values, which had to be optimized for each machine. The notch filter was adjusted to the revolution frequency fr while the other filters to give the 3 dB cut-offs at about 0.1 fr and 0.5 fr. The only difference was the front-end version built for the Tevatron, which operates at tunes very close to 0.5. In this case, the low cut-off could be higher and adjustment of the high cut-off required particular attention. The frequency characteristic of the Tevatron front-end is shown in Fig. 5-4. Note that the attenuation of the revolution frequency is ≈100 dB (i.e. five orders of magnitude). This attenuation level was achieved for all machines. The front-end frequency characteristic was measured without detectors, which add important additional fr suppression. Furthermore, the measurement in Fig. 5-4 does not reflect the fr suppression due to the fact that the beam is usually quite close to the centre of the pick-up. This further reduces fr since the resulting common mode signal from opposite electrodes is eliminated to a large extent by the first stage differential amplifier. These two additional contributions suppress fr by another two or three orders of magnitude (40-60 dB), giving an overall fr suppression of 140-160 dB. This enormous attenuation of the revolution line allows the system to have a large gain after the filters, so bringing the betatron oscillation amplitude up to a level which can be efficiently digitized. The SPS BBQ system was used with a 375 mm
D
Cf
-
Rf
Notch Filter
High Pass
Low Pass
A Gain Control
Cs
Fig. 5-2. Diode detector box. One such a box is installed on each PU electrode.
A
To Observation System
Fig. 5-1. A functional block diagram of the BBQ prototype. PU – pick-up; DA – differential amplifier; A – amplifier.
Fig. 5-3. BBQ front-end box. The left compartment accommodates a power supply, and the right one two identical processing channels.
– 21 –
stripline PU, with the detectors installed on the upstream ports and downstream ports terminated. The detectors were connected to the front-end using semi-rigid coaxial cables, reinforced with a heavy copper braid aimed at minimizing the shield resistance of the connection between the detectors and the front-end. This was to try to eliminate any voltage drop induced by possible stray currents between the tunnel installation and the acquisition system located on the surface, a few hundred meters away from the PU. Such a voltage would appear on the differential amplifier input and on the bias current of the diodes. The BBQ installation in the SPS tunnel is shown in Fig. 5-5 and Fig. 5-6. The SPS BBQ signals were acquired using a low cost 24-bit, 96 kS/s, USB sound card. All SPS measurements presented in the next chapter were acquired with a fixed sampling rate of either 96 kS/s or 48 kS/s, i.e. above the revolution frequency of 43.4 kHz. The main advantage of such a card was the ability to acquire the signals continuously for several hours, with the data stored as a few GB of WAVE files. Analysis was subsequently carried out off-line. The prototype BBQ system installed on the PS used only one channel, connected to two diode detectors installed on the H plane electrodes of a shoe-box electrostatic PU, normally used by the trajectory measurement system. Due to relatively high radiation levels (in the order of 10 kGy/year at the vacuum pipe level) the front end box was hidden in a hole in the floor. The diode detectors, however, were exposed to an integrated dose of several kGy, with no measurable degradation in their performance. The acquisition system for the PS machine had to have a sampling rate of at least 477 kHz, corresponding to the maximum revolution frequency. At first a sampling scope was used to perform these acquisitions, but the record length was limited to a few tens of ms. To overcome this problem and to increase the resolution, the oscilloscope was replaced by a 12-bit VME digitizer SIS 3100. The module allowed the acquisition of up to 256 kS per channel, stored in two memory banks. In order to obtain continuous acquisitions, software was used to read out each bank alternately, while still acquiring samples in the other bank. Due to the fact that the revolution frequency was not available for the digitizer, one of its internal sampling rates was used. To economize memory and processing speed, the rate was chosen to be close to the revolution frequency. Such a frequency was achieved by combining the internal sampling clock of 100 MHz, with the summing of 64 consecutive samples (a feature of the digitizer) and the fact, that only every 3rd sample was stored in the front-end computer memory. This resulted in an equivalent sampling rate of 100/64/3 MHz, i.e. about 521 kHz.
\
Fig. 5-4. Tevatron front-end frequency characteristic. Since Tevatron operates at tunes close to 0.5, building a filter for this machine was the most challenging. The notch filter is adjusted to the revolution frequency of 47.7 kHz.
Fig. 5-5. BBQ installation in the SPS – pick-up view. The detector boxes are inside the copper braids, on the pick-up outputs.
Fig. 5-6. BBQ installation in the SPS – front-end view.
– 22 –
6.
SPS results
6.2.
Throughout 2004 the SPS BBQ prototype system yielded a lot of data, stored as a few GB of WAVE files. In this paper only a very small fraction of this is analysed, to document most important measurements. Further measurements and their corresponding WAVE files can be found on the 3D method web site [1]. Since these measurements were obtained, the signal to noise ratio of the system has been further improved, and should lead to even better sensitivity in the future. Betatron oscillations were observed with the BBQ system without any explicit externally applied excitation throughout all SPS machine cycles for all types of beams. The only exception being a single LHC pilot bunch, where no signal was seen in the vertical plane once injection oscillations had died down. 6.1.
Mains ripple in the SPS beam spectrum One of the largest surprises after installing the first BBQ prototype on the SPS was the fact that the mains ripple harmonics were present in the beam spectrum. Many narrow lines were seen in the vicinity of the tune paths and these lines did not follow any tune changes. Such a case is shown in Fig. 6-1. This data was taken with no explicit beam excitation during a coast with an LHC type beam, consisting of 72 bunches separated by 25 ns. A control room driven tune jump by some 40 Hz, i.e. ≈0.001 fr, is clearly seen some 12 s into the acquisition. Before this jump the tune was sitting right on a mains ripple harmonic at 5.55 kHz, with the neighbouring harmonic at 5.5 kHz not very pronounced. Once the tune was changed, the level of the 5.55 kHz line is seen to drop substantially (≈25 dB), while that of the 5.5 kHz line is increased (≈10 dB, see Fig. 6-1e and 6-1f). The fact that the amplitude of these lines change with the proximity of the tune line suggest that the ripple is acting on the beam, and is not introduced by the BBQ electronics. This effect can even be noticed in the time domain record of Fig. 6-1a, where the signal is seen to decrease when the tune moves off the mains harmonic. These resonances are thought to be driven by the magnetic filed ripple of the SPS dipoles. It is difficult to imagine how mains ripple harmonics can have important amplitudes at a few kHz. However, since the effect is caused by a large number of magnets and the beam is very sensitive at frequencies close to the tunes, even a very small field ripple can make the beam to oscillate with amplitudes in the micron range. Mains ripple harmonics in the BBQ beam signal spectrum have also been seen in the PS, Tevatron and RHIC. This phenomenon is being extensively studied at RHIC to try to quantify its impact on the beam emittance and lifetime, as well as its effect on a 3D-based PLL tune tracker. It remains to be seen what, if any, influence of such effects may have on LHC tune measurements and performance. Notice the large tune peak amplitudes yielded by the BBQ prototype with no explicit beam excitation. In Fig. 6-1e the tune peak is more than 40 dB above the noise floor.
Observation of transverse damper noise The high sensitivity of the BBQ system allowed the unprecedented observation of noise introduced by the transverse damper system. The BBQ signals and their spectra are shown in Fig. 6-2 for a single LHC pilot bunch of ≈5⋅109 protons. Figure 6-2a shows the signal acquired with the H plane damper off, while Fig. 6-2b presents the corresponding measurement with the damper on. The triangular tune change requested for this particular studies is clearly visible in both cases. As expected, the injection oscillations (around 0.3 s) and tune kicker oscillations (around 0.5 s) are seen to last much longer when the damper is off than when it is on. What is, however, also noticeable are the much higher signal levels which are obtained with the damper on. This is a direct effect of the noise introduced by the damper system. The effect of the damper is also seen in the corresponding spectra shown in Fig. 6-2c and 6-2d. When the damper is on, the tune path becomes much wider and betatron oscillations smaller, but they are not removed completely, and are still clearly visible by the BBQ prototype. This is a very important observation, confirming that such a BBQ system could be used in the LHC to observe single pilot bunch operation underneath the noise floor of the transverse damper system. Another feature that can be seen in the spectra is the presence of synchrotron sidebands, which are particularly visible in Fig. 6-2d. The two narrow lines seen in Fig. 6-2c at 5.4 kHz and 5.7 kHz correspond to the 18th and 19th harmonics of 300 Hz, the component which is present at the output of 3-phase power rectifiers. Traces of these harmonics are believed to be present in the magnetic field of the SPS dipoles, giving rise to measurable, forced beam motion in the vicinity of the betatron tune. Most SPS beams necessitate the use of the damper to maintain transverse stability. In spite of this, measurements performed on these beams using the BBQ system, clearly shown that it is capable of observing betatron tunes with no explicit excitation. This suggests the possibility of co-existence between a transverse damper and a 3D-based PLL tune tracking system, provided that the beam oscillations are small enough to be undetected by the damper system. In the case of the SPS, the remaining oscillation amplitude was in the micron range. This was already a very comfortable amplitude for the BBQ system, yielding a signal to noise ratio, which was usually well above 20 dB. 6.3.
Tune changes induced by a collimator During 2004 a prototype of the LHC collimators was tested in the SPS. One important measurement to be performed during these tests was that of the impedance introduced by the collimators. Collimator impedance is currently thought to be one of the limiting factors to increasing LHC luminosity. For this reason it was very important to measure the impedance of the prototypes to check if the design fits the theoretical predictions. The imaginary part of the impedance can be estimated from
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(a)
Time domain record (sound card, sampling at 48 kS/s).
(b)
Spectrum evolution of the recorded signal.
(c)
Spectrum evolution, logarithmic colour coding.
(d)
Spectrum evolution, linear colour coding.
Normalized spectrum
Normalized spectrum [dB]
1
10 s 15 s
- 10 - 20 - 30 - 40 - 50
10 s 15 s
0.8 0.6 0.4 0.2
- 60 0
5.35
(e)
5.4
5.45 5.5 5.55 Frequency [kHz]
Spectra at 10 s and 15 s, logarithmic scale.
5.6
5.65
5.35
(f)
5.4
5.45
5.5 5.55 Frequency [kHz]
5.6
5.65
Spectra at 10 s and 15 s, linear scale.
Fig. 6-1. Measurement with one LHC batch of 72 bunches coasting at 270 GeV, ≈1011 protons per bunch, RF on, done on 4/10/04 at 22h04 with no explicit beam excitation. At 12 s the tune was changed by some 40 Hz, removing it from the resonance at 5.55 kHz caused by magnetic field fluctuations due to ripple in the output of magnet power supplies. The signal was acquired with a sound card at the rate of 48 kS/s. Each spectrum segment is calculated on 48000 Hanning windowed samples, overlapping by 50 % with the adjacent sample set, i.e. there are two spectrum segments per second on the contour plots.
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(a)
H plane time domain signal with the transverse damper off (10h01).
(b)
H plane time domain signal with the transverse damper on (10h07).
(c)
Tune path with the transverse damper off.
(d)
Tune path with the transverse damper on.
Fig. 6-2. Measurement with a single LHC pilot bunch at 26 GeV, intensity ≈5⋅109, done on 2/11/04 with no explicit beam excitation. The H plane BBQ signal shown in Fig. (a) was measured with the transverse damper off (10h01) and in Fig. (b) – with the transverse damper on (10h07, damper maximal gain of 23 dB, one of the two units working). Plots (c) and (d) show the evolution of the signal spectra. Spectrum segments are normalized separately with logarithmic colour coding.
the tune change introduced by the collimator jaws as they approach the beam. The prototype BBQ system proved to be the ideal tool to accurately measure these small tune changes. The measurements presented below were all carried on 12/10/2004 with a single LHC bunch of 1011 protons, coasting at 270 GeV with the RF on. In all cases no explicit beam excitation was applied. A series of measurements were taken for various collimator jaw settings. In each case the jaws were cycled between their “out” and “in” positions at ≈21 second intervals (corresponding to one SPS supercycle period). Figure 6-3 shows the results for 5 different gaps between the collimator jaws. The spectra seem to suggest that the time spent in the “in” and “out” positions are not identical. This is due to the fact that the tune only changes when the jaws get close to the beam, i.e. during the final few mm of its movement from the 51 mm fully open position. The longer parts of the traces therefore correspond to the time
when the collimator jaws are open or not fully closed. The tune path width was crucial for the measurement and was minimized by the control room to some 10 Hz FWHM. Since beams with more bunches gave much wider tune paths, they were not adequate for this measurement. It is seen that the tune variations are from ≈2 Hz (≈5⋅10-5 fr) for the largest gap to ≈10 Hz (≈2⋅10-4 fr) for the smallest one, i.e. from 20 % to 100 % of the tune path width. Such tune variations were evaluated with a specially designed algorithm with errors ≈0.5 Hz (≈10-5 fr). This again shows the very high sensitivity and resolution of the BBQ system, allowing such measurements to be taken passively without any added external excitation.
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(a)
Gap of 4.86 mm (3h36).
(b)
Gap of 3.86 mm (3h45).
(c)
Gap of 2.86 mm (4h01).
(d)
Gap of 2.26 mm (4h22).
Fig. 6-3. Tune changes induced by the LHC collimator prototype with no explicit beam excitation. Measurements done on 12/10/04 with a single LHC bunch of ≈1011 protons coasting at 270 GeV. Spectrum segments are normalized separately with logarithmic colour coding.
(e)
Gap of 1.96 mm (4h40).
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7.
PS results
Measurements were performed with many types of PS beams of which a few are presented below. Further measurements and their corresponding WAVE files, produced from the acquired data, can be found on the 3D method web site [1]. Contrary to the SPS, betatron oscillations without explicit beam excitation were much more difficult to observe in the PS and were often visible only for parts of a cycle. In such cases the PS tune kicker was therefore used to enhance the betatron signal. However, huge signals on the BBQ prototype were obtained even when the kick amplitude was set to minimum. 7.1.
Measurements without kicking Figure 7-1 shows two measurements performed on the AD cycle, taken one minute apart. The beam consisted of four bunches of ≈4⋅1012 protons each,
injected on harmonic 8 and occupying four consecutive RF buckets. The difference in these measurements is clearly seen when comparing the time domain records of Fig. 7-1a and 7-1b. The tune path is visible throughout the whole cycle in Fig. 7-1c, while in Fig. 7-1d the tune trace disappears half way through the cycle. This coincides with the appearance of increased wideband noise in the spectrum, leading to a masking of the tune peak. The source of this noise is believed to be related to the RF beam gymnastics, which are performed when the beam is moved from harmonic 8 to harmonic 20 in steps of 2, so that the bunches eventually occupy four consecutive RF buckets on harmonic 20. The procedure is done in the longitudinal plane, but also gives rise to variations in the bunch amplitude, resulting in the phenomenon observed. Two extreme cases of betatron motion without external excitation are shown in Fig. 7-2. On the SFTPRO cycle (Fig. (a) and (c)), preparing a fixed target proton beam for the SPS, the betatron tune is only visible for very short periods of time. The beam consisted of 8 bunches of about 3⋅1012 protons each, which are
(a)
H plane time domain signal measured at 14h50 on an AD cycle.
(b)
H plane time domain signal measured at 14h49 on an AD cycle.
(c)
Spectra of the signal in Fig. (a).
(d)
Spectra of the signal in Fig. (b).
Fig. 7-1. Two H plane measurements taken on an AD cycle on 15/11/04 at 14h50 (left) and at 14h49 (right) with no explicit beam excitation; acceleration 1.4 – 26 GeV of 4 bunches of 4⋅1012 protons each. Acquisitions with SIS3300 VME digitizer started 20 ms before injection (C = 151), sampling at 520.8 kS/s. Each spectrum segment is calculated on 5208 Hanning windowed samples, overlapping by 50 % with the adjacent sample set, i.e. there are 200 spectrum segments per second. Spectrum segments are normalized separately with logarithmic colour coding.
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(a)
H plane time domain signal measured on an SFTPRO cycle.
(b)
H plane time domain signal measured on an MDSPS cycle.
(c)
Spectra of the signal in Fig. (a).
(d)
Spectra of the signal in Fig. (b).
Fig. 7-2. Plots (a) and (c) show the H plane time domain record and spectra for an SFTPRO cycle, measured on 12/11/04 at 9h49 with no explicit beam excitation. Plots (b) and (d) show the H plane time domain record and spectra for an MDSPS cycle, measured on 5/11/04 at 18h29 during machine development studies with no explicit beam excitation. Spectrum segments are normalized separately.
accelerated from 1.4 GeV to 14 GeV and split into 420 bunches. The time domain picture shows that the beam contains many components which are unrelated to the tune, and end up swamping any residual betatron oscillations. The other extreme is the MDSPS cycle, accelerating 8 bunches from 1.4 GeV to 26 GeV, for which the betatron tune is clearly visible throughout, with a signal to noise ratio of between 40 dB and 60 dB. These were uncommonly large betatron oscillations, which occurred during a PS machine development session where the machine was probably not fully optimised. 7.2.
Measurements with small kicks Figure 7-3 shows two measurements performed during AD and SFTPRO cycles in the PS. In both cases the standard tune kicker was fired at regularly intervals (10 ms for AD and 5 ms for SFTPRO), with the strength adjusted manually to give the minimum kick possible during the whole cycle. The measurements can be compared to the corresponding ones performed without kicking (AD cycle shown in Fig. 7-1, SFTPTO in Fig. 7-2). Since the kicker strength remains constant during the acceleration period, the amplitude of the resulting oscillations decrease throughout the cycle. In this mode of operation the tune is easily observed with the BBQ system throughout the cycle. This is not the
case with the current PS tune measurement system which requires an increasing kick strength with energy, often leading to important beam losses. 7.3.
Mains ripple in the PS beam spectrum Mains ripple in the beam spectrum was also observed in the PS. This is illustrated most clearly during an LHC cycle accelerating 6 bunches of ≈8⋅1012 protons from 1.4 GeV to 26 GeV, while splitting them into 72 bunches. This cycle includes two injections, spaced by 1.2 s, during which time the circulating beam parameters remain fairly stable, allowing fine details to be observed in the beam spectrum. In order to make the effect more visible, the tune kicker was fired every 10 ms with the kick strength adjusted manually to its minimum setting. The H plane time domain record and its spectra are shown in Fig. 7-4. The mains harmonics, spaced by 100 Hz in the case of the PS, are clearly visible in Fig. 7-1c. Note that the lines do not move with the tune frequency, but get amplified once the tune moves close to or coincides with their frequency. This behaviour is that of a resonator (the beam) subjected to forced oscillations at high harmonics of 100 Hz, exactly as was observed in the SPS. The PS has its own DC power generator, but is operated as an AC machine with rectifiers, giving rise to these harmonics at 100 Hz. It is believed that the very
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(a) H plane time domain signal measured on an AD cycle with the tune kicker fired every 10 ms.
(b)
(c)
(d)
Spectra of the signal in Fig. (a).
H plane time domain signal measured on an SFTPRO cycle with the tune kicker fired every 5 ms.
Spectra of the signal in Fig. (b).
Fig. 7-3. Plots (a) and (c) show the H plane time domain record and spectra for an AD cycle, measured on 15/11/04 at 14h54, with the tune kicker fired every 10 ms. Plots (b) and (d) show the H plane time domain record and spectra for an SFTPRO cycle, measured on 4/11/04 at 17h31, with the tune kicker fired every 5 ms. Spectrum segments are normalized to the largest value with logarithmic colour coding.
small residual harmonics of 100 Hz situated at around 100 kHz give rise to enough magnetic deflection when in or near the tune resonance to produce these measurable oscillations of the beam, with amplitudes in the micron range.
length variations during the RF beam gymnastics and bunch splitting is also clearly visible. Total intensity also affects the DC voltage observed and beam loss probably accounts for some of the slower decreases seen during the cycle.
7.4.
DC detector voltages The BBQ front end output amplifier can be connected directly to the detectors using DC coupling, to allow the observation of the voltage evolution on the detectors (see the block diagram in Fig. 5-1). This feature was used for diagnostic purposes to check the peak detector voltage, as well as giving a measure of the beam position, which can be obtained by subtracting the detector voltage measured on opposite electrodes. An example of detector voltage evolution is shown in Fig. 7-5 for an AD cycle and in Fig. 7-6 for an SFTPRO cycle. It can be seen that the voltage rises after injection, as the bunches get shorter during acceleration. The bunch – 29 –
(a)
(c)
H plane time domain signal for an LHC cycle, kicking every 10 ms.
Zoom on the spectra of Fig. (b), linear colour coding.
(b)
Spectra of the signal in Fig. (a), logarithmic colour coding.
(d)
Segment at 0.8 s of the spectra in Fig. (c).
Fig. 7-4. H plane measurement performed during a PS LHC cycle on 15/11/04 at 13h58, with tune kicker fired every 10 ms. Mains harmonics spaced at 100 Hz are clearly seen on and around the betatron tune.
Fig. 7-5. Low pass filtered detector voltage for a part of an AD cycle.
Fig. 7-6. Low pass filtered detector voltage for an SFTPRO cycle.
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8.
Experience with beam sounds
10. Acknowledgements
During BBQ operation on the SPS it was found that listening to the signal yielded by the system can give a lot of online information. A trained person using headphones can distinguish between the sound of betatron and synchrotron frequencies, as well as many important events during an acceleration cycle, such as injection, acceleration start, transition and ejection. Once the sound of a normal machine cycle is imprinted on the mind, it is very easy catch any anomalies which can occur during machine tuning or due to various failures. This is mainly thanks to the logarithmic characteristic of human hearing, which makes it possible to capture small details which are nearly impossible to observe by eye when presented in a graphical way. Listening to the beam can easily be done in parallel to other activities, such as operating a computer and could give a new dimension to the work of machine operators! Some beam sounds for various SPS and PS beam types can be found on the 3D method web site [1].
9.
The authors would like to thank P. Cameron (BNL), who has greatly influenced this development though his valuable comments and his willingness to try out many new ideas at RHIC. Without this solid basis this development would not have been possible. In addition we would like to thank C.-Y. Tan (FNAL) for his corresponding tests in the Tevatron. We are indebted to J. Belleman for his continuous help throughout the whole development as well as in preparing this paper. Thanks also to J.L. Chouvet for his very careful design of the 3D front-end PCBs. A final word of thanks goes to all those who participated in the evaluation of the SPS and PS BBQ prototypes, with a special mention for G. Arduini, F. Caspers and R. Steerenberg.
11. References [1] www.cern.ch/gasior/pro/3D-BBQ/3D-BBQ.html.
Summary and conclusions
The development of the direct diode detection (3D) technique was triggered by the need to find a highly sensitive and robust tune measurement system for the LHC. This report has shown in detail the principle behind this method, which has proved to give more than an order of magnitude improvement in signal to noise over current tune measurement devices. The BBQ prototypes based on the 3D method, installed in the SPS and PS machines, have been demonstrated to be capable of measuring tunes without the need for explicit excitation, even for single low intensity (5⋅109 charges) LHC pilot bunches. It has been experimentally confirmed that such oscillations, with amplitudes in the micron range, are permanently present in the beam motion. The exact way by which these oscillations are produced is, however, still not fully understood. Due to the sensitivity obtained using this simple and cheap 3D detection method, it will be gradually introduced on all CERN circular machines, to consolidate all the existing tune measurement systems. The SPS BBQ prototype will be made fully operational during 2006, with an almost identical system planned for the LHC. As well as providing a highly sensitive tune measurement system, the BBQ has also been shown to be very useful for understanding and optimising machine performance. In addition it gives a new window on the previously unknown world of sub-micron coherent oscillations in large circular machines. Copies of the BBQ prototypes are currently being tested in BNL-RHIC and FNAL-Tevatron as part of the US-LARP collaboration. The latest results from both of these machines confirm the sensitivity of the system as well as its suitability for eventual PLL tune tracking.
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