Noise Performance Studies at the HERA-p Ring

DESY HERA 03–02

Noise Performance Studies at the HERA-p Ring

S. Ivanov, O. Lebedev

Institute for High Energy Physics (IHEP) Protvino, Moscow Region, 142281, Russia

Abstract

This document is a report on experimental studies of noises inherent in the 208 and 52 MHz RF systems of the HERA-p ring of DESY. The observations were accomplished during 2002. Resolution threshold of noise power spectrum detection equal to a few units of 10–13 12/Hz was attained. The paper describes measuring technique itself and mathematics behind signal processing. An al-bum of representative data plots for each of 4 + 2 RF cavities installed is in-cluded. Phase noise in 52 MHz RF system is claimed to be the most probable candidate responsible for longitudinal degrade of beam in a luminosity run.

The work was performed in collaboration with DESY under Attachment #88 to the Agreement between DESY (Hamburg, Germany) and IHEP (Protvino, Russia), re: Noise Performance of the HERA-p Ring, April 2002.

February 2003

ii

Table of Contents 1 Preamble ............................................................................................1 2 Principles of Data Acquisition and Analysis .....................................2

2.1 I/Q Decompositions .........................................................................................................2 2.2 Spectral Power Densities ................................................................................................. 3 2.3 Cross-Correlation Function ............................................................................................. 5 2.4 Structure of Diffusion Coefficient ................................................................................... 5 2.5 Diffusion due to 208 and/or 52 MHz RF Systems .......................................................... 8 2.6 Discrete Fourier Transform ............................................................................................. 9 2.7 Appearance of the Mains Frequency Lines ................................................................... 10 2.8 Averaged Magnitude of Fourier Spectrum.................................................................... 11 2.9 Example of Signal Processing ....................................................................................... 11

3 Two-Channel Differential Amplifier ...............................................14 3.1 What for? ....................................................................................................................... 14 3.2 Design Features ............................................................................................................. 15 3.3 Performance Data Measurements .................................................................................. 17

4 Measurements of Noise in RF System.............................................19 4.1 Generalities .................................................................................................................... 19 4.2 Background Conditions ................................................................................................. 20 4.3 Cavity A, 208 MHz........................................................................................................ 22 4.4 Cavity B, 208 MHz........................................................................................................ 25 4.5 Cavity C, 208 MHz........................................................................................................ 28 4.6 Cavity D, 208 MHz........................................................................................................ 31 4.7 Cavity A, 52 MHz.......................................................................................................... 34 4.8 Cavity B, 52 MHz.......................................................................................................... 37 4.9 Beam Measurements...................................................................................................... 40

5 Summary..........................................................................................43 5.1 Diagnostic Equipment ................................................................................................... 43 5.2 Noise Measuring Capability .......................................................................................... 43 5.3 Overall Noise Performance of the RF System............................................................... 43

6 Acknowledgements..........................................................................44 7 References........................................................................................45

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1 Preamble Study [1] of coasting-beam halo effect in the HERA-p ring of DESY suggests that longitudi-nal stochastic mechanisms (noises in accelerating systems or random ripple in dipole mag-netic field) might be responsible for degrade of beam during a luminosity run. The most probable candidate is a phase (frequency) noise in the RF system. One of goals of the activity launched by [2] is experimental verification of this hypothesis. This document constitutes the final report on the studies thus performed.

To accomplish the task, the analog I/Q demodulation technique about the 208 and/or 52 MHz references is employed. There are two non-intervening ways to proceed (in priority or-der):

1. To measure noises in 208 and 52 MHz RF cavity voltages, both with and without beam loading. Each of the 4 + 2 cavities should be diagnosed individually since major bulk of noise might well be inherent in a single cavity.

2. To diagnose beam response forced by the noise. The dipole and quadrupole modes of beam coherent in-phase oscillations (the coupled-bunch mode n = 0) are of practical in-terest.

To save time and investment cost, components and raw signals that are already available in the beam instrumentation system of the HERA-p ring [3] are employed. Only a few dedi-cated circuits were developed anew or borrowed from elsewhere.

The anticipated outcome of the study was to detect the presence of noise, spot the noisy sub-system, reveal the nature of noise, and start elaborating adequate cures. The greater part of these plans is fulfilled.

This report is organized as follows. Section 2 describes mathematics behind the signal processing procedure implemented. Section 3 specifies the dedicated low-noise differential amplifier manufactured to meet the particular demands of the study. Section 4 deals with the factual measurements of I/Q noises in 208 and 52 MHz RF systems and observations over the forced response of beam to the noise. Section 5 summarizes outcomes of the experimental efforts.

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2 Principles of Data Acquisition and Analysis Measurements were performed in the HERA-p RF Control Room, building 50/room 601 at the DESY site. Raw signals were taken from the exit sockets of 208 and 52 MHz I/Q de-modulation units [3]. The data was acquired with a digital oscilloscope Tektronix TDS 684A. Its built-in mathematics could not meet the specific demands of the study. To this end, digi-tized signal was dumped from the oscilloscope onto a 2.5-inch diskette via the scope’s inter-nal FDD and stored externally for the subsequent off-line post processing. Data samples from the I/Q channels were acquired pair-wise and simultaneously to allow for the subsequent cross-correlation analysis that is essential to perform the task.

The inherent capacities of the TDS 684A were used for a routine on-line quality control of signals, dedicated processing of the acquired data samples being accomplished off-line, with a PC by the purpose-written codes in the MathCAD PLUS 6.0 [4].

In this Section, we describe mathematics behind the signal processing procedure.

2.1 I/Q Decompositions

Consider the HERA-p ring at flattop. Attach observation point to azimuth Θ = 0 of the ma-chine. Let reference particles of beam be traversing Θ = 0 at time t = 0 and, then, at integer multiples of 2π/ωRF where ωRF is circular RF frequency (208 MHz). The fundamental RF harmonic of beam is seen ∝ cos(ωRFt) at Θ = 0.

Assume that the net accelerating RF field be applied to an equivalent narrow gap located at Θ = 0 as well. Across this gap, the accelerating field is seen as –VRFsin(ωRFt) since the HERA-p ring is operated above transition. Here, VRF is accelerating voltage amplitude per turn. Time dependence is taken as ∝ exp(–iωt).

A pair of orthogonal oscillations ( ) ( ) tt RFRF ω−ω− cos;sin (1)

constitutes a natural I/Q basis, referred to as the gap frame, that may be used to expand error δV(t) of the accelerating voltage into the two-term series,

( ) ( ) ( ) ( ) ( ).cossin ttuttutV RFQRFI ω−ω−=δ (2)

Here, uI(t) and uQ(t) denote the in-phase (I) and quadrature (Q) amplitudes that may be writ-ten down as (uI(t); uQ(t)) in a two-component vector notation.

Gap frame (1) is a special basis since the relevant I/Q voltages can be put in terms of the customary amplitude and phase excursions of the accelerating voltage –VRFsin(ωRFt),

( ) ( ),tutV IRF =δ ( ) ( ) ( ) .RFQ

t

RF Vtutdtt ≅′′δω=δϕ ∫ (3)

Cavity voltage diagnostics is accomplished in a frame imposed by the local oscillator, the LO frame. Its phase is not necessarily locked to (1) due to technical reasons. Therefore, the observation basis is

( ) ( ) ,cos;sin φ−ω−φ−ω− tt RFRF (4)

where φ denotes a phase shift between the net RF voltage and the local oscillator. The meas-ured I/Q voltages thus stand for expansion of δV(t) about another carrier oscillations,

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( ) ( ) ( ) ( ) ( ).cossin 00 φ−ω−φ−ω−=δ ttuttutV RFQRFI (5)

Vector (uI0(t); uQ0(t)) is transformed into (uI(t); uQ(t)) through a plain rotation ( ) ( ) ( ) ,sincos 00 φ+φ= tututu QII

( ) ( ) ( ) .cossin 00 φ+φ−= tututu QIQ (6)

Fig. 1 and Fig. 2 sketch relative positions of the gap and local-oscillator frames in a vector diagram.

Fig. 1: Relative position of the two I/Q frames.

Fig. 2: Conversion from the local-oscillator to the gap frames.

The 208 MHz RF system of the HERA-p ring comprises four cavities labeled with letter indices A, B, C, and D. We assume that their voltages add coherently at beam. Then, cavity voltage vectors, reduced to the equivalent accelerating gap placed at Θ = 0, are collinear to the VRF-vector in Fig. 1. Therefore, cavity voltage amplitude V and phase φ between the gap and LO frames can be deduced from the DC-coupled measurements of VI0 and VQ0, refer to Fig. 2, as

,20

20 QI VVV += .tan 00 IQ VV=φ (7)

Similar procedure is applied to the 52 MHz RF system of the HERA-p ring comprising two cavities labeled with indices A and B.

2.2 Spectral Power Densities

Let I/Q voltages uI(t) and uQ(t) be stationary random signals with a zero expectation value. Statistical properties of such noises in time domain can be described in terms of the cross-correlation matrix

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

τ+τ+

τ+τ+

tutututututututu

QQIQ

QIII (8)

that depends upon time offset τ, not t, due to the assumed (weak) stationarity of noises. An-gular brackets ⟨…⟩ denote averaging over the noise ensemble. Fourier transform over τ yields a power spectrum matrix which images (8) into the frequency domain,

( ) ( )( ) ( ) .

ωωωω

QQQI

IQII

PPPP

(9)

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Four matrix elements Pab(ω) are defined as double-sided Fourier transforms

( ) ( ) ( ) ( ) .exp τωττ+=ω ∫∞

∞−

ditutuP baab (10)

Either of indices a, b is allowed to run over I, Q independently. Functions Pab(ω) obey sym-metry conditions

( ) ( ) ( ).* ω=ω−=ω babaab PPP (11)

On inspecting equations (9) – (11), one finds out that: 1. Both PII(ω) and PQQ(ω) are purely real, non-negative, and even functions of ω.

2. RePIQ(ω) = RePQI(ω) and are both even functions of ω that are allowed to run over posi-tive and negative values.

3. ImPIQ(ω) = – ImPQI(ω) and are both odd functions of ω.

Therefore, it is sufficient to use the domain of ω ≥ 0 and diagnose no more than four spectral power density functions: PII(ω), PQQ(ω), RePIQ(ω), and ImPIQ(ω) there.

By virtue of (6), going from the LO to gap frames transforms spectral power densities ac-

cording to equations ( ) ( ) ( ) ,sin2sinRecos 2

002

0 φ+φω+φω=ω QQIQIIII PPPP (12)

( ) ( ) ( ) ,sin2sinRecos 200

20 φ+φω−φω=ω IIIQQQQQ PPPP (13)

( ) ( ) ( ) ( )( ) ,2sin212cosReRe 000 φω−ω−φω=ω QQIIIQIQ PPPP (14)

( ) ( ).ImIm 0 ω=ω IQIQ PP (15)

Sum PII(ω) + PQQ(ω) and imaginary parts of the non-diagonal spectra ImPIQ(ω), ImPQI(ω) are thus invariant with respect to rotation from (4) to (1).

All the more, it will be shown in Sect. 2.4 that narrow-band gap-frame functions PIQ(ω) and PQI(ω) would bring about a trivial contribution to the diffusion coefficient D(J) that gov-erns over the beam behavior under noise. To this end, the invariant imaginary parts of the spectra would apply no impact on beam and, as such, may be excluded from the task list.

Actually, it is thus necessary to diagnose only three spectral power density functions: PII(ω), PQQ(ω), and RePIQ(ω) in the domain of ω ≥ 0.

There exists a particular exemplar of noise (uI0(t); uQ0(t)) with

( ) ( ) ( ),00 ω=ω=ω PPP QQII

( ) ( ) ,0ReRe 00 =ω=ω QIIQ PP (16)

( ) ( ) ( ).ImIm 00 ω=ω−=ω QPP QIIQ

Transformation through (12) – (15) keeps such spectra unchanged,

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( ) ( ) ( ),ω=ω=ω PPP QQII

( ) ( ) ,0ReRe =ω=ω QIIQ PP (17)

( ) ( ) ( ).ImIm ω=ω−=ω QPP QIIQ

This is a special case of a random process with the net voltage error δV(t) behaving as a band-pass noise signal that is stationary in the lab frame. Its statistical properties do not depend on a rather arbitrary choice of basis, either (1) or (4), used for the I/Q decomposition. Stationar-ity (essentially, in beam frame) of the low-pass base-band I/Q voltages (uI0(t); uQ0(t)) is thus a necessary but not a sufficient condition for the stationarity of δV(t). Save for (16) and (17), net error δV(t) would rather be a periodically non-stationary random process. A more detailed discussion of this question can be found in [5].

2.3 Cross-Correlation Function

A variance (covariance) of random voltages uI(t) and uQ(t) can be found as the value of in-verse Fourier Transform (10) at τ = 0,

( ) ( ) ( ) ( ) .Re121

0

2 ωωπ

=ωωπ

==σ ∫∫∞∞

∞−

dPdPtutu ababbaab (18)

In the rightmost equation, integration goes over positive frequencies ω that are displayed by an oscilloscope.

Cross-correlation property of noises is commonly parameterized through a correlation co-efficient

.2

QQII

IQIQ σσ

σ=ρ (19)

Still, this coefficient convolutes the entire frequency-domain distribution of the noise power that is of practical interest here. Instead, one can turn to the differential interpretation of (18),

( ) ( ) ,0,Re12 ≥ωωωπ

=ωσ dPd abab (20)

and apply definition (19) to the local values of dσab2(ω). With this motivation, a spectral (dif-

ferential) cross-correlation function can be introduced,

( ) ( )( ) ( )

.Re

ωω

ω=ωρ

QQII

IQIQ PP

P (21)

It would mirror the degree of linear cross-correlation between frequency components of sig-nals in the I/Q channels. Perfect correlation in [ω, ω+dω] makes ρIQ(ω) = ±1, statistically in-dependent components having ρIQ(ω) = 0.

Function (21) is more suitable for plotting the experimental cross-correlation data since its inherent normalization reduces the span of ρIQ(ω) to ±1.

2.4 Structure of Diffusion Coefficient

Let us put down the two I/Q decompositions (2), (5) of the noise signal in a similar compact notation as

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( ) ( ) ( ),sin 0,

aQIa

a thtutV φ−ω′−=δ ∑=

,0=φI ,2π−=φQ (22)

( ) ( ) ( ),sin 00,0

aQIa

a thtutV φ−ω′−=δ ∑=

,0 φ=φI .20 π−φ=φQ (23)

Here, ω0 denotes angular rotation frequency (2π × 47.317 kHz). h′ may be set equal to either h = 4400, harmonic number of the 208 MHz RF or to 1100, harmonic number of the 52 MHz RF, particular option depending on the context. The fundamental radio frequency (208 MHz) is ωRF = hω0.

Longitudinal evolution of a bunch subjected to noise is described by a diffusion equation [1], [5] which reads

( ) ( ) ( ) .,,

∂∂

∂∂

=∂

∂J

tJfJDJt

tJf (24)

Here, D(J) is a diffusion coefficient, f(J, t) is a distribution function, J is action variable in the longitudinal phase-space plane (ϕ, dϕ/dt) where ϕ is phase in 208 MHz RF radians, J = JS = 8Ω0/π at RF separatrix, Ω0 is a circular frequency of small-amplitude synchrotron oscilla-tions.

Diffusion coefficient inside RF buckets (J ≤ JS) is a weighted sum over power spectra (9) sampled at various around-the-orbit (rotational, ∝ nω0) and within-the-bunch (multipolar, ∝ mΩs(J)) harmonics,

( ) ( )( ) ( ) ( )∑ ∑∞

−∞=

Ω+ω

Ω=

ba mn

bmn

amnsab

RF

JVJVJmnPV

JD, ,

)*()(0

220

21 (25)

with weight factors

( ) ( ) ( ) ( ) ( ) ,exp'

exp'2

*',

*',)(

φ−

−−φ+

+= −+

ahnm

ahnma

mn ihnJI

ihnJImhJV (26)

where Ωs(J) denotes frequency of non-linear synchrotron oscillations, Ωs(0) = Ω0. Summa-tion over a, b and particular values of phases φa, φb should be taken in compliance with either (22) or (23). Equations (25) and (26) hold true for both these expansions of δV(t).

In a smooth approximation, lumped noise ∝ δV(t) is seen as a plane wave of random ac-celerating field propagating around the orbit. On the other hand, the inherent dynamics of a bunch proceeds in a periodic way. It is described with a cyclic variable ψ that is a phase of synchrotron oscillations canonically conjugated to J, and dψ/dt = Ωs(J).

Therefore, to find weight factor (26), one has to calculate decomposition coefficients of a plane wave into Fourier series over the within-the-bunch multipole modes,

( ) ( ) .,exp21* ψ

ψ−ψϕ

π= ∫

π

π−

dimJhniJImn (27)

From this standpoint, the overall structure of D(J) becomes more apparent. Indeed, multi-plication of a base-band signal ua(t) by sin(h′ω0t – φa) in (22) or (23) translates frequency components of ua(t) from nω0 to a higher-frequency domain (n ± h′)ω0, their phases depend-ing on the phase φa of the up-mixing carrier. These up-converted components of noise are now capable of affecting the bunch resonantly. They excite coherent multipole oscillations of

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particles via (27) as transfer functions. To the final end, such a procedure would result in equation (26), [5].

Now, let us introduce a few simplifications: 1. Let phase ψ of synchrotron oscillations be launched so as to ensure ϕ(J, ψ=0) = 0.

Then, coefficient (27) would turn into a purely real function, and ( ) ( ).* JIJI mnmn = (28)

Otherwise, Imn(J) would have acquired a phase factor that is, nevertheless, insignificant physically.

2. Consider low-pass base-band noises ua(t) with bandwidth of their power spectra << ω0 or correlation time τc >> 2π/ω0, where 2π/ω0 is the period of beam rotation around the orbit. In this case, we can retain the single item with n = 0 in a sum over n of series (25).

3. At the flattop, protons exist in a symmetric potential well, say, in U(ϕ) = Ω02⋅(1–cosϕ)

for a single-RF regime. In this case,

( ) ( ) ( ).1, JIJI mnm

nm −=− (29)

Now, on applying to symmetry property (29) and putting phases (22) specific to gap frame (2) into equation (26) with n = 0, one finds out that a weight factor

( )

====

≠, & oddfor 0, &even for 0)(

0 QamIam

JV am (30)

and is equal to 0 otherwise. Therefore, the non-diagonal terms ∝ Vm0(I)(J)Vm0

(Q)*(J) and Vm0

(Q)(J)Vm0(I)*(J) in series (25) are nullified identically in the gap frame (1) — it constitutes

the unique basis indeed. The diffusion coefficient written for the gap frame now simplifies to

( ) ( )( ) ( ) ( )( ) ( )[ ]∑∞

−∞=

Ω+Ω

Ω=

m

QmsQQ

ImsII

RF

JWJmPJWJmPV

JD )()(22

0

21 (31)

with conventional weight functions resulting from (26)

( ) ( ) ( ) ( )( ) ,1141

odd

even,222

22),(0

),(

δδ

∝−±×

′== ′

m,

mmhm

QIm

QIm JI

hhmJVJW (32)

where δi,j is Kronecker’s delta-symbol. The non-diagonal power spectra PIQ(ω), PQI(ω) residual in the gap frame thus prove to

have no systematic effect on the beam diffusion by themselves. However, diagnostics of real parts of these spectra might be instructive from the technical viewpoint since they represent the degree of cross-correlation between random signals at the gap and can assist in revealing nature of the noises observed.

Diagnostics of the non-diagonal spectra RePIQ0(ω) and/or RePQI0(ω) in the LO frame is mandatory since these participate in converting PII0(ω) and PQQ0(ω) to the gap-frame’s PII(ω) and PQQ(ω) which functions are effective in D(J) (31), refer to (12) – (14).

Diagnostics of the non-diagonal spectra ImPIQ0(ω) and/or ImPQI0(ω) in the LO frame is less important since these are essentially stand-alone functions that 1. do not participate in driving beam diffusion altogether being an invariant part (15) of

PIQ(ω) and PQI(ω) whose entire contribution to D(J) cancels out,

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2. are not involved into cross-conversions (12) – (14) between PII(ω), PQQ(ω) and RePIQ(ω) which spectra are of practical interest.

2.5 Diffusion due to 208 and/or 52 MHz RF Systems

As a faithful approximation, one can presume that beam bunching at flattop of the HERA-p ring be accomplished due to the single-harmonic 208 MHz RF system. Longitudinal oscilla-tions inside a single-RF bucket can conveniently be parameterized through elliptic functions of modulus k with

( ),2sin ϕ= Ak ,10 ≤≤ k (33)

where Aϕ denotes amplitude of oscillations along ϕ, and 0 ≤ Aϕ ≤ π. Frequency of non-linear oscillations and action variable read

( ) ( ) ,20 kK

ksπ

Ω=Ω (34)

( ) ( ) ( ) ( )( ),1 2 kKkkEJkJ S ⋅−−⋅= (35)

where K(k) and E(k) are complete elliptic integrals of the 1st and 2nd kinds. Fourier expansion coefficient (27) can now be reduced to a form best suitable for calcula-

tions,

( ) ( ) ( ) ( )( ),odd,

even,,

sincos

,2snarcsin2sincos2 2

0' =

=ψψ

ψ

π′

π= ∫

π

mQmI

dmkkKkhhkImh (36)

where sn(x, k) denotes one of the Jacobi elliptic functions — sinus of elliptic amplitude. Functions Ωs(J) and Imh’(J) entering D(J) are given parametrically through (34) – (36). In case of h′/h = 1 when the I/Q noises are imposed via the main 208 MHz RF harmonic,

integration in (36) can be accomplished analytically [1], [5],

( ) ( ) ( ) ( ) ( )( ) .odd,

even,,1

2coshsinh

22 2

1

12

==

−

π

×

π=

−

−

mQmI

kKkK

mkK

mm

kImh (37)

The case of h′/h = 1/4, accounting for the I/Q noises imposed via the 52 MHz RF har-monic, is studied through a numerical integration of (36).

In a parabolic potential well U(ϕ) = Ω02ϕ2/2 one gets

,2ϕ= Ak ( ) ,0Ω=Ω ks ( ) 22 20

20 ϕΩ=Ω= AkkJ (38)

and

( ) ,2

′

=

′

= ϕ′ AhhJk

hhJkI mmhm (39)

where Jm(x) is the Bessel function of the m-th order. Equation (39) serves as a small-amplitude asymptote for both (36) and (37) holding for J ≤ (0.4 – 0.5)⋅JS approximately.

Fig. 3 and Fig. 4 plot weight functions (32), which determine efficiency of excitation of bunch multipole oscillations with the I/Q noise carried by either 208 MHz (h′/h = 1) or 52 MHz (h′/h = 1/4) RF harmonics. Factor 2 in the ordinate axis accounts for equal contributions to D(J) from the multipole harmonics m and –m.

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Fig. 3: Weight functions inside 208 MHz RF bucket, excitation through the 208 MHz harmonic.

Fig. 4: Weight functions inside 208 MHz RF bucket, excitation through the 52 MHz harmonic.

The 52 MHz RF sub-harmonic exhibits much a poorer non-linear content to the bunch. Therefore, it is highly effective in driving the dipole oscillations in the entire range of J ≤ JS, the impact increasing towards outskirts of the bunch. To this end, noise PQQ(ω) of the 52 MHz RF system in the dipole side band of synchrotron oscillations is much more hazardous inherently than its 208 MHz counterpart.

On the contrary, the 208 MHz RF system is more effective in exciting bunches through the higher synchrotron side bands (quadrupole, sextupole etc.).

Weight functions 2W1(Q) of dipole oscillations have the same slope at J = 0 where they are

tangent to straight line 8J/πJS irrespective of ratio h′/h, refer to Fig. 3 and Fig. 4.

2.6 Discrete Fourier Transform

For a stationary noise, the alternative to definition (10) is

( ) ( ) ( ) .,,1lim *

ωω=ω

∞→TUTU

TP baTab (40)

Here, Ua(ω, T) denotes a Fourier harmonic of ua(t),

( ) ( ) ( ) ,exp,2

2

dttituTUT

Taa ω=ω ∫

−

(41)

calculated through data samples clipped beyond |t| ≤ T/2 (the rectangular window function). In practice, to get the limiting value of (40) it is sufficient to adopt a finite, though large,

record length T >> τc, where τc is a correlation time of the noise in question. Assume that the data acquisition interval [–T/2, T/2] be broken by N equidistant points ti

with time increment ∆t, and let N be an even number for simplicity. Thus,

,NTt =∆ ,itti ⋅∆= .12

to2

for −−=NNi (42)

Let us approximate (41) with its partial integration sum (formula of rectangles)

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( ) ( ) ( ).exp,12

2i

N

Niiaa titutTU ω∆=ω ∑

−

−=

(43)

Now, take the N-point equidistant frequency array with an increment (frequency resolu-tion) ∆ω,

,2 Tπ=ω∆ ,jj ⋅ω∆=ω 12

to2

for −−=NNj (44)

and use these ωj as the arguments for (43). It would slice this function into a complex fre-quency-domain array of length N. It is nothing but the conventional Discrete Fourier Trans-form (DFT) Ua(ωj, T) of the real discrete time-domain waveform ua(ti) acquired at a sampling rate R = 1/∆t = N/T. There exists the exact inverse DFT from Ua(ωj, T) to ua(ti), though such a property is not demanded for by this study.

We have thus arrived at the N-point discrete estimate of (40)

( ) ( ) ( )TUTUT

P jbjajab ,,1 * ωω=ω (45)

that extends over a frequency span |ω| ≤ ωNyq where ωNyq/2π = R/2 is Nyquist frequency. Averaging ⟨…⟩ over noise ensemble is treated in frames of ergodic hypothesis. It allows

for interpretation of subsequent data captures of a given lasting random process as of individ-ual noise samples. Suffice these data samples, subscribed with (s), be separated by a dead time exceeding correlation time τc of the noise. Thus, it is implied that

( ) ( )∑−

=

ω=ω1

0

)( ,1 S

s

sxS

x (46)

where S is the number of data samples (records, sweeps) acquired.

2.7 Appearance of the Mains Frequency Lines

In practice, data records would be affected by a close-to-periodical interference due to oscil-lations at the mains cycle frequency (50 Hz) and its higher harmonics. One must be aware of how they would show up upon the signal processing.

To this end, assume that the data records be contaminated with a nonrandom signal ( ) ( ).cos ϑ+ω= tAtu M (47)

Here, ωM/2π = 50 Hz or its higher integer multiples, A is amplitude, and ϑ is phase. Applying procedure (45) to a deterministic signal (47) results in

( ) ( )( )( )

22sin

4

22

ωω

ωω≅ω

TTTAP

Mj

Mjj

m

m (48)

holding true at frequencies ωj around ±ωM, respectively. By virtue of (44), it is thus advisable to take T such as to confine a whole number of the mains fundamental perturbation periods (1/(50 Hz) or 20 ms). In this case, 1. only two frequency bins (at ω > 0 and ω < 0) would sample the central lobes of (48)

yielding a representative peak value A2T/4 that may be used for a straightforward estimate of amplitude A in (47) judging from the power spectrum data,

2. all the other frequency bins would correspond to nodes between side lobes of P(ω), and the discrete power spectrum (48) would then appear as a double-peaked histogram that

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resembles two delta-functions centered at ω = ±ωM — the factual amplitude spectrum of a monochromatic signal (47) at T → ∞.

2.8 Averaged Magnitude of Fourier Spectrum

Digital oscilloscope Tektronix TDS 684A employed for measurements has a handicapped set of the built-in mathematical options. Say, it cannot calculate power spectra compliant with (45). The only task-relevant function available is the averaged magnitude of Discrete Fourier Transform that, according to the TDS User Manual, reads

( ) ( ) ( ) ( ) .,,1,1 * TUTUT

TUT

M jajajaja ωω=ω=ω (49)

Since permutation between mathematical operations of raising to the 2nd power and ensem-ble averaging is not allowed, one cannot use the frequency domain records Ma(ωj) even to calculate the main-diagonal spectra Paa(ωj), saying nothing about the off-diagonal Pab(ωj).

However, functions (45) and (49) do have much in common. They both rely on complex arrays Ua(ωj, T). It allows for using the calculated Ma(ωj) to control quality and consistency of the data records ua(ti) acquired. Vector Ma(ωj) is compared to average magnitude of the DFT yielded by the oscilloscope itself. This DFT spectrum is archived prior to each session of the data acquisition. To comply with specifications of the TDS scope, the calculated spec-trum (49) is plotted in units of dB relative to 1 VRMS , that is as

( ).0at

2Volt 1log20 ≠ω

ω jaM (50)

The DC signal would have required for a different normalization. But this signal is ignored during the AC-coupled measurements in question.

2.9 Example of Signal Processing

To demonstrate and benchmark the signal-processing procedure, consider a simple example. Assume gap voltage be affected by noises whose discrete read-outs are

( ) ,iIiI Atu ξ= ( ) ,iQiQ Atu ζ= (51)

where ξi and ζi are independent random numbers from a uniform distribution in [–1,1], ± AI,Q is peak-to-peak span of (51). These read-outs can be interpreted as samples of a (white) noise with power spectra

( ) ,32 tAP III ∆=ω ( ) ,32 tAP QQQ ∆=ω ( ) 0=ωIQP (52)

flat over base-band frequencies |ω| << ωNyq = π/∆t. In the LO frame, voltages (51) would be seen through a rotation transform, inverse to (6),

( ) ( ) ( ) ,sincos0 φ−φ= tututu QII

( ) ( ) ( ) .cossin0 φ+φ= tututu QIQ (53)

Thus, we can generate simultaneous data records (uI0(ti); uQ0(ti)) that would imitate the cavity voltage measurements on the desktop. These records can be subjected to procedure (45), (12)–(15) to recover the prescribed spectra (52) and, thus, verify the post-processing code developed.

12

Table 1 specifies data record parameters used to illustrate the post-processing procedure. The underlined upper part of the Table lists parameters that coincide with those taken for the actual measurements (refer to Table 3). Data entries in the center of Table 1 are specific to the particular example in question.

Table 1: Data record parameters, test of signal post-processing Record length, T 1 s Time increment, ∆t 2⋅10–4 s Sampling rate, R 5000 samples/s Frequency resolution, ∆ω/2π 1 Hz Nyquist frequency, ωNyq/2π 2.5 kHz Observation frequency span 0–250 Hz Size of one record file in MathCAD format about 50 kbytes Number of I & Q samples acquired, S + S 50 + 50 LO-to-gap phase shift, φ 30 deg Power spectra in gap frame, PII = AI

2∆t/3 5 V2/Hz PQQ = AQ

2∆t/3 10 V2/Hz PIQ = 0 V2/Hz

with a superimposed harmonic perturbation (47) I: ωM/2π = 50 Hz, A2T/4 = 20 V2/Hz Q: ωM/2π = 150 Hz, A2T/4 = 15 V2/Hz

Fig. 5 shows spectral properties of the imitated acquired data set. They exhibit a non-zero cross-correlation. Fig. 6 plots results of rotating the spectra by φ back to the gap frame. Ex-pectation values for the power spectra comply with the preset parameters given in Table 1.

Fig. 5: Test of post-processing, spectra ac-quired in the LO frame.

Fig. 6: Test of post-processing, spectra ro-tated to the gap frame.

Fig. 7 and Fig. 8 show the similar example, but with the superimposed mains frequency lines that are specified in the bottom part of Table 1. Amplitudes of the harmonic signals are taken in compliance with (48) such as to ensure equal peaks of net power spectra reaching 25 V2/Hz at both 50 Hz (I) and 150 Hz (Q) lines. The harmonic content is diagnosed with a rea-sonable accuracy conforming to the frequency resolution adopted.

13

Fig. 7: Test of post-processing, spectra ac-quired in the LO frame with mains fre-quency lines added.

Fig. 8: Test of post-processing, spectra ro-tated to the gap frame with mains fre-quency lines added.

Plots in Fig. 5 – Fig. 8 are prone to noticeable fluctuations due to a finite number of sam-

ples S used in averaging (46). Still, increasing S beyond 25 – 50 proves to be a pointless step since the statistical ripple just slowly decreases ∝ S1/2, computations becoming too time-consuming.

14

3 Two-Channel Differential Amplifier In this Section, we specify the dedicated low-noise base-band differential amplifier manufac-tured for the noise detection.

3.1 What for?

Two identical channels of data acquisition are necessary for simultaneous monitoring the I/Q signals allowing for their subsequent cross-correlation analysis.

Either channel must have two inputs — the direct and inverted ones. Use of the subtrac-tion (differential) technique intends to suppress the supposedly dominant in-phase spurious electromagnetic fields affecting locally the base-band I/Q signals to be monitored.

This technique is explained in Fig. 9. External stray fields, say, at 50 Hz and its multiples affecting high-frequency cavity signals coming from the tunnel (left to section 1 in Fig. 9) would be translated by multiplication with the LO signal to a higher-frequency domain around ωRF. Hence, they would be suppressed by the subsequent low-pass filtering. Lengthy cables transporting base-band signals (between nodes 2–3 in Fig. 9) would be handled with the subtraction technique proper, and the in-phase common-mode signal will be eliminated with the differential amplifier. The two circuit sections are left unattended: 1–2 between fre-quency multiplier and the exit node of the demodulation unit, and 4–5 from the amplifier exit node to the entry circuits of the scope, inclusively. These sections should be kept as short as possible. The active device itself, amplifier 3–4, must be shielded.

Fig. 9: Subtraction (differential) technique to suppress external electromagnetic interference.

The transfer function of a channel is designed to yield

( )ω−ω

ω⋅

ω−ωω−

±=ωii

iKGH

H

L

(54)

with parameters specified in Table 2. The AC-coupling is mandatory to eliminate the heavy DC signal due to a high operating

RF voltage in the cavity. Higher cut-off in (54) performs a double task:

1. It allows for using G(ω) as a purpose-built lower-noise and higher-gain substitute for a low-pass amplifier finalizing the issued I/Q detection unit, thereby lowering noises inher-ent in the entire frequency down-mixing procedure.

2. It minimizes bandwidth of analog signal prior to its digitizing through the TDS thus ena-bling lower sampling rates and easing demand in acquisition memory to store representa-tive data records.

Use of the simplest cascade of the first-order passive differentiating and integrating cir-cuits minimizes the number of circuit elements implemented in (54). Since each of them is plagued with an inherent noise, it improves the overall noise performance of the amplifier.

15

Table 2: Technical specifications of the differential amplifier Gain, K (@ RLoad = 1 MOhm) 300 Lower roll-off frequency, ωL/2π 5 Hz Higher cut-off frequency ωH/2π 1.1 kHz Suppression of in-phase signals @ 20 Hz –70 dB @ 1 kHz –80 dB Input resistance 20 kOhm Output resistance 50 Ohm Long-term drift of DC off-set at output < ±0.1 mV AC spurious voltage, peak-to-peak @ the output node < ±3 mV reduced to the input node < ±10 µV

3.2 Design Features

The amplifier was manufactured from the circuit components provided by DESY. Low-noise monolithic amplifiers OP37 and AD797 by Analog Devices were employed. The circuit em-ploys high-precision (0.5%) resistors. Occasionally, the resistors were selected pair-wise to ensure even better precision. The output cascades use complementary transistors to increase power yield of the amplifiers.

Electrical circuit diagram of a channel is shown in Fig. 10.

+C18

200uF

Ch1(Ch2)

C4

6.8n

C8

6.8n

+VCC1

-VCC

G=100

0

Q2

KT3107E

1

2

3

-VCC

+VCC

+VCC1

U

C14

6.8nF

C100.15u

R220k

0

W

0

D(V)

R120k

D1

KC512A

31

R

V

0

R13

50

Fc=1.1kHz

0

-15V

0GND IN SYSTEM A

R14

50

0

R20

10k

0

C2

1.5uF

-VCC1

B(T)

C7

6.8n

+IN Ch1.2

21

E(W)

0

+VCC

0

+C16100uF

Channel 2

U1

OP37

3

2

74

6

8

1+

-

V+

V-

OUT

T2

T1-IN Ch1.1

C1

1.5uF

R10

200

C11

6.8n

16

D(V)

R7

100k

R85k

R95k

R61k

GND IN SYSTEM B

C5

6.8n

14

GND

D1

KC512A

31

7(21)

0

0

-VCC1 0

GND IN SYSTEM A

U2

OP373

2

74

6

8

1+

-

V+

V-

OUT

T2

T1

U3

OP373

2

74

6

8

1+

-

V+

V-

OUT

T2

T12(16)

C12

6.8n

0

+15V

T

U4

AD797 6

3

2 8

74

1 5

OUT

+

- DCMP

V+

V-

N1

N2

R2110k

R11

120

R18

1kC(U)

C9

6.8n0

R19

10kR4

1k

C13

6.8nF

Fc=5.2Hz

G=3

R5100k

+C15100uF +C17

200uF

-VCC1

P110K

V

Q1KT3102E

1

2

3

R22

10k

-VCC1

Channel 1

R12

50

-VCC

-VCC1

0

0

OUT Ch1

+VCC

C6

6.8n

R17

1k

Fc=5.2Hz

+VCC1

+VCC1

+VCC1

R31k

Fig. 10: Electrical circuit diagram of the differential amplifier.

16

The assembly board is equipped with a 44-pin connector that mounts the board inside a metallic enclosure equipped with an intake slot. The enclosure serves as an electro-magnetic screen as well. Entry RF connectors (LEMO) are planted into insulating spacers. Exit connec-tors (RKBN17) are mounted directly to the enclosure. Special care is taken to filter-out any in-leakage of ripple through the 15 V power supply. Fig. 11 plots schematics of in/out con-nectors and the power supply assembly. Fig. 12 shows photo of the entire device.

16

10n

D

white wire

X

Box

D2

KD106A

12

Front Panel of Box

2m

22 6

14

13 89 1

B

-IN Ch1.2

LEMO211

2+15V

-15V

10

V

T

-IN Ch2.2

2

BW

T

LEMO311

2

+15V

21

OUT Ch1

2m

U

V P

RKBN1711

2

D

7

Box

yellow wire

20

C

W

LEMO111

2

C

D1

KD106A

1 2

C3

A

17

10n

R

GNDC2

F

5

Ground

Z M

white

21

+IN Ch2.1

Back Panel of Box

14

LEMO411

2

11

CONNECTOR INTO BOX

H

4

OUT Ch2

E

16

+IN Ch1.1

Y

yellow

E

red

JN

319

S K

Box

RKBN1711

2

C17

Box

D

-15V

L

red wireR

V

10n

15

U

3000HH

12Box

2m

18

2

Fig. 11: Schematics of in/out connectors and power supply feed-through assembly.

Fig. 12: Differential amplifier. Electrical circuit board slotted inside the shielding enclosure (left), general operational view from the face side (right).

17

3.3 Performance Data Measurements

The amplifier was carefully benchmarked prior to measurements. Parameters of Table 2 are confirmed. Fig. 13 plots experimental data points and a graph of |G(2πf)| from (54). Fig. 14 plots the measured common-mode signal suppression factor.

Fig. 13: Amplitude-frequency characteristic of the differential amplifier.

Fig. 14: Suppression of a common-mode signal, experimental data points.

The last-moment checkout was performed in building 50/room 601 with the aid of 1 kHz meander signal yielded by the TDS scope for calibrations. Fig. 15 plots this reference signal measured to have about 5 mV peak-to-peak magnitude, with an unsuppressed DC content (see curve 1). Curve 2 shows the measured sustained response of the amplifier. Smooth curve 3 plots the calculated response of the design transfer function (54). There is a fair agreement between the curves 2 and 3 which confirms the prescribed status of the device.

Fig. 15: Calibration of the differential am-plifier.

Fig. 16: Scatter plot of spurious AC volt-ages at exit from the amplifier (I – channel 1, Q – channel 2).

Fig. 16 shows scatter a plot of the inherent spurious voltages accumulated over 25 s long observations. The spot size at 1σ is ±0.90 mV (I, channel 1) and ±0.75 mV (Q, channel 2). Since gain K = 300, the value attained of the self-noise reduced to the entry node is as small

18

as ±(2.5–3.0) µV at 1σ. It seems to be close to the bottom level feasible via standard technical solutions. Spectral properties of noise inherent in the amplifier are specified below as back-ground conditions for the measurements, refer to Sect 4.2.

19

4 Measurements of Noise in RF System

4.1 Generalities

Block diagram of the measuring circuit is shown in Fig. 17. Both, 208 and 52 MHz RF sys-tems are attacked with the identical techniques.

OUT Ch2

0

50

-IN Ch2

1

2I/Q Detection of

0

OUT Ch150

12

TDS Scope

Q OUT

1

2

HPF

DC

-1.1

kHz

+IN Ch1

+IN Ch2

208 MHz Cavity

HPFI OUT

50

Gain=300, Rin=20kOhm, Rout=50Ohm,Suppression Inphase Signals in Bandpass -70 dB(20Hz), -80dB(1kHz), Vaml=10V

Rin=1MOhm

1

2

HPF-IN Ch1 IN Ch1

1

2

HPF

DC

-1.1

kHz1

2

Two Differential Amplifiers

1

2

1

2

50

Cav IN

300

1

2

1

2

1

2

IN Ch2

50

LO IN

50

Bandpass 5Hz -1.1kHz, max5V@ 50Ohm

1

2

300

Fig. 17: Block diagram of the noise-measuring circuit, a 208 MHz cavity.

The built-in mathematics of the TDS 684A scope that was made available for measure-

ments could not meet the specific demands of the study. To this end, the scope was, basically, employed as a two-channel 8-bit ADC with a capability of converting data records into the MathCAD format. Parameters and a number of the data records must 1. comply with the higher cut-off in (54), 2. conform to the TDS performance, 3. produce files of a reasonable size, 4. be sufficient to accomplish a reasonable statistical averaging, and 5. be manageable via PC-based post-processing under a moderate CPU-time consumption.

Adopted parameters of the data records meeting these demands are listed in Table 3. The goal is to monitor noise spectra in 0–250 Hz base-band that confines 6 harmonics of the small-amplitude synchrotron frequency (< 40 Hz ca). The procedure to test the post-processing code used the similar data record parameters (refer to Table 1).

Table 3: Data record parameters, actual data acquisition Record length, T 1 s Time increment, ∆t 2⋅10–4 s Sampling rate, R 5000 samples/s Frequency resolution, ∆ω/2π 1 Hz Nyquist frequency, ωNyq/2π 2.5 kHz Observation frequency span 0–250 Hz Suppression of under-sampled signals towards the moni-tored 0–250 Hz base-band under a white-noise input < –10 dB Size of a record file in the TDS MathCAD format about 40 kbytes Dead time between subsequent acquisitions 60–70 s Number of I & Q samples acquired, S + S 25 + 25 per a session

20

Quality of the data set acquired has been routinely verified via comparing the averaged amplitude of the DFT amplitude spectrum calculated with equations (49) and (50), proceed-ing from the 25 data samples, versus the DFT amplitude spectrum provided by the TDS itself, averaged over > 300 samples. Archiving these DFT readouts was carried out prior to each session of data acquisition.

4.2 Background Conditions

They were monitored in the actual working environment of building 50/room 601. All 2×2 input sockets of the differential amplifier (54) were fed by idle paired 1.5 m long 50-Ohm terminated cable connectors. In the mean time, the two output sockets were loaded with DC-coupled 1-MOhm inputs of the TDS scope. Channel 1 is assigned to service I-signals, while channel 2 — the Q-ones. Date of measurements shown here is Dec 11, 2002.

Fig. 18 plots averaged amplitudes of the DFT, both calculated and directly acquired via the TDS (> 300 samples). There is a fair agreement between the curves confirming the con-sistency of data records and sufficiency of the 25-sample averaging.

Fig. 18: Averaged amplitude of DFT, calculated (upper curve) and directly acquired via the TDS (lower curve). Background in channel 1 (left) and channel 2 (right).

Fig. 19 shows the relevant power spectra calculated with equation (45). Fig. 20 plots the cross-correlation function (21) for the background self-noises in question.

Lines 50 and 150 Hz of the mains frequency are counter-correlated. The other frequency components in the background signal seem to be non-correlated.

Scatter plot of the background AC voltages accumulated over 25 s long observations is shown in Fig. 16. The spot size at 1σ is 0.90 mV (I) and 0.75 mV (Q). The conventional cor-relation coefficient (19) is equal to –0.12 that is likely to occur due to the counter-correlated contributions from lines at 50 and 150 Hz disclosed by Fig. 20.

Fig. 19 shows that the wide-band base line of the self-noise power spectrum runs at about (1–1.5)⋅10–10 V2/Hz. With the cavity field-probes and cable communication tracts available by now, the 190 kV voltage across, say, cavity A 208 MHz is seen as around a 70 mV ampli-tude of the DC-coupled signal (7). It is measured directly with the TDS, on bypassing the AC-coupled differential amplifier of gain 300. Thus, the estimated resolution threshold for the noise diagnostics constitutes

21

( )( ) ( )

( ) ,Hz11032

mV701

3001

HzV105.11

213

22

2102 −− ⋅−≅⋅⋅⋅−≅RFVP (55)

where 12 stands for units squared of fractional voltage deviation in either I or Q directions. Increasing the overall transfer sensitivity (in mV/kV) of cavity field detection is a straight-forward way to lower the resolution flat-bottom (55).

Fig. 19: Spectral power densities. Background in channels 1 (left) and 2 (right).

Fig. 20: Cross-correlation function of self-noise in channels 1, 2.

Fig. 19 shows that discrete lines at 50 Hz and its multiples in the self-noise spectrum are topped at about (0.02–1)⋅10–7 V2/Hz. Thus, by virtue of (48), resolution threshold for mains harmonic fractional amplitudes in the cavity voltage diagnostics constitutes

( )( ) ( )

( ) 5

21

22

27 1034.0

mV701

3001

HzV10102.04 −− ⋅−≅

⋅⋅⋅−≅

TVA RF . (56)

22

4.3 Cavity A, 208 MHz

As an example, every step of the data acquisition and post-processing procedure is explained here for the cavity A 208 MHz. Its status data is listed Table 4. For the sake of compactness, all the other cavities will be presented with intermediate comments skipped over.

Table 4: Status of cavity A 208 MHz during measurements Cavity # A Radio-frequency, fRF 208.195 MHz Date of measurements Dec 12, 2002 Beam energy 920 GeV Average beam current 34.3 mA Cavity voltage V 186.6 kV Detected DC voltages, (I; Q) in the LO frame (50.1; 41.4) mV Detected DC amplitude v in the LO frame 65.0 mV Phase φ of gap versus LO frames 39.6 deg Sensitivity v/V of gap voltage detection 0.35 mV/kV Scatter plot data in the LO/gap frame σII0/σII 5.7/5.7 mV σQQ0/σQQ 7.3/7.4 mV ρIQ0/ρIQ –0.22/0.21

Fig. 21 plots averaged amplitudes of the DFT, both calculated from the data set and ac-

quired directly from the TDS. Again, there is a fair agreement between the curves confirming consistency of the data records.

Fig. 21: Averaged amplitude of DFT, calculated and directly acquired via the TDS. Cavity A 208 MHz, the LO frame.

Fig. 22 shows spectral power densities acquired from the cavity signals and compared to the background read-outs. (The latter are the same as plotted in Fig. 19.) Useful data is well distinguishable against the background level, although the Signal-to-Background Ratio (SBR) cannot be referred to as high. It can be improved by going to higher sensitivities of the cavity field detection in mV/kV, refer to Table 4. None of the background spectrum discrete lines is more pronounced than the lines seen in the cavity signal. Discrete-line content seems being inherent in the cavity voltage ripple.

23

Fig. 22: Acquired spectral power densities against resolution threshold (lower curve). Cavity A 208 MHz, the LO frame.

Fig. 23 shows cross-correlation data required to rotate power spectra from the LO to the gap I/Q axes and, then, interpret them as amplitude and phase noises respectively.

Fig. 24 shows the scatter plot of the cavity voltage data in the LO frame. Abscissa data la-bels are given in a descending order so as to comply with the orientation of I-axis in Fig. 2. Statistical parameters of the spot are listed in Table 4. Comparing the spot dimensions to those of the background self-noise (see the darker spot in the center of Fig. 24) again con-firms the availability of useful data in the detected cavity signal.

Fig. 23: Cross-correlation function of noise. Cavity A 208 MHz, the LO frame.

Fig. 24: Scatter plot of spurious AC volt-ages in channels I, Q of the LO frame. Cav-ity A 208 MHz — the outer spot, self-noise in acquisition channels — the inner spot.

Table 4 lists parameters of detected DC-coupled signals and position of gap frame with re-spect to the LO frame. The I/Q coordinate axes of the gap frame are drawn in Fig. 24 for ref-erence.

Further data processing deals with the acquired spectral power densities shown in Fig. 22 and Fig. 23 and foresees the following three steps:

24

1. Normalization by v2, see Table 4, to reduce the spectra to fractional voltage deviations measured in 12/Hz.

2. Rotation by φ to move from the LO to the gap frame. 3. Multiplication by 1/|G(ω)|2 to correct for the amplitude-frequency distortion caused by the

acquisition channel (54) itself. Finally, the thus normalized, rotated and corrected spectra are plotted in Fig. 25. Cross-

correlation retained in the gap frame between amplitude (or I) and phase (or Q) noises is shown in Fig. 26. The latter plot has no direct relevance to the noise-driven diffusion of bunches — the beam fails to see the non-diagonal gap-frame spectra PIQ on average, refer to Sect. 2.4. It is topical to study the origin of noises.

Fig. 25: Spectral power densities. Cavity A 208 MHz, the gap frame.

Fig. 26: Cross-correlation function of noise. Cavity A 208 MHz, the gap frame.

25

4.4 Cavity B, 208 MHz

Table 5: Status of cavity B 208 MHz during measurements

Cavity # B Radio-frequency, fRF 208.195 MHz Date of measurements Dec 17, 2002 Beam energy 920 GeV Average beam current 45.2 mA Cavity voltage V 184.6 kV Detected DC voltages, (I; Q) in the LO frame (67.8; 22.8) mV Detected DC amplitude v in the LO frame 71.6 mV Phase φ of gap versus LO frames 18.5 deg Sensitivity v/V of gap voltage detection 0.39 mV/kV Scatter plot data in the LO/gap frame σII0/σII 12.2/10.5 mV σQQ0/σQQ 6.9/9.3 mV ρIQ0/ρIQ –0.58/–0.71

Fig. 27: Averaged amplitude of DFT, calculated and directly acquired via the TDS. Cavity B 208 MHz, the LO frame.

26

Fig. 28: Acquired spectral power densities against resolution threshold (lower curve). Cavity B 208 MHz, the LO frame.

Fig. 29: Cross-correlation function of noise. Cavity B 208 MHz, the LO frame.

Fig. 30: Scatter plot of spurious AC volt-ages in channels I, Q of the LO frame. Cav-ity B 208 MHz — the outer spot, self-noise in acquisition channels — the inner spot.

27

Fig. 31: Spectral power densities. Cavity B 208 MHz, the gap frame.

Fig. 32: Cross-correlation function of noise. Cavity B 208 MHz, the gap frame.

28

4.5 Cavity C, 208 MHz

Table 6: Status of cavity C 208 MHz during measurements

Cavity # C Radio-frequency, fRF 208.195 MHz Date of measurements Dec 08, 2002 Beam energy 920 GeV Average beam current 19.9 mA Cavity voltage V 189.7 kV Detected DC voltages, (I; Q) in the LO frame (–32.5; –28.6) mV Detected DC amplitude v in the LO frame 43.2 mV Phase φ of gap versus LO frames 41.4+180 deg Sensitivity v/V of gap voltage detection 0.23 mV/kV Scatter plot data in the LO/gap frame σII0/σII 4.4/4.1 mV σQQ0/σQQ 4.4/4.6 mV ρIQ0/ρIQ –0.12/–0.03

Fig. 33: Averaged amplitude of DFT, calculated and directly acquired via the TDS. Cavity C 208 MHz, the LO frame.

29

Fig. 34: Acquired spectral power densities against resolution threshold (lower curve). Cavity C 208 MHz, the LO frame.

Fig. 35: Cross-correlation function of noise. Cavity C 208 MHz, the LO frame.

Fig. 36: Scatter plot of spurious AC volt-ages in channels I, Q of the LO frame. Cav-ity C 208 MHz — the outer spot, self-noise in acquisition channels — the inner spot.

30

Fig. 37: Spectral power densities. Cavity C 208 MHz, the gap frame.

Fig. 38: Cross-correlation function of noise. Cavity C 208 MHz, the gap frame.

31

4.6 Cavity D, 208 MHz

Table 7: Status of cavity D 208 MHz during measurements

Cavity # D Radio-frequency, fRF 208.195 MHz Date of measurements Dec 18, 2002 Beam energy 920 GeV Average beam current 30.5 mA Cavity voltage V 30.8 kV Detected DC voltages, (I; Q) in the LO frame (–21.3; –6.5) mV Detected DC amplitude v in the LO frame 22.2 mV Phase φ of gap versus LO frames 16.9+180 deg Sensitivity v/V of gap voltage detection 0.72 mV/kV Scatter plot data in the LO/gap frame σII0/σII 4.8/4.3 mV σQQ0/σQQ 3.4/4.1 mV ρIQ0/ρIQ –0.41/–0.50

Fig. 39: Averaged amplitude of DFT, calculated and directly acquired via the TDS. Cavity D 208 MHz, the LO frame.

32

Fig. 40: Acquired spectral power densities against resolution threshold (lower curve). Cavity D 208 MHz, the LO frame.

Fig. 41: Cross-correlation function of noise. Cavity D 208 MHz, the LO frame.

Fig. 42: Scatter plot of spurious AC volt-ages in channels I, Q of the LO frame. Cav-ity D 208 MHz — the outer spot, self-noise in acquisition channels — the inner spot.

33

Fig. 43: Spectral power densities. Cavity D 208 MHz, the gap frame.

Fig. 44: Cross-correlation function of noise. Cavity D 208 MHz, the gap frame.

34

4.7 Cavity A, 52 MHz

Table 8: Status of cavity A 52 MHz during measurements

Cavity # A Radio-frequency, fRF 52.049 MHz Date of measurements Dec 16, 2002 Beam energy 920 GeV Average beam current 41.2 mA Cavity voltage V 70.0 kV Detected DC voltages, (I; Q) in the LO frame (23.9; –26.7) mV Detected DC amplitude v in the LO frame 35.8 mV Phase φ of gap versus LO frames –48.3 deg Sensitivity v/V of gap voltage detection 0.51 mV/kV Scatter plot data in the LO/gap frame σII0/σII 17.8/4.0 mV σQQ0/σQQ 18.1/25.1 mV ρIQ0/ρIQ 0.96/–0.39

Fig. 45: Averaged amplitude of DFT, calculated and directly acquired via the TDS. Cavity A 52 MHz, the LO frame.

35

Fig. 46: Acquired spectral power densities against resolution threshold (lower curve). Cavity A 52 MHz, the LO frame.

Fig. 47: Cross-correlation function of noise. Cavity A 52 MHz, the LO frame.

Fig. 48: Scatter plot of spurious AC volt-ages in channels I, Q of the LO frame. Cav-ity A 52 MHz — the outer spot, self-noise in acquisition channels — the inner spot.

36

Fig. 49: Spectral power densities. Cavity A 52 MHz, the gap frame.

Fig. 50: Cross-correlation function of noise. Cavity A 52 MHz, the gap frame.

37

4.8 Cavity B, 52 MHz

Noise behavior of this cavity is found to be the most offending one. To the present knowl-edge, it dictates the overall noise performance of the HERA-p ring RF system.

Table 9: Status of cavity B 52 MHz during measurements Cavity # B Radio-frequency, fRF 52.049 MHz Date of measurements Dec 16, 2002 Beam energy 920 GeV Average beam current 41.2 mA Cavity voltage V 70.0 kV Detected DC voltages, (I; Q) in the LO frame (6.4; –38.9) mV Detected DC amplitude v in the LO frame 39.5 mV Phase φ of gap versus LO frames –80.7 deg Sensitivity v/V of gap voltage detection 0.56 mV/kV Scatter plot data in the LO/gap frame σII0/σII 87.8/18.4 mV σQQ0/σQQ 31.1/91.3 mV ρIQ0/ρIQ 0.92/–0.78

Fig. 51: Averaged amplitude of DFT, calculated and directly acquired via the TDS. Cavity B 52 MHz, the LO frame.

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Fig. 52: Acquired spectral power densities against resolution threshold (lower curve). Cavity B 52 MHz, the LO frame.

Fig. 53: Cross-correlation function of noise. Cavity B 52 MHz, the LO frame.

Fig. 54: Scatter plot of spurious AC volt-ages in channels I, Q of the LO frame. Cav-ity B 52 MHz — the outer spot, self-noise in acquisition channels — the inner spot.

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Fig. 55: Spectral power densities. Cavity B 52 MHz, the gap frame.

Fig. 56: Cross-correlation function of noise. Cavity B 52 MHz, the gap frame.

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4.9 Beam Measurements

The goal of beam measurements is to observe forced response of beam to the external RF noise. The beam signal is taken from the resistive gap monitor and is then subjected to I/Q demodulation with the 52 MHz carriers. It is the same technique that is used for RF cavity voltage detection. Beam energy is 920 GeV, average beam current is 23.2 mA, and date of measurements is Dec 20, 2002.

Fig. 57: Averaged amplitude of DFT, calculated and directly acquired via the TDS. Resistive gap monitor, demodulation @ 52 MHz, the LO frame.

Fig. 57 plots averaged amplitude DFT spectra acquired with the TDS and calculated from data records. Fig. 58 shows the customary power spectra that look similarly for the I/Q chan-nels, having a pronounced positive cross-correlation, see Fig. 59.

Fig. 58: Acquired spectral power densities of beam signal. Resistive gap monitor, demodulation @ 52 MHz, the LO frame.

The reliable DC-coupled beam signal is not available. Therefore, to find phase shift φ be-tween the LO and beam frames, one has to involve a few extra assumptions: 1. On the one hand, cavity noise measurements indicate that noises in 52 MHz RF system do

strongly dominate over their 208 MHz counterparts in the accelerating field ripple.

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2. On the other hand, Fig. 4 shows that forced response of the 208 MHz bunches in question to the Q-noise of 52 MHz RF voltage must be stronger by about an order of magnitude than the response to the I-noise of the same very 52 MHz RF voltage.

Therefore, it is natural to presume that the beam-frame Q-axis be directed along the largest extent of the scatter plot, Fig. 60.

Angle of rotation φ can be found formally from the scatter plot data itself. Indeed, on inte-grating (14) in compliance with (18), one gets

( ) .2sin212cos 2

02

02

02 φσ−σ−φσ=σ QQIIIQIQ (57)

Assuming that σIQ2 = 0 in the beam frame, one gets equation for the rotation angle φ that

reads

.22

2tan0000

02

02

0

20

IIQQQQII

IQ

QQII

IQ

σσ−σσ

ρ=

σ−σ

σ=φ (58)

For the data records captured, σII0 = 24.6 mV, σQQ0 = 34.8 mV, ρIQ0 = 0.98, and φ = –35.0 deg. The beam frame thus found is drawn in Fig. 60.

Fig. 59: Cross-correlation function of beam signal. Resistive gap monitor, demodulation @ 52 MHz, the LO frame.

Fig. 60: Scatter plot of beam signal in chan-nels I, Q of the LO frame. Resistive gap monitor, demodulation @ 52 MHz.

Spectral power data rotated to this beam frame is shown in Fig. 61 and Fig. 62. All the spectra are divided by the factor |G(2πf)/K|2 (refer to (54)) to correct close-to-DC region for the differential amplifier DC roll-off.

Contrary to Fig. 58, the I/Q spectra in Fig. 61 have got quite a distinct appearance, the I-spectrum being simpler and lower. Apart from mains cycle frequency lines, the Q-spectrum exhibits frequency lines at 13 and 29 Hz. Peak at 29 Hz can be interpreted as a forced beam response to noise signal from cavity B 52 MHz, refer to Fig. 55. Line at 13 Hz might be due to synchrotron frequency of 52 MHz outer bunch in the composite double-RF bucket of the HERA-p ring. The small-amplitude synchrotron frequency (around 37 Hz) of the central 208 MHz bunches can only barely be seen in the Q-spectrum.

Both I/Q spectra show lines of 5 and 64 Hz. The latter one can be traced back, say, to cav-ity A 208 MHz where it can be seen in the both channels, refer to Fig. 25. Origin of the 5 Hz frequency line is not clear at the moment. All the more, it is in this region where the acquired

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data set shows a visible discrepancy with the averaged DFT amplitude spectrum yielded by the TDS scope, see Fig. 57.

Fig. 62 shows that the discrete frequency lines in beam spectrum tend to a linear cross-correlation, while the wide-band components seeming well non-correlated.

In general, beam response data complies with observations over noise in 208 and 52 MHz RF systems.

Fig. 61: Spectral power densities of beam signal. Resistive gap monitor, demodulation @ 52 MHz, the beam frame.

Fig. 62: Cross-correlation function of beam signal. Resistive gap monitor, demodulation @ 52 MHz, the beam frame.

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5 Summary

5.1 Diagnostic Equipment

Noise diagnostics in the actual environment of the HERA-p RF Control Room, building 50/room 601, proves to be barely feasible, if not impossible at all, without adopting dedicated measures that comprise:

1. Implementation of the purpose-built low-noise base-band amplifiers to follow immedi-ately the down-converting multipliers in the I/Q demodulation modules available by now.

2. Thorough shielding these low-noise and high-gain amplifiers against stray electromag-netic fields.

3. Use of the subtraction (differential) technique to suppress the dominant in-phase envi-ronmental interference along the transport paths of the base-band demodulated signals.

Also, it is advisable to adjust phases of local-oscillator signals in the I/Q demodulation mod-ules so as to make the LO-frame signals collinear to their gap-frame counterparts. In this case, noise data essential to beam survival could be extracted directly, cross-correlation data analysis being left optional.

5.2 Noise Measuring Capability

We have succeeded in observing and measuring noise signals from 208 and 52 MHz cavities. The resolution base-line threshold attained of noise power spectrum detection is as low as a few units of 10–13 12/Hz.

We have definitely observed alteration of noise behaviour in response to resetting opera-tional parameters of an RF cavity. Say, plunging from around 190 kV to a sustainable 3–4 kV minimum in a cavity 208 MHz was found to be accompanied with an about an-order-of-magnitude rise of noise power density in absolute units of V2/Hz. Noise signal from cavity B 52 MHz was observed up-surging noticeably prior to the 52 MHz RF system failure accident during a machine run, etc.

In a whole, beam-response observations comply with the noise diagnosed in 208 and 52 MHz RF systems.

5.3 Overall Noise Performance of the RF System

Level of I/Q noises at 208 MHz is found to be somewhat lower than it has been expected for. In a whole, 208 MHz RF system shows quite a satisfactory noise behaviour. Fig. 63 collates the relevant spectra for all 4 cavities. Base-line wide-band (0–250 Hz) constituent of noise runs at ≤ 10–12 12/Hz given 180–190 kV per cavity A, B or C. It increases to ≤ 10–11 12/Hz at 30 kV per cavity D.

Continuous stretch of noise at 52 MHz is noticeably higher: ≤ 10–11 12/Hz in cavity A, and ≤ 10–10 12/Hz in cavity B, both driven at 70 kV per cavity, refer to Fig. 64.

There is always a significant presence of discrete frequency lines in the noise spectra. None of the background spectrum frequency lines is more pronounced than the lines seen in the cavity signal. High-quality-factor harmonic content seems being inherent in the cavity voltage ripple itself.

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Amplitude (or I) and phase (or Q) noises rotated back to the gap frame usually retain a strong residual cross-correlation.

The worst noise performance is proper to cavity B 52 MHz in the lower-frequency band of 10–40 Hz. It falls into the dipole side band of incoherent longitudinal tune spread and is, thus, capable of exerting a strong adverse effect on the HERA-p bunches. The noise spectrum has a pronounced resonant shape. Its magnitude and central frequency (around 30 Hz) are not un-wavering in time. This peak occurs irrespective of presence or absence of beam in the ma-chine. (Accidentally, similar noise spikes were observed in the other RF cavities, albeit with much a smaller magnitude.) The nature of such a noise is not comprehensible at the moment.

Fig. 63: Spectral power densities. Cavities A, B, C and D 208 MHz, the gap frame.

Fig. 64: Spectral power densities. Cavities A and B 52 MHz, the gap frame.

6 Acknowledgements We thank F. Willeke for suggesting the subject matter of the study. We are grateful to E. Vo-gel, W. Kriens, R. Wagner, and C. Kluth for their prompt assistance in resolving various rou-tine questions emerged in performing this task.

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7 References [1] S. Ivanov and O. Lebedev, Coasting Beam in HERA-p Ring, Report DESY HERA 01–03,

Hamburg, October 2001.

[2] Attachment #88 to the Agreement between DESY, Hamburg and IHEP, Protvino, re: Noise Performance of HERA-p Ring, April 2002.

[3] E. Vogel, Fast Longitudinal Diagnostics for the HERA Proton Ring. Ph. D. Thesis, DESY–THESIS–2002–010, Hamburg, March 2002.

[4] MathSoft Engineering & Education Inc., 101 Main Street, Cambridge, MA, USA.

[5] S. Ivanov, Longitudinal Diffusion of a Proton Bunch under External Noise, IHEP Pre-prints 92–43 & 93–14, Protvino, 1992–93.